Fourpole properties of electro acoustic delay line with conventional and unconvectional transducers

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iii lil» ÜUii

I iii II

ELECTRO ACOUSTIC DELAY LINES

WITH CONVENTIONAL AND

UNCONVENTIONAL TRANSDUCERS

PROEFSCHRIFT

TER V E R K R I J G I N G VAN DE G R A A D VAN DOCTOR IN D E TECHNISCHE WETENSCHAPPEN AAN D E T E C H N I S C H E H O G E S C H O O L D E L F T OP GEZAG VAN D E RECTOR M A G N I F I C U S DR. IR. C. J. D. M. VERHAGEN, HOOGLERAAR IN D E A F D E L I N G DER TECHNISCHE N A T U U R K U N D E , VOOR EEN COMMISSIE UIT DE SENAAT TE VER-D E VER-D I G E N OP WOENSVER-DAG 30 OKTOBER 1968

OM 16 U U R

D O O R

LEO JOHAN van der PAUW

N A T U U R K U N D I G I N G E N I E U R GEBOREN TE HILLEGERSBERG

PROF. DRS. D. POLDER.

deze studie in de vorm van een proefschrift te doen verschijnen.

De directie van het Natuurkundig Laboratorium der N.V. Philips' Gloei-lampenfabrieken ben ik zeer erkentelijk voor de mij geboden rust dit werk te kunnen voltooien.

Den Heer R. W. van den Oever dank ik voor de talentvolle wijze, waarop hij de experimenten heeft voorbereid en uitgevoerd.

1. INTRODUCTION AND OUTLINE OF THE THESIS 1

1.1. Introduction 1 1.2. A simplified model 1 1.3. Some types of unconventional transducers 3

1.4. The delay line as a blackbox 6

2. PHENOMENOLOGICAL RELATIONS BETWEEN THE ME-CHANICAL AND ELECTRICAL VARIABLES OF A

PIEZO-ELECTRIC MEDIUM 9 2.1. Introduction 9 2.2. Phenomenological relations 9

3. THE GENERATION OF ELECTROACOUSTIC WAVES IN A

PIEZOELECTRIC MEDIUM 12

3.1. Introduction 12 3.2. The plane-wave approach 12

3.3. Some general properties of the piezoelectric wave impedance

Z(w,k) 14 3.4. A power expansion for small values of/! 15

4. A ONE-DIMENSIONAL THEORY OF THE

ELECTRO-ACOUS-TIC DELAY LINE 18 4.1. Introduction 18 4.2. The one-dimensional reflectionless model 18

4.3. The fourpole properties of the one-dimensional reflectionless

model 20 4.4. The impulse response of the one-dimensional model 21

5. THE THREE-DIMENSIONAL REFLECTIONLESS MODEL . . 23

5.1. Introduction 23 5.2. The space-charge distribution function and the weight function;

the homogeneous approximation 23 5.3. An approximate expression for the beam-divergence factor . . 25

5.4. The influence of beam divergence on the fourpole parameters . 30 5.5. Impulse response of a delay line with circular or square

ELECTRODES 34 6.1. Introduction 34 6.2. The influence of multiple reflections on the fourpole parameters 34

6.3. The impulse response of a delay line with reflecting electrodes 38

6.4. Some experimental results 40 7. SOME TYPES OF UNCONVENTIONAL TRANSDUCERS WITH

LARGE FRACTIONAL BANDWIDTH 43

7.1. Introduction 43 7.2. The transducer with exponentially decreasing polarization . . . 43

7.3. Fourpole properties of the planar transducer 45 7.4. Impulse response of a delay line with planar transducers . . . 51

7.5. Some experiments with planar transducers 53 7.6. Fourpole properties of a delay line with plane-concave electrodes 56

7.7. Estimate of the error introduced by the gaussian approximation 61 7.8. Result of an experiment performed on a delay line with

plane-concave transducers 62

APPENDICES 64 A l . Estimate of the error introduced by assuming a homogeneous

distribution function for the case of circular, plane-parallel

trans-ducers 64 A2. Estimate of the error introduced by the paraxial approximation

and the one-dimensional approximation for the isotropic case . 65 A3. Estimate of the error introduced by the isotropic approximation 70 A4. Analysis of the properties of a delay line with plane-concave

trans-ducers 73 A5. Estimate of the error introduced by the gaussian approximation 77

REFERENCES 78 Summary 79 Samenvatting 83

1.1. Introduction

In many instances it is necessary to give an electrical signal a time delay. For example in radar it is often desirable to compare a received echo with a previously received one.

In certain colour-television systems, such as the PAL system and the SECAM system, the signal containing the colour information is delayed a line time (64 [is) and combined with the incoming signal.

An alternative application is the storage of digital information for an arbitrarily long period. This is done by reshaping the output of the delay line and feeding it back again to the input. Because of the high propagation velocity of electrical signals the purely electrical delay line will mostly be of impractical length. In this case a better solution for obtaining the desired time delay, typically of the order of 1 (xs to 10 ms, is obtained by transforming the electrical signal first into an acoustic signal. The desired delay time is obtained by choosing an acoustic medium of appropriate length. Finally the acoustic signal is transformed again into an electrical signal.

Depending upon the specific application special attention is paid either to a minimum loss of energy or a maximum electrical bandwidth, requirements which are in some respects contradictory. These contradictory requirements will be investigated in this study. Particularly it will be shown that with some types of unconventional transducers an in principle unlimited relative bandwidth can be obtained, although at the expense of the energy efficiency.

1.2. A simplified model

The essential properties of a conventional electroacoustic delay line can be understood from a simplified model. This simplified model consists of a system of four plane-parallel electrodes which are embedded in a piezoelectric medium of infinite extension (see fig. 1.1). One pair of electrodes, denoted in fig. 1.1 by A and B, constitute the input transducer and the second pair C and D the output

A B C 0

Homogeneous piezoelectric medium

Input electrodes Output electrodes

Fig. 1.1. A simplified model of a delay line, consisting of 4 plane-parallel electrodes, embedded in a homogeneous piezoelectric.

transducer. In chapters 2, 3 and 4 it is discussed how, as a result of the electric charge, applied to the input electrodes, together with the piezoelectric properties of the material, acoustic waves will be generated at the electrodes A and B. These acoustic waves, which, due to the piezoelectric properties of the medium are accompanied by an electric field, will generate an electric-potential difference between the output electrodes C and D.

Due to the interference of the waves emitted by the electrodes A and B, characteristic zeros in the transferred energy as a function of the frequency will occur. More precisely formulated, this occurs if the distance between the elec-trodes A and B is equal to the acoustic wavelength or a multiple of this distance. For much the same reason, such a zero occurs also if the distance between the output electrodes C and D is equal to an integer times the acoustic wavelength. For this reason the "transfer impedance", defined as the open output voltage, divided by the current supplied to the input, will contain a frequency-dependent factor, given by {1 — exp(—/ w TAB)} {1 — exp(—; w TCD)}- Here T^B and TCD

are the acoustic transit times from electrode A to electrode B and from elec-trode C to elecelec-trode D, respectively. Apparently the transfer impedance will be zero if either co T^B = 2 JT n or CO T^D = Inn, where n is an integer. It is said, that the transfer impedance exhibits discrete "pass bands".

In practice the thicknesses of the transducers are preferably chosen so that the desired operating frequency band coincides with the lowest pass band, rather than with a pass band of higher order. This is so because we want to keep the purely reactive part of the input impedance and output impedance as small as possible in order to obtain the highest possible electrical bandwidth. For this reason the distances between the electrodes A and B and between C and D are the important parameters, determining the operating frequency band of the delay line.

A second important parameter is the distance between the electrodes B and C because this distance determines the "delay time" of the delay line. Mathemati-cafly this time delay corresponds to a factor exp(—/ m T) so that the transfer impedance will contain the frequency-dependent factor {1 — exp(—/o) T^B)}

{1 — exp (—/ CO TCD)} exp(— i cu T). The remaining parameter of our simplified mode! is the diameter of the electrodes. This diameter should be large compared to the acoustical wavelength in order to obtain an approximately parallel beam of energy, directed towards the receiving transducer. In other words, the diam-eter of the electrodes is chosen sufficiently large in order to obtain acceptably low diffraction losses.

As will be shown in chapter 5, the diffraction losses depend only on one parameter, the so-called "Fresnel number", provided the diameter of the elec-trodes is many times the acoustic wavelength, a condition that will be satisfied in most practical cases. The Fresnel number is for the case of circular electrodes defined as N = a> a^ll nlv, where a is the radius of the electrodes, / is the

distance between the electrodes B and C and v is the acoustic-propagation velocity. In order to have low diffraction losses, the Fresnel number should be about unity or higher. Physically the parameter a^/2 / v can be interpreted as an effective dispersion in acoustical transit time between nearest points and most remote points on electrodes B and C. A rather crude way of taking this time dispersion into account is the "single-relaxation-time" approximation, which approximates the beam-spreading effect by a factor (/ w a^/2 / f) (1 -f + i CO a} 12 I v)"' in the transfer impedance. In chapter 5 the accuracy of this single-relaxation-time approximation is investigated and also a more accurate approximation is discussed.

Contrary to the simplified model, in practice different materials are usually applied for the transducers and the delay medium and for the absorbing medium, the "backing". As a result of such an inhomogeneous structure two complications can arise. First acoustical reflections will in general occur at mechanical discontinuities. This effect will in an actual transducer give rise to multiple reflections between the electrodes. The influence of these multiple reflections on the electrical behaviour of the delay line is investigated in chapter 6. In chapter 6 also a simple experimental method for measuring the reflection coefficients at the electrodes is presented and illustrated by a practical example. The second complication can arise if at some place, where the piezoelec-tric properties are inhomogeneous, a dielecpiezoelec-tric displacement is present. From the phenomenological relations presented in chapter 2 it follows that also at these places acoustic waves will in general be generated. In chapter 7 it is shown that explicit use can be made of this effect in order to obtain transducers with favourable electric properties. We shall return to this question in the next section.

1.3. Some types of unconventional transducers

As pointed out in the previous section the transfer impedance of a delay line with conventional transducers exhibits as a function of frequency typical zeros, which will limit the useful bandwidth. In chapter 7 we shall show that uncon-ventional transducers that do not exhibit these typical zeros and which for this reason can have a much larger relative bandwidth, can be constructed.

One type of such a tranducer is a transducer with plane-concave electrodes. Such a transducer is obtained if in fig. 1.1 the plane electrode A is replaced by a concave electrode so that the convex side is facing the receiving transducer. In the same way, the electrode D is replaced by a concave electrode with its convex side facing the emitting transducer. As a result of this concave shape the energy emitted by electrode A will not be a parallel beam but rather a diverging beam, so that only a relatively small portion of the energy emitted by electrode A will arrive at the receiving transducer. Consequently only a small interference of the waves emitted by the electrodes A and B will occur. The same effect occurs

at the receiving electrode which, by virtue of the reciprocity principle, has an analogous influence on the transfer impedance. It turns out that the factor

1 — exp(—/ o) TAB), which is characteristic for a plane-parallel transducer, changes into the factor

e x p ( — / a> Tj)

I )

1 + (• CO T2

due to the concave shape of electrode A. This expression is a good approxima-tion, provided TJ > TJ. Here T^ is the acoustic transit time, corresponding to the thickness of the transducer in the middle and TJ = a b^j2 v, where a is the curvature of the concave electrode and b is the "effective" radius of the transducer. Physically, TJ can be considered as a time-dispersion constant. The time dispersion is due to the dispersion in the thickness of the transducer. An additional advantage of the plane-concave transducer is, that also the effect of multiple reflections between the electrodes of the transducer is greatly suppressed, due to the diverging action of the concave electrode.

As is shown in chapter 7 an alternative way to obtain a transducer that has no zeros in its transfer characteristic is to make use of a ferroelectric with an inhomogeneous permanent polarization. Let us for instance assume that in fig. 1.1 the permanent polarization of the medium is zero to the left of elec-trode A. Between A and B the permanent polarization shall have a value ^P to the left of a plane M and a value P to the right of the plane M. The plane M shall be halfway between A and B. Now not only are the electric charges the sources of piezoelectric waves, but also the plane M, where the polarization changes discontinuously. In fact the transducer AB can be effectively considered as two transducers, i.e. transducer AM and transducer MB. The "electrodes" of transducer AM should, however, be given a relative weight factor ^, due to the reduced polarization between A and M. From this consideration it follows that the frequency-dependent factor 1 — exp(—/ w T^B) should be replaced by the factor 1 — ^ exp(—/ co T^B) — i exp(—/ w T^B)- Though this new frequency-dependent factor is still oscillatory in nature, it does not become zero any more, except at cy = 0.

Still better properties are obtained, if the polarization of the material decreases continuously from P to zero, when going from B to A. If for instance this polarization decreases exponentially with distance, the frequency-dependent factor can be shown to become 1 — 1/(1 + / co T), where r is the transit time, corresponding to the characteristic distance at which the polarization has de-creased with a factor e = 2-7. For wide-band applications such a frequency dependence, which can also be written as / co T/(1 + i CO T), is particularly favourable because it can be corrected in a simple way electrically by means of an RC network. A drawback of this type of transducer is that it still suffers from

multiple reflections between the electrodes. These multiple reflections will reduce the bandwidth of the transducer.

A new type of transducer which has an inhomogeneous polarization and which does not suffer from multiple reflections is the so-called "planar trans-ducer". In fig. 1.2 the structure of such a planar transducer is sketched. Let A

Interface A medium Mi % medium M2

r

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Fig. 1.2. The structure of a "planar" transducer.

be the interface between a ferroelectric medium Mi and a non-piezoelectric medium M2. At the interface two comb-shaped electrodes P and Q are present (see fig. 1.2). The permanent polarization of the medium M1 is obtained once and for all by means of the electrodes P and Q by applying a sufficiently large potential difference at an elevated temperature. In chapter 7 it is shown that the behaviour of a planar transducer as a function of frequency can also imately be described by the factor /' co T/(1 + ; CO T), by virtue of the approx-imately exponential decrease of the polarization of the medium Mj. The characteristic length attached to this exponential decrease and hence T are proportional to the periodic distance of the electrodes. Because there is only one interface, no multiple reflections can occur.

An alternative structure is drawn in fig. 1.3. Here the complete delay line consists of a piece of ferroelectric material with two plane-parallel end faces. On the end faces the comb-shaped electrodes are applied, for instance by means of a photo-mask technique. Also in this case the (permanent) polarization is applied by means of the electrodes. The frequency dependence of such an "inverted" planar transducer is, however, somewhat different and appears to be equal to 2 o:i^ T^/(1 + m^ r^), where the time constant r is equal to that of a normal planar transducer Finally in chapter 7 some experiments performed on delay lines with plane-concave transducers and planar transducers are compared with the presented theory.

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W-/ '

7"

Fig. 1.3. The structure of a delay line with "inverted" planar transducers.

1.4. The delay line as a blackbox

The electrical behaviour of a delay line can be described by the two linear fourpole relations

f I = Z i i 7 i + Z i 2 72,

l'2 = •^2171 + ^22

J2-The input current 71, the output current y'2, the input voltage Vi and the output voltage V2 are assumed to vary harmonically with time. The polarities are defined as shown in fig. 1.4. The delay lines that we shall study belong to the class of reciprocal fourpoles, i.e. Z12 = Z21. We shall call Z , , the input impedance, Z22 the output impedance and Z12 = Z21 the transfer impedance of the four-pole.

Let us assume that the input test signal is a wave packet of a duration long enough to consider it as "monochromatic" and on the other hand short enough to separate it completely in time from any signal caused by reflections from the receiving transducer. In this way it is possible to distinguish between the useful direct signal and the spurious reflected signal. In this thesis we shall not in the first instance be interested in these spurious reflections and therefore we shall define the "direct-signal" fourpole impedances Z n and Z21 with reference to this direct signal. The "direct-signal" impedances Z,2 and Z22 are defined in a similar way by interchanging input and output of the fourpole. In practice

the spurious reflections will in most cases be relatively small so that the con-ventional fourpole impedances Z ^ , Z,2 and Z22 will differ only slightly from our "direct-signal" fourpole impedances. If necessary, additional terms, due to spurious reflections, will be dealt with separately.

If a high power efficiency is of primary interest the optimum power effi-ciency »/i, defined as the direct-signal power delivered to the load, divided by the power supplied to the input of the fourpole under optimum load conditions, is a relevant parameter. In accordance with our definition of the "direct-signal" fourpole impedances, this optimum direct-signal power efficiency is given by

Z12 Z i 2 *

Vi = • (1-1) 4 Re(Z„) Re(Z22)

The optimum value of the load impedance Z, is in that case given by Z, = Z22*. If, on the other hand, a large bandwidth is of importance, the condition Z, = Z22* is not preferable, since especially if Z22 is dominantly reactive, this load condition would give rise to a sharp resonance effect. For this reason, for large-bandwidth applications the optimum power efficiency under purely resistive source-impedance and load-impedance conditions is a more relevant parameter. More precisely we define the optimum wide-band direct-signal power efficiency »?2 as the maximum direct-signal power that can be extracted from the output with a resistive load, divided by the maximum power that can be extracted from the source, which is assumed to have a resistive source imped-ance. We assume that by means of an ideal transformer also the source resist-ance can be matched to the input impedresist-ance. The optimum matching conditions turn out to be in that case: R^ == \Zn\ and R, = [Zzji, where R, and /?, are the source resistance and the load resistance, respectively. Under these con-ditions the power efficiency turns out to be

4 Z , 2 Z i 2 *

rj^^ '-^-^ . (1.2) iZii (1 + z,j\z,,\r (1 + Z22/1Z22I)' Z22I

From (1.2), the inequality

12

Z12 Z12 Z12 Z12

< » y 2 < 0-3)

4 | Z i i Z 2 2 l |Z,,Z22l

can be derived. If Z n and Z22 are dominantly reactive, as will often be the case in practice, the optimum "wide-band" power efficiency rj2 will be approx-imately given by

Z12 Zi 2 *

V2<^-~——- 0-4) 1-^11 •^22!

Though optimum matching conditions will in general only be fulfilled for one frequency, Z, 1 and Z22 will under normal conditions vary only moderately with

frequency, so that for the whole considered pass band the power efficiency will be approximately given by

Z_{1 2 • ' ^ 1 2} 7 *

r] f^ . |Zii Z22I

A slightly more convenient parameter, relevant for the wide-band properties of a delay line, is the transfer function O defined by

0 = Z , 2 / ( Z , i Z 2 2 ) " ^ (1.5) We shall make (Z, 1 Z22)''^ definite by taking the real part positive. We

remark that 0 is the geometric mean of the "open-output"-voltage amplifi-cation and minus the "short-circuit"-current amplifiamplifi-cation of the fourpole.

The bandwidth of the device we shall relate to the transfer function O. Let / i and ƒ2 be the two frequencies, at which \0\ has fallen off by a factor of 2"^'^

with respect to its maximum value,/i being the lower frequency. Then we define the bandwidth as the difTerence ƒ2 —fu and the relative bandwidth we define as the ratio (ƒ2 —fi)/fi- Unless stated otherwise, these quantities will refer to the lowest pass band.

For some applications, for instance in digital applications, the response of the delay line to an input step voltage or an input current impulse is also of interest. Furthermore the experimental determination of the fourpole param-eters is conveniently carried out by photographing from an oscilloscope for instance the input voltage and the open-output voltage versus time, caused by an input current impulse. For this reason we shall also calculate this response in several cases.

2. PHENOMENOLOGICAL RELATIONS BETWEEN THE MECHANICAL AND ELECTRICAL VARIABLES OF A

PIEZOELECTRIC MEDIUM

2.1. Introduction

As pointed out in the introduction, the electrical signal applied to the input of an electroacoustic delay line is converted into a mechanical signal by means of an electromechanical transducer, called the input transducer. The opposite takes place at the output transducer. The mechanism of this conversion is based upon the piezoelectric properties of the transducer material. In this chapter we shall derive the phenomenological relations between the mechanical and elec-trical variables of a piezoelectric material.

2.2. Phenomenological relations

The phenomenological relations between the mechanical and electrical variables can conveniently be derived from the potential energy per unit volume, which we shall denote by H. As independent mechanical variable we choose the mechanical-displacement vector s = Ji i + .^2 J + •'3 k, where i, j and k are the unit vectors in our frame of reference x^,X2,Xi. As independent electrical variable we choose the dielectric-displacement vector D = Dj i + Z)2 j + I>3 k. The mechanical-deformation tensor a is defined as the gradient of s (o = Vs) and is characterized by the 9 components 5'i 1, >S'i2, . . ., ^33, and we shall write o = Sn ii + •S'12 ij + . . . + 5*33 kk, where

Sij = i>Sj/i)Xi.

It turns out experimentally that the potential-energy density / / is a quadratic function of the components of o and D:

H = S ( i c,,„ Su S,, + i /?,,. D, Dj + htj, S,j D,}. (2.1)

ijkl

Equation (2.1) is in principle valid only for small deviations from equilibrium and for homogeneous deformation and dielectric displacement.

If the variables are inhomogeneous, terms containing the gradients and higher derivatives of the variables will also enter into (2.1). Especially near a boundary the variables may change rather abruptly so that piezoelectric phenomena of a different nature could be expected there from a phenomenological point of view. Experimentally it turns out, however, that these effects can be neglected and that eq. (2.1) is also valid for inhomogeneous variables under normal exper-imental conditions.

We shall now discuss some symmetry properties of the tensors c, j3 and h. From (2.1) it follows that i / i s insensitive to interchanging S^ and 5^,. For this

reason we may put Cu,^, = Ci,„j without loss of generality. For the same reason we may put Pu = (iji. If we give the material a rigid rotation, keeping the dielectric displacement attached to the material, H should not change either. If we take this rotation sufficiently small, only the antisymmetric part of the deformation tensor will change as a first-order term, the other terms being products of the components of the rigid-rotation deformation and the original variables. From this it follows that the symmetry relations c.-jj, = Cjj^, and hijk — hjik must also hold. These conditions reduce the number of elastic constants to 21 and the number of piezoelectric constants to 18. The number of independent dielectric constants is 6. The number of independent material constants is greatly reduced if the material has symmetry properties.

A simple situation arises, for example, if we consider a polycrystalline ferro-electric with a weak permanently built in polarization. The permanent polariza-tion, briefly called the polarizapolariza-tion, can be characterized by a vector P. If P is sufficiently small, H will contain P only to the first power. Because the unpolarized material possesses the inversion-symmetry property, the expression for the potential-energy density will be of the form

ƒ/ = S {i cok, S,j Sk, + i /3,, D, Dj + flij,, S^j D,P,}, (2.2)

ijkl

where the tensors c, /3 and a will, apart from a constant tensor, contain P at least to the second power. Furthermore, for P = 0 the material will be iso-tropic and hence the tensors c, /3 and a should be invariant for any rotation of our frame of reference. It can be shown that in that case both c and a con-tain only two independent constants. The tensor /S is a constant in that case. H can then be written as

/ / = 2 {i c, Su Sjj + i C2 iS,j + Sj,y + \(iD,' +

IJ

+ a, Sa Pj DJ + fl2 {Su + Sjd (P, DJ + Pj Dd). (2.3) We shall now go back to the general formula (2.1). The stress tensor and the

electric field follow from the potential-energy density H according to the rela-tions

T,j =i>H!i>S,j, . ^ . E, = bH/i)Di. ^ '

Substituting this in (2.1) we obtain:

Tij = Cijki Siii + htjic D^, Ek = hijk ^1} + f^ki Dt. In tensor notation this can be written as

T = c : o + /! • D,

(2.6) E = /! : o + i3 • D. ^ ^ The double-dot product indicates that summation has to be carried out over two indices.

It should be noted that the symmetry relations Cu^^i = Cjn,, and /jy^ = /j^j^ imply that T is a symmetric tensor.

3. THE GENERATION OF ELECTROACOUSTIC WAVES IN A PIEZOELECTRIC MEDIUM

3.1. Introduction

In chapter 1 we presented a simplified model of a delay line, consisting of infinitely thin plane-parallel electrodes, embedded in a piezoelectric medium of infinite extension. In such a homogeneous medium the electric space charges can be considered as the sources where the electroacoustic waves are generated. Making use of the results derived in chapter 2 we shall in this chapter calculate the electric potential generated in a piezoelectric medium of infinite extension by a space-charge distribution that depends harmonically on time and space. This elementary solution will be the building stone for calculating the electric potential generated by electrodes that are embedded in a homogeneous piezo-electric medium of infinite extension.

3.2. The plane-wave approach

Because a space-charge distribution that is an arbitrary function of space and time is too complicated to solve analytically, we shall decompose the arbitrary distribution into a superposition of plane waves, according to

1

e(r,r) = ƒ dA:i dA:2 d^:3 dco e(co,k) exp(/cüf — ik- r), (3.1) 16 7z:*_<„

with

OO

p(o>,k) = ƒ dxi dA:2 dXi dt Q(r,t) exp{—icot + / k • r). (3.2)

— 00

Our task is to find the piezoelectric plane waves, generated by the "plane-wave" space-charge distribution Q(T,t) = Q exp(/co? — J k • r). If we neglect irrelevant forces, such as gravitational forces and coulomb forces, the force exerted on a volume element d F is given by the divergence of the stress tensor, multiplied by d V. Hence the equation of motion will be

m bh(T,t)/bt^ = V • T(r,0, (3.3) where m is the specific mass of the material. Substituting (3.3) in (2.6) and

putting o = Vs, we obtain the equations

m ö2s(r,0/öf 2 = V • ^ : Vs(r,0 + V • A • D(r,0, E(r,0 = h : v s ( r , 0 + p • D(r,0.

Furthermore we shall assume that the propagation velocity of acoustic waves is very small compared with the propagation velocity of light. This assumption permits us to write

V x E ( r , 0 = 0. (3.5) The last equation we need is Poisson's equation:

V • D(r,0 = eir,t). (3.6) We remark that the equations (3.4), (3.5) and (3.6) are also valid for

inhomo-geneous materials. We shall now make use of the homogeneity and the infinite extension of the material by assuming that also s and D shall vary harmonically in space, according to

s(r,/) = s exp(/co? — / k • r) and

D(r,0 = D exp(/cor — / k • r).

Furthermore (3.5) can be satisfied by putting E = ik0, where 0 is the electric potential. Substituting V = —ik and ö/ör = /co in (3.4) and (3.6), we obtain the algebraic equations

(—w co^ + k • c • k) • s + / k • /i • D = 0,

—/ ^ : k s + /3 • D = /• k 0 , (3.7) — / k • D = Q.

The simplicity, created by the introduction of the plane-wave approach, be-comes now apparent. For from the algebraic equations (3.7) the variables D and s, which are not of interest to us, can be eliminated by purely algebraic (matrix) manipulations and in this way a scalar relation between 0 and Q will be obtained. Carrying out this elimination we find the equation

e(co,k)

^(co,k) = . (3.8) k- { ( 3 - ^ - k . ( - m c o 2 + k - c . k ) - i - k . / i } - i - k

Recapitulating, eq. (3.8) expresses the electric potential 0(co,k) as a function of the space-charge density p(co,k), the frequency co, the wave vector k and the material constants. The elastic constants are given by the tensor c, the dielectric constants by the tensor jS and the piezoelectric constants by the tensor h. Equation (3.8) applies to a space-charge distribution which depends harmonically on the position coordinates and on time. The electric potential generated by a space charge that is an arbitrary function of space and time, is found by a superposition of plane waves, given by

1 r

<P(r,0 = / dkidk2dk3do}0{(ü,'k)Qxpiicot — ik-T); (3.9) 16 7t* J

this with the aid of (3.8) we calculate 0(cD,k). Note that, for economy of sym-bols, we have used the same symbols Q and 0, denoting a density function in r,t space as well as a density function in k,co space.

3.3. Some general properties of the piezoelectric wave impedance Z(co,k) Definition

We define the piezoelectric wave impedance Z(co,k) by

Z(co,k) = 0(io,k)/i CO e(co,k). (3.10) In view of (3.8) Z(co,k) is given by

Z(co,k) = [(cok- {^-h-k-i-mco^ + k - c - - k ) - ' • k • / ? } - ' • k ] - ' . (3.11) We shafl now investigate the behaviour of Z(co,k) as a function of k, leaving the direction of k unchanged and parallel to the unit vector u, so that k = ^ u. When the condition {Z(co,k)}~' = 0 is written out explicitly, an equation of the third degree in m co^jk'^ is obtained. From this it follows that, apart from A: = 0, there are in general three additional values oi k^ for which {Z(co,k)}~' becomes zero. Hence we can write Z(co,k) in the form

1 f l x,\n)v,\M) 1

Z(co,k) = - + . (3.12)

/ CO e(u) \k^ 10^ — k^ i^i^(u)Ji = 12.3

The scalar e is identical with u • / 3 " ' • u. The scalars x^ and 1;, will in general depend in a more complicated way upon the direction of the unit vector u. The term 1// w e(u) k^ is of purely dielectric nature and has nothing to do with the piezoelectric properties of the material. The three terms x-^ vt^/(o)^—k^Vi'^)icoe will, as we shall see later, give rise to three modes of propagation. We shall call the scalars Xi the piezoelectric coupling coefficients. The scalars i;, will appear to be the phase velocities of the corresponding mode of propagation.

We shall now derive some general conditions that the scalars e, Xi and Vi must satisfy. These conditions follow from the consideration that the nett energy delivered to the medium by setting up any space-charge distribution should not be negative. Let us more specifically assume a space-charge distri-bution given by

Q{T,t) = co&{kiXi) cosikzXi) cosikiXi) {exp(/'coO + exp(—/co*?)}. We assume that k^^, kj and ^3 are real constants and that co is a complex variable, of which the imaginary part is zero or negative. The electric potential created by this charge distribution, will then be given by

0{r,t) = cos(/riXi) co%{k2X2) cos(A:3X3) {/ m Z exp(/co/) + — /" CO* Z* exp(—/co*?)}.

The energy density delivered to the medium is given by ' öp(r,/o)

W(T,t)=f0(T,t,)- d/o.

-00 Ö/o

This integral can easily be evaluated and we find:

W{r,t} = cos^ikyXi) cos^(A:2X2) 005^(^3X3) exp{—2 Im(co) t} x X ( —Icoj^ Re(Z)/Im(co) + Re[/ m Z exp{2 / Re(co) t}]

Apparently if the imaginary part of co is negative, the real part of Z should necessarily be positive, because otherwise W{T,t) would certainly become nega-tive for certain values of /. As a consequence, the impedance function can have neither poles nor zeros in the lower half co plane, because at a pole or a zero the argument of the function is indeterminate. From this it follows that neither the point k v, nor the point —k v, can be in the lower half cj plane and hence Vj must be real. Per definition we shall take D, non-negative. From an investiga-tion of the behaviour of the impedance funcinvestiga-tion near the points co = 00, o) = ± k Vi and co = 0, we can easily convince ourselves that furthermore the conditions

e(u) > 0,

V ( u ) > 0 , (3.13)

S V < 1

must be satisfied.

3.4. A power expansion for small values of h

As already mentioned, Xj and Vf will in general depend in a complicated way on the direction of u. A qualitative insight into the behaviour of x, and Vj as a function of u can be obtained if we assume «j < 1. In that case an approximate value of Xi and v, can be obtained by applying a series ex-pansion of (3.11) and retaining only the first term. In this way we obtain for Z(co,k) the approximate expression

Z(cü,k) <^(kUcou- ^-' -u)-' -(u- p-'-ïi-u)- i-m co^ + k • c • k)-^ •

• (u • /! • / 3 - 1 . u) (/ co)-i (u • i3-». u ) - ^ (3.14) Next we write down the identity

-w co^ + k • c • k ) = q<qi

C(' k^ — m co^

where qi is a set of three orthogonal unit vectors, pointing in the principal directions of the matrix u • c • u and c/ are the three corresponding eigenvalues. Substituting this in (3.14) we obtain the expression (3.12) with

t;,(u) = {c/(u)/m}l/^

»<((") = q i - ( u - A • / 3 - i - u ) ( c / u - ; ö - i . u ) - i ' ^ ^ • ' A very simple example is obtained in the case of a weakly polarized poly-crystalline ferroelectric. Using the expression (2.3) for the potential-energy den-sity, we then have

Cijki = Ci dij d„i + 2 C2 (Ö,* dji + ell öjk) and

hijk = aidijP„ + 2a2 {Pi dj„ + Pj ó,»), so that

u • c; • u = (ci + 2 C2) u u + 2 C2 /

and (3.16) U • /! = fli U P + 2 02 (u • P / + P u),

where I is the unit matrix and d is the Kronecker symbol.

Apparently one of the principal directions of the tensor u • c • u coincides with u, and the corresponding eigenvalue, say Ci', is equal to c / = c^ + 4 €2-The two other eigenvalues are both equal to 2 C2 and, because of this de-generacy, the corresponding principal directions are indeterminate. Let us choose as unit vectors qi the set

qi = u,

q2 = u x P / | u x P | q3 = u x u x P / | u x P | . Hence, in view of (3.15) and (3.16), we obtain the result

?<.' = («1 + 4 02)' P^ cos^(P,u)/(ci + 4 C2) yS, X2^ = 0,

X3^ = 4 02^ P^ sin^(P,u)/2 C2 p, (3.17) Vi^ ={ci +4c2)lm,

^2^ = f s ' = 2c2/m.

Here (P,u) denotes the angle, included by P and u. The mode with index 1 we shall call the "longitudinal mode". By analogy with the electromagnetic modes, we shall call the mode with index 2 the "ordinary" transverse mode and the mode with index 3 the "extraordinary" transverse mode. Apparently the ordinary transverse mode is not excited in this case. This is a result of the axial symmetry of the material and will be retained if the polarization is no longer weak. The axial symmetry requires in fact that the unit vector qj shall be perpendicular to the axis of symmetry P and the unit vector u. On the other hand, the vector u • /; • /3~^ • u wdl lie in the plane of P and u, so that, according to (3.15), the corresponding coupling factor X2 will be zero. From a similar argument it follows that the terms of higher order

in the expansion of (3.8) will also be zero, so that even for relatively large piezo-electric coupling coefficients the ordinary transverse mode cannot be excited electrically. It can, however, be excited mechanically or by mode conversion, for instance at boundaries. Its propagation velocity will in general be different from that of the extraordinary mode.

We conclude this chapter by remarking that in a conventional delay line the angular dependence of the scalars Xi, e and v, of the piezoelectric material is unimportant. This is due to the fact that the dimensions of the electrodes are large compared with the acoustical wavelength. As a result, only plane waves with a k vector normal or almost normal to the electrodes will be emitted by the electrodes, so that for x„ e and Vi the constant value corresponding to a direction of k normal to the electrodes can be taken. This fortunate circumstance will allow us to treat the piezoelectric material as if it were isotropic. The boundary conditions at the electrodes are also greatly simplified by this circum-stance. We shall make use of this circumstance in the next chapters.

4. A ONE-DIMENSIONAL THEORY OF THE ELECTROACOUSTIC DELAY LINE

4.1. Introduction

In this chapter we shall apply the results derived in chapter 3, to a very simple model. This model is even simpler than the model presented in chapter 1, in that it assumes not only that the electrodes are plane-parallel and embedded in a homogeneous medium of infinite extension, but also that the electrodes are infinitely large. In that case the emitting electrodes can be considered as a pair of infinitely large, oppositely charged planes.

The decomposition of such a space charge into a set of plane waves is par-ticularly simple and the resulting k vectors, characterizing these plane waves, will be normal to the electrodes, so that the model is essentially one-dimensional. Hence the impedance function Z(co,k) can be considered as a function of the modulus of k only. As a result of the resonant character of Z(co,k) as a function of k the charged electrodes will emit electroacoustic waves. The discussion of these waves and the resulting electric behaviour of the fourpole will be the subject of this chapter.

4.2. The one-dimensional reflectionless model

We consider a homogeneous piezoelectric of infinite extension in a frame of reference x,y,z. At x = —d, x = 0, x = / and x = / + e four infinitely thin electrodes are embedded. The electrodes will be denoted by the subscripts A, B, C and D, respectively. We shall assume that, by a suitable choice of the orientation of the material, only one true piezoelectric mode will be excited. This last assumption is not an essential simplification, but it is convenient for our notation. Furthermore, the electrodes shall have negligible mass and negligible electrical resistance. The four electrodes, embedded in the ferro-electric medium, form an electroacoustic delay line. The ferro-electrical behaviour of this delay line can be characterized by two "fourpole" relations:

I'l = Z i i y ' i + Z i 2 72, (4.1) «2 = Z,27l + Z2272- (4.2) The "input" voltage Vy is defined as the potential difference v^-v,^. The

"output" voltage V2 is defined as I'C-*'D- The current ji is the current fed in at X = 0 and extracted at x = —d. Also the current J2 is the current, fed in at the electrode at x = / and extracted at x = / + e. We shall now calculate the "fourpole" impedances Z , i , Z]2 and Z22- Let us assume a harmonically varying input current y"i = Jexp(icot). The output current we set equal to zero. Hence the space-charge distribution is given by

/

e(x,co) = {(3(x) — d(x + d)}, i co A

where A is the area of one electrode. According to (3.2) the Fourier transform of this function is

/

Q{a),ki) — {1 — exp(—/ ki d)}. i a> A

In order to find the piezoelectric potential generated by this charge distribu-tion, we substitute the expression obtained for Q{o),ki) in (3.10) and we apply the Fourier transform (3.9), which gives:

/ / " / 1 x^ v^ \

0(co,x) = / — + — —— {1 - exp i-izd)} exp(-/zx)dz. 2n I CO e A J V z'' co"^ — zv^

(4.3) In (4.3) we have replaced the variable ^, by the complex variable z. However, the integral (4.3) is not uniquely determined unless we specify the path of inte-gration in the complex z plane. Let us first consider the first term 1/z^ of the factor 1/z^ + x^ v^Kco^ — z^ t;^). As remarked in the preceding chapter, this term will give a purely dielectric contribution to the electric potential. In this case it is not of interest whether we choose our path of integration above or below the singularity at z = 0, because the obtained difference in electric potential will be J d/i co e A and hence independent of x. The contribution of this term to the input impedance, which part we shall denote by Z^, is given by

Z j = d/i CO e A. (4.4) Expression (4.4) is none other than the well-known expression for the impedance

of a plane-parallel condenser. The contribution of the dielectric term to the transfer impedance Z,2 is of course zero, because outside the emitting elec-trodes the electric potential is constant.

We next consider the remaining, piezoelectric pait of (4.3). Now the path of integration is no longer arbitrary but has to be chosen in such a way that the boundary condition, called "Ausstrahlungsbedingung", is satisfied. This con-dition can be formulated in the following way. If we make the time factor exponentially increasing with time by giving cj a negative imaginary part, the function 0{x,t) should go to zero for !x| —> oo. Let us first assume that co is real and positive. Then one pole lies on the real axis at z = co/v and the other pole lies at z = ~co/v. If we now give a> a negative imaginary part, the pole on the positive real axis will shift into the fourth quadrant and the pole on the negative real axis will shift into the second quadrant. Apparently the Aus-strahlungsbedingung is satisfied if the path of integration is chosen along the

z plane

f^~\*C.}/V -U)/V

—•—_^ I i ' l

Fig. 4.1. The path of integration in the complex z plane, consistent with the "Ausstrahlungs-bedingung".

real axis. When shifting the poles back to the real axis, <Z>(co,x) should not change discontinuously, so that the path of integration should deviate from the real axis in such a way that it is above the pole at z = co/v and underneath the pole at z = —co/v, as indicated in fig. 4.1. Carrying out the integration along this path we obtain for the piezoelectric part of the electric potential:

J x^ V

0p(co,x) = {exp(—/ co|x|/i;) — exp(—/ co|x + d\/v)]. (4.5) 2 (/ co)^ e A

4.3. The fourpole properties of the one-dimensional reflectionless model The transfer impedance Z12 follows directly from (4.5) as

Zi2 = { ^ > , / ) - ^ p ( c o , / + e)}//, so that we obtain:

x^ V

Z12 = {1 — exp(—/coTi)} {1 — exp(—/C0T2)} exp (—ia>T), (4,6) 2 e (/ coy A

where we have put TI = d/v, TJ = e/v and T = l/v. For the piezoelectric part of the input impedance we find similarly

x^ V ^p = - 7r~r^—7 0 - exp(-/coTi)}, {i my E A so that and similarly d r x"- -| Z i i = 1 {1 — exp(—/coTi)} I oi e A \_ i CO T^ J (4.7) Z 2 2 / CO s - Fl — {1 - exp(-/coT2)}'|. (4.8) A ]_ I CO T2 J

We remark that, due to the absence of spurious reflections, in this model the "direct-signal" fourpole impedances are identical with the conventional fourpole

impedances. The optimum power efficiency rji, which we defined in chapter 1 as j?i = Zi2 Z,2*/4 Re(Zii) Re(Z22)

follows from (4.6), (4.7) and (4.8). We find for rj^ the value 1/4, independent of CO. Physically this should be interpreted in the following way. We apply an electric signal of a duration shorter than T to the input. Then of the electric energy applied to the input, 50% disappears into the negative x direction. Of the remaining 50 %, under optimum load conditions 50 % is dissipated in the electric load, 25% is reflected and 25% is transmitted into the "backing" to the right of the receiving transducer. For the transfer function 0{oj) which was defined as

0(CO) = Zi2/(Z„ Z22)^'' we find the approximate expression

0(0)) f=^ —x^ {1 — exp(—/coTi)} {1 — exp(—/c-jT2)}/2 / co (TI T 2 ) ' ' ^ . (4.9) If for instance we take TI = T2, the bandwidth turns out to be about 1/3TI.

4.4. The impulse response of the one-dimensional model

A convenient experimental technique is to study the behaviour of input and output voltage, resulting from the input current impulse j \ = Q d{t). The cor-responding input voltage v^ follows from the formula

e f

Vi{t) = / Zu(ö>) exp(/coO dco. 2JI J

— 00

An alternative approach is to interpret the factor l//co in (4.7) as an integration with respect to time and the factor exp(—/COT) as a time delay equal to r. From these considerations it follows that the term Q dji oj e A corresponds to a step voltage with height Q d/s A, starting at f = 0. The term ~Q d x^/ (/ cpy BTi A corresponds to a voltage linearly decreasing with time and with slope —Q dx^/e Xy A, also starting at ï = 0. The third term can be derived from the second by reversing the sign and starting a time TJ later. This input voltage as a function of time is drawn in fig. 4.2.

According to (4.6) the output voltage 1^2 consists of four terms. The first term starts at / = 7". It decreases linearly with time with slope —Q x'^ v/2 e A. Let us assume that TI < T2. Then the second term starts at ? = T + TJ with opposite slope, so that the resulting slope is zero. The third term starts at ? = r + T2 with slope Q x^ v/2 E A, until, at ? = T + TJ + T2, the fourth term sets in and the output voltage remains zero. The output voltage as a function of time is drawn in fig. 4.3.

"/

M

n->^)Si

Fig. 4.2. The input voltage, caused by an input current impulse for the one-dimensional reflectionless model. -\ Oit'y '2 rt 2eA v ^ 1 1 T-t-Tt 1 1 7 + T 2

N

Fig. 4.3. The output voltage, caused by an input current impulse for the one-dimensional reflectionless model.

Though the one-dimensional, reflectionless model is still rather unrealistic, it has already a characteristic feature, common to delay lines with plane-parallel homogeneously polarized transducers. This property is the presence of zeros in the transfer impedance Zi2. As follows from (4.6) the zeros occur for those frequencies for which COTJ or C0T2 is a multiple of 27i. In physical terms this is caused by the interference of the two waves generated at the two transmitting electrodes. For actual delay lines, also for co = 0, the transfer impedance will be zero. This property does not, however, follow from our one-dimensional model, but will turn out to be a diffraction effect.

5. THE THREE-DEVIENSIONAL REFLECTIONLESS MODEL

5.1. Introduction

In this chapter we shall, as in the preceding chapter, assume that the plane-parallel electrodes of the delay line are embedded in a homogeneous piezo-electric of infinite extension. Contrary to the model in the preceding chapter, however, we shall now assume that the electrodes aie of finite extension and, more specifically, of identical circular or rectangular shape. We shall further-more assume that the electrodes are located in opposition and that the phase velocity has an extreme value in the direction normal to the electrodes. This last condition is necessary in order to obtain a beam of energy in the direction normal to the electrodes. The line of thought in this chapter will be the same as that in the preceding chapter. Mathematically the analysis will be more involved, due to the more complicated space-charge distribution.

5.2. The space-charge distribution function and the weight function; the homo-geneous approximation

The method followed in the previous chapter for calculating the fourpole parameters was based upon the fact that for infinitely large electrodes the space-charge distribution in the medium is actually known. For finite electrodes, on the contrary, a rigorous calculation of the space-charge distribution is extremely complicated. Fortunately in most practical cases the thickness of the transducers is very small compared with the dimensions of the electrodes so that in that case the charge distribution on the emitting electrodes can be approximated by a homogeneous distribution. Let us assume that the receiving transducer is elec-trically "open", so that the nett charges on the receiving electrodes are zero. However, due to the electric charge applied to the input electrodes, charge dis-placements on the receiving electrodes will be induced. These charge displace-ments will in general be very difficult to calculate. We shall now show that, instead of trying to calculate these charge displacements explicitly, there is a more convenient way of taking these charge displacements into account. Let us apply a current _/i to the input of the delay line. The output current 72 shall be zero. The resulting charge distributions of the electrodes A, B, C and D shall be g^. QB, QC and QQ. The output voltage V2 will then be given by

00

^2(0)) = i(ü f {QA((^X) + ?B(w,k) + ec('«,k) + eD(ft>,k)} Z(co,k)x — 00

X {exp(—/ k • Tc) — exp(—/k • r^)} d/ci d^j dA-3. (5.1) Here Z(co,k) is the piezoelectric wave-impedance function defined in (3.11), and fc is any point on electrode C and r^ is any point on electrode D. For our argument it will be convenient to write (5.1) in an alternative, though

equiv-alent way. To that end we introduce "Green's function" G(co,rlro) which is defined *) as the electric potential generated at the point r, by a point charge of unit strength, applied at the point TQ. AS follows from its definition and the equations (3.2) and (3.9), G(co,rIro) is related to the impedance function Z(co,k) by the equation

00

i CO f

G(co,r|ro) = — - /Z(co,k) exp{/ k • (ro - r)} dk, dki dk^. (5.2) %n^ J

— 00

We remark that Green's function satisfies the relation G(co,rlro) = G(co,ro|r). In this case this "reciprocity property" follows from the relation Z(co,k) = Z(co,—k). It is however of a much more general nature. In terms of Green's function, (5.1) can then be written as

V2{cü) = ƒ e(co,ro) {G(co,rclro) — G(co,rD|ro)} dS'o, (5.3)

A + B + C + D

where TC denotes some arbitrary point on C, r^ some arbitrary point on D and d^o a surface element of one of the electrodes. Because the electrodes are sur-faces of equipotential, instead of taking some arbitrary point on C and D, we may also take the "average" potential, applying an arbitrary weight function which should, however, be normalized to unity. A convenient weight function will appear to be the weight function H'c(co,r) for electrode C and Wo{oi,t) for electrode D, defined in the following way. We apply a currenty2(co) to the output taking the input electrodes non-conducting. Then the surface charge on elec-trode C shall be equal to J2 Wc(cci,r)/i oo and the surface charge on elecelec-trode D to —J2 w^{co,x)/i CO. So we obtain

V2= ƒ e ( r o ) G ( r i | r o ) d S o { / w c ( r i ) d S i - / M r J d ^ J . (5.4)

A + B + C + D C D

Now it is easy to show that of the integration region SQ, which extends over all electrodes, only the region A + B gives a contribution to f 2. To prove this, we replace in (5.4) the function G(ri|ro) by G(ro|ri) and next we notice that the function

ƒ G(rolri) wc(ri) dS. - ƒ G(ro,ri) ^'^(r,) d^i (5.5)

C D

is independent of /-Q on the electrodes C and D. This must be so because (5.5) corresponds to the electric potential generated at VQ by a charge 1 to C and —1 to D, the electrodes A and B being absent (non-conducting). Furthermore both

ƒ Q{TO) dSo = 0 and ƒ Q{TO) dS^ = 0,

so that the contribution of the part C + D of the region A + B + C + D to 1^2 vanishes. Hence by our particular choice of the weight function we have properly taken into account the charge displacements induced by the electrodes A and B on the electrodes C and D.

We conclude that if the charge distribution on the emitting electrodes can be approximated by a homogeneous distribution, for much the same reason the weight function for the receiving electrodes may be assumed to be homogeneous too. In appendix Al we shall give an estimate of the error introduced by this "homogeneous" approximation.

5.3. An approximate expression for the beam-divergence factor Definition

We define the transfer impedance ZMN between two electrodes M and N as the voltage generated by the electrode M on the electrode N, divided by the current supplied to electrode M. In this chapter the charge-distribution function and the weight function for the electrodes M and N shall be assumed to be homogeneous. We remark that, as a result of the reciprocity principle, Z^N =

Z N M '

The fourpole parameters of the delay line with electrodes A, B, C and D can conveniently be written in terms of transfer impedances Z ^ N and we have:

Z i i = ZAA + ZBB 2 ZAB,

Z22 ^ Zee ~r ZDD — 2 ZcD, (5.6) Z12 ^ Zee + ZAD Z^e

Zgo-Transfer properties of circular electrodes

We shall first calculate the transfer impedance Z ^ N for two circular elec-trodes with equal radii a. The electrode M shall intersect the x axis at x = 0 and the electrode N shall intersect it at x = /. The charge-distribution function QtA^k) for such a homogeneously charged circular electrode, as follows from (3.2), is given by

u 2lt <> A »

QM (k)= — ^ /'• dr \ exp (/ K r cos cp) dcp = - — - ^ J^K a), (5.7) Tia^ J J k.a

0 0

where ÖM is the total charge on the electrode at x = 0, k^ is the component of k perpendicular to the x axis, and Jj is the Bessel function of the first order. Similarly averaging the factor exp(—/ k • r) over the electrode N will give a factor

2

exp(—/ ki I) Ji(kr a). (5.8) krO

Hence, in view of (3.9), (5.7) and (5.8) we find:

1 f'JAKa)

ZMN = [ — — Z(co,k) exp(—/ A:, /) dk^ dk2 dk^, (5.9)

— 00

where Z(co,k) is given by (3.12). We shall for the moment only consider the true piezoelectric part Zp(c<j,k) of Z(co,k). This part is given by

x\u) v\u)

Zp(co,k) = , (5.10) / 00 e(u) {oo^ — k-^ i'^(u)}

so that the true piezoelectric part Zp MN of -^MN is obtained by substituting (5.10) for Z(co,k) in (5.9).

We remark that if we take the limiting case a/l -^ co, co a/v —>• oo, we find, by making use of the identity

00

ƒ {JAz)/z} dz = 1/2,

lim Zp MN = — - ; ; exp(-/ o) l/v), (5.11)

a/i-,ao 2 ( / c o ) ' ' 7 t £ a^

aa/v^ oo

in accordance with the result found in the previous chapter. The scalars x, v and E corresponds in that case to a direction of u, parallel to the x axis.

We now introduce the approximation that Zp(cy,k) shall be independent of the direction of k. We shall call this approximation the "isotropic" approxima-tion. This assumption will introduce only a small error because, as will appear presently, only those plane waves with k vector parallel or nearly parallel to the X axis will contribute appreciably to the integral (5.9). We shall discuss the extent of the error, introduced by this "isotropic approximation" in appen-dix A3.

Introducing cylindrical coordinates and integrating (5.9) towards k^ and the cyhndrical angle, we obtain the expression

Jy^ {oo a z/v) f i 0) I

Z p M N - - i 7 ri - Z'V'^ ^""^

0

( 1 - z ^ ) 2 \ l / 2 dz. (5.12) The form (1 — z^)''^ is defined as I for z = 0. Furthermore the path of inte-gration can be chosen along the real axis, above the point 2 = 1 , provided oo is real and positive. We remark that the integration variable z in (5.12) is identical with sin 6, where 6 is the angle included by the wave vector k and the X axis. Apparently if either the condition co a/r > 1 or the condition CO //i; > I is fulfilled, only the integration region 0 < z < I will give an

appreciable contribution to the integral (5.12). If the condition oo a/v 2> 1 is fulfilled, this is due to the rapid convergence of the factor {Ji^ (oo a z/v)}/z and if on the other hand oo l/v > 1, this is because the exponential factor becomes very oscillatory if co / z^/v becomes of the order of unity. We shall assume that at least one of the conditions to a/v > 1, co l/v ^ \ will be ful-filled, so that the "isotropic" approximation will be justified. We now introduce a further simplification also based upon the same assumption, that only small values of z will give an appreciable contribution to the integral (5.12). This simplification is obtained by omitting the factor (1 — z^)~''^ and developing the exponent up to the second power in z in (5.12). We then obtain the approxi-mate expression: 00 x^ r r Ji {oo a z/v) Z J J M N ^ I exp(—iwl/v + iaolz^/2v)dz. (5.13) 71 a^ (/ co)^ e J z 0

We shall call the approximation (5.13) the "paraxial approximation". The parax-ial approximation is of specparax-ial importance for the calculation of the transfer im-pedance Zi2. In appendices Al, A2and A3 expressions are derived for the error in Z,2 introduced by the homogeneous approximation and by the paraxial approximation. It is shown there that, for the typical parameter values l/a = 30, CO a^/l V = 2 the estimated error introduced by the homogeneous approxima-tion is by far the largest, an upper limit being 7%. The calculaapproxima-tion of the input and output impedance is greatly simplified by the condition a/d'> 1, respec-tively a/e » 1. In that case the even more drastic one-dimensional approxima-tion (5.11) can be applied. Also in this case the error introduced by the homo-geneous approximation turns out to be the largest. For the parameter value a/d = 25 the error, introduced at the centre-band frequency ƒ = v/2d by the homogeneous approximation turns out to be less than 8% and the error, introduced by the one-dimensional approximation turns out to be less than 4%, giving in total an upper limit of 12% (see appendices Al, A2 and A3).

Returning to the paraxial approximation, we remark that (5.13) can be written as

ZpM^^- , , „ exp(-/ oo l/v) F, {oo a^/l v), (5.14) 2 71 a {i co)-' E

where

F, {co a^/l i;) = 1 - {Jo {co a'^/l v) + / 7, (co a^/l v)} exp(-/ oo a^/l v). (5.15) For a proof of this result the reader is referred to appendix A2. We notice that (5.14) exhibits a "delay" factor exp{—i oo l/v) and a "beam-divergence factor" Fc, which depends only on one parameter.

However, before discussing (5.14) more in detail, we shall first give the expression for rectangular electrodes. We assume again that the electrodes are of equal size and located in opposition. Let the sides of the rectangular elec-trodes be 2a and 2b. In that case the distribution function ^•(k), which for circular electrodes is given by (5.7), has to be replaced by the expression

sin(/:2 a) sm{k3 b)

QM{k) = ÖM — V 4 • (5.16)

k2ak3 b

Also the weight function (5.8) has to be replaced by the weight function sin(A:2 a) sin(A:3 b) exp(— iki l)/k2 ak^ b. In this case, too, we shall apply the paraxial approximation. The only situation where this approximation is not allowed will be if both one of the sides of the electrodes and the distance between emitting and receiving electrodes is of the order of the wavelength or smaller. This situation is not, however, of practical interest to us. By a proce-dure quite analogous to that used in the case of circular electrodes we arrive at the expression

^^ MN ^ - , ƒ " / , , exp(-/ CO l/v) {Floo a'/l v) F,{o} b^/l v)y'\ (5.17) t, ab{i co)-' e

Here Fs{oo a^/l v) is the beam-divergence factor for square electrodes and is given by

fl /"sin^x 1^ F,{cü a^/l v) = \- I — — exp(/ x^ / v/2 m a^) dx . (5.18)

— CO

Expression (5.17) is made definite by the convention [{F,{ooa^/lv)YYi^ = F,{ooa^/lv).

We notice that apparently the beam-divergence factor is the geometric mean of the two numbers obtained by taking square electrodes with sides respectively 2a and 2b. For this reason we can without loss of generality consider only the case of square electrodes with sides 2b. For numerical calculations it is con-venient to make use of the identity

1 °° • 2 1 I r sin^x r

- I exp(/ x^JTi z)dx = 2 (i zY'^ / (1 - x) e x p ( - ^ z / x^) dx. (5.19) n J x^ J

— 00 0

This identity can be established for instance by expanding the exponential factors in a power series of x^ and integrating term by term. The function G{z) which is defined by

1

can also be written as

2 sin (jr z) f 2 I — cos {71 z)\ (2z)l'2 71 z 1(2 z)i'^ 7tZ J where

z z

C{z) = ƒ cos(:7r x^/2) dx and 5 (z) = ƒ sin(7r x^/2) dx

o 0

are "Fresnel" integrals, which are tabulated for instance in ref. 2. We shall introduce the "beam-divergence time constant" T3 by the definition

r^^ A/2 711V, (5.21) where A is the area of one electrode,

/ is the distance between receiving and emitting electrode, and V is the propagation velocity in the delay medium.

For the case that / is large compared with the linear dimensions of the elec-trodes, a simple physical meaning can be attached to the beam-divergence time constant T3. In that case T3 can be considered as a time dispersion in the transit time between a point of electrode M and a point of electrode N. Furthermore for the case of circular electrodes the dimensionless quantity oo T3 is related to the so-called "Fresnel number" N by the relation:

N =00 T3/2 71. (5.22) In terms of 00 TJ the beam-divergence factor for circular and square electrodes

is in paraxial approximation given respectively by *)

Fc{m T3) = 1 — {^0(2 o) T3) + / J i ( 2 00 T3)} exp(—2 / oo T3) (5.23) and

Floo T3) = / CO T3 {G(co T3)}^ (5.24) where G{z) is defined by (5.20). In fig. 5.1 the modulus of the beam-divergence

factor F{oo T3) is plotted as a function of 00 T3 both for circular and square electrodes. Because the plot has a logarithmic scale this modulus in the case of rectangular electrodes can be obtained simply by intersecting the vertical with abscissa co T3 with the line connecting the points with abscissa co T3 a/b and CO T3 b/a. In fig. 5.1 we have also plotted the modulus of the function Fo(co T3) = / 00 Xi/{\ + / 00 T3). The last function can be used as a rough approximation, for instance in error estimations. Anticipating the results that

*) The calculation of the diffraction losses of a delay line, formulated in a somewhat dif-ferent way, has been treated by several authors 3,4,5,6,15^ Because the referred authors have used an alternative power expansion their results are presented in a difl"erent and more involved form.

A

Fig. 5.1. The modulus of the "beam-divergence factor"; I for square electrodes, II for cir-cular electrodes, HI for electrodes with a gaussian distribution function.

we will obtain in chapter 7, we remark that this function corresponds to a pair of electrodes with a gaussian charge distribution and weight factor.

5.4. The influence of beam divergence on the fourpole parameters

Before calculating the fourpole impedances, we shall first deal with the di-electric contribution to the fourpole parameters. We shall make the simplifying assumption that the electric field corresponding to the dielectric solution is homogeneous between the electrodes of the emitting transducer and zero out-side the electrodes. Because the diameter of the electrodes is large compared with the thickness of a transducer, this assumption will be justified in practice. Hence the dielectric contribution to the input impedance Z, 1 will be equal to d/i 00 E A, where A is the area of one electrode, and e is the dielectric constant in the direction perpendicular to the electrodes. The dielectric contribution to the transfer impedance Z12 can be made negligible by a proper unsymmetric drive, i.e. connecting the inner electrodes to earth. As pointed out in the previous section, for the input impedance and the output impedance the one-dimensional approximation is applicable, so that also for our three-one-dimensional model the expressions (4.7) and (4.8) for Zj 1 and Z22, respectively, are valid.

We now come to the transfer impedance Z12. From the approximate expres-sion F(co, T3) f%^ / CO T3/(l +/CO T3) it can easily be shown that the beam-divergence factor for the nearest electrodes may be taken identical with that for the most remote electrodes. The fractional error, due to this approxi-mation turns out to be of the 01 der of {l/d + i a^/d^)~^ for the relevant fre-quency region. The result is that the beam divergence simply introduces a multiplicative beam-divergence factor F{oo T3) in the transfer impedance Z12. Hence we obtain for Z j j the expression

x-' V

Z12 = {1 —exp(—/ coTi)} {1 —exp(—/(üT2)}F(coT3) exp(—/coT").

The function F{co r^) is given by (5.23) or (5.24). Expression (5.25) is the generalization of (4.6). The influence of beam divergence on the optimum power efficiency r]i will be that the value 1/4, as found for //, in the preceding chapter, will be reduced by a factor F* F. The influence of beam divergence on the bandwidth will be relatively small. Taking an explicit example, let us assume, that A = 1 cm^; thicknesses of transducers d = 200 y. and e = 200 [x; distance between transducers / = 16 cm; acoustical-propagation velocity of the medium t; = 2 5 0 0 m s ~ ' . For the centre-band frequency, defined by CO Ti =71, the beam-divergence parameter co r^ will have the value o r^ = A/2 d I, which is about 1-5. From fig. 5.1 we read that for this value \F\ is about 0-75, which corresponds to an additional energy loss of about 2-5 dB, due to beam divergence.

5.5. Impulse responce of a delay line with circular or square transducers

We shall not deal with the input and output impulse response because this impulse response is very nearly equal to that of the one-dimensional model, discussed in the previous chapter. For the transfer impulse response the di-electric contribution is very small so that we shall neglect this term,justaswe have done in the expression for Z,2.

As an intermediate step we shall first calculate the voltage generated at an electrode N by the application of a current impulse j = Q ö{t) to an elec-trode N. The resulting voltage v^{t) is none else but the Fourier transform of ZMN. multiplied by Q:

0 0

I'N(0 = (Ö/2 7f) / Z M N ( « ) exp(/ 0) t) dco. (5.26 — 00

We introduce the dimensionless time variable 0 = {t ~ l/v)/r-i and the dimen-sionless voltage variable ƒ (?) = —ATI E lvfj{t)/x^ Q. Substituting these varia-bles in (5.26) and substituting tor ZMN the expression (5.14) we obtain the expression

1

f {6) = / exp(/ z Ö) dz, Im(z) < 0. (5.27) 2 7t J (/z)^

— 00

For circular electrodes F{z) is given by

FXz) = 1 - {Jo{2 z) + / J,{2 z)} exp(-2 / z) and for square electrodes F{z) is given by

1

Fs{z) = 4 / z {ƒ (1 — x) exp(—7r / z x^) dx}^.