Malocha, D.C. “Surface Acoustic Wave Filters” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000 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 electro- magnetic wave velocity of 3210 8 m/s and an acoustic wave velocity of 3210 3 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 ? 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/v a , where DL is the electrode period and v a 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 (LiNbO 3 ), and lithium tantalate (LiTa 2 O 5 ). 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 v SAW = 3000 m/s and v EM = 3 2 10 8 m/s Parameter SAW E&M F 0 = 10 MHz l SAW = 300 mm l EM = 30 m F 0 = 2 GHz l SAW = 1.5 mm l EM = 0.15 m Delay = 1 ns L SAW = 3 mm L EM = 0.3 m Delay = 10 ms L SAW = 30 mm L EM = 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). ? 2000 by CRC Press LLC The coupling coefficient, k 2 , 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, v 0 , 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 LiNbO 3 128° YX LiNbO 3 YZ LiTa 2 O 3 k 2 (%) 0.16 4.8 5.6 0.72 C s (pf/cm-pair) 0.05 4.6 5.4 4.5 v 0 (m/s) 3,159 3,488 3,992 3,230 Temp. coeff. of delay (ppm/°C) 0 94 76 35 ht H ed jt () / ()= -￥ ￥ ò 12pww w Hhtedt jt () ()w w = - -￥ ￥ ò ? 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, f s = 2f 0 , is sufficient, while nonsymmetric frequency responses require sampling at twice the highest frequency of interest. A very popular approach is to sample at f s = 4f 0 . The SAW frequency response obtained is the convolution of the desired frequency response with a series of impulses, separated by f s , 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 f 0 . This periodic, time-sampled function can be written as (47.3) where a n represents the sample values, t n = nDt, n = nth sample, and Dt = time sample separation. The corresponding frequency response is given by (47.4) where f s = 1/Dt. The effect of sampling in the time domain can be seen by letting f = f + mf s , where m is an integer, which yields G(f + mf s ) = 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 B 1 , the transition bandwidth is B 2 , the inband ripple is R 2 and the out-of-band sidelobe level is R 1 . gt a t t nnn N N () ( ) / / =×- - ? d 2 2 Gf gt e gt e n jft N N n jnff N N ns () () () / / / / / == - - - - ?? 2 2 2 2 2 2 pp ? 2000 by CRC Press LLC 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 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, C s = electrode pair capacitance per unit length (pf/cm-pair), and f i (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, f i (t) = f 0 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, v in (t) = d(t), then V in (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 conduc- tance. Consider an interdigital transducer (IDT) with uniform overlap electrodes having N p 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 = 4kf 0 3/2 and f 0 is the carrier frequency. The corresponding frequency response is given by (47.9) where x 1 = (w – w 0 ) · t/2 and x 2 = (w + w 0 ) · 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) ht kCf t t t f d si i t () ()sin[()] () () / == ò 42 32 0 qqptt where EVG G aa () () () ()www w=×=× in 2 1 GH a () ()ww=×2 2 ** ht t rectt() cos( ) (/)=× ×kw 0 C s H x x x x () sin() sin() w kt =+ ì í ? ? ? ü y ? t ?2 1 1 2 2 DDDDt f tntNt Nt np = × =× ×= ×= 1 2 2 0 , ,,/tt ? 2000 by CRC Press LLC Assuming a frequency bandlimited response, the negative frequency component centered around –f 0 can be ignored. Then the frequency response, using Eq. (47.9), is given by (47.11) where The conductance, given using Eqs. (47.6) and (47.10), is (47.12) This yields the frequency-dependent conductance per unit width of the transducer. Given a uniform transducer of width, W a , the total transducer conductance is obtained by multiplying Eq. (47.12) by W a . Defining the center frequency conductance as (47.13) the transducer conductance is (47.14) The transducer electrode capacitance is given as C e = C s W a N p (47.15) Finally, the last term of the SAW transducer’s equivalent circuit is the frequency-dependent susceptance. Given any system where the frequency-dependent real part is known, there is an associated imaginary part which must exist for the system to be real and causal. This is given by the Hilbert transform susceptance, defined as B a , where [Datta, 1986] (47.16) where “*” indicates convolution. These three elements compose a SAW transducer equivalent circuit. The equivalent circuit, shown in Fig. 47.4, is composed of one lumped element and two frequency-dependent terms which are related to the substrate material parameters, transducer electrode number, and the transducer configuration. Figure 47.5 shows the H N x x p n n () sin() wk p w = ì í ? ? ? ü y ? t ? × 0 xN ff f N npp = - = -()()ww w pp 0 0 0 0 Gf N f x x kfCN x x a p n n sp n n () sin() sin() = ì í ? ? ? ü y ? t ? =×2 2 8 2 0 2 2 2 2 0 2 2 2 k p p Gf G kfCWN asap () 00 2 0 2 8== Gf G x x a n n () sin() 00 2 2 =× B Gu u du G a a a () () () ()w pw ww= - =* -￥ ￥ ò 1 1/ ? 2000 by CRC Press LLC 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 BCDF, 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 a small perturbation in the form of a slope or dip over the normal bandwidths of interest. The BCDF also predicts the expected harmonic frequency responses. FIGURE 47.4 Electrical equivalent circuit model. FIGURE 47.5 (a) Theoretical frequency response of a rect(t/t ) time function having a time length of 0.1 ms 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. ? 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) ht ht i i I () ()= = ? 1 HH htedt i i I i jt i I () () () / / ww w t t == ì í ? ? ? ü y ? t ?= - - = ? ò ? 1 2 2 1 E H IH i I i I = × = = ? ? () () w w 0 1 2 2 0 1 HHHHH ii i i I i i I i I 12 1 2 1 1 2 11 () () () () ()wwwww=×1× === ??? ? 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 two- transducer filter is given as TTE ? 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 mini- mized 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 propa- gation 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. ? 2000 by CRC Press LLC 47.7 Multiphase Unidirectional Transducers The simplest SAW transducers are single-phase bidirectional transducers. Because of their symmetrical nature, SAW energy is launched equally in both directions from the transducer. In a two- transducer configuration, half the energy (3 dB) is lost at the transmitter, and reciprocally, only half the energy can be received at the receiver. This yields a net 6-dB loss in a filter. However, by adding nonsymmetry into the transducer, either by electrical multiphases or nonsymmetry in reflection and regeneration, energy can be unidirectionally directed yielding a theoretical minimum 0-dB loss. The most common SAW UDTs are called the three-phase UDT (3PUDT) and the group type UDT (GUDT). The 3PUDT has the broadest bandwidth and requires multilevel metal structures with crossovers. The GUDT uses a single-level metal but has a narrower unidirectional bandwidth due to its structure. In addition, there are other UDT or equivalent embodiments which can be implemented but will not be discussed [Morgan, 1985]. The basic structure of a 3PUDT is shown in Fig. 47.8. A unit cell consists of three electrodes, each connected to a separate bus bar, where the electrode period is l 0 /3. One bus bar is grounded and the other two bus bars will be driven by an electrical network where V 1 = V 2 D 60°. The transducer analysis can be accom- plished similar to a simple IDT by considering the 3PUDT as three collinear IDTs with a spatial phase shift, as shown in Fig. 47.8. The electrical phasing network, typically consisting of one or two reactive elements, in conjunction with the spatial offset results in energy being launched in only one direction from the SAW transducer. The transducer can then be matched to the required load impedance with one or two additional reactive elements. The effective unidirectional bandwidth of the 3PUDT is typically 20% or less, beyond which the transducer behaves as a normal bidirectional transducer. Figure 47.9 shows a 3PUDT filter schematic consisting of two transducers and their associated matching and phasing networks. The overall filter must be analyzed with all external electrical components in place for accurate prediction of performance. The external components can be miniaturized and may be fabricated using only printed circuit board material and area. This type of device has demonstrated as low as 2 dB insertion loss. FIGURE 47.6Complete transfer function of a SAW filter including the acoustic, electrical, and delay line transfer functions. The current generator is I s , and R s and R L are the source and generator resistances, respectively. FIGURE 47.7Electrical network analysis for a SAW transducer. I G and R G represent the generator source and impedance, L T is a tuning inductor, C H and L H are due to the package capacitance and bond wire, respectively, and R P represents a parasitic resistance due to the electrode transducer resistance. The entire network, including the frequency-dependent SAW network, is solved to yield the single-port transfer function. ? 2000 by CRC Press LLC 47.8 Single-Phase Unidirectional Transducers Single-phase unidirectional transducers (SPUDT) use spatial offsets between mechanical electrode reflections and electrical regeneration to launch a SAW in one direction. A reflecting structure may be made of metal electrodes, dielectric strips, or grooved reflectors which are properly placed within a transduction structure. Under proper design and electrical matching conditions, the mechanical reflections can exactly cancel the electrical regeneration in one direction of the wave over a moderate band of frequencies. This is schematically illustrated in Fig. 47.10 which shows a reflector structure and a transduction structure merged to form a SPUDT. The transducer needs to be properly matched to the load for optimum operation. The mechanical reflections can be controlled by modifying the width, position, or height of the individual reflector. The regenerated SAW is primarily controlled by the electrical matching to the load of the transduction structure. SPUDT filters have FIGURE 47.8Schematic of a unit cell of a 3PUDT and the basic equivalent circuit. The 3PUDT can be analyzed as three collinear transducers with a spatial offset. FIGURE 47.9Schematic diagram of a 3PUDT which requires the analysis of both the acoustic transducer responses as well as electrical phasing and matching networks. ? 2000 by CRC Press LLC exhibited as low as 3 dB loss over fractional bandwidths of 5% or less and have the advantage of not needing phasing networks when compared to the multiphase UDTs. 47.9 Dispersive Filters SAW filters can also be designed and fabricated using nonuniformly spaced electrodes in the transducer. The distance between adjacent electrodes determines the “local” generated frequency. As the spacing between the electrodes changes, the frequency is slowly changed either up (decreasing electrode spacing) or down (increasing electrode spacing) as the position progresses along the transducer. This slow frequency change with time is often called a “chirp.” Figure 47.11 shows a typical dispersive filter consisting of a chirped transducer in cascade with a uniform transducer. Filters can be designed with either one or two chirped transducers and the rate of the chirp is variable within the design. These devices have found wide application in radar systems due to their small size, reproducibility, and large time bandwidth product. 47.10 Coded SAW Filters Because of the ability to control the amplitude and phase of the individual electrodes or taps, it is easy to implement coding in a SAW filter. Figure 47.12 shows an example of a coded SAW filter implementation. By changing the phase of the taps, it is possible to generate an arbitrary code sequence. These types of filters are used in secure communication systems, spread spectrum communications, and tagging, to name a few [Mat- thews, 1977]. SAW devices can also be used to produce time impulse response shapes for use in modulators, equalizers, and other applications. An example of a SAW modulator used for generating a cosine envelope for a minimum shift keyed (MSK) modulator is shown in Fig. 47.13 [Morgan, 1985]. 47.11 Resonators Another very important class of devices is SAW resonators. Resonators can be used as frequency control elements in oscillators, as notch filters, and as narrowband filters, to name a few. Resonators are typically fabricated on piezoelectric quartz substrates due to its low TCD which yields temperature-stable devices. A resonator uses one or two transducers for coupling energy in/out of the device and one or more distributed reflector arrays to store energy in the device. This is analogous to an optical cavity with the distributed reflector arrays acting FIGURE 47.10Schematic representation of a SPUDT which is a combination of transduction and reflecting structures to launch a SAW in one direction over moderate bandwidths. FIGURE 47.11A SAW dispersive filter consisting of a uniform transducer and a “down chirp” dispersive trans- ducer. The high frequencies have a shorter delay than the low frequencies in this example. ? 2000 by CRC Press LLC as the mirrors. A localized acoustic mirror, such as a cleaved edge, is not practical for SAW because of spurious mode coupling at edge discontinuities which causes significant losses. A distributive reflective array is typically composed of a series of shorted metal electrodes, etched grooves in the substrate, or dielectric strips. There is a physical discontinuity on the substrate surface due to the individual reflectors. Each reflector is one-quarter wavelength wide and the periodicity of the array is one-half wavelength. This is shown schematically in Fig. 47.14. The net reflections from all the individual array elements add synchronously at center frequency, resulting in a very efficient reflector. The reflection from each array element is small and very little spurious mode coupling results. Figure 47.14 shows a typical single-pole, single-cavity, two-port SAW resonator. Resonators can be made multipole by addition of multiple cavities, which can be accomplished by inline acoustic coupling, transverse acoustic coupling, and by electrical coupling. The equivalent circuit for SAW two-port and one-port resonators is shown in Fig. 47.15. SAW resonators have low insertion loss and high electrical Q’s of several thousand [Campbell, 1989; Datta, 1986; Morgan, 1985]. Defining Terms Bidirectional transducer:A SAW transducer which launches energy from both acoustic ports which are located at either end of the transducer structure. Interdigital transducer:A series of collinear electrodes placed on a piezoelectric substrate for the purpose of launching a surface acoustic wave. FIGURE 47.12Example of a coded SAW tapped delay line. FIGURE 47.13A SAW filter for implementing an MSK waveform using a wideband input transducer and a cosine envelope apodized transducer. FIGURE 47.14(a) SAW reflector array illustrating synchronous distributed reflections at center frequency. Individual electrode width (a) is 1/4 wavelength and the array period is 1/2 wavelength at center frequency. (b) A schematic of a simple single-pole, single-cavity two-port SAW resonator. FIGURE 47.15(a) Two-port resonator equivalent circuit and (b) one-port resonator equivalent circuit. ? 2000 by CRC Press LLC Surface acoustic wave (SAW): A surface acoustic wave (also known as a Rayleigh wave) is composed of a coupled compressional and shear wave. On a piezoelectric substrate there is also an electrostatic wave which allows electroacoustic coupling. The wave is confined at or near the surface and decays away rapidly from the surface. Triple transit echo (TTE): A multiple transit echo received at three times the main SAW signal delay time. This echo is caused due to the bidirectional nature of SAW transducers and the electrical and/or acoustic mismatch at the respective ports. This is a primary delayed signal distortion which can cause filter distortion, especially in bidirectional transducers and filters. Unidirectional transducer (UDT): A transducer which is capable of launching energy from primarily one acoustic port over a desired bandwidth of interest. Related Topics 2.1 Step, Impulse, Ramp, Sinusoidal, Exponential, and DC Signals ? 5.3 Distortion ? 10.2 Ideal Filters ? 49.2 Mechanical Characteristics References D.S. Ballintine, Acoustic Wave Sensors, San Diego, Calif.: Academic Press, 1995. C. Campbell, Surface Acoustic Wave Devices and their Signal Processing Applications, San Diego, Calif.: Academic Press, 1989. S. Datta, Surface Acoustic Wave Devices, Englewood Cliffs, N.J.: Prentice-Hall, 1986. C.S. Hartmann, D.T. Bell, and R.C. Rosenfeld, “Impulse model design of acoustic surface wave filters,” IEEE Transactions on Microwave Theory and Techniques, vol. 21, pp. 162–175, 1973. H. Matthews, Surface Wave Filters, New York: Wiley Interscience, 1977. D.P. Morgan, Surface Wave Devices for Signal Processing, New York: Elsevier, 1985. S.M. Richie, B.P. Abbott, and D.C. Malocha, “Description and development of a SAW filter CAD system,” IEEE Transactions on Microwave Theory and Techniques, vol. 36, no. 2, 1988. Further Information The IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control provides excellent information and detailed articles on SAW technology. The IEEE Ultrasonics Symposium Proceeding provides information on ultrasonic devices, systems, and appli- cations for that year. Articles present the latest research and developments and include invited articles from eminent engineers and scientists. The IEEE Frequency Control Symposium Proceedings provides information on frequency control devices, systems, and applications (including SAW) for that year. Articles present the latest research and developments and include invited articles from eminent engineers and scientists. For additional information, see the following references: IEEE Transaction on Microwave Theory and Techniques, vol. 21, no. 4, 1973, special issue on SAW technology. IEEE Proceedings, vol. 64, no. 5, special issue on SAW devices and applications. Joint Special Issue of IEEE Transaction on Microwave Theory and Techniques and IEEE Transactions on Sonics and Ultrasonics, MTT-vol. 29, no. 5, 1981, on SAW device systems. M. Feldmann and J. Henaff, Surface Acoustic Waves for Signal Processing, Norwood, Mass.: Artech House, 1989. B.A. Auld, Acoustic Fields and Waves in Solids, New York: Wiley, 1973. V.M. Ristic, Principles of Acoustic Devices, New York: Wiley, 1983. A. Oliner, Surface Acoustic Waves, New York: Springer-Verlag, 1978. ? 2000 by CRC Press LLC