9

Cavity model theory [ I ] ..._.. ’ Experiment [ I ]

L-- 0 0 5 1 1 5 2

Airgap width (mm) Figure 2 The resonant frequency of microstrip antenna with an airgap, E , = 2.32, a = 5 cm. and d = 1.59 mm

resonant frequencies and half-power bandwidth are displayed as a function of airgap width, respectively. It is seen that the agreement between theory and experiment is good.

4 , I

Full wave method

t

0 1 I I I I 0 0 5 1 1 5 2

Airgap width (mm) Figure 3 The half-power bandwidth of microstrip antenna with an airgap. e , = 2.32, a = 5 cm, and d = 1.59 mm

CONCLUSIONS A full wave method is used to analyze to the resonant fre- quency and half-power bandwidth of a microstrip antenna with an airgap. The method predicts the behavior of the struc- ture accurately.

3. K. Araki. H. Ueda, and T. Masayuki, “Numerical Analysis of Circular Disk Microstrip Antennas with Parasitic Elements,” IEEE Trans. Antennas Propagat., Vol. AP-34, 1986. pp. 1390-1394.

4. K. Araki, D. I. Kim, and Y. Naito, “A Study of Circular Disk Resonators on a Ferrite Substrate.” IEEE Trans. Microwave The- or>’ Tech., Vol. MTT-30, 1982, pp. 147-154.

Received 7-9-90

Microwave and Optical Technology Letters, 3/11. 39 1-393 G 1990 John Wiley & Sons, Inc. CCC‘ O895-2477/90/ $4.00

SOLUTION OF THIN-WIRE INTEGRAL EQUATIONS BY NYSTROM METHODS John S. Kot CSIRO Division of Radiophysics Epping New South Wales 2121, AUSTRALIA

KEY TERMS Wire unletinus. integral equation, numerical method

ABSTRACT Numerical soliitions to the electric-field integral equation (EFIE) for a thin wire ore obtained by a Nystroin method. The singular EFIE is treated by subtraction of the singularity and a thin-wsire integral equation convenient for numerical solution is obtained. Solutions for exact and approximate EFIE kertzelv nre compared.

1. INTRODUCTION

The electric-field integral equation (EFIE) for the current induced on a thin wire illuminated by an incident electric field has been used extensively to study wire antennas and scat- terers. It has also been applied to scattering by wire grids and wire-grid models of solid metal objects [l]. Solutions to the EFIE for thin wires have been obtained typically by the method of moments [2]. A comprehensivc review of moment-method solutions to the EFIE for thin wires is given by Imbriale [3], for various choices of basis and testing functions. In this article a different approach is taken. A numerical solution to the EFIE is obtained by a Nystrom, or quadrature, method [4]. This leads to a very simple formulation of the problem, and the singularity in the EFIE kernel can be dealt with in a straightforward way, resulting in well-behaved solutions. Use of exact and approximate EFTE kernels is compared.

2. EFIE FOR THIN WIRES The EFIE for thin wires is well known, for example. see 151. In this section a straight wire is used to define the geometry and to discuss the treatment of the singularity in the EFIE kernel. The extension to piecewise straight wires and grids of straight wires is straightforward. Extension to more complex wire shapes is also possible.

Figure 1 shows a straight wire of length I and radius u. orientated along the z-axis of a cylindrical-polar coordinate system. It is assumed that the radius a is small compared to the wavelength of any illuminating field. Then the only sig- nificant component of electric current on the wire is the z- directed component J ; , which is assumed to be constant around the circumference of the wire. The electric field due to all sources other thanJ, is termed E,. Harmonic time dependence

REFERENCES 1. K. E Lee, K . Y. Ho. and J . A . Dahele, “Circular-Disk Microstrip

Antenna with an Air Gap.” f E E E ?tans. Antennas Propagut..

2, K. Araki and T, Itoh, c ‘ ~ ~ ~ k ~ l Transform nomain ~ ~ ~ l ~ ~ i ~ of Open Circular Microstrip Radiating Structures,” IEEE Trans. ~ n -

Vol. AP-32, 1984. pp. 880-884.

1enna.s Propagat., Vol. AP-29, 1981, pp. 84-89.

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol 3, No 11, November 1990 393

i In the Nystrom method the integral over 1 is approximated by a N-point numerical quadrature rule. Applying this to Eq. (3) we obtain:

Figure 1 Geometry for straight-wire EFIE

of the form eku' is assumed. The EFIE for J, is

where 2 is a z-directed unit vector, k, = 2n/& and

R(2 , z ' , 4') = [ ( z - z')2 + 2a2(1, - cos 4')]'!*.

If the current flowing on the wire is approximated by a filament of current flowing along the wire axis, Eq. (1) sim- plifies to

2 r* dz' (5 + k;) g(&, z'))J,(z') = f . E,(z), (2) jws)

where

R(z, z') = [ ( z - z')* + a y .

Equation (2) is commonly referred to as the thin-wire in- tegral equation. As a consequence of the approximation, the singularity in the exact EFIE kernel is circumvented.

To obtain numerical solutions to Eqs. (1) and (2) we use the Nystrom method, as described by Delves and Mohamed [4]. Firstly, Eqs. (1) and (2) are written in the following form, where the superscripts (1) and (2) refer to Eq. (1) or (2), respectively.

1 P ( z ) , z ' )J(z ' ) dz' = S(z); i = 1, 2, (3)

where

S(Z) = i . E,(z).

The set of points {z,,; 1 n 5 N) are the N abscissae for the quadrature rule, with corresponding weights w,. In the results presented in this article we have used n-point Gauss- Legendre quadrature rules which are exact for polynomials of order 2n - 1.

To find the N unknowns J(zn), Eq. (4) is written for each of the set of points {z = 2,; 1 5 m 5 N). The resulting set of N linear equations is written as the matrix equation:

where

The solution of Eq. ( 5 ) gives the set of N current values {j,, = Jz(z,J; 1 5 n 5 N}.

3. NYSTROY METHODS FOR SINGULAR KERNELS The kernel function W ( z , 2') is singular at z = 2' . As a consequence the simple Nystrom method outlined above fails completely if applied to Eq. (1). The kernel function r(2)(2, 2') is not singular, but is sufficiently "badly behaved" so that the Nystrom method will converge only very slowly to a so- lution of Eq. (2). To deal with the problem of a singular or badly behaved kernel, we use a modified form of the method of subtraction of the singularity described by Delves and Mo- hamed [4], where successively higher-order terms are sub- tracted from the integrand to leave a well-behaved integral.

Firstly a function r&'(z, 2') is constructed, having the form

The function g$)(z, z') is chosen so that r a ( z , z ' ) - I'p(z, z ' ) is a well-behaved function of z' for IzI < 112, lz'l < 112. Then Eq. (3) is written in the form

1 [ r y z , Z ' ) J , ( Y ) - riyz, Z ' ) J ~ ( ~ ) I dz'

+ JZ(z)Q(I)(z) = S(z); i = 1, 2, (7)

where

i = 1, 2.

Now the integral containing the unknown J,(z') is well behaved by hypothesis, and the Nystrom method can be ap- plied directly to Eq. (7) with an expectation of reasonable convergence. Provided the function Q(')(z) can be evaluated, a scheme for the numerical solution of Eqs. (1) and (2) has been obtained.

The function g/;'(z, z ' ) is chosen to ensure sufficient reg-

394 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 3, No. 11, November 1990

ularity of the remainder function P ( z , z ' ) - Th"(z, z ' ) , and also to allow the function Q(!)(Z) to be conveniently evaluated. A number of different choices are possible. The functions used here are

(9)

Again, superscripts (1) and ( 2 ) refer to the functions rel- evant for Eqs. (1) or (2), respectively. For the case of gf'(z, z ' ) , the function Q'*'(z) may be evaluated in a closed form. For g")(z, z'), the 4' integral must be evaulated numerically. This may be done efficiently by noting that once the z' integral and differentiation with respect to z have been evaluated in a closed form, the resulting (6' integral is well behaved apart from a logarithmic singularity. This term may then be ex- tracted and integrated analytically, leaving a well-behaved remainder which can be integrated numerically. Other than at points very close to the wire ends, this may be done with a six-point Gauss-Legendre rule to typically relative ac- curacy.

4. APPLICATIONS For numerical calculations a simple gap source model was chosen, having the form

This approximates a urlit voltage source applied across a gap of width 6 in the wire, located at the point z = z,,,,~,.

Half-Wuve Dipole. The first example is a half-wave dipole, with 1/2a = 12.5, which was studied by Tmbriale [3]. This antenna was analyzed by solving Eq. (I), the exact EFIE kernel, using a Nystrom method based upon a three-panel Gauss-Legendre quadrature rule, with n points in each of the two intervals {1/2 5 121 5 6/2}, and a single point in the source gap. Results for the modulus of J I ( z ) are shown in Figure 2 for 13-, 17-, and 21-point solutions. The results show good convergence. Interestingly, the moment method used by Im- briale shows poorer convergence for thicker wires, while the Nystrom method used here shows the opposite behavior.

I I . . . I I -0.1 0 0.1 0.2

Distance Along the Wire in Wavelengths

- 21-point solution + 17-point solution o 13-point solution

Figure 2 Modulus of current on half-wave dipole antenna

TABLE 1 Input resistance R,,, for half.wave dipole

Order of Solution R$'(CL) Rf'(CL) R,. from (31 (a) 13 125.7 133.8 17 123.0 133.0 } 126 21 123.5 134.2

The same antenna was also analyzed using Eq. ( 2 ) , the approximate thin-wire kernel. Results for the input resistance R,, for the antenna for the two equations are shown in Table 1, and compared with the result obtained by lmbriale using a 50-subsection moment method with complete integration of the self term. R$.(*) refers to results obtained from the re- spective equations. Although the solution for Eq. ( 2 ) shows reasonable convergence, there is some discrepancy in the value of input resistance, and some problems occur for J, near the ends of the wire.

L-Shaped Wire Antenna. The second example chosen is the application of the method to a more complex wire shape, an L-shaped wire with equal-length arms. The arms are orien- tated parallel to the x and z axes, and join at the point z = 112, x = 1/2, (see inset in Figure 3). The source is located in the center of the z-directed arm. The length of each arm is i,,, and the wire radius is 0.041,,. Results obtained for the real parts of J, and J , using 31 points on each arm are shown in Figure 3. The equation used was Eq. (I). The currents at x = 112, z = I12 are seen to approach equal amplitude step functions, showing that continuity of current is being ap- proximately satisfied at the wire junction, although there is no explicit constraint to enforce continuity in the formulation. To demonstrate the convergence of the solution, results for the real part of J , for 21 points on each arm and 31 points on each arm are shown in Figure 4.

5. CONCLUSIONS The Nystrom method is a very simple method for the nu- merical solution of integral equations. However, when used with high-precision quadrature rules such as the Gauss-Le- gendre rules, it is capable of giving well-behaved solutions to quite "difficult" equations such as EFIE for wires, provided due attention is paid to the treatment of the behavior of the kernel. By extracting the contribution due to the singular part of the EFIE kernel and integrating it separately, a numerical

I . . I . . . I . . . . , _ _ . . I . _ I _ _ I . . I . . I . . . . -0.2 0 0.2 0.4 -0.05

-0.4 Distance Along the Wire in Wavelengths

- Current on z-directed arm ..._.. Current on x-directed a m

Figure 3 tenna

Real current on x- and z-directed arms o l L-shaped an-

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 3, No. 11, November 1990 395

0.15 . I . . . . , . - . . I . . . . , . I . - , . I . , I I

-0.4 4 . 2 0 0.2 0.4 Distance Along the Win2 in Wavelengths

-0.05

- 31-point solution + 21-point solution

Figure 4 Real current on z-directed arm of L-shaped antenna

solution scheme showing good convergence is obtained. The method is currently being applied to wire-grid models of planar antennas.

ACKNOWLEDGMENTS I would like to thank Paul Clark for his assistance with this work whilst a summer vacation student.

REFERENCES 1. J . H. Richmond, “A Wire Grid Model For Scattering By Con-

ducting Bodies,” IEEE Trans. Antennas Propagat., Vol. AP-14,

2. R. F. Harrington, Field Computation By Moment Methods, Ca- zenovia, New York, 1968.

3. W. A. Imbriale, “Applications of the Method of Moments to Thin- Wire Elements and Arrays,” in R. Mittra, Ed., Numerical and Asymptotic Techniques in Electromagnetics, Springer-Verlag, New York, 1975, Chap. 2.

4. L. M. Delves and J . L. Mohamed, Computational Methods for lntegral Equations, Cambridge University Press, 1985.

5. D. S . Jones, Methods In Electromagnetic Wave Propagation, Clar- endon Press, Oxford, 1987.

1966, pp. 782-786.

Received 7-9-90

Microwave and Optical Technology Letters, 3111, 393-396 0 1990 John Wiley & Sons, Inc. CCC 0895-24771 901 $4.00

ON QUASI-STATIC SOURCE MODELS FOR WIRE DIPOLE ANTENNAS Daniel J. Janse van Rensburg and Derek A. McNamara Department of Electronic and Computer Engineering University of Pretoria Pretoria South Africa 0002

KEY TERMS Dipole antennas, input impedance, moment method, source models

ABSTRACT The moment method computation of the input impedance of dipole antennas, obtained using numerically generated Source models which are more closelv related 10 the actual physical shape of the antennu feed region. are discussed.

INTRODUCTION In many instances the primary aim of an electromagnetic anal- ysis is to obtain the values of a few important quantities (eg., input impedance) with a rather high accuracy. If we cannot compute such observables with sufficient reliability to enable decisions to be based on them, then any attempt at their computation is of doubtful value. Miller [l] has perhaps best summarized the situation as follows: “Accuracy must be con- sidered the foremost attribute required . . . Inaccurate but efficiently and easily obtained results have no value.” The input impedance computation of dipole antennas considered in this article has been a point of discussion for many years [2-41, and has centered on attempts to properly incorporate the physics of the excitation region in the source model used [4, 61.

THE QUASI-STATIC APPROXIMATION If we wish to refer to the concept of input impedance at all, then we must be able to uniquely define the concepts of volt- age and current in network terms at some point on the ra- diating structure (i.e. the voltage between the antenna ter- minals and current into and out of these terminals, for whatever point on the antenna we decide to call its terminals). We usually wish to interface the antenna with receiving equipment circuitry which will have a set of terminals comprising a region of space small enough in terms of wavelength to allow a net- work voltage and current to be defined there as the potential difference between the terminals (Faraday’s law without a significant contribution from the magnetic displacement cur- rent term) and the integral of the magnetic field around any conductor (Ampere’s law without a significant contribution from the electric displacement current term), respectively. (Such conditions are, of course, tacitly assumed to be extant whenever we apply network theory to electrical circuit prob- lems at any frequency other than zero). This implies that the region we choose to consider to be the antenna terminal or feed (source) region must also be small in terms of wave- length. But this at once means that the spatial distribution of the electric field in such a source region can be determined from quasistatic considerations [5]-that is, a solution for the spatial distribution of the electric field in this region may be found as if it were an electrostatic field, but remembering that this distribution has an ep‘ time dependence. Thus, the form of the electric field in the feed gap of a dipole may be found by assuming an impressed potential difference between the dipole arms in this region and solving the associated electro- static problem for the electric field distribution there. Appli- cation of the equivalence principle then permits replacement of the problem by an equivalent one consisting of a dipole without a gap at its feedpoint, but with a ribbon of impressed magnetic current M , at the surface of the “gapless” dipole at the position of the gap in the original problem. The spatial form of M , is determined from the above quasistatic solution for the electric field in the gap. This impressed magnetic cur- rent density M , is then considered to generate the impressed field over the entire antenna structure, and thus it is used in the determination of the elements of the excitation vector in the moment method solution for the antenna current distri- bution. Such numerically generated source models (i.e., mag- netic current ribbons) allow accurate modeling of rotationally symmetric dipole antenna source regions, their precise forms depending as they do on the geometrical detail of the feed

396 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 3, NO. 11, November 1990