I fit2 IEEE TRANSACTIONS ON MAGNETICS. VOL. 31. NO 3 , MAY 1995

A Numerical Computation of External Q of Resonant Cavities

Hajime Igarashi, Yasuyuki Sugawara and Toshihisa Honma Department of Electrical Eng., Faculty of Eng., Hokkaido University

N13, W8, Kita-ku, Sapporo 060, Japan

Abstract - This paper describes a simple numerical method based on the finite element method for the analysis of the resonant frequencies and external Q values of a waveguide loaded cavity. The present method solves a second order proper equation with a damping term, which can be reduced to a linear one. It is shown that the present method provides the reasonable resonant frequencies and corresponding Q values for wide range of the electromagnetic coupling.

I. INTRODUCTION

In general, the electromagnetic cavity resonators are open systems and are coupled to external energy sources through, e.g. , coaxial lines and waveguides. The strong electromagnetic coupling, which has attracted considerable attention especially in heavily dumped cavities for particle accelerators [ l l , [21, disturbs the electromagnetic fields in the cavities and then significantly modifies the resonant frequencies. The electromagnetic fields and resonant frequencies of closed cavities can be effectively analyzed by various numerical methods. There are, however, few methods by which a cavity coupled to waveguide is effectively analyzed. Kroll et al. have reported a useful, highly sophisticated numerical method for computing the external Q, which characterizes the degree of external coupling, and resonant frequencies of the waveguide loaded cavities [ Z ] . Although their method, which is based on the theoretical relation between the phase change of electric fields and the frequency, can give the above quantities only using conventional computational codes for closed cavities, it requires physical consideration on the electric field distribution. Moreover, the method results i n a nonlinear equation, including several unknowns, which does not seem easy to handle.

This paper describes a simple, new approach based on finite element method for the analysis of the resonant frequencies and external Q values of a waveguide loaded cavity. In this method, the Robin- type boundary conditions is imposed on the virtual boundary which divides the whole region into the f ini te cav i ty -wavegu ide a n d in f in i t e ly long waveguide regions. The introduction of this boundary condition leads to a second order proper equation with a damping term, which can be reduced to a linear one solvable by standard numerical techniques.

Manuscript received July 6, 1994. This work was supported in part by Inoue Foundation of Science.

11. FORMULATION

In this paper, we restrict our consideration to the two dimensional problem. Fig. 1 shows a schematic of the waveguide loaded cavity with a metallic wall. Provided that the cross section of the system in the xy plane is independent of z, the z component of the electric field, a, in this cavity-waveguide system is governed by the two dimensional scalar Helmholtz equation

V2@ + k2@ = 0, where k denotes free space wavenumber. The functional F for (1) is expressed in the form

where SZ is the two dimensional solution region and rv is the virtual boundary at x=x,, which separates the finite cavity-waveguide region SZ to be discretized and the infinitely long waveguide region. We consider here the incident TEIo mode wave $*" coming from the right hand side of the model. The electric field is decomposed into the incident and reflection wave components as

(3)

The reflection wave Qref near rV is composed of only the principal mode if rV is located sufficiently far from the cavity so that the higher modes attenuate. Thus, near rv, the incident and reflection waves can

Waveguide

Fig. 1 Waveguide loaded cavity

0018-9464/95$04.00 0 1995 IEEE

1643

be expressed in the form where [K'I = [Kl - k: [MI and y=jJ. We see here that the problem results in the second order proper equation which governs a damping oscillation sys t em. F u r t h e r m o r e , i n t r o d u c t i o n of t h e (4a)

transformation aref (x , y ) = R E , exp C-jpx: ) cos ( EY) , (4b)

where Eo is the amplitude of the incident wave, p the propagation constant of the incident wave, a the width of the waveguide, and R denotes the reflection constant. Differentiating (3) with respect to x and

(94 = [ :;} = [ ;}. in (9) then leads to a linear eigenvalue problem,

using (4a) and (4b), we have the Robin-type boundary condition on rv

10 1 [I1 I -IMJ-'[Kq -[MJ]-',C, I i :; 1 = y i :: 1 (11) where [a denotes a unit matrix. After solution of (11) by a standard numerical technique, we can readily obtain the complex eigenvalues k, eigenfrequencies f and corresponding external Q values as follows :

a+( xo, Y ) ax + j ~ + ( x,, y 1 = 2 j ~ + ~ " ( x,, y ) . (5)

The insertion of the boundary condition -(5) to (2)

k = [(rc/a)2-y 2 3 U2 , f = c Re (k)/2n,

Q = Re (k) /[2 Im (k)l ,

(12a) (12b) (12c)

where c denotes velocity of light in vacuum. The ex terna l def ined by (12c) t h e electromagnetic coupling between the cavity and

Moreover, the finite element discretization of (6) gives the matrix equation,

and {Ne} and {NI) are the shape function vectors for the field and boundary elements, respectively, and {Neu} denotes the derivative of {Ne} with respect to U.

By varying the frequency of the incident wave and solving (71, we get the frequency response of the waveguide loaded cavity. At resonance, however, there is an outgoing wave but no incoming wave [21. Thus, the driving term in the right hand side of (7) can be eliminated at resonance. In addition, the dispersion relation, p2 = k2 - k,2, where k, is the cut-off wavenumber of TElo mode, holds in the waveguide. Equation (7) then becomes

[K'I {+I + Y2 [MI {+} + Y E 1 {@I = (01 9 (9)

waveguide, and also characterizes the energy loss through the waveguide [21, which is similar to the well known definition for a cavity with a lossy wall [31. We can easily find the resonant frequency, for the principal cavity mode, which gives the largest Q value. The linearization mentioned above increases the number of unknowns by factor two and produces full submatr ices i n t h e second row of (11) . Nevertheless, the present method has an obvious advantage that it does not require choice of adequate frequency-length pairs through observation of field plots, in comparison with the method by Kroll et al. [21. Moreover, the present method automatically gives the field distribution as eigenfunctions of (11). These advantages compensate for the drawbacks in the present method.

111. NUMERICAL RESULTS

We here apply the method introduced in the previous section to the analysis of a waveguide loaded cavity, shown in Fig.2, which has been taken up as TEAM workshop problem 18. In the analysis, the region 52 was discretized by t r i angu la r f in i te elements of the second order. The number of nodal points, which depends on the choice of D and t , is about 600.

I644

---------- FIT . TE niodewave sa a : 10-

z c L 2 Virtual boundary rv

Fig. 2 Waveguide loaded cavity with an iris

The analytical values for this simple configuration are obtained from the equivalent circuit shown in Fig. 3 [2]. Thus, the equation for p i s given by

where B represents the normalized susceptance and is approximated for a thin iris t = 0 as t41

sin2( f )] . (14) When the iris has finite thickness, we estimate B using the following formula [51 numerically obtained by the transmission equation method,

I l+{8 - ( j?aln)2}u2- 3 4

). (15) i l k - t r t 0 . 9 8 5 d - 1 . 2 5 t B = ( )2Cot2( - 0.51kc - t 2 0.5/kc-t

The nonlinear equation (13) can be easily solved by a standard root finding technique. Of course, this equivalent circuit technique can not be applied to the analysis of arbitrary shaped cavities, in contrast to the present method.

Fig. 3 Equivalent circuit. Yo denotes the characteristic admittance of the waveguide.

We also compute the resonant frequencies by the use of the three-parameter formula given by Kroll et al. 123. In this method, one finds the complex resonant frequency from the theoretical, nonlinear relation, including three unknown parameters, between the phase change i3 along the waveguide and frequency f. To determine this relation, we have to solve three eigenvalue problems to find the 0-fpairs for the three different closed regions formed by inser t ing a shorting plane in the waveguide. The shorting planes are located near rV in our analysis.

Tables I and I1 show the dependence of the external Q on the choice of the waveguide length D for t = O and t=5a/144 when the normalized iris opening width dJa is chosen as 0.5. I t is seen that the results by the present method and Kroll's method agree well with the results by the equivalent circuit, whereas the variation of D yields small changes in Q in the results by the latter method. These fluctuations in Kroll's method may be improved by employing the four-parameter formula, instead of the three- parameter formula. However, note that it is not easy to find adequate 0-fpairs for the accurate fitting of the 0-f curve . On the o ther hand , t he s l igh t discrepancies between the results by the equivalent circuit and present method may be due to either discretization error i n the present method or insufficient accuracy in the susceptance (14) and (15).

Figs. 4 and 5 show the dependence of t he normalized resonant frequency [, and Q on the iris opening width dla for the case of D =(5/9)a. We see that the present method gives the values consistent with those by the equivalent circuit, for the wide range of electromagnetic coupling.

Table I Dependence of external Q on waveguide length D for

t = o a

External Q

D Present Kroll

0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50

21.90 21.90 21.91 21.91 21.91 21.91 21.91 21.91 21.91

21.70 23.84 20.68 22.24 21.99 22.10 20.97 22.21 22.73

a The iris opening width d u is taken to be 0.5. The external Q computed by the equivalent circuitwith (13) and (14) is 21.70.

The three-parameter formula in I21 is employed.

I645

1.0 C U

h 8 $ ' 1

w

U c

8 0.8 bl

a N 4 rl

Table I1 Dependence of external Q on waveguide length D for

t=5af144 a

External Q

D Present Kroll

0.50 34.3 1 33.88 0.75 34.3 1 36.37 1 .oo 34.31 32.85 1.25 34.3 1 34.57 1.50 34.31 34.12 1.75 34.31 34.44 2.00 34.31 33.19 2.25 34.31 34.56 2.50 34.31 35.38

a The iris opening width d a is taken to be 0.5. The external Q computed by the equivalent circuit with (13) and (15) is 33.32.

The three-parameter formula in [2] is employed.

-

0 Present (t=O)

0 Present (t=5a/144)

lo4 -

103 - a .-i

d

bl U

4 102 -

10 -

1 L I I I

0.2s 0.5 0.75 Normalized iris opening width d/a

Fig. 4 Plot of the Q value against the iris opening width da. 'Present', 'Kroll' and 'Circuit' represent the results by the present method, the three-parameter formula [21 and the equivalent circuit shown in Fig. 3, respectively.

It may be worthwhile to note here that the Q values computed by the present method depend especially on the finite element discretization near the iris. This is due to the singularity in c$ at the corner of the iris. Thus, we finely discretized the region in the vicinity

- 0 Present (t=O)

OPresent (t=5a/144)

AKroll (t=O)

AKroll (t=5a/144) Circuit (t=O)

0.6 z 0.75 0.25 0.5

Normalized iris opening width d/a

Fig. 5 Plot of the resonant frequency f,,, normalized with respect to that for closed cavity, against the iris opening width d a .

of the iris employing elements of edge size 1 - d50, to obtain the results shown here.

Since the boundary condition (5) is valid even if the cavity has three, dimensional electromagnetic fields, the present method can be extended to analyze cavities with three dimensional structure, provided that the spectrum pollution, due to inadequate numerical treatment of vector fields, is appropriately avoided by, e.g., the edge elements. In addition, we can also use the boundary condition ( 5 ) for the analysis of characteristics of waveguide discontinuity such as an inductive window and bend.

V. SUMMARY We have described a new method for the analysis of

the resonant frequency and Q value of waveguide loaded cavities. The present method has been shown to give the reasonable external Q values for wide rang of coupling, only by solving a s t anda rd eigenvalue problem. We plan to extend out method for the three dimensional problem.

REFERENCES 111 R. B. Palmer, SLAC - PUB - 4542,1988. [21 N. M. Kroll and D. U. L. Yu, "Computer determination of the

external Q and resonant frequency of waveguide loaded cavities," Particle Accelerators, vol. 34, pp. 231-250,1990.

[31 R. E. Collin, Field Theory of Guided Waves, 2nd ed., IEEE Press, NewYork, 1991, pp.387-389.

I41 R. E. Collin, Foundations for microwave engineering, 2nd ed., McGraw-Hill, Singapore, 1992.

I51 F. Ishihara, T. Sibazaki, T. Suga and S. Iiguchi, "Formulae for Estimating the Wide Band Reflection Characteristics of Induct ive Window wi th Thickness i n R e c t a n g u l a r Waveguide," (in Japanese), Trans. IEI(=E, vol. J75-C-I, pp.92- 100,1992.