IEEE TRANSACTIONS O N SONICS AXD ULTRASONICS, VOL. SU-18, NO. 3, JULY 1971 155

Mechanical Filters-A Review of Progress ROBERT A. JOHNSON, MEMBER, IEEE, MANFRED BORNER, AKD MASASIII KONKO, MEMBER, IEEE

Abstract-This paper is a review of electromechanical bandpass 0 - filter, resonator, and transducer development. Filter types discussed include intermediate and low-frequency configurations composed of rod, disk, and flexure-bar resonators and magnetostrictive ferrite g and piezoelectric ceramic transducers. The resonators and trans- ducers are analyzed in terms of their dynamic and material char- 2 2o

acteristics. The paper also includes methods of realizing attenuation 2 poles at real and complex frequencies. The last section is a look at future developments. 30

IXTRODUCTIOX

ORE T H A N 20 years h a w passed since the first practical mechanical filters were introduced by Adler [ l] , Roberts [2], and Doelz [3]. These

fibers met an existing need for greater selectivity in the IF stages of AM and SSB receivers designed for voice communication. Rlodern receivers and telephone commu- nication equipments carry not only voice but data mes- sages as well, thus requiring lower passband ripple and well-defined and stable passband limits. I n addition, greater selectivity and lon.er loss are often required.

Mcchanic:~l filter development has kept pace with the demands of communication engineers. For instance, filters having passband-~,ipple requirements of 0.5 dB or less or 60/3-dB bandwidth ratios of 1.3/1 are not unusual. Package size has also heen reduced; a 10-pole filter can be designed into a l-c1n3 package. In addition, the lower frequency limits of mechanical filters are now in the audio rangr; the upper limit is tha t of an AT-cut crystal. Although the mechanical filter was conceived in the U S , a great' amount of cleyelopmental work is now being conductet1 in Europe and .Japan.

Early interest in the use of 111cchanical elements to provide passhand cllaracteristics resulted from the de- velopnlent, of the electrical bandpass filter by Campbell and Wagner in 1917 [4]. A number of patents relating to spring-mass systcms [ 5 ] were filed soon after, but the most significant work in terms of practical deyices was done by Maxfield and Harrison on phonograph

1 I I I l I I 1 100 2W 300 500 1 OW 2000 30W 5ooo

FREGUENCY (Hz1

Fig. 1. Improvement in phonograph design through the use of filter design metl~ods [61.

n m / F L E X U R E ( F "

FREOUENCY - Fig. 2. Distributed elemrmt mecllanicnl filtcr (Mnson, 1941).

filter shown in Fig. 2 [TI. Pome of the significant, features M:~nuscripl rrceivetl .J:ln11:ly F, 1971. of this design included the use of piezoelectric trans- R . A . Jnhnson is w i t h the Collins Radio Company, Newport ducers, distributed-Flement llalf-wavelength resonators, M. F3iirner is with AEG-Telefunken Rescarcl1 Institute, Ulm; and an attenuation-pole-producing coupling element.

Bcnch, Calif.

Crrmnny.

Pnm:lgata University, Yonemws, Japan. ical filter, shown in Fig. 3(a)! was developed by Adler. M. Iconno is with the Dcpartmcnt, of Electrical Engineering, Just a fev7 years later, the first practical IF mechan-

156 IEEE TR.INSACTIONS ON S O S I C S A N D ULTRASONICS, JULY 1971

TRANSDUCER COIL

COUPLING Wi R E

This fi1tc.r Inrltle w e of l d f - m r c l c n g t h platw coupled hy short small-tli:unctrr wires. The plate resonators can be representcc1 bp p:~raIlel tuned circuits, the coupling wires by seriw inclurtors, as shon-n in the electrical equivalent circuit of Fig. 3 (b). The nickel end plates act’ a.: magnet.o- strictive input and output transducers.

The frcqucncp response cllaracteristics are monotonic, that is there a r c no finite attenuation poles. Adler’s filter is typic:ll of most modern mcchanical filters in that distrihutetl parameter resonators arc coupled through nonrcsorlnnt lincs. ‘I‘here is a striking similarity betmeen the plnte-\\-irc filter a n t ! the latest t;vi)c of mech:anical filter, thc n~onolithic or q u a r t z mechanic:zl filter shown in Fig . 4 ( :~’ l .

T h e quartz nlcclmlical filter cleveloped by Beaver and Sykcs [S1 and K:‘:lkaza~:l [91 makes use of the cnergy- t,rapping concept; whcre standing w a ~ w are set 11p under each eloctrode pair. 111 t h e regions hetn.ecn the elec- trodes, tllc, acoustic, x i v c clcc‘ays oxponcntinlly, the electro(]ctl rc,gions act like r(~sonntOrs, and the non- clcctrodcd regions act like coupling clcmcnts. The elec- t,ric:il cyuivnlcnt circnit of t I 1 v platc-n-i1,c and the quartz mechanirnl filtcxr sllonn in Fig. 4(b) are itlentical except t,llat in the plate-wire case the input and output capac- itors are replaced by coils. The monolithic filter also has monotonic frequency response characteristics. A con- struction similar to that of the monolithic filters rle- scribed abore 11as been used by Borner and Schussler

BASIC FILTER STRCCTURES

Monotortic Designs Adler’s tlevclopn~cnt of the plate-wire mechanical filter

was soon followed by Doelz’s disk-wire filter [ 111, shown in Fig. 5 ( a ) . The disk-wire filter makes use of flexural modes of vibration of the disk resonators. The two most

STATIC CAPACITY COUPLING

REGION

ELECTRODE PAIR

uset1 modes are the one-nodal circle (-50-200 kHz) and the two-circle mode (200-600 kIIzl. The resonators are coupled h>- small-diameter w i r e t h t :11’(’

?pot n.cltlcrl around the circumftwnce of the disks. As a rwult of the yery complex displacement at the ctlgc of the tlisk, the coupling mode is a con1hinat)ion of extension and flexure and is, therefore, very difficult to analyze. The earliest designs made u w of small-diameter odd- nun1l)er quarter-wawlengtll iron-nickel alloy transducers. Latclr dcsigns, such 5 s tllat shown in Fig. 5 ( a ) , use onc- half and full-wawlcngth ferrite transducers.

Ferrite transducers are also used on the rod-wire filt,er shown in Fig. 5(h’l. The torsional rod-wire filter was developed indepcndcntly hy Biirner in Gerrnnny [lZ], [ 131, and Tanaka in .Japan [l41 ; the Tanaka design uses Langeyin metal alloy/ceramic transducers in place of the ferrite.. The cylindrical rod resonators are de- signed to vihrate in R half-marclength torsional mode at frequencies up t o 250 kHz and in a half-wavelength longitudinal mode when the filters are designed to operate at 4.55 kHz. Use of the torsional mode results in longitudinal coupling between resonators, whereas a t 455 kHz the coupling involves bending or flexure.

J O H K S O N e t d . : REVIEW' O F MECHANICAL FILTERS

Use of a wire to connect the transducer to the end resonator, as shown in Fig. 5 ( h ) , results in a reduction of spurious or unwanted responses, particularly if it is nt tacl~td a t :I 11od:d point of the strongest, unwanted modes. This technique, which is often used on disk-wire filters, rcsults in r1)urious modes being suppressod more than 60 dB, as shown in Fig. 6 ( a ) . In contrast to the excellcnt spurious response rc>jection of the torsional rod-wire filter is the response [Fig. Gcb)] of the rod- neck filter of Fig. 7 (a ) , which is driven in a similar manner, but, i.4 subject) to bro:ld banclwidth hn(1ing modes.

The rocl-ncrk fi1tc.r was tlcwloped hy Roberts [2] a t the same time the disk-wire filter was being developed. The basic clcl-ign concept is that of coupling half- wavelength tol,sionnl or lorrgituclinal resonators with c ~ u : ~ r t e r - w : ~ r c l ( ~ ~ l ~ t l ~ n c ~ k s : . T l l c , two m j o r probltwls of the early designs were the easily excited spurious bend- ing modes [ l 51, :~nd the difficulty in construct,ion and tuning due to h i n g turned out of a single rod. In adtli- t,ion, a t lon.er frcquencicu like 100 kHz, the filters become extremely long. In oulcr to reduce t h e length, Tanakn. developed tllc folrlccl line filter show1 in Fig. 7(h). Thc resonators vihrntt: in n I~alf-n-:~wlength longitudinal mode and are coupled hy relatively large-diamctcr short wires. The usc of 1ongitutlin:ll-mode Langevin trans- ducers plus the rcduced ovcrall length rcsults in :L greater rejection of unwanted moden. Like the rod-wire and folded designs, the disk-wire filter has a relatively low spurious response levc.1.

All four filters discussed so far operate at' frequcncies above 50 k H z . With the exception of some work reported hy MaPon and Konno, tllcre was little activit,y a t fre- quencies hclow 50 kHz hefore 1960.. One of the first practical tlesigns was that of Mason and Thurston [ 161, who used antisymmetric mode flexural resonators coupled in t,orsion as sllon-11 in Fig. 8(a) . The antisymmetric mode makes this clesign less susceptible to rnicrophonic excitation. A more widrly used design is the symmetrical mode filter shown in Fig. 8 h ) . Work by Iconno [ 171, [ IS] , Y,zkuwa [19], and Alhsmeier [20], h w e made this a wry practical tleyice in the frequency range of 300 Hz to 30 kHz. The symmetric-mode filter is driven hy pictzoelectric-ceramic transducers. The coupling wires :m attached to tlx resonators at the nodal points, which resuks in torsional coupling. This type of filter is w r y sensitive to chngcs in t,he position of the coupling wires so conderahle care is taken in manufacturing to ensure that bending modes are not propagated. The microphonic problenl has been solved, in part, by supporting the filter with high-damping silicon-ruhher supports.

In addition to the flexural-har/wire lorn-frequency mechanical filters, a considerable amount of work has been done in ,Japan on the development of tuning-fork mechanical filters [21], such as those shown in Fig. 9 (a ) and (b) . The three-prong filter is interesting in that it acts like a coupled two-resonator filter, thus providing a

157

14@ r 12c t -- -

2 GC U

t 4@

1'- -l- ---- - -

2:o--.- 200 400 G@@ 8W l0U0

FREOUENCY :kHz,

(:L)

I

(h) Pig. 8. Low-frcqrlrnry meehnnicnl filters. (a) Antisymmet.ric

mode [161. (b) Fundamental mode [181.

158 IEEE TRANSACTIONS o x SONICS ASD ULTRASONICS, JULY 1971

(b) Fig. 9. Tuning fork mechanical filters. (a) U-shaped coupling

type. (b) Two-pole three-prong type.

greater amount of selectivit'y than a simple two-prong device. The tuning fork and flexural-bar filters are widely used in Japan for remote and automatic control, selective calling and paging, telemetry, measuring instruments. and pilot and carrier signal pickup.

Finite Attenuation Poles. By using as many as 15 rcsonntors, highly selective

mechanical filters of the bar-wire, disk-wire, and folded- resonator types are realizable. Although the resultant passband amplitude and delay responses are sat'isfactory for voice communication, the ripple amplitude and clif- ferential delay variations may be excessive when data are to he transmitted. By making use of finite-frequency attenuation poles, fen-er resonators are needed, thereby reducing the differential delay, size, cost, and, most often, the pwsband ripple.

One of the first practical del-ices t o realize finite attenuation poles was the crystal-plate filter with cnpac- itive hridging [X!], [23], shown in Fig. lO(a). Although t,he platc configuration was described in hdler's basic patent [24], the itlea of using quartz and capacitive bridging was new. Connecting the capacitor from the top electrode of the input plate to the bottom electrode of the output plate is the same as adding a phase inverter across the out,put terminals of the equivalent circuit of Fig. lO(h) and results in a pair of attenuation poles as shown in Fig. lO(c) . The phase inverter adjacent to the coupling clement represents the 180-dcg phase shift' be- tween resonators a t the lowest nntural nlotle of the me- chanical system. The resonators vihrate in a length- extension modc ( 3 ' ' - S or -18.5' - S ) or face shear mode (CT) and are coupled through so-called Poisson coupling. Although t h e x filters have been designed and built with more than two resonators, most currently manufactured filters are de.4gnecl as 455-kHz 2 poles in cascade for use in SS13 and FM receivers.

The use of wires to couple mechanically alternate

ATTENUATION v resonators as iuclc~penclcntly considered by B6rner [25] and .Johneon [26]. Bijmer's filter shown in Fig. l l ( a ) makes use of l~nlf-~~:~velengtll torsional-mode resonators ant1 extensional mode-coupling elements. The disk-wire filter of Fig. 11 (1)) is compos(d of flexural mode disks and basicnlly extensional mode-coupling elements. When the spacing between the resonators is such that the hridging wire is less than one-half of an acoustic wave- length long, the ideal transformer of Fig. l l ( c ) a c t s like a simple one-to-one transformer and an attenuation pole is realized on the high-frequency side of the filter passband. This can be understood by converting the T

network, composccl of the two adjacent resonator cou- pling wires (inductors) and the bridging wire, to its T equivalent circuit,. The inductor in series with the center resonator produces an ilnpedance zero, which results in an attenuation pole above the filter passhand, as shown in Fig. l l ( d ) .

When the bridging n-ire is hctwecn one-half and a full- wa~elcngth long, the transformer in the electrical equiv- alent circuit acts like a phase inverter and an attenuation polc is realized below the filter passband. An alternate method of realizing the phase inversion in the rod-wire filter is ehoxm in Fig. l1 (a) by the dashed lines. In this case, out-of-phase regions of the alternate resonators are coupled. A similar mcthod is used with disk-wire filters where one of the alternate disks vibrates in a diameter mode (rather tllan a nodal-circle mode), the t,hree adjacent diPk wires being connected to in-phase sectors, and the bridging mire to an out-of-phase portion of the resonator [27].

JOHNSON e t al.: REVIEW OF MECHAXICAL FILTERS 159

TABLE I ATTENUATION POLE LOCATIOXS FOR VARIOUS

BRIDQING CONFIGURATIONS ~ ____ -. . . . . . -

Direct (1 : 1) Inverted ( - 1 : 1) m

l ATTENUATION

FREQUENCY - (d)

Fig. 11. Singlr-rrsonntor aconstir bridging. (a) Rod-wire, 1251. (h) Disk-wire (Johnson, 19664). (c) Electrical erlnivalcnt circuit. (cl) Frrqurncy response.

If a high degree of selectivity is needed both above and below the filter passband, a coupling wire can be used to bridge two resonators producing a ,symmetrical pair of attenuation poles. This type of design is subject to being ahle to realize a phase inverter with each bridging wirc. When no phase inversion takes place, the resultant frequency response is less selective than the monotonic case but delay compensation due to right-half plane attenuation polcs is possible (see Table I ) . The moat common method used to realize the phase inverter, in the ease of the rod-wire filter, is to space adjacent rcvonators a quar.ter-wavelengtl1 apart, which results in the hridging wire being three-quarters of a wavelength long. The advantages of the quartcr-wavelength coupling are that thc resonators are all tuned t o the center fre- quency of the filter and, in addition, the coupling is relatively insensitive to variations in mire length.

I n order to maint,ain a small package size, the coupling wire length between disk resonators is usually less than one-eighth of a wavelength. At 455 kHz, due to the thickness of each disk being on the order of a quarter- wavelength, simple tm-0-disk bridging results in sym- metrical finite-nttrnuatiol1 poles. Fig. 12 shows the fre- quency response of the 12-disk filter shown in Fig. 13. Note that the passband ripple is quite low. This is in part due to the fact that the disk-wire as well as the rod- wire finite-pole filters are designed with modern insertion- loss techniques. Included in the design method are trans- formations such as that shown in Fig. 14(a) where a low-pass or bandpass ladder network is converted to an equivalent bridged form [28]. Because of the narrow-

One resonator Vpper stopband pole Lower stopband pole

Two resonators Delay correction Upper and lower stopband poles

50 -

I

m 0 I

z 40 - 0 3

30 - U

20 -

10 -

454 455 456 457 458 459 460

FREOUENCY (kHz)

Fig. 12. Frequency responsc of :L double-disk bridging mechani- cal filter.

Fig. 13. hlrvhanical filter with wire bridging across two rrsona- tors (Collins Radio).

converted to a X network conlposed of two negative inductors and one positive inductor, all having the same absolute value [29]. As an example, the lorn-pass network of Fig. 14(h), which may be a Cauer-type filter [30], can be converted to the bandpass double-resonator bridging topology al1on.n in Fig. 14(c). The use of tloublc (coincident) polcs rcsults in a physically sgnmetrical mechanical configuration, ~ h i c h is helpful in reducing nmnufacturing costs. Thc short coupling wire length be- tween resonators, a,s shown in Fig. 13, not only decreases the package size but, increases t'he strength of the filter. The ahilit,y of the structure to withstand high ehork and vibration levels is also due to the coupling wires being located away from the centroid of t'he st'ructure.

160 lEEE TR4NSACTIONS ON SONICS A N D ULTRASONICS, JULY 1971

( c ) Fig. 14. Double bridging trnnsformntions. (n,) Gcnmnl trnns-

bandpass electrical equivalent circllit, formation. (h) Lorv-pass prototypr. (c) Fmal hridging-wire

have been designed to make use of spurious modes of vibration to control stophand selectivity. Although the theory was not understood until recently [31], i t was found that by varying the coupling-wire orientation around the circumference of the disks the slope of one side of the response could be increased a t t he expense of the other side. For inst'ance, in Fig. 15(a) each disk has a natural resonance near-frequency PI in the pass- band. Each disk also has a natural resonance (actually two as well as many others at different frequencies) near F , above the filter passband. Making use of the sim- plified equivalent circuit shown in Fig. 15 (b ) , me see that an attenuation pole is produced a t F , between F1 and F,. This results in a steeper response above the filter passband, which can be controlled by varying the coupling wire orientat,ion, which in turn controls the effect of resonator Fa. The same technique has been used in the design of plate filters [32] D-here the length and width dimensions control the frequencies Fl and F2 as shown in Fig. 15 (c ) .

By removing a segment from the edge of a disk rrsonator, two controllable diameter modes correspond- ing to each pair of degenerate modes can be produced [33]. This technique has been used to design a variety of multiple mode filters 1341-[36], most of which are still only laboratory models. An example of this type of design is shown in Fig. 16(a). This particular filter employs one-circle one-diameter modes of vibration, as well as piezoelectric ceramic transducers and has a fre-

(b)

F 2 i q y 7 j - 1

(C)

Fig. 15. Multiple-mode finite pol? configuration. (a) Disk-wire. (p) Simplified electrical cquirn,lcnt circuit. (c) Platc-typr dc- slpn.

F 2 x F1 8

( 1 ) )

Fig. 16. Multiple resonant modc filters. (a) Disk-wire design. (b) Lom-frcquency flexural hnr.

quency rwponse equivalent to a four-resonator design. The soIid nodal lines correspond to tJhe highcst natural mode.

In the case of low-frequency filters, a similar kchnique can be used where the corner of a flexural bar can be removed to produce two natural modes by destroying Fymmetry of the nloments of inertia. An example of this type of resonator is shown in Fig. 16(b) where the arrows show the displacement directions of the two modes [37]. Fig. 17 shows an early filter that makes use of this technique to obtain a total of six natural resonances as well as additional attenuation poles [38]. -4 large variety of devices h a w been designed using this method, includ- ing a t,hrce-resonance two-attenuation pole filter con- structed from a single bar [39].

JOHNSON e t al.: REVIEW OF >IECIIA~-IC.IL FILTEHS +(-----l 161

D 0 I 2 4 3 5 6

ATTEYUATION I l 1

U F R E O U E V C Y -

a

-%,,.l- 1 Q21 - ' (C)

Fig. 18. Two-port two-mode disk rcsonntor. (a) Driving point imprdancc modc.1. ( h ) Disk nodal patterns. (c) Single-mode b o - port cquivnlrnt circuit.

11010gy) i = 1. Tllc resonator could be :I rod or bar vibrating in a flexural and a longitudinal mode or, as s1~on.n in Fig. 18(b) , a disk vibrating in x two-diameter flexure Inode and a single-circle flcsurc motlc. If ordy one of the motles, but tn.0 point:: on the disk are con- sidered, the force-vclocity relationsllips can I K found from the schenlatic diagram of Fig. 1 8 ( c ) . I n gcneral, n resonator having 31 ports and S nat11ral rno(1es ran be descrihctl by the matrix equation

.v

211 = [ZI I1 211 = c d J k h j 2 i . ( 2 ) , -1 I n ( 2 j , v ] and f ] arc column n~nt~, icci . : of ortlcr 111 ancl

zl,l is nn element in the X X A1 Z matrix. A gcncmlized equiralent circuit I m e d on (21 is shown in Fig. 19.

162

by making use of a linear combination of independent waves that satisfy the differential equations of motion and boundary conditions on the major surfaces of the disk as well as approximately satisfying boundary con- ditions on the lateral surfaces [43]. Cnlike the exact solutions that are expected when analyzing an electrical network, solutions for frequency and equivalent mass of a mechanical resonator are only as good as the number of higllcr ordcr ~vaves that' are taken into account. The greater the number of '17'aves, the more accurately the boundary conditions can he satisfied.

The earliest work on finding the equivalent mass of a thick disk resonator was pcrformed by Sharma for the case of axisymmetric mode3 [44]. This analysis was based on the nlethod of finding the cquivalent mass by dividing the tot>xl kinetic energy in the system (disk) by one-half of the square of the velocity in a specified direction atj a point on the disk [45]. This same tech- nique couplcd with that of Onoe's 1431 has been used to calculate the equivalmt mass of disks vihrating in both synmctric and nonsymmetric modes [46]. Examples of equivalent mass versus position on the surface of the disk are shown in Fig. 20 for two adjacent vibration motlcs. Note that the equivalent mass at the disk edge ( T = 1.0) is lomcr in the case of the adjacent two- diameter one-circle mode than the two-circle mode. As a rule, the loner the impedance of a mode, the broader is its coupled response and the more difficult it is t o suppress.

Within the frequency range of interest (50-600 kHz) in the case of disk-wire filters, there are various other unwanted modes present such as radial and concentric shear modes. These can often be troublesome in the case of wideband filter designs where the modes sometimes fall in the range of the paszband. These particular modes, called contour modes because they involve no tran- -verse (flexural) vibration but only a change in the shape or contour of a disk, have studied by Onoe [47].

Bar Resonators and Coupling Elements Flexural-mode resonators are also used a t low fre-

quencies. In thc 500-Hz-to-50-kHz frequency range, the resonators are in the form of bars such as those shown in Fig. 8. Analysis of the resonant frequencies of thick flexural-bar resonators has been performed by Mason [45] and Niiser [48]. If the resonator is treated as a thin bar, equations similar to those describing a trans- mission line can 1)c writtcn :LT; [ 1.51, [49]

- II,(cu/l)

IZ,

Hlo(jwlz/Kaz)

Hi(jwl/Ka)

IEEE TRANSACTIOSS ON SONICS AND ULTRASONICS, JULY 1971

I f M

l " 1

1 I

H, = s.s H , = S.c + C.s S = sinh CY

I€, = c.c H 7 = S + S S = sin CY

H , = C % - 1 H , = s - S C = cosh a

H, = C.C -4- 1 B, = c + c c =cos CY

H , S . C - C.S H,, = C - C

CY* = ( p A / K ) w 2 1 4 I (Cy2 / ( jW1*) = (p-4K)1'z/j.

I n (3) , wllich relates to Fig. 21 (a ) , I< reprcxnts the product of Young's modulus times the moment of inertia of the cross section of the bar, and A , and l are, re- spectively, the density, cross sectional area, and length of the bar.

Using (3) , we can represent :I flexural-mode resonator by the equivalent circuit, shown in Fig. 21. This equiva- lent circuit represents a one-resonator two-port system similar to that shown in Fig. 1 8 ( c ) . Note that in this case the representation (where force is an across vari- able) is used resulting in a dual formulation where the resonator is represented by a series-tuned circuit. In acldi-

JOHNSON e t d.: REVIEW OF 41ECHASIC.4L FILTERS 163

G -

C -

0 -

0 -

0- a 1 l I I I 7 4 6 8 1 0

R A D I A L POSITION ir.

(a)

'\

tion, the two ports or points on the resonator are actually represented in the equivalent, circuit by four ports, two for linear motion and two for rotation. As in the earlier disk case, a general equivalent circuit can be drawn with- out difficulty by parallel connecting networks of the form shown in Fig. 21(b) [49]. Konno's normalized function E is similar to the 4 function of (2) when the impressed bending moments are equal to zero.

The equivalent circuits of Fig. 21(b) and (3) are very useful in the case of a rod-wire filter where the rod resona.tor vibrates in an extensional mode [50]. Fig. 22(a) shows a coupling wire attached to the ends of two rcsonators. The coupling wire is driven in flexure, but there is no rotation of the wire a t the pointas of contact with the resonator, i.e., O1 = e2 = 0. After some minipulation of 131 me can write

(a)

-E;yjq F1

"1

(b)

Fig. 22. (a) Longitudinal bar-rcsonntor wire-coupling element (case of zcro rotation). (b) r equivalent circuit.

The ABCD matrix of (4) can be transformed to a y matrix, which, in turn, can be used to calculate and y b of the a-equiwtlent circuit of Fig. 22 ( h ) . We find tha t

Y , = ~ ( I w w [ ( H ~ + m / & ] (5 )

Y , = - j(Ka3/uZ3) ( I17/H3) . The coupling wire acts as a quarter-wavelength line

when 116 = 0. I n this caw, we w e from (5) tha t ga = -yb ~~ ~~~ ~ ~~ . __

164 IEEE TRANSACTIONS ON S O N I C S A N D ULTRASONICS, JULY 1971

and from (4) after some manipulation that [50]

An analysis of an extensional-mode resonator or cou- pling wire is somewhat less difficult, than that of a flexural element. A rod or wire vibrating in an extensional mode acts as a simple transmission line and, t.hus, can be described by the ABCD matrix

1- 1,' I = ; i- cos ( D l ) i ~ , sin ( p / ) : rJ!r2 1 (7) LF,.~ L j(sin (PO/Z,,) cos ( P I ) A

where = W dzj antl Zo = (9 da). I!: is Young's modulus. Exteltsional-motle resonators and coupling elements h v e been described in consitlernble detail in [l41 : L n d [30].

Resonator Xater ia ls

Bccause of transduccr bandwidth limitations, the pres- ence of unwanted modes of vibration and competition with 7,C' and ceramic filters, the rntio of 1)andn-irlth to center frequency in the case of nxxllarlical filters is usually less than 10 percent but more commonly about I percent. This small fractional bandwidth requires the resonators to have a temperature coefficient of frequency of 1-10 ppm/"C, a corresponding low aging rate, and a Q value of at least 10 000.

The major contribution to the variation of frequency in :L metallic alloy with temperature is the change in the stiffllcss or Young's nlodult1s of the material. Iron-nickcl alloys that contain either 27 or 44 percent Xi hare a low- temperature coefficient. of stiffness I m t are relntively 1111- stn1)le with regard to changes of the percentage of Ki. By adding cl~romium, the stahility can he improvccl consider:lbly, but the aging charactctristics and Q values of t t m e so-callecl Elinvnr materials are not acceptable. The addition of titanium or beryllium to the Fe-Ni-Cr improves h t h aging and (2 antl, in addition, nlakes it possible to vary the tcmpcrature coefficient of resonant' frequency material by cohl work and lleat treating [51].

In order to obtain a low tcnlpcrature coefficient, thc Fe-hTi-Cr and BC or Ti nmterial is f i r z t solution an- ne:ilc[l, thcw qncrrcllc~(1, and tllen cold worked 15-50 per- cent. Kext, the rnatcri:d is precipitation hartlcncd (A? is precipitated from a supcwaturated solution by the BC or Ti ) by heat treatment at 40Oo-675"C for a t least two hours. The amount of roltl ~ o r k and temperature time determines the temperature coefficient. of the material. Fig. 23(a) and (b) show temperature curves of Ther- malast 5409 (Be) and Xi-Span C (Ti) after having lxen acljudcd for best temperature coefficiont. Both materials S ~ O W fl,ccluency shifts of 1 ~ s than 50 Hz Over a tempera- ture range of 100°C a t 500 kHz.

The addition of Be or T i has the cffcct of reducing the aging rate to leas than 1 X lo-' ppnl/week or approsi- matcly 25 IIz : I t 500 k H z over a period of 10 years. The

N I .-SPAN C

FREOUtNCY = 455 L H I

COLD W O R K ~ 33'.

y o 40 + A n 1 R 0

FREOUtNCY = 455 L H I

COLD W O R K ~ 33'.

y o 40 + A n 1 R 0

Be or Ti also improves the Q of the resonators, which may vary from 10 000 t o 25 000, depending on the amount of cold working and precipitation hardening. A value of 20 000 is typical for most applications and cawes only a small amount of loss a t t he pasband edgcs of a 1 percent, f1xctional h n d w i d t h filter. For in- stance, the response of a 3-kHz bandwidth filter at 200 kHz is practically that of a lossless network.

TT'hen adjusted for the best temperature coefficient, thc resonator frequency Fhifts have little effect on the passband ripple. The pni;s~)antl-ripple variation mill, for the most part, be determined by the characteristics of the transducer.

TRASFI)T:CERS A S D TR.4iXSI)CCER RI.ATEIZIAIS

Transducer Configurations The most. n-itlely used transducers h a m been the sirn-

plc mngnctostrictive ferrite rod transducer (Fig. 5) : m 1 the 1,anger-in ceralnic-nlc4nl alloy transtlllcer [Fig. 7 (b) ] for frcqucncies above 60 kHz and the compositr or sanclnic.l~-typc~ of ccr:~mic-rnetnl t~ranstiuca~r : I t lorn frc- quenciw (Fig 8). Altllough thcnc are the most popular, filt'ers arc Ix3ing manufacturetl that use longitadinal mode ceramic rock;, iron-nickel alloy wires, and various qu :~ r t e crystal cuts.

An electron~ecl~a~~icnl tranatlucer, regarcllcss of the type, can be char:lcterizccl by its resonant frequency, coupling coefficient, and static rcnctancc. As an example, thc nlagnetostrictivc ferrite tr:mducer shown i n Fig. 24 can be defined by the mechanical reuonant frequency f l (actually the mechanical resonance with the electrical terminals open circuited)! the electromechnnical coupling coefficient, which relates the electrical and mechanical parameters. and the inductnnce of the transducer coil L1. If the t,ransducer is directly att'achcd to the end reso-

165

nafor, it affects the repon:ult frequency of the comhina- tion in direct proportion t o its relative equivalent mass. In t he case of the electronwchanical coupling coefficient, i t is important that it be constant because it dircctly determines the transformation of the electrical terminat'- ing rePistance into a mechanical termination tha t must be matched to the nwchanical impetlance of the filter. In most applications, the elcctricd inductallre is resonated with a c:lpncitor, which is u d for temperature compen- sation in critical c:lscs.

T l ~ c t filter designer has tlle task of finding the most stnhle transducer that will, in addition, propc!rly match the clcctrical and n1cchanic:tl network.<. This involves cl10041lg :L ~ n a t c ~ i n l w i t h :l st:~l)lc coupling coefficient first and then :I configurntioll that retluco~ the trans- ducer cquiv:~lcnt 111:is t o a minimum so as to recluce thc frcqu(~ncy shift n.it11 tcw~pcwture. Thib c m hc clone 1)y decreasing t h r tliamctcr of tho frrritv roc1 o r the tlliclmcsb of tho ctbramic in n 1,angevin tlxnsclucer. The sim reduc- tion is, in turn, limited by tlw sensitivity of the coil or static capacity, or in thc c:\sc of ~ ~ i d e - b n n r l ~ ~ i r l t h filters by the fact that, in the limit, the electrical inductol.-c:1pacitor corn1)inations actually become the end resonators.

Mngnetostrirtive Ferrite Transducers

Thc most widely used filter transducer material is a cobalt-ruhstit~~tc.tl iroll-nic1;c.l ferrite, which w : ~ . ? spe- cially tlevclopotf for use in elect,romerhanical filters [fi2]. Addition of 0.6 percent cobalt results in a highly stable coupling coefficient, whereas 1.0 percent cobalt results in n very small variation of frequency fl with temperature. Fig. 25 shows the variation of coupling coefficient ant1 frcqucmcy f. with change in tclnperat,ure for B practical drsipn at 200 kHz [53]. I n order to be able to reduce the effect of the transducer on the end resonator fre- quency, opt in~um vnluc~ of pcrmancnt magnet bias and coil dimensions must be chosen so as to maximize t'he coupling coc.fficicnt. The coupling coefficicnt of 10 per- cent shown in Fig. 2.5, which is typical for R n-ell- designed trnnsduccr, is also thr limiting ralue of the filter fractional bandwidtll. The coupling coefficient yaria-

.1530.106 40 - 2c 20 40 60 80

TtMPERATU9E lDCl

(a)

166

transducers. The barium-titanate material used in early designs suffered from large changes in coupling coefficient and frequency with temperature, which, in turn, resulted in large passband-ripple variations with temperature. The development of lcad-zirconate-titanate (PZT) ma- terials [ 5 5 ] , which show vastly improved temperature charact,eristics, has made it possible for piezoelectric ceramic materials not only to compete with, but in a number of cases to replace: magnetostrictive ferrite transducers.

The principal advantage of the piezoelectric ceramic is its excellent electromechanical coupling coefficient. Whereas a bar-type ferrite transducer has a coupling coefficient of 10 percent, a side-plated bar of temperature- stable piezoelectric ceramic has a coupling coefficient between l 5 and 25 percent. The higher coupling coeffi- cient therefore makes it possible to use a lninimum amount of ceramic material as compared to a highly temperature-stable metal disk alloy in the design of composite end resonator/transducer assemblies. In addi- t'ion, the higher electrical Q of the piezoelectric material results in lower filter insertion loss and its greater inherent linearity results in lower intermodulation dis- tortion. Fig. 26 shows the variation of frequency and coupling with temperature for the case of a PZT ceramic that has a coupling coefficient kIz3 of approximately 23 perccnt.

T h e most widely used of the ceramic transducers is the Langcrin type, which is in the form of a rod com- posed of a ceramic disk sandwiched between two metal alloy rods, Fig. 7 (b) . The high planar coupling coefficient of a thin disk, which is on the order of 25-40 percent (for stable materials), and the Langevin configuration make i t possible to design resonator/transducer assem- blies that have high mechanical Q and good temperature characteristics with only a small reduction in the coupling coefficient [54]. Development' of this type of transducer for filter applications wa.s started approximately 15 years ago by Tanaka [l41 and Tagawa 1561 in .Japan. Because of the instability and low Q of most bontling materials such as epoxy, coupled with the fact that the bond has a.n appreciable thickness, the ceramic/metal alloy attnch- ment prohlcm is one of importance and haas required a great deal of effort in its solution.

In the past few years, there has hem a consider- able amount of analytical work clone in Germany on Langevin-type transducers [54], [57] ~ and [as]. This work includes equations that describe the variation of coupling, frequency, Q, and temperature coefficient with changes in the relative thickness and position of the ceramic disk and has been applied to some ncw filter types at' 45.5 kHz.

One of the m o d important Characteristics of piezo- electric. transtluccrs is the capability of operating in various modes of vihration, as m 1 1 as use in a large nunlhcr of mechanical configur a t' 1011s.

A good examplc i3 shon-11 in Fig. 8 where flexure modes

IEEE TR4NS.4CTIONS O X SONICS A N D ULTR.%SONICS: JULY 1971

3000X10~ 6r

3CCOX -40 + 40

TEMPERATURE l°Cl

(a)

232,-

c I

1 I I - 40 0 *40 +EO

TEhlPERATURE 1%)

(b)

Fig. 26. Characteristirs of PZT-type piezoelectric transducers (Murata). (a) Frequcncy variation. (b) Coupling Coefficient variation with rhsnge of trmprrxture.

are excited by composite ceramic/metal alloy transducers. In addition, the high electromechanical coupling coeffi- cient of the ceramic makes it possible to design very stable wide-bandwidth (10-20 perc,ent) low-frequency filters of this t>ype.

A great deal of analytical work has been accomplished in ,Japan in the area of describing the electromechanical characteristics of conlposite bending transducer reso- nators. Konno's and Kusakabe's published work on this particular subject alone totals more than 200 pages and includes twisted bars, mass loaded bars, and bars that vibrate in nondcgenerate modes, as well as simple bars driven with ceramic plates that only partially cover the upper and lower surfaces [59]. 9 very detailed analysis of a fully covered bar has been achieved by Okamoto et al . [GO], and is summarized in [ 191.

LOOKIXG TO THE FUTURE We will look at the nest few years based on the direc-

tion of our present technology. There will be break- throughs of course such as the monolithic filter of some years ago, hut these are difficult to predict, even if one is playing an active part in the development. Some of the more prctlictahle areas are size reduction, an expanded use of lan--frequency filters, an improvement of delay and ripple characteristics, the realization of multireso- nator monolithic filters with finite attenuation poles, as well as various in~proverncnt,~ in the discrete element

JOHNSON et al.: REVIEW OF MECKINICAL FILTERS

mechanical filter as a result of the large amount of effort being expended on monolithic filters.

167

Size Reduction In terms of mcch:mical filters, size reduction is still a

relatively uncsplorcd technology. In the case of disk-wire mechanical filtcm a size reduction from 5.0-1.3 cm3 was made with relatively little effort and no breakthroughs in t,ecImology othclr than thc use of a lower order mode of vibrat'ion. By making use of ceramic transducers and small diameter disks the volume can be reduced by 2 : l without a great deal of difficulty. ilt the present time, the longitudinal-mode bar, flexural-coupling rod design inde- pendently developed by Biirner [54] and Iionno [IS], (see Fig. 27) is packaged in less than 1 cm". Although it may be somewhat more difficult (in comparison with thc tlisk-wire filter) to reduce the size of this design because of the fixed length of the resonators, it is pos- sible to do so by decreasing the diamctcr of the rods, but only to the point where flexural modes bcconle a problem. Discrete element filters such as the disk or rod types actually have an advantage over monolithic struc- tures (Fig. 28) in that relatively large plating sur- faces are not needed to reduce both the filter impedance level and the effects of stray capacity. Disk and rod- type filters are low-impctlance devices because of the high dicllectric constant of PZT transducer materials, or in the case of magnetostrictive fcrrites, the ure of IOTY- turn coils. In addition, only t,wo transducers are needed. resonator ?pacing is conmlonly 0.025 cm rcgnrdless of filter hnndwiclth! and none of the volume is usecl for. the p1lrpow of rr(1ucing rrbflections from a boul1cl:iry ant1 very littlc is u w ( 1 for support.

I t n111rt l)e Fait1 tl1:lt tllcre arc :1ls0 some basic limita- tions :IS to hen, nn:~ll TW ran hilt1 a discrete component mechnnic~:~l filter. One: of co~~rr;c', is that of fnhricating the parts antl nw~tnhling then1 without cwessivc fre- rlucncy shift antl Ini>coupling. A wcond limit:ltion, as w e ' tli.srusscc1 earlicr, is t h a t of spurious no dos due to having to fix one of the tlimcnsions thus tlrcrcasing the frequency spacing of nearby hcnding modes 1541. & h -

athcv size-retlllction linlitntion relates to the nonlinearity of the transtluccr and resonator nlaterinls a. a function of size. Y:lkuma has matlc a very interc,sting comparison of ferrite co~'e inductors, piezoclcctric ceramics, nickel alloys, antl quartz, showing the relationship between various loss factors and minimum size for a specific handwiclth filter [61]. A final consideration is the three- dimensional shape of most discrete element mechanical filters, which not only prevents the use of planar manu- facturing proccsscs but ultimately may be the most serious volume-rccluction limitation.

Low Frequencies The large amount of development work on low-

frequency filters in Japan will most probably be felt in ot8her parts of the world. Fig. 29 shows, for example,

Fig. 27. I\ilininture 155-kHz rod-wire met.hnnicnl filter ( l cn?).

Fig. 2 5 . Eight-pole monolithic clunrtz mec,hnnic:Il filtcr

s o n ~ e of the wi(1e variety of filters k i n g manufactured by Fujitsu 1,td. IIost applications of low-frequency mechanical filters involve selecting single tones, which in turn means t h a t only one or two resonators at'e needed. Therefore, the selectivity is dependent solely on resonators that are coupled to the external mounting structure. This results in lower resonator Q and micro- phonic responses due to external vibrations.

The microphonics problem has been solved t o some degree by the use of highly damped supports. This has made it, possible to use funclamental modes of vibration,

168 IEEE TIL\NS.KTIORTS ON BONICS A N D ULTRASONICS, JULY 1971

0.9538 Arf-l

l98 200 202 204

FREOUENCY !kHz)

0 I \ I

200 207

- O o 2 i r ’

W 204

,,,7/,;<<<; ~ , 7 L < / ’ : n x4<,,;;:, >;,’.:,,’; FREOUENCY (kHz1

(l> 1 Fig. 30. Mcasurcd (a ) stopbnnd amplitude and ( h ) passband

response of a high-performance rod-wire mechanirnl filter t h a t emplavs douhlo rcsonator bridging of the type shown in Fig. 14 (Tclrfunkcn).

which redurc both the complexity of the filter and its physical size. The problem of minimizing unwanted out- puts due to impulse noise may be solved to some degree hy the use of H-shaped resonators [ G 2 ] . This typc of filter consists of two idcnticnl vertical masses connected by a flexible web (the horizontal elcment of the H ) . -4 piezoelectric ceramic transducer causes the web to flex and the masses to rotate. Each mass is supported near its centroid, thus preventing the ~ r c h from flexing when subjected to n shock impulse through the support.. A detailed analysis of the frequency response chnracterist’ics of this device hns hccn macle by Konno [631.

I n p - o v e n ~ e n t o j Delny and Passband Ripple

I t is possihlc to intornally delay equnlize a ~nechan- icnl Incider filter by making use of bridging u-ires C2.51. This 1x1s p m w l t o hc a ~ o r y practical nwthotl [ G $ ] , [65). This ticsign ia h a w 1 on the mc of very exact clllar.tcr-~~avelength long coll1)ling nircs. No frequency acljustnlcnts arc ncctlctl nftcr assembling the pretuned ro.mn:ltor.q:, in the case of chnnncl filters with pairs of a t t t~n~lnt ion poles (Fig. 30 :~nd [ M ] ) and in the case of rllanncl filtcrs n-it11 tlrl:zy equalization (Fig. 31) . Anothcr practical method is t o w e mechanical resonators ancl coupling clomrnts in an electrical lattice structure in much t h e same way :E quartz resonators are usetl [67]. The work on lattice filter,? has resulted in some new

uuuuuuuuuuu

L c a 3 z U c

12

1 1 L I

l 8 ~ 199 200

FREUUZNCY ikHr)

computer-niclctl tuning tt~chniques, which have also been rcportc(1 1)s Ynno [68]. In ortlcr to further improve the excellent response of voice chmnel filters, it will be advantageous in some c a s c ~ t o ~ n a k e u s e of on-line com- puters in the factory. The tlevelop~nent of practical computcr-a..,~isted tuning methods may result from work presently h i n g done on quartz monolithic mechanical filters.

Quartz Jlonolithic Allechanirnl Filters In much the s:tnlc n’ay 11s the t l i m t t c elenltwt me-

chanical filtw may benefit from monolithic mechanic:d filter tccllnology, R number of concepts ,such :M ac!onstic hridging will he applied to the monolithic filter. A t the present timc, attenuation poles are realized in t,wo- resonator str11ctarcs through capacitive bridging. The tn-o-resonator sections :ire capacjt,ively coupled t o form n multiwronnnt filtcr having a somewhat equal-ripple response [SS]. It is pmhnhle that exact, designs with four or more re.sonators 1wr plate ant1 electrical bridging will soon hccome availal~le.

JOHNSON et al.: REVIEW OF MECHAXICAL FILTERS

It has alrezttly been shown that thickness modes can bc trapped in much the same way as the shearing modes [70], which leads us to suspect that other new configura- tions will be dcvelopcd in the next, few ye:~rs.

A large amount of technology has been devoted t o the nlonolithic mechanical filter manufacturing process^ for inst:lnce in :~reas such as the w e of a laser to vary the coupling between electrode p i r s and the fahrication of extremely flat and parallel crgstd blanks. It is certain that this work will continue for a long period of time and shoultl he app1ic:tt)lc to nmhnnical filter technology in gencr:~l.

neve1opwcnts in ( I the / . C’orcntries

~ h ~ l ~ a n i c a l filtcr dcvelopnent~ outside of ,Japan, West Germany, and thc Is. S. llas 1)rimarily bccn concen- trated on distributed linc, filters of the type shon.11 in Fig. 7 (a) . This has Ileen the case in Great Britain [71] , Poland [ 721, the USSR [ 731, and East’ Germany, where they have becn Innnufncturecl for some years. Recent papers publiwhctl in Englnnd have reflected an interest in transducer dcaign [74] antl low-frequency flexure mode resonators 1751. A study of flexure modes in mechanical filtcrs is also a subject of interest in Czechoslovakia [ E ] . At this time, channel filters of the configuration of Fig. 511)) arc manufactured in East Gennany (200 kHz) [771 antl in Czechoslovakia (64-108 kHz) [78].

[31 M . L. Doclz and J. C. Ra,thnwny, “How to use mechnnicnl

I41 ( i . A . C:lmpbell, ”Physicnl theory of the rlectric ~vavc-filter,” I-I.’ filters,” Electronics, vol. 26, Mar. 1953: pp. 138-142.

151 R . V . L. Hartleg, “Frequency selective transmission system,” B e l l S y s t . Yech. J., vol. 1, Nov. 1922, pp. 1-32.

I61 J . 1’. Mnxfield and H. C. Hnrrkon, “Methods of high qunlity L.S. Pntcmt 1654 123, Dec. 1927.

rcvording and rt,producing of music and speech based on kle- phone research,” Bell Syst . Tech. J., vol. 5, July IWG, pp.

[TI JV. 1’. Mason, “W:lve transmission network,” U S . Patent 493-523.

2 345 491. Mar. 1944. [SI R . A . Sykrs and W. D. Beavcr, “High frequency monolitllic

filtexs with possible application to singlc freclucwcy a n d sin- gle side band u s ~ , ” Proc. Annu. Fwquencv Control Sum-

[91 l’. Nnkaznwa, “High frequency crystal t ~ l ~ ~ ~ t r o n ~ c c . l ~ n ~ ~ i c ~ a l f i l - p o s i ~ m , Apr. 1966, pp. 28S-308.

ter,” PTOC. A ~ t n u , Frequency Control Sgrmposirtrn, Bpr. 1962, on. 373-300.

1121 31. Btimvr. E , Tict,icl, ant[ H. Olln.;orgc. “hlcclrnnisrhe filter Ovt. 1952.

fiir die nnc.hriclrtentcrhnik,” Tekjz lnkeu J.; vol. 31, June

[l31 31. Kjrncr : E. Diirre, and I€. Schiislrr, “RI(~ll:lniscl1r

272-280. I~~i~wc.iten-\~:u~~tliilter,” Tekfunken J., vol. 36, h h y 1933, pp.

[l41 T. Tanaka Rntl T. Inoguchi. “Studies on electronics materials and their npplic:~tion.;,” Division of Elrctronics Mntr,ri:ils. Inslitule for Chemical Research, Kyoto Univ., Kyoto, Japan,

1958. IIP. 105-114.

1959, p1J. 22-25.

IEEE TRANSACPIOSS ON SONICS AND ULTRASONICS, JULY 1971 170

L431

1441

L451

M. Onoe and T. Yano, “Analysis of flexural vibrations of a circular disk,” ZEEE Trans. Sonics Ultrason., vol. SU-15,

polfilter,” Diplom-Arbeit, Institut fiir HF-Technik und HF- Physik der Technischcn Hochschule Karlsruhe, Munich,

R. L. Sharma. “Enuivalent circuit of a resonant. finitr,. iso- l661 H. Schiisler. “Filter nlit mechanischen resonatoren.” Bull. Germany, 1%. J u I ~ 1968, pp. 182-185.

tronic. elastic’circular disk.” J . Acoust. Soc. Am‘er.. vol. 28. xot . - i956 , pp. 1153-1158.

~, -

W. P. Mason, Electromechanicnl Tramducers and Wave Fil- ters. New York: Van Nostrand, 1948, pp. 291-297.

1461 E. Frymover, J. Klovstad, and R. A. Johnson, “Equivalent mass of a thick vibratine disk.” unoublished.

1471 M. O I ~ K , “Contour vibrations’ of i$otropic circular plates,” J . A C O ~ L S . SOC. AWLET., vol. 28, N O V . 1056, pp. 11561162.

I481 J. NIscr, “Exakt,e t)rrcchnung dcr biegcresonanzen rechtecki- ger und zylindrischer stLbe,” Bochfreq. und Elektroakust.,

L491 M. Konno and H. Naknmura, "Equivalent, elcctrirnl nrtwork vol. 74, 1965, pp. 30-36.

for the transversely vibrating uniform bar,” J . Acoust. SOC.

L501 M . Borncr, “Mechanische filter mit biegekopplung,” Arch. Amer., vol. 38, Oct. 1965. pp. 614422.

Elek. tibertragung, vol. 15, Dec. 1961, pp. 175-180. l511 -, “Magnetischc werkstoffe In e1ectromechanisc.hr.n

2. Sept,. 1966, pp. 613-620. rcsonatoren nnd filtern?” ZEEE Trans. Magnetics, vol. M A G -

1521 C. M. van der Burgt,, “Performance of ceramic ferrite resona- tors as transducers and filter elements,” J . Acuusi . Soc. Amer.,

L531 S. Srhweitzerhof, “Fber ferrite fiir magnetoslriktive schwingcr in filterkreisen,” niaChrichte?ltec/k Z., vol. 11, Apr. 1958, pp. 179-1 8.5

vol. 28, XOV. 1956, pp. 1020-1032.

I541 M . Biirner and H. Schiissler, “Miniaturisi~rllng mechanischrr

1551 I). Bcrlincwurt. B. Jaffe. H . Jaffe. and H. H. A. Kruemr.

- . . - - - .

I’ilt,er:” Telefztnken J . , vol. 37, 1964, pp. 228-246. .~

“Transducer properties of lead titanate zirconnte ceramics,” I R E I n t . Convention Xec., vol. 7, pt. 6! pp. 227-240.

1561 Y. Tagllwa and T. Hatano, Design of Mechnnical Fil ler and Crys tn l Filter C’irc~tits (in JaIt:mcse). ‘I‘ok.yo : Ohm, 1964,

1571 H . Srhiissler, “Merhnnichr Filtcr mit piezoel(~ktriscl1en pp. 11-12.

\I’:mdlern,” Telef7cnken J . . vol. 39, 1966, pp. 429-439. 1581 I T . Albsmeier. ‘‘Chrr Antricbe fiir elektrorncchnnische hnnd-

piisse mit piezoelcktrischen Wandlern,” Freguenz, vol. 19, Apr. 1965, pp. 125-133.

1591 C. Iiusakahe, M. Konno, and Y. Tomikawa, “Resonant fre- mwncies of transvorselv vihmtine bar excited bv elertro- sirirtive transducer,” Electron. Cokmnn. Jap., V O ~ 48, Iiov. 1965. pp. 18&101.

I601 T . Okxmoto. K . Y:dmwa: and S. Okuta, “1,ow frequency elrr- tromwhnnical filter,” Fuj i f su Sci. Tech. J., vol. 2, May 1966, 1111, 53-86.

. -

r611 E;. Y:!~IIIKI and S. Okuta, “Mininturization of the mrchani-

I n [ , Congress on Acnrtstics, vol. 4, Aug. 1968, pp. 97-100. wl vhrntor used in an clectro-mechanical filter,” R e p . C f h

1621 H. Ihkrr and ,J. R . C r c w y , “H-qhaprd rrsonntors signal np-

1631 M . Konno, K. Aoshima, and H. Nakamura, “H-shaped reso- iurn in tone tclmxtering,” ElPctronics, Oct. 1967. pp. 99-106.

nator and its application to electro-mechanical filters” (in .Jnpancsc), Eng. Bull. Yamngata Univ., vol. 10, Mar. 1969,

[641 R. Kohlhammer, “Entwnrf kanoni~cher koppclnctzwerke mit,

Terhnische Hochschule. Miinchen, Germany, 1970. Anwendungrn anf mechanische Filter.” Ph.D. dissertation,

1651 -, “Brrechnung, Ball und Untersuchung mechanischrr

pp. 261-285.

Schweizerischer Electrotechn. Verein (Zurich), Ma;. 1969, pp. 216-222.

1671 R. il. Johnson, “New single sidchand mechanical filters,” 19’70 WESC‘ON Tech. Papers, Aug. 1970, pp. 1-10,

I681 K. Shibayama, “Elcctromechanical filters,” Eleclron. C o m - mun. Jap., vol. 48, Kov. 1965, pp. 6672.

1691 H. Yoda, Y. Xakazama, and N. Kobori, “High frcqucncy crystal monolithic (HCM) filters,” Proc. 2Srd Annu. Symp.

1701 S . Fujishima, S. Xosnka, and I. Ishiyama, “10 MC ceramic Freguency Control, May 1969,. pp. 76-92.

filters hy trapped energy modes,” Proc. A C O I L S ~ . Soc. Jap., h’ov. 1966, p. 3.

L711 W. Struszynski, “h thcorctical analysis of the torsional electron-mechanical filters,” Marconi Rev., 3rd quart., vol. 22: 1956. pp. 119-143.

1721 F. Kaminski. “The synthesis of elrctromechanical chain filtcrs by means of equivalent circuits with distrihuted pnrametrrs” (in Polish), Rozpr . Elektrotech., vol. 15, 1969, pp. 717-749.

1731 A. IC. Losev, “Filters with multielemrnt coupling,” Tele- cornrnun. Radio Bng. (USSR), Jan. 1964, pp. 1-8.

L741 R. P. Wa.lters, “Balanced actlon toroidal transducer for clrc- tromechanical filters,” Electron. Eng., Mar. 1969, pp. 238-245.

1751 P . Kagaaa and G. M. L. Gladwell, “Finite clenlrnt analysis of flrsure-type vibrators with elcctrostrictive transducers,” ZEEE Trans. Sonics Ulirason., vol. SU-17, Jan. 1970, pp. 41- 49.

C761 Z. Faktor, “Spurious vibrations in component parts of clcc-

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