KEK Report 89- 4 June 1989 A

A Simple Method using MAFIA to Calculate External Q Values of Waveguide-Loaded Cavities

Tatsuya KAGEYAMA

NATIONAL LABORATORY FOR HIGH ENERGY PHYSICS

National Laboratory for High Energy Physics, 1989

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A SIMPLE METHOD USING MAFIA TO CALCULATE EXTERNAL Q VALUES

OF WAVEGUIDE-LOADED CAVITIES

Tatsuya KAGEYAMA

National Laboratory for High Energy Physics 1-1 Oho, Tsukuba-shi, Ibaraki-kcn, 305, Japan

Abstract

A fundamental theory on waveguide-loaded cavities in the book of "Microwave Electronics" by Slater was applied to numerical calculation with MAFIA of the external Q value of a klystron output cavity. This method is also applicable to accelerating structures with slots for damping out higher modes as proposed for future linear colliders.

KEYWORDS: MAFIA, Accelerating Structure, RF Cavity, Klystron, Waveguide, Linear Collider

1. Introduction Oesigning waveguide-loaded cavities has been usually performed

by cutting and measurement of cold models etc. In a sculptural manner like this, it takes a long time to complete a design. The recent development brought by the three dimensional codes of MAFIA [1] in the field of numerical solution of Maxwell's equations enables us to make Computer-Aided Design of waveguide-loaded cavities.

In a paper [2] from SLAC, two methods using MAFIA for calculating the loaded Q values of klystron output cavities are proposed. In one of the two methods, the coupling between an output cavity and a waveguide is expressed in terms of equivalent lumped circuit parameters for simplifying the waveguide-loaded cavity structure. In the other method, the outgoing rf power through a waveguide is estimated from an analysis of standing wave patterns in the closed waveguide-cavity structure.

This report deals with another simple method using MAFIA to calculate the external Q values of waveguide-loaded cavities. This method is based on a tuning characteristic of an rf cavity coupled with a waveguide which is terminated with a movable shorting plunger. The theoretical treatment of this tuning characteristic is fully described in 'Microwave Electronics' [3] by Slater. According to this theory, the external Q value of a resonant mode of the waveguide-loaded cavity is related to the frequency variation of the resonance when the shorting plunger is slightly moved in the vicinity of a "detuned" position along the waveguide. The word of "detuned" means that when the shorting plunger is located there, the closed waveguide is most detuned from the resonance of the cavity. As an example to test this method, we chose the output cavity of the relativistic klystron SL4 [4] developed at SLAC. This output cavity is also treated in the paper [2] mentioned above.

2. Calculation with MAFIA Dimensions of the output cavity of the relativistic klystron SL4

are shown in Fig.l. The cavity is coupled with a WR90 waveguide through a rectangular coupling aperture of 0.9779 cm (in width) by 0.8534 cm (in length parallel to the beam axis). A three dimensional

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plot of the shape generated by MAFIA is shown in Fig. 2. . This shape represents a quarter part of the output cavity with the waveguide closed by a short-circuiting plane. Considering the geometrical symmetry of the SL4 output cavity shown in Fig. 1, the quarter part in Fig. 2 is enough for the input of MAFIA calculation. The total number of mesh points was about 18,000. The typical mesh size was 0.05x0.05x0.0625 cm3 around the nose of the cavity, and 0.1x0.1x0.1 cm3 in the waveguide. MAFIA calculations were done as the position of the short-circuiting plane was moved along the waveguide by a step of 0.2 cm.

The calculated tuning curve of the SL4 output cavity is shown in Fig. 3, where the frequency is expressed in terms of its wave length (Xg) in the waveguide, and the position of the shorting plunger (short-circuiting plane) in terms of the distance (d) from the beam axis. In Fig. 3, the vertical dashed line represents a mode of TMoio of the output cavity, and the other two dashed lines represent modes of TEioi and TE102 of the waveguide terminated with the shorting plunger when the cavity and the waveguide are isolated from each other. The vertical dashed line means that the resonant frequency of the TM010 mode of the output cavity is independent of the plunger position when there is no coupling between the two resonators. As seen from Fig. 3, the two slant dashed lines cut across the vertical axis on the left side at the point of d = dshort, which is about 0.7 cm. This leads to the simple result that the resonant frequency of the waveguide terminated by the shorting plunger is related to the plunger position by d - dshort = kg/2 for the TE101 mode and by d - dsh

In Fig. 3, the tuning curve is observed being pushed away from the points where the two slant dashed lines cross the vertical dashed line. This behavior of the tuning curve is due to the electromagnetic coupling between the output cavity and the waveguide. At the intersections in Fig. 3, the resonant frequency of the waveguide terminated with the shorting plunger coincides with that of the TMoio mode of the output cavity. In other words, considering the degree of freedom of the waveguide-loaded cavity structure, there are two oscillators with the same frequency at each of the inters' otions. The coupling between the two resonators causes the resonant frequency to depend on the relative phase difference between the resonators, and splits the degenerate frequency level into two levels with different frequencies as indicated ^ g i + and A-gi- (or X,g2 + and Ag2-) in Fig. 3. The electromagnetic field patterns of the ^.gi- mode are shown in Fig. 4, and those of the Xg-\+ mode in Fig. 5. As seen from these figures, the coupled electromagnetic fields in the output cavity and in the closed waveguide for the Xgi- mode oscillate in the same phase with each other, and those for the Xgi+ mode oscillate in the opposite phase with each other.

Fig. 6 shows the electromagnetic field patterns when the short-circuiting plane is located 3/4 Ag(0 away from the "detuned short" surface stated above. The symbol X,g ^ represents the wave length in the waveguide corresponding to the frequency of the TMoio mode of ihc output cavity. In Fig. 6, most of the electromagnetic field energy is stored in the form of TMoio in the output cavity because the closed waveguide is most detuned from the cavity resonance. The frequency of the TMoio mode was calculated to be 11.158 GHz, from which the value of Xgco was calculated to be 3.31 cm.

The external Q value of the fundamental mode TMoio of the SL4 output cavity coupled with the waveguide can be estimated from the differential coefficients dd/dX'g of the tuning curve at the detuned positions of 1/4 Xgtl), 3/4 A,g(a and 5/4 A,go) away from the "detuned short" surface. However, the detuned position of 1/4 "kg^ was not adopted because this position is near the coupling aperture, where higher order modes, which decay exponentially along the waveguide, other than the dominant propagating mode of TEio are

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not negligible. At detuned positions far away from the coupling aperture, where the electromagnetic wave propagates in the dominant TEio mode, the differential coefficient dd/dX.g of the tuning curve is related to the external Q value Q c x l of the output cavity by the following equation [3]:

dd 2n+l QextVq . , n .

-rr~ = -. + * , n=integer, (1) dkg 4 KV & w

where Vg and v are the group velocity and the phase velocity in the waveguide, respectively. Data points near the detuned positions of 3/4 A.go) ( n = 1 in equation (1) ) and 5,4 X.gm ( n = 2 ) were used to calculate the differential coefficient dd/dXg. Three data points in the vicinity of each detuned position were picked up, and fitted by a line as shown in Fig. 7. From these linear fits, the differential coefficient dd/dXg was estimated to be 5.82 and 6.06 for the detuned positions of 3/4 XgW and 5/4 Xg, respectively. According to the equation (1), the external Q value was calculated to be 24.2 and 22.9 from these differential coefficients. The difference between these two Qext values can be considered as a measure of the accuracy of this method, which was estimated to be about 5 %. The unloaded Q value Qo of the output cavity was also calculated with MAFIA to be 4,030. Using this value of Qo and the values of Qcxt obtained above, the loadeo Q value QL of the output cavity was calculated to be 24.1 and 22.8. Compared with a value of Q L = 2 8 . 2 calculated by another method in the paper [2], these values of QL show better agreement with a measured value of Q L = 2 0 [4] for the SL4 output cavity.

3. Conclusion This study has proved that Slater's theory on the tuning

characteristic of an rf cavity coupled with a waveguide is very useful in numerical calculation using MAFIA of external Q values of waveguide-loaded cavity structures such as the output cavity of the relativistic klystron SL4. This method is simple and straightforward in using the frequency-domain MAFIA code. The error of this method comes only from errors of MAFIA calculation results, and is

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free from other systematic errors in simplifying a waveguide-loaded cavity structure by a lumped equivalent circuit. This method is also expected to be useful in calculating the external Q values of higher modes in damped cavities proposed by Palmer [5] for future linear colliders.

Acknowledgmen t The author would like to express his thanks to Professors K. Takata

and Y. Yamazaki for valuable suggestions and continuous encouragement. Thanks are due also to Dr. T. Higo for useful discussions.

References [1] R. Klatt et al., Proceedings of the 1986 Linear Accelerator

Conference, SLAC-Report-303, pp.276-278. [2] Y. Goren and D. U. L. Yu, SLAC/AP-73, February 1989. [3] J. C. Slater, Microwave Electronics, Van Nostrand, 1950. [4] M. A. Allen et al., SLAC-PUB-4650, June 1988. [5] R. B. Palmer, SLAC-PUB-4542, July 1988.

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Fig. 1 Dimensions (in cm) of the output cavity of the relativistic klystron

SL4, ( from Ref [2] ).

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Fig. 2. A three dimensional plot of the MAFIA input geometry. This is a

quarter part of the waveguide-loaded cavity structure of the relativistic klystron SL4.

Tuning Curve of the SL4 output cavity

ane

CU ao a +-

O (-. * ^ - i o l-i O on

to

o G O CO O

axis

a e3 4>

ID -a

S

fro

u

stan

* w TJ II

6 o

P< XI

5 -5/4 X

TE101

short

1/4 X

g to

TM010 in output cavity

OH'l>r->i'i'l'I'II'i-0 1 2 3 4 5 6 7 8 9 1 0

*. q (cm)

Fig. 3. The tuning curve of the SL4 waveguide-loaded output cavity

obtained from MAFIA calculations. The short-circuiting plane was moved along the waveguide by a step of 0.2 cm in the range from 1.6 cm to 5.2 cm away from the beam axis.

(MPIA MCTUHl CONTENT AKRON PLOT

HODt MO, t

rncoUENcr

XI- I

MAK AMBON * 7.9E-01

OUTPUT CAVITV TOTAL RMNOC M ( < , 9.mr< - 1 t . PLOTTING WAJNOft C I . M T t C - ! .

X

Electric Field .

OUTPUT CmJITY M F I A anjaft / f to M xo. i

PltTUHE CONTCNT ARKOM PLOT

S-FtCLD MO DC NO, t rPEOUCNCY

| 0 3 9 * . * 7 MESM PLANE*

I 2 1

MAX ftft.ft.ON s 3 . 9 C - 0 3

u X

Magnetic Field

TOTAL ftAMOC * , C PLQTTXNO RrtNO&t . Jt f .C - . 1 .

Fig. 4. The electromagnetic field patterns in the closed cavity-waveguide

structure for the X,gi- mode in Fig. 3 when the waveguide is tuned to the TMoio mode of the output cavity. The coupled electromagnetic fields in the cavity and the waveguide oscillate in the same phase with each other.

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PICTURE C O N T E N T

flNMOH PLOT

t-rttia HODC MO, B rneoutNCT HC*M PLANCi

MAK A**DH * B.st-et

OUTPUT CAVITY TOTM. *WNOC X t l . . T . < - . < . PLOTTING H^NOOt

1W1TC - .

Electric Field

(g>(g> g)

OUTPUT CAVITY

PtCTURE COMTCMT ARROH PLOT

f - M C L O RODC HO. K rncoucNcr

I I 1 3 7 . 5 7 HCSH PLANK;,

1Z 1

HAX A R R O H * 7.BC-B1

X

Electric Field

TOTM. RAWOC * PLOTTING RAMO& t

e

I T f t -O .Vt i . 3

OUTPUT CAVITY IMP I A x / a * / a a i * 3 i lis

PICTURE COMTCWr ARROH PLOT

a - r i c L O nooc N O . e FRCOUCNCr

I I I 9 7 . 3 7 MCSH PLAMC*

1 2 - 1 Z*

HAK ARROH s 2 . 4 E - 0 3

u K

Magnetic Field

TOTAL RANOE M I C PLOTTING RANOe t

Fig. 6. The electromagnetic field patterns in the closed cavity-waveguide

structure when the waveguide is detuned from the TMoio mode of the output cavity. We find most of the elctromagnetic fields in the form of the TMoio mode in the output cavity.

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X

> u u

JS

a P

3.5

3.4

3.3-

3.2-

2 3.1 -*i 6 u

TM010 in output cavity

3.0'

2.9

detuned position of 3/4 Xgco

3 25

= 5.82

3.30 3.35

5.1

x

> cd o

o J3

5.0-

4.9-

4 .8-o

a 4.7 M

6 o

4.6-

4.5

TM010 in output cavity

detuned position Of 5/4 Xgoj J^

3.25

= 6.06

3.30 X (cm)

9

3.35

Fig. 7. Data points in the neighborhoods of the detuned positions of

3/4 kgoj and 5/4 Xg away from the "detuned short" surface. They are fitted by a line to estimate the differential coefficient dd/d^g.

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