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 r 1 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