916 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTr-25, NO. 11, NOVSMBER 1977

P. GUILLON AND YVES GARAULT

Abstract—The applications of dielectric resonators are currently

of considerable interest in microwave integrated circuits [1]–[4].

The design of a dielectric resonator depends on its natural resonantfrequencies. Since exact solutions of dielectric rectangular and

cylindrical resonators cannot be rigorously computed a new and moreaccurate method has been developed to solve this problem. In thismethod all the surfaces are simultaneously considered as imperfectmagnetic walls.

Theoretical values agree very well with experimental results. Thedifference being less than 1 percent.

I. INTRODUCTION

D IELECTRIC RESONATORS made from low-loss

high-permittivity dielectric material offer the possibi-

lity for miniaturization of high-Q microwave filters [5]. Bell.

Laboratories has recently developed a consistent processing

technique for fabricating titanate ceramic having the desired

loss tangent in the microwave frequency region [6].

The resonant frequency of rectangular and cylindrical

dielectric resonators has been traditionally analyzed by

using the magnetic wall model [5], [7]–[10]. This method

gives a difference of about 10 percent between theoretical

and experimental results.

A more accurate method based on the variation method

has been reported by Konishi [11]; theoretical and exper-

imental results agree with an error less than 1 percent.

Recently Itoh and Rudokas [12] reported a numerical

procedure based on this analysis of the propagation charac-

teristics of the three-dimensional cylindrical resonator

structure. Although this method was less accurate than that

of Konishi it provides a good agreement between exper-

imental and theoretical results.

In this paper a new procedure is reported for predicting

the resonant frequencies of lower and higher modes of

isolated and shielded resonators of rectangular and cylindri-

cal shapes.

II. METHOD OF ANALYSIS

Let us consider a homogeneous, 10SS1CSScircular dielectric

resonator: relative permittivit y E2, radius a, height II.

Four approximations may be defined to evaluate the

resonant frequencies of dielectric resonators.

Approximation 1: Assume that all the surfaces are perfect

magnetic walls:

iixti=o

~.~=’)

Fig. 1. The isolated cylindrical resonator with its lateral surface, which isa perfect magnetic wall.

z

*

I

so

i I I

Fig. 2. The isolated cylindrical resonator with its flat surfaces, which areperfeet magnetic walls.

where ii is the unit vector normal to the boundary and ~ and

H are, respectively, the electric and magnetic field vectors.

Approximation 2~: Assume that the lateral surface SO

(Fig. 1) is a magnetic wall and S1 is an imperfect magnetic

wall.

Approximation 2=: S1 is a perfect magnetic wall and SOis

an imperfect magnetic wall (Fig. 2).

Approximation 3: All the surfaces (S.,S ~) of the dielectric

resonator are simultaneously imperfect magnetic walls.

To evaluate the resonant frequency with such approxima-

tions, we have to solve a system of two coupled characteristic

equations: one is obtained by assuming that only the lateral

surface (Sl ) is a perfect magnetic wall (approximation 2.);

the other, by assuming that only the flat surfaces (S.) are

perfect magnetic wall (approximation, 2~).The method used to improve the accuracy of the resonant

frequency consists of two steps.

1) Successively apply these approximations in order to

determine an effective dielectric resonator: radius a,, height

He, relative permittivit y 82. Then apply approximation 1 to

obtain the resonant frequency j= of the effective resonator.

2) Apply approximation 3 to the real resonator (a,H,&J

to obtain its resonant frequency ~.

The accurate theoretical resonant frequency ~ of the

dielectric resonator (a,H,ez) is obtained by taking the mean

Manuscript received August 2, 1976; revised April 12, 1977.value of ~. and ~.

The authors are with Laboratoire d’Eleetronique des Microondes,To define the effective resonator, we proceed as follows.

U.E.R. des Seienees, University de Limoges, Limoges, Cedex, France. Eflective Radius: We solve approximation 2~ to obtain

GUILLON AND GARAULT : RESONANT FREQUENCIES OF DIELECTRIC RESONATORS 917

the resonant frequency of the (a,H) resonator. Letjl~be this

frequency. Using approximation 1 to obtain a resonator

which has an ~1H resonant frequency and a radius a this

resonator should have a height Ill.

With approximation 2., the resonant frequency of an

(a,H1) resonator is~l=.

The radius of a resonator of height H ~resoniiting at~la is

a. in approximation 1.

Eflective Height: We solve the eigenvalue ec~uation 2. to

obtain the resonant frequency~z. of the dielectric resonator

(a,H).

Now assuming the first-order approximaticm 1 we may

determine the radius a ~ of the resonator of hei,ght H whichresonates at f2a. Considering the new resonator (a 1H) we

apply approximation 2~ to determine its resonant fre-

quency. Let f~H be this frequency. The first-order approx-

imation gives the effective height He of a resonator of radius

al and of resonant frequency ~Zff.

Frequency of the Efiective Resonator: f, is’ obtained by

assuming that the walls of the effective resonator are perfect

magnetic ones. The resonant frequency obtained with such a

consideration is f..

To outline the method we worked out a talble which is

available for all the resonant modes of a dielectric disk

resonator.

Determination of a.:

(a,H) + 2~ ‘f,H

(flH,a) + 1 a Hi

(a,H1) + 2. -f,.

(fl.,Hl) + 1 ~ a..

Determination of H.:

(a,H) + 2. + f,.

(fz.,H) + 1 -+ al

(al,H) + 2~ -+f2~

(f2H,al) + 14 H..

Determination off,:

(a.Hc) + (1) ~f..

Determination of~ :

(a,H) + (3) ~j.

Determination of the Frequency f of the Real Resonator:

f= *(f, +J).

This method outlined for a cylindrical resonator is still

valid for analyzing the resonant frequencies of rectangular

resonators. We shall study the eigenvalue equations which

are necessary to determine the resonant frequencies of themodes of cylindrical and rectangular resonatcms.

III. THE CYLINDRICAL RESONATOR

A. The Isolated Cylindrical Resonator (a,H,s2)

1) Definition of Approximation 2~: We write the magnetic

and electric longitudinal components (Appendix). Substitut-

,

ing these expressions into Maxwell equations and matching

the solutions at z = + H/2, we obtain the characteristic

equation which gives the resonant frequencies:

(1)

kO= mfi ct2=k~-k; /12 = k;&2 - k; (2)

where kC is given by

J.(k,a) = O, for TEH,~,P modes (3)

Ju(kCa) = O, for TMH,~,P modes (4)

and Jfl(kCa) is the nth order Bessel’s function of the first kind.

2) Definition of Approximation 2.: We assume that the

flat surfaces (Sl ) satisfy the open-circuit boundary condi-

tions, but the cylindrical surface does not.

a) Hybrid EH~,~,P modes: All the modes, except thosewith cylindrical symmetry are hybrid EH modes.

The expressions for E and H longitudinal fields compon-

ents are given in the Appendix. Using these relations, (20)

and (21 ), and Maxwell’s equations, we have a complete set of

field expressions,

Matching the electric and magnetic fields components at

r = a, gives-the secular equatio~ [13]

{

&2J~(pi) K:(P.)

1{

‘n(Pi) K~(pO)

~iJn(pi) – P. “ Kn(Po) “ PiJn(Pi) + PoK.(Po) }

(._..._Ik~O+ k~, 2_ n2F2 .

k; k;Ok;i ~

where

pi = kC,a PO= kCOa

k;, = k;e2 – /Y2 k~O= /Y2 – k:, P=g

(5)

(6)

(7)

i designates the mode dielectric, and o designates the outside

dielectric.

b) TEo,~,P and TMo,m,p modes: For the case of TEO,M,P

and TMO,~,p modes, relation (5) reduces to

‘o(Pi) K?(PO)—= –:PO~

‘i ~~(pt) KO(pO)

for TMO,~,p modes

L&i) _ VI KO(PJ

‘i ‘~(Pt) 142 ‘0 K;(P )0

(8)

(9)

for TEO,~,P modes.

3) Definition of Approximation 3: To explain the principle

of this approximation we consider the case of the dipolar

TEO,~,P mode.

We consider that all the surfaces (cylindrical and flat

surfaces) of the dielectric resonator (a,H,e2 ) are simultan-

eously imperfect magnetic walls. So, in order to obtain theresonant frequency, we have to solve the coupled equations

~). K.(P.)

‘i ‘~(Pi) ‘p” %L)(lo)

(-)(HP 2

)(1

~+tm2~fl _ 2zf2

2— (&2 - 1).

2–c(11)

918 IE?EE TRANSACTIONS ON MICROWAVE THEORY AND ~CHNIQUES, VOL. Mrr-25, NO. 11, NOVEMBER1977

f D)210’4(;zCm)2

e1

CH--- 2 ---

D

DIH

Fig. 3. Resonant frequencies of the TEOIP mode of isolated cylindricalresonators.

Equation (11) is obtained by substituting a’s value into (l).

4) Experimental Results: Using approximations 1,2., 2~,

and 3, we have realized a computer program which permits

us, by applying the method described in (7), to obtain the

resonant frequencies as a function of the parameters of the

dielectric resonator. The results obtained for an isolated

resonator are given in Figs. 3 and 4.

B. The Shielded Cylindrical Resonator

1) Definition of Approximation 2~ for a Resonator Placed

in a Microstrip Structure (Fig. 5): The effect of the metallic

walls is to modify the exponential decay of field components

outside the resonator, The tangential components of an

electric field must vanish on perfectly conducting surfaces.

Matching tangential components of electric and magnetic

fields at the boundaries results in a set of linear equations.

The characteristic equation which gives the resonant

frequencies of TEn,~,P modes is then

(/3tan/?~” tanha, H-a,

)(“ al tan fl~+~tanh ~d

)

(+ /?tan~~”tanhcq “d–al

)

( )“ a3tan~~+/?tanha3h =0 (12)

where

fD)’10’4(HzCm)2

20

10C* .100

21 2 3 DIH

Fig. 4. Resonant frequencies of the HE I 1P and HE1 ~ 2P modes of anisolated cylindrical resonator for ez = 100.

\ —ground

Fig. 5. The shielded cylindrical dielectric resonator.

plane

The method described in Section II is applied but now we

use (12) and (9) to calculate~, and~ and thus~of the TEti,~,P

modes. Theoretical results are presented in Figs. 6 and 7.

Fig. 6 indicates a very important property of the resonant

frequency behavior of the two lowest order modes with D/H.

The HE1lP is the lowest frequency mode for (D/H)s 1.42.Above (D/H) = 1.42, the TEOIP mode becomes the lowest

frequency resonanee. In the vicinity of (D/H) = 1.42, wherethe modes overlap, it is possible to use a single dielectric

resonator as a dual-mode resonator in order to do two-pole

bandpass filtering for wider bandwidths.

IV, THE RECTANGULAR RESONATOR

A. The Isolated Rectangular Resonator

The method outlined above is also available to determine

the resonant frequencies of a rectangular resonator.

1) Definition of Approximation 2~

a) TE.,~,P modes: To determine approximation 2 ~ weassume that the a and b surfaces satisfy the open-circuit

GUILLON AND GARAULT: RESONANT FREQUENCIES OF DIELECTRIC RESONATORS 919

1

1(

:fD)210’4(HzCmf

h,O.6ss rnrn,HE1, P mode

TEOIP,

DIH1 1:42 2

Fig. 6. Resonant frequencies of the TEO~, and ~El ,lP modes of ashielded cylindrical resonator for ez = 100.

A~MHz4 _

E,=l

~,=803,5 .

c,= 9,5

2,5 .

d mm1 I I I1 2 3 4 5

>

Fig. 7. Resonant frequencies of the TEOIF modes of a shielded cylindricalresonator for different thickness of the substrate.

boundary conditions (Fig. 8) while the fields decay exponen-

tially along the surfaces perpendicular to the Ozpropagation

axis. The eigenvalue equation, (1), is still available to

J//

/k/2

Xk’

Fig. 8. The isolated rectangular resonator.

Ax

II

flagnetic walls

o~– – j—-— –fb ‘Y

I

Fig. 9. The isolated rectangular resonator with two perfect magneticwalls.

determine the resonant frequencies of a dielectric rectangu-

lar resonator:

(13)

b) Hybrid modes: More generally and for modes other

than TE and TM modes, we can study the case for which two

surfaces of the section satisfy the open-circuit boundary

conditions.

As shown in Fig. 9, let y = O, b and z = O, and the H

surfaces be perfect magnetic walls. The longitudinal E and H

fields obtained with these conditions are given in the

Appendix. Substituting these components’ values into Max-

well equations and matching at x = t a/2, we obtain the

characteristic equation

This relation can be applied to the special case of LSEX

and lX’kfX modes, respectively, with longitudinal electric

and magnetic fields and with EX and HX equal to zero.

The characteristic equation (14) becomes

2 ka=_tan–l Ak

X2k

=2

for LSEX modes2 1

a=—

k C2 kxl‘2 tan-l — —

cl kxz

for iXMX modes.

(15)

(16)

lEI?E TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTr-25, NO. 11, NOVEMBER1977920

1(

s

:

:

fGHZ

\\.,,

a=5mm90100

b.6mm

H mmI I I f

1 5 10 15>

Fig. 10. Resonant frequencies of the TEI 1Pmode of an isolated rectangu-lar resonator.

fimper@t magnetic walls

1Fig. 11. The isolated rectangular resonator with two imperfect magnetic

walls.

We have given in the Appendix the eigenvalue equation

for hybrid modes when the surfaces x = ~ a/2 and z = O, H

satisfy the open-circuit boundary conditions, (28).

Using the method described previously (Section II) we

have evaluated the resonant frequencies of an isolated

rectangular resonator (Fig. 10).

2) Definition of Approximation 2a,~: Only terminal sur-

faces (z = O, z = H) (Fig. 11) of a rectangular resonatorsatisfy open-circuit boundary conditions.

We write the expressions of both the longitudinal and

tangential components in each medium (Appendix), and,

matching them simultaneously at the planes x = o,a and

y = o,b, obtain a determinant D (Appendix).The characteristic equation which gives the resonant

frequencies of hybrid modes of such a resonator is obtained

from the requirement that D vanishes.

B. The Shielded Resonator

1) Definition of Approximation 2~:

a) TE.,~,P modes: If the rectangular resonator is in a

microstrip structure, relation (12) is also available but it is

IJ

metallic wall

Fig. 12. The shielded rectangular resonator.

TABLE I

Experimental f f, - fex

Lmm Dmn F‘2 e f MHz f

p%

MHz exp exp

7 10 3s 4705 4660 I

S,4 15 3s 3293 3315 0,7

7 15 3s 3420 3450 I

4 10 3s 5326 5372 0,9

5 15 66 2S60 2850 0,4

7 10 66 3540 3506 I

now necessary to replace (xnm/a)2 by n2((n2/a2).+ (m2/b2))

in the al and ajs values.

b) Hybrid modes: We consider the geometrical struc-

ture shown in Fig. 12 and determine the fields in each

medium (Appendix). After having determined the tangential

E and H components and matching them at the boundary,

we obtain the characteristic equation necessary to calculate

the resonant frequencies of the hybrid modes of a shielded

rectangular resonator:

k2 k;, kX1kX,&2++81#~

kC2 c1 c1 C2

“{ El

tan kX,a/2 tan k,, 1 }– 82 tan kXza/2 tan kX1l = O.

(17)

A. Isolated

We give

V. EXPERIMENTAL RESULTS

Resonator

results for the dipolar TEO,. mode of the._ecylindrical resonator in Table I.

A comparison shows that experimental and theoretical

results agree within 1 percent.

Cohn ‘has shown in his report [5] that to obtain good

resonance frequencies by using approximation 2~ it is

necessary to multiply the relative permittivity of the resona-

tor &2by 0.875. Our method confirms this value since we find

a factor equal to 0.870.

B. Shielded Resonators

Resonator in a Microstrip Box: In Fig. 13 we have

GU3LLON AND GARAULT : RFSONANT FREQUENCIES OF DIELECTRIC RESONATORS 921

outside the rod, r > a,

E,O = Cl K“(kCOr) cos nOe-j6’z

HZO= DIK~(kCOr) sin nOe-jP’z. (21)

A’ K.(kCOr) is the modified Bessel’s function of the second kind

or the hyperbolic Bessel’s function of the second kind._theo

AA Ae~p B. The Rectangular Resonator

D.10mm1) The Isolated Resonator:

resonator H ❑ 5~~ a) Dejnition of approximation 2. for hybrid modes:

C3=100 Inside the dielectric sample, – a/2 < x < a/2,

E=, = El COS k;, x COS kYy350

Hz, = HI sin kX, x sin kYy

outside the dielectric sample, x < – a/2 and

Ezo = Eze-kx x cos k)y

320 HZO= Hze-kx x sin kYy.d mm

(22)

x > a/2,

(23)

I I I

5 20 30> b) Secular equation when the surfaces x = t a/2 and

Fig. 13. Experimental resonant frequencies of the TE,,,. modes of a z = O,H are perfect magnetic walls:shielded cylindrical resonator. “‘“

represented the experimental and theoretical variations of

the resonant frequencies of a dielectric resonator as a

function of the distance between the resonator and the

ground plane. Experimental and theoretical results agree

very well.

VI. CONCLUSION

The methods allow us to obtain with good accuracy the

resonant frequencies of isolated and shieldecl resonators;

this is important for the development of microwave in-

tegrated filters using stable dielectric resonatcms.

APPENDIX

A. The Cylindrical Resonator

1) The Isolated Resonator:

a) Definition ofapproxirnation 2H: Inside the dielectricsample, – H/2 < z < + H/2,

E=, = JH(kCr) sin n~{~l cos @z + X2 sin flz}

HZ, = .l.(k=r) cos nd{~~ cos jz + Xz sin /?z}. (18)

Outside the dielectric sample, z < – H/2 and z > H/2,

EZO= ~~J.(kCr) sin n6e-az

H,O = ~6Jn(kCr) cos n6e-Mz. (19)

Xi (i = 1,2, . . . . 6) are constant values, with a andl B as given in

relations (3) and (4).b) Definition of approximation 2=: For hybrid modes

EH.,~,P, inside the rod, r < a,

~’k’(i+i)k’’’i+%tankyb)(k;, k,, k,, 1

–k;&l ~+~.1 kC1kC, tan k,, b1

= O (24)

with

k;, = k~cz – (k; + flz)

k)?, = (k; + l?’z) – k~&l.

c) Definition of approximation 2a,~: Inside the

resonator

Hz, = H2 sin kX, x sin kY2y

E=, = E2 COS kX, X COS kY,y. (25)

Outside the resonator, x > a/2 and y > b,

HZ, = Hle-kx’x sin kY,y

E,l = E1e-kX’x cos kY,y

Hz~ = H3e-kY3y sin kX.x

E=, = E3e-kY’y cos kX,x. (26)

2) The Shielded Resonator:a) Definition of approximation 2. for hybrid modes:

Inside the resonator, – (a/2)< x < (a/2),

EZ, = Ez COS kX, X cosky y

Hz, = H2 sin kX,x sin kyy. (27)

Outside the resonator, (a/2) < x < (d – 1),

E=, = A ~J.(kC,r) cos n6e-jo’z

Hz, = B1 Jn(kcir) sin nee - jb’z (20)

E=, = El sin k.,(d – x) cos k,y

Hz, = H, cos k.,(d – x) sin kyy. (28)

922

C. Determinant D

with

l.a

~ ‘&z‘x’‘ln‘x’2flk,,—sin kY, bk;,

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-25, NO. 11, NOVEMBER 1977

1 a

~cos ‘“z

–1~ flky,e-k’’a12

C2

1

co&2kXze–kx2a/2.—

k~,

1 – kx2a/2

~ ‘Pzkx’e

&I e-kx2a/2

k~2

o.)p . /3ky, cos ky, b e-k.2a12 cop~k,l sin kY, b ~

~ ‘ln ‘y’ b k?, k?,COSk.,;

asin kX~~

kx2a/2

f3J~3 Cos kylb e–k,.a/2 flkY3 ‘in ‘Y,b ~-kx,a/2—7.2 1.2

Kc1 %3COS k.l ; “3 sin kXi ~

2

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[1] T. D. Iveland, “Dielectric resonator filter for applications in micro-wave integrated circuits,” IEEE Trans. Microwaue Theory Tech., vol.MTT-13, pp. M3-652, July 1971.

[2] Harrisson, “A miniature high Q bandpass filter employing dielectricresonator; IEEE Trans. Microwave Theory Tech., vol. MTT-16, pp.210-218, April 1968.

[3] G. I. Wofff, “Offene Mikrowellenresonatorem” Nachrichten Tech-nische Zeitschrisft, no. 6, pp. 2?9, 1971.

[4]

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Muller, “Dielectrische Resonatoren und ihre Anwendung alsMichrowellenfilter~ Zeitschrift fir angewandte Physik, vol. 24, no. 3,pp. 142, 1968.S. B. Cohn, “Microwave bandpass filters containing high Q dielectricresonators,” IEEE Trans. Microwaue Theory Tech., vol. MTT- 16, pp.218-227, Apr. 1968.Plourde and Lin, “Ba Ti~ 020 as a microwave dielectric resonator; J.American Ceramic Society, vol. 58, no. 9-10, pp. 418.Y. Yee, “Natural resonant frequencies of microwave dielectric reson-ators: IEEE Trans. Microwave Theory Tech., p. 256, Mar. 1965.Chow, “In the solution of field pattern of cyfindncal dielectric reson-ators; IEEE Trans. Microwave Theory Tech., p. 439, Sept. 1966.Sethares et al., “Design of microwave dielectric resonators; IEEETrans. Microwave Theory Tech., vol. MTT-14, p. 2, Jan. 1966.Okaya and Barash. “The dielectric microwave resonator.” Proc.I. R.E.,Oct. 1962.Konishi, Hoshino, and Utsuml, “Resonant frequency of a TEO16dielectric resonator,” IEEE Trans. Microwaue Theorv Tech., D. 112.Feb. 1976.

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Itoh and Rudokas, “New method for coupling the resonant frequen-cies of dielectric resonator; in E.M.C. 76 (Rome, Italy, Sept. 1976).