Optimization of ultrasonic transducers

Optimization of ultrasonic transducers

Optimization transducers J.M. THIJSSEN, of ultrasonic W.A. VERHOEF and M.J. CLOOSTERMANS The so-called KLM-model for ultrasonic transducers is empl...

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Optimization transducers J.M. THIJSSEN,

of ultrasonic


The so-called KLM-model for ultrasonic transducers is employed to optimize transducer design. Some new performance characteristics are defined which change monotonically with design parameters. These characteristics are based on the area of the envelope of the echo waveform produced by the transducer and of the corresponding amplitude spectrum. The efficiency of the transducer is defined by the round trip energy factor. The performance characteristics are used in a composite performance measure, which is then employed as a criterion in the optimization procedure. Two transducers are investigated: for medical imaging purposes and for spectral analysis of clinical echograms. The influence of electrical matching, backing impedance, matching layer impedance, bond line thickness and series induction on the optimized transducers is investigated. KEYWORDS:

ultrasonics, transducers,


List of symbols

Acoustic material


of backing


Acoustic WClayer


of intermediate


Amplitude spectrum of echo waveform from perfect reflector


Series induction, relative to value corresponding to clamped capacitance


Minimum bandwidth comprising 96.8% of amplitude spectrum (-20 width of Gaussian)


Round trip energy factor (integrated over whole echo spectrum)

Minimum bandwidth comprising 99.6% of amplitude spectrum (-40 width of Gaussian)








trip insertion impedance

loss (atfc) of source



of intermediate


Dynamic range, ratio in dB of main pulse peak to after-ringing



of piezoceramic





Minus 20 dB duration envelope


Frequency amplitude

at maximum spectrum



Centroid spectrum

Minimum duration comprising 96.8% of echo envelope area (-20 dB of Gaussian)



Amplitude excitation

Minimum duration comprising 99.6% of echo envelope area (-40 dB of Gaussian)




of echo

of echo amplitude

spectrum pulse

of electrical

In a recent paper’ a procedure has been described to calculate the electrical response of an ideal ultrasonic transducer, that is, the echo from a flat, perfect reflector placed in the far field. The procedure was The authors are in the Biophysics Laboratory Ophthalmology, University of Nijmegen, The May

of the Department of Netherlands. Paper received


0041-624X/85/01 ULTRASONICS.





of echo

based on the so-called KLM-model*, which is a transmission line description of the transducer. This model allows an easy adaptation to realistic demands for a complete system including backing, quarterwavelength matching layer(s), electrical tuning and impedance matching. The usefulness of the computer algorithms devised was demonstrated’ by a simulation which incorporated the parameters of a real transducer. The correspondence of the waveform of the







Et Co (Publishers)

Ltd 41

electrical impulse response spectrum of the simulation transducer was fairly good. as round trip insertion loss not investigated.

and of the amplitude to those of the real Other characteristics such and dynamic range were

In this paper the application of the devised methods to the design of a transducer yielding optimal performance will be shown. The primary problem discussed is the choice of the performance characteristics in the optimization strategy (see also Refs 3, 4. 5) as well as their relative weights in the optimization criterion. The strategy devised is based on the heuristic method of systematically changing the design parameters that are relevant to the construction of a transducer and the subsequent calculation of a weighted summation of the chosen transducer performance parameters. The latter parameters had to be suited to an automatic optimization procedure by computer, that is, discontinuities while changing the input parameters should be prevented. For this reason some unconventional performance parameters have been introduced. Furthermore, the frequency at maximum power, which is often used to characterize the time domain behaviour, is no longer considered to be an adequate parameter. The centroid frequency (centre of gravity of the power spectrum) is employed instead and its value is kept within the rather restricted limits of + 7% during the optimization procedure. The input parameters involved in the optimization are the acoustic impedance of the backing material, the thickness of the PZT ceramic, the impedance and the thickness of the inner of the two quarter-wavelength matching layers and finally the series induction. To investigate the influence of electrical matching the source/receiver impedance has been systematically changed for the case of the optimal transducer. An additional simulation has been performed to study the influence of the bond line thickness. The performance characteristics employed will be presented in two categories: the conventional characteristics as used in the previous paper’, that is, fractional bandwidth (l/Q), roundtrip insertion loss (RTIL), - 20 dB duration of the envelope of the echo waveform (TD,), the centre frequencyf, (at maximum power) and the dynamic range (DR); the so-called unconventional characteristics will be defined in the next section of this paper.

New performance


The various kinds of performance characteristics are used in a single valued performance measure. The weighting factors applied to the characterization have to be chosen according to the different fields of application, for example, non-destructive testing, medical echographic imaging and echographic tissue characterization. When restricted to the medical field the main application is the production of gray scale B-mode (two dimensional) images. This demands high resolution to obtain an adequate display of the anatomical structures, a large dynamic range’ and at the same time a high echo sensitivity. The axial resolution is determined by the duration of the acoustic transmission pulse, that is, both by the amplitude and the phase spectrum. Since it is not easy to define a simple quantitative measure for the linearity of the phase








J 1,f


s I






Fig. 1

a -


(or the


Bimodal as compared



as shown

+n 1 c and

continuously minimum



envelope, is rather

to that

of the total

changing bandwidth


in 1 b. The


1 d and,




due to a low minimum

of the curves.

is more



or amplitude long,







duratton is much

TizO, less different,

it is a parameter

In the case of spectra

of the area of the



is used

spectrum yielding a short duration temporal waveform*, the criterion is defined in the time domain. The centroid frequency is kept within rather restricted limits in this study (7.0 + 0.5 MHz). Because both in the time and frequency domains multimodal curves may occur. a simple criterion such as full-width half-maximum or -20 dB width is not continuous as a function of the design parameters. In this study new criteria are applied, that is. the smallest width enclosing 96.8% or 99.76% of the surface under the echo envelope curve, which are equivalent to the -20 dB and the -40 dB width. respectively, in the case of a Gaussian curve (Fig. 1). These durations are termed TI,, and TI,,. respectively. The echo envelope is calculated by means of a col7iplex-clemodulation algorithm9. A similar measure is used for characterization of the width of the amplitude spectrum corresponding to the RF echo waveform (BW,,). The dynamic range is not used in the performance measure because it does not apply to simple Gaussian-like echo envelopes and. moreover. the duration criteria defined above make it superfluous. The echo sensitivity is commonly expressed by the socalled round trip insertion loss (RTIL). which is





fc Fig. 2 The round trip insertion loss (RTIL) is defined as the ratio of the amplitude at the centre frequency f, to the amplitude at f, of the input spectrum: hence curve 2 has a larger RTIL than curve 1 (RTIL, > RTIL,). For the round trip energy factor (RTEF) the integrated power (that is, the energy) is used and, therefore, this measure expresses the situation correctly, that is, RTEF, > RTEF, means that transducer 2 is superior to transducer 1

defined as the ratio of the electrical echo signal amplitude [email protected] and the electrical input amplitude I(/) at the frequencyf, corresponding to the maximum value of the amplitude spectrum, that is, RTIL






This may be an adequate criterion for simple and narrow-band spectra, but for multimodal spectra it may become ambiguous. A further reason is that a spectrum with a somewhat lower peak value may contain equal, or even higher energy (cf. Fig. 2). Therefore, in this study the echo sensitivity is expressed by the so-called round trip energy factor (RTEF):

The other medical application, ultrasonic tissue characterization, demands a broadband transducer, high echo-sensitivity and to a lesser extent a short duration temporal waveform. The phase spectrum and the dynamic range are therefore not so important in the optimization procedure. The performance characteristics involved are the minimum bandwidth (BW,) enclosing 99.76% of the area of the spectrum (-40 dB of a Gaussian amplitude spectrum), the round trip energy factor, and the minimum duration of the temporal envelope enclosing 99.76% of its area (TIN). A final characteristic has been added to ensure that the above bandwidth criterion does not produce very narrowly peaked, wide base spectra. This criterion is the fractional bandwidth: FB


.i 0

In summary, optimization


the performance procedures are:


devised for the

Imaging transducer: X


-10 (Yilog,, (TIZO) -

10 pi log,, (TIM)

+ yi RTEF


Tissue characterization


In real measurements the frequency band will be limited to the region where Zy) is significantly non-zero, to prevent the integrand becoming singular due to instrument noise.




Y =



where (Aj)_l da is the bandwidth at -3 dB from spectral amplitude maximum, andf,, is the centroid frequency of the amplitude spectrum:




lOat log,,(BW,,)

+ yt RTEF


+ 10jYtlog,o(FB)



The procedure was applied to the program devised for the KLM model’. The design parameters as listed in Table 1 are systematically changed over the indicated ranges.



IMPB (O-30

x lo6

THIP (240-270 IMP1



kg mm2 s-l)

pm) 2.5 x lo6

THI 1 (80-100 INDU

Weighting factors

performance characteristics



FB at fc


pi = 2





New performance characteristics

Design parameters

kg mm2 s-l)


Imaging transducer



fc Tissue characterization transducer



ff, = 2









f Ce






k Phase


PIEZO-MATERIAL IWEDWCE 3.eeE+Bl SURFPCE 2.!ME-04 THICKNESS: 2.48E-84 OIEL.CON.: 1.60E-88 COUPL.CO : 7.0&z-81 PRO.SPEEO 3.70E+83 BACKIt& IWEWE: &WE-81 l/4 LWER NR 1 If’PEWE: 7.WE+BB THICK&SS 8.7’~~85 PRO.SPEED: 3.3xtEte3 l/4 LAYER NR 2 IHPEMNCE : 2.3eE+eQ THICKNESS: 1 .eeE-84 PRo.SPEE0. 3.3eE+e3 ELECTR P0RT. so. IMP : 1.4eE+86 SER. IN0 1. BBE+B9








::T. I.‘ 2808 OUR ~‘Z!z

impulse response

k Phase



6.54E-01 3 64E-01 J.JlE-07 7.59E+06 3 20E+01




[email protected] pRo.sFlx0: 3.78E+e3 8fxKIK I_: 9.886-81 l/4 LWER NR 1 IWEOWD: 7.wE*ee THICKNESS~ 1.2x-84 pRo.SPEEo: 3.x*83 1~4 LCIYERM? 2 IHPEMfCE. [email protected] THICKNESS- l.BBE-84 PR0.SPEE0. [email protected]+Q3 cOUPL.CO


b Fig. 3 a - Curves for optimal imaging transducer, with weighting set a, to yi: (3. 2, 1); b - curves for optimal tissue characterization transducer, with weighting set a, to 6; (2, 1, 1, 1). The input parameters, and the quality. or performance, characteristics are given to the right of these figures (transducer ceramic PZT, Z = 3 x 10’ kg rn-’ s-’ (30 Aayl))

Resu Its Optimization

The previously defined performance measures were used in two optimization simulation runs. The sets of weighting factors ((Wi,fit, yi) and (at through SJ applied are to be considered as an a priori subjective choice based on a compromise between optimal time and frequency domain behaviour, respectively, and the sensitivity of the transducer to be designed. The optimal transducers obtained for the sets of chosen weighting factors are shown in Fig. 3. The most evident difference in the frequency domain is the width at -3 dB down from the maximum of the spectra. Also evident is the shift of this maximum to higher frequencies for the lower case (the tissue characterization transducer) from 7.6 MHz to 8.5 MHz. The round trip insertion loss (RTIL) at theft values is less than 1 dB in both cases. Of course, this high sensitivity is achieved by using air backing. The phase spectrum is, as expected, almost straight in the relevant frequency band in the upper case (the imaging transducer). In the time domain the best imaging transducer shows an almost Gaussian envelope with low afterringing (dynamic range, DR = 32 dB!); the lower case displays considerable after-ringing, which is accompanied by a corresponding bimodality of the amplitude spectrum and of course a much lower DR (13 dB). The RF waveform in both cases is approximately at 7.0 MHz, which strongly supports the previously proposed definition and use of the centroid frequency (&), instead of the frequency at the maximum of the amplitude spectrum (f,). Influence




Fig. 4 Simulation of the imaging transducer according to: a - the optimum input parameter set of Fig. 3; b - with a 50 flsource/receiver imepdance. Simulation of the tissue characterization transducer according to: c - the optimum input parameter set of Fig. 3; d - with a 50 fl source/receiver impedance. Note the considerable degrading of pulse duration. The RTIL in both cases increased by 15 dB

of electrical


The simulations are based on the assumption that an electrical transformer is used to create an effective source/receiver impedance matched to the electrical input impedance of the transducer. The influence of a complete electrical mismatch, that is, the removal of the transformer and a direct 50 s1 connection to the transducer, is shown in Fig. 4. The upper two temporal waveforms are identical to those in Fig. 3: the lower two are mismatched. It is quite evident that the pulses are much longer, but it must be remarked that the RTIL is increased by approximately 15 dB in both cases as well! The amplitude spectra of both lower waveforms are correspondingly decreased in width (by a factor of two). Interestingly enough, a separate optimization using a 50 Sz source impedance yielded the same optimum parameter set as the one for the matched source impedance. The effect of the cable length in the case of an impedance mismatch has not been investigated but the kind of effect to be expected is quite similar to that of a mismatch of a finite length transmission line. of design


The influence on the performance characteristics of the imaging transducer by a systematic change of one of the optimized design parameters, while keeping the other four at their optimal values, will now be presented. Impedance

of the backing material. The results with regard to the conventional performance characteristics are shown in Fig. 5. The dynamic range is optimal at a backing impedance of 10’ kg m-* s-r (10 Rayl), while the





F .4a

2 8


i 030 Backing impedance (IMPB) [x

IO6 kg mm2s-r]

Frg. 5 Change in ‘normal’ performance characteristics of the optimal imaging transducer when changing the backing impedance. FB = fractional -3 dB bandwidth, RTIL = insertion loss at maximum frequency, TD,, = -20 dB pulse duration, DR = dynamic range of main pulse relative to after-ringing I



the layer material it is more beneficial to choose an impedance that is too high, at the expense of some efficiency. The dynamic range climbs to very high values, because the after-ringing almost completely disappears at 6 x lo6 kg me2 s-l. Thickness of the middle matching layer/bond

line. The most interesting aspect of this layer is in the case when it is degraded to a very thin layer, which then may simulate a bond line. For this purpose the outer matching layer is given its optimum value of 4.1 x lo6 kg me2 s-r and the inner layer is fixed at an impedance of 2.0 X lo6 kg mm2 s-r. In Fig. 8 the thickness of this bond line is given in fractions of the wavelength at 7.5 MHz and while using a propagation speed of 3300 m s-r. It can be seen from this figure that at thicknesses above x/50 (” 10 pm) the performance parameters start changing (cf. Ref. 4). It appears that the changes in pulse duration and centre frequency are negatively related to increasing thickness, and the changes in bandwidth and RTIL positively related. At large values of the thickness (x/4), the dependencies have changed considerably. It may be concluded that careful manufacturing may prevent significant degradation of the transducer performance. Series induction. The influence


IO Backing impedance (IMPB)



[x IO6 kg me2 s-r]

of the value taken for the series induction on the newly detined performance characteristics is shown in Fig. 9. The series induction is given relative to the induction corresponding in impedance to that of the clamped capacitance of the transducer. It is apparent from Fig. 9 that the main effects of tuning by a series induction are an enhanced time and frequency behaviour. The optima of the displayed curves do not exactly coincide with the

Fig. 6 Change in newly defined performance characteristics for the same transducer as in Fig. 5 when changing the backing impedance. Tlzo and TI,, are minimum duration covering 96.8% and 99.76% of the area under the envelope of the RF waveform BW,, is the minimum bandwidth enclosing 96.8% of the area of the amplitude spectrum. RTEF = round trip energy factor

other characteristics show an, almost, monotonic change. The changes of -20 dB duration (TD) and the fractional bandwidth (FB) are of the order of a factor three. The insertion loss displays a 12 dB increase (a factor of four) when going from air backing to the most heavy backing Comparing these results to those given in Fig. 6 reveals that the newly defined pulse durations, TI,, and TI,,, display a minimum at 15 and 22 X lo6 kg m-* s-r, respectively. The bandwidth BW,, almost saturates at 20 X [email protected] kg rnT2 s-l and the RTEF changes over a 10 dB range. It may be concluded that the change in dynamic range in Fig. 5 is mimicked by TI,, and that the RTIL and RTEF are quite similar in this case of a monomodal amplitude spectrum. The overall conclusion from Figs 5 and 6 is that bandwidth and pulse duration have to be traded off against the efficiency of the transducer. The use of the TIZO pulse durtion has the advantage of incorporating both the actual pulse duration and the dynamic range. of the middle matching layer: The results are shown in Fig. 7. For this case only the commonly used performance characteristics will be discussed. It is apparent that the performance of the transducer is distinctly optimal around 8.5 x 106 kg me2 s-r, although the extrema of the various curves are not exactly coincident. This impedance value follows from transmission line theory’O. The bandwidth increases almost continuously at higher impedances. so when selecting


Impedance of first



5 X/4

layer (IMP,)


IO6 kg mS2 s-‘I

Fig. 7 Change in ‘normal’ performance characteristics when changing the impedance of the middle of the two X/4-matching layers







A/200 “/loo “fso A/s6&


Thickness of bond line Fig. 8 Influence of the thickness of a bond line between the piezomaterial and a single-layered matching plate


performance explicitly and quantitatively. Moreover, the buyer of transducers may obtain an unambiguous means of comparing transducers from different manufacturers.



75’ 0





’ ‘03 1.5

Relative series induction (INDU) Fig. 9 Change of some performance characteristics when changing the relative series induction. The reference Induction is tuning away the clamped capacitance of the transducer ceramic

theoretical optimum at point 1 of the ordinate. The involvement of the source impedance, which was not incorporated in the optimization procedure, was investigated in a separate simulation. The best transducer resulted in the case of a relative induction of 1.0 and a source impedance of 1.5 a. The latter value is slightly different from the value taken in the main optimization described earlier.

Discussion In this paper it has been shown that the conventional performance characteristics of ultrasound transducers may be replaced by others. The use of centre frequency (frequency at the maximum of the amplitude spectrum) and round trip insertion loss is not ideal because of problems in applying these in the case of bi- or multimodel spectra. The new parameters, the centroid frequency and the round trip energy factor, have been shown to be adequate alternatives. Other new characteristics describing the waveform and the amplitude spectrum have been introduced for the sake of continuity with input parameters to the optimization algorithm. It may be worthwhile to investigate an extension to other applications. The weighting factors introduced in the equation describing the overall performance measures have been obtained partly on a purely subjective, or a priori basis and partly from experience with the simulation program The great advantage of the concept of an overall performance measure of course is that the designer, or manufacturer. of transducers has to express his experience and expectation about the

The optimization strategy described has led us to a set of two transducers, which are not much different when looking at the input parameters. This fact may illustrate once more the importance of knowing the effects caused by small deviations in the various constituent parts of the transducer as a whole. Furthermore, if sufficient computing power is available one could imagine involving all the design parameters in the optimization strategy.

Acknowledgement The authors would like to thank R Merks, Ph.D. (Oldelft Inc) for stimulating discussions and valuable criticism to the manuscript.

References I


Kernel, S.J.H. van, Thijssen, J.M. A Calculation Scheme for the Optimum Design of Ultrasonic Transducers, (iltrusonics 21 (1983) 134-140 Krimholtz, R., Leedom, D.A., Mathaei, C.L. New Equivalent

Circuits for Elementary Piezoelectric Transducers. Left 3





8 9 IO


6 (1970) 398-399

Hue, J., Gazaleh, M.G., Bruneel, C., Torquet, R. Optimization Criteria for the Piezoelectric Transducers used in Acoustical Imaging. In: Acoustical Imaging. Vol. 10, P. Alais and RF. Metherell. editors. Plenum Press, New York (1982) 751-759 Silk, M.G. Predictions of the Effect of some Constructional Variables on the Performance of Ultrasonic Transducers. Ultrasonics 21 (1983) 27-33 Hunt, J.W., Arditi, M., Foster, F.S. Ultrasound Transducers for Pulse-echo Medical Imaging. Ib:b:b.’ Tram. BME 30 (19X3) 453-48 1 Thijssen, J.M., Kervel, S.J.H., Cloostermans, M.J. A Calculation Scheme for the Design of Optimum Ultrasonic Transducers (abstract). Ulrrasound Med. Viol. 8, suppl. I (1982) 195 Kossoff, G. The Transducer. In: Handbook of Clinical Ultrasound. M. de Vlieger. et al. editors. Wiley. New York. (1978) 25-30 Pohlig, SC. Signal Duration and the Fourier Transform. Proc. I,SrX 68 ( 1980) 629-630 Whalen, A.D. Detection of Signals in Noise. Academic Press. New York (1971) Desilets, C.S., Fraser, J.D., Kino, C.S. The Design of Efficient Broad-band Piezoelectric Transducers. IkEb: Truns Sonic.~ U/tm.sonics

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