microondas magnéticas en el espacio vectorial

7/28/2019 microondas magnticas en el espacio vectorial

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MICROWAVE ANTENNA

THEORY AND DESIGN

Ediied by

S AMU E L S I LVE R

ASSOCIATE PROFESSOR OF ELECTRICAL ENGINEERING

UNNEB.SITY OF CALIFORNIA, i3EP.KELEY

OFFICE OF

SCIENTIFIC RESEARCH AND DEVELOPMENT

NATIONAL DEFENSE RESEARCH COMMITTEE

FIRST EDITION

N E W Y OR K , T OR ON T O L ON D ON

McGRAW-HILL BOOK CO,MPANY, INC.

1949


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.,

,.,

MICROWAVE . xTEN\ -. THEC R Y .ISD DESIGN

(hPYRIGH,T,

1949,

B>- THE

hlC~RA W-HILL

BOOK

~(II IP.I.NY,

lKC.

P31XTEI) lx THE U>-lTEI) STATES OF AMERICA

.111 rights

Testwed.

his book, or

parts

th r of

HI(IY

not be

reproduced

i n any f orm r i l ho? l prr r l i s s i on of

/he , L //i shers ,

THE MAPLE PRESS COMPANY, YORK,

PA,

*ienCe


3/635

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y,,

,,

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lf[CRO JV.4 VE A NTE.VNA TfZEOR Y

EDITORIAL STAFF

S AMU EL S I LVE R

H U B E RT M. J AME S

AND DESIGN

CO.VTRIB LTI.VG A PTHORS

J . E . llATON

R. hf. R IZD H E F FE R

L. J . I ;YG E S

J . R. RISSER

T. J . K E ARY

S . S ILVE R

H . K RU TTE R

O. A. TYS ON

(2. G . hlAcF ARL.4NE

L. C . \ AN ATTA


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Foreword

T

HE tremendous research and development effort that ~vent into the

development of radar and related techniques during }Vorld IJ ar II

resulted not only in hundreds of radar sets for military (and some for

possible peacetime) use but also in a great body of information and ncm

techniques in the electronics and high-frequency fields. 13ecause this

basic material may be of great value to science and engineering, it seemed

most important to publish it as soon as security permitted.

The Radiation Laboratory of 311T, ~vhich operated under the super-

vision of the National Defense Research (ommittec, undertook the great

task of preparing these volumes. The ~vorl{described berein, ho\\-eyer,is

the collective result of ~vork done at many laboratories, Army, Xavy,

university, and industrial, both in this country and in JZngland, (


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Preface

T

H E

need that arose during the ]var for utilizing the microwave region

of the radio frequency spectrum for communications and radar stimu-

lated the development of nelv types of antennas. lhe problems and

design techniques, lying as they do in the domain of both applied electro-

magnetic theory and optics, are quite distinct from those of long-wave

antennas.

It is the aim of the present volume to make available to the

antenna engineer a systematic treatment of the basic principles and the

fundamental microwave antenna types and techniques. The elements

of electromagnetic theory and physical optics that are needed as a basis

for design techniques are developed quite fully. Critical attention is

paid to the assumptions and approximations that are commonly made

in the theoretical developments to emphasize the domain of applicability

of the results. The subject of geometrical optics has been treated only

to the extent necessary to formulate its basic principles and to sho~v its

relation as a short wavelength approximation to the more exact methods

of field theory. The brevity of treatment should not be taken as an

index of the relative importance of geometrical optics to that of electro-

magnetic theory and physical optics.

It is in fact true that the former

is generally the starting point in the design of the optical elements

(reflectors and lenses) of an antenna. However, the use of ray theory

for microwave systems presents no new problems over those encountered

in opticson which there are a number of excellent treatisesexcept

that perhaps the law of the optical path appears more prominently in

micro~vave applications.

In the original planning of the book it was the intention of the editors

to integrate all of the major wQrk done in this country and in Great

Britoin and Canada. This proved, however, to be too ambitious an

undertaking. Nfany subjects have regrettably been omitted completely,

and others have had to be treated in a purely cursory manner.

It \vas

unfortunately necessary to omit two chapters on rapid scanning antennas

prepared by Dr. C. V. Robinson.

The time required to revise the

material to conform ~vith the requirements of military security and yet

to represent an adequate exposition of the subject would have unduly

delayed the publication of the hook. Certain sections of Dr. Robinsons

material have been incorporated into Chaps. 6 and 12.

i x


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x

P R E F A C E

I take pleasure in expressing here my appreciation to Prof. Hubei-t

M. James who, as Technical Editor, shared with me much of the

editorial work and the attendant responsibilities.

The scope of the book,

the order of presentation of the material, and the sectional division within

chapters were arrived at by us jointly in consultation with the authors.

I am personally indebted to Professor .James for his editorial Ivork on

my own chapters.

The responsibility for the final form of the book, the errors of omission

and commission, is mine. A word of explanation to the authors of the

various chapters is in order.

After the close of the Office of Iublications

and the dispersal of the group, I have on occasions made use of my

editorial prerogative to revise their presentations. I hope that the results

meet ~vith their approval. The policy of assignment of credit also needs

explanation. The interpretation of both Professor James and myself of

the policy on credit assignment formulated by the Editorial Board for

the Technical Series has been to the effect that no piece of work discussed

in the text would be associated with an individual or individuals.

Radi-

ation Laboratory reports are referred to in the sense that they represent

source material for the chapter rather than individual acknowledgements.

References to unpublished material of the Radiation Laboratory note-

books have been assiduously a~oided, although such material has been

dramm upon extensively by all of us.

In defense of this policy it may be

stated that the ]vorlc at the Radiation Laboratory was truly a cooperative

effort, and in only a few instances would it have been possible to assign

individual credit unequivocally.

The completion of the book was made possible through the efforis of

a number of people; in behalf of the editorial staff and the authors I wish

to acknowledge their assistance and contributions. Mrs. Barbara Vogel

and Mrs. Ellen Fine of the Radiation Laboratory served as technical

assistants; the production of figures and photographs \vas expedited by

hlrs. Frances Bourget and Mrs. nary Sheats. It proved impossible to

finish the ]t-orl


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Ch:lp. 11. The S:1( iomd Rcsc:wch (ouncil of Can:&~ :md the llrit isll

(entnd Radio 13urw~u h~~vc ~rwiously granted us permission to ti~li(.

m:ltcrial from ( unudi:m :md I;ritish reports in accord:mcc ~~ith mlrrrnt

security U3glllotioms. l>hc I?wII Telephone I.abora,twy supplied the

photographs of mct:d lens antennas.

S .4 MU E L kh LVlil{.

K : \ v\ T, ll l)s l. \ 1i 1lI T. WI M WI I Y,

f lslllxlm)x, l). (.,

:lpr d, 1947.


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.

ontents

FORE WORD BY L. A. DUBRmGE . . . . . . . . . . . . . . . . . . vii

PRE1744CE. . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

CHAP.1. SURVEY OF MICROWAVE ANT~~NNADESIGN PROBLEMS

1

1.1. The Wa velcngt hRegion. . . . . . 1

1.2. .Lnt enna P a t t erns . . . . . . . . . . . . . . . . . . . ..2

1.3. Types of }I icrow a ve B ea ms. 6

1.4. lI icrow a ve Tra nsmission I ,ines . 7

1.5. Ra dia t ing llernent s . . . . . . . . . ..8

16. .4 S urvey of kllcrow a vc }.nt en na Types 9

1.7. I mpeda nce S pecifica t ions. 13

1.8. P rogra m of t he P resent Volum e 14

C H .4P . 2. C I RC U I T RJ 31J ATIOIW, Rf3C I P R0C iTY TH W3RF ~ >fS . 16

21.

22.

23.

2.4.

2.5.

2.6.

2.7.

2.8.

2.9.

I n t roduct ion . . . . . . . . . . . . . . . . . . . ...16

The F our-t ermina l ~ et w ork. 17

The Ra yleigh Reciprocit y Theorcnl 19

Th6venins Theorem a nd t h e Nfzxim um-pow er Th eorem 20

The Tw o-w ire Tra nsmission I ,ine 21

The H om ogeneous Tra nsmission I ,ine 23

The LosslessLin e..... ..26

Tra nsforma t ion C h a rt s. 29

The F our-t ermina l Net w ork l:quiva lent of a S ect io]l of Tra ns-

missiOn Line.... . . . . . 36

TRANS MI ~ E VG ANn R E C E I VI NG ANTE NNAS . . 37

2.10.

2.11.

2.12.

2.13.

2.14.

2.15.

2.16.

2.17.

2.18.

The Ant ema a s a Termina t ing I mpeda nce 37

The Receiving Ant enna S yst em 40

The Tra nsmit t er a nd Receiver a s a C oupled S yst em

45

Reciprocit y bet w een t he Tra nsmit t ing a nd I t ecei\ ,in g P at t erns of

a n Ant enna . . . . . . . . . . . . . . . . . . . ...48

The .kvera ge C ross S ect ion for a Ma t ched S yst em 50

D ependen ce of t he C r oss S ect ion on Ant enna Misma t ch 51

The F our-t ermina l Net w ork Represent a t ion. 53

l)evelopment of t he Net w ork E qua t ions 56

The Reciprocit y Rela t ion bet w een t he Tra n sfer I m peda nce

C oet licient s, . . . . . . . . . . . . . . . . . . ...59

X111


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xiv CO.YYfil.VT S

C H AP . 3. R.\ II I .iTIOS FR031 C U IU WXT I ) IS TRI I)I T1OS S . (iI

31. Tllc Field I lq ua t ions. 6]

32. The (;onst it ,lt ive P a ra mct r rs;

I .inc:w it y mid S I lperpo sit ion . 6,5

33. I lou])da ry C ont lit ions. 66

3 ~ . The Field ~ q ~ la t io]ls for H :mnonic TI I n c ])(, p[,lld(>I 1[W 68

3.5. I a ynt ings Thmreln . . . . M)

36. The ll-a \ ,c k;q ll:lt ions. 71

3.7. S imple l~ a vc S olllt ions. 73

38. G enera l S ollt t ion of t he Field I }q ,,~ tiol,s in Tcr],,s of t l)r fk),,r t cs,

for a Ti]llc-pcr iodir lr i(ld . 8(I

3.9, F ield ]),,e t o S ollr t t ,s in a n U ]Ih oI uI (lt Yi I i(,gion 84

3.10. F ield in a licgion Rot m dcd h y S (lr fa ((,s of I ]lfilllt cl}- (OI )(l\ l(t I vr

llc,clia . . . . . . . . . . . . . . . . . . . . . ..8fi

311. Tl](, F a r-zone Fields 87

312. I ola r iz a tion .

)1

3~ 13. The I ;lcct r ic l)ipolc )2

314. Tllc I I t ignet ir lh ]mlc 95

3.15. The F :lr-zon r Fir l(ls of I ,i]l(,-rurr(,n t l)ist r illllt if )ls )(i

316. The H :df-~ 1 I V(, l)ilmlt ,

)8

317. S llpcrpm]t ion of l~ icl{ls ) 1

31S Th e 1)0111>1(, -(111)01(y s t em

101

319. I {cgldur S pa ce .I r r :iys

104

C ][.I P . 4. ll-.l;l; F I {O>-TS :~ ~ 1) RAYS 107

41. TI I C I I \ lygrn s-C rccn F orn~ lll:L for t h r I ll((,t r (l]]]:Lg]l(t i[, lit l(l 107

42, G col]lct r ica l ()~ ]t ir s : l~ a vefront s a nd l{:lys 110

43. C 1lr~ :it llr r of t he I lmys in a n I nllo]]I ogc,l](,[~ ls I lcdiunl 111

4+ . E n ergy FlOIv in (lcomet r ica l Opt ics 112

45. (;comct r ica l opt ics :is :L Zero-~ ra ~ clcngt h I ,in]it 114

46. The H ~ lygen s-F rrsnr l P rinciple a nd G conlct r ira l Opt ics: The F a r-

z on c .k ppr ox im a t i on

116

47. The P rin ciple of S t a t ionm-y I ha sc 11{)

48. F t = rnla ts I r incip]e.

122

4.9, The I ,a ,v of t he opt ira l P a t h

125

(]~ \l,, 5, S (.LTTE RIS C, .kN-T) D IFF R.AC TIOX. 12 )

51. (;cn cra l (onsidcrnt ion s

129

52. B ol]nda ry (o]ldit ions

130

53. I ieflect ion hy a n Infillit e P la ne S (lr t a rc; t he lr inril)lv of I t I I :igrs 132

AP P R OXI MATE J I E TH O D S F OR R E F LE C TO I+ S OF ;lI L B I TM AR Y S ] J .\ P E

54,

55,

56,

57,

58.

59.

510.

511.

512.

The G eomet rica l-opt ics I Ict hor l

C a lcula tion of t he S ca tt r r rd F ir l[l

S uperposit ion of t he S o~ lrcc l~ icl(l :I II d t l)(, S c:\ t t ercd l~ icld.

Th e C ur ren t-dist ril, ~ l tion 31ct hod

C a lc(lla t ion of t he S ca tt r r rd F ield

Applica t ion t o P oint -source md I J ir l{J -sollr re l;ecds.

Rea ct ion of a Reflect or on a P oint -source F eed

Th e .Aper t ur e-fi elci I fet h od

The F ra unh ofer Region.

,..

137

138

139

143

144

146

149

155

. .

158

,. . 160


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.

coYTAl

s

xv

l) I F1.ll.\ (r0N. . . . . . ...,,162

S .I S, (i,r ,cr t LI (~)])si,l,,r:,iit)])s mI tht, .ipproxim:lt c I lt ,t llods 162

514. l{t t lll~ ,t lt J J l t o :1 S [a lilr l)illr :~ (t lf)n I I o I )lcI N 164

515. l)~ ~ lli]lct s lr l]l{ipl~ i for t ll( l-TI,;X-XA IATTE RNS

169

61. lr il]l:~ ry a nd S [,co]lt lxry lt lt t rms 169

62. Tll(, l)iffr :t r t ion Fielt l 169

(i 3. I our i(r Int egra l li(,l)rc,s(,]lt a t i{jl~ of t he Fra unhofcr lie~ ion

174

64, (+ C J I (M1 I :ca tur t s of t ht , S ccolld:w y l~ tt vm

175

6,5. TI I C l{rct mlg~ ll:~ r .fpor t ure ]80

6.t i. Tl o-(lilllt ,n sio]la l P r ol)lcm s

182

67. I l]:i.w -error kYfect s. . 186

08. TI I C (irc(llm .ipcr t urc

192

&9. Th(, Field o]] t he Axis in t hr F rcsnr l ]t cgion

] 96

(l[,\ ],, 7. l[I (:ROI V,.fV}; TRAA-S ~ f I S S IOA I ,I N-E S 200

71

llicro~ j :ivc nnd I ,ong-~ va ve Trimsmission I ,illm 200

72. lrop:l~ a t io]l in ~ f:~ vcgllidcs of ln iform (;ross S w t ion 201

73. or t hogo]la llt y Rcl:lt ions a nd P ow er Flow . 207

74. Tra nsnlissiun I ,inr (onsidcra t ions in l~ :lvrguidrs 209

75. Xct J t ork Kqlliva lent s of J unrt ions a nd ohst a clcs 214

76. 7/.l/-modc Tra llsmission I ,ir lcs 216

77. (ozxis,l I ,incs: ?~ . f-rnodc 217

7.8. (oa xia l I .ines: T.I f - a nd T]i-n lodcs 219

7. ),

(:Is,.acIc

Tmnsformcrs: TJ ~ .lf-mode 221

71o. I a ra llel S t ~ lhs a nd S eries I Lea ct a ncm.

223

711. licct a ng~ lla r }Va vcguidcs: I A- a nd ?,lf-modes 226

712. I mpcda nrc Tra nsformers for I iect a ngula r (;uidcs 229

713. C ircula r ll-a veguide: T~ - a nd TJ f-modcw . 233

7.14. I vindow s for LTS Cin C ircula r G uides 235

715. I a ra llel-pla t e i~ a veguide. . 235

716.11esignN Totes . . . . . . . . . . . . . .. 238

C H AF . 8. lfI C ROWAVE D IP OLIl A3J TE ~ ~ AS ANI ) F13f?D S 239

81. C ha ra ct er ist ics of Ant enna Feeds 239

8.2, Coa xia l I ,ine Termina t ions: The S kir t D ipole 240

83. Asymmet ric D ipole Termina tion.

242

84. S ymmet rica lly 13nergizcr i D ipoles: S lot -fed S yst ems 245

85. S ha pe and S ize of t he D ipole . 248

86. lVa veguidc-line-fcd D ipoles. . 250

87. D irect ive D ipole Feeds, . . . 250

88. D ipole-disk F ords . . . . . . . . . . . . . . . . . . ...251

89, D ouble-dipole Feces, .,.... . . . . . . . . . . . . .253

8.10. Lfult i-dipole S yst ems, . . . 256

C H AP . 9. LI NI ?AR ARRAY AXTE iYNAS AND FF23DS 257

9.1. (kmcra l C onsidera t ions. . 257


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XVI

C ON T E J V T S

P A~ E E N TH E ORY . . . . . . . . . . . . . . . . . . . . . . ...256

92. G enera l Arra y Formula . . . 258

93. The Associa t ed P olynomia l 261

9.4. U 1liformArra ys . . . . . . ..264

9.5, B roa dside 13e:~ nls . . . . ...267

9.6. E r l(l-t ire I ~ ea n]s . . . . . . . . . ...274

9.7. 13ca 1u S ynt hesis . . . . . . . . . . . . . .. 279

RAD I ATI NG E I .E MMNTS . . . . . . . . . . . . . . . . . .. 284

9.8. llipole Ra dia t ors . . . . . . . . . . . . . . 284

9.9. S lot s in J va veguide }Ta lk. . 286

9.10. Theory of S lot Ra dia t ors. . 287

9.11. S lot s in Rect a ngula r J $a veguide; 1~ ,,-mode 291

912. E xperiment a l D a t a on S lot l{a di~ t ors 295

913. P robe-fedS lot s . . . . . . . . . . . . .

299

9.14. fVa veguide Ra dia t ors 301

9.15. Axia lly S ymmet rica l Ra dia t ors . 303

9.16. S t rea mlined Ra dia t ors . . . 310

ARRAYS . . . . . . . . . . . . . . . . . . . . .. 312

9.17. L oa ded-line Ana lysis. . 313

9.18. E nd-fire .4rra y . . . . . . . . . . . . . . . . . . . . . . 316

B ROAD S I D E ARE AYS . . . . . . . . . . . . . . . . . . . . . . .. 318

9.19, S uppression of E xt ra neous Ma jor I ,ohcs . 318

920. Resona nt Arra ys . . . . . . . . . . . . . . . .. 321

921. B ea con Ant enna S yst ems. 327

922. T$onresona nt Arra ys . . . . . 328

923, B roa dba nd S yst ems w it h >Torma l B ea ms 331

CH P

10.

WAVEGUIDE AND HORN FE~;~S. .

334

10.1. Ra dia t ion from Wa veguide of Arhit rmy C ross S crt ion 334

10.2. Ra dia t ion from C ircula r ~ ~ a v(guide 336

103. Ra dia t ion from Rect w w la r G uide. 341

10.4. Wa veguide Ant enna Feeds

347

105. The D ouble-slot Feed .

348

10.6. E lect roma gnet ic H orns. . . . . . 349

10.7. hrodes in lpla ne S ect ora l H orns 35o

108. J fodes in I f-pla ne S ect ora l H orns

355

109. Vect or D iffra ct ion Theory Applied t o S rct ora l H orns. 357

10.10. C ha ra ct er ist ics of Observed I t a dia t ion lt it t mns from H orns of

Rect a ngula r C ross S ect ion 358

10.11. Admit t a nce of Wa veguidc a nd H orns . 366

10.12. Tra nsforma tion of t he L-pla I~ c H orI I .idulit t a n cc f mm t he Throa t

t ot he U niform G uide . . . . . . . . . . . . . . . . ...369

10.13, Admit t a nce Cha ra ct er ist ics of H -pla ne S ect ora l H orms

374

1014. C ompoundH orns. .,.. , . . . . . . . . . . . . ...376

10.15 .The B ox H orn... . . . . . . . . ...377

1016. B ea m S lla ping hy ~ lca ns of Obst a clrs in H OH I :md \ Va veK{lidv

.4pert mcs . . . . . . . . . .

380


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C H AP . 11. D I I :I ,I cC TILIC i~ X1) lI J :T:lI ,-lI .. fTI s in 31icro~ v~ vc ,~ nt cmms. 388

I)ll:l,l:C,rRICIEXSF;S. . . . . . . . . . . . . . . . . . . . . . .

112. lr ill{ilJ lrso f I>rsign, .

113, S inlplc I ,cllses IVlt llo(it Zoning

114. Zoned l)lclrct r ic I ,cnscs . .

11,5. U sc of lIa t cr i:t ls ~ v]t h H i~ ll l{cfmct ivc In dcxrs

11.6. I )lclcct r ic I ,osscs mnd Tolcra nccs on I rons I a ra mct crs.

11.7. I t cflect ions from L)iclcct r ic S ur fa ccw

389

389

390

395

398

399

401

hfE TAL-I ,L~ TE LEXSE S . . . . . . . . . . . . . . . . . . . . . ...402

11.8. P a ra llel-pla t e I ,ensm, . . . . 402

11. ). Ot her 31et a -lcr ,s S t ruct ures. 406

11.10. l[ct a -pla t e I .cns Tolrra ncrs 407

11.11. B a nd,vi(lt h of l[et a l-plzt c I ,cnscs; Achroma t ic D oublet s 408

11.12. I t eflect ions from S urfa ces of P a ra llel-pla t e I ,enses 410

CH AP . 12. P E NCILB F Ahf AND S IhfP LE

FANAl;D-BEAN1 ANTEKAAS 413

PENCIL-BEAMANTENNAS. . . . . . . .,.........,...413

12.1. Pencil-beam Requirements and Tcchniqlles 413

12.2. Gcomctriral Parameters 415

123. The Surface-current and Aperture-firld I)istrih(ltions. 417

12.4.

The ILa dia tion Field of t he Reflect or

420

12.5. The Ant enna G a in., . . . . . . . . . . . 423

126. P rima ry P a t t ern D esigns for hI :mimizing (la in 433

127. E xperiment a l I t csldt s on %condm-y la t t mns 433

128. I mpeda nce C ha ra ct erist ics 439

12.9. The Vert ex-pla t e lf:~ t ching Trchniq uc 443

12.10. I t ot a t ion of I ohu-iza t ion Techniq ue 447

12.11. S t ruct ura l D esign P roblems. 448

S IMFLE FANNE D -B E AM ANTE NNAS . 45o

12.12. Applica t ions of Fa nned B ea ms a nd Nfct hods of P mdllr t ion 45o

1213. Symmet rica lly C ut P a ra hololds 451

12.14. Feed Offset a nd Cont our C ut t ing of Reflect ors 453

1215. The P a ra bolic C ylinder a nd Line S ource 457

12.16. P a ra llel-pla t e S yst ems 459

12.17. P dlbox D esign P roblems 460

C H AP . 13. S H AP E D -B E AM ANTE NNAS. .

131.

132.

133.

134.

135.

136.

137.

138.

13.9.

Sha ped-bea m Applica t ions a nd Req uirement s

E ffect of a D irect iona l Ta rget Response

S urvey of B ea m-sha ping Techniq ues.

D esign of E xt ended Feeds. . . .

C ylindrica l Reflect or Ant enna s .

Reflect or D esign on t he B a sis of Ra y Theory .

Ra dia tion P a t t ern Ana lysis.

D ouble C urva ture Reflect or Ant enna s

Va ria ble B ea m S ha pe.

465

465

468

471

487

494

497

500

502

508


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,..

XV1ll

COA71{.V 7s

CH AP . 14. ANTk;XNA IXSTALLAT1ON IROB I ,E 31S 510

G E ~ ~ I t .4L S IKVl~ Y OF I ~ S cr .4LLArIO~ I ~ OB LE MS . 510

14.1. (:rolm(l .lllt (,llna s 510

14.2.S llilJ .illt er lI]a s,,, . . . . . . . . ... ..511

143,

~lir(,r:ift


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.

C 11.IP TI ;R 1

S U RVE Y OF MI C ROWAVE ANTE NNA D E S I G N P ROB LE MS

]]Y s. S I I ,V1;ll

1.1. The Wa velengt h Region .I he designa t ion of t he bounda ries of

t he micro m ve region of t Iw r lcct rom a gm ct ic spect rum is pllrcly a rbit ra ry ,

Tile I ong-]va velcngt h limit I Ia s I xxm set v:~ r iously a t 25 (J r 40 cm, even

a t 100 cm. F rom t he point , of vie\ v of a nt enna t heory a nd design t echn-

iques, t he 25-cm va l~ le is t he most a ppropria t e choice, The short -

w a ~ elengt h limit t o )ihich it is possible t o ext end t he present t erhniq(les

ll:~ s not ~ e tl)ec>r ~ ra clle(i; it isin th cn ciglll)or }loo(lof lmm . Accor din gly

\ vesha ll cunsi(lcr t he microlva ve region t o ext end in w a velengt h from 0.1

t o 25 cm, in frcq llcncy from 3 X 105t o 1200 31c/see,

This is t he t ra nsit ion region bet \ \-een t he or(lina ry ra dio region , in

}vhich t he \ \ -a velengt llis ver y k~ r ge compa rw l w it h t he dimensions of a ll

t he component s of t he syst em (cxccpt perhops for t heh~ r ge a nd cumber-

some a nt enna s), a nd t he opt ica l region , in ]t -hich t he \ va vclengt hs a rc

excessil-ely sma ll. I .ong-\ va vc concept s rm(l t echniq ues cont inue t o be

useful in t he micro \ va ve region , a nd a t t he sa me t ime cer t a in devices

used in t he opt ica l r egion sllr h a slen se sa n dn ~ ir ror sa r cem ploy eci.

From

t he point of vie}v of t he a nt enna designer t he most import a nt cha ra ct er-

ist ic of t his fre(~ u cncy r egion is t ha t t he w a ~ ~ elengt hs a re of t he order of

ma gnit ude of t he dimcmsilmsof convent iona l a nd ea sily ha ndled mecha n-

ica l devices. This lea ds t o ra dica l modifica t ion of ea r lier a nt enna

t echniq ues a nd t o t he a ppea ra nce of nefv a nd st r iking possibilit ies,

especia lly in t he const ruct ion a nd use of complex a nt enna st ruct ures.

I t follow s fr om element a ry diffra ct ion t heor y t ha t if D is t h e m a xim um

dimension of a n a nt enna in a given pla ne a nd k t he iva vr lengt h of t he

ra dia t ion, t hen t he minimum a ngle }vit hin w hich t he ra dia t ion ca n be

concent ra ted in t ha t pla ne is

(1)

Wit h microw a ves on e ca n t hus pr oduce highly direct ive a nt enna s such

a s ha ve no pa ra llel in long-w a ve pra ct ice; if a givendirect ivit y is desired,

it ca n be obt a ined \ vit h a microw a ve a nt enna ]vhich is sma ller t ha n t he

eq uiva len t lon g- va v e a n t en na .

Th e ea se w it h w hich t hese sma ll a nt enna s

ca n be inst a lled a nd ma nipula ted ina rest rict ed spa ce cont ribut es grea t ly

t o t he pot ent ia l uses of microw a ves.

In a ddit ion , t he convenient size of

1


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2 S UR V E Y OF M I C R OI V . t J E AN TE XAV A D E S I G .V PR OB L E M S [SEC, 1.2

microw a ve a nt enna element s a nd of t he complet e a nt enna st ruct ure ma kes

it fea sible t o con st ruct a nd use a nt enna s of ela bora te st ruct ure for specia l

purposes; in pa rt icula r, it is possible t o int roduce mecha nica l mot ions of

pa rt s of t he a nt enna w it h r espect t o ot her pa rt s, w it h conseq uen t ra pid

mot ion of t he a nt enna bea m.

The microw a ve region is a t ra nsit ion region a lso a s rega rds t heoret ica l

met hods. The t echniq ues req uired ra nge from lumped-const a nt circuit

t heory , on t he low -freq uency side, t hrough t ra nsmission-line t heory , field

t heory , a nd diffra ct ion t h eor y t o geomet r ica l opt ics, on t h e high-fre-

q uen cy side. Th ere is freq uent need for using severa l of t h ese t h eor ies

in pa ra llelcombining field t heory a nd t ra nsmission-line t heory , sup-

plement ing geomet r ica l opt ics by diffra ct ion t h eory , a nd so on. Opt ica l

problems in t he microw a ve a nt enna field a re rela t ively complex, a nd

some a re of q uit e novel cha ra ct er : F or inst a nce, t he opt ics of a curved

t w o-dimensiona l doma in finds pra ct ica l a pplica t ion in t he design of

r a pid -s ca n n in g a n t en na s .

1.2. Ant enna P a t t erns.-B efore under t a king a survey of t he mor e

import a nt t ypes of micr ow a ve a nt enna ,

it w ill be necessa ry t o st a t e

precisely t he t erms in w hich t he performa nce of a n a nt enna w ill be

described.

Th e Ant enna a s a

Radiating Device:

The Gain Function.The

field

set up by a ny ra dia t ing syst em ca n be dir ided int o t w o component s:

t h e induct ion field a nd t he ra dia t ion field. The induct ion field is impor-

t a nt only in t he immedia t e vicinit y of t he ra dia t ing syst em; t he energy

a ssocia t ed w it h it pulsa t es ba ck a nd for t h bet w een t he ra dia t or a nd

nea r-by spa ce. At la rge dist a nces t he ra dia t ion field is domina nt ; it

represen t s a cont inua l flow of en er gy direct ly out w a rd from t he ra dia tor ,

w it h a densit y t ha t va ries inversely w it h t h e sq ~ ia rc of t h e dist a nce a nd,

in genera l, depends on t he direct ion from t he source.

In eva lua t ing t he performa nce of a n a nt enna a s a ra dia t ing syst em

on e considers only t he field a t a la rge dist a nce, w her e t he induct ion field

ca n be n eglect ed. The a nt enna is t hen t rea t ed a s a n effect ive point

source, ra dia ting pow er t ha t , per unit solid a ngle, is a funct ion of direc-

t ion only . The direct ive proper t ies of a n a nt enna a re most con~ enien t ly

expressed in t erms of t he ga in funct ion G (6,O). I /et 6a nd @ be r espec-

t ively t h e cola t it ude a nd a zimut h a ngles in a set of pola r coordina t es

cen t ered a t t he a nt enna . Let F (O,@) be t he pow er ra dia t ed per unit

solid a ngle in direct ion 0, @ a nd P~ t he t ot a l pow er ra dia ted.

The ga in

funct ion is defined a s t h e ra tio of t h e pow er ra dia ted in a given direct ion

per unit solid a ngle t o t he a vera ge pow er ra dia ~ ed per unit solid a ngle:

47r

(2)


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S E C . 1.2]

AN TE N N A PATTE R N S

3

Thus G (L9, ) expresses t he increa se in pow er ra dia t ed in a given direct ion

by t he a nt enna over t ha t from a n isot ropic ra dia t or emit t ing t he sa me

t ot a l pow er ; it is independent of t h e a ct ua l pow er level. The ga in

funct ion is conven ient ly visua lized a s t he sur fa ce

r = G(f3,@) (3)

dist a nt from origin in ea ch direct ion by a n a mount equa l t o t h e ga in

funct ion for t ha t direct ion . Typica l ga in-funct ion surfa ces for micro-

w a ve a nt enna s a re illust ra ted in Fig. 1.1.

The ma ximum va lue of t he ga in funct ion is ca lled t he ga in; it

w ill be denot ed by G M. The ga in of a n a nt enna is t he grea t est fa ct or

by w hich t he pow er t ra nsmit t ed in a given direct ion ca n be increa sed

by using t ha t a nt enna inst ea d of a n isot ropic ra dia tor .

The t ra nsmit t ing pa t t ern of a n a nt enna is t he sur fa ce

(4)

it is t hus t he ga in-funct ion surfa ce norma lized t o unit ma ximum ra dius.

A cross sect ion of t h is sur fa ce in a ny pla ne t ha t includes t he origin is

ca lled t he pola r dia gra m of t h e a nt enna in t his pla ne. The pola r

dia gra m is somet imes renorma lized t o unit ma ximum ra dius.

W-hen t h e pa t t ern of a n a nt enna ha s a single principa l lobe, t h is is

usua lly r eferr ed t o a s t he a nt enna bea m. This bea m ma y ha ve a

w ide va riet y of forms, a s is show n in F ig. 1.1.

The Antenna as a Receiving Dwice: The Receiving Cross Section .The

performa nce of a n a nt enna a s a receiving device ca n be descr ibed in

t erms of a receiving cross sect ion or receir ing pa t t ern .

A receiving a nt enna w ill pick up ener gy from a n incident pla ne w a ve

a nd w ill feed it in t o a t ra nsmission line w hich t ermina tes in a n a bsorbing

loa d, t he det ect or . The a mount of en ergy a bsorbed in t h e loa d w ill

depend on t h e or ient a tion of t he a nt enna , t he pola riza tion of t he w a ve,

a nd t h e impeda nce ma t ch in t he receiving syst em.

In specify ing t h e

performa nce of t he a nt enna , w e sha ll suppose t ha t t he pola r iza t ion of

t he w a ve a nd t he impeda nce cha ra ct erist ics of t h e det ect or a re such t ha t

ma ximum pow er is a bsorbed. Th e a bsorbed pow er ca n t hen be expressed

as t he pow er incident on a n effect i~ -c a bsorbing a rea , ca lled t he receiving

cr os s sect ion ,

or a bsorpt ion cross sect ion A, of t he a nt enna .

I f S is

t he pow er flux densit y in t he incident w a ve, t he a bsorbed pow er is

P, = ASA,

(5)

The receiving cross sect ion w ill depen d on t he direct ion in w hich t h e

pla ne w a ve is incident on t he a nt enna .

We sha ll w rit e it a s A, = A,(d,I$),

w here o a nd @ a re t he spher ica l a ngles, a lrea dy defined, of t he direct ion


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4 S L R J E 1 OF J I I C lK )I V.4 J E .4.V7E .\ .VA D I


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S E C . 12] AN T E N N A PATTE R N S

5

Thus, if it is possible t o ign ore t h e effect of t he ea rt h on t he propa ga t ion

of t he w a ve a nd if G , is const a nt , it w ill be possible t o oper a te t he receiving

syst em sa t isfa ct or ily ever yw her e w it hin t he sur fa ce

(13)

w h er e t he t ra nsmit t er is t a ken t o be a t t h e or igin .

This sur fa ce w ill be

ca lled t he free-spa ce cover a ge pa t t ern for on e-w a y t ra nsmission.

Coverage Pattern, Two Ways. -

-I n most ra da r a pplica t ions t he sa me

a nt enna is used for t ra nsmission a nd r ecept ion .

One is h ere int erest ed

in det ect ing a t a rget , w hich ma y be cha ra ct er ized by it s ( sca t t er ing

cross sect ion u. This is t h e a ct ua l cross sect ion of a sph ere t ha t in t he

sa me posit ion a s t he t a rget w ould sca tt er ba ck t o t he r eceiver t h e sa me

a mount of en ergy a s is r et urn ed by t h e t a rget . F or t his fict it ious iso-

t r opic sca t t erer , t he effect ive a ngle subt ended a t t he t ra nsmit t er is

U/R2

a nd t h e t ot a l pow er in t er cept ed is

(14)

S ca t t ered isot ropica lly , t his pow er w ould a ppea r ba ck a t t h e t ra nsmit ter

a s a pow er flux, per unit a rea ,

(15)

Act ua lly , t h e sca tt er in g of most t a rget s is not uniform. The sca tt er ing

cr oss sect ion of t he t a rget w ill in a ny ca se-be defined by E q . (15), but it

w ill usua lly be a funct ion of t h e or ient a tion of t h e t a rget .

The pow er a bsorbed b:- t h e r eceiver from t h e sca tt ered w a ve w ill be

P,= A+S=R

(16)

since h er e G , = G ,. I f t h e effect of t he ea r t h cm t ra nsmission of t h e

w a ves ca n be neglect ed, it w ill be possible t o det ect t h e t a rget only w hen

it lies w it hin t he surfa ce

(17)

a bout t h e t ra nsmit t er a s a n origin .

This sur fa ce w ill be ca lled t he free-

spa ce covera ge pa t t ern for t w e-w a y t ra nsmissi,m.

The ext en t of t he covera ge pa t t erns is det ermined by cha ra ct er ist ics

of t h e syst em a nd t ar get out put pow er , receiver sensit ivit y , t a rget size

t ha t a re n ot under t he con t r ol of t he a nt enna design er . The form of

t h e cover a ge pa t t erns is det ermin ed by but is not t he sa me a s t h e for m

of t h e a nt enna t ra nsmit t ing a ,nd receiving pa t t erns; in t h e covera ge

pa t t erns, r is propor t iona l t o [G ,(o, r J )]J fia t her t ha n t o G ,(o, + ). Th e


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6 S UR VE Y OF M I C R OWA F E AI V7E iVA44 D E S I G .V PR OB L E WS

[SEC. 13

desired form of t he covera ge pa tt ern is la rgely det ermined by t he use t o

be ma de of t he syst em. F rom it , one ca n derive t he req uired form of t he

t ra nsmit t ing or receiving pa tt ern of t he a nt enna ; it is usua lly in t erms of

t his t ype of pa t t ern t ha t a nt enna performa nce is mea sured a nd specified.

I t is t o be empha sized t ha t t he discussion of covera ge pa tt erns gi~ en

(b)

(c)

(d)

FI G .I. I.Typica lga in-functionur fa cesor microwa vea nt enna s. (a ) Toroida l(omni-

directiona l)pa t t ern;(b) pencil-beampa t t ern;(c) fla t-t opfla redbeam ; (d) a sym metr ica lly

flaredbeam.

here a ssumes free-spa ce condit ions. In ma ny import a nt a pplica tions,

covera ge is a ffect ed by int er ference a nd diffra ct ion phenomena due t o

t he ea rt h , by met eorologica l condit ions, a nd by ot her fa ct ors. A det a iled

a ccount of t hese fa ct ors, w hich ma y be of considera ble impor t a nce in

det ermining t he a nt enna t ra nsmit ting pa t tern req uired t ora given a ppli-

ca tion, w ill be found in Vol. 13 of t he Ra dia tion I ,a bora tory S er ies.

103. Types of Microw a ve B ea ms.The most import a n t t ypes of

microw a ve bea ms a re illust ra ted in Fig. 1.1.

The lea st direct ive bea m is t he t oroida l bea m, 1 w hich is uniform in

1S uch a bea m is a lso r efer red t o a s om nidir ect ion a l.

(I RE S t a nda r ds a n d

Definitions,1946.)


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S E C .

1.4]

MI C R O WAV E TR AN S MI S S I ON L I N E S

7

a zimut h but direct ive in eleva t ion .

S uch a bea m is desira ble a s a ma rker

for a n a irfield beca use it ca n be det ect ed from a ll direct ions.

The most dir ect ive t ype of a nt enna gives a pencil bea m,

in which

t he ma jor por t ion of t he en er gy is con fined t o a sma ll con e of nea r ly

cir cula r cr oss sect ion . Wit h t he high direct ivit y of t his bea m goes a

ver y high ga in, oft en a s grea t a s 1000. In ra da r a pplica t ions such a

bea m ma y be used like a sea rchlight bea m in det ermining t h e a ngula r

posit ion of a t a rget .

Alt hough t he pencil bea m is useful for precise det ermina t ion of ra da r

t a rget posit ions, it is difficult t o use in loca t ing ra ndom t a rget s. F or

t h e la t t er purpose it is bet t er t o use a fa nned bea m, w hich ext en ds

t hrough a gr ea t er a ngle in on e pla ne t ha n it does in a pla ne perpendicula r

t o t ha t pla ne. Th e grea ter pa rt of t h e en er gy is t hen dir ect ed int o a con e

of roughly ellipt ica l cross sect ion , w it h t h e long a xis, for exa mple, ver-

t ica l. B y sw eeping t his bea m in a zimut h, on e ca n sca n t he sky more

ra pidly t ha n w it h a pencil bea m, decrea sin g t h e t ime during w hich a

t a rget ma y go undet ect ed. S uch a fa nned bea m st ill permit s precise

loca t ion of t a rget s in a zimut h , a t t h e expense of loss of informa t ion

con cer nin g t a r get eleva t ion .

Ot her a pplica t ions of microw a ve bea ms req uire t he use of bea ms w it h

ca refully sha ped pola r dia gra ms.

These include one-sided fla res, such

a s is illust ra t ed in F ig. 1I d, in w hich t h e pola r dia gra m in t he fla re

pla ne is roughly a n obt use t ria ngle, w herea s in t ra nsverse pla nes t he bea m

rema ins na rrow . In ra da r use, such a bea m a t t he sa me t ime permit s

precise loca t ion of t a rget s in a zimut h a nd a ssures most effect ive dist r ibu-

t ion of ra dia t ion w it hin t he ver t ica l pla ne of t h e bea m. Toroida l bea ms

w it h a one-sided fla re in eleva t ion ha ve a lso been developed.

No t heoret ica l fa ct ors limit a ny of t h e a bove bea m t ypes t o t he micro-

w a ve region , but ma ny pra ct ica l limit a tions a re imposed on long-w ave

a nt enna s by t he necessa ry rela t ionship bet w een t he dimensions of t he

a nt enna element s a nd t he w a velengt hs.

104. Microw a ve Tra nsmission Lines.-The form of microw a ve

a nt enna s depen ds upon t he na ture of t h e a va ila ble ra dia ting element s,

a nd t his in t urn depends upon t he na ture of t h e t ra nsmission lines t ha t

feed energy t o t hese element s. We t h erefor e prefa ce a survey of t he

ma in t ypes of microw a ve a nt enna s w it h a br ief descr ipt ion of microw a ve

t ra nsmission lines; a det a iled discussion of t hese lines w ill be found in

C ha p. 7.

U nshielded pa ra llel-w ire t ra nsm ission lines a r e not suit a ble for micro-

w a ve use; if t h ey a re not t o ra dia t e excessive y , t he spa cing of t he w ires

must be so sma ll t ha t t he pow er-ca rry ing ca pa cit y of t h e line is severely

limited.

U se of t h e self-shielding coa xia l line is possible in t h e microw a ~ t ~


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8

S (


22/635

coa xia l lines lend t hemselves t o sllch t ermina t ions.

Ma ny long-w a ve

a nt enna idea s ha ve lw en ra rr-ied uver int o t he micro\ ra ve region, pa r-

t ic~ ~ hw lyt hose con nect ed w it h t heha lf-]ra ve dipde; t he t ra msit iorr , ho\ v-

ever , is r iot r ner eiy a ma t t rr of w ovelcmgt h sca ling.

In a microl}a ve

a nt enna t l~ e cross-ser t iona l dimensions of t he t ra nsmission line a re com-

pa rt ih lc t o t he dimensions of t he ha lf-~ va vc dipole, a nd conseq uent ly , t he

coupling lmt w een t he ra dia tor a nd t ile line becomes a mor e significa nt

pr ol)lem t lia n in a cor resp(jn clin ~ I on g-iva ve sy st em . Th e cr oss-sect ion a l

dimensions of t h e dipole element a r e dso compa ra ble t o it s lengt h. A

t ypi~ a l microw a ve dipole is show n in Fig. 12c; t h e a na lysis a nd undt = r -

st ancling of S 1lC }Imicro}va ve dipoles is a t best st ill in a q ua lit a t ive st a ge.

The ose of hollow ~ va veyuide lines lea ds t o t he employment of en t irely

(L ff c,r en t r a d ia t i n g s ys t em s.

The simplest ra dia t ing t ermina t ion for such

a line is j~ lst t he open end of t he g~ lir le, t hrough w hich t h e en er gy pa sses

int o spa ce. The dimensions of t h e mout h a per t ure a re t hen compa ra ble

t o t he w a velengt h ; a s a result of diffra ct ion , t he en er gy does not cont inue

in a lw a m corr esponding t o t h e cross sect ion of t he pipe but sprea ds out

considera bly a bout , t h e direct ion of propa ga tion defin ed by t he guide.

The degr ee of sprea ding depends on t he ra tio of a per t ure dimensions t o

w a ~ ekmgt h . On fla ring or const r ict ing t he t ermina l r egion of t he guide

in order t o con t rol t h e direct ivit y of t he ra dia t ed en ergy , on e a rr ives a t

elect roma gnet ic horn s ba sed on t h e sa me funda ment a l pr inciples a s

a coust ic horns (F ig. 1.20 ).

An ot her t ype of element t ha t a ppea rs in microw a ve a nt enna s is t he

ra dia t ing slot (Fig. 1.2r). Th ere is a dist r ibut ion of cur ren t over t he

inside w a ll of a w a veguide a ssocia ted w it h t h e w a ve t ha t is propa ga ted

in t he int er ior . I f a slot is milled in t h e w a ll of t h e guide so a s t o cut

a cross t he lines of current flow , t he in t er ior of t h e guide is coupled t o

spa ce a nd energy is ra dia ted t hrough t he slot . (I f t he slot is milled a long

t he line of curr ent flow , t h e spa ce coupling a nd ra dia t ion a re negligible. )

I slot w ill ra dia t e most effect ively if it is resona nt a t t h e freq uency in

q uest ion. The long dimension of a resona nt slot is nea r ly a ha lf \ \ -a ve-

iengt h , a nd t h e t ra nsverse dimension a sma ll fra ct ion of t h is; t h e per im-

et er rJ t h e slot is t hus closely a w a velengt h.

1.6. A S ur vey of Microw a ve Ant enna Types.We a r e n ow in a posi-

t ion t o ment ion br iefly t he pr incipa l t ypes of a nt enna s t o be considered

in t his book.

Antennas jo~ Toroidal Beams.A

t oroida l bea m ma y be produced

by a n isola ted ha lf-w a ve a nt enna .

This is a useful a nt enna over a la rge

freq uency ra nge, t he iimit being set by t he mecha nica l problems of sup-

por t ing t h e a nt enna a nd a chieving t he req uired isola t ion. Th e bea m

t hus pr oduced, how ever , is t oo broa d in eleva tion for ma ny purposes.

A simple syst em t ha t ma int a ins a zimut ha l symmet ry but permit s

con t r ol of direct ivit y in eleva t ion is t he biconica l horn , illust ra t ed in


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10 AS UR VE Y OF M I CR OW.41E

.4,17fl,V.VAI)E L7[G .N-

R O13L J Y.tf,9 [S m.

16

Fig, 13. The prima ry driving element bet w een t he a pexes of t he coues

is a st ub fed from a coa xia l line. The sprea d of t he energy is det ermined

by t he fla re a ngle a nd t he ra t io of mout h dimension t o w a velengt h.

Alt hough t his a nt enna is useful ov~ r a

la rge freq ~ lency ra nge, ma ximum di-

rect ivit y for given a nt enna ~ veight a nd

size is obt a irmble in t he microw a ve

region , w here t he la rgest ra t io of

a per t ure t o w a velengt h ca n be

realized.

Increa sed direct ivit y in a t oroida l

bea m ca n a lso be obt a ined w it h a n

a rra y of ra dia t ing element s such a s

dipoles, dot s, or bimnica l horns built

up a long t he symmet ry a xis of t he

bea m. The direct ivit y of t he a rra y is

det ermined by it s lengt h mea sured in

~ va velengt bs; high direct ivit ies

a rc

convenient ly obt a ined by t his met hod only in t he microlva ve region . .1

t ypica l microw a ve a rra y of t his t ype is shoum in Fig. 1.4.

Pt,ncil-brum A nfrnnas.-Bearr~s

that

ha re direr t ivit y bot h in eleva -

t ion a nd a zimut h ma y be pr(xlllccd by a pa ir of dipole element s or by a

dipole w it h a reflect ing pla te.

The ma jor port ion of t he energy is con-

t a ined in a cone ~ rit h a pex a ngle somew ha t less t ha n 180.

F IG . 14 -.4. m ir mw a ~ c lm a ron a r ra y .

S imila r bea ms a rc prodllced by horn a nt enna s t ha t permit cont rol

of t he direct ivit y t hrollgh choice of t he fla re t ingle a nd t he n~ {)llt l] dimen-

sions. H orns a re useful a t lo\ ver freq uencies a s J VC 1l s in t he rnicrolra ve

region; indeed, t he ea r ly w or k on horn s Iva s don e for \ ~ t i\ -elen gt hsra nging

from 50 t o 100 cm.

More direct ive hea lns-t r lic pencil bemns-ca n be prt d~ lced b,v

building up spa ce a rya ys of t he a lmw syst ems. T\ \ -,)-t iimensior la la rra ys

(ma t tress a rra ys) a n{i mldt ,i,mit h orn syst ems a r c I IS C(Ia t l,,,-er freq uen-

cies. Their dircct ivit y is severely limit ed, ho\ \ -ever , hy t l~ e ]nrt ll:mica l

problems occa sioned by t he rcc(llired ra t io of (I imrnsions t o }f :L , t , -

Iengt hs. S uch a rra ys ha ve not been employe(l in t lie micro~ ~ -a ve regif )n.


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$.Ec. 1.6]

A S UR VE Y OF MI C R OWAVE AN TE N N A T Y PE S

11

At t hese w a velengt hs it becomes fea sible, a nd indeed very convenient ,

t o repla ce t he t w o-dimensiona l a rra y t echniq ue by t he use of reflect ors

a n d len ses .

(a )

(b)

FIG . 1.5.P enci l-beam ant ennas .

(a ) P a r a b oloid a I m ir ror ; (b) m et a l-pla t e len s. (Met cd -

plate lens photo~aph courtes~ of the B ell Telephone Labordori e8.)

H ighly direct ive pencil bea ms a re produced by pla cing a pa rt ia lly

direct ive syst em such a s t he double-dipole unit , dipole-reflect or unit , or


25/635

horn a t t he focus of a pa ra boloida l reflect or or x ccnt rosymrnet r ic lens,.

The use t )f t hese devices is ba sed

(J I I

t h e r {)n cr I>t s of r a y opt ics, a (cor (lin x

t o lvhich t hr reflect or or Ims t a kes t he dilcrgr t ]l ra ~ s fr


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S E C ,

1.7]

I M PE D AN CE 8PE CI F I CA T I ON S

13

On e of our ma jor problems w ill be t o est a blish t he rela t ionships a mong t he

prima ry pa t t ern of t he a nt enna feed, t he proper t ies of t he opt ica l ele-

ment s, a nd t he seconda ry pa t t ern ,

(a )

(b)

F m. 1.6.An ten na s for pm a ucm g ifa r ed bea ms.

(a ) S im ple fla r ed -bea m a n t en na ; (b)

on e-s id ed f la r e d -h a m syste m.

1.7. I mpeda nce S pecifica t ions.-The a chievement of a sa t isfa ct ory

a nt enna pa tt ern is by no mea ns t he only problem t o be considered by t he

a n t en na design er .

I t is import a nt t ha t t he a nt enna pick up ma ximum

pow er from a n incident w a ve a nd t ha t it ra dia t e t he pow er delivered t o

it by a t ra nsmission line w it hout reflect ing a n a pprecia ble por t ion of it

ba ck int o t he t ra nsmit t er . In ot h er w ords, it is import a nt t ha t t he

a n ten na h a ve sa t isfa ct or y im peda n ce ch a ra ct er ist ics.


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The impeda nce problem in micro~ va ve a nt enna design t a kes on a

some~ vha t specia l cha ra ct er beca use of t he cha ra ct er ist ics of ot her ele-

ment s of t he syst em, pa rt icula rly t he t ra nsmit ting t ubes.

Conventional

t riode-t ube oscilla tors a re not genera lly useful in t he microw a ve region .

This is due t o inherent limit a tions in t he t ube it self a nd t o t he fa ct t ha t ,

element s in t he t a nk circuit no longer beha ve like lumped impeda nces.

The self-resona nt freq ~ lency of t he ordina ry t ube is considera bly below

t he microw a ve ra nge, a nd it is t herefore impossible t o design a pra ct ica l

circuit , t ha t w ill oscilla te a t t he req uired high fre


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t rea tment of w a vefront s a nd ra ys.

C ha pt er 5 dea ls w it h t he in t era ct ion

bet w een elect roma gnet ic ~ va ves a nd obst a cles; t he genera l t h eory of

reflect ors is h ere developed a s a bounda ry-condit ion problem, a nd a

discussion is given of t he rela t ion bet ~ veen t his t heory a nd convent iona l

diffra ct ion t heory , w hich a lso finds a pplica tion t o microw a ve a nt enna

problems. Fina lly, C ha p. 6 a pplies t his t heory in t rea t ing one of t he

funda ment a l problems of a nt enna designt he rela t ion bet ween t he field

dist r ibut ion over t he a per t ure of a n a nt enna (such a s a lens or reflect or)

a nd it s seconda ry pa t t ern.

C ha pt er 7, on microw a ve t ra nsmission lines, serves a s in t roduct ion

t o t he cha pt ers on a nt enna feeds: dipole feeds, linea r a rra ys, a nd horns.

Of t hese t ypes a ll but t he first ha ve found a pplica tions a lso a s complet e

a nt enna s; t hese a pplica t ions w ill be indica ted in t hese cha pt ers.

A cha pt er on lenses precedes t he t rea tment of more complex a nt enna

syst ems w hich is orga nized a ccording t o t he t ype of bea m t o be produced:

pencil bea ms, simple fa nned bea ms, a nd more complexly sha ped bea ms.

When a n a nt enna is inst a lled on ground or a ship or a irpla negenera lly ,

enclosed in a housingit s performa nce is modified from t ha t in fr ee

spa ce by it s enclosure a nd neighboring object s. The subject of a nt enna -

iust a lla tion problems is discussed briefly t o a cqua int t he engineer w it h

t he phenomena t ha t ma y be expect ed t o occur a nd some of t he current ly

know n solut ions of t he problems.

The concluding cha pt ers provide a st a t ement of t he ba sic t echniq ues

of a nt enna mea surement s a nd a descr ipt ion of cert a in t ypes of mea sur-

ing eq uipment t ha t ha ve given sa t isfa ct ory service in t he Ra dia t ion

Laboratory .


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2.1.

C H AP TE R 2

C I RC U IT RE LATI ONS , RE CI P ROC I TY TH EORE MS

B Y s. S I I .VE R

I nt rod uct ion . h ch c cir cu it t h eor y con sid er a t ion s a n d t ech niq ues

cha ra ct er ist ic of low -f req ~ lency ra dio vw rk do not ca rry over in a simple

ma nner t o t he microlra ve region .

Thus, for exa mple, in t rea ting a cir-

cuit element a s a lumped impeda nce,

it is a ssumed t ha t t he curr ent

(a nd volt a ge) a t a ny given inst a nt ha s t he sa me va lue a t ever y point in

t he clement . This a ssumpt ion is va lid if t he dimensions of t he circuit

element a re sma ll compa red w it h t he w a velengt h , w it h t he result t ha t

t he ph ase differen ces bet w een sepa ra t ed point s in t he element a re negligi-

ble. I f, ho~ vever , t he w a velengt h becomes compa ra ble t o t he dimensions

of t he element , t hese pha se differences become significa nt ; a t a given

inst a nt t he cur rent a t one point in t he element ma y be pa ssing t hrough

it s ma ximum va lue, ]vhile a t a not her point it is zero.

In such ca ses t he

circuit element must be rega rded a s a syst em of dist r ibut ed impeda nces.

The ext ension of convent iona l circuit t heory t o microl$-a ve syst ems

is fur t her complica ted by t he use of circuit element s such a s w a veguides,

in w hich volt a ges a nd current s a re not uniq uely defined. The a na lysis

of t hese element s must be a pproa ched from t he point of view t ha t t hey

ser ve t o g~ lide elect roma gnet ic \ va ves; a t t ent ion is cent ered on elect r ic

a nd ma gnet ic fields ra ther t ha n on volt a ge a nd current . The fina l result

of t he field t heor y a na lysis is t ha t under s~ lit a ble condit ions~ ~ h ich a re

genera lly encount ered in pra ct icea lva veguide ca n be set in t o eq uiva -

lence w it h a t w o-w ir e t ra nsmission line in ~ vhich t he funda ment a l qua n-

t it ies a re volt a ge a nd current .

The la t t er a re direct ly rela t ed t o t he

w a veguides elect r ic a nd ma gnet ic fields, respect i~ -ely . 1 B y mea ns of

t his equiva lence t he concept s of impeda nce, impeda nce ma tching, a nd

loa ded lines a re ca rkied over t o ~ v avegllides.

A w aveguide ca n it self be t rea ted a s a syst em of dist r ibut ed imped-

a nces. i)ist r ibut ,ed impeda nces a re t rea ted in t he sa me ~ va y a s lumped

impeda nces, by use of K irchhoffs current a nd volt a ge la lm for net works.

A syst em of dist r ibut ed impeda nce ca n, in fa ct , be repla ced by a net lv(jrk

of lumped-impeda nce element s. The la t t er differ from t he convent iona l

ra dio-circuit element s in t ha t t heir impeda nce is a t ra nscendent al func-

I Th e su bject is t rm t ed in ( h a p. 7.

A fl[ll t r r a t m m t of t h e ext [~ lw i [,]]f r iw I I I t

t h w r y t o w : lv (g ui(lc+ \ r i

11LL foun{i i,] }-:)1 8 of this .wui{s,

16


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t ion of freq lleucy ra ther t ha n a n a lgeljra ie funrt i{)n. B y mea ns of t llesc

equ iv a lent l ll rr lpr {i-eler r lell t net \ \or k s,

t he net \ \(J r fi t heorems t ha t a re

a pplica b le t u lolr -fr efl~ len r y l~ llll])ecl-elelll[t lt n et lvor fis a r e ca r lied ov er

t o syst ems \ \ it h dist ril)ut ed impeda n ce

1he frost pt ir t of t his cha pt er

\ vill r ev iew s ever a l n ct }i-or k t h eor em s

LLnd t}le

t ]vo-}rir .e t r:L r]sr]lissi(jI1-liIle

t heory t ha t a re I lsed in micr t jj :~ vc circuit t heory.

1he s (l ljject s \ \ i llbe

t rea ted br iefly , t he rea der I )eing refer red t t ) st a nd:lrd t ext s for mor e

com plet e discussion s :~ n (l pr oofs of t ile result s q llot ed h em .

The rela t ion Ix%veeu a t ra nsmit t ing a nd u mceir ing a nt enna a lso

ca n be expresse(l in t erms of a n eflllivt ilcnt net ll or k.

In t his \ ra y on e

ca n a rr ive a t a reciprocit y t heorem J I hich rel:~ t es t he t ra nsmission cha r-

a ct erist ics of a n a nt enna t o it s receiling cll:t r~ lct t list i(s. of pa rt icula r

import a nce t o a nt enna ([esigr l is t ile fa ct , proved I )y I Iscef t he r eciprocit y

t heorem, t ha t t he t ra nsmit t ing pa t t ern of a n a nt enna is t he sa me a s it s

receiving pa tt ern. The reciprocit y t heor em \ vill be discussed in t he

I a t t er pa rt of t his cha pt er .

2.2. The Four-t ermina l Net work. I .et usconsider a n a rl)it ra ry net -

nw rk, fr ee from genera tors, ma de 11P of linea r bila t era l element s. A

linea r bila t era l element is one for

~ 1

\ ~ hich t he rela tion bet ~ veen ~ olt a ge ~ .

i2

Oc

a nd current is linea r:

V =

IZ,

(1) ~ j

1

a2

OD

w her e t he va lue of t he impeda nce Z

FI O.21.-IJ our-t cr]rlird net w ork.

is independent of t he direct ion of t he

volt age drop a cross t he element .3

F or con ven ien ce t he n et ~ ~ ork \ vill be

pict ured a s enclosed in a box a nd present ing t o t he out side only a pa ir

of input a nd a pa ir of out put t ermina ls.

Th is is illu st r a t ed s ch em a t ica l ly

in Fig. 2.1. A boxed net ]~ ork of t his t ype is r efer red t o a s a four-t ermina l

or t w o-t er min a l-pa ir n et w or k.

The net w ork a s a unit involves four qua nt it ies: t he current i,, t he

volt a ge drop VI from :4 t o n, t he mlrrent L, a nd t he volt a ge dr op Vz

from C t o D . In conse(luence of t he linea r proper t y [F ;q . (l)] of ea ch

component element of t he net \ vork, t he rela t ions bet ween t he, volt ages

Vl, Vz a nd t he current s il, i, a re linea r:

VI = Z1lil Z12i2j

V2 = Z21i1 Z22i2.

}

(2)

1 W. L . E v er it t , C om m un icd fon E n g in per ir q ,

l lfcCr :Lw-Hill ,New York, 1937;

E. A. G uil lcmin,CowLmunica I ionVe~mrks,Vols. 1, 11, l~ ilr y , Xm- Yor k , 1931;T. E .

S hea ,

T ransmi ssi on V dw;or ks md I T -W. F i ller ,

Y : in Nostra nd,New lork, 1 129,

z S ee C hap . 1 for t he def in it ion .sof t hesep at t er ns

3I t is a s su medt h a t w e a r e d ea l in g w it h a sin glefr eq u en cy ,t h a t bot h t h e volt a g e

a n d cu r ren t d epen d on t im e t h r ou gh t h e s a m e fa ct or e~ u ~ .


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18

C I R C U I T R E L A 7 ON J S R E C I PR OC I T Y 7H hOE E M S

[SE (. 22

The impeda nce coefficient Z,, is t he input impeda nce a t All w hen CD

is open-circuit ed (zZ = O); simila rly ZZZ is t he input impeda nce a t

CD

w hen Al? is open-circuit ed. The qua nt it ies Z,, a nd Z21 a rc know n a s

t he t ra nsfer impeda nce coefficient s of t he n et w ork. As a result of t he

bila tera l proper t y of t he component element s of t he net w w rk, t he t ra nsfer

im peda n ce coefficien ts sa t isfy t he r ecipr ocit y r ela t ion l

Z,2 = Z21,

(3)

As a n a l~ erna t ive t o t he rela t ions expressed by 13q . (2), t he current s

ma y be expressed a s linea r funct ions of t he volt age:

iI = Y1l V1 F12V2,

iz = Y21V1 Y2212. } (4)

The a dmit t a nce coefficient Y,, is t he input a dmit t a nce a t All w hen t he

terminals

CD

a re short -circuit ed; Yz2 is t he a dmit t a nce a t C D \ vhen

A B

is short -circuit ed; a nd Ylj, YZ1 a r e t he t ra nsfer a dmit ta nce coefficient s.

The la t ter coefficient s sa t isfy a reciprocit y rela t ion

Y,2 = Y2,

(5)

in t he ca se of bila tera l element s. Th e im peda nce a nd a dm it ta nce coeffi-

cient s of t he net w mi-k a re rela t ed:

Y ,, = y;

y,2 = :;1; yz, = ylz = ~,

(6)

where

A = Z,IZ22 Z,ZZZ,.

(7)

B y vir t ue of t he reciprocit yy rela t ions, [E qs. (3) a nd (5)], t he net w ork

ha s only t hree independent pa ra met ers, C onseq uent ly it ca n be repla ced

by a net w ork of t hree lumped-im-

TYA:DZ

peda nce element s a rra nged in t he

form of eit her a T- or ~ -sect ion a s

show n in Fig. 22. The imped-

a n ce elem en ts of t he T-sect ion a r e

B

T-sect,on

D B

r -

sect,..

D

designa t ed by Z,, Z2, Z~ . In t he

I]o. 2.2.h-a nd m-sect ionequiva lent sof :L

ca se nf t he H -sect ion it is m or e con -

four-terminalnetwork.

venient t o (Ise a dmit t a nces; t he

element s a re designa t ed by YA =

l/ZA,

YE = l/Z3, Yc = l/ZC. The

r ela t ion s bet w een t he elem en ts of t h e r educed n et w or ks a n d t h e coefficien ts

of E qs. (2) a nd (4) a re

a . T-sect ion :

ZI = = Zll Z,2,

Z2 = Zn z,,,

1

(8)

Z3 = Z12,

1~ . ,1. G uillerllill, op.

cd.,

Vols. I , I I, Wiley , N ew Yor k, 1331, pa r t icula r ly

Vol. 1, ~ h a D . I V.


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S E C .2.3]

T H E R A Y L E I C H I { E C I I K WI T I T H J ?OR E i bl

19

b.

II-section:

Y,4 = YI1 Y12,

Yc = Y22 Ylz,

/

(9)

Y, = Y12.

The rela t ions bet w een t he T- a nd I I-sect ion element s for one a nd t he

sa me four-t ermina l n et w ork a re gi~ en by

(10)

w here t he qua nt it y A is t ha t defined in E q . (7).

The net w or k ca n a lso be cha ra ct er ized by a ny t hr ee of t he follow ing

mea sura ble q ua nt it ies: t he input impeda nce a t A B \ vhen C D is short -

circuit ed, t he input impeda nce a t A B \ vhen C D is open-circuit ed, t he

input impeda nces a t C ~ w hen AB

is open-cir cu it e d or s hor t -cir cu it e d,

The rela t ions bet w een t hese qua n-

t it ies a nd t he impeda nce coeffi-

cient s or t he I - a nd I I-sect ion

elem en ts ca n ea sily be der ived fr om

E q s. (2) a nd (8) or (9); t hey a r e

given explicit ly by E ver it t . 1

2.3. The Ra yleigh Reciprocit y

Th eor em .Th e r ecipr ocit y r ela t i on

bet w een t he t ra nsfer impeda nce co-

efficient s given in E q. (3) is funda -

ment a l t o t he va rious recipr ocit y

t heorems pert a ining t o net ]vorks.

Ml of t hese t heorems a re va ria nt s

of t he genera l t heorem der ived by

Ra yleigh. The pa rt icula r form of

(a )

a32

b)

l:l


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20 CIR~l 17 R E L A T1OV,S , R I ?(lPR ()([71 T I I l


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S E C .25]

T H E T WO-WI R E TR AN S MI S S 1O.V L I N E

21

C onsider t hen t he ca se in w hich t he loa d Z. is fed by t h e gen era t or t hrough

a four-t ermina l net }vor k, t h e genera t or emf being l~ a nd t s int erna l

impeda nce Z. (Fig. 2.4). The four-t ermina l n et w or k ma y be r epla ced

by it s T-sect ion eq uiva lent a s show n. B y Th6venins t heorem t he S YS -

t em is eq uiva lent t o a gener a t or of emf

VCZIJ(ZI, + Z.)

a nd int erna l

impeda nce 212 Z~ J (Zli + ZG )

feeding t he loa d impeda nce Z. di-

V(z

v(z+ r12)

rect ly . I t follow s t hen t ha t ma xi-

~

m um -pow er t ra n sfer w ill be a ch ieved

w it h a loa d t ha t is t he complex con -

; ; ,(z)~

+ Z

juga te of t h e in t erna l impeda nce of

,_7(z + (/2)

t h e effect ive g en er a t or :

1

(2:2)

z. = 2;2 ~ +

z:

(12)

2.5. Th e Tw o-w i re Tr a nsm ission

Line.-0ne of t he most import a nt

d ist ribu ted -im ped a nce sy st em s fr om

t he point of view of a nt enna t h eor y

is t he t w o-w ire t ra nsmission line. 1

-7or t he pr esent t he line ~ vill be con- - ~ ,

sidered in it s convent iona l form, a s a

I

I

I : lG.25.-rw w irere line,

pa ir of linea r conduct ors in a pla ne,

w hich support t h e propa ga t ion of a w a ve of w a velengt h sma ll com pa red

w it h t he lengt h of t he lines The problem of in t erest is t he dist r ibut ion

of volt a ge a nd curren t a long t he line for a w a ~ ~ eof single freq uency , in

w hich t he volt a ge a nd curr ent va ry w it h eiui.

The line is show n schema t ica lly in Fig. 2.5 as a pa ir of pa ra llel \ vires.

In genera l, h ow ever , t he spa cing bet lveen t be }vires ma y va ry a long t h e

line; t he only rest r ict ion imposed is t ha t t he line ha ve a n a xis of sym-

met ry . P osit ion a long t he line is specified by t he coordin a t e z a long

t he symmet ry a xis. I t is fur t her a ssumed t ha t t he ]ille is isola t ed from

pert urbing object s, so t ha t a t a ny posit ion a long t he line t he curr en t s

a t ever y inst a nt ma y be eq lml a nd opposit e in t he t ]vo component , lines.

The pr oper t ies of t he line a re specified by it s dist r ibut ed pa ra met ers:

(1) t he ser ies impeda nce per unit lengt h,

3(2) = I i(z) + jd(z),

(13a)

w here R(z) is t he ser ies resist a nce a nd L(z) t he ser ies induct a nce per

unit lengt h, t a king bot h com ponent lines t oget her , a nd (2) t he shunt

W. L. E vcr it t , op. cit . F or a ver y com plet e t rea t men t th e m a dcr is rcfm rx,d

t o I t . WT.K in g, H . I t . Nfim no,.\ . H . \ Vin ,q ,Transmission

Li7Les , .I rttenrzas , ad

11w e

C(,ifk.s,

MrGra v-Hil l, \Tcw York, 1945, (ha p. 1.


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22 C I R C U 17 I f l/ 1T I O.Y S , 1{1{611l{U C1 I Y lH E Oli 8.%l,?

[s);,, 2.5

a dmit ta nce per unit lengt h,

71(z) = G (z) + juC (.z),

(13b)

w h ere G (z) is t he t ra nsverse conduct a nce a nd C (z) t he ca pa cit a nce per

unit lengt h bet ween t he component members of t he line.

Th ese pa r a m-

et ers ma y be funct ions of posit ion beca use of va ria t ions in t he cond~ lct ors,

in t he spa cing bet lveen t he la t t er , or in t he st ruct lu-e of t he surrounding

d ielect r ic m ed iu m.

Ta king eit her conduct or for reference, let i(z) be t he current a t t he

point 2 a nd V(z) t he volt a ge drop from t he referen ce conduct or t o t he

ot h er member a t t he sa me point .

To obt a in t he spa ce dependence of

i(z) a nd V(z), consider a sect ion of line of lcngt h dz a bout t he point z.

Applying Ohms la w, \ ve ha ve

V(2 + dz) J (z) = i(2)m3(z) dz

a nd

i(Z + dZ) i(Z) = ~ (Z)y?(Z)

dz

for , respect ively , t he ser ies a nd shunt rela t ions a cross t he element of

line. The t erms on t he left -ha nd side, by use of Ta ylors t heorem,

b ecom e (d V/d z) dz a nd (di/dz) ck respect ively . Thus t he different ia l

eq ua t ions of t he line a re found t o be

dV =

z

(~(z)i(z),

di

dz =

m(z) v(z).

(14G)

(14b)

S e con d -or d er d iffer en t ia l

obt a ined by elimina ting

t h es e eq u a t ion s:

a :y _ [

eq ua t ions for volt a ge a nd current a lone a re

volt a ge or current from one or t he ot her of

(15a)

(15b)

From a genera lized p~ ~ int of view , E qs. (14) ra n be rega rded a s t he

defin it ion of a t w o-w ire t ra nsmission line. Tha t is, given a physica l

syst em support ing a w a ve w it h t ime dependence ew ~ ,t he propa ga tion

of w hich is expressible in t erms of a single coordina t e z a nd t w o qua n-

tities

(i, V)

rela ted by equa tions of t he form of E qs. (14), it is possible t o

set up a t ~ vo-w ire line represent a t ion for t he syst em.

The volt a ge a nd

current of t he eq uiva lent line a re direct y proport iona l t o t he w a ~ e qua n-

t it ies ent er ing t he different ia l equa tions, a nd t he ser ies impeda nce a nd

shunt a dmit t a nce per unit lengt h of t he eq uiva lent line a re proport iona l

t o t he coefficient s of t he w a ve qua nt it ies in t he different ia l equa tions.


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S m. 26] 1H E H O M OG I lN E (31i ,V 71{.4 .V ,V , ll,~,~I O.V I ,l:vl 1,

/

~ = ~ o if

R

1


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32

C I R C U I T R E L A TI Oi V S , R J 7CI PR OCI T Y T H E OR E M S [SEC. 2.8

This follow s from t he fa ct t ha t r is rea l w hen ~ is rea l a nd from t he rela -

t ions of E q . (47) bet w een t h e va lue of ~ w h en it is a rea l num ber a nd t h e

st a ndin g-w a ve ra t io.

To illust ra t e t hese rela t ionships let us suppose t ha t t he st a nding-w a ve

ra t io r ha s been mea sured on a given line, t oget h er w it h t he posit ion of

a volt a ge minimum; t he reflect ion coefficien t a nd impeda nce a re desired

a t a point a dist a nce 1 from t he minimum posit ion a w a y from t h e gen -

erator .

I t ~ rill be reca lled [E q . (46 b)] t ha t a t a ma ximum posit ion t he

pha se of r is equa l t o m; r is t hen direct ed a long t he nega t ive rea l axis

The im peda nce a t t his point is rea l, being R/Z~ = l/r . The vect or ~

t hus ext encls from t h e origin t o t h e circle correspondin g t o

R/ZO =

l/T-.

C oun t erclockw ise rot a tion of t his vect or t hrough a n a ngle 2@ ca rr ies us

t o t he desired poin t on t h e cha r t ; t h e com ponent s of f a t t ha t point a re

rea d off from t he pa ir of in t ersect ing circles. I t w ill be not ed on F ig.

27 t ha t t he per iph ery of t he cha rt ca r r ies a pha se a ngle sca le w it h t h e

pha se designa ted by t he ra tio of line lengt h t o w a velen gt h .

Th e S mit h cha rt ca n a lso be used t o st udy t he a dm it t a nce t ra nsform a -

t ion . F irst it should be n ot ed t ha t t h ere a re t w o conven t ions for t h e

defin it ion of a dmit t a nce. The con vent ion a dopt ed in t his book defines

t h e norm a lized a dmit t a nce q = (G / Yo) + j(B / I 0) t o be t h e reciproca l

of t he norma lized impeda nce { = (R/ZO) +

j(X/Zo);

posit ive suscepta nce

B t hus correspon ds t o nega t ive (ca pa cit ive) rea ct a nce. Th e ot h er

conven t ion defines t he a dm it t a nce t o be t he conjuga te of t h~ reciproca l

impeda nce, in or der t ha t posit ive suscept a nce (like posit ive rea ct a nce)

should be in duct ive.

The use of t h e la t t er con vent ion cha nges t h e use

of t he ch a rt in w a ys w hich t h e rea der ca n ea sily develop.

E q lla t ion (36) gives t he rela t ion bet lreen t he a dm it t a nce a nd t h e

v olt a g e r eflect ion coeiiicien t :

lr

= l+ r

I ,et us define a ne]r coefficien t

T= r (51)

a nd a ssocia te I r it h it a com plex pla ne ~ vit h a xes ~ R e a nd 11~ .

(Actually

t h e sa me com plex pla ne serves for bot h I_a nd T, t he t w o vect ors m a king

a n a ngle of 180 ~ ~ -it hea ch ot h er . ) The vect or T is, in fa ct , t he curren t

rejection coefficient,

expressing t h e ra t io of t he a mplit ude of t he reflect ed

curren t ~ va ~ -et o t h e a mplit ude of t h e inr iclent current w ave.

Th e la w

of t ra nsforma t ion of 1 a lon~ t h e line is )w ciw ly t h e sa me a s t ha t given

for r by E q . (49). On subst it ut ing 11~ . (51) in t o t he i-ela t ion bet w een q

a nd r , w e obt a in

1+ r

= lT

(52)


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G /Y, = const a nt a re a fa mily of circles t ha t coincide w it h t he const a nt

R/ZO fa mily in t he ~ -~ t r t iusfornmt ion a nd t ha t t llc clu~ mB /}-O = con-

st a nt coincide ~ r it h t he .YjZO circles.

I i-it h rw pcct ([) t l]e l:~ t t er it

should be not ed (in using t he cll:w t for a dmit t a nce) t ha t t he curves lying

in t he upper ha lf pla ne r epresent ca pa cit ive slw cept a m,e.

he dis-

t inct ion t ha t need be ma de l)et ~ w xm t he use of t ile cha rt for impml:mce

a nd a dmit t a nce ca n he m:de clea r by considering t ile pr(]l)lem of fin[ling

t he a dmit t a nce a t a point dist :mt 1 from a volt a ~ c nlinin~ ~ l]n in t ile (lirm-

t ion a lva y from t he gener :lt or , t he st a ndin~ -i~ :hvc mt io :L*a in I w ing r .

At a v ()lt a g e n lin in lllr ~ l llliesa l c~ r lg t }len cg :~ t ilel e:ll:~ \ i~ ;lle1l(e ~ ext en ds

a long t he posit i~ -e r ed a xis t o t he cirr le

G

l,

= r .

The st a rt ing point t hus lies on t he posit ire rra l a xis, inst ra d of on t lw

nega tive a xis, lIoving a long t he line :L\ Ya yfrom t llc gcnr);lt ol :L~ :lin

r ot a t es T in t he posit ive sen se (collr lt c,r (,lt i(k}t isr ) t llrollgl~ a n :L ]lglr 21j~ .

The a dmit t a nce a t t he ne~ r point is (lct ernliuml from t ile ll:~ ir t )i in t (,r-

sect ing coordina t e curves, jllst a s in t ile c:w c of t lw inllmi:lncr . I t

should be clea r t ha t t he a dmit -

A

t ance a nd impeda nce point s on t he ~ ~ , (%)

S mit h cha r t for one a nd t he sa me

pnint on t he line a re dia met r i-

ca lly opposit e t o one a not her ,

Th e S mit h ch a r t is p:llt ic[lla r ly

suit ed t o t he st lldy of :m imlw d-

a nce misma t ch t ha t a rises fr t )rn

t he superposit ion of rcflrct ions.

&

Forexample, thereln:~ yl)(, fi~(, li rs

Ir l

.,

of dlscont mu it ies on a t r :~ nsnlis-

sim lin e; t he over a ll r efler(i(}n cc)-

efficient a t a given point ) is, t o a

good a pproxima t if)n, t l~ e ler t (jr

sum of t he reflw ,t ion c(mffi(irn t s

11


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34 C I R C C I T R E L A T I O.V S , R E CI PR OCI T Y T H E OR E M S

[S E C 2 t 3

ha lf pla ne cont a ining t h e posit ive rea l a xis comes under considera t ion .

The impeda nce (a dmit t a nce) is represent ed in t his pla ne by a vect or

from t he origin. Wit h r efer ence t o t he a dmit t a nce w e not e a ga in t ha t it

is t a ken her e t o be t he reciproca l of t he impeda nce.

One a nd t he sa me

pla ne serves for bot h impeda nce a nd a dmit t a nce; F ig. 2.8 show s t he

rela t ion bet w een t he impeda nce a nd a dmit t a nce point s in t he pla ne

for a given point on a t ra nsmission line.

Th e im peda n ce t r a nsfor ma t ion

(2.30)

does not t a ke so simple a form in t he ~ -pla ne a s did t h e reflect ion coeffi-

cient t ra nsforma tion in t he T-pla ne. D ispla cement a long t he line pro-

duces a cha nge in bot h t he ma gnit ude a nd pha se of t he impeda nce.

The geomet r ica l t ra nsforma t ion is simplified by in t roducing t w o

fa milies of circles: t he curves ]r l = const a nt a nd t h e curves r-pha se =

constant .

These curves a re obt a ined from t he 17-~ t ra nsforma t ion

r = (~ 1)/({ + 1) of E q . (30). Writ ing r = {r]e@, w e find t ha t

a nd

,r,2=(:-)2+EY

(E+)2+(9

()

an(a:h

hese ca n be rew r it t en a s

respect ively . I t w ill be seen t ha t t he curves I I = const a nt a nd@ = con -

st a nt a re circles. The circle for a given II 1 ha s it s cen t er on t he rea l

a xis a t a dist a nce (1 + I r l 2)/(1 Ir] 2, from t he origin ; it s ra dius is

21rl/(1 Ir lz). C urves of const a nt I r I a re a lso curves of const a nt

st a nding-w a ve ra t io. B y E q . (43 b), w e find t ha t t h e cen t er of t he circle

is a t (T2 + 1)/2r a nd t ha t it s ra dius is (rz 1)/2r. The circle in t ersect s

t he rea l a xis a t t he point s l/r a nd r , corresponding t o t he va lues t ha t w e

obt a ined previously [E q . (47)] for t h e impeda nce a t t hese point s on t he

line w her e it is rea l. These t w o point s on t he cha rt t hus correspond t o

point s on t he line a t w hich t h e volt a ge minima a nd volt a ge ma xima ,


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%c. 2.8]

71{.1 .Y,SI ,OR .\ fA 11O.1- (H A 1


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36

C I R C C I T R E I ].4 T I O.J -,S , R I


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a ge a nd current a t t he out put end z = 1.

F rom t h elin e eq ua t icm slI ;q s.

(18) a nci (19)] l~ e ha ve t hen

2=():

v, = A+ . 42,

Zl =

,1-

.41 AZ);

Ao

2=1:

T2 = .4,(7~ + A2c7~ ,

1

i2 = r-- (41@ ~ 2(j7/),

k

U sing t he t w o current eq ua t ions t o solve for (I , a nd Aj in t erms of i,

a nd

iz

a nd subst it ut ing in t o t he volt a ge eq lmt ions, ~ ve obt a in

VI =

Z,lil

Z,2i2,

V2 = Z21i~ Z22i2,

1

(55)

wi th

Z,, = Z,, = Z,cot h(~ l), (56a )

Z12 = 22, = Z, csch (71).

(56b)

We t hus find direct ly t ha t t he net w ork is linea r a nd t ha t t he t ra nsfer

impeda nce coefficien ts sa t isfy t he recipr ocit y r ela t ions.

S ince t he line

is homogeneous, t he net work is symmet rica l ~ vit h respect t o it s t I ro ends;

hence Z,, = Z,z. F or a nonlossy line ~ = j~ ; on subst it ut ion in t o t he

a bove, t he net w ork pa ra met ers a re found t o be

Z,, = ZZ2 = jZO cot fil,

(57a)

Z12 = Z,, = jZO csc fit .

(57b)

TRANSMITTING AND REC EIVING ANTENNAS

2.10. The Ant enna a s a Termina ting I mpeda nce.The impeda nce

rela t ions bet w een a t ra nsmit t ing or receiving a nt enna a nd it s t ra nsmis-

sion line a re of pa rt icula r in t erest .

In t he follo~ y ing sect ions severa l

genera l idea s t ha t a re a ssocia ted w it h t he a na lysis of t hese rela t ions w ill

be discussed. I ,et us consider first t he ca se of a line feeding a t ra nsmit -

t ing a nt enna . I t w ill be a ssllmed for t he presenk t ha t t he a nt enna is

isola tedin pa rt icula r, t ha t it is removed from a ll ot her a nt enna sso

t ha t int era ct ions w it h ot her syst ems need not be considered. The

a nt enna fllnct ions like a dissipa t ive loa d on t he line in t ha t it ext ra ct s

pow er from it ; pa r t of t his energy is ra dia t ed int o spa ce, a nd pa rt is

dissipa ted int o hea t in t he a nt enna st r {lct ~ n-e. In genera l, t he a nt enna

does not a bsorb a ll of t he pofver incident on it from t he line b~ lt gives r ise

t o a reflect ed I ra ve in t he line; in effect t lw line is t m-mina t ml bv a n

im ped :m ce d iffr r en t fr om it s cli:~ l:~ (t cl.is t it i m ped a n re.

I loivever , t he

definit ion of t he t rrmina l imped:mre reprrw nt ing t he a nt enna is not fr ee

fr om a m bigllit y a n d r eq llir es som e C on si(lr mt ion .

I t is t o be not rd first t h :lt t he definit ion of a t erminfil impw la nrr


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,

38

CI RCC71T R E I ,A TI O.VS , R I ? C PR OC T1 T H E OR E MS

[S E C . 2.10

implies t he ident ifica t ion of a driving point , or set of input t ermina ls, for

t he a nt enna . In some ca ses, such a s t he ha lf-w a ve dipole or rhombic

loop a nt enna s fed from a t w o-w ire line a s illust ra t ed in Fig. 1.2, t he

st ruct ura l discont inuit y bet }f-een t he line a nd t he ra dia tor sllggest s a

driving point . lhis, how ever , is not enough; it is necessa ry t ha t t he

current dist r ibut ion in t he line be t ha t cha ra ct er ist ic of a t ra nsmission

line up t o t he a ssigned driving point .

At long w a velengt hs t his condi-

t ion is rea lized w it h t he a nt enna s cit ed a bove: t he int era ct ion bet w een

t he a nt enna a nd line ca n be represent ed by a lumped rea ct ive impeda nce

a cross t he driving t ermina ls in pa ra llel w it h t he impeda nce cha ra c-

t er ist ic of t he a nt enna it self.

At short w a velengt hs, how ever , t he in t er-

a ct ion bet w een t he ra dia t ing syst em a nd t he line ca uses a per t urba tion

of t he current dist ribut ion on t he la t t er t ha t ma y ext end ba ck over a n

a pprecia ble dist a nce; elect rica lly t her e is n o poin t of t ra nsit ion fr om t ra ns-

mission-line cur ren ts t o a nt enna curren ts.

This is a pa rt icula rly cogent

point in t he ca se of microw a ve syst e,ms t ha t ma ke use of w a veguide

lines, in w hich t he elect roma gnet ic fields exist in t he form of a number of

modes.1

A w a veguide is equiva lent t o a t w o-w ire line only w hen it is

support ing propa ga t ion of a w a ve in a single mode.

Microw a ve lines

a re, in fa ct , genera lly so designed t ha t t hey ca n support free propa ga tion

of only on e mode. Nevert heless, t hough a single mode is incident on t he

a nt enna , t he a nt enna it self excit es ot h er modes, in a ddit ion t o giving rise

t o a reflect ed w a ve in t he incident mode.

I t is only a t poin t s so fa r

from t he a nt enna t ha t t he ot her modes ha ve been a tt enua ted t o negligible

a mplit udes t ha t a w a veguide is eq uiva lent t o a t w o-w ire line.

Attention

should a lso be ca lled t o t he a bsence of a unique driving point in ca ses

w here t he t ra nsit ion from t he line t o t he ra dia t or is effect ed by a con-

t in uous st ruct ur a l t ra n sit ion . An exa mple of t his is a w a veguide fla ring

gra dua lly int o a horn w it hout st ruct ura l discont inuit ies in t he w a lls.

In t hese ca ses, a ga in, t he t ra nsit ion from t ra nsmission-line current s t o

a nt enna current s ca nnot be loca lized t o a point .

The a ct ion of a n a rbit ra ry a nt enna a s a t ermina l loa d on t he line ca n

be specified in t erms of t he reflect ion coefficient r mea sured in t he t ra ns-

mission line, a t a point so fa r from t he a nt enna t ha t it s only effect is t he

product ion of t he reflect ed t ra nsmission-line w a ve. At a ny point in t he

t ra nsmission-line r egion a n impeda nce (or a dmit ta nce) ca n be det er min ed

from t he mea sured r , by mea ns of E q . (30); t his ca n be t a ken a s t he loa d

impeda nce t ermina t ing t he line a t t ha t point .

F urt hermore, a ny such

point ma y be rega rded a s t he junct ion bet w een t he line a nd t he input

t ermina ls t o t he a nt enna in so fa r a s t he pra ct ica l a na lysis of t he syst em

is concerned .

This ra ises t he quest ion of t he represent a t ion of a n a nt enna by a n

equiva lent net ~ vork. There is no unique net w ork a ssocia t ed w it h a

1Se e C ha p . 7.


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SE


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40

C I R C U I T 12E L A T I ON S , R E C I PR OC I T Y T H E OR E M S

[SEC.

211

point of t he line) t o t he genera t or .

I t follow s from S ees. 24 a nd 2.9,

t he line w ill t ra nsfer ma ximum pow er from t h e genera tor t o a t ermina l

loa d of im peda n ce

z; Csc

@

ZL = jZO cot @ + z: + jzo

cot@

(60)

w here Zc is t he int erna l impeda nce of t he gen era t or a nd 20 is t he cha r-

a ct er ist ic impeda nce of t h e line.

I f t he a nt enna impeda nce Z. is

d iffer en t fr om Z., it is possible t o int roduce a rea ct ive n et w ork bet w een

t he input t ermina ls of Z. a nd t he line, w hich (a t on e freq uency a t lea st )

t ra nsforms Z. in t o Z.; t his net w ork w ill effect ma ximum-pow er t ra nsfer

t o t he a nt enna .

I t is t o be not ed t ha t in m icrow a ve syst ems a not her ma t ching prob-

lem exist s: The cha ra ct er ist ics of t he gen era t or a re such t ha t t he reflect ed

w a ve in t he line must be elimina t ed. This req uires t ha t t he a nt enna

impeda nce Z~ be t ra nsformed int o ZOin genera l a different t ra nsforma -

t ion from t ha t req uired by t he ma ximum-pow er-t ra nsfer condit ion . In

t hese syst ems t he genera tor must be independent ly ma tched t o t he line;

t he gen era t or int erna l impeda nce Z. is t ra nsformed int o ZO w it h t he

result Icf. E q . (60)] t ha t t he ma ximum-pow er condit ion t hen coincides

w it h t he condit ion for elimina ting t he reflect ed w a ve in t he line.

2.11. The Receiving Ant enna S yst em.The eq uiva lent circuit repre-

sent a t ions used in discussing receiving a nt enna s a lso need exa mina t ion .

C onsider a n a rbit ra ry a nt enna it ma ybe a cer~ t er ;dr iven dipole, a horn ,

or a combina tion of such element s w it h reflect ors a nd lensesfeeding

int o one end of a t ra nsmission line t ha t a t t h e ot her end is t ermina t ed

in a pa ssive loa d impeda nce. (Tha t is, t h e receiving circuit is free from

genera tors.) When a n ext erna l elect roma gnet ic field fa lls on t he receiv-

ing-a nt enna syst em, t he int era ct ion bet w een t he a nt enna a nd t he field

gives r ise t o a w a ve in t he line.

The a nt enna ma y be rega rded a s a

device t ha t t ra nsforms energy ca rr ied by a free w a ve in spa ce int o en ergy

ca rr ied by a guided w a ve on t he t ra nsmission line.

F rom t he point of

view of t he t ermina l loa d, how ever , t he a nt enna funct ions a s a genera tor ,

a nd it is cust oma ry. t o repla ce it by a genera t or in discussing t he efficiency

of t he receiving syst em a s it depends on t he a nt enna , line, a nd loa d

impeda nces. I t is our purpose t o discuss t he na t ure of t he equiva lent

genera t or . In t his connect ion t he problem of modes in microw a ve sys-

t ems a ga in a r ises. The field excit ed in t he line by t h e a nt enna a lw a ys

consist s of a number of t he modes t ha t a re possible in t he given line.

I t w ill be a ssumed t ha t t he line is designed t o support free propa ga t ion

of a single m ode a nd t ha t t he lengt h of line bet w een t he a nt enna a nd

loa d is m ore t ha n sufficient t o a t t enua t e t he ot her m odes t o negligible

a mplit udes; t here w ill t hen be a n a pprecia ble region over w hich t h e guide

is eq uiva lent t o a t w o-w ire line.


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s m. 2.11] T H E R E C117 V I N G A .V 7l?.V .TA .9 J S 71E ilf

41

B efor e discussing t he eq uiva lent gener a tor represent a t ion, it w ill be

w ell t c consider br iefly t he physica l processes of t he int era ct ion bet w een

t h e receivin g syst em a nd t he ext erna l field. F or t his purpose it w ill be

a ssumed t ha t a n essent ia lly pla ne w a ve from a ver y dist a nt source is

fa lling on t he receiving a nt enna .

In t h e neighborh ood of t he receivin g

a nt enna t he incident w a vefront ma y be rega rded a s a pla ne sur

microondas magnéticas en el espacio vectorial

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