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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
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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
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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
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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
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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
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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].
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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
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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)
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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)
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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)
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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-")
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(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
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• (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
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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
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_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)
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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
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-- . 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)
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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)
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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
Page 21
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
Page 22
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
Page 23
YX ,_ ElectricCurrent
X_ource InsideWaveguide
Figure 1Electric currentsourceinsidearectangularwaveguide
21
Page 24
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
Page 25
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
Page 26
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
Page 28
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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
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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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