Dispersion characteristics and field structure of an axially magnetized ferrite loaded rectangular waveguide

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