modeling and measuring all the elements of an ultrasonic nondestructive evaluation system ii:...

Res Nondestr Eval (2002) 14: 177–201 DOI: 10.1007/s00164-002-0005-6© 2002 Springer-Verlag New York Inc. Online publication: 30 August 2002

Modeling and Measuring All the Elements of an UltrasonicNondestructive Evaluation System II: Model-BasedMeasurements

C. Dang,1 L. W. Schmerr, Jr.,2 A. Sedov3

1Panametrics, 221 Crescent Street, Waltham, MA 02453, USA2Center for NDE and Department of Aerospace Engineering and Engineering Mechanics,Iowa State University, Ames, IA 50011, USA3Department of Mechanical Engineering, Lakehead University, Thunder Bay, Ontario,Canada, P7B 5E1

Abstract. In Part I: Modeling Foundations [1], a comprehensive model of an ultrasonic non-destructive evaluation (NDE) flaw measurement system was developed. Here, it will be shownthat this comprehensive model can be used to completely characterize commercial measurementsystems where all the elements of the system can be either modeled explicitly or measured, usingpurely electrical measurements. When combined, these models and measurements are shown tobe able to predict accurately the measured signals in an ultrasonic test.

1. Introduction

In a companion paper, “Measuring and Modeling All the Elements of an UltrasonicNondestructive Evaluation System I: Modeling Foundations” [1], (hereafter referred toas Part I), it was shown how models of the acoustic/elastic wave fields present in anultrasonic nondestructive evaluation (NDE) measurement can be combined with mod-els of all the electrical and electromechanical components of a measurement system toproduce a new, comprehensive ultrasonic system model called an electroacoustic mea-surement (EAM) model [1]. The EAM model was then used in a transducer design studyto examine the effects on the transducer output waveform of changing the thickness ofthe bonding layer between the transducer crystal and a front surface facing plate [1].When modeling systems with commercial transducers, however, it is not possible touse the EAM model of Part I directly in this fashion, since many of the transducer de-sign parameters are not known. Thus, it is necessary to be able to model commercialtransducers in terms of directly accessible parameters. Here, we will reformulate theentire EAM model in terms of a reduced set of transducer parameters, impedance andsensitivity, that can be obtained experimentally through purely electrical measurementsin a standard calibration setup. Using this new reduced form of the EAM model, wewill measure and model all the components of an ultrasonic measurement system and

Correspondence to: L. W. Schmerr, Jr.

178 C. Dang et al.

show that such models and measurements can be combined to predict the system outputvoltage accurately. Some important future applications of the reduced EAM model willalso be discussed.

2. The EAM Model

Consider the ultrasonic NDE immersion flaw measurement system shown in Fig. 1. TheEAM model for this system derived in Part I expressed the frequency components ofthe measured output voltage, VR(ω), in terms of three transfer functions [tA(ω), tG(ω),

tR(ω)] and an equivalent driving voltage, Vi (ω), in the form

VR(ω) = tR(ω)tA(ω)tG(ω)Vi (ω). (1)

The transfer function tG(ω) contains the electrical characteristics of the pulser, the proper-ties of the cabling between the pulser and the transmitting transducer, and the transmittingtransducer properties itself. A model of each of those elements in the sound generationprocess is shown in Fig. 2 [1]. The pulser is represented by the Thevenin equivalent volt-age source, Vi (ω), which is the driving voltage appearing explicitly in Eq. (1), and anequivalent pulser electrical impedance, Ze

i (ω). The cabling is modeled as a transmissionline that is characterized by a 2 × 2 transfer matrix, [T ], as is the transmitting transducerA in terms of a 2 × 2 transducer transfer matrix, [T A]. The radiation of sound into thesurrounding fluid in this model is accounted for by terminating the acoustical outputport of transducer A with an acoustic radiation impedance, Z A;a

r (ω), as shown in Fig. 2.If the transducer is modeled as a piston (i.e., constant velocity) source of ultrasound,then at the high frequencies of ultrasonic NDE systems the acoustic radiation impedanceis the constant Z A;a

r = ρ f cf SA, where ρ f is the density of the surrounding fluid, cf isthe compressional wave speed of the fluid, and SA is the area of the transducer outputport [1]. As shown in Part I, all these elements of the sound generation process can be

flaw

Transducer(transmitter)

cablingcabling

Transducer(receiving)

Pulser Receiver

flaw signalV(t)

Fig. 1. Components of an ultrasonic immersion NDE measurement systemfor testing flawed parts.

Modeling and Measuring an Ultrasonic NDE System II 179

Zi

transducer

cabling

[T] Ft[TA]

vte

pulser

ZrA;a

Ft[TA]

vt

Z r

A;a

(a)

(b)

I1

V1

Vi

V

I

Fig. 2. (a) Combined models of the pulser, cabling, andtransducer used in the generation of ultrasound, and (b) theinputs and outputs of the transmitting transducer, showingthe acoustical radiation impedance present at the transduceroutput port.

combined in the form of a transfer function, tG(ω), given by

tG(ω) = Ft

Vi= Z A;a

r

(Z A;ar T G

11 + T G12) + (Z A;a

r T G21 + T G

22)Zei

, (2)

where the transmitting cable transfer matrix, [T ], and transducer transfer matrix, [T A],have been combined to form up a “global” 2 × 2 transfer matrix,

[T G] =[

T G11 T G

12

T G21 T G

22

]=[

T11 T12

T21 T22

][T A

11 T A12

T A21 T A

22

]. (3)

As indicated in Eq. (2), this transfer function is the ratio of the force output of thetransducer at its acoustical port, Ft , to the Thevenin equivalent pulser driving voltage, Vi .

The transfer function representing the components involved in the reception of ultra-sound (Fig. 3) can be similarly obtained in the form [1]

tR(ω) = VR

FBLK= Ze

o(ω)K (ω)

(Z B;ar RG

11 + RG12) + (RG

21 Z B;ar + RG

22)Zeo(ω)

, (4)

where Ze0(ω) is the electrical impedance of the receiver and K (ω) a receiver voltage

amplification factor (see Fig. 3a). The acoustic radiation impedance of the receivingtransducer B is Z B;a

r . Again, the 2 × 2 transfer matrix, [R], representing the cablepresent between the transducer and the receiver, as well as the 2 × 2 transfer matrix,

180 C. Dang et al.

Fig. 3. (a) Combined models of the receiving transducer, cabling, and receiver cir-cuits present in the reception of ultrasound, and (b) the inputs and outputs of the receiv-ing transducer, showing the blocked force source and acoustical radiation impedancepresent at the transducer input port.

[T B], for the receiving transducer B have been combined together in Eq. (4) in terms ofa single 2 × 2 “global” transfer matrix, [RG], i.e.

[RG] =[

RG22 RG

12

RG21 RG

11

]=[

T B22 T B

12

T B21 T B

11

][R22 R12

R21 R11

]. (5)

This reception transfer function is defined as the ratio of the receiver output voltage,VR , to the “blocked” force, FBLK, at the face of the receiving transducer, where theblocked force is the force present at the receiving transducer produced by the incidentand scattered waves when the face of the transducer is held rigidly fixed [1].

Finally, the transfer function, tA(ω) = FBLK/Ft , represents all the acoustic and elasticwave propagation and scattering processes present in the measurement system. In Part I,a general reciprocity relationship was used to obtain a model of this transfer functionin terms of the fields present at a flaw for the flaw measurement system of Fig. 1.Similarly, an explicit model was obtained in Part I for the two-transducer calibrationsetup shown in Fig. 4. Thus, unlike the generation and reception transfer functions,which involve many components whose properties must be obtained from experimentalmeasurements, the transfer function tA(ω) can be found directly from acoustic and elasticwave propagation/scattering models.

By modeling tA(ω) and measuring Vi (ω) and all the unknown system component pa-rameters contained in tG(ω) and tR(ω), all of these elements can in principle be combined,through the use of Eq. (1), to obtain the measured voltage response. Equations (2)–(5)


Fig. 4. A calibration configuration consisting of two circular, immersion piston trans-ducers in a pitch–catch setup in which the center axes of the two transducers are alignedin the fluid.

show that a complete characterization requires measurements of the Thevenin equiva-lent voltage source, Vi (ω), and electrical impedance, Ze

i (ω), of the pulser, the receiver’selectrical impedance, Ze

0(ω), and amplification factor, K (ω), the eight transfer matrixcomponents of the cabling transfer matrices, [T ], [R], and the eight transfer matrix com-ponents of the transmitting and receiving transducer transfer matrices, [T A], [T B]. All ofthe pulser, receiver, and cabling parameters can be obtained through standard electricalmeasurement procedures. However, to date no practical measurement procedures areavailable to obtain the transducer transfer matrix elements of commercial transducersin which the underlying transducer properties (crystal properties, backing, etc.) are un-known. Fortunately, as shown in the next section, these transfer functions can be writtenin terms of a different set of transducer parameters that can be obtained by purely electri-cal measurements in standard calibration setups, making the complete characterizationof commercial ultrasonic systems possible.

3. The “Reduced” EAM Model

Consider first the transfer function for the sound generation process [Eq. (2)]. As shownby Eq. (3), the global transfer matrix appearing in this transfer function is just the productof the cabling and transducer transfer matrices. Thus, we have (see Fig. 2a)

VI

=[

T G11 T G

12

T G21 T G

22

]Ft

νt

=[

T11 T12

T21 T22

][T A

11 T A12

T A21 T A

22

]Ft

νt

. (6)

However, at the output port the force and velocity are related through the acousticradiation impedance, i.e., Ft = Z A;a

r νt , so that we find

[T G

11 T G12

T G21 T G

22

]Z A;a

r νt

νt

=[

T11 T12

T21 T22

][T A

11 T A12

T A21 T A

22

]Z A;a

r νt

νt

(7)

182 C. Dang et al.

or, equivalently, by canceling out the common velocity term, νt , on both sides,(Z A;a

r T G11 + T G

12)

(Z A;ar T G

21 + T G22)

=[

T11 T12

T21 T22

](Z A;a

r T A11 + T A

12)

(Z A;ar T A

21 + T A22)

. (8)

The terms appearing on the left side of Eq. (8) are just those terms appearing in theexpression for the generation process transfer function [Eq. (2)]. Now, we will show thatthe terms involving the transducer on the right side of Eq. (8) can be written in termsof only two transducer parameters—the transducer’s electrical input impedance and itsopen-circuit, blocked force receiving sensitivity, both of which will be defined shortly.

First, consider the transmitting transducer only (Fig. 2b). We haveV1

I1

=[

T A11 T A

12

T A21 T A

22

]Z A;a

r νt

νt

, (9)

so that the transducer’s electrical input impedance, Z A;ein = V1/I1, is given by

Z A;ein (ω) = Z A;a

r T A11 + T A

12

Z A;ar T A

21 + T A22

. (10)

Next, consider the same transducer acting as a receiver. We can analyze this transducerusing Figs. 3a and b if we simply make the replacements Z B;a

r → Z A;ar and [T B] → [T A]

in those figures. As shown in Part I, the equivalent source term for this receiver is the“blocked force,” FBLK(ω), acting on the face of the transducer, where, as mentionedpreviously, this blocked force is the force generated by the incident and scattered acousticwaves when the face of the transducer is held fixed [1]. Similarly, in Part I it was shownthat the equivalent acoustic impedance acting at the input of the receiver is just theacoustic radiation impedance, Z A;a

r , of the transducer when it acts as a transmitter [1].If, as shown in Fig. 3b, we have an open-circuit condition at the transducer electricalport (I = 0, V = V ∞), then in terms of the open-circuit voltage, V ∞, we find

Fi = T A22V ∞,

νi = T A21V ∞.

(11)

However, the blocked force is given by FBLK = Z A;ar νi + Fi , so that from Eq. (11) we

find

FBLK = (Z A;ar T A

21 + T A22)V ∞. (12)

Thus, for the transducer open-circuit, blocked force receiving sensitivity, M A;∞VFBLK

, defined

as M A;∞VFBLK

= V ∞/FBLK, we obtain

M A;∞VFBLK

(ω) = 1

(Z A;ar T A

21 + T A22)

. (13)

From Eqs. (8), (10), and (13), therefore,(Z A;a

r T G11 + T G

12)

(Z A;ar T G

21 + T G22)

=[

T11 T12

T21 T22

]Z A;e

in /M A;∞VFBLK

1/M A;∞VFBLK

, (14)


which, when placed into the expression for the generation process transfer function[Eq. (2)], gives

tG(ω) = Z A;ar M A;∞

VFBLK

(Z A;ein T11 + T12) + (Z A;e

in T21 + T22)Zei

. (15)

Equation (15) shows that the generation process transfer function depends on the trans-mitting transducer properties only through the transducer’s electrical input impedanceand open-circuit, blocked force receiving sensitivity. This is an important result, sincewe have shown that both this input impedance and sensitivity can be obtained by purelyelectrical measurements in a set of standard calibration setups [2]. As mentioned previ-ously, all the other elements in Eq. (15) can also be obtained by well-known electricalmeasurement procedures, so Eq. (15) can be used to determine this transfer functioncompletely.

An examination of the reception process transfer function [Eq. (4)] shows that theglobal receiving matrix terms appear in the same combinations as for the sound generationprocess. Thus, following exactly the same steps as just outlined, this transfer functioncan also be written in terms of the receiving transducer input impedance, Z B;e

in , andopen-circuit, blocked force receiving sensitivity, M B;∞

VFBLK. The result is

tR(ω) = Ze0 K M B;∞

VFBLK

(Z B;ein R11 + R12) + (Z B;e

in R21 + R22)Ze0

. (16)

Equations (15) and (16), when used in Eq. (1), give us a “reduced” EAM model interms of directly measurable transducer (and other) parameters. In the next section wewill discuss the experimental measurements of all of these system parameters for thecalibration setup shown in Fig. 4.

4. Complete Characterization of an Ultrasonic Measurement System

Table 1 lists all the system components appearing in the reduced EAM model and thenumber of voltage and current measurements required to obtain those component pa-rameters. As can be seen from that table, there are a total of 25 voltage and currentmeasurements needed to fully characterize the entire ultrasonic system. To facilitatemaking such a large number of measurements, a special voltage–current probe was con-structed [2]. Using this voltage–current probe and other standard voltage measurements,Dang et al. [2] described procedures for obtaining the cabling transfer matrix elementsand the transducer input impedance and open-circuit, blocked force receiving sensitivityfor commercial transducers. The transducer input impedance was measured directly withthe voltage–current probe and the results compared with two other methods [2] (one thatrequired several voltage measurements and another that employed an impedance ana-lyzer). In general, all three impedance measurement procedures gave consistent resultsthat were in agreement, but the voltage–current probe was the simplest to implement.The sensitivity measurement used purely electrical measurements and a three-transducercalibration procedure originally developed for lower-frequency acoustic transducers [2].

184 C. Dang et al.

Table 1. Model parameters that describe an ultrasonic measurement system and the mea-surements required to characterize those parameters experimentally.

Number and type ofComponent Model parameters independent measurements

Pulser a. Equivalent voltage source a. 1 voltageb. Electrical impedance b. 1 voltage

Cabling (on transmission) Four cable transfer matrix 2 voltage, 2 currentcomponents

Transmitting transducer a. Input impedance a. 1 voltage, 1 currentb. Open-circuit, blocked b. 3 voltage, 1 currentforce receiving sensitivity

Receiving transducer a. Input impedance a. 1 voltage, 1 currentb. Open-circuit, blocked b. 3 voltage, 1 currentforce receiving sensitivity

Cabling (on reception) Four cable transfer matrix 2 voltage, 2 currentcomponents

Receiver a. Amplification factor a. 2 voltageb. Electrical impedance b. 1 current

However, it was shown that at ultrasonic NDE frequencies, frequency-dependent cablingeffects are important and cannot be neglected, requiring that the cabling transfer matrixpresent in the sensitivity calibration setup also be determined. Again, the voltage–currentprobe was frequently used in making those measurements. Since the details of obtain-ing transducer and cabling parameters are described in [2], here we will discuss themeasurement of the remaining pulser and receiver parameters appearing in Table 1.

Pulser Characterization

The pulser is represented in the EAM model by an equivalent voltage source, Vi (ω),and an equivalent electrical impedance, Ze

i (ω) (see Fig. 2a). In [3] these pulser param-eters were obtained experimentally for a Panametrics 5052PR pulser/receiver, whichcontains a broad-band “spike” pulser. A typical measured output voltage pulse of thispulser/receiver is shown in Fig. 5. On the front panel of the 5052PR are four energy-level settings, ranging from 1 to 4, and 11 damping settings, from 0 to 10. The Theveninequivalent source and impedance of the pulser are functions of both these settings. Wechose to model the pulser at all four energy settings and three damping settings of 0, 5,and 10, respectively. Thus, the pulser source strength and impedance were obtained fora total of 12 combinations of settings.

At each setting, we first measured the open-circuit output voltage of the pulser, V (t) =V ∞(t), which theoretically is equal to the source strength, Vi (t), of the pulser. A knownload (Fig. 6), Ze

L , was then connected to the output port of the pulser and the voltage drop,V (t) = VL(t) was measured. All of the voltage measurements were taken with a wide-band sampling oscilloscope (LeCroy Waverunner, LT 342 series) having a 500-MHzsampling frequency. Taking the fast Fourier transform (FFT) of both V ∞(t) ≡ Vi (t)and VL(t) then gave the frequency-domain parameters, Vi (ω) and VL(ω). From Fig. 6 it


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-200

-150

-100

-50

0

50

Time (µs)

Am

plitu

de (

V)

Fig. 5. A typical output voltage measured for a Panametrics5052PR pulser/receiver.

follows that

VL(ω) = ZeL

ZeL + Ze

i (ω)Vi (ω), (17)

which can be solved for Zei (ω) as

Zei (ω) =

[Vi (ω)

VL(ω)− 1

]Ze

L . (18)

Actually, we used three different external impedances to measure the internal electricalimpedance. The external impedances were a 50- terminator, an 82- resistor, and a 220- resistor, respectively. The reason for choosing these different external impedances wasto study whether the internal impedance of the Thevenin equivalent circuit is dependenton the external impedance. Theoretically, the Thevenin equivalent circuit of a pulser

Fig. 6. Representation of the pulser as a Theveninequivalent voltage source and electrical impedance,where the pulser is terminated by an electricalimpedance, Ze

L .

186 C. Dang et al.

0 5 10 15 200

5

10

15

0 5 10 15 200

5

10

15

0 5 10 15 200.5

1

1.5

2

2.5

Frequency (MHz)

0 5 10 15 200

20

40

60

80

0 5 10 15 2010

20

30

40

50

0 5 10 15 206

7

8

9

10

Frequency(MHz)

(a)

Vol

tage

(µV

/Hz)

Impe

danc

e (Ω

)

(b)

(c)

Fig. 7. Measured properties of the pulser section of thePanametrics 5052PR pulser/receiver. The left column showsthe magnitude of the equivalent pulser source term versusfrequency and the right column shows the magnitude ofthe equivalent pulser electrical impedance versus frequency.Source and impedance measurements for: (a) energy setting 2and damping setting 0; (b) energy setting 2 and damping set-ting 5; (c) energy setting 2 and damping setting 10. Impedancemeasurements were taken at three different terminating condi-tions: a 50- cable terminator, an 82- resistor, and a 220-resistor.

should not depend on ZeL if the pulser acts as a completely linear device. However, as

other studies have shown [4, 5], the nonlinear internal diode protection circuits in thepulser can make such linearity assumptions questionable.

Figures 7 and 8 show the measured Thevenin equivalent sources and impedances forenergy settings 2 and 4 (and damping settings of 0, 5, and 10). Dang, in his thesis [3],also gave the corresponding results for energy settings 1 and 3, which behave similarly tothose shown. The left column in each figure shows the magnitude of source strength andright column shows the corresponding magnitude of the internal electrical impedancesmeasured by using the three external impedances. From these figures, the followingconclusions can be drawn.

1. The internal electrical impedance is a function of frequency. This behavior is notconsidered in many modeling studies, which either ignore the pulser impedanceentirely and simply drive the transducer directly with a specified voltage pulse, ortreat this impedance as a pure resistance.

2. For specific energy and damping settings, the internal impedance is dependentsomewhat on the external impedance.


0 5 10 15 200

10

20

30

40

0 5 10 15 200

20

40

60

80

0 5 10 15 200

10

20

30

0 5 10 15 200

10

20

30

40

50

0 5 10 15 200

2

4

6

0 5 10 15 204

6

8

10

12

Vol

tage

(µV

/Hz)

Impe

danc

e (Ω

)

Frequency (MHz) Frequency (MHz)

(a)

(b)

(c)

Fig. 8. Measured properties of the pulser section of the Panamet-rics 5052PR pulser/receiver. The left column shows the magni-tude of the equivalent pulser source term versus frequency and theright column shows the magnitude of the equivalent pulser electri-cal impedance versus frequency. Source and impedance measure-ments for: (a) energy setting 4 and damping setting 0; (b) energysetting 4 and damping setting 5; (c) energy setting 4 and dampingsetting 10. Impedance measurements were taken at three differentterminating conditions: a 50- cable terminator, an 82- resistor,and a 220- resistor.

3. At a given energy level, increasing the damping lowers the internal impedance andthe equivalent source strength.

4. A higher damping gives a wider bandwidth for the source strength.5. At a given damping setting, a higher energy level gives a higher source strength.

The internal impedance also changes with energy level.

Receiver Characterization

As shown in Fig. 3a, the receiver is modeled in the EAM model by its equivalentreceiving load, Ze

0(ω), and its amplification factor, K (ω). For the Panametrics 5052PRpulser/receiver, these parameters are affected by gain, attenuation (Attn), and dampingsettings on the front panel. The gain control is a coarse amplification factor control. Ithas two positions, 20 dB and 40 dB. The Attn control is a finer control that changes theattenuation of the received signals from 0 to 60 dB with a step of 2 dB. The combinationof these two controls gives many possible amplification factor settings for the receiver.

188 C. Dang et al.

Fig. 9. Measurement setup used to obtain the equivalent electri-cal impedance and amplification factor for the receiver section of aPanametrics 5052PR pulser/receiver.

The third control is the damping, which, as mentioned previously, has 11 positions from0 to 10. In a pulse-echo mode experiment, the damping controls the receiving load. In apitch–catch mode experiment, which is the type shown in Fig. 1, the circuit diagrams ofthe 5052PR indicate that it should not affect the receiving load. There is also a frequencyfilter control on the front panel of the 5052PR pulser/receiver, but this was left “off” inall of the following studies.

Since the pulser itself generates a wide-band response, it is not difficult to obtainthe pulser characteristics over a wide range of frequencies. However, if the receivercharacteristics are obtained in a given measurement setup, such as shown in Fig. 9, wherea receiving transducer is excited by incident acoustic waves, the receiver characteristicscan only be obtained over the bandwidth of that receiving transducer. Thus, to examinethe receiver parameters over a broad frequency spectrum, it is necessary to use multipletransducers with different frequencies. In [3], receiver characteristics were obtained forthe 5052PR pulser/receiver using 2.25-, 5-, and 10-MHz transducers. Here, however,we will discuss only the results using the 2.25-MHz transducer, since the other casesdemonstrated similar behavior.

The measurement setup is shown in Fig. 9, where the measured quantities are thevoltage, V0(t), and current, I0(t), at the receiver input and the output voltage, VR(t), ofthe receiver. The voltage, V0(t), and current, I0(t), were measured using the voltage–current probe, and the sampling oscilloscope was used for characterizing the pulser. Theamplitude of the current, I0(t), is usually small, so a wide-band amplifier was used toamplify that signal before it was sampled. To guarantee that V0(t) and I0(t) were theactual input voltage and current imposed on the receiver, the voltage–current probe waslocated directly at the receiver input terminal. The output voltage, VR(t), was also thenmeasured with the sampling oscilloscope.

After the signals Vo(t), I0(t), and VR(t) were sampled and stored, the FFT wastaken to obtain their corresponding frequency-domain values, V0(ω), I0(ω), and VR(ω),respectively. Then the amplification factor was obtained from

K (ω) = VR(ω)

V0(ω)(19)


0 0.5 1 1.5 2 2.5 3 3.5 40

200

400

600

800

1000

1200

-150

-100

-50

0

50

100

Frequency (MHz)

Am

plitu

de (

Ω)

Phas

e (D

eg)

amplitude

phase

Fig. 10. Measured complex electrical impedance (amplitude and phase) ver-sus frequency for the receiver section of a Panametrics 5052PR pulser/receiver,using a 2.25-MHz transducer in the measurement setup of Fig. 9. Gain setting20 dB, Attn setting 0.

and the receiver impedance from

Ze0(ω) = V0(ω)

I0(ω). (20)

Figures 10 and 11 show the receiver characteristics obtained with a 2.25-MHz receivingtransducer, operating in a pitch–catch mode. The gain for this case was set at 20 dB andthe Attn at 0 dB. The damping setting, which should not affect the receiver characteristicsin this pitch–catch setup, was set at 0. In general, these results and the results similarlyobtained with 5- and 10-MHz transducers (see [3]) showed that:

1. The receiver impedance exhibited very little structure in either its magnitude orphase. In general, its magnitude varied from 500 at 1 MHz to 1000 at 12MHz.

2. Similarly, there was very little variation in the magnitude and phase of the amplifi-cation factor as a function of frequency. The magnitude of this measured effectiveamplification ranged from 19 to 23 dB over a frequency range of 0–20 MHz, whichis very close to the desired 20-dB gain setting.

Ultrasonic System Characterization

By combining the pulser/receiver measurements described above with the cabling andtransducer measurements detailed in [2] and [3] and an explicit model for the acoustictransfer function, tA(ω), we can completely characterize the entire ultrasonic measure-ment process. The specific ultrasonic immersion measurement system we will charac-

190 C. Dang et al.

0 0.5 1 1.5 2 2.5 3 3.5 48

9

10

11

12

13

14

15

Am

plitu

de

Frequency (MHz)

-40

-20

0

20

40

60

80

100

120

140

Pha

se (

Deg

)

amplitude

phase

Fig. 11. Measured complex amplification factor (amplitude and phase)versus frequency for the receiver section of a Panametrics 5052PRpulser/receiver, using a 2.25-MHz transducer in the measurement setupof Fig. 9. Gain setting 20 dB, Attn setting 0.

terize is the two-transducer calibration setup shown in Fig. 4. The system has sevencomponents: a commercial pulser, cabling from the pulser to the transducer, a commer-cial transmitting transducer (transducer A), an acoustic transmission/reception config-uration, a commercial receiving transducer (transducer B), cabling from the receivingtransducer to the receiver, and a receiving termination, Ze

L . As seen in Fig. 4, the acousticconfiguration is simply the two transducers aligned along their axes and separated bya distance, D, in a water bath. At the receiving end, we used two known terminations(50- resistor and open-circuit) in place of the receiver section of the pulser/receiver.This was done in order to demonstrate the validity of our approach under very differentreceiving conditions. As just shown in the previous subsection, the commercial receivermeasured there acted to first order like a constant (in frequency) receiving impedanceand gain factor, so that it could only provide a single receiving termination condition.

The commercial pulser used was again a Panametrics 5052PR. All of the measure-ments were taken with the energy level of the pulser set to 2 and the damping set to 5.Following the procedures outlined above, both the source strength, Vi (ω), and equivalentimpedance, Ze

i (ω), of the pulser were measured. Since, as just shown, the impedanceobtained does depend somewhat on the electrical load connected to the pulser, the mea-surements of Ze

i (ω) were made here with the transmitting cable connected to the pulserand with the other end of the cable connected to the transducer. Figures 12 and 13 showthe measured source strength Vi (ω) and its internal electrical impedance Ze

i (ω). Bothof these measurements are consistent with what one would expect from the capacitivedischarge of a “spike” pulser.

The transmitting cabling was a single flexible cable with the length of 5.5 ft (1.68 m)connected directly between the pulser and the transmitting transducer as shown in Fig. 4.This cable can be described by a 2 × 2 transfer matrix T whose four elements, T11, T12,


0 2 4 6 8 10 12 14 16 18 201

2

3

4

5

6

7

8

9

10

11

Frequency (MHz)

0 2 4 6 8 10 12 14 16 18 20-200

-100

0

100

200

300

400

500

600

700

Am

plitu

de (

µV/H

z)

Phas

e (D

eg)

amplitude

phase

Fig. 12. Pulser source strength (amplitude and phase) measured at an energysetting of 5 and a damping setting of 2.

T21, and T22, are functions of frequency. Using two voltage and two current measurementsunder different termination conditions, as described in detail in [2] and [3], these fourelements of the transmitting cabling matrix were obtained. They are shown in Fig. 14,where the left column gives the amplitudes of the elements and the right column theirphases. The behavior of these elements is in general in agreement with what one wouldexpect from a simple transmission line model [3].

The transmitting transducer, transducer A in Fig. 4, was a commercial transducer(Panametrics V310, serial no. 257916), having a nominal frequency of 5 MHz and a di-ameter of 0.25 in. (6.35 mm). Assuming this transducer can be modeled as a planar pistontransducer, as shown previously it can be completely described by its input electrical

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-20

0

20

40

60

80

Frequency (MHz)

Phas

e (D

eg)

Am

plitu

de (

Ω)

amplitude

phase

Fig. 13. Pulser electrical impedance (amplitude and phase) measured atan energy setting of 5 and a damping setting of 2, with the pulser terminatedby the cabling and transducer used in this ultrasonic system characterizationstudy.

192 C. Dang et al.

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1

1.5

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4

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0.015

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1

1.5

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0

5

10


Amplitude Phase

T11

T12

(Ω

)T

21 (

1/Ω

)T

22

(Deg

)(D

eg)

(Deg

)(D

eg)

Fig. 14. Measurements of the four transfer matrix compo-nents for a 1.68-m flexible cable used on the transmitter side ofthe ultrasonic setup shown in Fig. 4. Left column, magnitudeversus frequency; right column, phase versus frequency.

impedance, Zein(ω), and its open-circuit, blocked force receiving sensitivity, M∞

VFB(ω).

The impedance, Z A;ein (ω), was obtained by measuring both the voltage and current at the

transducer electrical port with the voltage–current probe (the “direct method” in [2]). Theopen-circuit sensitivity M A;∞

VFB(ω) of transducer A was obtained using reciprocity-based

relations and a three-transducer calibration procedure [2, 3], requiring a total of threevoltage measurements and one current measurement, as shown in Table 1. Those mea-surements were all performed in the same configuration shown in Fig. 4, using differentcombinations of sending and receiving transducers [2, 3]. Results of those sensitivity andimpedance measurements are shown in Figs. 15 and 16. The amplitude of the sensitivity(Fig. 15) had a peak near the nominal center frequency of the transducer (5 MHz) butalso had a secondary peak at approximately 12 MHz. The electrical impedance of thetransducer (Fig. 16) showed a very capacitive-like behavior, which is to be expectedsince a piezoelectric crystal transducer to first order does behave like a capacitor. Thismay not be true, of course, if the commercial transducer contains additional “tuning”elements in its construction.

These same measurement procedures were used to also determine the open-circuit,blocked force receiving sensitivity and input electrical impedance of the receiving trans-ducer, B, in Fig. 4 (Panametrics V310, serial no. 184577), which had the same nominalfrequency and size as transducer A, i.e., 5-MHz center frequency and 0.25-in. (6.35-mm) diameter. The sensitivity, M B;∞

VFB(ω), and input electrical impedance, Z B;e

in (ω), of


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0.25

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0.35

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0

200

400

600

800

1000

Frequency (MHz)

Phas

e (D

eg)

Am

plitu

de (

V/N

)

amplitude

phase

Fig. 15. Measured open-circuit, blocked force receiving sensitivity (am-plitude and phase) versus frequency for a 5-MHz, 6.35-mm-diameter com-mercial transducer used as the transmitter in the setup of Fig. 4.

this transducer are shown in Figs. 17 and 18, respectively. Comparing Figs. 16 and 18,we see that the two transducers have very similar input impedances. Their open-circuit,blocked force sensitivities also are similar (Figs. 15 and 17) but do show some differ-ences in their detailed behavior. We expect that the two transducers should be similarsince they came from the same family (Panametrics V310 series) of probes.

The receiving cabling in the measurement system consisted of two components: aflexible cable and a fixture rod. The fixture rod had a length of 2.5 ft. (0.77 m) and wasused to hold the receiving transducer. The flexible cable had a length of 5 ft. (1.52 m) andwas connected to the fixture rod at one end and the terminating impedance, Ze

L , at the

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1500

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100

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120

130

140

150

Frequency (MHz)

Phas

e (D

eg)

Am

plitu

de (

Ω)

amplitude

phase

Fig. 16. Measured electrical input impedance (amplitude and phase) versusfrequency for a 5-MHz, 6.35-mm diameter commercial transducer used asthe transmitter in the setup of Fig. 4.

194 C. Dang et al.

0 5 10 15 200

0.05

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0.25

0.3

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0

100

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400

500

600

Frequency (MHz)

Phas

e (D

eg)

Am

plitu

de (

V/N

)

amplitude

phase

Fig. 17. Measured open-circuit, blocked force receiving sensitivity (ampli-tude and phase) versus frequency for a 5-MHz, 6.35-mm diameter commercialtransducer used as the receiving transducer in the setup of Fig. 4.

other (Fig. 4). Similar to the transmitting cabling, the receiving cabling was characterizedcompletely by four transfer matrix elements (R11, R12, R21, and R22) that were obtainedwith two voltage and two current measurements. The results are shown in Fig. 19. Theleft column in that figure gives the transfer matrix amplitudes and the right column theirphases. These results show more clearly than Fig. 14 (for the transmitting cable) the sineand cosine behavior (and associated phases) expected from a transmission line cablemodel. This is because the receiving cabling used here was one and a half times longerthan the transmitting cabling.

As Eq. (15) shows, knowledge of the pulser impedance, Zei (ω), the transmitting

transducer input impedance, Z A;ein (ω), open-circuit, blocked force receiving sensitivity,

0 5 10 15 200

500

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1500

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90

100

110

120

130

140

150

Frequency (MHz)

Phas

e (D

eg)

Am

plitu

de (

Ω)

amplitudephase

Fig. 18. Measured electrical input impedance (amplitude and phase) versusfrequency for a 5-MHz, 6.35-mm diameter commercial transducer used as thereceiving transducer in the setup of Fig. 4.


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0.5

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1.5

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0

100

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100

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0

100


Amplitude Phase

(Deg

)(D

eg)

(Deg

)(D

eg)

R22

R21

(1/

Ω)

R12

(Ω

)R

11

Fig. 19. Measurements of the four transfer matrix compo-nents for the 0.77-m fixture rod and 1.52-m cable used on thereceiving side of the ultrasonic setup shown in Fig. 4. Leftcolumn, magnitude versus frequency; right column, phaseversus frequency.

M A;∞VFB

(ω), and radiation impedance, Z A;ar , coupled with knowledge of the four cabling

transfer matrix elements [T11(ω), T12(ω), T21(ω), T22(ω)] are sufficient to completelydetermine the generation transfer function, tG(ω). All of these quantities were obtainedexperimentally, with the exception of Z A;a

r . The transducer radiation impedance wasfound directly from the expression Z A;a

r = ρ f cf SA, a result that is valid at megahertzfrequencies for a piston transducer [1]. In the present case ρ f = 1,000 kgm/m3 (thedensity of the surrounding water), cf = 1,480 m/s (the wave speed of water) andSA = πa2 (with a = 3.175×10−3 m). Combining all these parameters through Eq. (15),we obtain the measured generation transfer function shown in Fig. 20.

On the receiving side of the system shown in Fig. 4 there is no amplification present(K = 1) and the impedance of the receiver is now just the termination impedance(Ze

0 = ZeL), so Eq. (16) becomes

tR(ω) = ZeL M B;∞

VFBLK

(Z B;ein R11 + R12) + (Z B;e

in R21 + R22)ZeL

. (21)

This reception transfer function is determined from the measurements of M B;∞VFB

(ω),

Z B;ein (ω), and (R11, R12, R21, and R22) once the terminating impedance, Ze

L , is specified.Figure 21 shows the results of combining all these measurements for the case Ze

L = 50 .

196 C. Dang et al.

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0.02

0.04

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0.1

0.12

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400

600

800

1000

1200

1400

Frequency (MHz)

Phas

e (D

eg)

Am

plitu

de (

N/V

)

amplitude

phase

Fig. 20. Generation transfer function (amplitude and phase) versus fre-quency obtained from Eq. (15) and measurements of all the componentspresent for the ultrasonic setup of Fig. 4.

Comparing Figs. 20 and 21, it is obvious that the general characteristics of these twotransfer functions are similar in terms of peaks around the transducer center frequen-cies, etc., but that many of their detailed characteristics are also quite different. This isto be expected since, while the transmitting and receiving transducers do share simi-lar behaviors (recall Figs. 15–18), they are operating under different electrical loadingconditions.

Unlike the generation and reception transfer functions, the acoustic transfer function,tA(ω), can be obtained in an explicit fashion from models. For the testing configuration

0 5 10 15 200

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900

1000

Frequency (MHz)

Phas

e (D

eg)

Am

plitu

de (

V/N

)

amplitude

phase

Fig. 21. Reception transfer function (amplitude and phase) versus fre-quency obtained from Eq. (21) and measurements of all the compo-nents present for the ultrasonic setup of Fig. 4. Terminating impedanceZe

L = 50 .


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Frequency (MHz)

Phas

e (D

eg)

Am

plitu

de

amplitude

phase

Fig. 22. Acoustic transfer function (amplitude and phase) versus fre-quency for the setup of Fig. 4, where two 6.35-mm-diameter circularpiston transducers are separated by a distance of D = 0.444 m in awater bath.

of Fig. 4, for example, where two piston transducers of the same radius (a = b) arealigned along their axes, the effects of attenuation of the water can be combined with aloss-free model [6] to give

tA(ω) = 2 exp(ikD) exp[−α(ω)D]

×

1 − exp

(ika2

D

)[J0

(ka2

D

)− iJ1

(ka2

D

)](22)

where, for the specific system under consideration, a = 3.175 × 10−3 m is the radius ofeach transducer, k = ω/cf is the wave number, D = 0.444 m is the separation distancebetween transducers, and the water attenuation coefficient α(ω) = 25.3 × 10−15 f 2

Np/m with f = ω/2π the frequency (in hertz) [6]. Figure 22 shows a plot of this transferfunction versus frequency. This function behaves like a band-limited filter because of thefrequency-dependent attenuation of the water and the beam-spreading effects in Eq. (22),yielding a peak in the transfer function at a frequency of about 6.8 MHz.

If the generation and reception transfer functions are combined with the acoustictransfer function and the pulser source term according to Eq. (1), then the frequencyspectrum of the received signals, VR(ω), can be obtained. This spectrum can be invertedinto the time domain with an inverse FFT to yield the received voltage signal VR(t),which can then be compared with the actual measured signal. Figure 23 shows theseresults when the terminating impedance Ze

L = 50 . The signal synthesized fromEq. (1) and the measured waveform are both very close, with the synthesized signalbeing of slightly higher amplitude. If the terminating impedance instead is taken asZe

L = ∞ (open-circuit condition), the synthesized and measured waveforms are as shownin Fig. 24. Although the open-circuit responses are quite different from the previous case,again both synthesized and measured responses are quite close, with the synthesizedresponse slightly larger in amplitude than the measured response. Both cases demonstrate

198 C. Dang et al.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.15

-0.1

-0.05

0

0.05

0.1

Time (µs)

Am

plitu

de (

V)

Fig. 23. The synthesized and directly measured output volt-age versus time for the setup of Fig. 4. Terminating impedanceZe

L = 50 . The synthesized voltage curve is slightly higherin amplitude than the measured voltage.

remarkably good agreement given the complexity of the overall system and the largenumber of measurements required.

Another representation of the total system response can be obtained by combiningthe generation and reception transfer functions together with the pulser source term intoa single system factor, s(ω), where (see [1, 3])

s(ω) = tR(ω)tG(ω)Vi (ω). (23)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Time (µs)

Am

plitu

de (

V)

Fig. 24. The synthesized and directly measured output voltageversus time for the setup of Fig. 4 under open-circuit terminationconditions. The synthesized voltage curve is slightly higher inamplitude than the measured voltage.


0 2 4 6 8 10 12 14 16 18 200

2

4

6

8x 10-8

0 2 4 6 8 10 12 14 16 18 20-1000

0

1000

2000

3000

Frequency (MHz)

Frequency (MHz)

Phas

e (D

eg)

Am

plitu

de (

V/H

z)

synthesized

measured

synthesized

measured

Fig. 25. The synthesized and directly measured system factor (am-plitude and phase) versus frequency for the setup of Fig. 4. Termi-nating impedance Ze

L = 50 .

From Eq. (1), therefore, it follows that

VR(ω) = s(ω)tA(ω). (24)

On one hand we can obtain the system factor, s(ω), by simply multiplying the gener-ation and reception transfer functions and the voltage source term previously obtained,as shown in Eq. (23). On the other hand, we could directly measure the received outputvoltage, V m

R (t), of the system shown in Fig. 4 and, with our known model of the acoustictransfer function, tA(ω) [Eq. (22)], obtain a measured value of the system factor, sm(ω),through the process of deconvolution, i.e.,

sm(ω) = V mR (ω)t∗

A(ω)

|tA(ω)|2 + ε2, (25)

where ( )∗ denotes the complex conjugate and V mR (ω) are the frequency components

of the measured output voltage. Equation (25) represents a Wiener filter deconvolutionexpression where the Wiener filter constant, ε, is a parameter that is used to desensitizethe deconvolution process to noise [6]. Figures 25 and 26 show the system factors forthe two cases previously considered, i.e., Ze

L = 50 and ZeL = ∞, respectively. For

both cases, the Wiener filter constant ε was taken as 2% of the maximum amplitude of|tA(ω)|. We see that the amplitudes of the synthesized and measured system factors showvery similar behavior, with the synthesized results somewhat higher than the measuredresults and the phases matching very well.

200 C. Dang et al.

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8x 10-8

0 2 4 6 8 10 12 14 16 18 20-500

0

500

1000

1500

2000

2500

Phas

e (D

eg)

Frequency (MHz)

Frequency (MHz)

Am

plitu

de (

V/H

z)

synthesized

measured

synthesizedmeasured

Fig. 26. The synthesized and directly measured system factor (am-plitude and phase) versus frequency for the setup of Fig. 4 underopen-circuit termination conditions.

Obtaining system factors by deconvolution is an important capability, since the factorcan then be obtained through a single measurement. However, synthesizing that factorfrom the individual components is also important, since it allows us to analyze individu-ally the effects of individual system components such as cables, transducers, etc., whichare contained in the system factor. The reduced EAM model allows us, for the first time,to obtain s(ω) through such a synthesis process.

5. Summary and Conclusions

In Part I and in the present work we have developed a comprehensive model of an ultra-sonic NDE system. This model can serve as the basis for performing quantitative systemstudies at many different levels, ranging from an analysis of an individual component,such as a transducer crystal, for example, to the entire system itself. In developing thiscomprehensive model, several significant accomplishments have been made:

1. Explicit models of the generation and reception transfer functions have been givenin a form [see Eqs. (15) and (16)] that now makes it possible to characterizecommercial systems experimentally, using purely electrical measurements. Theimportance of these forms is that they show clearly what properties of the pulser,receiver, cabling, and transducer(s) need to be measured in order to characterize theelectrical and electromechanical components. In the case of the transducer(s), this


is especially important, since these equations demonstrate that it is the transduceropen-circuit, blocked force receiving sensitivity and the transducer input electricalimpedance that are the fundamental transducer quantities needed to model thegeneration and reception processes.

2. For the first time it has been shown that it is possible to measure all the compo-nents of an ultrasonic measurement system and then combine those componentmeasurements to predict the overall system response. Also, for the first time it hasbeen shown that a system factor (which combines all the electrical and electrome-chanical elements together into a single term) can be obtained from these samecomponent measurements instead of from a “black box” deconvolution process.This capability opens up new possibilities for characterizing and optimizing theseimportant elements of the ultrasonic system.

All of these accomplishments can serve as a foundation for some significant extensionsand future work. For example, the present modeling has dealt exclusively with immer-sion systems. However, contact ultrasound systems are widely used, and this modelingapproach should be applicable to them as well. One potential use of a contact modelthat could have far-reaching implications is its use in compensating for variations inmeasured response due to changes in the coupling layer between the transducer and thepart being inspected [7]. It appears possible that a couplant compensation system couldbe developed which relies on a monitoring of the input impedance of the transducer (aquantity that is demonstrably quite sensitive to couplant changes) and corrects for thecouplant variations. To make this possible, however, a model such as the EAM model isneeded so that the effects of the other system elements (cabling, etc.) can be accountedfor explicitly.

Acknowledgments. For C. J. Dang and L. W. Schmerr, this work was supported by The NationalScience Foundation Industry/University Cooperative Research Center Program at the Center forNDE, Iowa State University. A. Sedov was supported by the Natural Sciences and EngineeringResearch Council of Canada.

References

1. C. J. Dang, L. W. Schmerr, Jr., and A. Sedov. Res. Nondestr. Eval. DOI: 10.1007/s00164-002-0004-7(2002).

2. C. J. Dang, L. W. Schmerr, Jr., and A. Sedov. Res. Nondestr. Eval. DOI: 10.1007/s00164-002-0006-5(2002).

3. C. J. Dang. Electromechanical Characterization of Ultrasonic NDE Systems, Ph.D. Thesis, Iowa StateUniversity, Ames, IA (2001).

4. L. F. Brown. IEEE Trans. Ultrasonics, Ferroelectrics, and Frequency Control, 47:1377–1396 (2000).5. A. Ramos, J. L. San Emeterio, and P. T. Sanz. Ultrasonics 38:553–558 (2000).6. L. W. Schmerr, Jr. Fundamentals of Ultrasonic Nondestructive Evaluation—A Modeling Approach. Plenum,

New York (1998).7. G. Canella, Br. J. NDT, 179–182 (1974).

modeling and measuring all the elements of an ultrasonic nondestructive evaluation system ii:...

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