Technical memorandumHIGHER MODE RESONANCES OF HIGHPERMITTIVITY SQUARE CUBOID DIELECTRICRESONATORS ON INTEGRATED CIRCUITSUBSTRATES

range of values of substrate permittivity. The rectangularcuboid has dimensions 2A x 2B x H, where the dimen-sions 2A and 2JB of the rectangular shape are chosenalong OX and OY respectively, as shown in Fig. la,

Indexing terms: Millimetre wave devices and components, Dielectricmaterials, Microwave integrated circuits

Abstract: The resonant frequencies of low andhigh order modes of square cuboid, high permit-tivity dielectric resonators have been studied, boththeoretically and experimentally for the case of aresonator on a substrate, such as an integratedcircuit. A previous analysis for an isolated rec-tangular resonator has been extended for thepresent case and Fortran IV programs have beenwritten to calculate the resonant mode frequenciesas a function of the aspect ratios of the resonators.The dimensions of resonators for various aspectratios ranging from 0.2 to 0.9 were calculated,keeping the frequency of the fundamental TE11<5

mode constant at 8.2 GHz. The resonators weremade from zirconium titanium stanate (er = 34.3)for the experimental determination of resonantfrequencies. There was a good agreement betweenthe theoretical and experimental values of the res-onant frequencies of the various modes. It wasshown from the determination of the values of Qof the various modes that the E13(5 mode offers ahigher value of Q than the corresponding funda-mental TE11(5 mode.

Introduction

Dielectric resonators are used for several microwave andmillimetre wave functions since they are rugged, inexpen-sive, small, temperature compensated and tunable (ifnecessary) and their low loss, high permittivity propertiesoffers high values of Q, especially for integrated circuits.This paper is concerned with the modes of high permit-tivity square cuboid dielectric resonators on lower per-mittivity dielectric substrates which are backed byground planes. This is a very common configuration,used in the important circuit topology of microstrips forexample, and the results may be extrapolated to othercircuits, such as coplanar and suspended stripline and soon.

Early work, such as the design of MIC bandpass filters[1] used the magnetic wall waveguide model for the cal-culation of the resonant frequency of the fundamentalTE11<5 mode. Later, more accurate analyses [2], [3] onlyconsidered isolated resonators or the fundamental mode[4]. This present work, which is both theoretical andexperimental, investigates the resonant frequencies of thevarious modes and determines the Q of certain selectedmodes, which have more confined waveguide fields, someof which are given in References 5 and 6.

Theoretical analysis

The frequencies of TE, TM and E modes have been cal-culated for a range of resonator aspect ratios, and a

IEE PROCEEDINGS, Vol. 134, Pt. H, No. 6, DECEMBER 1987

Fig. 1 Cuboid dielectric resonatora Dimensions of cuboid dielectric resonator. The square cuboid has 2A = 2Bb Cross section of square cuboid dielectric resonator on microstrip substratedivided into six field regions, numbered 1 through 6.

which corresponds to a square cuboid. The fundamentalmode of the rectangular dielectric resonator is the TE11(?

mode where q is the integer (0, 1 ,2 , . . .) number of fieldvariations in the Z direction. The TE1 1 0 mode is usuallycalled the TE11(5 mode, where 5 is a fraction of a halfwavelength in the Z direction.

This present theoretical and experimental study con-siders the low and high order hybrid modes for the caseof a resonator on a substrate. In the computation of thefrequency of the fundamental TE11(5 mode for the rec-tangular dielectric resonator we have made the assump-tion that the TE1X mode for an infinite square cuboiddielectric waveguide (2A = 2B) has a very similar dis-tribution of the electric and magnetic fields to the TE01

mode in the circular dielectric waveguide. Therefore, it ispossible to find the propagation constant /? (= Xz) of thesquare dielectric waveguide by using the value of /3 for a

563

circular dielectric waveguide of an appropriate equivalentdiameter [5]. The propagation constant of the T E 0 U

mode in such a circular waveguide enclosed in the perfectmagnetic conductor is

(1)

(2)

(3)

and for square waveguide

fa = kit,-(n/lAf-

The eqns 1 and 2 will be equal when

D = 2.164

For a given dielectric resonator of square cross section,this expression gives the equivalent diameter of a circulardielectric waveguide that has the same propagation con-stant. The calculations for the TEomn and TMomn modes ofcircular resonators [8] can be obtained by combining thetheories presented in References 9 and 10.

For the hybrid E modes we have modified the designprocedure for the isolated rectangular dielectric resonator[3] to include the case of a resonator on a dielectric sub-strate with a metalised ground plane and a metal cover.The resonant structure is shown in Fig. \b. The guidedwaves are characterised by their transverse wavenumbers kx and ky inside the high permittivity mediumand their longitudinal propagation constant /?. Becauseof the high permittivity most of the electromagneticenergy is stored in the resonator and the fields arematched only on the lateral boundary by assuming aguide of infinte length (waveguide treatment). Two typesof hybrid modes can exist, Ey or Ex, depending upon thedominant electrical field polarisation along Oy or Ox

respectively. Both have identical field patterns except thatthe electric and magnetic fields are interchanged. For theEy type it is supposed that Hy = 0 everywhere. We shallconsider only the Ey type, which we will call the E type[4]-

By matching the fields on the lateral boundary, i.e. atx = ±A, and at y = ±B, and by assuming a guide ofinfinite length, we obtain the following two characteristicequations [3]

tan 2krA =

tan 2kr B =

2k\ - (sr - l)k2

2erkyl(8r-l)k20-k

2yy'2

k2(l + a2) - 82(er - l)k2

The propagation constant is given by

(4)

(5)

(6)

where k0 denotes the free space wave number = 2nf/c, cis the velocity of light and/ i s the resonant frequency.

Now, by matching the dominant transverse field com-ponents at the resonator terminations Z = 0, and H, weobtain the next characteristic equation

TJ qn, - i (PJXZ)

(7)

where q = 0, 1, 2, . . . is the number of field variations inthe axial Z direction. Pl and P2 are given by

Px = X3/tan hfTj \

P2 = XJtan h(t2) J(8)

where

*i =(9)

The attenuation constants X3 and X4 (for the two wave-guide regions below cut-off, viz. region 3 and 4 of Fig. lb)are given by

2 - k2

2V2 _A 4 —

+ J C J -+ kl -t

(10)

As an example of mode designation, consider the E12

mode: the subscripts 1 and 2 mean respectively the firstroot kr and the second root ky of the transcendental

and ky weequations (eqns 4 and 5). After solving for kx

find the value of /?(= xy) from eqn. 6. Now by solvingeqns. 8, 9 and 10, the height of the resonator can befound from eqn. 7. Similarly, we can solve for otherhybrid modes.

Theoretical results

A universal mode chart for a square cuboid dielectric res-onator of permittivity er = 34.3 (zirconium titaniumstanate) on a substrate of permittivity er — 9.8 (alumina)is shown in Fig. 2. It shows the graph of k0 • A versus

2.00

1.80

1.60

1.40

1.20

.fi.OO

80

60

40

0.20

0 20 0.40 0.60 0.80 1.00 1.20 140 1.60 1.80 2.00 2.202.402.60

A/H

Fig. 2 Universal mode chart for dielectric resonator

Permittivity er = 34.3, on a microstrip substrate of permittivity e2 = 9.8 and thick-ness H2 = 0.64 mm, / / , = oo. Plot of fc0 • a against A/H, where fe0 is the free spacewave number, 2/1 is the side of the square base and H is the resonator heightX

+DOVAA

Ell(5TE11<5E21SE128, TMl l^E225E31c5E135

[>Q•O

•A

E325E235E416E335Ell(lTE11(E21(l

+ 5)1 +5)

A/H. Fig. 3 shows the graph of resonant frequencies ofvarious modes versus the aspect ratio of a square cuboid(H/2A) keeping the frequency of the fundamental T E 1 U

mode resonant at 8.2 GHz.

Experimental confirmation of mode frequency anddetermination of mode Q-factor

A series of resonators was made with various aspectratios which all gave a theoretical resonant frequency of8.2 GHz for the fundamental TE11(5 mode. Using thecurve for the T E 1 U mode in Fig. 2, the dimensions of theresonators for different aspect ratios varying from 0.2 to0.9 were found and are given in Table 1.

The resonator material had a Q = 7000 at 7 GHz,er = 34.3 and a temperature coefficient xf = 0 ppm/°C.The frequency of 8.2 GHz was chosen so that the effect of

564 IEE PROCEEDINGS, Vol. 134, Pt. H, No. 6, DECEMBER 1987

aspect ratio on various higher modes could be studiedover the frequency range of the sweeper (8.0 to12.48 GHz). For each resonator, the resonant frequencies

24.00

2200

20.00

i 18.00

$ 16.00

% 14.00

t 12.00

10.00

8.00600

a b c d

0.00 0.20 0.40 0.60 0.80

aspect rat io, H/2A

1.00 1.20

Fig. 3 Comparison of theoretical mode frequencies with experimen-tally determined resonances (dots) for various values of H/2A (resonatoraspect ratio).Points a, b, c, d, e, f and g are high Q points. er = 34.3, e2 = 9.8, line-resonatorseparation = 0. H2 = 0.64 mm, Hl = oo, line impedance = 50 ilX

+•OVAA

E115TE11.5E21SE12<5, TMU8E225EMSE\3S

AOA

•A•

E32(5E23<5E41.5El 1(1 +d)TE11(1 +5)E21(l + 6)

Table 1 : Dimensions of the square cuboid resonators for theresonance of the TE11S mode at 8.2 GHz

Aspect ratio 0.2 0.25 0.3 0.4 0.5 0.7 0.8 0.9Resonator height, mm 1.96 2.22 2.41 2.86 3.28 4.11 4.56 4.97Resonator base, mm 9.81 8.95 8.03 7.16 6.57 5.88 5.70 5.53

of the modes and the corresponding value of loaded Qwere determined and the experimental values of the res-onant frequencies are shown by dots in Fig. 3. There wasa good agreement between theoretical and experimentalvalues. It was seen that from the experimental values ofQL and the resonances observed that the E13(5 mode is ahigh-Q mode, as is the TEU ( 1 + ( 5 ) mode. The experimentalvalues of QL for the fundamental TE11(5, E13(5 andTEll{1+S) modes are shown in Fig. 4. The Q values ofthese modes were increased by lifting the resonator abovethe microstrip using 0.5 mm PTFE spacers. By lifting theresonator the induced current loss due to the proximityof the ground plane is decreased, and hence the Qincreased. The Q values obtained for the E13(J mode forraised resonators are shown by the points a', b', d and d!in Fig. 4, whilst the points / ' and g' correspond to the

400.0

50.0

300.0O"S 250.0•o§ 200.0

150.0

100.0

TE||(,«6)on spacer

0.20 0.30 0.40 0.50 0.60 0.70 Q80 0.90 1.00

aspect r a t i o , H / 2 A

Fig. 4 Experimentally determined values of loaded Q for the high Qmodes as functions ofH/2Ax TE, 14 mode. Resonator on substrate.+ El3i and TE, 1(1 +i) modes. Resonator on substrate.* Ei3i and TE, , ( , +1) modes. Resonator on spacer.

IEE PROCEEDINGS, Vol. 134, Pt. H, No. 6, DECEMBER 1987

TE11(1+(5) mode. The raising of a resonator causes theresonant frequency to fall slightly. This is to be expected,as there is more of the field in the air rather than thesubstrate. The resonant frequencies corresponding to a, b,c and d were 10.95, 11.18, 11.60 and 12.25 GHz respec-tively whilst those of a', b', d and d! are 10.26, 10.6, 11.12and 11.9 GHz respectively. The frequencies of theTE11(1+<$) modes are: / = 12.04 GHz, / ' = 11.76,g' = 11.21 GHz. The frequency of d was 12.46 GHz.

The point e in Fig. 3 (corresponding to the resonatorof aspect ratio 0.5) is very close to the TE11(1+(5) modeand is also near the extreme end of the present sweepersystem. It is difficult to determine the QL at e. By liftingthe resonator using a PTFE spacer, the frequency can belowered. The QL at this frequency is about 450.

Conclusion

An analysis has been carried out of the resonant fre-quencies of various modes for the case of square cuboidresonators on a substrate. There was a good agreementbetween theoretical and experimental values of resonantfrequencies. From the determination of the values ofloaded Q for the various modes it was shown that theE13<5 mode offers higher values of Q than the fundamentalTE11(5 mode because there is less radiation.

D. SINGHG. B. MORGAN

22nd June 1987

University of Wales Institute of Science and TechnologyPO Box 25CardiffCF1 3XEUnited Kingdom

Mr Singh is on leave from the Defence Electronics Applica-tions Lab, Dehradun, India

Acknowledgment

We wish to thank Plessey Caswell for donating the ZTSmaterial and British Council for a grant for DeepakSingh.

References

1 IVELAND, T.D.: 'Dielectric resonator filters for application inmicrowave integrated circuits', IEEE Trans., 1971, MTT-19, pp.643-652

2 GUILLON, P., and GARAULT, Y.: 'Accurate resonant frequencyof a dielectric resonator', IEEE Trans., 1977, MTT-25, pp. 916-922

3 LEGIER, J.F., KENNIS, P., TOUTAIN, S., and CITERNE, J.:'Resonant frequencies of rectangular dielectric resonators', IEEETrans., 1980, MTT-28, pp. 1031-1034

4 SUTER, W.A.: 'Squaring off with dielectric resonators', Microwaves,1981,20, pp. 111-114

5 GOELL, J.E.: 'A circular harmonic computer analysis of rectangu-lar dielectric waveguide', Bell Syst. Tech. J. 1969, 48, pp. 2133-2160

6 MARCATILI, E.A.J.: 'Dielectric rectangular waveguide and direc-tional coupler for integrated optics', ibid, pp. 2071-2102

7 SETHARES, J.C., and NAUMANN, S.J.: 'Design of microwavedielectric resonators', IEEE Trans., 1966, MTT-14, pp. 2-7

8 SINGH, D.: 'Integrated circuit elements for short millimetre wave-lengths', PhD thesis, University of Wales, 1986

9 ITOH, T., and RUDOKAS, R.S.: 'New method for computing theresonant frequencies of dielectric resonators', IEEE Trans., 1977,MTT-25, pp. 52-54

10 RAMO, S., WHINNERY, J.R., and VAN DUZER, T.: 'Fields andWaves in Communication Electronics' (Wiley, New York, 1965)

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