NASA Contractor Report 201692

Analysis of Discontinuities in aRectangular Waveguide Using DyadicGreen's Function Approach inConjunction With Method of Moments

M. D. Deshpande

ViGYAN, Inc., Hampton, Virginia

Contract NAS 1-19341

April 1997

National Aeronautics and

Space Administration

Langley Research Center

Hampton, Virginia 23681-0001

https://ntrs.nasa.gov/search.jsp?R=19970019256 2018-08-29T03:58:37+00:00Z

Contents

List of Figures

List of SymbolsAbstract

1. Introduction

2

2

4

4

o Theory

Dyadic Green's function for an electric current source

in a rectangular waveguide

6

6

,

2.

3.

(a) Solution of inhomogeneous Helmholtz equation

(b) Electromagnetic field due to transverse currents

(c) Electromagnetic field due to longitudinal current

(d) Dyadic Green's function for electric field

Application

Analysis of cylindrical post in a rectangular waveguideNumerical Results

Conclusion

References

6

10

12

15

17

17

19

19

20

List of Figures

Figure 1 Electric current source in a rectangular waveguide 21

Figure 2

Figure 3

Figure 4

a

Rectangular waveguide with a cylindrical post located at x = _, z = -z I and

parallel to y-axis 22

Reflection coefficient of a y-directed post in a rectangular waveguide as a

function of frequency 23

Transmission coefficient of a y-directed cylindrical post in a rectangular

waveguide as a function of frequency 24

a,b

-->A

Ax, Ay, A z

.->E

ES

ez

E i

f

Ge

_eO

a xx , ayy, Gzz

gxx, gyy, gzz

g xx, gyy, g zz

H

l-ly,

List of Symbolsx-, y-dimensions of rectangular waveguide

magnetic vector potential->

x-, y-, and z-components of A, respectively

Electric field inside rectangular waveguide

scattered electric field

->x-, y-, and z-components of E, respectively

incident electric field vector

frequency in cycles per second

dyadic Green's function for magnetic potential

dyadic Green's function of electric-type

dyadic Green's function of electric-type for source free region

.->

xx-, yy-, and zz-components of G

functions associated with Gxx, Gyy, Gzz, respectively

Fourier transforms of gxx, gyy' gzz' respectively

Magnetic field inside rectangular waveguide-->

x-, y-, and z- components of magnetic field H, respectively

2

r

Io.->J

....>

JT

J

k o

k

kZ

k,m, n

T

Vyx, y, Z

X', y', Z'

X", y", Z"

_c,_,_

ZYY

F

V

_(.)

EO' _0

13m and E n

rl0

tp

CO

FEM

EM

EFIE

MoM

unit impulse current source-->

amplitude of current J

Electric current source

test surface current density

__>z-component of J

free-space wave number

= ko,_r_r

propagation constants along z-direction inside rectangular waveguide

propagation constant along z-direction

integer associated with waveguide modes

transmission coefficient

reaction of test surface current density with the incident electric field

coordinates of field point in cartesian coordinate system

coordinates of source point in cartesian coordinate system

dummy variables of integration

unit vector along the x-, y-, and z-axis, respectively

self impedance of the y-directed post current

reflection coefficient

gradient operator

delta function

permittivity and permeability of free-space

Neumann's numbers

free-space impedance

variable of integration

angular freqency equal to 2rcf

finite element method

electromagnetic

electric field integral equationmethod of moment

Abstract

The dyadic Green's function for an electric current source placed in a rectangular

waveguide is derived using a magnetic vector potential approach. A complete solution for the

electric and magnetic fields including the source location is obtained by simple differentiation of

the vector potential around the source location. The simple differentiation approach which gives

electric and magentic fields identical to an earlier derivation is overlooked by the earlier workers

in the derivation of the dyadic Green's function particularly around the source location. Numeri-

cal results obtained using the Green's function approach are compared with the results obtained

using the Finite Element Method(FEM).

I. Introduction

Analysis and design of dipole, monopole, or aperture radiator to excite high intensity

electromagnetic (EM) fields inside a reverberation chamber can be done using an integral

equation approach. The EM fields inside a reverberation chamber due to a radiator can be

determined by weighting an appropriate dyadic Green's function with an assumed antenna

current. The Electric Field Integral Equation (EFIE) is then set up by forcing the total tangential

electric field on the antenna surface to be zero. Using the Method of Moments (MoM), EFIE is

then reduced to a matrix equation which can be solved for the antenna current. From the current,

the EM field radiated by the antenna inside a reverberation chamber is determined. Also the

input impedance of the antenna as a function of its location and frequency can be determined.

This work is divided into two parts. In the first part we derive the appropriate dyadic Green's

function for an electric current source located inside a rectangular waveguide and cavity. Detailed

steps involved in this derivation are reported in this document. The second part of this work,

which will be reported in subsequent documents, consists of an application of the dyadic Green's

4

function to analyze a dipole antenna placed in a reverberation chamber.

Knowledge of a dyadic Green's function for cylindrical waveguides and cavities is

essential for analyzing and designing antennas and arbitrarily shaped objects placed inside a

cylindrical waveguide and cavity [1,2]. A detailed derivation of a dyadic Green's function for the

rectangular waveguide was presented by Tai [3]. In deriving these dyadic Green's function valid

for both source and source free regions, an additional term must be added to the classical

representation of the field expressions [4]. To include the additional term in the classical

representation, Tai [5] has presented an approach based upon the use of eigenvector functions. In

[6], an electric-type dyadic Green's function is obtained through a magnetic-type dyadic Green's

function obtained using the theory of distributions.

The purpose of this communication is to present a simple method using the vector

potential approach to determine the dyadic Green's function valid in the entire region of a

cylindrical waveguide. For an arbitrarily oriented electric current source in a rectangular

waveguide, expressions for the magnetic vector potential are obtained by solving the

inhomogeneous Helmholtz equation. The electric fields and hence the dyadic Green's function of

the electric-type is then obtained by taking the derivatives of the magnetic vector potential. In the

process of finding the electric field, if the derivatives of the vector potential are carefully defined,

the additional term discussed in [4-6] automatically follows. Reflection and transmission

coefficients due to a y-directed cylindrical post placed in a rectangular waveguide and excited by

a dominant mode are derived and numerical results are compared with the results obtained by the

Finite Element Method [7].

II. TheoryDyadic Green's Function for an Electric Current Source in a RectangularWaveguide

(a) Solution of Inhomogeneous Helmholtz Equation:->

Consider an infinite rectangular waveguide with electric current source J as shown in

-->figure 1. The electromagnetic fields inside the waveguide due to J can be determined from

--)H(x,y,z) = 1Vxf4

go(1)

E (x, y, z) - -Tk o

where the assumed time variation 2 °t has been suppressed. The magnetic vector potential

-->

A (x, y, z) appearing in (1) and (2) statisfies the inhomogeneous wave equation

(2)

V2_ (x, y, Z) + k_ (x, y, Z) = -go J (x', y', Z') (3)

If G (x, y, z, x', y,' z") is the dyadic Green's function for the rectangular waveguide for a unit

impulse current source 1 (x', y', z') inside the waveguide, then the magnetic vector potential

->A (x, y, z) can be written in the form

.-> ..->

A(x,y,z) = I I [_(x,y,z,x',y',z'). J (x',y',z')dx'dy'dz'Source

Substituting (4) in (3) we get

V2a ( . ) + k2a (.) = --_0Jr_ (X -- X') _ (y - y') _5(Z - Z')

(4)

(5)

where i is an unit dyadic, defined as i = 22 + YS' + _'2. Equation (5) may be written in compo-

nent form as

6

V2Gxx (.) + k2oGxx ( . ) = -_to_ (x - x') 8 (y - y') 8 (z - z') (6)

V2Gyy ( . ) + k2Gyy ( . ) = -go 8 (x - x') 8 (y - y') 8 (Z - Z') (7)

V2Gzz ( . ) + k2oGzz ( . ) = -_o 8 (x - x') _ (y - y') _ (z - z') (8)

Because of the nature of the problem and the boundary conditions, the other components of the

dyadic Green's function G (.) will not be excited and hence are not considered. The solutions

of (6), (7), and (8) may be assumed in the following forms

oo oo

Gxx( " ) : _ Z gxx(X"Y"Z"Z) C°S_---d-)sin ---if- (9)m=On=l

(m x)(n yIayy (.) = Z Z gyy (x', y', Z', Z) sin T cos --if-

m= ln=O

(10)

, Imrcx)(nrcy) (11)Gzz(') = _ _gzz(X"Y"Z'z) sin --'a-- sin Tm=ln=l

Substituting (9) in (6), (10) in (7) and (11) in (8) we get

_z2gxx( . ) + igxx ( . ) cos sin = -_toS(X-X')8(y-y')8(z-z' ) (12)

{ d2 k2t(m_x'_fn_Y'_--_zZgyy (" ) + Igyy(" ) sin ---h--)cos_--ff-) = -_08(x-x')8(y-y')8(z-z') (13)

-_z2gzz ( . ) + igzz ( . ) sln_-_-)sin = -p.oS(X-X')_)(y-y')8(Z-Z ')(14)

2 2 2_where k x = k 0 - Multiply (12)by cos(_-_)sin(_-_)and integrate over

the cross section of waveguide we get

• )+ igxx( )tdz

Likewise the equations (13) and (14) yield

d . k 2 }--]--2gyy ( ) + lgee ( " )clz

gzz( ) + igzz(. )

m n mrCx . n:gy=--l.t0-----v-cos sm 8(Z-Z')

= -_o-Tff"u't,--7)'_'_t, --V-)" (z- z')

13mgn . f__) . nl_y'= --ktO--_ sin _, sin(---b--)8 (z - z')

(15)

(16)

(17)

where I_ m and I_ n are Neumann's numbers [7] and equal to 1 for m = 0 and 2 for m e 0. In

order to determine the solution of the inhomogeneous differential equation (15) let us

assume gxx ( " ) = I £x ( kz) eJk'Zdkz

--oo

(18)

-jkz'zSubstitution of (18) in (15), multiplying by e and integrating over z leads to

- ernen c°s (ma_X') sin ( _ ) -jk_z'

egxx ( ) = -k + k

Substitution of (19) in (18) yields

(19)

oo

m n mrcx . nlty 1 , e JkzZeJ zZdk

g°_a--b c°s --7 sm "-T- k-k z

The integrand in equation (20) has poles at kz = -I-kI , therefore the integral in (20) can be

evaluated using contour integration in the complex domain [8]. Hence

(20)

xx(, cos T m X sin( )e (21)

where + sign in the exponential is taken when (z - z') < 0 and - sign in the exponential is taken

when (z - z') > 0. Likewise, gyy ( . ) and gzz (z) are obtained as

gyy (.) = -_/--_sln_----_)cos (22)

-J_ogmgn. ('m_X')sin(n___)e+Jkz(z-z')gzz(. ) =(23)

Substituting (21), (22), and (23) in (9), (10), and (11), respectively, the x-, y-, and z-components

of the dyadic Green's function are obtained as

_ _ --JgoEmgn ('___) (ngy') (mgx) (nrcy) +Jk,(z-z ')Gxx(" ) = E Z _/_ cos_ sin ----if-- cos ----a-- sin --if- e (24)

m=0n=0

_ooo _j[.tOEmEn " ('m_x"_ ('n_y') (m_x) (n_y)+jk,(z -z')Gyy(. ) = E E "_-/'_'_-sln_7)c°s_'-ff-)sin ---a---cos --if--e (25)

m=0n=0

(mrcx') (nrcy') (_m__fff_). ('nrcy)+Jk,(z -z')_ -Jl'tOemen sin sin sin sln_ T)e (26)G zz(') = E E 2k Iab --'a--" T

m=0n=0

The x-, y- and z-components of the magnetic vector potential due to the x-,y-, and z-directed

currents are then obtained as

A x (x, y, z)oo _ _J_oEmEn ( "_ ( Y'_

E E mltx .n+)m=0n=0 _/ _ _.vS_ ----_) sin

I ttj "x' ' (m_x"_ . ('nrcy"_ +jlq(z-z')JJ x t , Y, Z') cos_, 7) sln_. T)eSource

dv (27)

Ay(x,y,z) = Z z-J_t°EmEn " (m_x_ fn_y_m=On=O'_i "-'_sln_-'-d'-)c°s_--b - )

racy'f f_Jy(x',y',z')sin(m_x--_')cos(---_)e+Jk'<z-z')dv (28)

Source

oo °

_-J_OEmEn. (m_x'_. (nrCy)

az(x'Y'Z) = ?=0n_0 "_/= _ sln_---_jsan_---_-)

_Jz (x',y',z')sin(-_)sin(n/ty') +-jk,<z-z')-ff--)e dv (29)Source

The expressions in (27)-(29) are the required solution of inhomogeneous Helmoltz equation given

in (3).

(b) Electromagnetic Fields Due to Transverse Currents:

The electric and magnetic fields due to A x (x, y, z) are obtained from (2) as

__ --jo) _ _ --J_oEmEn((.2 (mrc)2_cos(mrCX)sin(nrcy))

Ex(ax) - k'-_om___On_=O _I "_ _'_ko-k'--d-J J k,-'-a---.] k.-if-JJ

mxx' nrcy' z') dv (30)_ HJx(x',y',z')cos(--d--)sin(--_--le +-jk'(z-Source

--jfo _ _ -J_-l'o m n( m_Z'_nl_ . (mrcx_ (n_y'_

Ey (Ax) : -_o m_=On_=O _ii _ _---h--)-ffsln_'--d--)c°s_--ff -)

_Jx (x',y',z')cos(-_) sin(n_y') +-Jk'(z-z') dv--v-)eSource

(31)

-- --j(o _ oo _j_O_ m En

Ez(Ax) - _Om__On_=O_ 2k I ab (+-JkI)(-_)" (mrcX'_sin(_)sxn_,"-a-")

10

(mltx"_ . (n_y"_ +-Jk,(z-z') dvf ffJx(x',Y',Z')C°S[---d-)sln_.-'-_-)e

Source

(32)

H x(Ax) = 0 (33)

Hy(Ax) = @ @--J green cos(.______/sin(__ff_ )mk___On_=o2kl ab (++'JkI) mrcx nlry

f ffJx (x', y', z')cos(-_)-'-bm_,----_-)e('nrcY'_ +-Jk,(z-z')dvSource

(34)

oo o_ • E E= nrcy

"_ ,_J m n(nlry] (m_X)sin(______ ]Hz(Ax) ml_=On_=o2ki"_ _. Tjc°s_ T)

(mrrx'_ . (nrcy') "{'jkI(Z-Z') dv

f ffJx(x',Y',Z')C°S[,-'-_)sm_'--_-) eSource

(35)

Similarly, the electric and magnetic fields due to Ay (x, y, z) are obtained as

--jtt) oo oo _j_OEml_n( nrr)(mlt) (mrcx). (nrty)

Ex(ay) = _ E E _/ a-b _--_-)_--_)cos_--_--jsln_---_)k0 m=0n=0

mrcx' nrcy'f ffJy(x',y',z')sin(----d--]cos(---_)e+-Jk'(z-z"dv

Source

(36)

--jo) °° °° --J_oEml_n .2 __/COS (_)Ey(Ay) - _02,m_=0n___ 0 _// _-_ (k 0-(?)2)sin(

• (mrrx') (n_y"_ +jkI(Z-Z')

f ffJy(x',y',z')sm_,_)cos_,--ff-)eSource

dv (37)

oo oo

( nrc) . (mrcx] . (nrcy]-j03 --J_oE m En ----ff -'-ff'-Ez(Ay) = .--_-E E 2k I ab (+-Jkl) sin sin

kO m=On=O

11

• (m_x"_ (n_y"_ +Jkl(z-z').f ff,y(x,,y,z,)sin ,a)COS -r )e avSource

(38)

= _ _ 2 EmEn sifl(_-_/COS(_-_ /Hx(ay) m_= ln___02k I ab (+JkI)

• (mrcx'_ (nrcy'_ +-Jkl(Z-Z').ffJy(x',y',z')sln_--d--)cos_---ff--)e av

Source

(39)

By (Ay) = 0 (40)

_ -J_mEn( n_Y)sin(__h_._)sin(__.__ )H z (ay) = m_--On_=O_--_i-_ k,---if- J minx n_y

f ffJy(x,,y,,z,)sin(mrcx') fnrcY') +jk,(z-z')---S- jcos_ --b--)eSource

dv (41)

(c) Electromagnetic Fields Due to Longitudinal Current:

The transverse electric fields due to A z (x, y, z) are obtained from (2) as

--j(.o _ ,,o _j_OEm E n mrC m_x . nrcy

Ex(Az):--ST Z Z 2k 1 ab (++-JkI)(--a-)C°S(----a--)sln(--ff-)k0 m=0n=0

Z'ffJz(x',Y',Z') sm_-'-d- )Source

(42)

--jfD _ _ --J_oEmEn (n_'_ . ('m_x_ f nny'_Ey (az) = -:-2 Z Z 2k I ab (+-JkI) --_-)sln_.----_-jcos[ --ff-)

k:O m=On=O

• ('m_x"_. fnrcy"_ +jk,(z-z')_Jz (X',y', z') smt--S-)smt--u )e

Source

dv (43)

In obtaining the longitudinal electric field representation due to A z (x, y, z), special

attention is required in performing the differentiation with respect to z on A z (x, y, z) . Since

12

A z (x, y, z) is continuous as a function of z, the first derivative of A z (x, y, z) is straightforward,

and therefore causes no difficulty. Hence

oo oo_J_oEmEn . fmrcx) (nrcy)?AOzz(X'Y'Z) = _ _ _ -_ (+jki) sm_T)sin Tm=0n=0

( )(,, ,, mnx nny _-j ,( - z ) dv" (44)_Jz(x'Y'Z") sin ----a-- sin _ eSource

where double prime quantities are the dummy variables of integration. Clearly _-----?z(x, y, z) is

discontinuous at z = z" so in performing the derivative of __A (x, y, z) with respect to z' _Z z

around z = z" care must be exercised to account for the jump in __A (x, y, z) as one crosses' _z z

the z = z" point. The behavior at z = z" is properly accounted for by an impulse function at

the point whereas the differentiation throughout the rest of region poses no problem, therefore,

_)?Az(x,y,z ) = _--J_oEmEn 2_. (m_x_. (nrty)m : 0n : 0 _// _-_ -kI )sm_. ----d--) sin _ ---b--)

_ _Jz(x",y",z")sin(_-_)sin( nrCy'')+-jkI(z-z'')_)eSource

e_ _ _j_Oe m e n

+_ 2 abm=0n=0

mrcx nrcy(-2j) sin(--a-/sin(-ff- )

" " +'k z z"

ffJ ¢x" y",Z")_(z-z" sm_--_)sln_--ff-)e dv"(45)Jd z _ ' ) " fm_x) . ('nrcy ) _-J,( - )Source

Integrating on z" in the second term of equation (45) yields

13

_z2 A z _ 0o(x,y,z) _ _-J_tOEmEn( ,2_. (mXx'_. (nXy'_

m:O.:O 5-b'-t- ')slnt-ujs'nt'--b-)

( ___ ) . ( n rcy" "_ "t-j kI ( Z - "S fyJz(X",y",z")sin smt-b--)e Z)dv"Source

( ss ()()())4 . (mxx_ nxy m_x" nrcy" ,, ,,-g0 Jz(x"'Y"'z)-_ Z sln_--_)sin --if- sin ---a--- sin --if- dx dy

source m = On = 0

(46)

Expanding g (x - x") _5(y - y") in the Fourier sine series over the domains 0 < x < a and

0 < y < b where 0 < x" < a and 0 < y" < b [9], it can be shown that

8 (x - x") 8 (y - y")

oo oo

4 .(mrcX'_sin(__Y)sin(_)sin(_..___ )= a-b Z Z sin t--a---) nrcy"m=0n=0

(47)

Using (47), (46) can be written as

= -gOJz (x, y, z) + "' --J_oEmEn( ,2"_ . (mgx'_ . (__.y)Z Y-, _/ _-_ t,-_,)slnt,--a-jslnt,m=On=O

I II Jz(x'''y'''z'') " (mgx"). (nxy")+jk,(z-z")smt '--a---)slnt _)eSource

d_l! (48)

The longitudinal component of the electric field is then obtained using (2) as

• --jf, D °°@-J_Lo_m_nCk2i_.._ 2)(m.x)T (_-_)Ez(Az) :J_xgOJz (x'y'z) +--_-X _-'_ t, o-k! sin sinkO KOm- - ln=l "

I IfJz (x', y',z')sin(-_)sin(_-)e+-Jk'(z-z')dvSource

(49)

14

Themagneticfield componentsdueto A z are obtained as

oo oo "E E= "w' 'w, -J m nn_. (mlcx'_ (nrcy'_

Hx (Az) 50n+O 2k# ab -_-- sm _ -"a"--)c°s[ --if-)

• (m_x'). (nlcy')+-jk,(z-ffJz(x',Y',Z') Sln[--_) .sin[ _ )e z'_dv

Source

(50)

oo oo

_ j Eml_nmlr, (mrCx'). (

Hy(Az) : m___On_o2ki-_ a c°st-_)sln_-_)

mrcx' nrcy'f ffjz(X',y',z')sin(---d---)sin(---_--)e +-jk'(z-z')

Source

dv (51)

t-1z (Az) = 0 (52)

The total electric and magnetic fields inside the waveguide due to J is then obtained by

superpostion of the electromagnetic fields due to A x , Ay , and A z .

(d) Dyadic Green's Function for Electric Field:

It is instructive at this point to defined the dyadic Green's function for the electri field

formulation. To this end, we write the vector wave equation for the electric field as

--)

: qO oJ (53)

m

If the electric field in terms of the dyadic Green's function G e (x', y', z'/x, y, z) is given as

E (x, y, z) = -jo_ktofffG- e (x', y', z'/x, y, z). J (x', y', z')dv' (54)

Substituting (54) in (53) the wave equation for the dyadic Green's function of electric-type is

obtained as

15

-- . k 2-VxV×G e( ) - oGe(.)

From equations (30)-(32), (36)-(38),

written as

= i5 (x - x') _ (y - y') _ (z - z')

(42), (43), and (49), the dyadic Green's function can be

(x - x') _ (y - y') _ (z - z') _Ge ( " ) = Geo ( " )- 2

k o

where Geo ( . ) is given by

Geo( • )_ n_ 0 --j EmE n +jkt(z-z')= 2k Iab e

=0 =

([k 2 (_)2]cos(_f)sin(_-_)cos(mrcx'_'nrcY'^^- -7-) sm--b--xx

( mrc_nrt . fm_x_ [nrcy_ (m_x'_. (nrcy'_^^

+ _-T )-b'-sm_ ----a--)c°s_ --b-')c°s _ "-h--) sln_ T )yx

nrcy mrcx' nrcy' ,,,,+ (+Jki)(-_)sin(_f)sin(---ff--)cos(----d--)sin(---ff--)zx

(_nrc](mrc_ cos (_m_) sin(n__). ('mltx"_cos( _)yc_+\ b ]\ a ] \ sln_--_)

+ [ko2_(?)z] sin(-_f)cos (_-_)sin (-_--_)cos (_-_),,

m_x nrcy mrcx' nr_y'+ (+jk,) (-?) sin (----_) sin (-if-) sin (--d--) sin ( ---_-)_,

(_) f__) (nrcy_ . (mrcx"_ . (ngy')^^+ (+_Jki) cos sin --ff-)s,n_---_jsln_---_jxz

+(+_jk,)(?)sin(mnx'_ ('nny'_ .('nrcy")^^----d--)cos_--_--)sin(_)sln_"-b--)yz

+ [k 2 _ k_]sin(_-_f]sin(_-_)sin(mrcx")'(_-]'2?,---d-) sln_, )

(55)

(56)

(57)

16

The expression in (57) is identical to the Green's function reported in reference [1].

Ill. Application

Analysis of Cylindrical Post in a Rectangular Waveguide:

Consider a rectangular waveguide with a cylindrical post as shown in figure 2. It is

assumed that the waveguide is excited by the dominant mode from the right. For simplicity it is

assumed that the surface current density on the post as

J = _I0_ x'- _(z-z') (58)

_/-+) -_ -+Let E s J be the scattered electric field due to the current J and E i be the incident electric

field due to TEl0 mode. The total electric field inside the waveguide is then given by

+(+)+E s J + E i . Subjecting the total tangential electric field on the surface of the post to zero, we

get following electric field integral equation:

/+/+)+)Es J + Ei t --'-- 0 (59)

where the subscript t is for the tangential component. Selecting a testing surface current density

____>as Jr which resides on the cylindrical surface, Galerkin's procedure reduces equation (59) to

(E s_ )J OJT)+(EilJT) = 0 (60)

Equation (60) can be written in a algebric form as

ZyyI 0 + Vy -- 0 (61)

17

whereZyyI0 = (Es J i JT), Vy = (Ei i JT).,withtheindicatedintegration performed in

cylindrical coordinates. Using (54) and (56), the expression for Zyy is obtained as

Zyy -_ - ( (%to2b / a)

Ig

E l f -jklr°sin ((P) ( mTCr° )m = 1, 3, 5, kl-_je cos ,. a cos ((p) dq_

"" 0

(62)

..->

Assuming an unit amplitude dominant mode E i can be written as

•J(' (a"-;)-_ _bb (_) e-J k0- zE i = _ sin (63)

Using (63) the quantity Vy can be written as

• 2 ,/_ 2 _ . 2 /I: 2 .

¢ 2/_b2_e-'J(k°-(/))Z'_e-'J(k°-(I))r°sm(_°)COS(=rOCos (_))d(pVy = ',,4a_, / ',, a

0

(64)

The algebric equation (61) can be solved for I0 . The reflected amplitude of the dominant mode

field at a reference plane z = 0 is then determined from

• 2 ,/_ 2

-ko rlolo ye-'ff(k0-(a))2z1F = /(-"_-_-_--_'_2"_aa (65)

//k"_l _1 !_\ o _.aJ J

The transmitted amplitude of the dominant mode at the reference plane z = 2z I is obatined as

T __

l+

-korloI o /-_-b]e-JJ(kl- (i)=)2z_

41k o ka) ) )

(66)

18

IV. Numerical Results

To validate the Green's function derived in this report, a y-directed cylindrical post of

( a )radius r 0 = 0.1 cm placed at x = _, z = 0.5 in a rectangular waveguide with a = 2.25cm,

b = 1.02cm and excited by an unit amplitude dominant mode field is considered. The reflection

coefficient at the z = O.Ocm plane and the transmission coefficient at the plane z = 1.0cm due

to the presence of the probe are calculated using expressions (65) and (66) and presented in

figures 3 and 4 along with the numerical results obtained using the FEM method [7,8]. The close

agreement between the results obtained from two different numerical methods confirms the

validity of the Green's functions derived here.

V. ConclusionThe complete dyadic Green's function for a electric current source located inside a

rectangular waveguide is derived using the magnetic vector potential approach. The magnetic

vector potential for an electric current source in a rectangular waveguide is obtained by solving

the inhomogeneous Helmholtz's equation. The electric and magnetic fields are obtained from the

magnetic vector potential through spatial differentiation. The fields which are valid over the

source region are obtained by carefully differentiating the vector potential around the source

location. The electric and magentic field expressions obtained by the present method are found to

be identical with the expressions reported in the literature. Numerical results on the reflection and

transmission coefficients using the Green's function approach are in a good agreement with the

numerical results obtained using the FEM techniques.

19

References

[1] J.J.H. Wang, "Analysis of a three-dimensional arbitrarily shaped dielectric or biological

body inside a rectangular waveguide, "IEEE Trans. on Microwave Theory and Techniques,

Vol. MTT-26, No. 7, pp 457-462, July 1978.

[2] M.S. Leong, et al, "Input impedance of a coaxial probe located inside a rectangular cavity:

theory and experiment, "IEEE Trans. on Microwave Theory and Techniques, MTT Vol. 44,

No. 7, pp. 1161-1164, July 1996.

[3] C.T. Tai, Dyadic Green's function in electromagnetic theory, Scranton, PA: Intext

Educational Publishers, 1972.

[4] R.E. Collins, "On the incompleteness of E and H modes in waveguides," Can. J. Phys.,

Vol. 51, pp. 1135-1140, 1973.

[5] C.T. Tai, "On the eigen-function expansion of dyadic Green's functions, "Proc. IEEE, Vol.

61, pp. 480-481, March 1974.

[6] Y. Rahmat-Samii, "On the question of computation of the dyadic Green's function at the

source region in waveguides and cavities, "IEEE Trans. Microwave Theory and

Techniques, Vol. MTT-23, pp. 762-765, September 1975.

[7] M.D. Deshpande and C. J. Reddy, "Application of FEM to estimate complex permittivity of

dielectric material at microwave frequency using waveguide measurements, ", NASA

Contractor Report 198203, August 1995.

[8] M.D. Deshpande, et al, "A new approach to estimate complex permittivity of dielectric

material at microwave frequencies using waveguide measurements, "To appear in IEEE

Trans. on Microwave Theory and Techniques, March 1997.

20

YX ,_ ElectricCurrent

X_ource InsideWaveguide

Figure 1Electric currentsourceinsidearectangularwaveguide

21

Y

X

Probe

r 0

TEl o

Mode

Incident

aLoadZ

Z1

Figure 2 Rectangular waveguide with a cylindrical post parallel to y-axis placed at

x = a/2, z = z 1.

22

0.5

0.0O

O

-0.5

1.0 -

a-a-_ z_-a.a. Imaginary

"A _.& ,j__-A AAA •

Real Part_ Present Method

Imaginary Part J

• Real Part "_ FEM Method [_zx Imaginary Part J

-1.0 IIlllIFllllllrrlfllllllllllllllrlllllll

8.5 9.0 9.5 10.0 (G10'5Hz) 11.0 11.5 12.0Frequency

Figure 4 Reflection coefficient of a y-directed post in a rectangular waveguide as a functionof frequency.

23

1,0 -

0.5

6o.o

r_

r_

#

-0.5

-1.0

. A _'_"& A

- a "A'A'_ _ "_'_A- Imaginary

rX..A_A_A. sk

_ _A_A"/I

Real Part _A'A" -

Imaginary Part } Present Method zxAA't? Z" -

• Real Part } FEM Method [8]A Imaginary Part

llll lllll i lllllillpilillililillilllllli

8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0Frequency (GHz)

Figure 5 Transmission coefficient of a y-directed post in a rectangular waveguide asa function of frequency.

24

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April 1997 Contractor Report4. TITLEAND SUI_1I, LE 5. FUNDING NUMBERS

Analysis of Discontinuitiesin a Rectangular Waveguide Using Dyadic C NAS1-19341Green's FunctionApproach in Conjunctionwith Methodof Moments

6. AUTHOR(S)

M. D. Deshpande

7. PERFORMINGORGANIZATIONNAME(S)ANDADDRESS(ES)ViGYAN, Inc.Hampton, VA 23681

9. SPONSORING/ MONITORINGAGENCYNAME(S)ANDADDRESS(ES)

National Aeronauticsand Space AdministrationLangley Research CenterHampton, VA 23681-0001

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Langley Technical Monitor: Fred B. BeckFinal Report

WU 522-33-11-02

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10. SPONSORING / MONITORINGAGENCY REPORT NUMBER

NASA CR-201692

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13. ABSTRACT (Maximum 200 words)

The dyadic Green's function for an electric currentsource placed in a rectangularwaveguide is derived usingamagnetic vector potentialapproach. A complete solutionfor the electric and magnetic fields includingthe sourcelocation is obtained by simple differentiationof the vector potential around the source location. The simpledifferentiationapproach which gives electric and magneticfields identicalto an earlier derivationis overlookedby the earlier workers inthe derivationof the dyadic Green's function particularly around the source location.Numerical results obtained usingthe Green's function approach are compared with the results obtained usingthe Finite Element Method (FEM).

14. SUBJECT TERMS

Dyadic Green's Function; Waveguide Discontinuities;Method of Moments

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