Field theory analysis and design of circular waveguide components

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FIELD THEORY ANALYSIS AND DESIGN

OF CIRCULAR WAVEGUIDE COMPONENTS

by UMA B A LA JI

B.E.. M adurai K am araj University. M ad urai. 1985 M .E .. A nna U niversity. M adras. 1987

A D issertation S ub m itted in P artial Fulfillm ent of th e R equirem ents for th e Degree of

D O C T O R O F PH IL O SO PH Y in th e

D ep artm en t of E lectrical a n d C o m p u ter Engineering We a c ^ p t th is dissertatio n as conform ing

Squired stan d ard

Dr. R ^ a h f d ie c k . Supervisor. D ept, of E lect. & C om p. Engg.

_________________________________

Dr. J. B ornem ^nn. D e ^ r tm e n ta l M em ber. D ept, of E lect. &: C om p. Engg.

---Dr. VV. J . R. Hoefer. D ep artm en tal M em ber. D ept, of Elect. &: C om p. Engg. Dr. S. D ost. O utside M em ber. Dept, of Mech. Engg.

Dr. J. Uher. E x tern a l E xam iner. Spar .A.erospace L im ited. C an ad a © UMA B A LA JI. 199-

U niversity of V ictoria

A ll rights reserved. This dissertation m ay not be reproduced in whole or in part by photocopy or other means, without the perm ission o f the author.

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11

S upervisor: D r. R. V'ahldieck

ABSTRACT

R F / m icrow ave te rre stria l p o in t-to -p o in t and s a te llite co m m u n icatio n system s e m ploy a large n u m b e r of w aveguide co m p o n en ts o p e ra tin g a t m icrowave and m illim eter wave frequencies, .\c c u ra te design of these co m p o n en ts for o p tim u m perform ance of th e overall sy stem is critical. To achieve th is, co m p u tatio n ally efficient and ac c u ra te n u m e ric al m eth o d s are indispensable tools in th e design and o p tim iz a tio n of co m p o n en ts a n d su bsystem s.

.Among th e large variety of p o te n tia lly su ita b le num erical m eth o d s, th e m ode m a tc h in g m e th o d in con ju n ctio n w ith th e generalized sca tte rin g m a trix technique has been found to be one of th e m ost reliable and straig h tfo rw ard techniques to d esign w aveguide com ponents. In th e past th e m eth o d has been m ain ly applied to eigenvalue a n d sca tte rin g problem s in re c tan g u la r waveguides. In this thesis, the m ode m a tc h in g m e th o d is ex ten d e d to ridge w aveguide problem s in circular waveguides and th u s closes a g ap in th e lite ra tu re th a t has ex isted for a long tim e.

T h e th esis begins w ith a stu d y of th e basic principles of th e m ode m atch in g m eth o d as it is know n from re c tan g u la r waveguides. T hese principles are th e n applied to th e an aly sis of th e re c tan g u la r ridged and coaxial w aveguide, followed by an eigenvalue an aly sis of rid g ed circu lar waveguide. R ath er th a n rec tan g u la r ridges, ridges of un i form an g u lar thickness(conically shaped) are used in th e circular w aveguide to avoid a m ix ed c o o rd in a te sy stem which would ren d er th e m ode m atching m eth o d com pu ta tio n a lly very inefficient. On th e o th e r han d , conically shaped ridges are as easy to fa b ric a te as re c ta n g u la r ridges and are not d e trim e n ta l to th e electrical perform ance of th e c o m p o n en t.

T h e thesis th e n continues to tre a t th e d isco n tin u ity problem at th e interface be tw een th e e m p ty circu lar waveguide and ridged circu lar waveguide. To verify th e c o m p u te d s c a tte rin g p a ra m ete rs, m easu rem en ts were perform ed and good agreem ent

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I l l

was found. By cascading several discontinuities tran sfo rm ers and evanescent m ode filters were designed. fifth o rd e r filter Wcis designed and fa b ricated and also here

good agreem ent betw een m easured and com p u ted d a ta was found.

T h e final c h a p te r in the thesis analyses th e coupling betw een orthogonal m odes in th e presence o f an asym m etric discontinuity. D eterm in in g th e coupling factor betw een o rth o g o n al m odes is an integral p a rt of the design of polarizers and duéd m ode filters a n d . for conically sh ap ed ridges, has not been published in th e open lite ra tu re yet. To realize various coupling coefficients, a single o r double ridges m ust be placed at a n a rb itra ry angle to th e exciting wave. T he m o d e m atching m e th o d is ex ten d ed to in clu d e also th is case and various convergence te sts have been perform ed to validate th e alg o rith m . As a final exam ple, th e algorithm has been ap p lied to design a circu lar p o larizer w ith two ridges.

.Although only tw o-port problem s are tre a te d in this thesis, th e basic fram ew ork for th e m ode m a tc h in g m ethod in circuleir w aveguide has b een estab lish ed an d can now be ex ten d e d to three-port problem s. This will be th e su b je c t of fu tu re work to an aly ze and design power d iv id e s and ortho m o d e tran sd u cers.

E xam iners:

Dr. R. V a l^ ie c k . Supervisor. D ept, o f E lect. & C om p. Engg.

______________

Dr. J. B o rn em an n . D ep artm en tal M em ber. D ep t, of E lect. & C om p. Engg. Dr. W. J. R. Hoefer. D ep artm en tal M em ber, D ep t, of E lect. & C om p. Engg.

D r. S. D ost, O utside M em ber, D ep t, of M echanical Engg.

___________

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IV

List of Tables

2.1 Eigenvalues of single ridged rectan g u lar waveguide of d im en sio n a = I9 m m . u= 9.5m m . q = l.7 m m and t = 0 . I 5 m m ... 32 2.2 C haracteristic im pedance of R C L ... 35 2.3 C utoff frequencies of TEM cell o f dim ensions a = 6 .1 m. b = 7 .3 m , a2 = 4 .0 6 m .

b2=0.157cm 38

4.1 C alculated and m easured S p aram eters (radius of in p u t/o u tp u t

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V l l l

List of Figures

1.1 T h re e resonator evanescent m ode circular waveguide filter ... 11

1.2 C irc u la r ridged waveguide t r a n s f o r m e r ... 12

1.3 .A. tw o-pole dual-m ode f i l t e r ... 13

1.4 .A circu lar waveguide p o la r iz e r ... 14

1.5 (a) Ridged section in a dual-m ode filter (b) M odified ridged sectio n as used in the present work (c) C orrugated section in a circular w aveguide p o larizer (d) R idged section as used in the present work ... 15

2.1 H -plane step d i s c o n t i n u i t y ... 19

2.2 D ouble plane step discontinuity in rectangular w a v e g u i d e s ... 20

2.3 (a)S y m m etric double plane step discontinuity in rectan g u lar w aveguide (b) .Asymmetric double plane step discontinuity in rectan g u lar w aveguide 22 2.4 M ag nitud e of reflection coefficient from a input waveguide of dim en sio n 10.7m m X 4.32m m to a o u tp u t waveguide of dim ension 15.6mm x 7.9m m 28 2.5 M ag n itu d e of transm ission coefficient from a in p u t waveguide of d im e n sion 10.7mm X 4.32m m to a o u tp u t waveguide of dim ension 15.8m m X 7 .9 m m ... 29

2.6 Single ridged re ctan g u lar w a v e g u id e ... 30

2.7 S y m m etric inner conductor step discontinuity in R C L ... 33

2.8 (a)C ross-section of RCL (b) E xpanded view of half th e cross-section of R C L ... 34

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L I S T O F F I G U R E S ix

2.9 C h arac te ristic Im pedance of R C L ' s ... 36

2.10 . \ ju n c tio n betw een two RCL w ith an assu m ed waveguide section in betw een t h e m ... 40

2.11 .M agnitude of reflection coefficient of cascaded in n er co n d u cto r step d isco n tin u ity in RCL (50 O hm RCL in p u t/o u tp u t to 25 O hm o f length 2 0 m m ) ... 41

2.12 M ag n itu de of transm ission coefficient of cascad ed inner co n d u cto r step disco n tin u ity in RCL (50 O hm RCL in p u t/o u tp u t to 25 O hm o f length 2 0 m m ) ... 42

3.1 Single ridged circu lar w a v e g u id e ... 45

3.2 C u to ff ch aracteristics of single ridged circu lar w aveguide (t/D = 0 .0 4 ) . 49 3.3 D ouble ridged circu lar w a v e g u i d e ... 51

3.4 C utoff ch aracteristics of fu n d am en tal m ode of d o uble ridged circu lar w a v e g u id e ... 52

3.5 C utoff ch aracteristics of double ridged circu lar w a v e g u id e ... 54

3.6 T riple ridged C ircu lar w av eg u id e ... 55

3.7 C u to ff ch aracteristics of trip le ridged circular w a v e g u id e ... 56

3.8 (a )Q u a d ru p le ridged circular w aveguide (b) S lo tte d circular w aveguide 57 3.9 C utoff ch aracteristics of q u a d ru p le ridged c irc u lar w a v e g u id e ... 60

3.10 C h arac te ristic im pedance of dou b le ridged c irc u la r waveguide. b = 2 c m . ridge thickness(20)= lO d e g r e e ... 63

3.11 C h arac te ristic im pedance of q u a d ru p le ridged c irc u la r waveguide. b = 2 c m . ridge th ick ness= 10 d e g r e e ... 64

3.12 C h aracteristics im pedance of q u ad ru p le an d d o u b le ridged circu lar waveguide versus ridge d ep th . b = 2 c m . ridge th ic k n e ss= 10 degree . . . 65

4.1 D iscontinuity regions (a) circu lar waveguide (region I ) (b) double ridged c irc u lar waveguide (region I I ) (c) side-view ... 68

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L I S T OF F I G U R E S x

4.2 D iscontinuity regions (a) circular waveguide (region / ) (b) q u a d ru p le ridged circular w aveguide (region I I ) (c) sid e-v iew ... 74 4.3 M ag n itu d e of S1 2 o f a discontinuity from a circu lar w aveguide to a

double ridged circ u lar waveguide. b = 2 m m . a = 1.5m m . f= 55G H z. 6 = o d e g re e ... 76 4.4 S -p aram eters in dB o f a discontinuity from circular w aveguide to a do u

ble ridged circular w aveguide of finite length. b = 4 m m . a = 2 m m . ridge thickness =125/fm (0 = 1 degree). 1= 1.1mm. -I- m easu red . - c a lc u la te d 77 4.5 .\ C hebychev T ran sfo rm er in double ridged circular w aveguide . . . . 78 4.6 Response of an o p tim u m 3-section double ridged c irc u la r w aveguide

transform er, dim ensions in cm. section 1: b=2. a = l .7 . 11 = 1.633. sec tion 2: b = 2 . a= 1 .1 3 . 12=1.351. section 3: b= 2. a = 0 .7 . 13=1.191. sec tion 4: b = 2 a = 0 . 5 79 4.7 Response of an o p tim u m 3-section quad ru p le ridged c ircu lar w aveg

uide tran sfo rm er, dim ensions in cm. section 1: b = 2 . a = 1 .6 2 . 11 = 1.519. section 2: b= 2. a = 1 .0 9 . 12=1.404. section 3: b = 2 . a= 0 .6 S . 13=1.363. section 4: b = 2 a = 0 . 5 ... SO 4.8 .\ q u ad ru p le ridged c ircu lar waveguide transform er w ith ta p e re d o u te r

circu lar waveguide h o u s i n g ... 81 4.9 R esponse of an o p tim u m 3-section q u ad ru p le ridged c irc u la r w aveguide

transform er, dim ensions in cm. section 1: b=2. a = 1 .7 . 11 = 1.162. sec tion 2: b = 1.95. a = 1 .4 5 . 12=1.474. section 3: b = 1 .9 . a = 1 .2 5 . 13=1.59. section 4: b=1.85 a = 1 . 2 ... 81 4.10 T hree section evanescent m ode circular waveguide f i l t e r ... 82 4.11 C ircu lar waveguide sy m m etric step d is c o n tin u ity ... S3

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L I S T O F F I G U R E S xi

4.12 C h aracteristic of a single reso n ato r evanescent m ode filter. R adius of input and o u tp u t w aveguide= 4m m . rad iu s of evanescent waveg- u id e= 2 m m . d e p th of rid g e = I.6 m m . thickness of ridge d = ôdeg, length of evanescent sectio n s= 2 m m . (a):length of re so n a to r= 3 m m (b ):len g th of r e s o n a t o r = 2 m m ... So 4.13 C om puted response of a 3-reso n ato r evanescent m ode c ircu lar waveg

uide filter, dim ensions in m m . radius of in p u t/o u tp u t se c tio n = 4 . ra dius of evanescent s e c tio n = b = 2 . a= 0 .4 . 6 = Ideg. Ie l= le 4 = 1 .1 6 6 . I r l= lr 3 = 1.595. Ie2=le3=4.396. lr2 = 1 .7 S ... ST 4.14 Passband response of a 3-reso n ato r evanescent m ode c irc u lar wave

guide filter, dim ensions in m m . radius of in p u t/o u tp u t se c tio n = 4 . ra dius of evanescent sec tio n = b = 1 .9 7 . a = 0 .4 . 9 = 5.5deg. Ie l= le 4 = 1 .2 . lrl= lr3 = 1 .5 S . Ie2= le3= 4.4. lr2=1.7S. dashed lines('- m easured in sertion loss, '-t-' m easured re tu rn loss, solid lin es-co m p u ted response . SS 4.15 W ideband response of th e 3-resonator evanescent m ode c irc u lar wave

guide filter in Figure 4.14. d ash ed lines(‘- -')-m ea su red response, solid lines-com puted r e s p o n s e ... 89 4.16 Passband response of an o p tim u m 5-reso n ato r evanescent m ode c ir

cu lar waveguide filter, dim ensions in m m , rad iu s of in p u t/o u tp u t sec- tio n = 4 . radius of evanescent s e c tio n = b = 2 . a = 0 .4 . 9 = 5.5deg. l e i = .3 6 . lrl = 1.07. Ie2=3.09. Ir2=1.74. Ie3=3.36, lr3 = 1 .7 4 . solid lines-com puted response, d ashed lines ('- -*)-measured response ... 90 4.17 W ideband response of th e 5-resonator evanescent m ode circ u lar wave

guide filter, dim ensions in m m . radius of in p u t/o u tp u t se c tio n = 4 . ra dius of evanescent se c tio n = b = 2 . a= 0.4. 9 = 5.5deg. le l= .3 6 , lrl = 1.07. Ie2=3.09. Ir2=1.74. Ie3=3.36. Ir3=1.74. solid lines-com puted response, dashed lines('- - ^-measured r e s p o n s e ... 91

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L I S T OF F I G U R E S xii

5.1 ( .A.)Discontinuity regions I A: II o f th e ridged c ircu lar w aveguide section

of finite length (B ) Subregions I A: 2 of section I I ... 95 5.2 Incident and reflected wave a m p litu d es of a ortho g o n al p o la riz atio n

coupling n e t w o r k ... 96 5.3 Convergence of th e m agnitude of 5i2jc at a d isco n tin u ity from a c ircu lar

waveguide to a single ridged c ircu lar waveguide versus th e n u m b e r of

T E an d T \ I m odes of both polarizations. b = l.O c m . a = 0 .4 c m . o =

45deg, 6 = 5deg, 1=1.0m m . f = 1 0 G H z ... 104 5.4 M agnitude of S-param eters of a discontinuity from circu lar w aveguide

to single ridged circu lar waveguide of finite len gth . b = 1.0cm. a = 0 .4 c m . Q = 45deg. 9 = 5deg. 1=1.0m m ... 105 5.5 M agnitude of S-param eters of a discontinuity from circular w aveguide

to single ridged circu lar waveguide of finite length. b = 1.0cm. a = 0 .4 c m . o = 45deg. 9 = 5deg. 1= 1.0mm 105 5.6 M agnitude of S-param eters of a discontinuity from circular w aveguide

to single ridged circu lar waveguide of finite length. b = 1.0cm. a = 0 .4 c m .

9 = 5deg. 1=1.0 m m ... 107 5.7 M agnitude of S-param eters of a discontinuity from circu lar w aveguide

to single ridged circu lar waveguide of finite length. b = 1.0cm. a = 0 .4 c m . o = 30deg. 9 = 5deg. 1=1.0m m ... 107 5.8 M agnitude of S -param eters of a discontinuity from circular w aveguide

to single ridged circu lar waveguide of finite length, b = 1.0cm. a = 0 .4 c m . Q = 60deg. 9 = 5deg, 1=1.0m m ... 108 5.9 M agnitude of 5i25c of a discontinuity from circular w aveguide to single

ridged circular waveguide of finite length. b = 1.0cm. o = 45deg, 9 = 5deg. 1=1.0m m . for various values of a ' ... 108 5.10 (.A) D iscontinuity regions I A: II of th e double ridged circu lar w aveguide

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L I S T O F F I G U R E S xiii

5.11 M ag n itu de of 5i25c of a d isco n tin u ity from c irc u la r w aveguide to dou ble ridged circu lar waveguide of finite length. b = lc m . a = 0 .5 c m . a = 45deg. 9 = 20deg. 1=1.0m m , f= 10G H z ... 112 5.12 M agnitude of S -param eters of a d isco n tin u ity from c ircu lar w aveguide

to double ridged circular w aveguide of finite le n g th . b = lc m . a = 0 .5 cm . Q = 45deg. 9 = lOdeg. 1=1.0 m m ... 113 5.13 R e tu rn Loss of orthogonal po larization s of th e differential p h a se shifter

u n it of a polarizer D imension in cm . section I an d section 7: a= 0 .9 . b = 1 .0 . 11 = 1.338. section 2 a n d section 6: a = 0 .7 b= I.O . 12=1.025. sec tio n 3 and section 5: a= 0.56. b = 1 .0 . 13=1.118. section 4: a = 0 .5 . b=I.O . 14=3.680 ... 114 5.14 P h ase difference between th e o rthogonal p o larizatio n s of th e differential

p h ase shifter u n it of the above polarizer ... 115 6.1 .A. sep tu m p o l a r i z e r ... 119 6.2 Irises for dual m ode filters (a) R ectan gu lar iris (b) and (c) S ectoral irises 120 C .l C ascading tw o-port scatterin g m a t r i c e s ... 132 C.2 S c a tte rin g m a trix of a d isco n tin u ity of finite l e n g t h ... 133 C.3 S c a tte rin g m a trix of a d isco n tin u ity followed by a guide o f electrical

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XIV

A c k n o w le d g e m e n ts

I am very grateful to m y supervisor. Dr. R. Vahldieck of the D e p artm en t of Electrical a n d C o m p u te r Engineering, for th e co n stan t encouragem ent, valuable sug gestions a n d advice as well as continuous su p p o rt th a t he provided du rin g th e en tire d u ratio n of th is stu d y a n d research. His rem arkable patien ce and th e confidence he has expressed in me have m ad e it possible for m e to co m p lete th is m ajo r u n d e rta k in g successfully.

I would like to express m y acknow ledgem ent an d g ra titu d e s to the C an ad ian C om m onw ealth Scholarship a n d Fellowship p lan and its various a d m in istra to rs du rin g th e period of m y s tu d y for providing the financial assistance en ab lin g me to u n d e rta k e this work in th e U niversity of V ictoria. C an ad a. I also wish to acknowledge m y g ra titu d e s to th e G o v ern m ent of In d ia for nom inating me cis a c a n d id a te for this scholarship.

My special th a n k s to all th e m em bers of th e exam in in g co m m ittee for reviewing my d isse rta tio n and d e term in in g its value to th e field. I would also like to place on record my g ra titu d e s to th e m any researchers and c o n trib u to rs in th e field relevant to this work, from whose research p ap ers and books I have gained considerably and have been ab le to fu rth e r improve my u n d e rstan d in g of th e subject for com pleting this work.

word of g ra titu d e to all my friends a t the U niversity of V ictoria for th e ir encour agem ent a n d help du rin g th e course of m y study. My special than k s to th e m em bers of LLiMiC g ro u p for th e ir valuable c ritiq u e , discussions a n d support.

Finally. I would like to th an k my h u sb an d and my d a u g h te r for th e ir patience, forbearance a n d encouragem ent in th e p u rsu it of m y academ ic and career goals. I also wish to th a n k my m other-in-law a n d my m o th er for th e ir su p p o rt an d finally for th e en cou rag em en t a n d support th a t I received from m y fath er and father-in-law . who are b o th no m ore b u t whose blessings. I am sure, are always with m e.

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Chapter 1 Introduction

A dvances in th e field of telecom m unications have been very ra p id a n d su b sta n tia l in recent years. W ireless com m unication is progressing a t geom etric p ro p o rtio n s in re sponse to th e m assive inform ation exchange th a t is taking place globally. T h e active role of radio frequency and microwave com ponents in th e teleco m m u n ication in d u stry has d irectly lead to th e dem ands for m in iatu rized com ponents w ith im proved perfor m ance. This is p articu larly so in satellite system s w here specifications are strin g en t and com pactness and light-weights are assets. However, sm aller com p o n en ts satis fying th e specifications with great accuracy require precision in m a n u fa ctu rin g , thus increasing the in v estm ent in fabrication greatly. C onsequently, th e m icrow ave indus try relies greatly on efficient co m p u ter aided analysis and design tools th a t elim in ate th e costly and tim e consum ing cut and try pro ced u re in the fa b ric a tio n . In ad d itio n , th e c o m p u te r aided design process should be cap ab le of han d lin g o p tim iz a tio n , as a m easu re of m eeting th e stringent specifications.

T h e m icrow ave com ponents can be either passive or activ e. T h e present work focuses on passive com ponents only and therefore th e discussions here are lim ited to issues related to th e m . A. couple of decades ago, passive m icrow ave com p o n en ts such as filters were first designed on th e basis of low frequency p ro to ty p es. T he m icrowave filter was then realized by replacing th e lum ped elem en ts in th e m using d istrib u te d elem en ts ev alu ated from the equivalent circuit ap p ro x im atio n s. Som e of th e com

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m ercially av ailab le software packages like T ouchstone. Spice a n d S u p e r-co m p a c t still use this th eo ry for the design of com ponents. .Although slowly, d e p en d in g on the application, h eld -th eo ry based m ethods replace netw ork ap pro ach es. In m icro strip and sim ilar technology, where th e electrom agnetic held b eh av io r is q u a sis ta tic , the equivalent circ u it ap p ro x im atio n s are valid over a broad frequency b an d . For such cases, the b e h av io u r of th e m icrow ave realization is som ew hat close to th e expected perform ance. D espite this, a rigorous num erical analysis is s till w orthw hile in order to predict th e e x a c t behavior an d a num ber of th em have been developed recently. .Also, the co m p lex ity of such circu its has increased su b sta n tia lly due to th e m in ia tu r ization. P ro x im ity of such com ponents d u ring system in te g ra tio n can d e g ra d e their perform ance. U nd er such circum stances, num erical analysis of th e electro m ag n etic held is a key to m eet th e design goals.

L nlike for cjuasistatic cases, th e equivalent circuit ap p ro x im atio n for waveguides is valid over a narrow band of frequency alone. Hence, this tech n iq u e of realization of waveguide co m p o n en ts from th e low frequency p ro toty p es no longer m e e ts, the m odern days d e m an d s of a c c u ra te perform ance req u irem en ts. O ver th e last two decades, a v ariety of num erical techniques have been developed for rigorous analysis and design in o rd e r to predict accurately th e ch aracteristics of th e c o m p o n en ts over a wide band of frequency. Every num erical tech n iq u e has u n iq u e features a n d m erits th a t m ake th e m su itab le for c ertain type of com ponents. T h e search for th e best technique in te rm s of efhciency and accuracy has continued to be a challenge. The ch aracteristics of waveguide com ponents are sensitive to d im ensions, and a num erical technique th a t can handle o p tim izatio n of th e in itial design is an ideal c a n d id a te .

T he num erical technique used in this work for th e design a n d analysis of w aveguide com ponents is called th e M ode M atching M ethod (M M M ) an d it was first proposed for th e solution o f rectan g u lar waveguide discontinuities [I]. S u b seq u en tly a num ber of researchers have used this tech n iqu e to design various w aveguide co m p o n en ts. The fact th a t this tech n iq u e lends itself to the ap proach of b u ild in g co m p o n en ts from

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3

th e basic d isco n tin u ities in th e m makes it su itab le for h a n d lin g co m p o n en ts com posed o f m any successive discontinuities. .\lso . an in teg rated sy ste m can be analyzed rigorously once th e com ponents in it have been individually a n aly zed . T h e m ost im p o rta n t fe atu re of th e .VIMM is th a t it is rigorous and yet e n ab le s th e o p tim iz a tio n of th e p a ra m e te rs of th e initial design.

T h e first section in this c h a p te r gives a b rief overview of th e various num erical tech n iq u es th a t can be used to analyze different types of passive m icrow ave com po nents along w ith th e ir m erits and dem erits.

T h is research focuses on th e com puter aided analysis a n d design of c e rta in pas sive co m p o n en ts for a n te n n a feeds in satellite system s. T h e m a in o b jectiv e has been to develop c o m p u te r aided design algorithm s for the design o f co m p o n en ts in circu lar w aveguide technology using a held theoretical approach, .-\n ten n a b eam form ing netw orks in sa te llite system s, however, have recently used re c ta n g u la r coaxial lines (R C L) as an a ttra c tiv e a lte rn a tiv e to waveguides due to th e ir sm aller size. How ever. all co m p o n en ts in th e sate llite systems cann o t afford th e use of RCL as certain co m p o n en ts d em an d for ex am ple rectangular, circular or co ax ial w aveguides, hnlines and m ic ro strip lines for various reasons. Each transm ission line has its ow n d istin ct a d v an tag es and disadvantages and preference of one over a n o th e r d epends u po n th e req u ire m e n ts of th e system .

T h e present work begins w ith a study on th e analysis of so m e of th e well known s tru c tu re s in rec tan g u la r waveguides and RCL. T h e principles o f th is stu d y have later been u tilized to rigorously design com ponents in circular w aveguides w ith good nu m erical efficiency an d using original modeling techniques. S om e o f th e d isco n tin u ities discussed in th is work are not available in the published lite ra tu re an d hence m e asu re m ents have been carried out in o rd er to validate th e num erical alg o rith m s p resen ted . .\ d e ta ile d o u tlin e of th e d issertatio n follows a t th e end of th is c h a p te r. T h e c o n trib u tion of th is work is in th e area of filters, m atching networks a n d polarizers in circu lar w aveguide technology.

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1 .1

N u m e r ic a l te c h n iq u e s - A n o v e r v ie w

E lectric and m ag n etic fields th a t are independent of tim e are called s ta tic fields. T h e basic field problem in electro statics is to solve Poisson s or L aplace’s equation for a p o ten tial function th a t satisfies th e boundary condition. For som e boundary value problem s a d ire c t analytical solution of these is possible while for o th ers a num erical evaluation is th e only approach.

For tim e-varying fields th e electric and m agnetic fields are described by M axw ell's equations:

V x // =

J + ^

(1.1)

V X £ = ( 1 . 2 )

V ■ D = p (1.3)

V B = 0 (1.4)

For a hom ogeneous, isotropic, source free region Xlaxwell’s eq u atio n s can be reform u lated to obtain th e Helmoltz eq uation. T he H elm holtz eq u atio n is of th e form:

V + fc^ c' = 0 ( I..5)

w here k = For sim ple cases, it is possible to d e term in e c’ . th e vector po te n tia l. from w hich the field in a hom ogeneous source free region can be evaluated explicitly. However, for cases where analytic solutions are not possible a num erical evaluation is sought. The num erical techniques them selves can be classified as tim e d om ain and frequency dom ain techniques d epending on w h eth er th e solution is ob ta in e d using sp a tia l d iscretizatio n (frequency dom ain m eth o d ) alone or spatial a n d tim e d iscretizatio n (tim e d o m ain m ethod) of th e M axwelFs eq u atio n . Some of th ese m eth o d s are discussed below.

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F in ite D ifferen ce Frequency D o m a in m e th o d (F D F D ):

T h is m e th o d is one of th e oldest tech n iq u es to solve Maxwells e q u atio n s. T h e p a rtia l differential equations are solved by ap p ro x im atin g th e derivatives of functions o f con tin uo u s variables in space, by fin ite differences using Taylor's series. Thus, th e region inside a specified boundary w h ere th e electrom agnetic field p ro b lem needs to be solved is divided into a re c tan g u la r m esh. T he value of the fu n ctio n a t th e in tersectin g points of the mesh, called th e nodes, are d eriv ed from th e value of th e functions a t th e neighboring points. T h e n um b er of nodes th a t are used to form th e m esh are increased until a convergence in th e value of th e function is observed. tech n iq u e called the Richardsons e x tra p o la tio n can be used to increase th e accuracy of th e solutions. Examples of this m e th o d have been discussed in [2].

In o rd er to get accurate solutions th e num ber of nodes can be large for certain cases. W ith an increase in the num ber of nodes the so lu tio n using finite difference tech n iq u e needs more m em ory space a n d co m p u tatio n tim e . Hence, som e problem s can becom e unwieldy using this m eth o d . B ut this m ethod can handle a rb itra ry and c o m p licated shapes, like curved boundaries using stair step ap p ro x im atio n s. Because of th e general approach to solve problem s, com m ercial packages are available. M.\FI.A. is one such tool to solve electrom agnetic field problems.

F in ite D ifferen ce T im e D om ain m e th o d (F D T D ):

T his m eth o d to solve Maxwells e q u a tio n was first proposed by Yee [3]. Since th e electro m ag n etic fields vary both in tim e and space, in this m eth o d th e p artial deriv ativ es w ith respect to tim e and sp a c e are a p p ro x im ated by finite differences. T h is m e th o d requires a closed region in o rd er to have a finite num ber of nodes. To h an d le problem s involving open b o un d aries special abso rb ing b oundary conditions have been derived. Com plicated g eo m etries can be h andled here too as in FD FD . T h e frequency dom ain response is o b ta in e d from th e tim e dom ain c h arac te ristic s using Fourier transform s. Exam ples of th is m ethod have also been discussed in [2].

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T ra n sm issio n L in e M a trix m e th o d in T im e D o m a in (T D T L M ) and F re q u e n c y D o m a in (F D T L M ):

H ere, the electro m ag n etic field problem is co n v erted to a th re e dim ensional e q u iv alent netw ork p ro b lem based on H uygen s p rin cip le of wave p ro p ag atio n [4]. T h is m e th o d is very g en eral and can h an d le a rb itra rily sh ap ed stru c tu re s an d open b o u n d aries w ith special absorbing b o u n d ary conditions. O nce again h ere too. th e c o m p u ta tio n tim e and m em ory space needed can be large for some problem s.

T h e frequency dom ain TLM m e th o d was proposed by .Jin a n d V ahldieck [5]. T h e sam e d iscretizatio n schem e as in th e TD T L M is used. However, th e re is no need to use a Fourier tran sfo rm to o b ta in th e frequency dom ain response since th e e n tire a lg o rith m works in th e frequency dom ain. In p a rtic u la r, for narrow band pro b lem s, th is m eth o d is faster th a n th e T D T L M .

M e th o d o f M o m e n ts (M oM ):

T h is m ethod is m ore an aly tical [6] when co m p ared to the above th re e m ethods. sum of basis fu nctio n s with unknow n coefficients is used as an a p p ro x im a te so lu tio n to a differential o r integral eq u atio n . By ta k in g th e inner p ro d u c t of th e re su ltin g e q u atio n with a w eighting function, a set of lin ear equations is o b tain ed w hich is solved to get th e ap p ro x im ate solution to th e p ro b lem . The choice of basis fu n ctio n s d ep en d s on th e problem itself an d th e b o u nd ary conditions. T h e w eighting fu n ctio n can be a d elta fu n ctio n , pulse function or th e sam e as th e basis fu n ctio n . W hen th e choice of the w eighting function is th e sam e as th e basis functions, th e m e th o d is also called G alerk in 's m ethod. T h e general M oM is also equivalent to a v ariatio n al m e th o d . .A ju d icio u s choice of th e functions used in this ap p ro ach can m in im ize c o m p u ta tio n tim e an d m em ory an d y et produce num erically a c c u ra te results.

F in ite E le m e n t M e th o d (F E M ):

.A variational form ulation of th e differential eq u atio n for th e specified b o u n d a ry cond itio n is first o b tain ed . T h e region of in te re st is divided in to a n u m b er of subregions called finite elem ents. T h e elem ents can be any polygon. A trian g le is th e

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sim p lest o f th e su rface elem ents but it can a p p ro x im a te very well m ost of the a r b itra ry sh ap ed stru c tu re s. In this sm all elem ent, th e function is ap p ro x im a te d by a p o ly n o m ial with unknow n coefficients. T h e choice on th e o rd er of th e polynom ial is an a d d itio n a l degree of freedom while using this m ethod. R ayleigh-R itz p ro ced u re tra n s form s th e variational form into a linear system of alg eb raic eq u atio n s. T h e system of e q u atio n s is then solved to o b tain th e unknown coefficients of the functions. Using a n u m b e r o f sm all elem en ts w ith higher order polynom ials to a p p ro x im a te th e function, a c c u ra te solutions to field problem s can be obtained using this m e th o d . Eigenvalue analysis o f various s tru c tu re s like finlines in rectan g u lar and circular w aveguide hous ings [7]. ridged c ircu lar waveguides [8.9] have been perform ed using F E M . T h e analysis of re c ta n g u la r w aveguide com ponents using FEM is available in [10.11.12]

Since th e procedure is general and can be used on a rb itra ry sh ap es, com m ercial C.A.D packages like H FSS (High Frequency S tru ctu re Solver from H ew lett Packard) are available. However, m em ory space and co m p u ta tio n al tim e req u ired are high in this m e th o d as well.

O th e r m ethods like integral equ atio n m ethod (lE M ) [13] and b o u n d a ry elem ent m e th o d (B E M ) [14] have been frequently used to solve waveguide d isco n tin u ity prob lems. T h ese m ethods deal w ith th e discretization of th e boundary ra th e r than th e whole region of th e p ro b lem as o th e r discretization approaches do. H ence, they are co m p u ta tio n ally efficient. However, they require m ore a n aly tical effort.

S p e c tr a l D o m a in M e th o d (S D M ):

T his m eth o d is well suited for analysis of p lan ar tran sm issio n lines such as m i cro strip s. finlines an d co p lan ar waveguides [15]. Basis functions a re chosen to a p p ro x im ate th e c u rren t in th e strip s on th e transm ission line. G a lerk in 's m ethod is then used to yield a hom ogeneous system of equations to d eterm in e th e propagation co n sta n ts, current d is trib u tio n and th e ch aracteristic im p ed an ce o f th e transm ission lines. Since th e m e th o d is highly analy tical, th e co m p u ta tio n al efficiency is excellent.

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s

M e th o d o f L in es (M oL):

A lthough th is m ethod was available to solve problem s in th eo retical physics, it was first proposed by Pregia [16] to solve electro m agn etic field problem s. To solve a two dim ensional electrom agnetic field problem , d iscretizatio n of space is ap p lied in one dim ension while analytical form ulations are possible in th e o th e r. This sem iana- lytical procedure saves co m p u tatio n tim e a n d mem ory. In dealing w ith s tru c tu re s like m ultilayered transm ission lines, this m ethod is often used. .\s th e m ethod can handle a rb itra ry shapes, analysis of finlines in circu lar and elliptical w aveguide housings [17.

IS] have been published using th is m ethod.

M o d e M a tch in g M eth o d (M M M ):

As stated earlier, the M ode M atching M ethod was first proposed by VVexler to handle waveguide discontinuity problem s. In this m eth o d th e fields on b o th sides of th e discontinuity are first expanded in te rm s of th e ir respective m odal functions. L sing the c o n tin u ity condition of the field at th e interface o f regions along with application of orthogonality p ro p erty of th e m odal functions, resu lts in a set o f alge braic equations. T h e solution of these algebraic equations yields th e eigenvalues or th e scattering p aram eters of th e fundam ental and higher o rd er m odes. T h e m odal representation of th e fields d irectly provides th e generalized s c a tte rin g m a trix which enables one to ev alu ate the field at any p o in t close to the d isco n tin u ity including th e effects of evanescent higher order modes along w ith th e p ro pag atin g fu nd am en tal m ode. C onsequently, a rigorous analysis of th e com ponents is possible by cascading th e generalized scatterin g m atrices of subsequent discontinuities in close proxim ity. T he technique to cascade individual discontinuities is called th e G eneralized S cat terin g M atrix T echnique (G SM T) [19]. .\n extension of the conventional sca tte rin g m a trix technique, it was introduced by M ittra and Pace.

For an efficient design of com ponents, o p tim iz a tio n is essential. O p tim izatio n of any function can tak e several iteratio n s before th e function converges and therefore it can be tim e consum ing, especially when th e evaluation of th e objective fu n ctio n is

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c o m p u ta tio n ally intensive. But as MMM in conjunction w ith G SM T is a num erically efficient ap p ro ach , o p tim izatio n of com ponents is possible w ithin a rea so n a b le am o u n t of c o m p u ta tio n tim e. However, it should be noted th a t th e .VIMM can handle only those stru c tu re s where th e m odal functions can be expressed an aly tically . T his m eans th a t highly a rb itra ry sh ap ed stru c tu re s cannot be an aly zed using th is m eth o d . On th e o th e r h an d , most waveguide stru c tu re s are of re c tan g u la r or c irc u la r shapes, and therefore explicit expressions for m odal functions can be found an aly tically .

T he equivalent circuit app ro x im atio n of several waveguide d isco n tin u itie s have been discussed in great d e tail by M arcuvitz [20]. Using th ese closed form expressions. M atth aei. Young and Jo n es have presen ted practical designs of m a n y com ponents in [21]. .Accurate design of waveguide com ponents p resen tly relies m o re on MMM. N um erous p ap er quoted in the references have applied MMM to a n a ly z e an d design com ponents such as m atch in g netw orks, filters and polarizers which involve sc a tte rin g problem s in rectan g u lar waveguide technology [22-32]. hnlines [33-3.5]. a n d m icro strip lines [36.37]. The propagation ch aracteristics for a Online s tru c tu re , shielded m i crostrip lines and ridged waveguides have been investigated in [36-40]. However, only little work has been published on th e analysis of com ponents in c irc u la r waveguides using M M M . .Analysis an d design of m ode converters using circular w aveguides have been p resen ted in [41.42]. M etal insert circular w aveguide filter, analo g o u s to the E-plane filter in rectan g u lar waveguide technology, was first developed by Filolie and Vahldieck [43] where th e insert was ap pro x im ated by a bow -tie sh ap ed s tru c tu re . T he present work builds on th is idea to design filters, m a tc h in g netw orks a n d polarizers in ridged c ircu lar waveguide technology. .A m ore d etailed discussion w ill follow in the next section.

In view of th e advantages of MMM. m an y researchers recently have focussed a tte n tion on th e com bination of discretization schemes with M M M [44.45]. T h is schem e has been ap p lied when th ere are rectan g u lar elem ents in a c ircu lar housing o r. in general, to problem s which involve m ixed co o rd in ate system . For a co m b in atio n of FE M and

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10

M M M or FD and MMM. th e eigenvalue analysis of the a rb itra rily sh ap e d s tru c tu re is done using FEM or FD. Subsequently, th e discontinuity from a s tru c tu re th a t is well su ite d for a MMM analysis is m ade possible by a com bination of tech n iqu es. Since th e s tru c tu re th a t is discretized is sandw iched on eith er sides by a region w here fields can be expressed as m odes, ap p licatio n of th e m atching condition a n d o rth o g o n a lity of m odes results in integrals w hich can be perform ed as su m m a tio n . T h erefo re, th e m o d al rep resen tatio n gets d irectly in co rp o rated and a generalized s c a tte rin g m a trix is o b ta in e d . Such hybrid techniques are a ttra c tiv e due to th e ab ility to h an d le s tru c tu re s th a t are. a t least to som e degree, a rb itra rily shaped. In th is work, how ever, only such stru c tu re s are investigated and designed for which th e MMM an aly sis can be used entirely. .M easurem ents have been carried out to validate th e th e o ry proposed.

1 .2

O r g a n is a tio n o f t h e d is s e r t a t io n

C h a p te r 2 begins with th e stu d y of M M M . .A full-wave analysis of d o u b le plane ste p re c ta n g u la r waveguide d isco n tin u ity is first discussed as an e x am p le. Eigenvalue an aly sis in ridged rectan g u lar waveguides an d rectangular coaxial line have been p erfo rm ed . .A step discon tin u ity in re c tan g u la r coaxial line is also an aly zed and co m p ared w ith th e lite ra tu re .

C h a p te r 3 discusses th e eigenvalue analysis in ridged circu lar w aveguides using M M M . T h e analysis of one to four ridges placed equiangularly has been analyzed a n d co m p ared w ith th e lite ra tu re . T h e ridges have been sh ap ed conically in o rd er to fit th e cylin d rical coordinate system . It is w orth m entioning here th a t it is possible to fa b ric a te such shapes as well. Besides, a com parison w ith lite ra tu re shows th a t for v er\' th in ridges, it is possible to ap p ro x im a te the re c tan g u la r cross-section as conically sh ap ed ones in th e analysis.

T h e eigenvalue solutions are necessary before the sca tte rin g p a ra m e te rs of a dis c o n tin u ity can be determ ined . Using th e resu lts from C h ap ter 3. a d isc o n tin u ity from

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11

F ig u re 1.1: T h ree resonator evanescent m ode circu lar w aveguide filter

an e m p ty circular w aveguide to a ridged circular w aveguide has been analyzed in C h a p te r -I. Based on this analysis, evanescent m ode circu lar w aveguide filters using ridged c irc u la r w aveguide resonators have been designed, fab ricated an d m easured. .Also ridged circu lar waveguide transform ers have been designed. .All designs have been o p tim iz e d . Figures 1.1 and 1.2 show an evanescent mode filter and a d o u b le ridged c irc u la r w aveguide transform er, respectively.

C irc u la r w aveguides can support orthogonal fu n d am en tal p o larizatio n . .A sym m et ric s tru c tu re s couple energy from one polarization to th e other. Such stru c tu re s are an essen tial featu re in dual m ode filters and polarizers. .A typical d u al m ode filter and a p o larizer are show n in Figures 1.3 and 1.4. T h e d u al mode filter has a coupling

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12

Figure 1.2; Circular ridged waveguide tran sform er

screw placed asym m etrically (usually at 45 degree) to th e excitation. T h e re are two o th e r screws called the tu n in g screws placed horizontally and vertically. T hese screws are needed to tu n e the filter in order to overcome th e discrepancies th a t can occur in practise. However th e tu n in g screws can be e lim in ated if an accurate design is available for practical realization. Such filters, without th e use of tuning screws, have been designed using a combination of F E M and M MM [45]. T h e hybrid approach was needed because the coupling screw was simulated as rectangular post. C h a p te r 5 deals w ith th e analysis of discontinuities where the ridges in th e circular waveguides have been placed asym m etrically to th e excitation as in th e case of a dual m o d e filter.

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13

f - r

Figure 1.3: two-pole dual-m ode filter

.A.S the ridges have been shaped conically, a rigorous analysis of a discontinuity b e tween e m p ty circular waveguide and single ridged or do uble ridged c ircular waveguide oriented at any a rb itra ry angle has been derived using M M M . The ridged region of a dual mode filter and polarizer shown in Figures 1.3 and 1.4 has been modified to th e one as shown in Figure 1.5 for the analysis in the present work. T he conclusions a n d recommendations for fu rth e r work has been presented in C h a p te r 6.

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u

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15

(a) (b)

(c) (d )

F igu re 1.5; (a) Ridged section in a dual-mode filter (b) Modified ridged section as used in the present work (c) C o rru g a te d section in a circular w aveguide polarizer (d) R idged section as used in th e present work

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16

Chapter 2 Analysis of Rectangular Cocixial

Lines Using Mode Matching

Method

2 .1

I n tr o d u c t io n

T h e R ectangular Coaxial Line (RCL) has been used in beam forming networks [46] as a n altern ative to th e rectangular waveguide. T h e RCL has reduced dimensions c o m p ared to the rectangular waveguides which makes it attra ctiv e for satellite appli cations. T h e power handling capability and loss characteristics of RCL are com p ara ble to th a t of circular coaxial line. Besides, m anufactu rin g com ponents using RCL is m o re economical com pared to that in circular coaxial lines. C o m p u te r aided design of RCL components is im portant to fu rth er reduce m anufacturing cost and to reduce th e design-m anufacturing cycle.

This chapter begins with a description of the M ode M atching M ethod cis the "engine" for the analysis of a variety of discontinuities utilized in the design of filters, polarizers and beam forming networks. T h e double step discontinuity in rectangular waveguides is analyzed first. Since, th e Mode M atching M ethod analysis of step discontinuities in rectangular coaxial lines is an extension to this ty pe of discontinuity it will be discussed second. The m ode m atching analysis of com ponents in circular

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17

waveguides discussed in th e later ch ap te rs can be d erived straightforw ardly from discontinuities in rectang ular waveguides.

In o rd e r to characterize discontinuities using the M ode M atching M ethod, th e eigenvalues of the cross-section left and right from th e discontinuity plane must be known. T h e eigenvalues of th e RCL cross-section cannot be w ritten an alytically as in a rectang ular waveguide or a circular coaxial line. Hence, in o rd e r to analyze discontinuities in the rectang ular coaxial lines an eigenvalue analysis of the cross- section has to be done first. There are four possible sy m m e trie s in th e rectangu lar coaxial lines cross-section. For a p a rtic u la r sym m etry of RCL. th e s tr u c tu r e is t h a t of a single ridged re c tan g u la r waveguide which has been analyzed in th e lite ra tu re before. This sy m m e try is discussed first, followed by th e eigenvalue analysis of a RCL for various o th e r sym m etries. The Transverse Electrom agnetic cell(T E M cell) which is used in electrom agnetic interference a n d co m patibility tests has th e cross-section of a RCL with very th in inner conductor. The eigenvalues of th e RCL o b ta in e d using the M MM are co m p ared with the lite ra tu re for a T E M cell. T h e c h aracteristic im pedance of the RCL is an o th er im p o rta n t param eter for th e discontinuity analysis and has been obtained in this chapter using the Mode M atching M eth o d and th e finite difference m ethod. Finally, a step discontinuity in th e RCL has been analysed and com pared with m easurem ents from th e literature.

2 .2

F u ll w a v e a n a ly s is o f a d o u b le s t e p d is c o n t i

n u it y in r e c t a n g u la r w a v e g u id e

T he analysis of any discontinuity using th e MMM involves th e following steps.

1. T he fields on both sides of the discontinuity are e x p a n d e d in te rm s of a series of modes of incident a nd reflected waves.

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IS

3. The c o n tin u ity conditions for th e tangential com ponents of th e electric and m ag netic fields are imposed.

4. Using t h e principle of orthogonality of modes, th e eq uations of the continuity conditions are transform ed into matrices relatin g the expansion coefficients of incident a n d reflected waves at the discontinuity.

•5. The m a tric e s are rearranged and inverted s u ita b ly to o b ta in th e generalized s c a tte rin g m a trix which describes the discontinuity in te rm s of th e d o m in an t and h ig h e r order modes.

Theoretically, the generalized scattering m a trix is of infinite dim ensions corre sponding to th e infinite n um ber of eigenmodes. T h e m a trix is t r u n c a te d to a finite size for num erical calculations. Hence a larger m a tr ix has to b e inverted if more modes are in clu ded in th e analysis. The test for th e m a x im u m n u m b e r of modes to be included for th e most accu rate result or. in o th e r words, a te st of convergence of th e ev alu a te d sca ttering param eters is hence an im p o r ta n t p a rt o f every analysis using the M M M . Since there is a choice on the n u m b e r of modes included in th e two or more regions of the discontinuity, the best possible ratio betw een the n u m b e r of modes also has to be identified. This phenomenon of relative convergence has been extensively s tu d ie d in th e lite ra tu re [47] and a fu r th e r discussion will be presented later at the e n d of this section.

.\ fact t h a t is w orth observing before beginning a n analysis of a d iscontinuity is to find if th e fu n d a m e n tal mode th a t is excited in th e discontinuity couples energy to all the h ig h e r order modes included in the analysis. If the fu n d a m e n ta l m ode th a t is ex cited in the discontinuity does not couple energy to so m e of the higher order modes, then such modes can be eliminated from the analysis. By including these modes, however, th e scattering parameters of th e fun d a m e n tal m ode do not get altered. This certain ly involves more com putational effort in c alc u la tin g the elem ents of the m a tric e s or inverting th em . On the other h a n d , a t certain discontinuities, it

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19

(a)

z=Ol (b»

Figure 2.1: H-plane step discontinuity

is so m etim es necessary to include higher ord er modes t h a t are not necessary for the analysis of th a t discontinuity. This is so because at subsequent discontinuities these higher o rd e r modes may get excited and. if the two types of discontinuities are in close p rox im ity to each other, all modes th a t can possibly be excited will interact and affect th e perform ance of the component. . \ n example of such a situation is discussed in C h a p te r 4. .A. knowledge of which modes must be included in the analysis can be derived from the sym m etries of a discontinuity and the s y m m e try in th e fundam ental m ode th a t is incident. For instance, at a H-plane discontinuity, as shown in Figure 2.1. the fu n d a m e n ta l mode { T Ei^q mode) excitation will n ot excite modes o th e r than th e T Em.o modes (where m = 1.2.3. At a discontinuity th a t has no sym m etries w hatsoever, it is essential to include all th e higher order modes in the analysis and check for convergence to get the accurate values of the S-param eters.

T h e do ub le plane step discontinuity as shown in Figure 2.2 is a sym m etric one. T h e incident fu ndam ental TE\_q mode will excite only m odes with m agnetic wall and electric wall s y m m e try as shown in Figure 2.2. This implies th a t it is sufficient to include only modes with such sym m etry nam ely the TEm.n and T \Im.n w ith m odd and n even. However, at an asymm etric step shown in Figure 2.3(b) all th e modes have to be included in the m ode matching analysis. The analysis of th e double plane step discontinuity is well known and has already been been presented in a n u m b e r of

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20

mw ew

Figure 2.2: Double plane step discontinuity in rectan gu lar waveguides papers. [23.24]. It has been repeated here for introducing th e principles of MMM.

T he full-wave analysis presented in the following is based on the m atching of th e tangential field components at th e discontinuity. The fields are derived from the superposition of th e electric and m agnetic vector potentials. T h e two potential vectors are represented in term s of mode functions that are transverse electric { T and transverse m agn etic ( T M ^ ^ ) to eith er the z coordinate (usually the direction of wave propagation) or th e .\ [ T and „) or y ( T „ and T.V/^ „) coordinates. T h e alternative field descriptions o ther th a n the one in propagation direction is necessary e ith e r for im proving the efficiency of the m ethod or for analyzing certain types of discontinuities. However, in the following analysis the z-com ponent of the vector potentials is used to describe the field components as this is m o re ap p ro p ria te for the discussions in la te r chapters.

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21

T h e six field com ponents are ob tain ed from the superposition of th e potential —»(&) —f(^) vectors l' . and t ’ . . — » — t.i'*) 1 —».(«> E = V ---- X c - -i- T-V X V X L'z (2.1) - 1 H = V X V X {/'- + V X f - (2.2) _,(/*) _,(^)

.

T he p oten tial vectors and ii'- individually satisfy the H elm ho ltz equation from which the fields a re derived. Superscripts h and e denote th e m a g n etic and electric p otentials respectively. They can be written as an infinite sum of t h e incident and reflected normal m odes in b o th regions of th e discontinuity. T he coefficients Cp., and .4m.„ a re the incident wave a m p litu des of T M and T E . respectively, while Dp., and Bm.n a re the reflected wave a m p litu des, respectively, as indicated in Figure 2.3.

- . j . .- ) = £ f ; (2.3) ,{Rh) I (X. ÿ. r) = m = 0 n = 0 = £ £ ) A i f ’r i « ' ’( x . i , ) ( 0 - - ’- '= - '-) (2.4) p = l q = l

where. R £ 1. 11 are the two regions of th e discontinuity. T h e wave im p e d a n c e Z and a d m itta n c e V are as follows:

^{Rh) __

m n j{Rh) (-••^1

“^m.n

(2.6 )

It can be observed that th e coefficients of th e field co m ponents namely .4. B . C and D have two indices. This is a direct consequence of the sep aratio n of variable solution to

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00

z=0

l b )

z=0

Figure 2.3; (a )S y m m etric double plan e step discontinuity in re c tan g u la r waveguide (b) .Asymmetric double plane step discontinuity in rectangular waveguide

th e H elm holtz equation. However, th e se coefficients form a vector d u rin g th e analysis, since m . n or p.q represent one p a rtic u la r m ode with a certain value of propagation co n sta n t J^.n or T h e transverse com ponent of the potential functions in th e region [ I c an be w ritten as:

+ <5o.n sin(A;(^y)z) sin (t^ y )y )

(•2.7)

(2.8)

where = m - J w . = n - j h . A].” ' = pi^/w and = qTz/h when w is th e w id th a n d h is the height of the re c tan g u la r waveguide in region I I .

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23

Sim ilarly in region I the potential functions are as follows:

Y 1 + éo.m Y 1 + ®Q.n

' - p Ï ’ = ^p.J*sin(Â:ip’( x -

c^)) sm{k[[Hy ~

Cy)) ( 2 . 1 0 )

where = m - j a . = mr/b . k^.^J = pir/a an d k^^J = q-irjb with a as th e w idth

and b as the height of the rectangular waveguide.

T h e propagation constants in eith er regions (/?). can be evaluated from

= l a t i )

The power carried by each mode and in each of th e region is evaluated from

where 5 ' ^ ’ is the surface area of the cross-section of that region. T h e coefficients an d 7^^^) of th e potential functions are chosen so that the m ag nitu de of th e power carried in th e corresponding m ode is unity. Thus.

P =

Iff for propagating modes

-l-Jff for evanescent T E modes (2.13) —J f f for evanescent T’A/ modes

This vields.

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24 9

— " — (0

\ A ¥ 7 T 7 f ? v ^

At th e ju n c tio n of the discontinuity (z=0) formed by two re c ta n g u la r waveguides, with th e ir axis in line, as shown in Figure 2.3, the continuity condition is applied as follows

E ^ ^ \ x . t j ) = E ^ ^ ^ \ x .y ) z G yÇi[cy.dy\ ( 2 .IS)

= 0 otherwise (2.19)

H'^^Hx. y) = H'^‘^ \ x . y ) x e j c ^ r - d x ] . y Ç : [ c y . d y ] ( 2 . 2 0 )

T h e above two continuity conditions of tangential electric a n d m agnetic fields can be separated into four sets of coupling equations using the p ro p e rty of ortho go nality of modes. T h e resulting coupling equations after m atching t h e ta n g en tial electric fields at the discontinuity and applying orthogonality is of th e following form:

+ B i y = + D»>) C2.21)

c;."’ + A'.'/' =

K A c> '>

+ B<|i) + vu(A<;;„ +

12

.

2 2

)

It m ust be noted th a t though the coefficients of the field com po nents, nam ely A. B , C a n d D when th e y form a vector are arranged in ascending o rd e r of th e cutoff frequen cies of th e corresponding modes. The elem ents of th e matrices V),/,. Vé/,. Vhe and can be c o m p u te d from the equations below.

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lO \\ ^ { lj . p q ) = \ \ h { i j . m n ) — Vhe{ij.rnn) = )dxdy j(/e) , Ale) # r r < Alle) pg ( e )dxdy (2.2 5 , */Cx *'Cy I

-A^O Cx ^Cy )dxdy

(2.23) (2.24)

(2.26) T he vectors e are the ta n g en tial electric fields evaluated from the p o te n tia l functions and are o b ta in e d using .\ppendi.x .A. or th e equations below:

_^(M) ^ mn (2.27) — »(/f) == (2.28) ^ V = ; x T r c t '! / . (2.29) -yille) ^ V (2.30)

It can be verified that is zero. This has been a n alytically proved in [48]. M atching th e H-fields and applying orthogonality again results in

+ K U c H , ” - D j " ' )

C'.'J - - o % i )

(2.31) (2.32)

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26

It c an be verified th a t th e coupling matrices in th e above equations a re transposes of the Vhh • Vv/i and obtained in th e E-field m atching . R earranging th e four sets of e q u atio n s, we have / r 0 0 r 0 £)(/) —I /i/i 0 r 0 V — iç/i 0 _V V ( 0 0 E 0 q E) k/i/i 0 - r 0 B(ii) Vee 0 _{- r} \ j (2.33)

where C is th e unit m atrix .

T h e generalized scatterin g m atrix of a step d iscon tin uity is hence given by 5 = A ' , - ‘ A', (2.34) . \ n a lte rn ativ e scheme of inversion [49] is given in th e A ppendix B. T h a t scheme is c o m p u ta tio n a lly m ore efficient, since the size of t h e m a trix to be in v erted is smaller c o m p ared to th a t in th e equation (2.34). Hence, th e inversion technique in Appendix B has been used in all th e analyses presented in this work.

T h e sym m etric double step discontinuity can also be analyzed using the T „ approach , as th e fundam ental mode T E ^q is th e sam e as T E \q. In this approach,

the fields at th e discontinuity are evaluated based on th e magnetic vector potential (Aj- = U) alone. This approach is not a rigorous one as the c ' p o te n tia l vector is ignored due to th e fact th a t the E^ field is very sm all at th e discontinuity. .Although this ty p e of analysis produces a com putationally efficient algorithm , it has been shown in [50] th a t for certain types of com ponent analysis, like resonant-iris waveguide filters.