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A comp ehensi e o e iew o high-speed solid- o o induc ion machines:
Applica ions, classi ica ion, and mul i-physics modeling
Vladimi Bileka,∗, Jan Ba aa, Ma ek Tomana, Pe Losakb, Ge d B ame do e c
aDepa men o Powe Elec ical and Elec onic Enginee ing, B no Uni e si y o Technology, B no, 616 00, Czech Republic
bIns i u e o Solid Mechanics, Mecha onics and Biomechanics, B no Uni e si y o Technology, B no, 616 69, Czech Republic
cIns i u e o Elec ical D i es and Powe Elec onics, Johannes Keple Uni e si y, Linz, 4040, Aus ia
A R T I C L E I N F O
Keywo ds:
Fini e elemen me hod
High-speed elec ical machines
Induc ion machines
Mul i-physical op imiza ion
Solid- o o machines
A B S T R A C T
Solid- o o induc ion machines ha e gained signi ican a en ion in a ious indus ial applica ions due o hei
obus ness, eliabili y, and cos -e ec i eness. This pape p esen s a comp ehensi e o e iew o hese machines,
co e ing hei classi ica ion and a ious applica ions. The pape s a s wi h discussing he widesp ead usage
o solid- o o induc ion machines in nume ous indus y sec o s, including manu ac u ing, anspo a ion, and
enewable ene gy gene a ion. The abili y o ope a e unde ha sh en i onmen al condi ions and in sa e y-c i ical
se ings has made hese machines indispensable in many ields o enginee ing. Thei de ailed classi ica ion
based on di e en o o opologies is p o ided, highligh ing he unique design ea u es and pe o mance
cha ac e is ics o each ca ego y. Simple and hyb id con igu a ions and hei dis inc ad an ages and limi a ions
in speci ic applica ions a e included. This pape u he explo es he essen ial aspec s o mul i-physics modeling
o solid- o o induc ion machines, inco po a ing elec omagne ic, mechanical, and he mal conside a ions o
gain deep insigh s in o he complex in e ac ions be ween componen s and o guide he op imiza ion p ocess
o enhanced pe o mance and e iciency. This wo k is in ended as a aluable e e ence o esea che s
and enginee s seeking a comp ehensi e unde s anding o solid- o o induc ion machines, om hei di e se
applica ions o he in icacies o hei elec omagne ic, he mal, and mechanical modeling. By shedding ligh
on hese aspec s, his wo k con ibu es o he ad ancemen and u iliza ion up ake o hese machines in mode n
indus ial se ings.
Con en s
1. In oduc ion ...................................................................................................................................................................................................... 2
2. O e iew o solid- o o IM applica ions ............................................................................................................................................................... 3
2.1. Gas comp esso and u bine applica ions.................................................................................................................................................. 3
2.2. Flywheel ene gy s o age sys ems ............................................................................................................................................................. 4
2.3. High-speed spindle applica ions............................................................................................................................................................... 4
2.4. Tu bomolecula pumps ........................................................................................................................................................................... 4
3. O e iew o solid- o o IM opologies.................................................................................................................................................................. 5
3.1. Smoo h solid- o o IM............................................................................................................................................................................. 5
3.2. Axially sli ed solid- o o IM.................................................................................................................................................................... 5
3.3. Solid- o o IM wi h adial o o su ace g oo es........................................................................................................................................ 7
3.4. Squi el-cage solid- o o IM..................................................................................................................................................................... 7
4. Sui able ma e ials o solid o o s........................................................................................................................................................................ 8
5. Elec omagne ic modeling o solid- o o IMs ........................................................................................................................................................ 9
5.1. Analy ical me hods................................................................................................................................................................................. 9
5.2. Nume ical me hods................................................................................................................................................................................. 9
5.2.1. 2D modeling and nume ical elec omagne ic calcula ions o induc ion machines ........................................................................... 10
5.2.2. Co ec i e end-e ec ac o s o a solid- o o .............................................................................................................................. 11
∗Co esponding au ho .
E-mail add esses: [email p o ec ed] (V. Bilek), [email p o ec ed] (J. Ba a), [email p o ec ed] (M. Toman), [email p o ec ed] (P. Losak),
[email p o ec ed] (G. B ame do e ).
h ps://doi.o g/10.1016/j.ijepes.2025.110520
Recei ed 8 May 2024; Recei ed in e ised o m 21 Decembe 2024; Accep ed 4 Feb ua y 2025
Elec ical Powe and Ene gy Sys ems 166 (2025) 110520
A ailable online 19 Feb ua y 2025
0142-0615/© 2025 The Au ho s. Published by Else ie L d. This is an open access a icle unde he CC BY license ( h p://c ea i ecommons.o g/licenses/by/4.0/ ).
V. Bilek e al.
5.2.3. Co ec i e end-e ec ac o s o a smoo h solid- o o wi h coppe coa ing .................................................................................... 13
5.2.4. Co ec i e end-e ec ac o s o a solid- o o wi h highly conduc i e end ings ............................................................................. 13
5.2.5. Co ec i e end-e ec ac o s o a solid- o o wi h adial o o su ace g oo es ............................................................................. 13
6. Cooling and he mal modeling o solid- o o induc ion machines........................................................................................................................... 14
6.1. Cooling concep s o high-speed machines................................................................................................................................................ 15
6.2. O e iew o commonly used he mal modeling echniques ........................................................................................................................ 15
6.2.1. The mal models u ilizing compu a ional luid dynamics .............................................................................................................. 15
6.2.2. The mal models u ilizing a lumped pa ame e he mal ne wo k ................................................................................................... 15
6.2.3. The mal models u ilizing ini e elemen me hod ......................................................................................................................... 16
6.3. The mal modeling using a lumped pa ame e he mal ne wo k .................................................................................................................. 16
6.3.1. Conduc i e hea ans e ........................................................................................................................................................... 16
6.3.2. Con ec i e hea ans e ........................................................................................................................................................... 17
6.3.3. Radia ion hea ans e ............................................................................................................................................................. 18
6.3.4. Modeling o coolan low .......................................................................................................................................................... 18
7. Mechanical modeling o solid- o o induc ion machines ........................................................................................................................................ 19
7.1. S ess calcula ion.................................................................................................................................................................................... 19
7.1.1. Load om o a ion ................................................................................................................................................................... 19
7.1.2. Tempe a u e load..................................................................................................................................................................... 20
7.1.3. Unbalance ............................................................................................................................................................................... 20
7.1.4. Unbalanced magne ic pull (UMP) .............................................................................................................................................. 21
7.1.5. Ex e nal loads.......................................................................................................................................................................... 21
7.2. C i ical speed calcula ion ........................................................................................................................................................................ 21
8. Gene al o e iew o o o opology cha ac e is ics ............................................................................................................................................... 22
9. Conclusion ........................................................................................................................................................................................................ 23
CRediT au ho ship con ibu ion s a emen ........................................................................................................................................................... 24
Decla a ion o compe ing in e es ........................................................................................................................................................................ 24
Acknowledgmen s .............................................................................................................................................................................................. 24
Da a a ailabili y ................................................................................................................................................................................................ 24
Re e ences......................................................................................................................................................................................................... 24
1. In oduc ion
High-speed elec ical machines a e becoming inc easingly popula
in indus ial applica ions, which is e lec ed in a g owing numbe o
publica ions in ecen decades. High-speed, ypically means a ci cum-
e en ial speed, a cha ac e is ic pa ame e o o o s eng h, is highe
han 100 m/s [1]. To achie e high o a ion speed, one op ion is o use
a gea box combined wi h a s anda d elec ical machine [2]; ano he
op ion is he u iliza ion o a high-speed elec ical machine ha enables
a di ec connec ion be ween he machine and he load, such as an
impelle , which inc eases o e all sys em e iciency and eliabili y [3]
compa ed o a sys em wi h a gea box. O he bene i s o high-speed
o e con en ional machines include smalle size and educed ma e ial
consump ion [4,5]. These a e he main ac o s o inc easing indus y
in e es o high-speed elec ical machines o nume ous applica ions,
such as comp esso s, acuum pumps, mic o u bines, and machining
ools [3].
I has been shown ha synch onous pe manen magne machines [6,
7] a e ypically used o highe speeds and lowe powe s, while,
induc ion machines (IMs) a e used o lowe speeds and highe pow-
e s [3]. In e ms o ope a ing pe o mance, i can gene ally be s a ed
ha synch onous machines o e highe powe densi ies, highe powe
ac o , and be e e iciency han compa able IMs. Induc ion machines,
howe e also ha e ad an ages, especially in e ms o esis ance o
high empe a u e and simple con ol, and in conjunc ion wi h a solid-
o o wi hou squi el-cage, hey ha e he po en ial o ole a e high
mechanical s esses a low manu ac u ing cos s [8].
The solid- o o is a non-lamina ed o o ype o IMs ha o e s
high esis ance o cen i ugal o ces, empe a u e, and en i onmen al
in luences, which may include, o ins ance, a ious ha sh chemicals
subs ances. Solid- o o s hus o e high eliabili y and du abili y o
elec ical machines. Thei pe o mance can be imp o ed in a ious
ways, o example, by sli ing o by adding a coppe laye o he o o
su ace o he o o ends. In ecen yea s, a ious modi ica ions ha e
been discussed and s udied in a wide ange scien i ic publica ions,
e.g., [9–14]. These, howe e ocused mainly on he desc ip ion o one
o a ew pa icula designs wi hou gi ing an ex ensi e o e iew o all
exis ing design concep s.
In he pas , an ex ensi e o e iew o solid- o o machines was
p esen ed in pape s [3,4,15] and doc o al heses [9,16–18] in he
pas . Al hough hese wo ks p o ided aluable insigh s, hey did no
simul aneously and comp ehensi ely add ess all he exis ing concep s
o modeling app oaches. Mo eo e , since hei publica ion he s a e-
o - he-a has u he de eloped, d i en by a g owing in e es in high-
speed solid- o o machines. This pape aims o ill in hese gaps by
p esen ing an ex ensi e o e iew o he opic o da e, co e ing all he
majo solid- o o opologies, along wi h a de ailed summa y o hei
cu en applica ions, wi h a unique o e iew o he exis ing modeling
echniques compa ing he di e en co ec ion ac o s ha we e mos ly
in oduced sepa a ely in he pas .
The modeling o solid o o elec ical machines is di e en o he
usual p ac ice o con en ional machines wi h a lamina ed o o . The
solid- o o is no only pa o he magne ic ci cui h ough which
magne ic lux lows, i also se es as a conduc o o elec ical cu en .
Due o he non-linea o o ma e ial p ope ies, calcula ing he elec o-
magne ic cha ac e is ics o an elec ical machine wi h a solid- o o is
a complex ask. The design o such a machine mus ake in o accoun
i s mul i-physical model, which has been he subjec o a numbe o
publica ions [19–21]. These models o high-speed machines ypically
include elec omagne ic, he mal, and mechanical calcula ions.
Elec omagne ic calcula ion and design a e among he i s s eps in
he mul i-physics design o high-speed elec ical machines [22]. A lis
o equi emen s and a sui able machine opology a e usually known
o selec ed be o ehand. Fo p ope elec omagne ic calcula ion, he
p ecise empe a u e o he machine, especially o s a o and windings,
mus be known. This calcula ion is he e o e pe o med oge he wi h
he he mal calcula ion in se e al i e a i e loops. Subsequen ly, he
s ess dis ibu ion in he o o c oss-sec ion is usually e i ied and
he o o dynamic analyzed o de e mine how close o he c i ical
speed he machine will be ope a ing. The empe a u es mus also be
known o mechanical analysis, since – depending on o o opology
– he mal expansions can in luence he s ess magni ude inside he
o o s uc u e. I , he equi emen s o he machine canno be me
In e na ional Jou nal o Elec ical Powe and Ene gy Sys ems 166 (2025) 110520
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V. Bilek e al.
Fig. 1. Simpli ied mul i-physics i e a i e p ocedu e o designing a solid- o o IM.
in his p ocess, hen he machine mus be edesigned o a di e en
opology selec ed o he o o . The i e a ion o e hese calcula ion
s eps is shown in Fig. 1.
This e iew pape add esses he ac ha solid- o o IMs ha e
ad anced conside ably, as e idenced by he signi ican numbe o
scien i ic publica ions ha ha e deal wi h hem in ecen yea s. Based
on hese publica ions, a de ailed summa y o exis ing designs and asso-
cia ed modeling echniques is gi en. Fo each key modeling echnique
(elec omagne ic, he mal and mechanical calcula ion), his a icle p o-
ides a comp ehensi e e iew o commonly used me hods o e ec i e
calcula ion and design o solid- o o IMs.
This e iew is s uc u ed as ollows: Fi s , an o e iew o ypical
solid- o o applica ions is gi en. This is ollowed by an o e iew o
possible o o opologies, including a discussion o hei ad an ages
and disad an ages om an in e disciplina y poin o iew. The sub-
sequen sec ions desc ibe elec omagne ic, he mal, and mechanical
modeling, wi h an emphasis on he cha ac e is ics o solid- o o s, which
di e om hose o lamina ed ones. The inal sec ion summa izes key
messages.
2. O e iew o solid- o o IM applica ions
As poin ed ou in [23], due o con inuous de elopmen s in he
ield o powe elec onics, powe in e e s, manu ac u ing me hods,
and ma e ials [24,25], use o high-speed solid- o o elec ical machines
in high-speed applica ions has inc eased apidly in he las decade. In
gene al, he design o high-speed machines is e y complex, and hus
usually equi es mul idisciplina y op imiza ion [23]. The high-speed
machines a e classi ied acco ding o he ci cum e en ial speed o he
o o . In some indus ial high-speed applica ions, elec ical machines
ha e di ec ly eplaced exis ing mechanical sys ems (composed o a
gea box and a low-speed elec ic machine) [3], while in o he s hey
complemen exis ing mechanical sys ems. The gene al ad an ages o
combining a high-speed machine wi h powe in e e s a e he o e -
all compac ness, lowe weigh , and highe eliabili y o he sys ems,
compa ed o he combina ion o a low-speed machine wi h a gea box.
Due o he simplici y o hei manu ac u e, low cos , high accu acy, and
o o obus ness, high-speed solid- o o induc ion machines (HSSRIMs)
a e a e y popula choice o high-speed applica ions [26].
Based on [23], Table 1lis s he key applica ions o high-speed
elec ical machines. This pape co e s mainly he mos impo an appli-
ca ion a eas: oil, gas and ai comp esso s and u bines, lywheel ene gy
s o age sys em, high-speed spindles, and u bomolecula pumps.
Table 1
Powe and speed equi emen s o selec ed applica ions, acco ding o [23].
Applica ion Powe Speed
Oil & gas 3–15MW 5–15 k pm
Spindles 300W–60 kW 15–300 k pm
Tu bocha ge 1–3 kW/10kW 150 k pm/80 k pm
Ai comp esso 4kW–500 kW 15–80k pm
Mic o- u bines 30–400kW 15–120k pm
Tu bomolecula
Pumps 50W–3 kW 70–100 k pm
Fig. 2. Con en ional comp esso (a) and in eg a ed comp esso (di ec d i e) (b),
acco ding o [3].
Fig. 3. High-speed high-powe IM o an in eg a ed comp esso . Cu iew o a squi el-
cage o o (a) and de ail o a high-speed lamina ed o o [27].
2.1. Gas comp esso and u bine applica ions
Gas comp esso s and u bines a e gene ally needed in he chem-
ical, oil, and gas indus ies, mainly o ga he ing, ansmission, and
p ocessing he gas downs eam [3]. Speci ically o p essu ized ai o
gas, comp esso s a e used in many indus ial applica ions including
pneuma ic ac ua o s, sandblas ing, machining, e men a ion, ins u-
men a ion and, ai polishing [19]. In ecen yea s, due o ad ances in o
powe elec onics [28], he combina ion o powe in e e s wi h high-
speed elec ic machines has s a ed o eplace con en ional machines
wi h gea boxes. The clea ad an ages o his solu ion [29–31] a e
highe powe densi y and e iciency, smalle size, and lowe weigh o
he en i e d i e. In addi ion, he d i e is oil- ee, highly eliable, and
low-main enance [32]. Bo h d i e ypes a e illus a ed in Fig. 2. To im-
p o e pe o mance, li espan, and e iciency o he comp esso d i e, use
o ac i e magne ic bea ings is ecommended [19]. IM a e widesp ead in
In e na ional Jou nal o Elec ical Powe and Ene gy Sys ems 166 (2025) 110520
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V. Bilek e al.
Table 2
Speed in ypical milling applica ions, acco ding o [41].
Applica ions Speed
Me al 4500–12 000 pm
S ones 8000–12 000 pm
Glass/ma ble 8000–14 000 pm
Wood 18 000–25 000 pm
Aluminum 30 000–40 000 pm
his indus y, anging om 3 MW o 15 MW wi h co esponding speeds
o 20 k pm o 5k pm [23,33]. Fig. 3shows a high-speed high-powe
IM o an in eg a ed comp esso .
2.2. Flywheel ene gy s o age sys ems
Flywheel ene gy s o ages sys ems (FESSs) use a o a ing lywheel o
s o e mechanical ene gy which is hen con e ed in o elec ical ene gy
by an elec ical machine. The elec ical machine manages he ansi ion
om mechanical o elec ical and back o mechanical ene gy [34]. A
FESS s o es ene gy as o a ional ene gy in a o a ing mass. The amoun
o ene gy s o ed depends on he ine ia and speed o he o a ing mass.
To elimina e ene gy loss due o ai ic ion, he lywheel is placed
wi hin a acuum [35]. Ideally, i is suspended on ac i e magne ic
bea ings [36] o s able ope a ion and con inuous o a ion wi hou
added powe and wi h minimal ene gy loss.
FESSs can be classi ied, acco ding o [37] in o high-speed (10 k–
100 k pm) and low-speed (<6 k pm). Low-speed FESSs usually ha e
la ge diame e s and low powe and ene gy densi ies [3]. High-speed
FESSs ha e smalle diame e s and highe powe and ene gy densi ies,
bu , hei powe a ing is ypically cons ained by cos conside a ions.
High-speed FESSs can be up o i e imes mo e expensi e han low-
speed FESSs [34]. Fo FESSs, ei he synch onous o IMs a e used. IMs
a e he bes choice o high-powe applica ions, as hey o e high
o que, high eliabili y, and high obus ness [38]. Since no elec omag-
ne ic losses caused by spinning a e p esen , hey a e a g ea op ion o
low-loss FESSs.
FESSs a e becoming a iable al e na i e o adi ional chemical
ba e y sys ems. Acco ding o [39], hei ad an ages include highe
ene gy s o age densi y, lowe isk o o e cha ge/o e -discha ge, easy
measu emen o he le el o cha ge emaining, ope a ion o e a wide
empe a u e ange, longe li espan, and en i onmen al iendliness.
They a e sui able o swi ching be ween medium o high powe s (kW o
MW) o sho pe iods o ime wi h high ene gy e iciency (>85%) [40].
Gene ally, FESSs also allow a e y high numbe o cha ge/discha ge
cycles (hund eds o housands), independen ly o he empe a u e and
he le el o discha ge. Thei ypical li e ime exceed 20 yea s, wi hou
nega i e en i onmen al consequences. Howe e , hey can s o e ene gy
o only a couple o hou s a mos [35]. Fig. 4shows schema ically and
pho og aphically a ypical FESS and i s componen s.
2.3. High-speed spindle applica ions
Con en ional low-cos high-speed spindles use bel d i es, which
limi s hei maximum speed [3,23]. The e e -inc easing demand o
highe speed and speed con ol, low ib a ion le els, and highe powe
densi y and e iciency o his applica ion has esul ed in he adop ion
o high-speed machines. Speed limi and powe ange [44] o spin-
dle applica ions a y conside ably, anging om 9k pm o 180k pm,
wi h co esponding app oxima e powe om 24kW down o 1 kW.
E o is ongoing o inc ease he ope a ing speed o high-speed spindle
applica ions [41,45].
The e a e h ee main spindle applica ions [3]: milling, g inding, and
d illing. The maximum spindle o a ional speed o milling applica ions
depends on he ype o ma e ial p ocessed, as shown in Table 2. The
o a ional speed is highe in g inding han in milling applica ions, wi h
speeds up o hund eds o housands o e olu ions pe minu e [41].
Design examples o high-speed IMs wi h a squi el cage we e gi en
in [46,47]. An example spindle is shown in Fig. 5.
Fig. 4. S uc u e and componen s o a lywheel (a) (acco ding o [42]) and lywheel
sys ems kine ic ene gy eco e y sys em o a Fo mula (1) ca (b) [43].
Fig. 5. High-speed and high-p ecision mo o ized spindle o p ecision machine y
(Jingdiao) [48].
2.4. Tu bomolecula pumps
The high-speed u bomolecula pump [49] is – simila o he u bo
pump – a ype o acuum pump, o c ea ing and main aining high
acuum. The wo king p inciple is based on momen um ans e [50].
I has mul iple s ages, each o which consis s o a as -spinning o o
and s a o blades. The o o blades hi he gas molecules, which ha e
a high p obabili y o being ans e ed p e e en ially o he nex o o
s age a e passing h ough he ollowing s a o s age. The whole sys em
wo ks like a pump, pu ing ene gy in o he gas and pushing he gas om
he inle o he exhaus in o de o main ain high acuum.
These pumps cons i u e he mos eliable and cos -e ec i e means
o p oducing high acuum, hey a e po able and as -s a ing, and
hey equi e li le suppo [51]. Fo be e pe o mance, low main-
enance cos and ic ion-less ope a ion c ea ing ul a-clean oil- ee
acuum en i onmen s, u bomolecula pumps wi h magne ic bea ings
ha e been used in ecen yea s [52,53]. They a e ideal o use wi h
acuum echnology in high- ech ields such as ilm deposi ion, semi-
conduc o manu ac u ing, high-ene gy physics, op ical/glass indus y,
mass spec ome y, gene al ul a-high acuum esea ch, and usion
echnology [54].
In e na ional Jou nal o Elec ical Powe and Ene gy Sys ems 166 (2025) 110520
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V. Bilek e al.
Fig. 6. C oss-sec ion o a high-speed u bomolecula pump [56].
Tu bomolecula pumps a e used o achie e e y high acuum con-
di ions p essu es as low as 10−10 mba , wi h o a ional speeds up o
100k pm and only a ew hund ed wa s up o a ew kilowa s o ou pu
powe [49]. The mos commonly employed ype o machine o his
applica ion is he h ee-phase IM. Howe e , i s design is mo e complex
han ha o con en ional machines, especially in ela ion o ex emely
p oblema ic hea exchange because he d i ing machine is wi hin high
acuum. The s a o can gene ally be cooled by wa e , bu he o o is
cooled only by he mal adia ion, he he mal dissipa ion e iciency o
which is low [55]. The o que ipple mus be e y small o minimize
he isk o mechanical esonance o he o a ing sys em. An example o
a u bomolecula pump is shown in Fig. 6.
3. O e iew o solid- o o IM opologies
HSSRIMs a e usually made o a single piece o e omagne ic ma-
e ial o a e composed o e omagne ic and non-magne ic ma e ials.
The main ad an age o his machine is i s abili y o each e y high
speeds hanks o obus o o cons uc ion. In high-speed applica ions,
cen i ugal o ces and ci cum e en ial speed a e key ac o s de e mining
he choice cons uc ion ype. This ype o o o is usually p e e ed o e
a lamina ed o o o i s megawa ange powe and highe o a ional o
ci cum e en ial speeds. Acco ding o [16], he ad an ages o HSSRIM
as:
•high mechanical in eg i y, igidi y, obus ness, and du abili y,
•high he mal du abili y,
•ease o p o ec ion agains agg essi e chemicals,
•high eliabili y,
•easy, s aigh o wa d, and cheap manu ac u e in mos cases,
•easy scaling o wide powe and speed anges, and
•low noise and ib a ion le els (in he case o smoo h solid- o o s).
Consequen ly, HSSRIMs a e o en p e e ed in indus y o high-speed
and high-powe -densi y applica ions. Below, he mos impo an o o
opologies ha a e commonly used o widely discussed a e p esen ed.
All o o opologies lis ed could po en ially be combined.
3.1. Smoo h solid- o o IM
The simples solid- o o cons uc ion is a smoo h solid- o o (SSR),
acco ding o [16,58]. SSRs a e he mos mechanically obus , bu lack
in elec omagne ic pe o mance. Slip and losses o his ype o machine
end o be la ge, and usually he powe ac o is low due o poo
lux pene a ion [15] in o he o o caused by high o o slip and low
elec ical conduc i i y o he o o ma e ial. As a consequence, SSRs
exhibi high appa en esis ance, low magne iza ion induc ance, and
an o e -sa u a ed su ace o he solid- o o , which esul s in ex emely
poo elec omagne ic o que densi y [17]. An SSR has he ad an ages
o ou s andingly simple manu ac u e, low cos , and he bes mechan-
ical and luid-dynamical p ope ies o low ai ic ion. I s design is
illus a ed in Fig. 7(a).
To imp o e he elec omagne ic pe o mance and inc ease he
o que densi y o he machine, non-magne ic high-conduc i i y end
ings can be added o he SSR, as s a ed in [16,18,59]. These help o
imp o e cu en low in he solid s eel, which ends o be aligned mo e
in o he axial-pa allel di ec ion, hus inc easing he Lo en z o ce. I has
been epo ed, [18,58,60], ha a wo-pole SSR equipped wi h coppe
end ings p oduces wice as much o que a a pa icula slip han an
o dina y SSR. The o o con igu a ion is illus a ed in Fig. 7(b).
Ano he possible SSR modi ica ion is o use a coppe -coa ed smoo h
solid- o o (CCSSR), as shown in Fig. 7(c). A conduc i e laye en elops
he ou e su ace o he o o , spanning om one end ing o ano he , as
p oposed in [15,61]. The hickness o he coppe laye may be cons an
along he leng h o he o o o may be inc eased in he end ings
egion. I he coppe laye is hicke a he end ings, i enhances he
conduc i i y o he o o , hus imp o ing he low o elec ical cu en
wi hin he o o . Se e al au ho s ha e sugges ed, ha ei he a low-
o a high-conduc i i y non-magne ic ma e ial should be used o he
coa ing [18,58,60]. A highly conduc i e laye (i.e., coppe ) ac s like
bo h an in ini e numbe o o o ba s and end ings. Due o i s high
conduc i i y, i is he main ci cui pa h o he o o cu en s. I also
ac s as a high- equency shielding o ai -gap ha monics and s a o slo
ha monics, which do no pene a e h ough he coa ing laye [57,59].
This helps o educe o o eddy cu en s in he i on and s a o winding
losses.
3.2. Axially sli ed solid- o o IM
A much be e al e na i e o SSRs in e ms o elec omagne ic
pe o mance is an axially sli ed solid- o o (ASSR) design a simple
o o modi ica ion achie ed by axially sli ing he c oss-sec ion o he
o o . The axial sli s con ibu e signi ican ly o o cing he undamen al
componen o he lux deepe in o he solid- o o [12,58]. Fu he ,
axially sli ing he o o dec eases he low- equency impedance o
he o o , which esul s in highe o que and inc eased e iciency.
Addi ionally, i inc eases he high- equency su ace impedance o he
o o , which esul s in lowe o o eddy-cu en losses o undesi ed
ha monics [62]. Acco ding o [17], he main disad an ages o axial sli s
in a solid- o o a e he pa ially comp omised obus ness o he ASSR
a highe ci cum e en ial speeds and ha he ai ic ion o he o a ing
o o is signi ican ly inc eased. Howe e , he axial sli ing imp o es
o o cooling because i s su ace is inc eased. This opology is shown
in Fig. 8(a).
Selec ing he igh numbe o sli s is c ucial. To achie e smoo h
o que cha ac e is ics, he combina ion o s a o slo s and o o sli s
[60] should be chosen ca e ully o a oid synch onous o que dips. To
selec he numbe o o o sli s, an analogous me hod o he selec ion
o o o ba s o squi el cage IM can be used he e. Acco ding o
s udies [16,63], he machine ou pu o que inc eases wi h inc easing
numbe o o o sli s, as illus a ed by Fig. 9. Only e en numbe s o
o o sli s we e conside ed in [16] o minimiza ion o unbalanced mag-
ne ic pull (which may be g ea e han he o o weigh ) and mechanical
ib a ion o he o o . Despi e he isk o highly unbalanced magne ic
pull, he elec omagne ic o que ipple dec eases when an odd numbe
o o o sli s is used. I has been sugges ed [64] ha be ween 5 and 15
sli s pe pole pai is ideal.
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Fig. 7. Cons uc ion o a smoo h solid- o o o an IM: (a) simple smoo h solid- o o , (b) smoo h solid- o o wi h coppe end ings, and (c) coppe -coa ed smoo h solid- o o ,
acco ding o [16,17,57].
Fig. 8. Cons uc ion o an axially sli ed solid- o o o an IM: (a) axially sli ed solid- o o , (b) axially sli ed solid- o o wi h sli o a ious dep h, (c) axially sli ed solid- o o
wi h coppe end ings, and (d) axially sli ed o o coa ed wi h conduc i e/ esis i e ma e ial, acco ding o [17].
Fig. 9. Pe uni ou pu o que as a unc ion o he numbe o o o sli s, acco ding
o [16,63].
Two essen ial aspec s a e sli dep h and wid h. The sli dep h has a
signi ican e ec on machine pe o mance [60], and a dep h o 60%
o he o o adius would allow he bes machine elec omagne ic
pa ame e s o be p oduced. Howe e , such dep h comp omises he
o e all in eg i y and obus ness o an ASSR, possibly esul ing in o o
des uc ion o educed li e span. Thus, dep hs be ween 40%–50% o
he o o adius a e usually chosen [65], depending on size, machine
ci cum e en ial speed, and o o mechanical s eng h. The sli wid h
should be as hin as possible, bu i is also limi ed by he mechanical
s ess ha is gene a ed a he bo om o he sli co ne and ha
concen a es a ound he inne co ne s o he o o sli s. Signi ican
educ ion in his mechanical s ess can be achie ed by ounding he
edges a he bo om o he sli s, bu in p ac ice his solu ion canno be
implemen ed o all applica ions. The op imal sli wid h ha achie es
he bes elec omagne ic machine pa ame e s, acco ding o [13,66,67],
is be ween 9 and 15% o he o o oo h pi ch. Howe e , he sli wid h
depends hea ily on he echnological capabili ies o he manu ac u e .
The na owes sli s p oduced and epo ed o da e [67] amoun o
13.47% o he o o oo h pi ch. As a gene al ule, o o sli s should
be designed as deep and hin as possible.
The ASSR can be sligh ly modi ied by a ying he sli dep hs, as
shown in Fig. 8(b). The mo i a ion o such a design is o educe he
high sa u a ion o he ma e ial a he na owes poin be ween he sli s
and po en ially inc ease machine o que. Howe e , his o o opology
is also limi ed by he maximum sli dep h (40%–50%). In [17] ASSR
wi h sli s o a ious dep hs es ic ed by he maximum allowable dep h
was conside ed, bu he elec omagne ic pe o mance o he machine
did no signi ican ly imp o e.
As wi h SSRs, coppe ings can be added o ASSRs, as shown
in Fig. 8(c). This design combines all he ad an ages desc ibed in
Sec ion 3.1, wi h he added bene i o axial sli s ha help o align he
cu en low in he axis-pa allel di ec ion. To que and e iciency o he
machine a e hus inc eased [68].
As o he SSR, ei he a highly conduc i e o a esis i e non-
magne ic ma e ial can be added o he su ace o an ASSR, as shown
in Fig. 8(d). I an ASSR is coa ed wi h a highly conduc i e ma e ial
(e.g., coppe ) [17], he machine exhibi s he bes elec omagne ic pe -
o mance wi h he added bene i o axial sli s. I coa ed wi h a esis i e
non-magne ic ma e ial [69], he machine beha es simila ly. The esis-
i e coa c ea es high su ace impedance and ac s as a high- equency
shielding. Rega dless o he coa ing, o o losses a e signi ican ly e-
duced. The coa ing helps o educe he pene a ion dep h o highe
ha monic componen s om he ai -gap in o he o o and can, o
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Fig. 10. Cons uc ion o an axially sli ed solid- o o o an IM, wi h axial sli s skewedin angen ial and axial di ec ions: (a) iden ical sli dep h and (b) sli s o a ious dep hs,
acco ding o [11].
Fig. 11. Cons uc ion o a shielded axially sli ed solid- o o o an IM, acco ding
o [12,13].
designs wi h an odd numbe o o o sli s, educe ansien unbalanced
magne ic o ces. Addi ionally, i a sui able ma e ial is used, he coa ing
can help o mechanically ein o ce he o o and conside ably educes
ic ional mechanical losses, which a e signi ican o ASSRs a high
speeds. The g ea es disad an ages a e high p ice and di icul manu-
ac u e. The coa ing should be welded in o he o o ma e ial, and he
weld should ha e a highe ensile s eng h han he coa ing ma e ial.
One special ASSR modi ica ion is skewing o he axial sli s in he
angen ial di ec ion, as illus a ed in Fig. 10(a) and p esen ed in [11],
whe e he elec omagne ic o que was hus inc eased by 10.7%. An-
o he possible imp o emen is o a y he axial sli dep h as illus a ed
in Fig. 10(b), which was shown in [11] o imp o e he elec omagne ic
o que by 37.7%. Sli cu s skewed in he angen ial di ec ion can be cu
deepe because highe o o in eg i y is e ained. A compa ison o he
e ec s o a ious sli skew angles (15 ◦, 30 ◦, 45 ◦, and 60 ◦) ound ha
elec omagne ic o que inc eases sligh ly wi h inc easing skew angle,
bu ha he o que ipple inc eases by 40.3% in he case o 30 ◦skew
angle. Skewing he axial cu s by 30 ◦in he angen ial di ec ion and
by 26 ◦in he axial di ec ion signi ican ly educed he o que ipple o
1% o he machine ou pu o que [11]. The ad an age o his kind o
o o modi ica ion is imp o ed machine o que, and he disad an age
is subs an ially inc eased manu ac u ing complexi y.
A o o opology was designed ha is based on all he designs
desc ibed and combines hei ad an ages: he shielded axial sli ed
solid- o o (SASSR), illus a ed in Fig. 11. An SASSR consis s o an
ASSR, coppe end ings, and a coppe coa ing [12,13] and esembles a
squi el-cage solid- o o . Axial sli s lowe he low- equency impedance
o he o o and inc ease he pene a ion o lux in o he o o ; he
coppe coa ing on he o o ee h ac s like o o ba s; and end ings
collec he induced o o cu en . The coa ing on he o o ee h also
minimizes he impac o highe - equency ai -gap ha monics on he lux
inside he o o and helps o educe eddy cu en losses. This machine
exhibi s he highes elec omagne ic o que and educed o que ipple
and eddy-cu en losses. A compa ison [13] o ASSR, CCSSR and SASSR
using nume ical calcula ions showed ha he SASSR had he highes
o que, powe ou pu , and elec omagne ic e iciency. The s a ing
o que was ema kably highe han ha o he CCSSR. Howe e , o he
bes o ou knowledge his is a heo e ical concep ha has ne e been
p oduced, es ed, and measu ed.
Fig. 12. Cons uc ion o axially sli ed solid- o o wi h adial g oo es o an IM,
acco ding o [70].
3.3. Solid- o o IM wi h adial o o su ace g oo es
This is a special case o solid- o o modi ica ion wi h shallow adial
g oo es on he o o su ace ac ing as a coa ing. The adial g oo es
inc ease he su ace impedance o he o o and ac as a highly e-
sis i e ma e ial laye (o high- equency il e ), p o iding he same
p ope ies as a highly esis i e coa ing. To achie e he desi ed e ec ,
he in e -g oo e dis ance and he g oo e wid h in he ac i e o o
leng h mus be designed co ec ly. Thei dep h should be simila o
he pene a ion dep h o he high- equency ha monic componen s
o he cu en induced in he o o . P ope design should educe he
appa en conduc i i y o he ou e o o laye by 80%–90%. Acco ding
o [18,70], eddy cu en losses a he o o su ace and o al eddy
cu en losses in he o o can be educed by up o 60% and 30%,
espec i ely. This simple modi ica ion can be used wi h any ype o
solid- o o ha does no con ain coa ing ma e ial. I s main ad an age
is simplici y o p oduc ion leading o g ea ly inc eased e iciency. In
addi ion, his modi ica ion does no nega i ely a ec he mechanical
and he mal p ope ies o he machine. An ASSR wi h adial g oo es is
shown in Fig. 12.
3.4. Squi el-cage solid- o o IM
Squi el-cage solid- o o s (SCSRs) a e almos iden ical o con en-
ional induc ion machines wi h squi el-age o o s. The main mo i a-
ion o using an SCSR is ha con en ional lamina ed o o s a e in
mos cases unsui able o high-speed applica ions. Since a lamina ed
o o canno mechanically wi hs and high cen i ugal and high ensile
o ces [71], use o a solid- o o echnology wi h bo ed holes/cu slo s
and coppe ba s inse ed and he ba s connec ed by a sho -ci cui
coppe ing is p e e able. Mos o he cu en is induced in he o o
ba s, whe e a small pa o he cu en lows h ough he o o s eel [10,
72]. Compa ed o o he solid- o o designs, eddy cu en losses a e
signi ican ly educed because o much lowe esis ance o he o o ba s
ela i e o he esis ance o he o o s eel [73,74]. Such a design has a
lowe slip and a lowe powe ac o , and i has he highes e iciency o
all IM opologies. The simples SCSR design has ec angula open slo s
wi h a b azed squi el cage (Fig. 13(a)). A sligh ly be e design om a
mechanical poin o iew is a SCSR wi h a squi el-cage embedded by
isos a ic p essu e (Fig. 13(b)).
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Fig. 13. Cons uc ion o a solid- o o wi h squi el cage o an HSSRIM: (a) ec angula open slo s wi h b azed squi el cage, (b) squi el-cage embedded by isos a ic p essu e, and
(c) ound embedded coppe ba s inse ed in o a solid- o o wi h d illed holes acco ding o [14].
Table 3
An o e iew o possible and sui able ma e ials o solid o o s, along wi h hei cha ac e is ic p ope ies. This able was aken om [25] wi h
he pe mission o he au ho s.
Name Ma e ial Tensile s eng h [MPa] Resis i i y [μΩ cm]
a 20 ◦C (a 200 ◦C)
Sa u a ion lux densi y [T]
Imac o M 4C Mn16-4 1100 25 (37.3) 1.63–1.95
AISI430 X6C l7 600 25.7 (38) 1.66–1.74
Ma aging Fe-Ni-Co-Mo-Ti 2200 49 1.9
C15 – 440 15.9 (30) 1.9
Supe mendu Fe-V-Co 800 40 2.4
S355 – 520 25.7 (38) 1.9–2.1
Fe-Ni Fe-Ni 545 48.2 1.5
Fe-Cu Fe-Cu – 11 1.6
ARNOKRO ME III Fe-C -Co – 69 –
Vicalloy Fe-Co-V 880 61.5 1.3
HyMu 80 Ni-Fe-Mo 930 58 –
Consumen – 345 13 2.15
Vim Va (AISI M50) C-Si-Mn-C -Mo-V 315 13 2.15
MuShield – 620 58 1.9
AISI H13 (O a ) C -Mo-V 1820 52 (63) 1.9
X20C 13 – 880 55 (66) 1.7–1.9
MoCN315M 34C NiMo6 345 19 (33) 1.97–1.99
The p oblem wi h SCSRs is ha inse ing he o o ba s in o he
solid- o o inc eases he leakage induc ance o he o o conside ably,
which educes he pe o mance o he machine. I he o o slo s
a e closed (Fig. 13(c)), he s eel b idges be ween he o o ba s a e
sa u a ed, as desc ibed in [60], hus educing machine e iciency. This
can be sol ed by cu ing an opening abo e he o o slo s, he eby
inc easing he mechanical ic ional losses. Despi e all i s challenges,
i an SCSR IM is p ope ly designed, i exhibi s he bes elec omagne ic
pe o mance o all o o opologies desc ibed.
The main disad an age is ha manu ac u e is complex and chal-
lenging. Despi e i s nume ous ad an ages, SCSRs a e no he p e e ed
choice o high-speed applica ions. In [22,75], i was s a ed ha he
mos c i ical pa o he design is he connec ion be ween he o o
ba s and he sho -ci cui ing. Due o i s high ope a ing speed (and he
associa ed high cen i ugal o ces) and high empe a u e, p ope o e all
elec omagne ic, he mal and mechanical design o he machine is
c ucial. This applies gene ally o all o o opologies men ioned abo e,
bu , his opology is he mos complex o all in e ms o cons uc ion.
4. Sui able ma e ials o solid o o s
Each solid o o opology equi es he ca e ul selec ion o he co ec
i on ma e ial. Pu e solid o o s come wi h he disad an age o high
eddy-cu en losses and low powe ac o in compa ison o lamina ed
o o s. These disad an ages can be pa ially o e come wi h he addi ion
o a coppe laye , coppe ends, o e en, in some cases, a squi el
cage. Wi h a pu ely solid o o , i is ad isable o selec a ma e ial ha
has good elec ical conduc i i y ha allows he low o o o cu en s
wi hou high esis ance and allows he p oduc ion o high machine
o que a low slip. This can be g ea ly enhanced by equipping he solid
o o wi h coppe end ings. When addi ional coppe elemen s, such
as a squi el cage o coppe laye a e added, i is ad isable o choose
i on wi h highe esis i i y o limi o o joule losses. The maximum
sa u a ion alue o he ma e ial is also c ucial and has a signi ican
impac on he design o he machine and i s o e all pe o mance.
Besides elec omagne ic pa ame e s, he mechanical p ope ies o
he ma e ial mus also be aken in o accoun , i mus also mee he
he mal equi emen s o he machine. In high-speed applica ions, he
ma e ial should ha e a su icien ly high ensile s eng h o allow he
o o o wi hs and high ci cum e en ial speeds and mee he o o dy-
namic equi emen s. Table 3con ains a summa y o selec ed ma e ials
and hei basic p ope ies ha a e sui able o use in solid o o s. Some
o he ma e ials lis ed ha e been used in case s udies o e i y hei
sui abili y. In s udy [24], i e ma e ials we e selec ed and compa ed
when used in an IM wi h a solid o o and squi el cage wi h a a ed
powe o 1 MW and a a ed speed o 12 000 pm. A simila s udy [25]
was ca ied using se en di e en ma e ials used in an IM wi h an ASSR
wi h a a ed powe o 180 kW and a speed o 10 100 pm.
The las ma e ial g oup used o solid o o s a e coppe alloys and
hei use in coppe -coa ed SSRs. The coppe coa ing is applied o he
solid o o by explosi e pla ing. Modeling a coppe connec ion on a
solid o o using nume ical me hods is e y di icul , as in p ac ice he
connec ion may no be uni o m and ai pocke s may o m. In o de
o success ully apply a coppe alloy in a solid o o , some impo an
mechanical c i e ia needed o be me du ing he p ac ical es s ca ied
ou by he au ho s. In p ac ice, he ideal heigh o he coppe alloy
should be a ound 3–8 mm wi h a maximum o 10 mm. The inal
c i e ion is he elonga ion alue o bo h he coppe alloy and i on o
he solid o o . The elonga ion should be a leas 10% o bo h ma e ials.
Table 4lis s all he sui able coppe alloys ha can be used o explosi e
pla ing. No single alloy has he ideal p ope ies, alloy selec ion is a
ade-o be ween elec ical conduc i i y, ensile s eng h, and ma e ial
cos .
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Table 4
An o e iew o possible and sui able coppe alloys o explosi e coa ing o solid
o o su aces and hei cha ac e is ic p ope ies. The pe cen age alue o elec ical
conduc i i y is ela ed o he maximum conduc i i y alue o pu e coppe , espec i ely
58 MSm−1.
Name EN code USN code Tensile
s eng h [MPa]
Elec ical
conduc i i y [%]
Cu-OF CW008A – 240–360 100
CuNi2SiC CW111C C18000 650–780 40
CuC 1Z CW106C C18150 400–600 75
CuC 1 CW105C C18200 400–500 80
GlidCop AL-15 – C15715 500–700 55
5. Elec omagne ic modeling o solid- o o IMs
5.1. Analy ical me hods
Analy ical design usually conside s signi ican simpli ica ions o he
machine. In he pas , an unsa u a ed o o wi h cons an pe meabili y
was conside ed, which yielded e y poo modeling accu acy when
compa ed o eal machine measu emen s [76]. Resea che s hen s a ed
o conside he 3D na u e o solid- o o s, hus inc easing he o e all
accu acy o analy ical me hods [76]. Mode n analy ical me hods con-
side , solid- o o sa u a ion [77], and some esea che s e en spli he
o o in o se e al laye s o ob ain mo e accu a e esul s: Fo ins ance,
in [76] h ee-dimensional linea me hods we e combined wi h he
ans e ma ix me hod. Though i s esul s a e good, his app oach
is ela i ely complica ed and limi ed o SSRs only. Newe analy ical
me hods, some o which we e p esen ed in [78–86], aim o e en be e
and mo e accu a e esul s, bu hey a e all limi ed o SSRs.
Much simple and mo e lexible analy ical me hods a e equi alen
elec ical ci cui s (EECs) o s eady-s a e machine pe o mance calcu-
la ion, whose g ea es ad an age is ease o use. Mo e ecen me hods
p oposed in [49,87] use T-shaped EECs, wi h empi ical o mulas o
ensu e he bes esul s. In [88], addi ional pa ame e s we e p esen ed
o calcula ion, such as esis ance and end- egion leakage eac ance o
he solid- o o . A special case o single-phase CCSSR IM analysis using
an EEC was p esen ed in [89]: He e, he machine was analyzed by
means o d-q axes modeling, whe e each axis has i s own EEC.
O e all, analy ical me hods a e simple and as , bu hey ei he
equi e signi ican simpli ica ions, which ha e a nega i e impac on
compu a ional accu acy, o a e limi ed o speci ic solid- o o opolo-
gies; hey e en may no wo k o he same ype o machine wi h a di -
e en powe ange. The challenges o cons uc ing accu a e analy ical
me hods can be summa ized by he o mula o pene a ion dep h:
𝑑=√2
𝜔p𝜎 𝜇,(1)
whe e 𝜔pis he angula equency o he pene a ing ield, 𝜎is he
elec ical conduc i i y, and 𝜇is he magne ic pe meabili y o he
ma e ial. The pene a ion dep h (1) is cha ac e ized by h ee a iables
ha a y o e ime and space and a e di icul o desc ibe analy ically.
Fu he , all h ee a iables in e ac a leas pa ially. In addi ion, con-
duc i i y 𝜎changes wi h empe a u e, which is p oblema ic, because
he o o is no hea ed uni o mly h oughou i s olume. The angula
equency 𝜔psigni ican ly changes he beha io o he machine a high
equencies. I also g ea ly a ec s lux-lines pene a ion in o he o o .
The magne ic pe meabili y 𝜇 a ies conside ably wi hin he ma e ial
used and due o he nonlinea B-H cu e. In addi ion, he magne ic
cha ac e is ics exhibi hys e esis e ec s. Since he o o ma e ial is
di e en ly sa u a ed h oughou i s olume, analy ical me hods mus
ake in o accoun he non-linea na u e o he ma e ial. An example o
he ela i e pe meabili y dis ibu ion is shown in Fig. 14.
Fu he , he dis ibu ion o o o losses is shown in Fig. 15. Mos
o he o o losses a e concen a ed in he ou e laye o he o o and
a e caused by he highe ha monic componen s ha o igina e in he
Fig. 14. Dis ibu ion o ela i e pe meabili y in a smoo h solid- o o (a) and in an
axially sli ed solid- o o (b).
Fig. 15. Dis ibu ion o eddy cu en losses in a smoo h solid- o o (a) and in an axially
sli ed solid- o o (b).
ai -gap and di ec ly a ec he pene a ion dep h o o o su ace. In
gene al, a solid- o o IM is a complex mul iphysics p oblem ha is
di icul o desc ibe pu ely analy ically. Mode n nume ical me hods a e
mo e e ec i e in sol ing such challenging p oblems and a e hus used
mo e equen ly.
5.2. Nume ical me hods
The cu en ly mos popula mode n nume ical me hods o calcu-
la ing solid- o o IMs a e based on he ini e elemen me hod (FEM),
he esul s o which a e in e y good ag eemen wi h measu ed da a.
Thei main ad an age is ha he calcula ions a e non-linea , while hei
main disad an age is ha hey a e compu a ionally ime-in ensi e.
Solid- o o IMs a e complex elec omagne ic-mechanical sys ems o
which 3D calcula ions a e ecommended o cap u e all aspec s o
he machine. Howe e , 3D models may equi e a e y ine mesh (a
leas in some egions, such as he ai -gap o he machine and he
sa u a ed end egions o he solid- o o ), and o IMs a ime-s epping
ansien analysis is needed [90,91]. On he o he hand, co esponding
models a e e y ime-consuming o e alua e and equi e conside able
compu e memo y. On he o he hand, hey a e able o calcula e wi h
high accu acy all machine p ope ies, including he passage o magne ic
lux om one pole o ano he in 3D di ec ions [92], he complex
dis ibu ion o eddy cu en s, and changes in magne ic lux densi y
dis ibu ion [93], aking in o accoun he leakage induc ance o he
s a o end windings in 3D space [94] and conside ing he leakage
induc ance o he solid- o o ends [95].
Al hough hese 3D models yield accu a e esul s [96], hey a e
oo ime-consuming o mos applica ions [96]. To minimize un ime,
ime-ha monic analysis can be conside ed, bu his app oach is usually
is only sui able o synch onous machines and canno calcula e all
ele an aspec s o IMs. Analy ical me hods in combina ion wi h he 3D
meshed eluc an ne wo ks me hod (RNM) we e p esen ed as possible
In e na ional Jou nal o Elec ical Powe and Ene gy Sys ems 166 (2025) 110520
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V. Bilek e al.
indi idual nodes a e connec ed by he mal esis ances. Indi idual hea
losses o he analyzed machine a e in he LPTN ep esen ed by cu en
sou ces, which is based on an elec o- he mal analogy [1]. In ac , he
eal empe a u e ield in he machine is eplaced by only a ew nodal
empe a u es, which leads o a e y small numbe o equa ions o be
sol ed and hus o e y as compu a ions [154,155].
No e ha hese models also ha e d awbacks, which a e mainly
ela ed o he p oblema ic modeling o some esis ances o he LPTN.
Fo ins ance, he he mal esis ances o con ec i e hea ans e a e in
p inciple speed-dependen . Fo some simple cases, hese dependencies
can be de i ed using he undamen als o hea ans e heo y [156,
157]. Howe e , mos geome ies and cooling condi ions in elec ical
machines, especially in high-speed machines ( o ins ance, due o he
o o sli ing), a e oo complex, and he con ec ion he mal esis ances
mus be modeled using empi ical ela ionships o using CFD analysis.
LPTN models also use a numbe o simpli ica ions, e.g., eal complex
geome ies mus usually be eplaced by simple ones.
Due o i s ad an ages, his me hod is widely used o he mal
modeling o elec ical machines, including high-speed SRIMs [113,120,
121,158], and has a g ea po en ial o inco po a ion in o op imiza ion
p ocesses and mul i-physics modeling. The e o e, mo e de ails on his
me hod a e gi en below.
6.2.3. The mal models u ilizing ini e elemen me hod
As p e iously men ioned in he pape , he ini e elemen me hod
is a powe ul and popula ool o elec omagne ic calcula ions. This
me hod can also be used o a he mal analysis [159]. FEM-based mod-
eling is hus he hi d op ion o implemen ing an elec ical machine
he mal model.
In gene al, he FEM-based he mal models se e o calcula e he
empe a u e ield in he machine unde s udy. Compa ed o models
based on he use o LPTNs, he FEM-based models ha e se e al ad-
an ages. Fo example, hey a e able o simula e a bi a ily complex
geome ies. Since hey p o ide a comple e empe a u e ield o indi-
idual machine pa s, hey can be used o de ec empe a u e ho spo s.
Howe e , he FEM-based models equi e almos all he same inpu
pa ame e s as LPTN he mal models, including he hea ans e coe -
icien s, which mus be calcula ed by means o empi ical equa ions o
ob ained by CFD analysis (see Sec ion 6.2.2).
In e ms o compu a ional powe equi emen s, FEM-based models
a e less demanding han CFD-based models, bu s ill signi ican ly mo e
demanding han LPTNs. Examples o FEM-based he mal analyses can
be ound in [160–164]. These models a e also equen ly used o e i y
he simple LPTN models; see, o ins ance [122,165–167].
Fig. 25 shows an example o calcula ing he empe a u e ield in
an SSR and an ASSR using FEM. The esul s a e no malized o a clea
compa ison. Simila ai -gap cooling condi ions we e conside ed in he
calcula ions. In addi ion, hea dissipa ion om he side walls o he
ee h was aken in o accoun in he ASSR, which, despi e sligh ly highe
losses, esul s in lowe o e all empe a u es compa ed o he SSR, as
can be seen in he igu e.
6.3. The mal modeling using a lumped pa ame e he mal ne wo k
Up o his poin , we ha e p o ided a ho ough o e iew o elec ical
machine cooling and he mal modeling echniques ha can be em-
ployed in SRIM applica ions. Fu he , we p esen ed speci ic app oaches
o implemen ing a he mal model o an elec ical machine. As p e i-
ously men ioned, he LPTN he mal models a e cha ac e ized by low
compu a ional powe equi emen s, which makes hem sui able o
inclusion in mul i-physics analyses. This sec ion he e o e concen a es
on hese models, pa icula ly on hei basic p inciples and undamen al
aspec s o hei c ea ion. This in o ma ion may also be use ul when
using specialized comme cial so wa e ha employs LPTNs. [168].
Fig. 25. No malized empe a u e dis ibu ion in: (a) an SSR and (b) an ASSR.
Fo a gi en LPTN, he nodal s eady-s a e empe a u es a e calcu-
la ed as a sys em o linea equa ions, whe e he numbe o nodes in he
LPTN is equal o he numbe o equa ions o be sol ed. Fo a s eady-
s a e analysis, he unknown nodal empe a u es a e hus ob ained by
sol ing he ma ix equa ion
𝐓=𝐆−1𝐏,(45)
whe e 𝐓is he ec o o unknown nodal empe a u es, 𝐏is he powe
sou ce ec o , and 𝐆is he he mal conduc ance ma ix. The indi id-
ual pa ame e s in Eq. (45) mus be c ea ed o he speci ic LPTN as
desc ibed, o example, in [1]. Examples o comple e LPTN models o
SRIMs can be ound, o ins ance, in [113,120], and [121].
The accu acy o he empe a u es calcula ed using an LPTN he -
mal model depends on se e al ac o s, including he complexi y o
he speci ic LPTN. Fu he mo e, accu a e knowledge o hea losses
also plays an impo an ole. P ima ily, i is essen ial o accu a ely
calcula e he indi idual he mal esis ances o he LPTN. Compu a ion
o hese esis ances, which a e modeled as he mal esis ances by
conduc ion, con ec ion, o adia ion, depending on he ac ual hea
ans e mechanisms in indi idual machine pa s, is desc ibed below.
6.3.1. Conduc i e hea ans e
In solid pa s o elec ical machines, hea is ans e ed by conduc-
ion, which is a hea ans e mechanism ha occu s in a subs ance
due o he exis ence o a empe a u e g adien [156,157]. The ba-
sic equa ion associa ed wi h hea ans e by conduc ion is he hea
conduc ion equa ion. Sol ing his di e en ial equa ion wi h speci ic
bounda y condi ions yields he empe a u e ield in he geome y unde
analysis.
Fo a gene al h ee-dimensional case wi hin a medium wi h o -
ho opic p ope ies, he hea conduc ion equa ion o a s eady-s a e
analysis can be w i en in he ollowing o m [169]:
𝜕
𝜕 𝑥(𝜆x𝜕 𝑇
𝜕 𝑥)+𝜕
𝜕 𝑦(𝜆y𝜕 𝑇
𝜕 𝑦)+𝜕
𝜕 𝑧(𝜆z𝜕 𝑇
𝜕 𝑧)+𝑝gen = 0,(46)
whe e 𝑇is he empe a u e, 𝑝gen is he a e o in e nal ene gy gene a-
ion pe uni olume, and 𝜆𝑥,𝜆𝑦, and 𝜆𝑧a e, espec i ely, he he mal
conduc i i ies in he indi idual di ec ions o he Ca esian coo dina e
sys em. No e ha o many pa s o elec ical machines, i is mo e
con enien o wo k wi h he cylind ical coo dina e sys em ins ead.
Ano he impo an equa ion in he ield o hea conduc ion model-
ing is Fou ie ’s law by which hea ans e a es can be calcula ed. Fo
he gene al h ee-dimensional case, hea ans e a es 𝑃x,𝑃y, and 𝑃z
in indi idual Ca esian coo dina es a e calcula ed by [156]:
𝑃x= −𝜆x𝐴x𝜕 𝑇
𝜕 𝑥,(47)
𝑃y= −𝜆y𝐴y𝜕 𝑇
𝜕 𝑦,(48)
𝑃z= −𝜆z𝐴z𝜕 𝑇
𝜕 𝑧,(49)
In e na ional Jou nal o Elec ical Powe and Ene gy Sys ems 166 (2025) 110520
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V. Bilek e al.
whe e 𝐴x,𝐴y, and 𝐴za e hea conduc ion a eas no mal o he 𝑥-, 𝑦-,
and 𝑧-di ec ions o he Ca esian coo dina e sys em, espec i ely.
By using a known empe a u e ield and hea ans e a es ob ained
by means o Fou ie ’s law, he mal esis ances by conduc ion can be
de i ed and hen used in an LPTN. The p ocedu e o such a de i a ion
can, o ins ance, be ound in [1,170].
No e ha he use o he hea conduc ion Eq. (46) leads o a e y
complex ma hema ical solu ion, and hus se e al simpli ying assump-
ions a e usually aken in o accoun in LPTN models. Typically, i is
desi able o simpli y he mul i-dimensional empe a u e ield o some
pa s o he machine analyzed. This can, o ins ance, be done in he
cases whe e he hea ans e in one o mo e di ec ions is negligible
compa ed o he hea ans e in he o he di ec ions. Such a simpli-
ica ion can usually be assumed, o example, o he s a o i on o
elec ical machines due o lamina ion e ec s o , e en ually, also due
o poo cooling condi ions on i s end aces. Thus, a one-dimensional
empe a u e ield wi h hea ans e only in he adial di ec ion can be
conside ed in his pa o he machine in many p ac ical cases [171,
172].
Howe e , he e a e cases in which he mul i-dimensional empe a-
u e ield canno be simpli ied, and hea ans e in mul iple dimen-
sions mus be conside ed. This also applies o he modeling o hea
ans e by conduc ion in solid o o s whe e absence o lamina ion
makes mul i-dimensional empe a u e ields mo e likely. Fo hese
mul i-dimensional cases, some simpli ica ions a e in oduced in he
calcula ions wi h LPTNs: The mul i-dimensional conduc ion is sol ed
as a combina ion o one-dimensional solu ions in he co esponding
di ec ions. App oaches o his kind we e p esen ed o modeling he
h ee-dimensional hea ans e in a gene al cuboidal elemen in [170]
and o modeling he h ee-dimensional hea ans e in a gene al a c-
segmen elemen in [173]. In p ac ice, hese gene al elemen s, e e ed
o as model bodies, a e used o modeling he conduc i e hea ans e
in indi idual machine pa s due o he geome ic simila i y be ween
hese model bodies and eal machine pa s.
Fig. 26 shows an example o a body ha can be used o wo-
dimensional hea ans e modeling in cylind ical machine pa s, o
ins ance, in he solid- o o [121]. In pa icula , using his model body
allows he hea ans e s in he adial and axial di ec ions o be sol ed
sepa a ely. Fig. 26(a) illus a es all impo an pa ame e s, such as
dimensions, whe e 𝑅in is he inne adius, 𝑅ou is he ou e adius, and
𝐿is he leng h o he cylinde . The bounda y condi ions assumed a e
also indica ed in he igu e, whe e 𝑇 ,in is he a e age empe a u e o
he inne cylind ical su ace, 𝑇 ,ou is he a e age empe a u e o he
ou e cylind ical su ace, 𝑇a,le is he a e age empe a u e o he le
end su ace, 𝑇a, igh is he a e age empe a u e o he igh end su ace,
and 𝑃loss a e he hea losses gene a ed in he body. All pa ame e s
associa ed wi h he adial and axial di ec ions ha e indices ‘ ’ and ‘a’,
espec i ely. The co esponding he mal ne wo k ha enables calcu-
la ion o mul i-dimensional hea ans e sepa a ely in he adial and
axial di ec ions can be seen in Fig. 26(b). The indi idual esis ances
a e calcula ed acco ding o he ollowing equa ions [121,153]:
𝑅1, =1
4𝜋 𝜆 𝐿⎡⎢⎢⎢⎣
2𝑅2
ou ln (𝑅ou
𝑅in )
(𝑅2
ou −𝑅2
in)− 1⎤⎥⎥⎥⎦
,(50)
𝑅2, =1
4𝜋 𝜆 𝐿⎡⎢⎢⎢⎣
1 −
2𝑅2
in ln (𝑅ou
𝑅in )
(𝑅2
ou −𝑅2
in)⎤⎥⎥⎥⎦
,(51)
𝑅3, = −[𝑅2
in +𝑅2
ou −
4𝑅2
in𝑅2
ou ln(𝑅ou
𝑅in )
(𝑅2
ou −𝑅2
in)]
8𝜋 𝜆 𝐿(𝑅2
ou −𝑅2
in),(52)
𝑅0,a =𝐿
2𝜋 𝜆a(𝑅2
ou −𝑅2
in),(53)
Fig. 26. Model body o wo-dimensional hea ans e modeling in cylind ical machine
pa s: (a) geome y wi h impo an pa ame e s, (b) co esponding he mal ne wo k.
Sou ce: Adap ed om [121,153].
𝑅1,a = −𝐿
6𝜋 𝜆a(𝑅2
ou −𝑅2
in),(54)
whe e 𝜆 and 𝜆aa e he he mal conduc i i ies in he adial and
axial di ec ions, espec i ely. In gene al, hese pa ame e s may ha e
di e en alues: 𝜆 ≠𝜆a. In he case o a solid- o o , howe e , hese
he mal conduc i i ies a e ypically equal: 𝜆 =𝜆a.
6.3.2. Con ec i e hea ans e
In a quiescen luid, hea is ans e ed by conduc ion. Howe e ,
in he p esence o bulk luid mo ion, hea ans e h ough he luid
is enhanced and hea is ans e ed by con ec ion in his case [156].
P ope modeling o con ec ion is essen ial in SRIMs because he o o
hea losses, which ac ually achie e signi ican alues in hese machines,
a e mos ly dissipa ed by his mechanism.
The a e o hea ans e by con ec ion be ween a solid body and i s
su ounding luid, o ins ance, be ween he o o su ace and he luid
in he ai gap, is calcula ed by
𝑃=𝛼con 𝐴con (𝑇su −𝑇∞),(55)
whe e 𝛼con is he con ec ion hea ans e coe icien , 𝐴con is he
su ace a ea, 𝑇su is he su ace empe a u e o he solid body, and
𝑇∞is he ee s eam empe a u e o he su ounding luid. The mal
esis ance by con ec ion is calcula ed by
𝑅=1
𝛼con 𝐴con
.(56)
The su ace a ea 𝐴con can usually be calcula ed e y easily and
p ecisely om a pa icula geome y, which is also ela i ely simple
in he case o SRIMs. Howe e , de e mining he accu a e alue o he
hea ans e coe icien 𝛼con is challenging. This coe icien depends
on se e al ac o s, such as he physical p ope ies o he cooling luid.
The p ope choice o cooling luid can he e o e signi ican ly a ec he
e ec i eness o con ec ion cooling [126]. Fu he , he hea ans e
In e na ional Jou nal o Elec ical Powe and Ene gy Sys ems 166 (2025) 110520
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V. Bilek e al.
coe icien hea ily depends on he shape and loca ion o he solid body
ela i e o i s su ounding luid, and hus di e en equa ions need o
be used o calcula e hea ans e coe icien s o di e en su aces o
he machine analyzed. Fo hese pu poses, empi ical co ela ions based
on dimensionless analysis a e ypically used [145].
Speci ic ela ions o calcula ing he hea ans e coe icien in
he ai gap can be ound, o ins ance, in [121]. The li e a u e gi es
o mulas o bo h ypical cases o ai gap cooling, ha is, wi h and
wi hou axial low. An o e iew o o he ela ions o calcula ing he
hea ans e coe icien in he ai gap was gi en in [174,175]. Some
ela ions o hea ans e coe icien s o o he impo an machine pa s
can be ound in [143,156,157,176], and [141].
6.3.3. Radia ion hea ans e
Unlike hea ans e by conduc ion and con ec ion, adia ion hea
ans e does no equi e a medium and can ake place in a acuum,
since he mal ene gy is ans e ed ia elec omagne ic wa es he e. In
mos p ac ical cases, adia ion hea ans e is igno ed, as i is negligible
compa ed o conduc ion and con ec ion [171,172]. Howe e , he e a e
some speci ic si ua ions whe e adia ion can play a signi ican ole,
o ins ance, when he machine is employed in ai c a and ae ospace
applica ions, whe e e y low ai densi y o e en acuum condi ions
may occu [177].
The amoun o hea ans e ed by adia ion depends on se e al ac-
o s, including empe a u es, emissi i ies, shapes, and also he ela i e
o ien a ion o he su aces be ween which hea is exchanged. Values
o he emissi i ies o a ious ma e ials and hei su ace inishings, as
well as pa icula equa ions o calcula ing he amoun o hea ans e
depending on he a ious mu ual con igu a ions o he indi idual adi-
a ion su aces, can be ound in [156,157]. Fo example, he hea low
a e by adia ion be ween he s a o and he o o can be modeled as a
case o wo long concen ic cylinde s, calcula ed as [119,157]
𝑃=
𝜎SB𝐴in(𝑇4
in −𝑇4
ou )
1
𝜖in +𝑅in
𝑅ou (1
𝜖ou − 1),(57)
whe e 𝑃is he hea low a e be ween he inne and ou e cylinde s,
𝜎SB = 5.67 × 10−8 W m−2 K−4is he S e an–Bol zmann cons an , 𝑇in
is he su ace empe a u e o he inne cylinde , 𝑇ou is he su ace
empe a u e o he ou e cylinde , 𝜖in is he emissi i y o he inne
cylinde , 𝜖ou is he emissi i y o he ou e cylinde , 𝑅in is he adius
o he inne cylinde , 𝑅ou is he adius o he ou e cylinde , and
𝐴in = 2𝜋 𝑅in𝐿is he su ace a ea o he inne cylinde , whe e 𝐿is he
leng h o he cylinde s.
Including adia ion hea ans e in an LPTN he mal model equi es
knowledge o he he mal esis ance by adia ion. To ob ain his esis-
ance, he equa ion o calcula ing he adia ion hea low a e mus
i s be ew i en as
𝑃=𝛼 ad𝐴in(𝑇in −𝑇ou ),(58)
whe e 𝛼 ad is he adia ion hea ans e coe icien . Fo he case o wo
long concen ic cylinde s, compa ing Eqs. (57) and (58) yields o he
adia ion hea ans e coe icien :
𝛼 ad =𝜎SB
1
𝜖in +𝑅in
𝑅ou (1
𝜖ou − 1)(𝑇4
in −𝑇4
ou )
(𝑇in −𝑇ou ).(59)
Finally, he he mal esis ance by adia ion can be calcula ed by
𝑅=1
𝛼 ad𝐴in
.(60)
No e ha he empe a u es in he equa ions abo e a e, in ac ,
unknown empe a u es calcula ed by he he mal model. Since his
equi es i e a i e compu a ion, he inclusion o adia ion in he he mal
model should always be conside ed ca e ully.
Fig. 27. Example o a machine wi h h ough- en ila ed cooling: (a) coolan low pa h,
(b) empe a u e p o ile.
Sou ce: Adap ed om [121].
6.3.4. Modeling o coolan low
As p e iously men ioned, solid- o o induc ion machines equi e
e ec i e dissipa ion o hea losses gene a ed in he o o . In hese
machines, main aining o o empe a u es a accep able le els is o en
achie able only by using cooling s a egies ha include di ec hea
dissipa ion om he o o o a cooling luid lowing in i s p oximi y,
o example, h ough he machine’s ai gap (Figs. 24 and 27).
A comp ehensi e o e iew o coolan low modeling can be ound,
o example, in [1,178]. The mos impo an aspec s a e summa ized
below and may apply no only o he speci ic ai -gap cooling con-
igu a ion conside ed he e bu also gene ally o any cooling opology
wi h ci cula ing coolan , o example, o wa e jacke cooling and sha
cooling.
An example o a machine wi h h ough- en ila ed cooling is shown
in Fig. 27(a). Due o i s simplici y, his se up is con enien o explain-
ing he undamen al concep s o coolan low modeling, and was also
employed o his pu pose, o ins ance, in [121].
Fig. 27(b) shows a ypical empe a u e p o ile o he coolan as a
unc ion o he posi ion in he coolan low pa h. As he coolan passes
h ough he machine, i abso bs hea losses ha cause he coolan o
hea up om i s inpu empe a u e 𝑇in o he ou pu empe a u e 𝑇ou .
The o al empe a u e ise o he coolan depends on he olume low
a e, which can be calcula ed by
𝑞=𝑃 o
𝜌𝑐p𝛥𝑇 o
=𝑃 o
𝜌𝑐p(𝑇ou −𝑇in),(61)
whe e 𝑞is he olume low a e o he coolan , 𝑃 o a e he o al hea
losses abso bed by he coolan , 𝛥𝑇 o is he o al empe a u e ise o he
coolan , and 𝜌and 𝑐p, which desc ibe he coolan ’s physical p ope ies,
a e he densi y and he speci ic hea capaci y a cons an p essu e,
espec i ely.
The coolan empe a u e is usually analyzed in se e al impo an
machine pa s, and he e o e he empe a u e p o ile (Fig. 27(b)) is
di ided in o se e al co esponding sec ions. In each o hese sec ions,
a linea empe a u e ise is assumed. The empe a u es 𝑇1,𝑇2, and 𝑇3
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co espond o he a e age empe a u es o he coolan in he le end-
winding egion, in he ai gap, and in he igh end-winding egion o
he machine, espec i ely, while he empe a u es 𝑇in,1,𝑇end,1,𝑇in,2,
𝑇end,2,𝑇in,3, and 𝑇end,3 a e inpu and ou pu coolan empe a u es o
he co esponding sec ions.
The linea empe a u e ise in he 𝑖 h sec ion o he coolan low
pa h is desc ibed by
𝛥𝑇𝑖=𝑇end,𝑖 −𝑇in,𝑖 =𝑃loss,𝑖
𝜌𝑞 𝑐p
,(62)
whe e 𝛥𝑇𝑖,𝑇in,𝑖,𝑇end,𝑖, and 𝑃loss,𝑖 a e he coolan o al empe a u e ise,
he coolan inpu empe a u e, he coolan ou pu empe a u e, and he
hea losses abso bed by he coolan , espec i ely, co esponding o he
𝑖 h sec ion o he coolan low pa h.
The a e age empe a u e in any speci ic sec ion is equal o hal
o he o al empe a u e ise in ha sec ion plus he inpu coolan
empe a u e in ha sec ion, as can be seen in Fig. 27(b). This s a emen
can be exp essed as
𝑇𝑖=𝑇in,𝑖 +𝛥𝑇𝑖
2=𝑇in,𝑖 +𝑇end,𝑖 −𝑇in,𝑖
2,(63)
which, like Eq. (62), is gene ally applicable. Using Eqs. (62) and
(63) yields he a e age coolan empe a u es in he indi idual sec ions
(Fig. 27):
𝑇1=𝑇in +𝑅q𝑃loss,1,(64)
𝑇2=𝑇in +𝑅q(2𝑃loss1 +𝑃loss2),(65)
𝑇3=𝑇in +𝑅q(2𝑃loss1 + 2𝑃loss2 +𝑃loss3),(66)
whe e he esis ance 𝑅qis calcula ed as
𝑅q=1
2𝜌𝑞 𝑐p
.(67)
No e ha he empe a u es 𝑇1,𝑇2, and 𝑇3a e nodal empe a u es
o he co esponding LPTN o he machine analyzed and a e calcula ed
concu en ly wi h all o he nodal empe a u es. Fu he , when an LPTN
con ains nodes ha ep esen a coolan , he ma ix Eq. (45) mus be
ex ended by ma ices co esponding o he coolan nodes, as desc ibed
in [1].
Finally, no e ha he equi ed olume low a e o he coolan
mus be ensu ed by adequa e ans, pumps, e c. Fo his pu pose, he
hyd aulic ci cui o he machine mus be analyzed [141,145,179–182].
7. Mechanical modeling o solid- o o induc ion machines
In gene al, mechanical calcula ions a e used o es ima e o o sa e y,
de e mine c i ical speeds and calcula e bea ing o ces. While he e a e
usually no majo complica ions wi h low-speed machine design, o
he HSSRIM he mechanical design can be a challenging. Fu he mo e,
he equi emen s o an elec omagne ically and mechanically op imal
design a e o en con lic ing. Fo example, om a mechanical poin o
iew, i is p e e able ha he sli s o an ASSR design (Fig. 8) be as
shallow and wide as possible, wi h a la ge ille a he bo om, and
o SCSR (Fig. 7) he ba s should be as deep benea h he su ace as
possible. This gene ally con lic s wi h elec omagne ic equi emen s, so
some comp omise solu ion mus be ound. The mechanical s ess in he
o o mus be below he yield s eng h, and usually some sa e y ac o
is conside ed.
F om a o o dynamic poin o iew, he e a e equi emen s con-
ce ning he le el o ib a ions, because hey gene a e addi ional o ces
on he bea ings, p oduce excessi e noise, and can cause he en i e
machine o esona e. Vib a ions a e caused ei he by insu icien o o
balancing o by he ope a ing speed being close o he c i ical speed.
The la e can be de e mined based on he eigen equency o he i s
bending mode shape. I is e iden ha sho o o s he e o e ha e a
highe c i ical speed. The ope a ing speed o an HSSRIM is o en nea
he c i ical speed. Hence, he c i ical speed mus be de e mined wi h
su icien accu acy du ing he design phase.
The e a e se e al ypes o loads ha a e conside ed in mechanical
calcula ions, including hose esul ing om o a ion, empe a u e, o o
unbalance, unbalanced magne ic pull, and ex e nal o ces. Depending
on he ci cums ances, no all ypes o loads con ibu e signi ican ly o
o o s ess. The impo ance o a pa icula load mus be de e mined in
o de o decide whe he i is o be included in he analysis.
7.1. S ess calcula ion
Well-known s ess o mulas o a o a ing cylinde and o bending
o o sion loading can be used howe e , hese equa ions a e de i ed
o a smoo h o a ing cylinde , and inco po a ing de ia ions om his
shape can be e y challenging. The main ad an age o he analy ical
solu ion is i s sho e alua ion ime, which is ad an ageous when
i e a ing he s ess calcula ion in he op imiza ion loop. Fo mo e
complex designs, such as ASSR and SCSR, i is o en mo e e ec i e o
use FEA. In his case, again, a 2D model is equen ly used (e.g. [183],
and [184]), which has he ad an age o sho e compu a ion ime and
he disad an age ha only he s ess in a c oss sec ion su icien ly a
om he ends o he o o can be de e mined. Calcula ing he s ess in
he o o ends equi es a 3D model. I is e icien o apply symme y
condi ions and pe o m he calcula ion on a cyclic sec ion only, as
done, o example, in [75]. This signi ican ly educes he compu a ional
complexi y o he model, which is especially desi able when a nonlinea
p oblem mus be sol ed, o ins ance, o p ess- i ed o o pa s o a
i wi h clea ance (e.g., SCSR wi h he ba s inse ed in o holes).
7.1.1. Load om o a ion
The o a ional load is due o cen i ugal o ces and gene a es s ess
ha inc eases quad a ically wi h inc easing machine speed. The analy -
ical app oach is su icien ly accu a e o SSRs (Fig. 7), whe e he s ess
in he middle sec ion can be calcula ed using di e en ial equa ions
ob ained, o ins ance, in [185]. The equa ions a e de i ed o he plane
s ain condi ion, ha is, o a long o a ing cylinde (Fig. 28). They
desc ibe he s ess componen s 𝜎 ,𝜎 , and 𝜎zas unc ions o 𝑟. Based
on [185], hese s ess componen s can be calcula ed espec i ely by
𝜎 =3 +𝜇
8𝜌𝜔2(𝑅2
ou −𝑟2),(68)
𝜎 =3 +𝜇
8𝜌𝜔2𝑅2
ou −1 + 3𝜇
8𝜌𝜔2𝑟2, and (69)
𝜎z=𝜇(𝜎 +𝜎 ).(70)
In hese equa ions, 𝜇is Poisson’s a io, 𝜌is he o o densi y, 𝜔is he
o o angula eloci y, 𝑅ou is he o o ou e adius and 𝑟is he a iable
dis ance in he adial di ec ion.
The s ess alue ha is subsequen ly compa ed wi h he yield
s eng h is called an equi alen s ess. The mos commonly used ype o
equi alen s ess, on Mises s ess (𝜎 M), can be calcula ed by Eq. (71)
acco ding [186]. To gua an ee sa e ope a ion, he maximum alue o
he equi alen s ess mus be below he yield s eng h, which is a
ma e ial mechanical p ope y.
𝜎 M =√(𝜎 −𝜎 )2+ (𝜎 −𝜎z)2+ (𝜎z−𝜎 )2
2. (71)
A ypical plo o he s ess componen in he o a ing solid cylinde
is shown in Fig. 29. No e ha he s ess is no malized, which means ha
all alues a e ela i e o he maximum alue o he equi alen s ess
𝜎 M. F om Eqs. (68) h ough (71) i is clea ha he maximum alue o
he equi alen s ess is a he poin whe e 𝑟is in ini ely close o ze o,
which is a he axis o o a ion.
Each geome ic ansi ion ac s as a s ess concen a o , including he
ansi ion a he connec ion o sha and o o . To ob ain he s ess in
his egion, he co esponding s ess componen s mus be mul iplied
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Fig. 28. C oss sec ion o he long o a ing cylinde — main dimensions.
Fig. 29. No malized s ess componen s in a o a ing solid cylinde (based on Eqs. (68)
h ough (71)).
by he s ess concen a ion ac o . S ess inc ease due o geome ic
ansi ions has been in es iga ed o a e y long ime, and he alues
o he s ess concen a ion ac o s we e summa ized acco ding o o m
and shape o he s ess aise , load case and s uc u al membe ype, o
example, in [187]. Simila ly, bu wi h some unce ain y, he maximum
s ess alue a he bo om o he no ches can be de e mined, o ASSRs.
The e ec o o o sli dimensions on he s ess was in es iga ed, o
ins ance, in [183]. The slo s in SCSRs a e also s ess concen a o s. In
his case, addi ional o ces om he inse ed ba s ac inside he holes.
As men ioned, o mo e complex designs, such as ASSR and SCSR, i is
o en mo e con enien o use FEA.
A compa ison o s ess in a SSR, coppe -coa ed SSR, SCSR wi h
coppe ba s inse ed in d illed slo s, and ASSR is shown in Fig. 30. The
s ess le el is no malized, which means ha he con ou s in he igu e
ep esen mul iplie s o he e e ence alue, which is de ined as he
maximum equi alen s ess (𝜎 M) in a smoo h o a ing cylinde . The e
is a signi ican inc ease in s ess a he base o he sli s in he ASSR
design and a ound he holes in he SCSR design. Fu he examples o
equi alen s ess calcula ions can be ound in [184], whe e in addi ion
o s ess om o a ion he he mal s ess due o he di e en he mal
expansions o ma e ials in he con ac was included and he s esses
induced by any clea ance o in e e ence i s we e conside ed. Simila
analyses we e p esen ed in [75,188].
To sional s ess is usually no calcula ed, since he mos analyses
assume he cons an machine speed and he o que, especially in HSS-
RIMs, does no p o ide a signi ican load. Howe e , in cases whe e a
high momen o ine ia body (e.g. a lywheel) is a ached o he o o ,
signi ican o sional s ess may a ise du ing accele a ion and decele -
a ion o he machine. This s ess mus be included in he calcula ion
o he equi alen s ess (𝜎 M). A simple analysis o he o sional s ess
Fig. 30. No malized s ess om o a ion. Con ou s ep esen mul iplie s o he maxi-
mum equi alen s ess (𝜎 M) in he smoo h cylinde : (a) SSR, (b) CCSSR, (c) SCSR, (d)
ASSR. (The designs a e desc ibed in Sec ion 3).
o squi el-cage o o s was pe o med in [189]; he model analyzed
consis ed o a mo o , a sha and a disk-shaped lywheel.
7.1.2. Tempe a u e load
Tempe a u e s ess is caused by he mal expansion o he ma e ial.
In high-speed machines ha ope a e a high empe a u es, his s ess
can easily exceed he yield s eng h.
The basic ela ionship o he di e ence in leng h wi h empe a u e
change is gi en by:
𝛥𝐿 =𝐿0𝛼 𝛥𝑇 ,(72)
whe e 𝛥𝐿 is leng h inc emen , 𝐿0 he ini ial leng h, 𝛼 he coe i-
cien o he mal expansion, which is a ma e ial p ope y, and 𝛥𝑇 he
empe a u e inc emen .
I he o o is assembled om ma e ials ha di e in he mal
expansion (SCSR wi h he ba s inse ed in o holes), he ole ances
be ween he indi idual pa s should also be aken in o accoun . Hea ing
and cooling may educe he clea ances o cause con ac loss, which
may cause p oblems du ing ope a ion. The in e aces be ween di e -
en ma e ials (SSR wi h coppe end ings, coppe coa ed SSR, SCSR)
should also be analyzed ca e ully. This p oblem is no limi ed o o o s
ha con ain mul iple ma e ials wi h di e en empe a u e expansions.
The mal expansion can also c ea e signi ican s esses a geome ic
ansi ions (e.g., a he bo om o he sli s in an ASSR).
An example o he s ess caused by he combina ion o empe a u e
load and o a ion is shown in Fig. 31. The s esses in he igu e ha e
been no malized in he same manne as in Fig. 30. I can be seen ha
he s ess gene a ed by he mal loading can be e y high, especially in
he in e aces be ween wo ma e ials and a loca ions wi h non-smoo h
geome y. I can easily exceed he yield s eng h.
The mal s ess was analyzed in [75], whe e he clea ance was
op imized o minimize he he mal s ess, and in [184], whe e a 2D
model o he o o was loaded by o a ion and by empe a u e a he
lowes and highes le els sequen ially.
7.1.3. Unbalance
Unbalance is an inhe en p ope y o any o o . All o o s should
he e o e be balanced be o e he machine is assembled. Du ing he
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Fig. 31. No malized s ess om o a ion and empe a u e loads. Con ou s ep esen
mul iplie s o he maximum equi alen s ess (𝜎 M) in he smoo h cylinde : (a) SSR,
(b) CCSSR, (c) SCSR, (d) ASSR. (The designs a e desc ibed in Sec ion 3).
Fig. 32. Balancing an unbalanced o o .
balancing p ocess on balancing machines, masses a e added o emo ed
in he balancing planes un il he o o ine ia o ces a e elimina ed
(Fig. 32). The o o is conside ed balanced when he esidual unbalance
is below he limi gi en by he ele an s anda d. A desc ip ion o he
balancing p ocess can be ound, o example, in [190,191]. Unbalance
may also occu du ing ope a ion, o example, due o sliding o a pa o
he compound o o (SCSR wi h he ba s inse ed in o holes), co osion,
sha bend, o mechanical wea o he o a ing pa s.
In he case o excessi e unbalance, he o ces ac ing on he bea ings
a e inc eased, esul ing in excessi e ib a ion and noise. Unbalanced
o o ib a ion was discussed in [192], whe e he au ho s measu ed
bo h he ib a ion du ing o o s a -up ( ansien esponse) and he
s eady ib a ion a a cons an speed. The simula ion o he s a -up o
an unbalanced o o moun ed on lexible suppo s desc ibed by s i ness
and iscous damping was discussed in [193], whe e he au ho s used
a simple model wi h h ee deg ees o eedom o in es iga e he plana
oscilla ions o he o o . They showed ha he ine ial o ces ac ing on
an unbalanced o o p oduce a o que opposing he d i ing o que.
7.1.4. Unbalanced magne ic pull (UMP)
I o o and s a o a e concen ic, he adial magne ic o ces a e in
equilib ium, bu a p oblem a ises i he ai gap is no homogeneous.
Such an inhomogenei y can no only be caused by o o misalignmen ,
bu also by o a ion o an unbalanced o o , especially nea he c i ical
speed. In his case, he o o s uc u al displacemen inc eases, and
hus he ai gap becomes non-uni o m, which esul s in inc eased o ce
in he di ec ion o he de o ma ion on he side whe e he ai gap is
smalle .
7.1.5. Ex e nal loads
Ex e nal o ces, such as hose due o he gea o bel ansmission
and g a i a ional o ces, can ac on he o o o inc ease he load on he
bea ings. Conside ing hese o ces when selec ing bea ings is c ucial.
The di ec ion o hese o ces is s able in a non- o a ing coo dina e
sys em, bu on a o a ing sha hey appea as pe iodic loads ha
cause cyclic s ess. A la ge ampli ude o cyclic s ess can cause ma e ial
a igue.
The ex e nal load ac ing on he o o may also include o ces ha
a e due o misalignmen o coupled machines (as shown, o example,
in Fig. 2). The e a e wo basic ypes o misalignmen : angula mis-
alignmen , whe e he sha axes o he wo machines mee a an angle
ela i e o each o he , and pa allel misalignmen , whe e he sha axes
o he wo machines a e pa allel, bu ha e an o se . These can be
sou ces o ib a ion and appea in he equency spec um a ypical
mul iples o he o a ional equency. Mo e de ailed in o ma ion can
be ound, o example, in [194].
7.2. C i ical speed calcula ion
The c i ical speed is he machine o a ional speed a which he o o
ib a ions signi ican ly inc ease. I is gi en by he na u al equency
(also known as eigen equency) o he i s bending mode shape, whe e
he equency uni s (e.g., Hz) can simply be con e ed o uni s o
speed ( ypically, pm). The na u al equencies and mode shapes can
be de e mined by sol ing he equa ion
𝐌
𝐱+𝐂
𝐱+𝐊𝐱 =𝟎, (73)
whe e 𝐌,𝐂and 𝐊a e he mass ma ix, damping ma ix and s i ness
ma ix, espec i ely. I can be seen ha he esul s depend only on o o
geome y, s i ness, and damping cha ac e is ics. Any phenomena ha
a ec hese ma ices should be aken in o accoun in he calcula ion o
c i ical speeds.
Once he ma ices ha e been speci ied, he mode shapes and eigen-
equencies can be de e mined as desc ibed, o ins ance, in [195].
Figs. 33(a) and 33(b) show an examples o he i s bending mode
shapes o an SSR, and an ASSR, espec i ely. Al hough he shapes look
simila , he na u al equency (and he e o e he c i ical speed) o he
sli ed o o is highe (in his case by abou 5%) because he axial sli s
do no signi ican ly modi y he bending s i ness, bu dec ease he o o
mass. In gene al, educing mass inc eases he na u al equency, and
educing s i ness educes he na u al equency.
Analy ical me hods a e adequa e o calcula ing he c i ical speed
o simple a angemen s (SSR o ASSR). Fo mo e complex designs, i
is mo e con enien o use FEA. De e mining he c i ical speed o SCSR
designs, whe e he ba s a e inse ed in o slo s is somewha p oblema ic
because, due o manu ac u ing ole ances, he ex en o clea ance o
o e lap be ween ba and hole a e no accu a ely known, and addi ion-
ally hey a e a ec ed by empe a u e. I is he e o e unclea how much
he ba s con ibu e o he inal bending s i ness o he o o . The ba s
a e o en conse a i ely assumed o add only mass and no bending
s i ness. Since he equency o he i s bending mode shape mus be
calcula ed, a model o he ull o o mus be used.
The s i ness and damping ma ices 𝐊and 𝐂usually include he
p ope ies o he coupling elemen s (e.g., bea ings and seals), which
a e o en speed-dependen . The e ms o he ma ices can he e o e
be a ec ed by he o a ional speed. The o o -bea ing sys em beha es
like a se ies o sp ings. I is known ha he esul ing s i ness o a
se ies o sp ings is lowe han ha o indi idual sp ings. The c i ical
speed o such a sys em is hus lowe han in he case o o o s in igid
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Fig. 33. Example o he i s bending mode shape o SSR design (a) and ASSR design
(b).
Fig. 34. Campbell diag am showing he p ocedu e o de e mining he c i ical speed.
suppo s. In [196] he bea ing s i ness in ela ion o c i ical speed was
in es iga ed.
In cases whe e he na u al equency depends on o o speed, he
c i ical speed is usually de e mined om a Campbell diag am, some-
imes (e.g. in [197]) called a na u al equency diag am. An example o
a Campbell diag am is shown in Fig. 34. The c i ical speed co esponds
o he in e sec ion o he na u al equency cu e wi h a line ha
desc ibes he o a ional equency.
Vib a ions caused by he inhe en unbalance o he o o du ing
s a -up a supe c i ical speeds we e calcula ed in [198], whe e he
ib a ion ampli ude a a ious s a -up speed p o iles was de e mined.
Many aspec s ha in luence he c i ical speed a e o en no inco po-
a ed in o he compu a ional model; hese include he gy oscopic e ec ,
in es iga ed in [199], asymme ic s i ness p ope ies o he bea ing
o bea ing suppo s, in es iga ed in [200], o speed-dependen bea ing
p ope ies, s udied in [196], and a sha wi h dissimila momen s o
a ea (asymme ical sha s), e alua ed in [197,201]. In he las o hese,
he ma ices 𝐌,𝐂and 𝐊in Eq. (73) depend on he angle o o a ion.
This was e alua ed, o ins ance, in [202].
Ano he phenomenon ha a ec s he c i ical speed is he UMP,
desc ibed in Sec ion 7.1.4. Since i s inclusion in he calcula ion de-
c eases he c i ical speed ob ained, he UMP should be aken in o
accoun especially i he ope a ing speed is close o he c i ical speed.
A co esponding analysis was p o ided, o example, in [203], whe e
he impac o UMP on he c i ical speed was e alua ed by means o a
ans e ma ix me hod and FEA. A UMP analysis was also p esen ed
in [196], whe e he e ec o o o eccen ici y on UMP was s udied.
Simula ion and expe imen al e i ica ion o he UMP we e epo ed
in [204]; In he expe imen , he o o s a ic eccen ici y was adjus ed
by adding a shimming pla e be ween housing and mo o ca ie .
I was shown in [205] ha i is di icul o gain insigh s in o indi-
idual e ec s based on a e y complex model. I is mo e in o ma i e
o analyze a simple model ha includes all ac o s o in e es .
8. Gene al o e iew o o o opology cha ac e is ics
The o o opologies desc ibed in his pape a e summa ized in
Table 7, accompanied by a compa ison o hei key cha ac e is ics.
Ro o opologies wi h a squi el cage o coppe coa ing p o ide he
bes elec omagne ic pe o mance bu a e challenging o manu ac u e.
Mo eo e , solid o o s wi h squi el cages demons a e some o he
wo s mechanical p ope ies and poo o o obus ness. These sho -
comings a e no p esen in SSRs and ASSRs. Howe e , machines wi h
hese o o s exhibi signi ican ly poo e elec omagne ic pe o mance.
The pe o mance o he machine can be imp o ed by equipping he
solid o o wi h a coppe end- ing o , in he case o an ASSR, by
skewing he sli s. Despi e hese modi ica ions, i is s ill no possible o
achie e he same pe o mance as o machines wi h a squi el cage o
a coppe coa ing. I canno be clea ly de e mined which o o opology
is he bes . The manu ac u ing o speci ic o o opologies can pose
echnological and inancial challenges, which a ise om he desi abili y
o he excellen elec omagne ic pa ame e s and a o able mechanical
and he mal p ope ies inhe en in hese designs.
The no malized elec omagne ic cha ac e is ics o he ou base
opologies o solid o o induc ion machine a e compa ed o hose
o he lamina ed cage squi el cage induc ion machine, and shown in
Fig. 35. In e ms o o que s. slip (Fig. 35(a)), he simple SSR machine
has, as expec ed, he wo s o que s. slip cha ac e is ic. In con as , he
sli ed ASSR machine has app oxima ely h ee imes he o que alue
when compa ed o he simple SSR. A lamina ed and solid o o machine
wi h a squi el cage has almos he same nominal and s a ing o que.
The g ea es de ia ion in he o que can be seen om b eakdown o
he s a ing o que. The coppe -coa ed SSR machine has much highe
nominal o que han he o he SSR, and he highes s a ing o que o all
he machines. Due o he highe leakage eac ance in his machine he
b eakdown o que is lowe han squi el cage o o s. This may change
wi h di e en a design o powe a ing o he machine.
Rega ding cu en s. slip (Fig. 35(b)), he si ua ion is e y simila .
The simple SSR and sli ed ASSR ha e he lowes o e all alues o phase
cu en , ollowed by he coppe -coa ed SSR machine which has he
highes nominal phase cu en alue o all machines, bo h in he a ea o
low slip (highes no-load cu en ) and machine s a -up. This is mainly
due o he heigh o he coppe laye . He e, he coppe laye is ela i ely
high which equi es a high magne izing cu en o compensa e o
he ai -gap leng h and coppe laye . The lamina ed and solid o o
wi hin squi el cage machines, exhibi s almos iden ical phase cu en
alues and he lamina ed o o machine shows he highes alues o all
machines.
In conclusion, hese elec omagne ic cha ac e is ics a e only in o -
ma i e and do no ep esen he absolu e cha ac e o he s udied
machines. In speci ic cases and wi h machine designs u ilizing speci ic
o o opologies, hese cha ac e is ics may signi ican ly a y. Howe e ,
based on hese simpli ied cha ac e is ics, i can be assumed ha he
squi el cage machine, wi h ei he a lamina ed o solid o o , clea ly
displays he bes elec omagne ic pe o mance. Fo high-speed applica-
ions, a lamina ed o o canno be conside ed in mos cases, as he o o
would no mechanically wi hs and he high ci cum e en ial speeds. As
an al e na i e, a solid o o wi h a squi el cage can be conside ed,
In e na ional Jou nal o Elec ical Powe and Ene gy Sys ems 166 (2025) 110520
22
V. Bilek e al.
Table 7
Compa ison o all o o opologies in e ms o selec ed cha ac e is ics, based on he li e a u e e iew p esen ed.
Fig. 35. Compa ison o calcula ed no malized o que (a) and no malized phase cu en RMS alue (b) s. slip cha ac e is ics o 5 undamen al o o opologies o an induc ion
machine. The no malized alues in he igu es a e ela ed o he a ed o que and a ed cu en o he lamina ed induc ion machine wi h squi el cage.
which has simila elec omagne ic pa ame e s and signi ican ly be e
mechanical p ope ies a high ci cum e en ial speeds. The assembly
o he squi el cage in he end- ing egion is pa icula ly challenging,
whe e i ypically equi es ein o cemen , o ins ance, h ough slee es,
o ensu e s uc u al in eg i y. Despi e he complica ed ein o cemen o
he end- ings, his c i ical pa migh s ill no be able o wi hs and high
ci cum e en ial speeds, as epo ed in [75]. The e o e, coppe -coa ed
SSR migh be a sui able choice o high-speed applica ions, since i has
excellen mechanical p ope ies a high ci cum e en ial speeds, is easie
o manu ac u e, and is a comp omise be ween squi el cage machines
and simple SSR/ASSR in e ms o elec omagne ic p ope ies. Gene -
ally, he main c i e ion o he selec ion o a sui able o o opology is
a comp omise be ween he elec omagne ic, he mal, and mechanical
cha ac e is ics o he machine.
9. Conclusion
The design o induc ion machines, o high-speed applica ions is a
e y complex discipline ha equi es a mul idisciplina y analysis o
elec omagne ic, he mal, and mechanical aspec s.
Fi s ly, his pape sough o gi e a comp ehensi e o e iew o high-
speed solid- o o induc ion machine applica ions and a classi ica ion o
solid- o o opologies. Fo each o o opology, i s me i s, d awbacks
and gene al guidelines o p ope design we e highligh ed. Despi e all
he signi ican di e ences be ween he indi idual opologies, i is no
possible o selec any one as he bes .
Secondly, his pape ocused on elec omagne ic modeling o HSS-
RIMs, concen a ing on s a e-o - he-a echniques and hei compa -
ison, especially on analy ical calcula ion me hods and 2D/3D FEM
nume ical simula ions. The 2D FEM nume ical simula ion was ound
In e na ional Jou nal o Elec ical Powe and Ene gy Sys ems 166 (2025) 110520
23
V. Bilek e al.
o be he mos e icien me hod o HSSRIM calcula ion, o e ing a
good ade-o be ween accu acy o he esul s and compu a ional
complexi y. Fu he mo e, se e al me hods o ad anced 2D FEM ma-
chine modeling echniques we e p esen ed ha ensu e g ea e acco -
dance be ween simula ed and measu ed da a o a ious solid- o o
opologies.
Thi dly, since he mal aspec s play an impo an ole in he model-
ing o high-speed machines, especially a he design s age, his pape
also ocused on p esen ing and discussing a ious s a e-o - he-a cool-
ing and he mal modeling echniques. Signi ican a en ion was paid
o analy ical me hods and LPTN models, as hei low demands on
compu ing powe make hem sui able o inclusion in mul i-physics
analyses.
Finally, s a e-o - he-a echniques o simula ing he mechanical
beha io o o o s we e desc ibed. The analyses can be di ided in o
wo ca ego ies: s ess analysis, which is essen ial o sa e y assessmen ,
and c i ical speed analysis, which p o ides in o ma ion on he speed
a which signi ican ib a ion occu s. Va ious phenomena ha can be
included in he calcula ions we e desc ibed and hei in luence on he
esul s discussed.
The main con ibu ion o his s udy is a summa y o all s a e-o -
he-a high-speed solid- o o induc ion machines and co esponding
esea ch indings, which will p o ide guidance o selec ing a sui -
able solid- o o opology and o se ing up a ho ough mul i-physics
machine design p ocess.
CRediT au ho ship con ibu ion s a emen
Vladimi Bilek: W i ing – e iew & edi ing, W i ing – o iginal d a ,
Visualiza ion, Concep ualiza ion. Jan Ba a: W i ing – e iew & edi -
ing, Supe ision, Concep ualiza ion. Ma ek Toman: W i ing – e iew
& edi ing. Pe Losak: W i ing – e iew & edi ing. Ge d B ame do e :
W i ing – e iew & edi ing.
Decla a ion o compe ing in e es
The au ho s decla e he ollowing inancial in e es s/pe sonal ela-
ionships which may be conside ed as po en ial compe ing in e es s:
Vladimi Bilek epo s inancial suppo was p o ided by Linz Cen e
o Mecha onics GmbH.
Vladimi Bilek epo s inancial suppo was p o ided by Cen e o
Resea ch and U iliza ion o Renewable Ene gy.
I he e a e o he au ho s, hey decla e ha hey ha e no known
compe ing inancial in e es s o pe sonal ela ionships ha could ha e
appea ed o in luence he wo k epo ed in his pape .
Acknowledgmen s
This esea ch wo k has been ca ied ou in he Cen e o Resea ch
and U iliza ion o Renewable Ene gy (CVVOZE). Au ho s g a e ully
acknowledge inancial suppo om he Minis y o Educa ion, You h
and Spo s o he Czech Republic unde ins i u ional suppo and BUT
speci ic esea ch p og amme (p ojec No. FEKT-S-23-8430).
Fu he mo e, his wo k has been suppo ed by he COMET-K2 ‘‘Cen-
e o Symbio ic Mecha onics’’ o he Linz Cen e o Mecha onics
(LCM) unded by he Aus ian ede al go e nmen and he ede al s a e
o Uppe Aus ia.
Da a a ailabili y
No da a was used o he esea ch desc ibed in he a icle.
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