APPLICATION OF FERROELECTRIC MATERIALS IN ELECTRICAL FILTERS

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APPLICATION OF FERROELECTRIC MATERIALS IN ELECTRICAL FILTERS

M. de Jong

To cite this version:

M. de Jong. APPLICATION OF FERROELECTRIC MATERIALS IN ELECTRICAL FILTERS.

Journal de Physique Colloques, 1972, 33 (C2), pp.C2-33-C2-37. �10.1051/jphyscol:1972208�. �jpa- 00214944�

JOURNAL DE PHYSIQUE Colloque C2, supplkment au no 4 , Tome 33, Aaril 1972, page C2-33

APPLICATION OF FERROELECTRIC MATERIALS IN ELECTRICAL FILTERS

M. de JONG

N. V. Philips' Gloeilampenfabrieken

Physics Laboratory, Division Elcoma (BE 3) Eindhoven. The Netherlands

Rhumk. - D'abord on prksente une discussion brke sur les avantages des rbonateurs pikzo- ceramiques comme element sklectif dans l'application des filtres klectroniques. Ensuite on prksente un rksumne des types differents de rksonateurs. Une nouvelle espkce de vibration d'kpaisseur, sera introduite, c'est-a-dire le cisaillement ^cc radial ^)). Enfin le mkanisme de piegeage sdectif d'energie acoustique et ses applications dans les filtres monolithiques est traite a l'aide de l'analogue optique des rkflections internes totales.

Abstract. - After a short discussion about the advantages of piezoceramic resonators as selective element in electrical filters, a survey is given of the various types of resonators. A new mode of thickness vibration is introduced, the ⁽⁽ radial ⁾⁾ thickness shear mode. Finally the selective trapping of acoustical energy and its application in monolithic filters is discussed in view of the optical analogy of total internal reflection.

2) high stability of frequency, and 3) low component sensitivity. Moreover the techniques used to design such filters have become highly sophisticated [I].

However, these LC networks tend to become incompat- ible with the present-day trends in electronics as microminiaturization, integration and concentration.

This holds especially for the coils, which are rather bulky and have a too low quality factor. To what extent do piezoelectric devices replace the classic LC networks ?

LC networks for bandpass filters can generally be subdivided into elementary sections (tuned circuits) each of which is characterized by a resonance frequency Instead of implementing the sections by discrete L and C components, mechanical resonance [2] can be applied as replacement. Because we deal with an electrical filter, electromechanical transducers must be added to convert the electrical signal into a mechanical one and vice versa (Fig. lb). In this respect piezo- electric materials are attractive since they combine within the same device the mechanical resonance and the mechanism of energy conversion. Since one aimes here for materials with high coupling factors, the ferro- electric materials are most suitable. Often polycrystal- line ferroelectric ceramics like the lead-zirconate- titanates are used. They have the advantage that the direction and magnitude of the effective polarisation can be chosen at will so that specific vibrational modes can be excited and the coupling factor can be varied.

Some types of resonators will be indicated but their I. Introduction. - For over fifty years frequency

mechanical mechanical

\

'.

'

-

^I

FIG. 1. - Schematic diagram of : a) L-C network ^; 6 ) electro- mechanical ^{filter ; c)} resonator ladder network ; d ) monolithic

filter.

selection of electrical signals has been performed by a ) El

O-

-04

11

0-

incorporation into filters will be left outside the scope of this paper.

In LC networks the elementary sections are coupled through discrete inductors or capacitors. Using piezo- electric resonators the coupling is also generallg performed by capacitors [3], [4] (Fig. lc) but also here replacement by some mechanical means is an alternative. In the low frequency range (< 2 MHz) only a few examples of mechanical coupling have been reported [5].

In the high-frequency range (> 2 MHz) a significant breakthrough has been reached in the ⁽⁽ monolithic

lumped LC networks. Some major advantages of such ^{4 1 2}

networks are : 1) negligibly low dissipation of power ;

L - C network

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972208

filter >> [6], [7], [8], [9], [lo], [ll]. Here the transducers, the resonance sections and the coupling sections are all combined into one device (Fig. Id). It consists of an array of electroded and non-electroded regions on one piezoelectric wafer. The acoustical energy is mainly confined to the areas under the electrodes, areas representing the resonating sections. Only a small portion of the acoustical energy leaks through the non- electroded areas into the adjacent electroded parts, performing the mechanical coupling between the resonant regions. Here explicit use is made of selective trapping of acoustical energy. This phenomenon is analogous to internal reflection in optics and will be discussed in detail.

Full integration is also obtained in surface-wave filters [12]. Surface waves can be excited in piezo- electric materials by means of an interdigital array of electrodes. The filters are no longer derivatives of coupled resonance circuits. Instead, the insertion loss of the transmitter and receiver have bandpass caracter- istics. The shape of the filter is determined by the layout of the interdigital array of electrodes specifically the number of fingers, their length and their width ; their average pitch determines the midband frequency.

These devices are most suitable for broadband filtering in which relatively high bandpass losses can be accept- ed. This application will not be discussed here ; the reader is referred to recent review papers [13].

11. Discrete resonators. - Several types of discrete resonators are used in electrical filters. According to the mode of vibration one distinguishes : 1) flex- ural [14] modes (5-50 kHz) ; 2) radial [15], [16] and extensional [I41 types (100-800 kHz) and 3) thickness modes [17] (> 2 MHz). In the first and second group, one pronounced resonance is observed determined by the largest dimension of the resonating body. For the thickness modes, however, a series of resonances is observed, the so-called inharmonics. Because the fundamental mode is measurated on the smallest dimension of the resonating body, the thickness, this mode is mixed with modes corresponding to the other directions ; they all by very close to the fundamental one [IS]. The appearance of such a series of inhar- monic resonances is basic for thickness mode vibrators and is the crucial problem to be discussed in our paper.

It is illustrated in figure 2a showing the frequency response curve of a ⁽⁽ radial ^N thickness shear resonator.

This vibrator which to the author's knowledge has not been reported before, is explained in figure 2b. The particle motion is approximately radial and in opposite directions on upper and lower surface. The resonator has a node in the centre as is clear from symmetry considerations. Excitation occurs by applying an alternating electric field in the thickness direction of a radially polarized disk (Fig. 2b). For a disk of radius R and thickness d, of which the mechanical boundary conditions are only satisfied on upper and lower face and of which the piezoelectric effect is assumed to be

-f (MHz)

FIG. 2. - a) The impedance Z of a ⁽⁽ radial ⁾⁾ thickness shear resonator as a function of frequency, with the inharmonic modes (1,O) upto (1,6) ; b) Displacement pattern of a (( radial )> thick-

ness shear resonator.

negligibly small, the resonances f,,, of the fundamental thickness mode are calculated to be :

where c, is the velocity of the pure shear wave, a is Poisson's ratio and 5, are the calculated eigenvalues :

to

⁼ 2.05 ; (, = 5.39 ; etc. The values of

t,,

substi- tuted in equation 1 fit the resonance frequencies of figure 2a very well.

Thickness resonators as exemplified above cannot be applied in electrical filters. One requires a suppression of all resonances except the fundamental mode fi,o.

This is obtained by introducing selective trapping of acoustical energy [6], [7]. Below a critical frequency f,, the wave energy is trapped in localized areas eventually leading to resonance, whereas for f >

f,

all energy leaks away. In addition f, is chosen such that [19]

so that only the lowest thickness mode is trapped and all higher inharmonics disappear. This acoustical phenomenon will be discussed below using the concept of total reflection, which is familiar in optics.

111. Acoustical energy trapping and total reflection. - Total reflection occurs if a wafe strikes the interface

(*) In thickness resonators the odd inharmonics fi, 1 ; fi, 3 ; etc., cannot be excited piezoelectrically because of cancellation of induced charge on the electrodes. An exception is the

ct radial* thickness shear mode, so that here one must choose :fl,o < fc < ^f1.1.

APPLICATION OF FERROELECTRIC MATERIALS IN ELECTRICAL FILTERS C2-35

dense

I

^rare d e n s e

I

^rare

c, <

^{cz CI}

<

^cz

@

<

^{(PC @}

> ^mc

FIG. 3. - Reflection of a light beam at the interface between a dense and a rare medium. a) total reflection (a, < qc) ; b) partial

transmission and partial reflection (a, > PC).

between a dense and a rare medium from the dense side at an angle exceeding a critical one. Using the notation of figure 3a, the condition is :

The critical angle cp, is related to the difference in wave velocity in the two media, c, and c2, respectively (c, < c,). Using Snellius' law (c,/c, cos cp, = 1) one obtains :

We may point out that in case of total reflection the wave fronts in the rare medium are always perpendi- cular to the interface, whereas in the dense area they are oriented at an angle (90° - cp).

Consider now a wafer of thickness d with a dense area of width D surrounded by a semi-infinite rare region with interfaces at A and B (Fig. 4). A wave propagating in the dense area will be confined into this region if it strikes the interfaces A and B at angles q < q,. If, on the other hand, the beam strikes the interfaces at angles cp > cp,, part of the energy will be transmitted into the adjacent area at each subsequent reflection and eventually all energy will have been leaked away (Fig. 4b).

In acoustics the optical beam considered finds its correspondance in the thickness shear waves [17], [18].

In a piezoelectric wafer polarized in the plane of the wafer, these waves can be excited by applying the driving electric field in the thickness direction [7] ; the particle motion is in the direction of polarisation.

One can make the inner area A-B of our wafer in figure 4 to behave as being acoustically denser in three different ways.

1 . MASS LOADING. - Consider a roughly plane- parallel wafer with thickness d, in the inner area and a smaller thickness d, = dl - Ad in the outer area. This waveguide will show a cut-off below which waves can still propagate in the inner part, but not in the outer

A 8

I

rare 1 dense

i

I I I race

FIG. 4. -Optical interferometer in which a dense area is surrounded by a rare medium. a) Path of optical beam that strikes the interfaces A and B at angle a, ⁱ a , ~ ; b) case of 9 > pc ; c) path of optical beam in a wafer with non-parallel surfaces ; d) piezoelectric wafer with electroded area of thick-

ness dl.

part. In this case the wave fronts in the thinner part will be parallel to the surfaces of the wafer. Therefore the wave propagation can be described just as well by considering two media of different wave velocity, namely c, and c, = c,(d2/dl), respectively. With (4) it follows that q, = (2 Ad/d)%. In practice the extra thickness of the electrodes, corrected for density with respect to the wafer material, suffices giving cp,

-

^lo-'.

The acoustical energy is thus confined to the electroded areas [19] and the virtual boundary between the elec- troded and non-electroded area behaves as reflecting interface (Fig. 4d).

2. PIEZOELECTRIC LOADING. - Consider a wafer of uniform thickness d, having an area of width D with short-circuited massless electrodes. The wavelengths A, and A, of the lowest plane waves in the electroded and non-electroded areas, respectively, are solutions of the equations [21] :

where K is the effective piezoelectric coupling factor.

For K 4 1, it follows that (A, - Al)/A,

-

^{(4/n2) rc2.}

The plate thus behaves as a non-piezoelectric, non- electroded wafer of thicknesses dl =

3

^1, and

C2-36 M. DE JONG

d, =

5.

I,. This situation is described in the preceding section, so that cp, = (2 J2/n) ^IC. For piezoceramic materials like the lead-zirconate-titanates, ^IC is generally so high (K: 2 0.3) that the piezoelectric loading predo- minates over mass loading 1201.

3. GEOMETRICAL LOADING. - Consider a wafer of width D and average thickness d of which upper and lower surface make a small angle y (Fig. 4c). At the interfaces A and B the wafer is bound by a medium of lower velocity. A travelling wave starting at interface A at an angle cp will strike the interface B after 1 reflections, but then at an angle cp' = cp - ly. Although the wave may have started as non-trapped (cp > cp,), it may reach the interface B at an angle cp' < cp, and thus be totally reflected. This is called geometrical trapping.

Non-parallelism thus leads to trapping at the narrow side of the wedge and detrapping at the open end. For cp' = 0 the wave is reversed and trapping occurs without apparent difference in density. The correspond- ing critical angle cp, is given by : cp, = ((Dld) y)%

whereby use is made of the geometrical relation (Fig. 4) : 1 = D/(dcp). Here it is assumed that I is very large and that y is small so that d can be consi- dered to be the average thickness. Geometrical trapping is used in oscillator quartz crystals ; they are delibera- tely polished lenticular.

Besides trapping we are also interested in the conditon for resonance in such circumstances. As an optical analogon we use here a multiple beam inter- ferometer with conductive surfaces (Fig. 4a). Inter- ference occurs if after a total of 1 reflections at upper and lower surface, the wave of wavelength I finds itself again in phase, thus if

where L is the length of path from upper to lower surface and (L -

3

I ) denotes the lack of wavelength after each reflection ; 6 is the phase change between incident and reflected beam at the interfaces A and B.

From text-books on optics [23] it can be found that (n2 cos2 cp - I)%

tan

(i)

⁼ _{n sin cp}

Furthermore the conductive surfaces require for the electrical boundary condition : E = 0, which is ful- filled for I = 2 d cos cp. With the help of the geome- trical relations derived from figure 4 (L = dlcos cp and 1 = 2 D/(d tan cp)), whereby 1 is considered to be large, one finds for the condition of constructive interference :

provided cp and cp,

<

1. Here p = (Dld) cp, is called the trapping parameter. Equation (6) is graphically displayed in figure 5. The curves represent the subse- quent orders of constructive interference.

FIG. 5. - The subsequent interference maxima q / q c versus the trapping parameter p = (Dld) qc.

The translation into acoustical resonance is simple.

For a wafer of constant thickness d and eigenfrequen- cies f1 and f, in inner and outer region, one finds :

Similarly one can find that the resonance frequency f of the total wafer and its relative difference A to f, is simply determined by the resonance value of cp as given by (6) :

A combination of the eq. (8) and (7) gives the transla- tion of figure 5 into frequencies for the acoustical case.

The subsequent interferences are analogous with our series of inharmonics in thickness mode vibrators. It follows also that f, is the cut-off frequencyf, introduced in chapter 11, since for cp + cp, one finds that f + f,. To satisfy the condition that only one mode is trapped it can be seen from figure 5 that one requires : p < 2 (*), or with eq. (4) :

This condition and curves like those of figure 5 are also derived by Onoe and Jumonji [7], solving the mechanical differential equations. The desired trapping can thus be obtained by a proper choice of the width D of the electroded area and of the trapping load A,.

(*) The mode m = 1 cannot be excited piezoelectrically.

APPLICATION OF FERROELECTRIC MATERIALS IN ELECTRICAL FILTERS C2-37 IV. Monolithic filters. - Consider a plane wafer of where I is the wavelength of the trapped wave under thickness d with two electroded areas of width D and consideration, in case of resonance equal to :

separated by a distance 2a (Fig. 6). This configuration

A

= 2 d c o s p . will perform as a n electrical filter equivalent with two

coupled resonant circuits. The electroded areas act as The coupling factor k is then given by : resonators trapping the acoustical energy due t o

4 2 a ) loading ; the non-electroded area acts as coupler. The k =

4 0 ) X

El

0 0

^{E2 = exp - ( 2 n a - d tan p,}

J

^{1 -}

jtanp)'

^-- _{tan Yc}

1.

⁽¹¹⁾

L I , I I / I I I I ~

FIG. 6. - Configuration of two mechanically coupled thickness resonators.

4 d

bandshape of a filter is approximately determined by the coupling factor k between resonating sections [23].

k can be calculated readily from our optical model. At total reflection, namely, a surface wave of amplitude u(z) penetrates into the rare medium. From elementary text-books on optics [22] it can be found that for an infinite rare medium :

In our monolithic filter, surface waves can be found

U ( Z ) = u(0) exp -

( y

(n2 cos2 9 - 1 ) s

I

z (10)

V ' l l l l l f l d V / ~ ~ ~ ~ ~ ~ ~ l

I 1 1 1 I in the coupling area a t both virtual interfaces. There- 1,

0 L-O---J,ZC&

^{I I} - -D-J

O

I2 fore the effective coupling factor k,, is :

Expressions similar t o those of eq. (1 1) have been given by several authors [8], [9] for quartz, but not for ceramics and moreover these are partly empirical.

With eq. (6) and (1 l), the bandfilter is completely determined whereby the thickness of the wafer d specifies the midband frequency.

The concept of two adjacent electroded areas can be extended a t will, corresponding t o an increasing number of coupled resonating sections [l 11, [24], [25] ; such devices on quartz are introduced recently in telephone equipment [24], using eight sections.

References [I] ZVEREV (A. I.), I. E. E. E. Spectrum, 1966, 3, Nr. 3,

129.

[2] JOHNSON (R. A.), BORNER (M.) and KONNO (M.), Proc. I. E. E. E., Transactions on Sonics and Ultrasonics, 1971, SU 18,155.

[3] SAUERLAND (F.) and BLUM (w.), I. E. E. E. Spectrum, 1968. 5. Nr. 11. 112.

[4] SAUERLAND'(F.), ~lectronics, 1969,42, Nr. 11, 102.

[5] KONNO (M.) and TOMIKAWA (Y.), Electronics and Communicatioi~s in Japan, 1969, 52A, 19.

[6] SHOCKLEY (W.), CURRAN (D. R.) and KONEVAL (D. J.), Proceedings 17th Annual Symposium on Frequency Control, 1963, 17, 88.

[7] ONOE (M.) and JuMONJI (H.), Electronics and Communi- cations in Japan, 1965, 48, 84.

[8] SYKES (R. A.) and BEAVER (W. D.), Proceedings 20th Annccal Symposium on Frequency Control, 1966, 20, 288.

[9] BEAVER (R. A.), Proceedings 21th Annual Symposium on Frequency Control, 1967,21, 179.

[lo] SYKES (R. A.), SMITH (W. L.) and SPENCER (W. J.), Proceedings I. E. E. E. Internationul Convention Record, 1967, 11,78.

[l 11 SCHNABEL (P.), Acusticu, 1969,21, 351.

[12] MITCHELL (R. F.), PRATT (R. G.), SINGLETON (J. S.)

and WILLIS (W.), Mullard Technical Communica- tions, 1970, 108, 179.

[13] WHITE (R. M.), Proc. I. E. E. E., 1970,58,1238.

[14] KATZ (H. W.), Solid State Magnetic and Dielectric Devices, John Wiley Inc. New York, 1959.

[15] MASON (W. P.), Piezoelectric Crystals and their Application of Ultrasonics, Van Nostrand Co, New York, 1950, p. 486.

[16] MUNCK (E. C.), Philips Research Reports, 1965,20,170.

[17] MINDLIN (R. D.), Quarterly applied Mathematics, 1962, 20, 107.

[18] HEISING (R. A.), Quartz Crystals for Electrical Circuits, Van Nostrand Co, New York, 1952.

[19] MORTLEY (W. S.), Wireless World, 1951, 57, 399.

[20] BLEUSTEIN (J. L.) and TIERSTEN (H. F.), J. Acoustical Society of America, 1968,43, 1311.

[21] TIERSTEN (H. F.), J. Acoustical Society of America, 1963, 35, 63.

[22] BORN (M.), Optik, Springer Verlag, Berlin, 1932, p. 41.

[23] DISHAL (M.), Electrical Communicution, 1954,31,257.

[24] BYRNE (R. J.), Proceedings 24th Annual Symposium on Freqzrency Control, 1970, 24, 84.

[25] SCHNABEL (P.), 7th International Congress on Acous- tics, Budapest 1971, preprint.