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UCRL-NIA-109338 Pt. II

Numerical Electromagnetic Code – NEC–4

Method of Moments .

Part II: Program Description – Theory.

Gerald J. Burke

January 1992

A

DISCLAIMER

This document was prepared as an account of work sponsored by an agency of theUnited Sta~es Government. Neither the United States Government nor the University ofCalifornia nor any of their employees. makes any warranty, express or implied, orassumes any legal liability or responsibility for the accuracy, completeness, orusefulness of any information, apparatus, product, or process disclosed, or representsthat its use would not infringe privately owned rights. Reference herein to any specificcommercial product, process, or service by trade name, trademark, manufacturer, orotherwise, does not necessarily constitute or imply its endorsement, recommendation,or favoring by the United States Government or the University of California. The viewsand opinions of authors expressed herein do not necessarily state or reflect those of theUnited States Government or the University of California, and shall not be used foradvertising or product endorsement purposes.

This reporthas been reproduceddirectly from the best available copy.

Available to DOE and DOE contractors from theOffice of Scientific and Techtdcal Information

P.O. Box 62, Oak Ridge, TN 37831Prices available from (615) 576-8401, FIX 626-8401

Available to the public from theNational Technical [nfonnation Service

U.S. Department of Commerce5285 port Royd Rd.,

Springfield, VA 22161

Work performed undertheauspicesof theU.S. Departmentof Energyby LawrenceLRermoreNational Laboratory under Contract W-7405-ENG-48.

I‘L———

DISCLAIMER

This document was prepared as an account of work sponsored by an agency of theUnited States Government. Neither the United States Government nor the University ofCalifornia nor any of their employees, makes any warranty, express or implied, orassumes any legal liability or responsibility for the accuracy, completeness, orusefulness of any information, apparatus, product, or process disclosed, or representsthat its use would not infringe privately owned rights. Reference herein to any specificcommercial product, process, or service by trade name. trademark, manufacturer, orotherwise, does not necessarily constitute or imply its endorsement, recommendation,or favoring by the United States Government or the University of California. The viewsand opinions of authors expressed herein do not necessarily state or reflect those of theUnited States Government or the University of California, and shall not be used foradvertising or product endorsement purposes.

This report has been reproduceddirectly from the best available copy.

Available to DOE and DOE contractors from theOffice of Scientific and Technical Information

P.O. Box 62. Oak Ridge, TN 37831Prices available from (615) 576-8401, FTS 626-8401

Available to the public from theNational Technical Information Service

U.S. Department of Commerce5285 portRoydRd.,

Springfield, VA 22161

Work performed under the auspices of the U.S. Department of Energy by LawrenceLivermore National Laboratory under Contract W-7405-ENG-48.

Preface

NEC-4 is the latest version of The Numerical Electromagnetic Code — Method of Mo-ments that has been developed at the Lawrence Liverrnore National Laboratory, Liverrnore,California, under the sponsorship of the U.S. Army, ISEC and CECOM, the Naval OceanSystems Center snd the Air Force Weapons Laboratory. The development of the versionNEC-4 was sponsored by the U.S. Army ISECat Ft. Huachuca, AZ. The NEC Method ofMoments code started ss an advanced version of the Antenna Modeling Program (AMP)developed in the early 1970’s by MBAssociates for the Naval Research Laboratory, NavalShip Engineering Center, U.S. Army ECOM/Communications Systems, U.S. Army Strate-gic Communications Command and Rome Alr Development Center, under Office of NavalResearch Contract NOO014-71-C-0187.

The documentation for NEC-4 consists of three volumes:

Part I: NEC Program Description — TheoryPart II: NEC Program Description — CodePart III: NEC User’s Guide

The documentation for successive versions of NEC has been prepared by updating manualsfor previous versions of the code, starting from AMP. In some cases this led to minor changesin the original documents, while in many cases major modifications were required.

Over the years many individuals have been contributors to AMP and NEC, and areacknowledged here as follows:

R. W. Adams K. K. Hazard E. K. MillerM. J. Barth W. A. Johnson J. B. MortonH. Bosserman D. L. Knepp G. M. PjerrouJ. K. Breakall D. L. Lager A. J. PoggioJ. N. Bnttingham A. P. Ludwigsen E. S. SeldenG. J. Burke R. J. Lytle R. W. ZiolkowskiF. J. Deadrick

Special mention is due to E. K. Miller who has provided the main driving force behindthe development of the code and modeling techniques since the development of the codeBRACT which preceded AMP. A. J. Poggio has also made major contributions to the codedevelopment and documentation throughout the development of NEC. Dr. W. A. Johnsonprovided valuable assistance in validating the model for wires penetrating the ground andthe basic Sommerfeld integral evaluation, and Dr. R. W. Ziolkowski provided assistance andguidance in developing the asymptotic approximations for the field near ground.

i

The development of NEC has also benefitted greatly from suggestions and guidance fromindividuals at the sponsoring agencies. Particular mention is due to J. Logan, J. W. Rockwayand S. T. Li at the Naval Ocean Systems Center and J. McDonald and R. Franklin at ISEC,Ft. Huachuca AZ.

ii

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . ...”.””.”””””””.””””””””.””” ““o-o”.osoo”””o”s.o-””-l

2. The Electric and Magnetic Field Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...4

2.1 The Electri cFieldIntegra lEquation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...4

2.2 The Magnetic Field Integral Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...6

2.3 The EFIEMFIE Hybrid ~uation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...7

3. The Moment-Method Solution ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...8

3.1

3.2

3.3

3.4

Outline of the Moment Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...8

Current Expansion on W]res . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Cument~pusion on Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . ..o . . . . . . . . . . . . . . . . . . ..l5

The Condition on Chmgeat a Junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...16

4. Evaluation of the Field in Free Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...21

4.1 The Electric Field due to Wire Segments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...21

4.2 The Magnetic Field due to Wire Segments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...26

4.3 The Electric and Magnetic Fields due to Surface Cutients . . . . . . . . . . . . . . . . . . ...27

4.4 Amodelfor End Caps on Wires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...28

5. Evaluation of the Field Near Ground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32

5.1 TheSommerfeld/AsymptoticSolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...32

5.1.1 The Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...33

5,1.2 Numerical Evaluation of the Sommerfeld Integrals . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.1.3 Table-Lookup and Interpolation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...39

5.2 Asymptotic Approximations for the Field over Ground . . . . . . . . . . . . . . . . . . . . . . ...45

5.2.1 ~eStepest.Dmmnt Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...46

5.2.2 Steepest-Descent Evaluation of the Somrnerfeld Integrals . . . . . . . . . . . . . . . . . . . .47

5.2.3 First-Order Asymptotic Evaluation of the Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . .51

5.2.4 Higher-Order and Uniform Approximations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52

5.2.5 Accurwy of the Approximations “o. ..”.”..................”””. ‘ . . ..- . . ..<.055

5.3 The Reflection Coefficient Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...56

6. Solution of the Mattix E-quation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...70

6.1 Solution of the Matrix Equation by L-U Factorization . . . . . . . . . . . . . . . . . . . . . . . ...70

6.2 Solution for Symmetric Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...71

6.3 The Numetid Green' s~nction Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

...111

7. Extensions tothe Model for Antennas and Scatterers . . . . . . . . . . . . . . . . . . . . . . . . . . . ...76

7.1 Voltage Source Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

7.1.1 The Applied-Field Source Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

7.1.2 The Bicone Source Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

7.1.3 A Coaxial-Line Source Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .80

7.1.4 Comparison of Source Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...81

7.2 Lumped or Distributed Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

7.3 Nonradiating Networks and Thmxnission Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

7.4 Radiated Field, Directivity and Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

7.4.1 The Radiated Field in F& Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

7.4.2 Effect of a Ground Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

7.4.3 Elliptic Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

7.4.4 Gain and Directivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

7.5 Msximurn Coupling Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...93

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

APPENDICES

A. Evaluation of the Scalar Potential for Triangular and Constant Charge Ilmctions .100

B. Approximations for the Field of a Segment to Avoid Loss of Precision . . . . . . . . . . ...103

B.1 Approximation for SmaUpandlzl>6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...103

B.2 Approximation for Small@l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..lO4

B.3 Approximation for p >6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

B.4 Choice of the Method for Numerical Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

C. Evaluation of ~ eaR/Rd# . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

D. Comparison of Approximations for the Thin-WIre Kernel . . . . . . . . . . . . . . . . . . . . . . . . . 110

D. 1 Approximation as a Current Filament on the Wire Axis . . . . . . . . . . . . . . . . . . . . . . 110

D.2 Approximation as a Current Filament on the Wke Surface . . . . . . . . . . . . . . . . . ...113

D.3 Comparison of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

E.

F.

G.

Evaluation of the Field due to a Constant Charge on a Wire End Cap . . . . . . . . . ...118

Sommerfeld Integrals for the Field Near Ground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Conversion Formulas for Source and Observer Locations . . . . . . . . . . . . . . . . . . . . . . . . . . 123

iv

.,,...— ..--——-----... . .... .. ..- . .=,.—-—----.,

H. Singularity Subtraction inthe Sommerfeld Inte@ds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..l25

I. Numerid Solution forthe Saddle Points inthe Sommerfeld IntWds . . . . . . . . . . . . ..l3O

J. Second-Order Asymptotic Approximations for the Field Due to Ground . . . . . . . . ...132

J.1 Second-Order Approximation for the Saddle Point Contribution . . . . . . . . . . . . ...132

J.2 Approximations for Near Vertical Incidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...139

v

1. Introduction

The Numerical Electmrnagnetics Code (ATEC) – Method of Moments is a computerprogram for analyzing the electromagnetic response of antennas and scatterem. The oodeis based on the numerical solution of integral equations by the method of moments, andcombines an electric-field integral equation for modeling thin wires with a magnetic-fieldintegral equation for closed perfectly conducting surfaces. This manual specifically describesthe NEC-4 program which is the latest in a series of NEC-MOM programs, and includes anumber of changes to improve modeling accuracy. Other versions of NEC currently in useinclude NEC-2, which can model wire structures in fiwe space or over finitely conductingground, and NEC–3 which is the same as NEC-2 but can also model wires buried in theground or penetrating from air into ground.

These codes offer a number of features for modeling antennas or scatterers and theirenvironments, including excitation by voltage sources or plane waves, lumped or distributedloading, and networks and transmission lines. The code output includes current distributions,impedances, power input, dissipation and efficiency, and radiation patterns and gains orscattering cross section. The model description and instructions for running the code aregenerally entered as commands read from an input file, although independent pr~ and post-processors have been developed to streamline this process [1].

The modeling algorithms in NEC-4 have been revised to avoid loss of precision whenmodeling electrically small structures. The single-precision (32-bit) code will now give ac-curate results for many electrically small models that would have required double precisionwith NEC-3 or earlier codes. Also, a new treatment is used for wire radius and junctionsfor accurate modeling of stepped-radius wires and junctions of tightly coupled wires. Someother minor features have been added for convenience in modeling, and the insulated-wiremodel, that was available in a special version of NEC–3, is now included in the standardNEC-4 code. Also, the code structure has been extensively revised, with more use of Fortnm77 constructs, to make it more modular and easier to understand and maintain.

NEC-4 and earlier versions of NEC have been used to model a wide variety of anten-nas and scatterers including antennas in complex environments such as ships and aircraft.Homver, since the computer time snd resources required by the moment-method solution in-crease with increasing size of the model relative to the wavelength, the solution may becomedifficult or impractical for large stmctures at high frequencies. The approach used in NECis most applicable to the types of problems encountered at HF, VHF and lower frequencies.Problems such as a UHF antenna on a surface, or even a VHF antenna on a large aircraftmay be handled better with other techniques such as the GTD approach used in the NECBasic Scattering Code [2]. Also, finite difference codes [3] may be better suited to problemsinvolving volumes of many cubic wavelengths or for interior coupling problems.

This manual describes in detail the mathematical and numerical tedmiques used inthe NEC-4 program for modeling antennas and scatterem. A bfief mtiew of the inte~~equations and the method of moments technique for solving them is included in sections 2and 3, but these topics are not considered in general terms. The reader is referred to [4] for

1

... ,- m -. .,,-.— -—>... !!- ... . .. .. -- .“.., w=-—=”K.— -- -.W4-,.-,.*WC . . .. —. V---- .-W —..—— -..— a“.. . . . . . m. -

,.. }

details on the derivation of the integral equations and to [5] for a comprehensive discussionof the method of moments. There are also a number of other sources that cover these topics.Persons wsnting to learn to use NEC should start with the NEC-4 User’s Guide, Part IIIof the NEC documentation. Part II, the Code Manual, contains detailed descriptions of thecodbg of subroutines in NEC-4 and information on the overall code structure, and will beof interest to persons wanting to modify or extend the code.

The code NEC-4 is the latest in a series of antenna modeling programs, each of whkhhss built on the previous one. The first in the series was the code BRACT which wasdeveloped at MBAssociates in San Rarnon, California, with tiding from the Air ForceSpace and Missiles Systems Organization [6, 7]. The development of BRACT was based onthe early work of J. Richmond [8, 9] in modeling wires by the solution of a Pocklington’sintegral equation. The three-term sinusoidal current expansion then being employed by Yehand Mei [10] in solving Hall?m’s type integral equations was used in BRACT along withpoint matching of the boundary condition. BRACT was capable of modeling wire antennasand scatterers and could include the effect of interaction with a finitely conducting groundthrough the Fresnel reflection-coefficient approximation [11]. However, BRACT was usedmainly by its developers.

AMP [12, 13, 14] was the first code released as a tool for use by the general electromag-netic community. It was developed under Office of Naval Research Contract NOOO14-71-C-0187 with funding from the Naval Research Laboratory, Naval Ship Engineering Center,U.S. Army ECOM/Communications Systems, U.S. Army Strategic Communications Com-mand and the U.S. Air Force Rome Air Development Center. Like BRACT, AMP solvedPocklington’s integral equation for thin wires with a three-term sinusoidal current expansionand point matching. The current expansion was chosen so that the current on a segment,with the form Ai + Bj sin ks + Ci cm ks, when extrapolated to the centers of the adjacent seg-ments would coincide with the values of current on those segments. At a junction of severalwires the current was extrapolated to the center of a “phantom segment” whose length wasthe average of the connected segments. This extrapolation procedure smoothed the currentdistribution along wires, but still left dlscontinuities in current and charge density.

AMP included options to model lumped or distributed loading on wires, transmissionlines and networks, snd was capable of using disk storage for solving problems too large forthe primary computer memory. AMP also had an option so a partially completed solutioncould be stopped and saved on a file, and latter restarted. This restart capability hasbeen dropped from NEC, but is a valuable feature for running large problems. A flexibleinput language was developed in AMP for defining the model, selecting calculation optionsand requesting specific computations and output data. Perhaps most important, detaileddocumentation was written for AMP, consisting of a User’s Manual, Engineering Manualand Code Manual.

2

AMP was restricted to modeling thin wires and surfaces represented as wire grids.However, a later version of the code, AMP-2 [15], included a magnetic-field integral equationmodel for surfaces and allowed connection of wires to surfaces using a technique demonstratedin [16].

The code NEC-1 was developed fkom AMP–2 with support from the Naval OceanSystems Center and the Air Force Weapons Laboratory. NEC–1 included a new way ofimplementing the three-term sinusoidal current expansion so that current and charge densityexactly satisfied continuity conditions imposed at the junctions. The current was forced tosatisfy Kirchhoff’s cument law at the junction, snd the charge densities on wires were relatedto a function of the log of wire radius to provide approximate continuity of potential. NEC-1also included a bicone model for voltage sources and several other developments for increasedaccuracy and efficiency.

NEC-2 [17], developed with support of the Naval ocean Systems Center, added twosignificant new modeling features to the Capability- of NEC-1. The Numerical Green’sFunction option implemented a partitioned-matti Solution so that a solution for a basicmodel could be saved on a file. New antennas or other modifications could then be added tothe basic model and the solution obtained without repeating calculations for the part of themodel coming from the file. Also, a solution for wires over a 10SSYground was implementedusing the rigorous Sommerfeld-integd approach and a table-lookup algorithm to reducecomputation time.

In NEC-3, which was developed with support of the Naval Ocean Systems Center, theSornmerfeld-integral ground model was extended to wires buried in the ground or penetrating ,horn air into ground [18]. Table-lookup, least-squares parameter estimation and asymptoticapproximations were used to reduce computation time. NEC-3 will also model wires orsurfaces in an infinite dielectric or lossy medium.

NEC-4 retains all of the capabilities of NEC-3, with changes and sdditions to improvethe accuracy and add new features. Techniques to avoid loss of precision with electricallysmall models were developed with support of the Naval Ocean Systems Center [65, 66].The treatment for electrically small loops has not been included in the current NEC4 butmay be added in the future. The new treatment for stepped-radius wires and junctions wasdeveloped with support of the U. S. Army, ISEC, and the development of the code anddocumentation has been sponsored by ISEC.

3

-.- ... . .. .. .—

2. The Electric and Magnetic Field Integral Equations

The NEC program uses both an electric-field integral equation (EFIE) and a magnetic-field integral equation (MFIE) to model the electromagnetic response of general structures.Each equation has advantages for particular structure types. The EFIE is well suited forthin-wire structures of small or vanishing conductor volume while the MFIE, which failsfor the thin-wire case, is more attractive for voluminous structures, especially those havinglarge smooth surfaces. The EFIE can also be used to model surfaces and is preferred forthin structures where there is little separation between a front and back surface. Althoughthe EFIE is specialized to thin wires in this program, it has been used to represent surfacesby wire grids with reasonable success for far-field quantities but with variable accuracyfor surface fields. For a structure including both wires and surfaces the EFIE and MFIEare coupled. This combination of the EFIE and MFIE was demonstrated by Albertsen,Hansen and Jensen at the Technical University of Denmark [16] although the details of theirnumerical solution differ from those in NEC. A rbzorous derivation of the EFIE and MFIEis given by Poggio and Miller [4]. The equationsfollowing sections.

2.1 The Electric Field Integral Equation

The form of the EFIE used in NEC followselectric field of a volume current distribution J,

E(r) = ~J47rk v

J(r’) .

and their derivation are outlined in the

from an integral

G(r, r’) dV’

representation for the

(2-1)

whereG(r, r’)= (k2~ + VV) g(r, r’)

g(r, r’) = exp(-jklr – r’1)/lr – r’]

k=u-, V=dzz

and the time convention is e@~. ~ is the identity dyadic 6=6= + ~~ + 6Z6Z. When thecurrent distribution is limited tn the surface of a perfectly conducting body (2-1) becomes

E(r) = ~J4~k s

J~(r’) oG(r, r’) dA’ (2-2)

with J@ the surface current density. The observation point r is restricted to be off the surfaceSsothatr #/.

If r approaches S as a limit, (2-2) can be written

E(r) = ~f47rk S

J@(r’) . G(r, r’) dA’ (2-3)

where the principal value integral, ~, is indicated since. g(r, /) is now unbounded. Theintegral in (2-3) may not exist even in a principal value sense, due to the degree of singularity.In practice the approximations are generally made to avoid r coinciding with r’.

4

—--..—.-. —, —-.—...-— . . . ...

An integral equation for the current induced on S by an incident field El can be obtainedfrom (2-3) and the boundary condition for r c S,

ii(r) x [E’(r)+ E~(r)] = O (2-4)

where ii(r) is the unit normal vector of the surface at r and Es is the field due to the inducedcurrent J$. Substituting (2-3) for ES yields the integral equation

4~kfi(r)x~Js(r’) (k2~+V~)g(r,r’)dA’.–ii(r) x Ef(r) = ~ (2-5)

The vector integral in (2-5) can be reduced to a scalar integral equation when theconducting surface is that of a cylindrical thin wire, thereby making the solution mucheasier. This reduction is accomplished by neglecting circumferential current on the wire andalso neglecting variation of the longitudinal current around the wire circumference. Theboundary condition is then enforced on only the tial component of electric field.

In addition, approximations are usually applied in evaluating the integral equation ker-nel to avoid difficulties resulting from the singularity when F coincides with r in (2-5). Ina commonly used form of the thin-wire kernel, the current is located on the wire surface,while the boundary condition is enforced on the wire axis. This modification has a physicaljustification in the extended boundary condition proposed by Waterman [19]. Watermannoted that due to the analytic continuabllit y of the solution of the integrzd equation, forcingthe field to vanish over any region within a closed surface is sufficient to make it vanisheverywhere within the surface, and thus on the surface for electric field. Evaluating thefield on the axis removes the singularity and also simplifies the integration, since integrationaround the wire becomes trivial when the evaluation point is on its sxis. For evaluationpoints outside of the wire, the field is commonly approximated by treating the total currentas a filament on the wire surface at a position 90° from the direction of observation.

In an alternative form sometimes used for the thin-wire kernel, the total current istreated as a filament on the wire axis and the boundary condition is enforced on the wiresurface. This procedure has the apparent advantage that the current filament remains con-tinuous at a bend in the wire or a step in radius, so that point or ring charges are avoided.However, this approximation of the thin-wire kernel does not produce the correct behaviorof charge at a step in wire radius when used with a continuous current distribution and pointmatching. With the current treated w a filament on the axis, the solution tends to convergewith continuous linear charge density across a step in radius, while it is known that morecharge should concentrate on the wire with the larger radius. The point charges that mayoccur at a step in radks are generally not a problem in the MM solution. They can beeliminated by smoothing the current over the annular surface at a step in radius or, withmore difficulty, at a bend. However, as long as current continuity (Kirchhoff’s current law)is enforced in the current expansion, the field due to inadvertent point charges can simplybe dropped from the evaluation. Also, continuous testing functions or finite-difference eval-uation of charge will smooth the effect of point charges. The current was located on the wireaxis in NEC-3 and earlier codes, but NEC-4 puts the current on the surface and matchesthe boundary condition on the axis.

5

Other forms of the thin-wire kernel use approximations of the exact kernel, with currentand field evaluation on the wire surface. A tw-term series approximation of the exact kernelis used as the “extended thin-wire kernel” in NEC3 and earlier versions. An advantage ofapproximating the exact kernel is that the wire can have either open or closed ends, whileends should be closed when the extended boundary condition is used. The extended thhl-wire kernel is not included in NEC-4, since the thin-wire kernel with closed wire ends wasfound to give as good accuracy for thick wires.

With the approximation of an axial current with uniform distribution around the wire,the surface current JS(r) on a wire of rdlus a Cm be replaced by a filamentary current 1where

1(s)5 = 2raJ.(r)

withs the distance along the wire axis at r, and ~ the unit vector tangent to the wire axisat r. Equation (2-5) then becomes

/(

8 )–ii(r) x E1(r) = #ii(r) x ~ 1(s’) k23’ – Vm g(r, r’) h’ (2-6)

where the integration is over the length of the wire. Enforcing the boundary condition inthe axial direction reduces (2-6) to the scalar equation

–h /( az_~.E](r) = ~ ~ )I(s’) k25” & – ~ g(r, r’) dsl (2-7)

Since r’ is now the point at d on the wire sxis while r is a point at s on the wire surfacelr – r’! a a and the integrand is bounded.

2.2 The Magnetic Field Integral Equation

The MFIE is derived from the integral representation for the magnetic field of a surfacecurrent distribution Js,

(2-8)

where the differentiation is with respect to the integration variable r’. If the current JS isinduced by an external incident field H~, then the total magnetic field inside the perfectlyconducting surface must be zero. Hence, for r just inside the surface S,

Hi(r) + Hs(r) = O (2-9)

where H1 is the incident field with the structure removed, and Hs is the scattered field givenby (2-8). The integral equation for JS may be obtained by letting r approach the surfacepoint r. from inside the surface s30ng the normal ii(ro). The surface component of equation(2-9) with (2-8) substituted for HS is then

~ lim/

–fi(rO) x H~(rO) = ii(r) x Am,4=0 S Js(r’) x V’g(r, r’) dA’

6

where ii(ro) is the outward directed normal vector at ro. The limit can be evaluated byusing a result of potential theory [20] to yield the integral equation

–ii(r) x H~(ro) = -;J.(ro)+;fifi(rdx [J@xv’g@-w dA’ ~2-lo)

For solution in NEC, this vector integral equation is resolved into two scalar equationsalong the orthogonal surface vectors ~1 and ~2 where

~1(rO) x ;Z(ro) = ii(ro).

By using the identity u. (v x w) =(uxv). wandnoting that ~lxii=-t2and ~zxfi=~1,the scaIar equations can be written

;Z(ro)” Hz(rO) = –~~l(ro) “J~(rO) --&ro) [J.@’)xv’g@or’l ‘A’ (2-11)

–~l(ro) “ H’(rO) = –~~2(ro) sJs(ro) + +~wo) [J.@’Pv’g@o,r’l dA’o (2-12)

These two components suffice since there is no normal component of (2-10).

2.3 The EFIE-MFIE Hybrid Equation

Program NEC uses the EFIE for thin wires and the MFIE for surfaces. For a structurewith both wires and surfaces, r in (2-7) is restricted to wires, with the integral for Es(r)extending over the complete structure. The thin-wire form of the integral in (2-7) is usedover wires while the more general form of (2-5) must be used on surfaces. Likewise, ro isrestricted to surfaces in (2-11) and (2-12), with the integrals for Hs (r) extending over thecomplete structure. On wires the integral is simplified by the thin-wire approximation. Theresulting coupled integral equations are, for r on wire surfaces,

and for r on surfaces excluding wires

/[~z(r) “Hi(r) =~~z(r)” ~ 1(s’) 5’ x V’g(r, r’)] ds’ – ~~l(r) “ Js(r)

-&J1~2(r)[J.(r’)xv’9(rr’Jl ‘A’

snd

(2-13)

(2-14)

(2-15)

The symbol JL represents integration over wires w~lle & represents integration over surfacesexcluding wires. The numerical method used to solve equations (2-13), (2-14) and (2-15) isdescribed in section 3.

‘7

, .., .= =..em,m.—-. ........... .,,.--.........-m.—-.. s..,. . ..& ,-., -——.-.——-.-” . . .. —..——-. .——- . . ..—. --- --------

3. The Moment-Method Solution

The integral equations (2-13), (2-14) and (2-15) are solved numerically in NEC by themethod of moments. The bssic procedure for this solution is described in this section,starting with a brief review of the moment method. The NEC solution combines pointmatching of the fields with a three-term current expansion on wires and a pulse expansionon surfaces. The point-matching approach is particularly convenient for combhing differentmodelirw techniclues that lead to a Green’s-function representation. Hence, the EFIE forwires, M-FIE for surfaces and the solution for a point source near the ground are combinedin NEC. The evaluation of the fields due to the distributed sources for these cum forms thecomputational core of the code, together with the solution of the matrix equation, and isdiscussed in succeeding sections.

3.1 Outline of the Moment Method

II

An excellent introduction to the moment method was given by R. F. Barrington in thebook Fiekf Computation by Moment Mehds [5]. A brief outline of the method follows.

The method of moments applies to a general linear-operator equation,

Lf=e (3-1)

where e is a known excitation, j is an unknown response and L is a linear operator (anintegral operator in the present case.) The unknown fi-mction ~ is expanded in a sum ofbssis functions ~j as

Nf = ~LYjjj (3-2)

j=l

and a set of linear equations for the coefficients ~j is then obtained & t~iw the inner

product of (3-1) with a set of weighting functions {w~} as

(w, Lf)=(wi, e) i=l,..., ~.

The inner product is typically defined as

(f) 9)= / f(’)9(’) ~As

where the integration is over the structure surface. The number of weighting fi.mctions istaken here to be equsJ to the number of basis functions, so that the number of equations isequal to the number of unknown coefficients in (3-2). Due to the linearity of L, equation(3-2) substituted for ~ yields

~~j(~, Lfj) = (w, e),

j=l

This set of linear equations can be written in matrix

[G][A]= [E]

8

a=l, . . ..N.

notation as

where G~j = (~i, L~j), Aj = CYjad Ei = (w,, e). The solution can then be written interms of the inverse matrix as

[A] = [Gfj-’[E].

The choice of basis and weighting fimctions has an important role in determining theefficiency and accuracy of the moment-method solution. Each basis function can eitherextend over the entire domain of the current, or a subdomain. Common choices for thebasis functions are rectangular pulses, piecewise-linem or piecewise-sinusoidal functions, orpolynomials. When w~ = $i, the procedure is known ss Galerkin’s method. In NEC thebasis and weighting fimctions are different, Wi being chosen as a set of delta functions

w~(r) = f(r – ra)

with {r; } a set of points on the conducting surface. The result is a point sampling of theintegral equation, known as point matching. Wires are divided into short straight segmentswith a sample point at the center of each segment, while surfaces are approximated by a setof flat patches with a sample point at the center of each patch.

With point sampling, the derivatives on the Green’s function for the EFIE place par-ticular importance on a smooth representation of the current with continuous derivatives.Hence a three-term sinusoidal expansion is used for the current on wires in NEC. The MFIEfor surfaces can be solved with a simpler current expansion. Good results are obtained witha delta-function representation of the current on surfaces. Subdomtin basis functions areused on both wires and surfaces, with support of ja restricted to a region near ri. Thischoice simplifies the evaluation of the integrals and ensures that the matrix G will be wellconditioned. Details of the current basis functions are given in the following sections.

3.2 Current Expansion on Wires

Wires in NEC are modeled by short straight segments, with the current on each segmentrepresented in the form

lj(~) = Aj + Bj sink~(s - Sj) + C’j[coskS(~ – Sj) – 1], IS- Sj[ < Aj/2 (3-3)

when ~j is the value ofs at the center of segment ~, ~d Aj is the length Of the segment”This expansion was first used by Yeh and Mei [10] and has been shown to provide rapidsolution convergence [21, 22] , particularly on longer wires. It has the added advantage thatthe electric fields of the sinusoidal currents are easily evaluated in closed form when k~ isequal to the wave number in the medhun. In most circumstances ks is chosen equal to thewave number in the medium containing the wire, since (3-3) then represents the naturalform for current on a long wire. However, in some cases, such as insulated wires in dielectricor lossy media, a different choice of k~ is more suitable.

In NEC-3 and earlier codes the current expansion on each segment had the form

Ij(S) = A; + Bjsinks(~ – sj) + C’jmsh(s – sj), IS - sjl < Aj/2. (3-4)

This form is equivalent to (3-3) except that precision maybe lost in evaluating (3-4) on shortsegments, with ksAj <<1, due to cancellation of the constant and cosine terms. The form of

9

(3-3) can be used on very short segments without loss of precision, as long as the constantsare evaluated carefully.

While (3-3) involves three arbitrary constants per segment, two of these are eliminatedby imposing continuity conditions on the current and charge at the segment ends. Theremaining constant for each segment is determined by solving the moment-method equations.Continuity conditions at the segment ends are enforced on the current and the linear chargedensity g, which is related to the derivative of current through the continuity equation

dI(s)— = –j#q(s).

ds

At a junction of two segments with equal radius the obvious Conditions are that the currentand charge are continuous across the junction. At a junction of more than two wires, thecontinuity of current is generalized to Kirchhoff’s law that the sum of currents into thejunction is zero.

The appropriate condition on charge is not as obvious. at a junction of three or morewires, or a step in wire radius. The actual distribution of charge in the vicinity of such ajunction, or even on a single wire at a bend, wiIl be complex, with higher-order variationsalong the wire and around its circumference. Variations around the wire must be neglectedin the thin-wire approximation, and the basis functions cannot support the rapid variationor singularity that occurs at an edge. Hence, what is needed is a condition that results in thebest match of the boundary condition that is possible with the particular basis functions. Oneapproach, used in [23], is to allow an unknown discontinuity in charge density at a junction,and determine the value in the moment-method solution by includlng new equations involvingadditional weighting functions. Howeverj the increased matrix order with this approachcan significantly increase computation time for large models. In NEC, the approximatedistribution of charge at a junction is determined from conditions in the vicinity of thejunction through qussistatic approximations. These conditions are derived in section 3.4.For now the proportionality factor for charge on each wire at a junction, relative to otherwires at that junction, will be represented as a; or a:, respectively, for end 1 or end 2of segment i relative to the reference direction. Thus for all segments i connected to ajunction, the condition on the derivative of current is d~(S)/~lSzSjudb~ = &Q, where4?isanunknown constant related to the total chsrge associated with the junction, and the + or- is chosen for the end of the segment that connects to the junction.

At a free wire end the current must go to zero. However, if the current can flow onto anend cap it goes to zero at the center of the cap rather than at the edge of the wire cylinder. Anapproximate treatment of wire end caps is included in the NEC-4 wire model, ss describedin section 3.5. In the development of the general basis functions, the current at the end ofthe wire cylinder will be related to the derivative of current through a proportionahty factorXi for segment i, so that . {

where se~ is the value ofs at the edge of the cylinder on segment i, and the + or – signsapply for the free end at end 1 or end 2 of the segment, respectively. The value of Xi will

10

. , .. .. ,,. .. . ..... . ..... ._ _

+●

●/“/●‘-\ ‘+ <

●

●“/ a;@J=l

\2

j=l

Fig. S1 Segmentscoveredby the ith bssis function.

be determined from a quasistatic treatment of the end cap. If there is no end cap Xi can beset to zero.

The above conditions on current and charge are enforced on the three-term currentexpansion of equation (3-3) in defining the basis functions. The conditions are then satisfiedby the total current for any combination of these basis functions. The general form of thebasis function will be developed for the configuration of segments shown in Fig. 3-1. Thebasis function for segment i extends onto all segments connected to segment i, going to zerowith zero derivative at the outer ends of the connected segments. Thus it forms what couldbe considered a generalized B-spline function. The current on each segment has the form of(3-3), so that the current on segment i is

~~(s) = ~“ + B: sin ks(S – Si) + Cf[COSk.(S – Si) – 1], IS – sil < Ai/2 (3-5)

while on segments connected to end 1 of segment i,

$(s) = A; + l?: sink.(s – Sj) + Cj-[COS&(S – Sj) – 1], Is- Sjt < Aj/2 (3-6)

With j= l,..., iV-, and on segments connected to end 2 of segment i,

with l= l,..., lV+. The total current expansion for a wire model with iV segments is then

N

1(s) = ~ Cqj’’(s) (3-8)i=O

where the basis function centered on segment z is

N- N+

with the components of the basis function for(3-7).

4=1

segment 2 given by equations (3-5) through

11

Equations (3-5), (3-6) and (3-7), forming the complete basis function associated withsegment i, involve 3(IV- + IV+ + 1) constants. All but one of these constants are eliminatedby enforcing the conditions on current and its derivative at the segment ends. At the endsof segment i the conditions are

where Q; and Q~ are related to the charge associated with the junctions at end 1 and end2 of segment i, respectively. If end 1 of segment i is a free end (~- = O) equation (3-10) isrqhced by

while if end 2 of segment i is a free end (IV+ = O) equation (3-11) is replaced by

fi(S~ + Ai/2) = -~:f:(s)18=4+&,2.

On segments connected to end 1 of segment i the end conditions me

fj~(sj – Aj/2) =0

;fj:(s) =08=sj_Aj/2

while on segments connected to end 2 of segment i the conditions are

f;(s~ + &/2) =0

Lf+ =0& c (s) S=st+At/2

=kSa;Q~.$fiw ~=St-A4/2

(3-12)

(3-13)

(3-14)

(3-15)

(3-16)

(3-17)

(3-18)

(3-19)

Generalizing for connected segments with reference directions opposed to segment i, equa-tions (3-14), (3-15) and (3-16) are solved for the constants in (3-6) to yield

[

1 1A;=F .—

2 sin ksAj/2 1@QY (3-20)sin ksAj

a~QFB; = (3-21)

2 cos kSAj/2

~a~Q~C3- =

2 sin ksAj/2”(3-22)

12

. ...=. .. . %

and equations (3-17), (3-18) and (3-19) are solved for the constants in (3-7) to yield

(3-23)

(3-24)

(3-25)

The upper sign in these equations applies when the reference direction of the segment is asshown in Fig. 3-1, and the lower sign for the opposite reference direction.

Since there is one less equation than unknowns for all parts of the basis function, wearbitrarily set A! – ~ = – 1, since this results in a basis function with positive amplitude.The basis function could be normalized to unit magnitude, but this is not done to savecomputation time. The normalization is absorbed into the constants ~j in (3-2). WhenN- # O and N+ # Oequations (3-10) and (3-11) can be solved for the constants in (3-5) toyield

(3-26)

(3-27)

(3-28)

The constants Q: can then be eliminated by enforcing Kirchhoff’s law on the current at thejunctions

~j~(sj + Aj/2) = .f~(si - Ai/2)j=l

f’f’’(st-&/2)= j~(si + Ai/2)4=1

where ~-

P;=– X(cos k8Aj – 1

)

a*sin k8A j j

j=l

The upper sign applies when the reference direction of the connected segment is the sameas that of segment i and the lower sign is for opposed reference directions. The sum for

13

Pi- is over segments connected to end 1 of segment i, and the sum for Pi+ is over segmentsconnected to end 2.

For N- = O and N+ # O the constants in (3-5) are

C&lsin k8A~/2 cos kaA~/2 - Xi sin k8Ai/2

(XMk~Ai – Xi sin k8Ai+ a~Q~

(X)Sk8Ai - Xi sin k8Aicos k8Ai/2

+ a~Q~sin k8Ai/2 + Xi cos kaAi/2

(X)Sk8Ai - Xi sin k8Ai cos kaAi – Xi sin k8Ai

-L cos k~Ai – 1 – Xi sin k~Ai

(3-30)

(3-31)

(3-32)

(3-33)‘i’ = (at+ Xi~+) sink~Ai + (a~Xi - v) mhAi

For N- # O and N+ = O the constants are

~“=G&l– sin k8Ai/2 _ (X)S kBAi/2 – Xi sin k8Ai/2@ =

cos k.Ai – Xa sin k8Ai+ a;Qi

cos k$Ai – Xi sin k~Ai

~=COS ksAi/2 _ sin k~Ai/2 + Xi cos kBAi/2

cos k8Ai – Xi sin k8Ai – a; Qi(X)S kaAi - Xi sin kaAi

with

Q;= a.1 – cos kSAi + Xi sin k*Ai

( – X~Pi-) sin kaAi + (a,~Xi + Pi-) cos k.Ai “i

If N-= N+ = O, the complete basis function is

f:=cos h(s – a)

cos k8Ai/2 – Xi sin k.Ai/2– 1. (3-38)

TM function satisfies the conditions that have been defined for the current at iiee ends.The use of a single isolated segment cannot be expected to yield an accurate solution forcurrent on the segment, but the segment may be useful as a source for fields illuminatinganother part of the model, where the accuracy of the current on the source segment is notimportant.

When a segment end is connected to a ground plane or to a surface modeled with theMFIE, the end condition on both the total current and the last basis iknction is

_$Ij(s) =0S=a&

replacing the condition of zero current at a free end. This condition does not require a

separate treatment, but is obtained by computing the last basis function as if the last segmentconnected to its image segment on the other side of the surface.

14

(3-34)

(3-35)

(3-36)

(3-37)

,,

* ,,..,$

In the program AMP, a predecessor of NEC,the current expansion also had the form of (3-3).However, continuity of current was approximatedby requiring that when the current fimction on asegment was extrapolated to adjacent segmentsit coincided with tlie current at the centers ofthose segments. The resulting basis function forsegment i had unit magnitude at the center ofsegment i and went to zero at the centers of theadjacent segments. Hence, the amplitude of ba-sis finction i, a~ in (3-2), was the value of thetotal current at the center of segment i. Thisis not true in NEC, so the current at the cen-ter of segment i must be computed by summingthe contributions of all bssis functions extendingonto segment 2.

3.3 Current Expansion on Surfaces

c

A

Fig. 2-2 Detailof the connectionof a wiretoa surface.

Surfaces on whkh the MFIE is used are modeled by small flat patches. The surfacecurrent on each patch is expanded in a set of pulse functions with two orthogcnud componentsper patch. A special treatment is used in the region of wire connections. The pulse functionexpansion for Np patches is

(3-39)

where rj is the position of the center of patch j, ~lj and ~Zj are the orthogonal unit vectorson the surface at rj and Vj(r) = 1 for r on patch j and Ootherwise.

The constants Jlj and J2j are the average stiace-current densities over the patch,representing the ~j in (3-2), and sre determined by the Solution of the mat~ ~uation.The integrals for fields due to the pulse basis functions are evaluated numerically at a singlepoint, so that for integration the pulses could be reduced to delta functions at the patchcenters. That this simple approximation of the current yields good accuracy is one of theadvantages of the MFIE for surfaces.

A more realistic representation of the surface current is needed in the region where awire connects to the surface. The treatment used in NEC involves the four coplanar patchesabout the connection point, and is very similar to that used by Albertsen et al. [16]. In theregion of the wire connection, the surface current contains a singular component due to thecurrent flowing from the wire onto the surface. With the coordinates as shown in Fig. 3-2,the total surface current should satisfy the condition

where 6(x, y) is a two-dimensional Dirac delta function, V9 denotes the surface divergence,JO(Z,y) is a continuous function in the region ABCD and 10 is the current at the base of the

15

wire flowing onto the surface. The form used in NEC is

j=l(3-40)

where Jj = Js(x~, ~j), fj = f (~j, ~j) with

and ~j and yy are the values of x and y at the center of patch j. The interpolation func-tions gj (z, y) are chosen such that gj (z, U) is differentiable on ABCD, gj(xi, IA) = 6ij and

x~=l 9j(~, !/) = 1. The functions used in NEC are as follows:

91(% v) = (~+ ~)(d + ?4)/4d2 g2(z, ~) = (d – Z)(d + y}pkiz

93(Z, v) = (~ – Z)(d – #)/4d2 94(*, !/) = (d+ Z)(d – y)/4d?

Equation (3-40) is used when computing the electric field at the center of the connectedwire segment due to the surface current on the four surrounding patches. In computingthe field on any other segments or on any patches, the pulse-function form is used forall patches, inchdng those at the connection point. This saves integration time and issufficiently accurate for the greater source to observation-point separations involved.

3.4 The Condition on Charge at a Junction

Kirchhoff’s current law is implicit in the general NEC basis fbction for wires. However,the conditions on charge were left as unspecified proportionality constants. Determining acorrect condition on charge is not ss easy as for current, since the charge must distribute sothat the tangential electric field is minimized over the junction or, in the quasistatic form,the electric scalar potential is continuous across the junction. Various conditions may affectthe distribution of charge at a junction, including changing wire radius, the proximity ofwires at the junction, different media perrnittivities when a wire crosses an interface, or thetransition from insulated to bare wire.

Approximations have been developed in an attempt to account for wire radius in thebasis functions without the need to solve an integral equation. Early junction treatmentsassumed constant linear charge density (241 or constant surface charge density [25] at ajunction of wires with differing rachs. -By &mlysis of a wireKing [26] derived a condition that charge distributes so that

r /n\ 1-1

“UHa -0”5772J

with tapered radius, -W-u and

(3-41)

where q~ is the linear charge density at the junction onA similar condition was derived by Shulkunoff and Friss

16

wire i and aj is the wire radius.[27] by equating potentials while

treating each wire as infinite and isolated tiom adjwent wires. Ttils cmdtion was appliedat junctions in NEC-3 and prior versions of NEC.

By obtaining a highly resolved solution at a step in radius, using a surface moment-method code, Glisson and Wilton [28] have shown that charge is singular at the outer edgeof the step, and goes to zero at the inner edge, as occurs on linear wedges. Outside of theregion of edge effects, the charge on opposite sides of the step differs approximately by theratio predicted by (3-41). This result suggests that (3-41) is not an appropriate conditionto apply precisely at the junction of wires with different radii. The charge will generallyhave higher-order variations at a step, bend or multiple wire junction, but these usually donot need to be included in a thin-wire model. What is needed is a condition on charge thatcomes as cbse to the correct charge behavior as is possible with the given current expansion.

Popovi6 et al. [23] obtained suitable conditions on charge by requiring that an approxi-mate integral of electric field along wires crossing the junction is zero. This treatment takesaccount of wire radius as well as the configuration of wires at the junction. At a junction ofA4 wires the resulting A4– 1 equations are appended to the moment-method equations, alongwith the equation expressing Kirchhoff’s law. However, increasing the size of the matrix isundesirable due to the M3 dependence of solution time.

In NEC-4, charge distributions for the basis functions are determined with minimalcomputational overhead by executing a small moment-method solution for each junction.Any junction on which the charge cannot be determined as uniform due to symmetry isconsidered isolated from the rest of the structure with the wires extended to infinity awayfrom the junction. An integral equation bssed on continuity of scalar potential can then bewritten for the junction of M wires as

, ~-jk@(s,s’)

{~f’!lt(s) &.(@) ‘s’=c ::k’m:.4=1

(3-42)

where gt(~) is the charge density at s’ on wire 4 and Ru(s, ~) is the distance between thepoints at s on wire z and / on wire 4, and C’ is an arbitrary constant. The distance Smushould approach zero for continuity of potential, but a finite but electrically small value isused for the numerical solution.

Since the the current expansion of (3-3) results in an approximately piecewise linearrepresmtation of charge, a similar treatment is used in solving (3-42). As shown in Fig. 3-3,the charge is expanded in triangular basis functions terminated with a constant functionextending to infinity to avoid introducing effects of tmncation lengths for the wires. Thecharge on wire 1!is then

n

94s) = ~ Wjfij(s)j=l

where the first basis function is

j~l (s)= At – S for OSsSAt= o otherwise.

17

------ -.

.

al 82

‘i--=4Al

Fig. $3. Basisfimctionsandmatch

Forj=2,...,1, l,

83 Sn_l Sn

pointsforthe solutionfor chargeon wiret?.

t@) = s -(j - 2)At=jAt–s= o

while the last basis fhnction is

ft.(s) = &

Substitution of this current expansion

for (j-2)A~sss(j-l)A~

for (j – l)At ~ s < jAt

otherwise

for s >0.

into (3-42) with the match points on wire i of

{

A1/2 fork=lsk = (k–l)Ai fork= 2,...,n

leads to the set of equations

where

nex {i=l,..., rnqtj@ik,tj = c k=l,,.., n

&l jsl(3-43)

(3-44)

The evaluation of the integral in (3-44) is dkussed in Appendw A. The length At is thesegment length at the junction on wire L Hence S1 coincides with the match point in themoment-method solution for current.

Equation (3-43) is solved for the basis function amplitudes, from which the chargedensity on wire 4 at the junction is obtained as

Q = (9/1+ Wn)At,

18

Q4 is then used for the at in the basis functions. Since only the ratios of charge densitiesare needed, the number of equations in (3-43) couid be reduced by one by subtracting oneequation from the others and dividing by one of the unknowns. Little is gained by thishowever since the solution time for the small matrix is negligible compared to the time toevaluate the matrix elements.

Tets for stepped-radius wires and multiple wire junctions have shown excellent conver-gence for the charge density at the junction with n = 2 in (3-43), using only the half-triangleand semi-infinite basis functions. Hence, while the code to solve {3-43) has been writtenfor arbitrary n, n has been set to 2. The time to solve (3-43) for a junction of two wireswith differing radi, including the evaluation of the four elements of the @ matrix, is thenabout five times the time to evaluate a single matrix eIement in the fill impedance matrix.Since this solution for charge must be done only once for each junction the increase in timeis generally smaII. For a structure with ten segments and one step in radius the time to fillthe impedance matrix is increased by about five percent.

Several other conditions may occur at a wire end that call for alternate junction treat-ments. When a wire is connected to a perfectly conducting surface the charge, or derivativeof current, is set to zero at the wire end. If the wire meets the surface at an angle from nor-mal the charge will vary around the wire circumference. However, since the current and itsimage must form an even function about the plane, zero charge is the appropriate conditionin the thin-wire approximation.

When a wire crosses an interface between two different media the current remains con-tinuous and charge density is discontinuous as g+/q- = c+/c., where e+ and e- are thepermittivities of the upper and lower media, respectively [18]. This condition results fromthe requirement of continuity of radial electric field across the interface when the penetratingwire is norrmd to the interface. However, the condition has been used for wires tilted bymore than 60 degrees from the normal with apparently good results, based on small valuesof tangential electric field computed along the wire at the interface. For a 10SSYmedium thecondition on derivative of current is

I;/I: = z+/E- (3-45)

where ~+ and & are the complex perrnittivities. In the basis functions, the a; at a junctionof wires on opposite sides of the interface are set equal to t+ or Z- for the medium containingthe wire.

Well converged results have been obtained using (3-45) on a monopole on a buried radial-wire ground screen. While (3-45) does not include the interaction of horizontal screen wireswith the interface, it represents the dominant effect on charge when crossing the interface.However poor convergence is obtained when using (3-41) at a junction of a monopole and aradial-wire screen just above the interface. In the latter case, segment lengths at the junctionmust be on the order of the distance of the screen from the interface to obtain a convergedresult.

No simple condition has been derived for charge at a junction of a wire with a dielectricor conducting sheath and a bare wire. In such a case, the best approach may be to leave

19

—,.. .. w.--” .s——— ——, ,.””, . . . . ,,, .-. . . . . “ , . . . . . . . . ..-..- w.,,=. ..!,. . . . . . . . . . . . . . . . .. . . . . .,.* ———.-— .*.*. -—m..!..— ———-——....

,,:, .7

the charge condition as undetermined and include an equation minimizing tangential electricfield at the junction as part of the moment-method equations. The same appears true wheninteraction with an interface has a substantial effect on charge distribution at a junction.The charge discontinuity can be introduced as an unknown in the moment-method solutionby including an additional basis function ending at the junction with zero value but nonzeroderivative. A suitable basis function can be obtained fkom the derivation in section 3.2 byevaluating the basis function as if the end of the segment at the junction was a free end withxi = O. To apply this method at a junction of n wires the new basis functipns would beadded on n – 1 of the wires at the junction. This does not introduce sn asymmetry at thejunction, since the other bssis fimctions will adjust accordingly.

20

4. Evaluation of the Field in Free Space

i

I

The moment-method solution of equations (2-13), (2-14) and (2-15) requires the evalua-tion of near electric and magnetic fields due to source currents on wire segments and surfacepatches. The evaluation is considered first in this section for the case of sources in freespace or in an infinite dielectric or 10SSYmedium. The evaluation of the electric field near aninterface between free space and earth is covered in section 5. The evaluation of the electricfield due to a wire segment is the most involved, due to the three components of current,constant, sin kss and cus kss, and also because considerable care is needed to avoid loss ofprecision at low frequencies. An approximate treatment of end caps on wires is also includedin this section.

4.1 The Electric Fields due to Wire Segments

In evaluating the field due to the basis functions on wires, the fields due to the componentfunctions in (3-5), constant, sin kss and cos kss, are evaluated and then combhed using theconstants derived for the basis functions. Care is needed in the evaluation of the field toavoid loss of precision at low frequencies and in particular limits of the coordinates. Onecause of precision loss is the cancellation of the fields due to the point or ring charges atthe segment ends. These point charges are sn artificial result of treating continuous wiresss made up of discrete segments, and the resulting fields must cancel exactly. Since thecontinuity of current, as required by Kirchhoff’s law, is enforced in the basis functions, thefields due to the point charges can simply be dropped from the evaluation, thus eliminatingthis source of precision loss. These terms are dropped in NEC–4, but not in NEC-3 andearlier codes.

NEC–3 also differs from NEC4 in the evaluation of the thin-wire kernel. In NEC-3 thecurrent is treated as a filament on the wire axis and the boundary condition is matched on thewire surface. Also, NEC–3 includes an extended thin-wire kernel, which is an approximateevaluation of the field on the surface of the wire due to current on the surface. The extendedthh-i-wire kernel is not used in NEC4, since the extended boundary condition model, withend caps on the wires, gives at least as good results for thick wires.

As discussed in section 2, the NEC4 wire model employes the extended boundary con-dition in the thin wire approximation, so that the current is treated as a tubular distributionon the wire surface, while the boundary condition is enforced on the wire axis. In evaluatingthe field of the current on the wire surface the segment is considered on the z axis of acylindrical coordinate system, as shown in Fig. 4-1. The integral over z is evaluated firstand then the integral over d. Hence the field due to a current filament on the z axis will beconsidered first.

Although in general the integral for the near fields of a current distribution cannot beevaluated in closed form, such an evaluation is possible for a straight filament of currentwith sin ks(z – ZO)distribution when ks is equal to the wave number in the medium. Thisresult was derived by Brillouin [29] and is included in [30]. For a current filament extendingfrom Z1 to zz on the z axis of a cylindrical coordinate system, as shown in Fig. 4-la, with

21

a) E.

EP

b)

z

Z2

Z1

— Z2

— Z1

Fig. Al Coordinate for evaluationof the field;a) currentfilamenton the z axie,b) currenton a roundwiresegment.

current distribution

()sin kz’1(2’) =10 ~s kz,

where k is the wave number in the medium, the field components at the point (p, z) are

For a constant current of strength 10 the field components are

[

e-jkRZ’5Z2–jWh (1+ jJ@-

E:(w) = w 1R3d=zl{[ L=z,+k’l:$dz’}

,e-jkRZ’=Z2~ (l+jkR)(z-z)~E$(p, Z) = ~nk

where R = Id+ (z - 29’]’/?

(4-1)

(42)

(43)

(4-4)

Although the fields due to the sin(kz’) or cos(k~) currents are expressed as termsevaluated at the segment ends, the fields are due to current and charge distributed alongthe segment as well as point charges at the ends where the current is discontinuous. For

22

the constant current there is only the current on the segment and point charges at the ends.!Mncecontinuity of current is enforced in the NEC basis functions, the field components dueto point charges at the segment ends can be dropped. The point-charge field for the segmenton the z sxis is

(4-5)

where R = p~ + (z – z’)Z. Subtracting the components of E~ from equations (4-1) through(44) leaves

()jqIo e-~kR cos k~ Z’=*E;(p, z) = ~T

- sin kti ~=zl

E~(p, Z) = O

(47)

(4-8)

(49)

The above equations for fields due to sinks and cos ks currents are restricted to k in thecurrent expansion equal to the wave number in the medium. This is usually the desirablechoice since it results in a current expansion approximating the natural form of current on along wire. However, in some csses, such as on an insulated wire embedded in a dielectric orlossy medium, it is better to use a factor ks in the current expansion that is different fromk. In this case additional terms must be included in (41) and (4-2). The field componentsfor current distributions of

are

(4-lo)

(411)

Eliminating the components due to point charges at the segment ends, these fields are

23

~- jkR

}~ dz’ . (4-13)

The equations (46) through (4-9) are an improvement over those including the pointcharges, but still present problems for numerical evaluation in some cases. They are exact,however, with the restriction that the current is continuous, goes to zero at free ends and thesum of currents into any junction is zero. Since the current expansion in NEC satisfies theseconditions, equations (46) through (4-10) are used as the basic forms for evaluating themethod of moments matrix and near fields. When a segment connects to a surface modeledwith the MFIE the field of the point charge at the connected end must be included, to cancelthe corresponding point charge on the surface. Also, at a junction of wires on opposite sidesof the air-ground interface the point-charge fields must be included in the field for eachconnected end since the charges are located in different media. These point-charge fields areobtained by evaluating (45) for the particular segment end.

For a segment with length A = 26 extending from Z1 = –6 to Z2 = 6 the field compo-nents given by (46) through (49) become

_ j(e-@b + e-~k~l) sin[k6)}

E;(P, Z) ={[

* (Z-6)*e-~kRl sin ‘d

+(z+6)—RI ]()

+ j(e-jk~z – e-~kRl)cos(k6)}

(4-14)

(415)

(416)

(417)

(4-18)

where RI = ~2 +(z+6)2]1/2 and R2 = ~2 + (z- 6)2]1/2 N. umerical precision may be lost inevaluating equations (414) through (4-18) in the limit of small kR or when R is much larger

24

thsn A or for small p when IzI >6. Also, the subtrwtion of @ – E: in evaluating the fielddue to the cos k.(s – sj) – I term in (3-5) requires careful treatment. Approximations validfor these cases are given in Appendix B.

To evaluate the field of a tubular current distribution the field of a current filament isintegrated around the circumference of the cylinder, using the geometry of Fig. 4-lb. Takingaccount of the change in direction of the @ vector from the filament to the evaluation point,the fields of the current on the cylinder, in terms of the field components & and EP for thefilament with unit current, are

J

10 2XJv(P) z) = ~ o E.(p’, %)d#J

and

wherep’= (p2 + a2 - 2pacosfb)1j2

and

CCSCY=p–acosq$

N“

Using (46) through (49) for the field of the filament, the field of the tubular current distri-bution can be written

E; (p, Z) =

J& (p, z) =

E:(p, z) =

where 10 is the magnitude of the total current on the cylinder and

G1 =~/

z~ e-~~R— d~

2fioR

/

21rGz =~ =~d~

27ro ~ R

G3 =~1

2XCOS~ e_ jk~

2T07 @

2?r

H

n e-jkRG4 =~ — dz’ @.

2’KOZ, R

withR = (P2 + a2 + Z2 – 2apcos@)1f2.

25

(422)

(423)

(424)

(425)

-0 -,.~.”--. . . . . . . . . . . . . . . . ..!m.—— --.-..? --- . . . . . . . . . . . . . . . .,—.-— . . . . . .

,-*

When the evaluation point is on the axis of the segment, so that p = O, the evaluationof the integrals over ~ becomes trivial. The result can be obtained by simply lumping thecurrent in a filament at any position around surface of the cylinder. When p is not zerothe integrals cannot be evaluated in terms of simple functions. Approximations based onthe assumption of a thin wire are commonly used in this case. An approximation that issometimes used for evaluation points outside of the segment is to treat the total current asa filament located at the @ = 90° position. However, it is well known that the radial electricfield outside of a wire is well approximated by treating the charge as a filament on the wireaxis. The difference between the radial field with the filament on the surface at @ = 90° andon the axis can be significant when p is on the order of the wire radius.

Approximations for the integrals in (4-22) through (4-25) are derived in Appendix D.Series approximations are considered that involve either R to a point on the segment axisor to a point at @ = 90° on the surface. The approximations involving R to the segmentsurface are used in NEC–4, so that

(4-27)

G4 wG~ =J

Z2 e-ikRt— dz’

Rt(4-29)

z~

and Rt=~2+a2+(z - Z’)2]112. Since Gz and Gs involve factors of p-l rather than p’- 1the result is not the field of a current filament on the surface.

For p < a, the approximations for Gz and G3 in Appendix D do not have leading termsresembling the integrand evaluated at a point. Hence, it might seem reasonable, for the levelof approximation in (426) through (4-29), to set G2 and G3 to zero for evaluation pointsinside the wire surface. This question is important in the case of a bend in a thick wire, wherethe segment lengths are short enough that the match point on one segment is buried insidethe other segment. ThE is a pathological case that violates the thin-wire approximation.However it has been found that better results are obtained from the moment-method solutionby continuing to use (426) through (429) when p < a rather than setting the radial electricfield to zero.

4.2 The Magnetic Field due to Wire Segments

The magnetic field of a wire segment is zero on the segment axis and ~ directed outside ofthe wire. For a segment on the z axis with length A = 26 and sinusoidal current distributions

()sin kz’1(/) =10 ~s ~z,

26

. .. . ...

the field components are

H; (P, z)‘%[e-’k’’(~f:~e -k-’k’i(:~)

e-~kRz sin k~–j(z–6)~

() ‘(3:)1+j(z+6)~1COSk6

where El = ~2 + (z + 6)2]1/2 and R2 = ~2 + (z - 6)2]1/2. For a constant currentis

/

e- jkR@ 6(1+ jk@F dz.Hf(p, %)= 4X _6

(4-30)

10 the field

(4-31)

The integral in (431) is evaluated numerically using adaptive Rornberg quadrature.

When p ~ Izl and Izl >6 the evaluation of (4-30) may lead to numerical errors due tothe cancellation of large terms. Hence, for z >8 and p/(z – 6) <10-3 the field is evaluatedusing the approximation

For z <

H; (p,%)

4.3 The

The

-6 an% p/(lz] – 6) < 10-3 equation (4-32) is evaluated for –z and the relation

= TH~ (P, –z) is used.

Electric and Magnetic Fields due to Surface Currents

electric and magnetic fields due to each patch making up a surface are evaluatedby treating the surface current over the patch as a point source located at the center of thepatch. This approximation restricts the minimum separation of source and evaluation points,but is compatible with the mo~ent-method solution. The terms{” J x V’g(r, r’) in equations(2-14) and (2-15) vanish when t, J and r–r’ lie in the same plane. Hence the surface integralsfor magnetic field in these equations do not contribute to the self-interaction terms for flatpatches, where the contribution of the singularity has been taken out as a separate term.With the point-source approximation, the magnetic field due to patches is evaluated as

M ‘-jkRH(r)=~~— ~_l R2 (1 + jkR)(R x Ja)S~ (4-33)

where R = lr–r~l, R=(r– ra)/R, Ji is the surface current on patch i and S1 is the amaof the patch.

The electric field due to patches is evaluated ss

E(r)=~~~[(k2132– jkl? - 1) Ji – (k2R2 – 3jk~ – 3) (Ji “R) R] Si. (4-3i)

27

4.4 A Model for End Caps on Wires

In order for the extended boundary conditionused in NEC-4 to be strictly valid the wire mustbe modeled as a closed surface. Hence the freeends of wires must be closed with caps. Popovi6et al. [23] described a careful treatment of endcaps of flat or hemispherical shape. The cap wasincluded in the moment-method solution with sev-eral match points located on the cap surface. Thisapproach appears too cumbersome for large, com-plex structures, so a simpler treatment of end capswas used in NEC–4. The wire is assumed to havea flat end cap, as in Fig. 42, with a constantsurface charge density that maintains continuityof current and charge density with the wire. For

AzIII*a+

IIII

II1

continuity of current from the wire surface onto pc- ~ >

the end cap, a surface current density ofe

‘s lJo)pJc(fJ) = ~Fig. 4-2 A wireendwitha flatend cap.

is assumed on the end cap, where Iw is the total wire current and

{

1 if the reference direction for

sIw is away from the end cap=

– 1 if the reference direction forIWis toward the end cap.

Rom the continuity equation, the surface charge density is then

js I (0),~V. JC(p)=~WPc=;

For continuity of charge density from the wire onto the end cap, it is required that

which, with (4-35), yields the condition

IL(Z) I 2s=—

(4-35)

Z=o

~w(z) Izd a -(4-36)

This value of l~/lw supplies the value for Xi in equations (3-12) or (3-13) defining the bssisfunction. A similar condition is imposed at wire ends in NEC–3 where

1$(Z) = ~SJO(ka)

m .=0 m“

28

. .. .

With small argument approximations for the Bessel functions Jo and J1, this equation re-duces to (436). However, in NEC-3 the field due to the charge on the end cap was neglectedso the end condition did little good.

The field at a point r due to the constant charge density PCon the end cap with areaA. is

1

~- jkR

E=(r) = ~ V —AC R ‘A’

(437)

where R = lr – dl and r’ is the integration point on AC. Due to symmetry of the currentJ., the vector potential is zero on the axis and is greatly reduced by cancellation at typicalevaluation points off of the axis. Hence the vector potential is neglected in evacuating thefield. The evaluation of the integral in (4-37) is discussed in Appendix E. The field E= isincluded in the method of moments impedance matrix where it is added to the field due tothe basis function at the wire end.

Fig. 4-3 Voltagesourcewithendcapeon the excitationgap

End caps must also be added to segments with voltage sources to prevent these sourcesfrom exciting the inside of the wire. A schematic view of a segment with a voltage sourceis shown in Fig. 4-3. The source drives a current Iw along the wire and also supplies acurrent lUCto charge the capacitor formed by the ends of the source gap. Using a simpleapproximation for the capacitance of the source gap of C = na2e/A the current onto theend caps is

The total chargedistribution, is

on the cap at the

j7rwe42 ~IVC=T .

on the caps is CV, so the surface charge density, assuming a constant

(4-38)

“plus” end of the source and –pvc at the opposite end.

(439)

Neglecting the vector potential of the current, the field due to the charged caps on asegment j with source voltage Vj and cap Al on the negative end and A2 on the positive

29

end is

(440)

Since the charge is proportional to the known source voltage, the fieId that it produces isadded to the excitation termE’ in (2-13) or to the right-hand sideof the matrix equation(3-4). ~wthefield on*~ent ilocated atradue tothewurm onx~entjis

where 6ij is the Kronecker delta function. Without the end caps, only the first term ispresent, with zero field on all segments except the source segment.

A similar treatment is needed on segments with impedance loads, since a load is really acurrent-dependent voltage source. If segment j is loaded with impedance Zj, the gap voltagedetermining p.. in (4-40) is V~ = Ijzj, where lj is the current at the center of the segment.Thus W. = ljZj~/Aj. The charges on the end caps produce a field component iIia“ E~(r~)on segment z that must be added to the matrix element for each basis function that extendsonto segment j. Since only the coefficient of Aj in (3-5) is not zero at the center of thesegment only the component of field due to the constant current is affected.

The effect of including end caps on wire ends and voltage sources is shown in Fig. 4-4and 45 for a quarter wave monopole with a wire radus of 0.01-4. The monopole was dividedinto 80 segments so that the ratio of segment length to radius was 0.3125. Since the radhs isconstant, the thin-wire kernel (TWK) in NEC-3 is equivalent to putting the current on thesurface and the match points on the axis. The invalid condition of zero field on the axis of anopen cylinder results in non-physical oscillations of the current. These oscillations becomeapparent for segment lengths shorter than about two times the rdlus. The extended thin-wire kernel (ETWK) in NEC–3 represents a physically valid condition, with the current andmatch points on the surface of the cylinder. However, the field is evaluated with only thetit two terms of a series expansion in radius. Hence oscillations in the current are greatlyreduced but are still present. The end-cap treatment in NEC-4 reduces the oscillations atthe wire ends and essentially eliminates them at the voltage source. That the oscillationsare not completely eliminated at the end is probably the result of the approximation of aconstant charge density on the end cap, neglecting the actual singular behavior of charge atthe edge.

To completely close the wire surface, the field due to charge on the annular surface ata change in radius should also be included in the solution. This has been tested in a codeliited to straight wires. The surface charge density on the annular surface was linearlyinterpolated between values on the wires. Also, Kirchhoff’s law was modified to allow forcharge accumulation on the annular surface. Initial tests showed that these changes reducedthe invalid oscillations of current when segments with small length-to-radius ratio were *usedat a step in wire radus. However, this treatment has not been included in the present NEC–4due to the difficulty of extending it to handle junctions of several wires with differing radiiand directions. Also, there are other problems in applying the thin-wire approximation atsuch junctions.

30

1- =\fll$

0-

-1-

-2-

-3-

-4-

-5 NEC-3, TWK-6 I I I

0 0.1 0.2

Distance/~

-6 I I r0 0.1 0.2

Distance/A

4

T

----.

3-.

‘X.ll

\-1

/

1~

:-0 0.1 .

Distance/A

Fig. 44 Currenton a A/4 monopoleexcitedby anincidentplanewavewith 2 V/m. The monopolehad a radiusof O.01~and waamodeledwith 80segments.

-60- i1 NEC-3, TWK

-80 I I0 0.1 0.2

Dtiance/A

20 1

NEC-3, ETWK-15 1 I I

0 0.1 Q.2

Distance/~

20

1

0 0.1 0.2

Distance/A

Fig. 4-5 Currenton a A/4 monopoleexcitedby a1 V sourceat itsbaae.Themonopolehada radiuaof O.OIAandwasmodeledwith80segments.

31

,“. ,...,. ——-——..,..7,..,.. —...— ,“.., ~.,,...’..,..., ..,. .*—w — —.4.——. —.”.-6

I

5. Evaluation of the Field Near Ground

The previous discussion of field evaluation has dealt with currents in an infinite medium.The medium may be free space, dielectric or conducting with the proper choice of wavenumber k. NEC can also model antennas above or in a conducting half space with air as theupper medium and an arbitrary material such as earth or water below. The ground planechanges the solution in three ways: (1) by modifying the current distribution through near-field interaction; (2) by changing the field illuminating the stmcture; and (3) by changing thereradiated field. Effects (2) and (3) are easily analyzed in terms of plane-wave reflection, as adirect ray and a ray reflected from the ground. The rera.dated field is not a plane wave whenit refiects km the ground, but the strength of the field can be found from the reciprocalproblem with a plane wave arriving at the ground from the distant evaluation point. Analysisof the near-field interaction is more difficult, however. The free-space Green’s function in thekernel of the integral equation must be replaced by the Green’s function for a source nearan interface.

NEC offers three options for modeling grounds. For a perfectly conducting ground, thecode simply includes the image of a source when evaluating the fields. The image field isevaluated with the formulas in Section 4 for an infinite medium. The most accurate modelfor a bssy ground is bssed on the solution for a point source near an interface, in terms of theSommerfeld integrals [31]. Table-lookup and asymptotic approximations are used to reducecomputation time. This model will be called the Sommerfeld/Asymptotic method. Thethird option treats a Iossy ground with a modified image model using the Fkesnelplane-wavereflection coefficients. While specular reflection is not accurate for sphericsl-wave near fields,this Reflection Coefficient Approximation (RCA) has been found to provide useful resultsfor structures in air at least O.lAOor more above the interface [11, 32, 33]. The advantage ofthis method is its simplicity and speed of computation, which are the same as for the imagemethod for perfectly conducting ground. The Sommerfeld/Asymptotic model is presentlyimplemented only for wires, but the same technique could be extended to the MFIE patchmodel.

The remainder of this section describes the field evaluations for the Sommerfeld/As-ymptotic and RCA solutions. Other considerations in modeling a structure near a groundplane, including the effects on incident and radiated fields and the current expansion on awire penetrating the interface, are discussed elsewhere in this manual.

5.1 The Sommerfeld/Asymptotic Solution

The development of the Sommerfeld/Asymptotic model in NEC can be traced backto the code WFLLL2A [34] which implemented a numerical evaluation of the Sornmerfeldintegrals in evaluating the elements of the moment-method matrix. The double integration inthis approach, when the fields involving Sommerfeld integrals are integrated over the currentdistribution, can increase the computation time relative to that for a model in free space byas much as two orders of magnitude. A code SOMINT was then developed (35] that usedbhariate interpolation in a table of pre-computed Sommerfeld integral values to obtain thefield values needed for integration over current distributions. This method greatly reduces

32

the required computation time. NEC-2 used a similar approach, with the interpolationalgorithm optimized. The singularity at the source and the free-space phase factor wereremoved analytically to reduce the solution time and dlOW modeling wires very close to theinterface. For interaction distances beyond the range of the interpolation table, the Nortonapproximations [36] were used in NEC–2.

When the ground model was extended in NEC-3 for wires above, below or crossing theinterface, the interpolation procedure became considerably more complicated. When sourceand evaluation points are on the same side of the interface, the field values depend onlyon the sum of the distances of the points from the interface and their radial separation.Hence only these two parameters were needed in a Hlvatiate interpolation. When source andevaluation points are on opposite sides of the interface, the fields depend on the individualdistances from the interface, so the interpolation space involves three parameters. Also, forsome coordinate ranges, the field transmitted across the interface is not as well behaved asthe reflected field.

For small separations of source and evaluation points, the algorithms developed forNEC–3, which are also used in NEC--4, employ a three-parameter interpolation with thesingularity and phase variation suppressed. For somewhat larger separations, the field valuesare obtained by interpolation with a model in which terms suggested by the asymptotic formof the field equations are fit to computed field values with a Ieast-squares algorithm. Thisapproach, which can be termed model-based parameter estimation, allows a larger samplinginterval in the computed values than would be possible with polynomhd interpolation. AtstiIl larger distances, standard asymptotic approximations me used. The equations andnumerical methods used in representing the field near ground are described in the remainderof this section.

5.1.1 The Field Equations

The Sommerfeld solution for a point source near an interface is usually derived bymatching boundary conditions at bhe interface in a cylindrical-wave expansion of the Hertzpotential. The form used here is taken from Bafios [37] and involves U and V integrals.The geometry of the half-space problem is shown in Fig. 5-1. The relative perrnittivity andconductivity of the lower medium will be denoted c1 and al respectively, while in the uppermedium these parameters are q and oz, and both media have free-space permeability W.Thus the wave numbers in the lower medium kl and upper medium k2 are

In most of the development to follow Q and UQwill be arbitrary, although NEC is currentlyrestricted to Q = 1, 02 = O.

The notation for electric field will use a subscript to indicate the field component incylindrical coordinates (p, ~, z) and a superscript to indicate the source orientation, V fora vertical electric dipole and H for horizontal. If the horizontal source is located along thez axis, the components l?: and E: must be multiplied by cos @ to obtain the p and z

33

z I

IFw. &l Coordinatesfor evaluationof the fielddueto a buriedsourceat anelevatedreceiver

components of field at an angle @ from the z SX@ while ~.$ is multiplied by sin d. These@ dependent terms will be omitted for now, since we want to use the field components intable lookup schemes with a minimum number of parameters. Also, a dipole with unitcurrent moment Ig = 1 will be assumed for now. The rectangular components of field@e to an arbitrarily oriented electric dipole source with current moment 1/ and dh-ectiond = @ + ~~ + d.ii will eventually be obtained from these terms as

The form of the field equations depends on the location of the source and evaluationpoints relative to the interface. When source and evaluation points are on opposite sides ofthe interface the fields involve only the Sommerfeld integral terms. When source and eval-uation points are on the same side of the interface the equations include terms representingthe direct field from source to observer and the field of an image of the source, in additionto the Sommerfeld integrals. The form of these equations is not unique. In fact, NEC usesonly the equations for a source below the interface and observer above, with the other casesobtained from transformations given in Appendix G. The full set of electric field equationsfrom Baiios is included in Appendix F for reference.

For a source with 14= 1 at height z’ in medium 1 and evaluation point at height z inmedium 2 (d <0, z ~ O) the electric field components are [37]

E;2 =– jwpo 82V1Z——

49T apaz(5-2a)

“2=%(:+’’)”2 (5-2b)

(W+U12)

-jwpo 82VME;2 = ~ (5-2c)

34

. ..... ...———. .. .. ..,.,-,-.,,,.,”_._,__g

(5-2d)

(5-2e)

where V12and U12 are the Somrnerfeld integrals

with 71 = (A2 - k~)lt2 and ~ = (A2 - k:)lj2. A subscript of 12 has been added to indicatethat the source is in medium 1 and the evaluation point in medium 2. Using the symmetryproperties of the Hankel function, (5-3) and (5-4) can also be written

(5-5)

(5-6)

Using the integrak from (5-3) and (5-4) and evaluating the derivatives in (5-2) resultsin the equations for the field components

(5-7C)

(5-7d)

(5-7e)

which are evaluated by numerical integration. Similar equations result with the integrals(5-5) and (5-6).

5.1.2 Numerical evaluation of the Sommerfeld Integrals.

The Sommerfeld Integrals in (5-7) cannot be evaluated in closed form. Various tech-niques have been used to obtain numerical or analytic approximations of these integrals,

35

,,

including numerical integration, asymptotic techniques and quasistatic approximations. ofthese, only numerical evaluation hss been able to provide values with high accuracy for sep-arations of source and evaluation points on the order of a wavelength and for a wide rangeof ground parameters.

A number of approaches have been tried for numerically evaluating the integrals. Siegeland King [38] integrated along the real ~ axis and Bubenik [39] applied Shanks algorithmto accelerate the convergence of the oscillating integrand. Johnson and Dudley [40] used atransformation of the integration variable to make the real-axis integration easier. Johnson’swork was done as part of the development of NEC-3, and has been used to validate the NECalgorithms.

While integration on the real A axis offers the advantages of a real argument and easilydetermined zeros of the Bessel function, many of the troublesome features of the integrandcan be avoided by deforming the integration contour into the complex A plane. Lytle andLager [41] integrated along contours suggested by the form of the steepest-descent contoursfor the integrands. Parhami et al. [42] integrated on the actual steepest-descent contourswith source and evaluation points on the same side of the interface. Numerical integrationon the steepest-descent contour is a very efficient way of evaluating the integrals when theseparation of source and evaluation points is large, since the integrand converges rapidlywith minimum oscillation. However, when the separation is on the order of a wavelength,little is gained by following the steepest-descent contour in the vicinity of the origin, andmore benefit may be gained by staying away from the pole near ka. Also, when source andevaluation points are on opposite sides of the interface it is much more difficult to determinethe saddle points and steepest descent paths.

An alternate integral form, known as the exact image method, has been obtained byLinden [43] which is more easily evaluated numerically than (5-3) in some cases. In addition,linear filtering techniques have been applied in evaluating the integrals [44].

In NEC, the Sommerfeld integrals are evaluated by numerical integration along contourschosen to obtain rapid exponential convergence of the integrand. The bssic technique is thatused by Lytle and Lager [41], with some refinements to the contours. Since the integrandsof V12and Ulz and the derivatives of VIZ in (5-7) have similar properties over the complexA plane, the integral (5-3) will be considered as typical. The integrands have branch pointsat *kl and &k2 due to the square roots in VI and 72. The branch cuts will be chosen to bevertical, as shown in Fig. 5-2. The implications of this choice of branch cuts and the choiceof Riemann sheets are discussed in [41].

The key to rapid convergence in the numerical integration is to exploit the exponentialdecay of the exponential and Bessel functions for large A The integration contour is deformedfrom the real axis into the complex plane, without crossing branch cuts or the pole, tooptimize convergence. The contour used with the integrals in the form of (5-3) is shown inFig. 5-2. The dominant factor for convergence in this case is the exponential function asAR increases. The @ssel function oscillates with slow convergence for increasing AR andgrows exponentially as IAII incresses. Hence it is of little help in convergence but restricts

36

AIPo

AR

II

kl ?

I I

I I

I II

Fig. tL2 Contourforevaluationof theBesselfunctionformof the Sommerfeldintegrals.

the contour to small PIAII. The contour off of the real axis avoids the rapid oscillation andmssible sp~e in the integr~d in the region Of k2” The br~ in the COntOUrWaScho*n at~ =(1 +-j) min[l/p, l/@l + 12’!)].

When (Izl + lz’1)/p is small, excessive oscillation can be encountered before convergenceon the contour of Fig. 5-2. It is then easier to evaluate the integrals in the form of (5-5).This integd can be written as

where

andF(A) = ‘y2Z– ~lz’ + jAp.

The function G(A) is relatively slowly varying, and the rapid oscillation is introduced by theexponential term. This behavior of the integrand is used in deriving asymptotic approxi-mations for large p and z by the method of steepest descent. It also suggests an optimumcontour for numerical integration. The steepest descent path passes through a saddle point,whkh is the solution of the equation

-$F(A)=o

and follows a path on which the imaginary part of F(A) is constant. The exponential termin (5-8) provides rapid convergence without oscillation on the steepest descent path. Adifficulty in using the steepest descent contour for numerical integration is that the saddlepoints and each point on the steepest descent contour must be found by iteration whensource and evaluation points are on opposite sides of the interface. However, for small tomoderate values of p, z and z’, much of the contribution to the integral comes ihm regionswhere IA! > rnax(lkl 1,Ikz 1). In this case F(A) can be approximated as

F(A) = AIS*(IZ’I+ Id)+ ~Pl

37

PI Pa

9 II I1 II I

I

:I

Fig. !S3 Contoursfor

where, for Ire(J) <

PI Pa AR

kl I

e 1II

t Ifevaluationof the Hankelfunctionformof the Sommerfeldintegrals.

o

The asymptotic formapproximated by the

s* ={

1 for l%(~) > max[Re(kl), Re(kz)]–1 for Re(A) c min[Re(kl), Re(kz)].

of the steepest-descent contour, on which Im[F(J)] is constant, is thencondition

lm(A)— = –s*—.Re(A) Izl : Iz’1

The basic contour used with the Hankel function form of the integrals is shown in Fig.5-3a, where

‘=tan-’(blfkfl)The break points were determined by trial to reduce computation time. The points PI and~ are chosen to avoid the rapid variations of the integrand in the region of k2 withoutexcessively increasing the contour length. The values depend on the separation of sourceand evaluation points R = b2 + (I4 + 1%’1)2]1/2, since this parameter influences the cost ofincreasing the contour length. The values iised are

1)1=k2 + (–1 + j)D~n

~ =kz + (1 +j)Dmin

where

However, Re~) is not allowed to be greater than Re(kl + kz)/2. The point ~ is chosen as

~ = max[l.OIRe(kl), 1.2Re(kz)] + jO.951m(kl)

to just miss kl but maintain a minimum distance from kz. When R is large, Izl + Iz’1 issmall and Im(kl )/Re(kl ) is small the contour in Fig. 5-3b is used to avoid oscillations on the

38

,,,,,- k

path ffom n to PZ. In the numerical evaluation of the branch cut integral,‘&riable is &mg~ to s where A = Iq + js2 to remove the infinite derivative with respect toAat A=kl.

The integration along these contours is accomplished with an adaptive interval-widthWmberg integration routine [45] which evaluates three- and fivepoint Romberg quadratureformulas, and cuts or expands the interval size according to an estimate of the error. Thecontour in Fig. 5-2 is used for (lzl + Iz’1) > 0.5p and the contours in Fig. 5-3 are used withthe Ihnkel-function form of the integrals for smaller Izl + Iz’]. The algorithm in NEC canbe used with p = Oor Izl+ Iz’] = Oand for R approximately in the range 10-5< k@ e 200.The limitation for small R is due to the changes in the integrand near the origin, relativeto the large convergence distance, and the limitation for large R is due to oscillations in theintegmnd as a result of not following the exact steepest-descent contour. The accuracy hasbeen checked by comparing with independent numerical results from [40] and with seriesapproximations for small R.

5.1.3 Thble-Lookup and Interpolation Techniques

When the Sommerfeld solution for a point source nesr ground is used in a moment-method code the double integration, over the current distributions and over A, can lead toexcessive computation time that severely limits the size of structures that can be modeled.To reduce the computation time in NEC, the field near ground is evaluated by interpola-tion in tables generated in advance by an auxiliary program that evaluates the Sommerfeldintegrals. This table-lookup approach must cover the range of separations of source andevaluation points from essentially zero to the range at which asymptotic approximationsbecome accurate. Two different techniques were used to cover this range in NEC-3 and 4.For small separations of source and evaluation points the singular component of the field issubtracted in an analytic form, and linear interpolation is used on the remainder. At largerseparations, the most rapid spatial variation in the field is due to the phase change. Thephase fi.mctions can have a complex form, represented asymptotically by propagation alongtwo separate ray paths. For these larger distances, NEC uses an interpolation algorithm inwhich functions suggested by the asymptotic approximations to the field are fit to numeri-cally computed field values. This approach, which can be considered a form of model-basedparameter estimation, requires substantially fewer field evaluations than would polynomialinterpolation.

The linear interpolation technique in NEC-3 and 4 is similar to that in NEC–2. However,since NEC–2 was limited to source and evaluation points above ground, the field dependedon ordy two parameters: the sum of the distances of the points from the interface andtheir radial separation. The field transmitted across the interface depends on the individualdistances of the points from the interface, so a three-parameter interpolation is needed. Asin NEC–2, the singular terms are subtracted from the field in analytic form. This stepoffers two advantages. The remainder functions to be interpolated remain smooth as theseparation R goes to zero, so that the point p = z = z’ = O can be included in a table withuniform increments. Also, the integrals of the R-2 and R-3 terms, which have the sameform as in the free-space field, can be evaluated with high accuracy using the equations from

39

section 4. The accurate evaluation of these singular terms makes it possible to model wiresvery close to the interface. The remainders from the field components after subtracting thesingularities are

E;z = E;12 – S;12 – %2

~;~ = E:2 – S;2 – S;:2

J!$$ = E#2 – S;2 – S;72

E;~ = E;2 – S$2 – S;~2

E:: = E:2 – S~2 – S;72

where the R2 and R3 singularities in the fields are canceled by the

(5-9a)

(5-9b)

(5-9C)

(5-9d)

(5-9e)

terms

–jw~ 2

[( )

-jk2R

S;2 = —— (z- Z’)2 “~ – k:$+J 1–1-jk2R+~R2 ~4T k;+h#

–jwp~ 2

[(

23 .3k2 zS;2 = —— 42) 1

~-jhR

—+J~– –1–jk2R ~47r k;+k$ p R2

–jw~ 2S22= ——~-j~R

(1+ jkdl)~41r k; + k:

S:2 = =*[’(’-’’)(++k’)l=)l=which have the form of the free-space fields. The remaining R-l singularities are canceledby subtracting the terms

e-jk2R

R

with R = ~ + (z – #)2]112, sine = (z – z’)/R and

k:–~cl=—

z kf-k;

k:+k;’ C2 = ‘2 (k; + k;)2 “

40

When hl? is sufficiently small, the subtraction is done in the integrands of the Sornmerfeldintegrals, as shown in Appendw H, to avoid loss of precision. When lklRI is not small thesubtraction should not be done, since the subtracted terms do not include the attenuationthrough the earth, and precision may be lost by subtracting and adding the much largerterms.

In NEC-2 only the R-2 and R-3 components of the field were subtracted and theremainder was multiplied by R to get a finite value at R = O. When this is done, the limitas R goes to zero depends on the ratio of (z + z’) to p. ln NEC–2, the interpolation wasdone in R and OCoordinates so that a limit could be defined at R = O. For the field crossingthe interface it is difficult to choose a set of three parameters such that the limit for R -0can be defied, the parameters range between constant limits and the tables with differentinterval sizes can be fitted together. Hence the added complexity of subtracting the R–lterms was necessmy. The remainder is multiplied by R. so that the Va3ueis zero at R = O.

The interpolation space in p, z and z’ is dhided into three regions for increasing depthof the source. The ranges of z’ for these tables are O < Iz’] < 0.4Lg in table 1, 0.4Lg <Iz’I g Lg in table 2 and Lg < Iz’1 < L~ti in table 3, where Lg = min(lkz/kl 1,0.3)~o andLW = rnin(151kz/kl 1,1.)Ao. The limits of p and z in each table are O S p s LC and() < z < LG.Before being stored in the tables, the field values are divided by functions toreduce the variations and make interpolation essier. For table 1 the singular components aresubtracted during evaluation, leaving a remainder that is finite as R approaches zero, butsubtraction is not done in the other tables. The values are then multiplied by

F1 =&jhR

f72 =R2#[k@2+’2)1/2+kIl#\]

F3 =~2e~[h’+kl(p2+zn)1/~

for tables 1, 2 and 3, respectively, to reduce the phase variation and improve the accuracyof the interpolation. These transformations are reversed after the values have been obtainedfrom the tables by interpolation.

The transformed field values are then stored in tables, and the values needed are obtained by linear interpolation, using the formula

?(P, z, ~) =1

(Pi – Pj)(%k - Zt)(#YI – ~~)[(P-W- Zf)(z’ - Wih

+ (p –-pj)(z – zt)(% – Z’),f~~ + (p – ~j)(zk – Z)(z’ – z~)fih

i- (~ – Pj)(zk – Z) (Z; – Z’)f~n + (pi -p) (Z – %t)(~ – ~n)fjkrn

+ (Pi – P)[z – zf)(z~ – z’).f,,kn + (Pi – P)(zk – z)(z’ – z~)fj.frn

+ (Pi– P)(zk – Z) (ZA – Z’).fjtn 1where A S p s pjl zk < z S ZEand z~ ~ z’ ~ z: ~d fikrn = f(P~#zk) z~)” If the sin@lfiterms have been subtracted, the S’ components are added to the interpolated values, and the

41

:.,

result is integrated over the current distribution using adaptive WmbergThe integmls of the singular S terms are evaluated using the free-spacesince these terms have the form of the free space field.

In NEC–2, cubic interpolation is used in the tw~parameter tables.

quadrature [45].field algorithms,

~lS could givebetter accuracy than the linear interpolation in NEC-3, or alternately allow larger parameterincrements. However, a three-dimensional cubic interpolation was not used to minimizecomplexity and evaluation time.

Inte~otation Using Model-Based Pammeter Estimation

For larger separations of source and evaluation points than are used in the linear in-terpolation tables, particularly larger values of p, the phase variation cannot effectively besuppressed by dividing by a simple factor, since the phase has a more complex form. Inasymptotic approximations the field is represented by two rays, one reaching the interfaceat an angle less than the totally reflecting angle and diffracting into the upper half-space,and the other reaching the interface at an angle greater than the totally reflecting angleand producing an evanescent field above the interface and a reflected wave in the ground.This asymptotic solution suggests the forms for functions that can be used in a model-basedinterpolation procedure, in which functions chosen on physical or mathematical grounds tobe close to the expected field behavior are fit to computed field samples. If suitable model-based interpolation functions can be found, the number of stored data values can be greatlyreduced from that needed with simple polynomial interpolation.

In the asymptotic model, the rays have phase factors of the form

PI (p, z, z’) =e-F(A1)

Z3(p,z, z’) =e-F(A2)

whereF(A) = (Az – k~)l/2z - (A2 - k~)lf2z’ + j@

and Al and J2 are saddle points of the integrand, which are solutions of the equation F’(A) =O for the particular values of p, z and z’. The solutions are chosen so that Re(A1) < Re(k2)snd Re(kz) < Re(A2) < Re(kl). In addition, the field will involve spreading factors, such asR;” with P1 and-~:” with ~, which can be approximated as

RI =~z + (%– lk@ll%’)2]1/2

R2 =~z + %’2]1/2

and angle factors such as z/R, #/R, p/R, etc. The field maysum of functions

Nf

@(p, z, z’) = ~ A. fn(p, z, z’)n=1

then be approximated by a

where each Jn is a product of a phase fac~or, a spreading factor and an angle factor. Thecoefficients An are determined by fitting E to values of the field determined by numerical

42

t

evaluation of the Sommerfeld integrals at Nt or more points over p, z and z’. In NEC, thenumber of samples exceeds JV~. The over-determined system of equations is then solved bythe method of Q-R decomposition [46]. The resulting J??represents a Iesst:squares fit tothe computed field values. With the constants An determined in this way, E can be usedto interpolate or extrapolate from the computed points that were fit. If the f. are goodapproximations to the field behavior, then considerably fewer computed values should beneeded for a given accuracy than with polynomial interpolation.

In NEC-3 and 4, this least-squares approximation technique is used with the sum offunctions

(5-lo)

where

Fl(p,z, z’) =R;l(%) (i)

~-i(&+kl%)”2

F~(p, z, z’) =R;l(+)(:)2

~- j(~z+kl lZz)112

()F3(p, z, z’) =R;l@ ~ @pW+klR2)l/*

F’(P, z, z’) =R;l~-l(%) (+)

~-j&z+kll?2)112

()F5(p, z, z’) =Fl (p, z, %’) ;

()~!(~,~,d) =~2(/2, %,%’) ;

()F7(p, z, z’) =F3(p, z, %’) ;

()~8(~, Z, Z’) =~4(~, Z, %’) &

with @I = (k? - ~l)1J2, & = (~ - A1)1J2 and A = (k: – A~)112. The saddle point Al isfound by solving the equation F’(Al) = O for the solution between O and kz, while for AZthe approximate solution A2 = klp/(p2 + z“)l/2 is used. An additional factor of p/R1 isincluded in all terms of (5-1O) for fitting the field components E~ and E:. Also, a factor of

p/Rl is included in the coefficients of Al,rn,rafor fitting -@’. These forms were determinedby studying the interpolation errors in using (5-10) while varying the spacing of the fieldsamples.

43

1- 01

z/Ao

● ***

● @*o0000

L~ :f ● 000

o0 L~

o2p/A(J

Fig. S-4 The threesubregionsusedin the least-squaresapproximation.The samplepointsin p and z areshownfor eachsubregion.

In the present code, this least squares approximation is used from the outer boundary ofthe interpolation region out to 3A0 in p, 2A0 in z and to a source depth of Iz’I = 0.8L9, whereLg = min(l kz/kl 1,0.3). This region is divided into three subregions m shown in Fig. 5-4.The field sample points used in determining the constants in (5-10) in each subregion aregiven in the following table, and some of these points are indicated in Fig. 5-4. An additionalset of samples near the interface, with small spacing in p, is included for subregions 2 and3 to sample the wave traveling mainly through the ground. This wave is not significant inregion 1, and the functions Ft in (5-10) are dropped for this region.

The functional approximation using (5-10) over subregion 3 for a dielectric ground isshown in Fig. 5-5. Both the low frequency (PI) and high frequency (R) components arepresent and are matched by the approximation. As can be seen, this method is effective forextrapolation as well ss interpolation.

‘lhble 1. Rangesof samplepointsusedin determiningthe coefficientsin equation(S10) to approximatethe fieldovereachof threesubregionsin the p, z, z’ space.

Region 1:

Region 2:

l%gion 3:

General Samples

Start A N

o. 0.2 4L~ 0.2 40. 0.4Lg 2

L9 top=O.6 50. o.5Lg 40. 0.4Lg 2

0.6 0.2 50. 0.2 40. 0.4Lg 2

Surface Samples

Start A N

o00

L9 0.2L9 60. 0.4L9 20. 0.5Lg 3

0.6 0.2Lg 80. 0.4L~ 20. 0.5L~ 3

44

Io.

I

.5 -1-l+1.

i

‘3.

Fig.%5Field transmitted across the interface, E], with 61 = 16, ~1 = o S/m and sOurce depth OfZ/ = i).1~. A total of 88 points are fit (36 in the dane of this figure with visible ones shown) to cover thethree-dimensional region 0.6s dAo <3.0, 0< z/~0 S 2.0 and 0-< lz’1/k <0.25 with 1- than 4 percenterror.

5.2 Asymptotic Approximations for the Field over Ground

When the separation of source and evaluation points is greater than about a wave-length, asymptotic approximations for the Sommerfeld integrals become practical. Withh:gher-order terms these approximations can yield good accuracy for separations as small ashalf a wavelength for some parameter ranges. The resulting formulas for arbitrary locationof source and evaluation points are not simple. The evaluation times for general 6symp-totic approximations are substantial] y less than for numerical evaluation of the integrals,but greater than the time for evaluation by table-lookup and interpolation. The asymp-totic approximations become more accurate with larger separations of source and evaluationpoints, and hence can be used over the large region of space, out to infinity, that could notbe covered with lookup tables.

Asymptotic approximations for the Sommerfeld integrals are obtained by the methodof saddle-point integration on the steepest descent path. Uniformly valid approximationsfor all locations of source and receiver have not been attempted here, due to the difficultiesintroduced by branch points and poles in the integrand. A uniform approximation for sourceand evaluation points approaching the interface is developed, using the modified saddle-pointmethod to account for the pole. For small radial separations of source and receiver the higherorder approximations fail, but the first-order forms remain valid. Correction terms to improvethe accuracy of the first-order approximations for small p are obtained by interpolation whenpossible.

45

5.2.1 The Steepest-Descent Method

Forasyrnptotic evaluation of the field we will consider the case of a buried source atZ (/ < O) and an elevated receiver at z as shown in Fig. 5-1, and it will be assumed thatRdkl ) > R.dkn). Fields for other positions of source and receiver can be obtained throughthe formulas in

where VI= (A2

.-.‘ “Appendix G. The Sommerfeld integrals have the general form

S(p,z, %’) =r

T(A)H$) (Ap)e71ti-w’ d~ (5-11)

– k~)1J2, ~ = (J2 – k~~12 and the contour of integration is along the real-.axis; just below the axis for A S O and just above the axis for A > 0. Assuming that pis sufficiently large that IApl >> 1 for values of A making a significant contribution to theintegral, the Hankel function in (5-11) can be replaced by its large-argument asymptoticform

whkh yields

s(p, z, z’)

This integral has the form

which is a standard form for

()2j1/2

~-jAP (5-12)#)(AP) w ~

= (%)1’21:A-1’2T’A)e’l’’-mz-’ApdA”

J

m

I G(A)e-Ft~J d~=-00

evaluation by the steepest descent technique.

(5-13)

The procedure of steepest-descent evaluation of an integral is described in many text-books. The discussions in [47] and [48] address the problem of evaluating the field near aninterface. The basic procedure is to find a solution for ~~ such that ~’(~s) = O. The point& is then a saddle point or minimax point of F. For example, there will be a path through& along which the real part of F is a minimum and increases most rapidly away from As.Normal to this direction of maximum increase %(~) will decrease most rapidly away fromA.. Due to the properties of analytic functions, Ire(F) is constant along the paths on whichRe(F) increases or decreases most rapidly. Hence the direction of most rapid increase in

Re(F), with Ire(F) constant, is an optimum path for evaluating the integral in (5-13), sincethe exponential will decrease rapidly away from ~g, without oscillation. The objective isthen to deform the origimd integration path to pass through & along the steepest descentpath (SDP) of the exponential, taking proper account of any singular points encounteredduring the deformation.

When F’(&) = O and F“(As) # O, an asymptotic expansion for the integral can bedeveloped by making a transformation for the integration variable such that

F(A) = F’(A.) + S2. (5-14)

The integral (5-13) then becomes

I1 = e-~(~s) m g(s)e-” *

-ccl

46

(5-15)

where

g(s) = G(A):.

The SDP in the s-plane is the real axis, with the saddle point at s = O. For a sufficientlylarge parameter in F, the exponential in (5-15) will decay to a small value over a regionabout s = O in which g(s) is slowly varying.

An asymptotic expansion of the integral (5-15) is obtained by expanding g(s), within aradius limited by possible singularities, as

g“(o) s~ + 9(n) (0) Sn + . . .g(s) = g(o)+ 9’(0)s + ~ ““”+7 (5-16)

and integrating term by term. The first-order approximation is

[~ z g(qe-ms) m ~-’2 & = fiq(())e-w.J-m

From (5-14), the quantity cfA/ds occurring in g(s) is

Applying L’H6pital’s rule to evaluate the limit as s goes to zero yields

fith ~g({) ~ud to the argument of d~ on the SDP. Hence

{

2g(o)= G(~g) ~/f(A8)

and the first order approximation for I is

()1/2

I - $ e-FOGo (5-17)

where

F.= SF(A) and G~ = *G(A) .A=A# A=A,

5.2.2 Steepest-Descent Evaluation of the Sornmerfeld Integrals

Evaluation of the Sommerfeld integrals in the form of (5-13) by the method of steepestdescent is complicated by the presence of branch points, which occur in both F(A) and G(A),and poles in G(A) in the integrzd for VIQ. For all field components, the function F(}) is

F(J) = (A2 – &)112Z – (A2 – k~)l/2z’ + j~P.

47

Branch points are located at +kl and *k2, so that ~(~) is represented by a four-sheetedRiemann surface.

The saddle points are solutions of the equation

F’(A.) = A,(A!– k~)-1/2z – As(A: – k~)-1/2z’ + jp = O. (5-18)

The contribution to the field integral from the saddle point at & can be interpreted as aray traveling at an angle from the normal of 01 in the lower medium and $2 in the uppermedium, where kl sin 01 = kQsin 02 = As. Solutions of (5-18) can easily be obtained in thespecial cases of z’ = O, for which

A.‘k2&=k2sino2

and z = O, for which

A.= ‘1 & = ‘lsinol”

In general, (5-18) may have two solutions on the principal Riemann sheet that maybe foundby solving a quartic algebraic equation in A~. It is usually more practical to find the solutionsby numericaI means, however. The procedure used in NEC is described in Appendm I.

Loci of the saddle points are plotted in Fig. 5-6 for a typical case, where the uppermedium is free space and the lower medium has a complex relative permittivit y of 16 – j16.Some steepest descent paths through these saddle points are shown in Fig. 5-7. The SDPalways passes through a saddle point in the region between Oand k’. Itmay then go to +caon the principal sheet or cross onto a lower sheet at a branch cut. In the latter case, a SDPthrough the second saddle point is needed to get to +00 on the principal sheet. The saddlepoint between O and kQrepresents a ray leaving the ground at an angle less than the angle6== sin-l (k2/kl) for total reflection at the interface, and traveling mostly through air. As zgoes to zero this saddle point approaches kQ,and the SDP closes into a branch-cut integralin the direction of steepest descent from k2. The ray then represents a lateral wave when thesource and receiver are in the lower medium. The saddle point beyond k2 represents a raythat reaches the interface at an angle greater than dC,and the field produced in the uppermedium is exponentially attenuated for increasing z. As # goes to zero this saddle pointapproaches kl and the integration path closes to a branch cut. This is the lateral wave whenthe source and receiver are in the upper medium.

The branch points at *kl and &k2 also occur in G(A). However, they are not a limitationin the series expansion of g(s), since they are removed by the transformation in (5-14). Atransformation A = kQsin w or A = kl sin w is sometimes made to explicitly remove a branchpoint. These transformations do not change the result of the steepest descent evaluation,but put the result in a form more readily interpreted in terms of rays.

For the Sommerfeld integrals involving VIQthe function G(A) has a pole in the regionbetween A = O and kQ that approaches k2 as kl/k2 becomes iarge. This pole is not crossedin deforming the integration contour to the SDP. However, the pole does affect the steepest

48

.

k2 \ IK-. \ I\ \h 1

\ \I AR

I “\\ ‘~, ‘,II

.- \ \ \ I.x \.. . . . \ I

1I

/b x.-,1-

Fig. !5-6 Saddle point loci for transmission across an interface, where : = z/(z – z’).

(j=o”

I..----=--------

---”~---~#-..- Au~

/’<“-

/,..”

,..“...’

,./ ;

11

/’/\\ ““”K\\\\\\*\

\’\%.\

Fig. S-7 Steepest.-desmnt paths for a source at height z‘ in the ground and evaluation point at height z in

airwith radial separationp, whereZ= z/ (z – z’) = 0.33and O= tan-l [p/(z – Z’)].

descent evaluation when z and / are small,so that the saddle point is near to k2. The

contribution due to the pole is known as the surface wave.

The asymptotic evaluation used in NEC does not treat the general case of a functionwith two branch points and a pole. The contribution of the saddle point between Oand k2 isevaluated with second order terms, and including the effect of the pole. This wave is usuallythe dominant contribution to the field, and the evaluation yields accurate values down tothe interface for p as small as A/2.

49

The saddle point associated with kl is assumed to be located at & = klp/~w, SOthat the wave is assumed to propagate to a point on the interface directly below the receiverand then up to height z. This approximation for the saddle point is justified since the fieldattenuates exponentially with increasing z, and hence the contribution is negligible unless zis small. The contribution from this saddle point is only evaluated to first order. Hence, theterm vanishes as z and # go to zero. In fact the field traveling in the lower medium does getsmaller at grazing incidence than when z’ is nonzero, but it does not vanish. An asymptoticapproximation includlng the waves above and below the interface and the pole contributionhas been developed by King [49] for z = O and Iz’1 << p. King demonstrates the interferencebetween the two waves when the loss tangent of the ground is small. A general extension ofthis approximation for nonzero z and Y would be difficult, however.

In summary, when the source and receiver are in the lower medium the asymptoticapproximations in NEC will include direct, reflected and lateral waves, while for source andreceiver on the interface or in the upper medium only the direct and reflected field withsurface wave are included. This should be sufficient for most antenna applications, since thewave in the lower medium is usually strongly attenuated in a 10SSYground. However, theapproximation cannot be used when the lateral wave in the ground is needed, as for diagnosticpurposes. Of course, all terms are included over short ranges where the Sommerfeld integralsare evaluated numerically.

The asymptotic evaluation also becomes dWicult as the rays approach normal incidence.When z = O and kl and k2 are real, the saddle point & = klp/~~ approaches the

branch point kz, and coincides with it when p/{~#- = sin 0. = kz/kl. The raysthen merge in a caustic near 13C.Asymptotic approximations in this region are discussed in[48]. It is shown that the function G(J) in (5-13) cannot be treated as slowly varying atthe saddle point, since the transmission or reflection coefficients have an infinite derivativeat 8=. The asymptotic evaluation in [48], taking into account the rapid variation of thereflection coefficient near 8=, results in a modified saddle point, so that the reflected wavein the lower medium develops a lateral shift as it merges with the lateral wave. When thesaddle points do not merge, the effectiveness of the steepest descent evaluation may still bereduced, since when z and p are small relative to Iz’ I the function F(A) in (5-13) does notchange significantly until IAIbecomes greater than Ikl 1. As a result, higher order terms inthe asymptotic expansion may diverge until Iz’1 becomes very large.

!~ Another problem as p becomes small is that the large argument approximation for the;~ Hankel function can no longer be used. However, the first-order asymptotic approximation~~ is valid to p = O. This first-order result may be obtained from geometric optics, without~!~~ restriction on p, as the product of a transmission” coefficient and a divergence factor for1,,? the ray tube ckssing th; interface. Hence the first order approximation is used in NEC

. from angles somewhat greater than t9cto 0 = O. To improve the accuracy of this result,second order asymptotic terms are interpolated from an angle greater than 8= to second,,

,1 order terms for p = O and added to the first order approximation. The interpolated secondi!‘1

!order terms improve accuracy for small R when z is not too small relative to Idl, but cannotbe used when Iz’1 is large compared to z. Hence, only the first-order approximations are,.

used when z < IZ– it and p/ ~~ is approaching or less than sin Oc. The asymptotic

50

approximations used in NEC are derived below.

5.2.3 First-Order Asymptotic Evaluation of the Field

.The derivation of first-order asymptotic approximations for the Sommerfeld integrals

will be demonstrated for the function V12given by the integral

Substituting the large-argument form of the Hankel function from (5-12), this becomes

(5-19)

Equation (5-19) is then in the form of (S13) with. 1/2

()G(J) = ;

~1/2

k;y2 + k;-yl

imdJ’(A) = (A2 – k;)1i2z – (A2 – k;)112z’ + j~p

F’(A) = A(A2 – k;)-1/2z – A(A2 – k;)-1i2z’ + jp

F“(A) = -&(A2 - &)-3j2Z + kf(A2 - k~)-3j2z’.

From (5-17), the first-order approximation for V12is then

2~Ti4 m #lz’-rQ%-j~JPV12- 2 (5-20)

k1J7z+ ‘lrl (kfI’:3# - k;r;3z)1’2

where 1’1 = (A: – kf)1f2 and 172= (A? – k~)l/2 with & a saddle point obtained by solving($18). A term in (5-20) becomes indeterminate when p goes to zero. However, the resultdoes not blow up, since & also goes to zero in this csse. Likewise, 1’1 may go to zero with #and I’Qmay go to zero with z. To avoid these indeterminate terms (5-20) will be written as

V12 - 2

*#/4

k1r2+ &rlv (5-21)

whereI’1r2#” -r2z-jA.p

*= @(k;r;d/rl - k&;~/r2)1j2”(5-22)

The following formulas, obttined from equation (5-18), are then evaluated to avoid dividingby zero

z’-~+jf- for rl ~ 0

fi-r2 .z 2 .~ for r2 ~ OE=iz%.

()2

t=j 6–Efor & ~ O.

It is seen from (5-22) that the first order asymptotic approximation for V12is zero wheneither 171or 17zis zero. Hence the result will go to zero at the grazing angle, with z = z’ = O.Higher order terms are needed to obtain a valid nonzero result in this case.

~om the above result, the first order asymptotic approximations for the components of “electric field are

These equations are coded in subroutine GASY1.

5.2.4 Higher-Order and Uniform Approximations

Higher-order asymptotic approximations are found by retaining additional terms in theseries (5-16) for g(s). For integration from –00 to 00 through a saddle point, the terms withodd powers ofs vanish. Hence the second-order approximation is obtained with the S2 term.The explicit form in terms of F and G is given in [50], along with several higher order terms.FYom equations (6.2) and (5.20) in [50], the second-order approximation when ~’(~s) = Oand ~“(~~) # O is. . .

2X 1/2

~#w

() Ee-FO(Qo + Q2) (5-24)

whereQ.= Go

Q2 = & [GO(5F~ – 3FzF4) - 12G1FQFs + 12GQF;] .

Second-order approximations for the field components are given in Appendix J. These formscannot be used for small p due to the approximation of the Hankel function and otherrestrictions noted in Section 5.2.2.

Higher-order asymptotic approximations are also developed in Appendx J for the case

P = O, by integration on the SDP from the limit A = O. For coordinates (p, z, z’) with

O < p < sin OA~p2+(z– Z’)2, where 6A is somewhat greater than Oc, the first-order ap-proximations are evaluated. Second-order terms are then interpolated from p = O and

P = sin 8A ~ + (z – z’)~ and added to the first-order result. This interpolation improvesthe accuracy when the limitation is the approximation of the Hankel function. It cannot

52

.,.. . ,. ..,.........+.—.waw.x.x...,.- ,W, _,&,

help when the saddle points merge or the convergence of the exponential is slow due to pand z behig small relative to Iz’1. In the latter cases only the first-order approximations areused in the code.

The steepest descent approximations converge slowly when z and Iz’I are small relativeto p due to a pole in the integrand of VIZ. When this occurs, the function G(A) cannot beconsidered slowly varying relative to the exponential, and the radius of convergence of theseries (5- 16) is limited to the distance between the saddle point and the pole. An asymptoticapproximation taking correct account of the pole can be obtained by the modified saddle-point method, as described in (47] and other references.

In the modified saddle-point method, a singular term is subtracted from the integrandand evaluated separately. The integral (5-15) then becomes

1 /1 II= e-F(A’) mg(s)e-s’ ds = e-F(A’) mg(s) -& e-s’ ds + J. (5-25)

-00 –00

wherea = liib(s – b)g(s) = ;~im(~ – &JG(A)

and from (5-14)

b=-”

The singularity is then isolated in the term

r a1P= e-F(As) —

– be“2 ds

_~ s

which from equation 7.1.4 in [51] is evaluated m

fiae-$’(&)Q(+jb) for Im(~) ? o1P= 2kjxae-~(~p)e~c(+jb) = ~

where erfc is the complementary error function

/

2-

‘rfc(’) = 7 ze-t2 dt

andQ(z) = -fizez’erfc(z).

A uniform asymptotic expansion of 1, including the contribution of the pole, can be ob-tained by adding lp to the steepest-descent approximation of the integral in (5-25) involvingg(s) with the pole removed. A more convenient procedure, used in evaluating difhactionintegrals [52], is to use the steepest-descent evaluation with g(s) containing the pole and add

53

(5-26)

where @(z) removes the asymptotic terms of Q(z) of the order that are contained in thesteepestdescent evaluation. The asymptotic expansion of Q(z) is, fkom equation 7.1.23 in[51],

Q(z) -- l_~(_l)ml .3... (21)l)(2z2)m

Iarg ZI <:.rn=l

Thus for a first-order uniform approximation, & with o(z) = –1 is added to (5-17). Forsecond-order, & with ~(z) = –1 + 1/(222) is added to (5-24).

I

and

In the integral (5-19) for V12the pole at the zero of “ -the denominator is located at

For the first-order approximation, the pole contribution to VIQis

where

b = @p) - F{A.).

VP12is then added to (5-21). The terms added to equations (5-23) to obtain a first-orderuniform approximation for source and receiver approaching the interface are

The pole contribution added to the second-order approximation in Appendix J is

54

I

Since the second-order approximation of the Hankel function was used in the SDP evaluation,the terms added to equations (J-9) through (J-13) for a second-order uniform approximationare

IThe second-order uniform approximations are coded in subroutine GASY2.1

I

(5-28a)

(5-28b)

(5-28C)

(5-28d)

(5-2&)

5.2.5 Accuracy of the Approximations

Some tests of the accuracy of the asymptotic approximations are shown in figures Fig.5-8 through Fig. 5-16. The relative errors were determined by comparison with the resultsof numerical evaluation of the Sommerfeld integds. The result shown as a solid line is fromthe NEC subroutine GEASY which chooses an appropriate approximation for the givencoordinates of source and receiver. The basic approximation is the second-order uniformresult, including the surface wave. When 6 = tan-l (P/lz – z’]) is less than dA = 11.3 degreesthe first-order approximation is evaluated and higher-order terms are interpolated between0 = O and 8A. The error for the first-order asymptotic approximation is also shown on theplots. When the source is located at the interface (z’= O) the relative errors for the Nortonapproximations [36], used in NEC–2, are shown as a dashed line. The Norton approximationsare limited to source and receiver on or above the interface.

Errors for aground with:1 = c1–@l/wco = 16–j16 are shown in Fig. 5-8 through Fig.5-13. In most cases the errors from the NEC asymptotic approximations are substantiallysmaller than those from the Norton formulas or the first-order asymptotic results. TheNorton formulas sometimes converge to a constant error, but this error level would decrease

The convergence is slower when d and 2 = z/[z – z’I are both small. When Z is less than0.05, NEC uses the first-order approximations, since higher-order terms increase the error.The first-order result for l?: vanishes as O goes to zero, since it represents only the R-lterm and the field has only higher-order components at @ = O. Hence this importmt fieldcomponent is lost when source and evaluation points are both in the ground on a verticalline. The table-lookup algorithms are used to a depth of a free space wavelength for small0 to partially fill this gap. Interactions with the interface involving E: from greater depthin lossy ground can usually be neglected in antenna applications. For EI = 16 – j 16 theattenuation is about 10‘1° at R/Ao = 2.

55

The errors along the interface when ?I = 16- jO are shown in Fig. 5-14. Both NECand Norton approximations show poor convergence since they neglect the lateral wave inthe ground. The R-l component of this wave will appear in the NEC approximation asthe source is lowered into the ground. The reaI parts of the field components are plotted inFig. 5-15. It is seen that the lateral wave in the ground is the dominant term in E$, whichaccounts for the lack of convergence for this component. The errors for a dielectric grounddecrease rapidly with height, as shown in Fig. 5-16.

The algorithm presently used in NEC to evaluate the field near ground, combining linearinterpolation, least-squares approximation and asymptotic approximations, has been chosento supply accurate field values for typical antenna applications, such as antennss on groundstakes or with buried or elevated screens. It can be seen that there are cases in which largererrors can occur. One such case is the interaction between two points at a significant depthin the ground, where the interaction via the interface is near the totally reflecting angle.The attenuation through a Iossy ground should reduce the importance of this interaction.It should not be too difficult to add to or extend the algorithms to improve the accuracy.If necessary, a more extensive interpolation table, involving only two parameters, could beused for evaluating interactions between points that are both in the ground.

5.3 The Reflection Coefficient Approximation

The image equivalent of a source over a ground plane provides a simple and fast way tomodel the effect of ground on an antenna. If the ground is perfectly conducting, the structureand its image are exactly equivalent to the structure over the ground. Since the image onlydoubles the time to compute the field, it is always used with a perfectly conducting ground.NEC also includes an image approximation for a finitely conducting ground, in which theimage fields are modified by the Fresnel plane-wave reflection coefficients. Although thismodel is far from exact for a finite ground, it has been shown to provide useful results forstructures that are not too near to the ground [11, 32, 33]. When it can be used, the reflec-tion coefficient approximation is two to four times faster than the Sornmerfeld/Asymptoticmethod and avoids the need of computing the interpolation tables.

Implementation of the image and reflection coefficient methods in the code is relativelysimple. For a perfectly conducting ground the Green’s function in the kernel of the integralequation becomes the sum of the free-space Green’s function for the source and the negativeof the free-space Green’s function of the image. The negative sign results from the reversal~f the sign of charge on the image. For the electric field, with free-space Green’s dyadlcG(r, r’) defined in (1), the Green’s dyadlc for a perfect ground is

whereG1(r, r’) = ‘~R o~(r, ~R - r’)

(5-29)

(5-30)

and ~R .~+~. iiii is a dyadlc that produces a reflection in the z = Oplane when usedin a dot product. Similarly, for the magnetic field with free-space Green’s dyadic

F(r, r’) = ~ x V’g(r, r’)

56

1

.1 \*.=- ---- __ ----- ----- I

Norton

2nd order asymp.

I I 1 1

2 4 6 e 10

R/Ao

I

qI I

“’”r\\ d“= 90°

%-- -.. ---- ---- ----- .

I 1 I IE:

t?= 90°

------ ---------- --

1 I I II 2 4 6 e

R/Ao

lE-4 I I I I I

e z 4 6 e:

R/Ao

e = 90°

.01 1 t I I

0 2 4 6“9R/Ao

I

Fig. 5-8 Relative errors in the second-order asymptotic approximation including surface wave and theNorton approximation for z = d = O. The complex permittivity of the ground is 16 – j16. The error for

l?: is the same ss for E~.

57

1 I I 1 I I I..-. El-....

..... e = 50”

.1? r

g

.? .01: ---’.~

&

2nd orderb

Eg=16–j16lE-4 I I I I

a 2 4 6 e 10R/A.

1 1 I I I IE:

. 1

.001

lE-4

R/A.

1 I I I I I 1E?

e = 50”...

. 1 -“”.. r. ....\ .. .

~ ,&g .01:~

2

.EIElla

lE-4 I I I IB 2 4 6 e 10

1

. i

lE-4

lE-5~1

R/A.

1 I I I t 1 IE:

....... e = 500.... ...1- ........ ....... ... ..... ......... ... ... ...... .... .. .‘. -..

----j .O1- ---- ---- ----

$.-”+.eEii -IX

lE-4 -

lE-5 I I1 I I I

e 2 4 6 B 10R/A(I

Fig. 5-9 Relative errors in the first-order and the second-order plus surface wave asymptotic approximationsand the Norton approximation for 0 = tan-1 ~/(z – z’)] = 50° and z’ = O. The complex permittivity of the

ground is 16- j16.

1 I1 1 I I I

.... E;....

.17

.01 : ‘-.‘.

001-

lE-4 I I I I I 10 2 4 6 B 10

R/AQ

I.001 I I 1 I I

0 z 4 6 8 10

1

.1

. ml

1 I

JI 1 I I

@i

r\

\

IlE-4 I I 1 1 I

e i? 4 6 e 10Rj~o

Fig. &10 Relative errors in the first-order and the second-order plus surface wavessymptiticapproximationsand the Norton approximation for 8 = tan-l~/(z – z’)] = 20° and z’ = O. The complex perrnittitity of the

~ud is 16- j16.

59

la

i

. 1

.01

.001

1%

1

. 1

.01

.WI

zg=16–j16 E;

\

e = 10”

.....,.

\\ ~. Norton 2nd order

‘.---

-------- ----

I I 1 12 4 6 B 1

R/A.

0 2 4 6R/A.

10

1

. 1

.01

.001

B 10

I I II .................................................

.e = 10”

----- ---- --

i, 4 6 sR/~0

10 I

J

17

. 1:

.01;

.001 I I 1 I0 2 4 6 8 1

.$=1(J”

\\

k .%,W.-.

-“----- .-

}

R/A.

1 1 1 1 IE:

r

........

......\

\N .

‘.--~

--- ---------

iI I I 12 4 6 e 10

Fig. 5-11 Relative errors in the first-order and second-order asymptotic approximations and the Nortonwp~tion ~r e= tan-l~/(z-d)] = 10° and z’ = O. The complex permittivity of the ground is 16 –j16.

60

1

.i

1

. 1

lE-4

lE-50

I I I 1

Zg= 16 –j16 E?

0=0°---,~---- ----- ----- --

Norton

2nd order

2

I I I 1E;

.<.. 0=0”+... .... ~.=.., Istorder-.<..

-“< ..:.: .~.=.=.= ...................\ ----- -

I I I I2 4 6 e

R/Ao

4

R/Ao

.

6 6:

1

1

lE-4

lE-SI

I I I IE:

\. 0=0°“L\

- .- ‘---- ---- - ------ ._.- ._._ ----

I I I I2 4 6 e

Rf~o

Fig. 5-12 Relative errors in the first-order and second-order asymptotic approximation and the Nortonapproximation for O= tan-l ~/(z - z’)] = 0° and z’

= O. The complex permittivity of the ground is 16 –j16.

The f?~ and l?: components are zero.

61

..-. . . . . ..-. — -—.,amw.”e-.m... w.. -“! ,.., . . . , .“ . . . . . . . m— ~,...-, ,,” . . . . . . ,“., . . . . . . . ..!.h——.—— . . . ..—..—.w-”-—

w

1 1 I 1i~= 16- j16 E:

.01 I I 1e z 4

R/Ao

t I

ig = 16- j16 E:

I Iz 4

R/A.

8=1°

1 I 1 I0 2 4 I

R/A.

1; I 1 IbFg= 16- j16 Ef

0=1°

. 1- .

.01 I Ie 2 4 6

I 1

e=lo

R/AI

Fig. S-13Relative errors in the first-order asymptotic approxirnatioxw for O= tan-l [p/(z - z’)] = 1° andz= 2/(2 - z’) = 0.01. The complex perrnittivity of the ground is 16 – j16.

Rel

ativ

eE

rror

.b

ww

I

- —

Rel

ativ

eE

rror

Iw

wI

I4

Rel

ativ

eE

rror

“m.

*w

(

is

Rel

ativ

eE

rror

I.-

—--

-.— -5--

--- —

. ---.

.- A?-:-

--

---

.-

---=

----.-

.---=

--

I 1 1

200 I t 1 I 50a1

;g = 16 – jO E;

1- \

E;

40015El- 8 = 90° - 8 = 90”

leQJ-

La i >.w 5@- ? l@flYti 2

Ele-

-lea-50- -200

? I ~- ~-100

0 2 4 6 e 10Ri~o

50 I 1 1 I IE:

40- a304 I 8 = 90”

1

20-

le -k-a @-Q7K-l El-*

-28- 1-30--40--50-

-300 ! 1I I I 1

e 2 4 6 e laR/A.

40 i I 1 I

20-

0-

%: -20-

z~ -40-

-60-

-80-,

-60 ~ ‘I

I I 1 -100 /I I I 1 I 1 Ie 2 4 6 8 10 0 2 4 6 0 10

R/A. R/A.

Fig. tL15Red part of the field components computed by numerical evaluation of the Scxnrnerfeld integrals

( ) and the second-order asymptotic approximations (– - – -) for z = z’= O. The complexpermittitity of the ground is 16- jO.

64

la 1 I 1 I 1 I

Zg =16–jO E; E,.“”. 0 = 80° :..,“ .,

1: .. r

~l\,\

d l\w 1-: \..:” \

i -----.01 ---

2nd order1

.001 1 I I I Ia 2 4 6 e 10

Fie.

R/AcI

I I I 1E;

#1..r 9 = 80°

........ ... ....... .... ..... ......

I I I I2 4 6 0 1

1 I 1 4 1 I 1... E?........ .. ...

““”..... ..... ..... ,, .,,. 0 = 80°‘...,,. ...... ..,, .,,, ,,,

. 1:

g I\,\~

? .01?z!2

.001

J

lE-4 [1 I 1 I

0 z 4 6 e 10R/A.

1’. 1 1 1 I

“.., E:\ ...

..A ... 0 = 80°

“....1: ‘- . ““””””’”’.......... .

... .. ..... .. ..... .,. ..,,,,. ,.~ ---

----C5

---- ---

$ .017

32

.Elel:

lE-4 I 1 I I0 2 4 6 8 1

R/AII

1

. 1

lE-4

lE-5

I I 1 1...... E?

..........................................................

e = 80°

I I I 1e 2 4 6 e 18

R/Ao

R/AII

.16Relative errors in the first-order and the second-order plus surface wave asymptotic approximationsan~the Nortonapproximationfor 6 = tin-l ~/(z – z’)] = 80° anclZ’ = O. The complex perrnittivity of theground is 16- jO.

65

the Green’s dyadic over perfect ground is

~Pg(r, r’) = f’(r, r’) + ~I(r, r’) (5-31)

where

Fr(r, r’) = ‘~R - ~(r, ~R - r’). (5-32)

In the reflection coefficient approximation for finitely conducting ground the imagefields are multiplied by the Fresnel reflection coefficients. The coefficients are derived byconsidering an incident plane wave that producm a reflected wave and a wave refracted intothe opposite medium. The field is resolved into TE and TM components relative to the planeof incidence formed by the incident ray and the normal to the interface. The TE component,with E norrrd to the plane of incidence, will be termed horizontal polarization, and theTM component, with E in the plane of incidence, will be vertical polarization. Matching theboundaqy conditions on the field components at the interface yields the reflection coefficients[30]

(5-33)

RH -(zRms,-~-)=

-“m(5-34)

where cos O= –i 05 and ZR = (cl – jal/~~0) ‘1/2. The signs have been chosen so that both

coefficients go to 1 for perfectly conducting ground, and the reference directions for the fieldcomponents are shown in Fig, 5-17.

q, /41,61

xN

&

W ‘ke

Horizontally Polarized Wave VerticallY Polarized Wave

Fig. &17 Reflection of a plane wave from an interface.

66

S*

The reflection coefficients can also be written in terms of surface impedance and admit-tance for the TE and TM polarizations as

Y~E – Y,TERv=–

Y~E + Y$E

whereY$E = q;~ Cos$, Y*TE= q;l WS et

Z;M= qoCose, Zy = ql (x)SOt

with qo = -, ~ = _ and cosOt = [1 - (kO/k1)2sinfl]112.

The reflected fields due to incident fields Ev and EH, with vertical and horizontalpolarization respectively, are

E:= - RV (~~ . Ev)

E~ = – l?ffEH

and the reflected magnetic field components are

H~ =RvHV

‘~=RH (fR”HH)o

An incident field with arbitrary polarization must be resolved into horizontal and verticalcomponents to determine the reflected field. If ~ is the unit vector normal to the plane ofincidence, the reflected field due to an incident field E is

ER =R~ (E1 . 5) @ + Rv [El – (E1 . @) S]

=RvE1 + (RH – Rv) (E1 . ~) @ (5-35)

where E1 is the field from the image of the source. Use of the image field in (5-35) accountsfor the changes in sign and vector direction of the field that were shown explicitly for thevertically and horizontally polarized cases. For the ma~etic field,

where HI is the magnetic field fkom the image of the source. Dyadlc Green’s functions foruse in the integral equations can be constructed as

Gg(r, r’) ‘G(r) r’)+ &(r, r’) (5-36)

~g(r, r’) =F(I’, r’)+ ~R(r, I“) (5-37)

whereG~(r, r’) = RvG1(r) r’) + (RH – Rv) ~ [~. G1(r, r’)]

67

f’R(r,r’) = &~I(r, r’) + (RV – RH) fi [j5.i?~(r, r’)]

with @ = P/lP 1,P = (r – ~) x ~ and ~1 and ~1 are from (5-30) and (5-32).

The Fresnel reflection coefficients are exact when the wave reflecting horn the groundis a plane wave, as that from a distant point source. The plane-wave reflection formulas alsogive the correct result for the field at a distant evaluation point when the source is near theinte~~, ~ cm be seeq by applying the reciprocity principle. More generally, the sum ofthe distances of the source and evaluation points from the interface should be large for theplane wave reflection solution to apply, although it becomes exact in any case in the limit ofperfectly ,conducting ground. In practice, the reflection coefficient approximation has beenfound to @ve satisfactory results for antennas at least O.lAo above the interface [11, 32, 33].However, it is not suitable for modeling long wires over 10SSYground. In fact, it may producea solution in which the current grows exponentially away from the source [53].

The first-order asymptotic approximations obtained by steepest-descent evaluation ofthe Sommerfeld integrals include the same coefficients as (5-33) and (5-34). However, inthe first-order asymptotic solution the coefficients multiply only the R-l components of thefields, while in the reflection coefficient approximation they multiply the total near field.While there is no clear justification for including the near-field terms in the reflection coeffi-cient approximation, the result has been found to provide a usefid asymptotic approximationfor a field component such as l?~ with p = O, in which the R-l term vanishes. The reflectioncoefficient approximation is only used in NEC for source and evaluation points above theground. When source and evaluation points are in the earth the approximation was found toconverge very slowly, An approximation for evaluating the field when source and evaluationpoints are on opposite sides of the interface can be obtained from the Fresnel transmissioncoefficients multiplied by a factor to account for the change in divergence of a ray tubeon crossing the interface. The divergence factor is generally determined for conservation ofpower in a ray tube assuming a R- 1 field, so its application to the total near field may bequestionable. The transmission coefficient approximation is not in the present version ofNEC.

NEC also includes a reflection coefficient approximation for a radial-wire ground screen.This model is based on an approximation developed by Wait [54] for the surface impedance ofa wire mesh on the surface of a finitely conducting ground. The use of the surface impedancein determining the reflection coefficients in a moment-method solution was demonstrated byMiller and Deadrick [55]. Using Wait’s result, the surface impedance of the ground screen isapproximated as

Zg(p)= ()WIn ~NCO

(5-38)

where N is the number of radial wires in the screen and CO is the radius of the wires.Equation (5-38) is the surface impedance from Wait’s result for a grid of parallel wireshaving the spacing that the radial wires have at a distance p from the center of the screen.The impedance of the grid is combined in parallel with the impedance of the ground which

68

. .-. ————.———L...

for high conductivity is approximately

[ 11/2

(1=M

q)(q – j(7/Lq) “

The screen and ground together then have a surface impedance of

If the intrinsic impedance of the upper medium is ~ the reflection coefficients are

For this surface-impedance approximation for a ground screen to be valid the refractiveindex of the ground must be large compared to unity and the spacing of wires in the gridmust be much less than the wavelength. Due to the assumption of specular reflection,only the properties of the ground directly under a vertical antenna will affect its currentdistribution. At the origin of the radial-wire screen the impedance is zero (Z9 is not allowedto be negative) so the impedance and current distribution of a vertical antenna at the originwill be the same as over a perfectly conducting ground. The far fields will demonstrate theeffect of the screen as the specular point moves away from the origin, although the effect ofdiffraction from the edge of the screen is absent. For antennas other than a vertical monopole,it should be remembered that the anisotropic nature of the screen has not been consideredin this approximation. The extension of the approximation to account for anisotropy in amesh screen was considered in [55].

69

6. Solution of the Matrix Equation

The matrix equation resulting from the Moment Method solution for wires and patchesin NEC is solved by factoring the matrix into a product of upper and lower triangular matricesby Gauss elimination. The triangular matrices are saved and used to obtain the solutions forany number of excitations by a simple process of forwmd and backward substitution. Sincethe time to factor the matrix is proportional to the matrix order cubed, the solution timebecomes critical for large problems.

When symmetry occurs in the structure, such that the complete structure can be formedby reflecting or rotating a subsection, the solution time and required computer memory canbe reduced by decomposing the solution into a sum of eigenmodes. The solution is thenobtained by solving a smsller matrix equation for each mode. The code also can implementa partitioned-matrix solution so that the matrix for a basic model can ~be saved in LUfactored form, snd the solution with additions to that model can be obtained with minimumcomputation. This partitioned-matrix solution, termed the Numerical Green’s Function(NGF), can be used when moving an antenna around on a large model such ss a ship, andalso to take advantage of symmetry in part of a model when unsymmetric parts must beincluded. These matrix solution techniques are described in this section.

6.1 Solution of the Matrix Equation by L-U Factorization

The matrix equation in NEC is solved by factoring the matrix into a product of lowerand upper triangular matrices. The matrix A = [aaj] is factored so that A = LU = [li~][u~j]where L is a lower triarwular matrix with ones on the diagonal and U is an upper triangularmatrix. The matrix equation

which is easily solved by first

and then solving

[aij] [lj] = [Ei] then becomes

[~i~l[~~jl[~jl = [~il (6-1)

solving

[lz~][j~] = [Ei] (6-2)

[%j][~j] = [.fk~. (6-3)

The matrix is factored using a standard Gauss-elimination algorithm [56]. The elementsof the L and U matrices are evaluated by sequentially solving equations resulting from thematrix product. The elements of the first columns of L and U are obtained by solving theequations

all = ull

@l = &lUll, i=2,..., n.

The second columns are then obtained from

alz = U12

aQQ= E21?L12+ u22

(1-iQ= &lU12 + 4?i2U22, i=3,..., n

70

and the general form for column r is

..

* = &lulr + (,2U2, + “““+ tr,r-1%-l,r + ?&r

%, z l?~lUlr +&2U* + “.“+~i,~-lw-l,i +f?if.*, i = r + 1,... ,n.

The coefficients of the L and U matrices can be written over the matrix A in memory andsaved for use with multiple right-hand sides in (6-1). The solution is obtained by evaluating(&2) as

‘i=~(Ei-Eti’f’)‘=’y-n

and then evaluating (6-3) as

Ia=fi– ~u~jIj, i=n, n–1,...,l.j=i+l

In the algorithm used in NEC, row interchanges are applied at each step of the Gausselimination to obtain the pivot element of maximum magnitude. Using the maximum pivotelement scaled by the equation size would be a better strategy. However, the equations in themoment-method solution will usually be of similar size, so this should not be too important.The EFIE for thin wires generally produces equations with diagonal elements several ordersof magnitude larger than elements at least several rows off of the diagonal. Hence, the wirepart of the matrix should be well conditioned, and use of optimum pivot elements may notbe necessary. However, the propagation of roundoff errors in very large matrices, with ordersof several thousand, has not been studied, so ptitial pivoting is retained in the code.

6.2 Solution for Symmetric Structures

Structures to be modeled often have symmetries or repetition patterns within theirstructure that can be exploited to reduce the computation time and storage requirements.Types of symmetry include rotational symmetry of degree NP, where the structure repeatsexactly on rotation about an uis by an angle 2m/~p, ~d reflection symmetry, where thecomplete structure can be formed by reflecting a sub-unit in one, two or three orthogonalplanes. NEC can take advantage of overall rotational and reflection symmetry both inevaluating and solving the matrix equation. It can also take advantage of a situation wherepart of the structure is symmetric but unsyrnmetric parts are also present. A partitioned-matrix solution is used in the latter case as described in the next section.

A structure with rotational symmetry of order lVPconsists of NP identical sections withsection number i rotated from section 1 by an angle 27r(2– I)/ZVP. If the segments and patches

71

are numbered in the same order in each section the resulting matrix has the structure

[

Al A2 A3 & ..” AN,

1[:1 [:1

11 ElAN, Al A2 A3 “ ““ ANP-l 12 &

ANP-l ANP Al A2 “ ““ ANP_2 13 = E3 (6-4)..

is A3 A4 A5 ““” Al 1;, E;P

where, if the order of the complete matrix is N, each Ai is a submatrix of order N/NP.A matrix with this form, where each row of submatrices is the same as the previous ro-wwith a circular shift to the right, is termed block circulant. An immediate advantage ofthis structure is that only the entries in the first row of submatrices must be computed andstored. Hence the computation time to evaluate the matrix elements and the memory neededfor storage are reduced by a factor of NP,

The block-circulant structure offers a still larger advantage in inverting or factoringthe matrix, due to the N3 dependence of the computation time. The solution for an arbi-trary excitation can be obtained by expanding the excitation in eigenmodes of the structurethrough a discrete Fourier transform, and solving for the response to each mode separately.The components of the excitation vector are transformed as

Np

Ei=~Si&, Z=l,..., NP

k=l

Np

&a=$~&Ekj Z=l,..., P’PP k=l

where * indicates the conjugate of the complex number and

& = expvh(i – l)(k – l)/NP]

with j = ~. With this transformation, the total excitation vector becomes

E=

E2 NPE3 =

xk=l

~NpJ

slk~k&k&kJS3k&k . (6-5)

SNPk&k

In each component vector of (6-5) the excitation on sector i of the structure differs from theexcitation on sector i – 1 only by a uniform phsse shift of 27r(k– l)/Np. If the structure hssrotational symmetry, the response to each of these component vectors must repeat with thesame phase shift around the structure. Hence the current response must have the form

11Is

1

NP13 =

x.. k=l

I;p

slk&ssk~k)S3k~k .

...

SNPk~k

(6-6)

72

where each vector in the expansion of I is the response to the component for the same k in(6-5).

Substituting (6-5) and (6-6) into (6-4), it is seen that the solution can be obtained bytransforming the submatrices as

k=l

then solving for the current modes

and finally summing

~ = #&i,

the total current as

i=l,. ... hip

i=l, . . ..Np

i=lf. ... Np.

Thus the solution is obtained by factoring and solving & matrix equations of order ~/~P.In the code, Z is evaluated by factoring A into LU form and then solving.

The same procedure can be used for structures that have planes of symmetry. TheFourier transform is then replaced by decomposition into modes that are even and oddabout the symmetry planes. All equations remain the same except that the transformationmatrices[s~k]We, for one plane of symmetry,

for two planes of s~metry,

S2=

and for three planes of symmetryj

S3 =

11111 –1 1 –111–1–11 –1 –1 1

1 –11–11–111–1–1111 –1–111–11111–1–1

111111111 –1

–1 –1–1 1–1 –1

1 –11–1–11–1111–1–1–1–1111 –1–11–111–1

.

With either rotational or reflection symmetry, the solution is obtained by factoringNp matrices of order N/NP rather than one matrix of order N. For reflection symmetry

73

IVP= 2Nr where N, is the number of reflection planes. Since the time for factoring a matrixis approximately proportional to the order cubed, the relative computation time is reducedhorn N3 to NP(lV/NP)3, or by a factor of N;. me solution time for each right-hand side isreduced by a factor of NP.

A further saving can be achieved if it is known that the excitation includes only a limitnumber of the possible modes. Such cases include an azimuthally uniform excitation (k = 1mode) or a plane wave incident along the axis of rotation (k = 2 and k = NP modes.)The general NEC does not treat these special cases, but a code NEC-GS [57] has beendeveloped to take maximum advantage of the symmetry and azimuthally uniform excitationof a vertical monopole on a radial-wire ground screen.

6.3 The Numerical Green’s l?hnction Solution

NEC includes an option to generate and factor an interaction matrix and save theresult on a file. A later run, using the file, can add to the structure and solve the completemodel by a partitioned-matrix algorithm without unnecessary repetition of calculations. Thisprocedure is call the Numerical Green’s I?hnction (NGF), since it is the numerical equivalentof replacing the free-space Green’s function in the integral equation with a Green’s functionfor the model on the file. The NGF is particularly useful for a large structure, such as aship, on whkh various antennas will be added or modified. It also permits taking advantageof partial symmetry, since a NGF file may be written for the symmetric part of a structure,taking advantage of the symmetry to reduce computation time. Unsymmetric parts can thenbe added in a later run.

For the NGF solution, the matrix is partitioned as

(&6)

where A is the interaction matrix for the initial structure, D is the matrix for the addedstructure, and B and C represent mutual-interaction matrices.

In the first step of the NGF procedure the matrix A for the basic model is evaluatedand factored into LU form. The factors Atand AU are then written on a file along with othernecessary data, including the model geometry and frequency. When a solution is requiredwith an addition to the basic model, matrices B, C and D in (6-6) are evaluated. Then thepartitioned matrix is transformed so that

where ~1 and IIU are LU fadored matrices, SUChthat

~ttiU = D’ = D – CA-lB.

Multiplication by A-l is to be interpreted as solution for each column of the matrix on theright, using the LU factored form of A. For each excitation vector, the solution for current

74

is obtained as12=D’-1(E2 – CA-lE1)

II =A-lE1 – (A-lB)Iz.

A complete LU factored matrix could be obtained from the partitioned matrix, but then theadvantage of symmetry in storing and operating with &Ati would be lost.

Electrical connections between the new and old (NGF) structure require special treat-ment. If a new wire or patch connects to an old wire, the current basis function for the oldwire segment is changed by the modified condition at the junction. Since the elements in ma-trix A cannot be changed, the old bssis function is forced to have zero amplitude by addinga new equation having all zeros except for a one in the column of the old basis function. Anew column is then added in matrices B and D for the modified basis function in the NGFstructure. Hence, connection of a single new segment to a junction of Nj segments in the

NGF structure will result in Nj + 1 new unknowns in the new part of the partitioned matrix.When a new wire connects to a patch in the NGF structure the patch must be divided intofour new patches to apply the surface-junction condition. Hence two equations are neededto set the two current components on the old patch to zero, and eight new unknowns sregenerated for the current components on the four new patches. Hence connection to a patchin the NGF model results in ten new unknowns, plus those for the new segments.

75

7. Extensions to the Model for Antennas and Scatterers

Previous sections have dealt with the problem of determining the current induced on astructure by an arbitrary excitation. We now consider some specific problems in modelingantennas and scatterers, including models for a voltage source on a wire, lumped and dis-tributed loads, nonradiating networks and transmission lines. Methods of calculating someobservable quantities are also covered, including input impedance, radiated field and antennagain.

7.1 Voltage Source Models

Modeling voltage sources is probably the most critical step in the MoM analysis of wireantennas, since errors are directly reflected in the computed input admittance and hence inthe gain and other related quantities. The basic requirements for the source model are thatit accurately represent the desired source voltage and, as closely as possible, duplicate thephysical characteristics of the actual antenna source, including the width of the excitationregion on the wire. In addition the model should introduce a minimum of computationaloverhead. It is highly desirable that the MoM impedance matrix be independent of thelocation of the source so that solutions for several different excitations can be obtained witha single evaluation and inversion of the matrix.

NEC presently includes two models for voltage sources: the applied-field or gap model,and the bicone or charge-discontinuity model. These source models are described in this sec-tion and typical results are shown. A coaxial-line source model is included in this discussion,since this model is in some ways similar to the bicone source model and has been used suc-cessfully in codes similar to NEC. Each of these source models involves either a modificationof the boundary condition to force a non-zero electric field on the wire or a modification ofthe current expansion to produce a discontinuity in charge density across the source, or acombination of these factors. Of the source models now offered in NEC, it will be seen thatthe applied-field source yields generally stable results but, particularly at low frequencies,may have a much wider distribution than the actual source. However, the source width isusually of secondsry importance except for its effect on source-gap capacitance. The biconesource model produces a more localized excitation when used on thin wires, but may giveinaccurate results for input impedance when used on thicker wires or on segments with largeradius-tdength ratios. Hence the applied-field source model is recommended for generaluse.

7.1.1 The Applied-Field Source Model

The simplest and most reliable source model now available in NEC is the applied-fieldor gap source. For a source voltage Vi on segment i with segment length Aa the field at thematch point at the center of segment i is set to

The field at all match points withoutsegments on either side of the source

Ei = Vi/As. (7-1)

sources remains zero. It is recommended that thesegment have lengths equal to that of the source

i’6

,.: ,. . .... ,,., .,.. ..

a)‘*’r————l

Q-iJ 1$

:&l 10-2

b)

‘“4F’---’7102

1

10-2

10-’ ~o. 0.1 0.2 0.3 0.4 0.5

~()-’ ~o. 0.1 0.2 *.3 0.4 0.5

End z/L Center End z/L Center

Fig. 7-1Near ckct.ric field along the wire on a linear dipoleWithfl = 2ln21t/a= 15for: a) an applied-fieldvoltagesourceandb) a biconesource. (FromAdamset al. [58].)

segment. When this rule is followed, the electric field along the wire resulting from a solutionusing the applied-field source model will generally be found to have the prescribed value atthe center of the source segment, remain approximately constant over the segment and dropto smaJl values beyond the region of the segment ends. A typical near-field distribution isshown in Fig. 7-la. The shape of this distribution will depend on the wire radius, sincewith the thin-wire approximation the field cannot change significantly over a distance muchless than the wire radius. However, the line integral of field over the source region typicallyremains close to the source voltage over a wide range of wire radii.

When the source segment and adjacent segments have unequal lengths the voltageresulting from the applied-field model may differ from the prmcribed value, resulting ininaccurate evaluation of the input impedance. Sometimes unequaJ segment lengths can beused without significant loss of accuracy. Good results have been obtained with a shortsource segment on the end of a transmission line connecting to much longer segments on thetransmission line wires. However, in general caution is advised when equal-length segmentscannot be used in the source region. The actual voltage resulting from the model can beobtained by integrating the near field, although the computation will require additionaleffort.

Ideally, the applied-field source model introduces a voltage K between the ends of thesource segment. Hence the antenna input admittance could be computed using the currentat the segment ends, or in an unsynmetric case the average of the current at the two ends.In practice the segment is usually sufficiently short so that the current variation over itslength is small and the current at the center can be used rather than at the ends. Hencethe input admittance is computed as 1~/14 where 1; is the current at the center of segmenti. The use of the current value at the segment center may be a source of error when thecurrent is changing rapidly as at the base of a A/2 monopole. No modification of the current

?7

-.=7

}—za

Fig. 7-2 Thebieonical transrnisaion line source on a wire.

expansion is needed with the applied-field source model, since the normal expansion allowsthe charge density to change sufficiently rapidly over the width of the source segment.

7.1.2 The Bicone Source Model

The bhxme source was developed in an attempt to obtain a narrower distribution ofthe source field than is provided by the applied-field source. The bicone source model issimilar to one used by Andreasen and Harris [59], and its adaptation to a code similar toNEC was reported by Adarns et al. [58]. For this model the source region is viewed as abiconical transmission line, as shown in Fig. 7-2, with the apex of the cones at the junctionbetween segments. The electric field due to the applied source voltage has been shrunk toa delta-function distribution in the bicone, so it is not seen at any of the match points onthe wire. However, a discontinuity in the derivative of current is introduced at the source,as required by the transmission-line model, and this perturbation of the current produces afield that illuminates the entire structure.

The voltage between points at &z on the bicone can be related to the derivative ofcurrent through the transmission-line equation as

~(z) = jzo dI(z)Tx

where 20 is the characteristic impedance of the bicone with half-angle t?,

Z~ = ~ ln(cot 0/2) x 120 ln(2/0).r

The actual biconical shape of the conductor is not modeled in NEC. Hence, since the choiceof cone angle 8 is uncertain for the cylindrical wire, Adams et al. [58] averaged 20 for 6 inFig. 7-2 varying from zero at the source to A/2 at the match point, assuming the segmentson either side of the source have lengths A. The average impedance is then

ZaV,=~~A’212(’)ln(~) d6=120~n($) -1].

Using this value for characteristic impedance and allowing for an unsymmetric current dis-tribution about the source, the discontinuity in the derivative of current at the source for avoltage Vi is

dI(z) dI(z)——x %=0+ dz *=O_=

78

–jkVo

60~n(A/a) – 1]”(7-2)

This discontinuity in the derivative of current is introduced into the NEC solution bymodifying the current expansion on the wire. For a bicone source at the first end of segmentl!, the current expansion of equation (3-8) becomes

(7-3)

where j;(s) is a new basis function centered on segment 4, as defined in section 3.2, butcomputed as if the first end of segment 1 were a free end and the segment ra&us were zero.Hence f; go= to zero with non-zero derivative at the source location. If the part of fj on

segment 4 is

then

:f;(s) = l?; cos(kAe/2) + C; sin(kA4/2).S=.W-AC12

Since the sum of the normal basis functions has continuous values = St – At/2, the current in (7-3) has a discontinuity in derivative of

snd derivative at

[

ho .@ –:I(s) 1] =A[~fCOS(k&/z)+ c; sin(W/z)].S=81- A&/2f~ 8=Sp&/!&t

Hence, from (7-2), a source voltage of VOrequires a value of fl~ in (7-3) of

Pt = ~{Pn(At/at) - l][~~cos(kA@) +C’~sin(kA4/2)]}-1. (7-4)

While the introduction of a bicone source into a model changes the current expansion,it does not change the MoM impedance matrix. The amplitude of the ~~ function in (7-3)is a known value fixed by the source voltage, so the field due to this current function goeson the right-hand side of the matrix equation. The matrix equation then becomes

[Gij][~j] = [Es] + P4[5]

where ~i is the excitation for segment- or patch-equation number i due to the field of ~~, andE~ is the excitation field due to any other sources. Thus the matrix G is independent of thebicone source location, as it is for other sources. The solutionneeded in (7-3) is

[~i] = [Gij]-l{[~j] + B..[~j]}.

for the expansion coefficients

This implementation of the bicone source is easily extended to multiple sources. Theaddition of the basis function j’; to the current exp~sion appe~ to introduce ~ ~ymmetryin the current. However, this is not the case, since the other basis functions have the same

79

.. ... .. ,..,.,,.. . . ..,.w _=..w,=-

—-——..-—-,,,1, ———” -... ...4

II1

Fig. 7-3 The coaxial-line source and equivalent magnetic current frill.

form ss f; except for the discontinuous derivative, and their amplitudes can adjust to producean exactly symmetric current if appropriate.

On a thin wire, the bicone source results in an effective applied field that is much morelocalized in the source region than that of the applied-field source, as can be seen in Fig.7-I. The applied-field source results in a nearly rectangular field distribution over the sourcesegment while the field of the bicone source approaches a delta function. The integrals ofthese two source-field distributions will yield approximately the same voltages. However, asthe wire radius is increased both field distributions become more rounded. For larger wireradius the integral of the field produced by the bicone source will differ substantially fromthe required voltage due to approximations in deriving (7-2), so this source model shouldonly be used with very thin wires.

7.1.3 A Coaxial-Line Source Model

A source model can also be developed to approximate the common practical case ofexcitation of a wire by a coaxial transmission line through a ground plane. This model isnot implemented in the presently released versions of NEC, but results of tests are includedhere for comparison with the other source models. The coaxial source has been used in anumber of wire antenna codes. Thiele [60] uses it with point matching and entire-domainbasis fi.mctions. The treatment of Popovi6 et al. [23] is more compatible with NEC since ituses point matching and a polynomial basis in which a discontinuity can be introduced intothe derivative.

As illustrated in Fig. 7-3, the coaxisl line with electric field

EP(p) = ~ for a<p<bpln(b/a)

in the opening is replaced by the equivalent case of a frill of magnetic current J~@ = – EPover the plane. It is assumed here that only the TEM mode is present in the coaxial line

80

opening, a condition that Popovi6 et d. state is accurate for kb <0.1. The magnetic currentbill produces sn excitation field over the structure. On the axis of the frill the z componentof this field is

VI

(

~-jk% ~-jk&

‘z(z) = 2ln(tJ/a) T – & )

where ~ = (Z2 + l?)112 Thiele [60] gives the field at an arbitrary(z2 + a2)lJ2 and R~ = .location. This field illuminates the entire structure, and hence fills the excitation vector onthe right of the matrix equation.

At the frill location the discontinuity in EP requires a discontinuity in charge densityand hence in the derivative of current on the wire of

dI(z) I dI(z) I –j2~kV0——dz .=W dz .=o_= q ln(b/a)”

This discontinuity in derivative of current is introduced into the current expansion as wasdone with the bicone source. Hence the coaxial-line source involves both a modification ofthe current expansion and a direct excitation due to the impressed field. As the wire radius isreduced the field due to the magnetic-current frill becomes increasingly concentrated in thesource region. The difficulty in sampling this field is a limitation of the coaxial-line sourcefor thin wires.

7.1.4 Comparison of Source Models

Input admittances obtained with the applied-field, bicone and coaxial-line source modelswere compared with the second order King-Middleton results [61] for a dipole with kh = 1.6for half length h and for three thicknesses, Q = 8, 11 and 20. The King-Middleton values arenot necessarily “correct” but are a convenient and widely accepted standard. Second orderKing-Middleton is the result of two iterations of a solution of Halh%’s integral equation witha delta-function source model.

Results from the applied-field source model are shown in Fig. 7-4. For Q = 8, Gconverges, although ten percent fkom King-Middleton. B continues to change due, at leastin part, to the increasing susceptance as the source gap becomes narrower. It has beensuggested [62] that the applied-field source model could be improved by subtracting a gapsusceptance and then adding an empirically determined value for the actual antenna feed.King [63] applied a similar correction to the King-Mlddleton results using the messuredsusceptance at a single frequency. For 0 of 11 and 20 the NEC results show increasinglybetter convergence and agreement with King-Middleton.

The bicone source results, shown in Fig. 7-5, blowup for $1= 8 at 20 segments. A similarblow-up occurs for Q = 11 with 90 segments but is excluded from the plot. The resultsfor Q = 11 appear useable to about 40 segments. At Q = 20 the results me acceptablebut neither relative convergence nor agreement with King-M1ddleton is as good as withthe applied-field source. These results demonstrate the failure of the bicone model withincreasing wire radius or small A/a.

81

10

“IsAG = 1OO(GJVSC- G&)/t%

s ‘,g \ AB = 100(BNiw – B~)/Bk

0x \

\ & = 8.262mmhosCL -5 \3 \ Bk = –4.065 mmhosw -10 \

5 \o! -15

AG\I& \ .

‘3-=A-J20 IEI 20 30 4e 50

NUMBER OF SEGMENTS

6

-1

\I Gk = 7.915 mmhos

~3\\ Bh = -5.036 mmhos

x0

ait

-35ah. -6

.\”

-9>-1 $2=11

0 50 100

NUMBER Of SEGMENTS

2.0

1

\ Gh = 6.666 mmhos1.5 \

g \ Bk = –5.991 mmhoaQ 1.0 \

\& 0.5

u L

\\

ii \0 -+ --- AB----

5 ----% -Ia.s

x-1.0

AG‘-l”s fl=20

-2.0 I I 10 50 190

NUMBER Of SEGMENTS

Fig. 7-4 Convergen- of input admittance hornthe applied-field source relative to Kkg-Mlddle@nRs31dtGk + j& DipOk with ~ = 2hI(2h/a) Of8, 11 and 20 are shown for length &h= 1.6.

‘-iki-k=0 10 20 30 49

NUMBER OF SEGMENTS

AG = 100(GNEC – Gk)/Gh

AB = 1OO(BIWEIC- Bk)/B&

/

AGGk = 7.915 mmhos ABBk = –5.036 mmhos

n=u0 20 40 68 ~ Sa

NUMBER OF SEGMENTS

1

J 01 Gk = 6.666 mmhoa

●\”w -6

I & = –5.991 mmhos

I\\\

:20 AG‘-7 f-l=-e 1-

0 50 1eraNUMBER Of SEGMENTS

Fig. 7-5 Convergence of input admittance fromthe bicone souroa relative to King-Middleton re-sult ck + j&. Dipoles with Q = 2 ln(2h/a) of 8,11 and 20 are shown for length kh = 1.6.

82

70

h

AG = 100(G~Ec - GJ/Gk60 \

G AB = 1OO(BNEC– B~)/Bk~ 50 \x \a 40 1 Gk = 8.262mmhos

Iz

30 I Bk = –4.065 mmhoa

EI

W 20 \b. I \\

-10 II I I I

0 10 20 30 40

z50

M II

Gk = 7.915 mmhos

g 290 Bk = –5.036 mmhos

.\”u 50>

0

AG

sl=ll ----- _____ -! 1

0 50 100

NUMBER OF SEGMENTS NUMBER OF SEGMENTS

850 T~ 800- = 6.666mmhos

g 750- -5.991 mmhlxl

; 700-~ 650-

w 600-~ 550-k 500-.s- 450-> 400- i-l=20

AG

350 ~ ‘- I I0 50 100

NUMBER OF SEGMENTS

Fig. 7-6 Convergenceof inputacimittancefromthe coaxial-linesourcein NECrelativeto King-Mlddletonresultck + ~&. Dipoleswith~ = 21n(2h/a) of 8, 11and20 areshownfor lengthkh = 1.6.

Accuracy of the coaxial-line source model depends on a sufficient sampling of the fieldof the magnetic current frill. This field decreases by half in a distance on the order of thewire radius. As shown in Fig. 7-6, the convergence of the coaxial-line source is best for$-1= 8. Convergence is slow for Q = 11 and very poor for Q = 20. PopoviE et al. [23]recommend a nonuniform sampling with about four match points within a distance of 3ato 10a on either side of the source. The NEC model also yielded better results with smallsegments in the source region, tapering to larger segments away from the source. However,this approach requires extra effort and makes the model source dependent. For very thinwires, such as the C?= 20 dipole, it seems nearly impractical to sample sufficiently finelyto get good results from the coaxial source. However, the structure of the coaxial feed isimportant only when the radius of the wire snd the outer conductor are of significant sizerelative to the wavelength. For thin wires the applied-field source could as well be used.

For reasons demonstrated by the above results, it is recommended that the applied-fieldvoltage source model in NEC be used in most cases. For electrically thin wires the effectof the actual feed structure of sn antenna on its input impedance will usually be negligible,and the results of the applied-field source can be used directly. In complicated cases, such

83

I

as the feed of an AM broadcast tower, the source region may not resemble any of the simplenumerical models, and an empirical correction may be necessary.

7.2 Lumped or Distributed Loading

Thus far, it has been assumed that all structures to be modeled are perfect electricconductors. The EFIE is easily extended to imperfect conductors by modifying the boundarycondition in equation (2-4) to

ii(r) x [E’(r)+ E’(r)] = Z.(r) [ii(r) x J.(r)]

where Z~(r) is the surface impedance at r on the conducting surface. For a wire the boundarycondition is

~. [Es(r) +E%)] = zw(~)~(~)

with r and 9 the position vector and tangent vector ats on the wire and ZW(s) the impedanceper unit length at s. The matrix equation can then be written

N

x Gij~j =–Ei + ~Ii, i=l,..., Na

j=l

where crj is the amplitude of basis function ~~Ei is the incident field On segment ~) Ii iS thecurrent at the center of the segment, Zi is the total impedance on the segment and Ai is thesegment length.

The impedance term can be viewed as an applied-field model of a voltage source withthe voltage proportional to the current. It is assumed that the current is essentially constant,with value Ii, over the length of the segment, which is reasonable for the electrically shortsegments used in the integral equation solution.

The impedance term can be combined with the matrix by expressing Ii in terms of theCYjSS

NIi= ~ ~jA~

j=l

where A! is the coefficient of the constant term in the current expansion of equation (3-3)for any basis function j extending onto segment i. The matrix equation modified by loadingis then

N

E G{j~J=–Ei, i=l,..., Nj=l

whereG;j = Gij – A$Zi/Ai.

For a lumped circuit element, Zi is computed from the circuit equations. For atributed impedance, Zi represents the impedance of a length Ai of wire. In the case

dis-of a

84

round wire with finite conductivity a and permeability p the impedance for a segment i withlength Ai and radius G is

r[

jAi W/JZa=— —

Ber(q) + jBei(g)

& 2~a Ber’(q) + jBei’(q) 1where q = e~i and Ber and Bei are Kelvin functiom This expression takes account ofthe limited penetration of the field into an imperfect conductor.

7.3 Nonradiating Networks and !t%r.wrnission Lines

Antennas often include transmission lines, lumped-circuit networks or a combination ofthese connecting between different parts or elements. When the currents on transmissionlines or at network ports are balanced, the fields produced by these elements, both for nearfield and radiation, can often be neglected. The effect of the network or transmission line isthen simply to establish a relation between the voltage and current at each of the connectionpoints. The solution for the antenna currents can be obtained by solving the equationsdescribing the networks or transmission lines together with the moment-method equationsfor the radiating structure. This approach avoids the substantial increase in the number ofunknowns that would result from modeling the transmission line wires or network leads withsegments. For transmission lines with characteristic impedance on the order of 150 ohms orless the spacing of the wires relative to radius may be too small for a thin-wire model, sothe ideal transmission line model becomes the only practical approach.

The procedure used in NEC is to compute a driving-point-interaction matrix fromthe complete moment-method matrix to characterize the electromagnetic interactions. Thedriving-point-interaction equations are then solved together with the network or transmis-sion line equations to obtain the induced currents and voltages. In this way the largermoment-method matrix is not changed by addition or modification of networks or transmis-sion lines. Another advantage of this approach is that the connection of networks does notalter the matrix structure resulting from symmetries in the model, so the efficient solutionmethods for symmetric structures can be used with asymmetric network connections. As aresult, it may be advantageous in NEC to model a one-port load as a two-port network withappropriate parameters when the value of the load must be changed msny times or whenconnection of the load would destroy structure symmetry.

The solution described below assumes an electromagnetic-interaction matrix equationof the form

[Gij][lj] = -[Ei]

where Ei is the exciting electric field on wire segment i and Ijis the current atof segment j. In NEC the interaction equation has the form

[G~j][Aj] = -[&]

where Aj is the amplitude of the jth basis function .fj in the current expansion

1(s) = ~ Ajfj(~).

j=l

85

(7-5)

the center

M-Port

Network

Fig. 7-7 Voltage and current referencerectiona at network connection points.

The same solution procedure can be used, how-ever, by computing [lj] from [A~]whenever [lj] isneeded. This must be done when starting withthe matrix [G~j] and computing the elements of

the inverse matrix [G~j]–l representing the cur-rent on segment z due to a unit field on segment

A model consisting of N. segments will be as-sumed with a general A4-port network connectedto segments 1 through A4. The network is de-scribed by the admittance equation

di- M

j=l

where Vi and 1~ are the voltage and current at port i, with reference directionsin Fig. 7-7. The connection of a network port to a segment is illustrated in Fig.segment is broken, and the port is connected so that

~; = –Ii

(7-6)

as shown7-8. The

(7-7)

where ~ais the segment current. A voltage source of strength Vj is shown connected acrossthe network port at segment j so that

1$ = I; – Ij. (7-8)

The port voltage, either with or without a voltage source, will be related to the field onthe segment by the applied-field voltage source model of equation (6-l). This port voltagerepresents an additional unknown in the problem, unless its value is fixed by connection

//

/ // /

Fig. 7-8 Network connections to segments with and without a voltage source.

86

of a voltage source. We will assume that segments 1 through Ml are connected to networkports without voltage sources, and segments Ml + 1 through 114are connected to networkports with voltage sources. The remaining segments have no network connections but mayhave voltage sources. In addition, all of the segments may be excited by an incident fieldrepresented by I?/ on segment i. The totzd field on segment i is then

Ei=E/’+E~, i=l,..., N~

where E,p = U/Al, Viis the voltage due to either a network port or a voltage source andA~ is the segment length. An addltiond voltage drop can result from a finite impedanceof the segment. This voltage is outboard of the network port. It is not shown explicitlyhere since, beiig proportional to the unknown current on the segment, it is included in themoment-method matrix.

Equation (7-5) may be solved for current as

Ii=-~G~’Ej, i= I,...)N~ (7-9)

j=l

where G~l is the (i, j)th element of the inverse of matrix G. The equations from (7-9) forsegments that have network ports without voltage sources are then written, with knownquantities on the right-hand side, as

whereMl

B~=– xGz~lE~ – ~ G~lEj.

j=l j= A41+l

Similarly, the network equations (7-6) are written using (7-7) as

MI

x Y~E.f+Ii=Ca, i=l)... jMlj=l

where

M

ci=– E fijVj.j= A41+1

The current is eliminated between (7-10) and (7-11) to yield

MI

x( )G~l–~j E~=Bi– Ci, i=l,..., M1.j=l

(7-lo)

(7-11)

(7-12)

87

The solution procedure is then to solve (7-12) for E~ for j = 1,..., Ml. Then, with thecomplete excitation vector determined, (7-9) is evaluated to determine Ii for i = 1, ..., N~.Finally, the remaining network equations are used with (7-8) to compute the generatorcurrents as

MI; =

xYijVj + Ii ,“ i= Ml+l,... ,M, (7-13)

j=l

The currents I: determine the input admittances seen by the sources.%

In NEC, the general A4-port network considered here is restricted to muItiple two-portnetworks, each connecting a pair of segments. A transmission line is treated as a special caseof a two port network with admittance coefficients determined by the ideal transmission lineequations. Including optional terminating admitt antes of Y1~ and YH on ends one and twoof the transmission line, respectively, the admittance coefficients for a line with characteristicadmittance Yo and length 4 are

Yll = – jYo Cot(kt) + Yl~

Y22= – jYo cot (kE) + Y2~

Y12,=Y21 = +jYo CSC(M?).

The lower sign for Ylz is used when the transmission line undergoes a twist relative to thereference directions at the connection points, as occurs between elements in a log-periodicdipole antenna, and the upper sign is used otherwise.

It should be remembered that the implicit transmission-line model neglects interactionsbetween the transmission line and the antenna or its environment. This approximation isjustified if the currents in the line are balanced, as in a log-periodic dipole antenna, andin general if the transmission line lies in an electric symmetry plane. However, the balancecan be upset if the transmission line is connected to an unbalanced load or by unsymmetricinteractions. For example, a vertically polarized log-periodic dipole antenna over groundcould be unbalance by interaction with the ground, and the importance of this unbalanceshould be considered in deciding whether to use the ideal transmission-line model. If theunbalance is significant, the transmission line can be modeled explicitly with segments ifthe two-wire model with the correct characteristic impedance does not violate the thin-wire approximation. Another option would be to use the ideal transmission-line model torepresent the balanced currents, and model a single wire with segments along the path of theline to carry the unbalanced component of current. However, this approach raises questionsabout how to connect the single wire.

7.4 Radiated Field, Directivity and Gain

The radiated field of an antenna or reradiated field of a scatterer is found with relative ‘ease once the current induced on the structure has been evaluated through the moment-method solution. The integral to obtain the far-zone field of a straight wire segment orpatch can be evaluated in closed form, and the field reflected from or transmitted acrossan interface with the ground can be evaluated exactly from the plane-wave reflection andtransmission coefficients, without the need to evaluate Sommerfeld integrals. The far field

88

evaluation yields only the field components decreasing as R-l, so the surface wave overground is lost. To include the surface wave, the code offers an option to compute the fieldat large but finite distances using the asymptotic approximations for the field over groundgiven in section 5.

7.4.1 The Radiated Field in Fkee Space

For the far-zone evaluation it is assumed that the maximum dimension of the currentdistribution (actually the maximum distance from the origin) is small relative to the distanceto the evaluation point ro and also relative to the wavelength. When IrOl>> Id[ the factorIro–/l in the denominator of the Green’s function in (2-1) can be replaced by To = Iro[. Theplanar-phase front approximation is used in evaluating the phase, so that klro – ~1 in theargument of the exponential is approximated by kro + k or’. The vector k is the wave vectorat < for the reciprocal problem, when a source is located at the evaluation point r.. Whenpoints ro and r’ are in the same medium, k = –kfio. When these points are on oppositesides of an interface between different media, k must be found by applying Snell’s law at theinterface with an incident wave from a source at ro.

The ra&ated field due to a structure including wires with current distribution lW(S) andsurfaces with current JS(s) is evaluated as

(7-14)

where Fw is due to currents on wires

Fw =J

~~w ($) f?-~(k”r) ds (7-15)f

and FS is due to currents on surfaces

F, =I

J,(r) e-j(k”r) dA.s

(7-16)

The integrals are over the wire contour g and the surface S, where r is the vector to theintegration point on the surface or the point s on the wire. The dot product in (7-14), where

~ = i5zi5Z+ ~~ + 6*6Z, serves to remove the radial component from the vector integrals.This step is accomplished in the code by taking the dot products with the i5eand i34 unitvectors transverse to the dkction of propagation.

The wire current is represented on each segment in the form

l~(t) = Ai + Bi sin(kst) + Ci[cos(k~t) – 1] (7-17)

where Ai, Bi and Ci are obtained for each segmentThe vector r to a point on segment i is

r = ra + tiii

89

i from the moment-method solution.

..-,.

I

where r~ is the location of the segment center, iii is the unit vector in the direction of thesegment, and t varies from –Ai/2 to Ai/2 for segment length A~. Then for N segments, theintegral for Fw becomes

N

Ftu = ~fiiQie-’(kr*)

i=l

where

/

AJ2

Qi= Is(t).-~t(k”@)d.-4/2

For the current function in (7-17), this integral can be evaluated as

{

2sin(~iAi/2)_ ~BiQi= Ai ~i

[

sin[(k~ – ~i)Ai/2] _ sin[(k~ + ~i)Ai/2]

k.q– Wi k.+ Wi 1+ Cj

[

sin[(k8 – wa)Ai/2] + sin[(k$ + wi)Ai/2] _ 2 Sin(wiAi/2)k. – wi k.+ wi Wi 1}

where wi = k” tii. When lkAi I is small Qi is evaluated by the series approximation

Qi ~ AiAi – ‘Ai~)3 (@Ai + 2jk$wiBi + k~Ci)

to avoid loss of precision.

The contribution from surface patches is

M

F.= zJisie-Nc”’l)

i=l

where ri is the location of the center of patch i, Ji is the current associated with the patchcenter in the delta-function current expansion and Si is the area of the patch.

7.4.2 Effect of a Ground Plane

When a ground plane is present the total radiated field is evaluated as

E= ED+ ER+ET

where ED and ER are the direct and reflected fields, respectively, from sources on the sameside of the interface ss the evaluation point and ET is the field due to sources on the oppositeside of the interface from the evaluation point. ED is evaluated using (7-14), while ER isevaluated as

–jkq e-’~ ~ER(rO) = ~ --(FW + F~) . (~ - FOiO) (7-18)

90

Risadyadic reflection coefficient and~R=6Z6Z +6@Y–6Z6z. Forperfectly conducting

ground ~ = –~. For a finitely conducting ground, the reflection coefficients for TE and TMpolarization are

where YOTE= COS6~/~ and Z$M = qi cos Oi with di the angle between the incident ray andthe normal to the interf~e and % the intrinsic impedance of the medium in which the ray ispropagating. Y~TEand Z~M are the surface admittance and impedance of the ground for TEand TM polarization respectively, given in section 5.3. T~e reflected field is then evaluatedhorn the field of the image of the source ((7-18) without R) as

& =

[EIRTM + fi(E1 . fi)(RTE – RTM)1

where F is the unit vector normal to the plane of incidence.

The transmission coefficients for TE and TM polarizations are [30]

where & and kt are the wave numbers of the incident and transmitted waves, respective y.The z component of the incident wave vector k~ is kaz = –kj cos 0~, and the z component ofthe transmitted wave vector k~ is kt= = [k? – (k? – k~Z)]112.

The electric ~eld components of t~e transmitted TM wave can be obtained from therelations E~ =~kix Hiand Et= q~kt x Ht. The results for p and z components of theelectric field are

The electric field transmitted across the interface from an incident wave with strength E andarbitrary polarization is then obtained with a dyadlc transmission coefficient as

E is evaluated using (7-14), including only sources on the opposite side of the interface fromthe evaluation point and replacing k by k~ = (ki . 6=)6=+ (ki “6V)@Y+ ktZ&.

91

. ,....,..,,. ,.. ,.,______—-,~—...—x”..t.

,’,,,

7.4.3 Elliptic Polarization

In the code the 60 and 6@ components of radiated electric field are computed in sphericalcoordinates. In general, these components will be out of phsse, resulting in an ellipticallypolarized wave. The field components are described in time-varying form as

so that the wave is linearly polarized when c = O or when either Em = O or Ed = O.Otherwise, it is elliptically polarized with major and minor axis magnitudes of

E minor = [E& + (@o - E&) sin’ -y – 2EwE~ sin? cos~ cm<] 1’2

where

is the tilt angle of the major axis fkom the Godirection. The polarization ellipse is describedby the ratio of minor axis to major axis, the tilt angle ~ and the sense of rotation, which isright-hand if ~ <0 and left-hand if ~ >0.

7.4.4 Gain and Directivity

The power gain of an antenna in the direction of the spherical coordinates (0, ~) isdefined as

pi-do) 4)GP(8, ~) = 47rTin

where PQ(O, ~) is the power radiated per unit solid angle in the direction (0, ~) and ~n isthe total power accepted by the antenna from the source. Since quantities in NEC are peakvalues rather than rms, the input power is evaluated from the voltage and current at thesource as

~.= @.e(vI*)

and the radiated power density is

PQ(8, (b)= #R2Re(E X H*)=R’ ‘#El

where R is the evaluation distance. The power gain is then

where E’ = RE for R ~ 00 is the quantity computed by subroutine FFLD in NEC.

92

.. .-a. - -s, ., .. . . . . ..-.-..” .,+ ,.e,!

——.— —&.— -.. ,.- . . .. . . ... *-%. .,,, . . . . .. . . . . ... **w.*—,*_,~R

——$” . . . . —.—.

The directivity, or directive gain, is

where Pr~ is the total radiated power. Pr~ is evaluated in NEC as Pr~ = Pin – He where~~ is the power dissipated in ohmic loss in lumped and distributed loads on the wires andin non-rdlating networks. am does not include power dissipated in a 10SSYground.

The power gain and directivity are also calculated for orthogonal field components (ii,*)where ii and $ may be vertical and horizontal components ae, 64 or major-axis and minor-axis components. For the component gains IE( is replaced by IE ofit or [E o+1 in evaluatingPQ. The total gain is the sum of these two components.

The code also will compute average power gain by integrating the far-field radatedpower over a sector of space

Gav =

with solid angle fi

1

JE’QGP(O,~) dfl =

If the solid angle of integration O includes the entire sphere or a subsector of the spheresuch that the the radiation pattern of the antenna in that subsector repeats exactly overthe remainder of the sphere due to symmetry, then Pi~Gav will be the total radiated powerobtained by integrating the far field. This evaluation represents a variational from for theradiated power, so is generally more accurate than pr~ evaluated from the voltage andcurrent at the source and ohmic losses.power can provide a useful check on thevoltage source model.

When a structure is excited by an

Comparison of these two evaluations of r~iatedaccuracy of the solution, and particularly on the

incident plane wave the scattering cross sectionnormalized to square wavelength is computed instead of gain. The cross section is

where W-t is the scattered power per unit area at a distance R in a given direction andWiw is the power per unit area of the incident plane wave.

7.5 Maximum Coupling Calculation

Coupling between antennas is often a parameter of interest, especially when a receivingsystem must be protected from a nearby transmitter. Maximum power transfer betweenantennas occurs when the source impedance and receiver load impedance are conjugate-matched to their antennas. Determination of the conditions for simultaneously matchingto the terminal impedances, and the resulting coupling, is complicated by the antenna in-teraction, since the input impedance of one antenna depends on the load connected to theother antenna. NEC includes an algorithm for determining the matched loads and maximumcoupling by the Linville method [64], a technique used in rf amplifier design.

93

. . .... .. .......... =. .-.——.

The first step is to determine the two-port admittance parameters for the coupled an-tennas by exciting each port with the other port short-circuited, and computing the selfand mutual admittances from the MoM solution. In doing this it must be remembered thatsimple wire segments are shorted unless a voltage source or load is specified on the segment.However, segments with a network or transmission line connection represent open-circuitedports unless they are shorted with a zero voltage source or a shunt admittance on the trans-mission line. This shorting of network ports is taken care of in the NEC-4 code, but wasnot handled correctly in earlier versions of NEC.

The maximum coupling determined by the Linville method is

Ckax=+/=)where

L=

The matched load admittance on antenna 2 for maximum coupling is

‘L=(+i$+1)Re(fi2)-y22where

~ = Gmax(Y’2Y21)*lfi2Y’~1

and the cohespondhig input admittance of antenna 1 is

Y21Y12X* = Yll — —.

Y,+ YZZ

The coupling for any other terminations is easily obtained from the coefficients Y1l, Y22andY12.

94

[1] J. Strauch and S. Thompson, Intemctive Gmphics ~tilit~ jor Awny NEC Automation(lG~ANA), Naval Ocean Systems Center, Rept. CR 308, 1985.

[2] R. J. Marhetka and J. W. Silvestro, Near Zone - Basic Scattering Code User’s Manualwith Space Station Applications, Technical Rept. 716199-13, The Ohio State UniversityElectroScience Laboratory, March 1989.

[3] R. R. McLeod, Tempoml Stuttering And Response Software, Users’ Manual, Version2.1 (R), UCRL-MA-104861, Lawrence Livermore National Laboratory, CA, September1990.

[4] A. J. Poggio and E. K. Miller, “Integral Equation Solutions of Three-Dimensional Scat-tering Problems,” Chapter 4 in Computer Techniques for Electmmagnetics, R. Mittracd., Pergamon Press, New York, 1973.

[5] R. F. Barrington, Field Computation by Moment Methods, The Macmillan Co. NewYork, 1968.

[6] E. K. Miller et al., Final Technical Report on the Log-Periodic Scattering Army Pmgmm,MBAssociates Report No. MB-R-70/105, Contract No. F04701-68-C-0188, 1970.

[7] RCS Computer Progmm BRACT, Log-Periodic Scattering Anuy Progmm - II, MBAs-sociates Report No. MB-R-69/46, Contract No. F04801-68-C-0188, 1969.

[8] J. H. Richmond, “Digital Computer Solutions of the Rigorous Equations for ScatteringProblems: Pmt. IEEE, Vol. 53, p. 796, 1965.

[9] J. H. Richmond, “A Wire-Grid Model for Scattering by Conducting Bodks,” IEEEtins. Antennas and Propagation, Vol. AP-14, p. 782, 1966.--

[10] Y. S. Yeh and K. K. Mei, “Theory of Conical Equiangular Spiral Antennas: Part I -Numerical Techniques,” IEEE fins. Antennas and Propagation, Vol. AP-15, p. 634,1967.

[11] G. J. Burke, E. K. Miller and A. J. Poggio, Dielectric Ground Intemctions with a Log-Periodic Antenna, Final Report, ECOM-0285-F, U.S. Army Electronics Command, FortMonmouth, NJ, 1971 (Contract DAAB07-70-C-0285).

~12] Antenna Modeling Pmgmm - Composite User’s Manual, MBAssociates Report No. MB-R-74/46, 1974.

[13] Antenna Modeling Progmm - Engineering Manual, MBAssociates Report No. MB-R-74/62, 1974.

[14] Antenna Modeling Progmm - Systems Manual, MBAssociates Report No. MB-R-74/62,1974.

95

[15] Antenna Modeling Program - Supplementary Computer Progmm Manual (AMP2), MB-Associates Report No. MB-R-75/4, 1975.

[17] G. J. Burke and A. J. Poggio, Numerical Electromagnetic Code (NEC) - Method ofMoments, UCID-18834, Lawrence Livermore National Laboratory, CA, January 1981.

[18] G. J. Burke snd E. K. Miller, “Modeling Antennas Near to and Penetrating aImssyInterface: L?L?W2%NIS.Antennas and Pm@gation, Vol. AP-32, No. 10, Pp. 1040-1049,1984.

[16] N. C. Albertsen, J. E. Hansen and N. E. Jensen, Computation of Spacecraft AntennaRadiation Patterns, The Technical University of Denmark, Lyngby, Denmark, June 1972.

[19] P. C. Waterman, “Matrix Formulation of Electromagnetic Scattering,” Proceedings ofthe JEW, Vol. 53, pp. 805-812, August 1965.

[20] J. Van Bladel, Electromagnetic Fields, McGraw-Hill, New York, 1964.

[21] A. R. Neureuther et al., “A Comparison of Numerical Methods for Thin-Wire Antennas,”%sented at the 1968 Fail URSI Meeting, Dept. of Electrical Engineering and ComputerSciences, University of California, Berkeley, 1968.

[22] E. K. Miller, An Evaluation of Computer Progmms Using Integml Equations for theEleetmmagnetic Analysis of Thin Wire Structures, UCRL75566, Lawrence LivermoreLaboratory, CA, March 1974.

[23] B. D. PopoviL, M. B. Dragovi6 and A. R. Djordjevi6, Analysis and Synthesis of WimAntennas, Research Studies Press, New York, 1982.

[24] E. P. Sayer, “Junction Discontinuities in Wire Antenna and Scattering Problems,” IEEEtins. Antennas and Propagation, Vol. AP-21, p. 216, 1973.

[25] J. L. Lin, W. L. Curtis and M. C. Vincent, “Radar Cross-Section of a RectangularConducting Plane by Wire Mesh Modeling,” IEEE tins. Antennas and Propagation,Vol. AP-22, p. 718, 1974.

[26] T. T. Wu and R. W. P. King, “The Tapered Antenna and its Application to the JunctionProblem for Thin Wires,” IEEE i%ans. Antennas and Propagation, Vol. AP-24, pp. 42-45, 1976.

[27] S. A. Schelkunoff and H. T. Fkiis, Antennas: Theory and Pmctice, Wiley, New York,1952.

[28] A. W. Glisson and D. R. Wilton, Numericnl Pmcedums for Handhng Stepped-RadiusWim Junctions, Department of Electrical Engineering, University of Mississippi, March1979.

[29] Brillouin, l?adiod{ect~citd, April 1922.

[30] J. A. Stratton, Electro~agnetic Theory, McGraw-Hill, New York, 1941.

96

[31] A. Sornrrmfeld, “her die Ausbreitung der Wellen in der drahtlosen Telegraphie~ Ann.Phys~ Vol. 28, p. 665, 1909.

[32] E. K. Miller, A. J. Poggio, G. J. Burke and E. S. Selden, “Analysis of Wire Antennas inthe Presence of a Conducting Half Space: part I. The Vertical Antenna in Free Space,”Canadian J. Phys., Vol. 50, pp. 879-888, 1972.

[33] E. K. Miller, A. J. Poggio, G. J. Burke and E. S. Selden, “Analysis of Wire Antennasin the Presence of a Conducting Half Space: Part II. The Horizontal Antenna in FleeSpace,” Canadian J. Ph~s., Vol. 50, pp. 2614-2627, 1972.

[34] D. L. Lager and R. J. Lytle, F’ortmn subroutines jor the Numerical Evaluation of Som-merfeld lntegm~ Unter Anderem, UCRL-51821, Lawrence Livermore Laboratory, CA,May 21, 1975.

[35] J. N. Brittingham, E. K. Miller and J. T. Okada, SOMINT: An Improved Model forStud@ag Conducting Objects Near Lossy HaV-Spaces, UCRL-52423, Lawrence LivermoreLaboratory, CA, February 24, 1978.

[36] K. A. Norton, “The Propagation of Radio Waves Over the Surface of the Earth and inthe Upper Atmosphere,” Proceedings of the Institute of Radio Engineers, Vol. 25, No. 9,Sept. 1937.

[37] A. Baiios, Dipole Radiation in the Presence of a Conducting Half-Space, Pergamon Press,New York, 1966.

[38] M. Siegel and R. W. P. King, “Electromagnetic Fields in a Dissipative Half-Space: ANumerical Approach,” J. Appl. Phys., 41(6), pp. 2415-2423, 1970.

[39] D. M. Bubenik, “A Practical Method for the Numerical Evaluation of Sommerfeld Inte-grals: IEEE Tbs. Antennas Propagat., vol. AP-25, no. 6, pp. 904906, 1977.

[40] W. A. Johnson and D. G. Dudley, “Real Axis Integration of Sommerfeld Integrals: Sourceand Observation Points in Air”, Radio Science, vol. 18, no. 2, pp. 175-186, March-April1983.

[41] R. J. Lytle and D. L. Lager, Numerical Evaluation of Sommerfeki IntegmZs, UCRL51688,Lawrence Livermore Laboratory, CA, October 23, 1974.

[42] P. Y. Parhami, Y. Rahmat-Samii and R. Mittra, “An Efficient Approach for EvaluatingSommerfeld Integrals Encountered in the Problem of a Current Element Radiating Overa Lossy Ground,” IEEE 2kans. Antennas Propagat., vol. AP-28, no. 1, pp. 100-104, 1980.

[43] I. V. Linden, E. Alanen and H. Von Bagh, “Exact Image Theory for the Calculation ofFields ~ansmitted Through a Planar Interface of Two Media? IEEE tins. AntennasPmpagat., vol. AP-34, no. 2, pp. 129-137, Feb. 1986.

97

[44]

[45]

[46]

[47]

[48]

[49]

[50]

[51]

[52]

[53]

[54]

[55]

[56]

[57]

[58]

[59]

W.L.Anderson, Impmved Digital Filters jor Evaluating Fourier and Hankel TbansformIntegds, U. S. Geological Survey Report USGS-GD-75-012, Available fkom NTIS no.PB-242-800, 1975.

E. K. Miller, “A Variable Interval Width Quadrature Technique Bssed on Romberg’sMethod,” Journal of Comp. Phys., vol. 5, no. 2, April 1970.

J. J. Dongarra, C. B. Moler, J. R. Bunch and G. W. Stewart, LINPAK User’s Guide,Siam Press, Philadelphia, 1979.

L. B. Felsen and N. Marcuvitz, Radiation and Scuttm”ng of Waves, Prentice-Hall Inc.,Englewood ClifIs, New Jersey, 1973.

L. M. Brekhovskikh, Waves in La~end Media, Academic Press, New York, 1980.

T. T. Wu snd R. W. P. King, “Lateral Waves: A New Formula and Interference Pat-terns,” Radio Science, Vol. 17, No. 3, pp. 521-531, May-June 1982.

R. B. Dingle, Asymptotic Expansions: Their Derivation and Interpretation, AcademicPress, London, 1973.

M. Abramowitz and L A. Stegun, Handbook of Mathematical Functions with Formu-las, Gmph and Mathematical Tables, U. S. Dept. of Commerce, National Bureau ofStandards, Applied Mathematics Series 55, December 1972.

R. Ziolkowski, University of Arizona, Private Communication, 1983.

E. K. Miller, R. J. Lytle, D. L. Lager and E. F. Laine, On hwmzsing the RadiationE’ciency of l?evemge- ~pe Antennas, UCRL-52300, Lawrence Iiverrnore National Lab-oratory, CA, July 6, 1977.

J. R. Wait, “Characteristics of Antennas Over Lossy Earth,” Chapter 23 in AntennaTheory, Part 2, R. E. Collin and F. J. Zucker, Eds., McGraw-Hill, New York, 1969.

E. K. Miller and F. J. Deadrick, Computer Evaluation of Lorun-C Antennas, UCRL-51464, Lawrence Livermore National Laboratory, CA 1973.

A. Ralston, A First Course in Numetical Analysis, McGraw-Hill, New York, 1965.

G. J. Burke, “Modeling Monopoles on Radial-Wire Ground Screensj” Applied Computa-tional Ekctromagnetics Newsletter, Vol. 1, No. 1, February 1986.

R. W. Adams, A. J. Poggio and E. K. Miller, Study of a New Antenna Soume Model,UCRL-51693, Lawrence Livermore National Laboratory, CA, October 28, 1974.

M. G. Andreasen and F. G. Harris, Analgses oj Wire Antennas of Arbitw.ary Conjigum-tion by Pm&se Theoretical Numerical Techniques, Tech. Rept. ECOM 0631-F, GrangerAssociatesj Palo Alto, CA, Contract DAAB07-67-C-0631, 1968.

98

[60] G. A. Thiele, “Wire Antenn=,” Chapt. 2 in Computer Techniques for Electromagnetic,ed. by R. Mittra, International Series of Monographs in Electrical Engineering, Vol. 7,1973.

[61] R. W. P. King, Theory of Linear Antennas, Harvard University Press, Cambridge, MA1956.

[62] E. K. Miller, Los Alamos National Laboratory, Private Communication.

[63] R. W. P. King, E. A. Aronson and C. W. Harrison, Jr., “Determination of the Admittanceand Effective Length of Cylindrical Antennas,” Radio Science, Vol. 1, pp. 835-850, 1966.

[64] D. Rubin, The Linvtlle Method of High 1%.quency i%msistor Amplifier Design, NavalWeapons Center, Research Department, NWCCL TP 845, Corona Laboratories, Corona,California, March 1969.

99

APPENDIX A

Evaluation of the Scalar Potential for~iangular and Constant Charge Ftmctions

The matrix elements in (3-42) represent Pthe scalar potentials of triangular and semi-infinite charge distributions. Since the tri-angles overlap, their potentials are most ef-ficiently evaluated by computing the poten- R

tials of positive and negative ramp functionson a segment and storing the contributionsto the separate matrix elements. Consider-ing a segment on the z axis as in Fig. A-1,the matrix elements involve the integrals

—- 6

Fig. A-1 Coordinate for evaluating the potentialdue to a aegrnent.

1

J

~ e-]kR—(6 + z’) dz’ = 6%(P, Z, –6, 6) + @l(P, z, ‘~, ~) (A-1)@p(p, z) = ~ _~ R

I

~ ~-jkR@m(P) z) = & _&T(6 – Z’) dz’ = 6@o(p,z, –6, 6) – 4’1(p, z, –6, 6) (A-2)

where Z2e-jkR

@o(p,z,zl, %2)=*/

~ dz’Z1Z2e-jkR ,

4!1(/2, 2,21, %2)=*/

‘Z dz’Z1 R

and R= ~2 + (~ - z)2]li2. Since RdR = (z’ – z) di, the integral @l can be evaluated as

zz e-jkR@l(p, z, 21, 22) = &

J+2’ – Z)dz’ + Z@()(~,Z, 21, 22)

Z1

e-jkRl) + z@o(p, 2,21, %2= &(e-jkR2 -

)

where RI = ~2 + (21 - %)2]1/2and Rz = ~2 + (22 – 2)2]1/2. Thus

@p(p, z) = (8+ Z)%(p, z, %1,22) + &(e-jk~2 _ e-jk~l)

Q%(A z) =(6 – Z)QO(P,z, 21, 22) –&(e-jkR2 _ e-jkR1) .

with 21= –6 and 22 = b.

100

(A-3)

(A-4)

I The semi-infinite integral is evaluated as

1

/

00 e-jk~— dz’@C(~,z) = ~ ~ R

1

/

z ~-jkR—dzl+.~

I

CQe-jkR— dz’

‘~OR 4m z R

= @o(p,z,o, z) + +./

00 e-~kR~ dz’

-co

= @o(~,z,(), z) – &H$)(kp). (A-5)

For electrically short segments, both the integral for @o and the difference of exponen-tial in (A-3) and (A-4) are conveniently evaluated using the series approximation of theexponential

I euR =ea~ [1+u(R – h) + $(R – RO)2 + $(R– fi)3+ :(R - ~)a]

1 xea~(To + TIR + T2R2 + T3R3 + T4Z#)

~where a = -jk, & = {p2 + [(z1 + z2)/2 – z]2}li2 and

! a2R~ a3~! To=l–aRo+7— — , a4ti) 6 24

a3% a4R;T1=a–a2m+~– —

6a2 a3Ro

Tz=~–—j ad~

4a3 a~h

T3 d–~=—

U4T4 =Z.

Then

eaR2 _ eaR1 _ e a% [?’1 (R2 – RI)+ T2(@ - R;)+ T3(R; - @)+ T4(R4 - @)] (A-6)

I anda@(TO~l + T112 + T2~3 + T314 + T415)@o(p,z,zl, z2) x e (A-7)

where

101

., . . . ... .... .+ . ____

The integrals 1. are evaluated as

II = log[(R2 + s2)/(R~+ Sl)] for z < Z1

= log[(R~ - SI)(R2 + s2)/p2] for Z1<Z<ZZ

= log[(Rl – sl)/(R2 – S2)] for z ~ zz (A-8)

12=%2–%1 (A-9)

‘211 + ~(SzRz ,13=~ – SIRI) (A-1O)

1 %5;)14= p212 + #2 (A-n)

15= ;p213 + ;(S2R; – SIR;) (A-12)

where S1 = ZI - z, Sz = 22 – z, R1 = (p2 + S~)112 and R2 = (P2 + S#)112. The option for11 is intended to avoid loss of accuracy due to cancellation.

The relative error in the five term approximation of the exponential is less than 0.01 forIll– ml= 0.2A and decreases ss IR - R015. Since [R- RoI < (22 – 21)/2 and 22- ZI is thesegment length in (A-3) and (A-4) or the distance of the evaluation point from the junctionin (A-5), this approximation can be used for all matrix elements” in the solution for chargeat a junction.

102

. ...—... ..—=.—-.._,,.- -ea.-., ..%,

,I

APPENDIX B

Approximations for the Field of a Segmentto Avoid Loss of Precision

The exact equations for the field of a straight filament of current on the z axis werederived in section 4. The field components for currents 10sinks, 10cos ks and a constant. .current 10; denoted Es, Ec and EK, respectively; are

E;={[

- (%-6)~- (%+ 6) =1 cos(kd)

_ j(e-#@ + e-~k~l) sin(kb)}

E;=g {[(z -6)e- jkRl

e+ -+ (z+@ & 1+j(e-jkR2–e-jkRl)cos(ki$)

}

( — - *)c@(kb)jqIo e-lkRaE;=_47r R2

(B-1)

(B-2)

(B-3)

(B-4)

(B-5)

where the seY

ent has length A = 26 extending from Z1 = –6 to zz = 6 and RI =[P2 + (%+@ 11/2 ~d R2 = k2 + (z - 6)2]1/2. While these equations are exact, their~valuation CM result in loss of precision due to cancellation in the limit of small kll andwhen 1? is much larger than A. Equations (B-1) and (B-2) also fail for small p when Izl >6.Approximations valid for these cases are given in the remainder of this appendix. Equation(B-5) was written for E$– E: since this is the quantity needed in NEC, and the cancellationbetween @ and E# must be taken into account in the evaluation.

B.1 Approximation for Small p and IzI >$

When Izl >6 and p < Izl – 6 the p components of electric field are proportional top.Since (B-1) and (B-2) contain p-1 factors the terms must cancel as p2,leading to a numerical

disaster. A suitable approximation for this case is obtained by expanding R in a power series

{[

- “k6–.iTIOSzPe-jklzl ~ e 3

$=7

&k6

1 kt$2 - + ~ sin(k%) –~

(B-6)

(B-7)

103

where S= = z/lzl. Terms of order ~/(lzl - i$)]4were neglected in the expansion of R, andterms of order k2p4/(lzl - 6)2 were neglected in the expansion of the exponential. Equations(B-1) and (B-2) are to be used for small p when I.zI<6.

B.2 Approximation for Small Ikl?l

When lkl?l I <1 and lkRal <<1 cancellation OCCUI’Samong the terms of (B-1) through@-5). Cancellation is particularly severe for the red parts of the fields for real k. Whilethe real parts become small their accurate etiuation is essential to computing the inputresistance in the method of moments solution. For small lkl?l the exponential can beexpanded in power series in kR with the results

[

jqIo 1 1 k2&=_ 2jk3zb–(Rz - RI) - ~ 1+:(Z-R;) COS(k6)47r G–X–2

[

k2 k4# ~ jk314 + #5-hIok II _ ajk~ – .@+ —

47r 6 1{[

E@f=* *++-: (R2+R1)+3$(Z+R!)

1+~(R~ + R?) sin(kt$) + 2%

[

k2 k4-kI1- 1}~k314+#5 .@3+ ~

(B-8)

(B-9)

(B-1O)

(B-11}

(B-12)

The terms 11 through 15 are from the integration of e-~kR/R for small kR and are defined

din equations (C-3) through C-7) of Appendix C. Terms of order (kR)5 have been neglectedexcept in the real part of ~ where the (kR)G term was retained as the first nonzero contri-bution.

Equations (B-8) and (B-9) cannot be used when p is very small and Izl >6. Equations(B-6) and (B-7), which were derived for this case, lose accuracy when Iklll is small. Hence

104

(B-6) and (B-7) were approximated for small lkRl by expanding the exponential. The resultforlzl> 6,p<lzl –6mdlkRl<lis

{

Es ~ j~IOp~ lzIIQ+ k2(z2 – 262)] 2jk3_—P (Z2 - 62)2 }

(B-13)47r 3

– jqIoSZkp63 6 + k2(z2 - 262)

{

2jk51zlE;= Jr

3(Z2 – 82)2 ‘T }(B-14)

Terms of order (kR)4 have been neglected in the imaginary parts of the fields.

B.3 Approxi-tion for R >6

The last case needing approximation is R > & RI and R2 in (B-1) through (B-5) canthen be expanded in powers of b with the results

(B-15)

(B-16)

(B-17)

where RO= ~z + zz]l/z T. erms of order (6/fi)4 have been neglected in the expansion of 1?,and terms of order (k6)4 were neglected in the expansion of the exponential.

When R >6 and lkRl <1 accuracy is lost in the real parts of the fields evaluatedby equations (B-15) through (B-19). The problem can be seen when the exponential areexpanded in powers of jkRo. When the products of this expansion with the terms in paren-theses are evaluated the leading imaginary terms, contributing to the real parts of the field,cancel. The problem is worst for E: where (kR)5 becomes the first imaginary term. Hence,once again, approximations for R >6 and IkRl <<1 are

105

.... ..... . ... . .. .. ,.-.. ...... .!

(B-24)

Terms of order (kRo)s have bwn neglected.

B.4 Choice of the Method for Numerical Evaluation

For accurate evaluation of the fields, the appropriate equations must be chosen fromthose given above. The logic for making this choice was developed by running a program thatdetermined the error in the single precision evaluation of the exact equations (B-1) through(B-5) and the error for a selected one of the sets of approximate equations. Equations(B-1) through (B-5) evaluated in double precision were used as a reference in computing therelative errors.

The logic developed for choosing the evaluation method is outlined in Fig. B-1. Thevalues now being used in the tests are:

c1 = 10.

C2= o.ol(2m)

C3 = o.ol(27r)

(2’4= 0.01

C5 = 0.04

These constants were chosen for single precision evaluation of the equations. If the wde wereto be converted to double precision the exact equations could be used over a wider range ofparameters. Hence constant Cl could be increased and Ca through C5 reduced to delay theuse of the approximations until their accuracy is comparable with double precision results.Determination of the optimum values is complicated by the lack of an accuracy referenceunless (B-1) through (B-5) are evaluated in quadmple precision. A reasonable first guessmight be to increase Cl and decrease CQ through Cs by two orders of magnitude for doubleprecision.

106

.,

SUBROUTINE

If (a > C16) thenIf (lkRol > Gz) then

Use Eq. B15 through B19Else

Use Eq. B20 through B24End if

Eke if (]kfil > C3) thenUse Eq. B3 through B5If (p > C’4min(l?l, Rz)) then

Use Eq. B1 and B2Else

Use Eq. B6 and B7End if

ElseUse Eq. B1O through B12If (p > Cs min(Rl, Rz)) then

Use Eq. B8 and B9Else

Use Eq. B13 and B14End if

End if

}

}

}

EKSCMN

EKSCLR

EKSCMN

EKSCEX

ESMLRH

EKSMR

Fig. B-1 Logic for choosing the field equations for given coordinates. The names of the subroutines con-taining the code in NEC-4 are shown at the right.

107

I

APPENDIX C

Evaluation of ~ eaR/Rdz’

The z component of electric field due to a constant current element on the z axis involvesthe integral

J

6 eaR

I = — dz’ (c-1).6 R

where a = –jk, 6 = A/2 forsegmentlengthA and

This integral cannot be evaluated in closed form. In NEC-2 and NEC-3 it was evaluatednumerically using adaptive Romberg quadrature and subtracting an analytically integrablel/R hmction to smooth the inte.grand when the evaluation point was on the source segment.A more efficient method has been found to be the use of a series expansion.

The integral is written as

whereR() = ~z + 22]1/?

Then IR – RoI ~ 6 and for typical segment lengths, with kb less than about 0.4, the expo-nential is approximated to good accuracy by the series

a2 a3 adea(R-&J x 1 +a(R– Ro) + F(R – RO)2 + F(R – RO)3 + Z(R - RO)4.

After expanding the products and separating terms in R and m the approximation of theintegral is obtained as

a3+(% – ff#)r4

a4 1+215 .

108

(c-2)

—...-”_.__.——

The integrals

I

15I*= ~n-2 ~z!

-6

are evaluated as

II = Iog((liz+ s2)/(Rl+Sl)] for Sl>o

= log[(ll~ - s~)/(132 - S2)] for s~<() (c-3)

12 =26 (c-4)

‘211+ ;(S2R213=~ – SIR1) (c-5)

1 3-s;)14= p212 + #S2 (C-6)

Is = 3 213+ ~(SZR;~P - sl~) (c-7)

where S1=-6—z, Sz=b– z,R1 = (p2 + Sf)lJ2 and R2 = (p2 + S~)l/2. The option for 11is intended to avoid loss of accuracy due to cancellation. When S1 s O < Sg and p is smallcompared to Rg the second form is used with Rg - Sz evaluated as

At the maximum normal segment length of 0.12A equation (C-2) yields a result witha relative error of about 10-5 due to truncation of the series. This error is similar to thatwith the Romberg integration and decreases as (k6)5. However, for R much larger than 6precision is lost due to cancellation of terms in (C-2). Hence for & > 106 the integral in(C-1) is evaluated numerically by three point Gaussian quadrature with very good accuracy.The three point integration is about twenty percent slower than the series due to the need toevaluate three complex exponential. Either method is faster that the Romberg integrationby a factor of two to twenty or more, depending on p, z and 6. The overall time for fillingthe interaction matrix in NEC is typically reduced by twenty to forty percent over that withRomberg integration. The Romberg integration routine remains in the code, however, foruse in evaluating the fields due to a ground plane.

109

.. . .. -----—.-. - -----

APPENDIX D

- Comparison of Approximations for the Thin-Wire Kernel

The problem of evaluating the kernel of the electric field integral equationwire has been studied by a number of investigators. Several approximations to

for a thinthe kernel

are derived in [1] and in references cited in that report. Most often attention has beendirected to evaluation of the exact kernel on a straight wire and the singularity that resultswhen the integration point coincides with the evaluation point on the wire surface. Whenthe extended boundary condition is enforced with the evaluation points on the wire axis,w is now done in NEC, the singularity is no longer a problem. Questions remain, however,about the appropriate approximations for the field of the tubular current distribution andthe resulting accuracy of the field.

In the thin-wire approximation the current on a wire is assumed to be axially directedand uniformly distributed around the wire surface. The evaluation of the field of this currentthen involves a surface integral around the wire circumference and along the length of thewire segment. The integral along the wire was evaluated in section 4, and the integrationover @ was reduced to evaluating the four integrals

21r

G3 =~/

Cosa._e-~kR d~2T~@

21rG4 =~

11

z~e-jkR

— dz’ @27rozl R

(D-1)

(D-2)

(D-3)

(D-4)

whereR= (p2 + a2 + Z2 - 2apcos@i2

p’= (P2 + a2 - 2pacos 4)1/2

p–acost$COSCY=

P’”

Approximations for these integrals are developed below, with alternate forms representinga current filament on the wire axis or on the wire surface at a position ninety degrees fromthe direction of observation.

D. 1 Approximation as a Current Filament on the Wire Axis

An approximation in which the Ieadhg term represents a current filament on the wireaxis for evaluation points outside of the wire can be obtained by expanding the integrandsin Maclaurin series in the wire radius a. The result is equivalent to the extended thin-wire

110

approximation derived in (1] and used in NEC–3. The result is derived here for the specificintegrals of interest in (D-1) through (D-4). First, the exponential terms are expanded inMaclaurin series in a when p > a or series in p when a > p. For p > a the approximations

where RO= (p2 + z2)li2. When p <

e-M e-jklb

— s= +pacos@(lR

a the expansions are

e-jkb

+ jkfk)~a

+P[ I

~ a2 COS2#(3 + 3jk& – k2R~) – (1 + jk&)RZ ~a

where & = (az + Z2)112The integrals over # can then be evaluated using, for Gz and G3,the following integrals derived from Eq. (3.613-2) in [2].

1

1

27r p-aces@ ~d=~

% o p2+a2–2apcos@ pfor p>a

= o for p<a

1

/

2“ (p–acos@)cos@ ~d afor p>a

X o p2+a2-2apcos~ ‘p

–1for p<a

‘z

1

I

2X (p–acos@)cos2#dd= 1 a2@+ p) for p>a

5 0 p2+a2–2apcos4–P for p < a.

‘Q

Then, when p > a, the approximations for Gl, G2 and G3 are

a2

[ 1“~P2(3 + 3jkRo - k2ti) – 2(1 +jkfi)% ~

$($ +2)(3 + 3jkRo - k2@~o

(D-5)

(D-6)

(D-7)

(D-8)

(D-9)

(D-1O)

111

.. . ..,.,.,,.._.____-, .. ..... ‘ .. ---- -,,=” . . ~+., -m”. m.,” . ..q .

—-4..—*— &m.’.-,”,. ”,-.

When p < Qthe approximations me

~-jk&G1=G~=x+$

[ 1a2(3 + 3jk& – m:) – 2(1 +jk&)@ * (D-II)a

(D-12)

(D-13)

The results for Gz and G3 differ from those in [3] due to the use of the exact integrals in(D-5) through (D-7). In [3] series approximations of p’ were used to evaluate these integrals.Hence the approximations here should be more accurate than in [3).

An approximation for C4 can be obttined, as done in [1], by noting that (D-8) isequivalent to

[

(ka)2 a2 & e-jk&1Gl= l–~–———.4 8ZQ &

Then

(D-14)

The leading terms in (D-8) through (D-1O) and (D-14)

J

Z2~-jkh ,q= —

q R1 ‘z

represent the approximations to these integrals obtained by setting the wire radius to zerofor p > a. Hence, the resulting approximation to the field is that of a current filament onthe wire axis. Ihm the higher order terms, the approximate relative enors in ~, G$ andG$ for small k& are

3a2p2 a2q=—-

4R$ mjj IJ!i$=

3a2p2

4*

~ ka2p23 =-”

112

D.2 Approximation as a Current Filament on the Wire Surface

An approximation in which the leading term represents a filament of current on thesurface of the cylinder can be obtained by writing

R = (p2 +a2 + Z2 – 2bcostj)1j2

where b = up and expanding in powers of h This approximation can be used for p > a orp < a as long as ap << R; where

Rt = (P2 + a2 + z2)1i2.

The approximations for the exponential terms are

~- jkR e-jkRt e-jkRt 2 2

— *T+ apcos 4(1 + jkRt)—

e-~kRt

R 43 + ~ COS2 +(3 + 3jkRt – k2&2) -K

Then, integrating over @ the approximation for GL is

e-~kRt aQP2GIzG~=———

Rt—(3 + 3jkRt - k2@e$.

‘4

Forp>a

G2 = G;e-jkRt U2

=— + ~(1 + jkRt)~pRt t

+ 7(2+ <)(3+ 3jkRt – k2~2)~P

e-~k& + jka2 e-~k& e-jkRtjka2p(2 + <)(1 + jk&)~

G3 ~ G! =?2p Rt‘—+ 8 P2

while for p < a

G2 *G! =~(1 + jkRt):~- - ~(3 + 3jkRt - k2&2)e$

The leading terms in (D-15) through (D-17) for p > a

(D-15)

(D-16)

(D-17)

(D-18)

(D-19)

113

resemble the terms for a current filment on the wire surface since l?~ is the distance to apoint on the surface at ninety degrees from the direction of observation. However, pin theseterms is the distance to the wire axis. Use of the radl~ distance to the wire surface, as foran actual current filament on the wire surface, results in significantly larger errors than theabove terms. The relative errors in these approximations for small kR~ are

E;3a2p2

‘my

3a2p2 a2E;=~+—242

ka2~2 ka2E;

‘~+ ~.

D.3 Comparison of Results

Approximations have been derived for the integrals Cl, G2, G3 and G4 which, forevaluation points outside of the wire (p > a) represent filaments of current located on thetire =is (~, G$, ~, ~) or on the wire surface (G! y G! t G!)” The integral G4 Cm alSObe approximated with a filament on the surface as

G: =/

zz e-jkRt~ dz’.

Z1

For p < a there are no simple current-filament approximations for G2 or G3 in the expressionsfor the radial electric field. On the wire axis Gz and G3 are zero and the approximations~, @, G~ and G~ for the sxial electric field become exact.

For evaluation points outside of the wire surface the relative errors of the current-filament approximations were derived above. The properties of these error terms are sum-marized below. Conditions at p = O are noted as an indication of the trend for small p,slthough these approximations are not used

Efhasamaximum atp=Oof E~=Ofq= a2/(46z2). E! is zero at p = O3z2/[16(a2 + %2)].

for p C a.

a2/(2z2) and a second maximum at p = fizand maximum at p2 = a2 + Z2 where E! =

~iszeroatp= Oand maximum at p = z where ~J

= 3a2/(16z2 . @ has a minimum

atp=Oof E$= a2/[2(a2+z2)] and a maximum at p2 = (a2+z2)/5 of z = 25a2/[48(a2+z2)],

E$ is zero at p = O and has a maximum at p =l/Zzof E$= ka2/~z. E! hasa

maximum at p = O of E$ = ka2/(2_).

The actual errors in these approximations, including GA, are shown in Fig. D-1 throughFig. D-4 for a wire radius of a = 0.00 IA. In Fig. D-2 and B-3 additional approximations are

114

included which are~- jkRt

G;=—ptRt

e-jkRG;=—

m

where pt = (p2 + a2)1/2. These expressions represent the result of actually putting a current

filament on the wire surface, as opposed to G! and G! in which p is the distance to theqrire axis. In all cases the relative errors were determined from a reference value computedby numerically integrating over 4. It should be noted that, in evaluating the fields, theterms G1, G2 and G3 are evaluated at the segment ends. Hence z = O in these integralsoccurs when the evaluation point is aligned with the segment end. G4, however, representsan integral over a segment which in Fig. D-4 extends from Z1 = –0.005A to Z2 = 0.005AHence for G4, z = O corresponds to an evaluation point aligned with the segment center.

For evaluating the field outside of the wire surface the current filament on the wireaxis (P) is seen to yield smaller errors than the filament on the surface (d) over muchof the parameter range shown. The exceptions occur for G1 and G4 when p is less thanabout 2/2. For the radial field terms, Gz and Gs, the axial approximation G@ yields smallererrors than the G* approximation. The Gg approximations result in about 30 percent erroron the wire surface. The GA approximations, which are used in NEC-3 as the “extendedthin-wire kernel”, are seen to yield substantially smaller errors thszt any of the first orderapproximations at the expense of additional complexity. For large p all terms eventuallyconverge to the correct result, although in the plots shown here the convergence appearsslow due to the limited range of p.

[1]

[2]

[3]

References

A. J. Poggio and R. W. Adarns, Approximations for Terms Related to the Kernel inThin- Win Integrul Equations, Lawrence Livermore National Lab. Rept. UCRL-51985,December 1975.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integmls, Series and Products, AcademicPress, New York, 1965.

G. J. Burke and A. J. Poggio, Numerical Electromagnetic Code (NEC) - Method ofMoments, Lawrence Livermore National Laboratory, Rept. UCID-18834, Jan. 1981.

115

-.,-. .-.!... ‘.. —. -—-— -*.--W. ..-4 . . . . ‘.,!W .M.-w-?--w-e-,.-..,.. .> ,. ....4., ,.. .4. ,,.. .,-,-w.!.,,-..— --.-’*?—. . . ..—- -— ——?

Z=o.

- l\I \

* .lE-6 I I I I

-1

0 5 15 20E-3

;;A

17z = O.OIA

Fig. D-1 Relativeerror in approximation for Cl for a = 0.00IA

lE-6~a

E-3 p/A

z = O.OIA

: ““L“01- /~.....................................................&oQ@i- 11 /“&;-----------

~ u“ 2“j lE-4 -

z

L_ .\ . @

lE-5- \

I \lE-6 I I I I

0 5 10C-3

15 20

p/A

Fig. D-2 Relative error in approximation for Gz for a = 0.00IA

116

.

I

x!’2:::-”’””. . . ......................----/

G;-------\ G:

\\

lE-6 I I 1 I I0 5

E-3

Fig. D-SRelative error in approximation

1

d.-.01

5

$ .001

lE-6

Z=o.

‘--G; --------------

z = O.OIA

, G:

*,1

I \/- 4-\ ‘1’v -\

I 1 1

5 15 20E-3

;;A

z = O.OIA

1!! G’3I.....................................................

sE-3

20

for G3 for a = 0.00IX

IE-E

z = O.oo!!d

11

6.-8E~ .OQl- ,-----------------

0 G;P

.2 lE-4-

3 lE-S - Gf! -<

l=~aE-3

p/A

1-

G .1-Cl.-

.O1-8

S .OO1-V>

“~ lE-4 -

Z! lE-5 - Gf--4lE-6 I I I I

0 5 10 15 20E-3 p/A

Fig. D-4 Relative emorinappmxkatiom for Gt forzl=-O.~A, ~=O.WA anda= o.oolA.

117

.... .,.. .,..,. .s,..sm=m- =.,.,. -..%. . .. . . . . , .- . . ,m...e,a.__ =_, ,- . . . .. . . ,, ... ,,,,.. .,,,,,..,.-.=--— —=-,,-.........,,.... ... ,...-.,,,.,,.,............-,,e...---=m— —.. . ...~w,-,,,.

APPENDIX E

Evaluation of the Field due to aConstant Charge on a Wire End Cap

The wire end cap is represented as a flatdisk as shown in Fig. El. Since the currenton the cap has no azimuthal variation in thethin-wire approximation, the vector poten-tial is zero on the axis of the disk and greatlyreduced by cancellation at points off of themcis. Assuming a constant surface chargedensity p~ on the end cap, the scalar poten-tial is

where

R=(p2+pn+

Forh=(p2+

%2- 2pp’ Cos0)1/2

Z2)li2 >>a an approx-

Fig. E&lCoordinates for evaluation of the fielddue to a charged end cap.

imation for *(;, z) can be obtained by ap-proximating the integrand by a Maclaurinseries in # as

~-jkR e-jkh e-jk~

— == +pp’cosd(l +jkRo)-R %

+P[ 1

-jk%

; p2 COS2$(3 + 3jkRo – k2R#) – (1 + jkRo)@ ~. ~

Then

@(p,z) =E{:+&[ 1}

e-jkh(3+3jk& - k2@p2 - 2jk@ -2% ~

The field components are then

8* p.za2 e-jkh

{

a2Ez(p, z) = –Z =—

4e ~(1+ jkh) - ~[3(1 + jkh) - k2@]

-$$~k3@ + 6k2@ - 15(1 +jk&)]} (Bl)

118

and

84$ p.pa2e-~k~

{

a2f3p(p, z) = –~ =~—

R$(1 +jkRO) - ~(3(1 +jkRcl) - k2R~]

-#$jk3R$ + 6k2Rjj - H(1 +jk%)]} (E2)

On the axis of the disk the field has only a z component that can be evaluated in closedform. When p = O

and the integral for @ becomes

where Rk = (a2 + Z2)1i2. Then

(k! -jk(&-izi) _ 1E.(O, z) = –~e-jk’z’ ~e

261%1 )(E3)

Equation (El) should be used on the axis when Izl > a since accuracy is lost due tocancellation in evaluating (E3).

119

APPENDIX F

Somrnerfeld Integrals for the Field Near Ground

The exact solution for the field of a point source near an interface wss first published byA. Sommerfeld [1], and involves integrals over wave number, commonly called Sommerfeldintegrals. Only the solution for the source below the interface and evaluation point abovethe interface is used in NEC, with the field for other configurations obtained by applyingreciprocity and adding the direct and image tem when appropriate. The full set of electricfield equations from Btios [2] is given below for reference, however.

The geometry of the half-space problem is shown in Fig. 5-1, and the wave numbers inthe lower medium Icl and upper medium kz are

The notation for electric field will use a subscript to indicate the field compcnent in cylindricalcoordinates (p, 4, z) and a superscript to indicate the source orientation, V for a vertic~source and H for a horizontal source. In addition, subscripts will be used to indicate themedium in which the source and evaluation points are located. The horizontal dipole sourceis oriented along the x axis.

For the source below the interface in medium 1 and evaluation point above the interfacein medium 2 (~ <0, z a O) the field components are

E;2 =– jwpo 82 VIZ—.

47r O@z

E;2 =*(2+k’)v’2

(

8%2E;2 = ? cos 4 — + Ul$?@2

)

(

jdpo sin ~ 1 m42E$2 = ~ -—+U12p ap )

82VIZE~2 = 3% COS@—

spa%’

where VIZand Ulz are the Sommerfeld integrals

~m’-w—J()(Ap)AdA?’1 +72

120

with VI =(A2 _ ~’Jl/2 For the ~OurCe ~b~~ the interface in medium

(A2 - k~)1i2 and 72= .1 and evaluation point below the interfam in medium 2 (z’ z (3,z < O) the field is

where

!

m ~71z-?2z’

U21=2 o —J~(Ap)A A‘Y1 + ?’2

For source and evaluation points above the interfwe (z’ z O, z Z O)

where

!

m ~--n(z+z’)

U22=2 o —J&@)A 071 +72

G22 = ~-jhRD/RD, RD = [p2 + (z - ~’)2] 1/2

G21 = #2%/R*, RI= [p2 + (Z + 2’)2]1’2 .

121

,,—-., ,M.nm, -..,..., — .--!””, ,., ., . . . . . . . ..” ..,-———.-.. ——)..———,~.—!,. -..-! . . ~. .1..! . -.———

For source and evaluation points below the interface (z’ <O, z < O)

[1]

[2]

J

@ ~% (Z+z’)

U11=2 ~ -Jo(Ap)AdA

Gll = ~-jk%iRD, RD = [p2 + (Z – Z’)2]1’2

Glz = ~-jklRf/R1, RI= [p2 + (Z + Z’)2]1’2.

References

A. Sommerfeld, “Uber die Ausbreitung der Wellen in der drahtbsen Tklegraphie~ Arm.Ph@k, Vol. 28, p. 665, 1909.

A. Btios, Dipole Radiation in the Presence of a Conducting Half-Space, Pergamon Press,New York, 1966.

122

.-—.—

APPENDIX G

Conversion Formulas for Source and Observer Locations

The Sornmerfeld-integral and asymptotic field evaluations in NEC are coded for thecase of a source below the interface and evaluation point above the interface. Fields forother cases are obtained horn this buried-source elevated-observer evaluation by appropriateconversions, including addition of direct and image fields. The source is located at a height~ and the evaluation point at height z, with radial separation p. The conversions are derivedby applying reciprocity, boundary conditions at the interface and the fact that the directfield depends on (z – z’) while the image field depends on (z+ d).

The direct and image fields are evaluated for a source in a finite medium with wavenumber kn, where n is 1 or 2. The cylindrical components of the electric field, for a sourcewith current moment 1~ = 1, are

E$(p,%’,z) = -n

EgJp, z’, z)= =4xk;

E:(p)%’, z) = -n

E~(p, z’, Z) = -

E~(p, z’, z) = =47rk3

[

/?(%– z’) 1~-jknR

~2 (3+ 3jknR – k~R2) ~ (G-1)

[ 1~(3 +3jknR - k~R2) – 1 – jhR+k~R2 ~ (G-2)

[

2

1-jhR

%(3 + 3jknR – k~R2) -1 – jk~R + k~R2 ~ (G-3)

[1 +j&R - k;R2] = (G-4)

[

p(z - z’) 1~-j~R

T(3 + 3jknR - k~R2) ~ (G-5)

where R = ~z + z- 2?)2]1/2. For a horizontal solwce along the x axis, the factor COS4 wouldLbe included in EP and E#, md sin@ in E$’ for the ~mplete field.

The fields for various locations of source and observer will be written as a functionof (p, d, z) where the second argument is the source height and the third is the observer’sheight. For clarity, however, a and b, which are always positive numbers, will be substituted ~for the source and observer’s heights. For source above the interface and observer below:

E~21(p, a, -b) = –E52(p, –b, a) (G-6)

ELI (p, u, –b) = E~2(p, -b, u) (G-7)

E$l(P, a, –b) = E&(P, -b, a) (G-8)

E& (p, a, -b) = E&(A ‘b, a) (G-9)

E%(P) at –b) = – E$2(P) –b, a). (G-1O)

123

For source and observer below the interface:

E~l(p, -a, -b)= ~fiz(p, –a - ~,0+) + ~~(p, –a, -~) - Efi(p, –a, b) (G-11}

%(A –a, –b) = 1 v~Ez12(p, –a – b, O+)+ Eq(p, -a, –b) – E~(p, –a, b) (G-12)

Efil(p, -a, –b) = E~2(p, –a – b,0+) + E$(JI, –a, –~) – E~(p, –a, ~) (G-13)

E$ll(p, –a, –b) = E&2(p, –a – ~,o+) + E$(P, –a, –b) – E.&(P, –a, b) (G-14)

l?~l(p, –a, –b) = ~E~@, –a - b, 0+) + E5(p, –a, –b) - ES(p, –a, b)

=- q2(P! –a – b, 0+) + Efl(p, –a, -b) + l?~(p, –a, b) (G-15)

For source and observer above the interface:

E$2(p, a, b) = –Jl~2(p, O-, a + ~) + E;(P, a,b) – E%(P, a, –b) (G-16)

E~~(p, a, b) = FlE~2(p,0-,a + b) + EY2(P, a,b) – ~~(p,a, –b) (G-17)

~$2 (A )ah)= J3;2(PI O_)a +b) + E$(p, a,b) – E$(p, a,-b) (G-18)

E&z@, a, b) = E&2(p, 0-, a + b) + E$(P, a, b) – E&($l a? ‘b) (G-19).

E~2(p, a, b) = –Fl E~12(p,0-, a + b) + E3(p, a,b) – EZ(p, a, -b)

= Eg2(P, O-ta + b) + E~(p, a,b) – E%(P, –a, b). (G-20)

124

APPENDIX H

Singularity Subtraction in the Sommerfeld Integralsfor the Field Due to Ground

To facilitate numerical evaluation of the Sommerfeld integrals near the field singularity,the singular part is removed in an analytically integrable form. This is done by subtractingthe term V. from V12where

andR = [p2+ (z - Z’)2]1?

The remainder, which must be evaluated by numerical integration, is then

where VI = (A2 – k~)l/2 and 72 = (A2 – kf)l/2. When R is small the subtraction must be

done in the integrand with careful treatment of canceling terms, while for larger R, V. canbe subtracted from the numerical result for VIZ to retain the simpler integrand.

Substituting V~2into the equations [5-2) for electric field yields the remainder terms

(H-lc)

(H-2a)

S;2 = -jw~ 2

[(—— (z - Z’)2 “~ -k;

4?r k;+~ $+3R) 1“22~ (H-2b)-1–jkzR+~R

-jw~ 2S;2 = ——[(

3 ‘a-k’)-l-’~Rle=P2~+~R (H-2c)4X kf + ‘Z

12s

s&=–jw~ 2 ~-jbR

Tm(H-2d)(1+ jk2R)~

S:2 = =*[’(Z-Z’)($+’*-’’)I= (H-2e)

After subtracting Vo, Viz is finite at R = O, while the field components in equations(H-l) have l/Rsingularities. Thel/Rsingularities till besubtracted fromthesefield@rmsin a later step to leave a finite remainder at R = O, but first the evaluation of the integrandfor V(2 is considered.

Careful treatment of canceling terms in the integrand of V/z is needed when IJI is largeto avoid loss of precision. This is especially important with the adaptive quadrature used inNEC. The cancellation can be isolated in simple terms by writing

(H-3)

The derivative of V~2with respect to z brings out a factor of –72. However, the derivativewith respect to # changes the canceling terms, so that

For large IAI the differences can be evaluated with a series expansion in A-l. AssumingIre(A) <0 and vertical branch cuts,

Sz = Sign[Re(J – kz)], IA!B lk21.

The series for the exponential with large IAIis

where

%qk: – )C;)2a2 =

8

and

s{

1 if Re(A) > Re(kl)=–1 if Re(\) < Re(k2).

It is assumed here that Re(kz) < Re(kl). When Re(kz) < Re(A) < Re(kl) the terms inV[2 do not cancel. This is not a problem, since the integration contour used for numericalevaluation goes betwmm the branch cuts to large IAI only when R is large, in which casesingularity subtraction is not done. Using the series expansions for large IAl the difference

I

terms are

T1 s alSA-l + azA-2 + asSJ-3 + a4A-4 + a5SA-5

I 7’2 = S(al + 151A-1+ b2A-2 + &#3 + b4A-4 + b5A-5)

T3 x S(al + clA- 1 + QA-2 + C3A-3 + qA-4 + C5J-5)

and

The equations (H-1) retain R-l sinWlarities from the derivatives of V(2 and from Ulz.These singularities can be evaluated and subtracted, although this was not done within theintegrand. To get the singular terms, the difference factors are approximated by their leading

127

“,- .,...”.— ~-. . ..m.m— ——*-... ,.. “ ..—. ——~, .,.,. ,...,* . . . . ..—.—

.—...-—.-————

terms for large A and small z’, with Re(A) > Re(kl ), as

The A-l terms must be included in T2 and T3 when they are not multiplied by z’. Theintegrals (H-3) and (H-4) for Ikl RI <<1 then become

aJI

w z’(k; – k2 k; – k;) ~_2k?) A-l + 1(2—VI1

~-n(z-z’) J@) dA&J 12=2 o 2(k~ + k?) 2(k2 + k~)2

The R-l components of the E’ terms in (H-1) can then be evaluated using the integralsnom {1]

J

m #z-z’) Jo(~p) dA = +

o

J

sin Om #(z-z’)Jo(Jp)~ ~J = ~

o

/

sin O–1m ~_ A(z-z’)J@p) ~~ = m

o

J

–Cosem &(z-z’)J$(~p)~ fj~ = ~

where R = ~2 + (z - ~)2]1J2 jd sind = (z – z’)/R The results, for k2R <<1 are

[

l–sine 1dcose e-JhR–jwo C2X. Cl ~ —47r R

[

_ cl # sin O e-jkR1–jtdpo C2 — —

41r

1(–jw~ C2

;-’)~cli(sin’(l:::e) -l)+ll=

l–sine

41r

cl_k?-ki 24=%C2 = ‘2 (kl + k2)2kf+k;’

128

Terms of l/R have been added to S~~2and S$2 to cancel the singularity of U12. Although

the factor e-J~R does not result from the integrals, it is included since it is contained inother field components. Subtracting these terms from the E’ terms leaves the remainder

1The E“ quantities are finite at R = O, so the origin can be included in an interpolation table.

References

[1] I. S. Gradshteyn and 1. M. Ryzhik, Tables of Integmls, Series and Products, AcademicPress, New York, 1980.

.

129

APPENDIX I

Numerical Solution for Saddle Pointsin the Sommerfeld Integrals

For asymptotic evaluation, the Sommerfeldintegrals are written in the form

where G is a slowly varying function of A, rela- —tive to the exponential. With the coordinates asshown in Fig. I-1, the function F(A) is

z’

F(A) = (Y – ky/2z – (A2 – /+)1/2%’+ jAp

-LLand is the same for all components of the field. Fig. I-1 Coordinates

The saddIe points ~~, needed in the steepest de- saddle point.

scent evaluation, are solutions of the equation

P kl

for evaluating the

(I-1)

Whenz=Oor#= O, (I-1) can easily be solved for & For general values of p, zand z’ the value of AS can be found by solving a quartic equation in A:, but a numericalsolution is usually more practical. The numerical procedure is complicated by the existenceof multiple solutions that can mislead the solution algorithm. An additional problem with(1-1) is that in the limits as z or z’ approach zero the saddle points should approach kz or kl,respectively, with the steepest-descent paths becoming branch-cut integrals. However termsin (I-1) become indeterminate in these limits. Hence (I-1) was multiplied by the denominatorterms to yield

~(~~) = JS(A: - k~)li2z - ~.(~~ – k:)l/2z’ + j(~g – k!)l’2(~g – k;)l’2P = O (I-2)

This equation is then solved by Newton’s method. Using the derivative

(I-3)

the iteration equation is

~i+l = ~i – f(~i)/f’(Ji) = ~i - ~(~i)

where

D(A) =A(A2 - k~)(~2 – k;)112z - ~(~2 – @(~2 – k~)lt2& + jp(~2 – k~)(~2 – @)

(2A2 - kf)(A2 – k~)lJ2z – (2A* – k;)(A2 – k~)li2z’ + jpA(2A2 - k? – k;) “(1-4)

130

Since (I-2) can be scaled by a constant, the solution depends on only two distanceparameters. Thenorrnalized parameters

- = %/(% – z’)

~=~/J-=sino

will be used, since they can independently range from O to 1.

Solutions for ASfor a typical case are shown in Fig. 5-6. On the principal Riemann sheetthere is a saddle point between J = O and k2, and a second SDP through a saddle pointbetween kz and kl may be needed. When z’ = O, a saddle point lies on the line between Oand ~ at & = k2 sin 6, with a possible branch-cut integral horn kl. When z = O, a saddlepoint is at & = kl sin 6, and the branch cut integral from k2 may contribute. When z issmall, but not zero, the saddle point near the line from O to kl passes onto the lower sheetat the branch cut from kz, and a saddle point near kz moves onto the principal sheet andapproaches the branch point as ~ increases. This detail near k2 is not shown on the plot.

Since the asymptotic approximations used here do not uniformly include the two branchpoints and a pole that occur in the Sommerfeld integrals, there is some arbitrariness in the

choice of saddle points. Equation (I-2) is solved only for the saddle point between Oand k2 inNEC, since this solution represents the wave traveling mostly through air. The saddle pointbetween k2 and kl represents a wave traveling mainly through the ground (the lateral wavefor z, i z O). When this wave is considered important, the approximation AS= kl sin 9 isused.

To obtain a reliable solution for ~, between O and kz, a table is first generated for thegiven ground parameters. For a particular value of Z, the solution is started at @ = O wherei. = ‘O. The parameter ~ is then incre-mented and the iteration in (I-3) is startedwith the previous ~S to find the new solu-tion. This process is continued until @ = 1,and then repeated for a new .5. A plot froma typical table of ~~ versus ~ and 2 is shownin Fig. I-2. For particular values of p, z and2, an estimate of ASis obtained by interpo-lating in the table for ~ and Z Iteration of(I-3) is then used to improve the accuracy.If the iteration results in lle(~~) > RE (kz)then ASis set to kz.

-0.4i0 0:!5 1.0

Real(A,/k2)

Fig. I-2 Imci of saddle pointe between A = Oandh, tlom the table generated for starting the iter-ative eolution. The complex relative permittivityof the ground was 10- j20.

131

Second Orderfor the

APPENDIX J

Asymptotic ApproximationsField Due to Ground

In the first section of this Appendix, second-order asymptotic approximations, from thesaddle-point method, are developed for the field transmitted across the air-earth interface.Corrections to these terms to obtain a uniform approximation valid for source and receiverapproachhg the interface are included in Section 5.2.4. These approximations for generalsource and receiver coordinates fail ss the radial separation p becomes small. A uniformasymptotic expansion including p = Ois not attempted here. First-order approximations areused for small p, and higher-order terns are obtained by interpolation when possible. Thisinterpolation of higher-order terms is described in the second section of this Appendix.

J.1 Second-Order Approximation

For an integral of the form

I =

for

r00

the Saddle Point Contribution

G(A)e-F(A) chJ-m

a second-order asymptotic approximation with a saddle point As, where F’(A~) = O andF“(As) # O, is from equations (6.2) and (5.20) in [1]

()1/2

1- $ e-FO(Qo + Q2) (J-1)

whereQc) = GO

Q2=~[ ~ Go(5F: – 3~Z~A) – 12Gl~z~s + 12G2@]24F2

and the notation

F.= $F(A) and Gn = ~G(J)AalllA, IAA,

has been used. The square root in (J-1) is chosen so that the argument of F-1/2 is equalto the argument of d~ on the steepest descent path. The term Q. represents the first-orderapprcdnation, and Q2 is the next higher order term. A term Q1 is included for an integralfkom a limit, but is not present for an integral through a saddle point.

In the integrals for the field components due to ground, F is the same for each componentwhile the function G vanes, Hence (J-1) will be written in the form

132

In the Sornmerfeld integrals for a buried source at Z’ (.z’ < O), field evaluation point at heightz (z ~ O) and radhd separation p, = in Fig. 5-1, F and its derivatives are

F. = r2Z– rlZ’+ j~.pF1s r;lr;l(JsrlZ – ~.r2# +~r1r2~)

F2 = 17;317;3(-~~r~z + ~~r;~’)

F3 = 3Asri51’~5(k~17~z - @;2’)

F4 = 31’;71’;7 [–k~(4A~ + k~)r~z + k~(4A~ + k~)r~z’]

2 li2 and & is the solution of the equationwhere I?l = (A; - k~)l/2 and r2 = (~~ - lc2)

F’(AJ = A.(A:– k;)-1/2z - AS(A:– ky/2z’ + jp = o. (J-3)

The terms involving F in (J-2) are then ~

Care is needed in evaluating these expressions to avoid division by zero. Although (J-3)may have multiple solutions representing two ray paths crossing the interface, the second-order approximation here will be used only for the saddle point between Oand ka and is notvalid for small p. Hence we need only be concerned with indeterminate forms resulting whenz goes to zero and simultaneously 172goes to zero. To avoid division by zero, the ratio rz/zwill be evaluated as

r2Az=y=

A,

~.z’irl – jp

obtained born (J-3). Then the terms involving F become

(J-4a)

(J-4b)

133

.. .. ,— .—.-— — “_ .“,+-.,. ~.,. s...6 . . . . . . . . . . . . . . . . . . . . ,, . . . . . ...”... . ,,.. - .,.. . ..~-.—”————--—. ”.-m-.—-——”——’

Equations (J-4b) and (J-4c) are still singular as z goes to zero, but the singulsr terms cancelwhen combined in (J-2) to yield

5F$ – 3FzFA 9r:’2A;’2z{

k;r~ + k:A;z6zaF;’2 = (–k;I’; + k~A:z2z’)7/2

– k~k:I’1A2z’[10A~rfA;z2 - A:z4(4~~ + k;) – r;(4~~+ #)] } (J-5)

~uation (J-2) can then be written in a form that can be evaluated when z and 17zgo toze%, whi& is

I - Ge-Fo(foga

where go= Go, gl = zG1, gz = Z3GZ ad

5F~ – 3FzF4!o=&+F2/ 24F;’2

+ !191 + !292) (J-6)

(J-7a)

1 F3 3&r:’2A:’2(k:r; - kfA:z4z’)!1 = ‘-— -

(J-7b)2 zy = 2(–k;r~ + k;A~z%)5/z

r~/zA~/2

!2 = ‘L(J-7c)

223$12 = (–k;r~ + k;A;zzz’)s/z “

Equations (J-4a) and (J-5) are used in evaluating j’o.

In approximating the integrals for field components, the two-term asymptotic expansionfor the Hankel function will be used

H~) (Z) ‘~(1+*)~-lZ+jz’4

o ~&(l_j&)e-~z-~m,4H’(2)(Z)

#) (z) -_g(l_j&)e-,.+i./?

The asymptotic approximations for the field components are then obtained by substitutingthe appropriate expressions for the function G and its derivatives into (J-6). For Ulz the

124

approximation is

(J-8)

so that

G(A) =( )

~VZ + J&-lJz DU(J) and D~(A) ‘ -

with VI = (A2 - k~)112 The coefficients in (J-8) are then(A2 - k~)112and 72= .

GO =(

A:/2 + $A.)

-1/2 ~ugo =

(

-1/2 ~ ~~3/2g~ = zG1 =: As

)( )& + A;/2 + -$;l’2 d~

2 8P

–A.d~= ZD~(&) ‘r1A2(r1 + zA2)

The approximations for the field components involve the derivatives of V12 and wherenecessary the term U12, The derivatives of the denominator of Vlz, Dv(A) will be representedas 1

dv = DV(&) = k~zAz + k;rl

&v=–A.(kfA;l + k@;l)

ZDV(AS) = (k~zAz + k;rl)2

k~k;(z31’;3 + A;3) + 2A:z(k~A;1 + Hzr;1)2d; = z3~; (h) =

(k~zAz + k;h)2 (k?zA2 + @l)3 “

135

For field component E~

where

! For field component E~

I

~y ._(g+k+i2

(J-9)

(J-1O)

136

For field component E;

–jwpo=—47r

“[’(“‘~ ‘“14%fogo+ fi91 + f2g2)e-(r2’-rlz’+j~~) _ U12 1where

( )

7j 31* dvgo= GO= ~y ~~

8p

( )( )7j 3/2 ~v

5 3/2 21~ #2 dv + &’2 – &sg~ = ZGI =% 2A, 16p

3292 = Z3G2=—

4 (7j~;1/2)dv+z2@,,/2 -&A:/2)~t5A!’2– ~

(5/2

)

,j 3/2 &+ As – #.

For field component E~

(J-11}

[(~jup~ e-jn14 2y~ /OgO+jlgl +f2g2)e-(r2z-rlz’+j~~+u12

1

(J-12)47r

For field mmponent E!

jwpo~2v12E! = ——

4X aflaz’

where

( )3j #2 dv

“ 90= Go srl A:’2– ~

%3( )Sj 5/2 dv

92 =Z3GQ =-—i#3 ‘p~a

r?

128

(J-13)

J.2 Approximations for Near Vertical Incidence

l%e approximations developed intheprevious section fail when pbecomessmallrel-

ative to Iz – z’I. One reason for this limitation is the use of the large argument forms forHankel functions with an argument involving p. More fundamentally, when the propaga-

tion path is mainly in the lower medium, two rays (two saddle points) may merge when

(/4@ = sin-l p pz + (2 – z’)5,

is near the totally reflecting angle dc = sin ‘1 (k2/kI). This

case is discussed in [2], and it is shown that the tr~stission or reflection coefficients canno longer be treated as slowly varying relative to the exponential. A correct asymptoticevaluation in this case shows that the reflected ray in the lower medium develops a lateralshift as it merges with the lateral wave, and a caustic occurs in the field near f?C.

A higher-order uniform asymptotic approximation for small p hsa not been attemptedhere. Rather, the first-order asymptotic forms are used, since these remain valid for smallp. The first-order approximations can also be derived through geometrical optics, withoutrestrictions on p. However, they converge much more slowly to the correct result, withincreasing separation of source and receiver, when O is near OCand z is much less than[z – 21. When the accuracy is limited by the approximation of the Hankel function for smallp,higher-order terms are obtained here by interpolation between p = Oand a limiting valuegreater than tan Oclz– z’I and added to the first-order result. A higher-order approximationfor the field at p = Ois developed below for use in this interpolation.

“ Expressions for p = O can be obtained by letting p go to zero in the Bessel functionform of the Sommerfeld integrals. For the terms involving VIZ the results are

a2—VIZ(O, 2, z’) =2 lim

J

~ #yl-#ew’-w

apax’Jf(~p) d~ = O

P+o o kf% + k;m

where

(J-14)

An asymptotic expansion for the integral & can be obtained by applying Eq. (5.21)from [1], for an integral from a limit at which F has a quadratic dependence on the integrand.

139

and with

For IV from (J-14),F. =jk2z – jklz’

F1 =0

F2 = – jkglz + jk; lz’F3 =0

Fb = – 3jk~3z + 3jk<3z’

snd

[

k~k~(~r3 + ‘Y~3) + 2A2(k~# + k;?;l) 1 –j(k~3 + %-3)Gv2 =

(k~72 + k;~l)2 (kf~2 + k;71)3 ~=o= (kl + ‘2)2 “

Then (J-15) with a = 3 yields

[ 12 -FIO Gvo+~(IV-fV=~e 2GV2F2 – Gvo174) .

similarly, U12 with P = O iS

/

m AU12(0,Z,%’)= 2 0 we

-(w-d) ~~

so that

I –2jG(.Io‘~ ,=0 = k, + k2

140

(J-15)

(J-16)

,. :

Eq. (J-21) with a = 1 then yields

The

Ulz ~ Wlz= &e[

-Fo Guo + $( Gu2F2 – @JoF4)

2 1approximations for the field components for p = O are then

(J-17)

E~(O, Z,z’)=0 (J-18a)

E~(O, Z,–jwp02jv~’) -—

41r (J-18b)

–@l@ (–iv + 012)Efl(O, Z, %’)-~ (J-18C)

~~~o ( fv + 012)Ef(o, z, z’) -~ - (J-18d)

E~(O, z, z’) =0 (J-18e)

with & and 012 given by (J-16) and (J-17).

The differences between the field approximations in (J-18) and the first-order approxi-mations for general p, from (5-23), are needed for the interpolated correction of the first-orderasymptotic terms. For the general asymptotic approximation, the saddle point & for smallp is from (J-3)

:=-j(;-;)-l* (;_:)-’.

The first-order approximations for the components of the field for small p are then

02—vlZ(/?, z,%’) -–2p e-j(kz-klz’)

aptlz kl (kl + kz) (z/k2 – z’/kl)2

G ) 2p2+ @ v12(p, z, z’) -

~-j(kzz-klz’)

klkz(kl + kz) (z/k2 – z’/kl)3

82 –2p2—Vla(p, z, z’) -

~-j(hz-klz’)

~p2 klkz(kl + kz) (z/k2 - z’/k1)3

la-—vlz(p, z, z’) -

-2j ~-j(k2z-klz’)

pap klkz(kl + kz) (Z/k2– Z’/kI)*i32—V’z(p, z,z’) *

2p e-j(kw-klz’)

Opaz’ ~2(h+ ~2) (Z/k2 – z’/kl)2e-j(hz-klz’)

~12(p, Z, ~’) ‘& (Z/k2 – ~t/kl)”

Hence for p = O the only nonzero terms are $$VIZ and Ulz. Subtracting these from (J-18),

the higher-order terms for p = O are

Eg’(o, z, z’) =0

141

(J-19a)

-jwo2jv (J-19b)E:”(o, z,z’) -~

E:~(o, z, z’) -%(-iv + u(~) (J-19c)

jwp~

(

2jEj~(o, z, z’) *X –iv +o;~ +

~-j(bz-klz’)

klk2(kl + k2) (z/k2 – z’/kl)2)

(J-19d)

mz~(o, z, z’) =0 (J-19e)

where---# -“. 2 ~-j(k2z-klz’)

[1:.-. =lJIQ —-. . . . .

For p less than a limit p. = IZ– %’1tan 6,4, where tan 8A is currently set at ~.2, the first-order asymptotic expressions (5-23) are evaluated for the coordinates (p, z, z’). Higher-orderterms are then evaluated for p = O,using (J-19), and for p = p~ by subtracting the first-orderapproximations of (5-23) from the second-order approximations (J-9) through (J-13). Thesehigher-order terms are interpolated to (p, z, z’) and added to the first-order result. Theexponential factor is omitted from the higher-order terms used in the interpolation, and thecorrect exponential for coordinates (p, z, z’) is included in the final result. Since E~ andE: are odd functions of

Jthe second-order terms are interpolated using the formula The

y, EP and E$ are even functions of p, so their higher-order terms arefield components Einterpolated as

flP) = ~ [P2mJ - (P+ PZ)(P - Pz)f(o)].

When the propagation path is largely in the lower medium (z/lz – z’I < 1) the tworays, from the saddle point between Oand k2 and the saddle point between k2 and kl, mergewhen 0 = tan-l (p/lz – z’1) is near the totally reflecting, or critical angle 8= = sin-l (k2/kl ).In this region, the asymptotic approximations based on isolated saddle points converge muchmore slowly than normal as the separation of source and receiver are increased. The second-order terms tend to blow up in this region until the distance becomes very large, so only thefirst-order armroxirnation is used.. .

[1] R. B. Dingle, AsymptoticPress, London, 1973.

References

Expansions: Their Derivation and Interpn#ation, Academic

[2] L. M. Brekhovskikhj Waves in Layered Media, Academic Press, New York, 1980.

142