Ultrasonic Flow Metering w ith Highly Accurate
Jitter and Offset Com pensation

vor gelegt v on
Dipl. - Ing .
Assia Hamouda
geb. in Batna, Algerien

von der Fakultät IV
- Elektrotechnik un d In formatik –
der T echnischen Unive rsität Berlin
zur Erlangung des a kad emischen Grades

Doktor der Ingenieu r wissenschaf ten
- Dr .- Ing. -
genehmigte Dissert atio n

Promotionsaussch uss
V orsitzender: Prof. Dr .- Ing. Friedel G erfers
1. Gutachter : Prof. Dr . rer . nat. Otto Manck
2. Gutachter : Prof. Dr .- Ing. Roland T hewes
3. Gutachter : Prof. Dr .- Ing. Nour -Ed dine Bouguechal

T ag der wissens chaftlichen Aussprache: 12. M ai 2016

Berlin 2017

Acknowledge ments

First and foremost, all thanks and praises are to God the Almight y for his blessing which
made this work possible and allowed for it to be completed.
I would like to ex press my deepest and most sincere gratitude to my sup ervisor Prof. Dr .
rer . nat. Otto Manck f or his unconditional su pport, valuable and conti nuous advice,
guidance and assistance throug hout my Ph.D. program, and his ver y active part in
initiating this work. I am especiall y thankful for his help with review ing and his
constructive comments and continuous support during the preparation of this thesis.
W it hout his strong support and endless patience I would have never b een able to complete
this research work.
I would like to thank Pr of. Nour -Eddine Bouguechal at the Universit y of Batna, Algeria,
for his kind support, helpful sugge stions, and encouragement.
I would like to thank P rof. Roland Thewes, Head of the Institute of S ensor and A ctuator
Systems at TU -Berlin for reviewing my Ph.D . work. He has of fered man y helpful
suggestions and remar ks.
I am indebted to Mr . Helmut Manck, the Senior Manager of eonas GmbH , for funding this
research a nd his commitment to support the development of this project.
I would like to acknowl edge Dr . R uediger Arnol d, Boris Joesaar , Dr . Mohamed La mine
Hafiane and Redouane Djeghader with whom I had interesting conversations.
I also wish to thank Hadjer , Khadra, Souad, and L atifa for their great f riendship and for
helping me w ith va rious mathematical issues.
I would like to give spe cial thanks to Dr . P eter Grametba u er an d Dr . Sa bine Se yde l for
proofreading my thesis and helpful suggests and comments.

I would like to give my special thanks to my famil y , my lovel y mother , my father , my
brother , my sisters, my mother in law , my broth ers in law , and sis ters in law . My warmest
thanks go to my dearest husband for his encouragement and patient love that inspi red me
to complete this work and also to my little angel Zinedine.

Kurzfassung

Diese Dissertation schlägt eine neue Methode zur Messung des W asserdurchflusses mit
einem Durchlaufzeit-Ultraschall-Durchflussmessgerät vor . Die entwickelte Methode
ermög licht einem Ultrascha ll -Durchflussmesser eine genauere Erfassung von sehr
niedrigen Durchflussraten. Flüsse von weni ger al s zwei Liter p ro Stunde (2 l/ h) in einem
typische Haushaltswasserzähler sind möglich. Der Fluss einer gegebenen Flüssig keit in
einem Rohr wird durch die Laufzeitdif ferenzmessung von Ultraschalls ignalen mit und
gegen den Fluss ermi ttelt . Je ge rin ger der Durchfluss ist, desto kleiner ist die
La u fzeitdif ferenz. Die Dif ferenz liegt bei niedrigen Du rchflussraten im Bere ich von
wenigen Pikosekunden. Die vor geschlagene Methode erlaubt das Messen der Dif ferenz im
Bereich von weni gen Pi kosekunden und überwi ndet technische S chwierigkeiten anderer
Messmethoden.
Die piezoelektrischen W andler sind die kritischen Komponent en des Ultras chall-
Durchflussmessers. Sie können die Genauigkeit der Ultraschall-Durchflussmesser
erheblich be einträchtigen. Die W ahl einer geeigneten anal y tischen F unktion für die
Besc h reibung d es V erha l tens des piezoelektrischen W andler s (T ransducers) ist notwendi g
zur Ermittelung eines geeigneten Durchflussmessverfahrens, welches in der Lage ist,
ge n aue und robuste Me sser gebnisse zu liefern. Im Ersatz schaltbild wird ein T ransducer
durch einen Oszillator mit parallel geschalteter Kapazität dar gestellt. Die einfachste
analy tisch e Lösung der entsprechenden Differentialg leichung erg ibt sich, wenn der
T ransducer mit einer Sinusfunktion angereg t wird. Im erste n Mo ment reagiert der
T ransducer mit einer Schwingung bei seiner eigenen Reson a nzfrequenz, die aber nach
einiger Zeit abklin gt. Danach sch wingt d er T ransducer nu r noch mit de r aufgezwungenen
Frequenz. W artet man al so lange genug, dann wi rd der transiente T eil abkli ngen und man
wird den stationären Zustand erreichen, wo der T ransducer nur noch mi t der
Zwangsfrequenz schwingt.

Das vorgeschlagene V erfahren beruht da r auf, di e Dif ferenz de r Laufzeiten indirekt zu
messen, indem die Phasendif ferenz z wischen den stationäre n T eilen der empfangenen
Signale in der stromauf wärtigen und d er strom a bwärtigen Richtun g berechnet wird und
indem eine Sinus-Fitting-T echnik mit kleinstem qu adratischen Fehler verwendet wird. Dies
verringert den Effe kt des Jitters in der Laufzeit. Der Jitter begrenzt die Messgenauigkeit
bei sehr geringer Strömung s geschwindigkeit.
Der letzte T eil der A rbeit untersucht das Of fset -V erhalten. Der Offse t ist die Abweichung
der Dif ferenz der La u fzeiten von Null bei nicht -fließende m W asser . Er ist u. a.
temperaturabhängig. Au ch einige Parameter der T ransducer sind temperaturabhängig, in
erster Linie die Resonanz frequenz selbst. Bei einer Erwärmun g von 20°C auf 80°C
verändert sich der Offset entsprechend und z.B. erreicht W erte um 300 ps, wenn er vorher
bei 20°C auf „ Null“ abgeg lichen wurde. Di e Me ssanordnung e rlaubt eine neue, bisher in
der L it eratur nicht b ekannte Art des Offset-Abgleichs durch Anpassung d er
Zwangsfrequenzen bezogen auf die T empe ratur des Mediums. Die La n gzeitstabilität der
zur Einst ellung der Of fsetdrift verwendeten Zwa n gsfrequenz wurde b ei verschiedenen
T emperaturen experimentell nachgewiesen.
Die erhaltenen Messerg ebnisse verdeutlichen die Genauigkeit un d Robust heit des
vorg eschlagenen V erfahrens: die Dif ferenz der Laufzeiten zeigt im T empe raturbereich von
20°C bis 80°C bei nicht-fließendem W asser einem P eak- to -Pea k-J itter von nur 15 ps und
einen Offse t von w eniger als 5 ps. Dadurch k ann man im V erg lei ch z u früheren T echniken
kleinere Durchf lüsse mes sen.
Diese Arb eit liefe rt Ans ätze für mögliche zukünftige Ultraschall-Durchflussmesser mit
hoher Genauigkeit. Um die Ansätze kommerziell gut in zukünftigen Durc hflussmessern
nutzen können, bedarf es einer Integ ration der Messtechnik in einem integrierten
Schaltkreis.

Abstract

This thesis proposes a new method for measuring water flow with a transit time ult rasonic
flow meter device. The developed method allows the ultrasonic flow meter to re ach a
better performance than currentl y available commercial flow meters by accuratel y
detecting ver y low flow rates of less than two liters per hour (2 l/ h) in a t ypical household
water meter . In principle, the flow velocit y of a given liquid is obtaine d by measuring the
transit times of an ultrasonic signal in the upstream and downstream direc tions . The
dif ference betw een the transit ti mes is directly pr oportional to the flow velocity . However ,
the fainter the flow is, th e smaller the transit time dif ference (TT D) is. Thi s dif ference c an
be as low as a few picoseconds, which gives rise to many technical dif ficulties in
measuring such a small ti me diffe rence with a given accurac y .
The piezoelectric transducers are critical compo nents in ultrasonic fl ow meters since the y
can signifi cantly af fect th e accuracy of the ultrasonic flow meters. Choosing an app ropriate
analy ti c fun ction that describes the b ehavior of the basic p art of a pi ezoelectric tr ansducer
proved to be essential for defining a suitable flow mea surement method that y ield s
accurate and robust measurement results. The elec trical equivalent circuit of a transducer is
repre s ented by an oscill ator connected to a parallel capacitance. The si mplest analytical
solution of the corre spo nding dif f erential equation is obtained when th e tra nsducer is
excited by a sinus function. First, the transducer oscillates at its own reso nant frequency ,
and its oscill ations die awa y after som e ti me. If we wait long enou gh, th e transient p art
dies out and wha t is le ft is the steady- st ate part, where the tra nsduc er oscill ates at the
force d frequency . Th e proposed method relies on measuring the TT D indirectly by
computing the phase dif ference between the steady - state parts of the receiv ed signals in the
upstream and downstream directions and by usin g th e least squar es sine-fit ting technique.
This reduces the ef fect of the TTD -jitter of the measurement, which limits the
measurement acc u racy at ver y low flow velocit y .

The last part of the wor k a ddresses the issue of the TTD -offset, which refers to an y
deviation of the TTD from z ero at no-flow conditions. The behavior of the TTD -o f fset is
investigated over a t e mperature range. Some parameters of the transducer , such as
resonance fre qu ency , a re temperature-dependent. When the temperature of the medium
around the transducers incre ases by 80°C from ambient temperature, the TTD -of fs et
(adjusted to z ero at ambient temperature) changes accordingly and reaches approximately
300 ps. The novel proposed approac h allows the c ompensation of the TTD -offse t by
adjusting the forced frequenc y with respect to the temperature of the me dium. The long-
term stabilit y of the driving frequenc y used to adjust the TTD -o f fset drift ha s been
experimentally p roved at dif ferent temperatures. The obtain ed measurement results
illustrate the acc u racy and robustness of the proposed method since the TTD is mea sured at
no-flow conditions, with a peak- to -peak TTD-jitter as low as 15 ps and the TTD -of fset less
than 5 ps with in a temperature range from ambient t em perature to 80°C. This allows to
reac h a smaller minim um detectable flow in comparison to previousl y developed
techniques.
This work of fers some s uggestions to design an ultrasonic flow meter with high accuracy
in the future. However , the commercial aspe ct of the futu re flow meter requires an
integration of the proposed measurement technique in an integrated circuit.

Abbr e viations and Symbols

Abbr eviation s
ADC Analog- to -Digital Converter
CSV Comma-separated values
LSSF Least Square s S ine-Fitting
OP AMP Operationa l Ampli fier
SNR Signal- to -Noise Ratio
STD Standard Deviation
TR T ransducer
TTD -offset T ransit T ime Dif ference-of fset
TTD -jitter T ransit T ime Dif ference-jitter

Symbols
A Pipe cross section [m 2 ]
c Speed of sound [m s -1 ]
C Spring compliance [m N - 1 ]
C s Serial capacitance [F]
C p Parallel capacitance [F]
d V iscous damping coef ficient [N s m -1 ]
D Pipe inner diameter [m]
DR ADC d y namic ran ge [dB]
fp Parallel fre qu enc y [Hz ]
fs Serial fre qu enc y [ Hz]
fr Resonance frequency [Hz]
fdr Driving frequency [Hz]
f f For ced fr equency [Hz]
F 0 Applied force [N]
I Current [A]
k Stif fness coef ficient [N m -1 ]
K W ater compressibility [m 2 N -1 ]
L T ransducer separation [ m]
L s Serial inductance [H]
m Mass [Kg]

N Sample size
N bit ADC re solution [bits]
Q Flow rate [m 3 s -1 ]
q Char ge [ C]
qh(t) Homoge n eous solution [ C]
qp(t) Particular solution [ C]
R s Serial resistance [ Ω]
R Pipe Radius [ m]
t up Upstream traveling time (against the flow) [ s]
t down Downstream traveling ti me (with the flow) [ s]
T Disk thickness [ m]
TTD T ransit time dif ference [ s]
T emp T emperature [°C]
v Flow velocity [m s -1 ]
v min Minimum flow velocity [m s -1 ]
V V oltage [ V]
V FRS Full scale voltage range [ V]
V LSB Least significant bit [ V]
VRD1 First direc tion r eceiver voltage [ V]
VRD2 Second direction receiver voltage [ V]
VTD1 First direc tion transmitt er voltage [ V]
VTD2 Second direc tion transmi tter voltage [ V]
x Displacement [ m]

W ater density [Kg m -3 ]

up Upstream phase [rad]

down Downstream phase [rad]
 Standard deviation of the transit time diffe rence [ s]
Standard deviation error of the transit time dif ference [ s]

 Damping coef ficient
ω f Angular forced fre que ncy [rad s -1 ]
ω r Angular re son ance frequenc y [ rad s -1 ]

Content

Chapter 1: Intr oduction .................................................................................................. 1
1.1. Motivation ......................................................................................................... 1
1.2. State of the Art .................................................................................................. 2
1.3. Main W ork Objective ........................................................................................ 3
1.4. Thesis Outline ................................................................................................ ... 3

Chapter 2: The Principles of Ultrasonic Flow Meter ................................................... 5
2.1. Introduction ....................................................................................................... 5
2.2. Doppler Flow Meter .......................................................................................... 5
2.3. T ransit T ime Flow Meter ................................................................................... 6
2.3.1. T ransit T ime Flow Meter Configuration ................................................... 6
2.3.2. The Principle of T ransit T ime Flow Meter ................................................ 8

Chapter 3: Ultrasonic Piezoelectric T ransd ucer Theory and Modeling ................... 10
3.1. I ntrodu ction ..................................................................................................... 10
3.2. Piezoelectric Ef fect ......................................................................................... 10
3.3. Ultrasonic Piezoelectric T ransducer ................................................................ 11
3.3.1. Piezoelectric Disk .................................................................................... 12
3.3.2. Piezoelectric Backing a nd M atching Layer............................................. 13
3.4. Piezoelectric T ransd ucer Mode lin g ................................................................. 13
3.5. Butterworth-V an D yke Mode l ................................ ......................................... 18
3.6. T ransmitted and Rec eived Signa ls Modelin g .................................................. 24
3.6.1. T ransmitter ............................................................................................... 26
3.6.2. Receiver ................................................................................................... 27
3.7. Quiet Reg ion Aspect ....................................................................................... 29
3.8. Characteristic Frequencies of Piezoelectric T ransducer ................................. 32

Chapter 4: Determination of the T ransit T im e Differ ence (TTD) ............................ 33
4.1. I ntrodu ction ..................................................................................................... 33
4.2. Experimental Setup for Measuring the TTD ................................................... 34
4.3. Methodology for the Calculation of TTD ...................................................... 36
4.4. Least Squares Sine-Fitting ............................................................................. 37
4.5. Estimation of the Quiet Reg ion ...................................................................... 38
4.6. Algorithm for Computing the TTD ................................................................. 40

Chapter 5: TTD -jitter Reduction ................................ ................................................. 42
5.1. I ntrodu ction .................................................................................................... 42
5.2. The Ef fect of the Sa mpling Jitter on TTD ....................................................... 43
5.3. Sources of TTD-jitter ...................................................................................... 44
5.4. TTD -jitter Reduction T echnique ................................ ..................................... 45

Chapter 6: TTD -Offset Cancellation ........................................................................... 50
6.1. I ntrodu ction ..................................................................................................... 50
6.2. Sources of TTD-offset ..................................................................................... 50
6.3. The Impacts of T em perature on the T ra nsdu cer Resonance Frequencies ....... 51
6.4. Zero F low TTD-offset Correc tion ................................................................... 53
6.5. T emperature Dependence of the TTD -of fset Calibration ................................ 57

Ch apter 7: Results and Anal yses .................................................................................. 60
7.1. I ntrodu ction ..................................................................................................... 60
7.2. Hardware Design ............................................................................................. 61
7.3. Software Al gorithm ......................................................................................... 62
7.3.1. T ransmission of th e Sinusoidal Burst ...................................................... 63
7.3.2. Digitizing of the T wo Direction Signals ................................ .................. 63
7.3.3. TTD -of fset Cancellation Algorithm ......................................................... 64
7.4. TTD -offse t Compen sation Evaluation ............................................................ 66
7.5. The Ef fect of T emperature on TTD-of fset Measurements .............................. 68
7.6. Discussion ................................................................................................ ....... 70

Chapter 8: Conclusion .................................................................................................. 72

Chapter 1
Intr oduction

1.1. Motivati on
Flow measurement is often critical in the household sector of the domestic economy [1] .
Therefore, in order to make a profit rather than run at a loss , it is imperative to accurately
measure what flows through the measuring pipe under all circumstances.
Ultrasonic flow meters have been used successfull y in industrial applications for several
decades [2 -3]. They have gained approbation in wide metering applications such as dirt y or
clean water , petrochemical products, natural gas, and so on. This is due to the significant
operational and economic advantages that the ultrasonic flow meters offer in contrast to
conventional meters. From the economic point of view , the y a re eas y to install,
inexpensive, and re quire less maintenance than other me ters such as mec hanical flow
meters, which need to be checked periodicall y [ 2] . In terms of operation, ultrasonic flow
meters can be highl y sensitive and accurate, and they t ypica ll y have a broader flow rate
range, good measure men t repeatability , and bi -directional flow capability .
As an y othe r measurement technology , ult rasonic flow meters have their measu ring
limitations, which are mostl y caus ed by changes in the temperature of the medium and
severa l other factors, suc h as water viscosit y and compressibility [4 -7]. If there is no flow
in the measuring pipe ( i. e., sti ll water in the pipe ), the upstream and downstream transit
times of such meters are equal and the dif ference between the two transit times should be
neg li gible. However , this ma y not alwa ys be the case since the transit time dif ference
(TTD) can be influenced by jitter and of fset. Therefore, an y dela y o ff set between th e
upstream and downstrea m transit times directl y t ranslates into a zero fl ow error . This zero
fl ow TT D-o ff set limits measurement accurac y at low flow velocities. The lower this error ,
the higher the accuracy of the flow meter .

Chapter 1: Intr oduction 2

1.2. State of the Art
Following [ 7], the accuracy of the ultrasonic flo w meter at no-flow conditions - for water
or gas applications - depends essentiall y on the reciprocit y of the electro-acoustic
measurement s y stem. This reciprocit y can be ach ieved by perfect tr ansducer s y mmetr y or
perfect electrical s y mmet ry in th e ultrasonic flow meter s y stem. This means that either the
impedances of the electric loads for both transmitting and receiving trans ducers are equal
or both transducers are i dentical. The electro-aco ustical reciprocity princi ple as referenced
by P er Lunde ’ s paper [8 -16] provides possibilities for reducing or even neglecting the need
for the zero flow calibration of ultrasonic flow meters for both liquid and gas applications.
In 2010, Borg J ohan presented in his Ph .D. t hesis "On electronics for measurement
systems" [17-18] a new methodolog y bas ed on driving the transducer with a current source
rather than a voltage source in order to ac hie ve good im pedance matching between
transmitting and rece ivi ng circuits. This method achieved a significant improvement in
reducing the zero flow error compared to Lunde ’ s work.
What followed was the s tate-of-the-art publi cation of Y ang Bo in March 201 1 [19] , when
he presented a new approach to improve the acc urac y and the stabili t y of ultra sonic
transducer flow metering ov er a wide temperature r ange under non-reciprocal operation
conditions. This approach is based on driving b oth transduc ers at a spe cific frequenc y
outside their resonance frequencies throu gh a sinu s burst in order to eliminate the effect of
temperature dependence of the resonance fr equency and therefore redu ce the long-term
drift of the transit time diffe rence me asurements, which is caused by temper ature
variations. The dr awbacks of this method are, on the one hand, the reduced signal- to -noise
ratio (SNR) caused by w orking outside th e resonance frequency of the tr ansducer and, on
the other hand, the fact that the excitation voltage level must be high even in the presence
of an amplifier . This results in a hig h powe r consumption of the whole measurement
system which p resents a big disadvantage in batter y applications. Ne vertheless, this
approac h achieved improved results compared with the two pr eviously mentioned
methods.

Chapter 1: Intr oduction 3

1.3. Main W ork Objectives
The main objec tive of this work is to develop a new methodolog y , which combines
hardware with appropriate digital evaluation soft ware al gorithms and whi ch can o vercome
the ef fect of the TTD -jitter noise and the TTD -of fset drift. This methodology reduces the
false flow detection and improves the accurac y of the ultr asonic flow m eter at low and
no-flow conditions. The first part of this re search work focuses on the analy t ical modeling
or the mathematical pr esentation of the piezoelectric transducer , which is based on the
driven damped harmonic oscillator s y stem. This model provides a better understanding of
the behavior of th e piezoelectric tr ansducer and pr ovides the necessar y con cepts that allo w
to measure the TTD very accurately .
The main contribution of this wor k is summarized b y the two main steps that were
undertaken to minimiz e the ef fect of zero flow error and TTD -jitter on the measurement
results:
 Develop a software algorithm on the basis of the proposed approach, which woul d
be capable to ef fectively reduce th e TTD -jitter .
 Develop another softwar e algorithm on the basis of the proposed approach, which
would be capable to continuously correc t the zero flow TTD-of fset.
1.4. Thesis Ou tline
A significant amount of the current research focuses on ult rasonic flow meters, especiall y
in the field of me asuring of flow r ate in th e transmissi on of gas through p ipelines, in order
to develop more accurate measurement methodol ogies. W e will elaborate and refer to the
appropriate theoretical back ground when describing our ex periments. This dissertation is
org anized as follows:
1. Chapter 1 outlines the motivation behind this work and positions our work amongst
the re cent r esearch ef forts in ult rasonic flow meter me asurement methodologies
curre ntl y available.
2. Chapter 2 describes the available ultrasonic flow meters used for measuri ng th e
flow rate of a flowing fluid.

Chapter 1: Intr oduction 4

3. Chapter 3 presents the theoretical background of a piezoelectric transducer . W e
address the relevant theo ries to derive an anal y tic model of the tr ansducer that can
describe its beha vio r in both cases, transmitter a nd receiver .
4. T aking into ac count the analytic transducer model and all the a spects derived
through the mathematical repre sentation of the tr ansducer presented in Chapter 3,
Chapter 4 presents the anal y sis and evaluation of our proposed method can be used
to calculate the transit time differe nce. It also presents a detailed description of e ach
of the measurement setup system’ s bl ocks.
5. Chapter 5 describe s the applie d TTD-jitter analysis and methodology used to
understand the causes of TTD-jitter . The proposed TTD -jitter reduction technique is
also described.
6. The z ero flow TTD -of fset correction te chnique i s the main subject o f C hapter 6,
starting with a stud y o f the TTD -of fset sources and ending with the proposed
approac h to mit igate this TTD -of fset.
7. In Chapter 7, the experimental results are p resented and analyzed.
8. Finally , conclusions are drawn in Chapter 8.

Chapter 2
The Principles of Ultrasonic Flow Meter

2.1. Intr o duction
Most ultrasonic flow meters use one of the two main principles: Doppler ef fect or T ra nsit
T ime Differe n ce. When a fluid is in mot ion with a certain velocit y , the flow meter
measures the flow of t his fluid either by calculating the dif f erence between the two
traveling times of an ult rasonic si gnal propagating with and a gainst the flow direction or by
measuring the frequency shift using the Doppler principle.
2.2. Dopple r F low Meter
This t ype of flow meter is based on the Doppl er principle discovered in 1842. T y pi cally ,
one transducer is fitted in the pipe wall as shown in Figure 2.1. It continuousl y transmits an
ultrasonic signal at a constant frequency f 1 into the flowing fluid. Th e particles insi de the
fluid refle ct the transmitt ed signa l, and their movement shifts the frequency of the
ultrasonic signal to a frequency f 2 . The frequenc y shift is proportional to the speed v of the
particles and hence to the flow . It is given by the f ollowing equation [20-22] :

(2.1)
where ∆ f is the frequenc y shift (the dif ference between transmitted and rece iv ed
freque n cies), θ is the an gle of the transmitter and receiver crystal ax is with respect to th e
pipe axis, c is the sound v elocity , and v is the flow velocity .

Chapter 2: The Prin ciples of Ultrasonic Flow Met er 6

Fig. 2.1: Doppler flow meter
Due to man y drawbacks and limi tations of ultrasonic Doppler flow metering, thi s method
is now used onl y in a few specific applications such as for waste water t hat contains dirt
particles o r gas bubbles. It has be en replaced b y the tran sit time ult rasonic flow meterin g
because in addition to flow rate measurement thi s method can also provid e information on
the type o f liquid and the working temp erature on the basis of sound veloci t y measurement
[20].
2.3. T r ansit T ime Flo w Meter
2.3.1. T ransit T ime Flow Meter Configuration
The ultrasonic transit time flow meter consists of one pair of transducers facing each other ,
which are s eparated by a known distanc e. Th e transducers are mounted according to
dif ferent geometries dep ending on the application. For instance, the in -li ne configurations
shown in F igures 2.2 and 2.3, as we ll as the configuration with reflectors shown in
Fig u re 2.4, are most often used in applications where the diameter of the pipe is less than
25 mm, whereas the diagona l configuration shown in Figur e 2.5 is used in applications
where the di ameter of the pipe is up to 10 meters [4].
T wo flow meter pipes have been used in this work. The confi guration of the first flow
meter pipe contains one pair of 4 MHz ultrasonic transducers ( Figure 2.2). The dist ance
between the two transducers and the inner r adius of the flow meter body are L = 42.2 mm
and R = 4 mm, respectively . The configuration of the second flow meter pipe contains one
pair of 1 M Hz ultrasonic transducers with L = 49.5 mm (Figure 2.3). All experimental
work has been performed using these meters.

Chapter 2: The Prin ciples of Ultrasonic Flow Met er 7

Fig. 2.2: F low meter pipe with in -line transducers: (1) inlet, (2 ) cable connector ,
(3) the upstr eam transducer , (4) flow meter body , (5) the dow nstr eam transdu cer ,
(6) outlet

Fig. 2.3: Flow meter pipe with in -line transducer s

Fig. 2.4: F low meter pipe with re flectors

Fig. 2.5: F low meter pipe with d iagonal transduce rs

Chapter 2: The Prin ciples of Ultrasonic Flow Met er 8

2.3.2. The Princi ple of T ransit T ime Flow Meter
The transit time flow met er is based on the tr ansit time diffe rence (TTD) p rinciple and uses
two transducers. Ea ch transducer can alternatel y transmit and receive an ultrasonic signal.
This signal is generated when a piezoelectric cr ystal is subjected to an alternating volta ge.
Conversely , the piez oelectric crystal generates volt age wh en the ultrasonic signal im pacts
the transducer . In the case of simultaneous excitation, the two transducers emit and receive
the ultrasonic si gnals at t he same time. On e ult rasonic signal trav els throu gh the pipe in the
direction of the flow (d ownstream direction) and the other against the flow (upstream
direction). Eve r y signal needs a certain period of time (ca lled tr ansit time) passes before
the sig nal is rec eived by th e opposite transducer . This transit time depends on three
parameters: the speed of sound c, the ult rasonic path length L, and th e flow velocity v as
illustrated by the following equations [23-24]:

(2.2)

(2.3)
where t up is the upstream transit time, and t down is the downstrea m tr ansit time.
If there is no flow , then:

.
(2.4)
At no-flow conditions the transit times are equal. Once the fluid starts to flow , the sound
wave moving with the flow travels faster than t he sound wave moving against the flow .
The differe n ce b etween the two transit times is directly proportional to th e flow velo cit y .
This can be mathematicall y expressed as follows:

(2.5)
Since flow velocit y v is much smaller than the speed of sound c, it can be derived from
(2.5) as follows:

(2.6)
Since the internal cross-section of the pipe is kno wn, the volume flow rate Q is determined
by the following formula:

,
(2.7)

Chapter 2: The Prin ciples of Ultrasonic Flow Met er 9

where A is the inne r circular cross-section, and R is the inner radius of the flow meter pipe.
The ratio of  t to the tra v eling time t 0 measured at no-flow conditions is given by:

. (2.8)
Equation (2.8) can b e use d to calculate the ne eded measurement accu racy . F or instance, the
minimum flow v min calculated for a given flow rate of two liters per hour (2 l/h) and pip e
diameter of 0.8 cm is about v min = 10 mm/s, or approximately 7 ppm compare d to the
sound velocit y (th e speed of sound in pure water and room temperature is about
c = 1500 m/ s). Therefore, a ccording to Equation (2.5), thi s minimum flow variation o f
10 mm /s produces a tran sit time dif ference value of about  t = 380 ps, provided that the
travelling time of the us ed flow meter pipe is 28 µs (calculated for L = 42.2 mm usin g
Equation (2.4)). Therefo re, to achieve an accura cy of 5 %, the desired ultrasonic flow
meter must be able to accuratel y measure the trans it time dif ference of at l east 20 ps (which
then would be its minimum measure d value).
This thesis focuses on the transit time flow meter mostl y because of it s extensive industrial
usage [ 5]. Besides, it ha s the highest cost ef ficiency and, unlike the Doppler flow meter ,
does not require the fluid to contain particles or air bubbles in order to reflec t the ult rasonic
sound.

Chapter 3
Ultrasonic Pie zoelectri c T ransducer Theor y and
Modeling

3.1. Intr oduct ion
An accurate description of the behavior of ultrasonic transducer ’ s active el ement requires a
detailed investi gation of the transmitted acoustic wave, which travels thro ugh the pipe, and
of the received a coustic wave, whic h is picke d up by th e opposite transducer a fter a
predetermined time.
This chapter starts with an overview of the piezoelectric ef fect, which is followed by a
description of the real transducer geometr y and its dif ferent composit e layers. Thereafter ,
an analytical approach of the piezoelectric transdu cer developed throu gh th e solut ion of the
dif ferential equation of driven damped harmonic oscillator , is analy zed in detail.
In the remainde r of this chapter , the emulated transmi tted and received w avefor ms, which
are generated using Matlab, are compared to the real signals obtained ex perimentally . We
show that a full agree m ent between the theoreti cal description and ex perimental signals
can be achieved with an appropriate choice of only three model para m eters: resonance
freque n cy , d amping factor , and excitation frequency .
3.2. Piezoe lectric Effect
The word "piezo", of Greek ori gin, means "push". According to [ 25], the ef fect known as
piezoelectricity is a p ropert y exhibited by certain classes of crystalline materials that
consist of polarized molecules. It was discovered by the brothers Pierre and J acques Curie
in 1880. When a piezoelec tric material is subjected to a mechanical stress , it generates an
electrical charg e, which is proportional to the appli ed stress. This behavior is called the
direct piezoelectric e f fect. I nversely , when the piezoelectric material is subjected to an

Chapter 3: Ultrason ic Piez oelectric T ransducer The ory and Modeling 11

electric field, it changes dimension and becomes strained [ 25]. This stra in is again
proportional to the applied field. hese two ef fects are shown in igure 3.1.
Besides qu artz and R ochelle salts, man y othe r piezoelectric materials are available
nowadays, such as Barium T itanate ( BaT iO 3 ), Lead Metaniobate (PbN b 2 O 3 ) and L e ad
Zirconate T itanate (PZT) [26-27] .

Fig. 3.1: Piezoelectr i c effects: (a) dir ect, (b) inverse [25]
3.3. Ultra sonic Piezo electric T r ansducer
The ultrasonic piezoelectric transducer is emplo y ed to convert electrical energy int o
mechanical energy (sou nd wave) and vic e ver sa. The piezoelectric m aterial has th e
following three properties [25]:
• the elasticity p ropert y which defines the mechanical aspect of the material
• the piezoelectric propert y which defines the electromechanical aspect of the
material
• the dielectric property which define s the electrical aspect of the material.
An ultrasonic transducer consists of three ma in pa rts as shown in Figure 3. 2: piezoelectric
disk, the rear part or backing la y er , and fr ont or matching la y er [ 28-29 ]. Man y factors
determine the pe rformance of ultrasonic tr ansducers, the most important of them being the
material properties of th e transducer components, including housing and connections, the
external mechanical, and electrical load c onditions and damping.

Chapter 3: Ultrason ic Piez oelectric T ransducer The ory and Modeling 12

Fig. 3.2: T ypical construction of an ultrasonic tr ansd ucer [29]:
(1) matching layer , (2) piezoelectr ic disk , (3) backing layer ,
(4) cable connector , (5) transducer housing, (6) electr odes
3.3.1. Piezoelectric Di sk
The piezoelectric disk is the main component for generating and receivi ng the ult rasonic
wave and the active e lement of the ultrasonic transduc er . The angle at which the
piezoelectric c r y stal is cut in relation to its c r y stallographic axes defines its vibration mode
and the t ype of the generated ult rason ic wave, wh ich can be a longitudinal or a shea r wave
(Figure 3.3), depending on the application. For instance, in our application, the ultrasonic
wave propagates through the water by a longitu dinal motion (compression/expansion) as
the fluid does not suppo rt the she ar motion [30 ]. The surface of th e disk moves up and
down, and the wa t er has to follow this movement directly .

Fig. 3.3: Response of a piezoelectric element to an AC voltage:
(a) compr ession motion generating longitud inal waves,
(b) transverse motion generating shear waves
The active el ement of the transducer shows a t ypical resonance frequenc y whose value is
precisely related to the size and shap e of the piezoelectric transducer according to
Equation (3.1).

(3.1)

Chapter 3: Ultrason ic Piez oelectric T ransducer The ory and Modeling 13

where f r is the resonance frequenc y , T is the disk thickness (see Figure 3.4), and c
repre s ents the wave velo cit y of the lon gitudinal vibration inside the disk and depends on
the acoustic properties of the disk.

Fig. 3.4: P ie zoelectric disk with thickness T and diameter D [29]
3.3.2. Piezoelectric Ba cking and Matc hing Lay er
The backing la y er consists of a high-density m aterial and is used to control the vibration by
absorbing the energy radiating from the ba ck face of the active element. The piezoelectric
materials are characterized by hi gh acoustic im pedance in comparison to water and air .
Consequently , the b andwidth of the response function of the disk is l ow . The acoustic
impedance mismatch can be overcome by adding a matching la yer to enhance the
bandwidth and sour ce sensitivity . More reliable wide bandwidth tr ansducers are obtained
by adding a quarter -wave matching layer [31-33 ].
3.4. Piezoe lectric T ransducer M odeling
The transducer may be driven by a volta ge bur st consisti ng of a finite number of sinus
cy cles with a w ell-defined frequency , which is chosen near the resonance frequenc y of the
transducer . V ibrating at the same fre quen c y as the applied voltag e, t he piezoelectric
material generates a wave, which prop agates th rough th e water . In o rder to unde rstand
exactly how this waveform is generated, in particular the "transient behavior" and "stead y-
state behavior", an anal y t ical description is needed to emulate the excitation process.
The transducer , like man y oscillators, can be m odeled as a harmonic os cillator , which is
character iz ed by its "resonance frequency" and "damping coef ficient". The excitation of the
harmonic oscillator by a voltage causes the oscillator to vibr ate at the sa me frequency as
the applied voltage. It vibra tes with g reater ampl itude if the frequency of the e x cited
voltage corresponds to the resonance frequency of the oscillator .

Chapter 3: Ultrason ic Piez oelectric T ransducer The ory and Modeling 14

In the case of a harmonic ex citation, three basic parameters dominate the behavior of the
oscillator ’ s response: angular resonance frequency ω r , damping coef ficient ζ , an d angular
force d frequenc y ω f . Ho wever , as the free vibr ation (transient part) dies out over time, onl y
the forced frequenc y do minates the remaining stead y-state p art. W hen there is no ex citing
signal, the signal dies out gradually , which c auses the decay of the si gnal to z ero due to the
transfer of the energy to the water . This switching-of f behavior is characterized onl y by ω r,
ζ .
The driven damped h armonic oscill ator model attempts to develop a b etter understandin g
of the ultrasonic transducer . The disk, which represents the transducer ’ s active element, can
be described by both models depicted in Figure 3 .5. During resonance, t he series resonant
circuit R s , L s , C s shown in Figure 3.5.b (also called motional branch) f orms a damped
harmonic oscillation and resonates in a sim ilar wa y to the mechanical model shown in
Fig u re 3.5.a. Therefore, it is possible to compare the two models and use the electrical
parameters (inductan ce, capacitance, and resistance) to represent the mechanical
parameters (m ass, sti f fness, and damping) [ 25]. He nce, some mechanical and elec t rical
properties such as material density , elastic mechanical parameters, piezoelectricity , and
dielectricity define the values of the motional components. The impac t of the parallel
capacitance shown in the electrical model is explained in more detail in the next section.

Fig. 3.5: Equivalent m o dels of the piezoelec tric element:
(a) me chanical model, (b) Butterworth-V an Dyke electrical model

Chapter 3: Ultrason ic Piez oelectric T ransducer The ory and Modeling 15

For sinus excitation, the basic description of the damped externally dri ven oscillator is
provided b y the following second order dif ferential harmonic equation:

, (3.2)
where:
F = F 0 /m is the applied e x ternal force F 0 divided by m,
m is the mass,
ω f is the ang ula r forced frequenc y of the sinus burst,
d is the viscous damping coef fic i ent,
k is the system stif fness coef ficient,

is the damping c o ef ficient , and

is the angular transducer resonance frequency .
The mechanical displace ment is analogous to the elec trical charge that can be used to
reformula te the p revious dif ferential Equation (3.2). The summar y of the equivalent
quantities between the mechanical and electrical models is presented in T able 3.1 [34 ].
Electrical

Mechanical

Char ge Q

Displacement x

Current I = dQ/dt

V elocit y v

Applied voltage V

Applied force F 0

Resistance R s

Friction c

Inductance L s

Mass m

Capacitance C s

Spring compliance C = 1/k

T able 3.1: Comparison of equ ivalent electr ical and mechanical r esonant cir cuits
According to T able 3.1, Equation (3.2) of the mechanical motion can be rewritten as:

(3.3)
where:
V is the amplitude of the applied voltage,
R S , L S, and C S are the parameters of the pi ezoelectric disk,

describes the damping coef ficient, and

Chapter 3: Ultrason ic Piez oelectric T ransducer The ory and Modeling 16

is the transducer serial resonanc e an gular frequency .
Assuming < 1 (underdamped harmonic oscillat or), the complete solut ion of Equation
(3.3) is given by Equ ation (3.4). Generally , this function can be divided int o two parts: the
solution of the homogeneous differe ntial equ ation with a zero right side giving the transient
response, and the p articular solution of the non-homogeneous equation as t he answ er to the
externally applie d volta ge:
. (3.4)
The homogeneous solution of this equation is g iv en by:
(3.5)
where is the damped natural ang ul ar frequenc y , equal to , and the
constants A h and θ are dependent on the initial conditions.
The particular solution, the forced or stationa r y solution, is given by :
(3.6)
A p and can be determined simply if w e use the complex form of the dif ferential
Equation (3.3), which can be written a s:

(3.7)
Using Moivre’ s fo rmula, and
Q(t)=
, one can write:

. (3.8)
By dividin g the previous equation into re al a nd imag inar y pa rts, we e nd up with the
following equations that correspond to the amplitude A p and the phase , respectively:

(3.9)

(3.10)

Chapter 3: Ultrason ic Piez oelectric T ransducer The ory and Modeling 17

The amplitude of the forced os cillations depends on the dif ference between the forced
freque n c y 1 of the applied voltage and the resonance frequency of the piezoelectric disk.
The damping coef ficient has a strong influence on the maximum amplitu de reached when
both force d and resonance frequencies are simil ar . Figures 3.6 and 3.7 show the t ypical
behavior of the amplit ude and the ph ase, respectively , when r ealistic para met ers a re
applied.
Regarding the amplitude response shown in Figure 3.6, a Gaussian t ype distribution around
the reson ance point can be observed when realistic parameters are applied, especially for
the damping coef ficient. Considering the main goal of this work, which is re ducing the
TTD -jitter and TTD -of fset, this illustration clearly d emonstrates th e strate gy which has to
be applied: If both transducers are d riven using a forced frequenc y , which is in the direct
vicinity of their resonance point s, good SNR in the receiving pa rt can be ex pected. In this
work, the recommended ran ge is ±500 kHz or approxim ately 10% of the reson ance
freque n cy .

Fig. 3.6: A mplitude of a driven d amped harmonic oscillator at ζ =0.02 an d
f r = 4.05 MHz
Regarding the phase response shown in Figure 3 .7, it can be observed th at the ph ase shift
between the excitation and the mechanical response of the tr ansducer is constant and
depends only on the differe n ce " ω r - ω f ". It can be also observed in Figure 3.7 that the
phase tends either towards 0° or towards 180 ° at frequencies awa y from the resonance
freque n cy . This feature would allow to derive a reliable measurement method bec ause in

1 Also called the dr iving frequenc y .

Chapter 3: Ultrason ic Piez oelectric T ransducer The ory and Modeling 18

these two pa rts of the p hase frequenc y response the phase shifts are qui te steady and no
longer d epend on the transducer paramete rs. From thi s observation it can be dire ctl y
deduced that the temperature dependence of the transduc er parameters ef fects does not
influence the measurements (outside the well-known relationship of the speed of sound
versus temperature). Y a ng Bo [1 9], who applied this idea, re po rted an approximatel y
200 ps pe ak- to -peak TTD -jitter and − 185.3 ps T TD-of fset measured at no -flow conditions
due to drastica ll y reduced S NR.

Fig. 3.7: Phase of a driven damped harm onic os cillator at ζ =0.02 and f r = 4.05 MHz
3.5. Butter worth-V an D y k e Model
The comparison of the mechanical and electrical models shown in Figure 3.5 shows that
the dif ference between t he two models is given by the p arallel capacitance C p , which is
inherent in a transducer where met al electrodes a re separated by a disk. W ith regards to the
experimental setup prese nted in Figure 3.13, the function generator which can be
repre s ented by a voltage source with output resistanc e ( ≈ 50 Ω ) is ad ded to the total
electrical excitation circuitr y (F i gure 3.8 ). The transmitter transducer is connected to this
voltage source via an external r esistance (R 1 ) of a ty pi cal v alue of 200 Ω . In the case of
simultaneous ex citation (Figure 4.1), two resistances R 1 and R 2 are used. The total
symmetry of th e two electrical cir cuits can nev er be assumed. Th e values of R 1 and R 2 can
be dif ferent, and also the material parameters of th e two disks are clearl y no t identical. This
dif ference in the two res istances does not cause a systematic TTD -of fset error as long as
both directions are symmetric in terms of impeda n ce.

Chapter 3: Ultrason ic Piez oelectric T ransducer The ory and Modeling 19

The harmonic model, which is independ ent of either sin gle or simult aneous excitation,
does not take into account the parallel capacita nce C p . Onl y the motional bran ch of the
transducer' s electrical model (Figure 3.5.b) can be repr esented by the harmonic oscillator
model.

Fig. 3.8: Simulation cir cuit usin g the transducer electrical model
Regarding the "Vtrans" node, the AC analysis of the electrical circuit shown in Figur e 3.8
results in the frequency re sponse magnitude and p hase plots shown in Fig u res 3.9 and 3.10,
respec tivel y . Hence, these re sults show a differe nt behavior when compared to the
harmonic model findings depicted in Figure 3.6 and 3.7. The refore, du e to the parallel
capacitance C p , the AC sim ulation results obtained with the transducer's electrical model
(Figure s 3.9 and 3.10) clearl y demonstrate a restricted sim ilarity compare d to the findings
obtained with the harmonic oscillator model (Fi gures 3.6 and 3.7). However , the an al y tic
evaluation of the two parallel bran ches gives a rather complicated amp litude and phase
equations (Equations (3.1 1) and (3.12)) compared to those obtained previously du ring the
evaluation of the harmonic oscillator (Equations (3.9) and (3.10)). Thus, the transfer
function Vtrans/V in of the electrical cir cuit shown in Figure 3.8 has a magnitude and phase
shift, which are respectively given b y:

(3.1 1)

. (3.12)

Chapter 3: Ultrason ic Piez oelectric T ransducer The ory and Modeling 20

Moreover , in the transducer electrical model the two parameters ω r and ζ , which
character iz e the harmonic oscillator model, are extended to three pa rameters: ω r , ζ , and C p .
The amplitude response over frequenc y (Fi gure 3.9) display s follo wing qualitative
behavior: At low frequencies, the impedance of th e mot ional branch is extremel y high and
the voltag e "Vtrans" ris es with increa sin g frequency . As the fre qu ency increase s, the
capacitance C s in the serial branch lowers the re sulting impedance, and therefore the
voltage "Vtrans" dec reases as well. At the series resonance point the minimum impedance
(R s ) is r eached. At onl y slightl y hi gher f requencies (at the parallel reso nance point) the
inductance dominates the resulting impedance of the motional branch. It resonates with C p ,
causing the volta ge "Vtrans" to inc rease. How ever , as th e fr equency gets hi gher , the
parallel capac itan ce prevails so that "Vtrans" tends towards 0 V.

Fig. 3.9: A mplitude r esponse at "Vtrans" node at f s = 4.05 MHz and f p = 4.3 M Hz
The phase behavior shown in Figure 3.10 can be easil y explained as follows: At low
freque n cies the resulting tr ansducer impedance is given mainl y by the c apacitance C s ;
hence, when compar ed to the ex citation phase sh ift, the phase tends towa rds -90°. At the
point where forced freq uency increases and rea ches the serial resonance fr equ ency , the
ef fect of inductance and capacitance compensat e each oth er , and the resulting impedance is
given onl y by the resista nce R s , which r epresents the acoustical loss es of t he transducer to
the surrounding environ ment and the energy transferred to the water [ 35 ]. In thi s case, the
phase tends to 0°, and th e equivalent t ransducer electrical model is simplified to a pa rallel

Chapter 3: Ultrason ic Piez oelectric T ransducer The ory and Modeling 21

connection of R s and C p . However , as the frequency gets high, the induc tance dominates
the resulting impedance of the motional branch, and the phase tries to reach +90°. At the
operating point where the fr equency becom es higher , the influence of C p cannot be
avoided. At thi s point the phase makes a fast transition from +90° to -90°.

Fig. 3.10: P hase r esponse at "Vtrans" node at f s = 4.05 MHz and f p = 4.3 MHz
Fig u res 3.9 and 3.10 sho w that the transducer exhibits two resonance f requencies at whic h
it appears resistive. Th e first frequency , at which the impedance of the transducer is the
smallest, is called series resonance frequenc y f s . This resonant first poin t is created when
C s resonates with L s . The second frequency , at whi ch the impedance of the transducer is the
greatest, is called parallel resonance frequenc y f p . This resonant second point is crea ted
when L s and C s re son ate with the parallel capacitor C p . These two frequencies can be
computed as follows:

, (3.13)

. (3.14)
W it h respect to the last results shown in Figures 3.9 and 3.10, the assumption that the
transducer can be model ed as a h armonic oscill ator becomes valid onl y if the voltage ov er
R s is anal y z ed instead of "Vtrans" sinc e the acoustic ene rgy transmitted by th e tr ansducer
to the water is proportional to the ener gy dissipated in the serial resistance R s [36 ]. The

Chapter 3: Ultrason ic Piez oelectric T ransducer The ory and Modeling 22

following two fi gures show the am plitude and the phase at nod e "V r ", respectively . There
is a dire ct correspondence betw een th e harmonic oscillator responses sho wn in Figures 3.6
and 3.7 and "V r" responses pres ented in Figures 3.1 1 and 3.12. To sum up, these curves
clearly d emonstrate th at the mathematic modelin g b ased on the driven damped ha rmonic
oscillator can govern the transducer' s b ehavior .

Fig. 3.1 1 : A mplitude r es ponse at "V r" node at f s = 4.05 MHz and f p = 4.3 MHz

Fig. 3.12: P hase r esponse at "V r" nod e at f s = 4.05 MHz and f p = 4.3 MHz
Even unde r the condition that the ma gnitude and phase responses at th e "Vtrans" node do
not demonstrate the behavior of a d riven damped harmonic oscillator , th e "Vtrans" signal
can be separated in to three dif ferent re gions on the basis of the harmonic oscillator

Chapter 3: Ultrason ic Piez oelectric T ransducer The ory and Modeling 23

modeling aspects and according to the measured examples depicted in Figure s 3.16 and
3.17.
• The first region contains both the homogene ous and the particular solution because
the particular solution alwa y s represents a sine w ave, and an y deviations from sinus
oscillations are ca us ed by the homogeneous solution.
• The next region is ch aracterized b y the pure sinus wave, i.e., the hom ogeneous
solution dies out because of the damping, and the remaining part is presented by the
particular solution.
• The last re gion can be observed i f the excitation is switched off. Hence, the
homogeneous solution is active. This reg ion is chara cterized by damped
oscillations.
In this work, the transit time dif ference (TTD) measurements are restri cted onl y to the
second region (in further text quiet reg ion or steady-state region). This reg ion is
character iz ed by the pure sinus of the particular solution . In this region the influence of C p
cannot be a void ed. Hence, the current throu gh C p is g iven b y:

(3.15)
For a pure sinus burst, this current can be represented by a sinus oscillation as follows:
(3.16)
The total current flowing into the two branches of the ci rcuit shown in F igure 3.8 can be
calculated as the sum of the current flowin g thro ugh C p and the current f lowing into the
motional branch:
( 3.17)
where,
(3.18)

. (3.19)
The influence of the par allel ca pacitance on th e second region, which is characterized by
the particular solution of forced oscillation, results in a simple phase shift. Hence, instead

Chapter 3: Ultrason ic Piez oelectric T ransducer The ory and Modeling 24

of 0 , a slightly changed value 2 can be d etected. The amplitude undergoes a sim ilar
change. Since the t ransit time dif ference is comp uted as the phase di f ference between the
two received si gnals 2 , th e changes of the amplitude do not af fec t the measurement
accuracy .
3.6. T ransmitted an d Received Sign als Modelin g
Numerical solutions of the mot ion equations are obtained using Matlab in order to emulate
the measured transmitted and rec eived si gnals. These waveforms are subsequentl y
compared to the experi mentally me asured ones. The me asurement principle used in th e
experiments is shown in Figures 3.13 and 3.14. The transducer T R1 is ex cited through the
resistance R by a sinusoidal burst in order to genera t e an ultrasonic signal that can be
rece iv ed by the se cond tr ansducer T R2 after trav eling through the water within the dist ance
that separa t es the two transducers.

Fig. 3.13: Exper i mental setup used to generate and detect the ultrasonic waves

Fig. 3.14: The experimental setup

2 For further detail s see Chapte r 4.

Chapter 3: Ultrason ic Piez oelectric T ransducer The ory and Modeling 25

Fig u re 3.15 shows a screenshot of Digilent Analog Discover y di gital oscilloscope
displaying the transmitted and received si gnals measured with 1 MHz flow meter pipe.
Assuming that the soun d velocity at ambient te mperature is 1500 m/s, the sound wave
traveling time calculated for an ultrasonic path length b etween T R1 and T R2 of L  49.5 mm
is about 33 µ s.

Fig. 3.15: A scree nsh ot of the transmitted and r eceived waveforms
The two channel measured waveforms demon strate a loss in the amplitudes between
transmission and reception. The 1 MHz flow meter pipe shows that the ultrasonic received
signal is attenuated by a factor of 2.5 compared to the transmitt ed sig nal, whereas for the
4 MHz flow meter pipe the attenua tion fa ctor becomes nearl y 10 ( Figure 3.22).
3.6.1. T rans mitter
From the mod eling point of view , the t ransmitted signal can be divided int o two parts. The
first part is character ized by the fact that the homog eneous and the particular solution are
both active. The second part is provided only by the homo geneous solution since the
excitation is switched off. Hen ce, the si gnal dies of f gradually since there is no driving
force . According to our measurements, the de cay of the amplit ude lasts about 10 to 15
periods.

Chapter 3: Ultrason ic Piez oelectric T ransducer The ory and Modeling 26

Regarding the particular and the homo geneous solut ion (Equations (3.5) and (3.6),
respec tivel y ), three para meters - tr ansducer resonance frequenc y ω r , damp ing coef fi cient ζ,
and the angular driving frequency ω f - are used in order to emulate the transmitter signal.
Fig u re 3.16 shows the simulated as well as the measured transmitted waveforms for
27 periods of uniform sinusoidal applied voltage. The simulati on is performed at a
damping co ef ficient ζ = 0.05, a resonance frequenc y f r = 1 MHz (f r = 2  /ω r ), and a forced
freque n c y f f = 1 MHz (f f = 2  /ω f ), wh ereas the m easured signal is obtained at f f = 1 MHz.
The second example shown in Figure 3.17 is obtained by chan ging o nly the forced
freque n c y f f from 1 MHz to 1.1 M Hz for the simulated a nd for the measured signals.
The difference s caused by th e parallel ca p acitance can be observed, especially at the
beg innin g when the ex citation starts and in the mo ment when the excitation is switched of f.
It can be seen that the blue and red curves look very similar outsi de the switching-on and
-of f behavior . A 10% change in the driving frequenc y provides a remarkable dif ference in
the shape between the two waves depicted in F i gures 3.16 and 3.17.

Fig. 3.16: T ransmitter response for a 1 MHz flow meter p ipe and L = 49.5 mm
at f f = 1 MHz: (a) em ulated w aveform , (b) measur ed waveform

Chapter 3: Ultrason ic Piez oelectric T ransducer The ory and Modeling 27

Fig. 3.17: T ransmitter response for a 1 MHz flow meter p ipe and L = 49.5 mm
at f f = 1.1 MHz: (a) em u lated waveform, (b) measur ed wavef orm
3.6.2. Receiver
Assuming that the transmitted signal, which is directed towards the second transducer , is
defined by the complete solution of Equation (3 .4) (the homogeneous a nd the particular
solutions), this complete solution is used to replace the second part of t he second order
dif ferential harmonic equation (Equation (3.3)). The followin g equation can be used to
emulate the rece ive r waveforms:

(3.20)
According to the numerical solution of Equation (3.20) obtained using Matlab, five
parameters are now used in order to emulate the ult rasonic receiver signal: two parameters
of each of the two transducers ω r1,2 , ζ 1, 2 , and the angular forced frequency ω f . Hence, the
rece iv er wa veform is ch arac t erized by t wo intervals. During th e first time int erval, the
external force is given by "Q h (t) + Q p (t)", while during the second interval the external
force is given by onl y Q h (t). Figure 3.18 shows the result of the simulated receiver results
achieve d for th e paramet er values ζ 1 = 0.01, ζ 2 = 0.02, f r1 = 1.01 MHz, f r2 = 1.02 MHz. The
external force is activated at the s ame frequenc y of 1 MHz for 33 c ycles f or both simulated
and measured si gnals. T he shapes of both waveforms look ver y simil ar . The me asured

Chapter 3: Ultrason ic Piez oelectric T ransducer The ory and Modeling 28

curve shows a di f ference at the end after rou ghly 103 µ s, where the first reflected signal
already approaches the received signal, causing interference between both signals.

Fig. 3.18: Receiver re sp onse f or a 1 MHz flow meter pipe and L = 49.5 mm
at f f = 1 MHz: (a) em ulated w aveform , (b) measur ed waveform
Fig. 3.19: Receiver r esp onse for a 1 MHz flow meter pipe and L = 49.5 mm
at f f = 1.02 MHz: (a) emulated waveform at ζ 1 = 0.01, ζ 2 =0.018, f r1 = 1 MHz and
f r2 = 1.02 MHz (b) measur ed w aveform

Chapter 3: Ultrason ic Piez oelectric T ransducer The ory and Modeling 29

By compa ring the measured receiver and transmitt er signals, it can be deduced that the
switch-on and th e swit ch-of f beha vior of the excitation, which creates spikes in the
ultrasonic transmitted signals, is no longer seen in the received ones. Therefore, in contrast
to the transmitt er , the receiver is activated and de activated rather smoothl y . The flowing of
the current over C p does not produc e any signa l spikes since the ultrasonic transmitter
signal impacts the receiver transducers more or less continuously . The influence of C p is
corre spondin gl y small.
In summary , the model of the ha rmonic oscillat or is found to be the m ost ef fective to
describe and anal y ze the measur ed curves. The dif ferent ph y sical waveforms in Figures
3.16 to 3.19 can be ea s il y emulated b y an appropriate choice of three parameters (the
transmitter transducer parameters ω r , ζ , and the angular driving frequenc y ω f ) in the case of
the transmitted signal, and five parameters (the tw o transducers parameters ω r1, 2 , ζ 1, 2 , and
the angular forced freq uenc y ω f ) in the case of the re ceived signal. Obviousl y , th e
requirement sp ecified p reviousl y - an anal y tical description of the resul ting signals - is
fulfilled by modeling the piezoelectric transducer as a driven da mped h armonic oscillator .
3.7. Quiet Re gion As pect
After the decay of the tra nsient part of the ultrasonic sig nal, the result ing signal is
character iz ed by a constant phase, uniform ampli tude, and a f requenc y which matches the
force d frequenc y . H ence, the mathematic representation of this signal is given by the
particular solution equa t ion 3 . As mentioned previousl y , the evaluation of the measured
curves h as to be restricted to this "quiet" time interval. Therefore, it be comes necessary to
analy ze quantitatively the influence of the ho mogeneous solution (transient response):
what is the minimum number of c y cl es required in a burst excitation to make sure th e
transient region dies of f completely and the steady-state is reached? This can be deter mined
by estimating the f requenc y variation th rough the transmitted or the r eceived signals using
consecutive z ero-crossings. The good a greement betwe en the emulated and the measured
curves ( Figures 3.16 - 3.19) allows to use of the former ones to perfor m the frequency
variation estimation. Figure 3.20 il lustrates the fr equency variation, which is performed on
the basis of the emulated transmitter signal ex ample for f f = 1.02 MHz, ζ = 0.05, and fr =
1 MHz, and using the analysis of the zero-crossings.

3 See Equation (3 .6).

Chapter 3: Ultrason ic Piez oelectric T ransducer The ory and Modeling 30

Fig. 3.20: Fr equency variation thr ough the transmitted signal
The zero-crossings are ca lculated by the line ar interpolation betwee n ever y p air of
successive positive and negative signal values and their corresponding time on the x -axis.
As a result, the homoge n eous solution or the resonance effect dies out after about
25 periods, which allows the transducer to r each the so -called quiet region or the stead y -
state region [3 7 ]. In orde r to achieve hi gh accura cy of less than 20 ps peak - to -pea k TTD -
jitter and TTD -of fs et of zero, the TTD measurements ar e carried out only within thi s
steady - state region 4 .
It can be deduced f rom the results reported in Figure 3.20 that the transducer must be
excited by a burst of more than 25 cyc l es in order to ge nerate enou gh measurement points
in the quiet reg ion parts of the ultrasonic signals, i.e., t y picall y 50 to 70 periods of
excitation. The number of the allowed excited c ycles is strictl y limi ted by wave reflections
ge n erated by the other t ransducer on the opposite side of the pipe. This pr oblem becomes
more serious when the ex citation of the two t ransducers is don e simulta neously and the
ultrasonic path length is between 5 and 8 cm. Then, the number of c y cles is limited to only
about a hal f of th e sound wave traveling time. However , when a sin gle ex citation is used to
excite the transducers, the number of c yc les is limited to about a sound wave traveling time
since the transmitted and the received signals are acquired fr om di f ferent sc ope channels.
In ord er to reach a mi nimum TTD -jitter and TTD -of fset in the receive mod e, the transducer
has to recover from the transmit mode and achieve a state where it becomes completel y

4 See Section 4.5 of Chapter 4 fo r further discussion.

Chapter 3: Ultrason ic Piez oelectric T ransducer The ory and Modeling 31

calm after passin g the t ransient part, where fr ee vibration is sti ll valid. Therefore, it is
necessary either to ex tend the pipe length or to us e a pair of t ransducers with a suffic ientl y
high resonance frequency . For a 1 MHz flow meter pipe and within a t y pical dist ance of
about 5 cm, the numbe r of periods is limi ted to roughly 20. Ho wever , a 4 MHz flow meter
pipe allows 4 times more periods. In this work, we ch ose to switch from 1 MHz to the
4 MHz flow meter pipes (Figure 2.2), where t ypically a sinus bu rst of 70 c y cl es is used for
excitation.

Fig. 3.21 : Measur ed tran smitted signal of the 4 MHz flow me ter pip e

Fig. 3.22 : Measur ed r eceived signal of the 4 MHz flo w me ter pip e

Chapter 3: Ultrason ic Piez oelectric T ransducer The ory and Modeling 32

3.8. Character istic Fr equenc ies of Piez oelectric T ransducer
As mentioned in Section 3.5, there are two fr equencies which characterize the piezoelectric
transducer impedance: the serie s and the pa rallel freque nci es. In ord er to esti mate the
freque n cies which characterize the 4 MHz transducer pair located in the 4 MHz flow meter
pipe used in our experiments, a variation of transmitt er voltage s VT D 1 and VTD 2 in both
directions with respec t to the driving frequenc y is obtained experimentall y using the
measurement setup shown in Figure 3.13 and the 4 MHz flow meter pipe . The amplitude
voltage s are estimated in the steady- state re gion of transmitted signals in both directions
using least square sine- fi tting. This experiment result (Figure 3.23) shows the same general
character istics when compared to th e simulated magnitude response of th e electrical model
of the transduc er shown in Figure 3.9. The p iezoelectric series resonance frequenc y
corre sponds to the minim um impedance of the transducer (minimum amplitude in
Fig u re 3.23), and the parallel resonance frequenc y corresponds to the max imum im pedance
of the transducer (maximum amplitude in Figure 3.23). The transduc er is protected by its
high im pedance in o rder to transmit the max imum energy to th e wat er . The max imum
voltage results from the fact that the driver s ends a minimum of energy to the transducer ,
thus the current over R s (Figure 3.8) becomes minimal.

Fig. 3.23: Measur ed transmitter voltages versus driving fr equency , p erformed at
ambient tempe ra tur e
f p2

f p1

f s2

f s1

Chapter 4
Determinati on of T ransit T ime Differ ence

4.1. Intr oduct ion
The transducer model b ased on the driven dam ped harmonic oscillator model shown in
Chapter 3 allows an accu rate description of the b ehavior of a piez oelectric transducer . The
free oscillation of the transducer di es out when it is excited by a burst of a suf ficient
number of cy cles in order to a llow the transducer to re ach a stead y - state. We take
advantage of this phenomenon - the d y ing out of the tra nsient reg io n - to measure
indirectly th e transit time differe nce (TTD) by comput ing the phase difference b etween the
steady - state region of received sign als in the upstream and downstream d irections. Man y
factors si gnificantly influence the upstream and downstream transit ti mes, such as water
viscosity , temp erature-dependent transducer parameters, and th e speed of sound c, which is
related to the comp ressibilit y K and the densit y  of water by th e following equation:

(4.1)
All these factors impose great challenges to develop a new methodolo gy for achieving high
measurement a ccuracy with re duced TTD -jitter and compensated TTD -offse t at no-flow
conditions and within a temperature range from room temperature to 80°C.
The first part of this ch apter introduces the experimental setup for mea suring the TTD,
followed by the description of the transmission and reception electronics associated with
the transducers.
The second part of the chapter describes the methodolog y applied to compute the TTD .

Chapter 4: Determ ination o f T ransit T i me Di ff er ence 34

4.2. Exper imental Setup for Mea suring the TTD
The operational principle of the s ystem used in the experiments is illustrated in Figure 4.1.
As mentioned previously in C hapter 3, in order to allow the receiving trans ducer to reach a
steady - state where both amplitude and frequency are settled, a sinusoidal burst of 70 cycles
ge n erated over a time p er iod of 100 ms b y a function waveform generator (Hameg HMF
2550) is used to drive the two transducers (TR 1 and TR 2 ) si mul ta neou sly thro ug h t he
re si sta nce s R 1 a nd R 2 . T he se tra nsd uce r s co nver t elec tr ica l exc ita tio n i nt o a me c han ica l
wa ve . The dri vi ng fre quency of the bur st is se lec ted nea r the res on ance fr equ enc ie s of bot h
tr an sdu cer s in or der to a ch ie ve the ma xim um ou tput v ibra ti on w ith an en hanc e d SN R.

Fig. 4.1: Block diagram of the experime ntal setu p for m easuring the TTD
Fig u re 4.2 shows a picture that was taken while performing the measurements.

Fig. 4.2: Experimental setup for measuring the TTD

Chapter 4: Determ ination o f T ransit T i me Di ff er ence 35

In o rder to eliminate the high starting jitter , which is caused b y tri ggering and recording
ultrasonic signals from both directions separately (sequentiall y ), w e rel y on the
simultaneous (synchronous) ex citation approach. This mea ns that the common
sync h ronized clock of th e dual channel digital os cilloscope (Di gilent Analog Discovery) is
used to tri gger and acquire si gnals from bot h directions. Hence, w e elimin ate an y
dif ferential delay time between the two input cha n nels.
As shown in Figure 4.3, the sound waves a re received sim ultaneously b y b oth transducers
after a traveling time that is proportional to the distance between the transduc ers
(L = 42.2 mm). The received signals are attenuated due to the loss or a bsorption of
acoustical energy b y the medium [ 34]. Therefore, w e us e two low -n oise operational
amplifiers (OP A2846 ID), shown in Fig ure 4.1, in order to increa se the signal levels. The
amplified signals are acquired b y the digital osc illoscope, which has two 14 -bit channels
with a sampling rate of 5 0  10 6 Samples/s per channel. The samples a re s aved in comma -
separa t ed valu es (CSV) data fo rmat and importe d to the computer (PC). An embedded
(PC) software algorith m written in Matlab h andles the automatic TTD measurement
repetition, providing all necessar y p rograms to c ontrol the function generator and digital
oscilloscope.

Fig. 4.3: Acquir ed ultrasonic sign als in both dir ections
(measur ed using a 4 MHz dr ivin g fr equency)

Chapter 4: Determ ination o f T ransit T i me Di ff er ence 36

4.3. Method ology for the C alculation of TTD
The TTD can be d educed from the phase dif ference between the stead y-s tate parts of the
rece iv ed signals in the upstream and downstr eam directions b y estimating the two phases
with the least squares sine-fitting algorithm of Matlab (see Figure 4.4) [19] [38 ]:
(4.2)
(4.3)
where and are the phase shifts between the received and transmitt ed s ignals
measured in the stead y-state regions (se e Figure 4.5) in the upst ream and downstre am
direction, respectivel y . In the quiet region, the received si gnal frequency f matches the
force d frequency of the ex citing sig nal. Therefore, the TTD can be computed as follows:
(4.4)

Fig. 4.4: Phase differ ence between the steady-state r egion of re ceived sign als in both
dir ections

Chapter 4: Determ ination o f T ransit T i me Di ff er ence 37

Fig. 4.5: Response of the 4 MHz transducer to the incident ultrasonic wave
Assuming that the phase shifts between the electric and acoustic signals on both
transducers for transmitted signals are t1 , t2 and r1 , r2 for received signals, the total
calculated phase differe n ce ( T ) between the signals recorded from both directions is given
by [19]:
. (4.5)
Since t up and t d ow n are the same at no-flow con ditions a ccording to [7] and [19] under
electro-acoustica l reciprocal ope ration, in which t1 + r2 = t2 + r1 , the phase difference
between the upstream and downstream directions can be compensa ted. Such operation
requires that the equival ent electrical impedances of the electronics and transducers are
equal in both dir ections. Therefore, under thi s co ndition the unmatched transducers ar e no
longer the onl y r eason for the TTD -of fset drift produced at no-flow conditions. The
dissy mm etr y between th e two directions si gnal p aths (in te rms of the electrical im pedances
of the elec t ronics and transducers used in the mete r), which m ainly causes the TTD -offse t.
4.4. Least Squa re s Sine-Fittin g
The least squares sine -fitting ( L SSF) is a ver y accura t e method to estimate all three
parameters - amplit ude, phase, and freque n c y - that chara cterize a digitized sinuso idal
signal sampled at a well-defined sampling rate [39 ]. The sine-fitting technique is used as a
filter and can si gnifica n tly r educe noise such as the ADC quantiz ation noise [40 ]. Th is
procedure is ver y ofte n used to recover dist orted and noisy signals in tests and
measurements [4 1-45 ]. A three-parameter sine- fit algorithm is used for a reasonable
estimation of amplitude, phase, and frequenc y of each recorded sample. Assum ing that the
data of n samples ha s b een saved, the fitted function is given by:

Chapter 4: Determ ination o f T ransit T i me Di ff er ence 38

, (4.6)
where A and are the amp litude and the phase of t he signal, respectively , f is the driving
freque n cy , and t n is the discre te- time vector .
The sum of sq uares of th e error between these three ex tracted parameters and the measured
data is g iven b y:
. (4.7)
The algorithm chooses s uccessive values of A, f, and to minimize  , taking into account
the signal- to -nois e-and-dist ortion ratio [41] [45 ].
4.5. Estim ation of the Quiet R egion
As mentioned before, accurate phas e estimation from the acquired recorded d ata is
performed in the quiet p arts of received signals. As shown in Figure 4.6. b, relativel y hi gh
random fluctuations in volt age occur in the r egion between the transmitted and received
signals, which can influence measurement a ccuracy . Such voltage fluctuations arise due to
insuf ficient ti me, which leads to a complete d ecay of the t ransmitted signal to zero before
the arrival of the re ceived signal due to simultaneous excitation. In the case of single
excitation, the transmitted and the received signals are acquired from diffe rent scope
channels, as shown in Fig ure 4.7. It demonstrat es that only a few millivolts t y pical noise
floor of the di gital oscill oscope is observed before the start of the transmitted signal and
the a rrival of the r eceived signal (Figure 4.7.a and 4.7.c). This also shows that in our
chosen te chnique th e hi gher voltage noise arise d ue to the prox imi ty of th e transmitted and
rece iv ed signals in the same channel.

Fig. 4.6: V oltage fluctuations in the r egion betwee n the transmitted and r eceived
signals measur ed at simultaneous excitation

Chapter 4: Determ ination o f T ransit T i me Di ff er ence 39

Fig. 4.7: Noise of the oscilloscope measur ed at single excitation
In order to protect the phase measurement inte rval from these volta ge fluctuations, the
transducer should be excited for a sufficiently l ong p eriod of time. Figures 4.8 and 4.9
show the results of the cy cl e- by - c y cle time dela y of the transmitted and rec eived signals,
respec tivel y . These results are calculated throug h the estimation of phase shifts of the 70
periods between the electric and acoustic sig n als on both transmittin g and receiving
transducers in both directions by usin g sine-fittin g. These results show that after 50 c y cl es
of excitation the two transmitted and received signals look more steady and the transient
region (free os cillation of the transducer) h as completel y died out. Therefore, on the b asis
of these results we decided to use the last 20 periods to extract sine-fitting parameters in all
our experiments.

Fig. 4.8: The cycle- by -cycle time delays of both transmitted signals in both dir ections

Chapter 4: Determ ination o f T ransit T i me Di ff er ence 40

Fig. 4.9: The cycle- by -cycle time de lays of both re ceived sign als in both dir ections
4.6. Algor ithm for Com puting the TTD
The main components of the algorithm used for estimating the phase and computing the
TTD ar e shown in Fi gure 4.10. A sine- fitting technique is used to reduce the error in the
phase estimation. This technique estimates the sine wav e that best fits the r ecorde d samples
of the last 20 c ycles of signals re ceived fr om both directions. The al gorithm used for
computing the TTD value performs the following steps:
 Read the recorde d d ata saved in CSV f ormat.
 The DC of fset correction is performed to remove it from the acquired signal. This
correction is done b y co mputing the mean volta ge value of all recorded v oltage samples,
which is then subtracted from each da ta voltage sa mple.
 An estimation of both frequencies of sine waves from the recorded sampling data is
performed b y using the least squares sine-fitting algorithm of Matlab. This estimation is
carr i ed out in the steady-state parts of re ceived sig nals in upstre am and downstream
directions.
 An estimation of amplitude and ph ase, which characterize the di gitized sinusoidal
signals, is performed by using the previousl y obtained mean value of the fitted
freque n cies.
 Calculate the dif ference betwee n the two fitted phases.
 Calculate the TTD.

Chapter 4: Determ ination o f T ransit T i me Di ff er ence 41

In order to control the shape of the curve, the upper and lower bounds of the estimated
parameters (amplitude, phase, and frequenc y) tha t charac t erize the dig itiz ed sinusoidal is
introduced. For instan ce, the amplitude has bounds [0, 5], the phase has the bounds [ -   ] .
W it h this construction in place, the fitted curve accurately ex tracts parameters from the
measured data, especiall y the two phases that are used to calcula te the transit time
dif ference.

Fig. 4.10: F low chart for c o mputing the TTD
.

Read the CSV file
of both directions

DC of fset correction

Estimate both directions
freque n cies

Estimate A and  usin g the
freque n cies mean value
estimated previously

Calculate the pha s e
dif ference

Calculate TTD

Chapter 5
TTD -j itter Reduction

5.1. Intr o duction
The purpose of samplin g and digitizing the t ransmitted and received ultrasonic signals is to
perform advanced sign al processing and a ccurat ely ex tract the transient time diffe rence
(TTD). The continuous signal waveform is defined on the basis of a set of time -discre t e
samples over a specified period of time. How ever , clock jitter causes uncertaint y in the
sampling time during t he data a cquisition of ultrasonic waves. This leads to random
fluctuations in the corres ponding amplitude. Th erefore, th e recorded data are corrupted by
noise and distortion. As the TTD is computed on the basis of a finite set of sampled data,
such uncertainties in the sampling time produce random fluctuations of TTD
measurements, known as the TTD -jitter .
The TTD -jitter in flow measurement has a huge impact on its a ccuracy . It can be
experimentally evaluated by repeatedl y performing the same me asurement and using the
same measurement setup under the same conditions.
This chapter investigates the TTD -jitter noise and its impact on the accur ac y of flow meter
measurement result s. The approaches us ed in order to mitigate the TT D-jitter are also
presented here .

Chapter 5: TTD-jitter Re duction 43

5.2. The Effect of Sampling Jitter on T TD
In the sampling-process, the sa mpling time unc ertainty (also c alled the sampling time
jitter 5 ) is the random fluctua tions in time location of a given waveform sample.
Corresponding l y , this un certainty in s ampling ti me introduces errors in the amplitude of
the ultrasonic signal. As illust rated in Figure 5.1 , for a given sine wave of period T and
pe ak - to -peak amplitude Vpp the sampling time j itter Δ t causes a change in the measured
value equal to Δ V = Vpp π Δ t/T [ 46 ]. This contributes to the increase of the TTD -jitter . In
the present application, the TTD -jitter refers to the dispersion of the measured TTD around
the mean v alue (zero fo r no-flow). The latter is assumed to be a stationar y Gaussian noise,
which means that the standard deviation does not change over ti me. The T TD-jitter is thus
defined by the me an value and the standard deviation.

Fig. 5.1: Effect of the sampling jitter on me asured value
T y picall y , the transducers are excited by a sinus burst of 70 cyc les at 4 MHz frequency .
The ex perimental results in Figure 5.2 sho w the last 19 periods of a transmitt ed signal
superimposed upon each other . Fi gure 5.2.b s hows a magnification of Figure 5.2. a.
Interestedly , this figure shows that for consecutive periods of a measurement, the z ero-
crossings jum p f rom one period to the next period statis tically around the zero point. The
Gaussian (or normal ) probability distribution function can be used to statisticall y
character iz e the maximum time deviation of the periods around the zero point cau sed by
the sampling time jitt er . The maximum time deviation of the 19 consecutive periods around
the z ero point is abo ut 374 ps, which corresponds to 6 the standard deviations
(  = 62.4 ps ). Note that t he standard deviation result is obtained by evaluating 19 time data

5 Not to be confused with the TTD - j itter.

Chapter 5: TTD-jitter Re duction 44

samples around the zero point shown in F igure 5.2.b apply in g the pr obability densit y
function . The time deviation of the periods increa ses with increa sin g number of time data
samples.

Fig. 5.2: T ime deviation of the periods ar ound the z er o point:
(a) the last 19 periods of the tr ansmitted signal superimposed upon each other ,
(b) a magnified ar oun d the zer o point
5.3. Sour ces of TTD-jitter
The TTD -jitter noise arises from many sources which contribute to th e measurement
uncertainty . These sources are uncorrelated and may originate from an y where in the si gnal
path from transmitting to receiving waveforms. Some of these sources are dedicated t o the
electronics associated with the transducers such as [ 39 ]:
 transmitter wav eform phase noise or driving fre qu ency inst abilit y of the
transmitted signa l due th e function generator
 jitter in the sampling instanc e that may occur due to imperfect sample-and-hold
circuit sy n chronization
 crosstalk between the cables
 i rr egularities in the tra ns mission medium.
Other noise sources such as the ef fects of the side lobes also contribute to the TTD -jitter .
As shown in Fig ure 5.3, most of the acoustic ene rgy is transmitted perpendicularly from
the piezoelectric disk. Th e main lobe is the lobe containing maxim um sound pressure. This

Chapter 5: TTD-jitter Re duction 45

lobe is calculated b y finding the angl e at which the sound pressure is halved [ 47]. The
acoustic wave directivit y 6 is temperature-dependent [48] , especially at the side lobes l evel,
which increases from low temperature to high te mperature. This can enh ance the lev el of
reverbera tion in the me asuring pipe, producing unde sired echoe s a ctin g as noise. This
noise ma y corrupt the si gnal along th e signal pat h, causing errors in the measured transit
time.

Fig. 5.3 : An example of a sou nd dir ectivity pattern [47]
5.4. TTD -jitter R eduction T echnique
Since the recorde d data of the received waveform are sampled with 50 MS/s, each period
in the stead y -state r egion contains a round 13 sample points. The sine-fitting applied
approac h adjusts Equ ation (4.6) to the s et of the r ecorded sampling data in order to extract
the characterized parame ters of the trace, namel y its amplitude, phase, and the frequency .
The followin g equation, obtained from Equ ation (4.4 ), is applied to compute the TTD
using the upstream and downstream extracted phases ( , :

(5.1)
The precision to which the TTD is determined c an be influenc ed b y two para meters: the
amplitude of the received signal and the number of the samples used in the fitting
algor ithm. Concerning t he first parameter, there is a clear correlation be tween the TTD -

6 T he acoustic directivity descr ib es how the pressure o f a sound w a ve is trans mitted fro m a transducer .

Chapter 5: TTD-jitter Re duction 46

jitter level and th e amplitude of the received signal in the quiet region. In other words, th e
obtained TTD has a jitter that depe nds on the input d y namic ran ge of the acquisition
system. This is optimi zed by adjusting the gain o f the two amplifiers (Figure 4.1), aiming
to cover most of the inp ut range of the ADC. The results shown in Figure 5.4 (carried out
at room temperature, no -flow condition, N = 250 fitted samples, and di fferent r eceiver
voltage s 7 V RD1 and V R D2 ) suggest that the incr ease of the voltage hea droom in both
directions reduces the TTD-jitter.
The used ADC has 14-bit resolution and an input full-scale range of ±2 .5 V 8 . The least
significa nt bit (LSB) voltage can be expressed as:

(5.2)
where N bit is the ADC ’ s resolution in bits and V FS R is the full-scale voltage range of the
ADC. According to Equation (5.2), the ADC has a voltage resolution V LS B of about
300 µ V. The d y namic range of d ata acquisition system (DR), which is the ratio of the
maximum input voltage V FS 9 (signal amplitude) to the minimum voltag e V LSB , can be
expressed as:

(5.3)
Fig u re 5.5 illustrates the measurement pr ecision (represented by the standard devi ation)
versus the d y n amic range of the acquisition system. Note that the stand ard deviations are
calculated f rom TTDs of 50 captured ultrasonic w aveforms. It can be deduced on the basis
of these results that high d y namic range of the acquisiti on s y stem can be achieved usin g a
high gain amplifier to cover the input full-sc ale range voltage (V FSR = 5V) of the ADC,
which results in very low TTD -jitter .

7 T he amplitude voltage s V RD1 and V RD2 are esti mated in the quiet region o f both directions of received
signals using si ne-fitting
8 According to the datas heet of the used ADC (AD964 8) [49], it achieves SIN DR of 74. 3 dB, SNR of
75.4 dB, and ENOB of 12 -bit at 9.7 MHz.
9 V FS refers to the full -scale sin e w a ve of the received si gnal i n the quiet regio n.

Chapter 5: TTD-jitter Re duction 47

Fig. 5.4: TTD measur ed at d iffer ent r eceiver amplitudes

Fig. 5.5: The standard deviation of the transit time differ ence (TTD) versus the
dynamic range of the acquisition system (N = 250 f itted samples)

Chapter 5: TTD-jitter Re duction 48

As mentioned previously , the TTD -jitter can also be redu ced by in creasing the number of
the signal samples that are used to a djust the sine-fitting para met ers. Since the noise
present at each sampling sequence is uncorrelated, a given number of sa mples N reduce s
the timing jitter values by a factor of (according to the averaging princip le). This can
mathematically be expressed as:

(5.4)
where is the standard deviation error of the TTD (after averaging), σ is the standard
deviation of the TTD, and N is the size of the samples.
Fig u re 5.6 depicts a sin gle TTD measurement carried out at room temperature and no-flow
conditions for dif ferent numbers of the fitted samples.

Fig. 5.6: TTD r esults measur ed at d iff er ent num bers of fitted samples (N)

Chapter 5: TTD-jitter Re duction 49

Fig u re 5.7 illustrates the measured TTD standard deviation (STD) versus the number of
fitted samples as well as the theoretica l limits estimated using Equation (5.4). It can be
deduced on the basis of these re sults that the measured TTD -jitter exhibits the same
behavior as the calculated one. In other words, as the number of samples increases, the
standard deviation of the measure d TTD decreases according to Equation (5.4).

Fig. 5.7 : The standard deviation of TTD versus the number of fitted samples (N)

Chapter 6
TTD -offset Cancella tion

6.1. Intr o duction
As described pr eviously , in the ideal case when t here is no flow in the m easuring pipe the
upstream and downstream transit times should be identical. However , in reality this is not
the case as there are many factors that prevent the ult rasonic signals (see Section 4.1) from
reac hin g the r eceivers after exactly the same traveling time. Hence, an y deviation of the
measured transit time dif ference (TTD) from zero at no-flow conditions is referred to as a
TTD -of fset. This TT D-offse t, which limits the minimum measured flow , p resents a s erious
drawback in high accu racy measurements. Therefore, it is worthwhile to develop a
theoretica l anal ysis which provides a better understanding of the TTD -offset sources and
their contributions in order to develop a measurement strategy that allow s effec tive TTD -
of fset compensation.
6.2. Sour ces of TTD-Offset
In water flow meter applications, the environment temperature inside th e measuring pipe
may var y from ambient temperature to 80° C. This leads to a variation of the transducer's
resonance frequency , wh ich cannot withstand the change of the oper ating t emperature [19]
[50-52 ]. However , according to the harmonic os cillation model 10 , if a pair of transducers
has dif ferent resonance f requencies, then th e phas e dif ference between the two stead y-state
parts of the received signals is diffe rent from zero as well. Figure 6.1 shows an ex ample of
two phase responses versus the driving frequency . This two plots of pha se responses are
obtained from the phase equation of the p articular solution of a damped driven harmonic
oscillator (Equation 3.10) for two transducers wit h diffe rent resonance frequencies (for this
example, the resonance frequencies a re fr 1 = 4 MHz and fr 2 = 4.05 MHz for the first

10 T he analytic m od eling of the piezo electric transducer disk d escribed in Chapter 3.

Chapter 6: TTD-offset Ca ncellation 51

transducer and the seco nd transducer , re spectivel y ). The dif ference between these two
phases is shown in Figure 6.2, where it c an be observed that within the resonance
freque n c y ran ge the phase dif ference is si gnificantl y different from zero. Thus, the z ero
flow er ror c an drift according to the temperature changes, whi ch can cause si gnificant
TTD -of fset.

Fig. 6.1: Phase r esponse versus drivin g fre quen cy f or two diff er ent transducers

Fig. 6.2: Phase differ ence r esponse versus drivin g fr equency
6.3. The Impact of T emperatur e on the T ransducer Resonance
Fr equencies
In ord er to predict wh at ef fect temperature changes have on the series and parallel
resonance frequencies of the 4 MHz transdu cer p air , the v ariation of transmitt er voltages
(VTD 1 and VTD 2 ) with r espect to the driving freq uency is investigated experimentally (s ee
Section 3.8). The ex periment allows to obtain an approximate determination of both
freque n cies by estimating the max imum and the minimum transmitter amplitudes

Chapter 6: TTD-offset Ca ncellation 52

measured at different t emperatures. This ex periment is carried out by ex citing the
transducers with a sinus burst of 70 c y cl es at a s pecified frequency in the r ange b etween
3.8 MHz and 5 MHz. The results illustrated in Figures 3.23, 6.3, and 6.4 demonstrate the
variation of transmitter volt age amplitudes in both directions me asured at ambient
temperature (  25°C), 60° C, and 80° C, respectivel y , with respect to the dri ving fr eque n cy .
Note that the measuring pipe is heated usin g a thermostat. It can be de duced from the
obtained results that th e frequenc y at which the amplit ude becomes mi nimal (minimum
impedance) is close to serie s resonance fr equency f s , whereas the frequency at which the
amplitude be comes maximal (maximum impeda nce) is close to the pa rallel resonance
freque n c y f p . Howev er , b ecause the parameters of the two tr ansducers are n ot identical, the
variation of both amplitudes is also not identical. The temperature dependence of the
resonance frequencies of the pair of 4 MHz transducers is summarized in T able 6.1.

Fig. 6.3: T ransmitter voltages versus driving fr equency measur ed at 60° C

Fig. 6.4: T ransmitter voltages versus driving fr equency me asured at 80°C

Chapter 6: TTD-offset Ca ncellation 53

T emperatu re [°C]

Resonance frequency [ MHz]

T ransdu cer 1

T ransdu cer 2

f s

f p

f s

f p

25°C

4.02

4.615

4.01

4.595

60°C

4

4.49

3.96

4.475

80°C

3.97

4.465

3.925

4.43

T able 6.1: The depende nce of r esonance fr equen cies on temper atu r e
As shown in Figure 6.5, the series f s and the parallel f p resonance frequencies of the used
transducer pair decrease as the temperature goes up. Regardless of the mismatch between
transducers, the variation of the resonance frequenc y with the operating tem perature r ange
can cause a diss y mmetr y b etween th e two dire ctions of the ultrasonic si gnal paths. Thus,
the temperature variation of the medium is the m ain reason for the zero f low TTD -offset
drift.

Fig. 6.5: The effect of temperatur e on the transdu cer r esonance fre quencies
6.4. Z ero F low TTD - offset Corr ect ion
As mentioned before, in order to eliminate the possibilit y that th e mete r detects a false flow
under no-flow conditions, the upstream and downstream transit times should ideally be the
same, although this may not be the case unless special precautions are taken. Due to the
fact that ever y flow di rection exhibits a slightly di fferent electrical impeda nce comp ared to
the other, there is a dissymmetry b etween upstream and downstr eam si gnal paths [7] . This
effect provides a different amount of currents flowing throu gh R 1 and R 2 (see Figure 4.1),
and c auses dif ferent amplitudes in the upstream and downstream trans mitted signa ls.
Fig u re 6.6 shows the result of the measurement ca rried out at room tempera tur e and

Chapter 6: TTD-offset Ca ncellation 54

no-flow conditions. Both transducers are excited with the same sinus burst at 4 MHz forced
freque n c y . The plotted curves show that the measured diff erence between the two
amplitudes of the transmitted sig nals is about 250 mV.

Fig. 6.6: Acquired ultrasonic signals in the upstr eam and dow nstream
directions (me asur ed at 4 MHz driving frequency)
By computing the ph ase shifts t1 and t 2 between the excitation and the sound wave on
both transduc ers for transmitted sig nals, a delay time di fference of -2.4 ns is obtained
between transmitted signals in both dire ctions. This relatively hig h starting de la y time
differe n ce can be explained b y u pstr eam and dow nstream unmatched electrical impedance
values, which le ad to the difference between t he two transmitter amplit udes (of about
250 mV). The 250 mV transmitter amplitude d ifference shown in Figure 6.6 r esults in
about 150 ps zero flow TTD-offset.
In ord er to effectivel y elimi nate TTD-offset at no -flow conditions, it is necessary to match
the upstream and downst ream electrical impedance of both tr ansducers and their associ ated
electronic to reach a highly s y mmetri cal signal p aths. According to t he literature [25] , the
electrical im pedance of a transduc er can b e controlled b y a d riving frequenc y. We hav e
used this fea ture to eliminate the electrical impedance mismatch betwee n both directions
because a well-mat ched upstream-downstream signal path r educes the transmitter
amplitude differe n ce and results in a ver y small zero flow TTD-o ffset.
By changing the sinus burst frequenc y from 4 MHz to 4.19 MHz, the 150 ps zero flow
TTD -of fset measured pr eviously is subst antially reduced to less than 5 ps (Figure 6.7).
Moreover , comparing the amplitudes of the transmitted signals depicted in Fig ur es 6.6 and
6.7, it can be observed that the dif fe rence between the two transmitter amplitudes is
reduce d from 250 mV measure d at 4 MHz to less than 15 mV , which is achieved by
exciting th e transducers at 4.19 MHz driving frequenc y under the pr eviously mentioned

Chapter 6: TTD-offset Ca ncellation 55

conditions. In addition, the dela y time di f ference of -2.4 ns achieved bet ween tr ansmitted
signals in both dir ections at 4 MHz is reduced to -664 ps at 4.19 MHz driving fr equency .
Therefore, choosing an appropriate driving frequency withi n the resonance frequency
range of th e tr ansducer is the key to canc el the long-term d rift of the TTD c aused by
temperature varia tions.

Fig. 6.7: Acquired ultrasonic sign als in the upstr eam and downstr eam dir ections
(measur ed at 4.19 MHz driving fr equency)
Fig u re 6.8 shows TTD measurement results obtained in a d riving frequency r ange from
4 MHz to 5 MHz with 5 KHz step variation at room temperature and no-flow condition s.
The result shows that withi n the resonanc e frequency range of the two transducers TT D-
of fset compensation c an be achieved at 4.185 MH z driving frequenc y . Figures 6.9 and 6.10
repre s ent the same measurements as before carried out at 60°C and 80 °C tempera ture,
respec tivel y , and they cl early sho w that these TTD -of fsets compensation are achieved at
dif ferent driving frequen cies (about 4.095 MHz f or 60°C and about 4.075 MHz for 80° C)
due to the f act that th e resonance frequen c y of th e transducer is changing as a function of
temperature as well.

Fig. 6.8: TTD versus driving f r equency measur ed at ambient temperatur e

Chapter 6: TTD-offset Ca ncellation 56

Fig. 6.9: TTD versus driving fr equency measur ed at 60°C

Fig. 6.10: TTD versus driving fr equency measur ed at 80°C
To sum up, fo r a successful TTD -of fset compensation it is necessar y to adopt the two
following strateg i es:
• Of fline TTD -of fset calibration must be performed in the manufacturing process in
order to provide a table with driving frequencies and their corresp onding
temperature within a tempera tur e range from ambient temperature to 80°C.
• Online TTD-offse t calibration must be performed with control algorithms that use
adequate d riving frequenc y suitable to compens ate the TTD -of fs et according to
the measure d t emperature inside the measuring pipe.

Chapter 6: TTD-offset Ca ncellation 57

6.5. T em peratur e De pendence of the TTD -o ffset Ca li brat ion
As mentioned before, choosing an appropriate dri ving frequenc y can d rasticall y reduce the
zero flow TTD-of fset. T o attain automatic comp ensation, one needs to accurately set a
suitable driving frequency to eliminate the TTD -of fset caus ed b y the temperature variation
inside the pipe. I n o rder to ensure repeatabilit y or precision of the drivin g freque n c y used
to compensate the TTD -of fset, several experimental mea sur ements are performed over
three months at four randoml y chosen temperatures (35°C, 60°C , 70°C, and 80°C). These
experiments have lead to the results illust rated in Fig u res 6.1 1 and 6.12, where the heatin g
up and cooling down of the pipe is done by means of a thermostat. As it can be seen in
Fig u re 6.1 1, the first experiment starts b y a djustin g the t emperature inside the pipe to 35°C;
the calibration of the TTD -of fset to zero is reached at 4.153 MHz driving frequency . While
maintaining the same driving frequenc y , the heating up of the pipe to 80°C increases the
TTD -of fset to about 270 ps. B y chan ging the driving frequenc y from 4.153 MHz to
4.075 MHz, the TTD-offset is calibra ted once again to nearly zero value. The cooling down
of the pipe from 80°C back to 35°C at the same driving frequency d ecreases the TTD -
of fset to about -320.3 ps, while the TTD -of fset reache s again zero at the same starting
excitation frequency (4.153 MHz) since in that moment the measured temperature inside
the pipe is again 35°C. The relativel y hi gh m easured TTD -jitter that can be seen clearl y in
the measured plots, especiall y at 35°C, can b e ex plained by th e fact that a fu ll -scale voltage
range of the ADC is not reached completel y . This happens because the rece ived si gnal
attenuation is normall y proportional to the viscosity o f the water , whic h decreases with
increasing temperature. Therefore, the gain of the amplifier is controlled by four
resistance s (see Fi gure 7.1) in order to brin g th e received signal to a l evel that is high
enoug h to resist the attenuation that it will be subjected to.
The same setup is used to perform the second ex periment, apart from the fact that the
maximum reac hed temperature is 60°C instead of 80°C. I t can be se en from Figure 6.12
that heating up and cool ing down the pipe from 35°C to 60°C and from 60°C to 35°C,
respec tivel y , provides TTD-of fset values that var y between -220.3 ps and 338.8 ps.

Chapter 6: TTD-offset Ca ncellation 58

Fig. 6.1 1 : TTD measur ements performed over a temperatur e range
fr om 35°C to 80°C

Fig. 6.12: TTD measur ements performed over a temper a tur e range
fr om 35°C to 60°C
The results showing th e temperature dependence of the driving frequenc y are obtained in
five measurements carrie d out over a three-month period (see Table 6.2 an d Figure 6.13).

Chapter 6: TTD-offset Ca ncellation 59

Taking in consideration the suppression of the zero flow error, it can be noticed that the
driving frequenc y at whi ch the two transdu cers are excited is strongly inf luenced b y the
temperature. Therefore, we have shown that the re is a cle ar , simple, sta ble, and reliable
relation between the required excitation force d f requency and temperature.
Note that the measured driving frequencies with respect to the temp erature reported in
Table 6.2 cha n ge from one pair of the transducers to the other.
T emperatu re
[°C]

Driving frequency [MHz]

1 st
measurement

2 nd
measurement

3 rd
measurement

4 th
measurement

5 th
measurement

35°C

4.152

4.153

4.151

4.154

4.152

60°C

4.098

4.098

4.097

4.096

4.097

70°C

4.088

4.089

4.088

4.087

4.085

80°C

4.076

4.073

4.073

4.072

4.071

T able 6.2: T emperatur e dependence of the driving fre quency

Fig. 6.13: T emperatur e dependence of the driving f r equency used for the ze r o flow
TTD -offset c o mpensation

Chapter 7
Results and Analysis

7.1. Intr o duction
In this chapter , we demonstrate a pra ctical application of the de veloped measure ment
methodology described in the previous chapter . Hence, the electronic s y stem design
introduced in Chapter 4 and the software algorithm used for signal proce ssing are
described in more detail. Afterwards, the experimental results are presented an d analy zed.
This analy sis allows to highlight the robustness of TTD -jitter redu ction and TTD -of fset
correction methodologies, developed in this thesis. Note that all measurement results
reported in this chapter a re performed at no-flow conditions withi n a specifie d temperature
range from ambient temperature to 80°C
Fig u re 7.1 illustrates the electronic instrumentation sy stem utiliz ed to perf orm the
ultrasonic measurement with significantl y improved accuracy . The designed electronics
and the software al gorithm written in Matlab are capable of handling automatic
measurement repetition, providing all c ontrol s ub-programs, namely , the control of the
function generator and analog discovery device.

Chapter 7: Resu lts and Analysis 61

Fig. 7.1: Block diagram of the experimental setup
7.2. Hard war e Design
The hardware elements of the ex perimental setup presented in Figure 7.1 are:
• An arbitrar y waveform genera to r , which generates a sinusoidal burst of a finite
number of cyc l es in order to excite the transducers.
• A pair of 4 MHz transducers located in the flow meter pipe, which are capable of
transmitting and receiving an ultrasonic signal.
• DC source to power the low -noise amplifiers, whic h are us ed to amplify the
voltage levels of the tw o received si gnals. Amp lifiers gain is controlled by four
resistors R 3 , R 4 , R 5 , and R 6 in order to reach 5 V full -scale voltage of the A DC.
• Digital oscilloscope, which has dual -channel, 14-bit ADCs for digital signal
acquisition.
• An embedded (PC) software algorithm written in Matlab handles automatic TTD
measurement repe titi on.
• Additionally , th e flow meter pipe is put in a wa ter bath with a thermost at to
regulate the temperature of the transducers at given levels, assuming that the
temperature of the transducer c an be considered t o be the same as the temperature
of still water filling in the pipe.

Chapter 7: Resu lts and Analysis 62

7.3. Soft ware Algor ithm
Completely automatic m easurement process is achieved through the dev eloped software
algor ithm, which is used to automate manual tasks such as the configuration of the function
ge n erator and the digital oscilloscope to output the desired si gnal and save numeri cal data,
respec tivel y . The algorithm is also used to compute TTD and compensate the TTD -offset
error if necessary . This automatic procedure is summarized in Fi gure 7. 2, and it mainl y
consists of the following four sub-algorithms:

Fig. 7.2: F low chart of the automatic me asu r ement r epetition
• A sub -algorithm controls the arbitrary wavefor m gene rator in order to generate
sinusoidal burst.
• A sub- algorithm controls the two 14-bit ADCs i n order to digitize the u pstream
and downstrea m si gnals.
• The sub-algorithm described in Section 4 .6 is u sed to compute the TTD. The
proposed method described in Chapter 5 is applie d to attain reduced TTD-j itter .
• The last sub-algorithm is used to check whether the TTD-offset tends towa rds
zero value or not.
T ransmit the sinusoidal
burst

Digitize the two
direction sig n als

Determine the shift pha ses of
both rece ived si gnals using the
LSSF , and cal culate th e TTD
through the ph ase di f ference
between the received signals

Of fset adjustment

Chapter 7: Resu lts and Analysis 63

o If the TT D-offse t is not approximatel y equal to zero, the adjustment of th e
TTD -of fset to z ero is performed by choosing an appropriate excitation
freque n c y (se e Section 6.5), which is cap able of avoiding the TTD-of fs et
error .
7.3.1. T rans mission of the Sinu soidal Burst
A sinusoidal burst of 70 c yc les simultaneousl y ex cites the two transducers ov er a time
period of 100 ms at a driving f requenc y of 4 MHz (17.5 µs burst len gth) after the power up
of the system. The automatic configuration of the arbitrar y waveform generator is
performed usin g the algorithm shown in Figure 7.3. The transmission procedure starts b y
opening the seria l port so that the communication betwee n the P C and the function
ge n erator can be enabled. All burst signal parameters (function, amplitude, frequency , the
number of c ycles, and period) must be specified i n the program in order fo r them to be sent
to the function generator through the serial port.

Fig. 7.3: Configuration of the function generator
7.3.2. Digitizing t he T w o Dir ection Sig nals
The two 14-bit ADCs of the digital oscillosco pe operate up to 100 MHz maximum
sampling rate. The trigger controls en able us to capture a stable waveform to be digitized
and saved into a file containing numerical data regarding time and amplit ude. One of the
channels is used as a trigger source, wh ere the tri gger level and slope controls provide the
Open the serial port

Initialize the func tion
ge n erator

Burst func tion: S IN
Burst frequency : 4 MHz
Number of burst cy cles: 70
Burst amplitude: 20 V
Burst per iod: 100 ms

Chapter 7: Resu lts and Analysis 64

basic trigger point definiti on. Therefore, th e slope control, whi ch determines whether the
trigger point is on the rising or the falling edge of the indicated signal, must be indicated in
the control software al gorithm depicted in Figure 7.4 in such a wa y that the s y stem can
deliver direct digital signals via the two i nte grated USB pow ered ADCs. The c apture of a
single-shot waveform is controlled b y tri ggering; thereafter , th e pe rformed acquisitions are
transferred to the PC in C SV format via USB with the maximum sample storage length of
8192 samples in order to be processed in Matlab.

Fig. 7.4: The acquisition of both dir ection signals
7.3.3. TTD-Offse t Ca ncellation Algorit h m
The TTD -offse t compensation strategy presented in Chapter 6 can be accomplished
automatically by using an alg orithm. This compensation is done with re spect to the
temperature of the mediu m around the transduc ers (see Section 6.5) in order to extract the
required d riving frequency th at corresponds to the temperature of the me dium (T able 6.2)
so that the TTD -of fset err o r c an be compensated. Choosing of the correct drivin g
Power up

T ri gge rin g
activated ?

fr e que n c ies

T ri gge r configuration

Y es

No

Start the acquisition

Saving and exporting
data

Chapter 7: Resu lts and Analysis 65

freque n c y can be done a ccurately by first determini ng the temperature of the water inside
the measuring pipe. The measurement software system includes a conceptual TTD -of fs et
compensation algorithm depicte d in Figure 7.5, which allows continuous detec tion and
compensation of the zero flow TTD -of fset error .

Fig. 7.5: Com p ensation of the TTD -offset
The TTD -o f fset compensation algorithm is exec uted im mediatel y after computing the
TTD. This al gorithm starts by comparing the ob tained TTD v alue to zero. If the TTD-
of fset is almo st equal to zero, the n ew st atus information (drivin g frequenc y ) is stored in
the PC buf fer to be us ed to excite the transduce rs again (i.e., it is the return to the
transmission of the sinusoidal burst sub-algorithm). However , if the TTD -o f fset is differe nt
from zero, the adjustment of the TTD -of fset must be performed by choosing the correct
Start the TTD-of fset
adjustment

TTD -of fset  0
?

Y es

No

Determine the driving f requency
accor din g to the measured temperature
from the T able 6.5

Measure the temperature
inside the pipe

Excite the transducers using
the new driving freque nc y

Save status

Return

Return

Chapter 7: Resu lts and Analysis 66

driving frequenc y that corre sponds to the measured water temperature around the
transducers. This drivin g frequenc y , which can be used as new tr ansmission frequenc y , is
obtained from T able 6.2. Thereafter , the new driving frequenc y is stored and used to excite
the transducers (i.e., it is the return to the transmission of the sinusoidal burst sub-
algor ithm).
7.4. TTD -offset C ompensation E valuation
In order to evaluate the of fset compensation described in Chapter 6, the measurements are
carr i ed out using a 4 MHz flow meter pipe at no-flow conditions. As mentioned above, the
two received signals are sampled at 50 MS/s, around 13 samples per period. Si nce the
TTD -jitter can be reduce d b y a factor of (see Section 5.4), the TTD value is evaluated
on the basis of 250 samp les taken from th e last 20 cycles in the ste ad y-stat e region fo r both
rece iv ed si gnals. For ea ch measurement setup, the TTD is compute d seve ral times with a
repetition rate of about 1.6 s . Fig ure 7.6 shows that at 35°C when the transducers a re
excited at 4 MHz driving fr equency , the average v alue of TTD-offse t is approximatel y
−370 ps and the value o f peak - to -peak TTD -jitter is 30 ps. Th e TT D-jitter me asurement
precision, which refers t o the standard deviation, is about 6 ps. The stan dard devi ation is
calculated from TTDs of 250 captured ultrasonic waveforms. I n order to check the validity
of our TTD-of fset cancellation method, the first chosen 4 MHz frequency is chan ged to
4.153 MHz according to T able 6.2, considering t hat the tempe rature inside the measurin g
pipe is 35°C. Therefore, the measured -370 ps TTD-of fs et value is reduced to almost zero
as shown in Figure 7.7.
Since the dy namic range of the acquisition sy stem considerably influences TTD-jitt er
performance , it is n ecessar y to use controllable gain amplifiers in order to reach the
maximum amplitude of the received si gnal and therefore to cover input full -scale range
voltage (V FS R = 5V) of the ADC. However , as expected, the two voltage amplitudes
measured in the quiet r egion o f both received si gnals increase along wit h the temperature
due to the fact that the viscosit y of water decrease s as the temperature goes up. This can be
deduced from the r esults shown in Figure 7.8. This test is performed by changing the
temperature of the ther m ostat from 80°C to 35°C.

Chapter 7: Resu lts and Analysis 67

Fig. 7.6: U nco mpensated zer o flow TTD - offset measur ed at 35°C

Fig. 7.7: C o mpensated zer o flow TTD - offset measur ed at 35°C

Chapter 7: Resu lts and Analysis 68

Fig. 7.8: T wo re ceived amplitudes measur ed over a temperatur e range
fr om 80°C to 35°C
7.5. The Effect of T em peratur e on TTD -off set Measur em ents
Since the temperature is the main cause of the TTD -of fs et, additional TT D measurements
are carried out at different tempera tures (ambient temperature , 60°C, and 80°C), always
using the 4 MHz flow meter pipe. The first result is illustrated in Figure 7.9, where the
TTD measurements ar e performed continuousl y over 6 hours at ambient temperature, and
the zero flow TTD-o f fset is adjusted onl y at the b eginning b y appl y ing a driving frequenc y
of about 4.180 MHz. In this result, the deviation from z ero of the TTD -offset value is less
than 50 ps ove r the whole measurement ti me. This is due to about 3 to 6° C variation o f th e
ambient tempera tur e during the measurement period.
Under the same conditions as shown in Figure 7.9, ex perimental m easurements are
performed at dif ferent ly -regulated temp eratures. Fig u res 7.10 and 7.1 1 depict the TTD
measurements performed at 60°C an d 80° C, respectively . According to T able 6.2, the TTD -
of fset cancellation is reached by settin g the driving f requencies to 4.098 MHz at 60°C and
4.075 MHz at 80° C. As mentioned previously (see Section 7.4), the relatively high peak- to -
peak TTD -jitter of 30 ps measured at 35°C can be ex plained by the fact that the received
signals are not amplified enough to cove r the d yna mic range of the of the acquisition
system. However , the peak- to -peak TTD-jitter is reduce d to 20 ps and 15 ps when

Chapter 7: Resu lts and Analysis 69

measured at 60°C and 80° C, respectively . This is due to the fac t that the received signal
attenuation is proportional to the viscosit y of the water , which decreases with increasing
temperature as shown in Figure 7.8. The stand ard deviation (calculated fro m TTDs of 250
captured ultrasonic w aveforms) of the TTD-jitt er decreases from 6 ps at 35°C to 4 ps at
80° C.

Fig. 7.9: TTD measur ements performed over s ix hours at ambient temperatur e

Fig. 7.10: TTD measur ements performed at 60°C

Chapter 7: Resu lts and Analysis 70

Fig. 7.1 1 : TTD measur ements performed at 80°C
7.6. Discu ssion
The transducers suf fer from variations in their character istic pa rameters such as static
capacitance , electrical impedance, and resonance frequenc y th at is temperature dependent,
as demonstrated b y the ex perimental measurements shown in Section 6.3. Furthermore, th e
exact value of the transducer wo rking reson ance frequenc y depends on the input
impedance when the tran sducer acts as a transmitter and on the output impedanc e when the
transducer acts as a rec eiver . On the other hand, u nder the sam e input or output im pedance
conditions the variation of the tempe rature changes the transducer resona nce frequenc y as
well. The validity of the proposed methodolog y is checked b y evaluatin g the variation of
the transducer ’ s resonan ce frequencies and the driving frequency , which is used to adjust
the TTD-of fset with respect to the temperature changes ( Figures 6. 5 and 6.13) . These
results show that an increase in the tempe r ature leads t o a d ecrease in the re sonance
freque n c y o f the transducer and th e driving frequency at which the TTD -of fs et can be
compensated. Mor eover , the long-term stabilit y of the driving f requenc y us ed to adjust the
TTD -of fset error has bee n ex perimentally proved at four dif ferent temperat ures. As shown
in T able 6.2, the variatio n of the drivin g frequency , which is used to comp ensate the T TD-
of fset, is less than 4 Hz at the same temperature of the medium around the transducers.

Chapter 8
Conclusion

The main goal of this thesis is to improve the performance accurac y of the ultrasonic
transit time flow meters . This work also provid es a detailed description of the theoretical
development and the experimental validation of the ultrasonic s y stem and develops a new
water flow measurement approach that solves t he TTD -jitt er and TTD -offse t probl ems
raised by the ultrasonic flow meters. Th e thesis starts with a proposal for the anal y tic
modeling of the pi ezoelectric tr ansducer disk. This proposal was borro wed from damped
driven harmonic oscillat or theor y . By inv estigating the amplitude and phase responses of
the volt age across the resistive element of the transducer's electric al model, which
repre s ents the energy tra nsferred to the water , it is found that the harmonic oscillator model
is highl y suitable to effec tivel y capture the electrical behavior of t he piezoelectric
transducer disk, the most critical part of the ultrasonic water flow meter .
The numerical solution of the second order differe ntial equ ation of damped harm onic
force d s y stem includes the transient or the homogeneous solut ion and the stead y -state or
the particular solution. This steady-state can be reached onl y b y ex citing the transducer for
a suf ficientl y lon g time period. W e have deduced through these solut ions that the accuracy
of the flow meter can be further improved if we rely on the pa rticular sol uti on instead of
the homogene ous one. This stead y-state part of the si gnal is characterized by a fixed
amplitude, a phase shift, and a freque n c y that mat ches the driving frequency . Therefore, the
TTD is measured as the phase diffe r ence between the r eceived signals in both directions.
Further mo re, the phase shifts on both transducer for the received signals are estimated in
the stead y-state parts o f the signal s using th e least square sine -fitting, which filters out all
the noise relate d to the si gnal digitization carried out by the two 14-bit AD Cs.

Chapter 8: Conclu sion 72

The second development established through this work is the TTD -jitt er reduction
approac h, which consid erably enhances TTD -jitter performance. As it can be inferred from
the experimental results, the standard deviation of the TTD -jitter is reduced drastically b y
using the sine-fitting tec hnique. Furthermore, the system ac curacy is carried out one step
further b y exploring two techniques, nam ely , emplo y ing low -noise amplifiers to improve
the ac quisiti on system dy namic range and usin g an ade qu ate number of sample s in a
specific time interva l (st eady-state re gion).
Another achievement of this stud y is the extension of the TTD -jitter reduction technique to
TTD -of fset cancellation, which also degrades the ultrasonic flow meters. A novel approach
to the problem has been deve loped to cancel the long-term zero flow TTD -offse t drift
caused by temperature variations. This approach is based on the principle th at continuousl y
adjusting the driving f requency nearb y the resonance wor kin g area of the transducers
reduce s the zero-flow TTD -offset to zero. The TTD -o f fset dependence of the for ced
freque n c y finds a sim ple explanation in the theor y of the oscillator: if the f orced frequenc y
is in the range b etween the respective resonance frequencies, a slight shift of the frequenc y
produces an increase or a decrease in the amplitudes of the transmitted acoustic waves.
Choosing a correct forced frequenc y according to the temperature of the medium can
adjust the amplitude of the transmitted signals and c ompensate the TTD -of fset. The
robustness of the zero flow TTD -of fset cancellation methodolog y is experimentally
validated in harsh conditi ons, such as high t emperatures(up to 80°C).
.

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