《电子工程师手册》学习资料（英文版）Chapter 47 Surface Acoustic Wave Filters.pdf_大学文库

47 Surface acoustic Wave Filters 47.1 Introduction 47.3 Basic Filter Specifications 47.4 SAW Transducer Modeling The SAW Superposition Impulse Response Transducer Model. Apodized SAW Transducers 47.5 Distortion and Second-Order Effects 47.6 Bidirectional Filter Response 47.7 Multiphase Unidirectional Transducers 47.8 Single-Phase Unidirectional Transducers Donald c. malocha 47.10 Coded SAW Filters University of Central Florida 47.11 Resonators 47.1 Introduction A surface acoustic wave(SAW), also called a Rayleigh wave, is composed of a coupled compressional and shear wave in which the SAW energy is confined near the surface. There is also an associated electrostatic wave for a SAw on a piezoelectric substrate which allows electroacoustic coupling via a transducer SAw technologys two key advantages are its ability to electroacoustically access and tap the wave at the crystal surface and that the wave velocity is approximately 100,000 times slower than an electromagnetic wave. Assuming an electro- magnetic wave velocity of 3 x 108 m/s and an acoustic wave velocity of 3 X 10 m/s, Table 47. 1 compares relative imensions versus frequency and delay. The SAw wavelength is on the same order of magnitude as line dimensions which can be photolithographically produced and the lengths for both small and long delays are achievable on reasonable size substrates. The corresponding E&M transmission lines or waveguides would be Because of SAWs relatively high operating frequency, linear delay, and tap weight (or sampling) control, y are able to provide a broad range of signal processing capabilities. Some of these include linear and dispersive filtering, coding, frequency selection, convolution, delay line, time impulse response shaping, and others. There are a very broad range of commercial and military system applications which include components for radars, front-end and IF filters, CATV and VCR components, cellular radio and pagers, synthesizers and analyzers, navigation, computer clocks, tags, and many, many others [Campbell, 1989; Matthews, 1977 There are four principal SAw properties: transduction, reflection, regeneration and nonlinearities. Nonlinear elastic properties are principally used for convolvers and will not be discussed. The other three properties are present, to some degree, in all SAw devices, and these properties must be understood and controlled to meet device specifications. c 2000 by CRC Press LLC

© 2000 by CRC Press LLC 47 Surface Acoustic Wave Filters 47.1 Introduction 47.2 SAW Material Properties 47.3 Basic Filter Specifications 47.4 SAW Transducer Modeling The SAW Superposition Impulse Response Transducer Model • Apodized SAW Transducers 47.5 Distortion and Second-Order Effects 47.6 Bidirectional Filter Response 47.7 Multiphase Unidirectional Transducers 47.8 Single-Phase Unidirectional Transducers 47.9 Dispersive Filters 47.10 Coded SAW Filters 47.11 Resonators 47.1 Introduction A surface acoustic wave (SAW), also called a Rayleigh wave, is composed of a coupled compressional and shear wave in which the SAW energy is confined near the surface. There is also an associated electrostatic wave for a SAW on a piezoelectric substrate which allows electroacoustic coupling via a transducer. SAW technology’s two key advantages are its ability to electroacoustically access and tap the wave at the crystal surface and that the wave velocity is approximately 100,000 times slower than an electromagnetic wave. Assuming an electromagnetic wave velocity of 32108 m/s and an acoustic wave velocity of 32103 m/s, Table 47.1 compares relative dimensions versus frequency and delay. The SAW wavelength is on the same order of magnitude as line dimensions which can be photolithographically produced and the lengths for both small and long delays are achievable on reasonable size substrates. The corresponding E&M transmission lines or waveguides would be impractical at these frequencies. Because of SAWs’ relatively high operating frequency, linear delay, and tap weight (or sampling) control, they are able to provide a broad range of signal processing capabilities. Some of these include linear and dispersive filtering, coding, frequency selection, convolution, delay line, time impulse response shaping, and others. There are a very broad range of commercial and military system applications which include components for radars, front-end and IF filters, CATV and VCR components, cellular radio and pagers, synthesizers and analyzers, navigation, computer clocks, tags, and many, many others [Campbell, 1989; Matthews, 1977]. There are four principal SAW properties: transduction,reflection,regeneration and nonlinearities. Nonlinear elastic properties are principally used for convolvers and will not be discussed. The other three properties are present, to some degree, in all SAW devices, and these properties must be understood and controlled to meet device specifications. Donald C. Malocha University of Central Florida

TABLE47 1 Comparison of SAW and E&M Dimensions versus Frequency and Delay, Where Assumed Velocities ar VsAw =3000 m/s and vEM=3 X 10 m/s =300um 1.5m 入1=0.15m lay=1 ns =3 um LEM =0.3 m Delay =10 us =30mm FIGURE 47.1(a) Schematic of a finite-impulse response( FIR) filter.( b)An example of a sampled time function; the envelope is shown in the dotted lines. (c)A SAw transducer implementation of the time function h(t) A finite-impulse response(FIR)or transversal filter is composed of a series of cascaded time delay elements which are sampled or"tapped"along the delay line path. The sampled and delayed signal is summed at a nction which yields the output signal. The output time signal is finite in length and has no feedback. A schematic of an FIR filter is shown in Fig. 47.1 A SAW transducer is able to implement an FIR filter. The electrodes or fingers provide the ability to sample or tap"the SAW and the distance between electrodes provides the relative delay For a uniformly sampled SAw transducer, the delay between samples, At, is given by At= AL/vo where AL is the electrode period and va is the acoustic velocity. The typical means for providing attenuation or weighting is to vary the overlap between adjacent electrodes which provides a spatially weighted sampling of a uniform wave. Figure 47. 1 shows a typical FiR time response and its equivalent SAw transducer implementation. A SAW filter is composed of a minimum of two transducers and possibly other SAW components. A schematic of a simple SAw bidirectional filter is shown in Fig. 47. 2. A bidirectional transducer radiates energy equally from each side of the transducer(or port). Energy not being received is absorbed to eliminate spurious reflections 47.2 SAW Material Properties There are a large number of materials which are currently being used for SAw devices. The most popular gle-crystal piezoelectric materials are quartz, lithium niobate(LiNbO, ) and lithium tantalate(LiTa,O5). The materials are anisotropic, which will yield different material properties versus the cut of the material and the rection of propagation. There are many parameters which must be considered when choosing a given material for a given device application. Table 47. 2 shows some important material parameters for consideration for for of the most popular SAW materials Datta, 1986; Morgan, 1985] c 2000 by CRC Press LLC

© 2000 by CRC Press LLC A finite-impulse response (FIR) or transversal filter is composed of a series of cascaded time delay elements which are sampled or “tapped” along the delay line path. The sampled and delayed signal is summed at a junction which yields the output signal. The output time signal is finite in length and has no feedback. A schematic of an FIR filter is shown in Fig. 47.1. A SAW transducer is able to implement an FIR filter. The electrodes or fingers provide the ability to sample or “tap” the SAW and the distance between electrodes provides the relative delay. For a uniformly sampled SAW transducer, the delay between samples, Dt, is given by Dt = DL/va, where DL is the electrode period and va is the acoustic velocity. The typical means for providing attenuation or weighting is to vary the overlap between adjacent electrodes which provides a spatially weighted sampling of a uniform wave. Figure 47.1 shows a typical FIR time response and its equivalent SAW transducer implementation. A SAW filter is composed of a minimum of two transducers and possibly other SAW components. A schematic of a simple SAW bidirectional filter is shown in Fig. 47.2. A bidirectional transducer radiates energy equally from each side of the transducer (or port). Energy not being received is absorbed to eliminate spurious reflections. 47.2 SAW Material Properties There are a large number of materials which are currently being used for SAW devices. The most popular single-crystal piezoelectric materials are quartz, lithium niobate (LiNbO3), and lithium tantalate (LiTa2O5). The materials are anisotropic, which will yield different material properties versus the cut of the material and the direction of propagation. There are many parameters which must be considered when choosing a given material for a given device application. Table 47.2 shows some important material parameters for consideration for four of the most popular SAW materials [Datta, 1986; Morgan, 1985]. TABLE 47.1 Comparison of SAW and E&M Dimensions versus Frequency and Delay,Where Assumed Velocities are vSAW = 3000 m/s and vEM = 3 2 108 m/s Parameter SAW E&M F0 = 10 MHz lSAW = 300 mm lEM = 30 m F0 = 2 GHz lSAW = 1.5 mm lEM = 0.15 m Delay = 1 ns LSAW = 3 mm LEM = 0.3 m Delay = 10 ms LSAW = 30 mm LEM = 3000 m FIGURE 47.1 (a) Schematic of a finite-impulse response (FIR) filter. (b) An example of a sampled time function; the envelope is shown in the dotted lines. (c) A SAW transducer implementation of the time function h(t)

absorber Output piezoelectric substrate FIGURE 47.2 Schematic diagram of a typical SAw bidirectional filter consisting of two interdigital transducers. The transducers need not be identical. The input transducer launches waves in either direction and the output transducer converts the acoustic energy back to an electrical signal. The device exhibits a minimum 6-dB insertion loss. Acoustic absorber damps wanted SAW energy to eliminate spurious reflections which could cause distortions TABLE 47.2 Common SAw Material Properties Parameter/Material ST-Quartz YZ LiNbO, 128YX LiNbO, YZ LiTa, O, k2(%) 0.16 C (pf/cm-pair) 3,159 Temp. coeff. of delay (ppm/C) The coupling coefficient, k2, determines the electroacoustic coupling efficiency. This determines the fractional bandwidth versus minimum insertion loss for a given material and filter. The static capacitance is a function of the transducer electrode structure and the dielectric properties of the substrate. The values given in the table orrespond to the capacitance per pair of electrodes having quarter wavelength width and one-half wavelength period. The free surface velocity, vo, is a function of the material, cut angle, and propagation direction. The temperature coefficient of delay(TCD)is an indication of the frequency shift expected for a transducer due to a change of temperature and is also a function of cut angle and propagation direction The substrate is chosen based on the device design specifications and includes consideration of operating temperature, fractional bandwidth, and insertion loss. Second-order effects such as diffraction and beam G eering are considered important on high-performance devices [Morgan, 1985]. Cost and manufacturing tolerances may also influence the choice of the substrate material. 47. 3 Basic Filter Specifications Figure 47.3 shows a typical time domain and frequency domain device performance specification. The basic frequency domain specification describes frequency bands and their desired level with respect to a given reference Time domain specifications normally define the desired impulse response shape and any spurious time responses The overall desired specification may be defined by combinations of both time and frequency domain specifications Since time, h(o), and frequency, H(o), domain responses form unique Fourier transform pairs, given by h(r)=1/2T H(o)e/do (47.1) h(t)e"u dt (47.2) c 2000 by CRC Press LLC

© 2000 by CRC Press LLC The coupling coefficient, k2, determines the electroacoustic coupling efficiency. This determines the fractional bandwidth versus minimum insertion loss for a given material and filter. The static capacitance is a function of the transducer electrode structure and the dielectric properties of the substrate. The values given in the table correspond to the capacitance per pair of electrodes having quarter wavelength width and one-half wavelength period. The free surface velocity, v0, is a function of the material, cut angle, and propagation direction. The temperature coefficient of delay (TCD) is an indication of the frequency shift expected for a transducer due to a change of temperature and is also a function of cut angle and propagation direction. The substrate is chosen based on the device design specifications and includes consideration of operating temperature, fractional bandwidth, and insertion loss. Second-order effects such as diffraction and beam steering are considered important on high-performance devices [Morgan, 1985]. Cost and manufacturing tolerances may also influence the choice of the substrate material. 47.3 Basic Filter Specifications Figure 47.3 shows a typical time domain and frequency domain device performance specification. The basic frequency domain specification describes frequency bands and their desired level with respect to a given reference. Time domain specifications normally define the desired impulse response shape and any spurious time responses. The overall desired specification may be defined by combinations of both time and frequency domain specifications. Since time, h(t), and frequency, H(w), domain responses form unique Fourier transform pairs, given by (47.1) (47.2) FIGURE 47.2 Schematic diagram of a typical SAW bidirectional filter consisting of two interdigital transducers. The transducers need not be identical. The input transducer launches waves in either direction and the output transducer converts the acoustic energy back to an electrical signal. The device exhibits a minimum 6-dB insertion loss. Acoustic absorber damps unwanted SAW energy to eliminate spurious reflections which could cause distortions. TABLE 47.2 Common SAW Material Properties Parameter/Material ST-Quartz YZ LiNbO3 128° YX LiNbO3 YZ LiTa2O3 k 2 (%) 0.16 4.8 5.6 0.72 Cs (pf/cm-pair) 0.05 4.6 5.4 4.5 v0 (m/s) 3,159 3,488 3,992 3,230 Temp. coeff. of delay (ppm/°C) 0 94 76 35 h t H e d j t ( ) = / ( ) -• • Ú 1 2p w w w H h t e dt j t ( ) w ( ) w = - -• • Ú

→B2,,—B2"2 FIGURE 47.3 Typical time and frequency domain specification for a SAw filter. The filter bandwidth is Bu the transition bandwidth is B2, the inband ripple is R, and the out-of-band sidelobe level is RI it is important that combinations of time and frequency domain specifications be self-consistent. The electrodes of a SAW transducer act as sampling points for both transduction and reception. Given the desired modulated time response, it is necessary to sample the time waveform. For symmetrical frequency responses, sampling at twice the center frequency, fs 2fo, is sufficient, while nonsymmetric frequency responses require sampling at twice the highest frequency of interest. a very popular approach is to sample at f 4f a The SAW frequency response obtained is the convolution of the desired frequency response with a series of pulses, separated by fs, in the frequency domain. The net effect of sampling is to produce a continuous set of harmonics in the frequency domain in addition to the desired response at fo. This periodic, time-sampled function can be written g(t) an…6(t-t where an represents the sample values, tn=nAt, n= nth sample, and At= time sample separation. The corresponding frequency response is given by G(=∑g(n)n=∑gtn)1x (47.4) where f= 1/At. The effect of sampling in the time domain can be seen by letting f=f+ mf, where m is an integer, which yields G(+ mf, )= G() which verifies the periodic harmonic frequency response. Before leaving filter design, it is worth noting that a SAW filter is composed of two transducers which may have different center frequencies, bandwidth, and other filter specifications. This provides a great deal of flexibility in designing a filter by allowing the product of two frequency responses to achieve the total filter 47. 4 SAW Transducer modeling The four most popular and widely used models include the transmission line model, the coupling of modes model, the impulse response model, and the superposition model. The superposition model is an extension of the impulse response model and is the principal model used for the majority of SAw bidirectional and

© 2000 by CRC Press LLC it is important that combinations of time and frequency domain specifications be self-consistent. The electrodes of a SAW transducer act as sampling points for both transduction and reception. Given the desired modulated time response, it is necessary to sample the time waveform. For symmetrical frequency responses, sampling at twice the center frequency, fs = 2f0, is sufficient, while nonsymmetric frequency responses require sampling at twice the highest frequency of interest. A very popular approach is to sample at fs = 4f0. The SAW frequency response obtained is the convolution of the desired frequency response with a series of impulses, separated by fs , in the frequency domain. The net effect of sampling is to produce a continuous set of harmonics in the frequency domain in addition to the desired response at f0. This periodic, time-sampled function can be written as (47.3) where an represents the sample values, tn = nDt, n = nth sample, and Dt = time sample separation. The corresponding frequency response is given by (47.4) where fs = 1/Dt. The effect of sampling in the time domain can be seen by letting f = f + mfs, where m is an integer, which yields G(f + mfs ) = G(f ) which verifies the periodic harmonic frequency response. Before leaving filter design, it is worth noting that a SAW filter is composed of two transducers which may have different center frequencies, bandwidth, and other filter specifications. This provides a great deal of flexibility in designing a filter by allowing the product of two frequency responses to achieve the total filter specification. 47.4 SAW Transducer Modeling The four most popular and widely used models include the transmission line model, the coupling of modes model, the impulse response model, and the superposition model. The superposition model is an extension of the impulse response model and is the principal model used for the majority of SAW bidirectional and FIGURE 47.3 Typical time and frequency domain specification for a SAW filter.The filter bandwidth is B1, the transition bandwidth is B2 , the inband ripple is R2 and the out-of-band sidelobe level is R1. gt a t t n nn N N () ( ) / / = ×- - Â d 2 2 G f gt e gt e n j ft N N n j nf f N N n s () () () / / / / / = = - - - - Â Â 2 2 2 2 2 2 p p

multiphase filter synthesis which do not have inband, interelectrode reflections. As is the case for most tech- nologies, many models may be used in conjunction with each other for predicting device performance based tal device data The SAW Superposition Impulse Response Transducer Model The impulse response model was first presented by Hartmann et al. [1973] to describe SAW filter design and synthesis. For a linear causal system, the Fourier transform of the device's frequency response is the device impulse time response. Hartmann showed that the time response of a SAw transducer is given by h(t)=4k C, f/2()sin(e(t) where 0()=2n f,(t)ch (47.5) and where the following defil k2= SAW coupling coefficient, C,=electrode pair capacitance per unit length(pf/cm-pair), and f(t)=instantaneous frequency at a time, t. This is the general form for a uniform beam transducer with arbitrary electrode spacing. For a uniform beam transducer with periodic electrode acing,f(r)=fo and sin e(r)=sin or. This expression relates a time response to the physical device parameters of the material coupling coefficient and the electrode capacitance Given the form of the time response, energy arguments are used to determine the device equivalent circuit parameters. Assume a delta function voltage input, vin(t)=8(t)h, then Vin(o)=1. Given h(t), H(o) is known and the energy launched as a function of frequency is given by E(@)=2-H( )?.Then E(o)=Va(o)·G(0)=1.G(0) (47.6) G(o)=2·|H(o) There is a direct relationship between the transducer frequency transfer function and the transducer conduc tance. Consider an interdigital transducer(IDT) with uniform overlap electrodes having N, interaction pairs. Each gap between alternating polarity electrodes is considered a localized SAW source. The SAW impulse response at the fundamental frequency will be continuous and of duration t, where t=N At, and h(t)is given by ht)=K·cos(ot)·ret(t/t) (47.8) where K=4k C, fMz H() sin(x) sin(x,) (47.9) where x=(0-0) t/2 and x2=(@+Oo).t/2. This represents the ideal SAw continuous response in both time and frequency. This can be related to the sampled response by a few substitutions of variables. Let 2.f, tn sm At =t,P 'At =t/ (47.10) c 2000 by CRC Press LLC

© 2000 by CRC Press LLC multiphase filter synthesis which do not have inband, interelectrode reflections. As is the case for most technologies, many models may be used in conjunction with each other for predicting device performance based on ease of synthesis, confidence in predicted parameters, and correlation with experimental device data. The SAW Superposition Impulse Response Transducer Model The impulse response model was first presented by Hartmann et al. [1973] to describe SAW filter design and synthesis. For a linear causal system, the Fourier transform of the device’s frequency response is the device impulse time response. Hartmann showed that the time response of a SAW transducer is given by (47.5) and where the following definitions are k 2 = SAW coupling coefficient, Cs = electrode pair capacitance per unit length (pf/cm-pair), and fi (t) = instantaneous frequency at a time, t. This is the general form for a uniform beam transducer with arbitrary electrode spacing. For a uniform beam transducer with periodic electrode spacing, fi (t) = f0 and sin q(t) = sin wt. This expression relates a time response to the physical device parameters of the material coupling coefficient and the electrode capacitance. Given the form of the time response, energy arguments are used to determine the device equivalent circuit parameters. Assume a delta function voltage input, vin(t) = d(t), then Vin(w) = 1. Given h(t), H(w) is known and the energy launched as a function of frequency is given by E(w) = 2·*H(w)* 2 . Then (47.6) or (47.7) There is a direct relationship between the transducer frequency transfer function and the transducer conductance. Consider an interdigital transducer (IDT) with uniform overlap electrodes having Np interaction pairs. Each gap between alternating polarity electrodes is considered a localized SAW source. The SAW impulse response at the fundamental frequency will be continuous and of duration t, where t = N · Dt, and h(t) is given by (47.8) where k = 4k f0 3/2 and f0 is the carrier frequency. The corresponding frequency response is given by (47.9) where x1 = (w – w0) · t/2 and x 2 = (w + w0) · t/2. This represents the ideal SAW continuous response in both time and frequency. This can be related to the sampled response by a few substitutions of variables. Let (47.10) h t k C f t t t f d s i i t ( ) ( )sin[ ( )] ( ) ( ) / = = Ú 4 2 3 2 0 q where q p t t E( ) w = V ( ) w × Ga a ( ) w = × G (w) in 2 1 G H a ( ) w = 2 × (w) 2 * * h(t) = × k w cos( t) × rect(t/t) 0 Cs H x x x x ( ) sin( ) sin( ) w kt = + Ï Ì Ô Ó Ô ¸ ˝ Ô ˛ 2 Ô 1 1 2 2 Dt D D D f t n t N t N t = n p × = × × = × = 1 2 2 0 , , t t , /

a(o)□ FIGURE 47.4 Electrical equivalent circuit model. 140 Frequency(MHz) Acoustic Conductance 10 FIGURE 47.5 (a) Theoretical frequency response of a rect(t/r) time function having a time length of 0. 1 us and a 200- MHz carrier frequency.(b) Theoretical conductance and susceptance for a SAW transducer implementing the frequency response. The conductance and susceptance are relative and are given in millisiemens. time and frequency response for a uniform transducer and the associated frequency-dependent conductance and Hilbert transform susceptance. The simple impulse model treats each electrode as an ideal impulse; however, the electrodes have a finite width which distorts the ideal impulse response. The actual SAw potential has been shown to be closely related to the electrostatic charge induced on the transducer by the input voltage. The problem is solved assuming a quasi-static and electrostatic charge distribution, assuming a semi-infinite array of electrodes, solving for a single element, and then using superposition and convolution. The charge distri bution solution for a single electrode with all others grounded is defined as the basic charge distribution function (BCDF). The result of a series of arbitrary voltages placed on a series of electrodes is the summation of scaled, time-shifted BCDFs. The identical result is obtained if an array factor, a(x),defined as the ideal impulses localized at the center of the electrode or gap, is convolved with the BCDE, often called the element factor. This is very similar to the analysis of antenna arrays. Therefore, the ideal frequency transfer function and conductance given by the impulse response model need only be modified by multiplying the frequency-dependent element factor. The analytic solution to the BCDF is given in Datta [1986] and Morgan [1985], and is shown to place small perturbation in the form of a slope or dip over the normal bandwidths of interest. The BCDF also predicts the expected harmonic frequency response c 2000 by CRC Press LLC

Apodized SAw Transducers Apodization is the most widely used method for weighting a SAW transducer. The desired time-sampled impulse response is implemented by assigning the overlap of opposite polarity electrodes at a given position to a normalized sample weight at a given time. A tap having a weight of unity has an overlap across the entire beamwidth while a small tap will have a small overlap of adjacent electrodes. The time impulse response can be broken into tracks which have uniform height but whose time length and impulse response may vary. Each of these time tracks is implemented spatially across the transducer's beamwidth by overlapped electrode sections t the proper positions. This is shown in Fig. 47. 1. The smaller the width of the tracks, the more exact the approximation of uniform time samples. There are many different ways to implement the time-to-spatial transformation; Fig. 47. 1 shows just one such implementation. The impulse response can be represented, to any required accuracy, as the summation of uniform samples located at the proper positions in time in a given track. Mathematically this is given by h(t)= ∑ h, (t) (47.17) H(o)=∑(o)=∑{J40-a (47.18) The frequency response is the summation of the individual frequency responses in each track, which may be widely varying depending on the required impulse response. This spatial weighting complicates the calculations of the equivalent circuit for the transducer. Each track must be evaluated separately for its acoustic conductance, acoustic capacitance, and acoustic susceptance. The transducer elements are then obtained by summing the individual track values yielding the final transducer equivalent circuit parameters. These parameters can be solved analytically for simple impulse response shapes(such as the rect, triangle, cosine, etc. but are usually solved numerically on a computer [Richie et al., 1988] There is also a secondary effect of apodization when attempting to extract energy. Not all of the power of a lly, not all of the a uniform SAW beam can be extracted by an apodized transducer. The transducer efficiency is calculated at H(0 E The apodization loss is defined as 0·log( (47.20) Typical apodization loss for common SAW transducers is 1 dB or less. Finally, because an apodized transducer radiates a nonuniform beam profile, the response of two cascaded apodized transducers is not the product of each transducers individual frequency responses, but rather is given by

© 2000 by CRC Press LLC Apodized SAW Transducers Apodization is the most widely used method for weighting a SAW transducer. The desired time-sampled impulse response is implemented by assigning the overlap of opposite polarity electrodes at a given position to a normalized sample weight at a given time. A tap having a weight of unity has an overlap across the entire beamwidth while a small tap will have a small overlap of adjacent electrodes. The time impulse response can be broken into tracks which have uniform height but whose time length and impulse response may vary. Each of these time tracks is implemented spatially across the transducer’s beamwidth by overlapped electrode sections at the proper positions. This is shown in Fig. 47.1. The smaller the width of the tracks, the more exact the approximation of uniform time samples. There are many different ways to implement the time-to-spatial transformation; Fig. 47.1 shows just one such implementation. The impulse response can be represented, to any required accuracy, as the summation of uniform samples located at the proper positions in time in a given track. Mathematically this is given by (47.17) and (47.18) The frequency response is the summation of the individual frequency responses in each track, which may be widely varying depending on the required impulse response. This spatial weighting complicates the calculations of the equivalent circuit for the transducer. Each track must be evaluated separately for its acoustic conductance, acoustic capacitance, and acoustic susceptance. The transducer elements are then obtained by summing the individual track values yielding the final transducer equivalent circuit parameters. These parameters can be solved analytically for simple impulse response shapes (such as the rect, triangle, cosine, etc.) but are usually solved numerically on a computer [Richie et al., 1988]. There is also a secondary effect of apodization when attempting to extract energy. Not all of the power of a nonuniform SAW beam can be extracted by an a uniform transducer, and reciprocally, not all of the energy of a uniform SAW beam can be extracted by an apodized transducer. The transducer efficiency is calculated at center frequency as (47.19) The apodization loss is defined as apodization loss = 10 · log(E) (47.20) Typical apodization loss for common SAW transducers is 1 dB or less. Finally, because an apodized transducer radiates a nonuniform beam profile, the response of two cascaded apodized transducers is not the product of each transducer’s individual frequency responses, but rather is given by (47.21) h t h t i i I ( ) = ( ) = Â 1 H H h t e dt i i I i j t i I ( ) ( ) ( ) / / w w w t t = = Ï Ì Ô Ó Ô ¸ ˝ Ô ˛ = Ô - = - Â Â Ú 1 2 2 1 E H I H i I i I = × = = Â Â ( ) ( ) w w 0 1 2 2 0 1 H H i i H H i H i I i i I i I 12 1 2 1 1 2 1 1 ( ) w = ( ) w × ( ) w ¹ × ( ) w (w) = = = Â Â Â

In general, filters are normally designed with one apodized and one uniform transducer or with two apodize transducers coupled with a spatial-to-amplitude acoustic conversion component, such as a multistrip coupler Datta, 1986 47.5 Distortion and Second-Order effects In Saw devices there are a number of effects which can distort the desired response from the ideal response he most significant distortion in SAW transducers is called the triple transit echo(TTE) which causes a to an electrically regenerated SAW at the output transducer which travels back to the input transducer, wher it induces a voltage across the electrodes which in turn regenerates another SAw which arrives back at the output transducer. This is illustrated schematically in Fig. 47. 2. Properly designed and matched unidirectional transducers have acceptably low levels of TTe due to their design. Bidirectional transducers, however, must be mismatched in order to achieve acceptable TTE levels. To first order, the TTE for a bidirectional two- transducer filter is given as TTE≈2·IL+6dB (47.22) where IL= filter insertion loss, in dB [Matthews, 1977]. As examples, the result of TtE is to cause a ghost in a video response and intersymbol interference in data transmission Another distortion effect is electromagnetic feedthrough which is due to direct coupling between the and output ports of the device, bypassing any acoustic response. This effect is minimized by proper design, mounting, bonding, and packaging. In addition to generating a SAW, other spurious acoustic modes may be generated. Bulk acoustic (BAW)may be both generated and received, which causes passband distortion and loss of out-of-band rejection BAW generation is minimized by proper choice of material, roughening of the crystal backside to scatter BAWs, and use of a SAW track changer, such as a multistrip coupler Any plane wave which is generated from a finite aperture will begin to diffract. This is exactly analogous to light diffracting through a slit. Diffractions principal effect is to cause effective shifts in the filter's tap weights and phase which results in increased sidelobe levels in the measured frequency response. Diffraction is mini- nized by proper choice of substrate and filter design. Transducer electrodes are fabricated from thin film metal, usually aluminum, and are finite in width. This netal can cause discontinuities to the surface wave which cause velocity shifts and frequency-dependent reflections. In addition, the films have a given sheet resistance which gives rise to a parasitic electrode resistance loss. The electrodes are designed to minimize these distortions in the device. 47.6 Bidirectional Filter Response A SAW filter is composed of two cascaded transducers. In addition, the overall filter function is the product of wo acoustic transfer functions, two electrical transfer functions, and a delay line function, as illustrated in Fig. 47.6. The acoustic filter functions are as designed by each SAW transducer. The delay line function dependent on several parameters, the most important being frequency and transducer separation. The propa gation path transfer function, D(o), is normally assumed unity, although this may not be true for high frequencies(f> 500 MHz) or if there are films in the propagation path. The electrical networks may cause distortion of the acoustic response and are typically compensated in the initial SAw transducer's design The SAW electrical network is analyzed using the SAW equivalent circuit model plus the addition of packaging parasitics and any tuning or matching networks. Figure 47.7 shows a typical electrical network which is computer analyzed to yield the overall transfer function for one port of the two-port SAW filter[Morgan, 1985. The second port is analyzed in a similar manner and the overall transfer function is obtained as the product of the electrical, acoustic, and propagation delay line effects. c 2000 by CRC Press LLC

© 2000 by CRC Press LLC In general, filters are normally designed with one apodized and one uniform transducer or with two apodized transducers coupled with a spatial-to-amplitude acoustic conversion component, such as a multistrip coupler [Datta, 1986]. 47.5 Distortion and Second-Order Effects In SAW devices there are a number of effects which can distort the desired response from the ideal response. The most significant distortion in SAW transducers is called the triple transit echo (TTE) which causes a delayed signal in time and an inband ripple in the amplitude and delay of the filter. The TTE is primarily due to an electrically regenerated SAW at the output transducer which travels back to the input transducer, where it induces a voltage across the electrodes which in turn regenerates another SAW which arrives back at the output transducer. This is illustrated schematically in Fig. 47.2. Properly designed and matched unidirectional transducers have acceptably low levels of TTE due to their design. Bidirectional transducers, however, must be mismatched in order to achieve acceptable TTE levels. To first order, the TTE for a bidirectional twotransducer filter is given as TTE ª 2 · IL + 6 dB (47.22) where IL = filter insertion loss, in dB [Matthews, 1977]. As examples, the result of TTE is to cause a ghost in a video response and intersymbol interference in data transmission. Another distortion effect is electromagnetic feedthrough which is due to direct coupling between the input and output ports of the device, bypassing any acoustic response. This effect is minimized by proper device design, mounting, bonding, and packaging. In addition to generating a SAW, other spurious acoustic modes may be generated. Bulk acoustic waves (BAW) may be both generated and received, which causes passband distortion and loss of out-of-band rejection. BAW generation is minimized by proper choice of material, roughening of the crystal backside to scatter BAWs, and use of a SAW track changer, such as a multistrip coupler. Any plane wave which is generated from a finite aperture will begin to diffract. This is exactly analogous to light diffracting through a slit. Diffraction’s principal effect is to cause effective shifts in the filter’s tap weights and phase which results in increased sidelobe levels in the measured frequency response. Diffraction is minimized by proper choice of substrate and filter design. Transducer electrodes are fabricated from thin film metal, usually aluminum, and are finite in width. This metal can cause discontinuities to the surface wave which cause velocity shifts and frequency-dependent reflections. In addition, the films have a given sheet resistance which gives rise to a parasitic electrode resistance loss. The electrodes are designed to minimize these distortions in the device. 47.6 Bidirectional Filter Response A SAW filter is composed of two cascaded transducers. In addition, the overall filter function is the product of two acoustic transfer functions, two electrical transfer functions, and a delay line function, as illustrated in Fig. 47.6. The acoustic filter functions are as designed by each SAW transducer. The delay line function is dependent on several parameters, the most important being frequency and transducer separation. The propagation path transfer function, D(w), is normally assumed unity, although this may not be true for high frequencies (f > 500 MHz) or if there are films in the propagation path. The electrical networks may cause distortion of the acoustic response and are typically compensated in the initial SAW transducer’s design. The SAW electrical network is analyzed using the SAW equivalent circuit model plus the addition of packaging parasitics and any tuning or matching networks. Figure 47.7 shows a typical electrical network which is computer analyzed to yield the overall transfer function for one port of the two-port SAW filter [Morgan, 1985]. The second port is analyzed in a similar manner and the overall transfer function is obtained as the product of the electrical, acoustic, and propagation delay line effects