ABSTRACT REDUCTION OF THE EDGE DIFFRACTION OF A CIRCULAR GROUND PLANE BY USING RESISTIVE EDGE LOADING by Rose Waikuen Wang Chairman: Valdis V. Liepa In many antenna measurements, a large flat circular conducting ground plane is a basic part of the measurement structure. To minimize the effects of edge diffraction, it is desirable to use as large a ground plane as possible. But in many instances this is not feasible due to the constraints imposed by structural limitations, such as mounting the antenna on a tower, or rotating the antenna on a pedestal to perform antenna pattern measurements. A large ground plane can also be cumbersome to use in laboratory where space is limited. The task here is to develop a finite size ground plane for an antenna whose electromagnetic characteristics resemble those on an infinite ground plane, that is, the antenna impedance and the radiation patterns approach those on an infinite ground plane. The basic problem is to reduce the ground plane edge diffraction effects over a wide range of frequencies. RL-796 = RL-796

Specifically, the problem addressed is that of a monopole located at the center of a circular ground plane whose edges are extended using resistive sheet material. The antenna impedance, radiation patterns, and currents on the ground plane are studied. The problem is formulated using the body of revolution technique and then solved numerically using the method of moments. Quantities studied for cases with and without resistive loading are the antenna impedances, the currents on the monopole and on the ground plane, and the far field patterns. To verify the computations, a monopole antenna was built and evaluated with both metal and resistive ground planes. The resistive material was made by spraying resistive paints of different conductivities onto a nonconductive material base to obtain the desired resistance variation. Since the resistivity of the sprayed sheet can not be accurately predetermined, non-destructive methods are devised to measure local resistivity at both DC and microwave frequencies.

REDUCTION OF THE EDGE DIFFRACTION OF A CIRCULAR GROUND PLANE BY USING RESISTIVE EDGE LOADING by Rose Waikuen Wang A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Electrical Engineering) in The University of Michigan 1985 Doctoral Committee: Associated Professor Valdis V. Liepa, Chairman Professor William J. Anderson Professor Chiao-Min Chu Professor Chen-To Tai Senior Research Scientist Dipak L. Sengupta

ACKNOWLEDGEMENTS The author wishes to express her gratitude to all the members of her Doctoral Committee for their helpful comments and constructive criticisms. She is especially indebted to her Chairman, Professor Valdis V. Liepa, for his invaluable guidance and encouragement. Appreciation also goes to Mr. Larry Champney for his help with the experimental work. Finally, the author wishes to thank her family for their encouragement, understanding and support during the course of this work. ii

TABLE OF CONTENTS ACKNOWLEDGEMENTS LIST OF TABLES LIST OF FIGURES LIST OF APPENDICES CHAPTER I. INTRODUCTION 1.1 Background 1.2 Outline of the Work 1.3 The Resistive Sheet Boundary Condition CHAPTER II. THEORETICAL BACKGROUND 2.1 Representation of the Electromagnetic Fields 2.2 The Method of Moments 2.2.1 General Procedure 2.2.2 Point Matching 2.2.3 Subsectional Bases 2.3 Application of the Method of Moments to Solve the E-Field Equations CHAPTER III. BODY OF REVOLUTION TECHNIQUES 3.1 Introduction 3.2 Application of the Method of Moments (M( Page ii v vi ix 1 1 3 5 8 8 9 9 12 12 OM) CHAPTER IV 4.1 4.2 4.3 3.2.1 Evaluation of the MOM Impedance Matrix 3.2.2 Evaluation of the Antenna Impedance 3.2.3 Evaluation of the Far Field r. RESISTIVE MATERIALS AND MEASUREMENTS Introduction Resistive Materials Measurement of the Resistivity of the Sample 4.3.1 DC Measurements 4.3.2 AC Measurements 13 17 17 18 18 28 28 32 32 33 41 43 51 iii

Page CHAPTER V. EXPERIMENTAL ANTENNA MODEL 58 5.1 Introduction 58 5.2 Construction of the Circular Ground Plane with Resistive Edge Loading 59 5.3 Antenna Impedance Measurements 61 5.4 Far Field Measurements 71 CHAPTER VI. NUMERICAL STUDIES 82 6.1 Introduction 82 6.2 Program Description 82 6.2.1 Initialization 83 6.2.2 Partition 85 6.2.3 Computation 87 6.2.4 Post-processing 90 6.3 Numerical Results 90 6.4 Comparison Between Experimental and Numerical Results 103 CHAPTER VII. CONCLUSIONS 112 APPENDICES 114 REFERENCES 141 iv

LIST OF TABLES Table Page 4.1 The Properties of Paints Used 34 4.2 Comparison of Resistivity Values Obtained Using DC Measurements 52 4.3 Comparison of AC and DC Measurements 56 4.4 Comparison of Shunt Capacitance at Different Frequencies 57 5.1 Number of Coatings and Mixtures Used in Preparing the Actual Model 60 5.2 Comparison of Impedance for a Monopole with Different Ground Planes 73 6.1 Comparison of Monopole Impedance - Theory and Experiment 107 v

LIST OF FIGURES Figure Page 3.1 A Line S Rotated about the Z-Axis Generates a Monopole Antenna on the Circular Ground Plane 19 3.2 Approximating of Generating Arc by Linear Segments for Strip of Revolution 22 4.1 Resistivity vs. Number of Coatings of Electrodag 110; Paper Base 35 4.2 Resistivity vs. Number of Coatings of Electrodag 109; Paper Base 36 4.3 Effect of Base Material on Resistivity; Electrodag 502 37 4.4 Resistivity vs. Mixture Ratio of Electrodag 110 & 502; Plastic Base, 2 coats 38 4.5 Resistivity vs. Mixture Ratio of Electrodag 110 & 109; Plastic Base, 2 coats 39 4.6 Resistivity vs. Number of Coatings of Electrodag 109 & 502 (1:4 Ratio by weight); Plastic Base 40 4.7 Spraying of Test Samples Using an Air-Brush Method 42 4.8 DC Measurement of Sample Using Direct Method 44 4.9 DC Measurement of Sample Using Two-Wire Line 46 4.10 Dimensions of Probe Geometries; (a) Coaxial Line, (b) Two-Wire Line 48 4.11 Equipment Block Diagram; (a) AC Measurement of Sample Using a Network Analyzer, (b) Equivalent Circuit 54 5.1 Making of Circular Resistive Sheet Using an Air-brush and a Phonograph Turntable 62 5.2 Resistivity vs. Distance from Center of Monopole Measured Using AC Method 63 vi

Figure Page 5.3 Photograph Showing the Contacts between the Ground Plane and the Resistive Sheet, the Ground Plane and the Monopole Antenna 65 5.4 Dimemsions of the Actual Model 66 5.5 Antenna Impedance Measurement Setup 67 5.6 Measurement of Impedance of the Monopole Mounted on a Finite Size Ground Plane with Resistive Sheet of Radius 12 cm 68 5.7 Measurement of Impedance of the Monopole Mounted on a Finite Size Ground Plane of Radius 12 cm 69 5.8 Measurement of Impedance of the Monopole Mounted on a Large Ground Plane of Radius 60 cm 70 5.9 Measured Monopole Impedance for Various Ground Planes 72 5.10 Block Diagram for Measuring the Far Field Patterns 75 5.11 Test Antenna Placement for Measuring the H-Field Pattern 77 5.12 Test Antenna Placement for Measuring the E-Field Pattern 78 5.13 Measured Far Field Patterns at 2.25 GHz 79 5.14 Measured Far Field Patterns at 2.50 GHz 80 5.15 Measured Far Field Patterns at 2.75 GHz 81 6.1 Structure of the Simulation Program 84 6.2 Diagram Showing how the Segments are Partitioned; (a) Old Method, (b) New Method 86 6.3 Effect of Gap Width on Impedance of a Half-Wave Dipole 92 6.4 Computed Monopole Impedance for Various Ground Planes vs. Frequency 93 vii

Figure Page 6.5 Impedance of Monopole (Height 0.223 Wavelength, Radius 0.004 Wavelength) vs. Ground Plane Size at 2500 MHz 95 6.6 Current Distribution on Ground Plane with Different Monopole Radii at 2500 MHz 96 6.7 Current Distribution on Ground Plane at 1875 MHz (Monopole Height 2.68 cm, Radius 0.048 cm, Excitation 1 volt) 98 6.8 Current Distribution on Ground Plane at 2500 MHz (Monopole Height 2.68 cm, Radius 0.048 cm, Excitation 1 volt) 99 6.9 Current Distribution on Ground Plane at 3000 MHz (Monopole Height 2.68 cm, Radius 0.048 cm, Excitation 1 volt) 100 6.10 Current Distribution on Ground Plane at 3750 MHz (Monopole Height 2.68 cm, Radius 0.048 cm, Excitation 1 volt) 101 6.11 Current Distribution on Monopole (Height 2.68 cm, Radius 0.048 cm, Excitation 1 volt) at 2500 MHz 102 6.12 Computed Far Field Patterns at 2.25 GHz 104 6.13 Computed Far Field Patterns at 2.50 GHz 105 6.14 Computed Far Field Patterns at 2.75 GHz 106 6.15 Comparison of Measured and Computed Far Field Patterns at 2.25 GHz 109 6.16 Comparison of Measured and Computed Far Field Patterns at 2.50 GHz 110 6.17 Comparison of Measured and Computed Far Field Patterns at 2.75 GHz 111 viii

LIST OF APPENDICES Appendix A B Page 115 Listing of the Computer Program Singularity Analysis of Self Terms for the Geometry of Revolution 137 ix

CHAPTER I. INTRODUCTION 1.1 Backaround In most antenna measurements, a highly conducting flat ground plane is a basic part of the measurement structure. Experimental work by Meier and Summers [1] indicates that a small ground plane may have appreciable effects on the measurements. It is often desirable to reduce and eliminate as much as possible the effects associated with the edges of a finite size ground plane and thus obtain the impedance and radiation characteristics of the antenna that would approach those when an infinite ground plane or a large ground plane is used. The input impedance of a monopole at the center of a metallic circular ground plane has been studied experimentally by Meier and Summers [1]. Theoretically the problem was studied first by Bardeen [2]. He considered the problem of an antenna placed vertically at the center of a circular ground plane and obtained an integral equation for currents on the ground plane. However, he did not solve the equation exce-:- for the case when a ground plane is small in comparison with the wavelength. Leitner and Spence [3] obtained a solution for this problem in the form of spheroidal functions. Unfortunately, 1

2 however, the series converges very slowly for large radii ground planes, and thus, for the practical case of a ground plane greater than ten wavelengths in diameter, this approach is limited to general applications. Storer [4,5] has solved the same problem using the variation method, and so did Fikioris [6]. Although, they both use a method that involves considerable complexity, no complete solution is applicable to ground planes of both small and large diameters. Theoretical comparison by Thiele and Newhouse [7,8] using the geometrical theory of diffraction, showed good agreement with experiments for both the circular ground plane and an octagonal one. However, as the number of sides increases, their method for the octagonal ground plane would not converge to the circular ground plane case and results based on this approach would be in error. Green [9] used a different analysis for the variation of input impedance of the monopole above a circular ground plane as a function of ground plane radius. For his method, the experimental value of the average input impedance had to be used and this was added to the calculated variations. Awadalla [10-12] made use of the fictitious edge current and the principle of magnetic ring current. His result is in good agreement with experiments for ground planes down to 0.6X in diameter. When the same technique is applied to a radiation pattern, it is found that

3 agreement is fairly good for large diameters, but poor when the diameters are small. Recently, Griffin [13] reported on the experimental study of a monopole on a circular ground plane with microwave absorbent material placed around the perimeter of the ground plane. His experiment was based on only one grade of absorbent material, the same as that used to line anechoic chambers. Since the wedge-shaped microwave absorbent material foam on the ground plane near the wedges attenuates the outward travelling wave as the electric field and the current pass through, the reflected components, if such still remain, are also attenuated. In the edge treatment study presented here, a tapered resistive sheet is used instead, where the edge itself is made into an absorbing structure by changing uniformly the resistivity from zero ohms per square to a large value (approximately 1000 ohms per square) at the outer edge. 1.2 Outline of the Work The task presented here is to develop a finite size ground plane whose electromagnetic characteristics on its surface and in the far field are approximate to those of an infinite size ground plane when excited by a monopole at the center. Specifically, the problem studied here is that of a monopole located at the center of a circular ground plane with resistive edge loading. The effect of such edge

4 treatment on the impedance and the radiation patterns of the antenna are of special interest. The problem is solved numerically by applying the method of moments to a suitable integral equation formulated for the surface of revolution. A computer program was developed to solve the currents on the monopole and the ground plane, for the antenna impedance, and for the far field patterns. The resistive sheet boundary condition is included and by choosing appropriate resistivity variation, the edge diffraction can be reduced over a wide range of frequencies. Using resistive paints, resistive sheet material was made whose resistivity can be varied by controlling the layers of paint applied and the choice of the conductivity of the paint. The antenna was then constructed and measurements were performed. The concept of the resistive boundary condition and its inclusion in the formulation [14-17] is discussed in the remaining section of this chapter. Chapter II is devoted to a representation of the E-field equations and the method of moments technique. Chapter III deals with the body of revolution technique in conjunction with the method of moments. Integral equations are derived and adapted for numerical assessment. The making of the resistive sheets and their resistivity measurements are discussed in Chapter IV. Chapter V deals with experimental studies in which a model is built and measurements are made for the antenna

5 impedance and the far field patterns. The numerical (computed) results for the current distribution and the antenna impedance as a function of frequency, antenna geometry, etc. are presented in Chapter VI along with some experimental data. A summary and the conclusions are provided in Chapter VII. 1.3 The Resistive Sheet Boundary Condition A resistive sheet is characterized by three unique properties. It is infinitesimally thin, carries only the electric currents, and these are proportional only to the tangential component of the (total) electric field. Mathematically, a resistive sheet is characterized by a parameter Rs as follows 1 R = lim (1.1) A-~0 o A O-~oo where R is the sheet resistivity (ohms/square) o is the conductivity of the material A is the thickness of the material. As A approaches to zero, o will be increased in such a manner that R s finite and non-zero in the limit. The result is an idealized (infinitely thin) electric sheet whose electromagnetic properties are specified by the single measurable quantity Rs. This definition is applicable

6 to a non-magnetic, conductive material whose thickness is small compared to the wavelength X and the penetration depth 6 The boundary conditions for an electrically resistive sheet of resistivity Rs are / + - n x (E - E) = (1.2) + - n x (H - H) = J (1.3) n x (n x E) = - R J (1.4) where n is a normal unit vector from the sheet J is the (total) electric current supported by the sheet. Next, let -i -i i E = t Et (1.5) t t J = t Jt (1.6) where t is tangential unit vector in the sheet. From Eq.(1.4) the resistive boundary condition on one side of the sheet becomes

7 Et(R) = Rs(R)Jt(R) (1.7) E(R) = E(R) + E(R) (18) Et(R)= Et t El S (1 9) E'(R) = R(R()Jt(R) - Et(R) (1.9) t S t t where Et is the total electric field in the t direction t i ^ Et is the incident field in the t direction Et is the scattered field in the t direction, and Et Jt is the total current in the t direction. Equation (1.9) expresses both the incident field and the scattered field in terms of resistivity Rs which need not be constant but can vary with the distance R in the sheet.

CHAPTER II. THEORETICAL BACKGROUND 2.1 Representation of the Electromagnetic Fields The total electromagnetic field can be represented as the sum of scattered and incident fields. A time harmonic field with ejWt time variation suppressed is assumed, where j = v/-T and w is the angular frequency. With the aid of electric and magnetic scalar potentials [33], the scattered field can be expressed by s 1 - E (R) = -jw A(R) - VX(R) - - V x A (R) (2.1) s - -* - * - 1 H (R) = -j A (R) - V (R) + - V x A(R) (2.2) where the vector and scalar potentials are defined as A(R) = V fs J(R)G(R,R') ds (2.3) * * X (R) = s ffJ J (' )G(R,R') ds' (2.4) 1 ~ (R) = fJ Pe (R')G(R,R') ds' (2.5) (R) = - ff p (R')G(R,R') ds' (2.6) S 8

9 and, they contain the free space Green's function -jkR e G (R,R) = (2.7) 4 T R where 2 R = - R' = [ I2+ '2 - 2rr'cos(c-' ) + (Z-Z') The quantities pe and Pm are the electric and the magnetic charge densities, respectively, and, are related to the surface currents through the continuity equation p(R') = -- [V(R' )] (2.8) 2.2 The Method of Moments 2.2.1 General Procedure The method of moments [18,19] is a numerical technique devised to solve the deterministic equation L(f) = g (2.9) where L is a linear operator, g is known, and f is to be determined. Let f be expanded into a series of functions, f f f3f f... in the domain of L as f = n fn (2.10) where the an are constants and the fn are called expansion

10 functions or basis functions. For exact solutions of f, Eq.(2.10) would be an infinite summation and the f would be required a complete set of basis functions. For approximate solutions, Eq.(2.10) is usually a finite summation. Substituting Eq.(2.10) into Eq.(2.9) and using the linearity property of L, one gets E an L(f ) = g (2.11) n A set of weighting functions or testing functions, {w1, w2, w3..} is then defined in the range of operator L. The inner product of Eq.(2.11) is taken with each wm, the result is an <WmLfn> = <wm, g> (2.12) n m=1,2,3.... This set of equations can be written in a matrix form as [lmn][an] = [gm] (2.13) where <wlLfl> <w1,Lf2> [mn] = <W2Lf1> <W2,Lf2>.. i * * * a * * * * *............................ (2.14)

11 [an]: (2.15) <W1, g> <w2, g> (2.16) [gm]... -1 If the matrix [1 ] is non-singular, its inverse [1 1] mn nm exists. The an are then given by [a] = [1-1] [g] (2.17) and the solution for f is given by Eq.(2.10). For a concise expression of the result, the transposed matrix of f is defined as [f] = [ f1t f2' f3..... ] (2.18) and, Eq.(2.10) can be written in matrix forms as f = [f] [a ] = If] [l] ] (2.19)

12 This solution may be exact or approximate, depending upon the choice of fn and wn. The particular choice f = w is known as the Galerkin's method (see Kantorovich n n and Krylov [20], Jones [21,22]) and is most often used in application of the method of moments to electromagnetic problems. 2.2.2 Point Matching The integration involved in the evaluation of lmn= < wm, Lfn > in Eq.(2.14) is difficult to perform for problems of practical interest. A simple way to obtain approximate solutions is to require that Eq.(2.11) be satisfied at discrete points in the region of interest. This procedure is called the point-matching method, which is equivalent to using the Dirac Delta Functions as the testing functions. 2.2.3 Subsectional Bases The method of subsections involves the use of basis function f n' each of which exists only in a subsection in the domains of f. Then, each an of the expansion function in Eq.(2.10) affects the approximation of f only over a subsection of the region of interest. This procedures often simplifies the generation of the matrix [1 mn]. Thus, it is convenient in our computation to use point matching in conjunction with subsectional bases method.

13 2.3 Application of the Method of Moments to Solve the E-Field Equations i The problem is formulated as follows. Let E denote s the impressed or the incident field and E the scattered field due to the currents and charges on the body. Then the total field E is the sum of the incident and the scattered fields, that is to say i s E = E + E (2.20) For a conducting surface S ( R=0 ), the boundary condition requires that the total tangential component of E vanishes on S. Hence i - Ean Etan (2.21) In the format of method of momemts, Eq.(2.21) can also be written as L(J) = E (2.22) tan From Eq.(2.1) L(J) = (jA + V4)tan (2.23) which follows from Eq.(2.1) and Eq.(2.21). In Eq.(2.23), L is an integro-differential operator and a subscript "tan" denotes the tangential component on S.

14 A solution of Eq.(2.22) gives the surface current J on S. Next, let the inner product of two arbitrary tangential vectors on S be defined by < F, G > = fs F * G ds (2.24) A set of expansion functions {Jj} is next defined for the expansion of currents on S by J = I J. (2.25) J ] J where Ij are constants to be determined. Because of the linearity property of L, when Eq.(2.25) is substituted into Eq.(2.22), it becomes -i Ij L(j) = Etan (2.26) J A set of testing function {Wi} is defined, and an inner product of Eq.(2.26) with each Wi is taken. This results in -i Ij<Wi, LJj> = <Wi, Etan i=1,2,3.... (2.27) For convenience and shorter representation, definitions from the circuit theory are introduced and the network matrices are defined as

15 [Z] = [<wi, LJ>] (2.28) -2.) [V] = [<Wi Etan>] (2.29) [I] = [Ii] (2.30) Eq.(2.27) then becomes [Z] [I] = [V] (2.31) The excitation matrix [V] is obtained from Eq.(2.29). It is either an incident field as in the case of scattering problems or a local source coordinate as in the case of radiation problems. In the radiation problem considered here, the term <Wi' Etan> in Eq.(2.29) is replaced by Vi/d, where V. is the locally generated voltage applied over a small gap centered at point i and d is the gap width. Now [Z] can be considered as a generalized impedance matrix. The impedance elements of Eq.(2.28) are explicitly given by j = Wfs i (jwA.+ V~) ds' (2.32) J~ which follows from equations (2.23) and (2.24).

16 Applying the Divergence theorem to the vector Wi4 on the surface, the following results f Vo Wi ds' = s V * w. ds' (2.33) and Eq.(2.32) can now be written as Zj = fJs (j Wi Aj - jV * Wi) ds' (2.34) Because the gradient of ~ has been eliminated in Eq.(2.32), Eq.(2.34) is now in a more convenient form for numerical evaluation.

CHAPTER III. BODY OF REVOLUTION TECHNIQUES 3.1 Introduction In this section, the formulation of the integral equations and the application of method of moments to the proposed problem are discussed using the body of revolution techniques. The body of revolution (BOR) geometry is the characteristic of many physical structures, such as rockets, missiles, satellites, raindrops and many types of biological cells. This method has the advantage of enabling one to apply the method of moments to three-dimensional structures which are fairly large with respect to the wavelength, yet requires only a fraction of the unknowns to be determined, as compared to a general three-dimensional method of moments formulation. Several authors have presented the techniques for treating problems involving radiation and scattering by perfectly conducting BOR. Andreasen [23], and, Mautz and Harrington [24 through 27] have employed the electric field integral equation (EFIE), whereas, Oshiro, Mitzner [28] and Uslenghi [29] have used the magnetic field integral equation (MFIE). Several extensions and refinements of the basic techniques of these authors have also been developed. 17

18 Recently, Glisson and Wilton [30,31] presented techniques which appear to have alleviated some difficulties previously encountered by others in the treatment of perfectly conducting and dielectric bodies of revolution. Their techniques are being adapted here. A special case of a body of revolution is a surface of revolution (SOR). Here, instead of determining the currents and charges throughout a body, they are determined on the surface only. The resistive sheet boundary conditions can thus be applied to the surface of revolution geometry, which may be closed (such as a spherical shell) or open (such as a coffee cup). Any line S revolving about the z axis will generate a surface of revolution geometry. Thus an antenna geometry of a monopole located at the center of the circular ground plane can be generated as a surface of revolution as shown in Fig.(3.1). Using the method of moments, the currents on the antenna and on the ground plane, the input impedance, and the far field patterns can be computed. 3.2 Application of the Method of Moments (MOM) 3.2.1 Evaluation of the MOM Impedance Matrix Consider a surface S generated by revolving a line about the z axis. The coordinate system is shown in Fig.(3.1). Here A,(,z are the usual cylindrical coordinate variables, and, t is the length variable along the

19 z A S 40 y x Fig. 3.1 A line S Rotated about the Z-Axis Generates a Monopcle Antenna on the Circular Ground Plane.

20 generating curve S. In general, the independent set of expansion functions of the J(t,4) on S [30] are defined as N J(t) = t [ n=1 n )(^ N+1 (n J n(t') + q [ J4 P2(t) (3.1) where n(t,) = 1 1 0 tn t' tn+ Otherwise (3.2) pn(t ) = 2 1 0 I t < t' < t n- = = n Otherwise (3.3) For the problem studied here, there is no 4 dependence, and hence the second term in Eq.(3.1) involving J% vanishes. The charge distribution is obtained from the derivatives of Jt with respect to t (c.f. Eq.(2.8)) and these can be approximated by d - [t (t')] = dt N+1 n=1 - -n -n-1 t t n P2(t') t - t n n-1 (3.4) where 2 2 Itn- tn-1 I = Atn [(n - n-1) + (Zn - Zn-) (3.5)

21 It is assumed that at the edges, the current Jt is zero, that is 0 Jt N+1 - j t 0 (3.6) Since each tn is common to two linear adjoining segments, it is convenient to approximate the incident field and the vector potential by their values at tn= t, Fig.(3.2). Integration of Eq.(3.2) in the variable t yields tq tq+2 q^ JtPl(t) t-U(t)dt = J t-U(t)dt + f tq-2 tq t-U(t)dt 1 = - (Atqtq-_ 2 + Atq+ltq+ ) U(tq) (3.7) where U is the vector quantity tested and tq_- is the unit vector describing the orientation of linear segments containing the points tq-1 and tq. The testing functions are defined as Wl(t) = W2(t) = 61(t) (3.8) (3.9)

22 — x — tN —x — tN+ —x —1 to! ' ' ' t n... N N+ z to t N+1 y x Fig. 3.2 Approximating of Generating Arc by Linear Segments for Strip of Revolution.

23 where, t = tq (3.10) 1(it) =, otherwise, t = tq — 62(t) = ( (3.11) 0, otherwise Substituting Eqs.(2.4), (2.8) and (2.33) into Eq.(2.34), one gets 1 e-jkR Zij = Jf ds'J.5ds[ J+ + (VW ) (3.12) 131 1 47rR Note, for body of revolution, in general N 27r Jsfds = dt f L(t) d4 (3.13) 0 0 An orthogonal triad of unit vectors ( n, 4, t ) can be associated with each coordinate point ( t, 4 ) where n, 4, t are defined as follows: n = cosy cos4 x + cosy sin4 y - siny z (3.14) = - sin4 x + cos% y (3.15)

24 t = siny cos4 x + siny sin4 y + cosy z (3.16) where y is the angle between the tangent to the generating curve t and the z axis, defined to be positive if t points away from the z axis and negative if t points towards the z axis. In this coordinate system, the surface divergence becomes 1 1 8 V * = - -(OJ ) +- (J) Lt t t ~ (3.17) and R becomes R = {2+,2- 2rL' cos(4-4') + (z-z')2} (3.18) To obtain the W.J term in Eq.(3.12), one writes Wi j = Up *q (3.19) where p and q represent the permutation of t and 4. The unit vector dot products, in terms of body coordinates (n, X, t) are ut, ut = sinysiny'cos(4-4') + cosycosy' (3.20) A A ui u, = -sin-'sin(4-' ) (3.21) u4, ut = siny sin(4-' ) (3.22) %' t

25, uv - cos(%-4') (3.23) For a resistive surface in operator form Eq.(1.9) becomes,i = ~ [j (R')] + R (R') t (R') t 1] t s t (3.24) where -1 = t jkZ f J(R') G(R,R') ds' 5 jZ 9 1 ff - - (I' J(R')) G(R,R') ds' k t s AI' 9t' + Rs(R') t (i') (3.25) It is desirable to express all the quantities of Eq.(3.25) in terms of local arc coordinates ( t, 4 ) on the body surface. Thus, Eq.(3.25) is written -1 E = t jkZo Sf Jt[sinysiny'cos(4-%')+cosycosy']G ds' s Z0 - ff - -(t' J t) G ds' k 3t s A + R (R') J (KI) 5 t (3.26) where

2 6 G (!~R!R ) = -jkR e 4 'TrR ( 3.27 ) and R R =[)+,2-2.u' cos(.4)+ (z-z') After some manipulation of Eq.(3.26), the impedance matrix such as Eq. (3.24) can be written as Z. -.R + 0siny Xs(tj-Yj)[Kl(ti~ t.; t.)1 + Z0csin XcAjY )[K (ti. t; t-)J kZ j + 0cosyj+ Xc(A6t j~YjflK(t i-!2 ti+; t.) 4 Tr 4 7Tk At. z 4Kiirk it i++1) K(ti r t i+1i; t j 2)JI t 3. 18)] where G 1 (t ill t') dt' ( 3.29 )

27 t2 K (tl, t2; tj) = f D t 1 Go(tj,t') dt' Tr G1 -= f - IT GO =J X-( Xs(Ltj,~fj) e-kR cos((-4') d4' R e-IkR d+' R = (Atj+lsinyj+l + Atj sinyj)/2 (3.30) (3.31) (3.32) (3.33) Xc(At j,y.) = (tj+1cosyj+1 + Atj cosyj)/2 (3.34) and i J is the field point index is the source point index. The matrix [Z]ij is the required MOM matrix to be evaluated. There are eight integrals to be evaluated in Eq.(3.28), which are basically the integration of the Green's functions for a given source and observation points. These integrals are defined in Eq.(3.29) and Eq.(3.30). Having the impedance matrix [Z]ij and given the excitation matrix [V], the current matrix [I] can be computed using the Gaussian elimination method. The current computed can then be used to evaluate the antenna impedance and the far field patterns.

28 3.2.2 Evaluation of the Antenna Impedance The input impedance, Zin, of an antenna is the impedance presented by the antenna at its terminals. In computation for the currents on the antenna one volt (rms) is applied across the gap and the impedance is determined from the equation in = in in (3.35) The current I. is defined as the total current at the input gap and is related to the current density Jt by In = 2 T ~ J n = 2t (3.36) Thus, the input impedance of the monopole antenna is Vin Z = in t2 (3.37) 3.2.3 Evaluation of the Far Field The scattered far field is an integral over the surface currents and can be written in the form e-jkR Es = A J (R') ------- ds' s R (3.38) where A is a constant

29 and R = IR -R' is the distance between a surface point R' and the far field observation point R. If the body is finite so that R is much greater than any of the body dimensions, then AejkR -jkR * R' J (R') e ds' R s (3.39) In terms of local arc coordinates ( t, 4 ) on the body surface, the dot products applicable to Eq.(3.12) are given by ut. ue = cosesinycos4 - sinecosy (3.40) u%- u0 = - cosesin4 (3.41) Then Eq.(3.39) becomes Et = _- Jt[cosesinycos- sinocosyle]ek(s in csz ds' R s (3.42) Using the integral representation for the Bessel function.m Jm () = - e-jcos e-m( d4 (2 0 (3.43)

30 one can analytically evaluate the ~ integration in Eq.(3.43), which results in A k Et = - f Jtejkzco~s[ jcosesinyJ1 - sinocosyJ ] dt' R0 (3.44) where = Jm(ksine) and = the Bessel function of 1st kind, 0th order = the Bessel function of 1st kind, 1st order. After the currents are evaluated using the method of moments and the body of revolution technique, it is a relatively straightforward task to compute the far field patterns from Eq.(3.44). Rewriting Eq.(3.38) in matrix form and dotting with u to obtain a (scalar) transverse component, one gets the following s ^ W -;kp E * u = -A' e [IkRz][I] (3.45) 4rR where [Z]i = 27j f n (t)ejkzcose[cososiny J1 + jsinecosyJ0] dt' (3.46) In Eq.(3.45), A' is a constant and 6 (t) are delta functions defined in Eqs.(3.10), (3. '1).

31 After some manipulation, the impedance matrix can be written as [Z]n 2T7 [jJ1cosXs- Jo sinexc] ekzcos(37) where Xs(AtnYn) = (Atn+1 sinn+1 + Atn sinyn)/2 Xc(AtnYn) = (Atn+1cosyn+l + Atn cosyn)/2 with n being the source segment index. Equation (3.47) requires essentially the evaluation of the zeroth-order and the first-order Bessel functions of the first kind. Since the unknown current distribution [I] is found by solving the MOM impedance matrix [Z]ij of Eq.(3.28), the far field patterns can be evaluated using Eq.(3.45) where [Z]n is obtained from Eq.(3.46).

CHAPTER IV. RESISTIVE MATERIALS AND MEASUREMENTS 4.1 Introduction The effects of edge diffraction can usually be reduced by adding absorbent materials around the edge, corrugating the edge or a combination of both. The latter would probably be more effective. Tapered resistive sheets are studied here primarily because the approach is new and it shows a lot of promise. Since the resistive sheets are not readily available and usually have to be custom made for a particular application, techniques are developed to make them in the laboratory, which consists of spraying resistive paints on plastic or other types of nonconducting base materials. The conductivity of the paint can be varied by mixing different paints in various proportions. The resistivity of the sheet can be controlled by the paint used and the layers of the paint applied. In this chapter, the making of resistive sheets is discussed. The properties of different types of paints are tabulated. The results of mixing different paints (by weight) and the effects on the sheet resistivity are plotted. Methods are devised to measure the resistivities of the sheets at DC and at microwave frequencies to determine if the resistivity of the sheets remains constant over the frequency range of interest. 32

33 4.2 Resistive Materials Thin resistive materials can be made by spraying resistive paints on plastic or paper material (Kimura [32]). The resistive paints contain finely processed carbon particles plus a bonding resin and solvent. The resistivity of the finished product can be controlled by selecting appropriate ratios of different types of paints to be mixed (Fig.(4.4) through Fig.(4.6)), the number and thickness of the coatings applied (Fig.(4.1) and Fig.(4.2)), and the drying time (type of solvent and temperature) as well as the type of the base material (Fig.(4.3)). For our study, lacquer base paints were chosen because they are easier to mix and can be redissolved even after drying. Also, lacquer thinner is readily available and can be used to clean the spraying equipment. The paints used were Electrodag~109, 110, 415 and 502. Their properties are summarized in Table (4.1). These paints can be directly applied by brush, dip or spray methods. The latter requires dilution with solvent. An airbrush was used to produce smooth and uniform coatings (c.f. Fig.(4.7)). To obtain the required spray consistency and eventual sheet material resistivity requires a lot of patience and practice. Paint thickness, air pressure, Electrodag~ is the trademark of Acheson Colloid Company, Port Huron, Michigan. 48060

34 Paint Pigment Density Solvent Resistance Type Ohms/sq. Electrodag Graphite 1.025 Lacquer Less than 109 Kg/t thinner 30 Electrodag Graphite 0.98 Lacquer 1.5-2.5K 110 Kg/t thinner Electrodag Silver 1.7 Lacquer Less than 415 Kg/t thinner 0.1 Electrodag Graphite 0.82 Lacquer Less than 502 Kg/t thinner 25,0 *0.001 inch Coating Table 4.1 The Properties of Paints Used.

35 Resistivity (ohms/square) 1600 1200 800 400 0 ' i I 1 0 2 4 6 8 10 Number of Coatings Fig. 4.1 Resistivity vs. Number of Coatings of Electrodag 110; Paper Base.

36 Resistivity (ohms/square) 25 20 15 10 5 L I I 1 I u I 0 2 4 6 8 10 Number of Coatings Fig. 4.2 Resistivity vs. Number of Coatings of Electrc-ag 109; Paper Base.

37 Resistivity (ohms/square) 6000 4000 2000 0 o Plastic Paper I I I I I I I I L - I 1 2 3 4 5 6 7 Number of Coatings Fig. 4.3 Effect of Base Material on Resistivity; Electr.dag 502.

38 Resistivity (ohms/square) 4000 3000 2000 1 - 1 000 I I I I I 1 0 I I I I - 100% 0% 90% 10% 80% 70% 20% 30% 60/ 40 % 50% ( 1 10) 50% (502) Mixture Ratio by Weight Fig. 4.4 Resistivity vs. Mixture Ratio of Electrodag 110 & 502; Plastic Base, 2 coats.

39 Resistivity (ohms/square) 250 200 - 150 - i o0 - 50 ' I I I I 1 AI I. 1 - 90% 10% 80% 20% 70% 30% 60% 40% 50% ( 110) 50% (109) Mixture Ratio by Weight Fig. 4.5 Resistivity vs. Mixture Ratio of Electrodag 110 & 109; Plastic Base, 2 coats.

40 Resistivity (ohms/square) 4000 3000 2000 1000 0 0 2 4 6 8 10 Number of Coatings Fig. 4.6 Resist.ivity vs. Number of Coatings of Electrodag 109 & 5"2 (1:4 Ratio by weight); Plastic Base.

41 the spraying distance from the brush to the sample, and the speed-of-hand motion all have significant effect on the final resistivity. Thus, by a combination of the paints and the coatings applied, the resistivity of the material can be controlled. For the actual ground plane model, where resistivities were needed to vary from 0 to 1000 ohms/square, Electrodag 109 was first used primarily because of its low resistivity, then a mixture of Electrodag 109 and 502 (ratio of 1:4, mixed by weight) was applied, see Table (5.1). In practice, it is simpler to measure liquids by volume, such as with a 10 cc syringe that was used. With the paints accurately weighed (they all came in the quart cans) and their densities calculated, the exact ratio for mixing by volume was obtained for the given ratio by weight. After the paints were mixed, a lacquer thinner was added to faciliate the spraying process with the air-brush. Depending on the drying time of different paints, it usually takes at least three to four days for the paints to be completely dried and stablized to obtain accurate resistivity measurements. 4.3 Measurement of the Resistivity of the Sample The resistance of the painted sample can be measured at DC and microwave frequencies. There are several

42 Fig. 4.7 Sprayinq of Test Samples Using an Air-Brush Method.

43 ways of making DC measurements. The most common approach (direct method) is to use a rectangular sample painted with silver electrodes on opposite sides, and measure the resistance with a multimeter (ohmmeter). A two-wire line is the other DC method used. A more accurate measurement of the effective resistivity can be obtained at frequency of operation using an open-ended coaxial sample holder and a network analyzer. 4.3.1 DC Measurements (a) Direct Method A resistive sample is cut into rectangular patches, then the opposite edges painted with silver paint (Electrodag 415) to provide the edge electrodes. After drying, the resistance of the sample is measured by an ohmmeter as shown in Fig.(4.8). The resistance Rm (ohms) and the sheet resistivity Rs (ohms/sq.) of the sample are related by n R = R s N m (4.1) or W R = R s m (4.2) where N = number of square cells in series n = number of square cells in parallel W = width of the sample

44 Ohmmeter Input Resistive w S-ilver \ \ Sheet \ Electrodes \ — 1 ----- v - Fig. 4.8 DC Measurement of Sample Using Direct Method.

45 I = length of the sample (b) Coaxial Transmission Lines and Two-Wire Lines The resistivity of a resistive sheet can also be measured by using a coaxial line or two-wire line geometry electrodes. A two-wire line is connected to an ohmmeter and is placed on the sample to be measured as shown in Fig.(4.9). The relation of sheet resistivity Rs (ohms/sq.) to the measured resistance R (ohms) is obtained next. The geometry of the problem is a planar one (all fields lie in the sheet) and the pertinent variables are the current density and the electric field within the sample. One can visualize this as a section of a coaxial line filled with conductive dielectric whose length approaches to zero in the limit. Thus, we start with Laplace's equation in cylindrical coordinates V2 = 0 (4.3) where 4 is the electric potential. Since there is no variation in the z or 4 directions Eq.(4.3) becomes 1 d d4 (} ) = 0 t do do (4.4) and its solution is

46 Two-Wire Line Resistive Sheet Fig. 4.9 DC Measurement of Sample Using Two-Wire Line.

47 In t - In a > = ( ) v in l a (4.5) where a and b are the inner and outer radii respectively, and V is the voltage applied, (see Fig.(4.10a)). The electric field intensity is 9a V E = - - a. ln() a(4.6) Next define sheet resistivity R 1 R = lim A O o A O-w00 where o is the conductivity A is the thickness. V and, R = - = Resistance measured in DC. m I Starting with resistive sheet boundary condition from Eq.(1.7), we have E = R J (4.7) I = 2TLJ (4.8) 2 mr LE I = (4.9) Rs

48 I (a) Coaxial Line (b) Two-Wire Line Fig. 4.10 Dimensions of Probe Geometries; (a) Coaxial Line, (b) Two-Wire Line.

49 Using Eqs.(4.6), (4.7) and (4.9), one then obtains v Aln(b) 0a IRs (4. 10) V 2 r 5s I ln(b) a 2Tr Rs= Rm b m ln( () (4. 1 1 ) (4. 12) Similarly, for the two-wire line, from Ramo, Whinnery and Van Duzer [33], we have = (4. 13) 3$ E = - 2 T Ex I = R ( (4. 14) (4. 15) where a = ln[p + {( D)2 - 11 }2 (x - a)2+ y2 S = ln[ 2 2 - (x + a) + y x - a g = ln[ 2 2 (x - a) + y x + a (x + a)2+ y2

50 Equating Eq.(4.14) and Eq.(4.15), one obtains V T R = s I (4.16) thus, 7T R = R 2 s m ln[D + {()2 - 1} (4 d d (4.17) D = Distance between the center of the two-wire line. d = Diameter of the two-wire line, (see Fig.(4.10b)). The sheet resistivity Rs can be measured using two-wire and coaxial geometry probes. When these probes are brought in contact with the resistive sheet, the resistance measurments are related to the sheet resistivities via Eq.(4.17) and Eq.(4.12), respectively. For both probe designs, the resistance measurements were found to vary significantly from one measurement to another, even when measured at the same point. This is a result of non-total contact of the sheet with the probe, and, indeed, Eq.(4.17) and Eq.(4.12) show that the resistivity measured is a contact geometry sensitive. Various approaches such as carefully polishing the probe tips to make them flat or varying the pressure applied did not alleviate the problem. The only alternative was to make a lot of measurements, and to average them.

51 Two sizes of two-wire probes were used and with each probe twenty measurements were taken for each of the five samples studied. Table (4.2) shows the averaged results which are compared to the values obtained by the direct measurement. Note, the deviations are from -12.59 to 32.75 percent from the direct measurement values. Measurements were also tried using the coaxial line probe, but here, the measurement variations were even greater, attributed to the fact that a uniform contact is difficult to achieve with the circular electrodes. Hence, no further measurements were made with this probe, nor are they reported herein. 4.3.2 AC Measurements Even though the coaxial probe method does not work well at DC, a similar technique works well at microwave frequencies (AC). This can be explained by the fact that the small non-contact spacing that gave errors at DC, has capacitance that at AC for all practical purposes provides a short. An important fact is that this is a non-destructive measurement technique, and can provide resistivities in the frequency range of interest. The concep:, is relatively simple. An open-ended coaxial transmission line provides an almost perfect open circuit, except for a small stray capacitance. If a resistive sheet is placed against the end, the impedance

52 Resistivity (ohms/square) Two-Wire Line Percentage Direct Avg. of 20 Measurements Error Method D= 0.088 cm D= 0.049 cm D= 0.088 cm D= 0.049 cm d= 0.032 cm d= 0.036 cm d= 0.032 cm d= 0.036 cm 3123 3384 2790 8.36 -10.6 2126 2444 2200 14.68 3.48 1215 1600 1062 31.68 -12.59 504 564 480 11.90 -4.76 58 77 66 32.75 13.70 Table 4.2 Comparison of Resistivity Values Obtained Using DC Measurements.

53 seen would then be due to the resistance plus the stray capacitance in parallel as shown in Fig.(4.11). The Hewlett Packard 8745A S-parameter test set with a model HP 8410A network analyzer was used. A 5 cm long, 7 mm air-line was attached to the test port and served as the probe. To make the reflection measurements - switch S11 was on. For calibration, a shunt was connected and the test channel gain and phase offset adjusted for zero dB amplitude and 180 degrees phase readings, respectively. With the short removed, the resistive sheet to be measured was then placed against the open-ended coaxial line, and pushed firmly with a styrofoam block. The amplitude and phase of parameter S11 which is also known as (voltage) reflection coefficient was then recorded. The parameter Sll is directly related to the complex impedance of the load by Z~ 1 + S11 Z0 1 S 1 (4.18) where Zo is the characteristic impedance of the coaxial line probe and for this setup Zo = 50 ohms. The expression relating measured resistance R at DC and sheet resistivity Rs for a coaxial line geometry still applies, and Eq.(4.12) becomes

54 Styrofoam Block Apply slight.- pressure Resistive Sheet t I R Coaxial Line Styrofoam Block (a) (b) Fig. 4.11 Equipment Block Diagram; (a) AC Measurement of Sample 'sing a Network Analyzer, (b) Equivalent Circuit.

55 2 7T R 2 S Z. b Rs Z ln( ) a (4.19) where the radii a and b for the 7 mm line are 7.01 mm and 3.05 mm, respectively. Table (4.3) shows the comparison between the DC and AC measurements for the eight samples at three different frequencies (1000 MHz, 1500 MHz and 2000 MHz). As observed, the resistivity (ohms/sq.) of the sample does not change significantly with frequency (or measurement). A variation of 5 to 10 percent is an acceptable result. A small capacitive component is also measured and varies from 0.02 pF to 0.08 pF for the frequencies measured (see Table (4.4)). This capacitance in part, is attributed to the outside fringing fields of the coaxial line and, in part, to the resistive paint and the base material used. However, the capacitance does not significantly influence the resistivity measurement of the sheets. As shown, the resistivity of the resistive sheet remains relatively constant from DC to 2 GHz.

56 Resistivity for various Resistivity Frequencies (MHz) Percent AC AVG DC Diff. (ohms/sq.) 1000 1500 2000 3123 3078 3038 3188 3101 -0.73 2627 2579 2651 2500 2577 -1.90 2126 2166 2036 2220 2140 0.65 1215 1299 1286 1280 1288 6.00,504 539 493 507 513 1.78 137 153 135 140 143 4.37 58 64.4 64 58 62 6.89 12 11.9 11.5 10.86 11.4 -5.00 Table 4.3 Comparison of AC and DC Measurements.

57 Resistivity Capacitance (pF) for various Average Frequencies (MHz) capacitance DC value (ohms/sq.) 1000 1500 2000 (pF) 3100 0.0448 0.0389 0.0249 0.0361 2600 0.0851 0.0779 0.0451 0.0693 2100 0.0445 0.0296 0.0496 0.0412 1200 0.0645 0.0512 0.0428 0.0528 500 0.0489 0.0452 0.0331 0.0424 140 0.0615 0.0509 0.0573 0.0565 60 0.0226 0.0319 0.0133 0.0226 12 0.0671 0.0842 0.0735 0.0749 Table 4.4 Comparison of Shunt Capacitance at Different Frequencies.

CHAPTER V. EXPERIMENTAL ANTENNA MODEL 5.1 Introduction The design and construction of resistive ground planes, monopole and the measurements are presented in this section. It has been shown by Senior and Liepa [16] that a tapered resistivity extension applied to a metal edge can drastically reduce its backscattering. The resistivity should vary from a low value (= 0 ohm/sq.) adjoining the metal edge to a large value (= 1000 ohms/sq.) at the outer edge. A quadratic resistivity taper that follows t2 form, where t is the distance measured from the edge adjoining the metal, is near optimum and was selected for use here. As shown in [16] the width of this taper should be 0.75 wavelength or wider to be effective. Using these design criteria, an edge treatment was chosen and the model constructed. Besides the resistive ground plane model, a similar metal ground plane of the same size and another large metal ground plane which was used to simulate an "infi: ite" ground plane were constructed. Measurements of antenna impedance and radiation patterns were made on these three models. The results are consistent with a simple reflection model concept. The 58

59 outward travelling wave on the ground plane is reflected by the edge of the ground plane to produce an inward wave of lower amplitude. Resistive material near the edge attenuates the outward travelling wave as well as the reflected wave. The antenna impedance curve of a finite ground plane with resistive edge appears to be very close to that of a large ground plane of five wavelengths in radius which can be considered, for all practical purposes, as an infinite ground plane, because the error in antenna impedance is only three percent (see Storer [4]). 5.2 Construction of the Circular Ground Plane with Resistive Edge Loadinq The resistive coatings can be made in the laboratory by appropriately blending conductive paints and spraying on a nonconductive base. This was presented in Chapter IV. A plastic sheet of 0.127 cm thick was chosen for the base material and a disc of twelve centimeters in radius was cut. Table (5.1) shows the proposed resistivity variation for the ground plane. Right at the base, from zero to three centimeters radius, the resistivity is zero which then proceeds to 1350 ohms/square in eleven steps. One can visualize this re istivity as being applied in bands using different paint mixtures and number of coatings as determined in Chapter IV. In practice this was accomplished by using a series of masks with circular holes cut from three to nine

60 Resistivity Distance from center Number of Paints used (ohms/sq.) (cm) Coatings 5 3 - 35 8 Electrodag 5 109 9 3.5 - 4 6 Electrodag 109 12 4 - 45 4 Electrodag 109 20 4.5 - 5 2 Electrodag 109 100 5 - 6 7 Electrodag* 109&502 1:4 109&502 = 1:4 175 7 - 8 5 Electrodag 109&502 = 1:4 250 - 9 4 Electrodag 250 109&502 = 1:4 380 9 - 10 3 Electrodag 109&502 = 1:4 700 10 - 11 2 Electrodag 700 1 109&502 = 1:4 1350 11 - 12 1 Electrodag 109&502 = 1:4 * By weight Table 5.1 Number o- Coatings and Mixtures Used in Preparing the Act-.El Model.

61 centimeters in radius. The resistivity within the unmasked region is controlled by the number of coatings applied. The portion of the band that is coated most, i.e., the central region, has the lowest resistivity. Figure (5.1) shows the actual painting of the material. The paint was sprayed with an air-brush onto the model which was placed on a phonograph turntable rotated at 16 rpm, Fig.(5.1). To get a consistent deposition or spray, it is sprayed slower at the outer edge and faster at the center. After the first band was sprayed, a new mask of larger radius was laid on the model to cover the portion that was not yet coated. Thus, by repeating the same process with nine different radii masks, a tapered resistivity variation on the circular model was obtained. After painting and letting it dry for two to three days, the resistivity of the resistive disc was measured using the AC method, where the sheet was brought against an open end of the coaxial line and its reflectivity was measured, as discussed in Chapter IV. The measurements were made at 2500 MHz and the results are shown in Fig.(5.2). Note, the resistivity variation is parabolic and follows closely to the procosed design given in Table (5.1). 5.3 Antenna Impedance Measurements To provide a means of mounting the monopole on the resistive ground plane, a 3 cm metal disc was mounted on top

62 II I Fig. 5.1 Making of Circular Resistive Sheet Using an Air-Brush and a Phonograph Turntable.

63 Resistivity (ohms/square) 1600 1200 800 400 0 0 2 4 6 8 10 12 Distance from Center of Monopole (cm) Fig. 5.2 Resistivity vs. Distance from Center of Monopole Measured Using AC Method.

64 of the painted surface at the center to which a rectangular flange mount SMA connector was attached. To assure a good electrical as well as mechanical continuity between the metal edge and the resistive material, a lacquer-based silver paint was used (c.f. Fig.(5.3)). The dimensions of the resistive ground plane and the monopole are given in Fig.(5.4). The monopole was made of silver-plated copper wire, 2.68 cm high and 0.048 cm in radius. A network analyzer was used to measure the antenna impedance and was set up as shown in Fig.(5.5). A 20 cm airline extension plus a 7mm-to-SMA adaptor were used to connect the antenna to the S-parameter test set. Figure (5.6) to Fig.(5.8) show photographs of the setup with the resistive (12 cm radius), metallic (12 cm radius), and metallic large ground plane (60 cm radius), respectively. Where needed, styrofoam blocks were used to support the antenna. For calibration of the network analyzer, a small circular copper tape disc was placed over the monopole, thus shorting it at its base to the ground. After each calibration, the copper tape was removed and the reflection coefficient S11 was measured. The impedance of the monopole was then evaluated using Eq.(4.18). Identical procedures were repeated for different frequencies and different models. Figure (5.9) shows the impedance measurements for the monopole antenna with three different ground planes at five different frequencies. In general, the impedance has

65 Fig. 5.3 Photogr ph Showing the Contacts between the Ground -lane and the Resistive Sheet, the Ground Plane and the Monopole Antenna.

66 z Metal Resistive Disc Sheet 1/ /268K 12 cmx Fig. 5.4 Dimensions of the Actual Model.

67 Ground Plane Monopole Antenna S Parameter Test Unit Fig. 5.5 Antenna Impedance Measurement Setup.

68;A * - - r At=L'J k LL' jpb 1 ' ~, r)'.tCr~,I: rr.::51 Ii, t, rJ"_7C" n Ir hIr 1, ,- -~T.l'f',ZPjffliII1IQgseCI.~,~.li:;.,; ':5;-cZ'i Fig. 5.6 Measurement of Impedance of the Monopole Mounted on a Firite Size Ground Plane With Resistive Sheet cf Radius 12 cm.

69 Fig. 5.7 Measurement of Impedance of the Monopole Mounted on a F i te Size Ground Plane of Radius 12 cm.

70 Fig. 5.8 Measurement of Impedance of the Monopole Mounted on a Large Ground Plane of Radius 60 cm.

71 similar behavior for all three ground planes, mainly because the antenna impedance is dictated more by the monopole height than by the ground plane. The monopole height is 2.68 cm and at 2606 MHz where the reactive component is zero (resonant condition), the equivalent antenna height is 0.233 wavelength. At 2500 MHz, the real part of the antenna impedance is 37.8 ohms with the (finite) metal ground plane, 33.1 ohms with the large metal ground plane, and 32.5 ohms with the resistive ground plane. Note, the antenna on the resistive ground plane has an impedance very close to that of the large ground plane model not only at 2500 MHz but throughout the frequency range measured. Table (5.2) gives the numerical values of the measured impedance so that more accurate assessments can be made if needed. 5.4 Far Field Measurements For the far field pattern measurements, both the E and H-plane field patterns were measured. The measurements were made in a relatively small antenna pattern range. There, a turntable provided a means of rotating the tested antenna about its center of radiation. The antenna under test was used as the receiving antenna. Attached to the antenna was a crystal detector, the output of which was fed into the pen amplifier of the antenna pattern recorder. The received signals as a function of test antenna rotation were recorded. For the H-plane pattern measurements, the monopole

72 Resistance (R) and Reactance (X) 300 Finite GP O v Finite GP w/r A O Large GP * O 200 1 - i00 - i 0 x -100 k I I I -200 I - - 1500 2000 2500 3000 3500 4000 Frequency MHz Fig. 5.9 Measured Monopole Impedance for Various Ground Planes.

73 Finite GP with Frequency Finite GP resistive Large GP (MHz) sheet 1875 7.2 - j 100 8.4 - 73 10.8 - 77 2000 10.1 - j 90 12.8 - j 58 14 - 62 2500 37.8 - j 5 32.5 - j 8.8 33.1 - j 8.5 3250 119 + j36 97.5 + 40 93.1 + 52 3750 208 + j 214 192 + j 170 194 + 187 Monopole Height: 2.68 cm Radius - Finite Ground Plane: 12 cm Radius - Finite Ground Plane with Resistive Sheet: 12 cm Radius - Large Ground Plane: 60 cm Table 5.2 Compa ison of Impedance for a Monopole with Different Ground Planes.

74 was mounted vertically so that when it was rotated in the horizontal plane, and the H-plane pattern was recorded. Conversely, the antenna was mounted on a side so that the monopole was horizontal and when rotated in the horizontal plane the E-plane pattern was obtained. A waveguide horn antenna was used at the transmitter. The separation distance between the transmitting antenna and the receiving antenna should be large enough to insure that the far field patterns are being measured. For this, the separation distance should be equal to or greater than 2 D2/x, where D is the maximum aperture dimension involved in either transmitting or receiving antenna. In this study for the test antenna, the ground plane was treated as part of the antenna, and the far field requirements were met in the measurements. To avoid errors due to reflections, radar absorbing material whenever appropriate was placed around the tested antenna. A block diagram of equipment used is shown in Fig.(5.10). The measurement frequencies used are 2.25 GHz, 2.5 GHz and 2.75 GHz. The measurements were made first with the monopole antenna section (consisting of the 2.68 cm monopole, SMA connector and the 3 cm radius metal disc) mounted on the resistive ground plane. Then the section was transferred and mounted on the same size metal ground plane and the far field patterns were measured. No measurements

75 TRANSMITTING ANTENNA ANGLE I NFORMATI ON AMPLITUDE I NFORMATI ON Fig. 5.10 Block Diagram for Measuring the Far Field Patterns

76 were made with the large 60 cm radius ground plane since such size could not be accommodated in the small chamber. Figure (5.11) and Fig.(5.12) show the mounting arrangement of the resistive antenna for the H-plane and the E-plane measurements, respectively. Styrofoam blocks and masking tape were used to support the antenna on the turntable. The recorded patterns are shown in Fig.(5.13) through Fig.(5.15) for 2.25 GHz, 2.5 GHz and 2.75 GHz, respectively. The H-field patterns are concentric circles, the larger circle is for the monopole on the finite ground plane (12 cm in radius), and the smaller circle is for the monopole on the resistive finite ground plane. The E-field patterns are similar to those of a monopole on an infinite ground plane but below (or spilling over) the horizontal axis. The side lobes are very dominant for the monopole antenna on the metal ground plane. The lobes do not exist for the monopole antenna with the resistive ground plane because the effects of edge diffraction have been minimized by the resistive treatment.

77 I ~Ak= A'* -7a__ L _b;E- i —, I I'I Fig. 5. 11 Test Ant,-enna Placement for Measuring the H-F i e Pattern.

78 Fig. 5.2 Test Atenna Placement for Measuring the E-Field Patternor Measuring the

79 7:H PLANE JMetal Resistive - - - E PLANE PL:~ E PLANE Fig. 5.13 Measured Far Field Patterns at 2.25 GHz.

80 H PLANE Metal -- Resistive - - - E PLANE E PLANE Fig. 5.14 Measured Far Field Patterns at 2.50 GHz.

81 a H PLANE Metal Resistive - - - 1 E PLANE Fig. 5.15 Measured Far Field Patterns at 2.75 GHz.

CHAPTER VI. NUMERICAL STUDIES 6.1 Introduction The best test of a computer program is to compare the (computed) numerical results with the experimental data. In this chapter a computer program is discussed that was developed to solve the electromagnetic problem of a monopole located at the center of a circular ground plane that can be metallic and/or resistive. The program computes the antenna currents on the monopole as well as on the ground plane, the far field patterns and the antenna impedance. The description of the program is discussed and numerical results relative to experimental data are presented. 6.2 Program Description The program is called RW.PROJECT which is based on Eqs.(3.28) through (3.47) to solve the current distribution on the monopole and on the ground plane, as well as the far field. The method of integration used in evaluating the integrals of Green's function in the 4 direction is the four-point Simpson integration. Special attention is given when the observation point falls within the source segment. Detailed analytical evaluation of such a segment is given in Appendix B. 82

83 The FORTRAN source program consists of 1454 lines of statements which include the main program and eleven subroutines. The structure of the program is shown in Fig.(6.1). The entire simulation process is controlled by the routine "PROCES". It governs four important steps, which are: 1) Initialization, 2) Partition, 3) Computation, and 4) Post-processing. 6.2.1 Initialization Subroutine INITAL This subroutine is used to initialize all the variables and the constants used in the computation process. Constants such as pi (r), imaginary number (j), mu (V), the conversion of degrees to radian (DTR), are defined. The variables SOUMAX, OBSMAX, corresponding to the maximum numbers of source points and observation points respectively, are known as programming parameters and are used to control the programming arrays. Variables such as wavelength, the beginning angle (THETA1) and ending angle (THETA2) and its increments (INC) for the far field computation are read. Other variables are used for logic control function, for example, the far field index (FARIDX) which controls if the the measurement of far field is necessary. The resistive segments are also determined in this subroutine.

84 Flow chart with the top-down approach: MAIN --------------------------------------------— * --- INITAL PROCES OUTPUT ----------- ----------------------------------—. --- READER COMPUT FARFLD ------- -------------------------- SEGMENT (LINE) ------ ------ ------ DI STAN GREENS MULTPY APPRMX Fig. 6.1 Structure of the Simulation Program.

85 6.2.2 Partition (a) Subroutine READER As the name implies, this subroutine reads all the input data, such as the beginning and the ending segments, voltage and impedance associated with each segment and the curve type (a line or a curve) and the index to calculate the variation of resistivity in each region in a parabolic manner. Since every segment must be defined continuously, it is also used to check for the discontinuous segments by giving an error message if such occur. (b) Subroutine SLINE This subroutine places the observation points, the source points and impedance associated with each segment in an array. The segments are partitioned in the way shown in Fig.(6.2), and are divided in such a manner that there are at least twelve points per wavelength. For example, if there are three segments, each has to be divided into t,m,n number of cells, according to a new way of partition. The beginning segment is divided in (1 + 1/2) equal divisions. The middle segment is divided into m divisions with the distance between tne cells at the end being one half the length of that on the middle. The end segment is also divided into (n + 1/2) equal divisions. This kind of partition has the advantage since the spacing between the two transition

86...... d d'... d"...... -------- X ------ -- — X —X —X —I --- —X ----- A B C D Segment 1 Segment 2 Segment 3 (a) e e e'.-.. e" 2 2 2 X -I.. -- — X x — X — x -- I — x X — -— x ----X ---A' B' C' D' (b) Where AB > A'B' CD > C'D' Fig. 6.2 Diagram Showing how the Segments are Partitioned; (a) Old Method, (b) New Method.

87 regions A'B' or C'D' is smaller than AB or CD as shown in Fig.(6.2). Also in the transition region (e+e')/2 is smaller than (d+d'). Though e and e' is a little larger than d and d' since Distance of each segment d = n (6.1) Distance of each segment e = n + 1/2 (6.2) and, for large n, d = e. With this kind of partition, a more accurate result is obtained as compared to the partition by the old method, as there is no discontinuity in the transition region between two adjacent segments. 6.2.3 Computation (a) Subroutine DISTAN The subroutine is used to compute the distance between the source point and the observation point. These distances are denoted as DS and DSS, which are defined as R1 and R2 in Eqs.(B.7) and (B.11). Thus, 2 2 DS = [(i -j) + (zi - z.) (6.3)

88 2 2 DSS = [(i + j) + j (z - zj) (6.4) There are eight integrals to be evaluated in Eq.(3.28). Using a three-point Simpson's integration, there should be at least twenty-four DSs and DSSs, since some distances are repeated, only fifteen of such values are required for each value of i and j. (b) Subroutine COMPUT This subroutine is the center of computation process which evaluates the MOM impedance matrix in Eq.(3.28). To compute the Green's function integrals, routine "GREENS" is called. After computation, the MOM [Z] matrix is solved by using routine "MULTPY". (c) Subroutine GREENS The integrals involving Green's function in Eq.(3.28) are evaluated in this subroutine. Since there are eight integrals to be integrated, they are denoted as G1 through G8 in the computation. Three point Simpson integration is used for the t integration whereas four-point Simpson's integration is used in the 4 integration, which enhance the accur-ay of the results. Special attention is given when the observation point lies within the source segment. If special treatment is needed, routine "APPRMX" is called.

89 (d) Subroutine APPRMX As IR - R'1 approaches to zero which makes the integrals of the Green's function in Eq.(3.28) to become singular, subroutine APPRMX is called upon, which is based on Eq.(B.1) through Eq.(B.16) in Appendix B. This routine also calls subroutine ELTKP if the elliptical function of the first kind is necessary in the computation. (e) Subroutine ELTKP Subroutine ELTKP is used to compute the elliptical function of the first kind K(m) where K(m) = a +a m +a m2+a 3m+a4m1-ln(m )(b +b 1m+b m2+b 3m+b4m4) (6.5) m = 1 = m (6.6) where a..... a4, b..... b4 are given in the Handbook of Mathematical Functions by Abramowitz and Stegan [34]. (f) Subroutine MULTPY The final phase of computation is to solve the [N by N] matrix. This routine is used for solving the matrix [Z] using Gaussian's elimination method to determine the current distribution [I] on the monopole and on the ground plane. Once [Z] 1 is known, the current is

90 obtained by [I] = [V] [Z]1 (6.7) The elements in the excitation matrix [V] given in Eq.(6.7) are usually zero except at the source point (which is the voltage across the gap) for the radiation problem discussed here. 6.2.4 Post-processing Subroutine FARFLD The computed current distribution obtained from the matrix inversion of [Z]ij. the MOM impedance matrix, can be used for calculating the far field. This is done by evaluating Eq.(3.47) in which the Bessel functions of the first kind (Jo and J1) are computed. The scattered far field are then obtained by multiplication of the current distribution [I] and the MOM matrix [Z]n, Eq.(3.45). Subroutine "FARFLD" is the final phase of the simulation process, it is an optional feature. Its operation is controlled by the index "FARIDX", which is initialized in the subroutine "INITAL". 6.3 Numerical Results In this section, numerical results are presented for the current distribution on the ground plane with the monopole located at the center of the ground plane. A gap voltage of one volt (rms) is applied between the ground

91 plane of size 12 cm and the monopole whose height is 2.68 cm with a radius of 0.048 cm. The gap width used in the computation is 0.048 cm, the same as the monopole radius. Comparisons are made between: i) the finite size ground plane (12 cm in radius), ii) our model, a ground plane (12 cm in radius) with tapered resistive sheet, and iii) a large size ground plane (60 cm in radius). Figure (6.3) shows the effect of gap distance on the input impedance of a half-wave dipole (height 0.461 wavelength, radius = 0.0053 wavelength). As noted, the input resistance is relatively independent of gap width which varies from 0.001 wavelength to 0.04 wavelength, but the input reactance changes significantly when the gap is shortened. A large negative reactance shows that the capacitive component is very dominant when the gap is small. For this study it was concluded that the gap width for the practical antenna should be about the same as the diameter of the antenna in order to escape the drastic capacitive effect. The impedance of the monopole (height 2.68 cm, radius 0.048 cm) on different ground planes, i) a finite size ground plane (12 cm in radius) with resistive edge, and ii) without resistive edge, and iii) a large ground plane (60 cm in radius) were computed for different frequencies from 1875 MHz to 3750 MHz and are shown in Fig.(6.4). Although improvement is no- very pronounced, the finite size

__ __ 80 RESISTANCE (ORHS) 75 H 100 50 -150 -350 -550 osa ID 70 -.000 0.010 0.020 0.030 0.040 Length of Dipole: 0.461 Wavelength Radius of Dipole: 0.005 Wavelength.000 0.010 0.020 0.030 0.040 Gap Width in Wavelength Fig. 6.3 Effect of Gap Width on Impedance of a Half-Wave Dipole.

93 Resistance (R) and Reactance (X) 300 200 100 0 -100 -200 1500 2000 2500 3000 3500 4000 Frequency MHz Fig. 6.4 Compute- Monopole Impedance for Various Ground Planes (Height 2.68 cm) vs. Frequency.

94 ground plane with resistive edge shows a good approximation to the large size ground plane. It is noted that, since the monopole is of height 2.68 cm (0.223 wavelength), it is shorter than 0.235 wavelength that typically would resonate at 2500 MHz. This explains why slight capacitive components are present at 2500 MHz. Figure (6.5) shows the monopole impedance as a function of the ground plane size. Computed results are for metal and resistively treated ground planes. For the metallic ground plane case, the results are compared with Meier & Summers' [1] experimental data. The metallic ground plane varies from 0.25 wavelength to 2.0 wavelength in radius in both Meier and Summer's experiments and our numerical computations. The resistive ground plane was made of metal of 0.25 wavelength radius, plus an added tapered resistive sheet (0-1000 ohms/sq.) whose width ranges from zero to 1.75 wavelength. The monopole is of height 0.223 wavelength and radius 0.003 wavelength. With the resistive ground plane at small radii, the resistive strip is narrow and hence the curve begins the same as for the metallic one. For ground plane radius one wavelength and larger, the impedance is almost constant as one would expect for the infinite size ground plane. This shows that tapered resistance can match the surface field. Figure (6.6) shows the comparison of the current on a metallic ground plane for different radii of the monopoles

Resistance (R) and Reactance (X) 45 Ir 35 /' ~\ -- / /\'r \ \.- /..... / A v Meier & Summer - Metal 20 [0 * Numerical - Metal <O Numerical - Resistive................................................................................................................................................................................ 5\ \ \\\ / / -25 0 0.5 1.0 1.5 2.0 Radius of Ground Plane in Wavelength Fig. 6.5 Impedance of Monopole (Height 0.223 Wavelength, Radius 0.004 Wavelength) vs. Ground Plane Size at 2500 MHz.

96 Amplitude (MA) 40 30 20 10 0 0 2 4 6 8 10 Distance along the Ground Plane (cm) 12 Fig. 6.6 Current Distribution on Ground Plane with Different Monopole Radii at 2500 MHz.

97 (0.003, 0.004, 0.01 wavelength; height 0.223 wavelength) at 2500 MHz. As can be seen, the current distribution on the ground plane does not change significantly for these different radii of the monopole. With the monopole (height 2.68 cm, radius 0.048 cm), at the center of a circular ground plane (radius 12 cm), the current distributions on the ground plane are compared in Fig.(6.7) through Fig.(6.10) for metallic and resistive ground planes. Consider first the metallic ground plane, at 1875 MHz and 2500 MHz the ground plane radii are less or equal to one wavelength, hence one current minimum is observed in Fig.(6.7) and Fig.(6.8). At 3000 MHz and 3750 MHz, the radii of the metallic ground planes are 1.2 wavelength and 1.5 wavelength respectively, and consequently two minima are observed in Fig.(6.9) and Fig.(6.10). With the resistive ground plane there are no minimum other than at the feed point (monopole) and the outer edge. Therefore, one can conclude that with resistive treatment the effects of travelling waves are minimized. Figure (6.11) shows the comparison of the current distribution on the monopole (height 2.68 cm, radius 0.048 cm), i) for a ground plane with resistive edge and ii) a ground plane without resistive edge (each has 12 cm in radius). The magnitude of current does not vary significantly. The phase of the currents on the monopole on the metallic ground plane and the resistive one have

Amplitude (MA) 12 4 Finite GP 0 Finite GP w/r A 0 1 1 1 i 1 0 2 4 6 8 10 12 Distance along the Ground Plane (cm) Fig. 6.7 Curren- Distribution on Ground Plane at 1875 MHz (Monopole Height 2.68 cm, Radius 0.048 cm, Excitation 1 volt).

Amplitude (MA) 40 30 20 Finite GP 03 Finite GP w/r A 00I I I I I 0 2 4 6 8 10 12 Distance along the Ground Plane (cm) Fig. 6.8 Curre, Distribution on Ground Plane at 2500 MHz (Monc-Ile Height 2.68 cm, Radius 0.048 cm, Excitation 1 volt).

Iu0 Amplitude (MA) 25 20 15 10 5 0 0 2 4 6 8 10 12 Distance along the Ground Plane (cm) Fig. 6.9 Current Distribution on Ground Plane at 3000 MHz (Monopole Height 2.68 cm, Radius 0.048 cm, Excitation 1 volt).

10 1 Amplitude (MA) 10 8 6 4 2 0 0 2 4 6 8 10 12 Distance along the Ground Plane (cm) Fig. 6.10 Current Distribution on Ground Plane at 3750 MHz (Monopole Height 2.68 cm, Radius 0.048 cm, Excitation 1 volt).

102 Amplitude (MA) / Phase (Degrees) 30 Finite GP O Finite GP w/r A Amplitude 20 O 10 k I I I 0 0 0.675 1.350 2.025 2.700 Distance in Wavelength Fig. 6.11 Curre..: Distribution on Monopole (Height 2.68 cm, REaius 0.048 cm, Excitation 1 volt) at 2500 MHiz.

103 positive phase which indicate that the impedance of the monopole is capacitive because the height of the monopole which is 0.223 wavelength at 2500 MHz is shorter than the resonance length. The computed far field patterns are shown in Fig.(6.12) through Fig.(6.14). As in the experimental cases, the side lobes are eliminated when a resistive ground plane is used. In the computation, the ground plane radius is 12 cm, monopole height is 2.68 cm and radius is 0.048 cm, the frequencies used for far field computations are 2.25 GHz, 2.50 GHz and 2.75 GHz. 6.4 Comparison Between Experimental and Numerical Results It is a good practice to use experimental data to verify numerical simulations, especially when computations are approximated to make them feasible. Here, comparisons are made between the experimental and numerical cases. The monopole impedance as a function of frequency as well as the far field patterns (E field patterns) are plotted and tabulated. Table (6.1) shows the comparison between the numerical and experimental results for the impedance of the monopole (height..68 cm, radius 0.048 cm) at 1.875 GHz, 2.5 GHz, and 3.75 GHz. Various ground planes are used, i) the finite size ground plane (12 cm radius), ii) the finite size ground plane with resistive sheet (also 12 cm in radius) and, iii) the large ground plane (60 cm in

35s 1CO 10' PZI 3' Ma -- Metal Resistive - - - E PLANE Fig. 6.12 Computed Far Field Patterns at 2.25 GHz.

105 13 '0' Metal 1 Resistive - - - E PLANE Fig. 6.13 Computed Far Field Patterns at 2.50 GHz.

106 Metal Resistive - - - E PLANE Fig. 6.14 Computed Far Field Patterns at 2.75 GHz.

Finite Ground Plane Finite Ground Plane Large Ground Plane Frequency with resistive sheet (Radius: 12cm) (Radius: 12cm) (Radius: 60cm) (MHz) -- - Theory Experiment Theory Experiment Theory Experiment 1875 7.4 -jll 7.2 - j100 8.9 - j85 8.4 - j73 10 - j76 10.8 - j77 2000 11 -j102 10.1 - j90 13 - j60 12.8 -j58 14.3 -j56 14 -j 62 2500 38 -j10 37.8 - j5 32.8 -j6.8 32.5 -j8.8 33.8 -j12 33.1 -j8.5 3250 115 +j40 119 + j36 103 +j 45 97.5 +j40 95 + j60 93.1 +j 52 ~~~.................. 3750 216 +j170 208 + j214 199 +j186 192 + j170 198.2+j195 194 + j187 II - - - - - - Height of Monopole: 2.68 cm Radius of Monopole: 0.048 cm Table 6.1 Comparison of Monopole Impedance - Theory and Experiment.

108 radius) which is used for comparison. Close agreement exists between the experimental model and the numerical cases. The monopole impedance on the finite size ground plane with resistive edge is a close approximation to that of a large ground plane. Figures (6.15) through (6.17) show the far field pattern of the same monopole on the finite size metallic ground plane and the resistive one at three different frequencies, 2.25, 2.5 and 2.75 GHz. Close agreement again exists between numerical and experimental data. The large size ground plane (60 cm in radius) was not used in the comparison because it was impossible to mount it and rotate it for the antenna measurements. Another factor was the size of the anechoic room, in which it was not feasible to obtain the far field criterion 2D2/X using the large ground plane diameter for D. It has been shown that good agreement exists between the numerical simulation and experimental data. Since the difference between the numerical and the experimental data of the antenna impedance and the far field patterns is typically only five percent or less, this provides a good verification that numerical computations or simulation codes are valid.

Measured Metal Resistive - - - Computed Fig. 6.15 Comparison of Measured and Computed Far Field Patterns at 2.25 GHz.

Measured Metal Resistive - - - Computed Fig. 6.16 Comparison of Measured and Computed Far Field Patterns at 2.50 GHz.

Measured Computed Metal Resistive - - - Fig. 6.17 Comparison of Measured and Computed Far Field Patterns at 2.75 GHz.

CHAPTER VII. CONCLUSIONS The problem of a monopole located on a finite size circular ground plane is solved using the surface of revolution technique and the method of moments. The resistive boundary condition is also included in the formulation. The numerical procedure was tested by comparison with the experimental measurements for impedance of the monopole and the far field patterns for both the metallic and resistive ground plane. Naor [17] studied the scattering of resistive plates, but his program has limitations as it handles only rectangular plate and the maximum area of this plate is restricted in practice to about a square wavelength. Since the body of:revolution geometry is the characteristic of many physical structures, this method has the advantage of utilizing three-dimensional structures which can be much larger in wavelength. In the modelling of an infinite ground plane, a ground plane of five wavelengths in radius is used. It has been shown that such a size would give at most three percent error in antenna impedance measurements. The impedance of the monopole on the metallic and the resistive ground planes are examined both experimentally and numerically. Close agreement exists between these results at the frequencies studied. 1 12

113 The current distributions of the monopole on different ground planes are also studied. It is observed that with the resistive edge, the standing wave pattern is eliminated. These standing waves which resulted from the edge diffraction, give rise to the side lobes in the far zone pattern. A monopole antenna was built and evaluated for both the metallic and the resistive ground plane. The measured impedances of the antenna with different types of ground planes have been found to be in good agreements with corresponding numerical results. The measured far field patterns have also been found to be in good qualitative agreement with numerical results. In some respect, the overall result may be regarded as a close approximation to the infinite ground plane case, but significant deviation may also exist. For example, even though the far field pattern of the edge treated monopole does not have any side lobe, it is still different from the pattern produced by a monopole above an infinite ground plane. For further study, the effect of dielectric coating of the resistive material on antenna characteristics can be investigated.

APPEND ICES 1 14

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 This is Rose Wang's program for emulating a scattering result. To use this program, files should be attached to the corresponding I/O units as follows: Logical unit 1: Input data file (First record must be the parameter) Logical unit 5: Terminal Logical unit 6: Terminal XS ARRAY YS ARRAY XB ARRAY YB ARRAY STORES THE X COORDINATES OF THE SOURCE POINTS. STORES THE Y COORDINATES OF THE SOURCE POINTS. STORES THE X COORDINATES OF THE OBSERVATION POINTS. STORES THE Y COORDINATES OF THE OBSERVATION POINTS. DS ARRAY STORES THE DISTANCE BETWEEN AN OBSERVATION POINT AND EACH OF THE SOURCE POINTS. REAL COMMON /SOURCE/ REAL COMMON /OBSERV/ REAL COMMON /DISTNS/ INTEGER REAL COMMON /VARIAB/ COMPLEX*8 COMMON /FOUIIS/ COMPLEX*8 COMMON /FOUIES/ INTEGER REAL COMMON /VARIAC/ COMPLEX*8 COMMON /VARIAD/ COMPLEX*8 COMMON /CNSTAN/ COMPLEX*8 COMMON /INPUT/ COMPLEX*8 COMMON /OUTPUT/ XS,YS,DIS,DSQ XS(100),YS(100),DIS(100),DSQ(100) XB,YB XB(100),YB(100) DS,DSS,THETA1,THETA2,INC DS(100,15),DSS(100,15),THETA1,THETA2,INC DOTNUM,CURTYP XS1,YS1,XS2,YS2,XB1,YBI,XB2,YB2 XS1,YS1,XS2,YS2,XB1,YB1,XB2,YB2,DOTNUM,CURTYP GAA,GA,GAAP,GAP GAA(15),GA(15),GAAP(15),GAP(15) G1,G2,G3,G4,G5,G6,G7,G8,GB,GBB G1,G2,G3,G4,G5,G6,G7,G8,GB(15),GBB(15) FOUIIN FK,FRQNCY,MEW,EPSILN,WAVE,DTR,BETA FOUIIN,FK,FRQNCY,MEW,EPSILN,WAVE,DTR,BETA(15) VOLTGE,IMPEDC,CURENT VOLTGE(100),IMPEDC(100),CURENT(100) IMAGI IMAGI VOLT,IMP VOLT,IMP Z Z(99,99) THE VARIABLES IN THE "MAXIMN" CONTROL THE ARRAY SIZES FOR THE ARRAYS IN THE COMMON "SOURCE", "OBSERV". INTEGER SOUMAX, OBSMAX COMMON /MAXIMN/ SOUMAX, OBSMAX THE VARIABLES IN THE "CONTAN" INDICATE THE CURRENT NUMBER OF VALID FNTRIES IN THE ARRAYS.

59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 1 11 112 113 114 115 116 117 118 119 120 121 122 123 124 125 INTEGER COMMON /ARRCTN/ REAL COMMON /CONSTN/ INTEGER COMMON /IOUNIT/ SOUCTN, OBSCTN,RECCTN,PTR,IM,FARIDX,LASTSG SOUCTN, OBSCTN,RECCTN,PTR,IM,FARIDX,LASTSG PI,RADIAN,ZO,XR1,YR1,XR2,YR2 PI,RADIAN,ZO,XR1,YR1,XR2,YR2 INPUTF, MESSGE, REPORT, TERMIN INPUTF, MESSGE, REPORT, TERMIN ------------------------------------------------------—. --- —-------. END OF COMMON --------------------------------------------------------. --- —------ - INTEGER KODE CALL INITAL(KODE) IF (KODE.NE.O) GO TO 999 WRITE(MESSGE,10) 10 FORMAT(1H,/,1H,10X,'*** CALL PROCES 999 STOP END SUBROUTINE INITAL(KODE) Result from the simulation X,* ) This subroutine initializes variables in the "COMMON" section. * * * * * ** * * * * * * * * * * * * ** * * * * * * * * * * * * * * * * * * * ** * * * * *w w * w * * * REAL COMMON /SOURCE/ REAL COMMON /OBSERV/ REAL COMMON /DISTNS/ INTEGER REAL COMMON /VARIAB/ COMPLEX*8 COMMON /FOUIIS/ COMPLEX*8 COMMON /FOUIES/ INTEGER REAL COMMON /VARIAC/ COMPLEX*8 COMMON /VARIAD/ COMPLEX*8 COMMON /CNSTAN/ COMPLEX*8 COMMON /INPUT/ COMPLEX*8 COMMON /'UTPUT/ INTEGER COMMON ".'AXIMN/ INTEGER COMMON /ARRCTN/ REAL COMMON /CONSTN/ INTEGER COMMON /IOUNIT/ XS,YS,DIS,DSQ XS(100),YS(100),DIS(100),DSQ(100) XB,YB XB(100),YB(100) DS,DSS,THETA1,THETA2,INC DS(100,15),DSS(100,15),THETA1,THETA2,INC DOTNUM,CURTYP XS1,YS1,XS2,YS2,XB1,YB1,XB2,YB2 XS1,YS1,XS2,YS2,XB1,YB1,XB2,YB2,DOTNUM,CURTYP GAA,GA,GAAP,GAP GAA(15),GA(15),GAAP(15),GAP(15) G1,G2,G3,G4,G5,G6, G7,G8,GB,GBB G1,G2,G3,G4,G5,G6,G7,G8,GB(15),GBB(15) FOUIIN FK,FRQNCY,MEW,EPSILN,WAVE,DTR,BETA FOUIIN,FK,FRQNCY,MEW,EPSILN,WAVE,DTR,BETA(15) VOLTGE,IMPEDC,CURENT VOLTGE(100),IMPEDC(100),CURENT(100) IMAGI IMAGI VOLT,IMP VOLT,IMP z Z(99,99) SOUMAX, OBSMAX SOUMAX, OBSMAX SOUCTN, OBSCTN,RECCTN,PTR,IM,FARIDX,LASTSG SOUCTN, OBSCTN,RECCTN,PTRIM,FARIDX,LASTSG PI,RADIAN,ZO,XR1,YR1,XR2,YR2 PI,RADIAN,ZO,XR1,YR1,XR2,YR2 INPUTF, MESSGE, REPORT, TERMIN INPUTF, MESSGE, REPORT, TERMIN 126 ----------------------------------------------—.

127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 END OF COMMON -------------------------------------------------------------------- KODE=O SOUMAX=100 OBSMAX=100 FOUIIN controls the iteration PI=3.1415927 EPSILN=8.85E-12 MEW=(4.0E-7)*PI IMAGI=CMPLX(O.O,1.0) RADIAN=57.29578 DTR=0.01745329 ZO =SQRT(MEW/EPSILN) INPUTF=1 MESSGE=6 TERMIN=5 REPORT=2 in the FOURIER'S function FARIDX=1 THETA1=0.0 THETA2=0.0 Read the parameters READ(INPUTF,10) WAVE,FOUIIN,THETA1,THETA2,INC,FARIDX,MCMFLG, *XR1,YR1,XR2,YR2 10 FORMAT(F8.5,I3,3F7.2,2I2,I,4F6.2) IF (FOUIIN.GE.O.AND.FOUIIN.LE.10) GO TO 20 WRITE(MESSGE,901) 901 FORMAT(1H,'-ERROR,: Fourier''s parameter out of * 'range.') KODE=-1 20 IF (WAVE.GT.O) GO TO 30 WRITE(MESSGE,902) 902 FORMAT(1H,'*ERROR*: Wrong wavelength.') KODE=-1 GO TO 999 30 FRQNCY=(3.0E8)/WAVE IF (MCMFLG.EQ.1) FRQNCY=(3.0E1O)/WAVE FK=(2*PI)/WAVE 999 RETURN END SUBROUTINE READER This subroutine reads in input data record, then partition them into intervals before processing. REAL COMMON /SOURCE/ REAL COMMON /OBSERV/ REAL COMMON /-iSTNS/ INTEGER REAL COMMON /VARIAB/ COMPLEX*8 COMMON /FOUIIS/ COMPLEX*8 COMMON /FOUIES/ INTEGER REAL XS,YS,DIS,DSQ XS(100),YS(100),DIS(100),DSO(100) XB,YB XB(100),YB(100) DS,DSS,THETA1,THETA2,INC DS(100,15),DSS(100,15),THETA1,THETA2,INC DOTNUM,CURTYP XS1,YS1,XS2,YS2,XB1,YB1,XB2,YB2 XS1,YS1,XS2,YS2,XB1,YB1,XB2,YB2,DOTNUM,CURTYP GAA,GA,GAAP,GAP GAA(15),GA(15),GAAP(15),GAP(15) G1,G2,G3,G4,G5,G6,G7,G8,GB,GBB G1,G2,G3,G4,G5,G6,G7,G8,GB(15),GBB(15) FOUIIN FK,FRQNCY,MEW,EPSILN,WAVE,DTR,BETA

195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 COMMON /VARIAC/ COMPLEX*8 COMMON /VARIAD/ COMPLEX*8 COMMON /CNSTAN/ COMPLEX*8 COMMON /INPUT/ COMPLEX*8 COMMON /OUTPUT/ INTEGER COMMON /MAXIMN/ INTEGER COMMON /ARRCTN/ REAL COMMON /CONSTN/ INTEGER COMMON /IOUNIT/ FOUIIN,FK,FRONCY,MEW,EPSILN,WAVE,DTR,BETA(15) VOLTGE,IMPEDC,CURENT VOLTGE(100),IMPEDC(100),CURENT(100) IMAGI IMAGI VOLT,IMP VOLT,IMP 2 Z(99,99) SOUMAX, OBSMAX SOUMAX, OBSMAX SOUCTN, OBSCTN,RECCTN,PTR,IM,FARIDX,LASTSG SOUCTN, OBSCTN,RECCTN,PTR,IM,FARIDX,LASTSG PI,RADIAN,ZO,XR1,YR1,XR2,YR2 PI,RADIAN,ZO,XR1,YR1,XR2,YR2 INPUTF, MESSGE, REPORT, TERMIN INPUTF, MESSGE, REPORT, TERMIN ----—. --- —------------------------------------------—. --- —-------- END OF COMMON ----—. --- —-------------------------------------------- ------------- REAL OLDX,OLDY INTEGER LINE,CIRCLE,CURVE,FLAG,KODE DATA LINE/'LINE'/,CIRCLE/'CIRC'/, CURVE/'CURV'/ PTR=1 OLDX=-1E10 OLDY=-1E10 FLAG=1 KODE=O RECCTN=O When "FLAG" is equal to one means that this is the first segment. 10 READ (INPUTF,20,END=995) XB1,YB1,XB2,YB2,VOLT,IMP,DOTNUM, * CURTYP,IM,LASTSG 20 FORMAT(4F9.5,2F6.2,2F8.2,I2,A4,I2,I1) RECCTN=RECCTN+1 IF (PTR.NE.1.AND.(XB1.NE.OLDX.OR.YB1.NE.OLDY)) GO TO 120 Find a proper subroutine to cut the line IF (CURTYP.NE.LINE) GO TO 50 CALL SLINE(KODE,FLAG) IF (KODE.NE.O) GO TO 990 FLAG=O OLDX=XB2 OLDY=YB2 GO TO 10 50 WRITE(MESSGE,60) RECCTN 60 FORMAT(1H,'*ERROR*: Unrecognizable curve type at record',I8) GO TO 990 120 WRITE(MESSGE,130) RECCTN 130 FORMAT(1H,'*ERROR*: Curve must be defined continuously', * /,1H, condition occurred at record',I8) 990 SOUCTN=O GO TO 999 995 SOUCTN=PTR-1 999 RETURN END SUBROUTINE PROCES *** *** * * *:**** * * ** * * * * * * * * * * * * * * * *** * * * * * * * ***** * * * ***** This subroutine is tne driver for the simulation process.

263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 REAL COMMON /SOURCE/ REAL COMMON /OBSERV/ REAL COMMON /DISTNS/ INTEGER REAL COMMON /VARIAB/ COMPLEX*8 COMMON /FOUIIS/ COMPLEX*8 COMMON /FOUIES/ INTEGER REAL COMMON /VARIAC/ COMPLEX*8 COMMON /VARIAD/ COMPLEXE8 COMMON /CNSTAN/ COMPLEX 8 COMMON /INPUT/ COMPLEXX8 COMMON /OUTPUT/ INTEGER COMMON /MAXIMN/ INTEGER COMMON /ARRCTN/ REAL COMMON /CONSTN/ INTEGER COMMON /IOUNIT/ XS,YS,DIS,DSQ XS(100),YS(100),DIS(100),DSO(100) XB,YB XB(100),YB(100) DS,DSS,THETA1,THETA2,INC DS(100,15),DSS(100,15),THETA1,THETA2,INC DOTNUM,CURTYP XS1,YS1,XS2,YS2,XB1,YB1,XB2,YB2 XS1,YS1,XS2,YS2,XB1,YB1,XB2,YB2,DOTNUM,CURTYP GAA,GA,GAAP,GAP GAA(15),GA(15),GAAP(15),GAP(15) G1,G2,G3,G4,G5,G6,G7,G8,GB,GBB G1,G2,G3,G4,G5,G6,G7,G8,GB(15),GBB(15) FOUIIN FK,FRQNCY,MEW,EPSILN,WAVE,DTR,BETA FOUIIN,FK,FRONCY,MEW,EPSILN,WAVE,DTR,BETA(15) VOLTGE,IMPEDC,CURENT VOLTGE(100),IMPEDC(100),CURENT(100) IMAGI IMAGI VOLT,IMP VOLT,IMP z Z(99,99) SOUMAX, OBSMAX SOUMAX, OBSMAX SOUCTN, OBSCTN,RECCTN,PTR,IM,FARIDX,LASTSG SOUCTN, OBSCTN,RECCTN,PTR,IM,FARIDX,LASTSG PI,RADIAN,ZO,XR1,YR1,XR2,YR2 PI,RADIAN,ZO,XR1,YR1,XR2,YR2 INPUTF, MESSGE, REPORT, TERMIN INPUTF, MESSGE, REPORT, TERMIN -------------------------------------------------------------------- END OF COMMON -------------------------------------------------------------------- CALL READER IF (SOUCTN.LT.3) GO TO 999 CALL COMPUT IF (FARIDX.EQ.O) GO TO 999 CALL FARFLD 999 RETURN END SUBROUTINE SLINE(KODE,FLAG) This subroutine puts the source points/observation points and the input voltage & impedance into proper position in the matrices. FLAG: is used to indicated if this is the first segment. Source segment and Non-source segment are processed in the same manner. However, the segments may have different partitioned length, depending on where they are on the curve. First segment: 2 points ------------- ---- ----------- Middle segment: 2 0oints

331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 -------— + — ---- ------------- Last segment: 2 points -------— + — ------— + --- —----- REAL COMMON /SOURCE/ REAL COMMON /OBSERV/ REAL COMMON /DISTNS/ INTEGER REAL COMMON /VARIAB/ COMPLEX*8 COMMON /FOUIIS/ COMPLEX*8 COMMON /FOUIES/ INTEGER REAL COMMON /VARIAC/ COMPLEX*8 COMMON /VARIAD/ COMPLEX*8 COMMON /CNSTAN/ COMPLEX*8 COMMON /INPUT/ COMPLEX*8 COMMON /OUTPUT/ INTEGER COMMON /MAXIMN/ INTEGER COMMON /ARRCTN/ REAL COMMON /CONSTN/ INTEGER COMMON /IOUNIT/ XS,YS,DIS,DSQ XS(100),YS(100),DIS(100),DSQ(100) XB,YB XB(100),YB(100) DS,DSS,THETA1,THETA2,INC DS(100,15),DSS(100,15),THETA1,THETA2,INC DOTNUM,CURTYP XS1,YS1,XS2,YS2,XB1,YB1,XB2,YB2 XS1,YS1,XS2,YS2,XB1,YB1,XB2,YB2,DOTNUM,CURTYP GAA,GA,GAAP,GAP GAA(15,GA5)GA(15),GAAP(15),GAP(15) G1,G2,G3,G4,G5,G6,G7,G8,GB,GBB G1,G2,G3,G4,G5,G6,G7,G8,GB(15),GBB(15) FOUIIN FK,FRQNCY,MEW,EPSILN,WAVE,DTR,BETA FOUIIN,FK,FRQNCY,MEW,EPSILN,WAVE,DTR,BETA(15) VOLTGE,IMPEDC,CURENT VOLTGE(100),IMPEDC(100),CURENT(100) IMAGI IMAGI VOLT,IMP VOLT,IMP z 2(99,99) SOUMAX, OBSMAX SOUMAX, OBSMAX SOUCTN, OBSCTN,RECCTN,PTR,IM,FARIDX,LASTSG SOUCTN, OBSCTN,RECCTN,PTR,IM,FARIDX,LASTSG PI,RADIAN,ZO,XR1,YR1,XR2,YR2 PI,RADIAN,ZO,XR1,YR1,XR2,YR2 INPUTF, MESSGE, REPORT, TERMIN INPUTF, MESSGE, REPORT, TERMIN ------------------------------------------------------—. --- —------- END OF COMMON --------------------------------------------------------—. --- —----—. INTEGER ENDPTR,FLAG,KODE REAL SARC,ARC,SPACIN,COMPEN,DOTPEN,EXPN ------------------------------------------ ENDPTR=PTR+DOTNUM CHECK IF THE ARRAY IS BIG ENOUGH TO HANDLE THESE NEW POINTS IF (ENDPTR.GT.SOUMAX) GO TO 100 COMPEN=O.O DOTPEN=O.O IF (LASTSG.EQ.1) DOTPEN=0.5 IF (CABS(VOLT).EQ.O.O) GO TO 30 IF (CASS(VOLT).EQ.0.0) GO TO 30 IF (FL^: EQ.O) GO TO 10 XB(1 I:. 1 XS(1 -- 31 YB(1)- 31 YS(1)=YB1 VOLTGE(1)=VOLT IMPEDC(1)=CMPLX(0.0,0.0) PTR=PTR+1 ENDPTR=ENDPTR+1 COMPEN=-0.5 DOTPEN=0.5

399 10 CONTINUE 400 401 SPACIN=SQRT(((X82-XB1)/(DOTNUM+DOTPEN))**2 402 +((YB2-YBI)/(DOTNUM+DoTPEN))-*2) 403 DO 20 I = PTR,ENDPTR 404 XB(I)=XBI+((XB2-XB1)/(DOTNUM+DOTPEN))*(I-PTR+0.5-COMPEN) 405 XS(I)=XB1I((XB2-XB1)/(DOTNUM+DOTPEN))*(I-PTR+0.5-COMPEN) 406 YB(I)=YB1+((YB2-YBI)/(DOTNUM+DOTPEN))*(I-PTR+0.5-COMPEN) 407 YS(I)=YB1#((YB2-YBI)/(DOTNUM+DOTPEN))*(I-PTR+0.5-COMPEN) 408 DIS(I)=SQRT((X82-XB( I))*2+(YB2YB( I))**2) 409 DSQ(I)=SPACIN 410 MID=I-1 411 VOLTGE(MID)=VOLT 412 IMPEDC(MID)=CMPLX(0.0,0.0) 413 20 CONTINUE 414 415 PTR=ENDPTR 416 GO TO 999 417 418 419 THE PRESENT SEGMENT IS NOT A SOURCE SEGMENT 420 421 422 30 IF (FLAG.EQ.O) GO TO 35 423 XB(1)=XBI 424 XS(i)=XBi 425 YB(1)=YB1 426 YS(1)=YB1 427 VOLTGE(1)=CMPLX(0.0,0.0) 428 IMPEDC(1)=IMP 429 PTR=PTRl1 430 ENDPTR=ENDPTR+1 431 COMPEN=-0.5 432 DOTPEN=0.5 433 434 35 CONTINUE 435 436 SPACIN=SQRT(((XB2-XBI)/(DOTNUM+DOTPEN))**2 437 a +((YB2-YBI)/(DOTNUM+DOTPEN))**2) 438 DO 40 I = PTR,ENDPTR 439 XB(I)=XBI4((XB2-XBI)/(DOTNUM+DOTPEN))*(I-PTR+0.5-COMPEN) 440 XS(I)=XBI+((XB2sXBI)/(DOTM PN IT0 C E 441 YB(I)=YB1#((YB2-YBI)/(DOTNUM+DOTPEN))*(I-PTR+0.5-COMPEN) 442 YS(I)=YBI+((YB2-YBI)/(DOTNUM+D OE *I R. MN 443 DSQ(I)=SPACIN 444 MID=I-1 445 VOLTGE(MID)=CMPLX(0.0,0.0) 446 IF (IM.LT.0) GO TO 36 447 SARC=(XR2-XB(I))**2+(YR2-YB(I))**2 448 GO TO 37 449 36 SARC=(XRI-XB(I))**2+(YRI-yB(I))**2 450 37 ARC=(XR2-XRI)**2+(YR2-YRI)**2 451 DIS(I)=SQRT(SARC) 452 IF(CABS(IMP).EQ.0.0.AND.IM.GT.0) 453 DIS(I)=SQRT((XB2-XB(l))**2+(YB2-YB(I))**2) 454 IF(CAES(IMP). EQ.OQANO.IM. LT.0) 455 * DIS(I)=SQRT((XBI-XB(I))**2+(YB-1YB(I))**2) 456 EXPN= 'FLOAT(IM)/10.0) 457 IF ( i. ~0) GO TO 38 458 IMPED' 111ID)=IMP 459 GO TC -iD 460 38 IMPEDC(MID)=IMP*((SARC/ARC)**EXPN) 461 40 CONTINUE 462 PTR=ENDPTR 463 GO TO 999 464 465 100 WRITE(MESSGE,110) SOUMAX 466 110 FORMAT(IH,'*ERROR* ARRAY SIZE NEEDS TO BE INCREASED,,

467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 * 'CURRENT SIZE:',I8) KODE=-1 999 RETURN END SUBROUTINE DISTAN(I) THIS SUBROUTINE CALCULATES THE DISTANCE BETWEEN A SOURCE POINT AND A OBSERVATION POINT. THIS PROCESS IS DONE FOR ALL SOURCE POINTS RELATIVE TO ALL OBSERVATION POINTS. I: varies from 1 to OBSCTN REAL COMMON /SOURCE/ REAL COMMON /OBSERV/ REAL COMMON /DISTNS/ INTEGER REAL COMMON /VARIAB/ COMPLEX*8 COMMON /FOUIIS/ COMPLEX*8 COMMON /FOUIES/ INTEGER REAL COMMON /VARIAC/ COMPLEX*8 COMMON /VARIAD/ COMPLEX*8 COMMON /CNSTAN/ COMPLEX*8 COMMON /INPUT/ COMPLEX*8 COMMON /OUTPUT/ INTEGER COMMON /MAXIMN/ INTEGER COMMON /ARRCTN/ REAL COMMON /CONSTN/ INTEGER COMMON /IOUNIT/ XS,YS,DIS,DSQ XS(100),YS(100),DIS(100),DSQ(100) XB,YB XB(100),YB(100) DS,DSS,THETA1,THETA2,INC DS(100,15),DSS(100,15),THETA1,THETA2,INC DOTNUM,CURTYP XS1,YS1,XS2,YS2,XB1,YB1,XB2,YB2 XS1,YS1,XS2,YS2,XB1,YB1,XB2,YB2,DOTNUM,CURTYP GAA,GA,GAAP,GAP GAA(15),GA(15),GAAP(15),GAP(15) G1,G2,G3,G4,G5,G6,G7,G8,GB,GBB G1,G2,G3,G4,G5,G6,G7,G8,GB(15),GBB('15) FOUIIN FK,FRQNCY,MEW,EPSILN,WAVE,DTR,BETA FOUIIN,FK,FRONCY,MEW,EPSILN,WAVE,DTR,BETA(15) VOLTGE,IMPEDC,CURENT VOLTGE(100),IMPEDC(100),CURENT(100) IMAGI IMAGI VOLT,IMP VOLT,IMP Z Z(99,99) SOUMAX, OBSMAX SOUMAX, OBSMAX SOUCTN, OBSCTN,RECCTN,PTR,IM,FARIDX,LASTSG SOUCTN, OBSCTN,RECCTN,PTR,IM,FARIDX,,LASTSG PI,RADIAN,ZO,XR1,YR1,XR2,YR2 PI,RADIAN,ZO,XR1,YR1,XR2,YR2 INPUTF, MESSGE, REPORT, TERMIN INPUTF, MESSGE, REPORT, TERMIN ----—. --- —------------------------------------------—, --- —--------- END OF COMMON THE OBSERVATION POINTS AND THE SOURCE POINTS ARE DEFINED AS: + --- —. --- —. --- —+ --- —+ --- — 0 1 2 3 4 XS(1) XS(2) XS(3) XS(4) XS(5) XB(1) XB(2) XB(3) XB(4) XB(5) INTEGER I,J REAL RQPH,ROMH REAL ZOPH,ZQMH REAL RNPH,RNMH,RNP1,RNM1,RNPQ,RNMQ REAL ZNPH,ZNMH,ZNP1,ZNM1,ZNPQ,ZNMQ SOUCTN OBSCTN USEFUL = 4 = 4 POINTS = 3

535 536 537 538 539 540 54 1 542 543 544 545 546 547 548 549 550 55 1 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 57 1 572 573 574 575 576 577 578 579 580 58 1 582 583 584 585 586 587 588 589 590 59 1 592 593 594 595 596 597 598 599 600 601 602 I END=SOUCTN RQPH=(XB( I)+XB(I1+1) )'*o.5 ROMH=(XB( I)+XB( I-1) )*05 ZQMH=(YB(I )-+YB(I-1 ))*os5 DO 50 JU 2, IEND RNM1=XS(U-1) ZNMI=YS(U-1) RNP1=XS(J+l) ZNPI=YS(J+1) RNPH=(XS(U)+XS(u~1))'*0.5 ZNPH=(YS(J)+YS(-+-I) )*Q.S -RNMH=(XS(J)4-XS(J-1))*0.5 ZNMH=(YS(J)+YS(J-1))'*0.5 RNPQ=(XS(J)*0.75+XS(J~+I)*0.25) ZNPQ=( YS(U ) 0. 75+YS( 1) *0.25) RNMQ= (XS(J) *0.754-XS(JU-I) *0 25) ZNMQ=(YS(J)*0.754+YS(U-1)*0.25) DS(U, 1) DS(U,2) D S (U. 3) DS(U,4) DS(U, 5) DS(U,G) DS(U,7) DS(U,8) DS( 3,9) DS(U, 10) DS(U, 11) DS(U, 12) DS(U, 13) DS(U, 14) DS(U, 15) DSS(U, I) DSS(U,2) DSS(J,3) DSS(U,4) DSS(U,5) DSS(U,G) DSS(U,7) D S S (J, 8 =SQRT( (RQPH-XS(u) )**2~ (ZQPH-YS(U) )**2) =SQRT( (RQPH-RNM1 )**2+ (ZQPH-ZNM1I)-*2) =SQRT( (RQPH-RNMH)**2+ (ZOPH-ZNMH) **2) =SQRT( (RQPH-RNP1I)**2+ (ZOPH-ZNP1 )**2) =SQRT( (RQPH-RNPH)**2+ (ZQPH-ZNPH)**2) =SQRT( (RQMH-RNMH)**2+ (ZQMH-ZNMH) **2) =SQRT( (RQMH-RNM1 )**2+ (ZQMH-ZNM1 )**2) =SQRT( (RQMH-XS(U) )**2+ (ZQMH-YS(U) )**2) =SQRT( (RQMH-RNPI )**2+ (ZQMH-ZNPI )**2) =SQRT( (RQMH-RNPH)**2+ (ZQMH-ZNPH)**2) =SQRT( (xB( I)-XS(u) )**2+ =SQRT((XB(I )-RNMH)**2+ (YB(I )-ZNMH)**2) =SQRT((XB(I )-RNMO)**2.(YB( I )ZNMQ)**2) =SQRT( (XB( I)-RNPH)**2+ (YB( I)-ZNPH)**2) =SQRT( (XB( I)~-RNPQ)**2.+ (YB( I)-ZNPQ)**2) =SQRT( (RQPH+XS(U) )**2~ (ZQPH-YS(U) )**2) =SQRT( (RQPH+RNM1 )**2+ (ZQPH-ZNMI )**2) =SQRT( (RQPH+RNMH)**2+ (ZQPH-ZNMH)**2) =SQRT( (RQPH+RNP1I)**2+ (ZQPH-ZNP1 )**2) =SQRT( (RQPH+RNPH)**2+ (ZQPH-ZNPH)**'2) =SQRT( (RQMH-.RNMH)**2+ (ZQMH-ZNMH)**2) =SQRT( (ROMH+RNMI )**2+ ( 71'AH-ZNM 1 )*2) =SQRT(( -4-X()*2

603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 66 1 662 663 664 665 666 667 668 669 670 (ZOMH-YS(U))**2) DSS(J,9) =SQRT((RQMH4RNP1)**2+ (ZQMH-ZNPI)**2) DSS(U,10)=SQRT((RQMH#RNPH)**2+ (ZQMH-ZNPH)**2) DSS(J,11)=SQRT((XB( I )+XS())*2+ (YB(I )-YS(J) )**2) DSS(J,12)=SQRT((XB(I)+RNMH)**2+ (YB(I)-ZNMH)**2) DSS(J,13)=SQRT((XB(I)+RNMQ)**2+ (YB(I)-ZNMQ)**2) DSS(J,14)=S0RT((XB(I)+RNPH)'*2+ (YB(I)-ZNPH)**2) DSS(J,15)=SQRT((XB(I)+RNPQ)**2+ (YB(I)-ZNPQ)K*2) 50 CONTINUE RETURN END SUBROUTINE COMPUT This subroutine computes the impedance and the current for the defined point on the "body. (Body of Revolution) REAL XS,YS,DIS,DSQ COMMON /SOURCE/ xS(100),YS(100),DIS(100),DSQ(100) REAL XE,YB COMMON /OBSERV/ XB(100),YB(100) REAL DS,DSS,THETA1,THETA2,INC COMMON /DISTNS/ DS(100,15),DSS(100,15),THETA1,THETA2,INC INTEGER DOTNUM,CURTYP REAL XSI,YS1,XS2,YS2,XB1,YB1,XB2,YB2 COMMON /VARIAB/ XSI,YS1,XS2,YS2,XBI,YB1,XB2,YB2,DOTNUM,CURTYP COMPLEX*8 GAA,GA,GAAP,GAP COMMON /FOUIIS/ GAA(15),GA(15),GAAP(15),GAP(15) COMPLEX*8 G1,G2,G3,G4,G5,G6,G7,G8,GB,GBB COMMON /FOUIES/ Gi,G2,G3,G4,G5,G6,G7,G8,GB(15),GBB( i5) INTEGER FOUIIN REAL FK,FRQNCY,MEW,EPSILN,WAVE,DTR,BETA COMMON /VARIAC/ FOUIIN,FK,FRQNCY,MEW,EPSILN,WAVE,DTR,BETA(15) COMPLEX*8 VOLTGE,IMPEDC CURENT COMMON /VARIAD/ VOLTGE(100),IMPEDC(100),CURENT(100) COMPLEX*8 IMAGI COMMON /CNSTAN/ IMAGI COMPLEX*8 VOLT,IMP COMMON /INPUT/ VOLT,IMP COMPLEX*8 Z COMMON /OUTPUT/ Z(99,99) INTEGER SOUMAX, OBSMAX COMMON /MAXIMN/ SOUMAX, OBSMAX INTEGER SOUCTN, OBSCTN,RECCTN,PTR,IM,FARIDX,LASTSG COMMON /ARRCTN/ SOUCTN, OBSCTNRECCTN,PTR,IM,FARIDX,LASTSG REAL PI,RADIANZO,XR1,YR1,XR2,YR2 COMMON -JNSTN/ PI,RADIAN,ZO,XRl,YR1,XR2,YR2 INTEGER INPUTF, MESSGE, REPORT, TERMIN COMMON.''-'NIT/ INPUTF, MESSGE, REPORT, TERMIN END OF COMMON COMPLEX*8 EQi,E02,EQ3,EQTNA,EQTNB,EQTNC,EQTND INTEGER I,J,M REAL XBMID,YBMID,XSMID,YSMID,DXYB1,DXYB2,DXYSI,DXYS2

671 REAL LAMB1,LAMB2,LAMS1,LAMS2,XBMM,YBMM,DISMM,DSOMM 672 REAL SIN1,SIN2,COS1,COS2,TSIN,TCOS 673 674 675 676 IEND=SOUCTN 677 M = FOUIIN 678 679 680 DO 100 I = 2, IEND 681 CALL DISTAN(I) 682 DO 80 J = 2, IEND 683 CALL GREENS(I,J,M) 684 XBMID=XB(I)-XB(I-1) 685 YBMID=YB(I)-YB(I-1) 686 XSMID=XS(J)-XS(J-1) 687 YSMID=YS(J)-YS(J-1) 688 DXYB1=SQRT(XBMID**2+YBMID**2) 689 DXYS1=SQRT(XSMID**2+YSMID**2) 690 IF (YB(I).EQ.YB(I-1)) GO TO 5 691 LAMB1=ATAN(XBMID/YBMID) 692 GO TO 6 693 5 LAMB1=90.0*DTR 694 6 IF (YS(J).EQ.YS(J-1)) GO TO 10 695 LAMS1=ATAN(XSMID/YSMID) 696 GO TO 15 697 10 LAMS1=90.0*DTR 698 15 XBMID=XB(I+1)-XB(I) 699 YBMID=YB(I+1)-YB(I) 700 XSMID=XS(J+1)-XS(J) 701 YSMID=YS(J+1)-YS(J) 702 DXYB2=SQRT(XBMID*2+YBMID**2) 703 DXYS2=SQRT(XSMID**2+YSMID**2) 704 IF (YB(I+1).EQ.YB(I)) GO TO 20 705 LAMB2=ATAN(XBMID/YBMID) 706 GO TO 23 707 20 LAMB2=90.0*DTR 708 23 IF (YS(J+1).EQ.YS(J)) GO TO 25 709 LAMS2=ATAN(XSMID/YSMID) 710 GO TO 30 711 25 LAMS2=90.0*DTR 712 30 SINI=DXYB1*SIN(LAMB1) 713 SIN2=DXYB2*SIN(LAMB2) 714 COS1=DXYB1xCOS(LAMB1) 715 COS2=DXYB2*COS(LAMB2) 716 TSIN=(SINI+SIN2)/2.0 717 TCOS=(COS1+COS2)/2.0 718 719 720 EQTNA=TSIN*SIN(LAMS1)*G7 721 EQTNB=TSIN*SIN(LAMS2)*G8 722 EQTNC=2*TCOS*COS(LAMS1)*G5 723 EQTND=2*TCOS*COS(LAMS2)*G6 724 725 726 EQ1=FK*0.5*(EQTNA+EQTNB+EQTNC+EQTND)*IMAGI 727 EQ2=(G1-G2)*IMAGI/(FK*DXYS1) 728 EQ3= -4-G3)*IMAGI/(FK*DXYS2) 729 730 731 Z(I-,J-1)=ZO*(EQ1+EQ2+EQ3)/(PI*PI*2.0)+SIN(LAMB1)* 732 (IMPEDC(J-1)/PI) 733 80 CONTINUE 734 100 CONTINUE 735 736 737 CALL MULTPY 738 FRQNCY=FRQNCY/(1.OE6)

739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 WRITE(MESSGE,101) FOUIIN 101 FORMAT(1H,'MODE NUMBER = ',12) WRITE (MESSGE,102)WAVE,FRQNCY 102 FORMAT(1H,'THE RESULTS FOR WAVELENGTH = ',F7.2,' CM ',4X, &'FREQUENCY = ',F7.2,'MHZ') WRITE(MESSGE,103) 103 FORMAT(/) IF (FARIDX.EQ.1) GO TO 999 WRITE(MESSGE,104) 104 FORMAT(8X,'DISTANCE',25X,'IMPEDANCE',10X,'CURRENT') WRITE(MESSGE,105) 105 FORMAT(/) WRITE(MESSGE,106) 106 FORMAT(5X,'RHO',7X,'Z',8X,'DIS',5X,'DSQ',5X,'RS',6X,'XS',7X,'MAG', w7X,'PHASE') IIND=SOUCTN-1 DO 110 MM=1,IIND AMP=CABS(CURENT(MM)) PHASE=RADIAN*ATAN2(AIMAG(CURENT(MM)),REAL(CURENT(MM))) DSQMM=DSQ(MM+1)/WAVE WRITE(MESSGE,107) XB(MM+1),YB(MM+1),DIS(MM+1),DSQMM, *IMPEDC(MM),AMP,PHASE 107 FORMAT(1H,3F9.4,F8.4,F10.4,F8.4,E11.4,F8.2) 110 CONTINUE 999 RETURN END SUBROUTINE GREENS(I,J,M) M: varies from 1 to FOUIIN ***** * * * * **** ****** *** **** * * * * ** * * 3* ** Xc* * * * * * *: * REAL COMMON /SOURCE/ REAL COMMON /OBSERV/ REAL COMMON /DISTNS/ INTEGER REAL COMMON /VARIAB/ COMPLEX*8 COMMON /FOUIIS/ COMPLEX*8 COMMON /FOUIES/ INTEGER REAL COMMON /VARIAC/ COMPLEX*8 COMMON /VARIAD/ COMPLEX*8 COMMON /CNSTAN/ COMPLEX*8 COMMON /INPUT/ COMPLEX*8 COMMON /OUTPUT/ INTEGER COMMON /MAXIMN/ INTEGER COMMON /ARRCTN/ REAL COMMON /CONSTN/ XS,YS,DIS,DSO XS(100),YS(100),DIS(100),DSO(100) XB,YB XB(100),YB(100) DS,DSS,THETA1,THETA2,INC DS(100,15),DSS(100,15),THETA1,THETA2,INC DOTNUM,CURTYP XS1,YS1,XS2,YS2,XB1,YB1,XB2,YB2 XS1,YS1,XS2,YS2,XB1,YB1,XB2,YB2,DOTNUM,CURTYP GAA,GA,GAAP,GAP GAA(155)GA(15),GAAP(15),GAP(15) G1,G2,G3,G4,G5,G6,G7,G8,GB,GBB G1,G2,G3,G4,G5,G6,G7,G8,GB(15),GBB(15) FOUIIN FK,FRQNCY,MEW,EPSILN,WAVE,DTR,BETA FOUIIN,FK,FRONCY,MEW,EPSILN,WAVE,DTR,BETA(15) VOLTGE,IMPEDC,CURENT VOLTGE(100),IMPEDC(100),CURENT(100) IMAGI IMAGI VOLT,IMP VOLT,IMP Z Z(99,99) SOUMAX, OBSMAX SOUMAX, OBSMAX SOU: '., OBSCTN,RECCTN,PTR,IM,FARIDX,LASTSG SCOT-N, OBSCTN,RECCTN,PTR,IM,FARIDX,LASTSG PI,RADIAN,ZO,XR1,YR1,XR2,YR2 PI,R"DIAN,ZO,XR1,YR1,XR2,YR2

807 INTEGER INPUTF, MESSGE, REPORT, TERMIN 808 COMMON /IOUNIT/ INPUTF, MESSGE, REPORT, TERMIN 809 810 -------------------------------------------------------------------- 811 END OF COMMON 812 -------------------------------------------------------------- 813 814 COMPLEX*8 DASC,DSFK,DASC1,DSFK1 815 INTEGER I,J,K,M 816 INTEGER FLAG1,FLAG8,FLAG11 817 REAL TNNM1,TNP1N,TNNMH,TNPHN 818 REAL GG1,GG2,GG3(8),GG4(14),PK,ELTKP 819 REAL RQ1,ROMH1,RQPH1,X,H,DT(15),TMPDS,TMPCS,TMPSN,TMPDV,TMPX 820 REAL RHN,RHNM1,RHNP1,RHNMH,RHNPH,RHNMQ,RHNPQ,TMPXX 821 822 823 824 RQ1 =XB(I) 825 ROMHI=(XB(I)+XB(I-1))*0.5 826 RQPHI=(XB(I)+XB(I+1))*0.5 827 RHN =XS(J) 828 RHNM1=XS(J-1) 829 RHNP1=XS(d+1) 830 RHNMH=(XS(J)+XS(J-1))*0.5 831 RHNMQ=XS(J)*0.75+XS(J-1)*0.25 832 RHNPH=(XS(J)+XS(J+1))*0.5 833 RHNPQ=XS(J)w0.75+XS(J+1)*0.25 834 835 DT(1) =2.0*(ROPH1*RHN) 836 DT(2) =2.0*(RQPH1*RHNM1) 837 DT(3) =2.0x(ROPH1IRHNMH) 838 DT(4) =2.0*(RQPH1IRHNP1) 839 DT(5) =2.0*(RQPH1*RHNPH) 840 DT(6) =2.0*(RQMH1IRHNMH) 841 DT(7) =2.0*(RQMH1*RHNM1) 842 DT(8) =2.0*(RQMH1XRHN) 843 DT(9) =2.0*(RQMH1*RHNP1) 844 DT(10)=2.0*(RQMH1IRHNPH) 845 DT(11)=2.0*(RQ1*RHN) 846 DT(12)=2.0*(RQ1*RHNMH) 847 DT(13)=2.0*(RQ1*RHNMQ) 848 DT(14)=2.0'(RQ1*RHNPH) 849 DT(15)=2.0*(RQ1*RHNPQ) 850 851 ITIME=(XS(J)*15.0)/WAVE 852 ITIME=(ITIME*3)+1 853 IF (ITIME.LT.4) ITIME=4 854 H=PI/(ITIME-1) 855 DO 100 K = 1, 15 856 GA(K)=CMPLX(0.0,0.0) 857 GB(K)=CMPLX(0.0,0.0) 858 TMPDS=DS(J,K)**2 859 GAA(K)=CMPLX(0.0,0.0) 860 GBB(K)=CMPLX(0.0,0.0) 861 862 --------------------------------------------------- 863 864 X=O.0 865 DO 35 K1 = 1, ITIME 866 867 USING FOUR POINTS SIMPSON INTEGRATION 868 869 IFAC=3 870 ICK =(K1-1)/3 871 ICK =K1-(ICK*3+1) 872 IF (ICK.EQO.O) IFAC=2 873 IF (K1.EQ.1.OR.K1.EQ ITIME) IFAC=1 874

875 TMPX=COS(X) 876 TMPXX=ABS(TMPX-1.0) 877 — - - - - - - - - - 878 IF: (DS(J,K).LE.1.OE-5.AND.TMPXX.LE.1.OE-5) GO TO 20 879 TMPDV=SQRT(TMPDS-DT(K)*(TMPX-1.0)) 880 TMPCS=COS(FK*TMPDV) 881 TMPSN=-SIN(FK*TMPDV) 882 DSFK=CMPLX(TMPCS,TMPSN)/(TMPDV*FK) 8830 GA(K)=GA(K)+(OSFK*IFAC) 884 DSFKI=(CMPLX(TMPCS,TMPSN)-1.0)/(TMPDV*FK) 885 GB(K) =GB(K)~(DSFK1I*IFAC) 886 - - - - - - - - - - 887 20 IF- (K.LE.10) GO TO 30 888 IF (DS(J,K).LE.1.OE-5.AND.TMPXX.LE.1.OE-5) GO TO 30 889 DASC=OSFK*TMPX 890 GAA(K)=GAA (K )+( DASC I FAC) 891 DASC1=(DSFK*TMPX)-( 1.0/(TMPDV*'FK)) 892 GBB(K)=GBB(K)-.(DASCI *IFAC) 893 -- - - - - - - - - 894 30 X:=X+H 895 896 35 CONTINUE 897 898 GAA(K)=GAA(K)*H*~0.75 899 GE3B(K)=GBB(K)*H*0.75 900 GA(K) =GA(K)*H*~0.375 901 GB(K) =GB(K)*H*0.375 902 100 CONTINUE 903 904 90C'5 906 907 TNNM1=SQRT( (XS(u)-XS(J- ) )**2+(YS(J)-YS(J-1 ) )*2) 908 TNPIN=SQRT( (XS(J-+-1)-XS(J) )**2+(YS(u#1 )-YS(.J) )**2) 909 TNNMH=SQRT( (0.5*~(XS(J)-XS(J-i1) ))**2+(0.5*(YS(J)-YS(Lj1) l) )'**) 910 TNPHN=SQRT( (0.5*(XS(J)-XS(J+1 )) )**2+(0.5*~(YS(J)-~YS(u-+1)) )**2) 911 912 913 BETA(1) =2*SQRT(RQPHI*RHN) /DSS(U,1) 914 BETA(2) =2*SORT(RQPHI*RHNMI)/DSS(J,2) 915 BETA(3) =2*SQRT(R0PH1I-RHNMH)/DSS(J, 3) 916 BETA(4) =2*SQRT(RQPH1*RHNP1)/DSS(J,4) 917 BETA(S) =2*SQRT( RQPH1I*RHNPH)/DSS( J,5) 918 BETA(G) =2*SQRT( RQMHI *RHNMH)/DSS(~J,6) 919 BETA(7) =2*SQRT(RQMHI*RHNM1)/DSS(J,7) 920 BETA(8) =2*SQRT(RQMHI*RHN) /DSS(J,8) 921 BETA(9) =2*SQRT(RQMH1I-RHNPI)/DSS(J,9) 922 BETA(10)=2*'SQRT(RQMHI*RHNPH)/DSS(u,10) 923 BETA(11)=2*SQRT(RQI*RHN) /DSS(u,11) 924 BETA(12)=2*SQRT(RQ1*RHNMH) /DSS(J,12) 925 BETA(13)=2*SQRT(RQI*RHNMQ) /DSS(J,13) 926 BETA(14)=2*SQRT(RQI*RHNPH) /DSS(J,14) 927 BETA(15)=2*SORT(RQ1*RHNPQ) /DSS(J,15) 928 929 Check if log function can be performed 930 931 GG4(1)=0.0 932 I[F(DS(. 1).NE.0.0) GG4(1)=DS(J,1)*ALOG(FK*DS(u,1)) 933 GG4(2')'- 0. 934 IF(OSV,, 2).NE.0.0) GG4(2)=DS(U,2)*ALOG(FK*DS(J,2)) 935 GG4(4)- -.0 936 1F(DS(J,4).NE.0.0) GG4(4)=DS(.J,4)*ALOG(FK*DS(J,4)) 937 GG4(7)=0.0 938 IF(DS(J,7).NE.0.0) GG4(7)=DS(U.7)*ALOG(FK*DS(J,7)) 939 GG4(8)=0.0 940 IF(DS(J,8).NE.0.0) GG4(8)=DS(U,8)*ALOG(FK*DS(J,8)) 941 GG4(9)=0.0 942 I[F(DS(J,9).NE.00) GG4(9)=DS(UJ,9)*ALOG(FK*DS(J,9))

943 GG4(11)=0.0 944 IF(DS(J,11).NE.o.0) GG4(ll)=DS(J,11)*ALQG(FK*DS(J,11)) 945 GG4(12)=0.0 946 IF(DS(J,12).NE.O.0) GG4(12)=DS(J,12)*ALOG(FK*DS(J.12)) 947 GG4(14)=0.0 948 IF(DS(U,14).NE.0.0) GG4(14)=DS(J,14)*ALOG(FK*DS(J,14)) 949 950 951 GG3(1)=(TNNM-GG4( 1)-GG4(2)) 952 GG3(2)=(TNNMI-GG4(7)-GG4(8)) 953 GG3(3)=(TNP1N-GG4(1)-GG4(4)) 954 GG3(4)=(TNP1N-GG4(8)-GG4(9)) 955 GG3(5)=(TNNMH-GG4(11)-GG4(12)) 956 GG3(6)=(TNPHN-GG4(11)-GG4(14)) 957 GG3(7)=(TNNMH-GG4(11)-GG4(12)) 958 GG3(8)=(TNPHN-GG4(11)-GG4(14)) 959 960 OS must be positive, all 15 entries must have valid value 961 962 963 If an entry of OS array does meet the condition below, 964 then, it will be processed by using direct calculation. 965 Otherwise, use the approximation method. 966 ~ i.e. If the observation point is within the source 967 region --- use approximation " 968 969 FLAGI=0 970 FLAG8=0 971 FLAG11=0 972 973 Flags are down, which means that GAl, GA8, GA11 have 974 not been approximated yet. 975 976 977 978 --------- Calculate "Gi" 979 IF ((DS(J,1).GT.I.OE-5).AND.(DS(J,2).GT.I.OE-5) 980.AND.(DS(J,3Y.GT.1.0E-5)) GO TO 106 981 982 FLAGI=1 983 CALL APPRMX(J,1,RQ1,RQMHIRQPH)1 984 CALL APPRMX(J,2,RQ1,RQMHI,RQPHI) 985 CALL APPRMX(J,3,RQ1,RQMHI,RQPH1) 986 102 G1=(GAP(1)4GAP(2)+4.0*GAP(3))*(TNNM1*FK/6.0)+GG3(1)*2.0/RQPH1 987 GO TO 110 988 106 G1=(GA(1)4GA(2)+4.0*GA(3))*(TNNM1*FK/6.0) 989 990 991 992 993 --------- Calculate "G2" 994 110 IF ((DS(%J,8).GT.I.OE-5).AND.(DS(J,7).GT.I.OE-5) 995 *.AND.(DS(U,6).GT.1.OE-5)) GO TO 116 996 997 FLAG8=1 998 CALL APPRMX(U,8,RQ1,RQMH1,RQPHI) 999 CALL APPPMX(U,7,RQ1,RQMH1,RQPH1) 1000 CALL APc' X(J,6,RQ1,RQMH1,RQPH1) 1001 112 G2=(GAPk~,+GAP(7)+4.0*GAP(6))*(TNNM1*FK/6.0)+GG3(2)*2.0/RQMHI 1002 GO TO 1-7 1003 116 G2=(GA(8 )i.GA(7)+4.0*GA(G))*(TNNM1*FK/6.0) 1004 1005 1006 1007 1008 --------- Calculate "G3" 1009 120 IF ((DS(, 1).GT.I.OE-5).AND.(DS(J,4).GT.1.OE-5) 1010 *.AND.(DS(J,5).GT.1.OE-5)) GO TO 126

1011 1012 IF (FLAG1.EQ.O) 1013 *CALL APPRMX(J,1,RQ1,RQMH1,RQPH1) 1014 CALL APPRMX(d,4,RQ1,RQMH1,RQPH1) 1015 CALL APPRMX(J,5,RQ1,RQMH1,RQPH1) 1016 124 G3=(GAP(1)+GAP(4)+4.0*GAP(5))*(TNP1N*FK/6.0)+GG3(3)"2.0/ROPH1 1017 GO TO 130 1018 126 G3=(GA(1)+GA(4)+4.0*GA(5))*(TNP1N*FK/6.0) 1019 1020 1021 ********** **** ***************** * 1022 1023 - Calculate "G4" 1024 130 IF ((DS(J,8).GT.1.OE-5).AND.(DS(J,9).GT.1.0E-5) 1025 *.AND.(DS(J,10).GT.1.OE-5)) GO TO 136 1026 1027 IF (FLAG8.EQ.O) 1028 *CALL APPRMX(J,8,RQ1,RQMH1,RQPH1) 1029 CALL APPRMX(J,9,RQ1,RQMH1,RQPHI) 1030 CALL APPRMX(J,10,RQ1,ROMH1,RQPH1) 1031 134 G4=(GAP(8)+GAP(9)+4.0*GAP(10))*(TNP1N*FK/6.0)+GG3(4)*2.0/RQMH1 1032 GO TO 141 1033 136 G4=(GA(8)+GA(9)+4.0*GA(10))*(TNP1N*FK/6.0) 1034 1035 1036 ***tt* ** * * 1037 1038 - Calculate "G5", "G7" 1039 141 IF (DS(, 11).GT.1.OE-5.AND.DS(, 12).GT. 1.OE-5 1040.AND.DS(J,13).GT.1.OE-5) GO TO 146 1041 1042 1043 FLAG11=1 1044 CALL APPRMX(J,11,RQ1,RQMH1,RQPH1) 1045 CALL APPRMX(J,12,RQ1,RQMH1,RQPH1) 1046 CALL APPRMX(d,13,RQ1,ROMH1,RQPH1) 1047 142 G5=(GAP(11)+GAP(12)+4*GAP(13))*(TNNMH*FK/6.0)+GG3(5)*2.0/RQ1 1048 G7=(GAAP(11)+GAAP(12)+4*GAAP(13))*(TNNMH*FK/6.0)+GG3(7)*2.0/RQ1 1049 GO TO 151 1050 146 G5=(GA(11)+GA(12)+4*GA(13))*(TNNMH*FK/6.0) 1051 G7=(GAA(11)+GAA(12)+4*GAA(13))*(TNNMH*FK/6.0) 1052 1053 1054 *************** ******** *** **** ********** 1055 1056 --------- Calculate "G6", "G8" 1057 151 IF (DS(J,11).GT.1.OE-5.AND.DS(J,14).GT.1.OE-5 1058 *.AND.DS(J,15).GT.1.OE-5) GO TO 156 1059 1060 IF (FLAG11.EQ.O) 1061 *CALL APPRMX(J,11,RQ1,ROMH1,RQPH1) 1062 CALL APPRMX(J,14,RQ1,RQMH1,RQPH1) 1063 CALL APPRMX(J,15,RQ1,RQMH1,RQPH1) 1064 154 G6=(GAP(11)+GAP(14)+4*GAP(15))*(TNPHN*FK/6.0)+GG3(6)*2.0/RQ1 1065 G8=(GAAP(11)+GAAP(14)+4*GAAP(15))*(TNPHN*FK/6.0)+GG3(8)*2.0/R01 1066 GO TO 161 1067 156 G6=(GA(11)+GA(14)+4*GA(15))*(TNPHN*FK/6.0) 1068 G8=(GAA(11)+GAA(14)+4*GAA(15))*(TNPHN*FK/6.0) 1069 1070 1071 1072 1073 *********************************************** 1074 1075 161 RETURN 1076 END 1077 SUBROUTINE MULTPY 1078

1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 THIS SUBROUTINE SOLVES THE EQUATION FOR THE MATRICES [Z][J]=[E] REAL COMMON /SOURCE/ REAL COMMON /OBSERV/ REAL COMMON /DISTNS/ INTEGER REAL COMMON /VARIAB/ COMPLEX*8 COMMON /FOUIIS/ COMPLEX*8 COMMON /FOUIES/ INTEGER REAL COMMON /VARIAC/ COMPLEX*8 COMMON /VARIAD/ COMPLEX*8 COMMON /CNSTAN/ COMPLEX*8 COMMON /INPUT/ COMPLEX*8 COMMON /OUTPUT/ INTEGER COMMON /MAXIMN/ INTEGER COMMON /ARRCTN/ REAL COMMON /CONSTN/ INTEGER COMMON /IOUNIT/ XS,YS,DIS,DSQ XS(100),YS(100),DIS(100),DSQ(100) XB,YB XB(100),YB(100) DS,DSS,THETA1,THETA2,INC DS(100,15),DSS(100,15),THETA1,THETA2,INC DOTNUM,CURTYP XS1,YS1,XS2,YS2,XB1,YB1,XB2,YB2 XS1,YS1,XS2,YS2,XB1,YB1,XB2,YB2,DOTNUM,CURTYP GAA,GA,GAAP,GAP GAA(15),GA(15),GAAP(15),GAP(15) G1,G2,G3,G4,G5,G6,G7,G8,GB,GBB G1,G2,G3,G4,G5,G6,G7,G8,GB(15),GBB(15) FOUIIN FK,FRQNCY,MEW,EPSILN,WAVE,DTR,BETA FOUIIN,FK,FRQNCY,MEW,EPSILN,WAVE,DTR,BETA(15) VOLTGE,IMPEDC,CURENT VOLTGE(100),IMPEDC(100),CURENT(100) IMAGI IMAGI VOLT,IMP VOLT,IMP Z 2(99,99) SOUMAX, OBSMAX SOUMAX, OBSMAX SOUCTN, OBSCTN,RECCTN,PTR,IM,FARIDX,LASTSG SOUCTN, OBSCTN,RECCTN,PTR,IM,FARIDX,LASTSG PI,RADIAN,ZO,XR1,YRI,XR2,YR2 PI,RADIAN,ZO,XR1,YR1,XR2,YR2 INPUTF, MESSGE, REPORT, TERMIN INPUTF, MESSGE, REPORT, TERMIN -------------------------------------------------------------------- END OF COMMON -------------------------------------------------------------------- COMPLEX*8 TEMPRA,TEMPRB,P INTEGER JDX,UP1,JP2,I1 N=SOUCTN-1 IF (N.GT.1) GO TO 10 CURENT(1)=VOLTGE(1)/Z(1,1) GO TO 110 10 NM1=N-1 20 25 30 DO 80 I=1,NM1 IP1=I+1 IF (CABS(Z(I,I)).NE.O.O) GO TO 50 DO 20 J=IP1,N JDX=J IF (CABS(Z(J,I)).NE.O.O) GO TO 30 CONTINUE WRITE (MESSGE,25) I FORMAT(1H,'Z MATRIX AT ',I3,' ROW HAS ALL ZEROS') GO TO 110 CONTINUE DO 40 K=1,N TEMPRA =Z(JDX,K) Z(JDX,K)=Z(I,K)

1147 40 Z(I,K) =TEMPRA 1148 TEMPRB =VOLTGE(JDX) 1149 VOLTGE(JDX)=VOLTGE(I) 1150 VOLTGE(I)=TEMPRB 1151 1152 50 CONTINUE 1153 DO 70 JP1=IP1,N 1154 P=Z(JP1,I)/Z(I,I) 1155 DO 60 K=IP1,N 1156 Z(JP1,K)=Z(JP1,K)-P*Z(I,K) 1157 60 CONTINUE 1158 VOLTGE(JP1)=VOLTGE(JP1)-P*VOLTGE(I)' 1159 70 CONTINUE 1160 80 CONTINUE 1161 1162 CURENT(N)=VOLTGE(N)/Z(N,N) 1163 I1=NM1+1 1164 85 I1=I1-1 1165 IF (I1.LT.1) GO TO 110 1166 IP1=I1+1 1167 DO 90 JP2=IP1,N 1168 VOLTGE(I1)=VOLTGE(I1)-Z(I1,UP2)*CURENT(JP2) 1169 90 CONTINUE 1170 CURENT(I1)=VOLTGE(I1)/Z(I1,I1) 1171 GO TO 85 1172 1173 110 RETURN 1174 END 1175 SUBROUTINE ELTK(PK,ELTKP) 1176 1177 1178 1179 THIS SUBROUTINE COMPUTES THE ELLIPTICAL FUNCTION 1180 OF THE FIRST KIND K(M) 1181 WHERE M1=I-M 1182 K(M)-AO+A1*M1+A2*M1**2+A3*M1**3+A4*M1**4 -1183 (BOB*Ml+B2*M1**2+B3*M1*3+B4*M1**4)*ALOG(M11) 1184 FOR MAGNITUDE(ERROR).LE. 2.0E-8 1185 1186 **t*^**********c***** **** 1187 1188 REAL ELTKP,PK 1189 DATA AO,A1,A2,A3,A4,BO,B1,B2,B3,B4/ 1190 $ 1.38629436112,.09666344259,.03590092383,.03742563713, 1191 $.01451196212,.5,.12498593597,.06880248576, 1192 $.03328355346,.00441787012/ 1193 1194 ------------ 1195 1196 A=AO+A1*PK 1197 B=BO+B1*PK 1198 IF (PK.LT.1.E-18) GO TO 10 1199 A=A+A2*(PK**2) 1200 B=B+B2*(PK**2) 1201 IF (PK.LT.1.E-12) GO TO 10 1202 A=A+A3*(PK**3) 1203 B=B+B3*(PK**3) 1204 IF (PK.LT.1.E-9) GO TO 10 1205 A=A+A4*(PK**4) 1206 B=B+B4*(PK**4) 1207 10 CONTINUE 1208 ELTKP=A-B*ALOG(PK) 1209 1210 ---------------- 1211 1212 RETURN 1213 END 1214 SUBROUTINE APPRMX(J,;,:RQ1,RQMH1,RQPH1)

1215 1216 1217 1218 1219 1220 1221 1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 Calculating the GAP value using the approximation. Calling routine --- GREENS REAL COMMON /SOURCE/ REAL COMMON /OBSERV/ REAL COMMON /DISTNS/ INTEGER REAL COMMON /VARIAB/ COMPLEX*8 COMMON /FOUIIS/ COMPLEX*8 COMMON /FOUIES/ INTEGER REAL COMMON /VARIAC/ COMPLEX*8 COMMON /VARIAD/ COMPLEX*8 COMMON /CNSTAN/ COMPLEX*8 COMMON /INPUT/ COMPLEX*8 COMMON /OUTPUT/ INTEGER COMMON /MAXIMN/ INTEGER COMMON /ARRCTN/ REAL COMMON /CONSTN/ INTEGER COMMON /IOUNIT/ XS,YS,DIS,DSQ XS(100),YS(100),DIS(100),DSQ(100) XB,YB XB(100),YB(100) DS,DSS,THETA1,THETA2,INC DS(100,15),DSS(100, 15),THETA1,THETA2,INC DOTNUM,CURTYP XS1,YS1,XS2,YS2,XB1,YB1,XB2,YB2 XS1,YS1,XS2,YS2,XB1,YB1,XB2,YB2,DOTNUM,CURTYP GAA,GA,GAAP,GAP GAA(15),GA(15),GAAP(15),GAP(15) G1,G2,G3,G4,G5,G6,G7,G8,GB,GBB G1,G2,G3,G4,G5,G6,G7,G8,GB(15),GBB(15) FOUIIN FK,FRONCY,MEW.EPSILN,WAVE,DTR,BETA FOUIIN,FK,FRQNCY,MEW,EPSILN,WAVE,DTR,BETA(15) VOLTGE,IMPEDC,CURENT VOLTGE(100),IMPEDC(100),CURENT(100) IMAGI IMAGI VOLT,IMP VOLT,IMP z Z(99,99) SOUMAX, OBSMAX SOUMAX, OBSMAX SOUCTN, OBSCTN,RECCTN,PTR,IM,FARIDX,LASTSG SOUCTN, OBSCTN,RECCTN,PTR,IM,FARIDX,LASTSG PI,RADIAN,ZO,XR1,YR1,XR2,YR2 PI,RADIAN,ZO,XR1,YR1,XR2,YR2 INPUTF, MESSGE, REPORT, TERMIN INPUTF, MESSGE, REPORT, TERMIN END OF COMMON -------------------------------------------------------------------- INTEGER REAL REAL GG1=0.0 GG2=0.0 J,KK RQ1,RQMH1,RQPH1 GG1,GG2,PK PK=1-BETA(KK) IF (KK.GT.5) GO TO 20 IF(DS(J,KK).GT.1.OE-5.AND.PK.GT.O) GO TO 15 GG1=ALOG(4.0)*0.5/(RQPH1*FK) GG2=ALOG(FK*DSS(J,KK))*0.5/(ROPHI*FK) GO TO 40 15 CALL ELTK(PK,ELTKP) GG1=ELTKP/(FK*DSS(J,KK)) GG2=ALOG(FK*DS(J,KK ) *0. 5/(RQPH1*FK) GO TO 40

1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326 1327 1328 1329 1330 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344 1345 1346 1347 1348 1349 1350 20 IF (KK.GT.10) GO TO 30 IF(DS(J,KK).GT.1.OE-5.AND.PK.GT.O) GO TO 25 GGI=ALOG(4.0)*0.5/(ROMH1*FK) GG2=ALOG(FK*DSS(J,KK))*0.5/(ROMH1*FK) GO TO 40 25 CALL ELTK(PK,ELTKP) GGI=ELTKP/(FK*DSS(J,KK)) GG2=ALOG(FK*DS(J,KK))*0.5/(RQMH1*FK) GO TO 40 30 IF(DS(U,KK).GT.1.OE-5.AND.PK.GT.O) GO TO 33 GG1=ALOG(4.0)*0.5/(ROQ1FK) GG2=ALOG(FK*DSS(J,KK))*0.5/(RQ1*FK) GO TO 35 33 CALL ELTK(PK,ELTKP) GG1=ELTKP/(FK-DSS(d,KK)) GG2=ALOG(FK*DS(J,KK))*0.5/(R1Q*FK) 35 GAAP(KK)=((GG1+GG2)*4.0+GBB(KK)) 40 GAP(KK)=((GG1+GG2)*4.0+GB(KK)) RETURN END SUBROUTINE FARFLD This routine is used to calculate the far field pattern. REAL COMMON /SOURCE/ REAL COMMON /OBSERV/ REAL COMMON /DISTNS/ INTEGER REAL COMMON /VARIAB/ COMPLEX*8 COMMON /FOUIIS/ COMPLEX*8 COMMON /FOUIES/ INTEGER REAL COMMON /VARIAC/ COMPLEX*8 COMMON /VARIAD/ COMPLEX*8 COMMON '`STAN/ COMPLEX' COMMON '.PUT/ COMPLEX COMMON /OUTPUT/ INTEGER COMMON /MAXIMN/ INTEGER COMMON /ARRCTN/ REAL COMMON /CONSTN/ XS,YS,DIS,DSQ XS(100),YS(100),DIS(100),DSQ(100) XB,YB XB(100),YB(100) DS,DSS,THETA1,THETA2,INC DS(100, 15),DSS(100,15),THETA1,THETA2,INC DOTNUM,CURTYP XS1,YS1,XS2,YS2,XB1,YB1,XB2,YB2 XS1,YS1,XS2,YS2,XB1,YB1,XB2,YB2,DOTNUM,CURTYP GAA,GA,GAAP,GAP GAA(15),GA(15),GAAP(15),GAP(15) G1,G2,G3,G4,G5,G6,G7,G8,GB,GBB G1,G2,G3,G4,G5,G6,G7,G8,GB(15),GBB(15) FOUIIN FK,FRONCY,MEW,EPSILN,WAVE,DTR,BETA FOUIIN,FK,FRQNCY,MEW,EPSILN,WAVE,DTR,BETA(15) VOLTGE,IMPEDC,CURENT VOLTGE(100),IMPEDC(100),CURENT(100) IMAGI IMAGI VOLT,IMP VOLT,IMP z Z(99,99) SOUMAX, OBSMAX SOUMAX, OBSMAX SOUCTN, OBSCTN,RECCTN,PTR,IM,FARIDX,LASTSG SOUCTN, OBSCTN,RECCTN,PTR,IM,FARIDX,,LASTSG PI,RADIAN,ZO,XR1,YR1,XR2,YR2 PI,RADIAN,ZO,XR1,YR1,XR2,YR2

1351 INTEGER INPUTF, MESSGE, REPORT, TERMIN 1352 COMMON /IOUNIT/ INPUTF, MESSGE, REPORT, TERMIN 1353 1354 -------------------------------------------------------------------- 1355 END OF COMMON 1356 -------------------------------------------------------------------- 1357 1358 COMPLEX*8 BZ1,BZO,FUNCTN 1359 COMPLEX*8 EOTNA,EQTNB,EQTNC,EQTND 1360 COMPLEX*8 FELD,EQ1 1361 REAL RFB,KPS,X,A,B,H 1362 REAL THETA,XSMID,YSMIDDXYS1,DXYS2 1363 REAL LAMF,LAMS1,LAMS2 1364 REAL SIN1,COS1,REALP1,REALP2 1365 1366 ------------ 1367 1368 IF (THETA1.LE.O.O.AND.THETA2.LE.O.O) GO TO 990 1369 WRITE(MESSGE,1) 1370 1 FORMAT(1H,/,1H,/) 1371 WRITE(MESSGE,5) 1372 5 FORMAT(1H,15X,'Far Field',/,1H,3X,'Theta',8X,'(Mag)',9X,'Phase') 1373 THETA=THETA1 1374 RFB = 10.0 * WAVE 1375 20 BZO=CMPLX(O.O,O.O) 1376 BZ1=CMPLX(O.O,O.O) 1377 FELD=CMPLX(O.O,O.O) 1378 1379 DO 100 J = 2, SOUCTN 1380 1381 XSMID=XS(J)-XS(J-1) 1382 YSMID=YS(J)-YS(J-1) 1383 DXYS1=SQRT(XSMID**2+YSMID**2) 1384 LAMF =DTR*THETA 1385 IF (YS(J).EQ.YS(J-1)) GO TO 30 1386 LAMS1=ATAN(XSMID/YSMID) 1387 GO TO 35 1388 1389 30 LAMS1=90*DTR 1390 35 XSMID=XS(J+I)-XS(J) 1391 YSMID=YS(J+1)-YS(J) 1392 DXYS2=SQRT(XSMID**2+YSMID**2) 1393 IF (YS(J+1).EQ.YS(J)) GO TO 40 1394 LAMS2=ATAN(XSMID/YSMID) 1395 GO TO 45 1396 1397 40 LAMS2=90*DTR 1398 45 SIN1 =SIN(LAMF) 1399 COS1 =COS(LAMF) 1400 1401 1402 KPS=FK*XS(J)*SIN(LAMF) 1403 A=0.0 1404 B=PI*2.0 1405 H=PI*2.0/30.0 1406 X=A 1407 IF (LAMS1.EO.(90.0*DTR)) GO TO 55 1408 BZO=((CI,-' -(COS(KPS*COS(A)),SIN(KPS*COS(A))))+ 1409 * (Cr,. -_(COS(KPS*COS(B)),SIN(KPS*COS(B)))))/2.0 1410 DO 50 K 1, 29 1411 X=X+H 1412 BZO=BZO-iCMPLX(COS(KPS*COS(X)),SIN(KPS*COS(X)))) 1413 50 CONTINUE 1414 BZO=BZO*H 1415 GO TO 70 1416 1417 1418 55 BZ1=(CMPLX(COS(KPS*COS(A)+A),SIN(KPS*COS(A)+A))+

1419 * CMPLX(COS(KPS*COS(B)+B),SIN(KPS-COS(B)+B)))/2.0 1420 DO 60 K = 1, 29 1421 X=X+H 1422 FUNCTN=CMPLX(COS(KPS*COS(X)+X),SIN(KPS*COS(X)+X)) 1423 BZ1=BZ1+FUNCTN 1424 60 CONTINUE 1425 BZ1=BZ1*H*IMAGI 1426 1427 --- —-------------------------------------------- 1428 1429 70 EQTNA=COSISIN(LAMS1)*DXYS1*BZI*IMAGI 1430 EQTNB=COS 1SIN(LAMS2)*DXYS2*BZ1 IMAGI 1431 EQTNC=SIN1*COS(LAMS1)*DXYS1*BZO(CMPLX(COS(FK*YS(U))*COS(LAMF)), 1432 * SIN(FK*YS(J)*COS(LAMF)))) 1433 EQTND=SIN1CCOS(LAMS2)-DXYS2-BZO(CMPLX(COS(FK*YS(J)*COS(LAMF)), 1434 * SIN(FK*YS(J)*COS(LAMF)))) 1435 1436 90 EQ1=(FK*ZO*0.5*IMAGI)*(EQTNA+EQTNB-EQTNC-EQTND)/ 1437 * RFB 1438 FELD=(EQ1*CURENT(J-1))/(PI*PI*4.0)+FELD 1439 100 CONTINUE 1440 1441 AMP=CABS(FELD) 1442 REALP1=AIMAG(FELD) 1443 REALP2= REAL(FELD) 1444 IF (REALP2.NE.O.O) GO TO 120 1445 PHASE=O.O 1446 GO TO 140 1447 120 PHASE=RADIAN*ATAN2(REALP1,REALP2) 1448 140 WRITE(MESSGE,110) THETA,AMP,PHASE 1449 110 FORMAT(1H,2X,F7.2,3X,E12.4,5X,F7.2) 1450 1451 THETA=THETA+INC 1452 IF(THETA.LE.THETA2) GO TO 20 1453 990 RETURN 1454 END

137 Appendix B. Singularity Analysis of Self Terms for the Geometry of Revolution When an observation point falls within the source segment, the integrals described in Eq.(3.28) may have singular integrands. A brief procedure of evaluating the integrals is shown here. A more detailed analysis can be found in reference [30]. Throughout this section, we employ the coordinate parameter valid for tj- < t < tj z = z j-l + t cosyj (B.la) I = r j_- + t sinyj (B.1b) 0 < - < At. Where L = t - tj_1 For self terms, we can apply equations (z - z') = (i - V') cosyj (B.2a) (t - 'I) = (L - ~') sinyj (B.2b)

138 For the self term, the M integral may be written as 2 T e-jkR M = J - cos(ma) da dt' t -r R and R = [(t - ' )2+ 2rt' (1 - cosa) + (z - z')2]2 (B.3a) (B.3b) As t -> t' and a -> 0, R -> 0, the integrand of (B.3a) is cleary singular. Then we can define M = I1 + I2 (B.4) where X-2 T I1= f~ IL-7T e-jkR 1 [ cos(ma) - - ] da dt' R R (B.5a) (B.5b) 1 2 2 L 1 = S J - da dl' I 1 -T R Since the integrand I1 is no longer singular, 12 may be evaluated numer -ally with a single change of variable. t2 1 I 2= 4 — R K(u) dl' 1 (B.6)

139 where R = [( + ' )2+ (z - z')2 ] (B.7) 2 and K(u) is the complete elliptical integral of the first kind defined by 1 K(u) = J - 2 2 d (B.8) 0 [1 - u sin] with 2 (tL a') u = --- (B.9) R2 The integrand of Eq.(B.6) is still singular. However, near the singularity at t = t', it varies as 1 1 K(u) -- > [(ln(4)+ln(R2)-ln(R1)] R2 tat' 2o (B.10) where R1 = [() ( - )2 (z - z)2 ](B.11) At this point, Jnly the last term is singular, we can add and subtract the singular term in Eq.(B.6) to obtain I2 = I + I" (B.12)

140 where 2 1 1 I' = 4 [- K(u) + ln(R1)] dt' (B.13) 1 R2 21 2 2 I =- -- ln(R1) d' (B.14) r 1 Now, the integral I' does not have a singular integrand, so the integral I2 can be evaluated analytically by the parameterization of Eq.(B.1) as follows 22 2 I2 = - — f in I t- ' I dt' 1i 2 = - [(t2- t1) - (12- l) ln(12- t) - - ) ln(t - )] (B.15) The integrals I1, I' and I" can thus be integrated numerically, and the M integral can be evaluated by M = I + I( + I1 1 2 (B. 16)

REFERENCES 141

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