Design Considerations on Wideband Longitudinally-Coupled Double-Mode SAW Filters

Transcription

1 Design Considerations on Wideb Longitudinally-Coupled Double-Mode SAW Filters Ken-ya Hashimoto, Tatsuya Omori Masatsune Yamaguchi Dept. of Electronic Mechanical Eng., Chiba University Abstract This paper discusses, both qualitatively quantitatively, the operation the design principle of current longitudinallycoupled double-mode SAW (DMS) resonators where the internal reflection within interdigital transducers (IDTs) is not negligible lower capacitance ratio is necessary. For the purpose, the p-matrix expression is used skillfully with a help of the coupling-of-modes theory. The internal reflection causes a) deformation of the IDT passb shape, ) frequency dependence of the effective SAW velocity within IDTs, ) suppression of higher-order resonances. It is shown that these features can be effectively used to enhance performances of DMS filters. In addition, under proper designs taking the internal reflection into account, most of all structural discontinuities can be removed, is most preferable for the reduced scattering loss at the discontinuity. Design criteria are also presented for DMS filters of wide bwidth, it is demonstrated how device performances are improved by proper design taking account of the criteria. I. INTRODUCTION From the first proposal in 97[], SAW resonators have been extensively studied, now are widely used as filtering /or frequency control elements in various consumer communication equipments[], [], [], [5]. For long years, the Fabry-Perot model[6], [7], [8] has been widely used to explain basic behaviors of SAW resonators. Although it offers physical insights for the operation of the resonators their fundamental design guidelines, actual operation of major SAW resonators is considerably different from that of this model. Namely, the internal reflection within interdigital transducers (IDTs) is not negligible, it plays an important role for the operation because IDTs are placed within the resonance cavity. Especially, in current RF SAW filters, the internal reflection is significant, because the thickness of the grating electrodes becomes relatively large at high frequencies. Let us consider longitudinally-coupled double-mode SAW (DMS) filters[9] which have widely been used in RFsections of recent hy phones. Since their passb shape is conformed by overlapped responses of multiple resonances, the location of resonance frequencies is very important for device design. Although the location is basically determined by the distance between the grating reflectors, it is also influenced by the IDT design. Clearly this fact makes the design of DMS filters complex, behavior of their characteristics on design parameters does not seem well understood. For taking the internal reflection into account, the numerical techniques have been applied extensively, presently seem to be well-established. However, results given by the procedure were expected to be too complex to underst their operation. This paper is aimed at revealing the operation the design principle of present longitudinally-coupled DMS filters where the internal reflection within IDTs is not negligible. To discuss its influence both qualitatively quantitatively, the p-matrix[] expression is used skillfully with a help of the coupling-of-modes (COM)[] theory. First, one-port SAW resonators are discussed to explain the influence of the internal reflection. It is shown that due to the deformation of the IDT passb shape the phase characteristic caused by the internal reflection, the resonator performance becomes inferior when the resonance frequency of the whole structure is designed to coincide with the synchronous (Bragg) frequency of the IDT. The minimum capacitance ratio γ is available provided that the IDT electrode pitch is chosen so that the whole structure resonates at the Bragg frequency for the grating reflectors, that the gaps between the IDT grating reflectors are removed. This is most preferable for the reduced scattering loss at the discontinuity. Secondly, two-port DMS filters are discussed. It is shown that although the internal reflection in the IDTs limits the achievable passb width should usually be suppressed, it is effectively used to control the location of the resonance frequencies in the conventional design. This suggests that the proper hling of the internal reflection would be crucial in realizing high performance DMS filters. Finally, design criteria are presented for DMS filters of wide bwidth, it is demonstrated how device performances are improved by proper design taking account of the criteria. A. p-matrix II. ANALYSIS Consider the behavior of an IDT with the acoustic length of L (see Fig. ). Since the IDT is equivalent to the shortcircuited (SC) grating when V =, the result of the following analysis on an IDT can be also applicable to shortcircuited grating reflectors. It is known that due to the reciprocity, the normalized

2 I(L) U+ U+ U- V x= x=+l Fig.. IDT unit cell amplitude U ± (x) of the SAW propagating toward the ±x direction, the applied voltage V the current I(L) are related to each other by the p-matrix[]; U () U + (L) I(L) = p p p p p p χp χp p U- U +() U (L) V () where p p are the SAW reflectivities at the acoustic ports (x = ) (x = L), respectively, for the SC grating, p is the transmission coefficient, p is the IDT input admittance Y I = G I + jb I when the IDT is acoustically isolated. The factor χ is four when V I are defined in the peak amplitude while U ± (x) by the RMS value, whereas χ = when all of them are defined in the peak amplitude[]. In the following analysis, the IDT is assumed to be bidirectional symmetric for simplicity. Under the assumption, the relations p = p p = p hold because the ports are equivalent to each other. In addition, the system is assumed to be unitary (lossless). Then the coefficients appearing in Eq. () should satisfy the following relations;, p + p =, () G I = χ p, () p p = p /p = p /p. () Here we define φ ± = (p ± p ). Since φ + φ correspond to the transmission phases when the acoustic ports of the SC grating are acoustically driven in-phase 8 out-of-phase, respectively[], the relation p ± p = holds thus φ ± is always pure real independent of the driving frequency. Then we get the following relation from Eqs. () (); p = p exp(+jφ + ). (5) B. Coupling-of-mode equations The behavior of the p-matrix elements is discussed by the following COM equation for bidirectional IDTs[]; U + (x) x = jθ u U + (x) jκ U (x) + jζv, (6) U (x) x = +jκ U + (x) + jθ u U (x) jζv, (7) I(x) x = jχζu +(x) jχζu (x) + jωcv. (8) Here, θ u = β u π/p I is the detuning factor corresponding to the mismatch of the wavenumber β u of the unperturbed SAW mode under the SC grating from the Bragg (synchronous) condition β u = π/p I, where p I is the periodic length of an IDT. And C, κ ζ are the static capacitance per period, mutual coupling coefficient transduction coefficient, respectively. It should be noted that πχζ p I /ωc corresponds to the conventional electromechanical coupling factor for unperturbed SAW mode[6]. Solving the simultaneous linear equations (6) (7) with appropriate boundary conditions, we obtain p = jχζξ L p = Γ ( E ) Γ, E (9) p = E( Γ ) Γ, E () p = ξ ( Γ ) E + Γ E, () [ ( E)( + Γ ] ) + jωcl, () jθ p L( + Γ E) where E = exp( jθ p L), θ p = θ u κ, Γ = (θ p θ u )/κ ξ = ζ/(θ u + κ ). C. IDT properties Fig. shows the normalized radiation conductance G I /χ(ζl) as a function of the normalized frequency (θ u p I /π). In the calculation, χζ p I /ωc =.8 L = p I. When κ p I is very small, the main lobe in G I exhibits typical (sin x/x) dependence. With an increase in κ p I, the main lobe becomes peaky its peak moves toward low frequencies. Although now shown, with an increase in L I, the peak amplitude becomes steep large while the position is scarcely changed. It is clear from Eq. (), G I takes a first zero at θ p L = π. On the other h, G I takes the maximum value at θ p L = π (θ u = (π/l) + κ ) when κ L. Since θ p is real for these frequencies, the SAW amplitude distributes as shown in Fig.. For the case (b), the SAW amplitude is antisymmetric with respect to the geometrical center, SAWs can not be excited detected by the IDT. Fig. shows p calculated by Eq. (9) for κ p I =.8π L = p I. Comparison of Fig. with Fig. indicates

3 lative conductance (db) κpi= κpi=.π κpi=.8π κpi=.π Fig.. Frequency dependence of G I = R(p ) IDT (a) at θpl=-π IDT (b) at θpl=-π Fig.. SAW amplitude distribution within IDT. that p is zero at θ p L = π where G I is close to its maximum. Within the stopb ( θ u < κ ), p = change in the phase is very gradual. On the other h, out of the stopb, small ripples appear the phase change is rapid. flection coefficient (db) -5 5 θu <κ +π/ π/ π/ Fig.. Frequency dependence of p. (κ p I =.8π, L = p I ) π flection phase (rad.) For the case where κ <, G I shown in Fig. becomes mirror image with respect to θ u =, the peak appears at θ p L = +π. And p shown in Fig. is unchanged, but (p ) shifts by π. Fig. 5 shows φ ± (= (p ± p )) for L = p I. It is seen that φ + is almost equal to φ, namely, the influence of the internal reflection is not significant for the SAW propagation out of the stopb ( θ p > κ ). On the other h, φ + φ = π within the stopb. Transmission phase (rad.) +π +π/ π/ θu <κ π Fig. 5. Frequency dependence of φ ± (κ p I =.8π, L = p I ) Fig. 6 shows φ ± for L = 5p I. Similar to Fig. 5, φ + is almost equal to φ when θ p > κ, gradient of φ ± is somewhat large in conjunction with increased L. On the other h, the gradient is not influenced by L when θ p < κ. Transmission phase (rad.) +π +π/ π/ φ- φ+ θu <κ φ+ π Fig. 6. Frequency dependence of φ ± (κ p I =.8π, L = 5p I ) Note that (φ ± )/ ω corresponds to the effective time delay passing through the structure, the value is small in the stopb. In other words, the effective SAW velocity is large in this region. This is because incident SAW is reflected near the structural edge hardly penetrate into the structure. On the other h, just below the stopb, the effective time delay for φ + becomes very large due to the multiple reflection between the edges. φ-

4 For various L κ, G I φ ± were also calculated. The result indicated that influence of the internal reflection is characterized by κ L; the influence significant the IDT behaves like a resonator when κ L > near the stopb whereas the influence is insignificant when κ L < /or a little far from the stopb. III. ONE-PORT SAW RESONATORS Let us discuss the behavior of one-port SAW resonators having a mirror symmetry structure (see Fig. 7). In the figure, L r L g denote the grating reflector length the gap length between the IDT grating reflectors, respectively. If the grating reflector IDT fingers are aligned continuously, then L g =. Grating Lr Gap Gap Lg LI Lg Grating Lr Fig. 7. One-port SAW resonator Here the design goal is to minimize the capacitance ratio γ defined by the resonance antiresonance frequencies ω r ω a, respectively; γ = { (ω a /ω r ) } = {(ωa /ω r )}. () Roughly speaking, achievable passb width insertion loss are limited by γ in filter applications. Basic design ideas are a) the SAW amplitude in the grating reflectors should be evanescent so that the SAW energy is trapped within the structure, b) the SAW amplitude in the IDT should be almost uniform should not be evanescent so that the SAW field penetrates deep into the IDT (see Fig. 8). flector IDT flector Fig. 8. SAW amplitude distribution within one-port resonator. From Eq. (), the input admittance Y of the IDT is given by χp Y = p + p + p Γ = jb I + jg I cot{(φ + + φ Γ )/} () = j Y I sin{(φ + + φ Γ )/ + φ I }, sin{(φ + + φ Γ )/} (5) where Γ (= Γ r exp( jβl g )) is the reflectivity of the grating reflectors observing from the IDT edges. Here Γ r is the reflectivity of the grating reflector itself. And φ I = tan (G I /B I ) φ Γ = j log Γ. Here, φ Γ = (Γ) when Γ =. It is interesting that Eq. () is equivalent to the result of the Fabry-Perot model[5] when φ + (= (p + p )) is regarded as the transmission phase βl between the grating reflectors. Eq. (5) suggests that the resonance frequency ω r could be determined by the condition of φ + + φ Γ = nπ where Y =. On the other h, the anti-resonance frequency ω a is given by the condition of φ + + φ Γ = (nπ φ I ) where Y =. Since the variation in φ I is usually gradual in the IDT passb, it is clear that large φ I small gradient of φ + + φ Γ with ω are preferable for achieving small γ. As a reference, let us discuss at first the SAW resonator design when the internal reflection within the IDT is ignored, namely, κ for the IDT region is set to be zero. Variation of (Γ r ) is nearly π within the stopb of the reflector grating (see Fig. ) independent of κ L r. Thus the single-mode resonance is realized when the variation of φ + within the stopb is less than π. Since φ + = π(ω/ω r )(L I /p I ) for this case, the single-mode resonance condition gives allowable maximum L I ; L I /p I <.5( ω/ω r ), (6) where ω is the stopb width. When κ p I =.8π for example, L I /p I < 8.75 since ω/ω r =.8. In this case, the traditional design[5], [6], [7], [8] gives the best result for both γ Q where the reflector position is displaced so that the sting wave pattern within the cavity coincides with the IDT finger location (L g = λ/8 p r /p I = ). Fig. 9 shows the frequency response of a design example. Here L I L r are chosen to be 6p I 5p r, respectively. Here, L I is chosen smaller than the value given by Eq. (6) so as to suppress adjacent ripples. Further suppression is possible by decreasing L I, though this results in increased γ. The ripple is suppressed also by weighting grating finger lengths appropriately with a penalty of increased grating length[7]. Next, let us discuss the resonator design when κ for the IDT is identical with that for the reflectors. Since ω/ω r = κ p I /π, maximum L I given by Eq. (6) is.5π/κ. This means that provided small γ is necessary, the internal reflection within the IDT is not always negligible because κ L I = 5. In the following three designs, L r is fixed to be 5p I L I /p I p I /p r are adjusted to minimize γ for giving L g ; Case When the device is designed to resonate at φ + = π (θ p L I = π): This design offers the largest GI then φ I is close to π/. However, Fig. 5 shows that

5 5 γ= Fig. 9. Design example when the internal reflection within the IDT is ignored (L g = λ/8). the gradient of φ + is steep. As a result, γ becomes considerably large. Fig. shows a frequency response of a device with L g = λ/8 designed so as to resonate at φ + = π. Comparing Fig. with Fig. 9, it is seen that γ is increased the ripple in the higher side becomes strong. γ= Fig.. Design example when L g = λ/8 (L I = 8p I p r/p I =.56) Case When the device is designed to resonate when φ + = π/ (θp L I = π/): Unless the peak value of G I is not very steep, φ I remains close to π/. In addition, the gradient of φ + is relatively small. Then, γ obtainable is usually smaller than that of Case. It should be noted that (Γ r ) = π/ at θ u =. This means that L g can be set to be zero when the device is designed so that the resonance takes place at the Bragg frequency for the reflectors. Making L g = is desirable for minimizing the SAW scattering to bulk waves at the gaps[8], therefore, enables to realize high Q resonators[9]. Fig. shows a frequency response of a device with L g = designed so as to resonate at φ + = π/. Comparing Fig. with Figs. 9, it is seen that the resonator possess small γ out-of-b ripples appear in the lower side. γ= Fig.. Design example when L g = (L I = p I p r/p I =.8). Case When the device is designed to resonate when φ + = (θp L I = ): From the same reason for Case, it is expected that smaller γ can be realized. Nevertheless, γ is scarcely improved because of small φ I. Furthermore, since φ + changes rapidly by π in lower frequencies, L I must be reduced significantly, this results in increase in γ. Fig. shows a frequency response of a device with L g = λ/8 designed so as to resonate at φ + =. Comparing Fig. with Fig., it is seen that achieved γ is limited strong out-of-b ripples appear. γ= Fig.. Design example of when L g = λ/8. (L I = 8p I p r/p I =.).

6 6 The discussion suggests that the optimal design could make L g zero adjust p I so that the whole structure resonates at the Bragg frequency for the SC gratings; all of our numerical attempts confirmed this suggestion. When higher Q /or lower motional resistance are desirable rather than lower γ, resonators are sometimes designed in the same way with Case where the resonance takes place at the frequency giving the peak in G I, where the steepest gradient of φ + the largest φ I are simultaneously achievable. However, since L g is nonzero under this design, the SAW scattering to bulk waves occurs, this may cause significant reduction in achievable resonance Q[9]. By the way, as shown in Fig. 5, variation of φ + within the stopb is less than π independent of L I. Hence, less than one resonance occurs within the stopb even when variation of φ Γ is taken into account. This means that the internal reflection within the IDT can be used to suppress higher-order resonances. For example, when the stopb for the IDT is chosen to almost coincide with that of the grating reflectors, only one resonance appears no matter how large L I is. Since φ I increases with L I, small γ extremely low input impedance are simultaneously achievable by simply increasing L I. In addition, L g can be set to be zero to suppress the BAW scattering because of φ + = φγ = near the lower edge of the stopb. This type is called the IDT resonator[], []. IV. TWO-PORT DMS FILTERS Next, let us discuss the behavior of longitudinallycoupled two-port DMS filters designed so as to support multiple resonances (see Fig. ). In the figure, L T denotes the gap length between the two IDTs. Because of the structural symmetry, the resonances are categorized into two types: one is the even-modes with symmetrical field distribution, another is the odd-modes with anti-symmetrical one (see Fig. ). this property enables to extend the distance between the resonance frequencies, in general this results in wider passb width than the Two-port case. Further increase in the number of IDTs may enable to further increase in the passb width through the increased number of degrees of freedom in the design parameters[]. Fig.. Three-port DMS resonator Hereafter only -port DMS filters will be discussed for simplicity since the basic procedures for the analysis the design of three-port DMS filters are the same with that of the -port ones. Fig. 5 shows its equivalent circuit near the resonances for the -port DMS resonators. In the figure, Y me Y mo are the motional admittances due to even- odd-modes, respectively. Although the circuit is similar to that for the one-port SAW resonator, the resonance circuit is involved as a shunt element between two IDTs. This is because the structure is equivalent to a one-port SAW resonator if two IDTs are electrically parallel-connected. When two IDTs are simply parallel-connected, only the even-mode resonance occurs whereas only the odd-mode resonance occurs when two IDTs are parallel-connected with polarity inversion. Ymo/ flector Gap Space Gap flector C Yme/ : C Lr Lg LI LT LI Lg Lr Fig. 5. Equivalent circuit for two-port double-mode SAW resonator. Fig.. Two-port DMS resonator Fig. shows the basic structure electrical connection of three-port DMS filters. Due to the electrical connection, only the even-modes are excitable for this case. Then When the relation between the currents I n voltage V n at the electric port n are expressed in the admittance matrix form; ( ) ( ) ( ) I Yi Y = t V. (7) I Y t Y i V

7 7 If V = V, the field distribution in the structure is symmetric, the following relation holds; I V = Y i + Y t Y e. (8) On the other h, if V = V, the field distribution in the structure is anti-symmetric, the following relation holds; I V = Y i Y t Y o. (9) By Y e Y o, the scattering parameter S for the corresponding two-port SAW resonator is given by G L (Y e Y o ) S = (Y e + G L )(Y o + G L ). () where G L is the characteristic impedance of the measurement system. It is clear that the equivalent circuit shown in Fig. 5 can be modified as shown in Figs. 6(a) (b) by the use of Y e (= Y me + jωc ) Y o (= Y mo + jωc ). Yo Ye Ye-Yo : Yo-Ye (a) Circuit A Ye-Yo Yo-Ye (b) Circuit B Fig. 6. Modified equivalent circuits for two-port double-mode SAW resonator. Let us express Y e Y o by the p-matrix elements. With a help of appropriate boundary conditions the relations of Eqs. ()-(5), we obtain the following results; Y e = p + χp p p Γ T (p Γ )(p T ) p = jb I + jg I cos ψ a e sin ψ b e () Yo Ye Y o = p + χp p p Γ + T (p Γ )(p + T ) p = jb I + jg I sin ψ a o cos ψ b o, () where T = exp( jβl T ), a e = cos φ Γ φ T b e = cos φ Γ φ T a o = sin φ Γ φ T b o = sin φ Γ φ T cos φ + φ, () sin φ + φ, () cos φ + φ, (5) sin φ + φ, (6) ψ = φ + + φ + φ Γ + φ T. (7) Figs. 5 6 indicate that (φ + φ )/ = π/ when the influence of the internal reflection is significant. Otherwise, (φ + φ )/ =. Therefore, it is clear that the variation of ψ against frequency is much larger than that of a e, b e, a o b o. Basic idea for DMS filter design is very similar to that of ladder-type SAW filter design[]. That is, it is clear from Fig. 6 that the resonance frequency of Y e (or Y o ) is set to coincide with the anti-resonance frequency of Y o (or Y e ). If we assume B I G I in Eqs. () (), this requirement is met when a e = b o or a o = b e. Accordingly, from Eqs. ()-(6), the condition reduces to φ + φ + φ Γ φ T = nπ. (8) At first, let us consider the case where the influence of the internal reflection is not significant. Since φ + = φ, we obtain b e = bo = from Eqs. () (6). Therefore, the resonance condition is given by ψ = nπ/ from Eqs. () (), requiring φ Γ φ T = nπ to satisfy the condition of Eq. (8). This means that the difference between the resonance frequencies is governed by L I their position is controlled by L T. On the other h, when the influence of the internal reflection is significant, the difference between the resonance frequencies can be adjusted by not only L I but also L T because b e b o are not small. Nevertheless, this design is not desirable because of poorer resonance characteristics resulted from smaller G I. For the operation over wide bwidth, current RF DMS filters are designed to satisfy the condition of Eq. (8) by more than two resonances. In practice, the first second resonances are placed in the frequency region where the internal reflection is not significant, L I L T are used

8 8 for adjusting their location distance. On the other h, the third resonance is placed close to the IDT stopb, its location is adjusted by the IDT grating reflector pitches. Since φ + φ G I /B I are finite frequency dependent, fine tuning for design parameters such as L I L T are necessary. One may also be able to adjust the location of resonances by L g. The same discussion in Section, however, suggests that making L g = could be preferable when the behaviors of G I /B I φ + + φ are taken into consideration. As an example, Fig. 7 shows Y e Y a of the designed DMS resonator with the condition of χζ p I /ωc =.8 κ p I =.8π. The optimization resulted in κ p I =.8π, L I /p I =.5, L r /p r = 55, L T =.6λ p r /p I =. G L =.9ωCL I. It is seen that two resonance frequencies coincide with two antiresonance frequencies. Ye Yo Fig. 7. Admittance characteristics of designed DMS filter Fig. 7 shows S when two DMS resonators are cascaded. It is seen that a very flat relatively wide passb is realized. By the way, Eq. () indicates that S = when Y e = Y o. From Eqs. () (), the condition of Y e = Y o can be met when φ + φ Γ = (n + )π. This shows the situation in which the SAW excited at one end of the IDT is canceled out by the SAW excited reflected at the other end. In Fig. 8, a notch response due to this can be seen at θ u p I /π =.. Note that the strong ripple at the rejection b is due to the multiple reflection at the grating reflectors, is suppressed also by weighting grating finger lengths appropriately with a penalty of increased grating length[7]. V. ADVANCED DMS FILTER DESIGN The above consideration gives the following criteria for further improvements; Scattering coefficient S (db) Fig. 8. Frequency response of the cascaded DMS filter (two-stage cascaded). A number of degrees of freedom should be increased in the design parameters which are effective in adjusting the location of resonance /or anti-resonance frequencies.. The IDT stopb width, namely the internal reflection in the IDT, should be reduced in order to place these resonances out of the IDT stopb.. The IDTs should be designed so that G I is almost flat relatively large over the filter passb. From these criteria, it is clear that precise control of the internal reflection in the IDTs is crucial for the optimal design of DMS filters. Current fabrication design technologies enable us to employ non-uniform IDTs /or gratings for increasing the number of degrees of freedom. Since the main task is to allocate small number of resonance /or anti-resonance frequencies properly, course global optimization is sufficient. As an example, Fig. 9 shows Y e Y a of the DMS resonator redesigned according to these criteria. The design conditions are identical with those for the original one shown in Fig. 7. It is seen that three resonance frequencies coincide with three antiresonance frequencies. Fig. shows S when two re-designed DMS resonators are cascaded. Comparing Fig. with Fig. 8, one can see that the passb is made wider the out-of-b rejection is improved. It should be noted that due to a sufficient number of degrees of freedom, not only L g but also L T are able to be made zero in this design. This enables further suppression of the bulk wave radiation from structural discontinuities, this results in further reduction of the device insertion loss. Of course, the analysis design principle discussed here are also applicable to more complex resonator struc-

9 9 Scattering coefficient S (db) Ye Yo Fig. 9. Admittance characteristics of re-designed DMS filter New Design Original Fig.. Frequency response of the re-designed DMS filter (two-stage cascaded) tures such as -IDT 5-IDT configurations. VI. CONCLUSIONS This paper has discussed the operation the design of current SAW resonators where the internal reflection in IDTs is not negligible lower capacitance ratio is necessary. In the qualitative quantitative discussion, the p-matrix expression was used skillfully with a help of the COM theory. It was shown that the internal reflection causes a) deformation of the IDT passb shape, ) frequency dependence of the effective SAW velocity within IDTs, ) suppression of higher-order resonances, these features can be effectively used to enhance performances of both one-port SAW resonators two-port DMS filters. In addition, under proper designs taking the internal reflection into account, most of all structural discontinuities can be removed, this is most preferable for the reduced scattering loss at the discontinuity. Design criteria are also presented for DMS filters of wide bwidth, it is demonstrated how device performances are improved by proper design taking account of the criteria through increasing the number of degrees of freedom in resonator design. REFERENCES [] E.A. Ash: Surface Wave Grating flectors sonators, Digest of IEEE Microwave Symp. (97) pp [] W.R. Shreve P.S. Cross:. Surface Acoustic Waves sonators, in Precision Frequency Control, Eds., E.A. Gerber A. Ballato, Vol. I, Academic Press, Orlo (985) pp. 95. [] P.V. Wright: A view of SAW sonator Filter Technology, Proc. IEEE Ultrason. Symp., (99) pp [] G.K. Montress, T.E. Parker D. Andres: view of SAW Oscillator Performance, Proc. IEEE Ultrason. Symp., (99) pp. 5. [5] C.K. Campbell: Surface Acoustic Wave Devices for Mobile Wireless Communication, Academic Press, Boston (998). [6] J.S. Shoenwald, R.C. Rosenfeld E.J. Steple: Surface Wave Cavity sonator Characteristics VHF to L-B, Proc. IEEE Ultrason. Symp. (97) pp [7] D.T. Bell R.C.M. Li: Surface-Acoustic-Wave sonators, Proc. IEEE, 6 (976) pp [8] P. Cross: Properties of flective Arrays for Surface Acoustic sonators, IEEE Trans. Sonics Ultrason., SU-, (976) pp [9] T. Morita, Y. Watanabe, M. Tanaka Y. Nakazawa: Wideb Low Loss Double Mode SAW Filters, Proc. IEEE Ultrason. Symp. (99) pp. 95. [] G. Tobolka: Mixed Matrix presentation of SAW Transducers, IEEE Trans. Sonics Ultrason., SU-6, 6 (979) pp [] D.P. Chen H.A. Haus: Analysis of Metal-Strip SAW Grating Transducers, IEEE Trans. Sonics Ultrason., SU- (985) [] O. Ikata, T. Miyashita, T. Matsuda, T. Nishikawa Y. Satoh: Development of Low-Loss B-Pass Filters Using SAW sonators for Portable Telephone, Proc. IEEE Ultrason. Symp. (99) pp. 5. [] K. Hashimoto: Surface Acoustic Wave Devices in Telecommunications Modelling Simulation, Sec..5, Springer Verlag () pp [] D.P. Morgan, private communication [5] K. Hashimoto: Surface Acoustic Wave Devices in Telecommunications Modelling Simulation, Chap. 7.., Springer Verlag () pp. 6. [6] K. Hashimoto: Surface Acoustic Wave Devices in Telecommunications Modelling Simulation, Chap. 5.., Springer Verlag () pp. 7. [7] P.S. Cross: Surface Acoustic Wave sonator Filters Using Tapered Gratings, IEEE Trans., Sonics Ultrason., SU-5, 5 (978) pp. -9. [8] R.C.M. Li, R.C. Williamson, D.C. Flers J.A. Alusow: On the Performance Limitations of the Surface-Wave sonator Using Grooved flectors, Proc. IEEE Ultrason. Symp. (97) pp [9] Y. Ebata: Suppression of Bulk-Scattering Loss in SAW sonator With Quasi-Constant Acoustic flection Periodicity, Proc. IEEE Ultrason. Symp. (988) pp [] K.M. Lakin, T. Joseph D. Penunuri: Planar Surface Acoustic Wave sonators, Proc. IEEE Ultrason. Symp. (97) pp [] Y. Koyamada, F. Ishihara, S. Yoshikawa: Narrow-B Filters Employing Surface Acoustic Wave sonators, Proc. IEEE, 6, 5 (976) pp [] M. Koshino, K. Kanasaki, N. Akahori, M. Kawase, R. Chujyo Y. Ebata: A Wide-B Balanced SAW Filter with Longitudinal Multi- Mode sonator, Proc. IEEE Ultrason. Symp. () pp