##### Document Text Contents

Page 1

Antennas in

Inhomogeneous Media

BY

J A N I S G A L E J S

Senior Scientist, Applied Research Laboratory,

Sylvania Electronic Systems, Waltham, Massachusetts, U.S.A.

4?

PERGAMON PRESS

OXFORD • LONDON • EDINBURGH • NEW YORK

TORONTO • SYDNEY PARIS • BRAUNSCHWEIG

Page 2

Pergamon Press Ltd., Headington Hill Hall, Oxford

4 & 5 Fitzroy Square, London W. 1

Pergamon Press (Scotland) Ltd., 2 & 3 Teviot Place, Edinburgh 1

Pergamon Press Inc., 44-01 21st Street, Long Island City, New York 11101

Pergamon of Canada Ltd., 207 Queens Quay West, Toronto 1

Pergamon Press (Aust.) Pty. Ltd., 19a Boundary Street,

Rushcutters Bay, N.S.W. 2011, Australia

Pergamon Press S.A.R.L., 24 rue des Ecoles, Paris 5e

Vieweg & Sohn GmbH, Burgplatz 1, Braunschweig

Copyright© 1969

Pergamon Press Inc.

First edition 1969

Library of Congress Catalog Card No. 68-21384

PRINTED IN GREAT BRITAIN BY A. W H E A T O N & CO. EXETER

08 013276 6

Page 151

§9.1 LINEAR ANTENNAS IN A STRATIFIED DIELECTRIC 143

satisfy the boundary conditions Exi = Exa, Eyi = Ey0, Hxi = Hx0, and

Hyt^Hyo+Jj., for z = 0. Substituting ^ and <$>} in (3.10) and (3.11), and

applying the Ex and Ey boundary conditions gives

A0( 1 + /?„,,) =Ai{\ + Rai),

y0B0(l-RM)=yiBi(-i+Rbi).

The / / x and Hv boundary conditions lead to

« M p J*3( »+**>) *?(l + KM)1

Y^o(1~My.(l-*») y«<l-*«)J

= ^+^/ /^ e " t a e " ! l ' y d x d y '

r i - « „ n i - /

^o(l+/?ao)

(9.1)

(9.2)

(9.3)

7o ! + /?„„ "1+/?„,.

= ^ M ^ / / * / " e " " " e " , W t k d > ' - ( 9 - 4 )

The amplitudes A j and B, are related to the linear current density of the

antenna Jx by (9.1) to (9.4). The scalar functions W j and <t>j and the field

components are thus uniquely related to Jx. The driving point impedance of

a flat strip antenna may be computed from the expression (1.30). The sur-

face current Js is assumed to have only an x component Jx, which in addi-

tion, is an even function about x = 0, y = 0; and it follows that

Z =

1<1>IX0

[2TT1(X = 0)]

where

OO 00

, / / F ( . . . >

— 0O — 00

I l Jx(x,y) cos ux cos vydxdy

K( \ V2 f l - f l q o p , 1 - f l q / J - 1

dwdi>, (9.5)

i r

w2 + i;2 .7OP (1—/?f t 0p) 7/Q (1

Rbiq)V

Rbiq)i

(9.6)

The designation of the dielectric layers by subscripts is seen from Fig. 9.1,

and in particular the subscripts Op and iq designate dielectric layers next to

z = 0 for z ^ 0 and z ^ 0 respectively. The other symbols are defined as

kit = (a)

2fi0eJi-\-ia)iJ,0crjl)

1/2, yn = i{k

2 ■v2Y

where j = 0 or / and / = 1, 2 . . . p or 1, 2 . . . q. The reflection coefficients of

Page 152

144 ANTENNAS IN INHOMOGENEOUS MEDIA Ch.9

the TE modes Rajl and of the TM modes Rbjl depend on the dielectric struc-

ture for |z| > 0. For stratified dielectric layers as shown in Fig. 9.1, the

reflection coefficient Rdjl (d= a or b) in the region of \zi+\\ ^ \z\ ^ \zi\ is

related to the reflection coefficient Rda-D of the region |z,| ^ \z\ ^ |z/-i| by

the expressions (8.18) and (8.19) if the double subscripts jl are substituted

for the single subscripts j . Also the absolute value of zj{ is considered for

\ZJ(\ < 0. The antenna current density is assumed to be representable by the

same trial function as in Section 2.3, but with an arbitrary value of k. This

gives

J A*, y) = [A sin [k(l-\x\)]+B{\-cos [k(l-\x\)]}]f(y) (9.7)

wi th / (y) = l/(2e) = const. Substituting (9.7) in (9.5) and using the station-

ary character of the impedance expression for determining the complex

amplitudes A and # , it follows that {AIB) and Z are given by (2.38) to

(2.41), with

FA = sin kl, (9.8)

FB= 1 - c o s it/, (9.9)

OC 00

y,M = ^ j dv(^fj j duF(u,v)gx(u)gM(u), (9.10)

0 0

SAW = 773-7(cos w/ -cos £/), (9.11)

#/>•(") = 75-^-2 f-sin 1//-sin A/V (9.12)

k~ — ir \u J

The impedance Z depends on the selection of the wave number k of the trial

functions (9.7). For a homogeneous dielectric, k should represent the wave

number of the dielectric kd. For antennas in a thick dielectric layer, k should

also be equal to the wave number of the dielectric layer kd, and for very thin

dielectric layers, k is approximated by the free space wave number k0. The

antenna impedance computed using these two values of k can be shown to

be continuous for dielectric layers of a thickness near X/100. The accuracy

of the computations may be verified by using

or other intermediate values of k for intermediate thickness of the dielectric

layer. For insulated antennas in a dissipative medium, the transmission line

theory may provide the first estimates of k. This is usually a satisfactory

guess, and small changes of k cause only negligible variations in the computed

impedance.

Page 301

2 9 4 SUBJECT INDEX

Transverse electric modes 6 power flow

Transverse magnetic modes 6 cylindrical 272, 278

distance dependence 278

spherical 276

Upper hybrid resonance frequency 243 Uniaxial medium

antenna impedance 239, 241, 259, 261,

280

effective resistance 272

Variational impedance formulation 18

Verification of antenna solutions 11, 249

Page 302

OTHER TITLES IN THE SERIES IN

E L E C T R O M A G N E T I C W A V E S

Vol. 1 FOCK — Electromagnetic Diffraction and Propagation Problems

Vol. 2 SMITH an

Vol. 3 WAIT — Electromagnetic Waves in Stratified Media

Vol. 4 BECKMANN and SPIZZICHINO-The Scattering of Electromagnetic Waves from Rough

Surfaces

Vol. 5 KERKER — Electromagnetic Scattering

Vol. 6 JORDAN — Electromagnetic Theory and Antennas

Vol. 7 GINZBURG—The Propagation of Electromagnetic Waves in Plasmas

Vol. 8 DU CASTEL —Tropospheric Radiowave Propagation beyond the Horizon

Vol. 9 BANOS — Dipole Radiation in the Presence of a Conducting Half-space

Vol. 10 KELLER and FRISCHKNECHT—Electrical Methods in Geophysical Prospecting

Vol. 11 BROWN — Electromagnetic Wave Theory

Vol. 12 CLEMMOW —The Plane Wave Spectrum Representation of Electromagnetic Fields

Vol. 13 KERNS and BEATTY—Basic Theory of Waveguide Functions and Introductory Micro-

wave Network Analysis

Vol. 14 WATT —VLF Radio Engineering

Antennas in

Inhomogeneous Media

BY

J A N I S G A L E J S

Senior Scientist, Applied Research Laboratory,

Sylvania Electronic Systems, Waltham, Massachusetts, U.S.A.

4?

PERGAMON PRESS

OXFORD • LONDON • EDINBURGH • NEW YORK

TORONTO • SYDNEY PARIS • BRAUNSCHWEIG

Page 2

Pergamon Press Ltd., Headington Hill Hall, Oxford

4 & 5 Fitzroy Square, London W. 1

Pergamon Press (Scotland) Ltd., 2 & 3 Teviot Place, Edinburgh 1

Pergamon Press Inc., 44-01 21st Street, Long Island City, New York 11101

Pergamon of Canada Ltd., 207 Queens Quay West, Toronto 1

Pergamon Press (Aust.) Pty. Ltd., 19a Boundary Street,

Rushcutters Bay, N.S.W. 2011, Australia

Pergamon Press S.A.R.L., 24 rue des Ecoles, Paris 5e

Vieweg & Sohn GmbH, Burgplatz 1, Braunschweig

Copyright© 1969

Pergamon Press Inc.

First edition 1969

Library of Congress Catalog Card No. 68-21384

PRINTED IN GREAT BRITAIN BY A. W H E A T O N & CO. EXETER

08 013276 6

Page 151

§9.1 LINEAR ANTENNAS IN A STRATIFIED DIELECTRIC 143

satisfy the boundary conditions Exi = Exa, Eyi = Ey0, Hxi = Hx0, and

Hyt^Hyo+Jj., for z = 0. Substituting ^ and <$>} in (3.10) and (3.11), and

applying the Ex and Ey boundary conditions gives

A0( 1 + /?„,,) =Ai{\ + Rai),

y0B0(l-RM)=yiBi(-i+Rbi).

The / / x and Hv boundary conditions lead to

« M p J*3( »+**>) *?(l + KM)1

Y^o(1~My.(l-*») y«<l-*«)J

= ^+^/ /^ e " t a e " ! l ' y d x d y '

r i - « „ n i - /

^o(l+/?ao)

(9.1)

(9.2)

(9.3)

7o ! + /?„„ "1+/?„,.

= ^ M ^ / / * / " e " " " e " , W t k d > ' - ( 9 - 4 )

The amplitudes A j and B, are related to the linear current density of the

antenna Jx by (9.1) to (9.4). The scalar functions W j and <t>j and the field

components are thus uniquely related to Jx. The driving point impedance of

a flat strip antenna may be computed from the expression (1.30). The sur-

face current Js is assumed to have only an x component Jx, which in addi-

tion, is an even function about x = 0, y = 0; and it follows that

Z =

1<1>IX0

[2TT1(X = 0)]

where

OO 00

, / / F ( . . . >

— 0O — 00

I l Jx(x,y) cos ux cos vydxdy

K( \ V2 f l - f l q o p , 1 - f l q / J - 1

dwdi>, (9.5)

i r

w2 + i;2 .7OP (1—/?f t 0p) 7/Q (1

Rbiq)V

Rbiq)i

(9.6)

The designation of the dielectric layers by subscripts is seen from Fig. 9.1,

and in particular the subscripts Op and iq designate dielectric layers next to

z = 0 for z ^ 0 and z ^ 0 respectively. The other symbols are defined as

kit = (a)

2fi0eJi-\-ia)iJ,0crjl)

1/2, yn = i{k

2 ■v2Y

where j = 0 or / and / = 1, 2 . . . p or 1, 2 . . . q. The reflection coefficients of

Page 152

144 ANTENNAS IN INHOMOGENEOUS MEDIA Ch.9

the TE modes Rajl and of the TM modes Rbjl depend on the dielectric struc-

ture for |z| > 0. For stratified dielectric layers as shown in Fig. 9.1, the

reflection coefficient Rdjl (d= a or b) in the region of \zi+\\ ^ \z\ ^ \zi\ is

related to the reflection coefficient Rda-D of the region |z,| ^ \z\ ^ |z/-i| by

the expressions (8.18) and (8.19) if the double subscripts jl are substituted

for the single subscripts j . Also the absolute value of zj{ is considered for

\ZJ(\ < 0. The antenna current density is assumed to be representable by the

same trial function as in Section 2.3, but with an arbitrary value of k. This

gives

J A*, y) = [A sin [k(l-\x\)]+B{\-cos [k(l-\x\)]}]f(y) (9.7)

wi th / (y) = l/(2e) = const. Substituting (9.7) in (9.5) and using the station-

ary character of the impedance expression for determining the complex

amplitudes A and # , it follows that {AIB) and Z are given by (2.38) to

(2.41), with

FA = sin kl, (9.8)

FB= 1 - c o s it/, (9.9)

OC 00

y,M = ^ j dv(^fj j duF(u,v)gx(u)gM(u), (9.10)

0 0

SAW = 773-7(cos w/ -cos £/), (9.11)

#/>•(") = 75-^-2 f-sin 1//-sin A/V (9.12)

k~ — ir \u J

The impedance Z depends on the selection of the wave number k of the trial

functions (9.7). For a homogeneous dielectric, k should represent the wave

number of the dielectric kd. For antennas in a thick dielectric layer, k should

also be equal to the wave number of the dielectric layer kd, and for very thin

dielectric layers, k is approximated by the free space wave number k0. The

antenna impedance computed using these two values of k can be shown to

be continuous for dielectric layers of a thickness near X/100. The accuracy

of the computations may be verified by using

or other intermediate values of k for intermediate thickness of the dielectric

layer. For insulated antennas in a dissipative medium, the transmission line

theory may provide the first estimates of k. This is usually a satisfactory

guess, and small changes of k cause only negligible variations in the computed

impedance.

Page 301

2 9 4 SUBJECT INDEX

Transverse electric modes 6 power flow

Transverse magnetic modes 6 cylindrical 272, 278

distance dependence 278

spherical 276

Upper hybrid resonance frequency 243 Uniaxial medium

antenna impedance 239, 241, 259, 261,

280

effective resistance 272

Variational impedance formulation 18

Verification of antenna solutions 11, 249

Page 302

OTHER TITLES IN THE SERIES IN

E L E C T R O M A G N E T I C W A V E S

Vol. 1 FOCK — Electromagnetic Diffraction and Propagation Problems

Vol. 2 SMITH an

Vol. 3 WAIT — Electromagnetic Waves in Stratified Media

Vol. 4 BECKMANN and SPIZZICHINO-The Scattering of Electromagnetic Waves from Rough

Surfaces

Vol. 5 KERKER — Electromagnetic Scattering

Vol. 6 JORDAN — Electromagnetic Theory and Antennas

Vol. 7 GINZBURG—The Propagation of Electromagnetic Waves in Plasmas

Vol. 8 DU CASTEL —Tropospheric Radiowave Propagation beyond the Horizon

Vol. 9 BANOS — Dipole Radiation in the Presence of a Conducting Half-space

Vol. 10 KELLER and FRISCHKNECHT—Electrical Methods in Geophysical Prospecting

Vol. 11 BROWN — Electromagnetic Wave Theory

Vol. 12 CLEMMOW —The Plane Wave Spectrum Representation of Electromagnetic Fields

Vol. 13 KERNS and BEATTY—Basic Theory of Waveguide Functions and Introductory Micro-

wave Network Analysis

Vol. 14 WATT —VLF Radio Engineering