Analysis and Synthesis of Transmission Lines and Wave Guides With Closely Spaced Discontinuities -...

8/3/2019 Analysis and Synthesis of Transmission Lines and Wave Guides With Closely Spaced Discontinuities - Byron Harris_ P

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E l e c t r o m a g n e t i c s L a b o r a t o r y R e p o r t No. 85-5

ANALYSIS A;VD SYNTHESIS OF TIWNSblISS I ON LINESAND WAVEGUIDES WITH CLOSZLY SPACED DISCONTI&XJLTIES

S c i e n t i f i c R ep or t

Byron L. Harrisand

P . E. Mayes

J u l y , 1985

Suppor ted byRCX, P u r c h a s e Order Yo. 6890897686

N o r r i s t o v n , N J 08057

Zlec t r o m a c ~n e t i c sL a b o r a t o r yDe Fa r t n e n t of E l e c t r i c a l and C o mp ut er E n z i n e e r i ~ g

Engine e r ing 5:qer i rnent S t a t i o nU n i v e rs i t y of I l l i n o i s

Ur b a r a , 11 61501


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TABLE OF CONTENTS

CHAPTER PAGE.NTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . 1DEVELOPMENT OF MULTIPLE-DISCONTINUITY ANALYSIS . . . . . . 42.0 Introduction . . . . . . . . . . . . . . . . . . . . 42.1 Analysis . . . . . . . . . . . . . . . . . . . . . . 42.2 Parallel-Plate Case . . . . . . . . . . . . . . . . . 102.3 Coaxial-Line Case . . . . . . . . . . . . . . . . . . 132.4 Validation for Parallel-Plate Case . . . . . . . . . 182.5 Determining the Number of Modes to Use . . . . . . . 23SOME APPLICATIONS OF MULTIPLE DISCONTINUITY AYALYSZS . . . 24

. . . . . . . . . . . . . . . . . . . ..0 Introduction 24. . . . . . . . . . . . ..1 Equivalent Length of a QWT 25

. . . . . . . . ..2 Reflection Coefficient Optimization 294 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . 37

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .EFERENCES 38


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CKAPTER 1INTRODUCTION

Microwave networks often contain abrupt changes in characteristic imped-ance of cylindrical or rectangular coaxial lines. These impedance changesare most commonly a result of a change in the size of either the inner orouter conductor. Such a step change will generate many higher-ordered modes(HOM'S) in addition to the original TEM mode supplied by the generator.These HOM's are needed in order to satisfy the boundary condition that thetangential component of the electric field on the conductor at the discon-tinuity junction be zero.

The generation of HOMts will, in general, alter the frequency charac-teristics of the scattering parameters as predicted by using a simpletransmission line model that ignores the HOM's. This thesis investigates

the analysis of lines with discontinuities taking the HOlit into account.Techniques, to be mentioned below, have previously been developed to

analyze a system with a single discontinuity taking the HOM's into account.For most systems the frequency of operation is such that the HOM's will beevanescent waves. Thus the amplitude of the HOM's created at the discon-tinuity will decay exponentially as a function of the distance from thediscontinuity. Therefore, if the discontinuities are far enough apart,then each discontinuity can be treated in isolation. Of course, if thediscontinuities are close enough, the fields at each discontinuity willinteract. In addition to taking the HOM's into account, this thesis willstudy the effect of closely spaced discontinuities.


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In 1944, Whinnery and Jamieson showed that the effect of a single dis-8

continuity at a given frequency can be represented in an equivalent circuitby a lumped capacitor located at the discontinuity [I], [ 2 ] . This givesthe engineer a relatively simply way of taking discontinuities into accountduring the analysis and design of transmission systems. There are, however,limitations to Whinnery and Jamieson's approach:

(1) Slowly converging series are used in the calculations.

(2) Finding equivalent circuits for two discontinuities closely spacedis a fairly complicated analytic procedure.

(3) Analyzing a line with more than two discontinuities closely spacedwould be prohibitively complicated.

More recently [7], a technique using the conservation of complex powerhas been applied to a single step discontinuity. This approach avoids theneed for using slowly converging series. tiowever, like the method ofWhinnery and Jamieson, it can only be applied to a single discontinuity.In Chapter 2, a technique is outlined that does not require the use ofseries and can, in theory, handle any number of closely spaced discontin-uities.

-As an example of applying the multiple discontinuity analysis, a coaxialquarter-wave transformer is considered in Chapter 3. A comparison is m d e


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between the simple quarter-wave transformer model and the actual situationwhen the HOM's are taken into account. In addition, the question of whethershaping the two impedance steps using closely spaced discontinuities canreduce the influence of the HOM's is considered.


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CHAPTER 2

DEVELOPMENT OF MULTIPLE-DISCONTINUITY ANALYSIS

2.0. I n t r o d u c t i o n

In t h i s c h a pt e r a t e c h ni q u e i s d e ve l op e d f o r t h e a n a l y s i s o f a g u i d e o rl i n e w it h m u l ti p l e d i s c o n t i n u i t i e s , T hes e r e s u l t s are t h en a p p l i e d t o t h ep a r a l l e l - p l a t e c a s e and t h e c o a x i a l c a s e. F o r v a l i d a t i o n of t h e p a r a l l e l -p l a t e c a s e , a numerica l compar ison i s made between th e tech niqu e give n h er ea nd p r e v i o u s l y p u b l is h e d r e s u l t s .

The a pp ro ac h g i v e n h e r e i n v o l v e s d e f i n i n g a m a t r i x r e l a t i n g t h e modesa t one s i d e of a s i n g l e d i s c o n t i n u i t y w it h t h e modes a t t h e o t h e r s i d e . Ana d d i t i o n a l m a tr i x i s u se d t o r e l a t e t h e modes a t t h e o u t p u t of a s e c t i o n ofa un i form gu ide t o t he modes a t t he i npu t . These ma t r i c e s c an t he n be manip-u l a t e d t o o b t a in a r e l a t i o n s h i p b et wee n t h e i n c i d e n t modes a t o ne s i d e o ft h e r e g i on wi t h m u l t i p l e d i s c o n t i n u i t i e s and t h e t r a n s m i t modes a t t h e o t h e rs i d e . This t echnique can be cons idered a g e n e r a l i z a t i o n of t h e t r a n s m i s s i o nm a t r i x a n a l y s i s t e c h n i q u e f o r s i n g l e waves ( s e e f o r e xa mp le [ 6 ] ) t o i n c l u d emany hi gh er -o rd er modes.

2.1. Ana lys i sThe development that fo l l ows w i l l be based on th e fo l lowing assumpt ions :

1) The guide i s a p e r f e c t c onduc to r .


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2 ) T h e coordinate z i s parallel to the direction of propagation.

3 ) The discontinuities occur only in z=constant planes.

4 ) Only a finite number of modes are needed to adequately specify thefields.

5 ) For a two-conductor line, o n e of the conductors remains flat throughthe discontinuity region.

T h e transverse components of the fields can be written in general as(see [ 9 ] for example)

where

$ satisfies the wave equation2 2 2vtJI + (Y, + B ) J I = 0

and $ vanishes on the conductor.


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At t h e d i s c o n t i n u i t y , t h e f o l l o w i n g b ou nd a ry c o n d i t i o n s must be s a t i s -

- -* t r , l H tr 2 o v e r sa (3b)

S r e f e r s t o t h e a p e r t u r e a r e a and S r e f e r s t o t h e co nd uc to r a r e a i na ct h e pl a ne of t h e d i s c o n t i n u i t y . ( see F igu re 2 .1 . )

U si ng t h e s e c o n d i t i o n s , t h e o r t h o go n a l p r o p e r t i e s o f e and ( I ) , thenmodes a t one s i d e o f t h e d i s c o n t i n u i t y can b e e x p r e s s e d as a l i n ea r combi-n a t i o n o f t h e modes o n t h e o t h e r s i d e . I n m a t r i x n o t a t i o n , a ss um in g t h et r an sv er se d imens ions o f R egion 1 t o be l a r g e r t h a n t h a t of R eg io n 2 ,

where t he matr ix e l e m e n t s a r e

T h e s u b s c r i p t s i n (5 ) r e f e r t o a p e r t u r e a r e a ( " a ") and t h e a pe r t u r z p l u sc o n d u c t o r a r e a ( "a+c") .


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Figure 2.1Cross section area and region definitions.


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The inner produc t used i s

and the norm i s g ive n by- 2 --Ilf II =

I t w i l l b e c on v en i en t t o d e f i n e a m a t r i x M c o n t a i n i n g b o t h t h e E - f i e l dand H-fie l d modes such th a t

Then ( 4 ) c a n b e w r i t t e n a s

w her e t h e d i s c o n t i n u i t y m a t r i x i s g i v e n by

For th e case where th e t ra ns ve rs e d imens ions of Region I a r e s m a l l e rthan tha t of Region 2 , ( 4 ) and ( 5 ) a r e no l o n g e r v a l i d . T he se e q u a t i o n sw i l l b e v a l i d i f t h e s u b s c r i p t s 1 and 2 a r e i n t e r c h a n g e d . S y m b ol i c a ll y , i fwe wri te D ( 1 $ 2 ) t o e x p r es s t h e d i s c o n t i n u i t y m a t r i x f o r t h e c a s e vhen>Region 1 i s l a r g e r t h a n R e g i o n 2 and D (1 ,2) f o r th e cas e when Region 1 i s

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