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Numerical Calculation of Transient
Field Effects in Quenching
Superconducting Magnets
vorgelegt von
Diplom-Ingenieur
Juljan Nikolai Schwerg
Von der Fakulät IV - Elektrotechnik und Informatik
der Technischen Universität Berlin
zur Erlangung des akademischen Grades
Doktor der Ingenieurswissenschaften
– Dr.-Ing. –
genehmigte Dissertation
Vorsitz der wissenschaftlichen Ausprache:
•Prof. Dr.-Ing. habil. Gerhard Mönich
Fachgebiet für Hochfrequenztechnik / Antennen und EMVt, Institut für
Hochfrequenz- und Halbleiter-Systemtechnologien, Fakultät
Elektrotechnik und Informatik, Technische Universität Berlin
Berichter:
•Prof. Dr.-Ing. Heino Henke
Fachgebiet für Theoretische Elektrotechnik, Institut für Technische
Informatik und Mikroelektronik, Fakultät Elektrotechnik und
Informatik, Technische Universität Berlin
•Dr.-Ing. habil. Stephan Russenschuck
TE-MSC-MDA-group, CERN Europäische Organisation für
Kernforschung, Genf, Schweiz
Tag der wissenschaftlichen Aussprache : 18.11.2009
Berlin 2010
D83
Numerical Calculation of Transient
Field Effects in Quenching
Superconducting Magnets
submitted by
Diplom-Ingenieur
Juljan Nikolai Schwerg
At the Faculty IV - Electrical Engineering and Computer Science
of the Technical University of Berlin
in order to receive the academic grade
Doktor der Ingenieurswissenschaften
– Dr.-Ing. –
approved dissertation
Chairperson of the Scientific Rehearsal:
•Prof. Dr.-Ing. habil. Gerhard Mönich
Chair of Antennas and EMC, Department of Computer Engineering and
Microelectronics, School of Electrical Engineering and Computer
Science, Technische Universität Berlin, Berlin, Germany
Scientific Board:
•Prof. Dr.-Ing. Heino Henke
Chair of Theoretical Electrical Engineering, Department of Computer
Engineering and Microelectronics, School of Electrical Engineering and
Computer Science, Technische Universität Berlin, Berlin, Germany
•Dr.-Ing. habil. Stephan Russenschuck
TE-MSC-MDA-group, CERN
European Organization for Nuclear Research, Geneva, Switzerland
Date of the Scientific Rehearsal : 18.11.2009
Berlin 2010
D83
Abstract
The maximum obtainable magnetic induction of accelerator magnets, relying
on normal conducting cables and iron poles, is limited to around 2 T because
of ohmic losses and iron saturation. Using superconducting cables, and em-
ploying permeable materials merely to reduce the fringe field, this limit can
be exceeded and fields of more than 10 T can be obtained.
A quench denotes the sudden transition from the superconducting to the
normal conducting state. The drastic increase in electrical resistivity cau-
ses ohmic heating. The dissipated heat yields a temperature rise in the coil
and causes the quench to propagate. The resulting high voltages and exces-
sive temperatures can result in an irreversible damage of the magnet - to
the extend of a cable melt-down. The quench behavior of a magnet depends
on numerous factors, e.g. the magnet design, the applied magnet protection
measures, the external electrical network, electrical and thermal material pro-
perties, and induced eddy current losses. The analysis and optimization of the
quench behavior is an integral part of the construction of any superconducting
magnet.
The dissertation is divided in three complementary parts, i.e. the thesis,
the detailed treatment and the appendix.
In the thesis the quench process in superconducting accelerator magnets is
studied. At first, we give an overview over features of accelerator magnets and
physical phenomena occurring during a quench. For all relevant effects nume-
rical models are introduced and adapted. The different models are weakly
coupled in the quench algorithm and solved by means of an adaptive time-
stepping method. This allows to resolve the variation of material properties
as well as time constants. The quench model is validated by means of measu-
rement data from magnets of the Large Hadron Collider. In a second step, we
show results of protection studies for future accelerator magnets. The thesis
ends with a summary of the results and a critical outlook on aspects which
could be subjected to further studies.
Common definitions and concepts in the design of superconducting ma-
gnets, derivations of electromagnetic models, and explanations of typical ef-
fects are collected in the detailed treatment. We introduce, e.g., the tempe-
rature margin to quench and the MIITs, and define the magnetic energy and
inductance in case of materials exhibiting hysteresis and diffusive behavior.
The momentarily dissipated hysteresis losses are derived for the critical state
model of hard superconductors. Furthermore, we review magnet protection
methods and the voltages occurring during a quench.
The appendix contains all information required for the reproduction of the
presented results. It comprises material properties such as the electrical resis-
i
ii
tivity or the heat capacity for a temperature range spanning from cryogenic
temperatures to some hundred kelvins. The model and simulation parameters
for the magnets used for this work are collected at the end.
Zusammenfassung
in deutscher Sprache
Die maximale magnetische Induktion in Magneten für Teilchenbeschleuniger
ist, aufgrund von Leitungsverlusten in den Kupferkabeln und Eisensaturie-
rung, auf ca. 2 T beschränkt. Durch den Einsatz von supraleitenden Kabeln
können jedoch Feldstärken von mehr als 10 T erreicht werden.
Als Quench wird der plötzliche Übergang vom supraleitenden zum normal-
leitenden Zustand bezeichnet. An dem sprunghaft vergrösserten elektrischen
Widerstand wird Wärme erzeugt, die die supraleitende Spule aufheizt und
zur Ausbreitung des Quenchs führt. Die auftretenden elektrischen Spannun-
gen und hohen Temperaturen können zu irreversiblen Schäden am Magneten
führen - im Extremfall zum Schmelzen des Kabels. Das Verhalten eines Ma-
gneten während eines Quenchs hängt von einer Vielzahl an Faktoren ab, wie
z.B. von der Konstruktionsweise des Magneten, den verwendeten Schutzmass-
nahmen, der externen elektrischen Beschaltung, den elektrischen und thermi-
schen Materialeigenschaften und den induzierten Wirbelstromverlusten. Eine
Analyse und Optimierung des Quenchverhaltens ist ein wichtiger Bestandteil
der Konstruktionsphase supraleitender Magnete.
Die Doktorarbeit gliedert sich in drei Teile: den Hauptteil, eine Sammlung
detaillierter Herleitungen und den Anhang.
Im Hauptteil der Arbeit wird der Quenchprozess in supraleitenden Be-
schleunigermagneten untersucht. Nach einem Überblick über die Bauweise
von Beschleunigermagneten werden die unterschiedlichen physikalischen Ef-
fekte, die im Verlauf eines Quenches auftreten, eingeführt. Für alle relevanten
Phänomene werden numerische Modelle vorgestellt und angepasst. Die ein-
zelnen Modelle werden im Quenchalgorithmus zusammengefasst und schwach
miteinander gekoppelt gelöst. Um die Variation von Materialeigenschaften
und Zeitkonstanten auflösen zu können, wird der Algorithmus mit einer ad-
aptiven Zeitschrittweitensteuerung ausgestattet. Das Quenchmodell wird an-
hand von Messungen von Magneten des Large Hadron Collider validiert. Im
Anschluss werden Ergebnisse von Studien für zukünftige Beschleunigermagne-
te gezeigt. Der Hauptteil endet mit einer Zusammenfassung der Ergebnisse
und einem kritischen Ausblick auf Aspekte, die im Rahmen weiterer Arbeiten
untersucht werden könnten.
In den detaillierten Betrachtungen werden Konzepte aus dem Bereich su-
praleitender Magnete, z.B. Temperaturtoleranzen und die Quenchlast, einge-
führt. Die Definitionen der elektromagnetischen Energie und der Induktivi-
tät werden im Zusammenhang von feldabhängigen und Hysterese behafteten
Materialien untersucht. Es wird ein Ausdruck für die Augenblicksleistung bei
Supraleiterhystereseverlusten hergeleitet. Weiterhin erfolgt eine ausführliche
Darstellung aller Spannungen, die während eines Quenches auftreten und eine
iii
iv
Beschreibung gängiger Magnetschutzkonzepte.
Der Anhang enthält alle Daten, die zur Reproduktion der vorgestellten
Ergebnisse nötig sind. Er umfasst Materialeigenschaften wie z.B. den elek-
trischen Widerstand oder die Wärmekapazität für Temperaturbereiche von
kryogenischen Temperaturen bis hin zu mehreren hundert Kelvin. Die Modell-
und Simulationsparameter der einzelnen Magnete sind ebenfalls im Anhang
aufgeführt.
Acknowledgement
The work presented in this thesis has been carried out in the framework of
the Doctoral Student Program at CERN in cooperation with the chair for
electromagnetism at the Technische Universität Berlin.
I am very thankful to my supervisor at CERN, Dr.-Ing. habil. Russen-
schuck, for assigning me to this rich and diverse subject. I got many inspi-
rations from our discussions, broadened my understanding on the design and
computation of superconducting accelerator magnets, and learned about the
high standards of scientific writing and notation.
I want to thank Dr. Auchmann who co-supervised my work for our discus-
sions and his support.
I am most grateful to Prof. Dr.-Ing. Henke from the Technische Universität
Berlin for supervising this external thesis. I am very glad for his support
during the last years, i.e. our discussions about field theory and the world of
science, his fast and profound feed-back and especially his positive spirit.
I want to thank Dr.-Ing. Bruns, Dr. Rodriguez-Mateos, and Dr. Verveij for
discussions on numerical methods, quench simulation at CERN, and cable
related aspects of quench.
I am very glad for all the support and advice I got from my first supervisor
at CERN, Dr.-Ing. Völlinger, since my time as a technical student.
During the start of his Ph.D.-project Mr. Bielert helped me conducting
the simulations for the inner triplet upgrade (used in Sec. 5.1). I very much
enjoyed our lunch and coffee breaks.
Over the course of the Ph.D.-project I was assigned to three groups, i.e. AT-
MEL, AT-MTM and TE-MSC. I want to thank the group leaders Dr. Mess,
Dr. Walckiers, and Prof. Rossi, as well as all my colleagues for the pleasant
work atmosphere.
Furthermore, I want to thank those people who helped me keeping my
personal balance by providing diversion and company: Thank you Deepali,
Ruxandra and Fabio for the years we spent together in Geneva and for adding
to my life.
Thanks to the people of the legendary Rue de Lyon 2: Thank you Diana,
Tamara, Ahmed and Emilia for the time we shared and things we have ex-
perienced. Thanks to Fiona for proofreading parts of the thesis and being
awesome!
My thanks to Steffen for our discussions on “gefangene Energie” and “Leis-
v
vi
tungsübertragung”. It was very comforting and helpful being able to share
with someone what doctoral students eventually have to go through...
“What is the area covered by one bushel of corn in a field?” - this question
led to numerous Sundays spent discussing with Swati in cafes and bars around
the city - instead of working on my thesis. We have never reached a conclusion,
but it was totally worth it!1
I was always lucky to enjoy the support of my family. Therefore I am very
grateful.
1A bushel is a unit of dry volume, usually subdivided into eight local gallons in the systems
of Imperial units and U.S. customary units. It is used for volumes of dry commodities,
not liquids, most often in agriculture. It is abbreviated as bsh. or bu. The name derives
from the 14th century buschel or busschel, a box.
•Wheat at 13.5% moisture by weight: 60 lb = 27.2155422 kg
(Wikipedia - http://en.wikipedia.org/wiki/Bushel - 01.08.2009)
Nullus est liber tam malus, ut non aliqua parte
prosit!
Gaius Plinius Caecilius Secundus
(AD 61/63 - ca. 113)
to my father
and our friends
vii
viii
Contents
1 Introduction 1
1.1 Superconductivity and Quench . . . . . . . . . . . . . . . . . . 1
1.2 QuenchSimulation......................... 3
1.3 Literature Overview . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Objectives of the Thesis . . . . . . . . . . . . . . . . . . . . . . 5
1.5 Structure of the Dissertation . . . . . . . . . . . . . . . . . . . 5
2 Magnet Features and Physical Phenomena 7
2.1 Superconducting Magnets . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 The LHC Main Bending Magnet . . . . . . . . . . . . . 8
2.1.2 Other Magnet Types and Design Features . . . . . . . . 12
2.2 QuenchProcess........................... 14
2.3 RelevantEffects........................... 19
3 Numerical Modeling 21
3.1 Coil Discretization and Numbering Scheme . . . . . . . . . . . 22
3.2 Superconductor Model . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 Magnetic Field Model . . . . . . . . . . . . . . . . . . . . . . . 25
3.4 InducedLosses ........................... 28
3.5 ElectricalNetwork ......................... 29
3.5.1 Lumped Network Elements . . . . . . . . . . . . . . . . 31
3.5.2 Magnet Representation and Introspection . . . . . . . . 33
3.6 ThermalModel........................... 36
3.6.1 Temperature Change . . . . . . . . . . . . . . . . . . . . 38
3.6.2 HeatTransfer........................ 38
3.6.3 Thermal Properties of Helium . . . . . . . . . . . . . . . 40
3.6.4 QuenchHeaters....................... 41
3.6.5 Total Dissipated Power . . . . . . . . . . . . . . . . . . 42
3.7 QuenchAlgorithm ......................... 43
4 Introspection 47
4.1 LHCMainDipole.......................... 47
4.1.1 Reproduction of Quench Measurements . . . . . . . . . 48
4.1.2 Introspection - Simulation of a Quench in the LHC Tunnel 52
4.2 3-D Thermal propagation . . . . . . . . . . . . . . . . . . . . . 54
4.3 Quenchrecovery .......................... 58
5 Extrapolation 61
5.1 Quench Protection Study Inner Triplet Upgrade Quadrupole . 61
ix
x Contents
5.1.1 Unprotected Quench . . . . . . . . . . . . . . . . . . . . 62
5.1.2 Quench Heater Protection . . . . . . . . . . . . . . . . . 63
5.1.3 Dump Resistor Studies . . . . . . . . . . . . . . . . . . . 66
5.1.4 Power Supply Inversion . . . . . . . . . . . . . . . . . . 70
5.1.5 Full Protection . . . . . . . . . . . . . . . . . . . . . . . 70
5.2 Fast-Ramping Dipole . . . . . . . . . . . . . . . . . . . . . . . . 71
5.2.1 Cooling Schemes . . . . . . . . . . . . . . . . . . . . . . 72
5.2.2 Quench Limits at Different Ramp Rates . . . . . . . . . 73
5.2.3 Quench Detection During the Up- and Down Ramp . . 74
5.2.4 Robust Magnet Design . . . . . . . . . . . . . . . . . . . 76
6 Conclusion and Outlook 77
7 Detailed Treatment 81
7.1 MarginstoQuench......................... 81
7.1.1 Current Density Margin . . . . . . . . . . . . . . . . . . 81
7.1.2 Temperature Margin . . . . . . . . . . . . . . . . . . . . 81
7.1.3 Margin on the Load-line . . . . . . . . . . . . . . . . . . 83
7.1.4 Enthalpy Margin/Energy Reserve . . . . . . . . . . . . . 83
7.1.5 Minimum Quench Energy . . . . . . . . . . . . . . . . . 86
7.2 MIITs ................................ 87
7.3 MagneticEnergy.......................... 89
7.4 Inductance ............................. 93
7.4.1 Self Inductance . . . . . . . . . . . . . . . . . . . . . . . 93
7.4.2 Mutual inductance . . . . . . . . . . . . . . . . . . . . . 95
7.4.3 Application of the Differential Inductance to Quench
Computation ........................ 96
7.5 Non-linear Voltage-Current-Characteristic of Superconductors . 100
7.5.1 Current-Sharing . . . . . . . . . . . . . . . . . . . . . . 101
7.5.2 Voltages Induced over a Superconductor . . . . . . . . . 104
7.5.3 Graphical Solution . . . . . . . . . . . . . . . . . . . . . 108
7.6 Cable Magnetization Losses . . . . . . . . . . . . . . . . . . . . 109
7.7 Superconductor Hysteresis Losses . . . . . . . . . . . . . . . . . 114
7.8 Modeling Rutherford-Type Cables . . . . . . . . . . . . . . . . 122
7.9 MagnetProtection .........................126
7.9.1 Quench Detection . . . . . . . . . . . . . . . . . . . . . 128
7.9.2 DumpResistor .......................130
7.9.3 Isolating the Magnet . . . . . . . . . . . . . . . . . . . . 132
7.9.4 Subdivision and Coupled Secondary . . . . . . . . . . . 132
7.9.5 Quench Heaters . . . . . . . . . . . . . . . . . . . . . . . 133
7.9.6 Quench Protection Sequence of Events . . . . . . . . . . 137
7.10 Voltages Occurring During a Quench . . . . . . . . . . . . . . . 138
7.10.1 Terminal Voltages . . . . . . . . . . . . . . . . . . . . . 138
7.10.2CoilVoltages ........................142
7.10.3 Voltages in the Coil Cross-Section . . . . . . . . . . . . 148
7.11 Runge-Kutta Method . . . . . . . . . . . . . . . . . . . . . . . 151
Contents xi
A Material Properties 153
A.1 Solids in Normal Conducting State . . . . . . . . . . . . . . . . 154
A.1.1 Temperature Levels . . . . . . . . . . . . . . . . . . . . 154
A.1.2 MassDensity........................155
A.1.3 Electrical Resistivity . . . . . . . . . . . . . . . . . . . . 156
A.1.3.1 Copper ......................159
A.1.3.2 Niobium-Titanium . . . . . . . . . . . . . . . . 161
A.1.3.3 Niobium-3-Tin . . . . . . . . . . . . . . . . . . 161
A.1.3.4 Insulators (Kapton, PVA, Epoxy) . . . . . . . 161
A.1.3.5 Other Materials . . . . . . . . . . . . . . . . . 162
A.1.4 Thermal Conductivity . . . . . . . . . . . . . . . . . . . 162
A.1.4.1 Copper ......................163
A.1.4.2 Niobium-Titanium . . . . . . . . . . . . . . . . 165
A.1.4.3 Niobium-3-Tin . . . . . . . . . . . . . . . . . . 165
A.1.4.4 Polyimide (Kapton) . . . . . . . . . . . . . . . 165
A.1.4.5 Other Materials . . . . . . . . . . . . . . . . . 165
A.1.5 HeatCapacity .......................167
A.1.5.1 Copper ......................169
A.1.5.2 Niobium-Titanium . . . . . . . . . . . . . . . . 169
A.1.5.3 Niobium-3-Tin . . . . . . . . . . . . . . . . . . 169
A.1.5.4 Polyimide (Kapton) . . . . . . . . . . . . . . . 171
A.1.5.5 Other Materials . . . . . . . . . . . . . . . . . 171
A.1.6 Permeability ........................171
A.1.6.1 Magnetic Iron . . . . . . . . . . . . . . . . . . 174
A.1.6.2 Other Materials . . . . . . . . . . . . . . . . . 175
A.1.7 Dielectric Strength . . . . . . . . . . . . . . . . . . . . . 175
A.1.8 Electrical Permittivity . . . . . . . . . . . . . . . . . . . 175
A.1.8.1 Insulators . . . . . . . . . . . . . . . . . . . . . 177
A.2 Solids in Superconducting State . . . . . . . . . . . . . . . . . . 177
A.2.1 Transition Temperature / Critical Field . . . . . . . . . 178
A.2.2 Critical Current . . . . . . . . . . . . . . . . . . . . . . 179
A.2.3 Then-Index.........................181
A.2.4 Critical Current Density . . . . . . . . . . . . . . . . . . 182
A.2.5 Critical Surface Parameterization . . . . . . . . . . . . . 183
A.2.5.1 Niobium-Titanium . . . . . . . . . . . . . . . . 183
A.2.5.2 Niobium-3-Tin . . . . . . . . . . . . . . . . . . 184
A.2.6 Electrical Resistivity . . . . . . . . . . . . . . . . . . . . 186
A.2.7 Thermal Conductivity . . . . . . . . . . . . . . . . . . . 186
A.2.8 HeatCapacity .......................186
A.2.8.1 Niobium-Titanium . . . . . . . . . . . . . . . . 187
A.2.8.2 Niobium-3-Tin . . . . . . . . . . . . . . . . . . 187
A.2.9 Permeability ........................188
A.3 Fluid/Gaseous Matter - Helium Properties . . . . . . . . . . . . 188
A.3.1 Temperature Levels / Phases . . . . . . . . . . . . . . . 189
A.3.2 Density ...........................189
A.3.3 LatentHeat.........................190
xii Contents
A.3.4 Heat Conductivity . . . . . . . . . . . . . . . . . . . . . 191
A.3.5 Volumetric Specific Heat . . . . . . . . . . . . . . . . . . 191
A.3.6 Dielectric Strength . . . . . . . . . . . . . . . . . . . . . 191
B Parameters 193
B.1 Geometrical Quantities . . . . . . . . . . . . . . . . . . . . . . . 193
B.1.1 Filament ..........................193
B.1.2 Strand............................193
B.1.3 Cable ............................194
B.1.4 Cable Twist-pitch in a 2D Approach . . . . . . . . . . . 197
B.1.5 Coil Cross-Section . . . . . . . . . . . . . . . . . . . . . 197
B.1.6 Magnet ...........................199
B.2 FillingFactors ...........................200
B.2.1 Length, Area and Volume Ratios . . . . . . . . . . . . . 200
B.2.2 Material Fractions . . . . . . . . . . . . . . . . . . . . . 201
B.2.3 Effective Electrical Resistivity, Thermal Conductivity
andSpecificHeat......................201
C Cases 203
C.1 LHC Main Bending Magnet . . . . . . . . . . . . . . . . . . . . 203
C.1.1 Strand............................203
C.1.2 Cable ............................203
C.1.3 MagnetData ........................205
C.1.4 External Electrical Circuit . . . . . . . . . . . . . . . . . 205
C.1.5 Magnet Protection . . . . . . . . . . . . . . . . . . . . . 205
C.1.6 Operating Conditions / Critical Values . . . . . . . . . . 209
C.2 LHC Inner Triplet Nested Dipole - MCBX . . . . . . . . . . . . 212
C.2.1 Strand and Cable . . . . . . . . . . . . . . . . . . . . . . 212
C.2.2 MagnetData ........................212
C.3 LHC Inner Triplet Upgrade Quadrupole - MQXC . . . . . . . . 215
C.3.1 Strand and Cable . . . . . . . . . . . . . . . . . . . . . . 215
C.3.2 MagnetData ........................215
C.3.3 Electrical Circuit . . . . . . . . . . . . . . . . . . . . . . 215
C.3.4 Magnet Protection . . . . . . . . . . . . . . . . . . . . . 215
C.3.5 Operating Conditions / Critical Values . . . . . . . . . . 215
C.4 Fast Ramping Dipole Magnet . . . . . . . . . . . . . . . . . . . 218
C.4.1 Strand and Cable . . . . . . . . . . . . . . . . . . . . . . 218
C.4.2 MagnetData ........................218
C.4.3 Electrical Circuit . . . . . . . . . . . . . . . . . . . . . . 218
C.4.4 Magnet Protection . . . . . . . . . . . . . . . . . . . . . 218
C.4.5 Operating Conditions / Critical Values . . . . . . . . . . 218
D Digressions 223
D.1 Surface-Charge of a Discontinuity of Resistivity . . . . . . . . . 223
D.2 An Arbitrarily Cut Cylinder . . . . . . . . . . . . . . . . . . . . 224
Contents xiii
D.3 Critical Current Density Relies on the Model of the Supercon-
ductor................................225
E Formulary and Constants 229
E.1 Formulary..............................229
E.1.1 Inductance of a Ring Conductor . . . . . . . . . . . . . 229
E.1.2 Field of a Long Solenoid . . . . . . . . . . . . . . . . . . 229
E.2 Constants..............................229
Bibliography I
List of Figures XVII
List of Tables XXIII
Notation and List of Symbols XXV
Curriculum Vitae XXXV
xiv Contents
1 Introduction
Widerstand
zwecklos
The maximum obtainable magnetic induction of accelerator magnets, re-
lying on normal conducting cables and iron poles, is limited to around 2 T
because of ohmic losses and iron saturation [Will 00, p. 55]. Using super-
conducting cables, and employing permeable materials merely to reduce the
fringe field, this limit can be exceeded and fields of more than 10 T can be
obtained.
For the new particle accelerator LHC [Lefe 95, Evan 09, Hein 07], currently
under commissioning at CERN [CERN 91], more than 8000 superconducting
magnets are used; some of them with a maximum field strength of more than
8 T. The coils of these magnets are mainly made of Nb-Ti strands where the
technological challenges are well mastered and a variety of tools for design
and calculation are available. With new materials available on an industrial
scale, e.g. Nb3Sn is able to carry much higher current densities [Devr 04,
p. 7], or with new applications like fast-ramped accelerators, new problems
occur and established design assumption do not hold any longer. Hence, the
methods for the simulation of quench in superconducting magnets have to be
revised.
1.1 Superconductivity and Quench
Superconducting accelerator magnets are made from hard, low-temperature
superconductors. The materials used for the superconducting cables exhibit
superconductivity if the temperature, the applied magnetic induction and the
current density are below critical values which are interdependent and form
the critical surface. The maximum current density in these superconductors
is in the order of some 1010 Am−2(Sec. A.2.5.1). This is a factor 1000 higher
than in standard copper wires (rated at about 107Am−2[Wils 83, p. 3]).
Exceeding locally the limits of the critical surface, the affected fraction
of the strand transits from superconducting to normal conducting state -
denoted as quench [Appl 69]. Following IEC [Supe 07a],
1
2 Introduction
Figure 1.1: Coil windings of the LHC main bending magnet MB3004 after a quench
at nominal current. The magnet developed an inter-turn short circuit which caused
a meltdown of the cable and the cold bore (the tube separating the coil from the
beam)[Walc 04, Siem 03].
[a] Quench [denotes] the uncontrollable and irreversible transition
of a superconductor or a superconducting device from the super-
conducting state to the normal conducting state.
With the development of a quench, several physically different but mutu-
ally dependent processes start: Due to the ohmic heating, the temperature
increases and the normal conducting zone expands over the superconductor.
If the device is short-circuited, the growing resistivity causes the current to
decrease. The changing current results in a change of magnetic flux and thus
induces eddy and magnetization currents in the metallic parts and cables, re-
spectively. These induced currents create losses and additional heating, which
distribute the normal zone even further.
A quench poses two different threats to a superconducting device: high
temperatures due to an unlimited ohmic heating, and excessive voltages due
to a steadily growing resistance at high currents [Iwas 05]. High temperatures
impair the electrical integrity of the insulation material or even cause a melt-
down of the cable. Excessive voltages can result in an electric arc between
adjacent coil windings punching holes into the insulation. In addition, the
high current density and temperature gradients during a quench can cause
an irreversible degradation of the current carrying capability of the supercon-
ductor.
The violence of an uncontrolled electromagnetic discharge shall be illus-
trated by an incident which happened to the LHC main bending magnet
MB3004 during cold testing on the test bench. During a quench at nominal
current the insulation of the magnet failed [Walc 04] and the magnet devel-
oped an inter-turn short circuit. The excessive heat deposition in the short
circuited turn caused a meltdown of the cable and the cold bore tube (the
tube between coil and beam) [Siem 03]. The integrity of the cryogenic system
was lost. Figure 1.1 shows the burnt-out coil windings after the incident.
Superconducting devices, where the occurrence of a quench cannot be ex-
1.2 Quench Simulation 3
cluded by the choice of operating conditions, need to be equipped with a
protection system. The protection either relies on an intrinsically safe design
or on active measures taken upon quench detection. In the former case, the
device is designed such that it can withstand a quench without overheating
and the quench is eventually able to recover. In the latter case, the current is
extracted actively - as soon as possible - from the device by either increasing
the internal resistance or switching in an external resistor.
The design of the protection system is an essential part of the construc-
tion of any superconducting device. Therefore, quench simulation programs
are needed in order to study the intrinsic quench behavior and define the
necessary protection, to test protection schemes and to analyze measurement
data.
1.2 Quench Simulation
The simulation of quenches in superconducting magnets constitutes both, a
multi-physics and a multi-scale problem. The accurate modeling consists of
a thermal, a magnetic and an electrical network problem coupled by means
of energy exchange and non-linear material properties.
The required spacial resolution spans from less than a millimeter for the
description of eddy currents in strands to several meters for an entire magnet.
The involved time constants reach from some microseconds for the resolution
of quench initialization to several minutes for the computation of full magnet
ramp-cycles. The minimum energy required to quench part of a strand may
be in the order of some micro joules while the total stored energy in a magnet
ranges at several mega joules.
Furthermore, quench computation constitutes a strongly non-linear prob-
lem. The relevant material properties, e.g. the thermal capacity and electrical
resistivity, vary over many orders of magnitudes during a quench. In addition,
superconducting materials exhibit hysteresis behavior.
On this account, the simulation of quench on accelerator type magnets
relies on the simplification of complex structures, homogenization of mate-
rial properties, and analytical sub models. Consequently, any quench model
is limited to a specific set of applications. This approach requires the cou-
pling of several computation methods, e.g. finite-differences, finite elements,
boundary-elements, Biot-Savart’s law, and network-models. The variation of
material properties can be encountered by using an adaptive time-stepping
approach or iteration methods.
Quench models published in the literature mainly differ in their choice of
simplification and their set of applied sub-models.
1.3 Literature Overview
Quench codes Many calculation methods for quench in superconducting
magnets have been published. Except for the very recent publication of Aird
4 Introduction
[Aird 06], none actually aimed at the modeling of the entire multi-physics
problem.
Wilson published the computer program QUENCH [Wils 68] modeling the
quench propagation due to ohmic heating in superconducting solenoids. A
concept of field and temperature dependent, anisotropic quench propagation
velocities is used to span the normal conducting zone over the coil. This
approach was further developed by Rossi, first in DYNQUE [Cana 93] and
later in QLASA [Ross 04]. A similar approach for accelerator type magnets
can be found in [Laty 97] by Latypov.
Krainz modeled and studied the protection of the LHC test string using an
electrical network model [Krai 97]. The simulation of the current decay in a
quenching magnet considering the external network, and a string of magnets
based on the commercial software SABER was done by Rodriguez-Mateos
[Rodr 97, Rodr 96].
Sonnemann and Calvi developed a finite difference model for the simula-
tion of the quench propagation velocity based on thermal material properties
and the local magnetic field called SPQR [Sonn 01b, Calv 00]. In [Sonn 01a]
Sonnemann presented a quench model and quench measurements for mag-
nets of the LHC. Kim published a finite difference model for the simulation
of quench in [Kim 01]. The quench behavior of a superconducting undulator
was simulated with a finite difference model [Bett 06] by Bettoni.
Caspi and Masson presented thermal models for the quench simulation
using commercial finite elements packages ANSYS [Casp 03] and COMSOL
[Mass 07], respectively.
The only program following an approach of hard-coupling of the different
physical effects is the program of VECTORFIELDS [Aird 06]. Due to their
focus on superconducting solenoids, with a huge number of fine windings, an
individual winding scheme is not defined and turn-to-turn voltages can not
be calculated.
Sources on superconducting magnets and quench phenomenon In addi-
tion to the publications on quench simulation models, the following stud-
ies of the quench process have to be mentioned. Iwasa,Mess and Wil-
son address quench, relevant effects and quench protection in their books
[Iwas 94], [Mess 96] and [Wils 83], respectively. Especially the book [Brec 73]
of Brechna provides extensive material for the understanding of supercon-
ducting magnets and quench.
Devred performed analytical quench studies for the SSC [Devr 89]. The
thesis of Verweij provides valuable information on the stability of supercon-
ductors and on induced losses in superconducting cables [Verw 95].
Background for the presented work The work presented in this thesis
builds on the CERN field calculation program ROXIE by Russenschuck
[Russ 98]. The program provides all necessary functionality for the electro-
magnetic design of superconducting accelerator magnets including the numer-
ical computation of the magnetic field produced by superconducting cables
1.4 Objectives of the Thesis 5
and parts made from non-linear iron. In the past years, ROXIE has been ex-
tended to simulate field disturbances and losses stemming from inter-filament
coupling currents (IFCC) [Wils 83], inter-strand coupling currents (ISCC)
[Mari 04] and persistent superconductor magnetization currents [Voll 02].
Equipping ROXIE with an electrical network and thermal model allows for
a coupled computation of all physical phenomena occurring during a quench.
1.4 Objectives of the Thesis
The challenge of quench simulation is to model all relevant physical phenom-
ena and effects with adequate accuracy, so that internal states of a quenching
magnet can be reproduced and analyzed in order to understand the quench
behavior. This leads to the following objectives:
•Analysis of the quench process in superconducting magnets focussing
on all physical effects and their relevance for the quench model.
•Study and test of (existing) models describing relevant effects. Imple-
mentation and adaptation of the numerical models into a global com-
putation environment.
•Development of a quench algorithm: Study of the interdependence of
the different sub-models. Implementation of a computational scheme
reflecting computational cost as well as the necessary coupling between
the sub-models. Finding a numerical method to resolve changes in ma-
terial properties and time constants.
The quench model has to be verified by reproducing measurement data with
all material- and model-parameters chosen within the range of uncertainty.
In a second step, the model can be used to extrapolate the magnet quench
behavior to different initial conditions and slight design changes.
1.5 Structure of the Dissertation
The dissertation is divided in three complementary parts, i.e. the thesis,
the detailed treatment and the appendix. The thesis is structured in 4 main
chapters:
•Overview over features of accelerator magnets and physical phenomena
occurring during a quench. We follow a typical quench process through
a magnet and discuss the physical phenomena in respect to the magnet
application. Definition of the scope for the quench model.
•Introduction of numerical models for all relevant effects occurring during
a quench. We give detailed explanations on the implementation and/or
adaptation of the used sub-models and their internal states. The dif-
ferent models are weakly coupled in the quench algorithm and solved
6 Introduction
by means of an adaptive time-stepping method. The such developed
algorithm allows to study quench propagation, magnetic quench back,
and voltages building up within the magnet winding.
•Simulation results reproducing quench measurements from the LHC.
The quench model provides a detailed view on the internal states of
quenching magnets. The presented reproduction of the coil voltage
signal including characteristic spikes clearly indicates the advantages of
the coupled approach.
•Extrapolation of the simulation results on new magnet designs and pro-
tection concepts. We show a detailed protection study for the LHC
inner triplet upgrade. For the case of fast-ramping magnets we are
exploring new protection possibilities.
Moreover, we summarize the presented approach and obtained results. We
give a critical analysis on aspects which could be subjected to further studies.
In order to reduce the complexity of the main part, detailed treatments,
e.g. common definitions and concepts of superconducting magnet design,
derivations of electromagnetic models, and explanations of typical effects are
collected in the last chapter. Notice that the different subjects are put down
such that they can be read independently from the main parts.
•Analytical hot-spot calculation by means of the MIITs-concept indicat-
ing the need for numerical methods.
•Definition of the magnetic energy and the inductance in case of materials
exhibiting hysteresis and diffusive behavior.
•Derivation of the instantaneously dissipated hysteresis losses for the
critical state model of hard superconductors.
•Modeling Rutherford-type cables including induced losses.
•Overview over magnet protection methods and voltages occurring dur-
ing a quench.
In order to provide the reader with all necessary information to reproduce
the presented results the appendix contains the following sections:
•Researched values are provided for all materials commonly used in su-
perconducting accelerator magnets. We briefly review the physical back-
ground to all relevant material properties.
•The definitions and formulae of model parameters describing the coil
geometry.
•Description of all magnets presented in this work. We describe all pa-
rameters of the superconducting strand and cable, the coil cross-section
and iron yoke, the winding scheme and electrical network, as well as
magnet protection.
2 Magnet Features and Physical
Phenomena
Dicebat Bernardus Carnotensis nos esse quasi
nanos, gigantium humeris insidentes, ut possimus
plura eis et remotiora videre, non utique proprii
visus acumine, aut eminentia corporis, sed quia in
altum subvenimur et extollimur magnitudine
gigantea.
John of Salisbury
(1120 - 1180)
In this chapter we introduce relevant features of superconducting magnets
and physical phenomena occurring during a quench. Due to the excessive
requirements for the modeling of all physical effects, we have to reduce our
approach and disregard a number of effects. The reduction is motivated by
our application and briefly compared to other approaches.
2.1 Superconducting Magnets
We distinguish between superconducting accelerator-type magnets and other
superconducting magnets. The first are used to guide the beam of a parti-
cle accelerator on their orbit, i.e. bending them on a circular trajectory or
focussing the beam. The latter kind comprises superconducting magnets for
MRI, the LHC experiments ATLAS and CMS, as well as fusion projects like
ITER.
The typical design features of superconducting accelerator type magnets
are introduced by means of the LHC main bending magnet. Major differ-
ences to other accelerator type magnets, e.g. superconducting correctors or
superferric magnets, are discussed in the following. The section ends with
a brief overview on the features of other commonly used superconducting
magnets, e.g. solenoids, toroids, and undulators.
7
8 Magnet Features and Physical Phenomena
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2.1.1 The LHC Main Bending Magnet
A typical superconducting accelerator magnet consists of a superconducting
coil formed around the aperture, a non-magnetic collar to hold the coil in
place, quench-heater strips, an iron yoke, and a helium vessel for the operation
at cryogenic temperature. Figure 2.1 shows a CAD image of the 15 m-long
LHC main dipole displaying all relevant features.
Aperture, beam pipe and beam screen The LHC operates with two beams
of identical electrical charges circulating in opposite directions. Therefore,
magnetic fields of opposite orientation are required to keep the particles on
their circular trajectory. The LHC MB is a double aperture magnet with two
dipolar coils in a common iron yoke and cryostat. The size of the aperture
denotes the inner diameter of the coil (56 mm).
The particle beams circulate in two beam pipes providing an ultra-high
vacuum in the inside and the inner wall of the helium-II-vessel. Inside the
beam-pipe a perforated tube (beam screen) protects the magnets from syn-
chrotron radiation and part of the beam losses.
2.1 Superconducting Magnets 9
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Figure 2.2: (left) CAD image of a double aperture coil configuration in cos θ-design with
a schematic representation of the magnetic field. Notice how the coil windings are bent
upwards/downwards in order to leave room for the beam pipe. Courtesy of CERN c
.
(right) Quadrant of one aperture of the coil cross-section of the LHC MB showing the
magnetic induction at nominal current level.
Superconducting coil Contrary to normal conducting magnets, where the
magnetic field in the air gap strongly depends on the contribution of the
ferromagnetic poles, the field inside the aperture of a superconducting magnet
is basically created by the current in the coil.
The superconducting coil consists of an upper and a lower pole, which are
typically race-track-type coils mounted below and above the beam pipe. By
reducing the distance of the windings to the aperture, the efficiency and the
magnetic main field can be increased. The windings are therefore pressed on
a winding mandrel resulting in a coil shape as shown in Fig. 2.2 (left).
The dipole field configuration is obtained by approximating an ideal cos θ
current distribution, e.g. [Schw 05a, pp. 36]. Therefore, the coil windings
(also denoted as conductors) are combined to blocks of different size and
distributed around the aperture. The space between two adjacent blocks is
filled with non-magnetic material in order to hold the conductors in their
positions and to improve the circular shape approximation. Due to their
shape the spacers are denoted as wedges.
The field of the magnet is further increased by nesting two or more coil
layers. The coil cross-section of the LHC MB is shown in Fig. 2.2 (right).
The figure indicates the field in the aperture and over the conductors. Due
to the higher magnetic induction close to the aperture, the two layers use
different cable types allowing to reduce the current density in the inner layer.
In the coil ends, the conductors are bent up/down in order to leave room
for the beam pipe. This yields additional bending strain to the windings and
results in a reduced mechanical support. The coil field in the ends is higher
than in the straight part of the coil. This is compensated by means of the
iron-yoke end design.
10 Magnet Features and Physical Phenomena
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Figure 2.3: LHC Main Bending Magnet (MB) strand and cable. (left) strand cross-section
featuring a copper core, Nb-Ti filaments (dark spots) embedded in the copper matrix and a
coper coat. (right) Rutherford-type cable. The filaments of one strand are separated from
the copper matrix by etching. Courtesy of CERN c
.
Superconducting cable The superconducting cables are made from several
superconducting strands in parallel, twisted along the cable axis and pressed
into a trapezoidal cross-section. The cable current of nearly 13 kA is dis-
tributed over the 28 or 36 strands (inner and outer layer cable). The strands
are in resistive contact. This cable type is denoted as Rutherford-type cable.
The superconducting strands consist of several thousands of superconduct-
ing Nb-Ti filaments dispersed in a copper matrix, which provides mechanical
stability and a parallel path for the current in case of a quench.
The cable is surrounded by three wraps of polyimide insulation tape provid-
ing the necessary electrical turn-to-turn insulation in case of a quench while
being sufficiently porous to allow the liquid helium to percolate.
Figure 2.3 shows the photos of the strand and Rutherford-type cable used
for the LHC MB.
Ground Insulation and quench heaters The outer layer of the coil is equipped
with two sets of quench-heater strips. In case of a quench a capacitor is
discharged over these resistive strips generating enough heat to quench the
covered conductors.
The coil is covered with several polyimide layers as an insulation between
the upper and lower pole, and to ground. Operating a string of magnets, the
potential to ground of each coil is defined by the other magnets of the string.
The collar, iron yoke, and helium-II-vessel are connected to ground.
In order to protect the coil from damage resulting from mechanical stress
applied by the collars, metallic coil protection sheets are mounted in the
outside of the coil.
The convection of liquid helium from the coil through the collars to the
heat exchanger is hampered by the coil protection sheet in radial direction,
and azimuthally around the beam pipe by the flaps of the ground insulation.
2.1 Superconducting Magnets 11
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Figure 2.4: Photo of the coil cross-section of the LHC MQY featuring the aperture, the
insulation to ground including flaps, the coil protection sheet, the conductors, and the
collar. Courtesy of G. Kirby CERN TE MSC.
Figure 2.4 shows a photograph of the typical cross-section of a supercon-
ducting accelerator magnet featuring the aperture, insulation to ground in-
cluding flaps, the coil protection sheet, the conductors, wedges, and the collar.
(Notice that lacking a photograph of LHC MB with comparable quality the
LHC MQY is shown).
Collars and iron yoke Consider the coil cross-section, the opposing currents
of the two sides of the coil yield enormous electromagnetic forces pressing the
coil apart. Therefore, the Lorentz forces are carried by a stiff collar made
from non-magnetic material. Furthermore, the collar provides prestress to
the coil preventing conductor movements. In the case of the LHC MB the
two coils are accommodated in a common collar made of stainless steel.
The iron yoke of a superconducting accelerator magnet serves two purposes:
•Enhancing the field in the aperture without increasing the excitation
current or the number of superconducting cables. Hence, increasing the
efficiency of the design.
•Reducing the magnetic field outside the magnet, i.e. the fringe field, in
order to provide safe operating conditions for electronic equipment in
close vicinity.
In the case of the LHC MB, the iron yoke accounts only for 3% of the magnetic
induction compared to an iron free design [Evan 09, p. 75]. However, the iron
yoke reduces the stray field outside the iron yoke to less than 50 mT [Russ 07,
p. 391].
Both, collar and iron yoke, are stacked in order to reduce induced eddy
current losses. In the case of the permeable iron this results in an anisotropy.
12 Magnet Features and Physical Phenomena
Helium-II vessel and heat exchanger The coil, the collar and the iron yoke
are immersed in liquid helium II at 1.9 K. Helium II features an extremely low
viscosity and percolates every cavity inside the coil windings and the stacked
collar and iron yoke. The heat exchanger pipe provides a heat sink.
The helium-II vessel is made of the shrinking cylinder on the outside, the
beam pipe on the inside and the end caps. The liquid helium is filled in to
the vessel via the bus-bar pipes. The helium-II vessel is wrapped in super-
insulation blocking heat radiation and placed in a vacuum cylinder preventing
convective heating.
Magnet protection and external network In case of a quench the magnet
needs to be protected from excessive voltages and temperatures. A quench
needs to be detected as soon and reliably as possible and the current has to
be extracted as fast as possible.
During operation the differential voltage over the two apertures is measured
and compared to a given threshold voltage. When the voltage exceeds the
threshold and the reading has been verified after a predefined delay, a quench
is detected.
The LHC MB is operated in strings of 154 dipoles all connected in series. A
current controlled power supply is setting the current in the string. In case of
a quench, the power supply is switched off and by passed by a free-wheeling
diode. A dump resistor is switched into the circuit in order to extract the
current.
A bypass diode is mounted inside the cryostat of every magnet. After
quench detection the quench heaters of the quenched magnet are fired. The
resistive voltage switches the by-pass diode and the magnet current begins to
commutate into the diode. Hence, the magnet disconnects from the rest of
the string allowing for a faster current decay.
Bus bars The connection terminals of the dipoles are on one side of the
magnet. For the series connection of the string of magnets the bus bar is
placed within the cryostat on the outside of the iron yoke. The cryostat of
the dipoles further more hosts small superconducting corrector magnets and
their powering, as well as the current leads for the main quadrupoles.
2.1.2 Other Magnet Types and Design Features
Design features less pronounced or relevant for the LHC main dipoles are
introduced by means of other magnet types. Due to the wide variety of appli-
cations and technical solutions, this list is in now way complete, nevertheless,
will give the necessary overview.
Corrector magnets In order to compensate for field errors of the main bend-
ing and quadrupole magnets, and for the correction of particle offsets, correc-
tor magnets are incorporated into the accelerator ring. The design of corrector
magnets ranges from spool-pieces, small superconducting multipole magnets
2.1 Superconducting Magnets 13
wound from a single wire and placed on the beam pipe, to double aperture
magnets of a few meters.
In the most basic design, the corrector is operated well below the critical
surface, leaving enough margin to withstand probable disturbances [Evan 09,
p. 76]. For the case of a quench, the magnets are provided with a parallel
resistor taking over the current as soon as the quench has grown wide enough.
Winding the coil with a single wire results in a significantly higher inductance.
Super ferric magnets The design of super ferric magnets is based on the
same principles as for normal conducting magnets. The magnetic field is
concentrated and formed by a ferromagnetic iron yoke in the center of the coil.
Instead of water-cooled copper cables the coil is wound from superconducting
cables. Owing to the high available current densities in superconductors and
the domination of the field quality by the iron yoke, super-ferric magnets can
be made with a small number of winding turns.
It is possible to use cable-in-conduit conductors; superconducting filaments
inside a metal tube cooled by a constant flow of liquid helium. The super-
conducting coil is hosted in a cryostat mounted inside the iron yoke. With
additional cooling, the magnet can withstand higher losses and be subjected
to faster current ramps.
Block-coil and double pancake Alternatively to coils designed following a
cos θ-approximation, in the block-coil or double pancake design, the coil cross
section consists of rectangular blocks arranged around the aperture.
Figure 2.5 shows a sketch of three non-accelerator type magnets: a toroidal
coil, a solenoid, and an undulator. Notice that all three magnet types are
based on completely different topologies and frames of reference. Hence, they
require a different description and calculation method.
Solenoid Superconducting solenoids can be wound from a single strand
(MRI) or a Rutherford-type cable (CMS).
In the first case, the coil features large number of windings and conse-
quently a high inductance. Such systems may be subdivided to steer the field
quality or for quench protection. All sub-coils are provided with a bypass
diode or shunt resistor constituting a inductively coupled system. Winding
the solenoid on a conductive central cylinder yields a coupled secondary coil
providing resistance and heating during a current ramp or quench.
In the case of CMS, the cable used for the solenoid is surrounded by high-
quality aluminum (many times the cross-sectional area of the cable) in order
to provide mechanical stability as well as a parallel path in case of quench.
Superconducting Undulator Equipping an undulator with superconducting
coils allows to operate with much higher current densities compared to copper
14 Magnet Features and Physical Phenomena
!"#$%
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+,-.$%%
!/$#(),0%
1"2,%
Figure 2.5: Sketch of other superconducting magnet configurations: part of a supercon-
ducting undulator featuring an iron yoke and two and a half poles, solenoidal coil wound
on a metallic cylinder, and a toroidal coil configuration.
coils. Nevertheless, the poles of the iron yoke saturate completely and the
field is mainly created by the coils [Hila 03].
The coils of the different poles may be connected in series and by-passed
by diodes, or powered individually. In both cases, the current profile in all
coils are coupled by the non-linear mutual inductance of the iron yoke.
Toroids In toroidal systems as for ATLAS or ITER, every coil is hosted in
its own cryostat. The cable-in-conduits can be used for the windings allowing
to cool effeciently. The coils are magnetically coupled.
2.2 Quench Process
The various physical effects occurring during a quench in a superconducting
magnet are introduced following a typical quench in one of the LHC main
bending magnets. Subsequently, we refer to other relevant phenomena.
Quench cause A superconductor quenches when the working point definedCritical
Surface:
Sec. A.2 by the local magnetic induction, the current density, and the temperature
exceeds the critical surface. The temperature difference between the working
temperature and the maximum temperature for the given field and current
density is denoted temperature margin; the conductor quenches for zeroTemperature
margin:
Sec. 7.1 margin.
A temperature increase in the magnet can be caused by conductor move-
ments and the associated friction, induced eddy current losses in the supercon-
ducting cables, radiation and beam particles entering the coil, quench heaters
and heat transfer from adjacent conductors. The current density in the su-
perconducting cables consists of the current provided by the power supply
and the current driven by the induced voltages. Both can result in a quench
in case of an over-current. Figure 2.6 summarizes the different phenomena
leading to a quench.
For accelerator magnets we can distinguish the following types of quench
[Mess 96, p. 119]:
2.2 Quench Process 15
Figure 2.6: Different phenomena in superconducting magnets which may cause a quench.
•We talk of a natural quench, when the working point on the load line is
moved across the critical surface by raising the excitation current and
thus, synchronously, the magnetic flux density.
•Quenches can also be initiated due to local heating or beam losses in the
accelerator. These disturbance quenches may happen with the working
point well below the critical surface.
•Often coil windings move slightly under the influence of the electromag-
netic forces and due to pressure changes during quenches. This may re-
sult in a successively higher field at subsequent quenches. The magnet
thus shows a training characteristic and the quenches are consequently
referred to as training quenches.
For the following explanations we consider a disturbance quench at a cur-
rent level close to the nominal current of the magnet. The initial normal
conducting zone extends only over some centimeters of a strand in the high-
field region.
Current-sharing, current re-distribution In the superconducting state the Electrical
resistivity:
Sec. A.1.3
& A.2.6
resistivity of Nb-Ti is many orders of magnitude smaller than the resistivity
of other commonly used conductors, e.g. copper or aluminum. In the nor-
mal conducting state the electrical conductivity of Nb-Ti is comparatively
small. Figure 2.7 shows the resistivity of most materials commonly used in
superconducting cables.
During the transition form the superconducting to the normal conducting Current-
sharing:
Sec. 7.5
state the current commutates from the quenched filaments into the surround-
ing matrix material. The commutation is governed by the voltage-current-
characteristic of the superconductor and Ohm’s law in the resistive matrix.
This phenomenon is called current-sharing.
The strands of the Rutherford-type cable are in resistive contact. With Current re-
dsitribution:
Sec. 7.8
growing resistivity of the quenched strand, the current starts to by-pass the
quench: The current flows over the cross-over and adjacent resistance to
16 Magnet Features and Physical Phenomena
Cu
Nb!Ti
Nb3Sn
0T
1T
5T
10 T
20 T
0T
1T
5T
10 T
20 T
0T
1T
5T
10 T
Al
Nb
Sn
Ta
x
Ti
1 2 5 10 20 50 100 200
10!12
10!11
10!10
10!9
10!8
10!7
10!6
Tin K
ΡE!T,B"
in #m
Figure 2.7: Electrical resistivity of the different materials used in superconducting cables.
For the superconducting materials a design current density is assumed of Jc(8.4 T,1.9 K) =
2.6·109Am−2for Nb-Ti and Jc(15 T,4.2 K) = 1.4·109Am−2for Nb3Sn. The resistivity
of tantalum, niobium, tin and aluminum are displayed by means of an arrow pointing from
the resistivity at room temperature to cryogenic temperatures. For titanium only a room
temperature value could be given. See also material properties in Sec. A.1.3 and Sec. A.2.6
neighboring strands. This is denoted as current re-distribution and happens
within some twist pitch lengths.
Ohmic heating, heat transfer, quench propagation, and quench recovery
The current flowing through the resistive part of the strand and cable causes
ohmic heating. The temperature starts to rise. The temperature gradient
causes a heat transfer along the strands, to the confined helium and over the
insulation to adjacent conductors as well as to heat reservoirs outside the
cable.
Below the λ-point helium II features a very high heat conductivity, al-Properties
of liquid
helium:
Sec. A.3
lowing to efficiently extract heat from the coil to the heat exchanger pipe.
Furthermore, the heat capacity of liquid helium is many orders larger than of
the materials used for the cable. Nevertheless, the heat transfer between the
quenched strand and the percolating helium features phase transitions, alter-
ing the thermal contact on the surface [Gran 08]. Therefore, the influence of
the helium strongly depends on the amount of heat and the time-span of its
release.
With the heat transfer the quench propagates along the cable and to ad-
jacent conductors. The propagation velocity and turn-to-turn propagation
2.2 Quench Process 17
delay depend on the temperature margin to quench as well as on the avail-
able cooling power and heat conductivity.
A superconductor recovers from a quench if the working point drops be-
low the critical surface. This can happen, e.g. due to the reduction of the
current density as a consequence of current-sharing/re-distribution or if the
ohmic heating in the surrounding matrix/strands is smaller than the available
cooling. Therefore, superconductors are provided with a maximum amount
of parallel copper for stabilization without increasing the strand cross-section
too much.
Quench load and MIITs Assuming adiabatic conditions disregarding any MIITs:
Sec. 7.2
heat transfer, the temperature increase can be estimated based on the current
profile and effective material properties. The integral over the profile of the
current squared represents the quench load or MIITs. For the given electrical
resistivity and heat capacity of the cable the quench load gives a conservative
temperature estimate.
Magnet protection With increasing temperature and quench propagation Magnet
protection:
Sec. 7.9
a resistive voltage builds up over the magnet. When the voltage passes the
threshold of the quench detection system, the power supply is switched off and
the current extraction is initiated by increasing the resistivity of the circuit.
Quench heater delay Upon quench detection, the quench-heater power-
supplies discharge capacitors over the resistivity of the heater strips in the
coil. The heat diffuses through several layers of insulation into the conductors
increasing the temperature above the quench limit. The time between firing
and quench in one of the covered conductors is denoted as the quench heater
delay. The quench heater delay depends on the amount of energy released,
the thermal properties of the contact between the heater strip and the coil,
and the margin to quench in the covered conductors.
Internal voltages and iron saturation The resistance of the conductors Voltages
during
quench:
Sec. 7.10
quenched by the quench heaters increases the voltage beyond the forward
threshold of the protection diode. The current of the string of magnets by-
passes the quenched magnet over the diode. The magnet is isolated from the
string and the terminal voltage is clamped to the diode forward voltage.
Consequently, the growing resistive voltage is matched by an inductive
voltage driving down the current in the magnet. The resistive voltage is
concentrated over the normal conducting parts of the coil. This voltage con-
centration can yield high electrical fields between adjacent turns and may
results in a dielectric break-down of the insulation.
Parasitic capacitances in the coil, i.e. between the windings and from
the windings to ground, influence the voltage distribution over the magnet.
Directly after the occurrence of a resistive voltage, the voltage is concentrated
18 Magnet Features and Physical Phenomena
around the disturbance thus yielding higher electric fields and even small
equilibration currents.
The distribution of the inductive voltage over the coil depends on theDifferential
inductance:
Sec. 7.4 linked flux over the different turns and is not homogeneous. In addition,
the magnetic induction in the magnet cross-section is subjected to saturation
effects in the iron yoke and induced eddy currents in the coil. Therefore, any
induced voltages or the differential inductance are in non-linear, diffusive and
hysteretic dependence of the magnet current.
Induced losses The Rutherford-type cables are subjected to three differ-SC
hysteresis
losses: Sec.
7.7
ent types of losses: Superconductor hysteresis losses in the filaments, inter-
filament coupling current (IFCC) losses in the strands, and inter-strand cou-
pling current (ISCC) losses in the twisted cable. The hysteresis losses result
from the persistent screening currents typically screening the inner of super-
conductors. The coupling currents are driven by the voltages induced overISCC and
IFCC: Sec.
7.6 superconducting loops closing over the resistive matrix of the strand or the
contact resistance between adjacent strands. The induced eddy current losses
increase quadratically with the time derivative of the magnetic induction.
Furthermore, losses are induced in the copper wedges between coil blocks
and in the laminated collar. The iron yoke shows small hysteresis losses.
Quench back Green distinguishes two different kinds of quench back in
superconducting magnets [Gree 84a]: thermal quench back and magnetic
quench back. In the first case, normal conducting regions are induced by
heat transfer from other parts of the magnet, e.g. from an aluminum cylin-
der in the center of a solenoid or copper wedges in a cos θmagnet. In the
second case, normal regions are caused by induced eddy current losses in the
superconductors due to changing magnetic fields.
Under normal operating conditions the amount of induced losses in a super-
conducting magnet has to be minimized and removed by cooling. However,
in the case of quench the magnetic quench back aids the built-up of resistivity
and fast current decrease. In the LHC MB wide parts of coil-cross section are
quenched by magnetic quench back due to the fast current decrease.
Quench of adjacent magnets, helium heat wave In the case of a string
of magnets, the temperature increase in one magnets can compromise the
cooling of the adjacent magnets. Furthermore, neighboring magnets can be
quenched by a front of warm helium traveling along the bus bars.
The following phenomena occur in other magnet configuration:
Convective cooling The heat stemming from induced losses or a quench
can be removed by heat conduction or the convection of liquid helium. In the
later case, the magnet design needs to be permeable for the helium flow or
have tubes where the helium can be pumped through. Superferric magnets
2.3 Relevant Effects 19
as well as the toroids for the ITER tokamak are cooled by a forced helium
flow.
Inductive coupling Consider an inductively coupled systems consisting of
two coils connected in series. Each of the coils is bypassed with a diode. After
a quench in one of the coils occurs, the resistive voltage growing due to quench
propagation and temperature increase, switches the diode and disconnects the
quenched coil from the rest of the circuit.
With the decreasing current in the quenched coil, a rising current is induced
over the other coil. When the current exceeds the critical current limit in the
second coil, the coil quenches instantaneously. Now the current decrease
of the second coil drives up the current in the first coil. This concept is
used for the protection of superconducting solenoids where a great number of
coils is coupled causing multiple current commutations, or of superconducting
undulators where the different coils are additionally coupled by non-linear iron
material.
In solenoids a central alumnium cylinder can be considered a coupled sec-
ondary, providing additional resistance and heating in case of a quench.
Mechanical stress and coil deformations The fast temperature increase
and the consequent thermal expansion of the different coil materials exposes
the conductors to mechanical stress. This can yield to reduction of the crit-
ical current density (degradation) or even conductor movement. Both may
influence the development of the quench and result in irreversible damage to
the magnet.
2.3 Relevant Effects
Accelerator type magnets can be distinguished by means of the following five
categories: Length of the magnet, margin to quench, magnetic induction level,
ramp-rate, and powering.
Length of the magnet: A magnet is considered short if the magnetic field
over the coil is dominated by 3D effects such as longitudinal iron satu-
ration or peak field increase in the coil end. Furthermore, a magnet is
considered short if quench propagation contributes significantly to the
resistance built-up compared to dump resistors or quench heaters.
In the case of long magnets the problem can be merely treated in a 2D
approach.
Margin to quench: The margin to quench is considered small if a quench
of one strand in a cable triggers a quench in all parallel strands. The
quench is immediate.
For magnets with a large margin, the effects of current-sharing, current
re-distribution and quench recovery have to be taken into account. This
20 Magnet Features and Physical Phenomena
allows to study the stability of the cables/magnet against disturbances
such as beam losses. A model of the cooling of the conductors is crucial.
Level of magnetic induction: The magnetic field level is high if iron satura-
tion effects have to be considered. This requires to apply the concept
of the differential inductance.
Ramp-rate: In magnets built for high ramp-rates design efforts are made in
order to reduce induced losses in the cables and to remove the additional
heat. Due to high induced voltages quench detection becomes more
difficult.
In magnets designed for slow ramping, quench back is more pronounced.
Powering: We distinguish between single powering and magnet systems. The
first kind can be protected individually. For the second kind, protection
and operation has to extended to coupled coils.
In the following we develop a model for the description of long magnets oper-
ating at high magnetic induction and small margin to quench. The magnets
are designed for a small ramp-rate and are individually powered. The model
can be extended to study effects in fast ramping magnets as well as to analyze
quenches in strings of magnets.
For the simulation of magnets requiring other features such as undulators
or solenoids, more specialized codes, e.g. as by Aird or Bettoni have to be
used.
3 Numerical Modeling
"Consider a spherical cow... ."
John Harte
(1988)
A quench is the resistive transition of a superconductor that occurs if the cur-
rent density, the magnetic field in the cable, or the cable temperature exceeds
a critical value. From this description it is evident that quench simulation
requires a multi-physics modeling approach. Figure 3.1 shows the different
physical models interacting during the quench simulation:
a) A geometrical model describing the magnet and coil and providing dis-
tances and the size of all elements.
b) A model of the superconducting state and the transition to the normal
conducting state.
c) Magnetic field computation consisting of analytical field computation of
line currents by means of the Biot-Savart law and a BEM-FEM model
for the numerical field computation of the magnetic induction in non-
linear iron parts.
d) Analytical models for the cable magnetization, i.e. induced cable eddy
current losses like inter-strand coupling currents and inter-filament cou-
pling currents.
e) An electrical network model describing the external electrical circuit by
means of lumped elements and determining the magnet current. Pro-
tection measures, e.g. bypass diodes and energy extraction resistors,
are included in this model.
f) A thermal model for the calculation of the temperature increase and
quench propagation. The model includes the quench protection heaters.
The solution of the individual sub-models is combined in the quench algo-
rithm. Depending on the computational effort and the grade of coupling,
the models are solved in two nested computation loops sharing their parame-
ters and results. The strong variation of model parameters is resolved by an
adaptive time-integration method.
21
22 Numerical Modeling
B
T
J
!TC
!JC
B1
B2
!"# $"#
%"#
&"#'"#
("#
Figure 3.1: Different models interacting in a quench simulation. Clockwise starting at
11h00: a) geometrical model and discretization of the coil, b) superconductor model and
quench decision, c) magnetic field computation with analytical field calculation and BEM-
FEM model of the nonlinear iron yoke, d) induced losses by means of inter-filament and
inter-strand coupling currents, e) electrical network of powering and protection elements,
f) thermal network model of heat conduction and quench heaters.
3.1 Coil Discretization and Numbering Scheme
A superconducting magnet basically consists of the coil, the iron yoke, and
the electrical circuit for powering and protection. While the electrical circuit
can be described by means of lumped elements, the description of thermal
and electromagnetic processes in the coil require a discretization.
We assume that a magnet is long, compared to the diameter of the magnet
aperture (generally true for dipole- and quadrupole magnets in accelerators)
and thus the numerical field computation can be carried out in two dimensions
(2D). Coil end effects are neglected. Where necessary, global quantities, such
as the inductance, are matched to measurements by geometric scaling factors.-
Thermal computations are carried out in three dimensions (3D). The coilMagnetic
and
average
winding
length:
Sec. B.1.6
cross-section is longitudinally extruded to the average winding length `w. The
connection of the conductors in the coil ends is seamless and modifications
to the local properties are not considered. We denote the approach as 2+1D
in comparison to a full 3D model also including the coil end region. The
approach yields the following discretization and numbering scheme:
•Transversally, the coil can be sub-divided by means of the different
conductors introducing the numbering i= 1, ..., Nc.
•Longitudinally, every winding length `wis discretized in Nzelements of
length `z=`w/Nzwith the numbering scheme j= 1, ..., Nz.
The coil thus consists in total of NcNz=Neelements; identified by the tuples
3.2 Superconductor Model 23
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#"
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%"
&"
%"
#'"
("
)"
%"
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%$" $"
+,-'./!"#$%&'()'*+,
Figure 3.2: Discretization and numbering of a coil: (left) Transversal or 2D discretization
by means of the conductors i. (center) Longitudinal or 2+1D discretization jover Nzcuts.
(right) Topological numbering ξalong the winding of the coil. The magnet terminals are
denoted ξ= 0 and ξ=Ne(Number of elements).
ij. The numbering scheme is illustrated by means of a schematic coil in Fig.
3.2 (left) and (center).
Topologically one index is enough to identify a specific element in the coil.
In order to represent the winding scheme of the coil, i.e. the connection of the
different conductors in the coil ends, a complementary numbering scheme is
introduced. The greek letter ξfollows the winding through the coil from one
terminal to the other, ξ= 1, ..., Ne. The mapping Ξlinks the two numbering
schemes:
Ξ : ξ→ij (3.1)
The numbering scheme Ξdefines a sense of orientation for the elements. If ξ
increases with the z-coordinate it is considered positive:
δi
Ξ=(1ξincreases with z
−1ξdecreases with z(3.2)
The magnet terminals are assigned to ξ= 0 and Ne. The current Ienters
the magnet at ξ= 0 in z-direction. Figure 3.2 (right) shows the topological
numbering scheme for a schematic coil.
In case of Ribbon-type conductors the magnet is discretized on the strand Ribbon-
type
conductor:
Sec. B.1.3
level. The grouping of the strands in cables is thus ignored.
Notice that the numbering schemes ij and ξare used through out this
chapter.
3.2 Superconductor Model
For the model of twisted Rutherford-type cables it is assumed that the current Rutherford-
type
cables:
Sec. 7.8
Iis evenly distributed over all strands of the cable. Ribbon-type cables are
modeled as cables consisting of one strand only.
24 Numerical Modeling
A strand consists of superconducting filaments embedded in a normal con-Electrical
resistivity:
Sec. A.1.3
and A.2.6
ducting matrix. Comparing the electrical resistivity of superconductors, e.g.
Nb-Ti and Nb3Sn, and common matrix materials, e.g. copper or aluminum,
in Fig. 2.7, it can be concluded: In the superconducting state, the current
is carried only by the superconducting part of the strand. In the quenched
state, the current is only carried by the normal conducting part of the strand.
During the transition from the superconducting to the normal conducting
state, the current is shared between both fractions.
Resistive transition An element of the coil is considered to be in the normalQuench
margins:
Sec. 7.1.1 conducting state when the current density margin (Sec. 7.1.1) equals zero:
∆Jij
c=(Jc(Bi
peak, T ij)−Ji
SCm Ji
SCm < Jc(Bi
peak, T ij)
0else ,(3.3)
where Jcis the critical surface parameterization, Bi
peak the peak field on theJcparame-
terization:
Sec. A.2.4 conductor iand Tij the temperature of the element ij. Here, Ji
SCm denotes
the current density in the superconducting filaments, if the applied current
Iis only distributed over the superconducting fraction of the cable Ai
cab,SC
(defined in Sec. B.1.3):
Ji
SCm =I
Ai
cab,SC
.(3.4)
For the visualization of the quench process in time and over the coil cross-
section, the current density and the temperature margin to quench can be
used.
Current-sharing When the critical current density is exceeded, i.e. the el-
ement is in the quenched state, the voltage across the superconductor starts
to rise rapidly. With the increasing voltage, part of the current through the
strand is drawn into the normal conducting material in parallel. The ohmic
losses and the consequent heating of the strand further decrease the critical
current density until the entire current is carried by the normal conducting
matrix. The commutation process is determined by the strongly non-linearNon-linear
behavior of
SC: 7.5 voltage rise over the superconductor.
Following Stekly [Stek 65], a linear approach is used to model current-
sharing: Exceeding the critical current density the excess current commutates
from the superconducting fraction of the strand into the normal conducting
matrix. The excess current is given byCurrent
sharing:
Sec. 7.5.1
Iij
NC =
0 ∆Jij
c>0
IM−Iij
c(∆Jij
c= 0) & (Iij
c>0),
IMelse
(3.5)
where IMdenotes the current through the conductor and Iij
cthe critical
current in the element for a the local magnetic induction and temperature:
Iij
c=Jc(Bi
peak, T ij)Ai
cab,SC.(3.6)
3.3 Magnetic Field Model 25
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Figure 3.3: Current sharing: The transition from superconducting to normal conducting
state can be subdivided into three phases. During the first phase, the conductor current
is carried by the superconductor (ISC =IM). The temperature in the conductor increases
only due to external heating or induced losses. The critical current in the conductor
decreases due to the increase of temperature. The second phase begins, when the critical
current drops below the conductor current and the excess current spills over into the normal
conducting part (ISC +INC =IM). The conductor is considered in the quenched state.
Ohmic heating in the normal conductor amplifies the temperature increase. The last phase
is reached, when the critical current has decreased to zero and the entire current is carried
by the normal conductor (INC =IM).
Only the current Iij
NC flows through the resistance of the element ij and
therefore causes ohmic heating. The conductor current IMflows entirely in
the normal conducting matrix as soon as the critical current is zero, i.e. the
critical field is zero. Figure 3.3 shows a schematic of the current transition
including current sharing.
3.3 Magnetic Field Model
Assuming that the magnet is long compared to the diameter of the magnet
cross-section, which is generally true for dipole- and quadrupole magnets in
accelerators, an 2D approach is used for the magnetic field calculation.
The magnetic field problem can be divided in 3 sub-domains, i.e. the
superconducting coil, magnetic materials as for the iron yoke, and empty
space, e.g. the beam pipe. The coil consists of superconducting cables which
are subjected to eddy currents and magnetization effects.
The quench model relies on the peak and average magnetic induction as
well as on the average magnetic vector potential on all conductors. Assuming
that induced eddy currents are small compared to the transport currents their
contribution to the magnetic induction is not taken into account. Therefore,
the field computation consists of the following two steps:
1. Calculation of the field created by the currents in the coils.
2. Computation of the secondary field of the magnetic materials as reper-
cussion on the coil field.
26 Numerical Modeling
!"#$%&'()*$
+"#,%&'()*$
-&).$
/()0$
/)0&*$
!1&).$
!)0&*$
!"
!1&).$
Figure 3.4: Magnetic field computation: The problem is subdivided into the coil, an iron
and an air domain. The magnetic induction Bis composed from the field of the coil and
the field of the iron domain.
The solution of both steps are iterated due to the field-dependent permeabil-
ity. Figure 3.4 shows the subdivision of the magnet and the field calculation
at an arbitrary position relying on coil and iron field. The total field is used
for the computation of the cable eddy currents and magnetization, and the
consequent losses.
Coil fields The magnetic field of the superconducting coils is calculated from
the field of ideal line currents. The cross-section of any conductor is evenly
subdivided into NDis patches, shown in Fig. 3.5 (left). Outside the cable,
more precisely outside any strand, the magnetic field is given by the sum
over the fields of the line currents placed in the center of the patches, ri
k.
If evaluating the field inside a strand, e.g. for the calculation of peak fields
on a conductor, the respective line current is replaced by a homogeneous,
cylindrical current distribution and singular expressions are avoided, see Fig.
3.5 (right).
The transposition of the strands along the axis of Rutherford-type cablesRutherford-
type cable:
Sec. 7.8 is not taken into account.
Magnetic Materials The influence of magnetic materials on the magnetic
field in the coils is calculated by means of the BEM-FEM-coupling method.
According to this method only domains containing magnetic materials are
meshed and described by means of finite elements (FEM). Empty space and
more relevant the coils are not modeled in finite elements but considered by
expressing the magnetic field on the boundary to the iron/FEM domain. The
field repercussion of the magnetic materials on the coil fields is calculated
using a boundary element approach (BEM). For field-dependent magnetic
permeability (iron saturation) the coupling between the two domains is solved
iteratively.
The BEM-FEM approach reduces the necessary number of finite elements
significantly. Furthermore, coil fields can be modeled more precisely and
independently of the used mesh.
Differences of the magnetic permeability, as shown in Sec. A.1.6.1 between
3.3 Magnetic Field Model 27
!!"
#"
!" !"#" #" #" #" #" #"
!"
!"
#"
$"
$"$"
%%"
$"&'($"
$"%()$"
*+,&-.#$%&'()$!*+,(!-./0.$%12-034+-5&6
Figure 3.5: Calculation of the field of a conductor. (left) Discretization of the conductor
i. The index kdenotes the patch number. (right) Peak field calculation. The field inside
a strand (shaded) is given by the field of all line currents, Bext, plus the field of the
homogeneous cylindrical current distribution inside the strand, Bstr.
various operation conditions and manufacturers, are negligible for the calcu-
lation of quench relevant quantities, i.e. for coil fields and the inductance.
Total magnetic induction - Superposition The magnetic induction as well
as the magnetic vector potential at any of the positions ri
kis given by the
sum of all field sources. In the present work only the coil field and the iron
repercussion are considered, thus
Bi
k=X
model
Bmodel(ri
k) = Bcoil(ri
k) + Biron(ri
k)(3.7)
Ai
z,k =X
model
Amodel(ri
k) = Az,coil(ri
k) + Az,iron(ri
k)(3.8)
Discrete field quantities For the present work two different cable types
are considered: Ribbon-type and Rutherford-type cables. The strands of a Cable
types: Sec.
B.1.3
Rutherford-type cable are twisted along the cable axis, i.e., every strand of
the cable takes every position within the cable cross-section over the twist-
pitch length. With a twist-pitch much shorter than the magnet length, the
magnetic field varies only over the cross section but not along the conductor.
Therefore each strand is exposed to the full variation of the field. For the
calculation of material properties the average field is used as discussed in Sec.
7.8.
For the calculation of field dependent material properties, e.g. magneto-
resistivity and specific heat of superconductors, the average modulus of the
field on the conductor is required. The quench decision depends on the peak
28 Numerical Modeling
field on the conductor:
Bi
av =PNi
Dis
k=1 Bi
k
Ni
Dis
,(3.9)
Bi
av =PNi
Dis
k=1 Bi
k
Ni
Dis
,(3.10)
Bi
peak = max{Bi
k}k=1,...,Ni
Dis ,(3.11)
The average magnetic vector potential in conductor iis used for the calcula-
tion of linked fluxes and induced voltages,
Ai
z,av =PNi
Dis
k=1 Ai
z,k
Ni
Dis
.(3.12)
We can assign a local frame to every conductor cross-section by defining
a vector parallel, e||, and orthogonal, e⊥, to the broad face. For keystoned
cables the parallel vector is approximated by the cable center as indicated in
Fig. 7.20. We define the average magnetic induction parallel and orthogonal
to the broad face as:
Bi
|| =Bi
av ·e||, Bi
⊥=Bi
av ·e⊥.(3.13)
Although the variation of the magnetic field in the coil ends is not directlyMagnetic
length taken into account for the computation of losses or material properties, it has
to be considered for the computation of induced voltages and the differential
inductance. In Sec. B.1.6, three different coil lengths are introduced, the
winding length `w, the magnetic length `mag and the inductance length `ind.
Comparing the inductance per length as well as the inductance of a magnet
measured or calculated in 3D gives the inductance length. For the calculation
of induced voltages over elements the factor
kmag =`ind
`w(3.14)
is used to scale the longitudinal extensions.
3.4 Induced Losses
Eddy-current losses in the cable are calculated from the local field sweep. WeCable eddy
currents:
Sec. 7.6 distinguish inter-filament (IFCC) and inter-strand coupling currents (ISCC).
Interfilament coupling currents are induced in the twisted superconductor-
copper matrix of a strand. Inter-strand coupling currents are induced in a
Rutherford-type cable in loops of superconducting strands and contact re-
sistances between strands. The two phenomena are summarized as cable
magnetization, see Sec. 7.6.
Superconductor magnetization hysteresis losses are disregarded as discussedHystersis
losses: Sec
7.7 in Sec. 7.7. Eddy currents in copper wedges between coil blocks and in coil
collars are not taken into account.
3.5 Electrical Network 29
Analytical magnetization losses Consider a Rutherford-type cable consist-
ing of Ni
sstrands. The loss density for a time-variant magnetic induction are
given by Eq. (7.63). Since every strand is fully exposed, the average loss per
strand can be calculated from the average magnetic induction over the cable
cross-section. The inter-filament coupling losses dissipated in the cable are
then given by
Pi
IFCC =Vi
e,strpi
IFCC,av,(3.15)
where Vi
e,str denotes the volume occupied by the strands in any element of
the conductor i.
The inter-strand coupling loss density calculated from the average magnetic
induction as given in Eq. (7.68). We get:
Pi
ISCC =Vi
e,strpi
ISCC (3.16)
In case of a quench the ohmic heating over the resistive zone, exceeds the
induced losses. Furthermore, the loss models are based on the assumption
of superconducting filaments and strands. Therefore, induced losses are only
taken into account while an element is in the superconducting state. The total
losses in one element are calculated from the sum of different loss models
Pij
0,losses =(Pi
IFCC +Pi
ISCC ∆Jij
c>0
0else (3.17)
Time dependence The analytic loss models are lacking the diffusive process,
i.e. the losses are an immediate response to any field change. In order to
avoid an over estimation, the coupling-current time-constants τlosses is a user
supplied parameter. The total losses are given to
Pij
losses =Pij
0,losses 1−exp −t−tfc
τlosses ,(3.18)
where tfc denotes the instance of the last field change.
Notice, for the quench algorithm it can be required to split the time interval
τlosses over several calculation steps. The steady continuation of exponential
decays is shown in Sec. 7.5.2.
3.5 Electrical Network
The magnet is connected to an external network represented by an generic
network model as shown in Fig. 3.6. It consists of a power supply, an energy
extraction system with switch and dump resistor, and a serial resistance and
inductance. The power supply is bridged by a free-wheeling diode in order to
conduct the current after it is switched off. The serial elements can represent
resistive current leads and the inductance of magnets connected in series to
the magnet. The magnet itself is represented by its electrical resistance and
30 Numerical Modeling
the differential inductance. It can be bridged with a parallel resistor or a
by-pass diode.
This generic electrical network allows to simulate quenching magnets in
single operation, e.g. on a test bench, or in a string, e.g. as for the main
dipoles in the LHC tunnel. Both configurations are explained in Sec. 7.10.1
and in Sec. 7.9 in respect of quench protection. The by-passing of the quench-Magnet
protection:
Sec. 7.9 ing magnet is explained in Sec. 7.9.3. The generic approach allows only to
simulate one current in the magnetic system. Therefore it does not permit
to simulate quench protection by magnet subdivision or a coupled secondary
winding (see 7.9.4).
Quench detection and validation A quench is detected as soon as the resis-Sequence
of events:
Sec. 7.9.6 tive voltage over the magnet Ures exceeds the detection threshold voltage Udet.
The quench is validated, if the resistive voltage remains above the threshold
level after a discrimination time ∆tDis. Quench protection measures are trig-
gered after quench validation, t=tval.
The numerical simulation automatically distinguishes the induced and resis-
tive voltage over the magnet. Therefore, compensation methods as described
in Sec. 7.9.1 are not required. Nevertheless, it is possible to monitor and use
any voltage within the magnet windings for quench detection.
Current change The generic electrical network model can be described by
means of three currents: the current through the quenching magnet IM, the
current through the by-pass diode IDand the current in the main circuit
IE. The current through the power-supply commutates instantaneously into
the free-wheeling diode after switch-off and therefore does not require the
definition of a fourth independent current. The three currents are related by
IE=IM+ID.(3.19)
The underlying mechanism of current decrease in the three branches, changes
with the switching of the two diodes. While the power supply is connected
to the network, the current function IPS(t)is imposed, i.e. IPS =IM=IE.
Quenching of the magnet or switching of the dump resistor show no effect.
The current change is given by,
dIE
dt=dIM
dt=dIPS
dt,dID
dt= 0.(3.20)
After the power supply is disconnected, the current decrease is defined by the
main circuit (IE=IM) and yields
dIE
dt=dIM
dt=−RQ+RsIM+UDR +UfDf
Ld+Ls
,dID
dt= 0.(3.21)
When the terminal voltage UTerminal reaches the threshold voltage of the by-
pass diode, the magnet current starts to commutate into the parallel path.
3.5 Electrical Network 31
!"#$%&'!()$*+,-"'.&/)#0'
!12'
"12'
#/'
34#"+56"%'7*%"#)'
"$#/' "6"8'
"9#$06"*:'
!3' #8'
"+1;'
$7'
$1'
!/'
.#$6*:'!:#0#")/'
<&=>*//'16-8#'
$!'
"?.'
";1;'
?-@#$'.4>>:&'
A$##=@5##:6"%'
16-8#'
7-8#:B!"#$%&'$(")#%*+&,-
Figure 3.6: Generic electrical network model: Quenching magnet, energy extraction
system with dump resistor and switch, power-supply with free-wheeling diode, and serial
elements. The magnet is represented by the resistivity of the quenched conductors and its
differential inductance. It can be bridged with a parallel resistor or a diode.
The current decrease in the main circuit thus depends only on the serial
inductance.
dIE
dt=−UfDf +UcDf +UDR +RsIE
Ls
,(3.22)
dIM
dt=−RQIM−UcDf
Ld
,(3.23)
dID
dt=−dIM
dt.(3.24)
If no serial inductance is present, the current in the main loop IEdrops to
zero instantaneously.
For a string of magnets, where the total inductance is orders of magnitude
larger than the inductance of the quenching magnet, protected by means of
quench heaters, the main current IEcan be considered as constant (see Sec.
7.10.1).
3.5.1 Lumped Network Elements
Power supply In the absence of a quench, the magnet current is imposed by
the power supply. The current ramping IPS(t)is determined by the magnet
operating conditions. The voltage over the power supply is controlled such
that the current change follows the specifications. This compensates for the
non-linear inductance and the resistive components in the circuit, i.e. resistive
joints or a quenching magnet.
After quench validation, the power supply switch-off is delayed by ∆tQT.
The current commutates instantaneously into the free-wheeling diode.
32 Numerical Modeling
!"!"#$%
!"!&#$%
!"!'%
#!%
!"!()*%
+$"!$%
!'!#$% !'!'%
,(--.&$--/012%%
!03*-%
!'!#$%
%'!'4%
!'!'%
)5.6788%!03*-%
9"3/*:%
!"!#$%
%"!'4%
!"!'%
;%
;%
#<%
=3*-/>!"#$%&'(#$%)*
Figure 3.7: (left) Current-voltage characteristic of the cold and warm diode. (right) Diode
model with lumped electrical elements.
Diodes The current flowing though a semiconductor diode is a an exponen-
tial function of the applied voltage [Elsc 92, p. 46]. The non-linear voltage
current characteristic can be simplified assuming zero conductance for an
applied voltage smaller than the diode threshold voltage and a differential
resistance for voltages above the threshold [Elsc 92, p. 86]. This linearization
can be represented by means of three lumped electrical network elements:
A voltage source providing the constant threshold voltage, the differential
resistance limiting the current for voltages above the threshold voltage and
an ideal diode. The ideal diode switches on for a voltage greater zero and
switches off when the current through the diode equals zero. Any current in
reverse direction is thus blocked. Note that in off-state the constant voltage
drops over the ideal diode.
Figure 3.7 (left) shows the voltage current characteristic for the two dif-
ferent diodes used in the model. The free-wheeling diode is described by
the forward threshold voltage UfDTh and its differential forward resistance
RfDfR =dU/dI.
The diode parallel to the magnet can be mounted in the magnet cryostatIsolating
the
magnet:
Sec. 7.9.3
and is then denoted cold diode. The threshold voltage of a cold diode is
significantly higher (around 6−8 V compared to ≈0.7 V) due to the cryo-
genic temperatures. After switching, the dissipated ohmic losses increase the
junction temperature and the forward voltage threshold drops to some volts
(see e.g. [Krai 97, pp. 136]). Therefore, the cold diode is modeled by the
forward threshold voltage UcDcTh in cold state, its differential forward resis-
tance RcDfR, the forward threshold voltage UcDwTh in warm state and the
time it takes to heat up the junction ∆tcDh. If the parallel diode is mounted
outside the cryostat under warm operating conditions, the parameters have
to be chosen as above.
Figure 3.7 (right) shows the lumped electrical element model of the free-
wheeling and by-pass diode. The ideal diode is marked with an asterisk. The
forward voltage is denoted by UwDf and UcDf, respectively.
With the simplified diode model, the terminal voltage of a magnet in a
3.5 Electrical Network 33
string as well as in single operation can be reproduced. Setting the forward
threshold voltage in the model to zero, the diode can be used to simulate a
parallel resistor.
Energy extraction system In order to extract energy from the magnetic Dump
resistor:
Sec. 7.9.2
system and to reduce the time constant of the current decay, a dump resistor
can be switched in. The current commutation from the switch into the resistor
RDR is assumed to be instantaneous. Current flows through the resistor after
a time interval ∆tDR after quench validation.
3.5.2 Magnet Representation and Introspection
In respect to the external electrical network the magnet is defined by the
lumped elements differential inductance Ldand the resistivity of the quenched
windings RQ. Internally, the resistance and the induced voltage is further
assigned to the individual elements. This allows for the computation of turn-
to-turn voltages as well as peak electrical fields.
Element Resistance The electrical resistance Rij
Eof an element depends
on the effective resistivity, the element cross-section and the element length
`z. The superconducting filaments are in parallel to the normal conducting
matrix material. In the superconducting state, the electrical resistance of the
element can be considered to be zero. After a quench, the electrical resistivity
of the superconducting filaments shows to be orders of magnitude greater
than of the matrix material (compare Fig. 2.7). Therefore, the resistance
is determined only by the matrix material and the normal conducting cross-
section area Ai
cab,NC. Different materials, used for stabilizing and coating Strand
parts: Sec.
B.1.3
superconducting strand, are in parallel along the strand.
The effective parallel electrical resistivity is defined in Sec. B.2.3. In gen-
eral, the electrical resistivity depends on the applied magnetic induction, Electrical
resistivity:
Sec. A.1.3
temperature and purity of the material. The element resistance is thus given
by
Rij
E=`z
Ai
cab,NC
ρeff
E(Tij, Bi
av, RRRi)(0 ∆Jij
c>0
1else ,(3.25)
where RRRiis the residual resistivity ratio of the conductor as defined in
Sec. A.1.3.
Resistive Voltages The resistive voltage over an element Uij
res is given by Current-
sharing:
Sec. 7.5.1
the local resistance and the current in the normal conducting fraction of the
conductor Iij
NC. In the present model the current in the superconducting
filaments does not yield a voltage drop over the element. The distribution of
the current over both parts is determined by means of current-sharing, see
Sec. 3.2. The voltage drop over the element ij reads:
Uij
res =Rij
EIij
NC.(3.26)
34 Numerical Modeling
Summing over all elements yields the total resistive voltage of the magnet
used for quench detection,
Ures =
Ne
X
ξ=1
Uξ
res.(3.27)
Consequently, the total voltage Ures differs from the voltage IPNe
ξ=1 Rξ
E,
which ignores current-sharing.
Dissipated power The current flowing through the normal conducting frac-
tion of the conductor causes ohmic losses and thus heating. The power dissi-
pated in each element is calculated from
Pij
ohm =Uij
resIij
NC =Rij
EIij
NC2(3.28)
Induced voltages: The induced voltage over an element is calculated from
the line integral of the magnetic vector potential. As shown in Sec. 7.4.3,
in a bulk conductor the vector potential can be split into an average part
yielding the induced voltage and a differential part causing eddy currents.
In case of Rutherford-type cables these eddy currents are accounted for by
the inter-strand coupling currents. The induced voltage Uij
ind over an element
reads
Uij
ind =`zkmagδi
Ξ
∂
∂tAi
z,av,(3.29)
where `zkmag represents the magnetic length of the element and Ai
z,av is the
average magnetic vector potential over the element ias as explained in Sec.
3.3. The induced voltage is defined positive along the winding for increasingScaling
factor kmag
defined in
(3.14)
vector potential respectively current. δi
Ξtakes into account the orientation of
the line segment in respect to the integration direction.
The total induced voltage Uind is thus given by the sum over all elements:
Uind =
Ne
X
ξ=1
UΞ(ξ)
ind (3.30)
Inductance As introduced in Sec. 7.4, the differential inductance is calcu-Inductance:
Sec. 7.4 lated from the flux change in the coil divided by the current change. With
the flux change expressed by the induced voltage Uind this results in
Ld=Uind
d
dtIM
.(3.31)
The fact that the differential inductance is calculated from the current
change as well as the current change is computed for the differential induc-
tance has to be resolved by either determining Ldin advance, an iteration
scheme, or an explicit scheme with small time steps. For zero current change
3.5 Electrical Network 35
!"#
!"#$%&'
()*+,-'
./0' $1'
Rij
E
Uij
ind
./2'
1#&,+3!"#$%&'$(")#%*+&,-.+/0'%12'/("3
Figure 3.8: Longitudinal cut through the discretized coil winding with lumped electrical
elements.
Eq. (3.31) is undefined and replaced by the apparent inductance. The ap-
parent inductance LΨis given by
LΨ=PNc
i=1 δi
ΞAi
z,av
IM
`mag,(3.32)
where `mag is the magnetic length of the magnet (defined in Sec. B.1.6). For
the present model no mutual inductances need to be considered.
Internal Voltage Calculation Without limitations we connect the magnet
terminal ξ= 0 to the ground. The electrical potential along the magnet
windings can be calculated by summing the inductive and resistive voltages
following ξ:
φξ=
ξ
X
ζ=1 Uζ
ind +Uζ
res(3.33)
The induced voltage is calculated in all turns of the coil from the time-
derivative of the linked flux. To evaluate resistive voltages we interpolate
the resistivities before- and after each time-step. This allows to define the
voltage between any two elements, as well as the electrical field between two
elements of identical longitudinal position j:
Uij,nk =φij −φnk,|E|ij,nj =|φij −φnj|
di,n
E
,(3.34)
with the distance of two parallel elements di,n
Ederived in Sec. B.1.5. Figure
3.8 shows the discretized coil winding with the lumped electrical elements
Uij
ind and Rij
E.
During the quench simulation the peak electrical field and its location are
constantly recorded.
36 Numerical Modeling
Parasitic capacitances Following the approach by Brechna [Brec 73, pp.
332], the voltage distribution over the coil of a superconducting magnet is
influenced by parasitic capacitances. If the capacitance to ground is much
larger than the winding capacitance of the coil, a sudden voltage rise over theParasitic
capaci-
tances:
Sec. 7.10.3
magnet is concentrated around the terminals instead of decreasing linearly
along the winding. This results in a significant increase of voltages and electric
fields between adjacent conductors.
For accelerator type magnets comparable to the SSC dipole, the voltage
increase can be estimated to a factor of two. As peak electrical fields are
at least an order of magnitude below critical limits, i.e. below the dielectric
strength of the insulation material, the effect of parasitic capacitances can beDielectric
strength:
Sec. A.1.7 neglected. Note that the electrical topology of the coil changes if capacitances
are considered.
3.6 Thermal Model
The thermal problem is described by the heat conduction equation for non-
stationry systems [Lewi 96, p. 4]:
ρD(T)c0
V(T, B)dT
dt=P+∇·(κT(T, B)∇T),(3.35)
where Tis the temperature and Bthe applied magnetic flux density. The
external heating power is denoted by P. The material parameters are the
volumetric specific heat c0
V, the thermal conductivity κT, and the mass density
ρD. Equation (3.35) does not consider convection, therefore cooling by heliumMaterial
Properties:
Sec. A mass flow can not be taken into account.
Finite volumes and linear approximations Introducing finite volumes, Eq.
(3.35) can be transformed into a network equation with lumped elements. In
the coil cross-section, each conductor constitutes one node in the network,
whereas the longitudinal subdivision is a user supplied parameter Nz(Sec.
3.1).
The temperature of each element is assigned to its center. The temperatureRutherford-
type cable:
Sec. 7.8 over the conductor cross-section is assumed homogeneous as shown in Sec. 7.8.
For the calculation of the temperature change, the element volume spans the
metal parts of the conductor, its voids and the surrounding insulation (see
Fig. 3.9 (upper row)). The thermal capacitance of the element is calculated
based on the node temperature (constant temperature approach) and the
conductor average magnetic induction.
For the calculation of heat fluxes, we assume a linear temperature varia-
tion along the conductor and over the insulation. Two adjacent nodes are
connected by means of a thermal resistor with the conductor cross-section or
the conductor surface (both without the insulation layer) as end planes. The
thermal conductivity between two adjacent elements is calculated based on
their average temperature and local average magnetic induction (see Fig. 3.9
(lower row)).
3.6 Thermal Model 37
!"#$%&!"#$%&'(#)*+$,-./01$#23&2+456
!"#
!"#
!"#$%&!"#$%&'(#)*+$,-./01$#23&2+4556
Figure 3.9: Finite volumes and linear/constant approximations for the thermal network
model. The left-hand side shows the cable cross-section and the right-hand side its longi-
tudinal extension.
Heat capacity (upper row): For the calculation of the heat capacity the volume spans over
the conductor cross-section including the insulation. The temperature is assumed to be
constant over the volume. The network node is situated in the center of the element.
Thermal conductance (lower row): For the calculation of longitudinal thermal conduc-
tances and heat fluxes, the conductor cross-section without the insulation is used as end
planes (orange) of a thermal resistor (blue). For transversal heat fluxes the conductor
surface planes (orange) are used. In between two planes a linear temperature distribution
is assumed (blue). Material properties use the average temperature of the two adjacent
elements.
!"#$%&'()*!+%
!",$'!-"(&%
."$+#/!"#$%&'(#)*+$,-!$&./0#$/&'1
Pij
tot
Cij
T
Gij,nj
T,trans
Gij
T,cool
!"#
!"#$%&'
!($)'&*+,"!%'
-()%$.!"#$%&'(#)*+$,-.+/01)231/&'4
Pij
tot
Cij
T
Gij
T,cool
Rξ,ξ+1
T,long
Figure 3.10: Lumped thermal network model in comparison to the the coil/conductor
geometry.
38 Numerical Modeling
Figure 3.10 shows the resulting thermal network model with lumped ele-
ments. The network elements are defined below.
Since in the transverse plane the temperature across each conductor is
assumed to be constant, current redistribution between the strands [Verw 06]
cannot be considered.
3.6.1 Temperature Change
The temperature change in an element is given by
dTij
dt=1
Cij
ThPij
tot −Sij
trans +SΞ−1(ij)
long +Sij
cooli,(3.36)
where Sij denotes the heat transfer from/to adjacent elements or to a coolant
(Sec. 3.6.2), and Pij
tot denotes the total dissipated power (Sec. 3.6.5). The
heat capacity of the element is given by
Cij
T=Vi
eceff
T(Ti,j, Bi
av)(3.37)
with the effective specific heat ceff
T(T, B)defined in Sec. B.2.3. Figure 3.11
(left) shows the specific heat of materials for superconducting cables. LargeVolumetric
specific
heat: Sec.
A.1.5
differences between copper and the superconductors as well as between the
superconductors and helium can be noticed at low temperatures. At room
temperature the differences are smaller. Liquid helium is dominant below
approximately 10 K and can be neglected beyond. The element volume is
given by
Vi
e=Ai
cab,eff`z(3.38)
with the cross-section of the insulated conductor Ai
cab,eff defined in Sec B.1.3.
3.6.2 Heat Transfer
Longitudinal Heat transfer The longitudinal heat transfer in the coil is
given by
Sξ
long =Tξ−Tξ−1kξ,ξ−1
T,long
Rξ−1,ξ
T,long
+Tξ−Tξ+1kξ,ξ+1
T,long
Rξ,ξ+1
T,long
.(3.39)
The heat flow between two elements can be modified by the parameter kξ,ξ+1
T,long.
This allows to steer the quench propagation velocity, to take into account
regions of lower or higher thermal conductivity within the coil, and to model
quench stoppers, i.e. barriers for the heat flux along the conductor. By
means of the numbering scheme Ξthe conductors are connected seamlessly
over the coil ends. The terminals of the magnet are perfectly insulated, i.e.
k0,1
T,long =kNe,Ne+1
T,long = 0.
3.6 Thermal Model 39
Nb!Ti
Nb3Sn
Cu
Kapton
Helium
1 5 10 50 100 500 1000
100
1000
104
105
106
107
Tin K
c!T"in
Jm!3K!1
Nb!Ti
Nb3Sn
Cu
Kapton
1 mT
1T
10 T
20 T
1 5 10 50 100 500 1000
0.01
0.1
1
10
100
1000
Tin K
ΚT!T,B"in
Wm!1K!1
Figure 3.11: (left) Volumetric specific heat of all relevant materials versus temperature.
(right) Thermal conductivity of all relevant materials versus temperature.
The longitudinal thermal resistance Rξ,ξ+1
T,long is given by
Rξ,ξ+1
T,long =`z
Ai
cab,xs
1
κeff,[=]
TTξ+Tξ+1
2, Bi
av.(3.40)
For the heat transfer all parts of the cable, i.e. superconductor, copper ma-
trix and insulation, are in parallel. Comparing the thermal conductivity of Thermal
conductiv-
ity: Sec.
A.1.4
the different materials in Fig. 3.11 (right), only the conductivity of copper
needs to be considered. The thermal conductivity of the two available super-
conductors and of the insulation material kapton are 2-4 orders of magnitude
smaller. The effective parallel thermal conductivity κeff,[=]
T(T, B)is defined in
Sec. B.2.3. The conductor cross-section without insulation, Acab,xs, is derived
in Sec. B.1.3.
Transversal Heat transfer Transversally, each element is subjected to the
heat transfer of all elements in the vicinity:
Sij
trans =
Nc
X
n=1,n6=iTij −TnjGij,nj
T,trans.(3.41)
The thermal conductance,
Gij,nj
T,trans =Ki,n
transκeff,[ | | ]
TTij +Tnj
2,(3.42)
40 Numerical Modeling
consists of a geometrical contact factor Ki,n
trans and the effective thermal con-
ductivity κeff,[ | | ]
T(T)(Sec. B.2.3). The geometrical contact factor is intro-
duced in Sec B.1.5. It represents the distance and area of the conducting
surface between two elements. Furthermore, it includes a parameter ki
T,trans
to adjust the heat flow and the turn-to-turn quench propagation velocity,
respectively.
Conductive Cooling In the presented model, cooling is included by an addi-
tional, global node of fixed temperature Tb. Therefore, the cooled face of the
cable and the distance to the cold surface have to be specified. The thermal
conductance to this heat sink is calculated in the same way as between two
conductors. This allows to simulate the heat extraction without considering
an increase of the bath temperature or the limitation of the heat exchangers
to extract more than a certain amount of heat.
Sij
cool =Tij −TbGij
T,cool (3.43)
The thermal conductance to the cold surface is given by
Gij
T,cool =X
α
Ki
cool,ακeff
TTij +Tb
2,(3.44)
where α {a, b, c, d}is any of the four faces of the conductor as introduced
in Sec. B.1.5.
3.6.3 Thermal Properties of Helium
The thermal model and its coefficients have to be modified in the presenceProperties
of liquid
helium:
Sec. A.3
of liquid helium in the voids of the cable, subsequently referred to as the
confined helium.
Superfluid helium features two phases. Phase I for temperatures above the
lambda-point (approx. 2.17 K at saturated vapor pressure) and phase II be-
low. The heat conductivity of phase II helium exceeds the conductivity of
copper by orders of magnitudes. Phase I liquid helium and super-critical he-
lium have negligible thermal conductivity. The specific heat of both, Phase I
and II helium exceeds that of copper or niobium-titanium by orders of magni-
tudes. Super-critical helium, however features a relatively small specific heat
compared to copper. Thus, for temperatures below the phase transition and
below the lambda-point, the confined helium needs to be considered.
We make the following assumptions on mass- and volume balance inside
the conductors: Below the lambda-point the mass density and total mass of
helium remains constant. At temperatures above this point the helium is
adiabatically compressed until the local pressure in the conductor rises up
to a limit p0. Then, the helium is heated at constant pressure, i.e., mass
flows out of the conductor (see Sec. A.3.5). The helium content and thus
the heat capacity of elements adjacent to a quench is reduced. Therefore, in
direct neighborhood of a quench helium properties are evaluated at a higher
3.6 Thermal Model 41
!"#$%&!"#$%&'!"#$%&(()
!"#$%&!"#$%&'!"#$%&()
!"#$%&!"#$%&'()!"#$%&*
Figure 3.12: Influence of liquid helium in the coil windings on the thermal model. (left)
Helium II: perfect heat conduction through the micro channels. Heat flow is limited by the
copper inside the cable and by the maximum permitted heat flux through the channels.
(center) Helium I: heat flow between conductors is dominated by the insulation material.
The huge heat capacity of the helium inside the cable limits the temperature increase.
(right) No confined helium.
temperature, e.g. 30 K above the element temperature. Figure 3.11 shows
the specific heat of helium for constant volume inside a conductor as derived
from measurements in [Van 86]. The helium content under nominal operation
conditions is a user-supplied parameter in the simulation and depends on the
compaction of the cable (see Sec. B.1.3).
Below the lambda-point, heat conduction between adjacent conductors is
determined by the helium percolating across the insulation and by the con-
ductivity of the conductors. The thermal conductivity of superfluid helium
being infinite at first approximation, the conductivity is determined by the
conductor material, i.e., mostly by the copper in the strands. The size and
availability of these helium channels depend on the used insulation material
and the applied pressure. If a maximum heat flux is exceeded the superfluid
helium “quenches” [Iwas 94, p. 117] and looses its perfect heat transfer prop-
erties. The maximum flux is user supplied and set to zero if no channels
are considered. Above the lambda-point or in the “quenched” state, the heat
conductivity of helium is neglected in our model.
Figure 3.12 shows how helium inside the cables influences the thermal model
of the coil.
The model does not consider the energy absorbed by the helium phase-
transitions. The confined helium is assumed to be in close contact with the
strands. Variations in the thermal contact due to nucleate and especially film
boiling are not considered.
3.6.4 Quench Heaters
Quench protection heaters are thin resistive strips colaminated with insulation Quench
heaters:
Sec. 7.9.5
material mounted on the superconducting coil. In case of a quench, the
quench heaters are fired heating up the covered conductors. The heating
causes further quench and thus significantly increases the coil resistivity.
In the present work, quench heaters are modeled as effective heat sources
inside the covered conductors, i.e. without constituting a node in the thermal
network model.
Each quench heater is described by three parameters, the effective initial
heating power PQH0, an internal delay ∆tQH0, and the discharge time con-
stant τQH. These properties may vary between quench-heater circuits and are
42 Numerical Modeling
!!" !#$%"
"&'("
)!&'("
*&'"
*&'+("
!&',-"
)!&'"
!#
"&'"
./0,%1!"#$%&'#()#*+,
Figure 3.13: Quench heater model depicting the relevant parameters as initial power
PQH0, decay constant τQH, and internal delay ∆tQH0. Setting the decay constant to zero
yields a constant power output.
therefore directly assigned to the covered conductor i:
Pi
QH =Pi
QH0
0t < tval + ∆ti
QH0
exp −t−tval−∆ti
QH0
τi
QH else ,(3.45)
where tval is the time of quench validation, i.e. the instance when the quench
is detected and validated after a discrimination time. The power is dissipatedQuench
detection:
Sec. 7.9.1 homogeneously along the covered conductor i.
The time constant τQH is determined by the electrical heater circuit and the
strip resistivity. As derived in Sec. 7.9.5, the dissipated power decays with
half the time constant of the discharge current or voltage. A time constant
set to zero allows to study steady heating of parts of the coil. Figure 3.13Sequence
of Events:
Sec. 7.9.6 shows the three parameters in the context of the events during a quench.
The parameters PQH0and ∆tQH0are chosen such that the observed/ex-
pected quench heater delay ∆tQH,i.e., the time between firing a heater and
the occurrence of a quench, can be reproduced. The quench heater delay de-
pends on the working point of the covered conductor and varies nonlinearly
over the full current range (see Sec. 7.9.5). Note, that both parameters are
model specific and have no direct meaning or measurable representation.
The quench heater model does not permit to compute the quench heater
temperature. Furthermore, it is not possible to calculate the effect of the lon-
gitudinal variation of the heater strip resistivity, i.e. heater cladding [Rodr 00]
and [Sonn 01a].
3.6.5 Total Dissipated Power
The following heat sources are considered in the thermal model: Ohmic heat-
ing Pij
ohm in the quenched parts of the superconductor, in the superconducting
parts induced losses Pij
losses and heating by quench heaters Pi
QH. Furthermore,
external heat sources, e.g. beam losses or other external heating mechanisms,
3.7 Quench Algorithm 43
can be applied by means of Pij
ext. The total dissipated power per element
yields:
Pij
tot =Pij
ohm +Pij
losses +Pi
QH +Pij
ext.(3.46)
Book-keeping of the individual heat sources and of the total heat transfer
permits to determine the cause of quench for each element in the thermal
model. It can be distinguished between local quench due to quench-back,
quench propagation, or quench heaters.
Combining the sequence of quench events with their cause, allows to deter-
mine the quench heater efficiency, i.e. does the quench heater cause a quench,
the quench heater delays ∆tQH, and the first time when quench back occurs
tQB.
3.7 Quench Algorithm
The description of the quench process is based on the afore introduced five
sub-models, i.e. the description of the superconducting state and its transi-
tion, the magnetic field computation, analytical models of induced losses, the
electrical network model, and the thermal model; and their coupled solution.
Start of Simulation We distinguish two different starting conditions for the
quench simulation. Dynamic conditions where a quench is triggered during
a pre-defined ramp-cycle or a quench starting from a specified steady-state
working point. In the latter case, the quench has to be actively triggered by
means of a heat pulse or by increasing the conductor temperature beyond
the critical temperature. In order to prevent an instant quench recovery the
temperature is raised at least 10% above the quench limit. The local extend
of the quench can be set to one element or an entire conductor.
In the former case, a quench can be caused by induced losses, by exceeding
the critical current limit or by using the active trigger after a pre-defined
delay.
Static computation The magnetic field model, i.e. the analytical calcu-
lation of the coil fields and the numerical calculation of magnetic materials
by means of the BEM-FEM-coupling method, can be performed under static
conditions. The time derivative of all field quantities is derived from the
change between two successive computational steps.
The analytical models of the induced cable eddy currents are also static. As
explained above, the results are interpolated in order to obtain the diffusive
characteristics of the losses. However, the loss models are evaluated together
with the magnetic field computation.
Time integration scheme The thermal-network-equations and the electrical-
network equation constitute coupled, non-linear, inhomogeneous differential
equations of first order. Both are solved with a classical Runge-Kutta method
of 4th order in an explicit time-integration scheme (Sec. 7.11). Within each
44 Numerical Modeling
computational step, we update all parameters by the most recently computed
values, i.e. in the case of the induced losses with the current value of inter-
polation.
The material properties vary significantly with temperature, e.g. the heat
capacity of copper increases by several orders of magnitude from cryogenic
to room temperature. Therefore, typical time constants of the differential
equation system are subjected to drastic variations. This can be taken into
account using adaptive time-stepping.
The Runge-Kutta method of 4th order can be equipped with a quality
factor [Coll 55, p. 68]. The quality factor gives indication whether to accept
or discard the result of the last integration step. If discarded, the step is
repeated with a shorter time-step width. In the case the step was accepted,
the time-step width can eventually be increased. The adaptive time-stepping
allows to cope with the highly-nonlinear material parameters, and to resolve
the growth of the resistive zone inside the magnet as well as the switching-in
of the protection resistor. The time step-width is limited by a minimum value
in order to ensure a finite computation duration.
Nested computation loops The update of the magnetic field is compu-
tationally more expensive than a time-step in the thermal model. Time-
constants in the thermal model, however, are much shorter than those of
the eddy currents in the magnetic model. It is therefore reasonable to fore-
see a weak coupling between thermal and magnetic computations, updating
magnetic values only when the excitation current has changed significantly.
Figure 3.14 shows the quench algorithm featuring two nested computation
loops. The outer loop contains the magnetic field and analytic loss computa-
tion. The inner loops consists of the Runge-Kutta algorithm for the thermal
and electrical model. The critical state of the superconductors is evaluated
in the inner loop - denoted quench decision.
Loop Termination Ohmic losses and coupling-current losses are driven by
the stored magnetic energy in the magnet. We use this to calculate the energy
decrease in the magnetic field. When it has decreased by a user-supplied
factor, the field is updated, i.e., the inner simulation loop is interrupted and
a magnetic field model computation is carried out. The quench simulation
then proceeds with updated values for the field- and coupling-current-loss
distribution.
End of Simulation The quench simulation ends, when the current or its
derivative have dropped below a predefined level, or the hot-spot tempera-
ture exceeds a maximum value of 1000 K.
In general, calculations have to be repeated with twice the minimum step-size
to check for numerical stability.
3.7 Quench Algorithm 45
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()**&*'
+,&%-./%0,'1&-2).3'
4)#&,'
56&.70,'4)#&,'
8$&"%6'9&%/*/)"':';$<&.%)"#$%-).'
!""&.'=)7<$-0>)"'())<'
?$-&.'=)7<$-0>)"'())<'
!"
#!"##$
40@"&>%'A/&,#'
=)7<$-0>)"'
!"
$" %"
&"
!"
'%$
'%$
&"
#&'($
#&'($
#)*$
$"
#!"##$
Figure 3.14: Block diagram of the quench simulation algorithm. We distinguish an inner
and outer computation loop. In the outer loop we perform the magnetic field and loss
computation. In the inner loop we evaluate the thermal model, the electrical field model
and the quench status of all superconductors.
46 Numerical Modeling
4 Introspection
The purpose of computing is insight, not numbers.
Richard W. Hamming
(1915-1998)
The quench model includes a number of parameters which need to be de-
termined in order to reproduce the measurements. Most parameters can be
derived from specifications given to the manufacturers or measurements of
magnet components. Other parameters evade direct measurement and are
determined indirectly, e.g. the thermal properties of the coil are adapted
to reproduce the measured quench propagation velocity, and quench-heater
parameters are validated on quench-heater delay studies.
Reproducing quench measurements, the simulation allows to review the
internal state of the magnets which is not accessible to measurements, e.g. the
temperature margin or the reason for a quench over all conductors. This way,
we can analyze quench propagation, heater efficiency, and quench protection
methods.
4.1 LHC Main Dipole
Figure 4.1: LHC MB.
Schematic coil cross-section
and iron yoke. Outer iron
diameter 570 mm.
The quench routine is demonstrated and validated
by means of measurements for the LHC main bend-
ing magnet (MB) [Brun 04]. We review relevant
properties of the magnet, and specify the used pa-
rameters.
The LHC main bending magnets guide particles
with a nominal energy of 7 TeV on an orbit of 27 km
circumference [Brun 04, p. 164]. With the maxi-
mum magnetic induction of 9 T they bend the tra-
jectory by 0.29 oover the magnetic length of 14.3 m
[Evan 09, p. 75].
The dipole features two apertures in a common
collar and iron yoke. The coils in each aperture are
composed of an upper and a lower pole. Each pole consists of 6 conductor-
blocks per quadrant, arranged in two layers. Figure 4.1 shows the magnet
47
48 Introspection
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!%&#
'()*+,-./#
'-01-2#
344-2## 561-2#708-2#
"&#
"9#
"%#
"!#
""#
":#
!;-2<(40.#
=2--+>*--.(4)#/(?/-#@?>-2#A6@@.8#
B?./#/(?/-#C(40BDE-F#
!G%H7# !G%HI# !G!H7#
!G!HI#
J0)4-1#
"#
Figure 4.2: (left) Coil blocks and quench-heater positions in one aperture of the LHC
main bending magnet. (right) Electrical circuit of the magnet measurement test station
after the current source is switched off. D1_L denotes the lower pole in aperture 1. D2_U
denotes the upper pole in aperture 2.
cross-section including the iron yoke. Saturation has a relatively small impactSaturation
influence
on LdSec.
7.4.3
on the differential inductance, which varies only by 5% with excitation. For
the thermal and resistivity calculations we use an average conductor length
between the two coil ends of 14.57 m.
The coil is wound from two different kinds of Rutherford-type Nb-Ti cable.
The critical current density in the strands is given by a fit to measurement
[Bott 00]. The magnet operates at a temperature of 1.9 K. The residual
resistivity ratio (RRR) of the copper matrix in the strands is in the range of
150 to 250 [Char 06].
The quench protection consists of a detection system, a cold bypass diodeMagnet
protection
Sec. 7.9 and quench heaters placed on the outer layer of the coil as indicated in Fig.
4.2. The threshold voltage of the detection system is 0.1 V [Denz 06]. Quench
heaters are fired after a delay of 10 ms for signal validation [Denz 06]. The
timing of the different heaters may vary by up to 10 ms [Sonn 01a]. A ca-
pacitor is discharged over the resistance of the heater strip, resulting in an
exponential voltage decay [Rodr 01]. The time constant for the dissipated
power is about 37.5 ms [Sonn 01a].
An exhaustive list of all parameters and properties used for the simulation
of the LHC main bending magnet can be found in Sec. C.1.
4.1.1 Reproduction of Quench Measurements
The quench model is gauged in order to reproduce the measured induced
losses and quench-heater delays. Finally, the the current decay measured at
the LHC test stand is reproduced and the voltages over the different coils are
analyzed.
4.1 LHC Main Dipole 49
!" #" $" %" &" '" (" )" *"
+,-./01"23,14"3-"5"
!#6"
!66"
#6"
%6"
'6"
)6"
7,0.,/"8,109":!";7"3-"<="
0.001
0.01
0.1
1
10
100
1000
0.001 0.01 0.1 1
Average losses in W/m
Ramp-rate in T/s
Hysteresis
ISCC
IFCC
Total
>?,/0@,"AB==,="3-"CD<"
E0<FG/0.,"3-"5D="
79=.,/,=3="
H2++"
HI++"
5B.01"
0.001
0.01
0.1
1
10
100
1000
0.001 0.01 0.1 1
Average losses in W/m
Ramp-rate in T/s
Hysteresis
ISCC
IFCC
Total
Figure 4.3: (left) Average cable losses for different ramp-rates. The vertical arrow high-
lights the nominal ramp-rate of the LHC. (right) Quench heater delays for the main bending
magnet. The arrows signify the range of measured values. Reprint of the quench heater
measurements by P. Pugnat [Pugn 07].
Induced Losses The cross-over contact resistances, Rc, of the inner and
outer layer cables are about 30 µΩand 60 µΩ, respectively [Lero 06]. The
adjacent resistance Rais set to 100 µΩfor both cables [Verw 07b]. Figure 4.3 Cable mag-
netization
losses Sec.
7.6
(left) shows the simulated dissipated cable losses for different ramp-rates. For
the graph, the losses are averaged over the coil cross-section and given per unit
length. In the case of the superconductor hysteresis losses the losses are aver-
aged over the complete ramp-cycle (as discussed in Sec. 7.7). The losses for
the nominal ramp-rate of 0.0066 T s−1(up-ramp in 1200 s) are highlighted by
a vertical arrow. The results are in good agreement with the values published
in [Verw 95, p. 153] (inter-strand and inter-filament losses) and [Voll 02, p.
135] (hysteresis losses).
As expected, in the double logarithmic plot the graphs for the induced
inter-filament and inter-strand coupling current losses show a slope of 2, and
the hysteresis of 1, respectively. The hysteresis losses are dominant up to a
ramp-rate of approximately 0.07 T/sand are disregarded during a quench.
From Eq. (7.69) we can calculate a maximum ramp-rate of approximately
1.5 T s−1.
Quench heaters The quench heater delay, i.e. the time between firing the Temperature
margin
Sec. 7.1.2
quench heaters and causing a quench in the coil, depends on the working point
of the magnet. For a lower current and thus greater temperature margin, the
delay is longer. Figure 4.3 (right) shows a reprint of the measured quench
heaters delays for the LHC MB [Pugn 07].
The quench heaters are described by the effective power injected into the Quench
heater
model Sec.
3.6.4
covered conductors. The number of conductors under the heater strips as
well as the time constant of the discharge are defined by the heater geometry
and powering circuit.
50 Introspection
0
2000
4000
6000
8000
10000
12000
14000
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0
100
200
300
400
500
Current in A
Temperature in K
Time in s
Measurements
RRR=200, Losses, no Ins., no He.
RRR=200, no Losses, no Ins., no He.
RRR=200, Losses, Ins., He.
RRR=100, Losses, Ins., no He.
RRR=100, no Losses, no Ins., no He.
Figure 4.4: Measured and simulated current decrease for varying simulation parameters.
Simulated peak temperature. Losses refer to cable eddy current losses such as inter-strand
and inter-filament coupling currents.
The simulation is gauged to the measurements by means of the initial power
PQH0and the firing-delay ∆tQH0. Both are determined by a parametric study
simulating quench heater firing for various values of power and initial magnet
current. After reproducing the characteristic shape of the function, the result
can be adjusted to the measured values by means of the additional delay.
For the following simulations we use an initial power of 20 W/mper covered
conductor and a delay of 23 ms.
Current We compare the simulations to the measured current and voltage
signals during a training quench of magnet MB2381 [Choh 07]. For this mea-
surement, the magnet was mounted on the LHC test stand and directlyLHC test
stand Sec.
7.10.1 connected to the power supply with a free-wheeling diode in parallel. After
quench detection, the power supply is switched off and the current commu-
tates into the diode. The protection diode inside the magnet cryostat is
consequently clamped to the forward voltage of the free-wheeling diode and
prevented form switching, see Fig. 4.2 (right). From the measured terminal
voltage we derive the diode properties, i.e., a diode threshold voltage of 0.7 V
and a forward resistance of about 390 µΩ. The quench starts at a current
level of 12.82 kA in conductor number 215 of pole D1_U.
Figure 4.4 shows the simulated current decays for different RRR values,
with and without induced losses in the conductors. Furthermore, the effective
heat capacity of the cable is varied by means of considering the influence of
the insulation and the confined helium. The results are compared to the
measured current decay.
It is possible to reproduce the measured current decrease for three different
sets of parameters grouped in two families:
4.1 LHC Main Dipole 51
-30
-20
-10
0
10
20
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Coil Voltage in V
Time in s
D2L
D1U
D1L
D2U
-25
-20
-15
-10
-5
0
5
0 0.05 0.1 0.15
Coil Voltage in V
D2U
Figure 4.5: Measured (solid) and simulated (dashed) voltage over the 2 poles of the 2
apertures of LHC 2381 during a training quench. (inset) The measured voltage, as well as
the simulation, show ripples during the first 150 ms.
1. Cable measurements indicate for the cables of the LHC MB a residual
resistivity ratio (RRR) of 200 [Char 06]. We can reproduce the current
decrease if we disregard the influence of the insulation material and the
confined helium.
2. Older specifications, e.g. [Lefe 95, p. 86], suggested a RRR value of
less than 100. In this case the measurements can be reproduced in two
ways; either including the heat capacity of the insulation or neglecting
all sorts of induced losses.
For the resulting hot-spot temperature we observe a temperature difference
of more than 100 K. We obtain the highest temperature in the second case
where no quench back takes place and the necessary resistance is built up
only by the quench heaters and temperature dependence of the resistivity.
We chose the first set of parameters for the following simulations.
In addition, we show the influence of the insulation material and content
of 1% confined helium. The current decrease is slower compared to the pre-
vious cases. The peak temperature, however, increases only by 20 K. The
measurement data cannot be reproduced for a RRR value of 200 if losses are
neglected.
Voltages The measured and simulated voltages across the four poles of the
two-in-one magnet are shown in Fig. 4.5. Small differences in the timing of
the quench heaters over the four poles cause asymmetric voltage distributions.
To reproduce the asymmetry, the quench heaters in our model were de-tuned
by less than 2 ms.
The measured voltage, as well as the simulation, show ripples during the
first 150 ms. We explain this by the fact that every conductor turning resistive
52 Introspection
causes a sudden increase in ohmic voltage. The magnet’s terminal voltage,
however, is clamped to the forward voltage of the diode. Disregarding para-
sitic internal capacitances, the increase in ohmic voltage must be distributed
evenly over the four pole inductances, resulting in a sudden change of all volt-
ages. Once all conductors are quenched, the ohmic voltage changes smoothly
with temperature which is reflected by the measured and simulated curves.
In order to distinguish from also existing numerical effects the voltage spikes
are studied analytically in Sec. 7.10.2.
The presented result underlines the advantage of a coupled approach for
the quench simulation: The thermal model as well as the models for cable
magnetization losses determine the growth of the resitive voltage. Owing
to the simultaneous calculation of the electrical network and magnetic field
problem, the voltage re-distribution can be resolved.
Interpretation The large number of different parameters, e.g. thermal prop-
erties, residual resistivity ratio, quench heater timings, and loss model param-
eters, offer a wide range of possibilities to reproduce a limited set of observ-
ables such as the measured current decrease and coil voltages. Nevertheless,
we are confident that if the parameters are chosen within a justifiable range,
we can gain deeper inside into the quench process and the interdependent
effects.
4.1.2 Introspection - Simulation of a Quench in the LHC
Tunnel
The validated model can be used to simulate a quench in the LHC tunnel
where no measurements are available.
We consider a quench at the nominal current of 11.85 kA. The quench
origin is located in the outer layer in conductor number 228, compare Fig.
4.2 (left). It is centered longitudinally. The forward voltage of the diode in
the magnet cryostat is assumed to be constant at 6 V [Verw 08a].
Figure 4.6 shows the temperature margin to quench in each conductor
at the magnet’s center as a function of time. Three different phenomena
can be distinguished: Quench propagation, quench-heater delay, and quench-
back. After the quench has been initiated, it propagates transversally and
longitudinally. The detection voltage is reached at 28 ms (marked with ’a’
in Fig. 4.6); 10 ms later the quench is validated and the heaters are fired
(b). After the pre-defined 23 ms, the dissipated heat starts to decrease the
quench margin in the conductors covered by heaters and cause a quench at
65 ms. With the additional quenched conductors, the diode threshold voltage
is reached and the current in the magnet starts to decrease (c). Eddy-current
losses create additional heating. The inner layer is quenched by quench-back
at approximately 132 ms (d). The quench propagation in the outer layer is
also accelerated by the induced losses. After about 300 ms the magnet is fully
quenched (f).
4.1 LHC Main Dipole 53
!"#$
%&'(&)*+,)&$
-*)./0$/0$1$
#"#$
2!$
23$
24$
25$22$
26$
#"#$ #"5$#"4$
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#"#!$ #"4!$ #"5!$
89:;<$=:"$
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@A0&$B7:9/&),0.$
#"2$
Figure 4.6: Temperature margin to quench over the conductors versus time for the pole
of quench origin. The numbers correspond to the block numbers in Fig. 4.2. From the
evolution of the temperature margin we can distinguish quench propagation, quench heater
delay, and quench-back. The position of the quench heaters is indicated by the vertical,
black bars. The quench origin is high-lighted by a black triangle.
!"#$%
&'()*$+,#-%
Figure 4.7: Illustrated temperature distribution over one aperture of the magnet. We
distinguish quench propagation, quench-heater firing, and quench-back. The coil is scaled
1:100 in axial direction. The pictures is taken after 40 ms (left), 100 ms (center) and 300 ms
(right).
Figure 4.7 shows the coil temperature at three different stages of the quench.
On the left, the initial quench zone propagates transversally and longitudi-
nally due to thermal conduction. In the middle, the quench heaters have been
fired and the covered conductors are resistive. The figure on the right shows
the magnet after quench-back. The coil is almost completely resistive.
54 Introspection
4.2 3-D Thermal propagation
Figure 4.8: LHC MCBX.
Schematic coil cross-section
and iron yoke. Outer iron
diameter 330 mm.
The 3-D thermal propagation of a quench is studied
for the LHC MCBX corrector magnet [Brun 04].
The MCBX consists of two nested, independently
powered dipoles, see Fig. 4.8. We consider the
outer dipole only. The magnet is wound from a 7-
strand ribbon-type conductor using a strand of the
LHC MB inner layer cable rolled into a rectangular
form. The strands are connected in one coil end,
such that each radial layer of strands is connected
in series. This connection allows for radial and az-
imuthal quench propagation within a coil block as
well as for longitudinal propagation within each ra-
dial layer of strands. The coil is fully impregnated
so that no cooling needs to be considered. The magnet is operated at 1.9 K
and the quench occurs at 734 A. Figure 4.9 (left) shows the series connection
of the strands in the magnet end. The soldering is covered by copper plates
and immersed in liquid helium; therefore, functioning as a quench stopper.
Figure 4.9 (right) shows the outer layer coil before mounting into the iron
yoke. For the present simulations, the magnet is neither protected by quench
heaters nor a dump resistor. The voltage across the power supply is neglected.
Plotting the temperature margin to quench over each conductor versus
time illustrates the quench process and allows to determine the turn-to-turn
quench propagation delay. Figure 4.10 shows how a quench originating on
the upper pole propagates through the coil. Both, measured and simulated
turn-to-turn quench propagation delay, yield 4 ms. The longitudinal quench
propagation velocity is calculated to be 18 m/swhich is in good agreement
with measurements [Karp 08c].
Please notice: The quench propagation velocity increases with decreasing
margin to quench. Close to the critical surface the quench would propagate
!"#$%&'()%*+!"#$%&'()*+',-./
Figure 4.9: MCBX Magnet. (left) Series connection of the strands in the magnet end.
The soldering is covered by a copper block and functions as a quench stopper. (right) Outer
layer coil before mounting into the iron yoke. Courtesy of M. Karppinen CERN TE MCS.
4.2 3-D Thermal propagation 55
!"#
$#
"#
%&'()*+&,#-&.#
"."# "."/#"."0# "."1# "."2# ".!# ".!/# ".!0# ".!1# ".!2# "./#
3456#4'#7#
36586,9+),6#
:9,;4'#4'#<#
"."/# "."0#"."#
,9(49=#
9>45)+?9=#
Figure 4.10: MCBX temperature margin over conductors versus time. (left) The quench
originated in the first conductor and spreads azimuthally and radially to neighboring
strands. The lower half of the plot covers the conductors in the upper pole of the magnet
and the upper half the lower pole. The conductors in the lower pole start quenching 130 ms
later. (right) The detailed view of the temperature margin plot gives one way to determine
the turn-to-turn propagation delay.
with infinite velocity. Induced losses cause an additional temperature increase
in the conductors and the margin to quench is reduced. The quench propa-
gation is accelerated by means of the smaller margin. Thus, in presence of
losses the quench propagation velocity depends also on the current decrease
rate. This process re-inforces itself since a faster growth of resistivity causes
a faster current decay which causes greater losses. At magnetic quench-back
the induced losses cause fractions of the magnet to quench which are dis-
connected from the initial quench zone. Here and for conductors covered by
quench heaters the quench propagation velocity becomes meaningless.
Figure 4.11 (left) shows the measured [Gilo 08] and simulated current de-
crease which match nicely up to 0.3 s. Then the measured current decreases
slower than the simulation. Due to the short overall length of the magnet, the
3-D saturation of the iron yoke is likely to influence the inductance and thus
the current decays in a different way than predicted from 2-D calculations.
This problem will have to be addressed in future simulations. The final tem-
perature distribution over the coil cross section is shown in Fig. 4.11 (right).
Figure 4.12 shows the temperature in the whole coil for different times.
A good match between simulation and measurements could be obtained
without gauging the material properties, since neither cooling nor helium
need to be considered.
The quench model allows to investigate critical voltages and electrical fields
in the magnet cross-section. Figure 4.13 shows the potential to ground over
all conductors versus time. The maximum potential stays below 180 V, which
corresponds to maximum electrical field of 3000 Vm−1. The voltage distribu-
tion over the coil cross-section is smooth and does not show high gradients.
In Fig. 4.14 (left) the voltage along the conductor winding is plotted for
56 Introspection
0
100
200
300
400
500
600
700
800
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0
50
100
150
200
250
300
Current in A
Peak Temperature in K
Time in s
Measurement
Simulation
Temperature
20
40
60
80
100
120
140
160
180
max Temperature in K
Figure 4.11: (left) Current decrease during quench in the MCBX: Simulation and mea-
surement match well up to 0.3 s. (right) Maximum temperature over the coil cross-section.
Figure 4.12: Quench propagation in the MCBX outer coil: The 3D model is displayed
from the non-connection side of the coil and longitudinally compressed by a factor of 10.
The quench started in upper pole, inner most turn, close to the coil end. (left) Temperature
distribution after 0.1 s. The quench propagated longitudinally and transversally in the up-
per most block. It spread to the block below over the connection-side. (right) Temperature
distribution at the end of the quench. The lower pole shows an elevated temperature due
to induced losses. The maximum temperature is 180 K.
the instance where the highest potential to ground could be observed (at
t= 200 ms). The left-hand side of the plot (conductor no. smaller than 800)
corresponds to the upper pole of the magnet. In the upper pole the ripple
on the voltage is caused by sequences of normal conducting (quenched) and
superconducting strands. Therefore, the voltage to ground is reduced by ap-
proximately 45 V by the winding scheme (Sec. 7.10.3). In the lower pole, the
variation in the potential is caused by changes of the confined flux over the
coil cross-section and therefore by the variation of the induced voltage.
By comparing the cumulated heat transfer from neighboring elements, in-
duced losses, and quench heaters, the reason for a conductor to quench can
4.2 3-D Thermal propagation 57
-20
0
20
40
60
80
100
120
140
160
Total Electrical Potential in
!"!# !"$# !"%# !"&# !"'# !"(# !")# !"*# !"+# !",#
-./0123.4#5."#
6789#7/#:#
!#
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;.39/<=>#3.#
?4.1/0#7/#@#
Figure 4.13: Potential to ground during a quench in the MCBX. (left) Variation in time.
(right) Potential to ground over the coil cross-section at t= 200 ms. The potential varies
smoothly over adjacent conductors.
0
20
40
60
80
100
120
140
0 200 400 600 800 1000 1200 1400 1600
Potential to Ground
in V
Conductor No.
1.8
2
2.2
2.4
2.6
2.8
3
2.0 = Heat transfer, 3.0 = Losses
Figure 4.14: (left) Voltage along the coil winding of the MCBX when the voltage to
ground is at its maximum (t= 200 ms). The left-hand side of the plot corresponds to
the upper pole of the magnet and the right-hand side to the lower pole. The voltage
ripple in the upper pole is caused by the alternating connection of normal conducting
and superconducting strands. (right) Cause of quench in the MCBX: The light colored
conductors quenched due to heat transfer from other quenched conductors, the dark colored
conductors quenched due to induced losses.
be derived. Figure 4.14 (right) shows that only a few turns are quenched by
induced losses. Especially the conductors in the low field region (see arrow),
where inter-filament losses are small, are quenched by heat transfer.
58 Introspection
4.3 Quench recovery
Figure 4.15: LHC MQY.
Schematic coil cross-section
and iron yoke. Outer iron
diameter around 495 mm.
During a test of the LHC MQY quadrupole mag-
nets [Brun 04], quench heaters were fired at a mag-
net current level of only 80 A (compared to a nom-
inal current of 3610 A). The current measurement
[Denz 08b] show clearly two different slopes indi-
cating a quench in the magnet and a later recovery.
The quench model is used to reproduce this effect.
The MQY is a double aperture magnet (see Fig.
4.15) with quench heaters on the outer radius of the
coils as well as between the second and the third
coil layer. The coils of both apertures are powered
independently. Figure 4.16 shows the electrical cir-
cuit. At low currents, the two apertures are mag-
netically decoupled. This allows to reduce the simulation effort by calculating
only one aperture and halving the values of the external electrical network
elements.
For the simulation, the effective power of the quench heaters has been varied
until a quench is triggered at 80 A and the slope of the current decrease
is consistent with the measurements. Furthermore, the cooling is set such
that a recovery from quench is within reach after few seconds. Figure 4.17
(left) shows both, measured and simulated current decay. Notice: In the
measurement the current of both apertures are shown. Figure 4.17 (right)
shows the change of temperature margin to quench over the conductors of
one pole of the magnet during the heater discharge. The quench occurs 0.1 s
after firing the heaters and the conductors recover 1−3 s later.
By means of this simulation details of the quench recovery which evade di-
rect measurements can be studied. The results have been obtained by gauging
model parameters within their reasonable range.
!!"
!#"
$%&'()"!" $%&'()"#"
*+(,)-,("#" *+(,)-,("!"
.(+,/0-12/'3!"#$%&'()&*+
Figure 4.16: The two apertures of the LHC MQY are powered independently. Neverthe-
less, the apertures of two adjacent MQY magnets are connected in series.
4.3 Quench recovery 59
!"#$
%"!$
#"#$
&'()*+,'-$.'"$
#"#/$ #"/$ /$ /#$
0123$1($4$
03253-6,*-3$
76-81($1($9$
Figure 4.17: (left) Current decrease in the MQY after triggering the quench heaters at a
current level of 80 A. The two dashed lines correspond to the measurement of the currents
of the two apertures. The full line represents the simulation. (right) Temperature margin
to quench for one coil of the MQY. Note the logarithmic time scale.
60 Introspection
5 Extrapolation
When you have a hammer,
all problems start to look like nails.
Robert Kagan
(Of Paradise and Power)
In the previous chapter the quench model was used to reproduce quench
measurements of different LHC magnets. Model parameters were thoroughly
determined and the inner state of the magnet could be analyzed.
In this chapter we show the quench analysis of two future accelerator mag-
nets. We study different conventional quench protection concepts, e.g. quench
heaters or dump resistor, for the inner triplet upgrade quadrupole. For a
fast-ramping dipole we investigate the influence of different quench detection
thresholds as well as a magnet design protected only by the copper stabilizer.
By nature, the design of future magnets lacks measurement data. There-
fore, we extrapolate from the results of existing magnets. The simulation is
based on the parameters determined above and adopted to values expected
from other simulations or experience.
5.1 Quench Protection Study for the Inner
Triplet Upgrade Quadrupole
Figure 5.1: MQXC
schematic coil cross-section
and iron yoke. Outer iron
diameter 550 mm.
In the framework of the LHC luminosity upgrade
a series of magnets in the interaction region near
the experiments ATLAS and CMS will be replaced
[Bagl 08]. The protection of one of the inner triplet
magnets, the MQXC superconducting quadrupole
magnet, is studied. The study has been carried out
in an early state of the project in order to demon-
strate feasibility and to explore the necessary pro-
tection efforts.
Specific attention is given to the evaluation of the
different quench protection methods, i.e. quench
heaters, dump resistor or both. The study consists
of the following five parts:
61
62 Extrapolation
•Simulation of an unprotected quench, where the magnet is short-circuited
upon quench detection and discharged over the normal conducting zone.
This motivates the need for quench protection and allows to study the
intrinsic quench behavior of the magnet, e.g. quench propagation within
the coil cross-section.
•Simulation of different quench-heater layouts. The chosen heater setup
is checked for reliability in case of a partial heater failure.
•Parametric study of the protection with a dump resistor. The study
allows to determine minimum and maximum values in order to guaran-
tee safe operation and to avoid quenching by quench back in case of a
regular discharge.
•Discharging the magnet over the power supply and feeding the energy
back to the grid.
•Comparison of the two different protection methods and their combina-
tion.
The MQXC quadrupole magnet is wound from the same two cables as the
LHC main bending magnet (MB), see Sec. C.1. A thorough selection of the
strands is expected to result in a 10% higher critical current density [Osto 08].
The magnet features one aperture in a circular iron yoke, see Fig. 5.1. For the
time being, no holes for heat exchangers or notches are considered in the iron
yoke. The coil consists of four poles in two layers wound from two different
cables. The magnet is 10.3 m long with a magnetic length of `mag = 10 m
[Fess 08]. Neither Helium nor cooling are considered. All parameters were
chosen based on the simulation and measurement results of the LHC MB.
The full list of parameters can be found in Sec. C.3.
5.1.1 Unprotected Quench
The quench behavior of a magnet depends on intrinsic features of the magnet
design, e.g. the inductance, quench propagation and induced losses, as well
as on the external electrical network. In order to study the intrinsic quench
behavior, a quench is simulated at three different locations assuming the
magnet to be short-circuited and unprotected. The quench either originates
in block 2 conductor 18, in block 1 conductor 14 or in conductor 6 of the same
block, see Fig. 5.2 (left).
Figure 5.3 shows the temperature margin to quench versus time for each
conductor of the first quadrant. Due to the isolated location of block 2 the
quench in conductor 18 can only propagate over the block of quench origin.
In the other two cases, the quench propagates also from the inner to the outer
layer. Nevertheless, the turn-to-turn quench propagation delay is smallest for
the quench in conductor 18. This is due to the higher magnetic induction and
the hence smaller temperature margin to quench.
5.1 Quench Protection Study Inner Triplet Upgrade Quadrupole 63
MQXC 120mmV3
!"
#" $"
%"
&"
!%"
!'"
(" &"
)"
'"
0
2000
4000
6000
8000
10000
12000
0 0.2 0.4 0.6 0.8 1 1.2
0
100
200
300
400
500
600
Current in A
Temperature in K
Time in s
Conductor 6
Conductor 14
Conductor 18
*+,-./0+1"""&"
*+,-./0+1"!%"
*+,-./0+1"!'"
2+31+04/5+,*+67819:+,;9<414,0*+,-./0+1:"
*.114,0"
Figure 5.2: (left) LHC MQXC Conductor and coil block numbering scheme for the first
quadrant. (right) Temperature and current change versus time for an unprotected quench
originating either in conductor 6, 14 or 18. The simulation of the quench starting in
conductors 14 and 18 was stopped when the hot-spot temperature exceeded the temperature
limit of 600 K.
The faster and further the quench propagates within the coil cross-section,
the faster the resistance of the magnet grows. Since the magnet is short-
circuited, the internal resistance is the only cause for the current decrease.
Figure 5.2 (right) shows the current and hot-spot temperature versus time.
The current decrease in case of the quench in conductor 18 begins much later
than in the two other cases. This is due to the fact, that the quench could
not propagate as far and therefore quench back sets in later (compare Fig.
5.3).
Although the current change is similar for the two quenches in block 1, the
hot spot temperature differs significantly. In case of the quench in conductor
14, the quench propagates radially to block 3. Due to its isolated position
the single conductor of block 3 cannot transfer any heat. Furthermore, the
conductor is located in a region of peak magnetic induction and is made from
a different cable than the inner layer. Both result in a higher resistivity and
therefore in larger ohmic losses for an identical current. The hot-spot moved
from the quench origin to block 3, see Fig. 5.4 (center). The effect of hot-
spot movement poses another limitation to the MIITs-concept (Sec. 7.2).
Although, the outer layer conductor receives a smaller number of MIITs, it
yields the higher temperature due to a different function Υ.
5.1.2 Quench Heater Protection
For magnets where the current discharge merely depends on its internal resis-
tance, i.e. a magnet connected to a power supply which is switched off upon
quench detection or a magnet which is by-passed in case of a quench, quench
heaters can be used for magnet protection (Sec. 7.9.5). For the MQXC two
64 Extrapolation
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2/.3'*#'*#4#
!"!#
5#
%#
6#
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8#
9#
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Figure 5.3: LHC MQXC Temperature margin over the conductors of the first quadrant
versus time in case of an unprotected quench. (left) The quench originated in conductor
18 (highlighted by a triangle) and spread azimuthally over block 2 only. The turn-to-turn
quench propagation delay is approx. 30 ms. (center) The quench originated in conductor 14
and spread radially from block 1 to block 3 and 4. Due to the block boundaries the quench
can only propagate azimuthally to one side. (right) The quench originated in conductor 6
and spread to both sides over block 1 as well as over block 4. For the last two cases the
turn-to-turn quench propagation delay is approx. 50 ms. Both show quench back in the
outer layer at around 0.35 s, compared to 0.4 s in case of a quench in conductor 18. Quench
back in the inner layer appears 0.025 s before.
0
100
200
300
400
500
600
700
max Temperature in K
!"#$%&"#'
()*&)+,#-+)''
./'0'
0
100
200
300
400
500
600
700
max Temperature in K
0
100
200
300
400
500
600
700
max Temperature in K
0
100
200
300
400
500
600
700
max Temperature in K
Figure 5.4: LHC MQXC hot-spot temperature over the coil cross-section at the end of
an unprotected quench. (left) The quench originated in conductor 18 (highlighted by a
triangle) and spread over the entire block 2. (center) The quench originated in conductor
14 and spread from block 1 to block 3 and 4. Note that the hot-spot is located in block
3 and not where the quench originated. (right) The quench originated in conductor 6 and
spread to both sides over block 1 as well as over block 4. This case shows the lowest
hot-spot temperature.
5.1 Quench Protection Study Inner Triplet Upgrade Quadrupole 65
!"#$%&'#()#*+,-./(*0+-$102#*#$),-$3"%)-*
+45
MQXC 120mmV3
MQXC 120mmV3
!'65
!'6(5
!'675
0
2000
4000
6000
8000
10000
12000
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0
50
100
150
200
Current in A
Temperature in K
Time in s
Setup 1
Failure Setup 1
Setup 3
,"**#$)50$585
9:;5
45
,"**#$)5
<#./#*()"*#5
Figure 5.5: (left) LHC MQCX Quench heater layouts. (top) Setup 1: The quench heater
strip covers 11 conductors. It is connected in series to the strip below the horizontal plane
as well as to two strips on the other side of the aperture. The connection scheme is shown
in Fig. C.8. (bottom) Setup 2: Each strip covers 4 conductors. Both belong to different
heater circuits consisting of 4 strips at similar locations. (right) Temperature and current
change versus time for 3 different heater setups and for two different initial current levels.
In addition to the heater setups 1 and 2 a heater failure in setup 1 is simulated. The initial
current levels are the nominal current and half the nominal current.
different heater designs were suggested as shown in Fig. 5.5 (left). The first
layout covers 11 conductors per heater strip and features two independent
electrical circuits; the second layout covers 4 conductors per strip and consti-
tutes 4 circuits [Fess 08].
The quench heater delays, i.e. the time between firing the heater cir-
cuits and detecting a quench in one of the covered conductors, is expected
to be identical to values measured for the LHC main bending magnet (MB)
[MQXC 08a], since the same cable and heater technology is used. The quench
heater model parameters (quench heater delay time and quench heater power)
have been determined as for the LHC main bending magnets (Sec. 4.1.1).
The quench heater performance is compared at nominal and half the nomi-
nal current. Setup 1 yields a faster current decrease and therefore a lower hot
spot temperature, see Fig. 5.5 (right). This is due to the larger resistivity
built up by in total 88 compared to 64 conductors covered by quench heaters.
Furthermore, quench back sets in approximately 0.02 s earlier in setup 1.
For setup 1, a quench heater failure is simulated. We assume that four
heater strips are connected in series to one quench heater power supply cov-
ering half of each pole, see Fig. C.8. The magnet is supposed to be protected
even if one power supply fails to fire. Figure 5.6 (center) shows the hot-spot
temperature at the end of the quench in comparison to the setup 1 (left) and
2 (right). For the heater failure the highest temperatures are obtained, but
considering a temperature limit of 300 K the magnet can still be regarded as
protected.
66 Extrapolation
In all three cases hot-spot movement can be observed - most pronounced in
case of the heater failure. The temperature plot in Fig. 5.5 shows a kink at
around 0.15 s for a quench at nominal current. The two slopes result from the
different thermal properties of inner and outer layer cable. For a quench at
half the nominal current the kink is harder to observe, but hot-spot movement
persists.
The temperature margin of a conductor increases with decreasing current.
Hence, more energy is required to quench a conductor. The efficiency of
quench heaters is reduced at lower currents what results in longer quench
heater delays. Figure 5.7 (bottom) shows the variation of the temperature
margin over all conductors of the first quadrant versus time for a quench at
half the nominal current and protection by setup 1. Compared to the cases
at nominal current (top) the quench process is much slower, e.g. quench back
sets in only after 0.55 s. Regarding the conductors covered by quench heaters,
the conductors with the smallest margin on the outside of the block quench
first. The hot-spot temperature of all three different cases is around half the
value of the quench at nominal, see Fig. 5.5 (right).
In the following, setup 1 will be used since it gives the lowest hot-spot
temperature, is reliable also in case of a heater failure and at half the nominal
current, and consists of a smaller number of electrical circuits which might
malfunction.
5.1.3 Dump Resistor Studies
Alternatively or complementary to quench heaters, the magnet can be pro-
tected by means of a dump resistor. Upon quench detection, the dump resistor
is switched into the electrical circuit causing the current to decrease. If the
current decreases fast enough, induced losses cause quench back, spreading
the normal zone over the entire magnet and therefore accelerating the cur-
rent decrease. If a dump resistor is used to discharge the magnet without a
pre-existing quench, the current change rate needs to be limited in order to
prevent quench back. For more details see Sec. 7.9.2.
For an analysis of the magnet quench behavior an energy extraction study
is performed. The magnet is discharged over a dump resistor of variable
resistance and the extracted energy, i.e. the energy dissipated in the dump
resistor, is compared to the total energy initially stored in the magnet. The
study is performed once with a quench initialized in conductor 18 and once
without pre-existing quench. The dump resistor is switched in after detection
without additional delay.
Figure 5.8 shows exemplarily the current decrease and temperature rise in
the magnet for two different dump resistors, 2 mΩ and 20 mΩ, respectively.
In case of the pre-existing quench, the dump resistor of 2 mΩ shows to be too
small to provide sufficient protection (the computation was stopped when the
temperature limit of 600 K was reached). The larger resistor reduces the hot-
spot temperature to 130 K. In absence of a quench, the magnet is quenched
after 3.9 s and 0.05 s, respectively. The induced losses and the absence of
5.1 Quench Protection Study Inner Triplet Upgrade Quadrupole 67
40
60
80
100
120
140
160
180
200
220
max Temperature in K
40
60
80
100
120
140
160
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220
max Temperature in K
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40
60
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100
120
140
160
180
200
220
max Temperature in K
40
60
80
100
120
140
160
180
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max Temperature in K
Figure 5.6: LHC MQXC hot-spot temperature in the coil cross-section at the end of the
quench for different heater setups and a simulated heater failure. (left) Setup 1 (center)
Setup 1, where one heater circuit fails, i.e. half the strips are not fired. (right) Setup 2.
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%#
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7#
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9#
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Figure 5.7: LHC MQXC Temperature margin over the conductor of the first quadrant
versus time for different heater setups and a simulated heater failure. The quench heater
positions are indicated by a black bar on the side. The failing heater is indicated by a
white bar. The quench origin is highlighted by a triangle. (top) Quench at nominal current
level. (left) Setup 1 (center) Setup 1, where one heater circuit fails, i.e. half the strips are
not fired. (right) Setup 2. (bottom) Quench at half the nominal current level for setup
1. The quench propagation and heater induced quench take much more time than in the
cases above.
68 Extrapolation
0
2000
4000
6000
8000
10000
12000
0 1 2 3 4 5
0
50
100
150
200
Current in A
Temperature in K
Time in s
Quench
No Quench
0
2000
4000
6000
8000
10000
12000
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0
50
100
150
200
Current in A
Temperature in K
Time in s
Quench
No Quench
Figure 5.8: LHC MQXC Dump resistor study. The magnet is discharged over a dump
resistor. Different resistances were studied in case of a quench and in case of no pre-existing
quench. (left) Dump resistor of 2 mΩ. For a pre-existing quench the resistor shows to be
too small to significantly reduce the hot-spot temperature. In absence of a quench, the
magnet starts to heat up while discharging and quenches due to losses at around 3.9 s. The
internal resistance of the magnet accelerates the discharge. (right) Dump resistor of 20 mΩ.
The dump resistor shows to be large enough to protect the magnet in case of a quench.
The discharge in case of no pre-existing quench causes a stronger temperature rise due to
losses compared to the 2 mΩ case. The magnet quenches after ca. 0.05 s. The hot-spot
temperature is lower due to the shorter heating duration and lower current at quench start.
cooling cause the coil temperature to rise above the quench limit.
Figure 5.9 (left) shows the current decrease for all simulated dump resistor
values in case of a pre-existing quench. For a dump resistor smaller than 5 mΩ
the simulation was stopped when the temperature limit of 600 K was reached.
For a dump resistor larger 10 mΩ the hot-spot temperature remains well below
300 K. In Fig. 5.9 (right) the same simulation was repeated without initial
quench. Only for a dump resistor of 1 mΩ and below, the magnet does not
quench while discharging. In both cases, initially quenched or not, the onset of
the current decrease becomes steeper and the current decay curve approaches
an exponential decay for an increasing resistor size.
Figure 5.10 shows the extracted energy normalized to the total stored en-
ergy for all simulated cases versus terminal voltage. Plotting over the ter-
minal voltage, i.e. in this case the voltage across the dump resistor, allows
to compare measurement/simulation results for different initial currents. If
the magnet is simply discharged over the dump resistor, the extracted en-
ergy ratio drops from 1, where no quench occurs, to 0.1 at around 5 mΩ, and
then increases again to nearly 0.5. In case of a pre-existing quench, the ratio
increases with dump resistor size.
Figure 5.10 also shows the hot-spot temperature for all simulated cases.
For a terminal voltage below approximately 100 V, the peak temperature is
significantly higher in case of a pre-existing quench (left) although a similar
amount of energy is extracted (see above). Compared to the confined pre-
existing quench, the quench caused by the induced losses is spread over large
fractions of the coil cross-section and thus results in a faster current decay. In
the right-hand plot, a significant dip can be observed at around 100 V. Here,
5.1 Quench Protection Study Inner Triplet Upgrade Quadrupole 69
0
2000
4000
6000
8000
10000
12000
0 0.2 0.4 0.6 0.8 1 1.2
Current in A
Time in s
2 mOhm
4 mOhm
6 mOhm
8 mOhm
10 mOhm
12 mOhm
14 mOhm
16 m Ohm
18 mOhm
20 mOhm
40 mOhm
60 mOhm
80 mOhm
100 mOhm
0
2000
4000
6000
8000
10000
12000
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Current in A
Time in s
2 mOhm
4 mOhm
6 mOhm
8 mOhm
10 mOhm
12 mOhm
14 mOhm
16 m Ohm
18 mOhm
20 mOhm
40 mOhm
60 mOhm
80 mOhm
100 mOhm
Figure 5.9: LHC MQXC dump resistor study for various dump resistors in the range from
1 to 100 mΩ. Current decrease versus time. (left) In the case of a preexisting quench: The
simulations were stopped when the temperature limit of 600 K was reached (1 to 5 mΩ).
Aiming at a maximum hot-spot temperature of less than 300 K a resistor greater than
10 mΩ has to be chosen. (right) For the case of no initial quench.
0
0.2
0.4
0.6
0.8
1
0 200 400 600 800 1000 1200
0
100
200
300
400
500
600
Extracted Energy versus Total Energy
Temperature in K
Terminal Voltage in V
Energy
Temperature
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 200 400 600 800 1000 1200
0
10
20
30
40
50
60
70
80
90
100
110
Extracted Energy versus Total Energy
Temperature in K
Terminal Voltage in V
Energy
Temperature
Figure 5.10: Extracted energy versus voltage over the dump resistor. (left) Temperature
and extracted energy as function of terminal voltage with initial quench. (right) Tempera-
ture and extracted energy as function of terminal voltage without pre-existing quench. The
terminal voltage corresponds to the dump resistor size via UTerminal =RDRInom.
the hot-spot moves within the coil cross-section from the inner to the outer
layer. Beyond 500 V or 40 mΩ, respectively, the results of both experiments
are similar. Notice the different scales on the temperature axis.
The maximum terminal voltage as well as the voltage over the dump resis-
tor was set to 500 V. Therefore, for the magnet protection a dump resistor of
40 mΩ is chosen since it yields the smallest hot-spot temperature with a ter-
minal voltage of 503 V for a quench at nominal current. The current decrease
and temperature rise are shown in Fig. 5.11 (left). For a dump resistor of
1 mΩ (corresponding to a terminal voltage of 12 V), the magnet can be safely
discharged without causing a quench. All stored energy can be extracted, see
Fig. 5.10 (right).
70 Extrapolation
0
2000
4000
6000
8000
10000
12000
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0
100
200
300
400
500
600
Current in A
Temperature in K,
Terminal Voltage in V
Time in s
503 V
40 mOhm
0
2000
4000
6000
8000
10000
12000
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0
50
100
150
200
Current in A
Temperature in K
Time in s
Dump Resistor
Quench Heaters
Both
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Figure 5.11: (left) Current extraction by inverting the power supply (dashed line). The
full line shows the protection by means of a dump resistor. The initial and maximum voltage
are identical 503 V. While for a dump resistor the voltage over the magnet decreases with
the current, it remains constant for the power supply and causes a faster current decay
at the end of the quench. (right) Comparison of the three different protection methods.
When quench heaters and dump resistor are used, the peak temperature is about the same
as before .
5.1.4 Power Supply Inversion
A power supply which can be operated in two quadrants, i.e. which can
provide a positive and negative output voltage for a positive current, can
be used to actively drive down the magnet current in case of quench. This
has three advantages to a dump resistor. The stored magnetic energy is not
dissipated in heat, but fed back to the power grid; no-cool down time for
the dump resistor has to be observed; and for a given maximum voltage, the
current decrease is faster.
As part of the study, the protection of the magnet with the dump resistor
chosen above is compared to the protection with a power supply. The power
supply provides constant −503 V over the magnet which is identical to the
initial voltage over the dump resistor. Figure 5.11 (left) shows the current
decrease and temperature rise in comparison. The discharge into the grid
results in a slightly smaller temperature due to the faster current decrease at
the end of the quench.
5.1.5 Full Protection
Figure 5.11 shows the current and temperature change for quench protection
by means of quench heaters, a dump resistor or the combination of both. The
quench is detected and validated after 10 ms. The dump resistor of 40 mΩ is
switched into the circuit after an additional delay of 10 ms.
All three temperature graphs show the distinct kink which indicates hot-
spot movement - most pronounced for the protection with quench heaters
only. It is therefore not surprising that the hot-spot temperature in case of
the combined protection is only slightly lower than for the dump resistor only.
5.2 Fast-Ramping Dipole 71
The quench heaters provoke a quench in the outer layer before quench back
does without heaters and therefore these conductors are subjected to ohmic
heating for a longer period.
Nevertheless, the combination of both methods has the advantage of a
higher redundancy since both methods separately proof to be sufficient.
5.2 Fast-Ramping Dipole
Figure 5.12: Fast-
Ramping Dipole. Schematic
coil cross-section and iron
yoke. Outer iron diameter
504.6 mm.
The quench protection of superconducting accel-
erators is usually based on a reliable detection of
a quench, the decoupling of the quenching mag-
net from the non-quenching magnets by means of
diodes or thyristors, an energy extraction system,
and the shut-down of the power converters. In re-
cent projects, e.g., the SIS300 dipole magnet for
the FAIR project [Henn 04], ramp-rates of 1 T/s,
i.e. about 1100 A/s, are proposed. The flat-top is
considered to last from 10 to 100 seconds. Consid-
ering an inductance of about 25 mH for a magnet
length of 2.9m, the ramp-induced voltage across
a magnet is 27.5V. The ring will be powered by
several power converters in order to deal with the
total voltage of e.g., 3000 V for an installation of 110 magnets.
Calculations below show that a quench detection threshold of 1 V will be
desirable during the ramp in order to protect the magnets. The precision of
the electronics must be at least a factor 10 better in order to cover unexpected
behavior such as parasitic transient effects. This is no technological challenge
at the flat-top. However, during the ramp the induced voltage across a magnet
rises in a short time to 27.5V, while the common mode voltage rises to ±
1500V (or 300 V if five independent powering sub-sectors are chosen). In
particular during the acceleration and deceleration of the ramps, the required
50dB signal to noise ratio (detection precision of 0.1 V over a signal of 27.5V)
"riding" on a rapidly changing 84 dB common mode background (Ratio of
precision over common mode voltage) is a challenge in a large installation
where reliability is the most important requirement.
Based on the fact that a magnet designed for fast-ramping operation can be
ramped down much faster than existing superconducting magnets, an alter-
native possibility can be considered. It would be sufficient to detect a quench
on the flat-top or at the injection plateau, if the magnets can survive an un-
detected quench during the ramp. Bypass diodes would not be needed in this
case, which themselves represent a technological challenge. If state-of-the art
power converters with capacitive storage are used, even the dump resistors
and switches would not be required.
In the subsequent sections we discuss four aspects of quench simulation for
fast-ramping magnets:
72 Extrapolation
Figure 5.13: The standard coil cross-section of the SIS300 two-layer dipole is shown on
the left-hand side of the figure (x < 0).
The modified coil cross-section with a wider cable and additional strands for a 50% increase
in copper content is shown on the right-hand side. In order to keep the main field, the field
quality and the quench margin unchanged, two conductors had to be added to the outer
layer. The axis shows the x-position in mm.
•We show the influence of different thermal models (Sec. 3.6) on the
simulated coil temperature during the powering cycle.
•We study the ramp-rate dependence of the quench current.
•We investigate the peak temperature in a magnet for undetected quenches
during the ramp phases. We compare the cases of quench detection
during ramp for a threshold of 1V and the 100 mV threshold that is
standard in slowly-ramping magnets.
•We discusse whether additional copper-content in the coil could protect
a fast-ramping magnet for an undetected quench during the ramps, or
even during an entire cycle.
For the purpose of this study we simulate the 2-layer design of a SIS300
dipole magnet as defined in [Kozu 06], see Fig. 5.12. The relevant data for
the quench calculation is summarized in Sec. C.4. The cross-section of the
magnet is illustrated in Fig. 5.13.
We consider conductive cooling via the Kapton insulation to the helium
bath across the inner and outer radial surfaces of the coil. The ramp cycle
is given in [Kozu 06] as an up-ramp from 1.6T to 6T in 4.4 s, followed by a
plateau at 6T of 11 s, and a down ramp to 1.6 T in 4.4 s.
5.2.1 Cooling Schemes
The temperature variation during a ramp-cycle can be calculated with three
different thermal diffusion models:
1. Adiabatic conditions, i.e., no cooling and no helium as it is appropriate
for potted coils.
2. Heat transfer across the inner and outer radial surfaces without consid-
ering confined helium inside the cable.
5.2 Fast-Ramping Dipole 73
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
0 2 4 6 8 10 12 14 16 18
4.7
4.8
4.9
5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
6
Current in A
Temperature in K
Time in s
Current
Cooling
Cooling + Helium
None
0
2000
4000
6000
8000
10000
0 1 2 3 4 5 6 7
0
10
20
30
40
50
Current in A
Hotspot temperature in K
Time in s
Current
Temperature
Figure 5.14: (left) Temperature variation during ramp-cycle of the SIS300 dipole mag-
net.(right) Current and peak temperature in a magnet during an up-ramp to the quench
limit and quench detection. The graphs correspond to ramp rates (from right to left) of
1.0 T/s, 1.5 T/s, 2.0 T/s, 2.5 T/s, and 4.5 T/s.
3. Modeling of conductive cooling (see above) and of the thermal capacity
of the confined helium (about 10%) inside the cable.
Figure 5.14 (left) shows the temperature variation during one ramp-cycle for
the three cases. The temperature variation per cycle is an order of magnitude
smaller for the wetted coil, as the heat capacity of helium is dominant at low
temperatures. In all cases the temperature decreases at the flat-top. In case
of the adiabatic model 1) this is due to the transversal thermal conduction,
which is, however, not sufficient to prevent a quench already in the second
cycle. In the following model 3 is used, which reproduces data published in
[Kozu 06].
5.2.2 Quench Limits at Different Ramp Rates
In Fig. 5.14 (right) we see the evolution of current and peak-temperature
inside the magnet for quenches near the flat-top. The magnet is ramped
until the quench limit is reached; and the quench protection system detects
a resistive voltage of 100 mV. The magnet is protected with quench heaters
and a dump resistor, see Sec. C.4. The ramp rate varies between 1.0 T/s and
4.5T/s.
The temperature in the magnet rises due to ramp-induced losses [Verw 95]
that reduce the temperature margin to quench. The quench current thus
decreases for higher ramp rates. The peak temperature after quench is also
reduced as a consequence of the lower quench current.
In Fig. 5.15 (left) the temperature margin to quench is shown for each
conductor in the cross-section as a function of time. The plot corresponds
to the fastest ramp-rate in Fig. 5.14 (left), i.e. a ramp-rate of 4.5 T/s. The
margin reduces steadily as a consequence of increasing current and field, and
due to induced losses. At t= 1.18 s the magnet quenches in block number
5. The quench is detected at t= 1.19 s and the quench heaters are fired at
74 Extrapolation
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Figure 5.15: (left) Temperature margin to quench versus time for an up-ramp with
4.5 T/s. Block 1, 2, 11 and 12 belong to the outer layer and are partially covered by quench
heaters. (right) Current and losses versus time during quench.
t= 1.24 s. At the same time the dump resistor is switched into the circuit
and the current drops sharply. The quench heaters are effective about 40ms
later.
The temperature margin in the inner layer coil increases after the quench.
As a matter of fact, a number of conductors recover from quench. This is
explained by the fact that the current density and field in the superconductor
drops quickly. The time constant of the induced eddy-current losses in the
cable, however, is of about 50ms, compare Fig. 5.15 (right). Moreover, the
heat capacity of the confined helium in the cable results in a long thermal
time constant. This explains why the temperature rise due to induced losses is
slow and the margin grows immediately after the dump resistor is switched in.
Eventually the inner layer is quenched by induced losses. The block number
6, see Fig. 5.13, does not get an equal share of induced losses. The reason
is that it is placed in parallel to the electromagnetic field. The Rutherford
cable used for the SIS300 magnet has a stainless steel core, which reduces the
cross-over resistance. Consequently, very little eddy-currents are induced in
a cable that is positioned in parallel to the field lines.
5.2.3 Quench Detection During the Up- and Down Ramp
We study the current decay and peak-temperature after a quench during
the up-ramp. Different detection scenarios are investigated. Quenches are
assumed to occur at either 50% or 75% of the plateau level. Quenches are
detected either only at the plateau, or when the resistive voltage has reached
a threshold of 100 mV or 1 V. Figure 5.16 (left) shows the current and peak-
temperature evolution for the six different cases.
An undetected quench during the up-ramp leads to a fatal temperature
rise in the magnet. The earlier the quench occurs during the up-ramp phase,
the higher is the peak-temperature. A detection threshold of 1 V during the
ramp is sufficient to protect the magnet against quenches occurring at 50%
5.2 Fast-Ramping Dipole 75
0
1000
2000
3000
4000
5000
6000
7000
8000
2 2.5 3 3.5 4 4.5 5
0
100
200
300
400
500
600
Current in A
Hotspot temperature in K
Time in s
Detection at Plateau
Detection at 1.0 V
Detection at 0.1 V
18.9 Applications
K
0
1
2
3
4
5
t
s
0. 0.5 1.0 1.5.
1
2
6
5
Fig. 18.21 Temperature margin to quench (one pole) as a function of
time for an up-ramp with 4.5 T·s−1. The outer-layer coil blocks No. 1
and 2 are partially covered by quench heaters. After the opening of the
protection resistor switch, the inner layer cables recover partially, be-
fore they are quenched due to cable eddy currents. Block numbering
scheme according to Fig. 18.24.
citation, since the quench propagation, and consequently the resistive voltage
rise, are slower.
Passively protected magnets Finally, we study how much copper is required
in the cables, so that the magnet can withstand an undetected quench during
the up- or down-ramp. Fig. 18.23 (left) shows simulations for I0=0.5 Inom
and I0=0.75 Inom. The baseline of 100% copper is defined by the cable param-
2 2.5 3 3.5 4 4.5 5
Flat-top
1 V
100 mV
1
2
3
4
5
6
7
0
100
200
300
400
500
600
T
max.
K
t
s
I
kA
T
max.
I
0.5 I
nom
2 2.5 3 3.5 4 4.5 5
Flat-top
1 V
100 mV
1
2
3
4
5
6
7
0
100
200
300
400
500
600
T
max.
K
t
s
I
kA
T
max.
I
0.75 I
nom
Fig. 18.22 Current decay and peak-temperature for quenches trig-
gered during the up-ramp at I0=0.5 Inom (left) and 0.75 Inom (right).
Detection at the plateau (dashed), at a resistive voltage of 1V (dash-
dotted), and at 100mV (continuous).
603
!""#$%#
!#%# &'()*(+#
0
1000
2000
3000
4000
5000
6000
7000
8000
2 2.5 3 3.5 4 4.5 5
0
100
200
300
400
500
600
Current in A
Hotspot temperature in K
Time in s
Detection at Plateau
Detection at 1.0 V
Detection at 0.1 V
18.9 Applications
K
0
1
2
3
4
5
t
s
0. 0.5 1.0 1.5.
1
2
6
5
Fig. 18.21 Temperature margin to quench (one pole) as a function of
time for an up-ramp with 4.5 T·s−1. The outer-layer coil blocks No. 1
and 2 are partially covered by quench heaters. After the opening of the
protection resistor switch, the inner layer cables recover partially, be-
fore they are quenched due to cable eddy currents. Block numbering
scheme according to Fig. 18.24.
citation, since the quench propagation, and consequently the resistive voltage
rise, are slower.
Passively protected magnets Finally, we study how much copper is required
in the cables, so that the magnet can withstand an undetected quench during
the up- or down-ramp. Fig. 18.23 (left) shows simulations for I0=0.5 Inom
and I0=0.75 Inom. The baseline of 100% copper is defined by the cable param-
2 2.5 3 3.5 4 4.5 5
Flat-top
1 V
100 mV
1
2
3
4
5
6
7
0
100
200
300
400
500
600
T
max.
K
t
s
I
kA
T
max.
I
0.5 I
nom
2 2.5 3 3.5 4 4.5 5
Flat-top
1 V
100 mV
1
2
3
4
5
6
7
0
100
200
300
400
500
600
T
max.
K
t
s
I
kA
T
max.
I
0.75 I
nom
Fig. 18.22 Current decay and peak-temperature for quenches trig-
gered during the up-ramp at I0=0.5 Inom (left) and 0.75 Inom (right).
Detection at the plateau (dashed), at a resistive voltage of 1V (dash-
dotted), and at 100mV (continuous).
603
!""#$%#
!#%#
&'()*(+#
,"#-# .,#-#
Figure 5.16: Current decay and peak-temperature for quenches during the up-ramp
phase. A Quench is initiated at 50% (left) or 75% (right) of the nominal current level.
It is assumed that quenches can only be detected at the plateau (dashed line), at a resistive
voltage threshold of 1 V (dash-dotted line), or 100 mV (continuous line).
1000
2000
3000
4000
5000
6000
7000
0 2 4 6 8 10 12 14 16 18
0
100
200
300
400
500
600
Current in A
Hotspot temperature in K
Time in s
Current
Plateau
90 %
75 %
50 %
Figure 5.17: Current- and peak-temperature evolution for quenches during the down-
ramp. Quenches start at the end of the plateau, at 90%, 75%, and 50% of the nominal
current level. It is assumed that quenches can only be detected at the injection level.
of the plateau level. The delays of the protection system can be deduced
from Fig. 5.16 by taking the time laps between the first rise in temperature
and the point where the current decays (and the energy is extracted by the
dump resistor). The detection delays are longer at lower excitation, since the
resistive voltage rises more slowly.
In Fig. 5.17 the same exercise is carried out for quenches occurring on
the down-ramp. It is assumed that quenches can only be detected at the
injection level. Comparison with Fig. 5.16 shows that quenches occurring
near the upper plateau are more critical during the down-ramp than during
the up-ramp. This was expected, since a current is forced to flow through the
quenched magnet during a longer period of time.
76 Extrapolation
1000
2000
3000
4000
5000
6000
7000
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0
100
200
300
400
500
600
Current in A
Hotspot temperature in K
Time in s
Current
100 %
110 %
150 %
0
1000
2000
3000
4000
5000
6000
7000
8000
0 1 2 3 4 5 6
0
100
200
300
400
500
600
Current in A
Hotspot temperature in K
Time in s
Current
Temperature
!"#$%"&%'%
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-)%,.01$%'*$,"2,.3)&%
1000
2000
3000
4000
5000
6000
7000
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0
100
200
300
400
500
600
Current in A
Hotspot temperature in K
Time in s
Current
100 %
110 %
150 %
456%
746%
Figure 5.18: (left) Peak-temperature and current evolution during quenches at 50% and
75% of the plateau level. It is assumed that the quench cannot be detected until the
plateau is reached. The simulation is carried out for the SIS300 conductor, as well as for
conductors that have a 10% to 50% higher copper content. (right) Peak temperature and
current evolution during a quench. The quench starts at 50% of the up-ramp. It is detected
at the plateau. Then the magnet is immediately driven down with the standard slope.
5.2.4 Robust Magnet Design
We can also investigate how much copper would need to be added to the
Rutherford-type cable so that the magnet survives an undetected quench
during a ramp. Figure 5.18 (left) shows quenches occurring at 50% and at
75% of the plateau for different copper content. The baseline of 100% is
given by the cable described in Sec. C.4.1. In case of 150% copper we added
strands and increased the copper-to-superconductor ratio for all strands, to
result in 50% more copper and the same amount of Nb-Ti superconductor. As
the cables become wider, the coil cross-section is re-designed with two more
conductors in the outer layer, in order to maintain the field quality, see Fig.
5.13 (right-hand side). The inductance of this coil increases by about 10%. It
can be seen in Fig. 5.18 (left) that this magnet could survive a quench that
occurs half-way on the up-ramp and that is detected only at the plateau.
Active protection may be reduced to a strict minimum, if a magnet contains
enough copper to survive a full powering cycle with an undetected quench.
We assume that a quench occurs at 50% of the up-ramp, that it is detected
at the plateau, and that the magnet is then ramped-down immediately with
the regular ramp rate. For the cable with 50% extra copper, Fig. 5.18 (right)
shows the temperature evolution. This cable had survived an undetected
quench during an up ramp, see Fig. 5.18 (left). We can see that the magnet
is not sufficiently protected for the shortened cycle. It follows that for a
magnet with only 50% additional copper a method of fast current shut-down
is indispensable.
6 Conclusion and Outlook
Rerum cognoscere causas
Virgil
(70 BCE - 19 BCE)
Most quench models published in the literature either focus on a single aspect
of the quench process, such as e.g. the propagation of the quench, or cover
parts of the quench process with constant parameters, e.g. a constant induc-
tance of pre-calculated field maps. An exception is the coupled approach of
Aird, which on the other hand does not permit to calculate internal voltages.
The quench algorithm presented in this thesis, solves the numerical models
of all relevant physical effects in a coupled approach. This allows to model
the interdependence of various physical phenomena, e.g. quench propagation,
induced losses, and quench back. Adaptive time-stepping is implemented to
resolve the highly nonlinear behaviour of materials such as the heat capacity
or the electrical resistivity.
The user is required to supply data for the characterization of the magnet
protection system, electrical and thermal properties of cable and insulator
materials, as well as geometrical description of the magnet. The data must
be available at a wide temperature range reaching down to cryogenic temper-
atures, and for a range of magnetic fields and pressures.
Given the large number of empirical parameters in the models, it must be
noticed that an important part of the simulation work consists of the deter-
mination of parameters such that the simulation matches the measurement.
It is important to realise that only when all relevant phenomena have been
taken into account and modelled accurately, the choice of material param-
eters is physically reasonable and within the range of measured values, the
simulation result matches the measured data within the uncertainty of the
measurement. Then we are able to reproduce the internal states of a quench-
ing magnet, i.e., observe quantities that evade measurement.
With this approach we could observe the movement of the hot-spot within
the coil cross-section and find an explanation for the voltage ripples during
the first phase of a quench. Under the aforementioned premise, the model can
be used for extrapolation; for the comparison of different protection methods
and for a study of design options for future magnets.
The special properties of the materials used in superconducting accelerator
77
78 Conclusion and Outlook
magnets, i.e. field dependent, diffusive, and exhibiting hysteresis, required a
review on the formulae for the electromagnetic energy and the concept of the
inductance. Furthermore, an expression for the instantaneously dissipated
power (in contrast to the dissipated energy) was derived for the superconduc-
tor hysteresis losses.
Critical conclusion Regarding the large number of parameters in comparison
to the limited number of observables, the famous quote on fitting an elephant
comes to mind1. However, the range and hence the impact of every parameter
is limited by the underlying physical models. Therefore, fitting means to
adjust the influence of all physical phenomena involved in the quench process,
and the interpretation of the necessary set of parameters offers additional
understanding of the process. Parameters which would need to be chosen
outside the physically reasonable domain in order to fit measurements indicate
an incomplete model2and give rise to further refinement.
Critical Outlook Not all aspects of the modeling of superconducting mag-
nets and quench simulations could be treated in this dissertation. Some had
to be disregarded because they exceed the scope of a thesis in electromagnetic
theory, some due to the limitation of computing power and some because ad-
equate models were not yet available. Therefore, future work should address
the following aspects:
•The magnetic field and cable magnetization model are static and calcu-
lated in 2D. The calculation of cable eddy currents relies on the assump-
tion of superconducting strands and is therefore only weakly coupled to
the magnetic field computation. This approach allows to describe the
magnetic field in long accelerator magnets operated close to the critical
surface.
With increasing computing power the following approach could be im-
plemented: Improving the magnetic field model to full, time-transient
3D. This allows to consider the variation of the magnetic field over the
coil ends, the change of magnetic length with iron saturation and in-
duced eddy current losses in copper wedges. Including the current and
temperature dependence of the superconducting strands into the mod-
els for cable eddy currents requires a cable model down to the level
1“Fitting an elephant” was attributed to C. F. Gauss, N. Bohr, Lord Kelvin, and E.
Fermi, and R. Feynman [Brow 03]:
Give me four parameters and I can fit an elephant. Give me five and I can
wag its tail.
2Consider an electromagnetic wave in an arbitrary medium. In a first approach we
describe the problem by means of the wave equation. If measurements can only be
reproduced applying complex values for the electrical permittivity, we conclude that the
wave exhibits losses in the medium, and that the problem was ill-posed: it is correctly
described by an electromagnetic wave equation with an additional term carrying the
first time derivative (comparable to the telegrapher’s equation on transmission lines).
Conclusion and Outlook 79
of individual strands resulting in longitudinal variations of the induced
magnetic fields. The magnetic field and eddy current models have to
be strongly coupled. This way, one could take into account the current
redistribution between adjacent strands, simulate the signal for quench
antennas and consider the voltages induced by eddy currents.
•The presented thermal model is designed for the calculation of quench in
magnets operating close to the critical surface and protected by quench
heaters. Heat transfer, cooling and the physics of liquid helium are only
implemented in a rudimentary fashion.
The thermal network should be extended using a Finite-Element Ap-
proach (with a higher resolution), modeling the cooling channels through
the coil and the iron yoke, and including a helium mass flow, i.e. con-
vective cooling. Furthermore, the strand/cable helium interface should
be studied for possible homogenization, i.e. how to include the limita-
tion of the heat transfer without resolving below the dimensions of a
strand.
This would allow to simulate magnets cooled by forced-flow helium, to
study magnets with big thermal margin and to analyze the impact of
heat depositions within the coil cross-section.
•The quench algorithm is based on the assumption that the time con-
stants of the magnetic field and cable magnetization model are much
longer than the ones of the thermal model. The electrical network model
works as an synchronization agent between the two time scales.
The coupling between the different sub-models should be made more
flexible allowing for changes of the individual time scales. This requires
a detailed analysis of the coupling between systems by means of input
values and variation of parameters.
•The presented model disregards all mechanical aspects of a quench, i.e.
the triggering of a quench by a sudden conductor movement and heat
of friction, or the mechanical tension in the coil cross-section due to
thermal expansion of the different components during a quench.
•The presented approach for the computation of the instantaneously dis-
sipated hysteresis losses in superconductors should be generalized. It
should include the field and temperature dependence of the critical cur-
rent density and be applied to circular geometries in order to describe
superconducting strands.
80 Conclusion and Outlook
7 Detailed Treatment
I didn’t have time to write a short letter, so I wrote
a long one instead.
Mark Twain
(1835-1910)
7.1 Margins to Quench
In order to secure safe and stable operating conditions of superconducting
accelerator magnets, margins to quench, i.e. intervals between working and
critical conditions, are defined and monitored during design and operation.
All margins to quench are derived from the critical current density. The
quenched state is reached when the applied current density exceeds the critical
current density, or the critical current is zero. This approach is conservative
since the transition from superconducting to normal conducting state is not
abrupt. Note that safe, but probably unstable, operation conditions can be
realized beyond the critical current density limit (Sec. 7.5.1).
It should be noticed that the definition (Sec. A.2.4) and derivation of the
critical current density (Sec. D.3) contain some uncertainty.
7.1.1 Current Density Margin
The current density margin ∆Jc(B, J, T)denotes the difference between trans-
port current density Jand critical current density Jc,
∆Jc(B, J, T) = (Jc(B, T )−J J < Jc(B, T )
0else .(7.1)
The current density margin is highlighted in Fig. 7.1 by means of a blue
vertical line for two different working points (B1and B2).
7.1.2 Temperature Margin
The transition temperature, or critical temperature of a superconductor,
Tc(B, J), is a function of the applied magnetic induction and current den-
sity. It can be calculated from the critical surface parameterization Jcby
81
82 Detailed Treatment
B
T
J
!TC
!JC
B1
B2
Figure 7.1: Margins to quench: Given are a constant operating current density (horizontal
plane) and temperature (red line). The temperature margin to quench for different fields
are highlighted by horizontal lines in the plane (green). The current margin is highlighted
by vertical lines (blue).
solving the equation
Tc(B, J) = min{T|∆Jc(B, J, T)=0}.(7.2)
Equation (7.2) poses an implicit root-finding problem depending on 3 quanti-
ties. The solution relies on numerical algorithms as, e.g.,Newton’s method
[Stoe 05, pp. 301]. Figure 7.3 (left) shows the critical temperature as a func-
tion of local magnetic induction for different values of applied current density.
The temperature margin to quench is then given by [Siem 05]:
∆Tc(B, T, J) = (Tc(B, J)−T T < Tc(B, J)
0else .(7.3)
The temperature margin to quench is highlighted in Fig. 7.1 by a horizontal
green line for two different working points. Figure 7.2 shows the temperature
margin to quench for the LHC main dipole, for injection and nominal current
level. At low excitation the temperature margin is 7 K and nearly no differ-
ences can be noticed over the cross-section. At high excitation the margin
reduces to 1 K where the magnetic field is highest, while the margin of 6 K
remains in the low field region.
Plotting the temperature margin to quench for each conductor versus timeAutobahn
plot gives a very good tool to analyze the sequence of events during a quench. Due
to the non-linear root-finding process this is computationally costly. Similar
results can be achieved plotting the current density margin.
7.1 Margins to Quench 83
97 107 117 127 137 147 157
1.612
1.865
2.118
2.370
2.623
2.876
3.129
3.381
3.634
3.887
4.140
4.392
4.645
4.898
5.151
5.404
5.656
5.909
6.162
6.415
Temperature margin (K)
Time (s) : 1200.
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97 107 117 127 137 147 157
6.930
6.946
6.961
6.976
6.992
7.007
7.022
7.038
7.053
7.069
7.084
7.099
7.115
7.130
7.145
7.161
7.176
7.191
7.207
7.222
Temperature margin (K)
Time (s) : 80.
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!"#$"%&'(%")*&%+,-)
,-).)
!"#$"%&'(%")*&%+,-)
,-).)
Figure 7.2: Temperature margin to quench, ∆Tcfor the LHC MB at injection current
level (left) and at nominal current level (right).
7.1.3 Margin on the Load-line
The margin on the load-line is a global criterium for a constant operation
temperature, Top =const.. It has little implication on quench computation
in accelerator magnets. It is mentioned here for the sake of completeness and
because of its common use in magnet design.
The performance of a magnet is limited by the peak magnetic induction
in the coil. The function of peak field versus excitation current density is
called load-line. For safe operation the working point PW,i.e. the pair of
applied current density Jand peak field Bpeak on the conductor, must not
exceed the limit given by the critical current density Jc(B). Extrapolating
the working point onto the function of the critical current density yields the
"quench point", PQ, an estimate for the quench field respectively quench
current density (see Fig. 7.3 (right)). The distance from the working point to
the quench point, divided by the distance of the quench point to the origin,
yields the margin on the load-line (usually expressed in per cent).
Margin-on-the-Load-line =OPQ−OPW
OPQ
(7.4)
7.1.4 Enthalpy Margin/Energy Reserve
Consider a volume Vof homogeneous volumetric specific heat ceff and tem- Definition
perature Ts. The volume is thermally isolated and the work done by the
system can be neglected, i.e. the pressure pdoes not change. An external
heat flux causes the temperature of the system to rise. Without heat trans-
fer, the amount of external energy and the change of the temperature in the
volume Vare related by
∆H(T) = VZT
Ts
ceff(τ)dτ, (7.5)
84 Detailed Treatment
J
0 2 4 6 8 10 12 14
0
2
4
6
8
10
Bin T
Tc!B,J"
in K
!"
#!
!"#$%!&'()!
$*!
$+!
,!"
,#"
%-%'(!
.((/01234566786497:;2064!"#$%&'&()*+,"-+%&*.
Figure 7.3: (left) Critical temperature Tcas a function of applied current density Jand
local magnetic induction B. The discontinuous (due to bad conversion of the root-finding)
line represents zero applied current density. The current density is increased between two
successive lines by 20% of the current density used to obtain a field of 8.4 T at 1.9 K.
The dotted lines represent the two common operating temperatures 1.9 K and 4.2 K. The
temperature margin to quench, ∆Tcis the vertical distance between the dotted line and
the line of Tc. (right) Margin on the load-line.
as can be easily derived from the continuous heat equation (3.35). The energy
reserve ∆His also called the change of the enthalpy of the system. For
an effective specific heat greater than zero, the change of enthalpy yields
a monotonous function and is invertible. The inverted function allows to
calculate the temperature change for a given external energy.
The change of energy density from the operating temperature Tsto the
critical temperature Tcis called the energy density reserve, or enthalpy density
margin,
∆hc=∆H(Tc)
V,[∆hc] = mJ cm−3.(7.6)
The stability of the operation of superconducting magnets against beamApproach
losses can be estimated by means of the energy density reserve [Jean 96].
Depending on the time and spatial distribution of the beam losses, different
parts of the superconducting cables are considered [Boci 06] and represented
in the applied effective specific heat (see Sec. B.2.3).
strand, very fast - ∆hstr
cThe beam interacts with part of one strand. The
pulse is too short to allow significant heat transfer along the strand or
transversally. The temperature increase is calculated from the strand
properties only. Exceeding the enthalpy margin causes the strand to
quench and the current to re-distribute into the neighboring strands.
cable, very fast - ∆hcab1
cThe beam losses interact with the cable homoge-
neously over the full cross-section and length. The beam particles in-
teract only with the metal part of the cable. The loss pulse is too short
7.1 Margins to Quench 85
Tb=1.9 K
Tb=4.2 K
1
5
10
50
100
500
1000
0.001
0.1
10
1000
105
Tin K
Dhcin
mJ cm-3
Injection
Nominal
Ultimate
0 2000 4000 6000 8000 10 000 12 000
1
2
5
10
20
50
100
200
Iin A
!hin
mJ cm"3
Figure 7.4: (left) Energy reserve density ∆hcbetween the bath temperature Tband the
elevated temperature T. The continuous line represents the energy change of a strand. The
dashed line represents the energy change for a cable with %10 insulation and 2% liquid
helium. (right) Minimum and maximum energy reserve density within the LHC main dipole
coil cross section for a strand (full), a cable with (dotted) and without (dashed) helium
and insulation.
for a significant energy transfer to the helium in the cable. The temper-
ature increase is calculated from the average properties of the strands
of the cable.
cable, fast - ∆hcab2
cAs for ∆hcab1
cbut with a longer duration so that the
heat can be transfered to helium and the insulation, but it is too short
for significant heat transfer along the cable or to neighboring cables. All
materials of the cable are considered for the effective specific heat. If
the cable temperature is below the vaporization temperature of helium,
the latent heat of vaporization has to be considered with approximately
2.6 J cm−3for helium (see Sec. A.3.3).
Note, in case of ∆hcab2
cit is incorrect to speak about enthalpy density margin
[Jean 96] since the enthalpy is defined for constant pressure. For helium the
pressure changes significantly when the temperature increases over the point
of evaporation (see Sec. A.3.2).
Example : Energy reserve of the LHC main dipole (MB) Consider-
ing a strand and a cable of the LHC MB outer layer and assuming a confined
helium content of approximately 2%, Fig. 7.4 (left) shows the energy reserve
∆hstr
cand ∆hcab2
cbetween the bath temperature Tband the elevated temper-
ature T. The helium creates a thermal buffer and increases the reserve by a
factor 100. The temperature of the cable does not change up to an external
energy of 40 mJ cm−3. The influence of the helium is much less pronounced
if the operation temperature is 4.2 K. Beyond 15 −30 K the influence of the
helium fully vanishes due to evaporation and the strong increase of specific
86 Detailed Treatment
97 107 117 127 137 147 157
160.0
160.3
160.6
160.9
161.2
161.5
161.8
162.1
162.4
162.7
163.0
163.3
163.6
163.9
164.2
164.5
164.8
165.1
165.4
165.7
Enthalpy Margin Cable 2 (mJ/cm3)
Time (s) : 80.
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97 107 117 127 137 147 157
35.61
35.84
36.08
36.31
36.55
36.79
37.02
37.26
37.49
37.73
37.96
38.20
38.43
38.67
38.90
39.14
39.38
39.61
39.85
40.08
Enthalpy Margin Strand (mJ/cm3)
Time (s) : 80.
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!"#$%&'()*%+,-").#+%"/))
-")012304)-")012304)
!"#$%&'()*%+,-")5%6&7))
97 107 117 127 137 147 157
59.69
62.69
65.69
68.70
71.70
74.70
77.71
80.71
83.71
86.72
89.72
92.72
95.73
98.73
101.7
104.7
107.7
110.7
113.7
116.7
Enthalpy Margin Cable 2 (mJ/cm3)
Time (s) : 1200.
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97 107 117 127 137 147 157
1.996
3.362
4.729
6.096
7.463
8.830
10.19
11.56
12.93
14.29
15.66
17.03
18.39
19.76
21.13
22.49
23.86
25.23
26.6
27.96
Enthalpy Margin Strand (mJ/cm3)
Time (s) : 1200.
ROXIE10.0BEMFEM *
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-")012304)
!"#$%&'()*%+,-")5%6&7))
!"#$%&'()*%+,-").#+%"/))
-")012304)
Figure 7.5: Energy margin to quench ∆hcfor the LHC MB (see Sec. C.1) with 2% helium
in the cable. The upper row shows the margins at injection current level and the lower row
at nominal current level. The left column shows the enthalpy margin in the strands for fast
losses and the right column the energy reserve of the cable for slow losses.
heat of the other solids.
Figure 7.5 shows the energy reserve of a strand (top) and cable with helium
(bottom) over the coil cross-section for injection (left) and nominal (right)
current. Due to the minimum of the magnetic field at the mid-plane of the
coil, strands in that region show only very little change of margin between
both current levels. Figure 7.4 (right) shows the minimum and maximum
energy reserves within the coil cross section over excitation.
7.1.5 Minimum Quench Energy
The energy reserve in Sec. 7.1.4 neglects all heat transfer, cooling and current
redistribution and is therefore a conservative estimated for most disturbances.
The minimum quench energy (MQE) takes into account the full thermal sys-
tem of the superconducting cables, including recovery from quench. It ex-
presses the minimum amount of energy and its distribution within the cable
in order to irreversibly drive a conductor into the normal conducting state.
The MQE can not be expressed analytically and requires measurements or
simulations as in [Will 08a, Will 08b].
7.2 MIITs 87
7.2 MIITs
Following the approach of Maddock and James the hot spot-temperature in
a superconducting magnet after a quench can be calculated from the recorded
current decrease [Wils 83, pp. 201]:
A time dependent current I(t)flows through a volume V=A`. The vol- Area ratio:
Sec. B.2.1
ume consists of a conducting fraction ηCond.
Awith electrical resistivity ρEand
a fraction of negligible conduction (see Fig. 2.7 for a comparison of the resis-
tivity of the different cable materials). The effective volumetric specific heat
of the total volume is ceff
V. Neglecting any kind of cooling or heat transfer out Effective
specific
heat: Sec.
B.2.3
of the considered volume, the dissipated ohmic power P=I2ρE`/ ηCond.
AA
results in a temperature rise:
ρE(B, T, RRR)`
ηCond.
AAI(t)2
| {z }
P
=A`ceff
V(T)dT
dt(7.7)
For constant magnetic induction B=B0, Eq. (7.7) can be re-arranged and
separated. Integrating the left-hand side over time from quench start tqto
t→ ∞,i.e. until the current decreased to zero, and the right-hand side over
temperature from operation or bath temperature Tbto the final temperature
Tyields:
MIITs =I2
0tval −tq+Z∞
t=tdet
I(t)2dt(7.8)
=A2ηCond.
AZT
Tb
ceff
V(˜
T)
ρE(B0,˜
T, RRR)d˜
T
| {z }
Υ(T)|B0
The integral on the left-hand side is denoted mega current-square time integral Introduction
of Υ
(short: MIIT-integral)1and its value MIITs ([MIITs] = MA2s). The integral
on the right-hand side constitutes a function Υdepending on the material mix
in the volume and the applied magnetic induction B0. For ceff
V/ρE>0for
all T, the function Υis monotonous and can be inverted giving the hot-spot
temperature Thot-spot:
Thot-spot = Υ−1MIITs
A2B0
.(7.9)
Figure 7.6 (left) shows the function Υfor different materials. The large
differences in resulting hot spot temperatures mainly stem from the differences
in electrical resistivity. Adding about 2% of helium to cable significantly
changes the function Υfor temperatures above 50 K.
1In the field of power electronics, the so-called “Grenzlastintegral” Ri2dtis used to esti-
mate the overload protection of semi-conductors for short over current pulses [Lapp 91,
p. 101].
88 Detailed Treatment
Cu!RRR!200"
Cu!RRR!100"
Cable "He
Cable
Nb#Ti, Nb3Sn
0 100 200 300 400 500
0
5.0 $1010
1.0 $1011
1.5 $1011
Tin K
%or
MIITs#Area2
in !kA"2sm#4
B!0.0T
B!2.5T
B!5.0T
B!7.5T
B!10.0T
45 MIITs
30 MIITs
0 100 200 300 400 500
0
2"1010
4"1010
6"1010
8"1010
1"1011
Tin K
#or
MIITs!Area2
in "kA#2sm$4
Figure 7.6: MIITs or Υcomputation. (left) MIITs versus temperature for different ma-
terials. In case of the cable, only the resistivity of copper is considered. In all other cases
the resistivity of the respective materials is taken into account. (right) LHC MB outer
layer cable with 2% helium, 10% kapton insulation and RRR = 200 for different applied
magnetic induction (dashed). The horizontal lines (dotted) correspond to 30 and 45 MIITs
related to the square of the cross-sectional area of the LHC MB outer layer cable. The full
line represents the analytic approximation ˜
Υ.
Example : LHC MB outer layer cable MIITs Figure 7.6 (right) shows
the function Υfor varying applied magnetic induction for the LHC MB outer
layer cable taking into account 2% helium and 10% kapton insulation (dashed
lines). The horizontal lines signify 30 and 45 MIITs, respectively. In the first
case the expected hot-spot temperature yields approximately 100 K for zero
field and 375 K for 5 T. In the second case the differences are even more pro-
found, i.e. 280 K and more than 500 K, respectively.
Following [Brun 04, p. 159] the function Υcan be anaytically aproximated
for the LHC. Applying the above terminology the approximation ˜
Υreads,
˜
Υ(T)=0.13 MA2s m−4·ηCond.
AT
375 K0.4
.(7.10)
The approximation is represented by a full line in 7.6 (right).
Discussion: The MIITs concept is expected to be conservative, yielding
higher temperatures than in reality as cooling and heat transfer are neglected.
This is only correct if the hot-spot location in the magnet coincides with the
quench origin and when additional heat sources do not contribute significantly
to the temperature rise. In a case where the hot-spot moves during quench
within the coil cross-section from the quench origin to a conductor with much
less favorable properties (see Sec. 5.1.1), e.g. higher resistivity, the hot-spot
temperature will be underestimated.
7.3 Magnetic Energy 89
The strong field dependence of the electrical resistivity of copper at cryo-
genic temperatures (Sec. A.1.3.1), and the field variation during current
switch off, cause a significant uncertainty in the application of Eq. (7.9) and
thus in the final hot-spot temperature. The uncertainties in determining the
helium content in the cable cross-section map directly on the uncertainties of
the hotspot-temperature.
A quench occurs at t=tqat a constant current I0. After quench detection
and validation at t=tval (see Sec. 7.9.1) the current is switched off as
fast as possible. The value of the MIIT-integral, and therefore the hot-spot
temperature, thus depend on the time it takes to detect the quench tdet −tq
[Brun 04, p. 159] as well as the time constant of the magnet current decay
[Schm 00]. Quench protection mainly addresses the minimization of these two
time constants (see Sec. 7.9).
7.3 Magnetic Energy
For the calculation of problem sets including materials with field dependent
material properties or materials exhibiting hysteresis, the formulae for the
magnetic energy have to be reviewed.
Poynting’s theorem: Starting from the general formulation of Maxwell’s
equations, Poynting’s theorem is derived. By multiplying the induction law
in differential form,
∇×E=−∂B
∂t ,(7.11)
by Hand using the vector identity ∇·(a×b) = b·(∇×a)−a·(∇×b)we
can write −H·∂B
∂t =H·(∇×E) = ∇·(E×H) + E·(∇×H).Substituting
with Ampere’s circuit law,
∇×H=J+∂D
∂t ,(7.12)
this yields Poynting’s theorem in differential form:
E·J=−∇·(E×H)−H·∂B
∂t −E·∂D
∂t .(7.13)
Integrating Eq. (7.13) over a volume Vand using Gauss’ law, yields Poynt-
ing’s theorem in integral form,
ZV
E·JdV+Z∂V
(E×H)·da=−ZV
H·∂B
∂t dV−ZV
E·∂D
∂t dV, (7.14)
where dadenotes the differential surface element on the volume boundary
∂V .
90 Detailed Treatment
!" !"#$%&'()*+,#%(
-'#.,$/%&01#2.()*+,#%(
3#."&14.&'()*+,#%(
-1#$0*(-5."&10#(6#,7##1()*+,#%+(
-1#$0*(8'95(/:#$(,"#(;/'9%#(</91=&$*((
!"
Figure 7.7: Energy exchange between different physical systems and different volumes.
We distinguish between energy flux over the volume boundary ∂V and energy exchange to
the thermal and mechanical system within the volume V.
Interpretation of the different terms We assume general electromagnetic
fields. We interpret:
RVE·JdVas an energy exchange to other physical systems, e.g. the me-
chanical or thermal system (see Fig. 7.7). For E·J>0, it describes
the work per time done on electric charges, e.g. ohmic losses. An inner
product smaller than zero means imposed current densities delivering
energy to the field.
It is important to notice that hysteresis and polarization losses are not
covered by this expression. The additional energy exchange to the ther-
mal system is introduced below.
H∂V (E×H)·daas the flux of electromagnetic energy over the volume bound-
ary ∂V . The term E×H=Sis denoted poynting vector and represents
the energy flux density (see Fig. 7.7); [S] = J s−1m−2.
Notice that in the presence of electromagnetic waves, the energy flux
over the infinite far-field boundary remains finite [Henk 01, pp. 283].
RVH·∂B
∂t dVas the change of magnetic energy stored in the volume Vand
the energy loss in hysteretic materials:
We integrate RVH·∂B
∂t dVover time and assume that the order of in-
tegration can be changed, i.e the integral over the volume does not
depend on time.
Zt1
t0ZV
H·∂B
∂t dVdt=ZVZt1
t0
H·∂B
∂t dtdV. (7.15)
Note that if H·∂B
∂t is a continuous function over the interval [t0, t1],
the integral can be solved by means of the fundamental theorem of
calculus and the antiderivative represents the magnetic energy Wmag.
7.3 Magnetic Energy 91
For piece-wise continuous functions the integral is given by the sum over
the continuous sub-intervals.
For general materials we assume the magnetic field is a continuous func-
tion of the magnetic induction H(B)and the magnetic induction B(t)
is piece-wise continuos differentiable over time. For a “path” B(t0)→
B(t1)we apply the rule for line integrals,
ZVZt1
t0
H·∂B
∂t dtdV=ZVZB(t0)→B(t1)
H·dBdV. (7.16)
The integral is generally not independent of the path B(t)and Hdoes
not represent a conservative field over B. Therefore it consists of both,
the magnetic energy and dissipated hysteresis losses. Due to the diffi-
culties to calculate hysteresis losses for anything else than a closed cycle,
it is not possible to extract the conservative part from integral (7.16).
The hysteresis losses over a closed cycle, γB:B0→B1→B0are given
by:
WV
hystγB
=ZVIγB
H·dBdV. (7.17)
Assuming HH·dB=HB·dH, and substituting B=µ0(H+M)as
well as γB→γH, the integral can be reduced to the volume VMwhere
the magnetization Mis defined.
ZVIγB
B·dHdV=ZVIγH
µ0H·dH
| {z }
=0, for closed cycle
dV+ZVIγH
µ0M·dHdV
=ZVMIγH
µ0M·dHdV
If Hcan be expressed as the gradient of a potential Wmag, with H=
∂Wmag
∂B , then the integral is path independent and the potential consti-
tutes the magnetic energy.
Wmag
B1
B0
=ZVZB1
B0
H·dBdV. (7.18)
For materials with constant properties, i.e. H= (µ)−1B, this further
simplifies to:
Wmag
B1
B0
=ZV
1
2(µ)−1B2
B1
B0
dV. (7.19)
RVE·∂D
∂t dVas the change of electrical energy stored in the volume Vas well
as the energy loss in hysteretic materials analog to above.
92 Detailed Treatment
Conclusion: Poynting’s theorem then reads: Consider a volume V, the
energy exchange from the electromagnetic system to other physical systems
and the flux of electromagnetic energy through the volume boundary equal
the increase of electromagnetic energy inside the volume plus all losses in
electric/magnetic hysteresis and polarization.
An expression for the magnetic energy inside the volume Vcan be defined
only for materials without hysteresis:
WV
mag
B1
B0
=ZVZB1
B0
H·dBdV. (7.20)
Application to slowly varying fields In case of slowly varying electromag-
netic fields, the term ∂D
∂t can be neglected. Introducing the magnetic vector
potential Awith ∇×A=B, the electric field in Eq. (7.11) can be expressed
by E=−∂A
∂t . Equation (7.14) changes to:
−ZV
∂A
∂t ·JdV−Z∂V ∂A
∂t ×H·da=−ZV
H·∂B
∂t dV(7.21)
At this point, we refrain from giving a physical interpretation of the surface
integral. For quasi-static problems variations of the magnetic field are instan-
taneously distributed over the entire domain and an energy flux density is not
defined.
Extending the volume to infinity V→ ∞, the integral over the far field
boundary can be dropped: The magnetic vector potential decays with 1/r
and the magnetic field with 1/r2while the surface increases with r2only.
For the integral containing the current density Jthe integration domain can
be reduced to the volumes where the current density is different from zero
V∞→VJ:
ZV∞
H·∂B
∂t dV=ZV∞
∂A
∂t ·JdV=ZVJ
∂A
∂t ·JdV(7.22)
We integrate over time and apply the rule for line integrals assuming that the
same conditions hold for Jand Aas explained above for Hand B:
ZV∞ZB0→B1
H·dBdV=ZVJZA0→A1
J·dAdV(7.23)
We obtain an alternative form for the hysteresis losses over a full cycle γA:
A0→A1→A0,
W∞
hystγA
=ZVJIγA
J·dAdV(7.24)
For materials without hysteretic behavior, the left-hand side of Eq. (7.23)
represents the magnetic energy in the entire domain. In the case of lossy
7.4 Inductance 93
materials, the volume VJon the right-hand side requires further subdivision.
We distinguish sub-domains with an imposed current density thus providing
energy to the electromagnetic system, and domains where energy is dissipated
in losses - to be identified by E·J>0. For loss and hysteresis free materials
we obtain an alternative form for the stored magnetic energy:
W∞
mag
A1
A0
=ZVJZA1
A0
J·dAdV(7.25)
7.4 Inductance
The magnetic flux in a superconducting magnet is in non-linear, diffusive
and hysteretic relationship with the applied current due to iron saturation,
induced eddy currents and superconductor magnetization. For the simulation
of quench, the electrical circuit is represented by lumped elements and the
current change in the loop is calculated from the voltage over the inductance.
Therefore, the inductance is derived for general materials and then applied
to quench computation.
7.4.1 Self Inductance
Consider a simple current loop connected to a current source as shown in
Fig. 7.8. The loop spans a surface Awith differential surface element da
and boundary ∂A. The differential line segment dsfollows the boundary in
a right-hand orientation.
The total magnetic induction Bconsists of the field created by the cur-
rent Iand the effect of any magnetic object with field-dependent, diffusive
or hysteretic properties. The dependence of the magnetic induction Bon
the current I, the current change ( dI
dt) and the current history (Rt
t0Idτ) is
modeled by the relation Γ,
B=Br,ΓI, dI
dt,Zt
t0
Idτ.(7.26)
For the general case, the following, commonly used equations give rise to
the definition of three different types of self inductance [Kurz 04].
Ψmag =LΨI, Uind =Ld
dI
dt, Wmag =1
2LWI2.(7.27)
In the stationary case, with constant material properties and no hysteresis,
all three inductances are identical: LΨ=Ld=LW.
Apparent inductance: For a given instance and current, the magnetic flux
ΨA
mag through the surface Ais given by
ΨA
mag =ZA
Bda=Z∂A
A·ds,(7.28)
94 Detailed Treatment
!"
!#"
!$"
!"
!"
#"$#$
Figure 7.8: Self inductance geometry. The current loop with surface Aand boundary ∂A.
The differential line segment and surface element are denoted daand ds, respectively.
where Adenotes the magnetic vector potential with ∇×A=B. The apparent
inductance [Deme 99] LΨis defined by
LΨ=ΨA
mag
I=R∂A A·ds
I.(7.29)
The magnetic flux shows the same dependence on the current as the magnetic
induction. With the limitation to a constant current, i.e. a steady state
situation, the apparent inductance may be non-linear and hysteretic.
Differential inductance: Assuming a current ramp-rate dI
dt, the voltage
induced over the terminals of the current source, Uind, is given by
Uind =dΨA
mag
dt=d
dtI∂A
A·ds=I∂A
∂A
∂t ·ds,(7.30)
and defines the differential inductance Ld[Naun 02, pp. 33] to:
Ld=Uind
dI
dt
=
dΨA
mag
dt
dI
dt
=
d
dtH∂A A·ds
dI
dt
.(7.31)
The differential inductance inherits the behavior of the magnetic induction
and may generally be non-linear, time-dependent and hysteretic.
Under the condition that dΨA
mag
dt=dΨA
mag
dI
dI
dt,i.e. that the system is time-
invariant, non-hysteretic and does not contain any secondary loops, the defi-
nition can be further simplified to:
Ld(I) = dΨA
mag
dI=d(ILΨ)
dI=dLΨ
dII+LΨ.(7.32)
Notice that a second current loop, which is magnetically coupled to the pri-
mary loop, would influence the induced voltage, but has no effect on the
apparent inductance.
7.4 Inductance 95
Energy inductance: For a system in virgin state, i.e. current and all fields
equal zero, the current is changed to I(t). The amount of energy Wsource
supplied by the power supply can be calculated from
Wsource(t) = Zt
τ=0
Uind(τ)I(τ)dτ=Zt
t=0
dΨA
mag(I(τ))
dτI(τ)dτ. (7.33)
Note that only for a loss and hysteresis free system, the energy supplied by
the source and the magnetic energy stored in the system are identical. The
energy inductance LWis then defined as time-dependent function:
LW(t) = Wsource(t)
1/2I(t)2=2
I(t)2Zt
t=0
dΨA
mag(I(τ))
dτI(τ)dτ. (7.34)
Under the following two conditions the expression for the energy inductance
can be further simplified. If again dΨA
mag(I)
dt=dΨA
mag(I)
dI
dI
dt, then the time
derivative can be split off. If the system is conservative, i.e. the energy
change does not depend on the current history, but only on start and end
current, then Eq. 7.34 reads:
LW(t) = 2
I(t)2Zt
t=0
dΨA
mag(I(τ))
dI
dI
dτI(τ)dτ(7.35)
=2
I(t)2ZI(t)
I=0
dΨA
mag(I)
dIIdI(7.36)
=2
I(t)2ZI(t)
I=0
Ld(I)IdI(7.37)
From the energy inductance the concept of inner and outer inductance
can be derived. Using the formula for the magnetic energy inside a volume
V, Eq. (7.20), the energy can be calculated separately inside and outside
the cable. Both energy fractions define complementary parts of the total
induction. For materials with constant properties where all three concepts of
induction are identical, the two parts can be conveniently approximated: The
outer induction is calculated from the geometrical induction of a loop along
the conductor. The inner inductance is given by the inductance per unit
length of a straight infinite long wire with the same cross-section multiplied
by the length along the loop.
7.4.2 Mutual inductance
Consider a second current loop of surface Band boundary ∂B. Surface and
boundary can be parameterized by daand ds, respectively. The two surfaces
Aand Bdo not intersect. Both loops are magnetically coupled, i.e. the flux of
loop Apenetrates loop Band vice-versa. The geometry may contain magnetic
objects with field-dependent, diffusive or hysteretic properties.
The concept of the geometrical and differential inductances are adapted to
the new situation:
96 Detailed Treatment
Apparent mutual inductance : For a given current I=IAin loop Aand
IBin loop B, the apparent mutual inductance of Bon A, is given by relating
the magnetic flux created by Bin loop Ato the current IB:
MAB
Ψ=ΨA
mag
IB
=H∂A A·ds
IB
.(7.38)
In general, the magnetic flux depends on both currents, i.e. on ΓAand ΓB
and so does the apparent mutual inductance. If the material properties are
constant, the flux can be separated into the contributions of either coil. The
apparent mutual inductance is then constant and symmetric, MAB
Ψ=MBA
Ψ.
Note that symmetry is not easy to define in case of varying material properties
when the mutual induction is given by a function.
Differential mutual inductance: The current in loop A shall be constant
IA. The voltage induced over loop Adue to loop Bis now related to the
current change in loop B:
MAB
d=UA
ind
dIB
dt
=
dΨA
mag
dt
dIB
dt
=
d
dtH∂A A·ds
dIB
dt
(7.39)
7.4.3 Application of the Differential Inductance to Quench
Computation
Since magnet components may be build from field-dependent, diffusive and
hysteretic materials the general definition of the differential inductance given
in Eq. (7.31) is applied.
Bulk or Rutherford-type conductors For a current loop made from a bulk
conductor, the confined flux varies over the conductor cross section. Isolating
the average linked flux, yields a differential flux over the cross section giving
rise to eddy currents. The change of the average flux induces a voltage over
the loop terminals (see Fig. 7.9 (left, center)).
For a Rutherford-type cable, all strands are connected in parallel and follow
a zig-zag trajectory along the cable (see Sec. 7.8). The linked flux on all
parallel paths is identical and the total induced voltage can be calculated from
the average inductance. The variation of the differential electrical field along
the strands also gives rise to induced eddy currents - inter-strand-coupling
currents (see Fig. 7.9 (right).
Field-dependent materials The permeability of the iron yoke is field depen-
dent and decreases strongly as soon as the magnetic induction exceeds the
saturation level. Therefore, the contribution of the iron yoke to the flux in
the magnet decreases with increasing current. If no other dependence needs
to be considered it can be shown that the differential inductance is smaller
7.4 Inductance 97
!"#$% !#&%
'!%
'!%
!#&%
Figure 7.9: Inductance of a bulk or Rutherford-type conductor. (left) and (center, bot-
tom) The confined flux varies over the conductor cross section. (center, top) The induced
electric field can be split in a an average and differential part. (right) The average induced
electric field is identical for each strand. The differential electrical field gives rise to induced
inter-strand eddy currents.
or equal to the geometrical inductance. Compare Eq. (7.32) where the flux
change with current is negative.
Notice that the magnetic field computation is carried out entirely in 2D, i.e.
over the magnet cross-section. The variation of the magnetic field in the coil
ends and the magnetic length are taken into account by scaling parameters.
For the induced voltages the magnetic length `mag is used (see discussion of
magnetic length and inductance length in Sec. B.1.6). The iron saturation
is different in the magnet ends. Hence, the magnetic length is a function of
excitation current.
Figure 7.10 shows the differential inductance for the LHC main bending
magnet considering saturation effects in the iron yoke.
The influence of induced eddy currents and superconductor magnetization
on the differential inductance are not taken into account in the present quench
model. Nevertheless, the general dependence is discussed allowing to analyze
peculiarities of inductance measurements.
Diffusive materials Induced eddy-currents or inter-strand coupling currents
show a diffusive characteristic. The influence on the differential inductance
is estimated by means of a simple network representation [Smed 93]. The
induced eddy-currents are considered as coupled secondary loops. The mag-
net is represented by the primary side of a transformer with inductance L1.
The sum of all inter-strand and inter-filament coupling currents is taken into
account by a secondary circuit with inductance L2and resistance R2. The
two sides are coupled by the mutual inductance M=kpL1L2, where kis
denoted the coupling factor, k[0,1). The inductances L1,L2and Mare
constant. At t= 0 a constant voltage U1is switched over the magnet.
98 Detailed Treatment
6.5
6.6
6.7
6.8
6.9
7
7.1
0 300 600 900 1200 1500 1800 2100 2400
0
2
4
6
8
10
12
Differential Inductance per unit length in mH/m
Current in kA
Time in s
Magnetic Iron (MI)
MI + Persistent Currents
Current
Figure 7.10: ROXIE simulation of the differential inductance of the LHC main bending
magnet considering field dependent and hysteretic materials. The differential inductance
only considering the field depending iron decreases by approximately 5% from injection to
nominal current. Considering persistent currents (hysteresis), the differential inductance
shows a significant dip after the current ramp-rate changed sign (at t= 0 and t= 1200 s,
assuming a pre-cycle for t < 0).
For a symmetrically oriented transformer (see e.g. [Flei 99, p. 494]) the
current in the secondary winding is given by
i2(t) = −U1
R2
M
L11−exp −t
τ, τ =L2
R2
(1 −k2).(7.40)
Calculating the differential inductance from the induced current change on
the primary side yields,
Ld(t) = U1
di1
dt
=L1
1
1 + k2
1−k2exp −t
τ(7.41)
The time constant and the initial value of the inductance are both simple
functions of the coupling factor k. Figure 7.11 shows the change of differential
inductance for different values of the coupling factor. It can be seen how the
differential inductance is reduced in the first instance after the voltage rise.
Hysteretic materials: In superconducting magnets the magnetic iron and
the superconducting filaments show a magnetization behavior which features
a hysteresis. The influence on the concept of the differential inductance is
highlighted by means of a simplified model.
Figure 7.12 (top, left) shows the magnetic flux stemming from the hysteretic
material in the magnet versus applied magnet current. The transition from
the lower to the upper branch as well as the shape of the minor loop are typical
for the superconductor magnetization. In case of ferromagnetic hysteresis, the
7.4 Inductance 99
k!0.99
0.9
0.75
0.5 0.25
0.0
Τ0.75
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
0.0
0.2
0.4
0.6
0.8
1.0
t!"L2!R2#
L
d!L1
Figure 7.11: Differential inductance of a magnet with time-transient effects, e.g. inter-
strand coupling currents. The plot shows the change of differential inductance over time
for different values of k: On the left we can read off the initial value. The time constant τ
corresponds to the ending of the full lines.
!"
#"
#"
$!"#$
!"
%&'($ %&'($
%"
Figure 7.12: Differential inductance by means of hysteretic materials. The hysteresis
loop represents a simplified superconductor magnetization loop including a minor loop.
Starting from the current, the magnetic flux in the magnet can be constructed by a simple
projection. The induced voltage is given by the flux change. Finally, the differential
inductance is calculated from the ratio of induced voltage and current ramp rate.
100 Detailed Treatment
transition occurs while increasing the current as compared to superconducting
materials where the transition occurs on the on-set of the current decrease.
Minor loops are followed horizontally instead of vertically.
Following the approach used in [Naun 02, pp. 36], the flux in the magnet
can be constructed graphically from the applied current. The flux belonging
to the geometry of the magnet winding is not considered here. The induced
voltage is calculated from the flux change. Both are shown in Fig. 7.12
(top, right). For this simplified model, the differential inductance is constant
and negative during the branch transition and zero everywhere else. In case
of ferromagnetic materials, the differential inductance is of constant positive
values during the transition (which occurs at different times). In first order,
this effect can be superimposed on the geometrical inductance and thus yields
lower/higher values during the transition.
Figure 7.10 shows the differential inductance for the LHC main bending
magnet considering the effect of superconductor magnetization currents.
7.5 Non-linear Voltage-Current-Characteristic of
Superconductors
The voltage-current-characteristic (VAC) of a superconductor is a stronglyNon-linear
VAC non-linear function of applied magnetic field, temperature and current (or
current density), see Sec. A.2.2. Following Wilson [Walt 74, p. 30], the
voltage over a superconductor, USC, can be expressed by a power of the
applied current ISC,
USC =UcISC
IcNSC
,(7.42)
where NSC serves as fit parameter and Ucas reference voltage (Sec. A.2.3).
The critical current Icdepends on the applied magnetic field and temperature
(see critical current density in Sec. A.2.4).
In an electrical network, superconducting elements constitute current-con-Current
controlled
voltage
source
trolled voltage sources, USC(ISC). Therefore, branch currents and voltages
cannot be determined by a simple inversion of the network matrix, but require
a (numerically expansive) iterative approach to resolve the interdependence
(non-linearity).
Consider the combination of a superconducting and a resistive element in
one branch of a simplified electrical network as sown in Fig. 7.13. Depending
on the type of connection, in parallel or in series, and the boundary condition,
current or voltage driven, four different elementary problems can be derived:
a) superconductor in parallel to a resistor driven by a current source: The
current splits over the two branches, the current through the supercon-
ductor defines the voltage over the branch and thus the current through
the resistor. This situation is called current-sharing and represents the
situation of a quenching copper-stabilized wire, see Sec. 7.5.1.
7.5 Non-linear Voltage-Current-Characteristic of Superconductors 101
!"
#!"#!!"$%
$&"%
!!"%
!&"%
!!"%
#!"#!!"$% $&"%
#"
!%
#!"#!$% $&"%
#!"#!!"$%
$&"%
!!"%
!&"%
#"
'$% ($%
)$% *$%
+,,-.*/01233'431&5.6/.-'78-9':/571!"#$#%
Figure 7.13: Simplified electrical networks containing superconducting and normal con-
ducting elements. The following situations can be identified: a) Current-sharing or the
voltage over a quenching copper stabilized wire. b) Voltage over a partly quenched magnet
winding. d) Voltage over superconducting current loop with resistive joints. The sub-
scripts “NC” and “SC” stand for normal conductor (ohmic resistive) and superconductor,
respectively.
b) superconductor in series with a resistor driven by a current source: The
current through both elements defines the total branch voltage. This
situation can be found in case of partly quenched magnet windings.
c) superconductor in parallel to a resistor driven by a voltage source: The
voltage over the branch defines the two branch currents.
d) superconductor in series with a resistor driven by a voltage source: The
voltage over the branch splits over the two elements. The voltage over
the superconductor defines the branch current and therefore the voltage
over the resistor. This situation can be found for induced voltages over
superconducting loops with resistive joints, see Sec. 7.5.2.
While the situations b) and c) are easily solved and implemented, the cir-
cuits a) and d) require either further simplifications, iteration techniques or
a graphical solution (Sec. 7.5.3).
7.5.1 Current-Sharing
Superconducting filaments are usually embedded in a normal conducting ma-
trix for stabilization. The matrix gives mechanical support, heat transfer to
the coolant and in longitudinal direction along the strands, and provides a
parallel path for the current in case of a quench.
102 Detailed Treatment
Such a superconducting strand can be described by the circuit shown in
Fig. 7.13 a). While the current is much smaller than the critical current
Ic, the voltage over the superconductor is negligible due to the usually large
n-index (in the order of 20-50). For an increasing current Ior a decreasing
critical current Ic,e.g. due to a temperature or magnetic field increase, the
voltage USC, given in Eq. (7.42), starts to grow. This voltage draws part
of the current into the path through the resistor - the external current Iis
shared between the resistive and superconducting path.
Neglecting the inductive coupling of the two current paths, the current inImplicit
equation of
current-
sharing
the superconductor ISC can be calculated from Eq. (7.42) with the implicit
equation,
RCu(B, T, RRR)(I−ISC) = UcISC
Ic(B, T)NSC
,(7.43)
where RCu denotes the temperature and field dependent resistance of the
copper matrix.
Figure 7.14 (right) shows the current commutation for different values of ap-
plied magnetic induction and transport current density. The current sharing
temperature range, i.e. the temperature interval where the current is shared
between the superconductor and normal conductor, decreases with decreasing
applied field and current density. The maximum range is approximately 3 K.
The current-sharing temperature range decreases with decreasing values for
NSC.
The voltage over the branch causes ohmic losses in the strand. In case
of insufficient cooling, the ohmic heat results in a temperature increase and
further reduction of the critical current. The fraction of the current through
the resistive part grows.
Current-Sharing Temperature The implicit equation for the current in the
superconductor, Eq. (7.43), causes a high computational effort, especially
for large numbers of NSC. Therefore, Stekly proposed an approximation
relying on the following assumptions [Stek 65]: 1.) While the current in the
superconductor is smaller than the critical current, it is entirely carried by
the superconducting fraction of the strand and no losses are dissipated. 2.)
When the current exceeds the critical current, the excess current “spills over”
into the surrounding copper matrix. The current in the normal conductor
causes ohmic losses. 3.) When the critical current density reaches zero, the
current is carried only by the normal conducting copper matrix.
This simplifies the solution of electrical networks containing superconduct-
ing elements significantly. The current controlled voltage source of Eq. (7.42)
has been replaced by a current source controlled by the external current I,
and both branches have been decoupled.
ISC(I) = (I I < Ic
Icelse (7.44)
Figure 7.14 (left) shows the commutation calculated according to Eq. (7.43)Stekly
current-
sharing
7.5 Non-linear Voltage-Current-Characteristic of Superconductors 103
B!8.4 T
B!2.5 T
B!0.0 T
J!Jnom
J!0.75 Jnom
J!0.50 Jnom
J!0.25 Jnom
2 4 6 8 10
Tin K
0.5
1.0
1.5
2.0
2.5
3.0
J
in kA mm"2
Tcs
!Tcs
1 2 3 4 5 6
Tin K
0.5
1.0
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Figure 7.14: Current-sharing: Consider a Nb-Ti superconductor (NSC = 40) embedded
in a copper matrix of identical cross-sectional area (λ= 1) and RRR of 200. The current-
sharing is shown for varying applied magnetic inductions and transport current densities.
The nominal current density Jnom = 2.34 kAmm−1is 10% below the critical current den-
sity at 8.4 T and 1.9 K.
(left) Different models for a quench and the consequent current commutation from the
superconductor to the copper matrix. The thick orange line represents the critical current
density Jcas a function of temperature for an applied magnetic induction of 8.4 T. The
temperature where the applied current density in the superconductor equals the critical
current density is highlighted by a circle. Beyond this temperature all three models show
a different behavior:
I. Step-function: The current commutates instantaneously from the superconductor into
the normal conductor (dashed line).
II. Current-sharing by Stekly: When the current density in the superconductor exceeds
the critical current density, the excess current “spills over” into the copper matrix. The
current density in the superconductor equals the critical current density. The temperature,
for which the transition sets in, is denoted current-sharing temperature Tcs. With increas-
ing temperature the critical current density decreases and more and more current is carried
by the normal conductor.
III. Current-sharing as defined by Eq. (7.43) and necessary iteration: The current com-
mutation from the superconductor (full line) to the normal conductor sets in at higher
temperatures compared to I and II. The temperature interval between the onset of current
sharing and the complete commutation into the normal conductor is denoted current-
sharing temperature range ∆Tcs. The current density in the normal conductor is given by
the dash-dotted graph.
(right) Current sharing (after model III) for different applied current densities and mag-
netic inductions. The current sharing temperature range decreases with field and current
density. The current density in the superconductor is given by the solid lines and in the
normal conductor by the dash-dotted lines.
104 Detailed Treatment
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Figure 7.15: (left) Model problem circuit. (right) Solution of the initial value problem
(if R > 0) depicting all relevant parameters.
in comparison to the critical current density. The approach after Stekly fea-
tures a longer current-sharing temperature range and smaller currents in the
superconductor. Therefore, dissipated ohmic losses are over estimated. For
small temperature rises and strong cooling, the stability of the superconduc-
tor can be under estimated. For the calculation of quench, where dissipated
losses and heat transfer from quench heaters and neighboring conductors ex-
ceed the cooling capability by many orders of magnitude, the differences can
be negelected.
The temperature where current-sharing sets in is denoted current-sharingCurrent-
sharing
tempera-
ture
temperature Tcs.Stekly furthermore linearizes the the critical current over
temperature for a fixed magnetic induction B0,
Ilin
c(T) = Ic(B0, Tb)Tc−T
Tc−Tb
,(7.45)
where Tbis the bath temperature and Tcdenotes the critical temperature for
the applied field B0. For I=Ilin
c(Tcs), this can be re-arranged and gives the
current sharing temperature as function of the applied current I[Mess 96, p.
120]:
Tcs =Tb+ (Tc−Tb)1−I
Ic(Tb, B0).(7.46)
7.5.2 Voltages Induced over a Superconductor
The model problem Given is a circuit consisting of a resistance R, an induc-
tance Land a constant voltage source Ue. The circuit is shown in Fig. 7.15
(left). At a starting time tsthe circuit carries an initial current Is. The value
of the resistivity may be zero or greater and the value of the voltage may be
negative, zero or positive. The value of the inductance is always greater zero.
After solving the initial value problem, it is investigated whether an arbi-
trary current level Ieis met by the current idepending on the model param-
eters and initial values. The time ∆tto reach Ieis calculated.
7.5 Non-linear Voltage-Current-Characteristic of Superconductors 105
The circuit can be described by the following linear, inhomogeneous differ-
ential equation of first order with constant coefficients:
Ldi
dt+Ri =Ue(7.47)
For a resistivity Rgreater than zero the solution of the initial value problem
reads:
i(t) = I∞+ (Is−I∞) exp −t−ts
τ,(7.48)
with τ=L/R and the current limit I∞=Ue/R. For t→ ∞ the current
convergences towards I∞. Figure 7.15 (right) shows the solution with all
relevant parameters. For a resistivity of R= 0 the equation greatly simplifies
and the solution of the initial value problem reads:
i(t) = Is+ (t−ts)˙
I, (7.49)
with the constant slope ˙
I=Ue/L. In this case, the current change is not
limited, thus I∞=±∞ (depending on the sign of ˙
I).
Equations (7.48) and (7.49) are rearranged in order to express ∆t=t−ts
as a function of the model parameters, starting values and the given current
level Ie:
∆t=(Ie−Is
Ue/L R= 0
τlog I∞−Is
I∞−IeR > 0.(7.50)
The current Iecan be reached, if the expression for ∆treturns a positive,
real value. For a positive voltage Uethe current level Ieneeds to fulfill
Is< Ie< I∞, while for a negative voltage it has to meet I∞< Ie< Is. In
the special case of R= 0 and Ue= 0 the current is constant.
Notice that sometimes it is necessary to split the period of observation/cal-
culation. Therefore, Eq. (7.48) or (7.49) can be joint repetitively using the
final current of one interval as the starting current of the preceding interval.
Currents Induced in a Superconducting Current Loop with Resistive Joint
Considering a superconducting current loop with a resistive joint in a time-
transient magnetic field. The joint has a resistivity Rjoint and the loop an in-
ductance L. The initial current is zero. The changing magnetic flux dΨm/dt
induces a voltage Uind over the loop. The geometry is shown in Fig. 7.16
(left).
Taking into account the non-linear resistivity of the superconducting ma-
terial, the maximum induced current is calculated.
The voltage over a superconductor is a strongly non-linear function of the
current through the superconductor (see Sec. A.2.2). In superconducting
state the voltage can be considered as infinitely small, while in normal con-
ducting state it is orders of magnitude greater than over comparable amounts
of copper. As shown above (Sec. 7.5), this causes the current to rapidly
commutate from the superconductor into a surrounding copper matrix. The
106 Detailed Treatment
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Figure 7.16: Currents induced in a superconducting current loop with resistive joint.
(left) Geometry. (right) Model of the superconductor and the associated network elements.
voltage-current characteristic can be simplified as shown in Fig. 7.16 (right)
[Verw 95, pp. 68]. While the current is below the onset current of the tran-
sition, Ic1, the resistivity is zero. In normal conducting state, i.e. above Ic2,
the superconductor is replaced by the resistivity of the normal conducting
matrix material RNC (or if not available, by its normal resistivity). During
the transition the superconductor is modeled by the differential transition re-
sistivity RTR =Uc(Ic2 −Ic1)and a virtual voltage source RTRIc1. The three
resistivities compare as Rjoint < RNC RTR.
The current rise can be expressed by the solution of the model problem (see
Sec. 7.5.2). Without limitations, the induced voltage shall be greater than
the critical voltage Uc(Uind > Uc). While the current rises all three regimes
of the simplified current-voltage-characteristic are passed, thus constituting
three different sets of parameters for the model problem. For each phase the
limiting current I∞and the time constant τare given by:
Phase Time Current I∞τ
10< t < t10< i < Ic1
Uind
Rjoint
L
Rjoint
2t1< t < t2Ic1 < i < Ic2
Uind+RTRIc1
Rjoint+RTR
L
Rjoint+RTR
≈Ic1 +Uind
RTR
3t2<t<∞Ic2 < i < I∞
Uind
Rjoint+RNC
L
Rjoint+RNC
Figure 7.17 (left) shows schematically the current rise over the three phases.
The current through the resistive joint and later through the transitioning
superconductor causes ohmic losses. The maximum dissipated power is then
given by Pmax =U2
ind/(Rjoint +RNC). If this heat is not transfered away from
the loop, it results in a temperature rise and a decrease in critical current. In
the simplified model, the two limits, Ic1 and Ic2 shift towards lower currents.
Therefore phase 1 and 2 are passed faster without changing the current rise
itself. This is shown schematically in Fig. 7.17 (right).
The model works for both, negative and positive induced voltages, in the
same way. For a negative voltage the orientation of the virtual voltage source
has to be inverted. For an induced voltage |Uind|/Rjoint smaller than Ic1 or
Ic2 not all regimes are passed.
7.5 Non-linear Voltage-Current-Characteristic of Superconductors 107
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Figure 7.17: (left) Superconductor with resistive joint. The induced current passes the
three regimes of the superconductor model. (right) Without a resistive joint the induced
current increases linearly over time until the critical current Ic1 is reached. The ohmic losses
yield a temperature rise and therefore the critical current to decrease. After the induced
voltage disappeared, the induced current reduces until it reaches the critical current Ic1
and flows without resistance.
Quench model: The analytical models for the cable eddy currents, i.e. inter-
filament and inter-strand coupling currents, are based on similar topologies.
Hence, without taking into account the U-I-characteristic of the supercon-
ductor, induced cable eddy currents and losses are over estimated for very
high ramp-rates.
Currents Induced in a Superconducting Loop without Resistive Joint Con-
sidering a seamless superconducting current loop, i.e. without resistive joint
Rjoint = 0, the approach has to be slightly modified. During the first phase
the current rises linearly with a slope of Uind/L and is not limited. As soon as
the current reaches Ic1 the differential resistivity RTR switches in and phase
2 and 3 follow as above. The current is therefore limited to I∞=Uind/RNC.
If now, in a preloaded state i>Ic2, the magnetic flux stops changing, then
the induced voltage drops to zero and the current starts decreasing. Phase 3
and 2 are followed in reverse direction. The current stops decreasing as soon
as it reaches Ic1. In this simplified model the total resistivity of the circuit
equals zero and the current is kept constant by the inductance.
Looking more accurately, the constant current actually decreases with a
time constant in the order of years. This is due to the tiny residual volt-
age over the superconductor as can be calculated from the realistic voltage-
current-characteristic (see Sec. A.2.2).
During this cycle ohmic losses are only dissipated during phases 2 and 3. If
the temperature rise at the end of the cycle is small enough to maintain a finite
critical current, the loop keeps a permanent current. If the superconductor
is cooled, the temperature decreases again due to the lack of any heating
and the critical current rises. Nevertheless, the permanent current does not
change.
Considering the the finite inductance of the current loop and the special
108 Detailed Treatment
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Figure 7.18: Graphical solution of the parallel and series circuit with a superconducting
and a normal conducting element, Fig. 7.13 a) and d). (left) parallel case. (right) series
case.
properties of the superconductor, we can give an explanation of the widely
known experiments where superconducting samples float in permanent mag-
netic fields. The inductance limits the slope of the current rise while the
superconductor limits the final current2.
7.5.3 Graphical Solution
If a superconductor is either placed in parallel to a normal conductor driven
by an external current source I(see Fig. 7.13 a)) or the superconductor is
placed in series to a normal conductor driven by an external voltage source
U(see Fig. 7.13 d)), the following non-linear equations have to be solved:
USC(ISC) = RNC(I−ISC), USC(ISC) = U−RNCISC.(7.51)
The graphical solution is shown in Fig. 7.18. For the parallel case (left), the
diagonal line represents the voltage over the resistance RNC and the inter-
section with USC defines the partition of the current Iinto ISC and INC. In
case of the series connection (right), the voltage over the resistance RNC is
represented by the space in between the horizontal and the dashed line. The
intersection therefore gives the partition of the voltage Uand determines the
current through the branch, ISC.
The voltage-current-characterisic depends strongly on the applied mag-
netic induction and temperature. Therefore, the graphical solution, obtained
above, has to be re-determined if any of these quantities changes.
2The explanation given is for type II superconductors. Most of the experiments are either
performed with type I materials or so called high-temperature superconductors. The
physics behind these kind of materials is different, but nevertheless the limitation of the
current holds.
7.6 Cable Magnetization Losses 109
7.6 Cable Magnetization Losses
Time varying magnetic fields induce eddy currents over superconducting cur-
rent loops and excite screening currents in superconducting filaments. In
superconducting cables, the former two are denoted inter-filament and inter-
strand coupling currents. The latter are denoted as persistent currents or
superconductor magnetization. All three effects are summarized as cable
magnetizations.
In the context of superconducting accelerator magnets, the three phenom-
ena are associated with induced field errors [Auch 08] and losses in the cables
[Kirb 07]. While we can ignore field quality for the quench simulation, the
model requires to include all relevant heat sources.
Inter-filament coupling currents Strands for superconducting accelerator
magnets consist of a large number of superconducting filaments embedded in
a copper matrix. The filaments are twisted around the strand axis. Eddy
currents induced over loops formed by superconducting filaments close over
the resistive copper matrix and cause heating. These currents are denoted
inter-filament coupling currents (IFCC).
The magnetization resulting from the eddy currents in these meshes can
be calculated analytically [Wils 83, pp. 176]. The calculation is carried out
in four steps: Calculation of the induced voltage over a segment along two
twisted filaments, determination of the current density in the surrounding
matrix material, calculation of the surface current density consituting the
return path and determination of the equivalent magnetization density.
Induced voltage: Consider two filaments along helical paths around the z-
axis, shown in Fig. 7.19. Without loss of generality they are exposed
to an applied field parallel to the y-axis; B=Byey.
The voltage induced over a segment of the two filaments is calculated
by means of Faraday’s law,
Z∂F
E·dr=−ZF
dB
dt·da,(7.52)
where Fdenotes the surface spanned by the double helix. The surface
is longitudinally delimited by ±z.
For the surface integral we can exploit that the integral is zero on faces
tangential to Band evalaute the integral in the xz-plane. The pro-
jection of F, denoted as ˜
F, is transversally bounded by ±r0cos 2π
psz
with r0the radius of the helical paths and psthe twist pitch length.
The change in magnetic flux is given to
Z˜
F
dB
dt·da=Zz
−zZr0cos(2π
psz0)
−r0cos(2π
psz0)
dBy
dtdxdz0
= 4r0sin 2π
ps
zps
2π
dBy
dt.(7.53)
110 Detailed Treatment
While the filaments are in the superconducting state, the voltage must
drop entirely across the resistance matrix. Assuming symmetry, the
contour integral simplifies to R∂F E·dr= 4U(z)and it follows:
U(z) = −dBy
dt
ps
2πr0sin 2π
ps
z.(7.54)
Current density in the matrix material: In the next step, we consider a rod
covered by filaments twisted along the axis and approximate the ensem-
ble by a thin superconducting layer at the radius r0. Disregarding any
mutual dependence between filaments, the induced voltage constitutes
a boundary condition for the potential inside the matrix material. The
potential φcan be expressed as a function of the ϕ-coordinate. With
ϕ= 2πz/ps, Eq. (7.54) can be rewritten as
φ(ϕ) = −dBy
dt
ps
2πr0sin ϕ=−dBy
dt
ps
2πy , (7.55)
and the gradient yields a uniform electric field. The current density in
the matrix can be calculated from Ohm’s law:
Jy=1
ρeff
Ey=1
ρeff
dBy
dt
ps
2π.(7.56)
Surface current density: The y-directed current density closes on the cylin-
drical shell by means of a helical surface current density JF. Owing
to the longitudinal invariance, the ϕ-component of the surface current
density can be determined from the continuity equation,
I∂V
J·da= ∆z Zϕ
π/2
Jyey·(−eρ)r0dϕ0+JF,ϕ(ϕ)!= 0 (7.57)
where dais a differential surface element on the closed surface ∂V (see
Fig. 7.19) and ∆zan arbitrary length in longitudinal direction. This
results in
JF,ϕ(ϕ) = Zϕ
π/2
Jysin ϕ0r0dϕ0=−1
ρeff
dBy
dt
ps
2πr0cos ϕ . (7.58)
As the filaments in the cylinder follow helical paths, the azimuthal and
longitudinal current densities are coupled. With one full transposition
over a twist pitch length pswe have
JF,z =−ps
2πr0
JF,ϕ.(7.59)
7.6 Cable Magnetization Losses 111
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Figure 7.19: Inter-filament coupling currents. (left) Geometry in the xy-plane. The
integral over the current density is evaluated over the closed surface ∂V . (right) two
twisted filaments along the z-axis and the projection on the xz-plane.
Magnetization density: After [Henk 01, p. 156], we can express a homoge-
neous magnetization Min a volume Vby means of a magnetization
surface current density JF,mag flowing on ∂V ,
M×n=JF,mag,(7.60)
where ndenotes the normal vector on the boundary. Identifying JF,mag =
JF,zezand disregarding the azimuthal component, the inter-filament
magnetization is given by
MIFCC =Myey=−1
ρeff
dBy
dtps
2π2
ey.(7.61)
Notice the negative sign. The magnetization opposes the change of
magnetic induction.
For a homogenous distribution of filaments within the strand, the result
can be scaled with the strand filling factor ηs. The field dependent resistivity
of the matrix material can be taken into account by
ρeff =ρIFCC +dρIFCC
dB|B|(7.62)
ρIFCC is the constant part of the effective resistivity and dρIFCC
dBthe slope of
the magneto-resistive effect; [dρIFCC
dB] = ΩmT−1.
The eddy-current losses are only dissipated in the matrix material. The loss
density can be calculated from the field components parallel to the applied
magnetic induction:
pIFCC =E·J=ρeffJ2
y=−MIFCC ·dB
dt=1
ρeff ps
2π2dBy
dt2
.(7.63)
The derivation is based on the assumption of filaments in superconducting
state. For a strand in the quenched state, the model is not valid. Due
112 Detailed Treatment
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Figure 7.20: (left) Model for inter-strand coupling currents. Courtesy R. de Maria
[Mari 04]. (right) Local frame and cable parameters for the derivation of the inter-strand
coupling current magnetization.
to the comparably low critical temperature of common superconductors we
do not need to consider the temperature dependence of the matrix material.
Furthermore, the induced eddy currents are not treated as a diffusion process.
Currents and losses appear instantaneously.
Inter-strand coupling currents Cables for superconducting accelerator mag-
nets consist of a number of strands connected in parallel and sharing the
transport current. The strands are twisted along the cable axis in order to
reduce the size of superconducting loops and to obtain a homogeneous current
distribution. Nevertheless, varying magnetic fields induce eddy currents over
the twisted strands which close over the contact resistance between neighbor-
ing strands. The current flow through the contact resistance causes heating.
These currents are denoted inter-strand coupling currents (IFCC).
A number of models for the calculation of inter-strand coupling currents can
be found in the literature. Electrical network models describing the strands
and their inter-connection by means of lumped elements [Devr 95], allow to
calculate induced coupling currents for various ramp-rates and field patterns
[Verw 95], see Fig. 7.20 (left). However, the numerical solution on the scale
of a full length magnet is computationally too demanding.
Analytical models describing eddy currents by means of the magnetization
M[Mess 96, pp. 102] and [Wils 72], reduce the computational effort signif-
icantly at the expense of accuracy. These models do not allow to compute
the time transient change of the magnetization after a field variation, or the
change of the resistivity in the contact resistances or the superconducting
strands with temperature and field.
The derivation of the analytical model by Wilson is lengthy and can be
found in [Wils 72]. We solely quote the result for the magnetization and
losses. Consider a Rutherford-type cable of mean width w, height hand
twist pitch length `p. We assume all strands in the superconducting state.
7.6 Cable Magnetization Losses 113
The cable defines a local frame consisting of one vector parallel e|| and one
vector normal e⊥to the broad face of the cable, see Fig. 7.20 (right). The
magnetization can be calculated from the average magnetic induction over
either cable face:
Mc
⊥=−1
120Rc
`pNs(Ns−1) h
w
dB⊥
dt,(7.64)
Ma
⊥=−1
3Ra
`p
h
w
dB⊥
dt,(7.65)
Ma
|| =−1
8Ra
`p
w
h
dB||
dt,(7.66)
where Nsis the number of strands and the indices c,a stand for the cross and
adjacent resistances, respectively. Notice that the magnetization opposes the
field change following Lenz’s law. The total inter-strand coupling current
magnetization is given by
MISCC = (Mc
⊥+Ma
⊥)e⊥+Ma
||e|| (7.67)
In this simplified model, neglecting the time constant of the induced eddy
currents, the loss power density can be calculated from the magnetization
and field change [Mess 96, pp. 101], i.e. from p=−M·dB
dt. This can be
motivated remembering that M∼Jand dB
dt∼Eand thus we get:
pISCC =−MISCC ·dBav
dt(7.68)
For the inter-filament coupling currents we could derive the same result ana-
lytically.
The analytical magnetization model assumes that the strands are in the
superconducting state and that the induced currents are well below the crit-
ical current. The maximum current induced in the superconducting strands
ISC,max can be estimated from the following equation [Verw 95, p. 75]:
ISC,max = 0.0415`phNs
Rc
dB⊥
dt,[ISC,max] = A. (7.69)
Superconductor magnetization The superconductor magnetization is fun-
damentally different to the former two induced currents. It does not depend
on the rate of field change and exhibits a hysteresis in a varying field.
The arising magnetic fields are accurately described by means of the inter-
secting-ellipses-model [Voll 02]. Although the model allows to calculate the
hysteresis losses, the dissipated power remains undefined. In Sec. 7.7 the
power is derived for a simplified analytical model. The results allow to con-
clude that losses stemming from persistent currents can be neglected for fast
ramping magnets and during a fast discharge as for a quench.
114 Detailed Treatment
7.7 Superconductor Hysteresis Losses
Cables in superconducting magnets are subjected to field dependent losses,
i.e. induced inter-strand and inter-filament coupling losses, and superconduc-
tor hysteresis losses. For the two kinds of induced eddy currents, analytical
and numerical models are available, allowing to calculate the dissipated power
under various operating conditions (Sec. 7.6). For the losses stemming from
the superconductor magnetization - as for hysteresis losses in general - the
literature only gives formulae for the dissipated energy over closed excitation
cycles.
The thermal stability of a cable to quench depends on the amount of dis-
sipated losses compared to the available cooling power. Whenever losses and
heating exceed the cooling capacity of the magnet, the cable temperature
rises and the conductor eventually quenches. Hence, the computation of the
cable stability against time transient losses, as well as the simulation of the
quench propagation process in a magnet, depend on the exact description of
all dissipated losses in time.
In the following, we derive an expression for the instantaneously dissipated
hysteresis losses during an arbitrary ramp-cycle using a simplified geometry
and hysteresis model.
Energy loss over a closed cycle We recall the approach for the calculation
of the energy loss over a full cycle. Consider a magnetic system exhibiting
hysteresis and subjected to a periodic input variation. In the steady state, we
can assume that the system is in an identical state at the start and end of any
closed cycle. Hence, all field quantities and especially the magnetic energy
are identical. The hysteresis energy loss is then given by (see Sec. 7.3),
WV
hyst =µ0ZVIM·dHdV. (7.70)
Based on the principle of energy conservation, the same result can be obtained
by integrating the energy supplied by the source (disregarding all other losses).
In the steady state, the net energy supply during a full period is dissipated
in losses. (The energy associated with the working point of the system is not
affected).
Approach For the calculation of the instantaneously dissipated losses, we
combine the principle of energy conservation with a physical model describing
the superconductor hysteresis.3. We make the following assumptions:
3Bertotti and Mayergoyz [Bert 06, pp. 347] write:
It should not be surprising that the expression for hysteretic energy losses has
been found only for the case of periodic input variations. The reason behind
this fact is that the energy losses occurring for periodic input variations can be
easily evaluated by using only energy conservation principle; no knowledge of
actual mechanisms of hysteresis or its model is required. The situation is much
more complicated when arbitrary input variations are considered. Here, the
7.7 Superconductor Hysteresis Losses 115
•The dissipated energy constitutes a monotone function in time. We can
therefore deduce the dissipated energy from the energy supplied by a
source by successively subtracting the energy stored in all sub-systems.
•If a model adequately describes the hysteresis measurements, the model
is exhaustive and no additional effects need to be considered. Therefore,
all relevant energy storages are established and can be monitored.
In the case of superconductors, the hysteresis can be well described by means
of mere electromagnetic models and the energy balance can be calculated by
means of Poynting’s theorem.
Simplified Problem The approach is demonstrated by means of a simplified
problem. Consider an infinite slab of superconducting material with thickness
2d. The slab is aligned parallel to the yz-plane and the center is placed at
x= 0, see Fig. 7.21 (left). A coil with N0windings per unit length is mounted
on the surface of the superconductor. The coil constitutes a surface current
density JF=∓N0Iez(at x=±d). The corresponding primary magnetic
field inside the coil is homogeneous
He=Heey=N0Iey,(7.71)
and cancels for |x|> d. The excitation current Iis shown in Fig. 7.21
(right), where I0and τdenote an arbitrary current level and time interval,
respectively. The ramp-cycle contains two closed cycles, i.e. 2→10τand
12 →14τ. The ramp-rate of the second cycle is twice the ramp-rate of the
first cycle.
Applying the approach to an infinite slab, reduces the complexity of the
analytical field calculation significantly. We merely deal with polynomials of,
at maximum, second order. The superconductor can be described by means of
the well established critical state model. With a given current ramp-cycle, we
can calculate the superconductor magnetization analytically without relying
on a numerical hysteresis model.
Critical state model The critical state model was developed by Bean for a
slab of hard superconducting material [Bean 62, Bean 64]. The model gives
a macroscopical description of the superconductor magnetization. It is based
on the following principles:
•The interior of a superconductor is shielded from an external magnetic
field by a layer of screening currents.
•The maximum current density in the superconductor is limited. The
limit is denoted critical current density Jc.
energy conservation principle alone is not sufficient, and an adequate model
of hysteresis should be employed in order to arrive at the solution to the
problem.
116 Detailed Treatment
!"
#"
$"
%!&"
'"
!"#
2
4
6
8
10
12
14
tΤ
-4
-2
2
4
II0
Figure 7.21: (left) Critical state model of a superconducting slab. The slab of thickness
2dis placed in a coil with current Iand N0windings per unit length. The coil constitutes
a surface current density JF=∓N0Iez. The magnetic field of the excitation coil, He,
is y-directed. Two screening current layers are shown. The inner layer was to screen a
positive external field. When the field starts decreasing the outer layer grows inwards. The
boundary is denoted xb. (right) Excitation current of the coil. We can distinguish two
closed cycles, 2→10τand 12 →14τ. The current ramp-rate of the second cycle is twice
the ramp-rate of the first cycle.
•Any magnetic field, albeit small, always induces the critical current
density independent of the ramp-rate. With increasing applied external
field, the screening layer grows inwards. The width of the screening
layer is called penetration depth.
•When reaching a certain field level, the screening currents cover the full
sample. This state is denoted as fully penetrated.
•Whenever the change of external magnetic field inverts direction, a new
screening layer of opposite polarity is created on the boundary of the
superconductor. The new layer annihilates the previous layer while
growing inwards, see Fig. 7.21 (left).
For the simplified model, the field and temperature dependence of the critical
current density (Sec. A.2.4) are not taken into account. For the demonstra-
tion of our approach, we arbitrarily set
Jc=He(t=τ)
d=N0I0
d.(7.72)
For the ramp-cycle given in Fig. 7.21 (right), this results in a fully penetrated
state as soon as the current changes by more than ∆I=I0.
Screening currents In the following, we determine the screening current
layers and their magnetic field for the given external field ramp. For the
sake of simplification, we give functions only for x > 0and note the kind of
symmetry in x,i.e. odd or even function.
The current cycle shown in Fig. 7.21 (right) defines 5 intervals. During
each interval a new current layer is created at x=dand grows inwards.
7.7 Superconductor Hysteresis Losses 117
Intervall i1 2 3 4 5
lower boundary ti
l0τ2τ6τ12τ13τ
upper boundary ti
u2τ6τ12τ13τ14τ
The ramp-cycle was chosen such that we do not need to be concerned with
merging current layers, i.e. layers of identical polarity growing into each
other. We define ˙
Hi
eas the derivative of the external magnetic field within the
interval i. The inner boundary xi
b(t)of each current layer can be calculated
by:
xi
b(t) =
d t −ti
l<0
d−|˙
Hi
e|
Jc(t−ti
l) 0 < t −ti
l<dJc
|˙
Hi
e|
0else
(7.73)
The screening current density as a function of the position xand time t, is
built from the individual current layers Ji. The function Jiis odd,
Ji(x, t) = Jc
˙
Hi
e
˙
Hi
e
ez(0x<xi
b(t)
1x≥xi
b(t).(7.74)
The superposition yields the total screening current density J,
J(x, t) = J1(x, t)+2
5
X
i=2
Ji(x, t).(7.75)
Notice that the screening current layers for i > 1are added twice. This is in
order to annihilate previous layers and does not violate the limitation of the
maximum current density in a superconductor.
Figure 7.22 (left) shows the screening current density distribution within
the slab cross-section over time. The slope of the “fish-bone” pattern depends
on the rate of change of the applied magnetic field He. For x > 0and
τ < t < 2τ, the current density is +z-directed. The center of the slab (x= 0)
remains current free until t= 1τ.
Electromagnetic Fields The screening field Hiis calculated from the screen-
ing current density J. Assuming symmetry we obtain
Hi(x, t) = −Zd
x0=x
J(x0, t)dx0ey.(7.76)
The function is even and opposing the external field change, see Fig. 7.22
(right). The magnetic induction in the superconducting slab yields
B(x, t) = µ0(He(t) + Hi(x, t)) .(7.77)
Due to the screening currents, the center of the slab remains field free (B=0)
until t= 1τ. Thereafter, the magnetic induction in the center of the slab lacks
118 Detailed Treatment
Out[378]=
Out[147]=
Figure 7.22: The system of the critical state model: (left) Screening current density
over the x-axis versus time. For an external field with piece-wise constant ramp-rate the
layers are of trapezoidal shape in the time-space contour plot. (right) Contour plot of the
internal magnetic field Hiover the x-axis versus time. The field increases by Jcdfor every
two contour lines. The function can be obtained by integrating the current density along
the x-axis from the outside to the inside.
the applied magnetic field. Integrating over the cross-section of the slab yields
the enclosed magnetic flux per unit length Ψ0.
Ψ0(x, t) = Zx
x0=0
B(x, t)dx0.(7.78)
The magnetic induction and magnetic flux are shown in Fig. 7.23. The
electrical field can be obtained either by integrating the change of magnetic
induction along the x-axis or by differentiating the total magnetic flux over
time:
E(x, t) = Zx
x0=0
∂B
∂t (x0, t)dx0ez=∂Ψ0
∂t (x, t)ez(7.79)
The change of magnetic induction and the electrical field are shown in Fig.
7.24.
Hysteresis loop Integrating the magnetic screening field Hiover the slab
cross-section, yields the internal magnetic flux per unit length Ψ0
i. Figure
7.25 shows the internal flux plotted over the applied external magnetic field
He. We can distinguish a “virgin curve” for 0→1τand two closed hysteresis
loops, 2→10τand 12 →14τ, respectively.
Apart from the first layer of screening currents, every new layer annihilates
the previous layer and creates a new layer of opposite polarity. This results
in a steeper slope than on the virgin curve.
Energy Balance We calculate the energy balance for the given current cycle
applying Poynting’s theorem in integral form (derived in Sec. 7.3) and inte-
grating the different terms over time. Due to the 1D-approach all quantities
are given per unit area and denoted by calligraphic characters, i.e. S,Wand
P, with [S]=[W]=[P] = Jm−2.
In the simplified model, the magnetic field is zero for |x|> d. We identify
the surface integral over the superconductor boundary as the electromagnetic
7.7 Superconductor Hysteresis Losses 119
Out[162]=
Out[184]=
Figure 7.23: (left) Contour plot of the total magnetic induction over the x-axis versus
time. The magnetic induction increases by µ0Jcd=µ0N0I0for every two contour lines.
The magnetic induction is given by the sum of the external magnetic field Heand the
screening field Hi(or magnetization M). Note that the field in the center of the slab is
completely shielded (equals zero) until t= 1τ. Further more, the field in the center of the
slab shows a time lack compared to the excitation, compare Fig. 7.21. (right) Contour
plot of the magnetic flux per unit length over the x-axis versus time. The confined flux
increases by µ0Jcd2for every two contour lines. The flux is given by the integral over the
magnetic induction from the center of the slab to the position x.
Out[320]=
Out[313]=
Out[373]=
Out[342]=
2
HysteresisForREAL.nb
Out[173]=
Figure 7.24: (left) Contour plot of the time derivative of the magnetic induction over
the x-axis versus time. The inner magnetic field varies only within the triangular shapes.
(right) Contour plot of the electrical field over the x-axis versus time. The electrical field
increases by µ0Jcd2/T for every two contour lines. This function can be obtained by
integrating over the time derivative of the magnetic induction from the center of the slab
to the position xor by differentiating the magnetic flux in Fig. 7.23 (right) over time.
Y
i
He
têt = 0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Figure 7.25: Hysteresis loop of the inner magnetic Ψiflux through the coil versus applied
magnetic field He. We can identify two closed cycles: 2→10τand 12 →14τ. The slope
of the “virgin” curve ,0→1τ, is different to the slopes during all following magnetization
changes.
120 Detailed Treatment
energy exchange with the current source (factor 2 because of the two surfaces).
This is denoted by S,
S(t)=2Zt
t0=0
[E(x=d, t0)×H(x=d, t0)] ·exdt0
= 2 Zt
t0=0
[E(x=d, t0)×He(t0)] ·exdt0.(7.80)
The superconductor magnetization is expressed by means of the screening
current density Jand therefore B=µ0H. Integrating ∂B/∂t ·Hover time
and the cross-section of the slab, we obtain the magnetic energy stored in the
slab Wat the time t:
W(t) = Zd
x=−dZt
t0=0
∂B
∂t (x, t0)·H(x, t0)dt0dx
=µ0
2Zd
x=−d
[He(t) + Hi(x, t)]2dx(7.81)
The energy loss per unit area of the slab is given by P,
P(t) = Zd
x=−dZt
t0=0
E(x, t0)·J(x, t0)dt0dx. (7.82)
The three quantities fulfill the following equation at all time,
−S =P+W.(7.83)
We conclude that the electromagnetic model of the superconductor hysteresis
is exhaustive, i.e. the critical state model comprises all energy storages and
sinks, and no further effects need to be considered.
Figure 7.26 (left) shows all three quantities plotted over time. The graph
for Pis monotonous, i.e. the accumulated energy is only growing and thus
constitutes the dissipated losses. The energy loss over either of the two closed
cycles can be read off from the graphs for Por S. It is given by the difference
of the start and end value.
Figure 7.26 (right) shows the time derivative of the three quantities. We
interpret the function dP/dtas the dissipated power P,
P=ZV
E·JdV(7.84)
Regarding the dissipated power in Fig. 7.26 (right), we can distinguish two
dependencies. During the built-up of a screening current layer the power
reduces to zero following the shape of a parabola. After the screening layer
is fully established, the dissipated power is constant. The magnitude of the
dissipated power is in linear dependence of the applied ramp-rate.
7.7 Superconductor Hysteresis Losses 121
W2Ø10 t
W12Ø14 t
W2Ø10 t
W12Ø14 t
!
-"
#
0
5
10
15
0
2
4
6
8
10
12
têt
-S
P
dWêdt
-
PSC
0
2
4
6
8
10
12
14
-15
-10
-5
0
5
10
15
têt
Figure 7.26: (left) Supplied, dissipated and stored magnetic energy over time: The dashed
line shows the integrated energy flux from the poynting vector, the dotted line the magnetic
energy and the solid line the integrated losses. The energy loss for the two closed loops,
2→10τand 12 →14τ, is indicated. (right) Energy change and power versus time. The
dashed line shows the energy flux, the dotted line the change of stored magnetic energy and
the solid line the dissipated power. For comparison, the dash-dotted line shows the negative
power as seen from the current source. Note that the power at the source only differs from
zero while the screening layers change. Furthermore, the maximum power depends on the
applied field level and is both, positive and negative.
Circuit approach For the sake of completeness, we give a description by
means of lumped circuit elements. In the 1D-approach, the voltage induced
over the excitation coil can be calculated from the electric field,
U00
ind =N0Ez,(7.85)
with [U00
ind] = Vm−2. This can be split into a part induced over the self
inductance of the coil and a part stemming from the superconductor U00
SC,
U00
SC =µ0N0∂Hi
∂t ,(7.86)
As shown in Fig. 7.26 (right), the dissipated power PSC =U00
SCIpresents four
distinct spikes during the time when the screening layers are changing. The
magnitude and the sign of the spikes differ. Therefore, PSC represents both
stored and dissipated energy and cannot be used to derive the momentarily
dissipated losses. An integration over the two closed cycles yields the energy
loss as above.
Quench model For the calculation of the heat load in superconducting mag-
nets, superconductor hysteresis losses can be considered by means of the av-
erage power, i.e. the total energy of a cycle divided by the cycle duration.
In the case of the LHC main bending magnet, the total hysteresis losses for
122 Detailed Treatment
a complete ramp-cycle, from injection to nominal and down to injection cur-
rent, account for 280 J/m[Voll 02, p. 135]. Over a ramp-duration of 2×1200 s
this constitutes an average heat load of 0.12 Wm−1.
For slow ramping magnets, the hysteresis losses can be neglected. Com-
pared to the quadratic ramp-rate dependence of induced eddy currents, hys-
teresis losses increase only linearly and are easily exceeded. In the case of the
LHC MB the induced eddy current losses are in the same order of magnitude
at a ramp-rate of approximately 0.1 T/s. The full discharge of the magnet
takes less than half a second.
Comparison to the literature Brechna [Brec 73, pp. 241] and Iwasa
[Iwas 94, pp. 274] derive the superconductor hysteresis losses for periodic
operation conditions also using Poynting’s theorem and the critical state
model. The problem of the momentarily dissipated losses is not addressed.
In his work on the model of the critical state of superconductors [Camp 07],
Campbell uses E·Jas an expression for the dissipated power density without
giving a derivation. The problem of the calculation of hysteresis losses is not
discussed.
Kanbara [Kanb 87] derives the hysteresis losses from the induced voltage
over a pick up coil yielding the power provided by the current source. As
shown above, the energy flux from the current source contributes to both,
stored energy and hysteresis losses. Therefore, it is possible to calculate the
energy loss for a complete cycle, but not to express the momentarily dissipated
power.
The approach of energy separation was applied to deduce the dissipated
losses from hysteresis measurements of ferromagnetic materials [Bozo 93, pp.
518], [Town 35] and [Okam 36]. Due to the number of different physical effects
interacting in ferromagnetic materials, the modeling of the different energy
storages is much more complicated. Furthermore, magnetostriction couples
the electromagnetic system to the mechanical system and the magneto caloric
effect requires to include the thermodynamic system.
7.8 Modeling Rutherford-Type Cables
Most superconducting accelerator magnets are wound from Rutherford-typeRoebel bar
cables, i.e. twisted multi strand cables resembling the Roebel bar [Roeb 15]
used in electrical machines.
The variation of the magnetic flux over the cable cross-section causes a
non-uniform current distribution in the different strands. By fully transpos-
ing all strands over a short length, the maximum loop size and thus the
induced eddy currents are reduced. In the 1970s it was shown at Rutherford
Appleton Laboratory (RAL) that a twisted hollow tube of superconducting
strands could be rolled into a rectangular or trapezoidal shape without sig-
nificantly reducing its current carrying capability (degrading). Cables with aKeystoning
trapezoidal cross-section are used to built cos θ-type coils and are denoted as
7.8 Modeling Rutherford-Type Cables 123
!!"
"#"
#"
Figure 7.27: Rutherford-type cable. (left) Trajectory along one strand of a 26 strand
cable over the twist pitch length `p. (right) Projections of the cable on each of the 3
coordinate planes. The strand from the figure left is highlighted in grey. Two loops for
induced eddy currents are indicated by a thick black line.
keystoned. Figure 7.27 (left) shows the trajectory along one strand of a 26
strand cable over the twist pitch length `p.
The strands of a Rutherford-type cable are in resistive contact. We distin- Resistive
contacts
guish between adjacent resistances and cross-over resistances. The resistance
depends on the strand coating material, the cable compaction, which may
vary over the cable cross section, and eventually on the cable core material.
Eddy currents induced in loops formed by superconducting strands close over
the resistive contacts and cause heating (losses). These currents are denoted
inter-strand coupling currents (ISCC). Figure 7.27 (right) shows the trans-
posed strands and highlights two superconducting current-loops.
In general two different types of coils have to be distinguished, potted Thermal
properties
coils, where the voids in between the strands of the cables are filled with
glue/insulation, and wetted coils, where the voids are flooded with liquid
helium. In the latter case, the heat capacity of the confined liquid helium
contributes significantly to the thermal stability of the cable. In both cases,
the resistive connection between adjacent and crossing strands allows heat
transfer between strands and a quench can spread over the cable cross-section.
Model for Quench For the simulation of quench in accelerator magnets the
numerical description of Rutherford-type cables on the level of strands is too
demanding. Not only that for every cable one node per strand has to be
considered, but also, due to the transposition of the strands, the cable has
to be discretized longitudinally at least with the same number of steps per
twist-pitch length.
In the present model, Rutherford-type cables are discretized with one node
in the center of the conductor cross-section and a user supplied number of
nodes along the length of the magnet. This reduces the number of elements
significantly. For the LHC main bending magnet, where already 320 conduc-
tors have to be considered in the cross-section, the number of elements can
124 Detailed Treatment
Figure 7.28: (left) Current distribution due to transport current, inter-strand coupling
currents and inter-filament coupling currents. (right) Current density over the broad face
of the cable. The difference between total current density and critical current density is
available for superconductor screening currents as described by the critical state model.
Note that the current density margin to quench ∆Jis much smaller on the left hand side
of the cable. The values are not to scale.
be reduced by a factor of approximately 302/`p,i.e. by 5000 in total.
For the quench model relevant properties, i.e. current density, temperature,
magnetic induction and loss power, we make the following assumptions:
Current density: The current density is constant over the conductor cross-
section and over the length of the cable.
The transport current is evenly distributed over all strands of the cable.
Between cables of different type the current density may vary due to
differences in cross-sectional area or copper-to-superconductor ratio.
Figure 7.28 illustrates how the current distribution over the cable and
strand cross-section is influenced by induced cable eddy currents, i.e.
inter-strand and inter-filament eddy currents. Under the assumption
that the induced current density is much smaller than the transport
current density, this effect is disregarded.
Current redistribution, i.e. the by-passing of quenched strands over the
cross-over and adjacent contact resistances, is not considered.
Temperature: The temperature is constant over the cable cross-section and
varies linearly along the cable.
The thermal conductivity of the strand materials is many orders of mag-
nitudes greater than of the surrounding insulation materials. Due to the
transposition of the strands along the cable axis, the thermal gradient
over the cable cross section is much smaller than between adjacent ca-
bles. Although, measurements show a gradient of up to 2.5 K over the
cable cross-section, in a worst case scenario a homogeneous temperature
distribution is assumed. In the case of confined helium II, the overall
temperature gradient is small.
7.8 Modeling Rutherford-Type Cables 125
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!"
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!"
!!"#$%
!&'(#)%
*%
*%
Figure 7.29: (left) Field and temperature over a twisted cable. Every strand is exposed
to the full variation of magnetic field (grey). The quench is calculated from the peak field
(thick cross). (right) Field decomposition.
Consequently, a quench always occurs immediately over the full con-
ductor cross-section.
Magnetic induction: The model relies on the average and peak value of the
magnetic induction and magnetic vector potential. These values are
constant along the conductor.
Magnetic induction and magnetic vector potential vary over the con-
ductor cross-section. Fully transposed, every strand is exposed to the
total variation of the field.
Field-dependent material properties, e.g. the electrical resistivity of
copper, are evaluated for the average of the magnetic induction yielding
mean properties. Based on the assumptions for current density and
temperature, a conductor quenches where the local field is maximum
- conductor peak field. Note that in case of cables with large cabling
angle β, we have to distinguish between longitudinal and transversal
field components, see Fig. 7.29 (right).
The inductance of the magnet is calculated from the integral over the
mean magnetic vector potential, see Sec. 7.4.
Induced losses: Induced losses are constant over the cable cross section. The
losses are constant along superconducting parts of a conductor and zero
where the conductor is quenched.
The loss models rely on the assumption of superconducting current loops
over the cables and within the strands. Therefore, the model and losses
change in case of a quench. The longitudinal temperature variation does
not need to be considered.
Figure 7.29 (left) shows the distribution of current density, magnetic conduc-
tion, and temperature over the broad face of the cable and along the cable
axis.
126 Detailed Treatment
Note that a twisted cable produces, additionally to the azimuthal field, a
longitudinal field on its axis. In a Rutherford-type cable, the transport current
follows a helical trajectory along the cable axis. We decompose the current
into a part flowing parallel to the cable axis and an orthogonal fraction,
i.e. a solenoidal component. The parasitic field along the cable axis can be
estimated by means of an ideal solenoid (Sec. E.1.2). For N/h = 28/115 mm
and IN = 11850 A (data for the LHC MB inner layer cable), we obtain a
central field of B= 0.13 T. This effect is small compared to the peak field
of approximately 9 T and hence does not influence the simulation of quench.
Nevertheless, in the low field region of the coil, this additional field can be of
interest for the calculation of persistent currents.
7.9 Magnet Protection
In case of a quench, a superconducting magnet has to be protected from highProtection:
high tem-
peratures,
excessive
voltages
temperatures and excessive voltages [Coul 96, Iwas 05]. High temperatures
can destroy the insulation material or even result in a meltdown of the cable.
Excessive voltages can result in arcs that could punch holes into the insula-
tion. Furthermore, high current density and temperature gradients can cause
irreversible degradation of the superconducting material reducing the current
carrying capability [Iwas 05].
The quench behavior of a magnet, i.e. the current decay and the hot-
spot temperature, depend on the magnet design, i.e. the inductance, the
thermal quench propagation velocity, its susceptivity to induced losses, and
the external electric circuit.
Following the overview given in [Iwas 05, Coul 94] quench protection canPassive
protection be sketched as shown in Fig. 7.30. Passive quench protection methods aim
at the optimization of the inherent features of the magnet; the choice of
materials and the design, such that safe operation can be guaranteed even in
case of a quench. Providing a parallel path to the current and strong cooling
to the conductors allows the current to by-pass the quench, and the quench
to recover. This can be applied on the magnet scale, e.g. for small corrector
magnets [Schm 00], or on conductor level: If, under all operating conditions,Cryostability
the cooling of the conductor exceeds the ohmic heating in the event of a
quench, the conductor is called cryostable or unconditionally stable [Dres 95,
pp. 56]. Cryostability was defined by Stekly in [Stek 65]. A magnet isSelf-
protecting called self-protecting if the quench propagation over the magnet is generally
fast enough to substantially decrease the current and protect the magnet from
over heating [Iwas 94, p. 326].
Active magnet protection relies on the timely detection of the quench, fol-Active
protection lowed by a rapid current extraction. Due to the inductance of the magnet
circuit, the current cannot be switched off instantaneously. Thus the power
supply is shortened by means of a free-wheeling diode. The current decay rate
is given by the inductance and resistivity of the remaining circuit, τ=L/R.
If the magnet is powered in a chain of magnets the time constant can be
7.9 Magnet Protection 127
!"#$%&'()*&%+,*$'
!"##$%&' "()%&'
(*+,#-".$/$-+'
*&(,%&*+'
.+0!"##' 1&-&(),2'
&3-*"(-'(4**&2-'
!,5&*'#4!!/+',6'
147!'*&#$#-,*'
.+0!"##'
#4.01$%$#$,2' 84&2(9'9&"-&*#'
:!!&21$3;<##"+#;=4&2(9>*,-&(),2;-+.%/%'
Figure 7.30: Scheme of quench protection methods.
significantly reduced by isolating the quenched magnet from the rest of the
chain by means of a by-pass diode or resistor. Triggering additional quenches Quench
heaters
in the isolated magnet by firing quench heaters and quench back further in-
creases the resistivity and accelerates the current decrease. In the case of the
remaining chain, and for individually operated magnets, the time constant
can be reduced by switching-in an additional dump resistor. By subdividing Dump
resistor
a magnet into several inductively coupled loops, the current in the quenched
loop decreases faster due to the reduced inductance and, as a positive side- Sub-
division
effect, increases the current in the linked loops and consequently drives them
resistive. Inductively coupling a secondary circuit with finite resistance al-
lows transfer of some of the stored energy into the secondary circuit. If the Coupled
secondary
resistor of the secondary circuit is in close contact with the primary windings
the dissipated energy can be used as a quench heater.
Two equivalent paradigms can be applied to active quench protection: Ex- Protection
paradigm
traction and dissipation of the stored magnetic energy, or the reduction of
the discharge time. Isolating the quenched magnet and subdividing the cir-
cuit reduces the inductance of the considered circuit and, therefore, both the
stored magnetic energy as well as the discharge time constant. Any increase
in resistivity either extracts energy from the quenched magnet into an ex-
ternal and less critical resistor or helps to evenly dissipate the energy inside
the magnet. In doing so, the time constant is further reduced. Nevertheless,
both paradigms rely on the rapid disconnection of the magnet from the power
supply.
The need for quench protection is often motivated by the amount of elec-
tromagnetic energy stored in superconducting magnets - in the case of the
LHC main bending magnet 7 MJ (at nominal current level). This number
sounds impressive if compared to the kinetic energy of a truck (10 t) driving
at a speed of 135 km/h, but somewhat underwhelming if we compare it to the
energy required to draw a warm bath (100 l of water heated by 18 K). The
danger results from the short duration of the energy release and from the
high energy density if the stored energy is dissipated only over the initially
128 Detailed Treatment
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+4%$,*4/'$
':$
!:$ !:$
Figure 7.31: Quench detection methods: (left) Floating bridge differential voltage. The
inductive voltage is fully compensated by the symmetric voltage taps. (right) Electronic
voltage compensation. The inductive voltage is measured by means of a reference system
and electronically substracted.
quenched zone.
7.9.1 Quench Detection
A quench in a superconducting magnet can be detected by means of an in-Resistive
voltage creasing resistive voltage. The voltage rises due to the propagation of the
quench along the cable and to adjacent conductors, as well as the rapid tem-
perature increase within the quenched parts (see temperature dependent re-
sistivity in Sec. A.1.3). The voltage over the magnet (or between two voltageInductive
voltage taps within the magnet) also carries the inductive voltage driving the magnet
current up or down. Therefore, for the quench detection the resistive voltage
signal must be extracted reliably from the measurement.
Figure 7.31 shows two different detection systems: In the first system the in-Compensation
methods ductive voltage is eliminated by triggering on the differential voltage between
two identical parts of the magnet. The induced voltage is fully compensated.
This system fails in case of a symmetric quench. For the second method, the
induced voltage over the magnet is measured by means of a reference system
and electronically compensated. The reference system can be an adjacent
magnet, a coil inside the magnet [Wils 83, p. 220] or a co-wound wire within
the cable [Borl 04]. Iron-saturation and dynamic effects result in a non-linear
or even hysteretic relationship between ramp-rate and induced voltage. If the
reference system does not experience an identical flux change, the electronic
compensation becomes more demanding.
The magnet system is considered quenched as soon as the resistive voltageQuench
criteria exceeds a given threshold voltage Udet for a given duration tDis. The time
discriminator reduces sensitivity to noise and reduces false triggers. For the
quench protection of the LHC main bending magnets, the differential voltage
between the two apertures is monitored. If this voltage exceeds the thresh-
old of 100 mV for longer than 10.5 ms the magnet is considered as quenched
[Denz 06]. The time it takes for the voltage to reach the threshold depends
on the physical properties of the superconducting device, the characteristics
7.9 Magnet Protection 129
of the electrical circuit and the working point of the quenched conductor
[Denz 01]. The time between quench start and quench validation constitutes
the first part of the quench load - independent of any active quench protection
methods (compare MIITs in Sec. 7.2).
The quality of the quench detection system depends on the following three
points: How reliably is a quench detected? Under what conditions can a
quench remain undetected, i.e. be missed? How likely is a false trigger,
which would unnecessarily cause the magnet to switch off? [Verg 02] While a
delay in the quench detection may cause excessive hot-spot temperatures, false
triggers and the successive fast de-excitation may be the major mechanism
for aging of the magnets [Evan 09, p. 84].
In the unlikely case [Coul 94], that a quench spreads out symmetrically Symmetric
quench
over the two apertures, the differential voltage remains zero and the quench
cannot be detected. The voltage would rise until the switching voltage of the
cold diode is reached. The magnet can be protected by comparing the voltage
over two adjacent magnets in addition to the voltages over the two apertures
[Coul 94]. The small variation of the cable properties due to the high qual-
ity requirements further contributes to the symmetry of quenches. Highly
symmetric quenches were observed in LHC dipole magnets during commis-
sioning [Verw 08a]. Note that, with an initial voltage rise of approximately
4−10 Vs−1[Verw 08a] and a cold diode switching voltage of 6−8 V (see
Sec. 7.9.3), the quench could remain undetected for up to 0.6−2 s, and the
hot-spot temperature would easily exceed the melting point of copper.
In the case of fast-ramping magnets with large inductive voltages, it is more Fast-
ramping
magnets
difficult to electronically detect a resistive voltage on a large background due
to the reduced signal-to-noise ratio [Denz 08a]. Higher threshold voltages
have to be used, and therefore a quench is detected later. This must be
considered during magnet design [Schw 09].
The quench detection system must also cover all superconducting current
leads [Coul 94], e.g. bus bars, even if they are designed to be cryostable.
It is also possible to detect a quench by the induced field change in the aper- Quench
antenna
ture. If a strand of a Rutherford-type cable quenches, the current through
the cable re-distributes over the remaining superconducting strands, thus by
passing the normal zone. The resulting field change can be detected by a
pick-up coil placed in the magnet aperture [Lero 93]. Sophisticated arrays of
pick-up coils covering the full length of the magnet - denoted quench anten-
nas - allow one to localize the quench origin and observe the transverse and
longitudinal quench propagation [Siem 95].
Example : Detection length of the LHC MB The resistive voltage Ures
over a quenched conductor depends on several parameters, e.g., the transport
current I, the longitudinal expansion of the quench `and the cable properties,
Ures =`I
ANC
ρeff
E(T, B, RRR),(7.87)
where ρeff
Eis the effective field and temperature dependent electrical resistivity
130 Detailed Treatment
Outer
Cable
Inner
Cable
RRR !50
RRR !100
B!3T
B!6T
B!6 T, RRR !100
0.0 0.2 0.4 0.6 0.8 1.0
0
10
20
30
40
50
I"!
10000 in A"m!10000
Tin K
!!"#$%!!"#$% !!"#&%"#&% "#&%
!!"%"%
Figure 7.32: (left) Voltage detection conditions for the LHC MB. The black lines refer to
the outer layer cable and the grey line to the inner layer cable. The solid lines are for zero
applied field and a RRR value of 200. For the dashed lines the RRR has been varied. The
dotted lines are for different applied magnetic fields. The dash-dotted line is for a RRR
value of 100 and an applied field of 6 T. (right) Reduction of the voltages to ground by
splitting the dump resistor and earthing in the middle.
of the normal conducting area ANC. This can be rearranged to calculate the
minimum longitudinal expansion of a quench to be detected.
The normal conducting area of the LHC MB inner and outer layer cables
are computed in Sec. B.1.3 based on the values given C.1.1 (strand) and C.1.2
(cable). The effective resistivity is given by the resistivity of copper, see Sec.
A.1.3.1. Figure 7.32 shows the necessary quench length `to detect a resistive
voltage of Udet = 0.1 V for different configurations.
For a quench in the outer layer cable with an estimated temperature of
20 K and a field of 3 T, the quench needs to propagate over 260 mm per 10 kA
transport current to be detected. This translates to a length of 3.4 m at in-
jection current.
7.9.2 Dump Resistor
The use of dump resistors serves two purposes: For a single magnet the
dump resistor is used to increase the circuit’s resistance, accelerate the current
decrease (eventually causing quench back) and thus to reduce the hot-spot
temperature in the magnet. In the case of a string of magnets, the quenched
magnet is by-passed by a cold diode and discharged over the coil resistance
after firing the quench heaters. Here, the dump resistor is used to extract the
current from the chain and therefore protect the cold diode and bus bar from
overheating.
7.9 Magnet Protection 131
!""#$%&'()**+,*(-.#$/01234#/53$(!"#$%&'(')*+,!(*-&.
Figure 7.33: (left and center) The 225 mΩ and 3.5 m long dump resistor used in the LHC
main dipole string. Courtesy K. Dahlerup-Petersen CERN TE MPE. (right) Cold diode
used in the main dipoles. Courtesy CERN c
The dimension of the dump resistor for a string of magnets is determined Dimensioning
for a
magnet
string
by the maximum allowable time constant of the cold diode and the maximum
voltage over each magnet respectively to ground. The time constant of the
current decrease is given by the total inductance of the string divided by
the dump resistance. The maximum dissipated energy in the diode therefore
determines the lower boundary for the resistor. The upper boundary is defined
by the maximum voltage to ground and by the maximum di/dt. If the string
current decreases too quickly, induced losses may cause further magnets to
quench [Coul 94] and eventually damage the cold diodes (see Sec. 7.10.1). The
maximum voltage to ground over the dump resistor can be reduced by limiting
the number of magnets per string [Dahl 01] and by using multiple dump
resistors at adjacent points of the loop [Dahl 00], see Fig. 7.32 (right). For
the LHC main bending magnets two sets of three parallel 225 mΩ-resistors,
resulting in a total resistance of 150 mΩ, are installed at adjacent points of
each string. One resistor fills 3 m3, weighs 1.8 t, and absorbs 230 MJ during
a full discharge [Dahl 00]. Figure 7.33 (left) shows the dump resistor used in
the main dipole string.
The current commutation from the switch over the dump resistor into the
resistor is not instantaneous. In the case of the LHC MB it takes around 8 ms
[Denz 06].
In the unlikely case, that the (redundant) dump resistor switches fail, the
current is driven down by firing the quench heaters of further magnets in the
string [Denz 01].
For the dump resistor of a single magnet, the maximum size is given by Dimensioning
for a single
magnet
the maximum voltage to ground and the maximum voltage over the magnet
terminals (see Sec. 7.10.1). Discharging a magnet over a dump resistor of
variable size, allows one to determine the maximum allowable di/t which does
not cause a quench by quench back. Furthermore, the amount of extracted
energy versus total energy can be optimized.
132 Detailed Treatment
Generally, the dump resistor can be described by its resistance RDR and
the switching delay after quench validation ∆tDR (the time for the current
commutation is neglected).
7.9.3 Isolating the Magnet
Connecting a diode in parallel to a superconducting magnet provides a parallelBy-pass
diode path for the current in case of a quench. When the voltage over the quenched
fraction of the coil reaches the threshold voltage of the diode, the chain current
starts to commutate into the diode. The magnet is isolated from the chain
and discharges independently over the diode (with a current flowing in reverse
direction). The diode threshold can be reached sooner, if quench heaters are
fired.
In case of the LHC MB cold diodes are used. The diodes are mounted inCold diode
the same cryostat and initially at bath temperature. Compared to diodes
at room temperature, the threshold voltage is significantly larger (6−8 V)
and decreases to some volts after switching, due to heating of the junction
[Verw 08a].
The maximum ramp-rate on the up-ramp of the magnet is limited by theRamp-rate
limitation diode threshold voltage. On the down ramp, the switching of the diode is
hampered by the voltage induced across the magnet (see 7.10.1). The used
diodes must be able to withstand the radiation level in the environment of the
magnet, carry the string current sufficiently long and provide a sufficiently
high back-ward voltage in order to hold during a fast-deexcitation [Coul 94].
Figure 7.33 shows the cold diode of the main dipoles.
Alternatively to a diode, a magnet can be by-passed by a resistor. The size
of the resistor is determined by the maximum leakage current permitted and
thus by resistive losses in the by-pass [Schm 00].
7.9.4 Subdivision and Coupled Secondary
The quench process of a magnet can be improved by subdividing the magnetSubdivision
in case of a quench; see Fig. 7.34 (left). Therefore, parts of the coil are
by-passed by shunts or diodes. In case of a quench, the by-pass isolates
the quenched segment and the current decreases faster due to the reduced
inductance. Furthermore, the falling current induces a current increase in the
other loops, due to the inductive coupling of the different segments [Wils 83,
pp. 226]. This may quench further segments by exceeding the critical current
and finally accelerate the current extraction.
The time constant of the current decrease can also be reduced by coupling-Coupled
secondary in a resistive secondary circuit [Wils 83, pp. 221], see Fig. 7.34 (right). The
method can be easily understood by considering it as an ideal transformer
with a resistor over the terminals of the secondary winding. If the resistance
of the secondary winding is in close contact to the superconducting coil, the
dissipated heat can heat up the conductors and cause thermal quench back
[Gree 84b] - e.g. observed in solenoids wound on an aluminum shell [Eber 77].
7.9 Magnet Protection 133
!!""#$
#$ %%&$
%%&$
&$
#'$ #&$
('$ (&$
'&$ !!""#$
#$ %$
&$
%)*+$ !)*+$
+,-./*0$)*+,10234$
"56*372/$
+,-./819#$
:..*108;%<))24)%!-*1+6=3,5*+>,1%!"#$%&%'%()*+(",-.$!./()$0123
Figure 7.34: (left) Subdivision. The magnet is sub-divided into several segments. In case
of a quench, the current in the quenched segment can decrease faster due to the smaller
inductance. The current in the residual segments increases due to the induced voltage and
may cause further quenches. (right) Coupled secondary. The resistance of the quenched
circuit is increased by means of an inductively coupled resistor. If the resistor is in close
contact to the superconducting coil thermal quench back accelerate the current extraction.
7.9.5 Quench Heaters
For a magnet which is disconnected from the power supply, the current de- Increase
resistivity
crease mainly depends on the resistance built-up. Additional resistance can
be created by heating up large fractions of the coil and thus spreading the
quench over the entire magnet. This way, the time constant of the discharge is
reduced and the stored magnetic energy is dissipated over a larger volume, re-
sulting in lower hot-spot temperatures. In a string of magnets, each by-passed
by a diode, the additional resistive voltage contributes to the switching of the
diode.
Figure 7.35 (left) shows the electrical circuit of a quench heater consisting Design
of a thin resistive strip, a switch and a capacitor. The parasitic inductance
can be neglected for most purposes. The capacitor is discharged over the
resistance of the strip. Figure 7.35 (right) shows a sketch of a heater strip with
the resistive central element, copper plating and the insulation. Although an
electrical insulation hampers the heat transfer from the heater to the coil, the
heater strip has to be insulated in order to block turn-to-turn voltages and
voltages to ground.
By plating the heater strip with well conducting copper, the resistivity Copper
plating
of the strip can be modified and the overall voltage reduced. The spot-like
heating is as efficient as full heaters due to quench propagation [Coul 94]. The
minimum length of the un-plated zones is given by the cable twist pitch length
and the minimum propagating zone. The first guarantees that all strands are
covered and the quench can not be by-passed and the latter ensures that
the normal zone does not collapse [Rodr 00]. Adapting the resistance of the
quench heater strips by plating, allows to use the same power supply for a
variety of different magnet types and lengths [Rodr 00].
The effectiveness of a quench heater, i.e. whether the quench heater can Quench
heater ef-
fectiveness
provoke a quench in the covered conductors or not, depends on the working
point of the covered conductors and the effectively dissipated power. The
working point, i.e. temperature, current and magnetic induction, defines the
134 Detailed Treatment
!!"# "!"#
#!"#
$!"$%#
%%# %&#
%'#
%(# &'
)*+,-./0*#
(1.2*-3++#(133-#
'04435#6-./*7#
6-./*7#6.835*#
9443*:2;<=++.>+<!,3*?@65013?/0*<!"#$%&'#()#*+,
Figure 7.35: (left) Quench heater electrical circuit. (right) Quench heater strip layout
with copper plating.
temperature and energy margin (Sec. 7.1.2 and 7.1.4) of the conductors. The
effectively dissipated energy depends on the initial power and time constant
of the quench heater circuit as well as on the insulation material between coil
and heater strip and cooling conditions.
The time between firing the quench heater circuit and the detection of aQuench
heater
delay quench in one of the covered conductors is denoted quench heater delay, ∆tQH.
For a given heater circuit, the quench heater delay increases with decreasing
current and field on the conductor. Below a certain current level, the quench
heaters can be ineffective and thus yield an infinite quench heater delay.
The protection with quench heaters is designed such that a maximumReliability
redundancy can be provided in case of a heater failure. Heater current
leads might accidentally be disconnected, e.g. as in the LHC main dipole
[Denz 08a], or the electronic system might fail to fire the circuit. Therefore,
multiple heater circuits can be used, able to protect the magnet in case of one
failing. Furthermore, heater strips can be mounted and connected such that
conductors of all poles are covered, e.g. instead of covering left and right of
one pole, cover only one side of the coil at top and bottom (see Fig. 7.44).
For the total loss of a quench heater strip, spare heaters can be foreseen in
the magnet. These additional heater strips would need to be mechanically
connected, but save the effort of opening the magnet, see e.g. low-field heaters
of the LHC MB.
For the LHC main dipole, the final heater strip temperature after firing isTemperature
estimates estimated to 280 K [Rodr 00]. The copper-plating of the heater strips results,
at first, in an inhomogeneous temperature distribution along the covered con-
ductor. Nevertheless, the difference reduces to less than 15% within the first
50 ms after firing [Rodr 01] due to longitudinal quench propagation.
Quench Heater Discharge The discharge of a quench heater can be modeled
by means of the RLC-series circuit shown in Fig. 7.35 (left). The power
supply is represented by the capacitance CQH with an initial voltage −U0. It
discharges over the heater strip at t= 0. The heater strip has an electrical
resistance RQH and forms a current loop with an inductance LQH. The voltage
over the capacitance, UC(t), is described by the ordinary differential equation
7.9 Magnet Protection 135
of second order,
d2UC
dt2+RQH
LQH
dUC
dt+1
LQHCQH
UC= 0.
The general solution is given by
UC(t) = U1exp ((−α+jωd)t) + U2exp ((−α−jωd)t),(7.88)
with
α=1
2
RQH
LQH
, ω0=1
qLQHCQH
, ωd=qω2
0−α2.
The damped frequency ωdtakes imaginary values if αexceeds ω0,i.e. if
RQH >2qLQH/CQH. The oscillation of the system is over-damped and Eq.
(7.88) can be re-written as
UC(t) = U1exp −t
τ1+U2exp −t
τ2,(7.89)
with
τ1=1
α+pα2−ω2
0
, τ2=1
α−pα2−ω2
0
, τ1< τ2.(7.90)
The two constants U1and U2are determined by the initial values for capacitor
voltage and circuit current I,
UC(t= 0) = −U0=U1+U2
I(t= 0) = 0 = CQH
dUC
dtt=0
=U1
τ1
+U2
τ2
,
which yields
U1=U0
τ1
τ2−τ1
, U2=−U0
τ2
τ2−τ1
,|U1|<|U2|.(7.91)
The current Irises with the time constant τ1followed by a decay with the
time constant τ2,
I(t) = U0CQH
τ2−τ1exp −t
τ2−exp −t
τ1.(7.92)
Figure 7.36 shows voltage, current and power versus time during rise and
decay.
For quench heaters where τ1τ2,i.e. where 2qLQH/CQH RQH, and
the current rise does not need to be taken into account, the system simplifies
significantly:
UC(t) = −U0exp −t
τ2, I(t) = U0
CQH
τ2
exp −t
τ2.
136 Detailed Treatment
012345
!1.0
!0.5
0.0
0.5
1.0
t!Τ2
U!"U0"
I!Imax
P
!Pmax
0 1 2 3 4 5
0.0
0.2
0.4
0.6
0.8
1.0
t!Τ1
I!Imax
P
!Pmax
Figure 7.36: (left) Quench heater voltage (full), current (dashed) and power (dotted)
versus time. The current rise is shown in more detail on the right. The rise time τ1is much
smaller than the current decay time constant τ2.
The power dissipated in the heater strips decreases with half the time constant
of the voltage and current:
P(t) = RQH U0
CQH
τ22
exp −t
τ2/2=Pmax exp −t
τ2/2.(7.93)
Example : Quench heaters of the LHC MB For the quench heaters
of the LHC MB the following values are given: U0=−900 V,RQH = 12.5 Ω,
and CQH = 7.05 mF (see Sec. C.1.5). The inductance of the high-field heater
loop is calculated by an approximation to LQH = 16.5µH, see Sec. 7.10.2.
U1= 0.0135 V U2=−900.0135 V
τ1= 1.3µsτ2= 88 ms (7.94)
Since the two time constants differ by 4 orders of magnitude, the peak current,
the maximum dissipated power, and the total dissipated energy are calculated
from the simplified model:
Imax = 72 A Pmax = 65 kW Etot = 5.7 kJ (7.95)
The energy is dissipated evenly along the length of the two heater strips
`QH = 15 m. Due to the plating the energy is mostly dissipated on the side
facing the conductors. The heater strip is wQH = 15 mm wide and covers
9 conductors for the high-field heaters. The surface power density is then
Smax =Pmax/A = 14.4 W/cm2and the power per conductor per length
240 W/m.
7.9 Magnet Protection 137
!"#$%&'
(#)#%*+$'
,-./0-*+$'
!1'
!0#)'
!2-.'
!34+5'
!(6)7/8'
!!9:7#'
!(6%"7'
!!9#5'
!(/+0#'
!(/+0#9+)'
!!;'
!+5'
!+5<-..'
3+=#7'4">>.?'@5'
4=/)%&A/$'("B>'6#C/C)+7'
D/7#'!"#$%&'9#-)#7C'
E"77#$)'/$'("B>'6#C/C)+7'
!"#$%&'9#-)#7C'#5#%*2#'
E+.0'(/+0#'C=/)%&#0'+$'
E+.0'(/+0#'&#-)#0'">'
!"#$%&';-%F'
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E&-/$'E"77#$)'H#7+'
I!(/C'
I!!J'
I!(6)0'
I!(6%0'
I!(6'
I!!9)0'
I!!9'
I!%(&'
I!!('
I!!3'
(#)#%*+$'(#.-?'
(/C%7/B/$-*+$'J/B#'
J7/88#7'(#.-?'
(6'J7/88#7'(#.-?'
(6'E+BB")-*+$'(#.-?'
!9'J7/88#7'(#.-?'
!9'(#.-?'
!"#$%&'("7-*+$'
(/+0#'9#-)A">'J/B#'
!"
Figure 7.37: Sequence of events during a quench with quench protection. The distances
are not up to scale. Trigger delays are in the order of milliseconds while the quench-heater
delays are between 30 to 100 ms. The duration of a quench is in the order of several 100
milliseconds.
7.9.6 Quench Protection Sequence of Events
Figure 7.37 shows the sequence of events during a quench and succeeding
quench protection actions. This example represents more or less the LHC
tunnel configuration, but can be easily adapted to other protection schemes.
After detection and validation, the power supply is switched off and the
current commutates into the free-wheeling diode. The beam is aborted, i.e
directed into a beam dump. The dump-resistor and the quench-heater power
supply of the quenched magnet are triggered. The current commutates into
the dump resistor and starts to decrease. After the quench heaters show to
be effective, the voltage over the quenched magnet rises and switches the cold
diode. The magnet is now decoupled from the chain and discharges over the
internal resistance. The diode heats up and the forward voltage decreases.
The current decrease induces losses in the magnet windings yielding further
quench, i.e. quench back. The current of the quenched magnet commutates
fully into the cold diode. The magnet chain is switched off.
138 Detailed Treatment
!""#$%&'()**+,*(-./0+1#*(!"#$%$&'()!*''"+,
Figure 7.38: (left) LHC magnet test station in SMA 18. (right) LHC main dipole magnets
in the tunnel. Pictures courtesy of CERN c
7.10 Voltages Occurring During a Quench
In case of a quench, superconducting magnets have to be protected fromProtection
from
excessive
voltages
excessive voltages in order to prevent permanent damage to the coil insulation
(see Sec. 7.9). The voltage build-up depends on the used materials, the
quench protection system and the working point of the magnet.
The voltages are analyzed for a genuine LHC main dipole (MB) as describedTerminal,
coil and
turn-to-
turn
voltages
in Sec. C.1. The differences between operation in the LHC tunnel, where the
magnet is connected to a string of identical magnets, and the operation on the
test bench are indicated for the magnet terminal voltage (see also Fig. 7.38).
By means of the voltages over the four poles, quench heater induced voltages
as well as the influence of asymmetric quenching can be shown. From the
turn-to-turn voltages, peak electrical fields can be calculated, and limitations
for the size of a dump resistor can be derived.
7.10.1 Terminal Voltages
LHC Tunnel: The LHC MB is operated in a chain of Nmag = 153 dipoleTunnel
setup magnets connected to a controlled voltage source. Each magnet is bridged
by a cold diode. In case of a quench the power supply is switched off and
the current commutates into the free-wheeling diode. A dump-resistor is
switched-in in order to extract the energy from the magnet chain (see Sec.
7.9 for the quench protection paradigm and Sec. 7.9.2 for protection method).
The electrical circuit is shown in Fig 7.39. The voltage development over the
different stages is shown in Fig. 7.40.
During the up-ramp of the magnet string the voltage of the power supplyNormal
up-ramp is evenly distributed over all Nmag magnets. The current ramp-rate dI
dtand
the magnet terminal voltage UTerminal are given by
dI
dt=UPS
NmagLd(I), UTerminal =UPS
Nmag
=Uup-ramp,(7.96)
7.10 Voltages Occurring During a Quench 139
!"#$%&'()(*+,%
!!&%
"!&%
#-%
.,''/01''2)34%!)+5'%
"6!71%!6!6&%
"89%
"6!6%
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"7',#)3;2%
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#!% B%
B%
C$$'35)DE-((;F(EG+2*;4'(E!"#$%"&'%(()*+
%%5%
"H+>%
9A%@;43'*%
Figure 7.39: Electrical circuit of the LHC tunnel configuration: Quenching magnet in
a string with Nmag other magnets and a dump resistor. One non-quenched magnet has
been exemplarily highlighted. The cold by-pass diode of non-quenched magnets has been
omitted. In case of a quench, the power supply is switched off and the current commutates
into free-wheeling diode. After the voltage over the quenched magnet has reached the
threshold voltage of the cold by-pass diode the magnet starts to discharge over the diode
and the internal resistance.
with UPS the voltage of the power supply and Ldthe differential inductance
of a single dipole (see Sec. 7.4). Note, that in order to follow a given current-
curve, the power supply has to be controlled due to the non-linear differential
inductance.
If any of the magnets in the chain quenches, a small resistive voltage starts Quench
detection
to develop. The quench is detected by monitoring the differential voltage over
the two apertures of each magnet (see quench detection in Sec. 7.9.1). The
detection is thus independent of the ramping of the magnet. The magnet is
considered quenched if the voltage exceeds the detection threshold Udet for a
given time ∆tDis.
In case of a quench, the power supply is switched off and the current com- Power
supply off
mutates into the free-wheeling diode. The terminal voltage of the quenched
magnet is given by
UTerminal =−UfDf +Ures
Nmag
+Ures,(7.97)
with Ures the resistive voltage of the quenched magnet and UfDf the forward
voltage over the free-wheeling diode of the power supply. This has to be dis-
tinguished from the voltage over each of the residual, non-quenched magnets
of the string denoted as UNoQ,
UNoQ =−UfDf +Ures
Nmag ≈ − UfDf
Nmag
.(7.98)
140 Detailed Treatment
!!" !#$%"!&'("
!)*+," !-./01" !23$,"!-4+#$" !-4+#$3+%" !+,"
5!-46"
5!-.%#"
5!-./#"
5!23" 5!/-7"
"0891':8"
"#$%"
";-;"
"-."
"/-/<7"
"/-=<7" "/-;."
!#
"<$1:4>'("
?88$>#4@AB66'C6AD+(%'E$6A!"#$%&'()*++'#,
Figure 7.40: Voltage development over the terminals of a quenched magnet in the LHC
string. Voltages and time intervals are not to scale.
The diode forward voltage consists of the diode threshold voltage UfDTh and
the voltage over the differential resistivity RfDfR giving only a few volts in
total. The current remains more or less constant due to the large inductance
of the chain. By switching-in a dump resistor the current decrease can beDump
resistor significantly accelerated. After switching, the current commutates from the
switch into the resistor during ∆tDRcd and yields approx. 1800 V (for a quench
at nominal current and a dump resistor of 150 mΩ) evenly distributed over
all magnets:
UNoQ =−UDR +UfDf +Ures
Nmag ≈ − UDR
Nmag
,(7.99)
UTerminal =−UDR +UfDf +Ures
Nmag
+Ures,(7.100)
where UDR =RDRIEand RDR the resistance of the dump resistor. With the
dump resistor, the current of the magnet chain decreases with a time constant
of approximately 100 s, see Sec. 7.9.2.
The quench heaters of the quenched magnet are fired upon quench detec-Quench
heaters tion. After the quench-heater delay ∆tQH (see Sec. 7.9.5), the first conductors
covered by quench heaters turn resistive, and the resistive voltage over the
magnet increases rapidly. As soon as the voltage across the magnet reachesCold diode
the cold diode forward threshold voltage UcDcTh, the cold diode switches
and decouples the magnet from the chain. The magnet terminal voltage is
clamped to the diode forward voltage:
UTerminal =UcDf =UcDcTh +RcDfRID≈UcDcTh.
The string current IEslowly starts to commutate from the quenched mag-
net into the diode. The current flow through the diode causes the junction
temperature to increase and the forward threshold voltage to drop to UcDwTh
7.10 Voltages Occurring During a Quench 141
after ∆tcDh. With increasing diode current IDthe forward voltage over the
diode rises and the terminal voltage is given by
UTerminal =UcDwTh +RcDfRID,(7.101)
with RcDfR the differential forward resistance of the cold diode. The volt-
age over the differential inductance of the quenched magnet, responsible for
decreasing the current, thus consists of
Uind =−UTerminal +Ures.(7.102)
The current decay in the remaining string is dominated by the dump resistor String
current
decay
and the string inductance is
dIE
dt=−UDR +UcDf
Nmag −1.(7.103)
The dump resistor has been chosen to give a time constant of around 100 s in
order to protect the cold diode and the busbars from overheating.
If the quench propagates along the string, e.g. by a heat wave propagat-
ing along the cooling system, the current decrease in the chain accelerates
following Eq. (7.103) and more magnets can be quenched by induced losses Multiple
quench
(quench back). Furthermore, the voltage of the dump resistor drops over
the cold diodes of the remaining magnets in reverse direction. If the reverse
blocking voltage of the cold diode is exceeded it can be damaged or destroyed.
For quenches at high currents, quenching of adjacent magnets after 30−300 s
has been observed [Verw 08a].
The ramp-rate of the magnet is limited by the forward threshold voltage Ramp-rate
limitation
of the cold diode.
The current in the magnet string changes very slowly compared to the cur- Constant
current
source
rent in the magnet where the quench does not take much more than half
a second. Therefore the external circuit could be modeled by means of a
constant current source. This simplifies the approach, but neglects the nega-
tive terminal voltage in Eq. (7.103). Due to the rapid voltage increase after
quench heater firing the error is small.
Test bench: On the test bench the magnet is connected directly to a power Test bench
setup
supply. In case of a quench, the power supply is switched off and the current
commutates into the free-wheeling diode. Eventually a dump resistor can be
switched-in. The electrical circuit is shown in Fig 7.41 (left). The voltage
development over the different stages is shown in Fig. 7.41 (right).
The quench detection and power supply switch-off happen similarly to the Terminal
voltage,
diode
setup in the tunnel. The terminal voltage of the magnet is clamped to the
free-wheeling diode forward voltage:
UTerminal =−(UfDTh +RfDfRIE).(7.104)
Note, that the cold protection diode parallel to the magnet is prevented from
switching since the voltage over the magnet is clamped to the warm diode
142 Detailed Treatment
!!" !#$%"!&'()" !*+,-."
"*+"
"/*/+"
"/*01"
!()"
!#
"0$.23456"
788$4#39:;<<5=<:>(6%5?$<:!"#$%"&'()&)&*+,-./0&*1()'()&2&*+,3
@-$4,134?"A5?4$%"
$;"
B.$$CD1$$634?"*3(#$"
"/*01"
%/*/+"
"&'"
*-28"+$<3<%(."
%*+"
".$<" "34#"
"0$.23456"
"/*/"
"*+"
E"
Figure 7.41: (left) Electrical circuit of a magnet on the test bench. (right) Voltage
development over the terminals of a quenching magnet on the test bench. In the case when
a dump resistor is switched into the circuit, the voltage is given by the dashed line. Voltages
and time intervals are not to scale.
forward voltage. It is therefore omitted in Fig. 7.41 (left). The voltage over
the differential inductance of the quenched magnet thus consists of
Uind =UTerminal −Ures =−(UfDTh +RfDfRIE)−Ures.(7.105)
Compared to Eq. (7.102) the forward voltage of the diode is of opposite
orientation and thus contributes to decrease the current (although only min-
imallly).
If a dump resistor is available, the terminal voltage after switching andDump
resistor commutation is given by
UTerminal =−(UfDTh + (RDR +RfDfR)IE)≈ −RDRIE,(7.106)
with RDR the dump resistor resistivity. The current decrease can be signifi-
cantly increased. Note that the voltage of the dump resistor drops over the
magnet terminals and therefore over the connection turns. The maximum
dump resistor size is limited by the highest allowable turn-to-turn voltage
(see Sec. 7.10.3).
7.10.2 Coil Voltages
Before installation in the LHC tunnel all magnets are tested for their quench
performance, i.e. whether they can reach a given current level without
quenching. In case of a quench on the test bench, current and voltages over
the magnet and the four poles are recorded. Although the terminal voltage is
clamped to the diode forward voltage (see above), the four coil voltages show
significant variations.
7.10 Voltages Occurring During a Quench 143
U1L
U1U
U2U
U2L
0.00 0.05 0.10 0.15 0.20 0.25
!40
!20
0
20
40
tin s
Uin V
Figure 7.42: Coil voltages recorded during a quench of an LHC main dipole on the test
bench.
As shown in Sec. 4.1.1, the coil voltages can be sucessfully reproduced byQuench
heater
signals
scattering
de-tuning the time when the different quench heaters show to be effective by
less than 2 ms. As stated in [Rodr 00] the quench heater trigger signal can
already vary by up to 10 ms.
Figure 7.42 shows the coil voltages recorded for an LHC dipole. The plot Voltage
shows
jumps and
spikes
shows sharp voltage spikes around t= 0.0 s not exceeding a duration of much
more than 10 ms. Note that the sampling rate of the measurement is only
5 ms. Following the voltage spikes, the four voltages slowly diverge. From
approximately t= 70 ms to t= 200 ms the voltages show a slight modula-
tion, or voltage jumps. The voltage jumps have also been reproduced in the
simulations.
In this section it is shown that the spikes result from the firing of the
quench heaters and the jumps stem from the asynchronous quenching of the
four poles.
Quench Heater Induced Voltages At the start of the quench, the total
voltage over the magnet is given by the resistive voltage of the quench and
the diode of the power supply (after switch-off, see above). The resulting
current change is small compared to the total current and the considered
time-frame of the spikes and is neglected. The electrical circuit of the magnet
and the four quench heaters is shown in Fig. 7.43. The four coils are connected
in series and are short-circuited. Each quench heater circuit constitutes an
over-damped RLC-circuit (see Sec. 7.9.5) coupled over the mutual inductance
MQH,C to the magnet circuit.
Since quench heaters are only modeled as effective heat sources inside the Approximation
covered conductors, the self and mutual inductances (see Sec. 7.4) have to
be estimated: Figure 7.44 (left) shows the coil cross-section of one aperture
with four additional conductors functioning as quench heaters. The quench
heater loops are mounted perfectly symmetrically on both sides of the two
144 Detailed Treatment
!!"#
"!"#
#!"#
$!"$%#
!!"# "!"#
#!"#
$!"$&#
%!"'(# %!"'(#
%!"'(# %!"'(#
!!"# "!"#
#!"#
$!")&#
!!"#
"!"#
#!"#
$!")%# %!"'(#
%!"'(#
%!"'(# %!"'(#
&$%# &'%#
&$*# &'*#
+,-./0.-#$# +,-./0.-#)#
!"$%# !"$&# !")%# !")&#
()
+,,-123456778975:;</8=-75!"#$%"$&'()*+!#,-.#&+/.)0-12)'&),*3
Figure 7.43: Electrical circuit model of the quench heater induced voltage spikes.
apertures. The mutual inductance matrix [M]/` is calculated with ROXIE
for nominal current,
[M]
M0`=
935.7 479.6 45.7 52.6 6.3−6.3−0.5−0.1
479.6 935.7 52.6 45.7−6.3 6.3 0.5 0.1
45.7 52.6 935.7 479.6 0.1 0.5 6.3−6.3
52.6 45.7 479.6 935.7−0.1−0.5−6.3 6.3
6.3−6.3 0.1−0.1 1.1 0.1 0.0.
−6.3 6.3 0.5−0.5 0.1 1.1 0.1 0.
−0.5 0.5 6.3−6.3 0.0.1 1.1 0.1
−0.1 0.1−6.3 6.3 0.0.0.1 1.1
,(7.107)
where [M] = (Mi,j)and i, j ={1L,1U,2U,2L,QH1L,QH1R,QH2L,QH2R}and
with M0= 1 µHm−1. The numbering can be taken from Fig. 7.43. Due to
symmetry, the mutual inductances between a heater and the coil have iden-
tical absolute values which shall be denoted MQH,C. The mutual inductance
between a heater and a coil of the other aperture is neglected. The self in-
ductance of each heater circuit has the same value and is denoted as LQH.
In Sec. 7.9.5 the quench heater discharge is calculated using the self induc-
tance LQH and neglecting any influence of the main circuit or other quench
heaters. The derivative of the heater current Eq. (7.92) is then given by
γ(t) = dI
dt=
U0CQH
τ2−τ1−exp “−t
τ2”
τ2+exp “−t
τ1”
τ1t≥0
0t < 0
,(7.108)
and is denoted γ. As shown in Sec. 7.9.5, the time constants τ1and τ2differ
by more than 4 orders of magnitude. During the regime of τ1the current
increases rapidly followed by the slow decay over τ2.
The four quench heaters follow the trigger signal after tQH1L = 0.0 ms,
tQH1R = 0.8 ms,tQH2L = 1.9 ms and tQH2R = 0.4 ms.
Due to the symmetric mounting, each quench heater induces identical volt-
ages over both coils of one aperture, but with opposite signs. Therefore,
7.10 Voltages Occurring During a Quench 145
!""#$%&'()**+,*(-./0+1#*(!"#$%"$&'()*+,-./01)'&)23"4)$5
0.000 0.001 0.002 0.003 0.004 0.005
!4000
!2000
0
2000
4000
tin s
U1Lin V
U1Rin V
U2Rin V
U2Lin V
Figure 7.44: (left) ROXIE model for the calculation of the inductance matrix in Eq.
(7.107). (right) Simulated voltage spikes.
the total voltage over the magnet remains zero and the magnet current Iis
constant. The induced voltages over each coil can be calculated from
{Uind}= [M]d
dt{i},(7.109)
with
d
dt{i}= (0,0,0,0, γ(t−tQH1L), γ(t−tQH1R), γ(t−tQH2L), γ(t−tQH2R))T.
Figure 7.44 (right) shows the induced voltages over the four coils. The
voltage level of the spikes depends on the rise time τ1of the heater circuits.
Overlapping spikes in the same aperture cancel each other out. The length
of non-overlapping spikes only depends on the heater rise time. The current
decay after the sharp rise induces only a very small voltage of opposite sign.
Voltages Arising from the Chronology of Quench Events As in the section Quench
heater
layout
before, the voltage over the initial quench and the diode of the power supply
are neglected. The magnet circuit is shown in Fig. 7.45. Each coil consists
of one quarter of the total inductance of the magnet and a time-dependent
resistance caused by the quench heaters. Each coil is covered by two different
quench heaters, one on each side.
A quench heater is effective as soon as the first conductor under the quench Model
heater quenches. Nevertheless, the covered conductors do not quench instan-
taneously due to differences in temperature and energy margin to quench (see
Sec. 7.1.2 and 7.1.4). The quench then propagates due to thermal conduc-
tion. At a certain point, the entire magnet quenches due to quench back.
This is modeled by the function Nhp representing the number of conductors
quenched by a quench heater as shown in Fig. 7.46 (left). The quench heater
146 Detailed Treatment
has a delay of tQHF.
Nhp(t, tQHF) =
0t<tQHF
NQHC
tHQC (t−tQHF)tQHF < t < tQHF +tHQC
dNC
dt(t−tQHF +tHQC)tQHF +tHQC < t < tQB
Nhp t>tQB
,
(7.110)
with NQHC conductors quenched within the time tHQC after tQHF. The
quench propagates with dNC
dtconductors per second. Quench back sets in
independently of tQHF at t=tQB. The used values are given in Tab. 7.1.
For the calculation of the time-dependent resistance of each half pole two
different models are applied. As assumed for the quench model, a conductor
quenches over the full cross-section nearly instantaneously. Therefore, the
function Nhp can only have discrete values. In magnets with a much greater
number of conductors, e.g. superconducting solenoids, the quench propaga-
tion is more continuous. Here Nhp is applied directly:
Rdisc.
hp (t) = R0roundNhp(t), Rcont.
hp (t) = R0Nhp(t).(7.111)
The resistance of one conductor is derived from the electrical resistivity of
copper (see Sec. A.1.3.1) at approximately 10 K and 3 T,
ρ0=ρcopper
E(T= 10 K, B = 3 T, RRR = 200) ≈4.9·10−10 Ωm,
and λ= 1.95,Ns= 36 and rs= 0.4125 ·10−3m(from Sec. C.1), yielding
R0=ρ0
`Magnet
λ
1+λNsπr2
s
= 0.58 ·10−3Ω.
Figure 7.46 (right) shows both models.
It is assumed that the quench heater delays follow directly the scattering of
the quench heater firing, i.e. that the resistance change in the circuits appear
with the same delays. A general delay of 20 ms is added. To reflect differences
Table 7.1: Assumptions made regarding the quench heater efficiency and the quench
propagation in the magnet
Quantity Unit Value
NQHC - 8
∆tHQC s 0.006
dNC
dt1/s 1
0.02
tQB s 0.2
Nhp - 40
7.10 Voltages Occurring During a Quench 147
!!"# !""#
!!$# !%$#
!&'()*#!&'()*#!&'()*#!&'()*#
#!"+$,# #!$+$,# #%$+$,# #%"+$,#
%&
-./0120/#!# -./0120/#%#
-../'(&3)455675)89:16;/5)!"#$%"$&'()*+!#,-.#&+/*01-2,"1".*34)1&*5
Figure 7.45: Model circuit for the explanation of voltages induced by asynchronous
quenching
!"
#!"#$
#!%$
&!!'(&#" !)*$
!!"+$
$#!"+$
#"
,**-.&/0(122342(567839-2(!"#$%"$&'()*+,)*#*&'-.)/"0)$+1*2-.34"-"5*67)-&*8
0.00 0.05 0.10 0.15 0.20 0.25 0.30
0
10
20
30
40
tin s
R
HC!R0
Figure 7.46: Model of the resistance growth in a half coil due to quench heater firing.
(left) Explanation of the different model parameters. (right) Comparison between discrete
and continuous approach.
0.00 0.05 0.10 0.15 0.20
!10
!5
0
5
10
tin s
U1Lin V
U1Rin V
U2Rin V
U2Lin V
0.020 0.022 0.024 0.026 0.028 0.030 0.032 0.034
!10
!5
0
5
10
tin s
U1Lin V
U1Rin V
U2Rin V
U2Lin V
Figure 7.47: Voltage jumps on coil voltages. (left) Voltage jumps due to all asynchronous
quench events. No jumps at the instance of simultaneous quench back. (right) Comparison
of the continuous and discrete approach. In case of the very drastic voltage change, the
continuous model shows similar behavior as the discrete approach.
148 Detailed Treatment
between upper and lower heater strips, an additional delay of ∆tlu = 0.5 ms
is applied to all upper heater strips. This results in 8 different starting times.
Since the voltage over the magnet is short-circuited, the total resistive volt-
age has to be matched by the voltage over the inductance. Due to symmetry,
the induced voltage is distributed evenly over the four coils. Although this
voltage causes the current of the magnet to decrease, the current change is
neglected to simplify the model. The voltage over each coil thus consist of
the individual resistive voltage plus one quarter of the total inductive voltage,
e.g.:
U1L(t) = Rhp(t, tQH1L) + Rhp(t, tQH1R+ ∆tlu)I+Uind/4
=IhRhp(t, tQH1L) + Rhp(t, tQH1R+ ∆tlu)
−1
4X
iRhp(t, ti) + Rhp(t, ti+ ∆tlu)#,(7.112)
with i={QH1L, QH1R, QH2L, QH2R}.
Figure 7.47 (left) shows the voltage over the four coils for the discrete
approach. Voltage jumps can be found around all asynchronous events, espe-
cially throughout the time where the quench heaters show to be effective. The
simultaneous quench of all coils due to quench back does not cause any volt-
age jumps. Figure 7.47 (right) shows the first and most pronounced jump in
higher resolution for both models. Although the quench propagates smoothly
in the latter model, the first voltage jump remains, and is of similar size. The
smaller, later jumps disappear. Without taking into account the up-down
asynchronous quenching ∆tlu the number of voltage jumps halves.
This model shows that the voltage jumps result from the redistribution of
the sudden changes of resistive voltage over the magnet. The “discretization”
of the quench events in terms of conductors accentuates the effect. This hy-
pothesis is supported by the observations made in [Verw 08a]: Kinks in the
differential voltage of the quench detection system (see Sec. 7.9.1) are iden-
tified by the propagation of the quench to adjacent conductors. In [Coul 96]
similar voltage redistributions were observed over magnets operated in the
LHC string and quenching asynchronously.
7.10.3 Voltages in the Coil Cross-Section
Based on the coil geometry and the winding scheme of the magnet, the lumped
quantities Ures and Uind (used above) can be mapped over the coil winding.
This allows for the computation of peak electric fields and excessive voltages
to ground.
Figure 7.48 (right) shows a schematic coil cross-section of a cos θ-magnetVoltage
along
magnet
winding
featuring two nested coil layers. The winding scheme is highlighted by the
dotted arrows. The magnet is connected to the external circuit at the conduc-
tors denoted 1 and 5. The external network consists of a power supply with
7.10 Voltages Occurring During a Quench 149
!"
#"
$" %"
&"
!" &"#"
$"
%"
!'("
!)*"
"+,-.,-/"
!#
!012"
$# !)*" %'"
!#
3441-.,567228926:;<=8/126!"#$%&'%"()*#+,$-.
Figure 7.48: (left) Voltage along the magnet winding length `winding. The dash-doted line
represents a magnet protected by a dump resistor. The solid line shows the same magnet
connected only to a power supply with a free-wheeling diode. The different slopes for the
induced voltage in the inner and outer layer are to represent the differences in inductance
within the coil cross-section. (right) Schematic coil cross-section of a cos θ-magnet featuring
two nested coil layers.
free-wheeling diode and eventually a dump resistor. The terminal denoted 1
is connected to the ground.
In case of a quench (in the conductor indicated with 4), a resistive voltage
rises over the magnet. This voltage, and the voltage across the external
circuit, are matched by the inductive voltage across the magnet. The resistive
voltage can be localized to the quenched conductor and the magnet terminals.
The induced voltage is distributed over the entire coil winding. Figure 7.48
(left) shows the voltage along the winding length `winding through the entire
magnet. The fact that the induced voltage varies over the coil cross-section
is represented by two slopes for the induced voltage (different in inner and
outer layer); compare [Wils 83, p. 203].
Two observations can be made. The voltage between subsequent turns in- Turn-to-
turn
voltages
creases with the gradient along the winding (the slope of the graph). Even
if the turn-to-turn voltage remains below critical values, the voltage between
adjacent turns of different layers can be excessive. An example is given by
means of the conductors 2 and 3. Although the voltage over the dump re-
sistor is distributed evenly over the winding, it drops over the two adjacent
conductors 2 and 5. Therefore, the resistance of the dump resistor needs to
be limited.
Consider the schematic coil cross-section shown in Fig. 7.49 (left). The Winding
scheme
letters A and B indicate two different winding schemes. In scheme A, the
conductors are connected row by row; in scheme B, column by column. For the
sake of simplicity, the magnet shall be short circuited. Quenched conductors
are shaded.
Figure 7.49 (right) shows the voltage distribution over the magnet winding
for both schemes. In the case of scheme A, the resistive voltage of the quench
150 Detailed Treatment
!"
#"
!"
#$%&'%&("
!"
#"
$"
%"
!))*&'%+,-../0.,1234/(*.,!"#$%"#&%"'()#*+$,-."/."+012$3$4
Figure 7.49: Influence of the winding scheme on the voltage to ground. (left) Schematic
coil cross-section with two different winding schemes. We consider a double-solenoid with
axis along the y-axis. The quenched turns are indicated by shading. In winding scheme
A the turns form interconnected radial disks. In scheme B the turns form connected sub-
solenoids. (right) Voltage along the magnet for the two different schemes
is concentrated at the start of the winding, resulting in one voltage peak. In
the case B, the resistive parts alternate with purely inductive parts of the coil.
The maximum voltage to ground is significantly smaller. Hence, the voltage
to ground is smaller if the quench propagates orthogonally to the coil winding
scheme, i.e. the quench spreads over turns which are not directly electrically
connected.
Electrical fields are calculated from the potential difference between two
conductors divided by the distance (taken form the coil geometry).
Capacitive Effects The approach presented above does not take into account
parasitic capacitances in the coil winding. The winding of the coil yields a
capacitance to ground as well as a series or winding capacitance. Fig. 7.50
(left) shows a schematic transmission line model of the coil winding with
parasitic elements [Brec 73, pp. 332]. Here Cg/` denotes the capacitance to
ground per unit length, Cs`the series capacitance and L, M/` the self and
mutual inductance per unit length. The end of the winding is connected to
the ground.
Consider a sudden voltage rise Udist due to a quench at an arbitrary position
inside the coil. This position is considered as terminal for the simplified
model. Following [Blum 19], the initial voltage distribution depends only on
the capacitances and the inductance can be neglected. The initial voltage
distribution along the winding coordinate zis given by
U
Udist
=
1−z
`winding α= 0
sinh “α“1−z
`winding ””
sinh (α)α > 0
,(7.113)
with
α=sCg
Cs
.(7.114)
After all transient effects decay the voltage distribution is given by the line
for α= 0. This is consistent with the approach presented above. During
7.11 Runge-Kutta Method 151
!!"#$%
"&"'!"'(%
#)$*"% #)$*"%
&#%"% &#%"%
&(%*"%&(%*"%
+"%
,--.'!"/*0##12#*345$1(.#*!"#$%&'%"()*#+,$-.+/+01#+"0$-2
6%
6%
Α"1
Α"2
Α"5
Α"10
steady state
Α"0
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
z!lwinding
U!Udist
Figure 7.50: (left) Transmission line representation of a the coil winding. (right) Voltage
over the coil winding for different values of α.
the transition, the voltage disturbance propagates along the coil winding and
may result in voltages greater than the initial disturbance [Brec 73, pp. 332].
Figure 7.50 (right) shows the initial voltage distribution over the magnet
winding. For increasing values of αthe voltage distribution becomes less
uniform and more concentrated at the location of the disturbance. This
increases the voltage gradient at the terminal by the factor α. As shown
above, turn-to-turn voltages can be estimated form the voltage gradient along
the winding, and consequently increase.
For small values of α, turn-to-turn voltages and voltages to ground well
below critical limits, the capacitive effects can be neglected. Capacitance
measurements for the SSC dipole [Smed 93], i.e. acos θdipole similar to
magnets used in the LHC, suggest a value for αof approximately 2. Therefore,
capacitive effects do not need to be considered while turn-to-turn voltages
remain orders of magnitude below critical values.
7.11 Runge-Kutta Method
The Runge-Kutta method denotes a family of implicit and explicit iteration
schemes for the numerical approximation of ordinary differential equations.
In this work we use the classic explicit Runge-Kutta method of 4th order.
We introduce the scheme following [Seld 73, pp. 210]. Consider the initial
value problem of the form
y0=f(t, y) with y(t0) = y0(7.115)
where y0=dy
dtand fis continuously differentiable. The basic idea of numer-
ical integration (the classical Euler method) is to approximate the increment
152 Detailed Treatment
by ∆y=h y0=h f(tn, yn)at a certain point, e.g., the initial value (t0, y0).
The step size of the integration is denoted h. A way to improve this method
is to calculate the derivative at the point yn+1 and average it with the deriva-
tive at ynbefore the actual integration step is performed (trapezoid method).
The Runge-Kutta method of 4th order is further refinement and reads:
k1:= f(tn, yn),(7.116)
k2:= ftn+h
2, yn+h
2k1,(7.117)
k3:= ftn+h
2, yn+h
2k2,(7.118)
k4:= f(tn+h , yn+hk3),(7.119)
With tn+1 =tn+hit yields
yn+1 =yn+h
6(k1+ 2k2+ 2k3+k4).(7.120)
As all explicit methods, the Runge-Kutta of 4th order is not unconditionally
stable and depends on the chosen time-step width h. The error can be con-
trolled by repeating the calculation with twice the step-size and comparing
the results [Seld 73, p. 212].
Adaptive Time-Stepping For systems with varying parameters, the typical
time-constants of the system may vary throughout the computation. This
way, an initially well adapted time-step width may end up being too short,
causing unnecessary high computational effort, or too long, yielding great nu-
merical uncertainty. The Runge-Kutta method of 4th order can be improved
by introducing an adaptive time-step width hn. A quality-factor [Coll 55, p.
68],
q=
k2−k3
k1−k2(7.121)
indicates if the Runge-Kutta time-step has been appropriate or whether it
should be shortened and repeated. If possible, the time-step size is increased
for the subsequent step:
0< q < 0.02 accept time step and increase the time-step width
for the following calculation step
0.02 ≤q < 0.05 accept time step
0.05 < q recalculate time step with shorter time-step width
In both cases, the new step-size can be conveniently calculated from the
quality factor,
hnew =hold
0.03
q.(7.122)
A Material Properties
“Ihr zahmen Täubchen, ihr Turteltäubchen, all ihr
Vöglein unter dem Himmel, kommt und helft mir
lesen, die guten ins Töpfchen, die schlechten ins
Kröpfchen.”
Aschenputtel
(Jacob Grimm and Wilhelm Grimm)
Low-temperature superconducting magnets most often consist of similar com- Materials
used in
accelerator
magnets
ponents and materials: Ferromagnetic low-carbon steel (magnetic iron) is
used to guide the magnetic flux, stainless steel or aluminum parts form sup-
port structures, and copper components are employed as spacers in the coil
cross section. The cables consist of uninsulated superconducting strands,
eventually surrounded by a stabilizer made of copper or aluminum, or con-
taining a core of copper or iron. The cables are insulated by wraps of Kapton1,
or impregnated with Epoxy resin or PVA2. The superconducting strands con-
sist either of Niobium-Titanium (Nb-Ti) or Niobium-3-Tin (Nb3Sn) filaments
embedded in a copper matrix. Nb3Sn-strands contain, in addition, fractions
of niobium (Nb), tin (Sn) and tantalum (Ta), due to the fabrication process.
The magnets are cooled with liquid helium (He).
The materials can be subdivided into fluid (respectively gaseous) and solid Classification
Phases
matter. The solid components can be further subdivided into chemical el-
ements (Al, Cu, Nb, Sn, Ti, Ta), alloys3(MI, SS, Nb-Ti) and compounds
(Nb3Sn, Kapton, Epoxy, G10). Some of the materials exhibit superconductiv-
ity at low temperature and low magnetic fields (Al, Nb, Ti, Ta, and especially
Nb-Ti, Nb3Sn). Liquid helium exhibits different phases at low temperature.
The simulation of superconducting accelerator magnets requires the follow- Relevant
Properties
ing material properties: The relative magnetic permeability µrof all compo-
nents of the magnet for the calculation of static magnetic fields. The critical
1Kapton is a polyimide film developed by DuPont. The chemical name for Kapton H and
HN is poly(4,4’-oxydiphenylene-pyromellitimide).
2As pointed out in [Lin 95] the abbreviation PVA can refer either to polyvinyl acetate
or polyvinyl alcohol. In the context of accelerator magnets PVA refers to polyvinyl
acetate.
3In the notation of alloys different materials are separated by a hyphen, e.g. niobium-
titanium: Nb-Ti. "NbTi" would represent a compound!
153
154 Material Properties
current density Jcis needed for the calculation of quenching and supercon-
ductor magnetization currents. The computation of quench propagation relies
on thermal properties, i.e., thermal conductivity κTand volumetric specific
heat cpof the coil and the coolant. The electrical resistivity ρEof all magnet
components is required for the calculation of induced eddy and magnetization
currents. For the calculation of internal voltages the relative permittivity εr
of the insulating materials is used. For practical reasons some of the material
properties are expressed per unit volume. The mass density ρDis required in
order to convert to quantities expressed per unit mass.
The material properties may depend on the applied magnetic induction B,
the temperature Tand the electrical current density J. A dependence on
the applied electrical field or on mechanical stress/strain is not taken into
account. In both cases, magnetic iron and helium, a dependence on the
applied pressure pmis considered.
The material properties given in the following sections are taken from a wideSources
range of publications. Some of the used publications are data compilations
and refer to original measurements. The work of Jensen et al. [Jens 80],
Floch [Floc 03] and Brechna [Brec 73] is in particular helpful.
As shown by Amati in [Amat 04], a comparison of the available measure-
ment data shows differences of up to 25% for the resistivity, up to 10% for the
specific heat and up to 25% for the heat conductivity. It is not always possible
to find adequate measurement data due to the difficulties measuring at very
low temperatures and/or very high magnetic fields. Furthermore, the variety
of different alloys is too big in order to always find measurements of material
properties. Therefore alternative data is given based on similarity or linear
combinations. This clearly disregards any metallurgical and microscopical
effects.
The material properties of the solids in the normal conducting state areStructure
of the text given first, followed by their properties in the superconducting state. At the
end the properties of liquid helium are presented. The text is organized by
the properties and not by the materials in order to facilitate comparisons.
When different sources deviate significantly, a comparison is made and the
final choice is justified.
A.1 Solids in Normal Conducting State
A.1.1 Temperature Levels
The term ambient temperature or room temperature is not strictly defined.
It can range from 273 K, over 293 K, and 295 K to 300 K [IEV 08]. There-
fore, in this text the term ambient temperature is used to refer to a generic
temperature accessible without technical aids and of common experience, i.e.
between 273 and 300 K.
Cryogenic temperature refers to temperatures only accessible by employingCryogenic
cryogenic cooling, i.e. temperatures below 120 K [Rade 02].
The standard temperature is defined as 293.15 K (or 20 oC). The meltingStandard
conditions
A.1 Solids in Normal Conducting State 155
absolute zero
0K
boiling point of He
4.22 K
freezing point
273.15 K
standard temperature
293.15 K
Cryogenic Temperature
120 K
Ambient Temperature
300 K
Copper Limit
1000 K
Solder Limit
600 K
Epoxy Limit
420 K T
Figure A.1: Temperature levels.
and boiling point of materials are given at normal pressure of 101.325 kPa
(both from the NIST publication [Wrig 03, p. 22]).
Furthermore, the following three fixed temperatures are referred to: The
freezing point of water (at normal pressure) 273.15 K (or 0oC), the boiling
temperature of liquid helium (at normal pressure) 4.22 K, and absolute zero
0 K (or −273.15 oC). Figure A.1 shows all temperature levels on one graph.
In order to use the material properties below for the solid state, it is im- Melting
tempera-
ture
portant to know the melting point. Table A.1 shows the melting temperature
TMfor materials found in superconducting magnets.
It can be seen from the melting temperatures that two limits of operating Thermal
limits
a magnet are given. The melting point of copper reduces the maximum plau-
sible temperature to around 1000 K. The melting point of solder, consisting
of lead and tin, reduces the maximum operation temperature to below 600 K.
The transistion temperature of epoxy and epoxy-composites, i.e., where the
crystal structure changes, is in the range of 410 −430 K [Imba 03, p.3-21].
A.1.2 Mass Density
Some material properties, e.g., the specific heat are given as quantity per unit
mass. In simulation programs the accelerator magnet is usually described
by its geometry, and thus by the volume of the different parts. In order to
Table A.1: Melting temperature of the some materials [Lide 06]
Material Symbol Melting Temperature
TMin K
Alumnium Al 933
Copper Cu 1358
Iron Fe 1811
Lead Pb 601
Niobium Nb 2750
Tantalum Ta 3290
Tin Sn 505
Titanium Ti 1941
156 Material Properties
convert between the two definitions the mass density ρD,i.e. the mass per unit
volume, is needed ([ρD] = kgm−3). Table A.2 shows the mass density of theDefinition
most common materials at ambient and cryogenic temperature. The changeSimplification
of density with temperature, ∆ρD/ρD= (ρD(300 K) −ρD(0 K))/ρD(300 K), is
less than 2% for most materials and can thus be neglected. For Nb3Sn the
change is in the order of 6%.
A.1.3 Electrical Resistivity
The electrical resistivity ρEis given by the relation between the electrical field
Eand the electrical current density J. The electrical conductivity κEis the
inverse of ρE.
E=ρEJ,[ρE] = VmA−1= Ωm.(A.1)
The electrical conductivity of metals depends on the length of the meanMotivation
of electrical
resistivity free path of the conduction electrons. This path length is limited by collisions
of electrons with either imperfections of the crystal or with lattice phonons
[Van 86, pp. 25]. The rate of collision is further increased by an applied
magnetic field by the Lorenz force acting on the electrons. This is referred to
as magneto-resistivity [Brec 73, pp. 430].
The resistivity of a metal at zero magnetic field consist of two additive partsMatthiessen’s
rule (Matthiessen’s rule [Brec 73, p. 423 and p. 428]): The residual resistivity
Table A.2: Mass density ρDof the most common materials at different temperatures
Material Symbol ρD(0 K) ρD(300 K) ∆ρD
ρDRemark
in kgm−3in kgm−3-
Aluminum Al 2.74 ·1032.70 ·103−1.5% [Jens 80]
Copper Cu 9.08 ·1038.96 ·103−1.4% [Jens 80]
Iron Fe 7.95 ·1037.83 ·103−1.5% [Jens 80]
Lead (chemical) Pb ≈11.3·103[Lide 06, 12-211]
Niobium Nb 8.61 ·1038.58 ·103−0.3% [Jens 80]
Tin Sn 7.42 ·1037.28 ·103−1.9% [Jens 80]
Tantalum Ta 16.69 ·103−4% 1&2
Titanium (com.) Ti ≈4.5·103[Lide 06, 12-211]
Niobium-Titanium Nb-Ti 6·103−0.2% 1,2&3
Niobium-3-Tin Nb3Sn 8.4·103−6% 1&2
Stainless Steel SS 8.00 ·1037.89 ·103−1.4% [Floc 08]
Polyimide 1.42 ·103
Epoxidharz (pure) ≈1.2·103[Schn 73, p. 87]
Polyester (pure) ≈1.1·103[Schn 73, p. 87]
Remark 1: The change of density is calculated from the linear contraction factor ∆L/L
assuming an isotropic contraction of the volume. Therefore the density change is given by
∆ρD
ρD(300 K) = 1 −(1 + ∆L/L)3.
Remark 2: The density is taken at room temperature from [Baue 07].
Remark 3: The sources quoted in [Baue 07, p. 5-2] vary strongly, i.e. between 5600 kgm−3
and 6500 kgm−3. Therein it is suggested to use 6000 kgm−3.
A.1 Solids in Normal Conducting State 157
ρo(RRR)and the intrinsic resistivity ρi(T),
ρE(T, RRR) = ρo(RRR) + ρi(T).(A.2)
The residual resistivity represents the temperature independent part of the
resistivity resulting from the scattering of the electrons at chemical impurities
or physical imperfections of the lattice [Hust 75]. The residual resistivity
thus only depends on the purity of the metal. At ambient temperatures the
electron-scattering process is dominated by electron-phonon-scattering, and
results in an approximately linear temperature dependence. This is denoted
as intrinsic resistivity [Hust 75]. In the intermediate temperature range the
two regimes are joined smoothly [Van 86, p. 25].
The ratio between the resistivity at ambient temperature and the residual RRR
resistivity is called the residual resistivity ratio (RRR). Since the residual
resistivity depends only on the number of imperfections in the lattice of the
metal the RRR can be used as a measure of purity [Van 86] and freedom from
strain [Hust 75]. RRR increases with increasing purity. The exact definition
is given below.
The dependence of the electrical resistivity on an applied magnetic field Magneto-
resistivity
can be added by means of
Kohler’s Rule: The following expressions are introduced in order to explain
Kohler’s rule: The change of resistivity due to a magnetic field ∆ρE/ρE,
∆ρE
ρE
=ρE(B, T, RRR)−ρE(T, RRR)|B=0 T
ρE(T, RRR)|B=0 T
.(A.3)
Note that this quantity depends on B, T, and RRR. A normalized magnetic
field B·S(T, RRR)is defined by
S(T, RRR) := ρE(RRR)|T=273 K
ρE(T, RRR)B=0 T
.(A.4)
Due to the minimal influence of the purity at ambient temperature the quan-
tity ρE(RRR)|T=273 K can be considered as constant.
According to Kohler’s rule [Brec 73, pp. 430] the change of resistivity
depends only on the normalized field B·Sand the angle ϕbetween the Anisotropy
applied field and the current/wire. The dependence defines a unique function
ffor every material.
∆ρE
ρE
=f(B·S(T, RRR), ϕ).(A.5)
The double logarithmic representation of the function fis known as the
Kohler plot. For most common metals the graph is a straight line [Seeb 98,
pp. 1070] or can be expressed by a polynomial in terms of log(B·S):
log ∆ρE
ρE=X
n
an(log(B·S))n.(A.6)
158 Material Properties
The electrical resistivity depending on temperature, magnetic field, and purity
can be obtained by solving (A.6) for ρE(B, T, RRR).
Residual Resistivity Ratio: The electrical resistivity ρEof normal conduc-
tors changes with temperature. For practical reasons, this can be expressed
by the residual resistivity ratio (RRR), i.e., the ratio of the resistivity at a
common and easily accessible temperature, i.e. ambient temperature, and
cryogenic temperature. Unfortunately the precise definition of the two tem-
perature levels varies among most common sources.
Table A.3 shows the definition of lower and upper temperatures according
to common sources. The error eRRR denotes by how much the RRR value has
to be modified compared to the NIST definition: eRRR =RRRother/RRRNIST.
Note that the error itself grows with increasing RRR. The quoted values are
for a RRR of 200.
Two definitions shall be highlighted: In [Voll 02] the two temperatures are
defined by phase-transition temperatures of water and helium. The lower
temperature by the boiling point of liquid helium at normal pressure and the
upper temperature by the freezing point of water. This way the tempera-
ture measurement is less difficult. The definition in [Char 06] with a lower
temperature of 10 K is a direct consequence of measuring the RRR of Nb-Ti
strands: Passing the critical temperature Tcthe Nb-Ti filaments become su-
perconducting and hence it is impossible to measure any resistivity. Therefore
a temperature above Tchas to be chosen (in line with [Supe 07b, p. 6]). In
case of copper this can result in a small error for very high RRR.
In the following the definition of NIST4shall be used [Prop 92]:Applied
definition
of RRR RRR =ρE(273 K)
ρE(4 K) B=0 T
.(A.7)
For materials or material combinations which exhibit superconductivity, the
lower temperature is chosen right above the transition temperature [Supe 07b,
4The National Institute of Standards and Technology (NIST), Agency of the United States
Department of Commerce.
Table A.3: Temperatures defining the RRR value according to different sources.
Source lower temperature upper temperature eRRR
in Kin K
NIST [Hust 75], Floch [Floc 03] 4 273 –
Verweij [Verw 05] 4 290 ≈7%
Van Sciver [Van 86] 4.2 273 <0.1%
Brechna [Brec 73] 4.2 293 ≈8.5%
Rossi [Ross 06] 4.2 295 ≈9%
Vollinger [Voll 02] 4.22 273.15 <0.1%
Charifoulline [Char 06] 10 293 >11%
A.1 Solids in Normal Conducting State 159
p. 6]. If the RRR is derived from the ratio of the bulk resistivity of a sample,
then the thermal contraction of the sample at low temperature has to be
taken into account.
A.1.3.1 Copper
For the resistivity of copper two different functions are shown. The first func-
tion is based on data from NIST. The second function is more approximative,
but favorable in terms of computation speed.
NIST: The electrical resistivity of copper for zero magnetic field ρNIST
E,Cu,0 is
taken from [Prop 92, pp. 8-4].
ρNIST
E,Cu,0(T, RRR)=(ρo+ρi+ρio) Ωm (A.8)
with:
ρo(RRR) = 1.553 ·10−8
RRR ,
ρi=
P1T
T0P2
1 + P1P3T
T0P2−P4exp −P5
“T
T0”P6!+ρc,
ρio =P7
ρiρo
ρi+ρo
,
ρc= 0.
The constants are:
T0= 1 K, P1= 1.171 ·10−17, P2= 4.49, P3= 3.841 ·1010,
P4= 1.14, P5= 50, P6= 6.428, P7= 0.4531.
The field dependence is taken into account by means of Kohler’s rule and the
following polynomial p(x)[Prop 92, pp. 8-23]:
x=B
B0·S(T, RRR)(A.9)
p(x) = −2.662 + 0.3168 log10(x)+0.6229(log10(x))2(A.10)
−0.1839(log10(x))3+ 0.01827(log10(x))4
ρE,Cu(T, B, RRR) = ρE,Cu,0(T, RRR)1 + 10p(x),(A.11)
with B0= 1 T. It is important to notice that the polynomial p(x)shows a
minimum for xmin ≈ −0.23. For values of x=B·S(T, RRR)smaller than
xmin the resistivity would increase with decreasing magnetic induction. This
160 Material Properties
50
100
200
500
1000
1 5 10 50 100 500 1000
10!11
10!10
10!9
10!8
10!7
Tin K
ΡE!T,RRR"
in #m
1 10 100 1000 104
0.001
0.01
0.1
1
10
100
BΡ"!273 K, RRR"
Ρ"!T,RRR"
Ρ"!T, B, RRR"# Ρ"!T,RRR"
Ρ"!T,RRR"
Figure A.2: Electrical resistivity of copper after [Prop 92]. (left) Electrical resistivity
ρNIST
E,Cu,0 versus temperature for different values of RRR. (right) Kohler plot.
translates directly to a minimum magnetic induction BNIST
min for which the
approach fails:
BNIST
min = 10xmin ρE(T, RRR)|B=0 T
ρE(RRR)T=273 K
(A.12)
For ambient temperature the minimum field BNIST
min is in the order of 1 T! At
cryogenic temperatures it is much smaller depending on RRR.
Figure A.2 (left) shows the electrical resistivity ρNIST
E,Cu,0 versus temperature
for different values of RRR. Figure A.2 (right) shows the polynomial p(x)in
the Kohler plot.
The NIST function is computationally very demanding. The accuracy to
measurements is estimated by NIST to be 15%.
Alternative function: The second approach is based on the zero field elec-
trical resistivity found in [McAs 88]:
ρE,Cu,0(T, RRR) = ρ0 1.545
RRR +1
P1
T5+P2
T3+P3
T!,(A.13)
with ρ0= 1 ·10−8Ωm and
P1= 74.697 K, P2= 98.550 K, P3= 162.74 K.
In Fig. A.3 (left) the electrical resistivity is plotted versus temperature for
different values of RRR and zero field.
The field dependence is taken into account by adding a linear term5de-
pending on RRR and magnetic induction B[Verw 05]:
ρE,Cu(T, RRR, B) = ρE,Cu,0(T, RRR) + a(b+RRR)B(A.14)
5see also [Krai 97] and [Floc 03]
A.1 Solids in Normal Conducting State 161
50
100
200
500
1000
1 5 10 50 100 500 1000
10!11
10!10
10!9
10!8
10!7
Tin K
ΡE!T,RRR"
in #m
1 10 100 1000 104
0.001
0.01
0.1
1
10
100
BΡ"!273 K, RRR"
Ρ"!T,RRR"
Ρ"!T, B, RRR"# Ρ"!T,RRR"
Ρ"!T,RRR"
Figure A.3: Electrical resistivity of copper after [McAs 88] and [Verw 05]. (left) Electrical
resistivity ρE,Cu,0 versus temperature for different values of RRR. (right) Kohler plot.
with a= 2 ·10−12 ΩmT−1and b= 74.
In Fig. A.3 (right) the Kohler plot for the function ρE,Cu(T, RRR, B)is
shown. This approach does not obey Kohler’s rule, but nevertheless, the lines
are within a relatively small band. The band is fully covered by the range
of uncertainty published in [Prop 92, p. 8-27]. The deviation of the zero
field resistivity to the NIST function is of less than 10% for a RRR of 200.
The advantage is the faster computation, since both parts of the function
are relatively simple to evaluate. The anisotropy of the magneto-resistivity is
neglected.
A.1.3.2 Niobium-Titanium
The electrical resistivity of Niobium-Titanium (Nb-Ti) in the normal con-
ducting state, ρE,Nb-Ti(T), is given by [Baue 07, p. 5-8]. Figure A.4 (left)
shows the resistivity versus temperature.
ρE,Nb-Ti(T) = (0.0558T+ 55.668) ·10−8Ωm.(A.15)
A.1.3.3 Niobium-3-Tin
For the electrical resistivity of Niobium-3-Tin ρE,Nb3Sn no analytic function is
available. In Tab. A.4 measurement data from [Baue 07, p. 4-11] is given for
interpolation. Figure A.4 (right) shows the resistivity versus temperature.
A.1.3.4 Insulators (Kapton, PVA, Epoxy)
The electrical resistivity of all insulation materials, e.g. Epoxy and polyimide
(Kapton), is considered to be infinitely high. For PVA a value of approx-
imately 2·1013 Ωm has been found [Lin 95]. This is nearly 20 orders of
magnitude greater than the resistivity of copper.
162 Material Properties
1 5 10 50 100 500 1000
2. !10"7
4. !10"7
6. !10"7
8. !10"7
1. !10"6
Tin K
ΡE!T"
in $m
1 5 10 50 100 500 1000
2. !10"7
4. !10"7
6. !10"7
8. !10"7
1. !10"6
Tin K
ΡE!T"
in $m
Figure A.4: (left) Electrical resistivity of Nb-Ti in the normal conducting state, ρE,Nb-Ti
after [Baue 07, p. 5-8]. (right) Electrical resistivity of Nb3Sn in normal conducting state,
ρE,Nb3Sn after [Baue 07, p. 4-11].
Nevertheless, the electrical field has to remain under a maximum field level,
the breakdown field strength (Sec. A.1.7).
A.1.3.5 Other Materials
The electrical resistivity of other common materials is given in Tab. A.5. For
comparison the resistivity of copper is given in the first line.
A.1.4 Thermal Conductivity
The thermal conductivity κTis given by the relation of the heat flux density
JTand the gradient of the temperature distribution T.
JT=κT∇T, [κT] = WmK−1.(A.16)
In solids heat is transported in two ways, by electrons and by lattice vibra-
tions. The total thermal conduction κTis the sum of electronic κeand lattice
Table A.4: Electrical resistivity of Niobium-3-Tin versus temperature from [Baue 07, p.
4-11]
Tin KρE,Nb3Sn in
WK−1m−1
22.33 ·10−7
72.36 ·10−7
12 2.39 ·10−7
22 2.46 ·10−7
32 2.52 ·10−7
42 2.59 ·10−7
Tin KρE,Nb3Sn in
WK−1m−1
52 2.67 ·10−7
102 3.07 ·10−7
152 3.47 ·10−7
202 3.81 ·10−7
252 4.06 ·10−7
297 4.10 ·10−7
A.1 Solids in Normal Conducting State 163
conduction κl:
κT=κe+κl(A.17)
In pure metals κeis much greater than κland thermal conductivity and
electrical resistivity are intimately related. According to the Wiedemann- Wiedemann-
Franz
law
Franz law, the thermal conductivity of metals can be calculated from the
electrical resistivity by:
κT=L0T
ρE
(A.18)
where L0is the Lorenz6number (L0= 2.44 ·10−8WΩK−2) and Tis the
absolute temperature. The Lorenz number is constant when the conduction Lorenz
number
electrons are scattered elastically. This is only true for very small and very
high temperatures [Brec 73, p. 434]. The Lorenz number varies most for high
purity (RRR) and is in addition also field dependent [Seeb 98, pp. 1086].
The thermal conductivity of metals thus depends on the same quantities as
the electrical resistivity, i.e., temperature T,RRR, and magnetic induction
B.
For insulators with a very high electrical resistivity, the thermal conduc- Thermal
conductiv-
ity for
insulators
tivity is dominated by lattice vibrations (phonon transport). Since the trans-
port of phonons is less efficient compared to the electron transport in metals,
the thermal conductivity of insulators is several orders of magnitude smaller
[Van 86, p. 29].
A.1.4.1 Copper
For the thermal conductivity of copper at zero magnetic induction, κT,Cu,0,
an analytic formula is given in [Prop 92, p. 7-16]:
κT,Cu,0(T, RRR) = 1
Wo+Wi+Wio
Wm−1K−1(A.19)
6Note that Ludvig Lorenz is spelled without a “t”.
Table A.5: Electrical resistivity ρEof some exemplary materials at 273.15 K.
Element/Material Symbol ρE(273.15 K) RRR Comments
in Ωm
Copper Cu 1.553 ·10−8up to 30000 [Prop 92],
[Seeb 98, p. 1076]
Aluminum Al 2.50 ·10−825-1000 [Jens 80, sec: X-B-2]
Niobium Nb 16.1·10−8≈30 [Jens 80, sec: X-M]
Lead Pb 19.2·10−81000 [Jens 80, sec: X-I-1]
Tin Sn 11.15 ·10−8100 [Jens 80, sec: X-Q-1]
Tantalum Ta 12.4·10−810 [Jens 80, sec: X-P-1]
Titanium Ti 39.4·10−8[Seeb 98, p. 1076]
Iron Fe 8.66 ·10−8100 to 200 [Jens 80, sec: X-H-1]
Stainless steel SS 304 1.02 ·10−62 [Jens 80, sec: X-X]
SS 304L 0.704 ·10−61.42 [Verw 07b]
164 Material Properties
with:
Wo=β(RRR)
(T/T0),
Wi=
P1T
T0P2
1 + P1P3T
T0P2+P4exp −P5
“T
T0”P6!+Wc,
Wio =P7
WiWo
Wi+Wo
,
Wc= 0.
Here, βis given by
β(RRR) = ρo(RRR)
L0
=1.553 ·10−8Ωm/RRR
2.443 ·10−8WΩK−2=0.634 mK2W−1
RRR ,
and ρo(RRR)denotes the above introduced residual resistivity of copper, see
Eq. (A.8). The constants are:
T0= 1 K, P1= 1.754 ·10−8, P2= 2.763,
P3= 1102, P4=−0.165, P5= 70,
P6= 1.756, P7= 0.838/β0.1661
r, βr=β/0.0003.
The thermal conductivity of copper versus temperature is shown in Fig. A.5
(left).
For the calculation of magnets, the considerable change of conductivity
[Prop 92, 7-20] has to be taken into account and another function has to be
used.
Using the Wiedemann-Franz law, it is possible to calculate the thermal
conductivity based on the electrical resistivity which itself is field dependent.
As mentioned above, the Lorenz number is not constant in the intermediate
temperature range, it depends on RRR and magnetic field. Figure A.5 (right)
shows the change of the Lorenz number with RRR as computed from the
thermal conductivity after [Prop 92, p. 7-16] and the electrical resistivity
after [Prop 92, p. 8-4]. A variation of a factor of 2can be easily noticed
(see also [Brec 73, p. 434] and [Van 86, p. 29])! Furthermore in [Aren 82] it
was shown that the Lorenz number for copper depends anisotropically on the
magnetic induction. The Lorenz number describing the thermal conductivity
transversal to the magnetic field, changes by 0.112·10−8WΩK−2per tesla, i.e.
approximately 5%. The longitudinal Lorenz number remains constant. There
is no analytic function available describing the Lorenz number as function of
temperature, RRR and magnetic induction.
For this work the thermal conductivity is calculated from the electrical
resistivity using a constant value for the Lorenz number. The Lorenz number
varies by a factor of ≈2over temperature (see Fig. A.5 (right)) and increases
by a factor of ≈2if the magnetic flux density is increased to 10 T [Aren 82].
A.1 Solids in Normal Conducting State 165
For the same field change the electrical resistivity changes by more than a
factor of 10 (see Fig. 2.7 for RRR = 200).
A.1.4.2 Niobium-Titanium
The thermal conductivity can be expressed as [Baue 07, p. 5-10]:
κT,Nb-Ti(T) = κ0aT6+bT5+cT 4+dT3+eT2+fT +g,(A.20)
with κ0= 1 WK−1m−1and
a=−5·10−14 K−6, b = 1.5·10−11 K−5, c = 6 ·10−9K−4,
d=−3·10−6K−3, e = 3 ·10−4K−2, f = 4.56 ·10−2K−1,
g= 6.6·10−2.
The thermal conductivity is shown in Fig. A.6 (left).
A.1.4.3 Niobium-3-Tin
For Niobium-3-Tin only measured data for interpolation is available [Baue 07,
p. 4-13] as shown in Tab. A.6 and Fig. A.6 (right).
A.1.4.4 Polyimide (Kapton)
The thermal conductivity of Polyimide (Kapton) is given by NIST [Nati 08]
to an exponential function of polynomial argument:
κNIST
T,Kapton(T) = κ010p(x),(A.21)
x=T
T0
,
p(x) = a+b(log10(x)) + c(log10(x))2+d(log10(x))3
+e(log10(x))4+f(log10(x))5+g(log10(x))6
+h(log10(x))7,
with κ0= 1 WK−1m−1,T0= 1 K and the following parameters:
a= 5.73101, b =−39.5199, c = 79.9313, d =−83.8572,
e= 50.9157, f =−17.9835, g = 3.42413, h =−0.27133.
The function is shown in Fig. A.7.
A.1.4.5 Other Materials
The thermal conductivity7of other common materials is given in Tab. A.7.
Since the thermal conductivity of many metals is not a monotonous func-
tion, the maximum value between 2and 500 K is given. For comparison, the
7In US publications often the unit BTU / oF FT HR is used. This can be converted to
SI units by
1BTU
oF FT HR = 1.731WK−1m−1(A.22)
166 Material Properties
50
100
200
500
1000
1 5 10 50 100 500 1000
1!104
100
200
500
1000
2000
5000
Tin K
Κ!T,RRR"
in Wm#1K#1
50
100
200
500
1000
1 5 10 50 100 5001000
1. !10"8
1.2 !10"8
1.4 !10"8
1.6 !10"8
1.8 !10"8
2. !10"8
2.2 !10"8
2.4 !10"8
Tin K
L0!T,RRR"
in W#K"1
Figure A.5: (left) Thermal conductivity of copper for zero magnetic induction. (right)
Variation of the Lorenz number of copper with RRR as calculated from the heat conduc-
tivity and electrical resistivity functions published in [Prop 92].
1 5 10 50 100 500 1000
1.00
0.50
5.00
0.10
10.00
0.05
Tin K
ΚT!T"
in W m"1K"1
1 5 10 50 100 500 1000
1.00
0.50
5.00
0.10
10.00
0.05
Tin K
ΚT!T"
in W m"1K"1
Figure A.6: Thermal conductivity versus temperature. (left) Niobium-Titanium
[Baue 07, p. 5-10]. (right) Niobium-3-Tin [Baue 07, p. 4-13].
Table A.6: Thermal conductivity of Niobium-3-Tin versus temperature from [Baue 07, p.
4-13]
Tin KκT,Nb3Sn in
WK−1m−1
2 0.01
7 0.184
12 0.98
17 2.02
22 2.53
32 2.78
42 2.74
Tin KκT,Nb3Sn in
WK−1m−1
52 2.64
102 2.13
152 1.79
202 1.63
252 1.53
297 1.41
A.1 Solids in Normal Conducting State 167
1 5 10 50 100 500 1000
0.50
0.20
0.10
0.05
0.02
0.01
Tin K
ΚT!T"
in W m"1K"1
Figure A.7: Thermal conductivity of Polyimide (Kapton), κT,Kapton. Function from
[Nati 08].
maximum thermal conductivity of copper (RRR = 200) is given in the first
line.
A.1.5 Heat Capacity
The heat capacity CTis defined as the amount of energy needed to raise
the temperature of a material by one degree. In terms of thermodynamic
quantities the heat capacity can be written as the derivative of the internal
energy UTunder either constant pressure, Cpor constant volume, CV:
CV=∂UT
∂T V=const.
, Cp=∂UT
∂T p=const.
,(A.23)
[CV] = hCpi= Jmol−1K−1.
Table A.7: Maximum thermal conductivity of most common materials
Material Symbol max(κT)Comment
in WK−1m−1
Copper Cu 4000.0see above
Aluminum Al 389.5Al 1100, [Schw 70, p. 43]
Niobium Nb 86.4[Baue 07, p. 6-14]
Tantalum Ta 69.0[Baue 07, p. 7-12]
Titanium Ti 24.0[Schw 70, p. 611]
Stainless steel SS 14.7316 LN, [Baue 07, p. 1-14]
Epoxy (pure) 0.11 −0.14 at 25 oC[Schn 73]
Polyester (pure) PE 0.2at 25 oC[Schn 73]
Polyvinyl acetate PVA 0.05 [Karp 08b]
168 Material Properties
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
0.0
0.2
0.4
0.6
0.8
1.0
T!ΘD
C!C"
Figure A.8: Normalized specific heat of solids. The value of C∞is given to 3R(see Eq.
A.24). The Debye temperature θDis given in Tab. A.8. The figure has been recalculated
after [Deby 12].
The internal energy consists of the energy associated with the motion of
molecules, the vibration of atoms in molecules and crystals, the electric energy
of atoms in molecules and lattices, the energy of all chemical bonds and the
energy of the free conduction electrons in metals. In solids the heat capacity
is dominated by the vibration of the lattice described in the Debye theory.
In metals below a temperature of 1 K the contribution of the free conduction
electrons has to be taken into account [Van 86, p. 21], but will not be
considered in this work.
Debye derives the heat capacity of solids as a function of temperature fromDebye
theory the spectra and associated energy levels of the lattice vibrations [Deby 12].
The heat capacity of all common solids essentially follows the same function
of normalized temperature T/θD. At low temperatures the function shows a
cubic dependence on the absolute temperature. Figure A.8 shows the nor-
malized function. The Debye temperature θDis a property of the material.
In Tab. A.8 the values of some materials are reported. For very high tem-Dulong-
Petit-limit peratures, i.e. T > 2θDthe heat capacity approaches the Dulong-Petit-limit
Table A.8: Debye temperature of some of the used materials
Material Symbol Debye Temperature θDin K
Alumnium Al 428
Copper Cu 343
Iron Fe 470
Niobium Nb 450
Tin Sn 200
Titanium Ti 420
A.1 Solids in Normal Conducting State 169
C∞[Peti 19].
C∞= 3NAkB= 3Rgas = 24.93 Jmol−1K−1.(A.24)
Here, NAdenotes the Avogadro number, kBthe Boltzmann constant and
Rgas = 8.31 Jmol−1K−1the gas constant.
For soilds and fluids it is difficult to measure the heat capacity for constant CV→Cp
volume (due to the huge forces of the thermal expansion!). Therefore the heat
capacity for constant volume and constant pressure have to be converted. For
metals at ambient temperature both capacities differ minimally [Van 86, pp.
18].
The heat capacity per unit mass is called specific heat and is denoted with Specific
heat
a lower case, dashed c0.[c0
V] = hc0
pi= Jkg−1K−1. For the computation of
accelerator magnets the so-called volumetric specific heat,i.e. the specific
heat per volume, is more practical. It is denoted with a lower case c. Where
not available from literature, it is calculated from the specific heat by means
of the mass density. For all solids the mass density is considered constant
(see Sec. A.1.2).
cV=ρDc0
V, cp=ρDc0
p,(A.25)
[cV] = hcpi= Jm−3K−1.
A.1.5.1 Copper
The volumetric specific heat of copper cT,Cu depends only on temperature
and is given by a fit to measurement [Floc 03]:
cT,Cu(T) = c0(aT4+bT3+cT 2+dT +e)(A.26)
with c0= 1 JK−1m−3. The fit parameters can be found in Tab. A.9. Figure
A.9 (left) shows the volumetric specific versus temperature.
A.1.5.2 Niobium-Titanium
The volumetric specific heat of Niobium-Titanium in the normal conducting
state is given by a fit to measurements [Verw 05]:
cT,Nb-Ti(T) = c0(aT4+bT3+cT 2+dT +e).(A.27)
with c0= 1 JK−1m−3. The fit parameters can be found in Tab. A.10. The
function is displayed in Fig. A.10 (left).
A.1.5.3 Niobium-3-Tin
For Niobium-3-Tin only measured data for interpolation is available, as shown
in Tab. A.11 and Fig. A.10 (right).
170 Material Properties
Table A.9: Fit parameters for the volumetric specific heat of copper [Floc 03]
Temperature ain K−4bin K−3cin K−2din K−1e
Range in K
T < 10 −3.08·10−27.23·100−2.13·1001.02·102−2.56·100
10 ≤T < 40 −3.05·10−12.99·101−4.56·1023.47·103−8.25·103
40 ≤T < 125 4.19·10−2−1.4·1011.51·103−3.16·1041.78·105
125 ≤T < 300 −8.48·10−48.42·10−1−3.26·1026.06·104−1.29·106
300 ≤T < 500 −4.8·10−59.17·10−2−6.41·1012.04·1041.03·106
500 ≤T < 1000 –1.2·10−4−0.215·1001.00·1033.18·106
1 5 10 50 100 500 1000
100
1000
104
105
106
Tin K
c!T"
in Jm!3K!1
1 5 10 50 100 500 1000
100
1000
104
105
106
Tin K
c!T"
in Jm!3K!1
Figure A.9: (left) Volumetric specific heat of copper [Floc 03]. (right) Specific heat of
Kapton after [Nati 08] using the mass density as given in Sec. A.1.2.
1 5 10 50 100 500 1000
100
1000
104
105
106
Tin K
c!T"
in Jm!3K!1
1 5 10 50 100 500 1000
100
1000
104
105
106
Tin K
c!T"
in Jm!3K!1
Figure A.10: Volumetric specific heat of (left) Niobium-Titanium [Verw 05] and (right)
Niobium-3-Tin [Baue 07, p. 4-13].
A.1 Solids in Normal Conducting State 171
A.1.5.4 Polyimide (Kapton)
The volumetric specific heat of Polyimide (Kapton) is given by NIST [Nati 08]
to an exponential function of polynomial argument:
cNIST
T,Kapton(T) = c010p(x)(A.28)
x=T
T0
,
p(x) = a+b(log10(x)) + c(log10(x))2+d(log10(x))3
+e(log10(x))4+f(log10(x))5+g(log10(x))6
+h(log10(x))7
with c0= 1.42 ·103Jm−3K−1,T0= 1 K and the following parameters:
a=−1.3684, b = 0.65892, c = 2.8719, d = 0.42651,
e=−3.0088, f = 1.9558, g =−0.51998, h = 0.051574.
The function is shown in Fig. A.9 (right). The published data on the specific
heat shows variations of around 100% [Floc 08].
A.1.5.5 Other Materials
The volumetric specific heat of other common materials is given in Tab. A.12
for comparison [Jens 80].
A.1.6 Permeability
Materials can be classified into one of three different groups by the force Classification
they experience in a gradient magnetic field. Diamagnetic material is weakly
repelled, paramagnetic material weakly attracted and ferromagnetic material
is strongly attracted [Purc 89, pp. 258].
The magnetic force acting on a material generally results from the spin and General
Mechanism
orbit magnetic moments of the electrons. The three classes differ in the way
the magnetic momenta compensate or couple.
In diamagnetic materials the spin and orbit magnetic moments cancel in Diamagnetic
the absence of an external magnetic field. An applied field causes the spin
moments to slightly exceed the orbit moments, resulting in a small magnetic
moment opposing the external field [Russ 07, p. 155].
In paramagnetic materials the individual spin magnetic momenta do not Paramagnetic
interact and take a random distribution. An external magnetic field causes
them to align with the field direction, resulting in a small magnetic moment
nearly independent of field strength and fully reversible [Russ 07, p. 155].
In ferromagnetic materials the spins are coupled and build clusters of iden- Ferromagnetic
tical orientation. The cluster size is given by an energetic minimum depending
on applied magnetic field, temperature and impurities. Increasing the mag-
netic field causes clusters with parallel orientation to grow. A further increase,
results in rotation of the orientation of whole clusters parallel to the field.
172 Material Properties
Table A.10: Fit parameters for the volumetric specific heat of Nb-Ti
Temperature ain K−4bin K−3cin K−2din K−1e
Range in K
Tc< T < 20 –1.624·101–9.28·102–
20 ≤T <50 −2.177·10−11.19838·1015.5371·102−7.846·1034.138·104
50≤T <175 −4.82·10−32.976·100−7.163·1028.3022·104−1.53·106
175≤T <500 −6.29·10−59.296·10−2−5.166·1011.3706·1041.24·106
If the change of specific heat in superconducting state is not taken into account the critical
temperature Tcis set to 0 K.
Table A.11: Volumetric specific heat of Niobium-3-Tin versus temperature from [Baue 07,
p. 4-3]
Tin KcT,Nb3Sn in
JK−1m−3
24.66 ·103
72.01 ·104
12 4.66 ·104
17 9.36 ·104
22 1.67 ·105
42 6.09 ·105
Tin KcT,Nb3Sn in
JK−1m−3
72 1.14 ·106
102 1.62 ·106
162 1.90 ·106
212 2.02 ·106
262 2.07 ·106
297 2.10 ·106
Table A.12: Volumetric specific heat of common materials for comparison [Jens 80]. All
values are given in JK−1m−3.
Material Symbol cp(2 K) cp(10 K) cp(100 K) cp(300 K) Comment
Copper Cu 0.25 ·1037.7·1032.28 ·1063.46 ·106Sec. A.1.5.1
Aluminum Al 0.3·1033.8·1031.3·1062.44 ·106
Niobium Nb 1.54 ·10319 ·1031.73 ·1062.3·106
Lead Pb 1.47 ·106Remark 1
Tin Sn 0.34 ·10358 ·1031.38 ·1061.62 ·106
Tantalum Ta 1.13 ·10319 ·1031.9·1062.33 ·106
Titanium Ti 2.35 ·106Remark 1
Iron Fe 3.52 ·106Remark 1
Stainless SS
steel
Polyvinyl PVA
acetate
Epoxy ≈2·106[Schn 73]
Remark 1: From [Lide 06, p. 4-127] in combination with Sec. A.2.
A.1 Solids in Normal Conducting State 173
!
!""#$%&'()*+#,&*-.,/"#,0#1(!"#$%&'()"&*'%+,-($./,.'(!0$*&12
!,2
"32
#4"$
#4%12
5/+*0/$2
6,,#7#,1&8-#2
9/:$%*,;2
<&1"-*3#=#$+2
5#7#,1&8-#2
9/:$%*,;2
<&1"-*3#=#$+2
"$
>&,?&$2
3:,7#2
Figure A.11: General BH-curve [Russ 07]. Increasing the magnetic field from zero, the
virgin curve passes the reversible, rapid irreversible and finally the saturation stage. Lower-
ing the field to zero, the magnetic induction follows a new curve to the remanence magneti-
zation field. Further decreasing the magnetic field (to negative values) causes the induction
to drop to zero at the coercitive field. This behavior is denoted hysteretic and associated
with losses. Field sweeps of smaller amplitude result in minor loops.
The magnetization Mis defined as the magnetic moment density, Magnetization
M=N¯
pm,(A.29)
where ¯
pmdenotes the average magnetic moment and Nthe number of dipoles
per volume [Henk 01, p. 152]. This way the magnetic induction is most
generally given by,
B=µ0(H+M).(A.30)
The magnetization of a ferromagnetic material Mrises with magnetic field
Hin three stages: initial reversible magnetization, rapid irreversible magne-
tization and slow approach of the saturation, related to the reversible shifts of
domain walls, irreversible rotation and shift of domain walls, and reversible
rotation of domains [Wlod 06]. Figure A.11 shows a schematic plot of the BH-curve
magnetic induction versus magnetic field of a hysteretic, ferromagnetic sam-
ple.
The magnetization can be anisotropic resulting from the microscopical crys- Anisotropy
tal orientation or macroscopical lamination of the material.
In case of non-hysteretic materials, the magnetic permeability µis defined Non-
hysteretic
by the relation between the magnetic induction Band the magnetic field H.
B=µH,[µ] = 1 VsA−1m−1.(A.31)
The relative permeability µris given by the ratio of the permeability and the Relative
permeabil-
ity
permeability of empty space, i.e. µ0= 4π·10−7VsA−1m−1:
µr=µ
µ0
.(A.32)
174 Material Properties
1.000.50 5.000.10 10.000.05
1
10
100
1000
104
Bin T
Μr
1!104
10 50 100 500 1000 5000
1.00
0.50
2.00
0.20
0.10
0.05
0.02
Hin A m"1
B
in T
Figure A.12: Relative permeability µras function of magnetic induction (left). Change of
magnetic induction with magnetic field (right). The operating conditions are: No pressure
at (full) 4.2 K, (dashed) 77 K, (dotted) ambient temperature, and at a pressure of 20 MPa
and 4 K (dash-dotted).
A.1.6.1 Magnetic Iron
The iron used for yoke laminations in accelerator magnets needs to meet the
following three requirements: Low coercitive force (for high premeability and
low hysteresis losses), high resistivity at low temperature (for low induced
eddy currents), and high saturation magnetization (reduction of the influence
of the iron-nonlinearity) [Shch 04].
For iron, the relative permeability increases with the purity of the material.Modification
Values of 7000 to 40,000 can be found while providing a saturation field
strength of Bs= 2.14 T. Nevertheless impurities, e.g. silicium (Si), are
introduced in order to increase the electrical resistivity (eddy currents) and
narrow the hysteresis loop [Schn 73, p. 92].
The iron used for the LHC lamination also consists of 0.02% nickel (Ni),LHC
0.02% sulfur (S), 0.02% tin (Sn), and 0.01 % phosphorous (P). In [Shch 04]
a large number of different magnetic steels have been measured and can be
consulted for comparison. Furthermore, the 4-digit steel grade nomenclatura
is explained briefly.
Measurements: Figure A.12 shows the BH-curve as well as the relative
permeability for different operating conditions and different sources of mea-
surement versus magnetic field respectively magnetic induction. It can be
seen that the maximum relative permeability varies with temperature and
applied pressure, but the saturation field, i.e. where the relative permeability
starts to drop significantly, is nearly identical.
A.1 Solids in Normal Conducting State 175
Analytical Formula: As shown in [Wlod 06] the magnetization of iron can
be approximated analytically by means of the following function:
LH
a= coth H
a−a
H(A.33)
M(H) = MaLH
a+Mbtanh |H|
bLH
b(A.34)
The parameters Maand Mbdenote the reversible and irreversible components
of the saturation magnetixation. The parameters aand bdetermine the rate
of their approach to saturation. The sum of Maand Mbequals the saturation
magnetization Ms
Figure A.13 shows the fit for a steel made by Cockerill and used at LHC
for 4 K and zero applied stress. The fit parameters are:
Ma= 3.711 ·105Am−1, a = 9.338 ·104Am−1,
Mb= 1.370 ·106Am−1, b = 8.943 ·101Am−1,
and µ0Ms=µ0(Ma+Mb)=2.188 T.
A.1.6.2 Other Materials
Table A.13 shows the magnetic properties of other materials used in acceler-
ator magnets.
A.1.7 Dielectric Strength
Insulation material withstands electrical fields up to the so-called breakdown
field strength or dielectric strength Ebt ([Ebt] = Vm−1).
According to [Schn 73, pp. 82] it is distinguished between a fast, purely Electrical
and
thermal
breakdown
electrical breakdown, and a thermal breakdown. The electrical breakdown
is caused by an electron avalanche. The thermal breakdown results from
a current flowing through the insulation. The resulting temperature and
conductivity increase causes a further increase of the insulation current. The
thermal breakdown is more important for the design of accelerator magnets.
The thermal breakdown field decreases with temperature (exponentially)
and insulation thickness d(∝1/√d). The values given in Tab. A.14 are
based on the data published in [Schn 73, p. 84]. They shall serve only as a
first orientation.
The breakdown strength of the insulation may vary if the magnet is cooled
by liquid helium. Therefore the dielectric strength of helium, especially in
the gaseous phase, has to be considered (see Sec. A.3.6).
A.1.8 Electrical Permittivity
The electrical permittivity Eis given by the relation between the dielectric
displacement Dand the electrical field E.
D=EE,[E] = AsV−1m−1.(A.35)
176 Material Properties
1.000.50 5.000.10 10.000.05
10
50
100
500
1000
Bin T
Μr
10 100 1000 104105
1.000
0.500
0.100
0.050
0.010
0.005
0.001
Hin A m!1
B
in T
Figure A.13: Analytical µr- (left) and BH-curve (right) for the Peiro Steel at 4K with no
stress.
Table A.13: Relative permeability of commonly used materials at room temperature and
zero magnetic field
Element/Material Symbol µrComments
Iron Fe 5000 [Henk 01, p. 162], see 1
Stainless steel - 1.005 [Peir 04], see 2
Aluminum Al 1.000017 [Henk 01, p. 162]
Copper Cu 0.999991 [Henk 01, p. 162]
The relative permeability of Niobium, Titanium and Tantalum is in the same order of
magnitude as of Alumnium. For Lead and Tin it is similar to Copper [Lide 06].
Remark 1: This value is stated for general, "back of the envelope" calculations not
taking into account saturation. For more details see Sec. A.1.6.1.
Remark 2: The relative permeability of stainless steel decreases from 1.05 at nearly zero
field to 1.005 at less than 2 T.
Table A.14: Breakdown field strength Ebt of common insulation materials [Schn 73, p.
84]
Material Ebt in kVmm−1
d= 3 mm 1 mm 0.25 mm 0.1 mm
Polyimide (Kapton) 30 52 104 165
Epoxy 16-23 27-40 56-81 88-127
Polyester 15 26 53 83
PVC 30-40 52-70 104-140 165-220
Glass 80 at 77K [Seeb 98, p. 1055]
polyvinyl acetate (PVA) 5-20 at 105 oC[Lin 95]
The values in the first column (d= 3 mm) are taken form [Schn 73, p. 84]. The values
given for thinner insulation thickness are calculated by scaling by p3/d.
A.2 Solids in Superconducting State 177
The relative permittivity ris given by the ratio of the permittivity and the relative
permittiv-
ity
permittivity of empty space, i.e. 0= 8.854 ·10−12 AsV−1m−1:
r=E
0
.(A.36)
A.1.8.1 Insulators
The relative electrical permittivity of Kapton (polyimide) at room tempera-
ture is in the order of 3-4 [DuPo 96]. For polyvinyl acetate (PVA) a value of
3.5 is given [Lide 06, p. 13-13].
A.2 Solids in Superconducting State
Below a certain temperature some materials exhibit superconductivity, i.e. Supercon-
ductivity
the total loss of resistivity. The transition between normal and superconduct-
ing state is abrupt. The transition temperature is called critical temperature
Tc.
According to the critical temperature level, superconductors can be clas- LTS, HTS
sified into low-temperature superconductors (LTS) and high-temperature su-
perconductors (HTS). The distinction can be made by the necessary coolant,
i.e. liquid helium for LTS and liquid nitrogen for HTS [Voll 02, p. 11]. The
boiling temperatures of both gases can be found in Sec. A.3.1. In this text
only LTS are considered.
Low-temperature superconductors are further distinguished by their behav- Different
types
ior in an external magnetic field:
Type I superconductors, e.g. elements as niobium and mercury, prevent Type I
any magnetic flux from penetrating. The interior of the superconductor is
perfectly screened from any external field by surface currents. The screening
disappears when the field exceeds the critical field level Bcand the super-
conductor transits to normal conducting state. As shown by Meissner and
Ochsenfeld the screening does not depend on how the superconductor was
exposed to the field, i.e. whether it was moved into a field or cooled down to
superconducting state in a pre-existing field. Superconductors of type I can
be considered perfect diamagnetics with fully reversible magnetization. The
magnetization over applied field is shown in schematically in Fig. A.14 (left).
Since superconductors of type I expel any magnetic field, they are not capable
of carrying volumetric currents and thus cannot be used for superconducting
cables and magnets [Voll 02, pp. 14].
Up to the lower critical field Bc1type II superconductors exhibit the same Type II
behavior as type I superconductors. Exceeding the lower critical field, flux
starts to penetrate the superconductor in flux tubes. Flux tubes consist of a
normal conducting center enclosed by supercurrents screening the residual su-
perconductor from the penetrating flux. The flux inside the tube is quantized
to
Φ0=hp
2em≈2.067 ·10−15 Vs,(A.37)
178 Material Properties
!"#$
%$
!"#$
%$
%!"
!"#$
%$
%!#" %!$"
%&&'()*+,-./'0*.1203&'04'5,!"#$%&'()"&*'%+,-($./,.'(!0$*&12
Figure A.14: Superconductor magnetization of type I (left) and type II (center), both
fully reversible. Type II superconductor with pining centers, so-called hard superconductors
(right).
where hpdenotes the Planck constant and emis the elementary electrical
charge (see Tab. E.1). As the external field increases, the number of flux
tubes through the superconductor consequently rises. A repelling force acts
between the flux tubes, moving them across the sample until their position
reaches an energetic minimum. This state of partly penetrated magnetic field
is called mixed-state. It persists up to the upper critical field Bc2(Bc1
Bc2) where the superconductor transits to the normal conducting state. The
magnetization is fully reversible and field dependent (Fig. A.14 (center)).
A current density is applied through the superconductor perpendicular toFlux
motion the external field. This results in a force on the flux tubes, perpendicular to
both field and current density, and the flux tubes start to move out of the
sample. Any motion of flux tubes causes heating in the superconductor and
can cause it to quench. Furthermore, the applied current density requires
a gradient of magnetic induction over the superconductor and therefore a
gradient of flux tubes.
The free motion of the flux tubes is hampered by lattice imperfections andFlux
pinning grain boundaries inside the microstructure of the superconductor. By intro-
ducing pinning centers, a counter-force is introduced and flux tubes can be
pinned up to a certain field respectively current density level. If the external
field is reduced again, the number of flux tubes would need to be decreased
by moving flux tubes out of the sample. Since a large number is trapped in
pinning centers, instead flux tubes of inverse orientation start to enter the
sample. The magnetization now shows a hysteresis, see Fig. A.14 (right).
Such superconductors are also denoted as hard superconductors.Hard
supercon-
ductors In this section, the conditions to exhibit superconductivity for different ma-
terials are listed. For the technical relevant low-temperture superconductors
all material properties are given in the superconducting state.
A.2.1 Transition Temperature / Critical Field
Table A.15 shows the transition temperature Tc0at zero applied field and
the critical field Bc0at zero temperature for some superconducting elements
(type I) [Jens 80]. Furthermore, the upper critical field Bc20 at zero temper-
ature of the two commonly used type II superconductors is given.
A.2 Solids in Superconducting State 179
It is important to note, that not only the alloy Nb-Ti and compound Nb3Sn
exhibit superconductivity, but also the elements they consist of as well as
elements used for stabilization and improvement of the strands. Niobium in
particular shows a rather high critical temperature and field.
For further information the common elements aluminum, lead and mercury
are provided. Mercury was the first superconducting element discovered by
Onnes[Buck 93].
A.2.2 Critical Current
The transition of a superconductor from superconducting to normal conduct-
ing state is not a step function, but a continuous transition [Clar 77]. Fig-
ure A.15 (left) shows the voltage-current-characteristic of a superconductor
(VAC), i.e. the plot of the voltage over the superconductor versus applied Voltage-
current-
characteristic
(VAC)
current for fixed magnetic induction and bath temperature.
From a certain current level upwards, the voltage starts to rise significantly
and ohmic heating sets in. When the heating exceeds the heat removal by
the coolant, the temperature of the strand starts to rise drastically and the
sample quenches. This is called thermal run-away. The quench current level
is denoted by Iq. It strongly depends on the measurement setup, especially
on the cooling conditions, and can therefore not be used as universal charac-
terization value for a superconductor [Clar 77].
For the characterization of superconductors and the definition of the super- Criteria
conducting state, 5 different criteria were suggested in [Bruz 04]. The current
level, where
1. the longitudinal electrical field equals Ec,e.g. 10−6Vm−1,
Table A.15: Transition temperature and critical field of superconducting elements (type
I) [Lide 06, p. 12-57] and the two technically applied hard superconductors Nb-Ti [Voll 02,
p. 21] and Nb3Sn [Devr 04, p. 44]
Element Symbol Transition Critical magnetic
temperature induction at zero
at zero field, temperature, Bc0
Tc0 in Krespectively Bc20 in T
Aluminum Al 1.175 0.0104
Mercury Hg 4.154 0.0411
Niobium Nb 9.25 0.2060
Lead Pb 7.196 0.0803
Tin Sn 3.722 0.0305
Tantalum Ta 4.47 0.0829
Titanium Ti 0.40 0.0056
Niobium-titanium Nb-Ti 9.2 14.5
Niobium-3-tin (binary)1Nb3Sn 16 24
Niobium-3-tin (ternary)1Nb3Sn 18 28
1Binary Nb3Sn consists only of Nb and Sn, while ternary Nb3Sn contains one more sub-
tance, e.g. Ti or Ta for stabilization and performance improvement.
180 Material Properties
Ec!0.1 ΜV cm#1
Pc!1000 W m#3
Ρc!10#14 %mIc
Quench
Iq
0 100 200 300 400
0
0.00001
0.00002
0.00003
0.00004
Iin A
U' in
V!m
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
I!Ic
U!Uc
Figure A.15: (left) Voltage-current-curve for a strand of the LHC MB outer layer cable at
1.9 K and 9 T. The first three definitions for the critical current Icare shown. Furthermore,
a thermal run-away is illustrated by means of a vertical arrow. It can be seen that the
quench current Iqis greater than the critical current. (right) Variation of the voltage over
a superconductor depending on the n-value Nsc. The lines are for Nsc =1, 5, 10, 15,
20, 30, 40, and 50. Niobium-Titanium strands have a value of approximately 40, while
Niobium-3-Tin strands show values between 10 and 20.
2. the resistivity onset equals ρc,e.g. 10−11 Ωm,
3. power per strand volume equals Pc,e.g. 103Wm−3,
4. the current equals the quench current (take-off point) Iq, or
5. the current equals linear down extrapolation of the slope before take-off
point,
shall be denoted critical current Ic.
The first 3 were investigated earlier by Clark and Ekin in [Clar 77]. It
was demonstrated that even if all 3 criteria intersect at one point for a given
temperature and applied magnetic induction, due to the nature of the VAC
they differ at most other working points. Therefore, the resulting critical
current strongly depends on the applied definition! The 4th criterium was
already ruled out in the introduction to this section and the last criterium
has been ruled out because not all superconductors reach a linear transition
part before take-off [Bruz 04].
In the international standard of the CENELEC8the critical current is de-
fined as the maximum direct current that can be regarded as flowing without
resistance [Supe 07a, p. 13]. The standard9allows one to use either the
critical electrical field criterion or the resistivity criterium. In the case of
8European committee for Electrotechnical Standardization, Brussels, Belgium
9The standard applies for Nb-Ti strands with λ > 1,df>1µm,NSC >12,Ic|B=0.7Bc20 <
1000 A, where λdenotes the copper to superconductor area ratio, dfthe filament diam-
eter. For varying strands the standard has to be adapted [Supe 07a, p. 11].
A.2 Solids in Superconducting State 181
the critical field criterium the current is measured for an electrical field of
Ec1 = 1 ·10−5Vm−1and Ec2 = 1 ·10−4Vm−1. For the resistivity criterium
levels of ρc1= 1 ·10−14 Ωm and ρc2= 1 ·10−13 Ωm are applied, where the
resistivity refers to the resistivity of the total strand cross-section [Supe 07a,
pp. 25].
Throughout this work, the electrical field criterion is used for calculations. Applied
definition
The critical current Icis defined as the current level where the longitudinal
electrical field over the sample reaches 10−5Vm−1.
The applied levels for critical electrical field as well as resistivity vary among Variety in
Literature
different sources. Sources and values are given in Tab. A.16 for comparison.
A.2.3 The n-Index
Following Wilson10 [Walt 74, p. 30], the voltage-current-characteristic (VAC)
can be fitted by an exponential function
U
`=EcI
IcNSC
(A.38)
where U/` denotes the voltage per sample length `,Icand Ecthe above
defined values and NSC a fit parameter. Equation (A.38) is shown in Fig.
A.15 (right) for different values of NSC.According to the CENELEC standard,
the fit is performed close to the critical current or between the two electrical
field levels Ec1and Ec2[Supe 07a, p. 27].
The exponent NSC is usually denoted as the "n-index" or quality index. Quality
index
This is due to the fact that the n-index increases with the quality of pro-
duction, i.e. filament size distribution and filament distortion [Devr 04, p.
43]. The n-index also depends on applied magnetic induction [Warn 86] and
temperature [Devr 04, p. 43].
Within this work, for Niobium-Titanium an n-index of NSC,Nb-Ti = 40
and for Niobium-3-Tin of NSC,Nb3Sn = 20 are used [Verw 08b]. Note that the
critical current and n-index of a strand can be degraded by mechanical strain,
10Most authors refer to the informal report of Walter for the introduction of the n-index.
In the report, Walter quotes unpublished work of Wilson giving correct reference.
Therefore, we cite Walter’s report while rightfully referring to Wilson.
Table A.16: Different limits for the critical current Icfrom common literature
Source Ecin Vm−1ρcin Ωm−1
Bruzzone [Bruz 04] 1·10−610−11
Boutboul [Bout 01] – 10−14
Clark [Clar 77] 3·10−410−13
Devred [Devr 04, p. 41] 1·10−510−14
CENELEC [Supe 07a] 1·10−510−14
1·10−410−13
182 Material Properties
e.g. bending and rolling. Therefore, the n-index of a complete Rutherford-
type cable can be significantly smaller (around ≈30% in case of LHC MB)
than of the strands before cabling [Verw 07a].
Alternatively to the fit by Walter a full field and temperature exponentialAlternative
expression approach was published by Dorofejev [Doro 80]:
E=Jρnexp T−Tc
T0
+B
B0
+J
J0.(A.39)
A.2.4 Critical Current Density
As previously mentioned, the critical current Icdepends on the applied mag-
netic induction Band temperature T. In the space of these three quantities,
the limit between normal conducting state and superconducting state consti-
tutes a surface. Instead of the critical current, most commonly the critical
current density Jcis used. The critical surface is thus given by Jc(B, T ).
The critical current density can be obtained in two different ways: EitherCritical
surface from the critical current or from the magnetization measurement. In the first
case the critical current is converted into the critical current density under
certain assumptions on the current distribution and influence of the conductor
self-field11 [Bout 06]. In the second case, the measured magnetic moment is
first converted into a magnetization density, and then into a current density
based on the assumption of a local current density distribution [Wils 83, pp.
243]. It should be noted that this approach strongly relies on the underlying
model assumption for the superconductor magnetization (Sec. D.3).
Both approaches suffer from some inherent and external limitations. ForLimitations
the critical current measurement, the influence of the self-field on the mea-
surement increases with current. The temperature can be stabilized very
accurately up to the boiling point of liquid helium. Beyond that point the
sample needs to be thermally stabilized by a heated and thermally controlled
sample holder [Wils 83, p. 249]. A lower temperature limit is given by the
increasing effort in refrigeration power. For the LHC, the maximum mag-
netic induction is limited to 1.5 T in the magnetization measurement station
[Le N 01] and to 11 T in the critical current measurements station [Bout 01].
For the critical current measurement, the maximum current is limited to
1000 A [Bout 01]. Figure A.16 shows the limitations to either measurement
method. The critical current density is only accessible in two disconnected
patches!
In order to overcome this limitation, Green presented in [Gree 88] a methodParameter-
ization to predict the critical current density based only on the critical field, tem-
perature, and one measured value for Jc. With a mathematical function
representing all underlying physical effects limiting the current density in a
superconductor, the critical current density could be parametrized by means
11 The CENELEC standard strongly encourages not to apply any self-field compensation
to the measurement results in order to allow an inter-laboratory comparison. In case it
is nevertheless necessary, a very simple approach with the field of a straight line current
is recommended [Supe 07a, Appendix B].
A.2 Solids in Superconducting State 183
m
Tc
Ic
Bmax Magnetization
Bmax Critical Current
Helium Boiling Point
Lower Temperature Limit
Current Source Limit
0 2 4 6 8 10 12
0
2
4
6
8
Bin T
Tin K
Figure A.16: Critical current density measurement limitations. All limits are plotted
with dashed lines, leaving two disconnected patches where measurements can be performed.
The full lines highlight where extensive measurement took place for the LHC. Remark: The
current source limit is given by the intersection of the critical surface (shading) with the
maximum current density and therefore curved line.
of a few fitting parameters12. This function is called the critical surface pa-
rameterization.
A.2.5 Critical Surface Parameterization
A.2.5.1 Niobium-Titanium
In [Bott 00] Bottura published a parameterization for the critical current
density as a function of the applied magnetic induction B, local temperature
Tand reference current density, Jref. The function relies on the four fit
parameters α,β,γand C0:
Jc(B, T) = Jref
C0
BB
Bc2(T)α1−B
Bc2(T)β1−T
Tc0 nγ
(A.40)
where Tc0 denotes the upper critical temperature for zero field. Bc2(T)is
given by:
Bc2(T) = Bc20 1−T
Tc0 n(A.41)
Here Bc20 denotes the upper critical field at zero temperature. The parameter
range was given to αbetween 0.5 and 0.8, βaround 1, and γin the range of 2.
12 This approach can be illustrated by the following example. Basic Newton’s physics
teaches: The trajectory of a falling object with a horizontal velocity greater than zero,
follows a parabola. Knowing the shape of the trajectory, the position of the object can
be predicted based on the position of the apex and any second point.
184 Material Properties
0
5
10
Bin T
0
2
4
6
8
Tin K
0
1. !1010
2. !1010
3. !1010
4. !1010
J
cin A m"2
0.1 T
1T
2T
4T
8T
0 2 4 6 8 10
0
1. !1010
2. !1010
3. !1010
4. !1010
5. !1010
Tin K
J
cin A m"2
Figure A.17: Critical current density of Niobium-Titanium after [Bott 00] and [Voll 02].
(left) Critical surface of Nb-Ti. (right) Critical current density versus temperature for
different values of magnetic induction.
For high fields, the critical current density is mainly influenced by the value
for the parameter β. As pointed out in [Schw 06], the nature of the function
changes from parabolic to root-like according to βgreater or smaller than 1.
As shown in [Ekin 81], the strain dependence of Nb-Ti is very small and is
thus neglected.
The parameters are taken from [Voll 02] and can be found in Tab. A.17.
The critical surface of Nb-Ti is shown in Fig. A.17.
A.2.5.2 Niobium-3-Tin
The parameterization presented by Summers in [Summ 91] allows the predic-
tion of the critical current density of Nb3Sn as a function of field, temperature,
strain and radiation damage, and relies on one fit parameter only: C0.
Table A.17: Fit parameters for the critical current density
Parameter Value Unit
Bc20 14.5 T
Tc0 9.2 K
Jref 3·109Am−2
C027.04 T
α0.57 −
β0.9−
γ2.32 −
n1.7−
A.2 Solids in Superconducting State 185
The critical current density Jcis given by
Jc(B, T, ε) = CNb3Sn(ε)
√B1−B
Bc2(T, ε)2 1−T
Tc0(ε)2!2
(A.42)
where Bdenotes the applied magnetic induction, Tdenotes the temperature
and εdenotes the compression of the superconductor.
The constant CNb3Sn(ε)consists of the free fit parameter C0and a strain
depending term.
CNb3Sn(ε) = C01−α|ε|1.70.5.
The constant αis given by Summers to 900 for compressive (ε < 0) and to
1250 for tensile (ε > 0) stress.
The upper critical field Bc2is given by
Bc2(T, ε)
Bc20(ε)= 1−T
Tc0(ε)2!"1−0.31 T
Tc0(ε)21−1.77 ln T
Tc0(ε)#
where the upper critical field due to strain Bc20(ε)is given to
Bc20(ε) = Bc20m 1−α|ε|1.7,
and Bc20m denotes the upper critical field at zero temperature and strain.
The upper critical temperature Tc0 for zero intrinsic strain is given by
Tc0(ε) = Tc0m 1−α|ε|1.71/3(A.43)
where Tc0m denotes the upper critcal temperature at zero field and zero stress.
The constants Bc20m and Tc0m depend on the number of materials in the
alloy, i.e. whether only Nb and Sn are used or Ta respectively Ti are added
for stabilization. The constants are given in [Devr 04, p. 44] to 24 K and 16 K
for binary and 28 T and 18 K for ternary compounds.
For the powder-in-tube (PIT) strand developed at the university of Twente
[Oude 01] for the Next-European-Dipole (NED) project [Devr 06] the criti-
cal current density has been fitted in [Schw 05b, Appendix] using the above
function. With εgiven to −0.0025 [Devr 04] and
Bc20(−0.0025) = 23.97 T,
Tc20(−0.0025) = 17.79 K,
the fit is performed by means of the Mathematica function NonlinearFit
and yields the constant
C0= 52.05 ·103T1/2Am−2.
The deviation from the measured magnetization is in the order of 25%.
186 Material Properties
0
10
20
Bin T
0
5
10
15
Tin K
0
5. !1010
1. !1011
1.5 !1011
J
cin A m"2
0.1 T1T
2T
4T
8T
0 5 10 15
0
1. !1010
2. !1010
3. !1010
4. !1010
5. !1010
Tin K
J
cin A m"2
Figure A.18: Critical current density of Niobium-3-Tin after [Summ 91] and [Schw 05b].
(left) Critical surface. (right) Critical current density versus temperature for different values
of magnetic induction.
A.2.6 Electrical Resistivity
As seen in Sec. A.2.2 the electrical field rises sharply across a superconductor
when the current density approaches the critical current density. Equation
(A.38) can be transformed to express the resistivity of the superconductor,
ρSC, during the transition
ρSC(B, J, T) = Ec
JNSC−1
Jc(B, T)NSC .(A.44)
Figure A.19 shows the resistivity of Nb-Ti and Nb3Sn versus temperature
and magnetic induction for different transport current densities.
A.2.7 Thermal Conductivity
Data on the thermal conductivity of superconducting materials is very scarce.
In [Kim 00] Kim suggests an extrapolation of data from the normal conduct-
ing state to the superconducting state. Even though electrical superconduct-
ing, thermally the heat conduction remains limited.
A.2.8 Heat Capacity
The volumetric specific heat for type II superconductors is given in [Dres 95,
p. 22] to
cSC(T, B) =
β+ 3 γ
T2
c0 T3+TB
Bc20 T < Tphonon
cNC(T)+3γT 3
T2
c0 −γT +γ+TB
Bc20 T > Tphonon
.(A.45)
Here, cNC denotes the specific heat in the normal conducting state, and βand
γare fit constants. The temperature Tphonon denotes the limit up to which
the specific heat is proportional to T3.
A.2 Solids in Superconducting State 187
normal-conducting resistivity
2
4
6
8
10
10-22
10-19
10-16
10-13
10-10
10-7
Tin K
r
Ein Wm
normal conducting resistivity
5 10 15 20
10-22
10-19
10-16
10-13
10-10
10-7
Tin K
r
Ein Wm
normal-conducting resistivity
2 4 6 8 10
10-22
10-19
10-16
10-13
10-10
10-7
Tin K
r
Ein Wm
normal conducting resistivity
5
10
15
20
10-22
10-19
10-16
10-13
10-10
10-7
Tin K
r
Ein Wm
!"#"
$"#"
%"#"
&'"#"
$"#"
!"#" %"#"
Figure A.19: Resistivity of Nb-Ti (left) and Nb3Sn (right) in superconducting state
versus temperature for different field levels. The current density is varied between 0.5 and
1.1 of the design critical current density. The design critical current density for Nb-Ti is
Jc(8.4 T,1.9 K) = 2.6·109Am−2and for Nb3Sn Jc(15 T,4.2 K) = 1.4·109Am−2: (dotted)
J= 1.1Jc, (full) J= 1.0Jc, (densely dashed) J= 0.9Jc, and (dashed) J= 0.5Jc.
It can be seen from the formula, that the specific heat in the supercon-
ducting state increases with applied magnetic induction. For T > Tphonon it
is greater than in the normal conducting state. Below that limit it can be
smaller.
When exceeding the critical temperature for an applied magnetic induc-
tion, Tc(B), the specific heat drops down to the specific heat in the normal
conducting state. This drop is typical for phase transitions without latent
heat [Dres 95, p. 22].
A.2.8.1 Niobium-Titanium
In the superconducting state, the volumetric specific heat cth,Nb-Ti of Nb-Ti
depends on the temperature and on the applied magnetic induction. In the
normal conducting state the heat capacity depends only on the temperature
[Verw 05]. Therefore, two separate fits to measurement are given: One for
temperatures above the critical temperature Tc(J, B)shown in Eq. (A.27),
and one for below in thus superconducting state:
cT,Nb-Ti(T, B) = c0(aT3+c TB),(A.46)
with c0= JK−1m−3,a= 49.1 K−3and c= 64 K−1T−1. The critical temper-
ature depends on the applied field and the current density in the supercon-
ductor. The specific heat is shown in Fig. A.20 (left).
A.2.8.2 Niobium-3-Tin
In [Kim 00, p. 5] the volumetric specific heat of Nb3Sn is given to
cT,Nb3Sn(T) = aT3+b, (A.47)
188 Material Properties
10 T
5T
0T
superconducting
normal
conducting
2 4 6 8 10
0
10 000
20 000
30 000
40 000
Tin K
c!T"
in Jm!3K!1
superconducting normal
conducting
5 10 15 20
20 000
40 000
60 000
80 000
100 000
120 000
140 000
Tin K
c!T"
in Jm!3K!1
Figure A.20: Volumetric specific heat in superconducting state. The transition is discon-
tinuous when the critical temperature is reached. (left) Nb-Ti [Verw 05]. The specific heat
is plotted for different applied magnetic induction. The transition is shown exemplarily for
the maximum temperature T= 9.14 K, but would happen in the same way for any lower
temperature. (right) Nb3Sn after [Kim 00] in superconducting state and after [Baue 07, p.
4-13] in normal conducting state.
with the constants a= 22.68 Jm−3K−4and b= 988.2 Jm−3K−1. Contrary
to the general formula (A.45) this function does not show any dependence on
magnetic induction.
A.2.9 Permeability
Type I superconductors expel any external magnetic induction up to the
critical field Bc1. Therefore they can be considered as perfect diamagnetics.
In type II superconductors the interior is partially screened from the ex-
ternal field while it also penetrates in form of flux tubes. The flux tubes are
trapped by so-called pinning centers preventing free movement. Therefore
type II superconductors show a hysteretic diamagnetic magnetization.
The magnetization is not expressed by means of a relative permeability
constant (as for magnetic iron), but by macroscopical models as in [Voll 02]
for type II superconductors.
A.3 Fluid/Gaseous Matter - Helium Properties
For the application of low temperature superconducting magnets the onlyHelium
available coolant is liquid helium13. Here, it is referred to helium-4 (4He), i.e.
4He
helium atoms with 2 neutrons, in comparison to the much rarer helium-3.
Apart from the coolant, all other materials are solid. Therefore this sec-
tion addresses only properties of helium. Other gases or liquids are quoted
13Hydrogen is in liquid state between 14.01 K (melting point) and 20.28 K (boiling point).
Within that temperature range it could be used as a coolant, but comes with certain
general disadvantages as, e.g., being highly explosive.
A.3 Fluid/Gaseous Matter - Helium Properties 189
01234567
0.01
0.1
1
10
100
!"#$"%&'(&))&
!"#$"%&'(&)&
*+,-.&
/-0"%&
1&!"2(&
3."45+0&&
6-"27&
89:;<9&=>&?:?@&A6+B&
8;:<C&=>&9D:<&A6+B&
8@:9&=>&9:9E&A6+B&
F(G,(.+7$.(&"2&=&
6.(HH$.(&"2&;?@&6+&
Figure A.21: Phase diagram of helium (4He) following [Van 86, p. 55].
occasionally to give a comparison.
A.3.1 Temperature Levels / Phases
Figure A.21 shows the phase diagram of helium-4. Helium features a gaseous, Phases
a solid and two liquid phases. The liquid phases separate the gaseous and the
solid state in the phase diagram. The two liquid phases are denoted normal
liquid helium or helium I (He I) and superfluid helium or helium II (He II).
They are separated by the vertical λ-line, i.e. the temperature of the lambda-
transition Tλ(p)as a function of pressure p. The λ-line spans form the lower
triple-point (gaseous) at (2.172 K,5·103Pa) to the upper triple-point (solid)
at (1.76 K,2.97 ·106Pa). The critical point of the transition between He I
and the supercritical helium is at (5.2 K,2.23 ·105Pa).
The transport properties of He II are very different to those of normal He II
liquids featuring very low viscosity and extremely high heat conductivity
[Van 86, p. 90].
Table A.18 shows the boiling points of different gases for comparison.
A.3.2 Density
The density of helium is a function of temperature and pressure. For increas-
ing temperature the density starts to decrease and drops by nearly an order
of magnitude at evaporation. The evaporation temperature increases with
increasing pressure. For a confined volume of liquid helium, the pressure and
temperature rise simultaneously when subjected to heating.
Saving the effort of modeling the helium liquid/gas transition and a full Simplified
model
model of the pressure distribution inside a magnet, certain assumption can be
190 Material Properties
1 5 10 50 100 500 1000
0
5
10
15
20
25
Tin K
p
in
bar
1 5 10 50 100 500 1000
5
10
20
50
100
200
Tin K
ΡDin
kg m"3
Figure A.22: (left) Assumption of the pressure rise over temperature. (right) Resulting
piece-wise linear approximation of the density of helium versus temperature.
made. After passing the λ-point, the pressure rises linearly with temperature
from the operating pressure to the maximum pressure in the magnet vessel
given by the pressure release valve (e.g 20 bar). From approximately 6 K
onwards, the pressure stays constant for the rest of the quench process. The
assumed pressure rise is shown in Fig. A.22 (left). This approach is in good
agreement with the experiences gained at the LHC protoype magnet string
[Chor 98].
Combining the helium property tables for constant pressure in [Van 86,
Appendix 2] results in an approximated helium mass density versus temper-
ature as shown in Fig. A.22 (right).
A.3.3 Latent Heat
The latent heat denotes the amount of energy released or absorbed by a
substance during the change of state or phase. The transition from helium
I to gaseous helium requires around 2·104Jkg−1up to a temperature of
4.5 K, and then drops to zero at around 5 K [Van 86, p. 65]. In [Iwas 94, p.
112] 2.6 J cm−3are given. Table A.19 shows the latent heat or enthalpy of
Table A.18: Boiling point of different gases (at room temperature) at normal pressure
[Rade 02]
Material Symbol Boiling point in K
Oxygen O290.2
Nitrogen N277.4
Hydrogen H220.3
Helium He 4.22
A.3 Fluid/Gaseous Matter - Helium Properties 191
vaporization of other common gases and liquids.
The phase transition of He II to He I is considered of second order and has
no heat of transformation [Van 86, p. 91].
A.3.4 Heat Conductivity
The thermal conductivity of helium II is orders of magnitude greater than of
other liquids and even of highly conducting metals as copper [Van 86, pp.
94]. In [Iwas 94, p. 116] a value of approximately 100 kWm−1K−1is given.
This 10 to 100 times more than the heat conductivity of copper (see Sec.
A.1.4.1) and 100,000 times more than of water under standard conditions.
The thermal conductivity of helium II is considered as perfect, i.e. without
any temperature gradient, but limited to a maximum heat flux. Exceeding
the lambda point or the critical flux He II transits to He I with very low
heat conductivity - this can be compared to the quenching of superconduct-
ing material [Iwas 94, p. 117]. The heat conductivity of helium I is only
0.02 Wm−1K−1[Iwas 94, p. 116].
A.3.5 Volumetric Specific Heat
Based on the pressure assumption in Sec. A.3.2 the specific heat (for constant
volume) can be taken from [Van 86, Appendix 2] as shown in Fig. A.23 (left).
The specific heat shows a peak at the λ-point, drops after transition and rises
to an approximately constant value after evaporation.
Combining the mass density of Sec. A.3.2 with the specific heat yields the
volumetric specific heat. Figure A.23 (right) shows the decrease of specific
heat due to the drop of density.
A.3.6 Dielectric Strength
Considering an electrical field over a gas filled void, free electrons are accel- Breakdown
in gases
erated. Collisions with neutral atoms or molecules ionize the gas and create
further free electrons. At a certain field level, the electrons gain enough en-
ergy between two successive collisions to cause further ionization: Townsend
avalanche breakdown [Cobi 41, p. 145].
The number of ionizing collisions depends on the free path length of the Paschen’s
law
electrons and their kinetic energy. Thus the electrical breakdown field depends
Table A.19: Heat or Enthalpy of Vaporization [Lide 06, pp. 6-94]
Material Symbol Hvin Jkg−1
Helium He 2.1·104
Hydrogen H24.55 ·105
Nitrogen N22·105
Water H2O 2.26 ·106
192 Material Properties
5 10 50 100 500
1!104
5!104
1!105
500
1000
5000
Tin K
c in
J kg"1K"1
5 10 50 100 500
5!104
1!105
5!105
1!106
5!106
1!107
Tin K
c in
Jm"3K"1
Figure A.23: Approximation of the thermal properties of helium with varying pressure.
(left) Specific heat of helium. (right) Volumetric specific heat.
on the distance between the two electrodes, as well as the density of the gas
(or the pressure). Plotted over the product of pressure and distance the
electrical breakdown voltage shows a minimum for each gas at characteristic
value: Paschen’s law [Cobi 41, pp. 162].
For helium at ambient temperature the minimum break down voltage isHelium
156 V at a pressure distance product of 5.33 Pa m. In the field of supercon-
ducting magnets with distances of approximately 0.2 mm between two con-
ductors and 1 mm between two layers, the pressure with minimum breakdown
voltage is given to 0.25 bar and 50 mbar, respectively. Operating the device
with pressurised helium II reduces the risk of an electric breakdown [Lebr 97].
For comparison the values for air are 327 V and 0.76 Pa m [Cobi 41, p. 165].Air
B Parameters
The Devil is in the details, but so is salvation.
Hyman G. Rickover
(1900-1986)
B.1 Geometrical Quantities
B.1.1 Filament
Most basically, a filament of a superconductor consists of the superconducting
material and is of circular cross-section. For the stacking process of the super-
conducting strands hexagonal filaments are used as well. The cross-sectional
area is approximated by a circle.
While this basic model holds well for common Nb-Ti filaments, Nb3Sn fil-
aments feature more details. Here, the superconducting fraction is produced
by a thermal diffusion process of niobium and tin resulting in unreacted frac-
tions of these materials, as well as diffusion barriers to protect the surrounding
matrix material from pollution. Therefore, the filament cross-section Afil,tot
consists of at least 3 fractions, i.e. the core Afil,core, the superconducting tube
Afil,sc and a coating Afil,coat.
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Afil,tot =πr2
f(B.1)
Afil,core =πr2
fc (B.2)
Afil,coat =π2rftfc +t2
fc≈2πrftfc (B.3)
Afil,sc =Afil,tot −Afil,coat −Afil,core
=πr2
f−2rftfc +t2
fc−r2
fc(B.4)
For even more complex filaments or different reaction processes, more shells
would need to be considered.
B.1.2 Strand
The superconducting filaments are embedded in a matrix of normal conduct-
ing material, e.g. copper or aluminum. During extrusion the strand is twisted
193
194 Parameters
!"#$%&' ()#*'
!"#$%&'
Figure B.1: Different cable types. (left) Ribbon-type cable with glued strands. (center)
Rutherford-type cable with metal core. (right) Braided strands.
along its axis, to minimize the loop surface area spanned by the filaments in-
side. Most often no filament is placed in the strand center, since it would
not be twisted, leaving a normal conducting core. For insulation purposes
between neighboring strands, the strands may be coated.
Apart from the filaments described above, the strand cross-section features
a core Astr,core, a matrix Astr,mat and a coating Astr,coat (maybe even more
than one layer).
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Astr,tot =πr2
s(B.5)
Astr,core =πr2
sc (B.6)
Astr,coat =π2rstfc +t2
sc≈2πrstsc (B.7)
Astr,inner =Astr,tot −Astr,coat −Astr,core
=πr2
s−2rstsc +t2
sc−r2
sc(B.8)
Astr,fil =NfAfil,tot (B.9)
Astr,mat =Astr, inner −Astr,fil (B.10)
If the materials used for core, matrix and coating are identical and the
filaments only consist of the superconducting material, the area fractions by
material can be easily calculated by the copper to superconductor area ratio
(see Sec. B.2.1).
B.1.3 Cable
Three different kinds of cables can be distinguished: Rutherford-type cables,
ribbon-type cables and braided cables, see Fig. B.1.
Rutherford-type cables consist of a number of strands, twisted along the
axis of the cable and rolled in a rectangular or trapezoidal form. The center of
the cable may contain a core in order to increase the electrical resistance be-
tween crossing strands. The cable is insulated by multiple wraps of insulation
tape, and/or may further be potted with a resin.
B.1 Geometrical Quantities 195
The transposition of the strands along the cable axis has the following
implications: The current follows a zig-zag path along the cable, resulting in
a small longitudinal field component (see Sec. 7.8) and a higher resistivity per
length [Imba 03, p. 3-22], i.e. `sp > `p/2. In the cross-section of the cable,
the cross-section of the strands is of elliptical shape. The volume covered by
the strands can be calculated by means of the formulas given in Sec. D.2.
!!"
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"#"
"$#"
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"
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!
#
"
"$#"
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1-23&"45#"6)&7"
`sp =s`p
22
+h2
c(B.11)
βp= arccos `p/2
`sp !
= arctan hc
`p/2!(B.12)
Astr,ell =Astr,tot`sp
`p/2=Astr,tot
cos βp(B.13)
Between the strands, as well as between strands and the insulation, voids
remain. The confined volume in between four circles with radius acan be eas-
ily derived to (4 −π)a2. Depending on the compression and the key-stoning
of the cable the voids might be smaller than estimated above, and especially
not evenly distributed over the cable width. The first disadvantage is over-
come by introducing two dimensionless tuning coefficients, kcab,void,inner and
kcab,void,outer. For a thermal model with homogeneous temperature distribu-
tion over the cable cross-section (see Sec. 3.6), the local distribution of the
voids can be neglected, if the total void volume is modeled correctly.
The effective cable cross-sectional area Aeff
cab consists of a core Acab,core,
the elliptically cut strands Acab,str, the inner and outer voids, Avoid,inner and
Avoid,inner, as well as the insulation Ains.
!"#$%
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Acab,xs =wci +wco
2hc(B.14)
Acab,str =NsAstr,ell
=NsAstr,tot
cos βp(B.15)
Acab,core =hctcc (B.16)
Avoid,inner =kcab,void,inner
(4 −π) (Ns/2−1)
cos βp(B.17)
196 Parameters
Avoid,outer =kcab,void,outer Acab,xs −Acab,str (B.18)
−Acab,core −Avoid,inner
Ains ≈(wci +wco)tia + 2hctir + 4tiatir (B.19)
Aeff
cab =Acab,str +Acab,core +Avoid,inner (B.20)
+Avoid,outer +Ains
This approach neglects the length of the strand changing between two layers
in the cable edges. Note that neglecting the twist pitch results in an error of
1/cos(βp)for the calculated longitudinal lengths and cross-sectional areas.
The superconducting as well as the normal conducing surface area perpen-
dicular to the current are given by (using the geometrical ratios λand ηfrom
Sec. B.2.1),
Acab,SC =NsNfAfil,SC =Nsπr2
sηA
SC =Nsπr2
s
1
1 + λ(B.21)
Acab,NC =NsAstr,tot −NfAfil,SC
=Nsπr2
sηA
NC =Nsπr2
s
λ
1 + λ(B.22)
Due to the keystoning, the long edge of the cable is longer than hcand
given by
`long-edge =swco −wci
22
+h2
c.(B.23)
Ribbon-type cables are most conveniently modeled by means of cables only
consisting of a single strand and an insulation layer. Braided cables are not
addressed in this work.
Example : Normal and superconducting area for the LHC MB
The LHC main dipole features two different cables used in the inner and
outer layer of the coil: The normal and superconducting surface areas are
calculated based on the values given in Sec. C.1.1 (strand) and Sec. C.1.2
(cable).
For the inner layer cable, with a strand radius of rs= 0.5325 mm, a copper-
to-superconductor ratio of λ= 1.65, and Ns= 28 strands, this results in:
ALHCinner
SC = 9.41 mm2, ALHCinner
NC = 15.53 mm2(B.24)
For the outer layer cable, with a strand radius of rs= 0.4125 mm, a copper-
to-superconductor ratio of λ= 1.95, and Ns= 36 strands, this yields
ALHCouter
SC = 6.52 mm2, ALHCouter
NC = 12.72 mm2(B.25)
B.1 Geometrical Quantities 197
!" #"
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!!"
&"'"
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."
/"
0"
122()$3456%+%7(*(+85"#$%&'%"#()'*($%+,
Figure B.2: Contact coefficients: (left) Conductor cross-section with named surfaces
and position vectors. (right) Different thermal contact coefficients for transversal heat
conduction and cooling (1) and (2). Minimum distance for the electrical field calculation
(3).
B.1.4 Cable Twist-pitch in a 2D Approach
Regarding the magnet as a 2D problem where 3D features are included by
extruding longitudinally, it is convenient to relate everything to properties of
the 2D cross-section. Therefore, only the cross-sectional areas of Eq. (B.14)
to (B.20) are used.
The cross-sectional areas perpendicular to the current in the strands, e.g.
in Eqs. (B.24) and (B.25), can be derived by applying the factor cos(βp).
For the calculation of the longitudinal resistivity of a cable, the overesti-
mated cross-sectional area as well as the underestimated path length have to
be compensated by a factor 1/(cos(βp))2. For longitudinal conductivity the
inverse is applied.
B.1.5 Coil Cross-Section
The coil cross-section of a superconducting magnet consists of Ncond arbi-
trarily distributed conductors. For the calculation of heat transfer between
adjacent conductors, it is necessary to define a measure of proximity. The
calculation of electrical fields within the cross-section relies on the minimum
distance between two conductors.
Figure B.2 (left) shows the cross-section of the conductor i. The four faces
are denoted by α {a, b, c, d}. Each face is characterized by the surface area,
Ai
α, the vector to the center of the face, ri
center,α, and the vector to the corner,
ri
corner,α.
Thermal Contact Coefficients Assuming homogeneous and especially con-
stant thermal properties, the heat flux along a cylinder depends on the ratio
of cross-sectional area and height - further on denoted as the transversal
thermal contact coefficient K. In case of the arbitrarily shaped volumes in
between two conductors, i.e. right tetragonal prisms, the approach is adapted
by considering mean areas and the distance of the surface centers.
198 Parameters
The distance between the center of two faces αand βof the conductors i
and nis given by
di,n
α,β =ri
center,α −rn
center,β.(B.26)
The area Ai
αis calculated from the longitudinal discretization length times
the conductor inner width wci (α=a), the conductor outer width wco (α=c)
or the long-edge llong-edge (α=b or c).
The contact coefficient ˜
Ki,n
α,β is calculated for any two faces of the coil cross-
section.
˜
Ki,n
α,β =
Ai
α+An
β
2
di,n
α,β
.(B.27)
Multiple connections between two conductors are eliminated by consider-
ing only the strongest of the 16 links. Connections between two conductors
passing through a third conductor are eliminated by a maximum distance
criterium: Only links over a distance smaller than a user-supplied maximum
distance dmax (smaller than the conductor thickness) are taken into account.
In order to adjust the heat transfer between two conductors, e.g. for the
implementation of quench stopper, the contact coefficient can be modified by
assigning a real constant ki
T,trans to each conductor.
˜
Ki,n =
ki
T,trans+kn
T,trans
2max n˜
Ki,n
α,βo di,n
α,β < dmax
0else ,(B.28)
with i6=nand α, β {a, b, c, d}. The number of links can be further reduced
by dropping all connections where the contact coefficient ˜
Ki,n is smaller than
a user supplied ratio kmax,trans of the maximum link, yielding
Ki,n
trans =
˜
Ki,n ˜
Ki,n > kmax,trans max n˜
Ki,noi,n=1...Ncond
0else .(B.29)
The contact coefficient for cooling is calculated in a similar way. Here, the
user supplies a distance di
cool,α between a specified face and the cold surface.
The cooling coefficient is then given to
Ki
cool,α =Ai
α
di
cool,α
.(B.30)
Here multiple connections are possible. Where no distance is specified, cooling
is not taken into account.
By means of the aforementioned approach it is possible to generate a
transversal thermal network for an arbitrary coil cross-section. The grade
of inter-linkage as well as certain variations to the thermal contact between
elements can be easily applied. For a common coil cross-section built of
conductor blocks, the turn-to-turn contact coefficients are reliably detected.
B.1 Geometrical Quantities 199
Because of the strong simplification of the heat transfer process and the pos-
sibility of intersecting “heat conduction prisms”, the approach is not valid for
strongly non-linear heat conductivity.
Minimum Distance between Conductors For the calculation of electrical
fields between adjacent conductors, the minimum distance is needed:
di,n
E= min nri
corner,α −rn
corner,βoα,β {a, b, c, d}(B.31)
The vector rcorner, and the faces a, b, c and d are defined in Fig. B.2.
B.1.6 Magnet
For a superconducting magnet five different geometrical lengths shall be dis-
tinguished: The overall length `total, the magnetic length `mag, the inductance
length `ind, the average winding length `w, and the coil length `c.
The magnetic length `mag is defined as [Brun 04, p. 165]
`mag(I) = R+∞
s=−∞ B0(I, s)ds
Bav(I),(B.32)
where Idenotes the excitation current, Bav the average field in the straight
part of the magnet and B0is the main component integrated along the orbit.
Approaching the coil end, the magnetic induction starts to change depending
on the design of the coil and the iron yoke. Outside the magnet, the magnetic
field decreases quickly to zero. Due to the difference of the iron saturation in
the coil ends compared to the magnet center, the current dependence in Eq.
(B.32) does not cancel out. The variation over excitation current depends
thus on the iron yoke design and the overall length of the magnet - for longer
magnets it is smaller. If not stated otherwise, the magnetic length is given at
operating temperature and nominal current [Brun 04, p. 164]. Notice that for
higher order multipole magnets the field in the center is zero, and therefore
Bav has to be replaced by the maximum field on the reference radius.
The inductance length is similarly defined as
`ind =Uind
di
dtL0
d
,(B.33)
where Uind is the measured voltage induced over the whole magnet for a
current ramp rate di
dt.L0
ddenotes the differential inductance per unit length
of an infinitely long magnet with identical cross-section (see Sec. 7.4). The
inductance length depends on the level of excitation and thus represents 3-
D saturation effects. The inductance length is used to map the induced
voltage onto the coil windings. It can be motivated that `mag ≈`ind, by
extending the expression for the magnetic length by an effective coil width
and differentiating by time.
200 Parameters
!!"
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Figure B.3: Definition of coil end and coil end length.
The winding length is defined as the average length along one half-turn in-
cluding the coil heads as shown in Fig. B.3. The winding length is used for the
calculation of resistive voltages as well as for the longitudinal discretization.
The coil end length is the distance from the coil end to the part of the coil
where the magnetic field problem can be assumed 2-D.
B.2 Filling Factors
B.2.1 Length, Area and Volume Ratios
Given an area Atot segmented into nsubareas An, then the area ratio ηA
nis
defined as
ηA
n=An
Atot
.(B.34)
The length ratio η`
nand volume ratio ηV
nare defined accordingly:
η`
n=`n
`tot
, ηV
n=Vn
Vtot
.(B.35)
For identical lengths ηV
n=ηA
nand for identical cross sectional areas ηV
n=η`
n.
For superconductors only consisting of a superconducting material and a
copper matrix, the copper to superconductor area ratio λis widely used. It
is defined as the ratio of the superconducting, ASC, and the copper, ACu,
cross-sectional area:
λ=ACu
ASC
.(B.36)
The copper to superconductor ratio can be transformed into ηA
SC and ηA
Cu by
means of
ηA
SC =1
1 + λ, ηA
Cu =λ
1 + λ.(B.37)
For strands consisting of more than 2 materials, e.g. most of the Nb3Sn
superconductors, an additional ratio can be used. The copper to non-copper
area ratio λnon-Cu,
λnon-Cu =ACu
Anon-Cu
.(B.38)
B.2 Filling Factors 201
Note that the cross-sectional area is now divided into 3 parts, and thus the
copper and superconducting parts do not add up to the strand cross-sectional
area:
ηA
Cu =λnon-Cu
1 + λnon-Cu
, ηA
SC =
λnon-Cu
λ
1 + λnon-Cu
, ηA
non-Cu =1 + λnon-Cu
λ
1 + λnon-Cu
.(B.39)
B.2.2 Material Fractions
By means of the following scheme materials are assigned to the different parts
of the filament, strand and cable cross-section:
Area Cu Nb-Ti Kapton He ...
Afil,core
Afil,SC
Afil,coat
Ns×Nf×PAfil,tot 1
Astr,core
Astr,matrix
Astr,coat
Ns×PAstr,tot 2
Acab,core
Avoid,inner
Avoid,outer
Ains
PAcab,eff 3
Combining areas of identical material filling and dividing by the total area of
the filament, strand or cable yields the respective material area ratios, e.g.
ηA
str,Cu. Due to the identical longitudinal length, the volume ratios are equal
to the area ratios.
B.2.3 Effective Electrical Resistivity, Thermal Conductivity
and Specific Heat
Consider a cuboid of length `and cross-sectional area A. It is transversally
segmented into Nsub-volumes of length `and resistivity ρEn(see Fig. B.4
(left)). The effective longitudinal resistivity of the cuboid is given by
ρeff,[=]
E=1
PN
n=1 ηA
n1
ρEn
.(B.40)
For a cuboid segmented longitudinally in Ncuboids of length `nand iden-
tical cross-sectional areas (see Fig. B.4 (center)), the effective longitudinal
resistivity of the cuboid is given by
ρeff, [ | | ]
E=
N
X
n=1
η`
nρEn.(B.41)
202 Parameters
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&$"
&%"
##"
#'"
#%"
!#" !$" !%"
&''()*+,-./0/1(2(03-!"#$%&#'()#*+(,-*./#*%#01
Figure B.4: Effective thermal and electrical properties. Electrical resistivity and thermal
conductivity for parallel sub-domains (left) and sub-domains in series (center). Effective
specific heat (right).
The effective thermal conductivity is then given by:
κeff, [=]
T=
N
X
n=1
ηA
nκTn, κeff, [ | | ]
T=1
PN
n=1 η`
n1
κTn
.(B.42)
A volume of homogeneous temperature, consisting of Nsub-volumes Vn
with specific heat cT(see Fig. B.4 (right)) shows an effective specific heat of
ceff
T=
N
X
n=1
ηV
ncTn.(B.43)
This approach is only valid if the material properties are homogeneous
within each sub-domain. It can be adapted for materials depending on an
external quantity and thus varying within the sub-domains. These domains
are sub-divided such that constant quantities can be assumed.
C Cases
On two occasions I have been asked,-"Pray, Mr.
Babbage, if you put into the machine wrong figures,
will the right answers come out?" [...]I am not able
rightly to apprehend the kind of confusion of ideas
that could provoke such a question.
Charles Babbage
(1791-1871)
This chapter serves as a collection of all parameters used for the simulations
presented in the main part of the thesis. The description of the LHC main
bending magnet is exemplary for all cases.
C.1 LHC Main Bending Magnet
The LHC main dipole features two apertures in a common iron yoke. The
coils in each aperture are subdivided in an upper and a lower pole. Each pole
consists of 6 conductor-blocks per quadrant, arranged in two layers. Both,
strands and cables, are different for the outer and inner layer resulting in a
grading of current density.
C.1.1 Strand
The parameters of the Nb-Ti filaments and the copper-stablized strands for
the inner and outer layer cables are given in Tab. C.1 and C.2, respectively.
Figure 2.3 (left) shows a photography of the strand cross-section.
The critical current density Jcin the strands is given by a parameterization
(Sec. A.2.5.1). Identical fit parameters are used for the inner and outer layer
strands (see Tab. A.17). The residual resistivity ratio (RRR) of the copper
matrix in the strands is in the range of 150 to 250 [Char 06].
C.1.2 Cable
The LHC MB coil is wound from key-stoned Rutherford type cables. The
cable parameters for the inner and outer layer cables are given in Tab. C.3.
203
204 Cases
Table C.1: LHC Main Bending Magnet (MB) Filament
Quantity Symbol Unit Inner Layer Outer Layer Ref.
Materials
Superconductor Nb-Ti Nb-Ti Sec. A.2.5.1,
Sec A.1.5.2
Core – –
Coating – –
Geometry
Radius rfµm3.5 3 [Brun 04, p. 157]
Core radius rfc µm – –
Coating thickness tfc µm – –
Table C.2: LHC Main Bending Magnet (MB) Strand
Quantity Symbol Unit Inner Outer Ref.
Layer Layer
Materials
Matrix Copper Copper Sec. A.1.3.1,
Sec A.1.4.1,
& Sec. A.1.5.1
Core – –
Coating Sn5wt%Ag Sn5wt%Ag [Brun 04, p. 157]
Geometry
No. of filaments Nf–8900 6500 [Brun 04, p. 157]
Radius rsmm 0.5325 0.4125 [Brun 04, p. 157]
Core radius rsc mm – –
Coating thickness tsc mm – –
Twist pitch psmm 18 ±1.5 15 ±1.5[Brun 04, p. 157]
Model parameters - Electrical / Inter-filament coupling losses
Copper/Super- λ–1.65 ±0.05 1.95 ±0.05 [Brun 04, p. 157]
cond. ratio
Copper RRR RRR – 150 −250 150 −250 [Char 06]
Copper matrix ρIFCC Ωm1.24 ·10−10 1.24 ·10−10
resistivity
Copper matrix dρIFCC
dB
Ωm
T9·10−11 9·10−11
resistivity B
dependence
Filling factor ηs–0.5 0.5
Specifiction
Critical current IcA
at 10 T,1.9 K ≥515 [Brun 04, p. 157]
at 9 T,1.9 K ≥380 [Brun 04, p. 157]
C.1 LHC Main Bending Magnet 205
Figure 2.3 (right) shows a photography of the cable. The filaments of one
strand have been extracted by etching.
C.1.3 Magnet Data
The magnet features a two-in-one design, i.e. two apertures in one common
iron yoke. To each aperture belong a lower and a upper coil, each built in
two-layers. The inner layer consist of 4 coil blocks and the outer layer of
two. Figure C.1 shows the magnet cross-section with cryostat, iron yoke and
superconducting coils. The winding scheme is shown in Fig. C.2. The block
and conductor numbering scheme used in ROXIE is shown in Fig. C.3. All
parameters are given in Tab. C.4.
At a ramp rate of 7.5 mT ·s−1, AC losses of 180 mW per meter can be
expected in the magnet [Brun 04]. During a quench, these losses increase by
many orders of magnitude, and cause quench-back [Rodr 01]. For a full ramp-
cycle from injection to nominal current level and down to injection, hysteresis
losses of 280−400 J per meter were calculated [Voll 02, p. 135] and [Verw 95,
p. 153].
C.1.4 External Electrical Circuit
For quench simulation two different electrical situations have to be distin-
guished: The operation in the LHC tunnel and the setup on the test bench.
All parameters are given in Tab. C.5.
Tunnel Setup For the LHC the main bending dipoles are operated in strings
of 153 magnets. Each magnet is bridged by a diode mounted inside the
cryostat. The forward voltage of the diode is assumed to be constant at
8 V. In case of a quench, the power supply is switched off and by passed by
thyristors [Dahl 01]. Two dump resistors are switched into the circuit. The
discharge time constant of the string is τ= 104 s resulting in a maximum
di/dt= 125 As−1[Dahl 00]. The electrical network is shown in Fig. C.4
Test Stand On the test bench the magnet is only connected to a power
supply with by-pass thyristor. The cold diode is prevented from switching by
the power supply bypass diode and can thus be omitted in this configuration.
C.1.5 Magnet Protection
The quench protection consists of a quench detection system, a cold by-pass
diode and quench heaters placed on the outer layer of the coil as indicated
in Fig. C.2. The resistive voltage over the magnet is detected by a double
floating bridge detector as shown in Fig. C.5. The threshold voltage of the
detection system is 0.1 V [Denz 06]. Quench heaters are fired after a delay
of 10.5 ms for signal validation [Denz 06]. The timing of the different heaters
may vary by up to 10 ms [Sonn 01a].
206 Cases
Table C.3: LHC Main Bending Magnet (MB) Cable
Quantity Symbol Unit Inner Outer Ref.
Layer Layer
Materials
Insulation kapton kapton Sec. A.1.4.4,
& Sec. A.1.5.4
Cable Core – –
Inner void filling helium helium Sec. A.3
Outer void filling – –
Geometry
No. of strands Ns–28 36 [Brun 04, p. 157]
Insulation Thickness
radial tir mm 0.15 0.15 [Gran 08]
azimuthal tia mm 0.12 0.13 [Gran 08]
Cable
Height hcmm 15.1 15.1[Brun 04, p. 157]
Width, inner wci mm 1.736 1.362 [Brun 04, p. 157]
Width, outer wco mm 2.064 1.598 [Brun 04, p. 157]
Core thickness tcc mm – –
Keystone angle φkey o1.25 ±0.05 0.9±0.05 [Brun 04, p. 157]
Aspect ratio 7.95 10.2 [Brun 04, p. 157]
Twist pitch `pmm 115 ±5 100 ±5[Brun 04, p. 157]
Cabling angle βpo
Model parameters - Thermal / Inter-strand coupling losses
Helium in voids – % 10 10 Study
Cross-over resistance RcµΩ30 60 [Verw 07b]
Adjacent resistance RaΩ 100 100 [Verw 07b]
Specification
Critical current IcA
at 10 T and 1.9 K >13750 [Brun 04, p. 157]
at 9 T and 1.9 K >12960 [Brun 04, p. 157]
Linearization dIc
dBAT−1>4800 >3650 [Brun 04, p. 157]
MIITs MA2s
at 300 K,8 T 45 [Brun 04, p. 157]
at 300 K,6 T 30 [Brun 04, p. 157]
C.1 LHC Main Bending Magnet 207
Figure C.1: LHC Main Bending Magnet (MB) dipole cross-section. Courtesy of CERN
c
.
Table C.4: LHC Main Bending Magnet (MB) Magnet Data
Quantity Symbol Unit Value Ref.
Material
Wedge material: copper
Quench stopper: none
Cooling surfaces: cooling to cold bore tube (distance
around 1 mm) and collar (distance
around 0.5 mm)
Iron yoke: magnetic iron Sec A.1.6.1
Number of apertures – – 2
Number of poles – – 2
Number of blocks per pole – – 6
Num. of conductors per pole Ncp – 40
Coil length `cm 15
Average winding length `wm 14.57
Magnetic length `mag m 14.31 [Brun 04, p. 164]
End part of coil `ce m –
208 Cases
A
B
Anode right
Outer (1)
Inner (2)
LO
LI
LI
LO
UO
UI
UO
UI
YT111
YT112
YT121 YT211
YT122 YT212
YT221
YT222
Figure C.2: LHC Main Bending Magnet (MB) winding scheme and quench heater layout
[Brun 03].
!"
#" #$"
#%"
&" #'"
(" #)"
%" #*"
$" &+"
!!"
!#" &$"
&%"
'" &!"
)" &#"
*" &&"
!+" &("
#+!"
(!"
!)$"
#$"
#!$"%$"
!$!"!"
Figure C.3: LHC Main Bending Magnet (MB) Numbering schemes for the right aperture.
(left) Block numbers. (right) Conductor numbers. The skipped numbers are located in the
left aperture.
MBB
MBA
MBB
MBA
MBB
MBA
Beam 2
Beam 1
F
D
F
D
F
D
A
B
A
B
A
B
A
B
A
B
A
B
RB (M3)
Field (positive)
53.45 m
C33L8 C32L8
downstream
IC3332L8 IC32L8_2 IC32L8_3 IC32L8_4
+
-
+
-
Figure C.4: LHC Main Bending Magnet (MB) connection scheme in the tunnel.
C.1 LHC Main Bending Magnet 209
The quench heater strips consists of 15.0×0.025 mm stainless steel with
copper plating insulated with 0.175 mm polyimide insulation and epoxy glue.
Two heaters covering 13 cables, 15 m long and 9 mm apart. Sufficient to
heat spot-wise due to quench propagation velocities of 15 to 20 ms−1. The
optimized pattern of 400 mm covered followed by 120 mm non-covered strip
results in a heater resistance of 0.35 Ωm−1[Rodr 00].
A capacitor is discharged over the resistance of the heater strip, resulting in
an exponential voltage decay [Rodr 01]. The time constant for the dissipated
power is about 37.5 ms [Sonn 01a]. Measurements indicate that a heater-
provoked quench at 1.5 kA occurs around 80 ms after the heaters are fired.
At nominal current the delay reduces to 35 ms [Pugn 07].
In case of a quench in the LHC tunnel, the diodes and busbars are protected
from overheating by extracting the current by means of a dump resistor (Sec.
C.1.4).
All parameters of the quench protection are given in Tab. C.6.
C.1.6 Operating Conditions / Critical Values
The operation conditions and current levels of the LHC MB are given in Tab.
C.7. For orientation some critical values are given, too.
210 Cases
Table C.5: LHC Main Bending Magnet (MB) Electrical Circuit
Quantity Symbol Unit Value Ref.
Power supply
Switch off delay ∆tQT ms 0
Crowbar forward resistance RfDfR Ω0
Crowbar forward voltage UfDTh V 1
Cold diode
Switching voltage UcDcTh V 6 [Verw 08a]
Forward voltage UcDwTh V 1-2 [Verw 08a]
Forward resistance / resistance RcCfR Ω–
Delay from switching voltage ∆tcDh ms –
to forward voltage
Serial elements - Tunnel
Inductance LsH 153 x Ld
Resistance RsΩ–
Serial elements - Test stand
Inductance LsH 0
Resistance RsΩ0
Table C.6: LHC Main Bending Magnet (MB) Protection
Quantity Symbol Unit Value Ref.
Quench detection
Detection voltage Udet V±0.1[Denz 06]
Discrimination delay ∆tDis ms 10.5 [Denz 06]
Quench heater
Material stainless steel, copper plating [Rodr 00]
Width wQH mm 15.0 [Rodr 00]
Height hQH mm 0.025 [Rodr 00]
Insulation thickness dQH mm 0.175 [Rodr 00]
Length `QH m 15 [Rodr 00]
Nominal resistance (at cold) RQH Ωm−10.35 [Rodr 00]
Capacity CQH mF 7.05 [Rodr 00]
Time constant (power) τQH ms 37.5 [Rodr 00],
Sec. 7.9.5
Initial voltage UQH V 900 [Rodr 00]
Switching delay ∆tQHtd ms 4 [Denz 06]
Quench-heater delay (at nom. current)
high field heaters ∆tQH ms 25 [Rodr 00]
low field heaters ms 35 [Rodr 00]
Scattering of switching delay ∆tQHs ms <10 [Rodr 00]
Quench Heater Model
Initial power density per conductor PQH0 Wm−120
Heat transfer delay ∆tQH0 ms 22-25
Dump resistor - Tunnel
Resistance RDR mΩ75 [Dahl 00]
Switching/commutation delay ∆tDR ms 11.5 [Denz 06]
Decay time constant τDR s 104 [Dahl 00]
C.1 LHC Main Bending Magnet 211
A
B
Inner (2)
Outer (1)
LO
LI
UO
UI
LI
UO
UI
Iron-yoke
Anode
right
015
014
013
012
213 212
219
211
113
119
112111
LO
U
U
U
U
AB
2
4
1
3
U
U
U
U
U
2L
2U
1L
1U
Figure C.5: LHC Main Bending Magnet (MB) Quench detection. The resistive voltage
over the LHC MB is measured redundantly. The two floating bridges with voltages U1−U4
and U2−U3overlap, such that the interconnection between the two apertures is covered
twice.
Table C.7: LHC Main Bending Magnet (MB) Operating Conditions and Critical Values
Quantity Symbol Unit Value Ref.
Operation temperature TbK 1.9 [Siem 05]
Injection,
current Iinj A 763 [Brun 04, p. 164]
peak field T 0.54 [Brun 04, p. 164]
energy TeV 0.45 [Brun 04, p. 164]
Nominal,
current Inom A 11850 [Brun 04, p. 164]
peak field T 8.33 [Brun 04, p. 164]
energy TeV 7 [Brun 04, p. 164]
Ultimate,
current Iult A 12840 [Brun 04, p. 164]
peak field T 9.0 [Brun 04, p. 164]
Maximum temperature Tmax K 600 Sec. A.1.1
Maximum ramp rate max{di/ dt}A/s -125 [Dahl 00]
Maximum turn to turn voltage Umax,tt V 75 [Brun 04, p. 159]
Maximum voltage to ground Umax,ground V 488 [Dahl 01]
Max. voltage between apertures V 150 [Verw 08a]
212 Cases
C.2 LHC Inner Triplet Nested Dipole - MCBX
The MCBX consists of two nested independently powered dipoles. We con-
sider the outer dipole only. The quench simulation is carried out for a magnet
of the pre-series, i.e. the MCBXT horizontal dipole [Karp 08a].
C.2.1 Strand and Cable
The magnet is wound from a 7-strand ribbon-type conductor using a strand
of the LHC MB inner layer cable (see Sec. C.1.1) rolled into a rectangular
form (no change of properties is assumed). Each strand with cross-section
1.53 x0.85 mm is insulated by a layer of 0.06 mm PVA. The strands are glued
on the short edge to the cable [Karp 08a].
C.2.2 Magnet Data
The strands are connected in one coil end such that each radial layer of strands
is connected in series. Figure C.6 shows the coil cross-section and the winding
scheme of the magnet [Karp 08a]. All magnet parameters are given in Tab.
C.8.
The connection of the strands allows for radial and azimuthal quench prop-
agation within a coil block and for longitudinal propagation within each radial
layer of strands. The coil is fully impregnated so that no cooling needs to be
considered. The magnet is operated at 1.9 K [Brun 04, p. 245].
The MCBX consists of two nested dipole coil layers surrounded by an alu-
minum shrinking cylinder, a laminated iron yoke, a stainless-steel outer shell
and an end plate, which supports the electrical connection [Brun 04, p. 244].
The magnet cross-section is shown in Fig. C.7. The inner radius of the yoke
is 90.2 mm and the outer radius 164.6 mm [Karp 08a].
For the present simulations, the magnet is neither protected by quench
heaters nor a dump resistor. The voltage across the power supply is neglected.
C.2 LHC Inner Triplet Nested Dipole - MCBX 213
Table C.8: LHC MCBX Magnet Data
Quantity Symbol Unit Value Ref.
Material
Wedge material: copper with thick layer of G10 as
insulation
Quench stopper: each layer is connected to the next
one on the outside. The connection
is covered by a copper plate and
immersed to liquid helium
Cooling surfaces: cooling to cold bore tube
(distance around 1mm)
Iron yoke: magnetic iron Sec A.1.6.1
Number of apertures – – 1
Number of poles – – 2
Number of blocks per pole – – (3)
Number of conductors per pole Ncp – 406 [Karp 08a]
Coil length `cm 0.5 [Karp 08a]
Average winding length `wm 0.5 [Karp 08a]
Magnetic length – m 0.38 [Karp 08a]
End part of coil `ce m –
!"#$%&'()*
Figure C.6: LHC MCBX corrector magnet coil cross-secion and winding scheme. (left)
winding scheme for one coil. The current enters the coil at the top in the inner layer.
From there the coil is wound down to the mid-plane in a radial layer. The next layer is
connected from the mid-plane to the coil top. This connection is done outside the magnet
and functions as quench stopper. (right) The second coil is identically wound and connected
on the mid-plane.
214 Cases
4
magnets were aligned to each other by means of dowel pins
in the head spacers and wrapped with fiber-glass. After
curing the completed assembly the outermost fiber-glass
bandage was turned to a precise dimension. The collapsible
mandrel was extracted and the aluminium shrink rings were
fitted on. The radial interference of 0.075 mm resulted in 20-
25 MPa circumferential stress in the coils. The shrunk
assembly showed some elliptic deformation and therefore the
ID of the yoke laminations was made 0.4 mm larger. These 2
mm thick Fe37 plates were stacked around the coil assembly,
the blocking keys (20 x 20 mm) were mounted, and a 15 mm
thick stainless steel cylinder was shrink-fitted around the
yoke with a radial interference of 0.23 mm. Each lamination
is designed to support the coils radially in one azimuthal
direction only [7]. This is made by off-centring the hole in
the lamination by 1 mm with respect to the outer boundary.
By sequentially stacking four laminations at angular
orientations of 0, 90, 180, 270 degrees respectively the coils
can be effectively supported and centred. The laminations
move inwards during the cooldown and the blocking keys
stop the movement at a pre-defined temperature building-up
a circumferential stress in the stainless steel outer shell.
Finally, the soldered series connections were made radially
on the G10 end-plate. Wires were brought together under
pressure and soldered within a groove on a pre-tinned Cu-
bar. A conservative overlap of 100 mm per joint was chosen
to minimise the heat load.
C.
FE-Analysis
The assembly parameters were studied with an FE-model
in ANSYS! [8]. The interferences were introduced using gap
elements between the coil assembly and the aluminium
cylinder, between the aluminium cylinder and the yoke, and
between the yoke and the outer shell. Friction was not taken
into account. The interference between the outer shrink ring
and the yoke lamination was simulated by defining initial
gap conditions along this boundary as a function of the
angular position of the gap element. One horizontal
lamination and one lamination acting in vertical direction
were modelled. Their “counter-laminations” with a
movement in opposite direction were simulated by coupling
the nodes on the outer radius of the aluminium cylinder and
on the inner radius of the outer shell respectively over their
diagonals. The design variables consisted of the interference
between the coils and the inner shrink ring, the offset of the
hole and the OD of the yoke laminations, and the play in the
blocking keys at RT.
The evolution of the azimuthal coil stress for 8 load steps
is illustrated in Fig. 5. The clamping system is designed to
sustain the magnetic forces at nominal field. It is not rigid
enough to resist the deformations at excitation levels close to
the short-sample current, where the forces are four times
higher. In this case the elliptic deformation of the inner coil,
in particular, would result in locally high tensile and
compressive stresses and would thereby risk to damage the
coils. It is therefore not advisable to train the magnets to
their critical current.
Fig. 4 Cross-section of the MCBX-magnet. 1. Inner Coil, 2. Outer Coil, 3. Bronze coil spacer, 4. Fiber-glass insulation, 5. Al. shrink ring, 6. Iron (Fe37) Yoke
laminations, 7a. Vertical, and 7b. Horizontal blocking keys, 8. St. Steel outer shell, 9. End plate for series connections (G10).
-100
-80
-60
-40
-20
0
20
40
After 1st Shrink
fit
After Yoke
Assembly
At 1.9 K
Inner I=511 A
Outer I=599 A
Inner I=362 A,
Outer I=424 A
Inner I=1057A
Outer I=1113 A
Inner I=511 A,
Outer I=599 A
""##[MPa]
Inner Max
Inner Min
Outer Min
Outer Max
Fig. 5. Evolution of minimum and maximum azimuthal coil stresses at RT,
after cooldown, and magnet energised
.
Figure C.7: LHC MCBX magnet cross-section (old design) [Karp 99]. 1. Inner Coil,
2. Outer Coil, 3. Coil spacer, 4. Fiber-glass insulation, 5. Shrink ring, 6. Iron yoke
laminations, 7a. Vertical, and 7b. Horizontal blocking keys, 8. Outer shell, 9. End plate
for series connections. Courtesy of M. Karppinen TE MCS.
C.3 LHC Inner Triplet Upgrade Quadrupole - MQXC 215
C.3 LHC Inner Triplet Upgrade Quadrupole -
MQXC
C.3.1 Strand and Cable
For the Inner Triplet Upgrade Quadrupole the same cables as for the LHC MB
will be used. The parameters are given in Sec C.1.1 and C.1.2, respectively.
By sorting and picking the best performing cables, the critical current is
expected to be larger than for the LHC MB. Therefore, the fit parameters of
the critical current density parameterization (see Tab. A.17) are modified.
The critical current as well as the updated fit parameters are given in Tab.
C.9.
C.3.2 Magnet Data
The MQXC features a 120 mm aperture. The magnet coil consists of 4 poles,
each built in two-layers. The inner layer consist of two coil blocks as well as
the outer layer (where one block is made from one conductor only). The coil
cross-section and winding scheme are shown in Fig. C.8. All parameters are
given in Tab. C.10.
For the first study a round iron yoke with inner radius of 129 mm and outer
radius of 275 mm is considered. The differential inductance varies less than
5% over excitation due to iron saturation.
C.3.3 Electrical Circuit
The magnet is connected to a power supply and a dump resistor. In case
of a quench, the power supply is switched of and bridged by a diode. All
parameters are given in Tab. C.11.
C.3.4 Magnet Protection
The quench protection consists of a quench detection system, a dump resis-
tor and quench heaters placed on the outer layer of the coil as indicated in
Fig. C.8. The threshold voltage of the detection system is 0.1 V. Quench
heaters are fired after a delay of 10 ms for signal validation. The dump resis-
tor is switched in the same instance. All parameters of the magnet protection
system can be found in Tab. C.12.
C.3.5 Operating Conditions / Critical Values
The operation conditions and current levels of the MQXC are given in Tab.
C.13. For orientation some critical values are given, too.
216 Cases
Table C.9: MQXC Cable Parameters. All other parameters are identical with the LHC
MB.
Quantity Symbol Unit Inner Layer Outer Layer Ref.
Critical current IcA
at 10 T,1.9 K 14800 [Osto 08]
at 9 T,1.9 K 14650 [Osto 08]
Linearization dIc/dBAT−15040 4050 [Osto 08]
Fit parameters,
C0– T 29.47 30.94
β– – 1.044 1.044
Table C.10: Inner Triplet Upgrade MQXC Magnet Data
Quantity Symbol Unit Value Ref.
Wedge material: copper
Quench stopper: none
Cooling surfaces: not considered
Iron yoke: magnetic iron Sec A.1.6.1
Number of poles – – 4 [Borg 08]
Number of blocks per pole – – 4 [Borg 08]
Number of conductors per pole Nc– 36 [Borg 08]
Coil length lcm 10.3 [Fess 08]
Average winding length lwm 10.3 [Fess 08]
Magnetic length – m 10.0 [Fess 08]
End part of coil Ice m 0.3 [Fess 08]
Table C.11: Inner Triplet Upgrade MQXC Electrical Circuit
Quantity Symbol Unit Value Ref.
Power supply
Switch off delay ∆tQT ms 0 [MQXC 08b]
Crowbar forward resistance RfDfR Ω0 [MQXC 08b]
Crowbar forward voltage UfDTh V 1 [MQXC 08b]
Table C.12: Inner Triplet Upgrade MQXC Magnet Protection
Quantity Symbol Unit Value Ref.
Quench detection
Detection voltage Udet V±0.1[MQXC 08a]
Discrimination delay ∆tDis ms 10 [MQXC 08a]
Quench Heater Model
Initial power density/conductor PQH0 Wm−130
Heat transfer delay ∆tQH0 ms 20
Time constant (power) τQH ms 37.5 [Osto 08]
Quench-heater delay see LHC MB [MQXC 08a],
Sec. C.1.5
Dump resistor
Resistance RDR mΩ40 [MQXC 08a]
Switching/commutation delay ∆tDR ms 0 [MQXC 08a]
C.3 LHC Inner Triplet Upgrade Quadrupole - MQXC 217
MQXC 120mmV3
!"#$
!"%$
!"&$
!"'$
Figure C.8: MQXC coil cross section [Borg 08], winding scheme and quench heater layout
[Fess 08].
Table C.13: Inner Triplet Upgrade MQXC Operating Conditions and Critical Values
Quantity Symbol Unit Value Ref.
Operation temperature TbK 1.9 [Siem 05]
Nominal current Inom A 12580 [Fess 08]
Ultimate current Iult A 15720 [Fess 08]
Maximum temperature Tmax K 600 [MQXC 08a],
Sec. A.1.1
Maximum turn to turn voltage Umax,tt V 75 LHC MB Sec. C.1.6
Maximum voltage to ground Umax,ground V 250 [MQXC 08a]
218 Cases
C.4 Fast Ramping Dipole Magnet
C.4.1 Strand and Cable
The strand consists of a Nb-Ti filament of 3.5µm embedded in a copper
matrix. The strand parameters can be found in Tab. C.14.
The parameters of the Rutherford-type cable with stainless steel core are
given in Tab. C.15. The geometry of the cable is identical to the LHC MB
outer layer cable (Sec. C.1.2).
C.4.2 Magnet Data
The fast ramping dipole coil consists of two poles, each built in two-layers.
The inner layer consist of 4 coil blocks and the outer layer of two. The coil
cross-section and winding scheme are shown in Fig. C.10. All parameters are
given in Tab. C.16.
C.4.3 Electrical Circuit
The magnet is only connected to a power supply with by-pass thyristor and
a dump resistor. In case of a quench, the power supply can be inverted in
order to drive down the current.
C.4.4 Magnet Protection
The quench protection consists of a quench detection system, a dump resistor
and quench heaters placed on the outer layer of the coil as indicated in Fig.
C.10. The threshold voltage of the detection system is varied from 0.1 to 1.0 V.
Quench heaters are fired after a delay of 10 ms for signal validation. The dump
resistor is switched in 50 ms after quench validation. All parameters of the
magnet protection system can be found in Tab. C.17.
C.4.5 Operating Conditions / Critical Values
The magnet is operated at 4.7 K. The ramp cycle is given in [Kozu 06] as an
up-ramp from 1.6T to 6 T in 4.4s, followed by a plateau at 6T of 11 s, and
a down ramp to 1.6 T in 4.4 s. Critical values, e.g., maximum temperatures,
are taken from the LHC MB in Sec. C.1.6.
C.4 Fast Ramping Dipole Magnet 219
Table C.14: Fast Ramping Dipole Strand
Quantity Symbol Unit Value Ref.
Materials
Matrix Copper Sec. A.1.3.1, Sec. A.1.4.1,
Sec. A.1.5.1
Coating Stabrite
Geometry
Number of filaments Nf– –
Radius rsmm 0.4125 [Kozu 06, p. 5]
Core radius rsc mm –
Coating thickness tsc µm 0.5[Kozu 06, p. 5]
Twist pitch psmm 5[Kozu 06, p. 5]
Model parameters - Electrical / Inter-filament coupling losses
Copper/Super- λ–1.4[Kozu 06, p. 5]
conductor ratio
Copper RRR RRR – 278
Copper matrix ρIFCC Ωm4·10−10 [Kozu 06, p. 5]
resistivity
Copper matrix dρIFCC
dBΩm/T 9·10−11 [Kozu 06, p. 5]
resistivity B
dependence
Filling factor ηs–0.5
Table C.15: Fast Ramping Dipole Cable
Quantity Symbol Unit Value Ref.
Materials
Insulation kapton Sec. A.1.4.4,
Sec. A.1.5.4
Cable Core stainless steel
Inner void filling helium Sec. A.3
Outer void filling –
Geometry
Number of strands Ns–36 [Kozu 06, p. 5]
Insulation Thickness
radial tir mm –
azimuthal tia mm –
Cable
Geometry see LHC MB outer layer Sec. C.1.2
Core thickness tcc µm 25 [Kozu 06, p. 5]
Keystone angle φkey o–
Twist pitch `pmm 100 [Kozu 06, p. 5]
Cabling angle βpo
Model parameters - Thermal / Inter-strand coupling losses
Helium in voids – % 10
Cross-over resistance RcmΩ 20 [Kozu 06, p. 5]
Adjacent resistance RaµΩ 200 [Kozu 06, p. 5]
220 Cases
Figure C.9: Fast ramping dipole iron yoke and magnet cross-section after [Kozu 06, p.
10].
Table C.16: Fast Ramping Dipole Magnet Data
Quantity Symbol Unit Value Ref.
Material
Wedge material: copper
Quench stopper: none
Cooling surfaces: cooling to cold bore tube and collar
(distance around 1mm)
Iron yoke: magnetic iron Sec A.1.6.1
Number of apertures – – 1
Number of poles – – 2
Number of blocks per pole – – 6
Number of conductors per pole Ncp – 71
Coil length `cm 1
Average winding length `wm 1
Magnetic length – m 0.75
End part of coil `ce m –
C.4 Fast Ramping Dipole Magnet 221
!"#$
!"%$
&'()*+,-.$
Figure C.10: Fast Ramping Dipole coil cross section, winding scheme and quench heater
layout.
Table C.17: Fast Ramping Dipole Magnet Protection
Quantity Symbol Unit Value Ref.
Quench detection
Detection voltage Udet V0.1−1.0Study
Discrimination delay ∆tDis ms 10
Quench Heater Model
Initial power density per conductor PQH0 Wm−1200
Heat transfer delay ∆tQH0 ms 60
Time constant (power) τQH ms 37.5
Quench-heater delay (at nominal current) ∆tQH ms around 90
Dump resistor
Resistance RDR mΩ200
Switching/commutation delay ∆tDR ms 50
222 Cases
D Digressions
Hier stehe ich, ich kann nicht anders
Martin Luther
(1483-1546)
D.1 Surface-Charge of a Discontinuity of
Resistivity
Given is a homogeneous current density Jflowing perpendicularly through
the interface of two half-spaces of different electrical resistivity ρE1and ρE2.
The entire space is of homogeneous electric permittivity 0and magnetic
permeability µ0(see Fig. D.1).
For a continuous current density J, the electrical field Eas well as the
electric displacement field Dshow a discontinuity at the interface, due to:
E1=ρE1J,D1=0ρE1J,(D.1)
E2=ρE2J,D2=0ρE2J.(D.2)
From Maxwell’s equation,
I∂V
D·da=ZV
qVdV=Q, (D.3)
this gives rise to a surface charge density qF,
qF=J0(ρE1−ρE2).(D.4)
Normalizing Eq. (D.4) to J, gives the charge per current, Q/I, independent
of the surface area and only depending on the material properties ε0,ρE1and
ρE2,
Q
I=0(ρE1−ρE2).(D.5)
Considering the interface between an ideal superconductor with zero re-
sistivity and copper (at ambient temperature, ρE,Cu = 1.553 ·10−8Ωm), the
charge per current yields only Q/I = 1.375 ·10−19 CA−1. Compared to the
elementary electrical charge, i.e. em= 1.602 ·10−19 C, this is less than one
electron!
223
224 Digressions
!" #"
$%&"
$%'"
!" #"
!" #!" $!"
$%"
#%"
!("
!("
)*+,-./"01-+2/"3/45467589":54.;<6<*589"
=>>/<?5@A:52+/445;<4A#$%$&'($)&*"
Figure D.1: Surface charge density due to a discontinuity of the electrical resistivity: (left)
homogeneous current density, (right) discontinuous electrical field and field of electrical
displacement.
Conclusion: Taking into account the discrete nature of the electric charge,
this example clearly exceeds assumptions of continuous, homogenized matter
in Maxwell’s theory! Nevertheless, the result shows that the surface charge
can be totally neglected for any material with a resistivity well below 1 Ωm.
D.2 An Arbitrarily Cut Cylinder
A circular cylinder of radius ais cut by a plane under an arbitrary angle
β. The geometry is shown in Fig. D.2 (left) and (center). In a cartesian
co-ordinate system parallel to the cylinder axis, the equation of the two in-
tersecting objects read:
1 = x
a2+y
a2,(D.6)
z
x= tan (β).(D.7)
The line of intersection is given by
x(ϕ) = acos (ϕ) (ex+ tan (β)ez) + asin (ϕ)ey
=ak cos (ϕ)ek+asin (ϕ)ey,(D.8)
with ekparallel to the plain of intersection and orthogonal to ey. Equation
(D.8) can be identified as the coordinates of an ellipses. The major half-axis
is given by b,
b=ak =aq1 + tan (β)2=a
cos (β),(D.9)
and the minor half-axis by a. The surface area of the ellipse is given by
Aellipses =πab =πa2
cos (β).(D.10)
The volume of a cylinder terminated by two parallel cuts, is either given by
the length of the axis `axis times the surface area of the circular cross-section
or by its height `height times the surface area of its elliptical base [Bron 00, p.
160]. Note that `height =`axis/cos (β). See Fig. D.2 (right).
D.3 Critical Current Density Relies on the Model of the Superconductor 225
!"
#"
$"
!"
$"
%"
!"
!"
#"
%"
!"
!"#$%&'()*+%
%"
,--'(.#"/0!1!2'*'1$/!""#$%&%'()*
Figure D.2: Arbitrarily cut circular cylinder. (left) elliptical cut shape. (right) volume
of a cylinder cut by two parallel planes.
D.3 Critical Current Density Relies on the
Model of the Superconductor
The critical current density Jcis not a unique property of the superconducting
material but a parameter of the model used to describe the superconductor.
In general, current density as such evades direct measurement. By measur-
ing the current flowing through special cross-sections it is possible to conclude
on the local current distribution. In case of superconducting filaments, the
small diameter of only some micro meters does not allow to perform such
studies. Hence the only quantities which can be measured are the critical
current and the superconductor magnetization.
As explained in Sec. A.2.4, the measurement of both quantities is difficult
and suffers from technical limitations. Typically only parts of the field and
temperature range can be accessed.
Under the assumption of a local current distribution the global quantity
critical current as well as the secondary quantity magnetic moment can be
converted to the critical current density, see Fig. D.3 (left). The current
distribution is a property of the used model of the superconductor. There-
fore, this conversion must be performed differently for every model and con-
sequently the same set of measurement data yields different results for the
critical current density!
Disregarding this fact, most often analytical formulae are used for the con-
version. Before showing a general method of conversion the analytical formu-
lae shall be derived and discussed.
For the critical current measurement a homogeneous distribution of current
over the conductor cross-section is assumed. This allows to calculate the
critical current density to
Jc=Ic
ASC
,(D.11)
with ASC the superconducting cross-sectional area. Calculating the self-field
226 Digressions
!!"
"#$%" &" !" #"
'()*+,(-($." /,)$*01,-)21$"
)**+-#$3")"
!+,,($."%#*.,#4+21$"
')3$(25)21$" 6,#2!)7"6+,,($."
6,#2!)7""
6+,,($."
8($*#.9"
:!"
:!"
;<<($%#=>8#3,(**#1$*>!"#$%&'!("")*+,)*-#+./01)'2&('+3
#$
%$
Figure D.3: (left) Transformation of the measured critical current and superconductor
magnetization to the critical current density. The conversion relies on an assumed cur-
rent distribution. (right) Current distribution in a superconducting strand exposed to an
external magnetic field for two different models: (top) Wilson model [Wils 83, pp. 165],
(bottom) intersecting-ellipses model [Voll 02].
of the applied current, the influence on the field dependence can be corrected
[Bout 06]. This approach neglects the influence of screening currents .
According to the Bean model [Bean 62, Bean 64] of superconductor mag-
netization, the superconductor is screened from an external field by a layer
of screening currents. The current density of the screening currents always
equals the critical current density. The layer thickness varies with the applied
external field. When reaching a certain field level the screening currents cover
the full sample. This state is denoted as fully penetrated.
The Wilson model [Wils 83, pp. 165] applies this approach to a circular
filament assuming screening currents with an elliptical inner boundary. With
increasing field the minor axis of the ellipses decreases and equals zero in
fully penetrated state. At this point the current distribution is given by to
half-circles of homogeneous current density. The magnetic moment per unit
length, m0, of this geometry depends only on the radius aand the current
density. Re-arranging yields the critical current density to
Jc=m03
4
1
a3,(D.12)
see [Le N 99] and [Voll 02, p. 35]
If the model of superconductor magnetization does not rely on a homoge-
neous current density as shown in Fig. D.3 (right), the magnetic moment has
to be computed from the integral of the local current density Jmag,
m=ZVsample
M(r0)dV0=ZVsample
r0×Jmag(r0)dV0,(D.13)
and the conversion can not be performed analytically. Here Mdenotes the
D.3 Critical Current Density Relies on the Model of the Superconductor 227
local magnetization1,Vsample the sample volume, r0a position within the
sample cross-section, and dV0a differential volume element. This poses
an inverse problem and the critical current density can only be determined
iteratively:
!"#$%&
!'()&
!(*()(+,-"*&
!)$,'&
!./*&
0(12$''("*3!"#$"%&'()&*'(+,-.*/''&01%&02(13*"''&,1.34
The function of critical current density would then be defined as the function
which minimizes the differences between measured and simulated magnetic
moment. The same applies for the critical current transformation.
The intersecting-ellipses model is based on the assumption of screening
currents varying over the strand cross-section with local magnetic induction.
Regarding a fully penetrated sample this variation is nevertheless small. The
error caused by a critical current density assuming homogeneous current den-
sities should be small. For lower fields that approach is inconsistent and might
result in undesired limitations of the model!
1Note: Magnetization Mis also a local quantity and evades direct measurements.
228 Digressions
E Formulary and Constants
E.1 Formulary
E.1.1 Inductance of a Ring Conductor
A conductor of radius bis bent to a circle of radius athus forming a single
turn. With aand bboth in cm, the geometrical inductance in µH is given by
[Grov 04, p. 143 (119b)]:
L= 0.004πa ln 8a
b−1.75,[L] = µH.(E.1)
E.1.2 Field of a Long Solenoid
The field along the axis of an iron-free, long solenoid can be estimated by
means of Ampere’s law:
B=µ0
N
hI, [B]=T,(E.2)
where Ndenotes the number of windings over the solenoid height hin meter
and Ithe winding current in ampere.
E.2 Constants
Table E.1: Constants used throughout this work
Constant Symbol Value Unit
Critical electrical field Ec10−5Vm−1
Permittivity of empty space 08.8541 ·10−12 AsV−1m−1
Permeability of empty space µ04π·10−7VsA−1m−1
Speed of light in empty space c01
√µ00= 2.9979 ·108ms−1
Elementary electrical charge em1.602 ·10−19 As
Planck’s constant hp0.626 ·10−34 Js
Flux quantum Φ0hp
2em= 2.067 ·10−15 Vs
Boltzmann constant kB1.38065 ·10−23 J K−1
Gas constant Rgas 8.31 J mol−1K−1
Avogadro number NA6.022 ·10−23 mol−1
229
230 Formulary and Constants
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List of Figures
1.1 Destroyed coil windings of the LHC main bending magnet
MB3004 after an inter-turn short. . . . . . . . . . . . . . . . . . 2
2.1 CAD image of the superconducting LHC main bending magnet. 8
2.2 CAD image of a double aperture coil configuration. Quadrant
of one aperture of the coil cross-section of the LHC MB. . . . . 9
2.3 LHC MB strand and cable. . . . . . . . . . . . . . . . . . . . . 10
2.4 Photo of the coil cross-section of the LHC MQY. . . . . . . . . 11
2.5 Sketch of other superconducting magnet configurations. . . . . 14
2.6 Different phenomena in superconducting magnets which may
causeaquench............................ 15
2.7 Comparison of the electrical resistivity of the different materials. 16
3.1 Different models interacting in a quench simulation. . . . . . . 22
3.2 Numbering schemes. . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 Currentsharing. .......................... 25
3.4 Magnetic field computation: Domain subdivision and field re-
composition.............................. 26
3.5 Calculation of the field of a conductor. . . . . . . . . . . . . . . 27
3.6 Generic electrical network model. . . . . . . . . . . . . . . . . . 31
3.7 Diodemodel. ............................ 32
3.8 Longitudinal cut through the discretized coil winding with lumped
electricalelements.......................... 35
3.9 Finite volumes and linear/constant approximations for the ther-
malnetworkmodel. ........................ 37
3.10 Lumped thermal network model in comparison to the the coil/-
conductorgeometry. ........................ 37
3.11 Comparison of the thermal conductivity and volumetric spe-
cific heat of all relevant materials. . . . . . . . . . . . . . . . . 39
3.12 Influence of liquid helium in the coil windings on the thermal
model................................. 41
3.13 Quench heater model. . . . . . . . . . . . . . . . . . . . . . . . 42
3.14 Block diagram of the quench simulation algorithm. . . . . . . . 45
4.1 LHC MB. Schematic coil cross-section and iron yoke. Outer
iron diameter 570 mm........................ 47
4.2 LHC MB coil layout and electrical circuit. . . . . . . . . . . . . 48
4.3 LHC MB average cable losses and quench heater delays. . . . . 49
4.4 LHC MB reproduction of current measurements. . . . . . . . . 50
XVII
XVIII List of Figures
4.5 LHC MB reproduction of measured pole voltages. . . . . . . . . 51
4.6 LHC MB temperature margin for quench in the tunnel. . . . . 53
4.7 LHC MB 3D temperature plot for a quench in the tunnel. . . . 53
4.8 LHC MCBX. Schematic coil cross-section and iron yoke. Outer
iron diameter 330 mm........................ 54
4.9 LHCMCBXMagnet. ....................... 54
4.10 LHC MCBX temperature margin. . . . . . . . . . . . . . . . . 55
4.11 LHC MCBX current decrease and temperature distribution. . . 56
4.12 MCBX quench propagation over outer coil. . . . . . . . . . . . 56
4.13 LHC MCBX potential to ground during a quench. . . . . . . . 57
4.14 LHC MCBX voltage along the coil winding and reason to quench. 57
4.15 LHC MQY. Schematic coil cross-section and iron yoke. Outer
iron diameter around 495 mm.................... 58
4.16LHCMQYmagnet. ........................ 58
4.17 LHC MQY current decrease and temperature margin to quench. 59
5.1 Inner Triplet Upgrade MQXC schematic coil cross-section and
iron yoke. Outer iron diameter 550 mm. . . . . . . . . . . . . . 61
5.2 Inner Triplet Upgrade MQXC conductor numbering and cur-
rent and temperature change for an unprotected quench. . . . . 63
5.3 Inner Triplet Upgrade MQXC temperature margin over con-
ductors versus time for an unprotected quench. . . . . . . . . . 64
5.4 Inner Triplet Upgrade MQXC temperature over the coil cross-
section for an unprotected quench. . . . . . . . . . . . . . . . . 64
5.5 Inner Triplet Upgrade MQXC quench heater layouts and per-
formance. .............................. 65
5.6 Inner Triplet Upgrade MQXC hot-spot temperature for differ-
entheatersetups........................... 67
5.7 Inner Triplet Upgrade MQXC temperature margin over con-
ductors versus time for different heater setups . . . . . . . . . . 67
5.8 Inner Triplet Upgrade MQXC dump resistor study in case of a
quench and no pre-existing quench. . . . . . . . . . . . . . . . . 68
5.9 Inner Triplet Upgrade MQXC dump resistor study in case of
quench for various dump resistors. . . . . . . . . . . . . . . . . 69
5.10 Inner Triplet Upgrade MQXC energy extraction with dump
resistor. ............................... 69
5.11 Inner Triplet Upgrade MQXC comparison of the three different
protectionmethods. ........................ 70
5.12 Fast-Ramping Dipole. Schematic coil cross-section and iron
yoke. Outer iron diameter 504.6 mm................ 71
5.13 Fast-Ramping Dipole standard and modified coil cross-section. 72
5.14 Fast-Ramping Dipole temperature variation during ramp-cycle.
Current and peak temperature in a magnet during an up-ramp
tothequench............................ 73
5.15 Fast-Ramping Dipole temperature margin to quench versus
timeforanup-ramp......................... 74
List of Figures XIX
5.16 Fast-Ramping Dipole current decay and peak-temperature for
quenches during the up-ramp phase. . . . . . . . . . . . . . . . 75
5.17 Current- and peak-temperature evolution for quenches during
thedown-ramp ........................... 75
5.18 Fast-Ramping Dipole peak-temperature and current evolution
duringquenches. .......................... 76
7.1 Marginstoquench.......................... 82
7.2 Temperature margin to quench. . . . . . . . . . . . . . . . . . . 83
7.3 Critical temperature as a function of applied current density
and local magnetic induction. Margin on the load-line. . . . . . 84
7.4 Energy reserve density as function of temperature and excitation. 85
7.5 Energy margin to quench for the LHC MB. . . . . . . . . . . . 86
7.6 MIITs computation. . . . . . . . . . . . . . . . . . . . . . . . . 88
7.7 Energy exchange between different physical systems and dif-
ferentvolumes. ........................... 90
7.8 Self inductance geometry. . . . . . . . . . . . . . . . . . . . . . 94
7.9 Inductance of a bulk or Rutherford-type conductor. . . . . . . . 97
7.10 ROXIE simulation of the differential inductance of the LHC
main bending magnet considering field dependent and hys-
tereticmaterials. .......................... 98
7.11 Differential inductance of a magnet with time-transient effects . 99
7.12 Differential inductance by means of hysteresis. . . . . . . . . . 99
7.13 Simplified electrical networks containing superconducting and
normal conducting elements. . . . . . . . . . . . . . . . . . . . 101
7.14Current-sharing...........................103
7.15 Model problem circuit and solution of the initial value problem. 104
7.16 Currents induced in a superconducting current loop with resis-
tive joint. Geometry and model of the superconductor. . . . . . 106
7.17 Superconductor with resistive joint. . . . . . . . . . . . . . . . . 107
7.18 Graphical solution for current sharing. . . . . . . . . . . . . . . 108
7.19 Inter-filament coupling currents. . . . . . . . . . . . . . . . . . 111
7.20 Model for inter-strand coupling currents. . . . . . . . . . . . . . 112
7.21 Critical state model geometry and excitation current. . . . . . . 116
7.22 Hysteresis losses: Screening current density and inner magnetic
field..................................118
7.23 Hysteresis losses: Magnetic induction and flux. . . . . . . . . . 119
7.24 Hysteresis losses: Time derivative of the magnetic induction
andelectricalfield..........................119
7.25Hysteresisloop............................119
7.26 Hysteresis losses: Energy and power versus time. . . . . . . . . 121
7.27 Rutherford-type cable. . . . . . . . . . . . . . . . . . . . . . . . 123
7.28 Current distribution over a Rutherford-type cable. . . . . . . . 124
7.29 Field and temperature over a twisted cable and field decompo-
sition. ................................125
7.30 Scheme of quench protection methods. . . . . . . . . . . . . . . 127
XX List of Figures
7.31 Quench detection methods . . . . . . . . . . . . . . . . . . . . . 128
7.32 Voltage detection conditions for the LHC MB . . . . . . . . . . 130
7.33 LHC MB dump resistor and cold diode. . . . . . . . . . . . . . 131
7.34 Quench protection by subdivision or coupled secondary . . . . 133
7.35 Quench heater electrical circuit. Quench heater strip layout
withcopperplating. ........................134
7.37 Sequence of events during a quench with quench protection. . . 137
7.38 LHC magnet test station and main dipoles in the LHC tunnel. 138
7.39 Electrical circuit of the LHC tunnel configuration . . . . . . . . 139
7.40 Voltage development over the terminals of a quenched magnet
in the LHC string. Voltages and time intervals are not to scale. 140
7.41 Electrical circuit and terminal voltage of a magnet on the test
bench ................................142
7.42 Coil voltages recorded during a quench of an LHC main dipole
onthetestbench. .........................143
7.43 Electrical circuit model of the quench heater induced voltage
spikes.................................144
7.44 ROXIE model for the calculation of the inductance matrix.
Simulated voltage spikes. . . . . . . . . . . . . . . . . . . . . . 145
7.45 Model circuit for the explanation of voltages induced by asyn-
chronousquenching.........................147
7.46 Model of the resistance growth in a half coil due to quench
heaterfiring.............................147
7.47 Voltage jumps on coil voltages . . . . . . . . . . . . . . . . . . 147
7.48 Voltage along the magnet winding. . . . . . . . . . . . . . . . . 149
7.49 Influence of the winding scheme on the voltage to ground. . . . 150
7.50 Capacitive effects on the voltage distribution over the coil wind-
ing...................................151
A.1 Temperature levels. . . . . . . . . . . . . . . . . . . . . . . . . . 155
A.2 Electrical resistivity of copper . . . . . . . . . . . . . . . . . . . 160
A.3 Electrical resistivity of copper . . . . . . . . . . . . . . . . . . . 161
A.4 Electrical resistivity of Nb-Ti and Nb3Sn in the normal con-
ductingstate ............................162
A.5 Thermal conductivity and Lorenz number of copper. . . . . . . 166
A.6 Thermal conductivity of Niobium-Titanium and Niobium-3-Tin.166
A.7 Thermal conductivity of Polyimide (Kapton). . . . . . . . . . . 167
A.8 Normalized specific heat of solids after Debye. . . . . . . . . . . 168
A.9 Volumetric specific heat of copper. Specific heat of Kapton . . 170
A.10 Volumetric specific heat of Niobium-Titanium and Niobium-3-
Tin. .................................170
A.11 General BH-curve. . . . . . . . . . . . . . . . . . . . . . . . . . 173
A.12 Relative permeability as a function of magnetic induction . . . 174
A.13 Analytical µr-andBH-curve. ...................176
A.14 Superconductor magnetization of type I, type II and hard su-
perconductor.............................178
List of Figures XXI
A.15 Voltage-current-curve for a strand of the LHC MB outer layer
cable and variation of the voltage over a superconductor de-
pending on the n-value. . . . . . . . . . . . . . . . . . . . . . . 180
A.16 Critical current density measurement limitations. . . . . . . . . 183
A.17 Critical current density of Niobium-Titanium. . . . . . . . . . . 184
A.18 Critical current density of Niobium-3-Tin. . . . . . . . . . . . . 186
A.19 Resistivity of Nb-Ti and Nb3Sn in superconducting state. . . . 187
A.20 Volumetric specific heat of Nb-Ti and Nb3Sn in superconduct-
ingstate. ..............................188
A.21 Phase diagram of helium. . . . . . . . . . . . . . . . . . . . . . 189
A.22 Assumption of the pressure rise over temperature and resulting
approximated density of helium. . . . . . . . . . . . . . . . . . 190
A.23 Approximation of the thermal properties of helium with vary-
ingpressure..............................192
B.1 Different cable types. . . . . . . . . . . . . . . . . . . . . . . . . 194
B.2 Contact coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 197
B.3 Definition of coil end and coil end length. . . . . . . . . . . . . 200
B.4 Effective thermal and electrical properties. . . . . . . . . . . . . 202
C.1 LHC MB dipole cross-section. . . . . . . . . . . . . . . . . . . . 207
C.2 LHC MB winding scheme and quench heater layout. . . . . . . 208
C.3 LHC MB numbering schemes. . . . . . . . . . . . . . . . . . . . 208
C.4 LHC MB scheme in the tunnel. . . . . . . . . . . . . . . . . . . 208
C.5 LHC MB quench detection. . . . . . . . . . . . . . . . . . . . . 211
C.6 LHC MCBX corrector magnet coil cross-secion and winding
scheme. ...............................213
C.7 LHC MCBX magnet cross-section. . . . . . . . . . . . . . . . . 214
C.8 Inner Triplet Upgrade MQXC coil cross section, winding scheme
and quench heater layout. . . . . . . . . . . . . . . . . . . . . . 217
C.9 Fast Ramping Dipole iron yoke and magnet cross-section. . . . 220
C.10 Fast Ramping Dipole coil cross section, winding scheme and
quench heater layout. . . . . . . . . . . . . . . . . . . . . . . . . 221
D.1 Surface charge density due to a discontinuity of the electrical
resistivity...............................224
D.2 Arbitrarily cut circular cylinder. . . . . . . . . . . . . . . . . . 225
D.3 Transformation of the measured critical current and supercon-
ductor magnetization to the critical current density. . . . . . . 226
XXII List of Figures
List of Tables
7.1 Assumptions made regarding the quench heater efficiency and
the quench propagation in the magnet . . . . . . . . . . . . . . 146
A.1 Melting temperature of the some materials . . . . . . . . . . . 155
A.2 Mass density of the most common materials . . . . . . . . . . . 156
A.3 Temperatures defining the RRR value according to different
sources. ...............................158
A.4 Electrical resistivity of Niobium-3-Tin versus temperature . . . 162
A.5 Electrical resistivity of some exemplary materials at the ice-
pointofwater............................163
A.6 Thermal conductivity of Niobium-3-Tin . . . . . . . . . . . . . 166
A.7 Maximum thermal conductivity of most common materials . . 167
A.8 Debye temperature of some of the used materials . . . . . . . . 168
A.9 Fit parameters for the volumetric specific heat of copper . . . . 170
A.10 Fit parameters for the volumetric specific heat of Nb-Ti . . . . 172
A.11 Volumetric specific heat of Niobium-3-Tin . . . . . . . . . . . . 172
A.12 Volumetric specific heat of common materials for comparison . 172
A.13 Relative permeability of commonly used materials . . . . . . . 176
A.14 Breakdown field strength of common insulation materials . . . 176
A.15 Transition temperature and critical field of superconducting
elements...............................179
A.16 Different limits for the critical current from common literature 181
A.17 Fit parameters for the critical current density . . . . . . . . . . 184
A.18 Boiling point of different gases . . . . . . . . . . . . . . . . . . 190
A.19 Enthalpy of vaporization. . . . . . . . . . . . . . . . . . . . . . 191
C.1 LHCMBFilament.........................204
C.2 LHCMBStrand ..........................204
C.3 LHCMBCable...........................206
C.4 LHC MB Magnet Data . . . . . . . . . . . . . . . . . . . . . . . 207
C.5 LHC MB Electrical Circuit . . . . . . . . . . . . . . . . . . . . 210
C.6 LHCMBProtection ........................210
C.7 LHC MB Operating Conditions and Critical Values . . . . . . . 211
C.8 LHC MCBX Magnet Data . . . . . . . . . . . . . . . . . . . . . 213
C.9 Inner Triplet Upgrade MQXC Cable Parameters . . . . . . . . 216
C.10 Inner Triplet Upgrade MQXC Magnet Data . . . . . . . . . . . 216
C.11 Inner Triplet Upgrade MQXC Electrical Circuit . . . . . . . . . 216
C.12 Inner Triplet Upgrade MQXC Magnet Protection . . . . . . . . 216
XXIII
XXIV List of Tables
C.13 Inner Triplet Upgrade MQXC Operating Conditions and Crit-
icalValues..............................217
C.14 Fast Ramping Dipole Strand . . . . . . . . . . . . . . . . . . . 219
C.15 Fast Ramping Dipole Cable . . . . . . . . . . . . . . . . . . . . 219
C.16 Fast Ramping Dipole Magnet Data . . . . . . . . . . . . . . . . 220
C.17 Fast Ramping Dipole Magnet Protection . . . . . . . . . . . . . 221
E.1 Constants used throughout this work . . . . . . . . . . . . . . . 229
Notation and List of Symbols
Roman Letters
A(generic) area
Acab,core cross-sectional area of
the cable core
Acab,NC normal conducting
part of the cable
cross-sectional area
Acab,SC superconducting part
of the cable cross-
sectional area
Acab,str cross-sectional area
occupied by the
strands of the cable
Acab,xs cross-sectional area of
the bare cable
Aeff
cab effective cable cross-
sectional area
Afil,coat cross-sectional area of
the filament coating
Afil,core cross-sectional area of
the filament core
Afil,SC superconducting
cross-sectional area of
the filament
Afil,tot total filament cross-
sectional area
Afil,coat cross-sectional area of
the strand coating
Astr,core cross-sectional area of
the strand
Astr,ell elliptical cross-
sectional area of
the strand core
Astr,fil cross-sectional area of
the strand covered by
filaments
Astr,fil cross-sectional area of
the strand covered by
matrix material
Astr,tot total strand cross-
sectional area
Avoid,inner cross-sectional area of
the inner voids of a in-
sulated cable
Avoid,outer cross-sectional area of
the outer voids of a in-
sulated cable
A(generic) magnetic
vector potential
B(generic) magnetic in-
duction
Bccritical field type I SC
Bc1 lower critical field type
II SC
Bc2 upper critical field
type II SC
Bpeak peak field on the con-
ductor
c0speed of light in empty
space
ceff
Veffective volumetric
specific heat
cTvolumetric specific
heat
ceff
Teffective volumetric
specific heat
CQH QH power supply ca-
pacitance
XXV
XXVI List of Tables
Cwkleines Weh am
grossen Zeh
di,n
Eminimum distance in
2D between the con-
ductors iand n.
dQH quench heater insula-
tion thickness
Ddielectric displace-
ment
emelementary electrical
charge
E(generic) electrical
field
Eccritical electrical field
Ebt electrical break-down
field
htime step-size of the
Runge-Kutta method
hccable height
hpPlanck’s constant
hQH quench heater height
∆hcenergy density reserve
/ enthalpy density
margin
H(generic) Magnetic
field
∆Henergy reserve
i(generic) current
inumbering of the con-
ductors in the coil
cross-section
I(generic) current
Iccritical current
Ilin
clinear approximation
of the critical current
dIc
dBlinearized critical cur-
rent change with field
IDcurrent in the by-pass
diode
IEcurrent in the string of
magnets
IMcurrent in the quench-
ing magnet
Iqquench current
ISC,max maximum induced
inter-strand coupling
current
jnumbering of the slices
in longitudinal direc-
tion of the coil
J(generic) current den-
sity
Jccritical current density
JFsurface current den-
sity
JTthermal flux density
∆Jccurrent density mar-
gin
kcoupling factor be-
tween two inductances
kBBoltzmann constant
kicoefficient of the
Runge-Kutta method
kcab,void,inner factor of the inner void
of a Rutherford-type
cable filled with he-
lium
kcab,void,outer factor of the outer void
of a Rutherford-type
cable filled with he-
lium
Ki,n
trans Geometrical transver-
sal contact coefficient
between conductor i
and nconsidering the
surface area of adja-
cent faces and the dis-
tance.
List of Tables XXVII
Ki
cool,α Geometrical transver-
sal contact coefficient
between conductor i
and a cold surface on
face αconsidering the
surface area the dis-
tance.
`(generic) length
`ccoil length
`ce end part of the length
`mag magnetic length
`pRutherford-type cable
twist-pitch length
`QH quench heater length
`waverage winding
length
`sp length of a strand in
a twisted Rutherford-
type cable
Lddifferential inductance
LQH QH parasitic induc-
tance
Lsinductance in series to
the quenching magnet
LWenergy inductance
LΨapparent inductance
M(generic) mutual in-
ductance
M(generic) magnetiza-
tion density
Mddifferential mutual in-
ductance
MIFCC inter-filament cou-
pling current magne-
tization
MISCC inter-strand coupling
current magnetization
MΨapparent mutual in-
ductance
NAAvogadro number
Ncp number of winding
turns per pole
Nenumber of elements in
the thermal model
Nfnumber of filaments in
a strand
Nmag number of magnets in
a string
Nsnumber of strands in a
cable
NSC n-index
Nznumber of longitu-
dinal discretization
steps
pIFCC inter-filament cou-
pling loss density
pISCC inter-strand coupling
loss density
psstrand twist pitch
length
P(generic) power
PQH0 quench heater initial
power
r(generic) distance, ra-
dius
rffilament outer radius
rfc filament inner/core ra-
dius
tfc filament coating thick-
ness
rsstrand outer radius
rsc strand inner/core ra-
dius
r(generic) position/lo-
cus
rffilament radius
rfc filament core radius
rsstrand radius
rsc strand core radius
XXVIII List of Tables
R(generic) resistance
RaRutherford-type cable
adjacent resistance
RcRutherford-type cable
cross-over resistance
RcDfR forward differential re-
sistance of the by-pass
diode
RDR value of the DR
RfDfR forward differential re-
sistance of the free-
wheeling diode
Rgas gas constant
RQresistance of the
quenched part of the
magnet
RQH quench heater strip re-
sistance (at cold)
Lsresistance in series to
the quenching magnet
qquality factor of the
Runge-Kutta method
qVelectrical charge den-
sity
Spoynting vector
t(generic) time
tcc cable core thickness
tdet time of quench detec-
tion
tDiode time when diode
switches
tDiodeHot time when diode
warmed up
tDRcur time when current
commutation to DR
ends
tDRtrig time of switch-in
dump resistor
tfc filament coating thick-
ness
tia azimuthal cable insu-
lation thickness
tir radial cable insulation
thickness
tPSoff time of power supply
off
tqtime when quench
starts
tQB time of quench back in
the magnet
tQHeff time when quench
heater shows to be
effective
tQHfire time when QH fire
toff time when current in
the quenched magnet
decayed to zero
toff,all time when current in
the string of magnets
decayed to zero
tsc strand coating thick-
ness
tval time of quench valida-
tion
∆tcDh time for the cold diode
to heat up
∆tDis discrimination du-
ration for quench
detection
∆tDR time between DR trig-
ger and end of current
commutation
∆tDRcd DR commutation du-
ration
∆tDRtd DR trigger delay
∆tQD time between quench
start and quench val-
idation
∆tQH QH delay, i.e. The
time between Þring
the quench heater cir-
cuit and the detection
of a heater quench in
one of the covered con-
ductors
List of Tables XXIX
∆tQH0 quench heater initial
delay
∆tQHs QH trigger scattering
∆tQHtd QH trigger delay
∆tQP duration of quench
process
∆tQT trigger delay
T(generic) temperature
Tbbath temperature
Tccritical or transition
temperature
Tcs current sharing tem-
perature
Thot-spot hot-spot temperature
Tmax maximum allowable
temperature in case of
quench
Top operating temperature
∆Tctemperature margin
∆Tcs current sharing tem-
perature range
U(generic) voltage
UcDf forward voltage over
the by-pass diode
UcDcTh threshold voltage of
the by-pass diode in
cold state
UcDwTh threshold voltage of
the by-pass diode in
warm state
Udet voltage threshold for
quench detection
UDR voltage over the DR
UfDf forward voltage over
the free-wheeling
diode
UfDTh threshold voltage
of the free-wheeling
diode
Uind induced voltage
Umax,ground maximum allowable
voltage to ground
Umax,tt maximum allowable
turn-to-turn voltage
UPS voltage over the power
supply
UQH initial voltage over
quench heater power
supply
UNoQ voltage over the non-
quenched magnets in
the string
Ures resistive voltage
USC voltage over a super-
conductor
UTerminal voltage over the mag-
net terminals
Vvolume
wci inner width of a key-
stoned cable
wco outer width of a key-
stoned cable
wQH quench heater width
WV
mag magnetic energy in the
volume V
WV
hyst hysteresis energy loss
over a full cycle in the
volume V
Wsource energy provided by
the source
Greek Letters
αface of conductor:
with α={a, b, c, d}.
βpcabling angle in a
Rutherford-type cable
γpath
Γfield dependence
on current, current
derivation and current
history
XXX List of Tables
δi
Ξlongitudinal orienta-
tion of the element
with index i
Eelectrical permittivity
0electrical permittivity
of empty space
rrelative electrical per-
mittivity
ηlength, area, or vol-
ume ratio
η`length ratio
ηAarea ratio
ηA
Cu area ratio of the cop-
per material to the to-
tal cross-section
ηA
SC area ratio of the super-
conducting material to
the total cross-section
ηdstrand filling factor
ηVvolume ratio
κTthermal conductivity
κeff,[=]
Tserial effective ther-
mal conductivity of
stacked materials
κeff,[||]
Tparallel effective ther-
mal conductivity of
stacked materials
λcopper to supercon-
ductor ratio
λnon-Cu copper to non-copper
area ratio
µmagnetic permeability
µ0magnetic permeability
of empty space
µrrelative magnetic per-
meability
ξindex of the topologi-
cal numbering of all el-
ements in the coil
Ξmapping function be-
tween the topological
numbering ξand the
numbering scheme us-
ing ij
ρDmass density
ρEelectrical resistivity
ρeff
Eeffective electrical re-
sistivity
ρeff,[=]
Eparallel effective elec-
trical resistivity of
stacked materials
ρeff,[||]
Eserial effective elec-
trical resistivity of
stacked materials
ρIFCC constant approxima-
tion of the electrical
resistivity for IFCC
dρIFCC
dBlinear approximation
of the electrical resis-
tivity for IFCC
ρsc electrical resistivity in
SC state
τ(generic) time/time
constant
τDR decay time constant of
a string of magnets
τQH Quench heater power
time constant
υfunction relating MI-
ITs to the hot-spot
temperature Thot-spot
ϕangle and polar coor-
dinate
φelectrical potential
φkey cable keystone angle
Φ0flux quantum
Ψmag (generic) magnetic
flux
List of Tables XXXI
Mathematical Notation
xvariable / cartesian
coordinate
eiunit vector in direc-
tion of the coordinate
i
Avector field / vec-
tor function (bold up-
right print)
fscalar field / scalar
function
f(a)value of the function f
evaluated for a
df
dxtotal differential of
scalar function fafter
x
df
dxx=avalue of the total dif-
ferential of the scalar
function fafter xat
x=a
∂f
∂x partial differential of
scalar function fafter
xnot considering vari-
ations in y, z, ...
∇Φgradient of scalar field
Φ
∇×Acurl of vector field A
∇·Adivergence of vector
field A
{x}algebraic vector
[A]algebraic matrix
|| parallel
⊥perpendicular
∂boundary operator
[...]operator returning the
unit of measurement
of the argument, e.g.
[B] = T
dsdifferential path ele-
ment
dadifferential surface ele-
ment
dVdifferential volume el-
ement
Hclosed surface / path
integral
Abbreviations
Al aluminum
ATLAS A Toroidal LHC Ap-
paratuS
CAD computer-aided design
CENELEC European comittee for
Electrotechnical Stan-
dardization, Brussels,
Belgium
CERN European Organi-
zation for Nuclear
Research, Geneva,
Switzerland
CMS Compact Muon
Solenoid experiment
Cu copper
DR dump resistor
FAIR Facility for Antiproton
and Ion Research at
GSI
Fe iron
GSI Gesellschaft für Schw-
erionenforschung,
Darmstadt, Germany
H hyrdogen
He helium
Hg mercury
H2Owater
HTS high temperature su-
perconductor
XXXII List of Tables
IEC International Elec-
trotechnical Commis-
sion
IFCC inter-filament cou-
pling currents
ISCC inter-strand coupling
currents
ITER the way to new en-
ergy. International fu-
sion program
LARP LHC Accelerator Re-
search Program
LHC Large Hadron Collider
LTS low temperature su-
perconductor
MB LHC main bending
magnet (guiding beam
on circular trajectory)
MCBX LHC inner triplet
dipole
MIITs mega current square
time integral
MQE minimum quench en-
ergy
MQX LHC inner triplet
quadrupole with wide
aperture (focussing
the beam for the
experiments)
MQXC sLHC inner triplet up-
grade quadrupole
MQY LHC wide aperture
quadrupole
MRI magnetic resonance
imaging
NC normal conductor,
normal conducting
N nitrogen
Nb niobium
Nb-Ti niobium-titanium
Nb3Sn niobium-3-tin
NIST National Institute of
Standards and Tech-
nology, Boulder, USA
O oxygen
Pb lead
PE polyester
Ph.D. philosophiae doc-
tor, lat. doctor of
philosophy
PVA polyvinyl acetate
QH quench heater
RAL Rutherford Appleton
Laboratory
ROXIE Routine for the Opti-
mization of magnet X-
sections, Inverse field
calculation and coil
End design
RRR residual resistivity ra-
tio
SC superconductor,
superconducting
sLHC LHC luminosity up-
grade
Sn tin
SS stainless steel
SSC Superconducting Su-
per Collider
Ta tantalum
Ti titanium
TQ LARP technology
quadrupole
TU-Berlin Technische Univer-
sität Berlin
VAC voltage-current-
characteristic
Translations of Quotes and
Background Explanations
Nullus est liber tam malus, ut non aliqua parte prosit!
There is no book so bad that it is not profitable on some part.
Widerstand zwecklos
Resistance is futile - also a catch phrase used by the Borg of the Star Trek
fictional universe.
Dicebat Bernardus Carnotensis nos esse quasi nanos, gigantium
humeris insidentes, ut possimus plura eis et remotiora videre, non
utique proprii visus acumine, aut eminentia corporis, sed quia in
altum subvenimur et extollimur magnitudine gigantea.
Bernard of Chartres used to say that we are like dwarfs on the shoulders of
giants, so that we can see more than they, and things at a greater distance,
not by virtue of any sharpness of sight on our part, or any physical distinction,
but because we are carried high and raised up by their giant size.
Rerum cognoscere causas
To know the causes of things - Head note of “Der Tagesspiegel” (Berlin).
Ihr zahmen Täubchen, ihr Turteltäubchen, all ihr Vöglein unter
dem Himmel, kommt und helft mir lesen, die guten ins Töpfchen,
die schlechten ins Kröpfchen.
O gentle doves, O turtle-doves, And all the birds that be, The lentils that in
ashes lie Come and pick up for me! The good must be put in the dish, The
bad you may eat if you wish.
Hier stehe ich, ich kann nicht anders
Here I stand. I can do no other.
Milk production at a dairy farm was low so the farmer wrote to the
local university, asking help from academia. A multidisciplinary
team of professors was assembled, headed by a theoretical physi-
cist, and two weeks of intensive on-site investigation took place.
The scholars then returned to the university, notebooks crammed
with data, where the task of writing the report was left to the
team leader. Shortly thereafter the farmer received the write-up,
and opened it to read on the first line: "Consider a spherical
cow... ."
XXXIII
XXXIV List of Tables
Curriculum Vitae
Nikolai Schwerg was born 1980 in Berlin. After finishing
high-school in 1999, he fulfilled his community service
in a nursing home. From October 2000 to April 2006 he
studied electrical engineering at the Technical Univer-
sity in Berlin. During his studies he specialized on field
theory, radio-frequency engineering and power electron-
ics. Nikolai worked for two years as teaching assistant at
the chair of theoretical electrical engineering. He spent
more than a year at the European Organization for Nu-
clear Research (CERN) writing his student minor and
diploma thesis. After graduating he began working as doctoral student at
CERN in April 2006.
Publications: Parts of the dissertation have been published in the following
scientific papers:
1. N. Schwerg, B. Auchmann, and S. Russenschuck. Quench simulation in
an integrated design environment for superconducting magnets. IEEE
Trans. Magn., 44(6):934-937, June 2008.
2. N. Schwerg, B. Auchmann, and S. Russenschuck. Validation of a cou-
pled thermal-electromagnetic quench model for accelerator magnets.
IEEE Transactions on Applied Superconductvity, 18(2):1565-1568, June
2008.
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