AN EFFICIENT INTEGRAL TRANSFORM TECHNIQUE OF A SINGULAR WIRE ANTENNA KERNEL. S.-O. Park

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1 AN EFFICIENT INTEGRAL TRANSFORM TECHNIQUE OF A SINGULAR WIRE ANTENNA KERNEL S.-O. Park Department of Electronics Engineering Information and Communications University 58-4 Hwaam-dong, Yusung-gu Taejon, , South of Korea Abstract-This paper presents an efficient integral transform tech- nique for evaluating the singular part of the scalar and vector potentials due to a uniform current subdipole. By performing several transformations, the original double integral 1/-Rs with a singular kernal can be represented as a finite one-dimensional integral whose integrand possesses an analytically integrable logarithmic singularity. The results of the newly driven integral are compared with those using other available numerical methods and checked the validity of the proposed method. 1. INTRODUCTION In the impedance matrix of a wire antenna, the diagonal or self-interaction terms [Zmm] I which contain a singularity in the integrand have more dominant values than the off-diagonal terms [Zmm]. Since the inversion of the moment matrix [Zmm] is very sensitive to small inaccu- racies in the diagonal terms [Zmm], accurate evaluation of singularity becomes an important requirement. The direct integration of the diagonal term [Zmm] converges very slowly, whereas the extraction of the singularity provides the fastest convergence toward the actual value, if this term can be solved analytically. We will now consider to evaluate efficiently the self-term double integral of on the cylindrical surface whose integrand contains Rs R, a singularity on the interval of integration. The general self-interaction term for evaluating the magnetic vector and electric scalar potentials due to a z -directed uniform current can be rewritten, which takes the

2 1106 where 2A and a represent one segment length and radius of wire antenna, respectively [2]. The application of standard quadrature rules to integral (1) whose integrand possesses a singularity requires considerable computational time. One possible way to overcome this problem is to subtract out the singularities. Thus, subtracting out the singularity leads to and could be defined as A... where the second term of the integrand in (3) has a slowly varying function. For this reason, this part can be easily evaluated with standard nu- merical integration. From early, many author have presented different solutions for solving the remaining first term #o. Their contributions of this problem were mentioned in [1, 3]. This communication presents another new integral transform technique for evaluating the double in- tegral of (4), which greatly facilitates the numerical integration with accuracy and efficiency. 2. INTEGRAL TRANSFORM TECHNIQUE OF A SINGULAR KERNEL To evaluate the double integral of (4) on the surface of the cylinder, can be rewritten in the form (eq. 6, in [1])

3 1107 where lm(ka) and Km (ka) are modified Bessel functions of the first and second kind, respectively. Upon integration with respect to z', (6) reduces to Using the formula (5, p. 429) in [4], the integrand Ko(ak). Io(ak) in (7) can be expressed as an integral form Substituting (8) into (7), it can be expressed as The infinite double integral of (9) can be further simplified if one of the integration has an analytic solution. Using the formula of [5], the integration of (9) with respect to k can be carried out analytically Thus, substituting (10) into (9), the infinite double integral (9) can be converted into only an infinite one dimensional integral as However, the obtained infinite integral formula of (11) is not suitable for evaluating numerical integration. The integral (11) over the q plane can be converted into an integration over the x plane by using Parseval's theorem.

4 1108 since and F2 (q) are an even function of q. Let us define Fi(ry) = [Jo(ary)]2 and (1 - Deduced from formula in [5], /l(x) can be solved as where Q_i (r) 2 is referred to as a spherical Legendre function of the second kind. For the convenience of calculation, can be represented as 2 (after corrections of formula in [6]) where K(m) is the complete elliptical function of the first kind. Using the above relation, Q_ i in (13) can be easily expressed as 2 With the aid of the formula in [5], can be easily obtained as Substituting (13) and (16) into (12), the infinite integral of (11) with respect to q results into the finite integral explicitly as The integrand of (17) contains an integrable singularity. After the singular parts of integral are extracted and calculated analytically, the

5 1109 remaining nonsingular integration can be evaluated using a numerical procedure because the integral varies smoothly. 3. EVALUATION OF THE INTEGRAL i#o Using the previously mentioned technique, the infinite integral in (7) has been transformed into a finite one dimensional integral (17). How- ever, the integrand of (17) possesses singularity at x = 0. In order to evaluate the integral at and near the singularities, first we need to describe the asymptotic behavior of f 1 (x) and f2 (r). To accomplish this, we introduce the following asymptotic behavior of the complete elliptical integral ( in [6]). where K(m) is the complete elliptical integral of the first kind. Using (18) and (15), the asymptotic behavior of can be approximated at and near the singularity = 0, by The asymptotic behavior of f2(x) at and near the singularity x can be represented by = 0 In order to evaluate the integral (17), we subdivide the interval of integration into the two regions; the region at and near the singularities ([0, 61), and the remaining region away from the singularity ([6, 2a] ) as follows; '--I At and near the singularity region, the original functions and f2 (r) are approximated by their asymptotic functions which allow the closed-form integration. A direct integration of the first integral in

6 1110 Figure 1. The values of f2 (r), and f i sy at A = a = 1. x (21) gives a following analytical solution Since the function 11 (x) and f2 (r) are well behaved in the regions 2a]), the second integral of (21) is carried out numerically by using the self-adaptive integration scheme. The values of f 1 (x), f2 (r), and are evaluated at the parameters of A = a = 1, and plotted in Figure 1. As seen in Figure 1, an asymptotic function is well describing the original function at and near x = 0. This gives a small freedom for choosing the value 6 in (21) whose small change doesn't significantly affect the accuracy of the overall value of the integral (17). 4. CONPUTATIONS AND COMPARISONS The integration of (17), which is valid for the entire region of A/a, was performed and tabulated in on the fourth column of Table 1. To

7 1111 check the validity of the one dimensional integral of (17), the values of other three different methods are evaluated and listed in Table 1. First, Butler's series form (10) in [2] which is valid only for A la > 2 was evaluated with sufficiently high degree of accuracy and the values are listed on the first column of Table 1. The series solution of (l0a) in [1] which is valid only for A/a < 1 was calculated with accuracy up to four significant digit and listed on the second column of Table 1. Table 1. The integration values of by using (17) and other methods. The singular treatment used in MININEC in [7], which is valid for the entire region of 0/a, was also evaluated and listed on the third column of Table 1. As can be seen in Table 1, the proposed method has an excellent agreement with other three comparison results across the entire range of 0/a. These results demonstrate that the newly derived transformed 1-D integral of (17) is valid without limitation, which greatly facilitates the numerical integration of the singular double self-term. _ 5. CONCLUSIONS In this contribution, the self-term double integral 1/Rs with a singular kernal in wire antenna was transformed into a finite 1-D integral which has many practical numerical advantages without limitation. The results of the proposed method were compared with the those of other available numerical and analytical methods. It has been demonstrated that there is good agreement with each other. This method can be

1112 applied to evaluate efficiently and accurately the diagonal matrix elements in Method of Moments solution of moderately thick radii wire antennas.

8 1112 applied to evaluate efficiently and accurately the diagonal matrix elements in Method of Moments solution of moderately thick radii wire antennas. ACKNOWLEDGMENT This material is based upon work supported by Korea Institute of Information Technology Assessment. REFERENCES 1. Park, S. O., and C. A. Balanis, "Efficient kernel calculation of cylindrical antenna," IEEE Trans. Antennas Progpagat., Vol. AP-43, , Nov Butler, C. M., "Evaluation of potential integral at singularity of exact kernel in thin wire calculations," IEEE Trans. Antennas Progpagat., , Mar Werner, D. H., "An exact formulation for the vector potential of a cylindrical antenna with uniformly distributed current and arbitrary radius," IEEE Trans. Antennas Progpagat., Vol. AP-41, , Aug Watson, G. N., A Treatise on the Theory of Bessel Functions, Cambridge at the University press, Gradshteyn, I. S., and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, Abramowitz, M., and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications, New York, Julian, A. J., J. C. Logan, and J. W. Rockway, "MININEC: a mini-numerical electromagnetic code," NOSC, Technical Document 516, Sept Seong-Ook Park was born in KyungPook, Korea, in December, He received the B.S. degree from KyungPook National University, KyungPook, Korea, in 1987, the M.S. degree from Korea Advanced Institute of Science and Technology, Seoul, Korea, in 1989, and the Ph.D. degree from Arizona State University, Tempe, AZ, in 1997, all in Electrical Engineering. From March 1989 to August 1993, he was a Research Engineer with Korea Telecom, Teajon, Korea, working with microwave systems and networks. He later joined the Telecommunication Reasearch Center, Arizona State University, until his departure in September Since October 1997, he has been with the Information and Communications University, Teajon, Korea, as an Assistant

9 1113 Professor. His research interests include analytical and numerical techniques in the area of microwave integrated circuits and MMIC's. Dr. Park is a member of Phi Kappa Phi Scholastic Honor Societies.

TRADITIONALLY, the singularity present in the method of

TRADITIONALLY, the singularity present in the method of 1200 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 54, NO. 4, APRIL 2006 Evaluation Integration of the Thin Wire Kernel Donald R. Wilton, Fellow, IEEE, Nathan J. Champagne, Senior Member, IEEE Abstract