DISPERSION CHARACTERISTICS AND FIELD STRUCTURE OF AN
AXIALLY MAGNETIZED FERRITE LOADED RECTANGULAR WAVEGUIDE
J. T. Ba a and D. M. Belle
Di ision o Enginee ing
B own Uni e si y
P o idence, Rhode Island
Abs Fac
Abasic limi a ion in an ea lie wo k on he axially ma ne ized,
e i e loaded ec angula
guide has led o ade ailed e-examina ion o his p oblem. Bo h ase ies solu ion and a pe u -
ba ional echnique a e used o ind dispe sion cu es and ield pa e ns.
(1)
In oduc ion
By assuming an ej(@-Bz) de endence we a e led
o he amilia coupled equa ions !,3
(, ~ +al)ez =-jBk ~hz
(V: +a2)hz =j6k &ez 1
Applica ion o a echnique desc ibed elsewhe e
leads o asolu ion o he o m3
e❑U1+u2, hz =qlul +q2u2,
z
whe e
@:+ J12*)ul ~ = o.
,Y
This yields asimple solu ion o he ci cula guide?~b
Fo he ec angula guide we pos ula e asolu ion
o he o m
U1,2= ~(A~’2cos~+ B~’2sin ~)(C~’2cosk~’2y
+~~’2sink~’2y)
whe e each e m sa is ies (l), and he eigh bounda y
condi ions a e imposed on he se ies. The e a e no
ela ions o o hogonali y be ween e ms, bu ou
se s o coe icien s can be exp essed as unc ions o
he emaining ou . This leads o an in ini e dimen-
sional homogeneous sys em, and, by unca ion, he
p oblem is educed o inding he ze os o acomplex
de e minan .
Al e na i ely, i is seen ha equa ions (l), o-
ge he wi h he bounda y condi ions, decouple o (3=0,
gi ing pu e TE o TM modes. In iew o his ac we
w i e ~he ields in he guid~ as
ez= ~ n(x,y)~n ,hz= ~gn(x,y)13n
n.o n=O
and =~a~~n. Simila exp essions a e w i en o
he emai~ing equency dependen pa ame e s.
The ollowing equa ions a e hus ob ained:
al al n-1
(V~+ao ) n+~ai n-i =-j 1a~gn-i-~
i=l i.o
‘2 n-1
(V~ao )gn+ ~ai ‘2 gn_ i=j1a~ n_i-~ ,
i=l i.1)
plus he condi ions a he bounda ies
n=o ,
n-2 agn-i_2 n-1 a
-j ~a:’ ~n-i-1 ●
+j~a~~ i~oaial ~ = O.
i.o i= O
Use o G een’s unc ions o elec ic and
ype lead o gene al exp essions o i and
i ❑~~ F1 sin ~sin ~
nm mn
gi =~~G;ncos~cos~ .
nm
The a; ‘s a e ob ained by applying G een’s
o ( * n+* )o (gn,gn+2), depending on
o modes n
I can be shown ha only powe s o ,82
he exp ession o , as expec ed, and ha
magne ic
gi 9
iden i y
he ype
en e in
h- con-
ains odd powe s o @when ez con ains e en”ones
(quasi-TM modes) and ice e sa (quasi-TE modes). I
is ob ious ha he me hod is good only o quasi-TE,
-TM modes.
Solu ion
Fo he i s me hod i was ound ha o n>3
e,h app oxima ed by mo e han 24 e ms) he
~&~ o i e esul ing de e minan a e e y uns able
as a esul o imagina y Kn ‘s leading o hype bolic
unc ions. Fo n=2 (8 x8de e minan ) he ze os a e
well de ined, bu he accu acy is poo and canno be
imp o ed.
In he pe u ba ional me hod one can w i e gene al
exp essions o any a bi a y e, m o o de nin
e ms o he p e ious ones. On he o he hand, con e -
gence op only alimi ed ange o 6is expec ed,
since e en o he dielec ic guide
OJ%OEO=k~+62 ,%21/2
%-[1+ (# 1
c
and is gi en as ase ies o powe s o $2 o
B’2<k’2. Howe e his ange can be ex ended by ana-
ly ic c& inua ionl.
No e ha his di icul y does no exis in he
dielec ic guide i we exp ess z = 2(62), bu in ou
p oblem bo h z and appea .
Resul s
Dispe sion cu es o he lowes modes o he ec-
angula guide as shown on Fig. (1) ha e been compu ed
om bo h me hods. In he se ies solu ion, he
quasi-TE/TM modes a e easily iden i ied o Bsmall.
O he modes a e unde in es iga ion.
Figu e 1shows he dispe sion cu es o he
quasi-TE1o modes, as ob ained om he pe u ba ional
equa ions, wi h
z: ~6 a~~n .
n=O
71
The se ies u ns ou o be al e na ing, hus p o-
iding e o bounds.
The egion o con e gence is O<$<n, 2.6 ad/~m
o Hdc =O,dec eases as we app oach esonance, and
becomes ai ly la ge (8 ~5 ad/cm) abo e esonance.
Wi hin his egion con e gence is as ; o example,
o B=2 ad/cm (Hdc =O), =7.2542 0.0005 Ghz,
and e en o $=2.4 ad/cm he e o is 0.01 Ghz.
This i s egion o con e~gence inc eases o highe
o de modes.
The pa abolic app oxima ion o =&loE
$2/(l+x) (do ed line) was ound >p o ide a alue
o accu a e o be e han a1% h ough he whole
egion o con e gence (This exp ession is equi alen
o llJ21Joc -62 =(n/a)2 o he dielec ic guide).
Figu e 2shows he o a ing na u e o he ans-
e se H ield e en o asi ua ion close o cu o .
The ans e se E ield is e y simila o ha o
he pu e TEIO mode.
The ields a he walls o he guide a e no
plo ed since he igonome ic se ies gi ing hem
do no con e se he e o he eal alues (Gibbs’
Bo h can be ex ended o include e i e losses by
as aigh o wa d modi ica ion o he e i e pa am-
e e s.
Re e ences
1. G. Ba zilai and G. Ge osa, “A Modal Solu ion o a
Rec angula Guide Loaded wi h Longi udinally Magne-
ized Fe i e”, Elec omagne ic Theo y and An ennae,
Edi o : E. C. Jo dan, pp. 573-590, Pe gamon P ess,
1963.
2. A. A. Th. M. Van T ie , “Guided Elec omagne ic
Wa es in Aniso opic Media”, Appl. Sci. Resea ch,
ol. 33, 1953.
3. M. L. Kales, I!Modes in Wa eguides Con aining
Fe i es”, Jou nal o Appl. Physics, Vol. 24, Numbe 5,
(May, 1953).
4. H. Suhl and L. R. Walke , “Topics in Guided Wa e
P opaga ion h ough Gy omagne ic Media”, Bell Sys em
Tech. J., ol. 33, 1954.
5. L1. G. Chambe s, “P opaga ion in aFe i e-Filled
Wa emide”, Qua . J. Mech. and Appl. Ma h., Vol. VIII,
phenomenon). Pa -4, Decembe , 1955.
Discussion
The pe u ba ional me hod is capable o being ex-
ended o wide anges o Band o o he geome ies
in adi ec manne . On he o he hand, o modes
o he han quasi-TE/TM we mus e u n o aconside -
a ion o he i s me hod ou lined abo e.
(Ghz)
18
16
14
12
10
8
6
10715
I2345 6 7~( ad/cm)
FIG. I
72
Ez
E+
Hz
H
‘
Hdc=o ,
IE I
z‘ax =0.02
‘ lmax
...............
...............
...............
...............
...............
...............
...............
...............
. . . . . . . . . . . . . . .
...............
...............
...............
I...............
...............I
...............
...............
...............
...............
. . .. . . . . . . . . . . . .
...............
...............
47TM~=1071 Gauss
Hz max =, 78
‘ lmax .
,4 ///.... ,/, ,,,
//////..>/ ,,, //
/ /////..,,,, ,,,
/ /////..,,,, ,,,
//////4.,,// ///
//////...,// ///
,(///...., ,,, ,,
.l lllllll li !.
.1 LIIIIIII1l 1$.
., 1111111111 1$.
., 111111111 11!.
l:; ,,111111 l,,’.]
/////....,,////
/////., . , / ////
/////...0/)////
/////..,*//////
//.///..,0//////
/////..,0)/////
/////...0//////
QJ =l /4
FIG.2
47TM~=1071 Gauss
,F=7.25Ghz
H max
‘ lmax XZTE=I.25
m
Jllll Il~ll
:Illil 11111.
11111
1‘
[1111.
:11111 Ill Il.
,11111 11111,
Jllll 11111.
:Jllil 11111.
I
. . . - - -——-- - - ...
.- - - ---—— -- - - - -
.-.- - ----- - - - .- -
------—-- -- - - . .
-..- - -———- -...-
. . - - - ------ - - - .
..-.----- -- - - .- -
(LJ =T/2
[-. . %-------- - . . .
....- ------ - ...
...- - ------ - - - .
. . - - - ----- - - - .-
.- - - ----- -- &. . .
...- - ----- -- ...
...#- ------ ~. . .
II
‘ max ~Z ~E=o.50
%max
FIG.3
73
