Abstract:
Surface acoustic wave resonators are used to provide band pass filters. The length of each resonator is related to the lumped constant inductor and capacitor values of the components of the low pass filter equivalent. This invention allows high frequency s.a.w. type filters to be constructed by reference to already available inductor and capacitor values from tables of low pass filter designs.

Description:
The Government has rights in this invention pursuant to Contract Number DAAB07-75-C-1346 awarded by the Department of the Army. 
    
    
     BACKGROUND OF INVENTION 
     The design of filters with lumped elements is a well advanced art. Darlington (J. Math. Phys., 18, 257-353, Sept. 1939) has shown how the elements of a filter may be found for a prescribed transmission versus frequency (power transmission ratio) subject to certain constraints so as to make the filter physically realizable with passive elements. A power ratio with prescribed maximum excursions within the passband is conveniently represented by a Tchebysheff polynomial of the first kind. There are many other filter types that realize given transmission characteristics (Zverev, Handbook of Filter Synthesis, John Wiley and Sons, 1967). Tchebysheff filters have been realized with microwave cavities by utilizing their lumped circuit representation (Microwave Transmission Circuits, McGraw Hill Book Co., Inc., 1948). 
     Surface Acoustic Wave (s.a.w.) resonators have been realized recently with very high Q&#39;s (Li, R. C. M., J. A. Alusow, and R. C. Williamson, 1975 Ultrasonics Symposium Proceedings, IEEE Cat. No. 75 CHO 994-4SO, 279-283, 1975). Identical resonators have been cascaded so as to realize a filter design with a (relatively) flat passband and steep falloffs into the cutoff bands (Haus, H. A., and R. V. Schmidt, IEEE Trans. On Sonics and Ultrasonics, V. SU-24, No. 2, March 1977). However, the preceding paper did not disclose the criteria for realizing specific filter designs with s.a.w. resonators. 
     SUMMARY OF THE INVENTION 
     This invention provides a method for realizing electrical filters with surface acoustic wave (s.a.w.) grating resonators. Specific examples of filters having Tchebysheff, Butterworth, and Gaussian transfer characteristics in the passband are used to illustrate the invention. The invention can also be applied to filters having a maximally flat delay, linear phase filters with equiripple error, and Legendre filters. In brief, any filter with a low pass prototype synthesizable with series or parallel inductors and capacitors can be realized as s.a.w. passband filter by a proper choice of the number of reflectors (e.g., grooves) in the individual cascaded grating sections. 
     More generally, any electrical passband filter consisting of a cascade of series and parallel branches with a series resonance circuit in each of its series branches and a parallel resonance circuit in each of its parallel branches may be realized with s.a.w. resonators. The procedure consists in choosing the number of reflectors (e.g. grooves) of the s.a.w. resonators in correspondence with the Q&#39;s of the LC circuits and the Bragg frequencies of the s.a.w. resonators equal to the resonance frequencies of the LC circuits. 
     BRIEF DESCRIPTIN OF THE DRAWINGS 
     Further objects and features as well as the construction of the novel s.a.w. bandpass filters of this invention can be more fully understood from the followng detailed description taken in conjunction with the accompanying drawings in which: 
     FIG. 1 is an electrical schematic diagram of a low pass prototype of a three element filter. 
     FIG. 2 shows a s.a.w. resonator constructed of two gratings spaced one-quarter of a wavelength apart with its equivalent electrical tuned circuits shown as dependent on the reference planes of the resonator. 
     FIG. 3 shows a s.a.w. filter made from three serially cascaded resonators. 
     FIGS. 4A, 4B, 5A, 5B, 6A, 6B, show the filter characteristics obtainable from the s.a.w. resonators assembled in accordance with this invention where the transfer characteristic in the passband are Tchebysheff, Butterworth and Gaussian respectively. 
     FIG. 7 is an electrical schematic of a two-section LC filter which does not have a lowpass equivalent. 
     FIG. 8 is a block diagram of the s.a.w. grating filter of this invention. 
    
    
     DETAILED DESCRIPTION OF INVENTION 
     Any filter, with a low pass prototype synthesizable with series or parallel Ls and Cs can be realized as a s.a.w. passband filter solely by a proper choice of the number of grooves in the individual cascaded grating sections. 
     Consider the low pass prototype of a three element filter in FIG. 1. To achieve a certain filter characteristic, one may pick the values of Ls and Cs from filter-design tables as in the book by Zverev. In all tabulations, the bandwidth ω b  of the design is picked to be 1 rad/s, and the characteristic impedance is 
     
         Z.sub.o =1Y.sub.o =1Ω. 
    
     When the prototype filter is converted to a passband filter, the transformation is made 
     
         ωC→[(ω/ω.sub.o)-(ω.sub.o /ω]ω.sub.o C=[(ω/ω.sub.o)-(ω.sub.o /ω)]Y.sub.o Q.sub.C 
    
     where Q C  is the quality factor of the parallel resonance circuit (as a one port), and Y o  is a characteristic admittance (also picked as 1Ω in the normalized design). For a particular value of C lifted from the table for the lowpass prototype, the Q C  is 
     
         Q.sub.C =(ω.sub.o C/Y.sub.o) 
    
     for 1 rad/s bandwidth. The transformation for the impedance of the inductor is 
     
         ωL→[(ω/ω.sub.o)-(ω.sub.o /ω) ]ω.sub.o L=[(ω/ω.sub.o)-(ω.sub.o ω)]Z.sub.o Q.sub.L 
    
     The corresponding value of Q for the series resonance circuit derived from the inductor is 
     
         Q.sub.L =ω.sub.o L/Z.sub.o 
    
     For a bandwidth of w rad/s all capacitors and inductances have to be decreased by a factor of w. Therefore, the Qs of the required resonance circuits are reduced by the factor w when the bandwidth is increased by the factor w: 
     
         Q.sub.C =(ω.sub.o C)/(wY.sub.o), and 
    
     
         Q.sub.L =(ω.sub.o L)(wZ.sub.o) 
    
     A s.a.w. resonator constructed as in FIG. 2 of two identical gratings each of length l separated by a quarterwave section resonates at the center of the stop band of the individual gratings (at the Bragg frequency). In the neighborhood of the resonance frequency, the equivalent circuit is a two port with a parallel resonance circuit 1 across two shorting bars 12, 13. This equivalent circuit applies when the reference planes 2, 3 are chosen λ/8 beyond the center planes of the last groove or finger on the edges of the gratings. The addition of two quarterwave sections 7, 8 to each side of a two port transforms the admittance matrix of the two port into its impedance matrix, the parallel LC circuit across the shorting bars into a series circuit 4 in one branch connecting the top input and output terminals 5, 6. The series resonance circuit 4 represented by the inductance in the low frequency prototype must have a Q given by ω o  L/wZ o . 
     The external Q, of the resonator, is given by 
     
         Q.sub.ext =(ω.sub.o sinh.sup.2 κl)(v.sub.g κ) 
    
     where κ is the coupling constant of the grating, v g  is the group velocity and ω o  the center, or Bragg, frequency. This external Q is defined as the Q of the transmission cavity, as a two port. 
     Now Q C  or Q L  is the Q of the LC resonance circuit as a one terminal pair element. Therefore, Q C  or Q L  =20 ext . For a given κ, i.e., a given reflection from each of the reflectors (e.g., grooves) making up the resonator, one may find the length l of each of the two gratings of the resonator, by inverting the equation for Q ext   
     
         l=1/κ1n{a+29 (a.sup.2 +1)}                           (1) 
    
     where 
     
         a≡(v.sub.g /ω.sub.o)·(Q/2)            (2) 
    
     where Q is Q C  or Q L  depending upon whether the parallel resonant circuit or the series resonant circuit is being realized by the s.a.w. resonator. 
     Consider a filter made up of three resonators and suppose their one-half lengths have been evaluated from the preceding formula to be l 1 , l 2 , and l 3 . Then the s.a.w. filter realization consists of four grating sections of lengths l 1 , l 1  +l 2 , l 2  +l 3 , and l 3 , respectively, as shown in FIG. 3. For example, the grating section of length l 1  +l 2  is formed of one-half of the length of the first and second resonators. Each section is a uniformly spaced grating. This gratifying feature of the resonator design results from the cascade of two λ/8 and one λ/4 sections: the reference plane 3 of the first resonator λ/8 beyond the center of the last groove 10, followed by the quarterwave section, λ/4, leads to reference plane 8 and followed by a section λ/8 between the reference plane 2 of the second resonator and the center of its groove 11. 
     The length of each grating section is adjusted from its calculated value to provide the nearest integer multiple of reflectors. 
     There are many bandpass filter designs which contain single Ls and Cs in each branch of the lowpass prototype. Every single one of these can be realized as a s.a.w. bandpass filter by the transformations indicated above. Among the filter types that can be realized in this manner are the following: Tchebysheff, Butterworth, maximally flat delay, linear phase with equirriple error, Gaussian and Legendre. 
     In their s.a.w. realization, all these designs utilize uniform gratings of varying lengths separated by quarterwave sections. The degree to which the ideal performance is retained in the s.a.w. realization is a function of the &#34;goodness&#34; of representation of the resonator in terms of a lumped L-C circuit. 
     FIGS. 4A, 4B, 5A, 5B, 6A, and 6B show the transmission function of three different filter designs for the Tchebysheff, Butterworth and Gaussian filter characteristics, respectively, as laid out by the lumped equivalent-circuit procedure, but computed from the full grating equations of the IEEE article referred to earlier for κ=4cm -1 , κ=0.0024 cm. δ≡ω/v g  -π/Λ where Λ is the grating period. 
     
                       TABLE 1______________________________________C.sub.1    l.sub.1 /λ              L.sub.2  l.sub.2 /λ                             C.sub.3                                    l.sub.3 /λ______________________________________a     1.8636   201     1.2804 183   1.8036 201b     1.000    170     2.000  205   1.000  170c     0.2624   110     0.8167 161   2.2262 210______________________________________ 
    
     Table 1 gives the parameters for Tchebysheff (a), Butterworth (b) and Gaussian (c) filter designs where the capacitor and inductor values C 1 , L 2 , C 3  obtained from filter tables such as in the book by Zverev and the lengths of the gratings l 1 , l 2 , l 3  of each resonator (normalized with respect to the wavelength λ of the center frequency w o ) corresponding to the value of C 1 , L 2  and C 3 , respectively, are calculated using the design equations of this invention. From these values of l 1 , l 2 , and l 3  a filter such as shown in FIG. 3 may be constructed by conventional grating fabrication techniques. 
     Although the invention has been described in detail with respect to the three element inductor and capacitor circuit as shown in FIG. 1, with its corresponding three resonator s.a.w. filter circuit in FIG. 3, it is apparent that the method of this invention may be applied to any low pass filter having additional series and parallel branches, with inductors on the series branches, and capacitors in the parallel branches by merely following the procedure provided in this specification to provide corresponding additional cascaded resonators. It should also be apparent to those skilled in the art, that the low pass filter circuit may be of the T-type rather than the π-type shown in FIG. 1. 
     Although the invention is described for synthesis of s.a.w. filters in analogy with L-C filters that possess a low pass equivalent, it can be extended to any s.a.w. filter design in correspondence with an L-C filter which does not possess a lowpass equivalent in that the resonant frequencies of the L-C circuits are not identical. See FIG. 7 for the example of a circuit with two branches. Each s.a.w. resonator can be made to possess a different resonance frequency ω 1  and ω 2 , respectively by, for example, different reflector spacings each of two s.a.w. resonators, so that the Bragg frequencies of the resonators are made to coincide with the resonance frequencies ω 1  and ω 2 , respectively. The lengths of the resonators are picked using formulae (1) and (2) in terms of the Q&#39;s of the resonance circuits, Q C  =√C/L/wY o  for the parallel branch, Q L  =√L/C/wZ o  for the series branch. 
     The techniques for the fabrication of s.a.w. resonators is well known to those skilled in the art. As an example, one form of resonator having a κ of 6.439 cm -1  is obtained from ion milled grooves of approximately 0.01 of a wavelength depth in LiNbO 3 . In another typical resonator made with 12νm 12μgratings, the gratings are typically formed by indiffusing 1000A° of Ti into LiNbO 3  for 30 hours at 1007° C. which provided a coupling coefficient κ of 2.53 cm -1 . The techniques for coupling signals into and out of resonators is also well known to those skilled in the art and is shown in FIG. 8 where the cascaded resonators 30 of FIG. 3 are shown in relation to an input transducer 81 and an output transducer 82 on a s.a.w. propagating substrate 83 to constitute the s.a.w. filter 80 of this invention. 
     It is evident that those skilled in the art, once given the benefit of the foreging disclosure, may make numerous other uses and modifications of, and departures from the specific embodiments described herein without departing from the inventive concepts. Consequently, the invention is to be construed as embracing each and every novel combination of features present in, or possessed by, the apparatus and techniques herein disclosed and limited solely by the scope and spirit of the appended claims.