THEORY OF MONOLITHIC CRYSTAL FILTERS … Bound...THEORY OF MONOLITHIC CRYSTAL FILTERS USING...

R 703 Philips Res. Repts 24, 331-369, 1969

THEORY OF MONOLITHIC CRYSTAL FILTERS USINGTHICKNESS-TWIST VIBRATIONS

by V. BELEVITCH and Y. KAMP

AbstractThickness-twist vibrations of piezoelectric plates are analyzed withoutusing power-series expansions in the thickness coordinate, and simplemodes are obtained by the method of separation of variables. In the caseof a uniform electric field, the modes are orthogonal and yield explicitsolutions for the forced vibrations of monolithic quartz filters; the uni-form-field approximation only affects the equivalent impedances byfourth-order terms in the piezoelectric coupling factor. Equivalent cir-cuits are derived from the uniform-field theory and the synthesis ofnarrow-band filters is achieved.

1. Introduction

At sufficiently high frequencies, a thin plate of anisotropic material acceptsmodes of vibration in which the displacement is a function of the coordinatealong the thickness dimension. Such modes are called thickness-shear when thedirections of propagation and of displacement coincide, and thickness-twistwhen the two directions are at right angles. Any mode of such a type only existsabove a cut-off frequency which is a function of the material density. By elec-troding some regions of the plate, one increases their equivalent density, so thatthe electroded regions have lower cut-off frequencies than the unelectrodedregions. At frequencies intermediate between both cut-offs, propagation with-out attenuation is possible in the electroded regions, but only evanescent modesexist in the unelectroded regions. By this, so-called energy-trapping, principlethe electroded regions behave as resonators whereas the unelectroded regionsproduce a coupling between adjacent resonators, so that the whole structureconstitutes a narrow band-pass mechanical :filter in the neighbourhood of thecommon resonance frequency of all resonators. If, in addition, the plate materialis piezoelectric, the resonators at both ends of the structure can be used astransducers, and an electrical filter is obtained.Such monolithic filters have been recently described in various engineering

journals, but no publication is available where their equivalent circuits andmethods of design are. seriously established and justified. The present paperattempts to fill this gap, for :filtersusing thickness-twist vibrations, by meansof an original and almost elementary treatment. The paper is self-contained;its main ideas and their relations to the work of others are summarized in thefollowing paragraphs.In the literature, the analysis of high-frequency vibrations of thin plates,

332 V. BELEVITCH and Y. KAMP

either mechanical+-") or piezoelectric 3.4), has always been based on power-series expansions in the thickness coordinate for all variables. For purelymechanical thickness-twist (but not thickness-shear) vibrations, such an ap-proximate procedure is unnecessary, since elementary modes are obtained bythe classical method of separation of variables, as shown in sec. 2. This is nolonger true for piezoelectric vibrations,' discussed in sec. 3. If, however, onereplaces the real physical problem (where a voltage is applied between theelectrodes) by a slightly different problem (where a uniform electric field isforced inside the material) one falls back to the same set of orthogonal modesas in the mechanical problem, and the forced vibrations of a general monolithicfilter can be analyzed by elementary methods (sec. 5). In the case of a uniforminfinite plate (pure thickness vibrations), the solution of the real problem,obtained by Tiersten 5) and discussed in sec. 4, is compared in sec. 6 to thesimilar solution of the uniform-field problem, with the result that the imped-ances deduced from both solutions only differ by fourth-order terms in thepiezoelectric coupling factor. As a consequence, only the uniform-field problemis treated in the remainder of the paper (except appendix B). In our approach,this problem is studied in its own right as a carefully defined and meaningfulphysical problem, which is simply slightly different from the real problem arisingfrom practice, but the mathematical solution of the modified problem is com-plete and rigorous. This is in contrast with the approach 6) where the uniformfield is taken as an approximate solution of the real problem, which leads toinconsistent mathematics.In secs 7 to 10, monolithic filters are analyzed by solving the uniform-field

problem in each homogeneous section of the structure, equivalent circuits arederived and their narrow-band approximation is discussed. Finally, secs 11-12are devoted to a complete solution of the synthesis problem. Appendix A dealswith the mechanical eigenfrequencies of an iterative filter structure, which havebeen computed and measured by Schnabel 7); our theory is simpler and leadsto closed-form analytical expressions for the case of weak coupling. In appen-dix B the general theory (non-uniform field) is developed for crystals of arestricted symmetry class.The geometry of the monolithic filter is defined in fig. 1. The plate, of thick-

ness H along X2' is assumed infinite in both other dimensions. Along X3 (the

~~-,X3~-#~--~~~~~

Fig. 1. Geometrical parameters of the crystal plate.

THEORY OF MONOLITHIC CRYSTAL FILTERS USING THICKNESS-TWIST VIBRATIONS 333

direction ofpropagation), electroded regions (shown shaded) of uniform lengthB alternate with unelectroded regions of (possibly different) lengths Al, A2' ....The problem is independent of x 1 (the direction of the displacement), but atypical slice of length L along x 1 is in fact often considered, so as to yield finitecapacitances for each region. In the thickness direction, the plate is defined byo ~ X2 ~ H rather than by -H/2 ~ X2 ~ H/2; although the second choiceis more convenient in some respects, the first choice is preferred because itleads to using only cosine functions in the variable X2'

2. Mechanical vibrations

We consider an anisotropic solid having a monoclinic symmetry with digonalaxis Xl and want to show that it accepts modes of vibration with displacementonly along that axis (ul =ft 0, U2 = U3 = 0) and independent of that coordinate(hence 'è)/'è)xl = 0). We use the notations of Mason 8) throughout.

The strain tensor has only two non-zero components:

(1)

For the monoclinic case *), the zero elements in the elastic tensor Cl} are suchthat the stresses and strains of subscripts 1 to 4 are completely decoupled fromthe ones of subscripts 5 and 6. Consequently, the only non-zero stress com-ponents are

(2)T6 = CS6 SS + C66 S6'

For a density e and a time dependence exp Uwt) of all variables, the onlydynamical equation is

(3)

for the equations in U2 and U3 are identically zero.By eliminating Ss, S6' Ts and T6 from (1), (2) and (3) one obtains the second-

order partial-differential equation

(4)

*) In ref. 8 (p. 386) this tensor is given for the case where the digonal axis is x2; one obtainsthe case treated here by permuting subscripts 1 and 2, and this also induces a permutationof 4 and 5.

and the boundary conditions to (JU/()X2 = O. Simple modes of the form

u = '1fJr(X3)cos (I' 'TtX2/H), (6)


For a plate of thickness H, infinite along x I and of arbitrary structure along x 3,the boundary conditions at the free faces X2 = 0 and X2 = Hare T6 = 0 forall X3' Equation (4) and the boundary conditions cannot be satisfied by simplesolutions with separated variables forming an orthogonal set with respect toX2 unless Cs6 = O. The latter case thus allows an elementary treatment; sinceit is also of major practical interest *), our discussion will be limited to thatcase. We further write u for UI'

Equation (4) now reduces to()2U ()2U

C66 -_ + Css -_ + e (02 u = 0()X22 ()X32

(5)

with I' = 0, 1,2, ... , are thus acceptable, and (5) reduces to the ordinary dif-ferential equation

(7)

withk/ = C66 (I' 'Tt/H)2 - e (02

Css(8)

By (1), the strains are

d'1fJr I' 'TtX2SS =-COS--,

dX3 H(9a)

I' 'Tt r st xS

. 26=-H'1fJrSm~. (9b)

Expressions (6) and (9a, b) show that for an even mode (I' even, and in partic-ular I' = 0), u and Ss have even symmetry and S6 odd symmetry with respectto the median plane X2 = H/2; for an odd mode, u and Ss have odd symmetryand S6 has even symmetry.

*) One has CS6 = 0 rigorously for stronger symmetry classes, and approximately for AT-cutquartz referred to suitably rotated axes. This simplification is used by most authors. ForCS6 '# 0, the mode r = 0 is not altered. For r '# 0, a solution of (4) satisfying the boundaryconditions is

exp [kr (X3 - CSS X2/C66)] cos (r 'Jt X2/H),

where kr2 is given by (8), but with Css replaced by

css' = Css - CS62/C66'

These solutions are not orthogonal with respect to x2' The orthogonality is essential toobtain explicit general solutions as will appear from sec. 5.

THEORY OF MONOLITHIC CRYSTAL FILTERS USING TIUCKNESS-TWIST VIBRATIONS 335

For r =1= 0, we rewrite (8) as

k, = r f1, [1- (wir We)2]1/2, (10)

with# = (c66Icss)1/2 niH,

We = (c66Ie)1/2 stlH,

(11)

(12)

and the propagation constant (10) is real for W < r We and imaginary forW > r We, so that the plate behaves as a high-pass filter of cut-off frequencyr We for mode r, The mode r = 0 is particular in more than one respect. First,(9a-b) give S6 = 0, Ss = o'IjJrlox3' so that the vibration is independent of thethickness dimension. Second, (10) gives

ko =jw h, (13)

withh = (elcss)l/2,

and propagation is possible at all frequencies.

(14)

3. Piezoelectric equations

We now consider a plate of piezoelectric material and denote by E the electricfield and by D the electric induction. For monoclinic symmetry with CS6 = 0,the piezoelectric relations are

Ts = Css Ss - e2S E2 - e3S E3'

T6 = C66 S6 - e26 E2 - e36 E3'

D2 = e2S Ss + e26 S6 + 822 E2 + 823 E3'

D3 = e3S Ss + e36 S6 + 823 E2 + 833 E3'

The electrical equations are

(15)(16)(17)(18)

E =-grad V,

div D = O.

By substitution of (15)-(16), (1) and (19) into (3) one obtains

(19)(20)

02U 02U 02V 02V 02VC66--+ CSS--+ e26--+(e36 + e2S)--- +e3S--+ w2 ç u =0.

OX22 ox/ OX22 OX20X3 OX32 (21)

By substitution of (17)-(19) and (1) into (20), one obtains02U 02U 02U

e26 -- + (e2S + e36) --- + e3S -- +OX22 OX20X3 OX32

02V 02V 02V- 822 -- - 2 823 --- - 833 -- = O. (22)

OX22 OX20X3 OX32


The mechanical and electrical oscillations are thus coupled in (21)-(22). Themechanical boundary conditions are T6 = 0 for all X3 on the faces (X2 = 0,X2 = H), the continuity of u and of Ts for all X2 at the boundaries betweenthe electroded and unelectroded regions, and Ts = 0 for all X2 at the freeends. On the other hand, the electric field extends in free space outside thepiezoelectric material, where it satisfies the Laplace equation; the boundaryconditions are V = constant and E3 = 0 on the electroded regions, and thecontinuity of the tangential component of E and of the normal componentof D on the unelectroded boundaries. Finally the charge of the condenserformed by an electroded region of length L along Xl' and extending fromX3 = a to X3 = b, is

b

Q =L Jadx3'a

where a is the charge density equal to the discontinuity Do of D2 on the lowerface (X2 = 0). One. thus obtains the upward current by

b

I=-jLw JDo dX3' (23)a

The problem thus stated is not solvable by the method of separation ofvariables, since the difficulty already appears in the electrical problem. Onemay try to get rid of the external electric problem by some means which definea separate problem (electrical as well as mechanical) for the intervalo ::::;;X 2 ::::;;H. This can be achieved by assuming that the dielectric constantof the piezoelectric material is much larger than the one of the surroundingmedium, and leads to the boundary condition ö V/()X2 = 0 on the faces of theA-regions. The problem thus defined is a realistic approximation of the trueproblem for thin plates because(a) the electrostatic field only leaks outside the electroded regions by small

end effects,(b) the dynamic field produced by the mechanical vibration in the A-region is

small owing to the evanescent nature of the vibration and to the smallnessof the piezoelectric effect,

(c) the stray field in the surrounding medium is further reduced by the differenceof the dielectric constants.

Unfortunately, the problem thus simplified is still not solvable by the methodof separation of variables because the boundary conditions on the faces of theA-regions and of the B-regions are different and cannot be satisfied by the samemodes. The discussion is continued in appendix B.

4. Thickness vibrations

In the case of a uniform, fully electroded plate the considerations of the endof sec. 3 are irrelevant. Moreover, the problem is independent of X3 and anelementary solution (to be called Tiersten's solution and denoted by subscript T)is easily obtained as follows. Equation (22) reduces to

d2--(e26 u- 822 V) = O.dX22

(24)

THEORY OF MONOLITHIC CRYSTAL FILTERS USING TIlICKNESS-TWIST VmRATIONS 337

Since V is assumed to take the values ± Vo/2 on the faces (where Vo denotesthe applied voltage), all functions will have odd symmetry with respect to themedian plane X2 = HI2 and the solution of (24) is of the form

e26 u - 822 V = a (X2 - HI2).

After eliminating V by (25), (21) reduces to

d2u--+ .2U =0d 2 'X2

(25)

(26)

with(27)

and(28)

where

r = ( )1/2822 C66

(29)

is the piezoelectric coupling coefficient. By (16), the condition T6 = 0 on thefaces is

du a e26

dX2 822 C66'(30)

for X2 = 0 and X2 = H. The solution of (26) and (30) is

a e26 sin {. (X2 - H12)}u=---------

822 C66' • cos (. H12)

One deduces V by (25) and determines the constant a by V = ± Vo/2 on thefaces. This yields

822. Voa= ,

2 Y/ tan (r H12) - • H(31)

wherey

(32)


and Tiersten's solution is finally defined by

e26 Vo sin {r (X2 - Hf2)}UT =-- . ,(33)

2 C66' Y/ sin (r Hf2) - (r Hf2) cos (. Hf2)

Vo • (X2 - Hf2) cos (r Hf2) - Y 12 sin {r (X2 - Hf2)}G=- . ~2 (r Hf2) cos (r Hf2) - Yl2 sin (r Hf2)

On the other hand, (17) reduces to

dD2 = -(e26 U- 822 V),

dX2(35)

hence to a by (25). If the electrodes are taken as finite of dimensions L along Xl

and B along X3, but continued to infinity by isolated guard electrodes at thesame potentials so as to avoid end effects, the current (23) is I = -j Q) L B a.By identifying it to the current j Q) C Vo in a capacitance C one obtains theequivalent capacitance as

CoC= ,1 - (2 Yl2f.H) tan (r Hf2)

(36)

whereCo = 822 BLfH (37)

is the electrostatic capacitance.

5. The uniform-field problem

In sec. 3 we have already idealized the problem posed by the analysis ofpractical filters. We now introduce an additional idealization which leads to aseparable problem in all cases. In the modified problem, we consider that auniform field

(38)

is impressed in all homogeneous sections (with Vo = Vlor V2 in the portsections and Vo = 0 elsewhere). Since the electric field is, however, now con-sidered as impressed in the thickness of the plate (by some fictitious distributedgenerators) instead of being applied through boundary conditions, the secondelectrical equation (20) does no longer hold because the distributed generatorsgive a distributed charge density corresponding to div D =F O. The problemthus defined has a simple physical interpretation. The condition E3 = 0 fora plane X2 = constant is imposed by inserting in the material a plane metallicgrid with infinitesimal meshes. The condition for all X2 thus assumes a densenetwork of such plane grids with infinitesimal separations continuing in the

THEORY OF MONOLmrrC CRYSTAL FILTERS USING THICKNESS-TWIST VIBRATIONS 339

whole material, both in the electroded and unelectroded regions. Since, how-ever, Ez is discontinuous at the region boundaries (zero outside, -Vo/H insidethe end sections) the grids are interrupted at the boundaries, and short-circuitedamong themselves outside. In a port region, the impressed field Ez correspondsto an applied voltage ~Ez dxz between adjacent grids at distance dxz, andthe field only takes the uniform value (38) if the adjacent grids are connectedto a generator of voltage Vo dX2/H. The connection to the real physicalgenerator Vo (the voltage applied to the condenser) is thus through an idealtransformer of ratio dX2/H as shown in fig. 2, and the primaries of all infinites-

Fig. 2. Electrode connection in the uniform-field problem.

imal transformers are in parallel on the same generator VOo The current ientering a slice of width dX2 is given by expression (23) but with the local valueof D2 replacing Do; it is multiplied by dXz/H to produce the primary currentdelivered by the generator and must be integrated in X2 to give the total currentin all parallelled primaries. The result is again (23), provided Do be defined as

(39)

The grid network thus simultaneously impresses a uniform field and smoothesthe electric induction to the equivalent value (39).

For the uniform-field problem thus defined, the potential resulting from (38)and the boundary values ± Vo/2 is

(40)

With (40), (21) reduces to (5), whereas (22), resulting from (20), is abandoned.The only boundary condition on the faces is

(41)

At the junction between sections, the continuity of E2 and D3 is no longerrequested by the underlying physical model, and only the mechanical variables uand Ts need be continuous. One thus obtains a well-posed mathematical problem,

340 v. BELEVrrCH and Y. KAMP

whose solution will first be discussed and then related to the solution of thereal problem of sec. 3.We first derive the analogue of Tiersten's solution for the uniform-field

problem, i.e. the solution satisfying (5) and (41) in an infinite plate with'è)/'è)X3 = O. Equation (5) reduces to

d2u__ T + a2 UT = 0,dX22

(42)

with(43)

and (41)-(42) only differ from (26) and (30) by notation. The solution is there-fore

e26 Vo sin {a (X2 - H/2)}UT=-

C66 a H cos (a H/2)(44)

and VT is (40).We now consider the general case ('è)/'è)X3 =1= 0) and subtract from the un-

known solution U the forced solution (44) in which Vo is taken as Vlor V2

in the port sections (input and output, respectively) and as zero elsewhere. Thesubstitution

(45)

defines a new homogeneous problem in the variable Uh' By linearity, u,, satisfiesthe same differential equation (5) but the boundary condition (41) also becomeshomogeneous:

(46)

because the term 'è)UT/,ÖX2 yields the value required by the right-hand memberof (41). The solution Uh is thus a linear combination of the modes of sec. 2.In counterpart, the conditions at the junctions, which are the continuity of U

and ofOU Vo

Ts = Css - + e2S -'è)X3 H

(47)

now introduce discontinuities generated by Vo and UT' Designating by <5 thediscontinuity (difference between the values inside and outside the port sectionsat the junctions) and using the fact that UT is zero outside and independentof X3 inside, one obtains the relations

(48)

(49)

THEORY OF MONOLffiiIC CRYSTAL FILTERS USING THICKNESS-TWIST VffiRATIONS 341

Since (44) has odd symmetry, its Fourier expansion only contains odd modesand is, by an elementary integration and the use of (43) and (8),

(50)

If the solution Uh is represented as a sum of modes (6), condition (48) thus gives

Vo51/J --4err - 26 1 2 H2

C55 'Cr

(51)

for r odd and 5'1Jlr = 0 for r even. On the other hand, the right-hand memberof (49) is constant and only excites mode zero by

5( d'lJlo) = _ e25 Vo •

dX3 C55 H(52)

Consequently the even modes with r =1= 0 are not excited at all, the mode zerois excited by (52) and

(53)

whereas the odd modes are excited by (51) and

O(d'lJlr) = o.dX3

(54)

Since each 'IJlr in a homogeneous section satisfies (7) it is of the form

'IJlr = A exp (k, X3) + B exp t=k; X3), (55)

with A or B = 0 in the end sections to make 'IJlr zero at X3 = ± 00, respec-tively. For n intermediate sections, there are thus (2 n + 2) constants. Sincethere are (n + 1) junctions and two conditions (on 'IJl" and on d'lJlr/dx3) ateach junction, the solution is completely determined.It remains to compute the current entering a port by means of (23), (39)

and (17) which reduces to

Owing to the integration (39), the odd part of 'è)U/'è)X3 does not contribute, andu can be replaced by its even part 'IJlo in the first term. The first two terms are

Since

d1po Vo e26Do = e25 -- - 822 - + - [u(H) - u(O)].

dX3 H H

U = 1po+ ~ 1prcos (r 'Jt x2fH) + UT1.3...

(56)

342 v. BELEVITCH and Y. KAMP

then constant whereas the last term is an exact differential, so that (39) becomes

where UT is (44), one finally obtains

d1po Vo 2 e26 I 2 e262 Vo

Do = e25 - - E22 - - -- 1pr- tan (a Hf2).dX3 H H C66a H2

(57)

1,3•••

By integrating (7) one obtains

(58)

and (58) applied to (57) then gives, for a port region of length B = b - a,

(59)

r= 1,3 ...

where

{2 y2 }

CT = Co 1+ a H tan (a Hf2) , (60)

with Co and y defined by (37) and (29). For pure thickness vibrations, U reducesto UT so that all 1prare zero in (56); the impedance of (59) then reduces to theone of CT'

6. Discussion

For thin plates, where the electrostatic end effects are negligible, the generalproblem of sec. 3 and the uniform problem of sec. 5 are identical if the plate isnot piezoelectric. Consequently, the uniform problem constitutes an approx-imation of the general problem in the case of small piezoelectric effects. Thisis easily checked on the case of thickness vibrations where both problems havebeen completely solved. By (27)-(28), (32) and (43), the parameters r and y 1

of the general problem only differ from the corresponding parameters a and yof the uniform problem by second-order terms in y, and the same differenceexists between the solutions (33)-(34) and (44), (40), respectively. Moreover, theresulting capacitances (36) and (60) formally differ only by terms in y4. Sincethe corresponding impedances present, however, resonances and antireso-nances, which differ between the two problems by terms in y2, the differencebetween the two impedances may be enormous at a given frequency.


By (43) and (12), the argument Z = (J H/2 of (60) is n w/2we• Consequently,(60) is infinite at the cut-off frequencies of all odd mechanical modes andreduces to Co at the even modes. The curve of the reactance versus frequencyis shown on fig. 3 where the full curve represents the electrostatic reactance (of

x

Fig. 3. Crystal reactance versus frequency for thickness vibrations.

Co) and the dashed curves represent the piezoelectric thickness resonances. Theseparation between adjacent poles and zeros is small (of the order of y2) andcan be evaluated by setting W = r We + 'I'J and computing

im W n'I'J 2 Wetan -- = -cot -- R::I _ -- •

2 We 2 We n 'I'J(61)

One thus obtains

(62)

By using the seriestan z "\' 1~ = Z:: (rn/2-)-2-_-z-2

r= 1.3...

(63)

one obtains the partial fraction expansion of the admittance corresponding to(60) and this yields the equivalent circuit of fig. 4 where all inductances havethe common value

n2

Lo=----8 Co y2 we

2 (64)

I,1/9Low~ I

Fig. 4. Crystal equivalent circuit for thickness.vibrations.

344--------------------------------------------------------V. DELEVITCH and Y. KAMP

By (12), (29) and (37), this can be rewritten as

eH3Lo = ,

8 e262 RL

a result mentioned by Onoe and Jumonji 9).The behaviour of the reactance deduced from (36) is very similar. The first-

order effect in y2 is 'to move the poles from r We to r We (1+ y2/2), but thedistance between adjacent poles and zeros is (62) within terms in y4.

For narrow-band high-frequency filters, absolute resonance frequencies mustbe determined with a very high accuracy which can only be deduced frommeasurements on experimental models, whereas the theory is mainly neededfor computing response characteristics on a relative frequency scale. For AT-cutquartz (y = 0,088), the uniform-field theory is practically satisfactory and willalone be pursued in the remainder of the paper.

(65)

7. Equivalent circuits

The classical equivalent circuit of Mason 10) for longitudinal vibrations isderived by means of the analogy where the current i and the voltage v arerespectively identified to the mechanical velocity j W "P and to the longitudinalforce resulting from Ts multiplied by the cross-section LH. We adopt theseanalogies for the mode r = 0 and thus write

iD =i co "Po, (66)

d"PoVo =LHCss--.

dX3(67)

For the modes r i= 0, the roles of"P and d"P/dx3are permuted in all boundaryconditions so that the inverse analogy is more convenient. We therefore definei, as proportional to j W d"Pr/dx3but, in order to keep to i; the dimension ofiD, we put

d"Pri,=i W H-

dX3

The Mason equivalent circuit contains an ideal transformer of ratio

(r i= 0). (68)

(69)

and (66) and (69) permit to rewrite the second term of (59) as

-j W L ezs ["Pr]: = -no [iD]: = no [io]:.

With the notation2Le26

n =------.~ H2 k 2r

(70)


and (68), a similar rewriting occurs in the last (summation) term and (59) be-comes

I = j co CT Vo + ~ n, [irf -r=O,1.3... b

(71)

By (67) and (69), (52) becomes <5vo = -no Voo With (70), we similarly trans-form (51) into

<5vr= -nr Vo, (72)

provided we adopt the definition

Vr = -L Css '1fJr/2 (I' =F 0). (73)

Denoting by subscripts e and u the values in the electroded and unelectrodedregions, respectively, and remembering the definition of the discontinuitysymbol <5, one rewrites (72) as

Vr,u = n, Vo + Vr,e' (74)

Let each separate homogeneous part of the filter, for one mode, be repre-sented by an electrical equivalent black-box (to be determined later) and con-sider the interconnection of an electroded section E to its adjacent unelectrodedsections U. It is clear that the equivalent circuit of fig. 5 correctly gives (74) asthe voltage constraints at the interconnected terminal pairs aa' and bb'. More-over, the currents ill and ib are continuous in fig. 5, and this yields correctly (53)_

~ ..lP- .------- Áa

Va,ei iVb,eb

U E U

_- L......-r--r-

Va,uia-i4 l1.,u

...L

nr IEI~.a' b'

Fig. 5. Equivalent circuit of one electroded region between two unelectroded regions.

by (66) for I' = 0, and (54) by (68) for I' =F O.Since Vo is common to the equiv-alent circuits of all modes (I' = 0, 1,3, ' , .), the partial circuits are parallelledat the terminals 00' of fig. 5. If a shunt capacitance CT is added to the overallresulting network, one correctly obtains the total current entering terminalOby (71), since the first term is the current in CT whereas the upward currentill - ib in the primary winding of the transformers contributes to a current


n; (in - ib) entering through terminalO. Finally, the boundary conditiond1pr/dx3 = 0 at the free ends, and the identifications (67) and (68) show thatthe ends of the electrical equivalent system must be shorted for the moder = 0 and opened for all other modes.. It remains to find the equivalènt circuits for each mode of the separatehomogeneous parts represented by black-boxes in fig. 5. Since each part obeysthe differential equation (7) for each mode, it is a transmission line of propa-gation constant k, and of some characteristic impedance Wr d~fined as the ratiovrfir for a wave exp (k; X3) attenuated in the positive direction of the current,hence for X3 < 0 (as results from figs 1 and 5) if Re k,> O. By (66)-(67) oneobtains Wo = L H Css kofjw, hence

Wo =L H (e css)1/2

by (13)-(14). By (73) and (68) one obtains, for r :;6 0,

(75)

L Cssw:=----r 2jwHkr

Finally, the classical T-equivalent circuit of a transmission line of length D,hence of a transfer constant Br = k, D for mode r, is shown in fig. 6 and hasthe element values

(76)

Za = Wr tanh (BrI2); z;, = Wr/sinh Br. (77)'Iza

Za 2Zb

l' 2'Fig. 6. T-network.

8. Energy trapping

In the following we only consider the mode r = 1, but will discuss in sec. 9under what conditions the other modes are not excited, or excited to a negligibleextent. V:fedenote by Wu the cut-offfrequency ofthe unelectroded regions. Sincethe equivalent density is larger in the eiectroded regions, one has Wu > We by(12), but we assume that the difference is relatively small and put .

. (78)

Moreover we only treat the behaviour of the system in the frequency intervalWe < W < Wu and introduce the transformed frequency variable cp by

W-We·cos- cp = . . 7

Wu-We .(79)


so that 4>decreases from nl2 to 0 with W increasing from We to WU' Conversely,(78) and (79) give

W = We (1+LI cos- 4». (80)

For a given oi, thepropagation constant k1 of (10), for the _electroded regionstakes an imaginary value to be notedj ke, whereas the characteristic impedanceW1 of (76) takes a real value to be noted R. With the narrow-band approxima-tion 1- w2/w/'~ 2 (1- w/we) throughout, the abbreviation

p = P, (2 LI)1/2

and the use of (80) one obtains the values

k, = p cos 4>,

(81)

(82)

L CssR=-----

2 W Hp cos 4>(83)

For the unelectroded regions, the propagation constant, to be noted ku, is realand deduced from (10) for r = 1 by replacing We by WU' One obtains similarly

ku = p sin 4>. . (84)'

The characteristic impedance is imaginary of the form j X. From (81), onededuces j XI R = j keik.; hence

XIR = cot 4>. (85)

We now consider an electroded region oflength B along X3 between infinitelylong unelectroded regions. The equivalent circuit of the electroded region in-volves the phase shift k; B = P B cos 4>by (82). In order to avoid the expres-sions 8/2 appearing in (77), we designate this phase shift by 2 {3and then put

{3= b cos 4>, (86)

withb =pBI2. (87)

By (77), the equivalent circuit of the electroded region is a T-network of im-pedances

Za = j R tan {3; Zb = -j Rlsin (2 (3). (88)

On the other hand the unelectroded regions, being infinite, are equivalent totheir characteristic impedances j X with X given by (85). If one temporarilydisregards the transformer ns of fig. 5 and the shunt capacitance CT of (60),one thus obtains the equivalent circuit of fig. 7 for the resulting one-port. Theresulting impedance seen from port 0 is

-jR jR(tan{3+cot4»' jRZ = + =-(cot4>-cot{3) (89)

sin (2 (3) . . 2 _ t " 2,

348 V. BELEVITCH and" Y. KAMP

jRcotrp

r--------,I jRtan{3 jRtan{3 i

jRcotrp

L __~ J

Fig. 7. Equivalent circuit of one resonator with energy trapping.

and is zero for cot 4> = cot {J. This is not affected by the fact that we havedisregarded the transformer and the shunt capacitance. The resonances arethus given by (J = 4> + k st: By (86) this yields

4> +knb='---

cos 4>(90)

and implicitly determines the roots 4> as a function of b as shown on fig. 8.The lowest resonance corresponds to k = 0 in (90) and always exists; higherresonances only exist for b > n,The following discussion is limited to the case of a single resonance (b < n)

b

I

2

10 20 30 40 50 60 70 80 90____ (/)(0)

Fig. 8. Graphical representation of eq. (90).

TIIEORY OF MONOLITHIC CRYSTAL FILTERS USING THICKNESS-TWIST VIBRATIONS 349

and we denote by CPo the corresponding solution of (90), which reduces to

b = CPo/cosCPo

and (80) gives the resonant frequency

Wo = We (1 + LIcos" CPo)-

(91)

(92)

With the notation(93)

(11) gives

fJ, = n Ä/H (94)

and (87) and (81) give

(95)

We shall also need the value of (89) in the neighbourhood of the resonance.We thus set cp = CPo + e, W = Wo + 'YJ with the relation

'YJ = -2 e We LIcos CPo sin CPo (96)

resulting from (80). One then has

cot cp = cot CPo - e/sin2 CPo (97)

and, by (86) and (91),

f3 = b cos CPo - e b sin CPo = CPo - e CPo tan CPo, (98)

e CPocot f3 = cot CPo + (99)

cos CPo sin CPoFinally (89) becomes

jR eZ =- - -- (1+ CPo tan CPo) (100)

2 sin? CPoand, by (96)

j R 'YJ [1+ CPo tan cpoJZ = 4 We LI cos CPo sin" CPo •

(101)

In order to obtain the impedance Z' seen from port 0 one must divide (101) byn12 where nl resulting from (70) and (82) is

2L e26 2L e26nl =--= .

H2 k/ H2 p2 cos" cP


It is equivalent to replace R by R' = R/n12 in (106). With the value of presulting from (81) and (94), one obtains

c Ä.3 :n;3 (2 Lt)3/2 cos" cPR' = _5_5 _

,8 wL e262 '(102)

where one can put approximately W = We and cp = CPo. One finally obtains

Z'=jN'YJ

where, after eliminating We by (12),

(103)

(104)

Near the resonant frequency Wo, the impedance of a series-resonant circuit ofinductance L is 2j L 'YJ.By comparison with (103), the equivalent inductanceis thus N/2, where Nis (104). If zl is eliminated from (104) by (95) and 'thenb by (90) one obtains

e H3 CPo (CPo + cot CPo)N = ---- -------

8 B L e262 sinê CPo

and the corresponding inductance value N/2 is mentioned by On oe andJumonji 9).

9. Effects of the other modes

In the computation of sec. 8 we have neglected the shunt effect of the capac-itance CT and ofthe equivalent impedances ofthe other modes. We now discussthese effects successively.

Since the trapped resonance Wo lies in the narrow interval We < Wo < wu,

and since the impedance of CT (represented in fig. 3) has a zero at We, its shunteffect will be disastrous unless its-adjacent pole almost coincides with Wo. Thecondition of exact coincidence is obtained by making Wo = We + 'YJl where'YJl is given by (62) with r = 1. By (92), this condition becomes

(105)

If one wishes a single resonance in the interval (we, wu), one must have b ~ :n;

in (91), hence 0 ~ CPa ~ 68° or 1 ~ cos CPo ~ 0·37. By (lOS) this yields thedesign restrietion

(106)

which, for AT quartz forces to choose Lt between 0·32 % and 2·35 %. On the


other hand, by substituting (l05) into the equality resulting from (91) and (95)one obtains

CPa= r A B V2/ H (107)

and the restrietion 1>0 ~ 68° gives

B 0·83-::;;;--;H yA

(l08)

with the value A = 0·65 for quartz, this gives B/H::;;; 14·4.We next estimate approximately the combined effect of all higher modes

r ~ 3. These are all strongly attenuated both in electroded and une1ectrodedregions, and each homogeneous region behaves as two uncoupled impedancesequal to the characteristic impedance Wr' The equivalent circuit of fig. 5 formode r thus reduces approximately to the one of fig. 9 and yields an impedance

w,. w,.

Fig. 9. Analogue of fig. 5 for higher modes.

Wr/n,z. Since this impedance is slowly variable it is sufficient to evaluate itat We' By (76), (70), (IO) and (94) one obtains successively

Wr j C55 H3 kr3 j C55 n3 A3 (r2 - 1)3/2

nr2 8 We L e26

2 8 We L e262

(109)

The overall impedance equivalent to the effect of all higher modes r = 3, 5, ...is the parallel combination of impedances (109). Since it is inductive it will bedenoted as M We; by (12) one obtains

nA (2 H2M=---

8 aL e262(110)

wherea = ~ (r2 - 1)-3/2.

r=3.5 .•• (111)

The constant (111) cannot be expressed explicitly in terms of elementary con-stants but can be estimated by various classical means as being ofthe orderof'


0·05. In any case the inductance (110) is much larger than (104) so that itsshunt effect is negligible in the frequency range of interest.It remains to consider the mode r = 0, which always propagates without

attenuation so that it is necessary to assume that the unelectroded regions onboth sides of the electroded region of length B have both some finite length C,and are then electrically shorted at their mechanically free ends. The corre-sponding circuit is thus the one of fig. 10 with the transmission line parameters(13) and (75). By classicalline theory, one transforms the circuit of fig. 10 intofig. 11 with

. WoZb=-----

jsin (w h B) {WhB}Za =i Wo tan (w h C) + tan -2- (112)

and the resulting input impedance is

Wo { WhB}z, + Z8/2 =; tan (w lz C)-cot-2- . (113)

Fig. 10. Analogue of fig. 5 for mode r = O.

Fig. 11. Simplification of fig. 10.

The mode r = 0 can be ignored when this impedance is infinite, and thisoccurs either for

w h C = (2 s + 1)n12, (114)

where s is an integer, or for

to hB =2sn. (115)

.The second condition, but not the first, makes Zbinfinite in fig. 11 and is tobe preferred when one considers several coupled resonators because it preventsany spurious transmission of mode zero between the ports, while simultaneously

·-=s.2H

(116)

THEORY OF MONOLITHIC CRYSTAL FILTERS USING TIllCKNESS-TWIST VIBRATIONS 353

ensuring an infinite impedance for mode zero at both ports. A similar analysis,for a more complex structure, shows in fact that it is only condition (115) thatcan ensure both advantages. As a conclusion, any practical design should bebased on condition (115) in the frequency range of interest. Approximating W

by We and using (12) and (13), one rewrites (115) asÀB

For quartz, the restrietion BfH ~ 14·4 mentioned above corresponds toÀ Bf2H ~ 4·8. Consequently only the integers from 1 to 4 come into consid-eration. The various possible design parameters are mentioned in table I,where BfH is deduced from s by (116), then rpo by (107) and LI by (105).Notethat cos" rpo determines Wo by (79).

TABLE IResonator-design data

s BfHrpo cos rpo cos- rpo LI(%)

(radians)

1 3·085 0·249 0·969 0·939 0·3352 6·170 0·498 0·878 0·771 0·4083 9·255 0·748 0·733 0·538 0·5854 12·340 0·997 0·543 0·295 1·068

10. Filter sections

The equivalent circuit of a composite filter results from the interconnectionof a number of separate equivalent circuits for the A- and B-type sections andof the particular port sections of type B. It is convenient to incorporate thecharacteristic impedances j X of the A-sections in series into the B-sectionsand to extract them correspondingly from the A-sections. The resulting portsection is represented in fig. 12 and the A-section in fig. 13. As regards theintermediate B-sections, their electrodes may be left open or shorted or, moregenerally, loaded by some impedance ZL; the resulting circuit is thus fig.14.In all sections we have disregarded the transformer of ratio nl; this is com-pensated by dividing all characteristic impedances by n1

2, which does not alter

the ratio (85) of the characteristic impedances.If one designates by 2 (f.1 the attenuation of the unelectroded region of length

Ah one has 2 (f.1 = k; A, hence (XI = (p Ad2) sin rp by (84), or

(XI = f3 ql tan rp, (117)where

(118)


jX

.__-4-----Ol'

Fig. 12. Port section with incorporated terminations.

Fig. 13. Intermediate unelectroded section with extracted terminations.

Fig. 14. Intermediate electroded section with incorporated terminations.

by (86)-(87). The transformed characteristic impedance of the unelectrodedregion is j R' cot cpby (85), where R' is (102). This gives the equivalent circuitof fig. 6 with the element values

Za = j R' cot cp (tanh al- 1); Zb = j R' cot cplsinh (2 al) (119)

for the section of fig. 13.The impedance values of the T-equivalent of a B-section are given in fig. 7

(part enclosed in dashed lines) except that R must be replaced by R' to accountfor the impedance transformation. We denote by

u=i R'lsin2fJ (120)

the negative of the value of the shunt branch. The series branches are alwaysassociated withj X' =j R' cot cp (transformed fromj X in figs 12 and 14), andtheir combined value is

Ze = j R' (tan fJ + cot cp). (121)

One thus obtains the circuits of figs 15 and 16 for the sections of figs 14 and12, respectively.Let us denote by Z' the resonant impedance deduced from (89) by the im-

pedance transformation from R to R'. This is the impedance which takes thevalue (103) near resonance. By (89) and (120)-(121) one has Z' = =U + Ze/2,hence

Ze =2Z' +2 U. (122)

TIiEORY OF MONOLmfiC CRYSTAL FILTERS USING THICKNESS-TWIST VmRATIONS 355

Fig. IS. Equivalent to fig. 14.

O~'

O'~"Fig. 16. Equivalent to fig. 12.

The decomposition of Ze according to (122) generates, in the centre of thenetwork of fig. IS, the T-network represented in fig. 17 and equivalent to theone of fig. 18, as easily checked by comparing impedance matrices. Conse-quently fig. 15 can be replaced by fig. 19.Similarly, the decomposition (122) generates, inside fig. 16, the T-network

of fig. 20, equivalent to fig: 21, and this changes fig. 16 into fig. 22.We have thus obtained the equivalent circuits of fig. 22 for the port sections

(fig. 12) and of fig. 19 for the intermediate B-sections (fig. 14). In the neigh-bourhood of the resonance, Z' is small in -fig, 22, whereas U of (120) does notresonate. Consequently one has IUI:» IZ'I in the filter pass-band, and U canapproximately be replaced by an open circuit in fig. 22, which then reduces tofig. 23 by a classical network transformation.

Fig.17. Part of fig. 15.

Fig. 18. Equivalent to fig. 17.


Fig. 19. Equivalent to fig. IS.

-u 2U

TFig. 20. Part of fig. 16.


O~I~1

O/~11


Fig. 23. Approximately equivalent to fig. 22.

In fig. 19, one can similarly replace U by an open circuit, and the same net-work transformation brings then ZL in series with the resonant impedances 2 Z'.This makes pass-band transmission impossible, unless ZL itself is small, andthe best design is, obtained when all intermediate electroded sections areshorted. With U = 00 and ZL _: 0, the transformer of ratio -1 in fig. 19 hasno effect on the transmission and the circuit of fig. 19 behaves as a single series


impedance 4 Z'. One may as well drop the transformer of ratio 2 in fig. 23,divide all impedances by 4, and consider that all resonators (intermediate, andin the port sections) behave as series impedances Z' of value (103) near Wo.Finally the complete monolithic filter is equivalent to a number of such identicalseries impedances separated from each other by T-networks according to fig. 6(representing the A-sections offig. 13)whose branch impedances have the values(119) divided by 4. Moreover, these impedances are non-selective and can beevaluated at Wo, hence with cp = CPo and f3 = CPo as results from (86) and (91).One thus obtains the values

jR'Za = - cot CPo (tanh IXI- 1);

4

j R' cot CPoZb=-----

4 sinh (2 lXi) ,(123)

with(124)

The replacement of the resonator impedance Z' by its approximation (103)leading to a lumped equivalent circuit is only valid for I BI « I -in (96), hencefor 1171«We L1. This gives IZI «R in (101), hence IZ/I «R' after transfor-mation by nl. It is only possible to achieve good pass-band behaviour if theremaining filter branches (123) have also magnitudes much smaller than R',and this requires ai» 1, i.e. small couplings between adjacent resonators.For lXi» 1, and using the approximations

tanh aR:! 1 - 2 exp (-2 a),

sinh (2 IX)~ exp (2 IX)J2,

(125)

(l26)

the elements (123) take the values

Zt, = -e; =i Rh (127)where

(128)

With the value (102) of R' taken at We and CPo, and using (10), (128) is rewrittenas

whereÄ? n2 H (e css)1/2

Ro=-------16L C262

(130)

A T-network of elements (127) is the equivalent circuit of a quarter-wave-length line of characteristic impedance RI as results directly from (88) withR = RI and f3 = -nJ4 (the phase shift is 2 f3 = -nJ2). Such a 2-port behavesas a dualizer, i.e. it transforms any shunt impedance Zt>on which it is terminated,


'~2

~tr 2'

.'~2jR;

1'~2'

Fig. 24. Impedance transformation by a dualizer.

into the dual impedance R?/Zt in series at the input, as schematically shownin fig. 24, and conversely.

11. Principles of filter design

If a monolithic :filter consists of a number of idèntical electroded sections(with the intermediate sections shorted) separated by (possibly different) un-electroded sections of sufficient relative lengths to allow the approximations(125)-(126), its equivalent circuit for the narrow-band approximation of themode r = 1 is a sequence of identical series resonators separated by dualizersof characteristic impedances RI' By applying successively the transformationoffig. 24 one produces a ladder :filterwith resonant circuits in the series branchesand antiresonant circuits in the shunt branches, all tuned at the same frequencyWo, and this is the classical configuration of a band-pass :filter deduced froman analogue low-pass with all attenuation poles at infinity (polynomial filter).The validity of the approximations (125)-(126)"is, however, essential to allowthe above derivation because, in the opposite case, one obtains a ladder struc-ture where all the resonant and antiresonant circuits are differently detunedfrom Wo, and this does not yield a satisfactory :filter characteristic. Also theidentical tuning of all resonators at Wo is essential to justify the commonnarrow-band approximation (103), hence the equivalent.circuit, and the validityof the approximation U = 00 in figs 19 and 22. Finally, the short-circuits onthe intermediate sections (ZL = 0 in fig. 19) are equally important. Any seriousdeviation from any of the above restrictions produces circuit configurations orrelative element values which cannot yield satisfactory band-pass behaviour.Since, however, the approximations (125)-(126)are only used in the expressionsof the branch impedances of the A-sections which produce small non-selectivecoupling effects between adjacent resonators, a relatively coarse approximationis sufficient to validate the further derivations. For a = 1, one has exp (-2 rx)= 0·135 and the first neglected term in (125) is exp (-4 «) = 0·018. Therestrietion rx> 1 (corresponding to a minimum attenuation of 2 Nepers isthus probably sufficient) .. Consider the polynomial low-pass ladder of fig. 25 operating between ter-

minal resistances T; an ideal transformer of ratio m (possibly in = 1) has beenforeseen 'at the :filter"output to allow the simulation' of the case of unequalterminations. In terms of ÇI. reference frequency Wo (to be considered as the cut-

THEORY OF MONOLITHIC CRYSTAL FILTERS USING THICKNESS-TWIST VmRATIONS 359

Fig. 25. Polynomial low-pass ladder.

off frequency in a sense to be defined) the element values of fig. 25 are propor-tional to the reference values

TLo=-;

Wo

1Co =--,

Two(131)

and the coefficient ft denotes the normalized value of the ith element. Wefurther put

(132)

and define the normalized frequency

x = w/wo. (133)

The total number of elements is noted r.With the above conventions, the series branches (for i odd) of fig. 25 have

the impedances j ft T x and the shunt branches have the impedances T/jftx.Between the first inductance /1Lo and the first capacitance 12 Co we inserttwo dualizers of impedance RI in cascade; their combination produces aphase shift of 'TC and is equivalent to an ideal transformer of ratio -1 whichdoes not affect the transmission. We now keep the first dualizer, but bring thesecond one to the output .of the 2-port while changing all following branchesinto their duals. This yields the 2-port of fig. 26 where the inductances Ll and£2 can be made equal by choosing RI so that the impedance j /1 T x of thefirst branch is equal to the dual R12 j 12 x/T of the impedance of the originalsecond branch. The condition thus obtained is R12 =11 T21f2 or

(134)

by (132). If we now consider the network of fig. 26 .as starting at port a, its ithseries branch (i odd) is the dual of the (i + l)th branch of the original net:work and thus has the impedance j R12 ft+ 1 x/T whereas its ith shunt branch

7~ ---IDR' 7 IC"l' --- L21a'

Fig. 26. Transformation of fig. 25.

Z =iI,Tx (136)


(i even) has similarly the impedance R12/jj;+lTx. We note these impedancesas j j;+ 1X Tl and T1Jjj;+ IX, respectively, where

(135)

The element values of the network starting with port a are thus formally similarto the original element values except that T is replaced by Tl and that all sub-scripts oîf,are increased by one. If one repeats the process (by inserting dualizersof impedances R2 after L2' etc.), the filter of fig. 25 becomes a sequence ofidentical inductances 11Lo, i.e. of impedances

separated by dualizers (between the ith and the (i + I)th inductance) of im-pedances

(137)

and followed by a cascade of dualizers and by the transformer I/m. The com-bined effect of these output networks merely alters the terminal impedance.Moreover, it is easily checked that there is no impedance alteration if the net-work is symmetric Cm = 1, fr-I+1 =Iè or antimetrie (fr-lH/ft = m2).A band-pass filter of relative bandwidth fJ around Wo, i.e. of cut-off frequen-

cies Wo (1 ± fJ/2), is obtained by the frequency transformation

W2-W02

x= ,Wo W fJ

which, for a relatively narrow band, is approximately

(138)

2 (w-wo) 2'YJx=----=--,

Wo fJ Wo fJ(139)

where the notation 'YJis the one of (96). This changes (136) into

2j/l T'YJz=---- (140)

which must be identified with (103). This gives

11 T =u»; fJ/2. (141)

This equation and (137), where RI is (129), are the basic relations for filterdesign. The relations are further simplified by eliminating zl by means of (105)and (29). This changes (104) into

(! A y 21/2 H2 cot CPo + CPoN =----------

8L e262 sinê CPo(142)

Hy . (2 (2)1/2s= -L 1(; 822 CS5 •

(144)


and the equality resulting from (129)-(130) and (137) into

!1 Tjg, = S exp (-2 0:1) cot 4>0' (143)

where

Finally, if Wo is approximated by We of (12), (141) becomes

(145)

The final relations are now (143)-(145). By eliminating j'[ T between (143) and(145) one also obtains

(146)

12. Design formulas

For an image-parameter filter, the normalized element values are fi = 2 forall i except j'[ = J,. = 1. This gives gl = 2 for all i except g1 = gr-1 = 21/2• Ifone denotes by 0: without subscript the half-attenuation of the intermediate A-sections (all identical) and by 0:' the half-attenuation of the two extreme A-sec-tions (adjacent to the port sections), one deduces from (143) that 2 0:' = 2 0: +- t log 2, so that the attenuation 2 0:' of the extreme sections must be smallerthan the one of the intermediate sections by 0·345 Neper. Finally the designrelations (143) and (146) reduce to

T = 2S exp (-2 0:) cot 4>0' (147)

16 y2 exp (-2 0:) sin (24)0)!5= •

1(;2 (4)0 + cot 4>0)(148)

The trigonometrie functions involved in (147)-(148) are tabulated in table Ilfor the values of 4>0 related to the possible choices for s resulting from table I.In particular, the relative bandwidth !5 is the largest (for a given attenuation 0:)

for s = 4. For quartz, (144) gives S = 19.104 (HIL) ohms. The resulting coef-ficients in (147)-(148) are also given in table Il. In any design 0: is arbitrary,provided it is large enough, and the ratio q is then computed by (124) yielding

0:q= .

4>0 tan 4>0(149)

362 v. BELEvrrCH and Y. KAMP

The coefficient appearing in (149) is also given in table II.

TABLE IIDesign data for image-parameter :filters

sin (2~o) ~ exp (2et) exp (2 et) Xs cot ~o xTLjH lNo tan ~o

~o + cot ~o co (MO)

1 3·929 0·114 0·144 1·494 15·82 1·837 0·360 0·453 0·699 3·693 1·078 0·546 0·687 0·41 1·444 0·646 0·554 0·698 0·246 0·648

Although the resulting filter is terminated in mid-series, the additional shuntreactance of fig. 4, originating from (60), loads each port, and one may try toincorporate it into the filter structure by considering that it represents approx-imately an antiresonant circuit of capacitance Co tuned at Wo with an inductanceequivalent to the remaining elements of fig. 4, this tuning being effectivelyachieved by condition (105). In an image-parameter filter of nominal imped-ance T and bandwidth ~ Wo, the required value for the capacitance of a mid-shunt antiresonant branch is C = 1ITwo~. From (147)-(148) and (37), andusing (12), (91), (95), (105) and (144), one deduces

ColC = 64,,2 exp (-4 et) ~o cos" ~oln2(~o + cot ~o) (150)

and Co is much too small. Consequently the antiresonance at Wo of the react-ance of fig. 4 is much wider than required and the shunt effect of the endreactance is in fact negligible throughout the pass-band.A Chebyshev filter of attenuation ripple Am (in Nepers) has the effective

attenuationA = t log [1+ H2 cos" (r arccos x)], (151)

where x is (138) andH = (e2Am - 1)1/2. (152)

The normalized element values are defined by recurrence 11) from2 sin (nI2r)

fl = sin (lJ12r) , (153)where

byf3 = log coth (Am/2) = 2 arcsinh (1/H)

2 {cos (njr) - cos (2 i nlr)}g,2 =fd,+ 1 = . h2 (IJ/2 ) + . 2Cl)sin r sin In r

(155)

(154)

and can be inserted into (145)-(146). Here again, only the relative values of theet, are determined,

THEORY OF MONO~ITHIC CRYSTAL FILTERS USING THICKNESS-TWIST VmRATlO~S 363

Appendix A

We compute the mechanical eigenfrequencies of a plate consisting of ridentical electroded regions B separated by r - 1 identical unelectroded regionsof lengths A and terminated by infinite unelectroded regions at both ends. Theequivalent circuit is a cascade of r sections according to fig. 27 (equivalent toB bordered by AI2 on each side) terminated at each end by the circuit of fig. 28

Fig. 27. Intermediate section in an iterative structure.

Za

DFig. 28. End section in an iterative structure.

(representing AI2 terminated on its characteristic impedance and thus repro-ducing the correct terminationj X' for the last resonator of the cascade, whencombined with the other AI2 section of fig. 27). ByBartiett's theorem, the 2-portof fig. 27 is equivalent to a lattice of impedances

2 z, (Za + Z'12)Zl = ; Z2 =2~.

2 z, + Za + Z'12(Al)

By (127), the first lattice impedance becomes

1 - Z'12~Z, =-2Zb 1+Z'12~ . (A2)

The image impedance Wand the transfer constant 8 of the section are relatedto the lattice impedances by

tanhê (812) = Z2/Z1>

W = Z2 coth (812).

(A3)

(A4)

For a cascade of r sections, the transfer constant is r 8. The eigenfrequenciesare obtained by expressing that the total driving-point impedance is zero atsome point, for instance at the input. Consequently, the cascade terminated


on the impedance Z of fig. 28 at its output must produce the impedance -Zseen from the input, and this yields the relation

or

Z + Wtanh (r 8)-Z=-------

1+ (ZfW) tanh (r 8)

-2 = (WfZ + ZfW) tanh (r 8). (A5)

The impedance Z of fig. 28 is approximately 2 Zb since this value is muchsmaller than X', for a weak coupling. With this value of Z, and the substitu-tion of (A4) and of the value (AI) of Z2 into (AS) one obtains

coth 8 tanh (r 8) =-1. (A6)

For 8 =1= 0, 8 =1= j n, (A6) is equivalent to tanh (r 8) = -tanh 8, hence tor 8 = -8 + j k n, or to

8 = j k nf(r + 1), (A7)

where k = 1, 2, ... , r since the extreme values k = ° and k = r + 1 leadto the excluded solutions 8 = ° and 8 = j n which obviously do not satisfy(A6). By (A7) and (A2), (A3) becomes

kn 1+ Z'f2Zb

1- Z'f2Zb

tan2---2 (r + 1)

which is equivalent to

Z'f2Zb = -cos {k nf(r + I)}.

By (103) and (127)-(128) one obtains the shifts 'Y/ of the eigenfrequencies withrespect to Wo. The r eigenfrequencies are finally Wk = Wo + 'Y/k with

n, sin- (2 <Po) k n (2 A )-- = cos -- exp - - <Po tan <Po ,zlWo 1+ <Po tan <Po r + 1 B

(A8)

with k = 1, 2, ... , r.

Appendix B

Thickness-twist vibrations in piezoceramics

Piezoceramics belong to the hexagonal class C6v (6 mm). By permuting sub-scripts 1 and 3, hence 4 and 6, in the tables ofMason 8) one obtains the followingsimplifications for the constants appearing in (21)-(22): e36 = e25 = 0,e26 = e35' 823 = 0, 822 = 833' css = C66, CS6 = 0. Subscripts can thus be


dropped in all coefficients without danger of confusion. Equations (21)-(22)then reduce to

c Llu + eLl V + (.02 a u = 0,

e Llu - e LIV = 0,

(BI)

(B2)

where LIis the two-dimensional Laplace operator in X2,X3. The boundary con-ditions on the faces, resulting from the general problem stated at the end ofsec. 3 are

V= ± Voj2;()

-(cu+ eV) =0()X2

(B3)

for the electroded regions, and

()u-=0()X2

(B4)

for the unelectroded regions. At the junctions, the variables which must becontinuous are u, E2 = -0V/OX2, Ts = (0/OX3) (c U + e V) and D3 =(OfOX3) (e u - e V). Equivalently, this requires the continuity of

ou ()V oVu,-,-,-.

OX3 ()X2 OX3(B5)

Finally, on the faces of the electroded regions, D2 = (OfOX2) (e U - e V)reduces to

ouDo = (e + e cje)-

OX2

by the second condition (B3), and (B6) is the expression to be used in (23) to

(B6)

compute the current.By eliminating LIV from (Bl)-(B2) one obtains

c' Llu + (.02 a u = 0, (B7)

where c' is defined by (28)-(29). Since the applied potential has odd symmetrywith respect to the median planes, and since all equations and boundary con-ditions are compatible with an odd symmetry for u and V, the even solutionsare not excited. For the unelectroded regions, where the density a will be notedau, the modes 'are (6) with r odd and 'Ijlr satisfying (7) with

(B8)

V = e ule (B9)

'366 v. BELEVrrCH and y, KAMP

resulting from (B?) and similar to (8). Moreover (B2) and the first conditions(B4) are satisfied with

and the modes in the unelectroded regions are thus determined.For the electroded regions, we defined a homogeneous problem by the sub-

stitution (45) and

(BlO)

where UT and VT are the Tiersten solutions (33)-(34). This replaces the firstboundary condition (B3) by ~, = O. In order to find simple modes, we postu-late a dependence exp (g X3) in X3 for all variables, and omit the correspondingfactor. A simple solution of (B?) having odd symmetry is then of the formu = sin {m (X2 - HI2)} with the relation

2_ 2 2 I'gm - m - (I) (le C , (BIl)

where (le is the density in the electroded region and where m is not yet deter-mined. With the assumed form of u, (B2) has a forced solution resulting from(B9) and a free solution satisfying LIVh = 0 with a dependence exp (gm X3),

hence with a dependence sin {gm (X2 - H12)} in X2 to ensure odd symmetry.The general solution of (B2) is thus

m cot (m H/2) = Y12 gm cot (gm H/2), (Bl3)

~,= A sin {gm (X2 - H/2)} + (e/a) sin {m (X2 - H/2)}. (BI2)

The boundary conditions on the faces give two relations from which A can beeliminated to yield

where Yl is (32). Finally, eliminating gm between (Bll) and (B13) one obtainsa transeendental equation in m defining the modes in the electroded regions.

Although the resulting characteristic equation in m cannot be solved explicitlyone can continue the formal analysis. At this stage, we know that, in the un-electroded regions, u is a combination of solutions (6), i.e.

u = ~ 1jJr(X3) cos (I' TCX2/H),r

(BI4)

where 1pr satisfies (?) with k,defined by (B8). Similarly, in the electroded regions,u is of the form

u = UT + ~ CPm(X3) sin {m (X2 - HI2)},RI

(BIS)


where UT is (33) and where rPmsatisfies

d2rP

d: -gm2 rPm= 0,

X3

'(B16)

where gm is deduced by (Bll) from the roots m of the characteristic equation.The constants involved in each 'lPr and rPmare determined, in principle, by thecontinuity of U and of (m/bx3 at all junctions and by the evanescence of thesolutions at infinity at both ends of the plate. The determination of V, whichinvolves additional constants to be defined by the continuity of the last twovariables of (BS), is unnecessary since the current can be computed by (23) and(B6) without knowing the detailed electric-field distribution. Consequently, werestrict the further discussion to the solution of the mechanical problem, i.e.to the determination of the constants in (BI4)-(BlS).The modes in the unelectroded regions form an orthogonal set, whereas the

modes in the electroded regions do not. One can, however, expand each modeof (BIS), and the forced term UT, into a Fourier series of the form (BI4). Allexpansions only involve the elementary integral (for r odd)

H2 f r n X2 • {( H)} 4 m cos (m H/2)Srm = - cos--sm m X2 -- dX2= .n H 2 H {m2 - (r nf H)2}

o

(BI7)

By (27), (33) and (B8), the conditions for a junction at some point X3 = a

are then'lPr(a) = Vr + ~ Srm rPm(a),

m

'IP/(a) = ~ s; rPm'(a),m

(BI8)

(BI9)

where the prime denotes derivatives with respect to X3 and where

4 e VoVr=-----------------------------

c' H2 kr2 {I - (2 Y//7:H) tan (7: H/2)}

Equations (BI8)-(BI9) constitute an infinite linear system in the arbitrary con-stants of the solutions 'lPr and rPm·

We further compute the current by (23), (B6) and (BIS). The term UT of(BIS) produces a current j (J) C Vo where C is (36). The remaining terms of(BIS) appear as m rPm(X3) cos (m H/2) in (B6), and the integration of (23) isreplaced by a differentiation as in (58). One thus obtains

(B20)

I =i (J) {C Vo -(L e/Y12) ~ (m/gm2) [rPm'tcos (m H/2)}. (B21)

m a

m =snfH+ Es (B22)


Relations (BI8), (BI9) and (B21) play the roles of (51), (54) and (59), re-spectively, in the uniform-field theory. One cannot establish a separate equiv-alent circuit for each mode because all modes are mutually coupled at everyjunction by (B18)-(BI9), but the original circuit for mode 1 is obtained (withmodified parameter values) if all higher modes are neglected, which is certainlylegitimate for thin plates where the higher modes are strongly attenuated andonly contribute to end effects near the junctions.To complete the discussion we verify how the solution tends towards the

one of the uniform-field theory for small y. For y = 0, the solutions of (B13)are m =snfHwhere s is an odd integer (we adopt a symbol different from rin order to avoid possible confusions in summations). For y small one thus has

and (B13) yields

(B23)

where gs can be deduced from (BIl) with the approximation of order zero form. Owing to (B22), the summations in m of (BI8)-(B 19)are now replaced by sum-mations in s and the coefficient (BI7), now noted Sw must be evaluateddifferently for s =1= r and for s = r because the denominator is of the orderof Es in the second case. By (B22) one has, within the first order,

cos (m Hf2) = 'YJsEs Hf2. (B24)where

'YJs= (_I)<s+1)/2, (B25)and one thus obtains

Srs = 2 H 'YJsEs sfn (S2 - r2) (r =1= s),

Srr = 'YJr(1 + Er Hf2 n r), (B26)

so that the matrix Srs is diagonal in the approximation of order zero, and thevarious modes are almost uncoupled. Since, in the same approximation, one has

sin {m (X2 - Hf2)} = 'YJscos (S:Tt x2fH) (B27)

it is apparent that Srr CPrtends to the mode "Pr of the uniform-field theory intheelectrodedregions, and that (BI8)-(B19) reduceto (51) and (54). On the otherhand, computing within the first order in y2, and using (B22)-(B24), one rewrites(B21) as

{"'\' cot (gs Hf2) b}

I =j (J) C Vo + L e .f.._; s, ['YJscp/Ja • (B28)

s

THEORY OF MONOLlTIiIC CRYSTAL FILTERS USING THICKNESS-TWIST VIBRATIONS 369

Since 'YJsCP.' tends to 'p.', and since cot (gs H/2) can be replaced by 2/gsH forH small, (B28) reduces to (59).

MBLE Research Laboratory Brussels, May 1969

REFERENCES1) R. D. Mindlin, Quart. Appl. Math. 20, 51-61, 196i.2) R. D. Mindlin and W. J. Spencer, J. acoust. Soc. Amer. 42, 1268-1277, 1967.3) H. F. Tiersten and R. D. Mindlin, Quart. Appl, Math. 20, 107-119, 1962.4) W. D. Beaver, J. acoust. Soc. Amer. 43, 972-981, 1968.5) H. F. Tiersten, J. acoust. Soc. Amer. 35, 53-58, 1963.6) P. Lloyd and M. Redwood, J. acoust. Soc. Amer. 39, 346-361, 1966.') P. Schnabel, to be published in Acoustica.8) W. P. Mason, Physical acoustics and theproperties ofsolids, Van Nostrand, New York,

1958; see appendix, p. 356.9) M. Onoe and H. Jumonji, Electr. Communie. Eng. (Japan) 48, 84-93, 1965.10) W. P. Mason, Electromechanical transducers and wave filters, Van Nostrand, NewYork,

1948, 2nd ed.; see p. 205, fig. 6.7.11) V. Belevitch, Wir. Eng. 29, 343, 106-110; Apri11952 .

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