the influence of filter parameters on response of...

Electrocomponent Science and Technology, 1981, Vol. 9, pp. 115-1240305-3091/81/0902-0115 $06.50/0

(C) 1981 Gordon and Breach Science Publishers, Inc.Printed in Great Britain

THE INFLUENCE OF FILTER PARAMETERS ON THERESPONSE OF CERAMIC FILTERS OPERATING IN THE

THICKNESS-TWIST MODE

S. LEPP,VUORI and P. RUUSKADepartment ofElectrical Engineering, University ofOulu, Oulu, Finland.

(Received July 3, 1980; in final form November 12, 1980)

Starting with common equivalent circuits, computer programs have been designed to enable the calculation of filterresponses from the parameters of the ceramic material and the dimensions of the electrodes. By varying theparameter values their effect on the filter response is shown. To test the calculated results a number of test filtershave been made and their responses measured. The filter responses have behaved as predicted by the calculations.It has been possible to calculate the response of thickness-twist mode filters with a centre frequency in the rangeto 5 MHz, a bandwidth of to 10%, a load impedance of 100 to 600 s2 and a comparatively linear phase response.

1. INTRODUCTION

Ceramic filters intended for the HF range make useof the thickness-vibration resonance mode. There aretwo principal classes of operation in the thicknessmode: Thickness-Dilation (TD) and Thickness-Twist(TT) or Shear (TS). The difference between thickness-twist and shear-vibration depends on which directionthe polarization axis and the electrodes are placed(Figure 1).

Thickness-dilation vibration, where the polarizingaxis is at 90 to the electrodes, is the most commonlyused mode of vibration in commercial filters. Filtersusing thickness-vibration modes have a practicalupper frequency limitation which is imposed becauseit is difficult to machine plates thinner than a halfwave-length. By making use of the harmonics of thebasic vibration frequency it is possible to producea filter with a centre frequency in the order of tensof megahertz.

TT TS

FIGURE The direction of the electrodes and polar-ization, (P), in thickness-twist, (TT), and shear, (TS), modevibration.

Thickness-twist mode vibration is theoreticallysimpler than thickness vibration in the dilation modeand there are no other vibration modes connectedwith it. Applications of the thickness-twist mode are,however, more limited than the thickness-dilationmode due to a narrower frequency range caused bya small frequency constant and also a high polarizingvoltage required during in manufacture. These filterscan be used in the frequency range to 5 MHz.

The conventional method of making ceramicfilters for lower frequencies is to connect singleresonators in lattice or ladder networks. Thickness-mode filters are usually constructed in monolithicform by placing more than one resonator on a singleplate. This results in a reduction in weight, cost andsize and also gives higher reliability. Monolithicfilters are based on creating two or more resonancesin a single resonator. This also makes it possible toavoid transformers which, in the case of conventionalfilters, are often required. ,2

The creation of more than one resonance is basedon the energy trapping phenomenon. By making useof this phenomenon it is possible to form the requirednumber of resonances and also to limit the vibrationenergy to the area near the electrodes, which isimportant in the realization of more complicatedconstructions.

In order to assist the design of ceramic filteroperation in the thickness-twist mode, the effectof trapped filter parameters on the filter response hasbeen examined by applying equivalent circuits to thebasic element of monolithic filters.

115

116 S. LEPP/VUORI AND P. RUUSKA

MATHEMATICAL MODELS OF THICKNESS-TWIST MODE FILTERS AND THEIRCOMBINATION

2.1 Thickness-Twist Mode Propagation

Wave movement on a ceramic plate can be prop-agated only above a certain cut off frequency(waveguide phenomenon). In the electrode part ofthe plate the cut off frequency is lower than on thepart without an electrode. If an electrical vibrationwhich has a frequency between the cut off frequenciesis guided onto the electrodes, the vibration reflectsoff the boundary and the energy is trapped in thevicinity of the electroded region. This results inmechanical and electrical resonances (Figure 2).

log zl

fl frlfr3 fr5 f2FIGURE 2 Resonances for wide electrodes. I"1 is the cutofffrequency in the electroded region and f2 is the cutofffrequency in the unelectroded region, fr: to frs are theresonant frequencies.

If a ceramic plate is assumed to reach infinity inthe y-direction and the wave is propagating in thex-direction (Figure 3), the upper cut off frequencya is

f= 2--’where h is the thickness of the ceramic plate, c4D4 isthe elastic shear compliance, p is the density of theceramic material and v4D is the velocity of thevibration propagation (Figure 3). The lower cut offfrequencya is

The wave number, , can be calculated from theequations:

h h 2 h2cot( ) kl, + R(j )R-

ph

FIGURE 3

Z

The dimensions of the filter.

0 0,5 1,0 1,5 20

a)

f>- fl1,0

0,8

0,6

0,4

0,2

FIGURE 4 a) Resonant frequencies of trapped modes as functions of resonator parameters: symmetricresonant frequencies, antisymmetric resonant frequencies, b) Resonant frequencies of two trapped-energyresonators on the same plate as functions of resonator parameters. The distance between the resonators is d, h isthe thickness of the ceramic plate and is the electrode width (Figure 3).

PARAMETER INFLUENCE ON CERAMIC FILTERS RESPONSE 117

where k is the electromechanical coupling factor ofthe thickness-twist mode, p’ is the density of theelectrode material and h’ is the thickness of theelectrode.

If two resonators are placed on one plate anacoustic coupling exists between them which resultsin the excitation of antisymmetrical resonances,fax, fa2, in addition to the basic symmetricalresonances, frl, fr2 (Figure 4a). The basicresonant frequency is split into two resonantfrequencies as the two resonators are brought closerto each other as shown in Figure 4b.

The basic element of a monolithic filter is shownin Figure 5. The dimensions are chosen so that tworesonances can exist. The construction where theupper electrode is divided is the equivalent oftwo closely placed resonators. The double moderesonator acts as a filter whose bandwidth andcentre frequency are dependant on the mechanicaldimensions.

Co and C’, the phase shift which takes place in thefilter by a transformer and the load impedance byZL. The circuit acts as a bandpass filter since onlyvibrations which have a frequency near the resonantfrequency can be passed. The capacitance values ofthe Onoe model can be calculated from the resonantand antiresonant frequencies of the filters impedanceversus frequency plot (Figure 6):

Co + ) 0

2CaCo +2C’+ =0(co4/col)G CaCo +C’+ + =0(o/) (o/)G GCo +C’+ + =0(o/o ): (6/:):

,,1t7_=I

: 1:1Co Z

FIGURE 5 The basic element of a monolithic filter, thestanding wave corresponding to a symmetric and anti-symmetric resonance and the equivalent circuit of the filter.

2.2 The Onoe Model

The basic element of a monolithic filter can bedescribed with an equivalent circuit presented byM. Onoe et al (Figure 5). In this model resonatorsare represented by series resonance circuits, Za andZs, the capacitances between all the electrodes by

12= 0

u=o(01 (.0 (0 OJ (.06 (.04 (.0

FIGURE 6 The impedance curve of the filter with outputopen and short-circuited (o 2rfr and o 2rfr2 ).

If the Q-factor of the resonant circuits is known, theinductances and resistances of the equivalent reson-ance circuits can be calculated from the capacitancevalues:

6oLaCaQ =LJRQ COu LaIRa

The values of the static capacitances can also becalculated from the filter geometry with the help ofthe values in Figure 3.4

118 S. LEPPVUORI AND P. RUUSKA

Co eow(1 K(m’) )C’ eoe w 2 K(m)

where

tanhOrd/4h)m

tanh(rr2/+ d)/4h)m’ x/f m2

K(m) is the elliptic integral of the second kind.

In the equations eo is the vacuum permittivity ande the relative permittivity of the ceramic material.

Unfortunately the component values of the Onoemodel resonance circuits cannot be calculated withoutfirst measuring the filter so its use alone when design-ing filters is not possible.

2.3 The Nakamura-Shimizu model

The component values for the resonant circuits ofthe Onoe model can be obtained by applying andanalysing the filter with the Nakamura-Shimizumodel, which is based on a physical analysis of themechanical vibration (Figure 7). In the case of thethickness twist model:

Co wl /hsin h/2

N wleh/

’e h De

W2 C4 4

Oe rle602

h

0s =rlsd7r2 (.02

In the equations:N is the transformation relation between the

electrical and mechanical vibrations

Ze is the wave impedance in the electroded region0e is the phase shift in the electroded regionr/e is the wave number in the x-direction, in the

electroded regionr/s is the wave number in the x-direction, in the

unelectroded regionis the wave number in the z-direction

vs is the propagation velocity in the unelectrodedregion

Zs is the wave impedance in the unelectrodedregion

0s is the phase shift in the unelectroded region

The model can also be applied in the thickness shearmode.

U

-jz,e,

jZesinee

.ctn -FIGURE 7 The Nakamura-Shimizu-model.2.4 Combining the Models

It is possible to calculate filter responses with theNakamura-Shimizu model but to be able to use anequivalent circuit with lumped elements to analyzethe circuit, the two models of the filter must becombined together. This is done by obtaining thecomponent values of the Onoe model’s resonantcircuits from the Nakamura-Shimizu model (Figure 8).

The filter admittance at the resonant frequenciesis

N=/coCo +,

Z,,u Zmek,,2where Zx is the impedance at the symmetricalresonant frequency and Zu is the impedance at theantisymmetrical resonant frequency. The mechanical


z -16

m -24Z

.< -32

-40

-48

0,80 0,86 0,92 0,98FREQUENCY

FIGURE 8 Filter response calculated from the Nakamura-Shimizu (/) model alone and from the Onoe model (k) usingthe NS model to calculate the lumped elements of theresonant circuits. The frequency is normalized to f

capacitance between the upper electrodes and theload admittance value. In order to compare the twomodels of the filter, a program was designed accord-ing to the NS model to calculate the filter responsefrom the input information. It is possible to linkthe program, which calculates the resonantfrequencies and the inductances as a subprogram,with the common circuit analysis programs. Withthis linking it is possible to do a more sophisticatedanalysis on the behaviour of the circuit.

3. THE INFLUENCE OF PARAMETERS ONTHE FILTER RESPONSE

3.1 Calculated Results

Figure 9 shows the effect of the width, l, and thedistance, d, between the electrodes on the numberand placement of resonant frequencies. The calcu-lations have been made for the PZT-7A material(Table I). The centre frequency is determined by thethickness, h, of the ceramic plate. The width of theelectrodes must be decided so that the relation l/h

impedance, Zmek, is derived from6

0eZmek,,2 fZeOe(1 + sin 0e

0 e

Oe Ze )-I 0e Ze Os c)-I(tan +s + (tan + Zsm (tanh)-where c for the symmetric case

c -1 for the antisymmetric case.The other components of the resonant circuits

can be obtained from the resonant frequencies andinductances.

2.5 Computer ProgramThe resonant frequencies and the correspondinginductances were calculated from the previousequation with the help of a Fortran program whichwas based on a stepping method. The program gavethe resonant frequencies and inductances and therequired width of the electrodes, the distancebetween them and the constant, a, as input data.In order to calculate the filter responses, a programwas designed based on the Onoe model. This programcalculated the poles and zeros of the filter transferfunction as well as the amplitude and phaseresponse of the triter. It requiredas input data theresonant frequencies, the inductances of the resonantcircuits, the quality factor, Q, of the filter, the

f/f2i

0-,oo

0,90

h-o,ao ,

1,0 2,0 3,0 4,0t/h

FIGURE 9 The shift of resonant frequencies as a functionof the resonator parameters, h is the first harmonic of thebasic resonance frequencies.

TABLEProperties of PZT-7A material

Frequency constant N (TT, TS-mode)

a

efl/eO

eCurie temperature

250 Hz m2 490 m/s

0.7757.6.103 kg/m

0.6746060060350C

The time dependence ofNThe temperature dependence ofNfor the temperature range -60C to +85C

-0.08%/decade2.9%

120 S. LEPPVUORI AND P. RUUSKA

o

oo

0,30o ,z,0

0,52

00 LIo-C

0

0, 770 0,830 0,890 0,950 1,010FREQUENCY

FIGURE 10 Filter responses with different electrode distances.

II t\...,,/... , o

0,776 0,836 0,896 0,956 1,016FREQUENCY

FIGURE 11 Filter responses with Q-factors of 60 and 250.


has a value which allows two resonances to exist.Figure 10 shows the filter response as the distancebetween the upper electrodes is varied. It can beseen that the passband width, the ripple and thelinearity of the phase response are all dependenton the distance d.

The Q-factor of the filter affects the height of theresonances, the attenuation, the passband ripple andthe phase linearity (Figure 11). The character of thephase response can be seen on the group-delay plot.The magnitude of the Q-factor depends on theceramic material used and can also be effected ifcovered with an energy absorbing material which isnecessary in order to reduce spurious responses inthe resonator. It is therefore possible only to estimatethe Q-factor of the final filter at the design stage. Theshifting of the dominant poles of the transferfunction in the s-plane, as the load impedancechanges, is shown in Figure 12. As the load resistancedecreases, the poles move closer to the imaginaryaxis and their real parts become almost equal(Figure 13). This means that the amplitude responsetends to become more symmetrical, the phaseresponse becomes nonlinear, the resonant peaksbecome sharper and the passband attenuation

o,2

5 10 2O

-0,04 -0,03 -0,02 -0,01

FIGURE 12 The shift of the dominant poles of thetransfer function in the s-plane as the load admittancechanges. The value of the normalized load conductance isshown in the picture.

1,00

0,90

| // i, ii": /

o"0,763 0,823 0,883 0,943 1,003

FREQUENCY

FIGURE 13 Filter responses with different load resistance RL.


Z

C)

,\ )./o\1-\/

0,770

\

\.!

0,830

00 LI.I

0

0.890 0.950 1,01FREQUENCY

FIGURE 14 Filter responses with different shunt capacitance C’.

increases. The capacitive part of the load impedancedoes not have a large effect on the filter responseso its value can be ignored.

The placement of the zeros of the transfer functionare determined by the capacitance C’, between theupper electrodes. This value can be made larger byconnecting a shunt capacitor across the filter. Bychanging the capacitance value it is possible to alterthe slope of the filter (Figure 14).

3.2 Experimental Results

In order to test the calculated results a number offilters were constructed from a commercially

TABLE IIThe symmetric, frl, and antisymmetric, fal, resonant

frequencies according to measurements and calculations asthe electrode width and the distance between them varies

(Figure 3).

Measured values calculated values

l/h d/h Jr, fa, fr,

1.21 0.57 0.842 0.878 0.863 0.9151.29 0.60 0.865 0.898 0.860 0.9181.38 1.00 0.861 0.877 0.863 0.8891.53 0.71 0.850 0.876 0.847 0.881

available material, PZT-7A (Table I). The electrodeswere prepared by etching an evaporated layer.

Table II shows the measured resonant frequenciesof the filters compared with those of the correspond-ing calculated values. The resulting differences can beexplained by inaccuracies in measuring the thicknessof the ceramic plate and in determining theresonant frequencies.

Table III shows the effect of the load impedanceon the lower resonant frequency according to the

TABLE IIIThe symmetric resonant frequency of the filter according to

measurements and calculations with different loadconductances (PZT-7 A, h 0.62 mm, d 0.35 mm,

0.75 mm, Q 55).

GL fm fc0.8609 0.8625

43 0.8620 0.862629 0.8630 0.862717 0.8635 0.863312 0.8660 0.86419 0.8696 0.86964 0.8742 0.87480 0.8762 0.8828

GL is the normalized load admittance, fm is the measuredresonant frequency and fc is the calculated resonant frequency.


V

0,770 0,830 0,890 0.950 1,01FREQUENCY

FIGURE 15 Calculated filter responses as the load impedance changes (a correcting factor of 0.25 had to be usedfor the load admittance value).

calculations and experimental results. It can be seenthat as the normalized load conductance changesfrom 4 to 43, the measured frequency change isonly 0.8% less than the calculated change.

Figure 15 shows the calculated responses with adifferent load resistance (considerably larger thanthat of Table III) and in Figure 16, are the corres-

ponding measured responses. When calculatingthese curves, a correcting factor of 0.25 has to beused for the value of load conductance. Since thecurves correspond so well to each other it ispossible to conclude that the equivalent circuitsand the results that were achieved from them canbe considered reliable.

RL= 150

dB

-0

--10

--20

--30

,--40

--50

1.6 1.8 2’.0 MHzFREQUENCY

RE= 330

dB

-0

RL= 620--502’.o’16 1.8 MHz .4 1.6 1.8

FREQUENCY FREQUENCY

FIGURE 16 The measured filter responses of the filter whose calculated responses are shown in Figure 15.


Although it is possible to reduce the number ofresonances to the required number by applying theenergy trapping phenomenon, it is almost inevitablethat some other method will have to be used in orderto keep the spurious resonances to a minimum. Incommercial monolithic filters the ceramic plate ismade thinner or even partly cut off between differentresonators and the packaging material is used as anattenuation material at the sides of the ceramic plate.In the experiments epoxy cement was used as anenergy absorbant material. The attenuation materialalso effects the Q-factor of the filter and its value canbe adjusted by the amount of energy absorbantmaterial applied.

Figure 17 shows a calculated electrode design for

17

FIGURE 17MHz filter.

r-J L-

The dimensions of the electrodes of a 1.85

a filter with a centre frequency of 1.85 MHz., a-20 dB bandwidth of at least 200 kHz and a phaseresponse as linear as possible. The pattern has beenevaporated onto a 0.52 mm thick polarized PZT-7Aplate. By choosing a load impedance of 150 2 afilter with measured and calculated responses asshown in Figure 18 was constructed that confirms thereliability of this method.

4. CONCLUSIONS

In this paper the effects of different parameters onthe response of a thickness-twist mode filter havebeen calculated. From measurements made fromtest filters it has been shown that the filter responses

" -10 mecsuredo -20

z -30

,x-40

-50

1,5 1,7 1,9 2,1 2,3 MHzFREQUENCY

FIGURE 18 The calculated and measured filter responsesof a 1.85 MHz filter (Figure 16).

behave as predicted by the calculations. In order todesign a filter the Q-factor must be measured fromthe test samples. From the calculations it is possibleto design a filter with a centre frequency of to5 MHz., a bandwidth of to 10%, a load impedanceof 100 to 600 2 and a characteristic phase responseto meet the requirements. The 20 to 30 dB.attenuation of the ceramic filter depends mainlyupon the character of the ceramic material.

REFERENCES

1. R. Holland and E. P. EerNisse, Design of resonantpiezoelectric devices. Research monograph No 56, TheM.I.T. Press Cambridge, Massachusetts and London,England, pp. 179-210, 1967.

2. H. H. Schuessler, Ceramic filters and resonators, 1EEETransactions on Sonics and Ultrasonics, vol. 21, pp.257-268, April 1974

3. M. Onoe and H. Jumonji, Analysis of piezoelectric resona-tors vibrating in trapped-energy modes. Electron. CommurJapan, vol. 48, pp. 84-93, Sept. 1965.

4. P. Wolfe, Capacitance calculations for several simple two-dimensional geometries, Proceedings of the IEEE, No 50,pp. 2131-2132, October, 1962.

5. K. Nakamura and H. Shimizu, Equivalent circuits forthickness modes of elastic waves propagating in the planeof piezoelectric plate, Trans. 1EEE Japan, vol. 55-A,pp. 95-102, Feb. 1972.

6. T. Uno, 200 MHz thickness extensional mode LiTaOmonolithic crystal filter, 1EEE Trans. Sonics and Ultra-sonics, vol. 22, pp. 168-174, May 1975.

7. Vernitron Ltd., Five modern piezoelectric ceramics,Bulletin 66011/F, Jan. 1976.

8. M. Onoe, H. Jumonji and N. Kobori, High Frequencycrystal filters employing multiple mode resonatorsvibrating in trapped energy modes, Proc. 20th Feb.Control Symp., pp. 266-287, 1966.

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