approach for solving [he problem Df If-pl ane ~lI\'l!gu ide junctiDns " ith lossy feni le posts Df arbi lrary shupe is proposed. llw jlllK'liuns are all o"-ed 11,1 have arbitrary eross ~Iion. The approach is a rOlllbin:ltio n I,If the n"ite-el", menl method and the analytica l method. To ..how Ihe "~Iid ity a nd usefull\esl> of [he method, V-June[ion eircu lalors with a (ir('uhl" ferrile paS[ are considered. Our results agree well wilh earl ier ~~puinll'nl~1 and l!teore licil resulls. The

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... lna~es 01 V-ju nctiun drculato,-.; "-llh a triangu lar cquilate rnl fenile pos[ Dr a triangular ferrite [IO't h~ving uepreswll sides Ire inveSli ga ted. The influences uf [he !eITite ~o;es un the performance are exami ned. IN'J'RODUCfIO N A PPLICATIONS OF waveguide junctions with ferrite posts have been of wide-ranging use in microwave devices and circuits, and research on them has been continued steadi ly. Davies [1] presen ted the theoretical treatmenl for a symmetrical waveguide junction circulator with a circular ferrite post. This met hod was extended to junctions with coaxial composite ferr ite posts which prod uced much larger bandwidths [2] -[41. This analysis, however, is limited for j unctions Ihat have geometrical symmet ry. Recently, the point-matching method was extended to the asymmetri cal junctions (5] and was applied to the junctions with a tria ngular ferri te post (6J. The point-matching technique is powerful for the waveguide junctions with arbitrarily shaped ferrite posts, but the ferrite losses are neglected. Okamoto [71 presented a method based on Ihe ~nlegral equations for solving the problem of waveguide Junctions with lossy ferrite posts. In his approach, the Junctions are allowed to have an arbitrary cross section and arbitrary number o f ports, but only the ferrite posts wi th S~l ooth boundaries such as a circular ferrite post and a trtangular ferrite post having rounded angles are studied. In Ihis paper, a fini te-element method for the analysis of the H-plane waveguide junctions with lossy ferrite posts of arbitrary shape is described. For the analysis of planar circulators, Lyon and Helszajn III] presented a method based on circuit theory and the r . lOne-element method. In their approach, the system is f1sSumed to be free of any losses and the fi nite-element n:'ethod is used for the computation of the eigenvalues and eIgenvectors o f the normal modes of a magnetized ferrite Manu>cripl received April 2, 1985; revised Ju ly 26, 198 5. k ~ author.; are with 1M Dcpanment o f Electronic Enginreri no Ho ka,uo U · . .,. lEE nlvC r.. lty, Sapporo, 060 Japan. : 'E log N umbcr S405l!13. , , w! :r, :r; , , y .L. r o r Fig. 1. Geometry of problem. resona tor, and then the ci rcuit parameters are determined by using these data of the normal modes. Their approach is very useful for the planar circulators using arbitrarily shaped resonators and it can be applied to H-plane waveguide junctions with arbitrarily shaped ferrite posts. However, the scallering coefricients are quite sensitive to the values of the circui t parameters, so it is necessary to ensure that sufficient significant digits are carried through in the computation of the normal modes. In our approach, on the other hand, it is not necessary to compute the eigenvalues and eigenvectors of the normal modes. Making use of the method, we fi rs! treat Y -junction circulators with a circular ferrite post for comparison with the previously published experimental and theoretical results [31, [51-[71 . The performance of Y -junction circulators with a lriangular equilatera l fe rrite post or a triangu lar ferrite post having depressed sides are nCA t investigated. The influences of the ferrite losses on the performance are examined. Fig. 1 shows lhe H-plane waveguide junction with a fu ll-height fe rrite post of arbitrary shape. The dc magnetic field is applied in parallel with the z axis. The boundaries r" (i' = 1', 2', 3') lie in the region {} with r, (i c:: 1, 2, 3) and the short-circuit bou ndary r, and the region surrounded by r " and r completely encloses the waveguide discontinuities. III general, the wavegu ides need not be symmetrically OOJ 8-9480/ 86/ 0JOO.{)!03$01.OO OJ 986 J EEE II. BASIC EQUATIONS