synthesis of acoustic matching networks by discrete space fourier transform method

Synthesis of acoustic matching networks by discrete space Fourier transform method

J. Stevenson Kenney Scientific Atlanta, Network Systems Group, 4356 Communications Drive, Norcross, Georgia 30093

William D. Hunt

School of Electrical Engineering and The Microelectronics Research Center, Georgia Institute of Technology, Atlanta, Georgia 30332

(Received 29 June 1990; revised 20 November 1990; accepted 6 December 1990)

This paper describes a method used to analyze and synthesize acoustic plane-wave matching structures between a transducer and a medium of arbitrary characteristic impedance. The analysis is applicable to multilayer structures with periodic discontinuities in characteristic impedance. It is shown that the frequency response of such structures is approximately equal to the discrete Fourier transform of the individual layer reflection coefficients. The synthesis procedure uses the inverse discrete time Fourier transform to generate a sequence of layer reflection coefficients, and applies a boundary condition to calculate the characteristic impedances. Using this method, wide bandwidth matching structures can be obtained using finite impulse response (FIR) filter topologies as a model. A numerical example is presented to show how the procedure is implemented, and a linear analysis is performed which models the layers of the synthesized structure as ideal transmission lines. The results of the analysis are in good agreement with the response predicted by the Fourier method.

PACS numbers: 43.20.Rz, 43.20. Fn, 43.40.At

INTRODUCTION

The matching of acoustic impedance between transducers and media is a problem that is common to a variety of ultrasonic applications, including medical imaging and non- destructive evaluation. For these applications, the power transfer efficiency directly affects the dynamic range of the system. Therefore, it is desirable to realize structures in which the power transmitted into a medium is maximized over some prescribed bandwidth. This paper addresses this problem and proposes a method by which such structures may be synthesized.

While the Fourier synthesis method is well established in the literature for electromagnetic transmission line and waveguide impedance matching problems, optical filtering, and digital filtering applications, it has not been directly applied to the problem of acoustic impedance matching. It is the authors' belief that the bulk of the method described in

this paper represents the first application of the discrete Fourier synthesis method to acoustic impedance matching.

I. PREVIOUS WORK

Acoustic impedance matching between piezoelectric transducers and various materials has been investigated for a number of years. Numerous methods have been devised to maximize the power transfer into a particular medium. Be- cause of the similarities between acoustic plane-wave propagation and electromagnetic plane-wave propagation, many of the concepts and methods devised for microwave engineering may be directly applied to this problem. ] A common technique used to match transmission lines is the use of quarter-wave transformers, which provide an impedance

transformation over a wide bandwidth. 2-5 While this meth-

od can be directly applied to acoustic transducers, 6 a major limitation is that the 3-dB bandwidth of the match is deter-

mined by the ratio of the source and load impedance. This may be improved upon somewhat by the use of multiple sections 2'7; however, the bandwidth is fundamentally limited to an octave, and the synthesis procedures become in- creasingly complex. 8'9

To improve the bandwidth of the matching structure a tapered transmission line can be employed, whereby the characteristic impedance of the line changes as a function of distance. Synthesis methods for such tapered transmission lines have been devised for use in microwave networks, 2-5 and for acoustic systems. •ø-•3 These methods are not so easily applied to the case of piezoelectric transducers since it is not practical to assume that the impedance of the medium can be varied continuously over space. Microwave circuits and acoustic waveguides have the advantage that the impedance is a function of the cross sectional dimensions of the

structure. This is not the case with plane-wave piezoelectric transducers. The impedance of a transmission line is a function of the material parameters only, and can be changed only in an abrupt fashion. TM For this reason, it is practical to consider only stepped impedance structures. Holte and Lambert have considered this case for the acoustic wave-

guide problem, and have devised a synthesis procedure for the realization of these structures based on infinite impulse response (IIR), or z-transform theory. • 5 This method is ver- satile in that conventi6nal filter functions, such as Butter- worth and Chebychev, may be realized by choosing the pole locations appropriately in the z plane. The major disadvan- tage is that the Bolinder approximation • is employed,

2123 J. Acoust. Soc. Am. 89 (5), May 1991 0001-4966/91/052123-08500.80 @ 1991 Acoustical Society of America 2123

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whereby incremental changes in the reflection coeficient are assumed to be proportional to the logarithm of the characteristic impedance at a point. This is only valid for small changes in impedance per unit length. In the limit as these changes become infinitesimally small, the solution to the problem becomes that of the continuously varying case. An impedance function is obtained by solving the Ricatti equation in one dimension. 2 Solving this nonlinear differential equation generally requires the use of numerical techniques, and is hence only an approximation to the stepped case. The effects of this approximation become apparent when rela- tively small numbers of sections are used to match very dif- ferent impedances.

The procedure described in this paper uses a discrete time Fourier synthesis method. This technique is adapted from previous work in the digital signal processing (DSP) area, whereby finite impulse response (FIR) methods are used to synthesize tapped delay line fitters. •6 While this technique is well established in the DSP literature, and has also found applications in the area of surface acoustic wave fitters, •7'•8 and optical fitters, •9 the constraints and requirements used to apply this method to acoustic matching networks have not been formulated. This paper will derive the conditions for which this theory may be applied to acoustic impedance matching.

II. FOURIER ANALYSIS OF LAYERED STRUCTURES

Consider the single reflection model shown schematically in Fig. 1. Here we assume plane-wave propagation in a lossless, linear medium, and normal incidence of the plane wave on the matching layers. While these conditions may not be perfectly realized in practice, they can be approximat- ed to some degree. By constraining the cross-sectional dimensions of the transducer and matching layers to be much greater than an acoustic wavelength the wave fronts are approximately parallel to the layer interfaces and propagating perpendicular to them over most of the radiation field. The use of tow loss materials does not greatly perturb the analysis from the lossless case. Linearity is ensured by constraining

z k

INCIDENT WAVE

Pk

REFLECTED WAVE

z k-1

TRANSMITTED WAVE

l+p k

Zk-Zk_ 1 ,Ok= Zk+Zk_ 1

the applied acoustic stress to a low level at which the stiffness matrix is considered linear. At an interface between a mate-

rial of acoustic impedance zk_ • and zk, the portion of the wave amplitude that is reflected is given by p•, where

p• = (z• -- z•_ • )/(z• + z•_ • ). ( 1 )

This definition assumes an incident plane-wave amplitude normalized to unity. Following the conservation of acoustic power, the transmitted wave will have an amplitude given by 1 d-p•. Since there is no mechanism for "memory" in the simple system, Eq. ( 1 ) applies for the entire frequency spec- trum. By memory we mean the dependence of the output signal on input signals which occurred previously in time.

The situation becomes more complicated when a single lossless layer of material is considered, as shown in Fig. 2. An incident monochromatic plane wave will produce a superposition of time delayed reflected and transmitted plane waves due to multiple reflections at each interface. We are mainly concerned with the power transfer function H through the layer. Since the multiple reflections provide a memory mechanism, H will be a function of frequency co. If the sum of all amplitude reflection coefficients is given by F (co), then the steady state power transfer is

H(co) = 1 -- IF(co)I (2) The time delay across one layer for each of the individ-

ual reflections can be expressed in terms of the phase shift in radians 0 given by

O = ( 1/2)fl = (h/V)co, (3)

where h is the thickness of the layer and Vis the propagation

2 -j2O Pk+l (1-Pk)e

2 '1 2. -j4O 'Pk Pk+l [ 'Pk ;e

2 3 '1 2.-j6O Pk Pk+l [ 'Pk ;e

FIG. 1. Reflection from a single interface. FIG. 2. Multiple reflections from a single layer.

2124 J. Acoust. Soc. Am., Vol. 89, No. 5, May 1991 J.S. Kenney and W. D. Hunt: Acoustic matching networks 2124


velocity of the acoustic wave. The scaled phase shift variable fi is introduced for convenience since only even multiples of 0 will appear in the following expressions. Summing the reflected phase shifted waves, we get

F(pk,Pk+ •,9,) =p• +p•+• (1 --p•,)e -•a

-- P kP•, +, ( 1 -- p•, ) e - • m

Rearranging, and collecting terms under the summation, we get

r(pk,Pk + 1, fi )

----Pk d-Pk+ l e--j•--p•pk+ l e--jn

d- (1--p•,) • (- 1)nlp•,+lp•,+•e -j(n2)a + ß

(5)

In solving the infinite summation, we obtain

r(pk,Pk+ l, fi) =Pk d-Pk+ l½--Jfi d- 6k(Pk,Pa+ l,fi), (6) where

6k (Pk,Pk + 1, fi) = -- (,Ok,Ok+ l ½--Jfi) (pk --,Ok+ 1 ½--jfi)

1 + PkPk + 1 e -j • (7)

Note that in Eq. (6), if we constrain fi to be constant for a given frequency, for all k, then the first two terms in the sum can be thought of as a discrete time Fourier transform (DTFT) ofthe sequence (-..0, O, pk,Pk +l, 0, 0'' .).16 The phase shift is held constant by keeping the ratio of thickness h to propagation velocity V constant. It is also important to note that the Fourier coefficients are of the order of p (p being the average value of the sequence P k ), whereas the error term coefficients ek are approximately of the orderp 3 if p is small. This seems reasonable given the fact that a multiply reflected wave must be reflected off of a minimum of three interfaces in order to propagate back toward the input. Thus we see that the reflection coefficient for a single layer is approximately equal to the DTFT of the reflection coefficient of the individual interfaces. We will now extend this to

the case of multiple matching layers. The problem of extending the result derived for the case

of a single layer, Eq. (6), to that of an N-layer structure (shown in Fig. 3 ) is two fold. First, like the previous case, we have to deal with multiple reflections between all interfaces in the structure. The superposition of these multiple reflections will distort the predicted frequency response of the

Z Z Z Z Z Z Z Z Z Z

S I 2 3 4 Zk-1 Zk Zk+l N-3 N-2 N-1 N , FIG. 3. Multilayer acoustic matching structure.

structure to some degree. As was the case when only a single layer is considered, we will show that the magnitude of these perturbations are small compared to the primary reflection; hence, they can be ignored in the synthesis procedure developed in the next section. The second problem encountered is that of the transmission loss of the incident and reflected

waves through the preceding layers in the structure. The errors incurred by neglecting this effect are in most cases too great to be ignored in the analysis of multilayer structures. While this effect is not easily treated in the synthesis procedure, a correction term can be calculated to offset these errors.

Consider the reflection at the input of an N-layer structure due to the k th layer. The incident wave will be multiplied by the factor 1 + pk as it passes beyond the k th interface. Similarly, the reflected wave will be multiplied by a factor of 1 --p• on the return. Thus each singly reflected wave will be attenuated by a factor of ( 1 -- p•) at each interface (indicated by the index 1) preceding the k th layer. This is easily handled by a product term applied to each p•. We must also account for the multiply reflected waves in a man- ner similar to that presented for the case of a single layer. In general, this is quite difficult to do since we must include the effects of N reflected waves being reflected and re-reflected off of N- 1 interfaces. Thus we have N[ (N- 1 )! ]2 multiply reflected waves. In addition to this, as in the case of the primary reflection, we must account for the transmission loss for each of the multiply reflected waves. Rather than rigorously evaluating the effects of the multiple reflections, we will let E• (p• ,P2 ,'",RN, fi) represent the ensemble amplitude of each of the N reflected waves. Reasoning inductively from the single layer case, we can conclude that the magnitude of this term is of the order ofp 3 if [p[ ,• 1. Thus given this assumption, the reflection coefficient referenced to the input of an N-layer structure F• can be expressed as

k--1

r k (•["•) --pk ½--jkf• ff (1 --p•) l=0

d- Ek (PO 'PI''"'PN' • )' ( 8 )

It is important to notice that the product in Eq. (8) can be represented as

k--1

II (1-p•) = 1 d- Tk(po,Pl,...,Pk_l ), (9) l=0

where T k is defined as the remainder of the product expression. The utility of substituting Eq. (9) into Eq. ( 8 ) is apparent when we sum the reflection terms of each interface as

given in Eq. (8). In doing this, we obtain an expression for the reflection coefficient at the input of the structure:

N N

= pke -jm + • Tk(Po,pt,...,pN)e --jm k =0 k=O

N

d- Z E•, (po,p/,...,pN,fi). (10) k=O

We see that the first term in Eq. (10) is the DTFT of the sequence of reflection coefficients, Pk. It is this term that we would like to retain for use in the synthesis procedure described in the next section. The second term results from the

transmission loss which occurs as the wave traverses the k th



layer in the forward and reverse directions. Given Pi, i = 0, 1,2,...,N, and solving for Tk in Eq. (9), we can calculate the value of this term as a function of 1•. In this sense, it can be thought of as a correction term to be applied to the first term which accounts for the effects of the transmission loss. It can

be seen from Eqs. (9) and (10) that the magnitude of this term is of the order of p3, where p is defined to be an average value of the sequencepk. The last term accounts for the multiple reflections between layers. Since each E• is of the order of p3, we can assume that the magnitude of the E•'s are sufficiently smaller than unity. It is then correct to say that the input reflection coefficient F (1•) is approximately equal to the DTFT of the sequence of individual reflection coefficients p•.

At this point, having established a relationship between F(11) and the DTFT of p•, we would like to develop an analogy between this situation and sampled system theory. The relationship between the continuous time frequency co (in radians per second) and what is called the discrete time frequency 1• (in radians ), is established by the sampling rate of the system. It is convenient to use the DSP nomenclature of discrete time frequency for the variable 1• rather than the phase shift as we have been discussing it. The reason for this is that, if the sampling rate of a system is held constant, the continuous time frequency variable co is simply a scaling of 1• over the interval 0<l• < 2rr. In this sense, the factor 2h/V can be thought of as the sampling period of our spatially sampled system. In fact this situation is analogous to that of a surface acoustic wave (SAW) filter, where an acoustic signal is sampled in time as it propagates past periodically spaced interdigitated electrodes. •7'•8 Figure 4 shows this situation schematically. It must also be noted that like any sampled continuous time system, the discrete time frequency response function F (1•) of the spatially sampled layered structure will be periodic in 1• with period 2rr rad. Equiv- alently, F (co) is periodic in co with period rrV/h rad/s.

In this section we have established a method whereby we can calculate the input reflection coefficient as a function of frequency co. In doing this, we use the relationship first presented in Eq. (3), 1• = 2hco/V, and assume that the ratio of the thickness h to the acoustic velocity V is a constant throughout the structure. Then, neglecting errors incurred by multiple reflections, we arrive at the analysis formula:

FIG. 4. Schematic representation of a transversal filter.

N

r(n) = k-----O

N

-• + • p• T• (Po,Pi , "' ,P•v )e-•m. k=0

(11)

T• is easily found from Eq. (9) to be k--I

= II /=0

We will use Eq. ( 11 ), neglecting the transmission loss term, as a basis for the synthesis procedure discussion in the next section.

III. SYNTHESIS OF MATCHING NETWORKS

The synthesis problem involves generating a sequence of layer impedances from a specified input reflection coefficient function F (ll). In the synthesis procedure, we neglect the effects of the transmission loss through the layers, and the superposition of multiple reflections. In doing this, the synthesis procedure involves taking the inverse DTFT of F (l•). The result of the inverse transform is a sequence pk which is given by

p• =-• _ F(ll)e•dll k=0,1,...,N. (13) The response of the synthesized sequence can be analyzed using Eq. ( 11 ), which applies a correction for the transmission loss. Oncep• is known, the impedances of the individual layers can be calculated by applying a boundary condition.

Before the synthesis procedure can be performed, several constraints must be applied to F (11) in order to generate a sequence p• that can be physically realized. The first criterion is that of finite length. Obviously, one cannot realize structures of infinite length, so a requirement must be placed on the sequence that it must end at some integer N. Related to this is the requirement of causality. For our purposes, this simply means that the sequence pk must begin at k = 0. These conditions are met by defining F (1•) appropriately. While general classes of functions exist whose inverse DTFT satisfy the above requirements, their treatment is difficult. What is more often done in designing FIR filters, such as SAW filters for instance, is to choose an ideal low pass filter function and modify the generated sequence to meet these conditions. 16 These modifications lead to predictable errors in the expected response of the structure.

Consider the case where the reflection coefficient is given by an ideal low-pass function

F(11) = {• ,11, < 11c Inl>nc' (14) This is shown in Fig. 5 (a). Since the available power gain of a lossless network is defined by

= -- II'l :, the above definition for F corresponds to an ideal high-pass filter. Of course for discretely sampled systems, the stopband is repeated in the l•-domain every 2rr rad, and in the co domain with a spacing of 2rr/T (where T = 2h/Vis the sampling period). Thus the ideal response of the sampled system will have a re-entrant stopband. We see that using Eqs. (14)

2126 J. A½oust. So½. Am., Vol. 89, No. 5, May 1991 J.S. Kenney and W. D. Hunt: Acoustic matching networks 2126


1.5

0.5

0

-0.5 0.00 2.00 4.00 6.00 8.00 10.00

FREQUENCY (MHz)

0.30

0.25 ß ß ß ß ß ß ß ß 0.20

0.15

O.lO

0.05

0.00

-0.05

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

LAYER NUMBER

1.5

0.5

0

-0.5 o

(d) 1.5

0.5

0

-0.5 0 00 2.00 4.00 6.00 8.00 10.00 00 2.00 4.00 6.00 8.00 10.00

FREQUENCY (MHz) FREQUENCY (MHz)

FIG. 5. (a) Ideal low-pass function used to generate individual layer reflection coefficients. (b) Layer reflection coefficients calculated from inverse DTFT, Eq. ( 13 ), of ideal low-pass function. (c) Input reflection coefficient neglecting transmission loss and multiple reflections, calculated from DTFT of (b). (d) Input reflection coefficient calculated using Eq. (12). Includes correction for transmission loss, but neglects multiple reflections.

and (15), we can obtain a power match over some finite bandwidth determined by 11c and T.

The inverse transform of Eq. (14) produces the sequence

sin lick /ok=• k=0,_l,_+2,.... (16)

rrk

We see that this sequence is noncausal and infinite in length, but the magnitude of the terms approaches zero as I kl gets large. In order to form a realizable network, the sequence must be modified so that it only has nonzero elements for 04k4N, where N is a fixed integer. The simplest form of modification is to simply truncate the sequence at k = 0, and k = N. The error produced by this operation is minimized if the function is tending to zero as k approaches 0 and N, 16 SO we simply shift the sequence to the middle of the "window," and truncate it. The shift will result in a linear phase shift in F (fi), but will leave the magnitude unchanged. The larger the truncation window is made, the smaller the error in F (ll) will be. This can be seen from the following argument. By the convolution theorem, •6 a multiplication in the k domain is equivalent to a convolution in the fi domain. The truncation of the sequence is equivalent to a multiplication by an ideal low-pass function. So the effect can be calculated

by convolving F (fi) with the DTFT of this function, which is a sinc (fi) function. As Nbecomes large the sinc (fi) function approaches a Dirac delta function •5(fi). Since this is the identity element for the convolution operation, we can see that as N becomes large, the errors incurred by the truncation become small. It might also be noted that other methods of truncation have been shown to produce smaller errors in ideal lowpass FIR filters. •6 These generally involve multiplication by a windowing function with a less abrupt transition to zero.

Performing the truncation procedure on Eq. (16), one obtains the sequence Pk, given by Eq. (17) and shown graphically in Fig. 5 (b).

sin[ (k- N/2)lic ] ,ok = k = 0,1,2,...,N. (17)

•r(k -- N/2) The errors to the ideal filter function which occurred be-

cause of the truncation can readily be seen by calculating the DTFT ofEq. (17). An example of this is shown in Fig. 5 (c).

The sequence P k uniquely defines some filter function F (fi); however, no mention of the actual source and load impedances Zs and ZL, as shown in Fig. 3, has been made. This implies that an additional constraint will be required. Using Eq. ( 1 ) we can calculate the characteristic impedance



of the N th layer if we know PN and ZL. Similarly, we can calculatezN_ • if we know zN andpN_ •. Following this process, the impedance of each section can be calculated iteratively, working back from the load, as shown in Eq. (18)

1 •tON Z N = Z L

1 +PN ß

ß

1 --PN 1 --PN--• 1 --p• Z i = Z L ß . .

1 •-PN 1 •-PN--1 1 •-Pi

1 --,o N 1 --,o N_ 1 1 --Po Z 0 =Z L ''' . (18)

1 +PN 1 +PN--• 1 +Po

If we follow this process, we find that the impedance Zo is not equal to Zs. This situation is similar to the theory of double-terminated ladder networks in lumped element filter design. 2ø Following this analogy, we find that we can multiply the sequence Pk by a constant b so that Zo equals Zs. It is shown in the Appendix that such a constant always exists for Zs,ZL > 0. The constant is found by setting Zo equal to Zs and solving the polynomial p(b) given as

N ZL N p(b)-II (1--bpm)=O.

m=O Z S m=O

(19)

Of the N roots found, we then choose the real, positive root which satisfies Eq. (20):

lb I<l/Pmax

tOmax = max{p•) i = 0,1,2,...,N. (20) Equation (20) also forces the sequence to satisfy one final realizability requirement that the magnitude of pk be less than or equal to unity. Once the value of b is calculated, the impedance of each layer can be calculated using Eq. (18), replacing p•, with bpm. Since the DTFT is a linear transformation, we find that the effect of this scaling is equivalent to scaling F (f•) by b. Since there is no requirement that lb I< 1, this may lead to a situation where IF (cz) I > 1 for some finite range of f•. We will find, however, that the transmission loss correction term given in Eqs. ( 11 ) and (12) will correct for this when applied. Figure 5(d) shows an example of the frequency response of the synthesized sequence as predicted by Eq. ( 11 ). The magnitude of F is still slightly larger than unity in the stop band because the effects of multiple reflections have been neglected.

IV. A NUMERICAL EXAMPLE

To illustrate the method described above, a typical problem will be discussed. All calculations and figures are obtained using a MATHCAD TM model generated by the authors to implement the synthesis procedure and analyze the errors. In addition to this, the results of a steady state AC model are shown for comparison. This model was generated using SUPER-COMPACT TM, a linear circuit analysis program, which includes all error effects discussed in this paper.

Consider a transducer fabricated from PZT-5A

(Zo = 33.6 MRayl). We wish to match this transducer to a medium which has a characteristic impedance close to that

of water (Zo = 1.5 MRayl). The required bandwidth is from 2 to 6 Mhz ( 1.58 octaves). We desire that the magnitude if the reflection coefficient be no greater than 0.2 in this bandwidth, which corresponds to a power transmission efficiency of 96%.

Given this set of constraints, we choose the lower cutoff frequency fc to be 1 Mhz to allow for the filter transition region. Since the passband must extend through 6 Mhz, we must choose the sampling rate f• = 1/T such that the re- entrant stopband is about 500 kHz above this point. Also, since the re-entrant frequency can be calculated from

fr =is --fc, (21)

a sampling rate of 8 MHz will set the re-entrant point to be 7 Mhz. This sampling rate corresponds to a phase length of 22.5 ø per layer.

The ideal filter response is shown in Fig. 5 (a). The number of layers N must be chosen such that this is not seriously distorted by the truncation. The distortion that we are pri- marily concerned with is that in the passband, and that in the transition band. Since limiting the number of layers is equivalent a multiplication by a window function in the spatial domain, one could estimate the effect of this operation by convolving in the frequency domain the step function shown in Fig. 5 (a) with a sinc function arrived at by computing the Fourier transform of the window funtion. In this example N has been chosen empirically to be 32, simply by examining the generated sequence p•. It is seen from Fig. 5 (b) that the maximum value ofp• is approximately one-tenth of the val- ues ofp• toward the end of the sequence, so we can assume that we are safe in choosing N as such. The reflection coefficient F(f•) is calculated using the DTFT plotted in Fig. 5 (c). It is seen that the predicted response (neglecting multiple reflections) meets the design criterion of a reflection coefficient magnitude of less than 0.2 in the passband. How- ever, we have not yet examined the error term to see if we will exceed this specification because of the transmission losses. To do this we must first calculate the normalization factor.

From the sequence p• we apply the normalization condition of Eqs. (19) and (20). To avoid the computational effort required to solve a 32nd order polynomial, we ap- proximate p(b) by including only the five most significant terms, Pko-2 tOpko+ 2, where k0 = N/2. Noting that the sequence is symmetric about k 0, we obtain a fifth order polynomial. With an initial guess of b = - 1, we solve by New- ton's method to obtain the normalization value,

b -- - 1.474. Now that the normalized sequence is known, we can calculate the error function given in Eq. (12). As stated in the last section, it is equivalent to sum from 0 to N, since no reflections occur past the N th layer. Also, we note that the last interface has no multiple reflections associated with it, so we then need only to sum from 0 to N-- 1. The corrected frequency response function is shown in Fig. 5 (d). It is seen that the response has been distorted from that shown in Fig. 5 (c), but does not exceed the design specification. We also note that the magnitude exceeds unity. This is because the stored energy in the multiply reflected waves has not been accounted for.

The impedance of each layer may be calculated



• 45

• 40

u 35 z

• 30

• 20

• •o

u 5

, , , , , , , , , , , , , , , , , , , , , , , , , , , ,

0 8 1.6 24 32 LAYER NUMBER

FIG. 6. Characteristic impedance of matching layers calculated using Eqs. (18)-(20).

iteratively as in Eq. (18). This is plotted in Fig. 6. As expected, we see the characteristic impedances of the layers vary about a line between Zs and zL, with the greatest differential change occurring about the middle of the structure.

We now analyze the synthesized network using a linear microwave circuit analysis program, SUPERCOMPACT TM, the results of which are shown in Fig. 7. This software pack- age calculates transmission matrices at each frequency from linear circuit models and cascades these to simulate the fre-

quency response of the circuit. The layers have been modeled as ideal lossless transmission lines, each having a phase length of 22.5 ø and an impedance as calculated in the MATHCAD TM model. The results of this analysis are closer to what we would see in physical measurements than that shown in Fig. 5 (d) because the effects of multiple reflections are included. Figure 7 shows a plot of the magnitude of the input reflection coefficient F (co). It is seen that, including the effects of the higher order reflections, the synthesized network meets the specifications presented earlier.

V. SUMMARY AND CONCLUSIONS

A method has been presented whereby a network of discrete layers of uniform phase delay can be synthesized to

'11 I I Vl o 0 1 2 3 4 5 6 7 8 9 10

FREQUENCY (MHz)

FIG. 7. Input reflection coefficient calculated by SUPERCOMPACT TM model. Includes effects of transmission loss and multiple reflections.

match an acoustic transducer to an arbitrary load medium. A numerical example is presented and analyzed using an ideal transmission line model. The results of the analysis agree well with the predicted results of the synthesis procedure, indicating the validity of the approximations used to arrive at this method.

In this analysis, we have assumed that several ideal situations exist that will not be perfectly achieved in practice. The first of which is that of plane-wave propagation, which allows for a one dimensional analysis. This assumption is valid when the dimensions of the transducer face are much

greater than an acoustic wavelength. When the plane-wave assumption is not valid, one might consider doing the analysis in two dimensions. However, in doing this, the synthesis procedure is greatly complicated, and may not be feasible. The other assumption is that lossless materials are used in the matching section. Again, the analysis can be augmented to include these effects, but the synthesis method is not long- er direct. One can make some generalizations as to the effect of lossy materials based on an analogous situation which occurs in lumped element filter synthesis. In general, the synthesis of electrical filters is done with lossless elements because of the simplicity of this approach. When one per- forms the analysis after including the effects of losses, the most noticeable degradation usually occurs in the transition band: the steepness is reduced. This might also be expected when one analyzes a structure composed of lossy materials where the impedances have been generated from the synthesis procedure described earlier.

Other possibilities for further research occur to one when considering the realization of the synthesized structure. The precision with which layer thickness can be fabricated is limited, and obtaining materials with exactly the correct characteristic impedances is not always possible. These effects can best be dealt with empirically by substituting the actual parameters in the model and observing the effects, or by assuming these variables to be stochastic and computing a histogram of the response variations.

APPENDIX: EXISTENCE OF A NORMALIZATION FACTOR

Equation (18) implies that a normalization factor is required when the impedance of the source differs from that of the load. This factor is calculated by solving the polynomial given in Eq. (20) and repeated here and parameterized with Ot -- ZL /Zs •/O

N N

rn =O m =0

(A1)

We wish to examine the roots of this equation as a varies from zero to infinity. Fortunately, the study of this problem has been developed extensively in control systems theory, and is known as the root locus method. 2• Note that when

a = 0, the roots of Eq. (A 1 ) are given by N

m=O



As a tends toward infinity, the roots of Eq. (A 1 ) approach those of the polynomial

N

N(b) = 1-[ (1 -- bpm ) = O. (A3) rn •---• 0

Since the roots of Eqs. (A2) and (A3) are all real, the root locus will originate from the roots of D(b), proceed along the real axis, and terminate at the roots of/V(b) as a is varied from zero to infinity. In the control theory literature, the roots of D( b ) and/V(b) are referred to as the open loop poles and zeros, respectively. In our case, the coefficients of the factors Pm are the same for both polynomials, so the individual roots progress from - 1/pro to d- 1/t3m as a is varied from zero to infinity.

In order to satisfy the physical realizability requirement that the magnitude of the reflection coefficient of an interface be less than or equal to unity, we must choose a root b which satisfies

Ibpml<l m = 0,1,2,...,N. (A4)

This implies that b must be in the range

1 1 --•<b<• for 0<a<oo

tOmax tOmax (A5)

Pmax = max•oi • i = 0,1,2,...,N.

Depending on the value of a, up to N roots exist that will satisfy this requirement. In addition to this, we know from the root locus that one root will satisfy this requirement for all a. This will be the smallest magnitude root of Eq. (A 1 ),

and is hence chosen to be the normalization factor to avoid

ambiguity.

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synthesis of acoustic matching networks by discrete space fourier transform method

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