ultrasonic transducer sensitivity and model-based transducer characterization

Res Nondestr Eval (2002) 14: 203–228 DOI: 10.1007/s00164-002-0006-5© 2002 Springer-Verlag New York Inc. Online publication: 30 August 2002

Ultrasonic Transducer Sensitivity and Model-BasedTransducer Characterization

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

1Panametrics, 221 Crescent Sreet, 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. It is shown that the role that an ultrasonic piezoelectric transducer plays in both gen-erating and receiving ultrasound in an ultrasonic nondestructive evaluation (NDE) measurementsystem can be completely described in terms of the transducer’s electrical impedance and open-circuit, blocked force receiving sensitivity. Furthermore, it is shown that both of these quantities canbe obtained experimentally via a model-based approach and purely electrical measurements. Themeasurement of sensitivity uses a method originally developed for lower-frequency acoustic trans-ducers. However, it is shown that at the higher frequencies found in ultrasonic NDE applicationselectrical cabling effects play an important role and must be compensated for in determining thetransducer sensitivity. Examples of experimental measurement results using these new approachesare given.

1. Introduction

Ultrasonic transducers are difficult devices to characterize, since they are complex elec-tromechanical devices that convert electrical energy into acoustical energy and viceversa. One-dimensional (1-D) transducer models, such as the Mason and KLM models[1], are useful in designing ultrasonic transducers, but they cannot be used effectivelyin characterizing commercial transducers, since many of the underlying parameters ap-pearing in those models are not obtainable. If one represents a transducer as a two-port“black box” relating the voltage and current at the electrical port to the force and velocityat the acoustic port, then in the frequency domain [i.e., assuming harmonic driving termsof exp(−iωt) time dependency], an ultrasonic transducer can be described in terms offour frequency-dependent terms that make up the elements of a 2×2 transfer matrix [2].Sachse and Hsu [3] refer to obtaining all of these transfer elements as “complete char-acterization” of a transducer. They suggest that a series of experimental mechanical andelectrical measurements can, in principle, determine all the transfer matrix elements.To our knowledge, a practical experimental procedure for obtaining such a complete

Correspondence to: L. W. Schmerr, Jr.

204 C. Dang et al.

transducer characterization has never been accomplished. However, it will be shown thatwhen a transducer is used in an ultrasonic measurement system (as either a transmitter ora receiver), it is sufficient to characterize two frequency-dependent parameters. These pa-rameters are the transducer’s electrical impedance and its open-circuit, blocked force re-ceiving sensitivity. Measuring the electrical impedance is relatively easy, since this can bedone with standard electrical measurement procedures. Measuring the transducer sensi-tivity, however, is another matter, since this quantity involves both electrical and mechan-ical (acoustic) variables. In the acoustics literature, a three-transducer calibration proce-dure has been developed that bypasses the need for any measurement of mechanical fieldsat the transducer acoustic port and obtains the open-circuit sensitivity of an immersiontransducer directly from a series of electrical measurements [4]. For transducers operat-ing at the kilohertz frequencies found in many acoustic studies, this approach is effectiveand well tested. However, for ultrasonic transducers operating at megahertz frequencieswe have found that implementation of this calibration procedure is more complex and theunderlying model-based relations need to be modified [5]. The primary source of thesedifferences comes from the fact that at the megahertz frequencies used in nondestructiveevaluation (NDE) tests the effects of cabling on the measured response becomes non-negligible and must be explicitly accounted for in the process of obtaining the transducersensitivity.

In the following section, we will define transducer sensitivities explicitly and thenotations that will be used to describe those sensitivities. In Sec. 3, we will relate a trans-ducer’s input impedance (when acting as a transmitter) and its open-circuit, blockedforce sensitivity (when acting as a receiver) to the underlying transducer transfer ma-trix and demonstrate that these impedance and sensitivity parameters completely de-fine the effects that an ultrasonic transducer has on the measurement process when thetransducer acts as either a transmitter or receiver (or both). Model-based relationshipswill be outlined in Sec. 4 that can be used for experimentally determining this open-circuit, blocked force receiving sensitivity, using a three-transducer calibration approachthat explicitly accounts for any cabling effects present. In the modeling process, wewill show that the knowledge of an acoustic transfer function plays a key role in thecalibration procedures and that a new generalized receiving transducer sensitivity pa-rameter appears, a parameter that is directly related to the open-circuit, blocked forcereceiving sensitivity. Also in Sec. 4, we will connect the acoustic transfer function de-fined in our model-based calibration procedure to the “reciprocity parameter” commonlyused in the acoustic literature for low-frequency transducer calibrations, and describethe relationship between our open-circuit, blocked force receiving sensitivity and othertransducer sensitivity parameters that can be defined. In Sec. 5, the model-based re-sults of Sec. 4 will be combined with experiments to demonstrate practical proceduresthat can be used to determine both an ultrasonic transducer’s input impedance and itsopen-circuit, blocked force receiving sensitivity. The significant errors that cabling canproduce in sensitivity measurements if cabling effects are ignored will be explicitlydemonstrated. Finally, in Sec. 6 we will discuss how characterizing an ultrasonic trans-ducer in terms of its impedance and sensitivity is a key element in making quantitativeultrasonic measurements and describe some potential uses of such well-characterizedtransducers.

Ultrasonic Transducer Sensitivity and Characterization 205

2. Transducer Sensitivities

It will be shown that transducer sensitivities are important quantities in modeling trans-ducer effects in an ultrasonic measurement system. In this section, therefore, we willdefine what is meant by sensitivity and describe the notations that will be used in latersections.

In general, sensitivity is simply defined as the ratio of an output quantity to an inputquantity. For an ultrasonic transducer characterized as a two-port network, there are themechanical quantities of force and velocity (F, v) at the acoustic port and quantities ofvoltage and current (V, I ) at the electrical port. Since an ultrasonic transducer can beused as ether a transmitter or a receiver, there are a variety of sensitivities that can bedefined from these quantities. In the acoustics literature, when the transducer acts as atransmitter, the transmitting sensitivity is denoted as the “speaker” sensitivity, Soi, whereo can be one of the output quantities (F, v) and i can be one of the input quantities (V, I ).We will follow that same notation here. Thus, for example, the speaker sensitivity of atransmitting transducer, A, S A

F I = F/I , represents the ratio of the force output by thattransducer to the input current that drives it. In the SI system this sensitivity would havethe dimensions of newtons per ampere. Similarly, we follow the notation of the acousticsliterature and denote a receiving transducer sensitivity as a “microphone” sensitivity,Moi. In this case, the sensitivity M B

VF = V/F , for example, would represent the ratioof the output voltage to the input force driving a receiving transducer, B. In the the SIsystem this sensitivity would have the units volts per newton. Six other pairs of speakeror microphone sensitivities could also be similarly defined.

In many cases these sensitivities are determined under general electrical and acousticalloading conditions at the two ports of the transducer. However, it will also be necessary todetermine sensitivities under specific limiting electrical loading conditions. In particular,we will use a superscript “∞” to denote an open-circuit electrical output condition sothat, for example, M A;∞

VF represents an open-circuit voltage sensitivity of a receivingtransducer, A.

3. Transducer Modeling

The key role that ultrasonic sensitivities play in transducer modeling is not immediatelyapparent when considering a model of the sound generation process such as shown inFig. 1a [6]. In that model, an immersion transducer, A, is represented as a two-portsystem, characterized by a 2 × 2 transfer matrix, [T A], as shown in Fig. 1b, so that thevoltage and current are related to the force and velocity through the components of thattransfer matrix in the form

V

I

=[

T A11 T A

12

T A21 T A

22

]Ft

vt

. (1)

At the acoustic port, the force and velocity terms are coupled by the properties of thewaves radiating into the adjacent fluid medium. This coupling can be modeled, as shown

206 C. Dang et al.

Vi

Zitransducer Acabling

[T] Ft[TA]

vte

pulser

ZrA;a

radiation into fluid

Ft[TA]

vt

Z r

A;a

transducer transfer matrix

(a)

(b)

I

Vacousticradiationimpedance

Fig. 1. (a) Equivalent circuit and transfer matrix models of the componentspresent in the sound generation process of an ultrasonic NDE system, and(b) the transducer transfer matrix and the (voltage, current) and (force, velocity)parameters present at the input and output ports, respectively.

in Fig. 1b, by placing an acoustic radiation impedance, Z A;ar , at the acoustic port [6]. As

a result, Ft = Z A;ar vt , and it follows from Eq. (1) that

V = (T A11 + T A

12/Z A;ar )Ft ,

I = (T A21 + T A

22/Z A;ar )Ft . (2)

The acoustic radiation impedance can be frequency dependent, i.e., Z A;ar = Z A;a

r (ω),and in general one would have to obtain this parameter experimentally for a giventransducer. However, as shown in [6], if the transducer acts as a piston source radiat-ing into water at megahertz frequencies, then Z A;a

r = ρ f cf SA, where ρ f , cf are thedensity and wave speed for the fluid and SA is the area of the transducer. In this case,the acoustic radiation impedance is just a known constant, a case that we will assumehenceforth. Note that the “a” superscript is used here in the notation for the radia-tion impedance to indicate explicitly that it is an acoustical impedance. Similarly, an“e” superscript will be used to indicate that an impedance parameter is an electricalimpedance.

The electrical impedance of the transducer is Z A;ein = V/I , so we find directly from

Eq. (2) that this impedance is given by

Z A;ein = T A

11 Z A;ar + T A

12

T A21 Z A;a

r + T A22

. (3)


Fig. 2. Model of a transmitting transduceras an electrical impedance and an ideal am-plifier.

We can also define a transducer speaker sensitivity, SFI = F/I , where from Eq. (2) wehave for a transmitting transducer A,

S AFI = Z A;a

r

T A21 Z A;a

r + T A22

. (4)

Then, we have

Ft = S AFI I = S A

FI

Z A;ein

V . (5)

These results show that the two-port transducer model can be replaced by a modelconsisting of only two quantities—the electrical impedance, Z A;e

in , and an ideal voltage“amplifier,” K A

t = S AFI/Z A;e

in (see Fig. 2), since these quantities, together with the pulserand cabling properties, are all that are needed to obtain the output force, Ft , or outputvelocity vt = Ft/Z A;a

r of the transducer. This reduction to these two quantities is impor-tant because while to date it has not been possible to develop a method based on purelyelectrical measurements to obtain the four transfer matrix elements of the transducer, it ispossible, as will be shown later, to obtain both Z A;e

in and K At through such measurements.

Similarly, we can consider the same transducer when it acts as part of a model of thesound reception process (see Fig. 3a). It has been shown [5, 6] that the acoustic wavesincident on the transducer can be modeled as a blocked force “source,” FBLK, togetherwith the acoustic radiation impedance, Z A;a

r , as found in the sound generation process.This blocked force is the total force that would be generated at the receiving transducerface if that face were held motionless [6]. In many studies in the acoustic literature,this blocked force is taken to be twice the force on the face of the receiving transducerdue to the incident waves only (i.e., as if the receiver were absent), a result that is validwhen the interactions of the received waves with the receiving transducer are treatedas plane-wave interactions. However, such plane-wave assumptions are not needed tojustify the model of Fig. 3a [6].

In modeling the reception process, it can be shown that Fi and vi , the total forceand velocity, respectively, acting directly on the transducer (as shown in Fig. 3b) can berelated to V and I , the output voltage and current, respectively, in the transfer matrix

208 C. Dang et al.

Fig. 3. (a) Equivalent circuit and transfer matrix models of the components present inthe sound reception process of an ultrasonic NDE system, and (b) the acoustic source andimpedance present at the receiving transducer and the (force, velocity) and (voltage, current)parameters present at the input and output ports, respectively, of the transducer transfermatrix.

form as [6] Fi

vi

=[

T A22 T A

12

T A21 T A

11

]V

I

. (6)

If, in Eq. (6) we take open-circuit (I = 0, V ≡ V ∞) conditions at the electrical port, wefind

Fi = T A22V ∞,

vi = T A21V ∞. (7)

Since FBLK − Fi = Z A;ar vi , we can define an open-circuit, blocked force receiving

sensitivity of the transducer as M A;∞VFBLK

= V ∞/FBLK and show, using Eq. (7), that thissensitivity is given by

M A;∞VFBLK

= 1

T A22 + Z A;a

r T A21

. (8)

Now, replace the blocked force, radiation impedance, and transducer transfer matrixof Fig. 3b by the Thevenin equivalent circuit shown in Fig. 4a. For this equivalent circuitthe voltage source, Vs , must be equal to open-circuit output voltage, i.e, Vs = V ∞ =M A;∞

VFBLKFBLK. We can obtain the equivalent impedance, Ze, shown in Fig. 3b by shorting

out the source, Vs , so we have simply Ze = −V/I (the minus sign arises because ofthe direction of I ). This means that we take, equivalently, FBLK = 0 in Fig. 3b, which


Fig. 4. (a) Thevenin equivalent circuitrepresenting the receiving transducer andthe acoustic source and impedance presentat its input port, and (b) the explicit val-ues of the Thevenin equivalent source andimpedance in terms of the receiving trans-ducer open-circuit, blocked force sensitiv-ity and input electrical impedance.

gives Fi = −Z A;ar vi (the minus sign arises from the assumed direction of vi ). Placing

this relationship into Eq. (6) then givesFi

−Fi/Z A;ar

=[

T A22 T A

12

T A21 T A

11

]V

−V/Ze

. (9)

Taking the ratio of these two equations to eliminate the Fi and V terms, we can solvefor Ze to obtain

Ze = T A11 Z A;a

r + T A12

T A21 Z A;a

r + T A22

, (10)

which, comparing with Eq. (3), shows that Ze = Z A;ein , i.e., the Thevenin equivalent

electrical impedance of the transducer acting as a receiver is the same as its electricalimpedance when used as a transmitter. The source and Thevenin equivalent impedancevalues of the receiving transducer circuit of Fig. 4a, therefore, are given explicitly by theterms shown in Fig. 4b.

Comparing Eqs. (4) and (8), we see that the previously defined speaker sensitivity, S AFI ,

and the open-circuit, blocked force receiving sensitivity, M A;∞VFBLK

, are simply related, i.e.,

S AFI = Z A;a

r M A;∞VFBLK

. (11)

210 C. Dang et al.

Using this result and the transducer models we have developed in Fig. 3b and Fig. 4b,we see that regardless of whether the transducer is used as a transmitter or a receiver,its effects on the generation or reception processes can be completely characterized byits electrical impedance, Z A;e

in , and its open-circuit, blocked force receiving sensitivity,M A;∞

VFBLK. Measuring a transducer’s electrical impedance can be done with standard voltage

and current measurements. However, the measurement of the open-circuit, blocked forcereceiving sensitivity is not as direct. In the next section a three-transducer calibrationtechnique is described that allows one to obtain this open-circuit, blocked force receivingsensitivity from purely electrical measurements.

4. Determining Transducer Sensitivity

Consider the calibration setup shown in Fig. 5, where two circular, planar immersiontransducers are aligned in the fluid along their central axes at a separation distance, D.A pulser drives transducer X and the waves generated by that transducer are receivedby transducer Y. In an immersion setup such as this one it is not practical to makemeasurements directly at the output ports of the transducers, which are usually locatedin the fluid at the ends of movable scanning arms, so measurements are typically made atlocations external to the immersion tank. In Fig. 5, we show the measurement point, P, onthe transmitting side, where the current, I m

P , can be measured and the point Q at the endof the cable on the receiving side, where the voltage produced by transducer Y, due totransducer X, V m

XY , is measured. Also, as shown in the detailed insert of Fig. 5, the end ofthe receiving coaxial cable is assumed to be terminated by a known impedance, Ze

L , at Q.

Acoustic Transfer Function

In Sec. 3 we have shown that knowledge of the transducer electrical impedance andopen-circuit, blocked force receiving sensitivity is sufficient to relate the voltage and

Fig. 5. The acoustic calibration setup for determination of transducer sensitivity.


current at the electrical port of the transducer, when it is acting as a transmitter, to theoutput force, Ft , at the transducer’ acoustic port. Similarly, these same impedance andsensitivity parameters are sufficient to relate the voltage and current at the electricalport of the transducer, when it is acting as a receiver, to the blocked force, FBLK, actingat the transducer’s acoustic port. Thus, if we can obtain an acoustic transfer function,tA(ω) = FBLK/Ft , it is possible to relate the input and output voltages and currents in asetup such as shown in Fig. 5. Such relationships are the basis for the development ofthe three-transducer calibration procedure defined in the following subsection. For thecalibration setup of Fig. 5, the acoustic transfer function can be obtained explicitly viaa variety of different methods [5, 7, 8] if the transducers are assumed to act as pistonsources. The result is

tA(ω) = 2

πa2

exp(ikD) − 16a2b2 ·

π/2∫0

sin2 u cos2 u

(a − b)2 + 4ab cos2 u

· exp[ik√

D2 + (a − b)2 + 4ab cos2 u]

du

, (12)

where

=

πb2, a ≥ b,

πa2, b ≥ a.(13)

Dang [5] derived Eq. (12) explicitly, using a model due to King [9], and discussed theevaluation of Eq. (12) in a number of different limiting cases.

Three-Transducer Calibration Method

Our goal in this subsection is to obtain an explicit expression for the open-circuit, blockedforce receiving sensitivity of a given transducer, A, in terms of measurable electricalquantities. This can be done by using transducer A in conjunction with two other trans-ducers, B and C, whose sensitivities are also unknown. These three transducers are thenarranged, in the testing configuration of Fig. 5, in three different setups as shown inFig. 6. In setup I, transducer C is acting as a transmitter (X = C) and transducer A as areceiver (Y = A). The measured voltage in this setup, therefore, is V m

CA, following thenotation established earlier, and we let the acoustic transfer function for this setup begiven as t I

A. We also denote the voltage acting directly at the electrical port of the receiv-ing transducer in this setup as VCA (see Fig. 6). Although this voltage is not measureddirectly, it is a quantity that is useful for expressing the open-circuit sensitivity in simpleterms and, as will be shown, it can be simply related to V m

CA. In setup II, transducer Cremains the transmitting transducer (X = C) and transducer B is used as the receiver (Y= B). The acoustic transfer function in this case is t II

A , which could be different from t IA

if either the distance between transducers, D, was different in the two cases (Fig. 5) orif transducers A and B were of different sizes. The measured voltage in this setup is V m

CBand the voltage acting directly at the receiving transducer is VCB. Finally, in setup III,transducer B is used as the transmitter (X = B) and transducer A is again the receiver

212 C. Dang et al.

Fig. 6. The three transmitting/receiving transducer combinations used forthe determination of transducer sensitivity.

(Y = A). The acoustic transfer function is denoted as t IIIA and the measured voltage is

V mBA. In this setup the voltage at the face of the receiving transducer is VBA and we also

let IB be the current at the electrical port of transducer B. Like VBA, in the calibrationconfiguration of Fig. 5 it is difficult to measure IB directly, but the current at P, I m

P (Fig. 5)is measurable and can also be related directly to IB .

Consider now the general testing configuration of Fig. 5 where transducer X is thetransmitter and transducer Y the receiver. From Eq. (5) and Eq. (11) the force transmittedto the fluid by the transmitter X can be written in the form

Ft = Z X;ar M X;∞

VFBLK

Z X;ein

VX , (14)

where VX is the voltage directly at the electrical port of the transmitting transducer X(see Fig. 6). Now consider the receiving transducer, Y. A model of this reception processis shown in Fig. 7a. The termination impedance, Ze

L , is assumed known and the fourterms of the cable transfer matrix, [R], can be easily found through purely electricalmeasurements [5]. Thus, since

VXY

IXY

=[

R11 R12

R21 R22

]V m

XY

V mXY/Ze

L

, (15)

it follows that

VXY

IXY= R11 Ze

L + R12

R21 ZeL + R22

≡ Ze0, (16)

which shows that the cabling and terminating impedance can be combined into a knownequivalent electrical impedance, Ze

0, as shown in Fig. 7b, where the transducer andits inputs (blocked force, radiation impedance) have also been replaced by a Thevenin


Fig. 7. (a) The measured voltage, V mXY , for any of the three setups

shown in Fig. 6 and the corresponding current and voltage, IXY , andVXY , respectively, directly at the electrical output port of the receivingtransducer. (b) The Thevenin equivalent circuit representing the receivingtransducer and the acoustic source and impedance present at its input portand the equivalent impedance, Ze

0, representing the cabling and termina-tion impedance, Ze

L .

equivalent circuit (see Fig. 4b). From Fig. 7b, then, it follows directly that

FBLK = VXY

MY ;∞VFBLK

(1 + ZY ;e

in

Ze0

)(17)

Dividing Eq. (17) by Eq. (14), we obtain an expression for the acoustic transfer functionin the generic setup shown in Fig. 5 as

tA = VXY(1 + ZY ;ein /Ze

0)

M X;∞VFBLK

MY ;∞VFBLK

Z X;ar

Z X;ein

VX. (18)

If we write Eq. (18) for each of the three setups configurations of Fig. 6, we obtain

Setup I (X = C, Y = A):

t IA = VCA(1 + Z A;e

in /Ze0)

MC;∞VFBLK

M A;∞VFBLK

ZC;ar

ZC;ein

VC; (19a)

Setup II (X = C, Y = B):

t IIA = VCB(1 + Z B;e

in /Ze0)

MC;∞VFBLK

M B;∞VFBLK

ZC;ar

ZC;ein

VC; (19b)

Setup III (X = B, Y = A):

t IIIA = VBA(1 + Z A;e

in /Ze0)

M B;∞VFBLK

M A;∞VFBLK

Z B;ar

Z B;ein

VB= VBA(1 + Z A;e

in /Ze0)

M B;∞VFBLK

M A;∞VFBLK

Z B;ar

1

IB. (19c)

214 C. Dang et al.

In setups I and II shown in Fig. 6, we have implicitly assumed that the pulser settingsand cables to the transmitting transducer are the same in both of these cases. This isreflected in the fact that we let VC be the same voltage in both setups on the electricalport of C in Eqs. (19a) and (19b). For setup III, we let the voltage on transducer B beVB but we combined this voltage with the input impedance of transducer B to write theacoustic transfer function in terms of the current, IB , at the electrical port of B [Eq. (19c)].

Using Eqs. (19a)–(19c) we can combine these acoustic transfer functions in the formt IAt III

A /t IIA to find

t IAt III

A

t IIA

= VCAVBA(1 + Z A;ein /Ze

0)2

VCB IB(M A;∞VFBLK

)2 Z B;ar (1 + Z B;e

in /Ze0)

, (20)

which can be solved for the open-circuit, blocked force receiving sensitivity of transducerA as

M A;∞VFBLK

=(

1 + Z A;ein

Ze0

)√VCAVBA

VCB IB

1

Z B;ar (1 + Z B;e

in /Ze0)

t IIA

t IAt III

A

. (21)

Following the approach of Dang [5], Eq. (21) can also be interpreted in terms ofa generalized receiving sensitivity of transducer A by considering Fig. 7b again anddefining the generalized sensitivity of A, M A

VFBLK, as the voltage at the transducer electrical

port (when that port is loaded with the electrical impedance, Ze0, due to the cabling and

cable termination conditions) divided by the blocked force, i.e.,

M AVFBLK

= VXY

FBLK. (22)

However, from Fig. 7b we find

MY ;∞VFBLK

FBLK − VXY = ZY ;ein IXY

= ZY ;ein

Ze0

VXY , (23)

so that the open-circuit, blocked force and generalized sensitivities are related in theform

MY ;∞VFBLK

= MYVFBLK

(1 + ZY ;e

in

Ze0

). (24)

Using this relationship in Eq. (21) with Y = A, we see that Eq. (21) can be interpretedalternatively as giving the generalized receiving transducer sensitivity,

M AVFBLK

=√

VCAVBA

VCB IB

1

Z B;ar (1 + Z B;e

in /Ze0)

t IIA

t IAt III

A

, (25)

where this generalized receiving sensitivity can be related to the open-circuit, blockedforce receiving sensitivity through Eq. (24).


Fig. 8. The currents and voltages at the input andoutput ports of the cabling used in the generation ofultrasound for setup III in Fig. 6 and the terminationof the cable output port in that setup by transducerB’s electrical impedance.

Either Eqs. (25) and (24) or Eq. (21) can be used to obtain the transducer open-circuitreceiving sensitivity experimentally, through purely electrical measurements. This ispossible because although the voltages and currents appearing in these equations are notdirectly measurable, we have from Fig. 7a

VXY = (R11 + R12/ZeL)V m

XY , (26)

where the voltages V mXY are the measured voltages shown in Fig. 5. To relate IB to the

measurable current, I mP , consider the transmitting cable and transducer model of Fig. 8.

From that figure we see thatV m

P

I mP

=[

T11 T12

T21 T22

]Z B;e

in IB

IB

, (27)

giving

IB = I mP

(T21 Z B;ein + T22)

. (28)

In addition to these voltage and current measurements, it is also necessary to mea-sure the transfer matrix elements of the transmitting and receiving cables that appear inEqs. (16), (26), and (28). This is not difficult, since the cables are simple passive electri-cal components [5]. Also, we see we need to measure the input electrical impedances ofboth transducers A and B used in the determination of the open-circuit receiving sensi-tivity. Such electrical impedance measurements are also not difficult to perform [5]. Theacoustic radiation impedance of transducer B is taken here as its high-frequency limitfor a piston transducer so that it is given explicitly as Z B;a

r = ρ f cf SB , and the acoustictransfer functions for each setup can be obtained from Eq. (12).

Comparison to Lower-Frequency Acoustic Measurements

If the open-circuit receiving sensitivity is obtained for transducers operating in the kilo-hertz range or less, as is the case in many acoustic applications, then cabling effectsare negligible (i.e., the cable transfer matrices are just unit matrices). In addition, if all

216 C. Dang et al.

three transducers used in the determination of sensitivity of transducer A have the samediameters and are placed at the same distance, D, apart, then the three acoustic transferfunctions are all the same and t I

A = t IIA = t III

A ≡ tA. Finally, if the measurements areall made at the ends of the cables with open-circuit conditions (Ze

L = ∞), then underall of these assumptions, which are commonly made in the acoustics literature, Eq. (21)becomes

M A;∞VFBLK

=√√√√V m;∞

CA V m;∞BA

V m;∞CB I m

P

1

Z B;ar tA

. (29)

Equation (29) is very close in form to the one used in acoustics applications. The maindifference is that in the acoustics literature the open-circuit receiving sensitivity is usuallydefined in terms of the average pressure, pinc, acting on the face of the receiver, M A;∞

V pinc,

where

M A;∞V pinc

= V ∞

pinc. (30)

This sensitivity can be related to the corresponding open-circuit, blocked force receivingsensitivity in Eq. (29) if the receiving transducer is assumed to act as a piston and theincident and reflected waves at the receiver behave locally as plane waves [5], which is areasonable assumption for the calibration setup of Fig. 5. In that case FBLK = 2 pincSA,where SA is the area of transducer A and we obtain

M A;∞V pinc

= 2SA M A;∞VFBLK

=√√√√V m;∞

CA V m;∞BA

V m;∞CB I m

P

4S2A

Z B;ar tA

. (31)

Equation (31) can now be compared directly to the analogous expression found in acous-tics [10, 11], given by

M A;∞V pinc

=√√√√V m;∞

CA V m;∞BA

V m;∞CB I m

P

J , (32)

where J = J (ω) is called the “reciprocity parameter.” Comparing Eqs. (31) and (32),this reciprocity parameter is given in terms of the acoustic transfer function as

J (ω) = 4S2A

Z A;ar tA

. (33)

In obtaining either Eq. (31) or Eq. (32), it was assumed that the transfer function isfor two transducers of equal size. In that case, the expression for the transfer function[Eq. (12) with a = b] can be approximately given in explicit form as [5, 12]

tA(ω) = 2 exp(ikD)

1 − exp

(ika2

D

)[J0

(ka2

D

)− iJ1

(ka2

D

)], (34)

where J0, J1 are Bessel functions of order zero and one, respectively. Now, consider thecase where the sending and receiving transducers are very close, so that ka2/D 1.


Then, since both Bessel functions in Eq. (34) go to zero in this limit, to first order onehas

tA(ω) = 2 exp(ikD) (35)

and the reciprocity parameter becomes, with Z A;ar = ρ f cf SA,

J (ω) = 2SA

ρ f cfexp(−ikD), (36)

which is called the reciprocity parameter for plane-wave calibration [13] in the acousticsliterature (note—only the magnitude of this reciprocity parameter is often given, sothe complex exponential term is absent in many discussions of this parameter). Onthe other hand, when D a we have J0(ka2/D) 1, J1(ka2/D) ka2/2D, andexp(ika2/D) 1 + ika2/D, so that placing these approximations into Eq. (34), we find

tA(ω) −ika2

Dexp(ikD). (37)

In terms of the wavelength, λ = 2π/k, and with SA = πa2, the reciprocity parameterin this limit is given by

J (ω) = 2iλD

ρ f cfexp(−ikD), (38)

which is called the reciprocity parameter for spherical wave calibration in the acousticsliterature [14].Thus, Eqs. (33) and (34) provide the generalization needed to performtransducer sensitivity measurements when the transducers are not in either the very nearfield, where a plane result is valid, or in the far field, where the transducers act as pointsources and receivers and a spherical wave limit is appropriate.

Relations between Sensitivities

We have expressed the behavior of a transducer, acting as a transmitter or receiver,in terms of its input impedance and open-circuit, blocked force receiving sensitivity.However, it is possible to use other transducer sensitivities in place of this open-circuit,blocked force sensitivity, as we have already seen in Eq. (11), where a “speaker” sen-sitivity was related to this open-circuit “microphone” sensitivity. Such a replacementis possible since all the different sensitivities are based on the same underlying trans-fer matrix of the transducer combined with various loading impedances. Dang [5] hasgiven the relationship of a variety of other sensitivities to the open-circuit, blocked forcereceiving sensitivity, which we list here:

MVFBLK = Ze0 M∞

VFBLK

Ze0 + Ze

in

(39a)

MIFBLK = M∞VFBLK

Ze0 + Ze

in

(39b)

218 C. Dang et al.

MV v = Zar + Za

in

Ze0 + Ze

in

Ze0 M∞

VFBLK(39c)

MIv = Zar + Za

in

Ze0 + Ze

in

M∞VFBLK

(39d)

SFV = Zar M∞

VFBLK

Zein + Ze

i

(40a)

SFI = Zar M∞

VFBLK(40b)

SvV = M∞VFBLK

Zein + Ze

i

(40c)

Sv I = M∞VFBLK

(40d)

Except for the blocked force, all the force, velocity, voltage and current quantitiesused to define the above sensitivities are quantities appearing directly at the transducerinput and output ports. The five impedances appearing in these equations are:

Ze0, the equivalent electrical impedance acting at the electrical port of a receiving

transducerZe

i , the equivalent electrical impedance acting at the electrical port of a transmittingtransducerZe

in, the equivalent electrical impedance of a transducer acting as a transmitterZa

r , the acoustic radiation impedance of a transducer acting as a transmitterZa

in, the equivalent acoustic input impedance of a transducer acting as a receiver

5. Measurements of Transducer Impedance and Sensitivity

When determining the impedance and open-circuit, blocked force receiving sensitivityof a transducer, a number of current and voltage measurements are required. To simplifythe measurement process, we have constructed a special voltage–current probe as shownin Fig. 9. The probe consists of a small portion of coaxial cable that is stripped of itsouter conductor. A commercial current probe is then clamped on the central conductor tomake a direct reading of the current passing through the cable. A voltage measurementis made at the same location through a T-connector as shown in Fig. 9. The currentprobe used was a Tektronix CT-2 probe. It has a bandwidth of 1.2 kHz to 200 MHz anda sensitivity of 1 mV/mA when terminated into a 50- resistance. Since the currentbeing measured is often small, the output of the current probe was fed to a wide-bandamplifier whose output was sampled with a LeCroy LT342 Waverunner oscilloscope.The voltage measurement was made by sampling the voltage across the T-connector withthis same oscilloscope. The current and voltage waveforms were stored and transferred toa computer, where a fast Fourier transform (FFT) was performed to generate the sampledfrequency components of the measured current and voltage signals. This voltage–current


voltage measurement

currentmeasurement

current probe

Fig. 9. The voltage–current probe used for makingmany of the electrical measurements required in thedetermination of a transducer’s impedance and sen-sitivity.

probe was used extensively in the measurements of transducer electrical impedance andsensitivity, and in the determination of cabling effects. Those measurements will now bedescribed in some detail.

Transducer Impedance Measurements

Dang [5] outlined three different methods for determining the input electrical impedanceof a commercial transducer. These he called (1) the direct method, (2) the voltagemethod, and (3) the impedance analyzer method. We will discuss and compare those threemethods.

In the direct method, an ultrasonic pulser (Panametrics 5052PR) was connected di-rectly to an immersion transducer with a very short cable (to avoid any cabling effects).The voltage–current probe was inserted between the pulser and the transducer, and volt-age and current waveforms were measured (Fig. 10a). Taking the FFT of those signalsgave the input voltage, V (ω), and current, I (ω), in the frequency domain. In this methodthe input electrical impedance of the transducer, Ze

in(ω), then was obtained simply bydividing these two measured responses, i.e.,

Zein(ω) = V (ω)

I (ω). (41)

In the voltage method, the same basic setup of Fig. 10a was used again, but withoutthe voltage–current probe. Instead, first two voltage measurements were made at theend of the short cable under different cable termination impedances, Ze

T , and with thetransducer absent. First, the open-circuit voltage, V∞(ω), was measured (Ze

T = ∞). Thisopen-circuit voltage is just the source strength, Vi (ω), of the Thevenin equivalent circuitof the pulser shown in Fig. 10b. Then, the voltage, VT (ω), at the end of the short cable

220 C. Dang et al.

Fig. 10. (a) The testing setup for the measurement of transducer electrical impedance by thedirect method. (b) The Thevenin equivalent circuit for the pulser and the terminating impedance,Ze

T , used in the voltage method. (c) The testing setup in which an impedance analyzer is usedto measure the transducer electrical impedance.

was measured when the cable was terminated with a known resistance (ZeT = Re

T ). Fromthese measurements, and the circuit of Fig. 10b, the Thevenin equivalent impedance ofthe pulser can be obtained:

Zei (ω) =

[V∞(ω)

VT (ω)− 1

]Re

T . (42)

Finally, the transducer was placed on the end of the cable and the voltage, Vin(ω),measured at the transducer electrical port. In this setup, if we assume that the Theveninequivalent impedance of the pulser is unchanged, we find, in the same manner as Eq. (42)was obtained, that

Zei (ω) =

[V∞(ω)

Vin(ω)− 1

]Ze

in(ω), (43)

so that, equating Eqs. (42) and (43), we obtain an expression for the transducer electricalinput impedance in terms of the measured voltages in the form

Zein(ω) = V∞(ω) − VT (ω)Vin(ω)

V∞(ω) − Vin(ω)VT (ω)Re

T . (44)

We should note that in performing electrical measurements of the Thevenin equivalentimpedance of pulsers like the Panametrics 5052PR used here, we have found somedependency of the pulser’s equivalent impedance values on the external load, so thatequating Eqs. (42) and (43) may not be completely true. However, here such loadingeffects are neglected.


0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

100

200

300

Frequency (MHz)

Am

plitu

de (Ω

)

Input electrical impedance of 2.25MHz transducer

Direct measurement

Voltage measurement

Impedance analyzer

0.5 1 1.5 2 2.5 3 3.5 4 4.5 575

80

85

90

95

100

Frequency (MHz)

Pha

se (

Deg

s) Direct measurement

Voltage measurement

Impedance analyzer

Fig. 11. Experimental measurements of the input impedance versusfrequency for a 2.25-MHz transducer by the direct method, by thevoltage method, and by use of an impedance analyzer.

An impedance analyzer was also used to measure the electrical impedance of anultrasonic transducer directly (Fig. 10c). In this case, the pulser was replaced by theimpedance analyzer, which itself generates swept frequency signals. At each frequency,the analyzer measures both the voltage and current and computes their ratio, which (inthe absence of cabling effects) is just the transducer electrical input impedance. Theimpedance analyzer used here was an HP 4194A. The transducer was connected to theanalyzer through a 16047D test fixture and a very short wire. The analyzer was controlledby a computer which also served to store the measured impedance values for further dataprocessing.

All three methods described above were used to measure the electrical input imped-ances of a number of commercial ultrasonic NDE transducers as discussed by Dang [5]in detail. Here, we will only present results of all three methods for a 2.25-MHz trans-ducer (Fig. 11). Generally, the amplitude of the impedance of this transducer decreasedmonotonically with increasing frequency (Fig. 11a) and the phase was slowly varyingover the bandwidth of the transducer. Such behavior is not unexpected, since to firstorder the plated crystal in an ultrasonic transducer behaves like a capacitor. However,if the transducer contains internal “tuning” circuits, as some commercial transducersdo, we do not necessarily expect to see such capacitor-like behavior. As can be seenfrom Fig. 11, the curves obtained by all three methods for the same transducer do showsome differences, but generally all the measurements were within 10% of each other. Interms of ease of use, the direct method using the voltage–current probe was definitely themethod of choice, since it required fewer measurements than the voltage method (anddid not rely on any assumption about the behavior of the pulser under different external

222 C. Dang et al.

Fig. 12. The testing setup used to measurethe elements of the cabling transfer matrix.

loads) and did not require any of the special fixturing needed when using the impedanceanalyzer.

Cabling Transfer Matrix Measurements

In measuring transducer input impedance by any of the three methods just discussed, itwas possible to use very short cables and so minimize any cabling effects. In measuringtransducer sensitivity by the three-transducer calibration procedure described previously,however, it is not practical to make measurements under such conditions. Instead, thesetup of Fig. 5 is required, where cabling effects are present at both the transmittingand receiving portions of the ultrasonic measurement system. Fortunately, cables aresimple passive electrical components, so it is possible to completely characterize themby obtaining all four elements of their 2 × 2 transfer matrices. This was done as follows.The cable to be characterized was connected to an ultrasonic pulser (Fig. 12). The voltagesand currents at both ends of the cable (ports 1 and 2) are labeled (V1, I1) and (V2, I2),respectively, as shown in Fig. 12, where, in terms of the cable transfer matrix, [T c],

V1

I1

=[

T c11 T c

12

T c21 T c

22

]V2

I2

. (45)

First, port 2 of the cable was left in an open-circuit condition (I2 = 0, V2 = V∞)

and the voltage and current at port 1 were measured with the voltage–current probe anda sampling oscilloscope. At port 2 the open-circuit voltage was also measured with thesampling oscilloscope. Taking the FFT of those pulse measurements and using Eq. (45)then gave two of the cable transfer matrix elements:

T c11(ω) = V1(ω)

V∞(ω),

T c21(ω) = I1(ω)

V∞(ω). (46)


Similarly, port 2 was short-circuited (V2 = 0, I2 = Is) and the voltage and current atport 1 measured with the voltage–current probe and the short-circuit current at port 2 wasalso measured with this probe. Then, the FFT was again taken of these measurementsand Eq. (45) used to obtain the remaining two cable transfer matrix elements:

T c12(ω) = V1(ω)

Is(ω),

T c22(ω) = I1(ω)

Is(ω). (47)

If the cable acts as a reciprocal device, then the determinant of the transfer matrix mustbe unity, i.e.,

det[T c] = T c11T c

22 − T c12T c

21 = 1, (48)

which is a criterion that can be used to check the consistency of the measurements.Figure 13 shows the results of making such measurements on a 1.5-m-long, flexible

50- cable having BNC connectors at its ends. Both amplitudes and phases of all fourtransfer matrix elements of the cable are shown in that figure. From Fig. 13 it can be seenthat at the megahertz frequencies found in typical NDE tests the cable does not simplyact as an ideal “pass-through” type of component. In fact, the frequency-dependentamplitudes and phases of Fig. 13 correspond closely in form to that expected from a

0 5 10 15 200

0.5

1

1.5Amplitude

0 5 10 15 20-5

0

5Phase

0 5 10 15 200

20

40

60

0 5 10 15 20-200

0

200

0 5 10 15 200

0.01

0.02

0 5 10 15 20-100

-50

0

50

0 5 10 15 200

0.5

1

1.5

Frequency (MHz)0 5 10 15 20

-4

-2

0

2

Frequency (MHz)

T21

(1/

Ω)

T22

T12

(Ω

)T

11

Deg

Deg

Deg

Deg

Fig. 13. Measurements of the transfer matrix of a cable. Left column,magnitude of the transfer matrix components versus frequency. Rightcolumn, component phase terms versus frequency.

224 C. Dang et al.

0 2 4 6 8 10 12 14 16 18 200.96

0.98

1

1.02

1.04

1.06

1.08

1.1

1.12

Frequency (MHz)

-5

-4

-3

-2

-1

0

1

2

3

4

Amplitude

PhaseA

mpl

itude

Phas

e (D

eg)

Fig. 14. Reciprocity check for the cable transfer matrix components of Fig. 13 interms of both amplitude and phase.

model of the cable as an ideal transmission line [15]:

[T c] =[

cos(kclc) −iZe0 sin(kclc)

−i sin(kclc)/Ze0 cos(kclc)

], (49)

where lc is the length of the cable, kc = ω/c is the wave number of signals propagating inthe cable, and Ze

0, c = √1/µε are the intrinsic impedance and wave speed for the cable,

where (µ, ε) are the cable permeability and permittivity. Evaluating the determinant ofthese experimentally determined transfer matrix elements, it can be seen from Fig. 14that the reciprocity condition of Eq. (48) is indeed well satisfied.

Transducer Sensitivity Measurements

The three-transducer calibration procedure outlined in Sec. 4 for obtaining a transducer’sopen-circuit, blocked force receiving sensitivity is more complex than the proceduresoutlined above for obtaining the transducer’s input electrical impedance or the transfermatrix elements of the cabling. Therefore, we will explain in detail, step by step, themultiple measurements required to determine this sensitivity.

As shown earlier, three transducers, A, B, and C, are required for this procedure,which is to determine the open-circuit, blocked force receiving transducer sensitivity oftransducer A. These three transducers are used in the immersion setup shown in Fig. 5. Forsimplicity, we chose three transducers of the same radius and held the separation distance,D, between the transmitting and receiving transducers fixed for all measurements. Thenthe acoustic transfer function was the same for all three of the setups shown in Fig. 6


(t IA = t II

A = t IIIA ≡ tA). The steps taken to obtain the transducer sensitivity were then as

follows:

1. The electrical input impedances Z A;ein (ω), Z B;e

in (ω) of transducers A and B, respec-tively, were measured, using the procedures outlined previously.

2. The transfer matrix elements,

[T ] =[

T11(ω) T12(ω)

T21(ω) T22(ω)

], [R] =

[R11(ω) R12(ω)

R21(ω) R22(ω)

],

for the transmitting and receiving cables, respectively (in the testing configurationshown in Fig. 5), were obtained, following the methods described previously.

3. For setup I shown in Fig. 6, where transducer C is the transmitter and transducer Ais the receiver, the voltage V m

CA(ω) was measured across the receiving impedance,Ze

L , which was a known resistance (ZeL = Re

L).4. For setup II (Fig. 6), transducer C remained in use as the transmitting transducer,

at the same pulser settings as in setup I , while transducer B was used as thereceiver. In this setup the voltage V m

CB(ω) was then measured across the sameknown resistance, Re

L .5. In setup III (Fig. 6), transducer B was used as the transmitter and transducer A as

the receiver. In this configuration the voltage V mBA was measured across the same

known resistance, ReL , and the current I m

P (ω) was measured on the transmittingtransducer side with the voltage–current probe.

6. Using the known resistance, ReL , the measured components of the receiving cable

transfer matrix, [R], and Eq. (26), the voltages measured in setups I, II, and IIIwere converted to the voltages present in those setups directly at the electricalports of the receiving transducers, i.e.,

VCA = (R11 + R12/ReL)V m

CA,

VCB = (R11 + R12/ReL)V m

CB, (50)

VBA = (R11 + R12/ReL)V m

BA.

Also, the current, I mP (ω), measured in setup III was converted to the current, IB(ω),

directly driving transducer B in that setup by using the measured input electricalimpedance of transducer B, Z B;e

in (ω), the measured components of the transmittingcable transfer matrix, [T ], and Eq. (28) to yield

IB(ω) = I mP

(T21 Z B;ein + T22)

. (51)

Finally, since the cabling and terminating resistance were not changed in the threesetups, these setups all saw the same equivalent receiving impedance, Ze

0(ω), givenin terms of the components of the receiving cable transfer matrix components, [R],and the resistance, Re

L , by Eq. (16):

Ze0 = R11 Re

L + R12

R21 ReL + R22

. (52)

226 C. Dang et al.

7. Using all of the previously described measurements and derived quantities, thegeneralized impedance, M A

VFBLK(ω), can be obtained from Eq. (25) as

M AVFBLK

=√

VCAVBA

VCB IB

1

Z B;ar (1 + Z B;e

in /Ze0)

1

tA, (53)

where the acoustic transfer function, tA, can be found from Eq. (34) since thetransmitting and receiving transducers were always of the same radius, and, asbefore, we take the acoustic radiation impedance of transducer B, Z B;a

r , as simplyZ B;a

r = ρ f cf SB , where ρ f , cf are the density and wave speed for fluid and SB isthe transducer area.

8. From the generalized sensitivity M AVFBLK

(ω), the measurement of the electrical in-

put impedance, Z A;ein (ω), of that transducer, and the value of Ze

0(ω) the open-circuit,blocked force receiving sensitivity of A can finally be obtained [see Eq. (24)] as

M A;∞VFBLK

= M AVFBLK

(1 + Z A;e

in

Ze0

). (54)

Using all these steps, Dang [5] determined the receiving sensitivity for a number ofdifferent commercial transducers. Figure 15 shows the results for a 2.25-MHz, 0.25-in.-diameter (1.27 cm) planar transducer. In that figure, three amplitude and phase curves areshown for (1) the open-circuit, blocked force receiving sensitivity obtained from Eq. (54),

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

Frequency (MHz)

(a) Open circuit

(b) Generalized

(c) No compensation

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-300

-200

-100

0

100

200

300

Frequency (MHz)

(a) Open circuit (b) Generalized

(c) No compensation

Am

plitu

de (

V/N

)Ph

ase

(Deg

)

Fig. 15. Receiving sensitivity measurements for a 2.25-MHz trans-ducer: (a) open-circuit, blocked force sensitivity (solid lines); (b) gen-eralized sensitivity (dashed lines); (c) the generalized sensitivity whencabling effects were not included (short–long dashed lines).


(2) the generalized receiving sensitivity obtained from Eq. (53), and (3) the generalizedsensitivity obtained from Eq. (53) when the cabling effects are simply ignored. It canbe seen that while all of these curves exhibit similar behavior in both amplitude andphase, there are significant differences in the magnitude of the amplitude responsesfor the open-circuit, blocked force sensitivity and the generalized sensitivity and evengreater differences between the open-circuit, blocked force sensitivity and the sensitivitythat would be obtained if cabling effects were ignored. These differences indicate theimportant role that cabling effects have in determining transducer sensitivity and that sucheffects cannot be neglected when making sensitivity measurements for typical ultrasonicNDE transducers. Similar behavior was found in the sensitivity measurements for 5- and10-MHz transducers [5].

6. Summary and Discussion

Model-based methods for determining the impedance and sensitivity of ultrasonic NDEtransducers have been described for planar immersion transducers. These methods gener-alize the techniques used previously to characterize lower-frequency acoustic transducersso that those techniques can now be applied to the higher-frequency transducers found inNDE applications. It should also be possible to use this approach for other types of NDEtransducers such as focused immersion transducers and contact transducers, extendingsignificantly the characterization methods presently available for those probes [16, 17].

Having the ability to characterize transducers in this manner can lead to a number ofsignificant new capabilities. For example, since the effects of a transducer in an ultrasonicmeasurement system can be completely described once that transducer’s impedanceand sensitivity are known, knowledge of these parameters, and that of other elementsin the measurement system (pulser/receiver, cabling, etc.), could allow one to predictthe effect on the measured system response when one of those elements are changed.This, in turn, could allow one to requalify ultrasonic NDE measurement systems moreeconomically, without requiring an extensive set of experimental performance validationtests. Knowledge of a contact transducer’s impedance and sensitivity could also bepotentially used to devise a means to measure and compensate for the effects of couplantthickness variations in contact testing applications. This type of application would havefar-reaching consequences, since such couplant variations currently severely limit thereliability and usefulness of such contact tests for quantitative NDE studies.

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. G. S. Kino. Acoustic Waves: Devices, Imaging, and Analog Signal Processing. Prentice Hall, EnglewoodCliffs, NJ (1987).

2. E. K. Sittig. IEEE Trans Sonics and Ultrasonics SU-14:167–174 (1967).

228 C. Dang et al.

3. W. Sachse and N. N. Hsu. Physical Acoustics—Principles and Methods, Vol. XIV, pp. 277–406. In: W. P.Mason and E. N. Thurston (Eds.), Academic Press, New York (1979).

4. R. J. Bobber. Underwater Electroacoustic Measurements. Naval Research Laoratory, Washington, DC(1970).

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

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

7. K. Yamada, and Y. Fujii. J. Acoust. Soc. Am. 40:1193 (1966).8. K. Beissner. Acustica 19:212–217 (1981).9. L. V. King. Can. J. Res. 11:135–155 (1934).

10. W. R. Mc Lean. J. Acoust. Soc. Am. 12:141–146 (1940).11. R. J. Bobber. J. Acoust. Soc. Am. 39:680–687 (1966).12. L. W. Schmerr, Jr. Fundamentals of Ultrasonic Nondestructive Evaluation—A Modeling Approach.

Plenum, New York (1998).13. W. J. Trott. J. Acoust. Soc. Am. 34:989–990 (1962).14. G. Chertock. J. Acoust. Soc. Am. 34:989 (1962).15. D. M. Pozar. Microwave Engineering, 2nd ed. Wiley, New York (1998).16. B. Fay, G. Ludwig, and H. P. Reimann. Acustica 69:73–80 (1989) .17. B. Fay and H. P. Reimann. IEEE Trans. Ultrasonics, Ferroelectrics, Frequency Control 41:123–129 (1994).

ultrasonic transducer sensitivity and model-based transducer characterization

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