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. Bara and D. M. Belle Division of Engineering Brown University Providence, Rhode Island AbstFact Abasic limitation in an earlier work on the axially ma netized, fferrite loaded rectangular guide has led to adetailed re-examination of this problem. Both aseries solution and a perturbational technique are used to find dispersion curves and field patterns. (1) Introduction By assuming an ej(@-Bz) de endence we are led to the familiar coupled equations !,3 (,v~ +al)ez =-jBkf~hz (V: +a2)hz =j6kf&ez 1 Application of atechnique described elsewhere leads to asolution of the form3 e❑U1+u2, hz =qlul +q2u2, z where @:+ fJ12*)ul ~ = o. ,Y This yields asimple solution for the circular guide?~b For the rectangular guide we postulate asolution of the form U1,2= ~(A~’2cos~+ B~’2sin ~)(C~’2cosk~’2y +~~’2sink~’2y) where each term satisfies (l), and the eight boundary conditions are imposed on the series. There are no relations of orthogonality between terms, but four sets of coefficients can be expressed as functions of the remaining four. This leads to an infinite dimensional homogeneous system, and, by truncation, the problem is reduced to finding the zeros of acomplex determinant. Alternatively, it is seen that equations (l), together with the boundary conditions, decouple for (3=0, giving pure TE or TM modes. In view of this fact we write ~he fields in the guid~ as ez= ~fn(x,y)~n ,hz= ~gn(x,y)13n n.o n=O and f=~a~~n. Similar expressions are written for the remai~ing frequency dependent parameters. The following equations are thus obtained: al al n-1 (V~+ao )fn+~aif n-i =-j 1a~gn-i-~ i=l i.o ‘2 n-1 (V~ao )gn+ ~ai ‘2 gn_ i=j1a~fn_i-~ , i=l i.1) plus the conditions at the boundaries fn=o , n-2 agn-i_2 n-1 af -j ~a:’ ~n-i-1 ● +j~a~~ i~oaial ~ = O. i.o i= O Use of Green’s functions of electric and type lead to general expressions for fi and fi ❑~~ F1 sin ~sin ~ nm mn gi =~~G;ncos~cos~ . nm The a; ‘s are obtained by applying Green’s to (f *fn+* )or (gn,gn+2), depending on of modes n It can be shown that only powers of ,82 the expression for f, as expected, and that magnetic gi 9 identity the type enter in hcontains odd powers of @when ez contains even”ones (quasi-TM modes) and vice versa (quasi-TE modes). It is obvious that the method is good only for quasi-TE, -TM modes. Solution For the first method it was found that for n>3 e,h approximated by more than 24 terms) the ~&~ of tife resulting determinant are very unstable as aresult of imaginary Kn ‘s leading to hyperbolic functions. For n=2 (8 x8determinant) the zeros are well defined, but the accuracy is poor and cannot be improved. In the perturbational method one can write general expressions for any arbitrary te,rm of order nin terms of the previous ones. On the other hand, convergence fop only alimited range of 6is expected, since even for the dielectric guide OJ%OEO=k~+62 ,%21/2 f%-[1+ (# 1 c and fis given as aseries of powers of $2 for B’2<k’2. However this range can be extended by analytic c&tinuationl. Note that this difficulty does not exist in the dielectric guide if we express fz =f2(62), but in our problem both fz and fappear. Results Dispersion curves for the lowest modes of the rectangular guide as shown on Fig. (1) have been computed from both methods. In the series solution, the quasi-TE/TM modes are easily identified for Bsmall. Other modes are under investigation. Figure 1shows the dispersion curves for the quasi-TE1o modes, as obtained from the perturbational equations, with fz: ~6 a~~n . n=O 71 The series turns out to be alternating, thus providing error bounds. The region of convergence is O<$<n, 2.6 rad/~m for Hdc =O,decreases as we approach resonance, and becomes fairly large (8 ~5rad/cm) above resonance. Within this region convergence is fast; for example, for B=2rad/cm (Hdc =O), f=7.2542 f0.0005 Ghz, and even for $=2.4 rad/cm the error is 0.01 Ghz. This first region of conve~gence increases for higher order modes. The parabolic approximation for =&loEf $2/(l+x) (dotted line) was found t>provide avalue of faccurate to better than a1% through the whole region of convergence (This expression is equivalent to llJ21Joc -62 =(n/a)2 for the dielectric guide). Figure 2shows the rotating nature of the transverse Hfield even for asituation close to cutoff. The transverse Efield is very similar to that of the pure TEIO mode. The fields at the walls of the guide are not plotted since the trigonometric series giving them do not converse there to the real values (Gibbs’ Both can be extended to include ferrite losses by astraightforward modification of the ferrite parameters. References 1. G. Barzilai and G. Gerosa, “A Modal Solution for a Rectangular Guide Loaded with Longitudinally Magnetized Ferrite”, Electromagnetic Theory and Antennae, Editor: E. C. Jordan, pp. 573-590, Pergamon Press, 1963. 2. A. A. Th. M. Van Trier, “Guided Electromagnetic Waves in Anisotropic Media”, Appl. Sci. Research, vol. 33, 1953. 3. M. L. Kales, I!Modes in Waveguides Containing Ferrites”, Journal of Appl. Physics, Vol. 24, Number 5, (May, 1953). 4. H. Suhl and L. R. Walker, “Topics in Guided Wave Propagation through Gyromagnetic Media”, Bell System Tech. J., vol. 33, 1954. 5. L1. G. Chambers, “Propagation in aFerrite-Filled Wavemide”, Quart. J. Mech. and Appl. Math., Vol. VIII, phenomenon). Part-4, December, 1955. Discussion The perturbational method is capable of being extended to wider ranges of Band to other geometries in adirect manner. On the other hand, for modes other than quasi-TE/TM we must return to aconsideration of the first method outlined above. f (Ghz) 18 16 14 12 10 8 6 10715 I2345 6 7~(rad/cm) FIG. I 72 Ez E+ Hz Ht ‘t Hdc=o , IE I z‘ax =0.02 ‘tlmax ............... ............... ............... ............... ............... ............... ............... ............... . . . . . . . . . . . . . . . ............... ............... ............... I............... ...............I ............... ............... ............... ............... . . .. . . . . . . . . . . . . ............... ............... 47TM~=1071 Gauss Hz max =, 78 ‘tlmax . ,4 ///.... ,/, ,,, //////..>/ ,,, // / /////..,,,, ,,, / /////..,,,, ,,, //////4.,,// /// //////...,// /// ,(///...., ,,, ,, .ltlllllll lit!. .1 LIIIIIII1l 1$. ., 1111111111 1$. ., 111111111 11!. l:; ,,111111 l,,’.] /////....,,//// /////., . , t / //// /////...0/)//// /////..,*////// //.///..,0////// /////..,0)///// /////...0////// QJt=lr/4 FIG.2 47TM~=1071 Gauss ,F=7.25Ghz Ht max ‘tlmax XZTE=I.25 m Jllll Il~lltt :Illil 11111. 11111 1‘ [1111. :11111 Ill Il. ,11111 11111, Jllll 11111. :Jllil 11111. I . . . - - -——-- - - ... .- - - ---—— -- - - - - .-.- - ----- - - - .- - ------—-- -- - - . . -..- - -———- -...- . . - - - ------ - - - . ..-.----- -- - - .- - (LJt=T/2 [-. . %-------- - . . . ....- ------ - ... ...- - ------ - - - . . . - - - ----- - - - .- .- - - ----- -- &. . . ...- - ----- -- ... ...#- ------ ~. . . II ‘t max ~Z ~E=o.50 %max FIG.3 73