Judah disk greens

601 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 39, NO. 3, MARCH 1991 Reduced Form of the Green’s Functions for Disks where and Annular Rings f(rk,ak) = Y;(ak)J,(rk)- J;(ak)Y,(rk) (3) Fayez A. Alhargan and Sunil R. Judah k2 = O’WE Abstrncr -Available Green’s functions for circular and annular ring microstrip circuits involve doubly infinite series. These series are com- putationally expensive in terms of the time necessary for summing the series and the memory required to hold the eigenvalues. In this paper the Green’s function is simplified to a single series using a new single- summation method. The resulting single series eliminates the need for the eigenvalues and increases the speed of computation. a proof Of (2), consider karJ,( rk)f( kr,, ku) 8, ka +- 4 J; ( ka ) FJk) = Ogrgroga r,fO where (4) I. INTRODUC~ION In the analysis of microstrip antennas of patch circuits the Green’s function is obtained using the modal expansion. This method gives rise to a doubly infinite series [l]. The mode matching method has also been used to obtain the impedance directly in a single series format; this method is applicable to ports around the periphery only. However, Carslaw [2] has used the result of Kneser [5] to show that the Green’s function for cylindrical coordinates can be analytically summed over the modes, reducing the Green’s function by one series. In this paper the Green’s function is simplified by a method different from that of [5]. Although there are a number of methods for obtaining this result, the method used here is direct in its approach. The Green’s functions for the circular disk and the annular ring are obtained in single series format and the two methods are compared for both accuracy and efficiency. 0 n=o n#O Note that no restriction is placed on n; it can be irrational, complex, etc. Now using Mittag-Leffler’s expansion theorem: m r1 1\ F(z)=F(O)+ C Rm (5) m=l where p, is the mth pole of F(z), and.Rm is the residue of F(z) at the mth pole. Now FJk) has poles at k = * k,, and F,,(O) = 0. Therefore Hence 11. GREEN’S FUNCTION FOR A CIRCULAR DISK The Green’s function for a circle of radius a is given by [l] But giving This can be simplified to Manuscript received June 21, 1990; revised October 16, 1990. This work was supported by the National Guard of Saudi Arabia through a study grant to F. A. Alhargan. The authors are with the Department of Electronic Engineering, Hull University, Hull, HU6 7RX, United Kingdom. IEEE Log Number 9041943. Therefore 0018-9480/91/0300-0601$01.00 01991 IEEE 602 IEiEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 39, NO. 3, MAKCH 1991 Multiplying through by jwpdmn cos n(4 - q50) and then sum- ming over n gives (2). For the special case r = ro=O, the Green’s function is given by G(074/0740) = ak2a2 + --J c As a proof of (15a), consider (16) 5, + ~kf( rk , ak)f( rok , bk ) FAk) = 4f’( bk, nk) k(b2 - U‘) -jwpd jwpd 1 a<r<r0<b m=l (k,2m-k2)J02(akOm) where which simplifies to f’(bk,ak) =Yi(ak)Ji(bk)- J,’(ak)YA(bk). (17) (lo) In the same manner as above, FJk) can be expandcd using Mittag-Leffler’s expansion theorem as follows. Fn( k) has poles - As a proof of (lo), consider at k = f k,, and Fn(0) = 0. Hence akYd(ak) 1 F(k) = +-. 4Jd(ak) auk Now F(k) has poles at k = Mittag-Leffler’s expansion gives kom and F(0) = 0. Therefore using where Subtracting from both sides l/~ka and then multiplying by jwpd/ak gives (10). 111. GREEN’S FUNCTION FOR AN ANNULAR RING af’( bk, ak) (21) ak . f”(bk,ak) = With some manipulation and the fact that 1 ( x’ - n’) %xx) (22) E’’( x) = - - %’( x) - ~ X X’ The Green’s function of an annular ring microstrip antenna of ar, 4/ro9 40) we have inner and outer radii a and b respectively is given by [l] Multiplying through by jupd cos rd4 - 4J/ak and then sum- ming over n gives (15a). This simplifies to IV. IMPEDANCE FORMULATION G( r, 4 / ro 3 40) IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 39, NO. 3, MARCH 1991 -( b2 - &)f2(bknm,ak,,)- Hence using the double series form, the impedance matrix elements are given as jwpdA,A, jopd z =- +-c c wu: 4a n=fjm=l 4~ k’a2 - 2d,2dJ (26) anJn(d,knm)Jn( d,knm) cos n+,, sin nA, sin nA, whereas the reduced form gives ppd 5 a,Jn( kd,)f( kd,, ku) cos n( sin nb, sin nA, nW,nW, 2d,2dJ z,, = - JL( ka) ~ 16 n=O O<d,gd,ga, d,#O where ~ 603 -- I / Number of Modes Fig. 1. The double-series form for the ring gives Timing for disk and ring. - jupd jopd anf’(bknm,akn,) Z1’=4k2a(b2-u2) +-c 4a n=Om=, c (k$,,-k’) nW A, =sin-’ (z) 4, = 4, - 4,. (x) cos n4,, sinc2 ’ [(b2-$-)f2(bknm,uknm)- In the above, it is assumed that the port connections are circular. The elements of the impedance matrix for a multiport annular ring are obtained in a similar manner, giving, for the double (32) series form, and, for the single series form, iwd z,, = - ’ 16 a,nf(d,k,uk)f(dJk,bk)cosn4,,sinnA,sinnAJ .E nwnW (29) n=O f’(bk,uk)- 2d,2dJ U< d, G d, < b. For the case where the ports are around the periphery the equations can be simplified further as follows. The double-series form for the disk gives PW -jupd jupd z,, = - +-c c 45rk’u2 4~ a, cos n4,, sinc2 (z) (30) and the reduced form gives ~~ ~ and the reduced form gives f nW\ where W Ai z - 26 V. COMPARISON OF THE Two FORMS Comparison of the equivalence and time taken for the compu- tation of the two forms was carried out by simulating two junctions: (i) A 4-port rat race circular disk of radius 15 mm and (ii) a 4-port annular ring of outer and inner radii 15 mm and 5 mm respectively. In both cases the ports widths were 4.4 mm and were positioned at O”, 90°, 135”, and 225”. The substrate was assumed to have a relative permittivity of E, = 2.5 and a height of 1.524 mm. The results are given for Z,, at a frequency of 4.5 (GHz). Fig. 1 shows the time ratios (time taken by double series/time taken by reduced form) against the number of modes used for the annular ring and the circular disk. Fig. 2 shows the values of Z,, 604 0.4 I Reduced Form 0.2 Number of Modes Fig. 2. Z,, for disk. Reduced Form Double Series 0.4 -0.3 -0.4 -0.5 1 0 20 40 60 80 Numaer Of Model Fig. 3. Zll for ring. for the circular disk for both the double series and the reduced form and the discrepancy between them. Fig. 3 shows the values of Z,, for the annular ring for both the double series and the reduced form as well as the discrepancy between them. VI. CONCLUSION The Green’s functions for the circular disk and the annular ring have been reduced to single-series forms in a mathemati- cally direct manner, eliminating the need for the eigenvalues and, as a consequence, improving the speed and accuracy of the computation. ACKNOWLEDGMENT The authors wish to thank Dr. M. Page for providing experi- mental results to validate the theory. REFERENCES [l] K. C. Gupta et al., Computer Aided Design of Microwave Circuits. Nonvood, MA: Artech Ijouse, 1981, p. 247. [2] H. S. Carslaw, “The Green’s function for the equation V2u + K‘U = 0,” Proc. London Math. Soc., vol. 15, pp. 84-93. Apr. 1916. [3] A. Benalla and K. C. Gupta, “Faster computation of Z-matrices for rectangular segments in planar microstrip circuits,” IEEE Trans. Microwave Theory Tech., vol. MTT-34, pp. 733-763, June 1986. [4] M. D. Abouzahra and K. C. Gupta, “Multiple-port power divider/combiner circuits using circular microstrip disk configura- tions,” IEEE Trans. Microwave Theory Tech., vol. MTT-35, pp. 1296-1302. Dec. 1987. [5] J. C. C. A. Kneser, “Die Entwicklung Beliebiger Funkticinen von beschrankter Schwankung nach Besselschen,” Math. Ann., Bd. 63, Cam- pp. 510-517, 1907. [6] G. N. Watson, A Treatise on The Theory of Bessel Functions. bridge, England: Cambridge University Press, 1966, p. 499. A Surface Integral Equation Method for the Finite Element Solution of Waveguide Discontinuity Problems Omar M. Ramahi and Raj Mittra Abstract -The surface integral equation method, which is typically employed in the finite element solution of open-region scattering proh- lems, has been applied in this paper to the solution of waveguide discontinuity problems. The major advantage offered by the surface integral equation approach over other available methods is that it allows the mesh-truncating boundaries to be brought as close to the discontinu- ity as possible, thus helping to reduce the size of the system matrix. In addition, unlike the mode matching technique, the surface integral equation formulation does not require the solution of any auxiliary matrix system. Numerical results are presented to illustrate the validity of the formulation. I. INTRODUCTION IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 39, NO. 3, MAHCH 1991 0018-9480/91/0300-0604$01 .OO 01991 IEEE When designing waveguide devices, it is often necessary to introduce discontinuities or loads that are used for different purposes such as phase shifting or power matching to a specific load or termination. The analysis of such waveguide junctions or discontinuities has traditionally been carried out using the mode-matching techniques and the integral equation method [ 11, [2]. However, when the discontinuities are irregularly shaped or involve inhomogeneous or anisotropic objects, the integral equation methods become quite laborious and difficult tc i apply. For such complex and irregularly shaped geometries, either the finite element or the finite difference method becomes the preferred choice. Additionally, the finite methods generate highly sparse and banded matrices which can be efficiently handled using special algorithms. When using the finite element (or the finite difference) method to solve boundary values problems such as waveguide disconti- nuities, two major consideration arise. First, it is always desir- able to bring the mesh-truncating boundary as close as possible to the discontinuity junction in order to reduce the nuniber of mesh points and, hence, the size of the associated matrix. Second, a boundary condition must be imposed on the turminal boundaries such that it accurately reflects the proper field behavior there. The task of devising an efficient solution proce- dure that accommodates the above two considerations is the principal subject of discussion in this paper. Conventionally, finite element formulations of the waceguide discontinuity problem are based upon the truncation of the finite element mesh region at a distance where the amplitudes of the evanescent modes become negligible, and then the impo- sition of a Dirichlet or a Neumann boundary condition (31, [41. Such construction offers the advantage of generating a sparse matrix system. In certain applications, such as the modeling of electromagnetic pulse simulators, the width of the simulator/ waveguide may range from a fraction of a wavelength to tens of Manuscript received June 5, 1990; revised October 15, 1990. The authors are with the Department of Electrical and Computer IEEE Log Number 9042351. Engineering, University of Illinois, Urbana, IL 61801.