antenna modeling for radio amateurs

1 S.D. Stearns, K6OIK ARRL Pacificon Antenna Seminar, San Ramon, CA October 17-19, 2008

Antenna Modeling for Radio Amateurs

Steve Stearns, K6OIKNorthrop Grumman

Electromagnetic Systems LaboratorySan Jose, California

[email protected]@arrl.net

mailto:[email protected]

mailto:[email protected]


Outline

History of electromagneticsComputational electromagnetics

Method of Moments (MoM)BRACT, WIRA, AMP, NEC 1-4, IE3D, WIPL-D, FEKO

Four main programs for amateursEZNEC4nec2WIPL-D LiteFEKO LITE

Advanced applicationsTerrain analysis by MoM

References, books, and software


Question for 2008

Is the current the same everywhere along a wire?

Iin Iout

outin II?=


Answer: It Depends

“Yes” for steady-state d-c current“Almost yes” for low-frequency a-c current or short wiresBut “no” for high-frequency a-c current because electrons can bunch upJ.C. Maxwell found a way to make the answer yes

Iin Iout

∫∫

∫∫

∂∂

=−

•⎟⎠⎞

⎜⎝⎛

∂∂

+=

∂∂

+=×∇

Snoutin

S

dSDt

II

dStDJ

tDJH

0


History


Who Does Not Belong in this Picture?


Answer: Leonardo Da Vinci and Isaac Newton

Isaac Newton1642-1727

Michael Faraday1791-1867

Georg Simon Ohm1789-1854

Carl Friedrich Gauss1777-1855

Andre-Marie Ampere1775-1836

James Clerk Maxwell1831-1879

Leonardo Da Vinci1452-1519


The Maxwellians

Oliver Heaviside1850-1925

George Francis FitzGerald1851-1901

Oliver Joseph Lodge1851-1940

Heinrich Rudolph Hertz1857-1894

John Henry Poynting1851-1914


Heaviside’s Vector Equations for Maxwell’s Theory

“And God said, Let there be light; and there was light.” Genesis 1:3

m

e

t

t

ρρ

=∇=∇

∂∂

+=×∇

∂∂

−−=×∇

⋅⋅BD

DJH

BME

HBED

με

==

HMEJ

m

e

σσ

==


Key Dates in Antennas

1842 Discovery of radiation – J. Henry1873 Treatise on electrodynamics – J.C. Maxwell1875-87 Early radiation demonstrations: Edison 1875;

A.E. Dolbear 1882; H. Hertz 18871889-06 Phased arrays – S.G. Brown, J.E. Murray, Artom1895-01 Radio communication, fan dipole, polar plots

– G. Marconi1897 Biconical dipole, loading coil, tunable LC matching

network, counterpoise, “impedance” – O.J. Lodge1907 Goniometer, electrical steerable array, radio direction-

finding – Bellini & Tosi1907 Ground losses, ground waves – Zenneck1923 Wave-tilt antenna – H.H. Beverage1928 Endfire array with parasitic elements – Yagi & Uda1947 Polyrod antenna – G.E. Mueller and W.L. Tyrell


Key Dates in Antennas continued

1947-75 Small antennas – H. Wheeler1948 Fundamental limit on antenna bandwidth – L.J. Chu1950 Antennas – J.D. Kraus1952 Advanced Antenna Theory – S.A. Schelkunoff1956 Theory of Linear Cylindrical Antennas – R.W.P. King1959 “Method of moments” – A.V. Kantorovich and G.P. Akilov1961 Antenna Engineering Handbook – H. Jasik1967 Matrix methods for fields problems – R.F. Harrington1974 Vivaldi antenna – L.R. Lewis, M. Fasset, and M. Hunt1976 Landstorfer antenna – F.M. Landstorfer2003 Metamaterial radomes – R.W. Ziolkowski and A.D. Kipple2005 Antenna Theory, 3rd ed. – C.A. Balanis2006 Electrically Small, Superdirective, and Superconducting

Antennas – R.C. Hansen


Old Style Antenna Analysis

Step 1 – Get the current distributionAssume sinusoidal – induced EMF methodSolve Hallen’s integral equation (1938)Solve Pocklington’s integral equation (1897)

How to do it – Mathematical solution Iterative and variational methods

– Approximation as a ratio of infinite series– King-Harrison (Proc. IRE, 1943)– Middleton-King (J. Appl. Phys., 1946)

Hill’s radiation pattern integration method (Proc. IEE, 1967)Limitations and complications

Not all antennas are wire antennasNot all antennas are made just of metalMath is hard


Induced EMF Method

Assumes sinusoidal current distributionMethod gives pattern, radiation resistance, and reactanceAccurate for pattern and impedance of dipoles up to half-wavelength and verticals up to quarter-wavelengthInaccurate for impedance of dipoles longer than half-wavelength and verticals longer than quarter-wavelengthUsed widely for the design of AM broadcast vertical towers


Induced EMF Method continued

Radiation resistance

Reactance

[ ]

[ ]⎭⎬⎫−−

⎪⎩

⎪⎨⎧

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−+

⎟⎠⎞

⎜⎝⎛

=

⎭⎬⎫⎥⎦

⎤⎢⎣

⎡−+⎟

⎠⎞

⎜⎝⎛++

⎩⎨⎧ −+−+

⎟⎠⎞

⎜⎝⎛

=

)(Si2)2(Si)cos(21

2Ci)(Ci2)2(Ci)sin(21)(Si

2sin2

)(Ci2)2(Ci2

ln)cos(21

)(Si2)2(Si)sin(21)(Ci)ln(

2sin2

2

2

2

klklkl

lkaklklklkl

klX

klklklCkl

klklklklklCkl

R

in

in

π

η

π

η

Terms vanish when l/λis a half integer

Wire radius term


Equations for Obtaining the Current Along a Wire

Pocklington’s equation (1897)

Hallen’s equation (1938)

General form

[ ]

gfL

zkCkzBjzdR

ezI

aEjzdzzGkz

zI

l

l

jkR

z

iz

l

lz

=

+−=′′

=−=′⎥⎦

⎤⎢⎣

⎡′⎟⎟

⎠

⎞⎜⎜⎝

⎛+

∂∂′

∫

∫

−

−

−

)(

|)|sin()cos(4

)(

)(),()(

11

2/

2/

2/

2/

22

2

με

π

ρωε

Linear operatorUnknown function

Driving function


Computational Electromagnetics


The Universe of Antenna Modeling Methods

Complexity of Materials

Elec

tric

al S

ize

Courtesy of EMSS


Computational Electromagnetics

Method of moments (MoM)A method for solving integro-differential equations such as Hallen’s or Pocklington’s equation at a given frequencyEarliest and longest legacy of software codes for antenna modelingBRACT, WIRA, AMP, NEC, NEC-2, NEC-3, NEC-4, MiniNEC, ELNEC, EZNEC, winNECPlus, 4nec2, FEKO, WIPL-D, Zeland IE3D

Finite element method (FEM)Best for design of small antennas of complex structureAnsoft HFSS

Finite difference time-domain method (FDTD)Time-domain methodBest for design of small antennas for broadband applicationsCST Microwave Studio, Zeland Fidelity, Faustus MEFiSTo

Geometric, physical, and uniform theories of diffractionBest for electrically large antennas


The Method of Moments


Originators

Leonid Vitaliyevich Kantorovich1912-1986

Gleb Pavlovich Akilov1924-1964

Roger F. Harrington1925-

Boris Grigoryevich Galerkin1871-1945

I.G. Bubnov1872-1919

Jack H. Richmond1922-1990


Method of Moments

Published by Kantorovich and Akilov in 1959 as a general method for solving linear integro-differential equationsIntroduced into electromagnetics by Roger Harrington in 1967

Currents are weighted sum of basis functionsSolve for the coefficients of the basis functions for all segmentsCalculate radiation pattern and feedpoint impedance from currents

Uses two kinds of mathematical functions: basis functions and test functionsBasis and test functions can be global or local (sub-sectional)Global basis functions expand the current on a wire in an infinite series, e.g. Fourier seriesLocal basis functions break antenna into small conducting line segments, surface patches, or volumesSubsectional basis functions appear to give better results when solving Hallen’s equation rather than Pocklington’sTest functions are best thought of as projections


Integro-Differential Equations Made Simple

Start with an equation. The analysis problem is to find f

Assume f can be expanded as a weighted sum of basis functions

Set all projections (via test functions) of left and right sides equal

Write as a matrix equation

( )

( )

( )

( ) ( )

( ) ( ) ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

•

•=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

••

••

•=•

=⎟⎠

⎞⎜⎝

⎛=

=

∑

∑

MNMNM

N

mn

mnn

nnn

g

g

a

a

fLfL

fLfL

gfLa

gfaLfL

gfL

φ

φ

φφ

φφ

φφ

MM

L

MOM

L 11

1

111


The Solution

Solve for the vector of expansion coefficients

Obtain f

( ) ( )

( ) ( )

[ ]( ) ( )

( ) ( ) ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

•

•

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

••

••==

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

•

•

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

••

••=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−

∑MMNM

N

Nn

nn

MMNM

N

N

g

g

fLfL

fLfLfffaf

g

g

fLfL

fLfL

a

a

φ

φ

φφ

φφ

φ

φ

φφ

φφ

M

L

MOM

L

L

M

L

MOM

L

M

11

1

111

1

11

1

1111


Principle MoM Computer Codes

BRACT & ANTBRACT – Developed late 1960’s at MBAssociates, San RamonWIRA – Developed early 1970’s by M. Andreasen, F. Harris and R. Tanner at TCIAMP/AMP2 – Developed mid 1970’s by G. Burke at MBAssociates, San RamonNEC-1 (1979) – Added more accurate current expansions; multiple wire junctions; thick wiresNEC-2 (1981) – Sommerfield-Norton ground interaction for wire structures above lossy ground; numerical Green's function allows modifying without repeating whole calculationNEC-3 (1985) – Buried wiresNEC-4 (1992) – Improved accuracy for stepped-radius wires and electrically-small segments, end caps and insulated wires, catenary-shaped wires, improved error detectionZeland IE3D (1992) – Adaptive meshing, developed by Dr. Jian-XiongZheng. Company in Fremont, CAWIPL-D (ca 2000) – Advanced MoM for wires, plates, and dielectrics based on work of A.R. Djordjevic, B.M. Kolundzija, U. Belgrade, SerbiaFEKO (ca 2000) – Hybrid method developed by U. Jakobus at EMSS, Stellenbosch, South Africa


The Development of NEC


L.B. Cebik, W4RNL, 1939-2008

Brought the joy of antenna modeling to Amateur Radio


EZNEC


EZNEC http://www.eznec.com/

Developed by Roy Lewallen, W7ELNow in version 5.0Six products available

EZNEC v.5 demo program $0 (free)EZNEC-ARRL v.3 & v.4 $45 (on ARRL Antenna Book CD-ROM)EZNEC v.5 $90EZNEC+ v.5 $140EZNEC Pro/2 v.5 $500EZNEC Pro/4 v.5 $650 (sold only to NEC-4 licensees)

EZNEC includes either the NEC-2 or NEC-4 enginesNEC-4 license for qualified US academic and noncommercial users can be obtained from Lawrence Livermore National Laboratory for $300. This probably includes you!

Form at: https://ipo.llnl.gov/technology/software/documents/NEC.pdf

http://www.eznec.com/

https://ipo.llnl.gov/technology/software/documents/NEC.pdf


Key Parts of EZNEC

Specifying the antenna modelWire geometry (including radials)Excitation sourcesWire loadsTransmission linesGround type and parametersFrequency or sweep range

Specifying the desired outputsRadiation pattern crossection at a given frequencyGain in a specific directionPattern beamwidthFront-to-back ratioFront-to-rear ratioImpedanceSWROutput data files for other programs


EZNEC Main Screen and Control Panel

“Model contains loss” warning


Wires and Segments

Each wire in an antenna is defined byLocation (coordinates) of both endsDiameter of wireNumber of segments (all equal in length)Material or conductivity

(x1, y1, z1) (x2, y2, z2)Diameter = AWG #12

No. segments = 7

Copper σ = 58 MS/m


Wire Table for UHF Discone Antenna

Steve Stearns, K6OIK, “All About the Discone Antenna,” QEX, Jan/Feb 2007


View Antenna

Steve Stearns, K6OIK, “All About the Discone Antenna,” QEX, Jan/Feb 2007


Discone SWR and Impedance Referenced to 75Ω

Marker at 460 MHz

UHF TV band highlighted blue470 to 710 MHz

1. Setup a frequency sweep 100 MHz to 1 GHz2. Write impedance data to MicroSmith .gam output file3. Use Word to reformat .gam to Ansoft .flp file format4. Import to Serenade SV project and design matching network


Improvement After Stub Matching Network


EZNEC Gain Patterns of Discone at 470 MHz


Tips for Getting Better Accuracy from NEC-2

Segment length to wavelength ruleSegment length < λ / 20

Segment length to diameter ruleSegment length > 2 × diameter

Equal segment length ruleAll segments in a model have equal lengthNever connect long segments to short segments

Acute angle junction ruleJunction angles or segment lengths large enough that middle 1/3 of joined segments don’t interpenetrate

Segment alignment rule for parallel wires

Closely spaced parallel and near parallel wires have their segments aligned (paired)

Wires near groundAll wires must be least two diameters above groundWires cannot touch ground


Excitation Sources

Ideal voltage and current sources can be inserted in any segmentSources are used to excite a feedpointMost antennas have a single feedpointA unit current source of 1 + j0 amperes generally works wellPhased arrays with multiple antenna feedpoints can be driven with a separate current sources at each feedpoint or with a single source driving a feed networkIf a segment contains both source and load, they are in seriesA voltage sources and load can be assigned to a segment when it is desired to simulate the Thevenin equivalent of a real generator – needed for mutual impedance effects in phased array feedsA “split-feed” consists of putting two half-voltage sources in adjacent segments to simulates a source at the junction between the segments


Wire Loads

Ideal, non-radiating point loads can be inserted in any segmentIf a segment contains both source and load, they are in seriesLoads are used to model coils, traps, and internally to NEC, wire conductivity (ohmic loss)Load types available are:

Constant impedance R + jXSeries RLC networkParallel RLC networkTrap networkLaplace impedancepositive-real rational functionup to 5th degree

012

23

34

45

5

012

23

34

45

5

)()()()()()()()()()(

)()()(

bjbjbjbjbjbajajajajaja

jQjPfZ

++++++++++

==ωωωωωωωωωω

ωω


Dielectrics and Wire Insulation

Dielectrics occur in antennas in bulk form or insulated wires, e.g. polyrod antennas, twin-lead folded dipoles, twin-lead J-poles, Butternut radials, buried radialsNEC-2 has no capability for dielectricsNEC-3 and NEC-4 handle dielectrics by accurate methods

NEC-3 handles wires in semi-infinite dielectric media, e.g. buried radialsNEC-4 handles insulated wires by accurate methodsNewer professional codes such as IE3D, FEKO, and WIPL-D handle dielectrics accurately by surface and volume equivalence principles

L.B. Cebik compared several ad hoc approximations for insulated wire (Note 83)EZNEC v.4 and above claim to do insulated wires but use NEC-2EZNEC Pro/4 v.5 uses the accurate NEC-4 dielectric capability


Insulated Wires Done Right !

Rigorous theoryJ.H. Richmond and E.H. Newman, “Dielectric Coated Wire Antennas,”Radio Science, vol. 11, no. 1, pp. 13-20, Jan. 1976J.P.Y. Lee and K.G. Balmain, “Wire Antennas Coated with Magnetically and Electrically Lossy Material,” Radio Science, vol. 14, no. 3, pp. 437-445, May-June 1979

Good quasistatic approximationB.D. Popovic and A. Nesic, “Generalisation of the Concept of Equivalent Radius of Thin Cylindrical Antennas,” IEE Proc., vol. 131, pt. H, no. 3, pp. 153-158, June 1984Larger effective radius with extra distributed inductance to offset

Equations are different from those of L.B. Cebik W4RNL, A. YurkovRA9MB, or D. Federov UA3AVR

⎟⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛ −×=⎟

⎠⎞

⎜⎝⎛=<⎟

⎠⎞

⎜⎝⎛×= −⎟⎟

⎠

⎞⎜⎜⎝

⎛ −

ab

a'aLb

aba'a

r

rr

r

ln1102ln2

701

εε

πμε

ε


Five Ground Types and Their Restrictions

Free spacePerfect ground

A lossless perfect electrically conducting (PEC) ground plane, i.e. a flat mirrorWires may touch groundGood for turning off ground losses to evaluate the ground losses of real ground

Real groundsHigh-accuracy (Sommerfeld-Norton) ground

– Most accurate and computationally expensive ground type– Best method for low horizontal wires down to λ/200 above ground– Wires may not touch ground

Fast ground– Uses complex reflection coefficient method– Horizontal wires should be at least λ/10 above ground– Wires may not touch ground

MININEC ground– Hybrid (compromise) calculation designed for early PCs– Assumes perfect ground for calculating currents, but switches to dielectric

ground for far-field pattern calculation – Hence no ground losses– Horizontal wires should be at least λ/5 above ground– Vertical wires may touch ground


Ground Parameters

Ground is specified by two numbers: conductivity σ and dielectric constant εrFor “average” ground, σ = 5 mS/m and εr = 13At HF frequencies, εr has more influence than σ; so don’t worry about the precise value of σ (stations near salt water excepted). Concentrate on getting a good estimate of εr

Ground Characteristics σ εrExtremely poor: cities, high buildings 0.001 3Very poor: cities, industrial 0.001 5Sandy, dry 0.002 10Poor: rocky, mountainous 0.002 13Average: pastoral, heavy clay 0.005 13Pastoral: medium hills and forest 0.006 13Flat, marshy, densely wooded 0.0075 12Pastoral, rich soil, US Midwest 0.010 14Very good: pastoral, rich, central US 0.030 20Fresh water 0.001 80Salt water 5 80


General Caveats Regarding Antenna Modeling

NEC is “blind” to current modes – computes total current, not resolved into common and differential current modes

Current modes are “noumena;” total current is “phenomena”Antennas that rely on interacting modes do not scale if λ/λgor vf changes

Insulation on wires affects common and differential current modes differently. Such antenna designs do not frequency scale easily.

Antennas made of insulated wire cannot be analyzed byNEC-2, but NEC-4 and EZNEC Pro/4 work okay

Twin lead folded dipoleTwin lead J-poleButternut radials


4nec2


4nec2 http://home.ict.nl/~arivoors/

A free full-featured GUI for NEC-2 and NEC-4Written and supported by Arie Voors, NetherlandsRuns under Windows 2000 and XPIncludes standard EZNEC models as .nec filesComes with NEC-2 executables but can use NEC-4 executablesComes configured for up to 11,000 segments but can be increased by to any number by recompiling the NEC-2 or NEC-4 source codesTwo versions

4nec2 – limited to machine memory4nec2X – uses virtual memory for bigger problems

Has 3D graphics and two optimizersGradient descent optimizerGenetic optimizer

Permits writing NEC script, thereby giving access to allNEC-2 and NEC-4 commands

http://home.ict.nl/~arivoors/


4nec2 Wire-Grid Models of Boeing 747 and Automobile


4nec2 Screen Displays

Main screen Geometry screen

Edit screen

Wire tab


4nec2 3D Pattern of Antenna on 747 – Vert Pol


WIPL-D

Wires, Plates, and Dielectrics


WIPL-D http://www.wipl-d.com/

Originated at University of Belgrade, Serbia (former Yugoslavia)Handles both 3D antennas and microwave circuitsMethod is MoM/SIE, e.g. surface equivalence principleFast multipole method (FMM) added to new version 7.0Capabilities include: lossy conductors, dielectric and magnetic materials, near and farfield calculations, optimizerPolynomial basis functions and curved bilinear quadrilateral surface meshing give high accuracy with small computationMeshed surfaces appear flat but are really curvedLacks infinite Sommerfeld-Norton ground, but has work-aroundTwo limited versions of interest to Radio Amateurs

WIPL-D Demo – Free download from http://www.wipl-d.com/WIPL-D Lite – More capable version from Artech House,ISBN 1580539653 ($399). Contact WIPL-D for replacement “bug-free”.exe file after installation

http://www.wipl-d.com/

http://www.wipl-d.com/


WIPL-D Models of Single and 4x4 Array of Polyrods


WIPL-D Model of Fighter Plane


FEKO


FEKO http://www.feko.info/

Developed and sold by EM Software & Systems (EMSS), South AfricaSwitches automatically among multiple “engines” like a Toyota PriusMain method is MoM/SIE, but has MoM/VIE, FEM, FMM, and several optics approximationsCapabilities similar to WIPL-D: lossy conductors, dielectric and magnetic materials, near and farfield calculations, optimizerCurved surfaces are approximated by many flat trianglesTriangle surface meshing and low-order basis functions give heavy computation burden, hence the need for multiple enginesHas infinite Sommerfeld-Norton groundLimited LITE version of interest to Radio Amateurs

FEKO LITE – Free download from http://www.feko.info/sales

http://www.feko.info/

http://www.feko.info/sales


FEKO Model of Global Hawk (RQ-4A)


FEKO Pattern of Horn Antenna in Wing Pod


Advanced Applications

Terrain Analysis by MoMMeshing Silicon Valley

A Rigorous Alternative to Ray Tracing


Earth Terrain Looking Down Saratoga-Sunnyvale Road

Surface meshed terrain of SaratogaColor indicates computed earth currentsZoom to see current direction arrows


Computer Used For Antenna Design and Electromagnetic Systems Analysis

Description 6 Xi NetRAIDer network servers

Processors 12 AMD Opteron 64-bit

Memory 96 Gbytes

Disk storage 12 Tbytes

Compute speed > 53 GFLOPs/sec


Required Computation

0

10

20

30

40

50

60

0 2 4 6 8

Frequency (MHz)

Ktr

iang

les,

Run

Tim

e, M

emor

y U

sed

KiloTrianglesHours

Memory GB

Frequency (kHz) Triangles Hours Memory (GB)

1,900 3,928 0.125 0.53

3,750 12,834 1.54 5.48

7,150 38,717 52.1 9.38


South

North

7.15 MHz Antenna Height = 50 m, Polarization = Horizontal

Earth’sContribution

Ground Currents


South

North

Frequency = 7.15 MHz Antenna type = 3 element YagiAntenna height = 164 ft (50 m)Antenna polarization = horizontalTotal field strength V+H shown

Courtesy of Keith Snyder, KI6BDR

3D Antenna Pattern


The Next Step – Modeling the Landscape


Landscaping Details

Details of leaves and branches

Models of trees

Courtesy of WIPL-D


Other Useful Antenna Software

winSMITH 2.0 by Agilent (formerly Eagleware)ISBN 1884932908 $127 from SciTech Publishing http://www.scitechpublishing.com/$149 from Amazon.comFor interactive design of ladder networks for impedance matchingExcellent tool for learning to use the Smith chartGrossly overpriced

MultiNEC by Dan Maguire, AC6LA, http://www.ac6la.comExcel/Visual Basic program; low cost but currently unavailablePuts NEC, EZNEC, and 4nec2 on autopilot for making a series of runsInexpensive alternative to a real optimizerDoesn’t work with EZNEC-ARRLTemporarily unavailable

http://www.scitechpublishing.com/

http://www.ac6la.com/


References

Current distribution in a wireH.C. Pocklington, “Electrical Oscillations in Wires,” Cambridge Phil. Soc. Proc., vol. 9, pp. 324-332, 1897.Erik Hallen, “Theoretical Investigations into the Transmitting and Receiving Qualities of Antennae,” Nova Acta Regiae Soc. Sci, Upsaliensis, ser. IV, vol. 11, no. 4, pp. 1-44, 1938.K.K. Mei, “On the Integral Equations of Thin Wire Antennas,” IEEE Trans. Antennas Propagat., vol. 13, no. 3, pp. 374-378, May 1965.

Bubnov and Galerkin methodsI.G. Bubnov, Stroitel’naia Mekhanika Korablia (Structural Mechanics of Shipbuilding), Tech. Rept., 1914.B.G. Galerkin, “Series Solution of Some Problems of Elastic Equilibrium of Rods and Plates,” Vestnik Inzhenerov i Tekhnikov, vol. 19, pp. 897-908, 1915. (English translation: NTIS Rept. TT-63-18924)

Method of MomentsL.V. Kantorovich and G.P. Akilov, Funktsional’nyj Analiz v NormirovannykhProstranstvakh (Functional Analysis in Normed Spaces), 1959. (English translation: Pergamon, 1964)R.F. Harrington, “Matrix Methods for Field Problems,” Proc. IEEE, vol. 55, no. 2, pp. 136-149, Feb. 1967.R.F. Harrington, Field Computation by Moment Methods, Macmillan, 1968.


References 2

Key papers in antenna modeling theoryJ.H. Richmond, “Digital Computer Solutions of the Rigorous Equations for Scattering Problems,” Proc. IEEE, vol. 53, no. 8, pp. 796-804, Aug. 1965.G.A. Thiele, “Calculation of the Current Distribution on a Thin Linear Antenna,”IEEE Trans. Antennas Propagat., vol. 14, no. 5, pp. 648-649, Sept. 1966.M.G. Andreasen and R.L. Tanner, Investigation of General Wire Antennas, Final Rept., DTIC AD0819198, TRG Division of CDC, Menlo Park, CA, August 1967.R.L. Tanner and M.G. Andreasen, “Numerical Solution of Electromagnetic Problems,” IEEE Spectrum, vol. 4, no. 9, pp. 53-61, Sept. 1967.A.R. Neureuther, et al., “Comparison of Numerical Methods for Thin Wire Antennas,” URSI meeting, Fall 1968.N.N. Wang and J.H. Richmond, and M.C. Gilreath, “Sinusoidal Reaction Formulation For Radiation and Scattering from Conducting Surfaces,” IEEE Trans. Antennas Propagat., vol. 23, no. 3, pp. 376-382, May 1975.K.S.H. Lee, L. Marin and J.P. Castillo, “Limitations of Wire-Grid Modeling of a Closed Surface,” IEEE Trans. Electromag. Compat., vol. 18, no. 3, pp 123-129, Aug. 1976.E.H. Newman and P. Tulyathan, A Surface Patch Model for Polygonal Plates, Tech Rept., DTIC ADA119682, Apr. 1981; also IEEE Trans. Antennas Propagat., vol. 30, no. 4, pp. 588-593, July 1982.Moment Methods in Antennas and Scattering, R.C. Hansen (ed.), Artech House, 1990, ISBN 0890064660.Computational Electromagnetics: Frequency-Domain Method of Moments, E.K. Miller, L. Medgyesi-Mitschang, and E.H. Newman (eds.), IEEE Press, 1992, ISBN 0879422769.M.M. Weiner, Monopole Antennas, CRC, 2003, ISBN 0824704967.


References 3

NECG.J. Burke and A.J. Poggio, Numerical Electromagnetics Code (NEC-2), Part I: Program Description – Theory; Part II: Program Description – Code; Part III: User’s Guide, Lawrence Livermore National Laboratory, Jan. 1981.J.K. Breakall, G.J. Burke, and E.K. Miller, “The Numerical/Electromagnetics Code (NEC-3),” EMC Symp., Zurich, March 5-7, 1985.G.J. Burke, Numerical Electromagnetics Code – NEC-4; Method of Moments; Part II: Program Description – Theory, Lawrence Livermore National Laboratory, Jan. 1992.

EZNECEZNEC User Manual, download current version from http://www.eznec.comL.B. Cebik, W4RNL, “A Beginners Guide to Modeling with NEC,” 4-part article, QST, pp. 34-38, Nov. 2000; pp. 40-44, Dec. 2000; pp. 44-48, Jan. 2001; and pp. 31-35, Feb. 2001.L.B. Cebik, W4RNL, ARRL Antenna Modeling Course, ARRL, 2002, ISBN 0872598721.ARRL Antenna Book, 21st ed., pp. 4-1 to 4-23, ARRL, 2007, ISBN 0872599876.

http://www.eznec.com/


References 4

4nec2http://home.ict.nl/~arivoors/

Zeland IE3DJ-X. Zheng, “A General Purpose 3D Electromagnetic Simulation and Optimization Package – IE3D,” IEEE MTT-S Int’l Symp., vol.1, pp. 373-376, May 23-27, 1994.

WIPL-DB.M. Kolundzija, J.S. Ognjanovic, and T.K. Sakar, WIPL-D Microwave: Circuit And 3D EM Simulation For RF & Microwave Applications –Software and User's Manual, Artech House, 2006, ISBN 1580539653.B.M. Kolundzija and A.R. Djordjevic, Electromagnetic Modeling of Composite Metallic and Dielectric Structures, Artech House, 2002, ISBN 0890063605.

FEKOU. Jakobus, “Review of Advanced Electromagnetic Modeling Techniques in the Computer Code FEKO based on the Method of Moments with Hybrid Extensions,” ACES Newsletter, vol. 18, no. 2, July 2003.D.B. Davidson, Computational Electromagnetics for RF and Microwave Engineering, Cambridge University Press, 2005, ISBN 0521838592.

http://home.ict.nl/~arivoors/


Good Reading

Bruce J. Hunt, The Maxwellians, Cornell University Press, 1991, ISBN 0801482348


Favorite Antenna Books

Books for antenna engineers and studentsAntenna Engineering Handbook, 4th ed., J.L. Volakis editor, McGraw-Hill, 2007, ISBN 0071475745. First published in 1961, Henry Jasik editor.R.C. Hansen, Electrically Small, Superdirective, and Superconducting Antennas, Wiley, 2006, ISBN 0471782556.C.A. Balanis, Antenna Theory, 3rd ed., Wiley, 2005, ISBN 047166782X. First published in 1982 by Harper & Row.J.D. Kraus and R.J. Marhefka, Antennas, 3rd ed., McGraw-Hill, 2001, ISBN 0072321032. First published in 1950.S.J. Orfanidis, Electromagnetic Waves and Antennas, draft textbook online at http://www.ece.rutgers.edu/~orfanidi/ewa/E.A. Laport, Radio Antenna Engineering, McGraw-Hill, 1952. http://snulbug.mtview.ca.us/books/RadioAntennaEngineering

Antenna research papersIEEE AP-S Digital Archive, 1952-2000 (2 DVDs), JD0351.IEEE AP-S Digital Archive, 2001-2003 (1 DVD), JD0301.IEEE AP-S Digital Archive, 2001-2006 (1 DVD), JD0304.

http://www.ece.rutgers.edu/~orfanidi/ewa/

http://snulbug.mtview.ca.us/books/RadioAntennaEngineering


Favorite Antenna Books continued

Books for radio amateursARRL Antenna Book, 21st ed., Dean Straw (N6BV) editor, American Radio Relay League, 2007, ISBN 0872599876.Practical Wire Antennas 2, Ian Poole (G3YWX) editor, Radio Society of Great Britain, 2005, ISBN 1905086040.J. Devoldere (ON4UN), ON4UN’s Low-Band Dxing, 4th ed., American Radio Relay League, 2005, ISBN 0872599140.J. Sevick (W2FMI), The Short Vertical Antenna and Ground Radial, CQ Communications, 2003, ISBN 0943016223.L. Moxon (G6XN), HF Antennas for All Locations, 2nd ed., Radio Society of Great Britain, 1983, ISBN 1872309151.

ARRL Antenna Compendium series – Volumes 1 through 7

ARRL Antenna Classics series – six titles


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