S.BROWN UNIVERSITY PROVIDENCE, R. I. · Maxwell's equations within the waveguide may be written in the form \7XE VXH = or as H (20) I I (21) We want to define a "ivoltage"l depending

$Page 1: S.BROWN UNIVERSITY PROVIDENCE, R. I. · Maxwell's equations within the waveguide may be written in the form \7XE VXH = or as H (20) I I (21) We want to define a "ivoltage"l depending$
S , S.BROWN UNIVERSITYPROVIDENCE, R. I.

I AFCRLd 949

EQUIVALENT CIRCUIT PARAMETER FOR ANINHOMOGENEOUS BIFURCATED WAVEGUIDE

BY

H. M. CRONSON MA._• DAXTU 9 AUJY OYDMPAUOI

ABBRDLEN I ROT'NG GWU= • XD'

6 ARF9(604)-4561 z .Orepo - 456113

S• 1961

AFCRI 949

IEQUIVALENT CIRCUIT PAR-METER FOR AN

3 INHOMOGENEOUS B ,URCATED WAVEGUIDE

By

I H. M. Cronson

U Scientific Report AF 4561A/3

3 DIVISION OF ENG1NEERING

BROWN UNIVERSITY

PROCVIDENCE, RHODE ISLAND

September 1961

Contract Monitor: Dr. Werner W. Gerbes

I "The research reported in this document has been sponsored in part by theElectronics Research Directorate of the Air Force Cambridge Research Lab-oratories, Office of Aerospace Research, and by the Office of Naval Re-search and the David Taylor Model Basin. The publication of this reportdoes not necessarily constitute approval by the Air Force of the findingsor conclusions contained herein."Contract title: iResearch Directed r th study of Radiation of Elec-

tromagnetic Wavesi

I Contract number: AF 19(604)-4561

I

I003i

III

"Requests for additional copies by Agencies of the Depart-ment of Defense, their contractors, and other Government agenciesshould be directed to the:

I ARMED SERVICES TECHNICAL INFORMATION AGENCYARLINGTON HALL STATIONARLINGTON 12, VIRGINIA"

"All other persons and organizations should apply to the:

U. S. DEPAR'ThfLW OF CO-ERCEOFFICE OF TECHNIICAL SERVICES

WASHINGTON 25, D.C."

IINIIIII

I

TABLE OF CONTENTS

PAGE

Abstract ....... ii

I. Introduction 0.0

II. Field Equations 000

A. Reduction to a Half-Space Problem ...... o ....... ... • 3

B. Field representation in the Half-Space Guide .......... 3

III. Equivalent Circuit Representation ...................... •• 7

A. The Transmission Line Analogy of a Parallel

Plate Waveguide................7............... •. 7

B. The Waveguide Junction as an Equivalent Circuit .*.... 9

IV. Determination of the Network Capacitances .... o. * ........ .... 14

V. Solution of the Equations ........ ................... •.. 20

A. Parameters 20

B. Method of Approxima~tion ,....o....... *........... ..... 20

C. Coiipaftfson with the Results Obtained by Marcuvitz

for Ej = 60 . . 25

VI. Scattering Description 29

References .0...... . .. ..0.000000000000.00000.0000000 ....... . . 35

I

I

I.

ABSTRACT

In this report an approximate solution for an equivalent circuit

representation of a two dimensional waveguide junction is obtained over

several ranges of Isignificant parameters. The junction is formed by a

semi-infinite, dielectric filled, parallel plate guide contained symmet,

rically within an infinite parallel plate guide. The structure is first

reduced to a half-space problem and formulated in terms of the appropriate

field modes in each of the parallel plate guides forming the junction.

These modes are matched at the plane of discontinuity and the resulting

equations rearranged to resemble ordinary circuit equations. The circuit

elements of the equivalent circuit representation are identified from the

equations. The determination of the values of these elements requires

the solution of an infinite set of inhomogeneous algebraic equations.

Solutions are obtained by approximate methods with the aid of an IBM 7070

computer. The approximate solutions are checked by comparing the results

obtained for the degenerate case, in which the dielectric vanishes, with

known exact results0 This check suggests that the approximate results

for the case of non-vanishing dielectric should be within a few per cent

I of the correct values.

I

II

I. Introduction

This report concerns the determination of an equivalent circuit rep-

resentation for a waveguide junction formed by a semi-infinite, dielectric

filled, parallel plate guide contained symmetrically within an infinite

parallel plate guide (see Fig. I). We will assume that the dimensions of

the guide are such that only the T.E.M. mode will propagate in each guide

at the excitation frequency, and that the exciting source is so polarized

that there will be no field variations in the x-directiono

The literature on waveguide discontinuities discloses the closely

related problem of a homogeneous E-plane bifurcation, which can be solved1

exactly using integral transform methods° However a preliminary inves-

tigation of these transform procedures applied to the problem considered

in this report revealed that the inhomogeneity introduced by the dielectric

in guide B made these methods impractical to apply. Another approach,

better suited to the type of discontinuity considered here, is an approx-

imate method originally due to Whinnery and Jamieson. 2 Their approach is

based on mode matching at the discontinuity. It has the advantage of

being relatively straightforward, inherently providing a clear physical

picture of the propagating and non-propagating waves in the guide. More-

over, it is particularly adaptable to the two dimensional equivalent

1N. Marcuvitz, "Waveguide Handbook!', M. I. T. Radiation Laboratory

Series, Vol. 10 (McGraw-Hill Book Company, Inc., 1951), pp. 160.167°

. RoWhinnery and H. W. Jamieson,. "Equivalent Circuits for Dis-

continuities in Transmission Lines"!, Proc0 IoR.Eo, 32, 98-114, Feb., 1944.

I* 2

circuit formulation of the problem.

In the structure examined here, the method consists of finding the

evanescent mode amplitudes excited in the guide of height 2a* After con-

siderable manipulation of the field equations at the plane Z = 0, we

obtain an expression for the equivalent capacitances of the junction in

terms of these non-propagating mode amplitudes which, in turn, are defined

by an infinite set of non-homogeneous equations. An approximate solution

to these equations over a convenient range of parameters is obtained by

machine calculation using an IBM 7070 computer. The procedure is checked

by comparing the results for the degenerate case, in which E = Eo ,

with the value s given by Marcuvitz. 1

II

II3

II. Field Equations

A. Reduction to a Half-Space Problem.

The given waveguide configuration can be simplified by the following

* considerations.

(1) Due to the method of excitation and the nature of the discontinuity,

there will be no H-modes in the guide..

(2) The symmetry of the structure allows us to place an infinite perfec-

tly conducting plane at the plane y 0 without disturbing the fields.

Thus instead of solving the original problem, it is sufficient to solve

the half-space problem shown in Fig. 2. (See Appendix of reference 3 for

the justification of these statements.)a

B. Field Representation in the Half4Space Guide.

I The field in each of the guides shown in Fig. 2 can be represented

by an incident And reflected T.EoM. mode plus an infinite sum of evanescent

E modes. Using harmonic time dependence, e and assuming T.E.M. sources

at frequency in guides B and C at Z =--oo and in guide A at Z = oo, the

y component of the electric field in guide B is given by:

for -and

where nd T k I are always real.

3 H. M. CronsonI "Equivalent Circuit Parameters for an InhomogeneousI Bifurcated Waveguide", Master's thesis, Brown University, (1961).

I •.

I 4

The x component of magnetic field in guide B can be obtained

from MaxwellIs Equation j = , using the fact that since

only E modes are present, Hy = z 0. This yields

F .DQ

I H=-4~ [ih9..~a j+jc*E M cIr (2)

for :! Aý and

Similarly, for the fields in Guide A we can write

E, Cj 0' + i,, 3

I ' I1

1 forO• o• and •> . .•iq .. .•...

where k " andq/ / .•.-

I Also, in Guide 0, letting y = y-b,

IA,

Alo in Gudllttn -(6)

I ...

I

II S.

for O C: and Z O

where C ( Z

For subsequent calculations it is convenient to define the following

I quantities,

,, AO 0,• .OA A],,I0 6-j (6a)

+ "I v)

II &- 2, , , (8)

I

II

Hence equations (1) through (6) can be written, at Z =0, as

Ey (11)

III!0

I0

I6I

fZ A 0 + A,(12)

*,-, CJ, ,, o + Y , o 0,,(13)

iw ( 0 (1=4 )

We now employ the condition of continuity of the tangential E and

to fields at Z = 0 to give,

I pa

I~

I

DO 00

Th aov reatoshp betee th A, n ilbeuiie ae

to prvd anA eqiaetcrutrpeetton for th wvegud juncion

I0

IC"I"ýrr Y

II7

III. Equivalent Circuit Representation

A. The Transmission Line Analogy of a Parallel Plate Waveguide.

Consider an infinite parallel plate waveguide of height d in which

the T.E.M. mode is propagating in the positive Z direction (see Fig. 3).

The T.E.Mo wave may be written for harmonic time dependence ejwt as

where D is a constant determined by the strength of the source.

Maxwell's equations within the waveguide may be written in the form

\7XE VXH =

or as H (20)

I (21)I

We want to define a "ivoltage"l depending on Ey and a "current"

depending on Hx so that equations (20) and (21) can be viewed as

transmission line equations. We define

* - E•, (22)

For the parallel plate guide

-y (23)

!8

We also define!A= kv o (24)

where -. is the current per unit length on the surface of the top plateA

(y = d), w is a length in the x direction, and Z is the unit vector

in the Z direction, (see Fig. 3). On the surface of a perfect conductor

A x H = 2 where for this case n is the normal pointing into the guide.

Similarly

We note

- A x

Therefore

We further define

Substitution of equations (23) and (25) into (20) and (21) yields

Sd V( I-jA ) (26)-- I -

II9

* -~V(~)(27)

I The appropriate transmission line equations are given by Marcuvitz

(op. cit.) in the form

ýý -J Z,..9 (28)I--

I ~-&/Y2 (29)

where 2 and J are the voltage and current respectively in the transmission

line, k - , and Z = i/Y is the characteristic impedance. In comparing

equations (26) and (27) with (28) and (30), we see that the T.E.M. mode in a

parallel plate waveguide can be completely represented by a transmission line

with characteristic impedance.

S(29-a)

Note that the power flow in the guide through a cross section w x d is

H9P P j4- uH~ Z4d0 W-&~ H1which is the expression for the power flow in a transmission line.

B. the Waveguide Junction as an Equivalent Circuit.

Since a parallel plate waveguide can be represented by a transmission

line, it is convenient to represent the junction by a lumped network. The

junction in Fig. 2 stores energy with no dissipation in a local region close

10

to the plane Z b. Thus it is reasonable to represent the junction by

a lossless, lumped network. In this section we will show that the field

equations at the junction, expressed in terms of our defined voltage and

current, correspond to the nodal equations for a certain lumped, lossless,

Iand passive network.

The voltage from 0 to b in guide B at the plane Z = 0 is determined

using equations (10) and (22):

* VC)/ :Vefo 6 (= E, +

The voltage from 0 to a in guide A and from 0 to c in guide C at

Z 0 are found to be

We note that, since thecosine dependent modes integrate to(9 •b)e.

Svoltages at Z = 0 are due only to the electric field of the dominant T.E.M.

i mode.

If equation .(16) is integrated with respect to y from 0 to b and

added to the result obtained when equation (17) is integrated with respect

to y from b to c, one obtains

SI + iCnc =A. a-or in terms of' voltages',

*O V 08 V ~VOA (30)

I, 11

The defined current on the top plate in guide B at Z ='0 can be

found from equations (25) and (13.) as

I 7Jjo)/ -jL- •4-hoc§4o)= . oo.-, -

Similarly, the defined current on the top plate in guide C and guide A

at Z = 0 can be written as

IWi C. ;(V

We designate the currents of the dominant mode asI

Integrating equation (18) with respect to y from 0 to y.. .re obtain

i- Ami, At; Ir

Substituting equation (7) we have

ax MM6(31-a)

We can also show that equation (19) can be written

, Y C.... .(32-a)

7]I.p rI oCA

12

Suppose the summation term in equations (31-a) and (32-a) is

multiplied by 16- CO and we define

• "O.- - ..t-a, (33)

We may then write equations (31-a)- and (32-a) as

Y~& =Y 0A-O6 +J'CYco C -Yo (Aa O +,6-L,0 .J60.-IFCO

Using these equations and the definitions of the currents in terms of the

admittances, we obtain

3 A W U , eU o (31-b)

Ic 16T A , WLIf UW +4 (32-b)

I Substituting VOC = VOA -VOB and VOB = VOA - VOC in equations (31-b)

and (32-b) and remembering that a b + c, we have

I1L3 1, -t-j UrUr

SI

V

I. TI C 'I ,,

313

Define

U F -

F3r CA (36)bc

Substituting these definitions in the previous equations yields

15 A I-A J (ACB8VO' +-J CA V4Ae4* 4QCV, + AeuCA VAA

which are the nodal equations of the network shown in Fig. 4. Thus, the

waveguide junction can be represented by a lumped, lossless, and passive

network.

IIII

I

I ... i.•'

_U

I

1 114

IV. Determination of the Network Capacitances

SFrom equations (34)., (35), (36), we see that the capacitances couldAm

be determined if F were known. To find F we must find B . We shall0 00

now determine equations for this unknown,

We multiply equation (16) by cos MTU and integrate with respecta+

to y from 0 to b. This equation is then added to the result obtained when

equation (17) is multiplied by cos m and integrated with respect toa

j y from b to c. These operations result in

SA,, I L7i- V (7

g=

where ,

Next we multiply equation (18) by cos and integratewith respect• .b

to y from 0 to b.o The result is

a-~ (38)

IEquation (19) is multiplied by cos q'E y' and integrated with respect

I

I

,i 15

to y' from 0 to C. This gives

I~i Y J39)~~A~nr

Substituting the B and C from equations (38) and (39) into (37) yields

r

4

A^ 1

0..

DO Do8

g•.

3 (equation continued on next page ).

I0 ,LZlIC

I°"16

I PI

•k .(o)

I.17

where EB E- and the encircled numbers designate the terms of

the equation for subsequent identification.

,Next, each term is rearranged so that Ap appears only under the

first summation. For instance term is rearranged by noting that

3 where is the wavelength in guide B and also by

changing the order of summationITerm becomes

T 6 Z9tCL4 2-)MCI)

b

In order to group all the Am terms together in equationto.G we also separate

term 0 into two parts. One including all the Ap save Am and the other

1 just containing Am. When all the terms have heen suitably rearranged

u equation (40) becomes

• i,'7#" 4..

2. A41'w +,,rX4) 7r_ t IN ,.r-

(equation continued on next page)UU

18I ..

o44

3 +

p~r#)

I - ® ® ,-

I m~_b 4]

m 1to Ao " 1 ifbmsa integer

Iwhere X(m) = J .and misa n

Cb7

ar'•[B•-CcI 0 if• m is not an integer

We nwwrite equation (33) as Fz Qw- L A¶•Al7 ()

I31U

I0Iz2

19

Using equations (36) and (42) we write

CA ~ r L 4K/ r (43)

From equations (35) and (36) we note

C B C-LA(4

I

3 The remainder of this report will be concerned with determining CA

by solving equation (41) by approximate methods.

I One check on the correctness of equations (t11) and (43) is provided

by noting in Fig. 4 that for the case E = C- G0 CA should not change

I when the dimensions of guides B and C are interchanged. In accordance

with this physical observation it can be shown (see Appendix of reference 3)

using equations (4l) and (43) for the case =1 that CA is invariant to

I the transformation replacing b by c and c by b. Thus in this respect the

equations check.I

* 20

V. Solution of the Equations

A. Parameters

3 The main objective of this report is to determine the guidecapacitances for various combinations of , B' and Since

I the analysis is based on only the ToEoMo modes propagating in each guide,

there is the restriction that 2a 2b .2 are not greater than 1.

The values chosen for this report are

I ~b-a= 0.1, 0.5, 0.9

E B ' 1I1 2.5, lo., loo

II = O. o06, 1.0

in order that results may be obtained over a wide range of parameters.

Since it can easily be shown that a , those

permutations of the parameters for which 2 b 1 will not be allowed.

I B. Method of Approximation

The infinite number of unknowns X(m) can, in principle, be determined

from the infinite set of simultaneous equations represented by equation (41).

5 The X(m) are then used to calculate the capacitance CA , by employing

equation (43). The other capacitances immediately follow from equations (44)

I and (45). In practice, however, equation (41) must be solved by some method

of approximation. The method utilized in this report attempts to approximate

the exact answer by solving only finite sets of simultaneous linear equations.

I

I1 21

I We will solve equation (41) as a set of simultaneous equations of rank 4., 5,

and 6 and from the behavior of these solutions determine an approximate

capacitance. The validity of this approximation will be examined by a com-

parison with the exact results for the case E B = 1

I The quantity we are interested in obtaining is the capacitance CA

given by equation (43). We define

CA 4K /Y Vi(4t6)n? - e #,L 0" •

where X(m) are the solutions to the infinite set of equations represented

by (Mi). Let the index m in equation (a1) be restricted to the integers

3 from 1 to 6. The solutions to this 6 x 6 set will be called /X6 (m) Thus

one approximation to C ,could be

C _W__ (47)

This approximation has 2 sources of error. The first is the error arising

from the fact that /K 6 (m) • X(m) for m = 1,6. The second is the error

3 due to the exclusion of

PO X&ft)I "•7

It is our aim to determine a refinement of the approximation given by

U equation (47) based on the computed results, which will reduce the errors

mentioned above,

The computer program was designed to solve the system of equations

Iof rank 4t, 5, and 6. It was felt that a plot of these results for a few

Io ak4 ,ad oI etta

I

I 22

Stypical cases and an extrapolation would indicate the probable solutions

for the first four unknowns (X(m), m = 1, 4) and also serve to exhibit

I the behavior of the unknowns for m > 4. As an example of this procedure

I for one set of parameters, Figs. 5, 6, and 7 show respectively the

dependence of the solutions on the equation set size, a rough estimate of

Sthe values ofA X(m) for m = 1, 4 extrapolated from the finite sets, and

the extrapolated values of X(m). An examination of the computed and

graphical results reveals the following information:

I (1) X(m) sin (, b) is always positive

I (2) The extrapolated value of X(m) is very close to x6(m)

(3) The extrapolated value of x(m) for m> 5 decreases

roughly as 1 . Here we must make an exception for

b = 0.5 because the results show (2m + 1) >> x (2m)Ia Ix II 1

But in this case the odd values of m decrease as 1 and these are them

only values that enter into the computation of C

Based on the above observations a suitable refinement to equation (47)

is

I () 4 5I. /,r/ (48)

The series is roughly the same if is 0.1, 0.5,

3 or 0.9.

I23

We note for b = 0.5 the value of this series can be determined

a

from the well known result

00 . 00S(2/M

Am=

From this we find

0= 0.826144

Our approximation is now given by

IC(2) =C(l) + o-413 K (5)IIor using equation (48)

-X C 652 (4~9)... •_ C•• _C_ _ __,__N c)! •

i>c-r £ur c 77'

where IX

The computed results are tabulated below. Figures 8 through 13 show

these results in graphical form,

I

ý.O co '0 H 24~trA 'LN V\ ' '\ \C) I 0C\J C) (NJ -:T I 04i 04

C ~ c I fu\ I\

0 * 0 0 5-40 0)

0 0)

Go Go o aai)4

('4 H 04J IA \0 ON r- coa IH ( H H '- C ON I H 1 I 4,

r- 04l H H (04 H- H 4 .0j 0 * 0

to 0 0

co H co \4'C 4 tr 0 ON Elb H 4 C CN 0( CH I IA I I 0

0 ~ '400 r- 040O 0

\ 0 0 0. '0

0 H 0 0 0 0 0

04 0 04 1.r C0aN 04 O .~ r.

0 0 0 0 0 m 0 0 T

_A cn V. \ oC - 0 n -I0 ON 0 - O'0 C) d

H0

I08 Hq H 0 H 0 0) ) 0) 0) 0

0 0 04 H H H 0) C 0

H H H .d to

H UN' O\ H IA' ON H C\' ON H L1' ONC.O cuI* C) C ) C C; CG C) C; C; 0 0; C3

25

C. Comparison with the Results Obtained by Marcuvitz for ea= Ce.

O On pages 353-355 of Marcuvitz (op. cit.) the E-plane bifurcation

problem is discussed and the results given graphically for an exact solution

of this problem using integral transform methods, These results can be

compared to those of this report for the case 6B = 1. We use the notation

on page 353 but to avoid confusion; since b denotes different dimensions

in this report and in Marcuvitz, we let the b of Marcuvitz be called

3 The dimension Marcuvitz labels b is the dimension of guide A in this

report. Comparing Fig. 4 in this report to Figure 6.4 - 2 in Marcuvitz

3 we observe

X =- co I ,

where X = 2-L

For this problem and the characteristic impedance is given

by

3 Therefore

/ _ -

Using = 2T we have

2nZ

""I- (50)S27T-

Ii

I1 26

When >0 we, have t(51)

Using the graph on page 354 we can construct the table below and compare

CA with the computed C42 of this report.

fiE ~E

II1IUI

27

0 Q co bK0\ 0\ 0' 'i

. C4 8 A Aý C

oC~j 0 cm c'j

to 0 H ,0 10 m

I; oC\J 0" \ 0 C'j tr

~\10I8 r

In ~ %0 'o ONj

a "0 %4D 0

'0

0 0 0 0 0 0

Ii 28

From the above results we see that in all but one case the error is within

4% of Marcuvitz' results. Thus C is a fairly good approximation toIOe f B 1. Since the structure ofB he equatiorsis the same for

CB 1 1, and the computed results manifest no drastic change foreB i 1, we

3 may expect that 66 will be within a few per cent of the correct value

for all BiIiiIIiiIiIII

29

VI. Scattering Description

This section will be concerned with transforming the impedance

description of the guide to a scattering description. Since we know now

the values of the capacitances in the equivalent circuit, all other wave-

guide quantities such as reflection coefficient, standing wave radio, etc.,

may be determined. We will assume, for the sake of simplicity, that only

guide B is excited and that guides A and C are terminated in their character-

istic impedance. To calculate the reflection coefficient at the junction,

I-r(0 ), we use equation (30) given on page 14 of "Waveguide Handbook" (op.

cit.).

+-(52)I + ('(s)

where Y'(Z) = Y(Z)/Y°

and Y(Z) is the admittance of the transmission line at Rý and YO

is the characteristic admittance of the transmission line.o

We note that the circuit shown in Fig. 4 corresponds to the half-space

problem. However, due to symmetry, the reflection coefficient of the

original guide in Fig. 1 is the same as the half-space guide. The circuit

we consider is shown in Fig. 14., where the characteristic admittances are

found from equation (29-a). We make the following definitions:

YA-YA JCA (53)

B (514)

30

YC= YA0 +>WcC (55)

We then have

Yo) Y- YS Y, (56)

YBO yol y

Using equations (29-a) and (45) we write equation (55) as

p F a - • CA • (57)

We make use of the quantity - CA computed in part V by defining the

positive number CV

CV-CA (58)

We now substitute equation (58) into (57) to yield

YC + i-li(59but since C) 2- 7r 2Tr equation (59) can be written asI

Y •- + J" lz CV (60)

Similarly equation (53) can be written as

I -, cv (61)

I- a

I31

After a few of these substitutions equation (56) may be written as

Y1J (I+~ A C 12-c)+ jCLC -)-~~CAL+ C) (62)

I where A -Z-L

and C- VISubstituting (62) into (52) we can calculate the reflection coefficient

r(o).

Let us now examine the behavior of the reflection coefficient for

two simple cases.

Case 1. - 0I .

If ___ = 0, then CL = 0 and equation (62) becomes_L

-i-O (63)

Substituting equation (63) into (52) we obtain

P ý- .(64)

We note when A 1

and when<.

rIo

32

B , equation (64) becomes,

A'Iso i 8 BP O ) = I - b(6 5)

This is in agreement with the physical result that as b ->l., the guide

3 .,ceases'to become bifurcated and no reflections occur.

Case 2 ~

A study of the Table of Computed Capacitance in section v-C shows

that this is a fairly good approximation when 20--/A 0.6. Since

A -. 1 ,we also have C2«< A. With this approximationb I L

equation (62) can be written

Y'(o=+f (66)

This tells us the obvious fact that the best match., i.e. Y'(o) = 1 occurs

when A-->1 and4TE --> 1.

It is also interesting to inquire how the wave entering from guide B

-:-is divi~ded between guides A and C. Using equations (6-a) and (29-b) we

obtain

VOAALOU c~ (67)

where A. and C~are the complex amplitudes of the outgoing waves

pr6pagating in guides A and C respectively. We have the relations

(68)VOA VAVO

I 33where I is shown in Fig. 14. Combining equations (60), (61), (67),

I and (68) we get -

Call I -JC (69)

and I'O

(70)Iwhich gives the result that the magnitudes of the transmitted waves in

guides A and C are always the same. To find the ratio of the powers flowing

into guides A and C, assuming them terminated by their characteristic:

I impedance we employ the equationI~~~ A=Jv2.L

PA !VOA 12YA C,(1

_ c

This is a general result independent of t' B and subject to the restriction

that 2 0-

We now give a table of computed I(a) for the range of parameters

considered in this problem.

34

Table of Computed j0)j

.b : 1B 00a •rc •l

0.1 1 .90 .90 .90015 1 .5o .50 .370.9 1 .10 .10 .100.1 2,5 .94 .93 .92o.5 2e5 .65 .61 .350.9 2.5 .32 .32 -011 10 .97 .98 .95o.5 10 .81 .77 -0.9 10 .59 -0.1 100 .99 .99 .99I 0.5 100 .94 - -0.9 100 .85 -

These results show that increasing E B and decreasing and

tends to increase MlCLIa

I

The dash -). indicates that F, 2- 0 and thus solutions'

with these sets of parameters are excluded from this analysis.

IReferences

1N. Marcuvitz, "Waveguide Handbook". M.I.T. Radiation LaboratorySeries, Vol. 10 (McGraw-Hill Book Company, Inc., 1951).

2. J. R. Whinnery and H. W. Jamieson, "Equivalent Circuit for Discon-

tinuities in Transmission Lines", Proc. I.R.E., 32, 98-114,February, 1944.

3. H. M. Cronson, "Equivalent Circuit Parameters for an InhomogeneousBifurcated Waveguide", Master's thesis, Brown University,(1961).

IIIII

IIIIIIi

II

IyI ~(conductor)

C(conductor)

IxI

I (conductor)

I (conductor)

I FIG. I A SEMI - INFINITE , DIELECTRIC FILLED, PARALLELPLATE GUIDE CONTAINED SYMMETRICALLY IN AN

INFINITE PARALLEL PLATE GUIDE.

IIIIII

I y(conductor)

I C guide C (conductor)

3b guide B guide A a

3 (conductor) x z

I FIG. 2 THE INHOMOGENEOUS BIFURCATED GUIDE

III,

I -"-"_ z___ __ J

I FIG. 3 ORIENTATIONS OF ,t, AND HX

Vooc

IoIol

' I

I -- i OA

ov _b cc 'Io / I-o

I I

108 IA

FIG. 4 EQUIVALENT CIRCUIT REPRESENTATION

Iw

I w

z0

DD

0 a

11 w

0 (D MIa I I IHD

cLi~o 31 zI 0 w 0

0w U)I--

0-w0

w

00I~~C _ _ _ __O

LL

(%j to LL 0,

0<

0_

0"

U-

(DU

II-(0L

x0

XILi

* (9

C~ ci0 0 0 0

(M* (0

I IL

w 6

flic

IM 0

(0 0

~ii-J-

00xw

I-

IcI2

I; 0JV

0w X

I _L _ _

a<nI - 4 < (D

L~w

00I ZO (D

I 0:

0 0

I> 4

CD0 ,Z

00

LOON I-,-

I0

0Iz W

IIw

101 0Iw>0

>0

zW

00I 0 00

>IIIwIUI

Io

IW <

I>

I-wz

WI-I (00LU 4

>0

I w0

W-

tO Cj0C

I0

0

Icz w

00

~i4 CDwc'w

'-0

W gi

*z 5~ CJ

G 6J

I CtD

*c 0,6IO

0 0 0

I0U0

*w 0

UW

04

-:> w

W W

I 0,

IIII

I CHARACTERISTIC CcADMITTANCE

YBI

Y--- CB

CA YA A

TA

II

I FIG. 14 EQUIVALENT CIRCUIT OF THE HALF SPACEGUIDE WITH GUIDES A AND C TERMINATEDIN THEIR CHARACTERISTIC ADMITTANCES

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