feko exampleguide

FEKO

Examples Guide

Suite 5.0

July 2005

Copyright 1998 – 2004: EM Software & Systems-S.A. (Pty) Ltd32 Techno Avenue, Technopark, Stellenbosch, 7600, South AfricaTel: +27-21-880-1880, Fax: +27-21-880-1936E-Mail: [email protected]: http://www.feko.info

mailto:[email protected]

http://www.feko.info

CONTENTS i

Contents

1 Introduction 1-1

2 CAD input examples 2-1

2.1 dipole plate PO: Dipole in front of a plate with physical optics . . . . 2-1

2.2 horn: A rectangular horn antenna . . . . . . . . . . . . . . . . . . . . 2-5

2.3 RCS DI sheet: RCS of a thin dielectric sheet . . . . . . . . . . . . . . 2-9

2.4 patch coupled: Proximity coupled patch antenna with microstrip feed 2-12

2.5 trihedral MLFMM: Using the MLFMM . . . . . . . . . . . . . . . . . 2-15

2.6 bio demo FEM: Exposure of muscle tissue using MoM/FEM hybrid . 2-18

3 Text input examples 3-1

3.1 Example 1: Half wavelength dipole . . . . . . . . . . . . . . . . . . . 3-1

3.2 Example 2: A dipole antenna in front of a plate . . . . . . . . . . . . 3-4

3.3 Example 3: Dipole in front of a plate with physical optics . . . . . . 3-10

3.4 Example 4: Dielectric sphere . . . . . . . . . . . . . . . . . . . . . . . 3-14

3.5 Example 5: Dipole in front of a dielectric beam . . . . . . . . . . . . 3-18

3.6 Example 6: Magnetic field probe . . . . . . . . . . . . . . . . . . . . 3-21

3.7 Example 7: Monopole on a finite circular ground plane . . . . . . . . 3-24

3.8 Example 8: A horn antenna . . . . . . . . . . . . . . . . . . . . . . . 3-27

3.9 Example 9: Dielectric cube . . . . . . . . . . . . . . . . . . . . . . . . 3-32

3.10 Example 10: Yagi-Uda antenna over a real ground . . . . . . . . . . . 3-36

3.11 Example 11: Dipole antenna in front of a PO plate with edge currents 3-43

3.12 Example 12: A metallic sphere coated with a dielectric . . . . . . . . 3-46

3.13 Example 13: Dielectric/magnetic sphere solved with volume currents 3-49

3.14 Example 14: Conducting cube modelled with PO and wedge correction 3-54

3.15 Example 15: Dipole antenna in front of a dielectric sphere . . . . . . 3-58

3.16 Example 16: Dipole antenna in front of a metallic Fock cylinder . . . 3-61

3.17 Example 17: Hertzian dipole in front of a parabolic reflector . . . . . 3-64

July 2005 FEKO Examples Guide

ii CONTENTS

3.18 Example 18: UHF antenna array . . . . . . . . . . . . . . . . . . . . 3-67

3.19 Example 19: Dipole antenna in front of a UTD plate . . . . . . . . . 3-75

3.20 Example 20: Monopole antenna on a metallic UTD plate . . . . . . . 3-79

3.21 Example 21: Mobile communications antenna on the roof of a building 3-83

3.22 Example 22: Planar dipole antenna (using wires) on a substrate . . . 3-88

3.23 Example 23: Dielectric cone on top of a metallic cylinder . . . . . . . 3-93

3.24 Example 24: Planar dipole antenna on a substrate (using triangles) . 3-97

3.26 Example 26: Input impedance of a two wire transmission line . . . . 3-100

3.27 Example 27: Yagi-Uda antenna in front of a cylindrical (UTD) mast 3-102

3.28 Example 28: Dipole antenna in front of a dielectric PO cylinder . . . 3-106

3.29 Example 29: Pin fed patch antenna on a finite dielectric substrate . . 3-110

3.30 Example 30: Patch antenna on a dielectric substrate . . . . . . . . . 3-114

3.31 Example 31: Wire antenna penetrating a real ground . . . . . . . . . 3-120

3.32 Example 32: RCS of a thin dielectric sheet . . . . . . . . . . . . . . . 3-124

3.33 Example 33: Shielding effectiveness of a thin hollow sphere . . . . . . 3-127

3.34 Example 34: Coaxial cable (modelled with surface triangles) . . . . . 3-130

3.35 Example 35: Replacing antenna with equivalent sources . . . . . . . . 3-135

3.36 Example 36: Example of S-parameter calculation above a ground plane 3-145

3.37 Example 37: Proximity coupled patch antenna with microstrip feed . 3-148

3.38 Example 38: Microstrip filter . . . . . . . . . . . . . . . . . . . . . . . 3-151

3.39 Example 39: Log periodic antenna . . . . . . . . . . . . . . . . . . . . 3-156

3.40 Example 40: Coupling between impedance matched dipoles . . . . . 3-160

3.41 Example 41: Using the MLFMM . . . . . . . . . . . . . . . . . . . . 3-166

Index I-1

EM Software & Systems-S.A. (Pty) Ltd July 2005

INTRODUCTION 1-1

1 Introduction

This Examples guide presents a set of simple examples which demonstrate most of thefeatures of the code FEKO. The examples have been selected to illustrate the featureswithout being unnecessarily complex or requiring excessive run times. The input filesfor the examples can be found in the examples\CAD_input and examples\text_inputdirectory under the FEKO installation. CADFEKO is the preferred method of geometryentry from FEKO Suite 5.0. In some cases it may be necessary to enter geometry usingthe text input method that was previously employed. All of the examples in the text inputsection are still relevant for users using CADFEKO for input, as they illustrate featuresof the solver. CADFEKO users should not be concerned if the geometry input section isnot entirely understood - the concepts in the solution control section are important. Foran introduction to the FEKO environment and the operation of FEKO, please consultthe Getting started manual.

Only time domain harmonic sources are supported; consequently calculation is done inthe frequency domain. However, FEKO includes the module TIMEFEKO, based on theFourier transform, to allow some time domain analysis. A discussion of TIMEFEKO, andan example of its use, is given in the User’s manual.

Another important feature of FEKO is the ability to describe the geometry in terms ofcertain parameters and to vary or optimise these with a specific goal in mind. This is donewith the optimising module OPTFEKO which is discussed in more detail in the User’smanual. Additional documentation on OPTFEKO can be found in the file example.optin the doc\optfeko subdirectory under the FEKO installation.

Changes in this manual with respect to the previous one of July 2004 (Suite 4.2) areindicated as follows:Sections that have changed from those in the previous version of the manual.Sections that were newly added to this version of the manual.

Running FEKO LITE

FEKO LITE is limited with respect to problem size and therefore cannot run the followingexamples from this guide

8, 9, 11, 12, 13, 14, 16, 17, 18, 21, 23, 28, 29, 33, 34, 35 and 41while these need to be modified

7 (last FF card), 30 (FR card), and 38 (FR card)as indicated in the *.pre files themselves.

For more information on FEKO LITE, please see the Getting started manual.


CAD INPUT EXAMPLES 2-1

2 CAD input examples

2.1 dipole plate PO: Dipole in front of a plate with physicaloptics

This example uses the same structure as the text input Example 2 shown in figure 3-3,but in this case the geometry is defined using CADFEKO, and the physical optics (PO)approximation is used to determine the currents on the surface of the plate.

One quarter of the plate is created using a polygon in CADFEKO. The upper half of thedipole is constructed using a polyline. The first section of the polyline is named feed andthe second is named top. The wire is constructed in sections so that the feed segment canbe uniquely specified to the solver.

The mesh is imported in three stages in the *.pre file. The first quarter of the plateis imported, and the first plane of suymmetry is applied. Next, the top of the dipoleis imported before the last plane of symmetry is applied. Lastly, the feed segment isimported. The PO card is used to define the PO region. The complete model controlinput file is given below.

** PREFEKO input file generated by CADFEKO

** Import quarter of plate

IN 8 31 "dipole_plate_PO.cfm" Plate1.Face1

** Mirror the plate in the plane y=0 (xz-plane) -- ideal magnetic wall

SY 1 0 3 0

** Import top half of dipole

IN 8 31 "dipole_plate_PO.cfm" Dipole.Top

** Mirror in the plane z=0 (xy-plane) -- perfect electric conducting plane

SY 1 0 0 2

** Import the dipole feed segment

IN 8 31 "dipole_plate_PO.cfm" Dipole.Feed1

** PO approximation for the plate with the label 2, ray search is switched off

PO: Plate1.Face1 : 1 : 0 : 0 : 0

** End of geometry

EG 0 0 0 0

** Set frequency

FR: 1 : : : : : #c0/#lambda

** Set source

** Excitation by means of a voltage gap (E-Field)

A1: 0 : Dipole.Feed1 : : : : 1


2-2 FEKO EXAMPLES: dipole plate PO

** Solution control

** Calculate the electric and magnetic near fields along the x axis

FE 3 70 1 1 0 -1.97 0 0 0.1 0 0

** Calculate the far field (horizontal plane)

FF 1 1 181 0 90 0 0 2

** End of file

EN

** CADFEKO Checksum: 45188c4f26a32c32546968e2e2106632

Some extracts from the output file dipole_plate_PO.out follow

DATA OF THE VOLTAGE SOURCE NO. 1

real part imag. part magn. phase

Current in A 9.7098E-03 -5.7193E-03 1.1269E-02 -30.50

Admitt. in A/V 9.7098E-03 -5.7193E-03 1.1269E-02 -30.50

Impedance in Ohm 7.6461E+01 4.5037E+01 8.8739E+01 30.50

Power in Watt: 4.85489E-03

VALUES OF THE ELECTRIC FIELD STRENGTH in V/m

in free space

LOCATION EX EY EZ

X/m Y/m Z/m magn. phase magn. phase magn. phase

-1.9700 0.0E+00 0.0E+00 0.000E+00 0.00 0.000E+00 0.00 7.655E-02 139.69

-1.8700 0.0E+00 0.0E+00 0.000E+00 0.00 0.000E+00 0.00 7.625E-02 149.18

-1.7700 0.0E+00 0.0E+00 0.000E+00 0.00 0.000E+00 0.00 7.582E-02 158.50

-1.6700 0.0E+00 0.0E+00 0.000E+00 0.00 0.000E+00 0.00 7.525E-02 167.64

VALUES OF THE MAGNETIC FIELD STRENGTH in A/m

in free space

LOCATION HX HY HZ


-1.9700 0.0E+00 0.0E+00 0.000E+00 0.00 1.977E-04 137.30 0.000E+00 0.00

-1.8700 0.0E+00 0.0E+00 0.000E+00 0.00 1.958E-04 146.53 0.000E+00 0.00

-1.7700 0.0E+00 0.0E+00 0.000E+00 0.00 1.935E-04 155.57 0.000E+00 0.00

-1.6700 0.0E+00 0.0E+00 0.000E+00 0.00 1.906E-04 164.39 0.000E+00 0.00

VALUES OF THE SCATTERED ELECTRIC FIELD STRENGTH IN THE FAR FIELD in V

Factor e^(-j*BETA*R)/R not considered

LOCATION ETHETA EPHI directivity in dB ...

THETA PHI magn. phase magn. phase vert. horiz. total


FEKO EXAMPLES: dipole plate PO 2-3

90.00 0.00 1.246E+00 170.01 0.000E+00 0.00 7.2683872 -999.9999 7.2683872

90.00 2.00 1.244E+00 169.91 0.000E+00 0.00 7.2547916 -999.9999 7.2547916

90.00 4.00 1.238E+00 169.62 0.000E+00 0.00 7.2139059 -999.9999 7.2139059

POLARISATION

axial r. angle direction

0.0000 180.00 LINEAR

0.0000 180.00 LINEAR

0.0000 180.00 LINEAR

Figure 2-1 shows the distribution of the near field along the x axis and figure 2-2 theradiation pattern in the horizontal plane. It is clear that there is good correlation betweenthe PO method and the method of moments results (figures 3-4 and 3-5). There is,however, a big difference in the computation time.

Figure 2-1: Electric near field along the x axis for a dipole in front of a plate, treatedwith PO


2-4 FEKO EXAMPLES: dipole plate PO

Figure 2-2: Radiation pattern (in dB) in the horizontal plane ϑ = 90 for a dipole in frontof a plate, treated with PO Example 3


FEKO EXAMPLES: horn 2-5

2.2 horn: A rectangular horn antenna

Figure 2-3: The meshed geometry of the horn example

A horn antenna structure, as shown in figure 2-3, is examined next. The geometry creationis very similar to that of the demo example (a copy of which is in the \docsubdirectoryof the FEKO installation. The most significant difference is that a different feed probeis used in the waveguide, and symmetry in two planes is exploited. This example differsfrom the demo example and the first getting started example in that gain compensationfor the antenna is discussed in this example. The input file is


** Import quarter of the horn antenna

IN 8 2 "horn.cfm"

** Mirror the quarter around the plane y=0 (xz-plane) -- ideal magnetic wall.

SY 1 0 3 0


2-6 FEKO EXAMPLES: horn

** Import the feedline

IN 8 1 "horn.cfm" HornAntenna.Line1

** Mirror around the plane z=0 (xy-plane) -- ideal electric wall.

SY 1 0 0 2

** Import the feed segment

IN 8 1 "horn.cfm" HornAntenna.Feed1

** End of geometry

EG 1 0 0 0 0

** Set the frequency

FR 1 0 #frequency

** Set the excitation

A1: 0 : HornAntenna.Feed1 : : : : 1 : 0

** Calculate the horizontal radiation pattern

FF 1 1 361 0 90 -90 0 1

** Calculate the vertical radiation pattern

FF 1 361 1 0 0 0 1 0

** Integration of the full 3-D pattern over a sphere to get the radiated power

** accurately (use symmetry, only 1/4 sphere, multiply power by 4).

** A test using a finer angular stepping has shown that a stepping of 5 deg.

** is fully sufficient for the dimensions under consideration (must be adjusted

** if a horn antenna with higher gain is modelled, since then more sidelobes occur)

#stepping = 5

#nthe = 90/#stepping + 1

#nphi = 180/#stepping + 1

FF 3 #nthe#nphi0 0 0 #stepping #stepping

** End of file

EN

** CADFEKO Checksum: 45188c4f26a32c32546968e2e2106632

Some extracts from the output file horn.out are given below.



Current in A 1.0511E-03 -1.8467E-04 1.0672E-03 -9.96

Admitt. in A/V 1.0511E-03 -1.8467E-04 1.0672E-03 -9.96


Inductance in H 1.5687E-08



FEKO EXAMPLES: horn 2-7





...

90.00 -4.00 1.293E+00 102.13 0.000E+00 0.00 16.6567 -999.9999 16.6567

90.00 -3.00 1.316E+00 102.70 0.000E+00 0.00 16.7980 -999.9999 16.7980

90.00 -2.00 1.333E+00 103.11 0.000E+00 0.00 16.8991 -999.9999 16.8991

90.00 -1.00 1.343E+00 103.36 0.000E+00 0.00 16.9597 -999.9999 16.9597

POLARISATION


0.0000 180.00 LINEAR

0.0000 180.00 LINEAR

0.0000 180.00 LINEAR

0.0000 180.00 LINEAR





0.00 0.00 1.212E-01 -151.71 0.000E+00 0.00 -3.3176 -999.9999 -3.3176

1.00 0.00 1.245E-01 -135.52 0.000E+00 0.00 -3.0824 -999.9999 -3.0824

2.00 0.00 1.272E-01 -119.72 0.000E+00 0.00 -2.8949 -999.9999 -2.8949

3.00 0.00 1.290E-01 -104.17 0.000E+00 0.00 -2.7700 -999.9999 -2.7700

4.00 0.00 1.299E-01 -88.71 0.000E+00 0.00 -2.7127 -999.9999 -2.7127

POLARISATION


0.0000 180.00 LINEAR

0.0000 180.00 LINEAR

0.0000 0.00 LINEAR

0.0000 0.00 LINEAR

0.0000 0.00 LINEAR



Integration of the normal component of the Poynting vector in the angular

grid DTHETA = 5.00 deg. and DPHI = 5.00 deg. ( 703 sample points)

angular range THETA angular range PHI radiated power

-2.50 .. 92.50 deg. -2.50 .. 182.50 deg. 1.74560E-04 Watt

0.00 .. 90.00 deg. 0.00 .. 180.00 deg. 1.26874E-04 Watt

Note that the source power is calculated from a single current and in some cases thismight not be very accurate. In this example this is 0.525569 mW. We can obtain a moreaccurate calculation of the radiated power by integrating the far field. (Note that thisgives the radiated power which is not the same as the source power for lossy antennas.)In this example, the integrated radiated power is 4 times 0.12687 mW (we only integratedover a quarter of the far field region). Thus the actual power is 0.152 dB higher than


2-8 FEKO EXAMPLES: horn

the calculated source power. Since the power is used to normalise the directivity calcula-tion, the actual directivity is 0.152 dB lower than the calculated values. The directivitymight therefore be compensated by specifying a -0.152 dB offset in POSTFEKO. Theseresults are shown in figures 2-4 and 2-5 for the horizontal and vertical radiation patternsrespectively.

2702402101801501209060300-30-60-90

20

15

10

5

0

-5

-10

-15

-20

-25

-30

Phi (degrees)

Dire

ctivity

(dB

)

Figure 2-4: Radiation pattern in the horizontal plane ϑ = 90

3603303002702402101801501209060300

20

18

16

14

12

10

8

6

4

2

0

-2

-4

-6

-8

-10

Theta (degrees)

Directivity

(dB

)

Figure 2-5: Radiation pattern in the vertical plane ϕ = 0


FEKO EXAMPLES: RCS DI sheet 2-9

2.3 RCS DI sheet: RCS of a thin dielectric sheet

z

y

x

a = 2 m

b = 1 m

i

i

iEi

Si

Hi

n

Figure 2-6: Geometry of RCS of a dielectric sheet with the incident plane wave

The geometry for this example is shown in figure 2-6 — a thin dielectric plate. The size,thickness and material parameters can be determined from the input file below. The plateis illuminated by an incident plane wave such that the bistatic radar cross section maybe calculated.

As indicated in the section “Dielectric solids” in the “General comments” chapter of theUser’s manual, there are a number of ways with which such a thin dielectric plate may betreated in FEKO. In principle we may use the volume equivalence principle, discretisingthe dielectric into small cuboids (as was done for the cube in the text input Example 9).However, it uses substantially less memory to realise the sheet with the SK card. Theinput file is as follows


** Import model

IN 8 31 "RCS_DI_sheet.cfm"

** Symmetry (Geometrical only due to arbitrary incidence direction)

SY 1 1 1 0

** End of geometry

EG 1 0 0 0

** Set frequency

FR 1 1 #frequency

** Set source

A0 0 1 1 1 0 #theta_inc#phi_inc #eta_inc

** Define the thin dielectric sheet


2-10 FEKO EXAMPLES: RCS DI sheet

SK: Plate.Plate : 4 : : : : 0.004 : 1 : 0 : : #tan_delta : #eps_r

** Bistatic RCS (vertical cut)

FF 1 181 1 0 0 0 2 0

** End of file

EN

** CADFEKO Checksum: 3c24b344689d85335fd3fe46af8e77a2

The geometry is discretised into triangular elements, similar to conducting plates. Thethin dielectric sheet formulation is then applied to all triangles with the given label. Weare interested in the calculated RCS. Some extracts from the output file follows

EXCITATION BY PLANE LINEAR POLARISED ELECTROMAGNETIC WAVE

Number of excitation: N = 1

Frequency in Hz: FREQ = 1.00000E+08

Wavelength in m: LAMBDA = 2.99792E+00

Direction of incidence: THETA = 20.00 PHI = 50.00

Dir. of polarisation: ETA = 60.00

Direction of propag.: BETA0X = -4.60764E-01

BETA0Y = -5.49117E-01

BETA0Z = -1.96945E+00

Field strength in V/m: |E0X| = 9.65425E-01 ARG(E0X) = 180.00

(Phase in deg.) |E0Y| = 1.96747E-01 ARG(E0Y) = 0.00

|E0Z| = 1.71010E-01 ARG(E0Z) = 0.00

DATA OF LABELS

Label 0: DOSKIN = 4 DOLAST = 0 DOCOVR = 0

Triangle thickness: 4.00000E-03 m

Eps_r = 7.000E+00 Sigma = 1.168E-03 S/m tan(delta) = 3.000E-02

All segments and triangles without a listed label are perfectly conducting

POWER LOSS (in Watts)

| in the segments | in the

Label| skineffect conc.load distr.load coating | triangles

0| 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 | 4.3373E-06

total| 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 | 4.3373E-06

Total loss in the segments: 0.0000E+00 W

Total loss in the triangl.: 4.3373E-06 W

Loss (total): 4.3373E-06 W



LOCATION ETHETA EPHI scattering cross sect. ...


FEKO EXAMPLES: RCS DI sheet 2-11

THETA PHI magn. phase magn. phase in m*m

0.00 0.00 1.489E-02 177.12 2.862E-03 -2.65 2.889E-03

2.00 0.00 1.472E-02 177.11 2.828E-03 -2.65 2.822E-03

4.00 0.00 1.450E-02 177.10 2.789E-03 -2.65 2.742E-03

POLARISATION


0.0008 169.12 RIGHT

0.0008 169.12 RIGHT

0.0008 169.12 RIGHT

Figure 2-7 shows the bistatic RCS as a function of the angle ϑ in the plane ϕ = 0.

Figure 2-7: Bistatic RCS of a thin dielectric sheet


2-12 FEKO EXAMPLES: patch coupled

2.4 patch coupled: Proximity coupled circular patch antenna withmicrostrip feed

Figure 2-8: Proximity coupled circular patch antenna. The lighter triangles are on a lowerlevel (closer to the ground plane). The dielectric layers are hidden to show the geometryof the triangular elements.

This example considers a proximity coupled circular patch antenna. The geometry of thetriangles is shown in figure 2-8 and the parameters of the dielectric layers can be obtainedfrom the GF card in the listing of the input file (patch_coupled.pre) below. The meshsize is related to the width of the strip to avoid having triangles with a large aspect ratio.Note that magnetic symmetry is used to reduce the number of unknowns. The AE cardis used to define a line between points as the strip line feed port — this line must extendto both sides of the symmetry plane. The same structure is solved using text input inExample 37.


#d = 1.590 ** half of the dielectric thickness

#er = 2.62 ** relative permittivity

#ur = 1.00 ** relative permeability

** Import model

IN 8 31 "patch_coupled.cfm"

** Magnetic symmetry plane x=0

SY 1 3 0 0

SF 1 #sf

** End of geometry

EG 1 0 0 0

** The dielectric layers

GF 10 1 0 1 1

2*#d #er #ur


FEKO EXAMPLES: patch coupled 2-13

** Excitation of the microstrip line

AE: 0 : Feedpt : Feedpt2 : 3 : : 1 : 0

** Frequency loop:

FR 8 0 #frequency 0.05e9

** Just calculate the impedance

OS 0

EN

** CADFEKO Checksum: bc7966c2ac08aef203047f2d69c33d97

Some extracts of the S-parameters as listed in the output file are given on the next page.Figure 2-9 shows the input impedance on the Smith chart. There is a small frequencyshift which can be reduced by using a finer mesh.

EXCITATION BY VOLTAGE SOURCE AT EDGE

Number of voltage source: N = 1


Wavelength in m: LAMBDA = 1.07069E-01

Open circuit voltage in V: |U0| = 1.00000E+00

Phase in deg.: ARG(U0) = 0.00

Electrical edge length in m: LEN = 4.37300E-03

Indices of the edges:

431 432

DATA FOR THE GREEN’S FUNCTION

Multilayer dielectric substrate

number of layers NLAYER = 1

ground plane present? GPLANE = Yes



Current in A 5.2283E-02 -3.7638E-02 6.4422E-02 -35.75

Admitt. in A/V 5.2283E-02 -3.7638E-02 6.4422E-02 -35.75





2-14 FEKO EXAMPLES: patch coupled

Figure 2-9: Reflection coefficient of the proximity coupled patch.


FEKO EXAMPLES: trihedral MLFMM 2-15

2.5 trihedral MLFMM: Using the MLFMM

Y

X

Z

Figure 2-10: Plane wave incident on an electrically large trihedral

In this example we consider a single plane wave incident (from ϑ = 60 and ϕ = 0)on a trihedral. The size of the trihedral (13.5λ2 surface area) was chosen such that wecan still solve it incore on a PC with 768 MByte of RAM. This is on the small side forthe MLFMM, but enough to demonstrate the advantage thereof. The same structure issolved using text input in Example 41.

The file trihedral_MLFMM_41.pre is listed below. Note the use of the FM card at theend of the geometry section in the input file and the wrapped EG card which includesthe Single precision field (the 1 in column 101).


** Import model

IN 8 31 "trihedral_MLFMM.cfm"

** Use the MLFMM instead of the standard MoM

FM

** End of geometry

EG 1 0 0 0 ...


2-16 FEKO EXAMPLES: trihedral MLFMM

1

** Set frequency

FR: 1 : : : : : #frequency

** Set source

** Excitation by means of an incident plane wave

A0 0 1 1 1 0 60 0 0

** Solution control

** Bistatic radar cross section in the vertical plane Phi=0

FF 1 91 1 0 0 0 2

** End of file

EN

** CADFEKO Checksum: ebed199a511d48e7ad97ba9319fd7b14

Some extracts from the output file trihedral_MLFMM.out are

DATA FOR MEMORY USAGE

Number of metallic triangles: 3915 ...

FAST MULTIPOLE METHOD (FMM)

Multilevel FMM is used

Storage of some elements with single precision (saves memory)

DATA FOR THE FMM

Level of the MLFMM: 5

Finest box size DELTA/LAMBDA: 0.2300

Number of nearfield matrix elements: 1437056 ( 4.2601 % of full MoM)

SUMMARY OF MEMORY REQUIREMENT FMM (in MByte)

Near-field matrix: 16.4458

Far-field matrix

Direction vectors Fourier trans: 0.0304

Fourier trans. basis functions: 8.5078

Transfer function: 3.0772

Interpolation and Filtering: 0.1733

Matrix-vector-multiplication: 2.8754

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

total: 31.1098 MByte

for comparison classical MoM: 514.7227 MByte


FEKO EXAMPLES: trihedral MLFMM 2-17

PRE-CONDITIONING OF THE LINEAR SET OF EQUATIONS

...

Memory requirement for preconditioner: 54.443 MByte

A total memory of 87.190 MByte has been allocated dynamically

(peak memory usage so far 95.295 MByte)

The solution required 47 seconds on a Pentium 4 2.4 GHz PC. For comparison the MoMresult required about 515 MByte of RAM and 240 seconds solution time. Even if onesets up the model to exploit the single plane of symmetry, the MoM requires 262 MByteor RAM and 66 seconds solution time. As the problem size increase, the difference willbecome more and more significant. Figure 2-11 compares the results obtained with theMLFMM with those obtained with the MoM.

Theta (degrees)

1801651501351201059075604530150

RC

S(d

Bs

m)

30

25

20

15

10

5

0

MoM MLFMM

Figure 2-11: Bistatic RCS of a trihedral. Comparison of the MLFMM and MoM results.


2-18 FEKO EXAMPLES: bio demo FEM

2.6 bio demo FEM: Exposure of muscle tissue sphere using aMoM/FEM hybrid

Figure 2-12: Sphere of muscle tissue illuminated by a dipole antenna. Note the air layeraround the sphere, to reduce the number of boundary triangles of the FEM region.

This example considers the exposure of a sphere of muscle tissue to the field created by adipole antenna. The geometry of the example is shown in figure 2-12. The point to noteis that an air layer is used around the sphere to reduce the number of triangles on theboundary of the FEM region. If this air layer was not used, the more highly discretisedsurface of the sphere would be on the MoM boundary. DI cards are used to specify theparameters of the FEM region. The solution control file is as follows.


** Import model

IN 8 31 "Bio_demo_FEM.cfm"

** End of geometry

EG 1 0 0 0

DI: muscle : : : : : #epsr : 1 : : : #tand : 1000

DI air 1 1

** Set frequency

FR 1 #freq

** Set source

A1: 0 : dipole.feed : : : : 1 : 0

PW 1 0 1


FEKO EXAMPLES: bio demo FEM 2-19

** Solution control

FE 3 1 1 31 0 0 0 -0.08 0.16/30

** End of file

EN

** CADFEKO Checksum: 5b38f94de08a5c479d10024b067cf819

The solver automatically switches to use the FEM/MoM hybrid solution when the meshcontains tetrahedral elements. Note that an iterative solver is used.

SOLUTION OF THE LINEAR SET OF EQUATIONS WITH CGS

Maximum number of iterations: MAXIT = 500

Stopping criterium (vector norm): PREC = 1.00000E-05



Stopping after 24 iterations with a minimum residuum of 4.31908E-06

CPU time for solving the linear set of equations: 16.391 seconds

In order that all voltage sources together result in a real power of 1.00000E+00 Watt,

all currents are multiplied by the factor PWFAKTOR = 1.46336E+01.



Current in A 1.3667E-01 -7.2075E-02 1.5451E-01 -27.81

Admitt. in A/V 9.3395E-03 -4.9253E-03 1.0559E-02 -27.81



Power in Watt: 1.00000E+00

POWER BUDGET IN THE DIELECTRIC (in Watt)

Integration of the normal component of the power density vector pointing into the

respective medium over the surface of the dielectric body (MoM surface equiv. princip.)

or volume integral for the corresponding medium (FEM and MoM volume equiv. princip.)

medium power dens. metallic network source power loss

integral losses losses power in medium

0 unknown 0.00000E+00 0.00000E+00 1.00000E+00 unknown

air 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00

muscle 2.41014E-02 0.00000E+00 0.00000E+00 0.00000E+00 2.41014E-02


2-20 FEKO EXAMPLES: bio demo FEM

SUMMARY OF LOSSES

Metallic elements: 0.0000E+00 W

Dielectric (surface equiv. princ.): 0.0000E+00 W

Dielectric (volume equiv. princ.): 0.0000E+00 W

Dielectric (FEM region): 2.4101E-02 W

Mismatch at feed: 0.0000E+00 W

Non-radiating networks: 0.0000E+00 W

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

Sum of all losses: 2.4101E-02 W

Efficiency of the antenna: 97.5899 %

(based on a total active power: 1.e0000E+00 W)

From the output file, it can clearly be seen that the CGS iterative method is used to solvethe matrix equation, and that 24 iterations were required for the solution. The lossesin the FEM region can also be seen. The electric field through the sphere is shown inFigure 2-13.

Figure 2-13: Electric field through the sphere of muscle tissue.


TEXT INPUT EXAMPLES 3-1

3 Text input examples

3.1 Example 1: Half wavelength dipole

∼B

C

D

A

Figure 3-1: Half wavelength dipole antenna

This example shows how to calculate the radiation pattern and input impedance for asimple half wavelength dipole shown in figure 3-1. The wavelength λ is 4 m, the lengthof the antenna 2 m, and the wire radius 2 mm. The input file is as follows

** A lambda/2 dipole antenna in free space

** Radiation at a wavelength lambda of 4 m

** Set the wavelength

#lambda = 4

** Segmentation parameters

#seg_len = #lambda / 20

#seg_rad = 0.002

IP #seg_rad #seg_len

** Define the points

#h = #lambda/4 ** Half the dipole length

#l = 0.4*#seg_len ** Half the length of the feed segment

** (Must be smaller than 0.5*#seg_len to allow only one segment)

DP A 0 0 -#h

DP B 0 0 -#l

DP C 0 0 #l

** Define lower half of the dipole without the feed

BL A B

** Mirror the lower half of the dipole upwards, about the plane z=0 (xy-plane)

** (with the same command electric symmetry about the plane z=0 is established)

SY 1 0 0 2

** Create feed segment with the label 1

LA 1

BL B C


3-2 FEKO EXAMPLES: 1

** End of the geometric input -- write complete geometry to output file

EG 0 0 0 0 0

** Calculate and set the frequency (approximately 75 MHz)

#freq = #c0 / #lambda ** #c0, the speed of light in vacuum, is predefined

FR 1 0 #freq

** Excite by means of a voltage gap (E-Field) on the segment with label 1

A1 0 1 1 0

** Calculate the far field in the vertical plane

FF 1 37 1 0 0 0 5 0

** End

EN

The first line of the input file must contain either the file name example_01 or it mustbe a comment line or an empty line. It is usually a comment line. This is followed bythe IP card which defines the wire radius and the maximum segment length. Here themaximum segment length has been set to λ

20 = 0.2 m. The dipole antenna is located onthe z axis between the two points A (at z = −1) and D (at z = 1). The excitation isplaced in the middle of the dipole at z = 0.

The dipole could have been constructed with a single BL card by connecting the points Aand D. (See the first example, dipole.pre, in the Getting started manual.) However, ifa single wire is used, the feed segment cannot be specified by label and no use is made ofthe symmetry. To ensure that the feed segment has a unique label, the antenna is createdin three sections namely A–B, B–C and C–D. The points B and C lie symmetric aboutz = 0 and the distance between them must be less than the maximum segment lengthsuch that only one segment is created.

First the points A and B are connected by means of the BL card. This generates thebottom half of the dipole. The plane z = 0 is then defined as a plane of ideal electricsymmetry (SY card). This mirrors the existing segments to create the top half of thedipole. It is, of course, also possible to create the top half of the dipole with another BLcard and thus not to utilise symmetry. All structures following an LA card will have thelabel specified by it. Since no LA card has been used yet, all segments created thus farhave the default label 0. The BL card connecting points B and C follows the LA card andthus creates a segment with a unique label (label 1 in this case) as no other segments arecreated after this LA card. This label is used in the A1 card to specify the feed segment.The EG card ends the geometric input.

The input files for the examples can be found in the examples\simple directory underthe FEKO installation. Solutions for these examples are obtained by running PREFEKOand FEKO as discussed in the Getting started manual.


FEKO EXAMPLES: 1 3-3

The user is advised to run FEKO on example_01 and compare the FEKO output fileexample_01.out with the extracts of the output file listed below.



Current in A 1.1032E-02 -4.3617E-03 1.1863E-02 -21.57

Admitt. in A/V 1.1032E-02 -4.3617E-03 1.1863E-02 -21.57







...

90.00 0.00 7.387E-01 66.12 0.000E+00 0.00 2.1751791 -999.9999 2.1751791

95.00 0.00 7.346E-01 66.12 0.000E+00 0.00 2.1260442 -999.9999 2.1260442

100.00 0.00 7.222E-01 66.13 0.000E+00 0.00 1.9785679 -999.9999 1.9785679

POLARISATION


0.0000 -180.00 LINEAR

0.0000 -180.00 LINEAR

0.0000 -180.00 LINEAR

The resulting input impedance is Z = (78.4+j 31.0) Ω and the maximum gain is 2.18 dB.



3.2 Example 2: A dipole antenna in front of a plate

Figure 3-2: Dipole antenna in front of a conducting plate

This example considers a λ2 dipole in front of a square conducting plate with side length

λ. The wavelength is 3 m which results in a frequency of approximately 100 MHz. Thedistance between the antenna and the plate is 3

4 λ = 2.25 m.

The horizontal radiation pattern and the near field are calculated for both a perfectlyconducting plate and a plate with losses. The complete input file is given below.

** A lambda/2 dipole antenna 3/4 lambda in front of a plate with

** side lengths equal to lambda.



#lambda = 3

#tri_len = #lambda / 7

#seglen = #lambda / 15

#segrad = 0.002

IP #segrad #tri_len #seglen


#a = #lambda/2

DP P1 0 0 0

DP P2 0 #a 0

DP P3 0 #a #a

DP P4 0 0 #a

#d = 3/4*#lambda

#h = #lambda/4

#gap = 0.45*#seglen



DP A #d 0 #h

DP B #d 0 #gap

DP C #d 0 -#gap

** A quarter of the plate is created in the quadrant y>0 and z>0 with label 2

LA 2

BP P1 P2 P3 P4


SY 1 0 3 0

** Create the top half of the dipole antenna (without excitation)

** Use Label 0

LA 0

BL A B


SY 1 0 0 2

** Create the excitation segment with label 1

LA 1

BL B C

** End of the geometry input

EG 0 0 0 0 0

** Calculate and set the frequency

#freq = #c0 / #lambda

FR 1 0 #freq


A1 0 1 1


FE 3 70 1 1 0 -1.97 0 0 0.1 0 0


FF 1 1 181 0 90 0 0 2

** Add losses to the plate -- skin effect

SK 2 3 0.005 1 1.0E5


FE 3 70 1 1 0 -1.97 0 0 0.1 0 0


FF 1 1 181 0 90 0 0 2

** End

EN

The comments at the start of the input file are followed by an IP card that sets themaximum segment length to λ

15 = 0.2 m, the wire radius to 2 mm, and the maximumtriangle edge length to λ

7 = 0.429 m.

Next, the points are defined, using DP cards, as shown in figure 3-3. A quarter plate iscreated from these points by using a BP card. This is mirrored by applying magneticsymmetry to the plane y = 0 (SY card). The top half of the antenna is then createdusing a BL card. The top half of the plate and the dipole are now mirrored by specifyingelectric symmetry in the plane z = 0. Thereafter the feed segment is created.



Here all triangles have label 2 (they follow the LA card that specifies label 2), the segmentsof the dipole have label 0 except for the feed segment (to which the voltage gap will beapplied) which has label 1. (Note that no label increase is specified by the SY cards.)

The program PREFEKO meshes the structure into 128 triangular patches and 9 segments,as shown in figure 3-3.

P1

P4

P2

C

P3

B

A

Figure 3-3: The meshed geometry of Example 2

After the end of geometry the frequency and excitation are set and a near and far fieldcalculation requested. This will yield the results for perfectly conducting triangles whichis the default. The SK card is then used to define skin effect losses. This is followed by asecond set of calculation requests.

Below are some extracts from the lossless section of the output file example_02.out



Current in A 9.7345E-03 -5.6445E-03 1.1253E-02 -30.11

Admitt. in A/V 9.7345E-03 -5.6445E-03 1.1253E-02 -30.11







(total field, incident and scattered)

LOCATION EX EY ...

medium X/m Y/m Z/m magn. phase magn. phase

0 -1.97000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00 0.00000E+00 0.00

0 -1.87000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00 0.00000E+00 0.00

0 -1.77000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00 0.00000E+00 0.00

EZ

magn. phase

7.03252E-02 139.87

6.98094E-02 149.19

6.91203E-02 158.31



LOCATION HX HY ...


0 -1.97000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00 1.90618E-04 140.39

0 -1.87000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00 1.89841E-04 149.82

0 -1.77000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00 1.88747E-04 159.07

HZ

magn. phase

0.00000E+00 0.00

0.00000E+00 0.00

0.00000E+00 0.00





90.00 0.00 1.240E+00 168.98 0.000E+00 0.00 7.2149 -999.9999 7.2149

90.00 2.00 1.238E+00 168.90 0.000E+00 0.00 7.2011 -999.9999 7.2011

90.00 4.00 1.232E+00 168.65 0.000E+00 0.00 7.1595 -999.9999 7.1595

POLARISATION


0.0000 180.00 LINEAR

0.0000 180.00 LINEAR

0.0000 180.00 LINEAR

The extract below shows the same results when losses are included on the trianglesDATA OF THE VOLTAGE SOURCE NO. 1


Current in A 9.7345E-03 -5.6445E-03 1.1253E-02 -30.11

Admitt. in A/V 9.7345E-03 -5.6445E-03 1.1253E-02 -30.11






POWER LOSS METAL (in Watt)


Label | skineffect conc.load distr.load coating | triangles

2| 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 | 3.0107E-07

total | 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 | 3.0107E-07



Loss metal (total): 3.0107E-07 W



LOCATION EX EY ...


0 -1.97000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00 0.00000E+00 0.00

0 -1.87000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00 0.00000E+00 0.00

0 -1.77000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00 0.00000E+00 0.00

EZ

magn. phase

7.02916E-02 139.86

6.97747E-02 149.18

6.90846E-02 158.30



LOCATION HX HY ...


0 -1.97000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00 1.90538E-04 140.39

0 -1.87000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00 1.89759E-04 149.81

0 -1.77000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00 1.88663E-04 159.06

HZ

magn. phase

0.00000E+00 0.00

0.00000E+00 0.00

0.00000E+00 0.00





90.00 0.00 1.235E+00 168.98 0.000E+00 0.00 7.2146 -999.9999 7.2146

90.00 2.00 1.233E+00 168.90 0.000E+00 0.00 7.2008 -999.9999 7.2008

90.00 4.00 1.227E+00 168.65 0.000E+00 0.00 7.1592 -999.9999 7.1592

POLARISATION


0.0000 180.00 LINEAR

0.0000 180.00 LINEAR

0.0000 180.00 LINEAR



Figures 3-4 and 3-5 show the near field along the x axis and the horizontal radiation.

Figure 3-4: Electric near field along the x axis for Example 2

Figure 3-5: Radiation pattern (in dB) in the horizontal plane ϑ = 90 for Example 2



3.3 Example 3: Dipole in front of a plate with physical optics

This example uses the same structure as Example 2 shown in figure 3-3, but in this casethe physical optics (PO) approximation is used to determine the currents on the surfaceof the plate.

The triangles on the plate all have label 2 and this is therefore be specified in the POcard to define the PO region. The complete input file is given below.

** A lambda/2 dipole antenna 3/4 lambda in front of a plate with

** side lengths equal to lambda.



#lambda = 3



#segrad = 0.002

IP #segrad #tri_len #seglen


#a = #lambda/2

DP P1 0 0 0

DP P2 0 #a 0

DP P3 0 #a #a

DP P4 0 0 #a

#d = 3/4*#lambda

#h = #lambda/4

#gap = 0.45*#seglen

DP A #d 0 #h

DP B #d 0 #gap

DP C #d 0 -#gap

** A quarter of the plate is created in the quadrant y>0 and z>0 with label 2

LA 2

BP P1 P2 P3 P4


SY 1 0 3 0

** Create the top half of the dipole antenna (without excitation)

** Use Label 0

LA 0

BL A B


SY 1 0 0 2

** Create the excitation segment with label 1

LA 1

BL B C

** PO approximation for the plate with the label 2, ray search is switched off

PO 2 1 0 0 0

** End of the geometry input

EG 1 0 0 0 0



** Calculate and set the frequency


FR 1 0 #freq


A1 0 1 1


FE 3 70 1 1 0 -1.97 0 0 0.1 0 0


FF 1 1 181 0 90 0 0 2

** End

EN

Note that the addition of a single PO card is the only change from the input file used forExample 2. (Besides the fact that the skin effect is not considered here.)

Some extracts from the output file example_03.out follow



Current in A 9.7098E-03 -5.7193E-03 1.1269E-02 -30.50

Admitt. in A/V 9.7098E-03 -5.7193E-03 1.1269E-02 -30.50




in free space

LOCATION EX EY EZ


-1.9700 0.0E+00 0.0E+00 0.000E+00 0.00 0.000E+00 0.00 7.655E-02 139.69

-1.8700 0.0E+00 0.0E+00 0.000E+00 0.00 0.000E+00 0.00 7.625E-02 149.18

-1.7700 0.0E+00 0.0E+00 0.000E+00 0.00 0.000E+00 0.00 7.582E-02 158.50

-1.6700 0.0E+00 0.0E+00 0.000E+00 0.00 0.000E+00 0.00 7.525E-02 167.64


in free space

LOCATION HX HY HZ


-1.9700 0.0E+00 0.0E+00 0.000E+00 0.00 1.977E-04 137.30 0.000E+00 0.00

-1.8700 0.0E+00 0.0E+00 0.000E+00 0.00 1.958E-04 146.53 0.000E+00 0.00

-1.7700 0.0E+00 0.0E+00 0.000E+00 0.00 1.935E-04 155.57 0.000E+00 0.00

-1.6700 0.0E+00 0.0E+00 0.000E+00 0.00 1.906E-04 164.39 0.000E+00 0.00







90.00 0.00 1.246E+00 170.01 0.000E+00 0.00 7.2683872 -999.9999 7.2683872

90.00 2.00 1.244E+00 169.91 0.000E+00 0.00 7.2547916 -999.9999 7.2547916

90.00 4.00 1.238E+00 169.62 0.000E+00 0.00 7.2139059 -999.9999 7.2139059

POLARISATION


0.0000 180.00 LINEAR

0.0000 180.00 LINEAR

0.0000 180.00 LINEAR

Figure 3-6 shows the distribution of the near field along the x axis and figure 3-7 theradiation pattern in the horizontal plane. It is clear that there is good correlation betweenthe PO method and the method of moments results (figures 3-4 and 3-5). There is,however, a big difference in the computation time.

Figure 3-6: Electric near field along the x axis for Example 3



Figure 3-7: Radiation pattern (in dB) in the horizontal plane ϑ = 90 for Example 3



3.4 Example 4: Dielectric sphere

Figure 3-8: Dielectric sphere with incident plane wave

Here a dielectric sphere with radius R = 1 m and a dielectric constant of εr = 36 is locatedat the origin of the coordinate system. The excitation is an electromagnetic plane wave(shown in figure 3-8) propagating in the z direction and polarised in the x direction. Thefree space wavelength is 20 m. The calculation is done using the equivalent surface currentmethod. (Examples 9 and 13 discuss using the volume current method for dielectrics.)** A lossless dielectric sphere with radius R=1m and Epsilon=36.

** Excitated by means of an incident plane wave with lambda 20m

** (3.33m in the dielectric)

** Set parameters

#lambda = 20

#r = 1

#epsr = 36


#tri_len = #lambda / sqrt(#epsr) / 8

IP #tri_len

** Create the sphere

** Define points

DP A 0 0 0

DP B 0 0 #r

DP C #r 0 0

** Assign the medium’s properties

ME 1 0



** Create an eighth of the sphere

KU A B C 0 0 90 90 #tri_len

** Mirroring in all three coordinates

** yz-plane: ideal electric conducting plane

** xz-plane: ideal magnetic conducting plane

** xy-plane: only geometric symmetry

SY 1 2 3 1

** The sphere is modelled with planar trianglular patches, allowing only

** an approximate representation of the real geometry. To improve the

** agreement with an exact solution, one can scale the sphere created so

** far so that the surface of the discretised sphere equals the surface

** of the real sphere:

** Without the TG card here below FEKO calculates the total surface of these

** triangles as 12.127642. Scale the sphere such that the total triangle

** surface is the same as that of the sphere.

#a_sphere = 4*#pi*(#r^2)

#a_triang = 12.127642

#s = sqrt (#a_sphere / #a_triang)

TG 0 #s

** End of the geometry

EG 1 0 0 0 0

** Assign the dielectric properties

DI 1 #epsr 1 0



FR 0 #freq

A0 0 1 1 1 0 -180 0 0

** Calculate near fields along the z axis

#delta = #r / 20 ** stepping

#zrange = 2*#r ** values between -zrange and +zrange are plotted

#nz = floor( 2*#zrange/#delta ) + 1

FE 1 1 1 #nz 0 0 0 -#zrange 0 0 #delta

** Radar cross section in the vertical plane PHI=0

FF 1 91 1 0 0 0 2

** End

EN

The meshed structure is similar to figure 3-8 but has 176 triangles. Note that here onlyone ME card is used as all the triangles lie on the surface of the dielectric and there areno conducting structures. Note also the use of the DI card to specify the parameters andthe fact that only one FE card is required to calculate the near field inside and outsidethe dielectric region.



When PREFEKO meshes a sphere, all the triangle corners lie on the surface of the sphere.Thus the meshed sphere is, on average, slightly smaller than the original sphere and theaccuracy may be improved by increasing the radius of the sphere as done here.

Some extracts from the output file example_04.out are given below.

Scaling by a factor 1.018E+00

Surface of the triangles in m*m: 12.5663706



LOCATION EX ...

medium X/m Y/m Z/m magn. phase

0 0.00000E+00 0.00000E+00 -2.00000E+00 9.13919E-01 39.25

0 0.00000E+00 0.00000E+00 -1.97500E+00 9.10146E-01 38.92

0 0.00000E+00 0.00000E+00 -1.95000E+00 9.06153E-01 38.60

...

1 0.00000E+00 0.00000E+00 -2.50000E-02 1.21019E-01 2.49

1 0.00000E+00 0.00000E+00 1.11022E-16 1.20985E-01 -1.29

1 0.00000E+00 0.00000E+00 2.50000E-02 1.21338E-01 -5.07

EY EZ

magn. phase magn. phase

0.00000E+00 0.00 0.00000E+00 0.00

0.00000E+00 0.00 0.00000E+00 0.00

0.00000E+00 0.00 0.00000E+00 0.00

0.00000E+00 0.00 0.00000E+00 0.00

0.00000E+00 0.00 0.00000E+00 0.00

0.00000E+00 0.00 0.00000E+00 0.00





0.00 0.00 1.105E-01 -0.95 0.000E+00 0.00 1.53332E-01

2.00 0.00 1.104E-01 -0.95 0.000E+00 0.00 1.53169E-01

4.00 0.00 1.102E-01 -0.95 0.000E+00 0.00 1.52681E-01

POLARISATION


0.0000 0.00 LINEAR

0.0000 0.00 LINEAR

0.0000 0.00 LINEAR

Figures 3-9 and 3-10 compare the near field along the z axis and the radar cross section asa function of the angle to exact (theoretical) results. Note that the radius of the meshedsphere is slightly larger than that of the theoretical one along the z axis.



Figure 3-9: Near field along the z axis

Figure 3-10: Bistatic radar cross section of the dielectric sphere



3.5 Example 5: Dipole in front of a dielectric beam

The problem of a dipole in front of a lossy dielectric beam, as shown in figure 3-11, isnext. The calculation is done using the equivalent surface current method, but couldjust as well be done with volume equivalent currents. The geometry and the electricalparameters (such as the dielectric constant) can be determined from the input file

A

E H

B

F G

I L

C

J K


** A Dipole antenna in front of a lossy dielectric cube.

** See article Sarkar (IEEE AP-37 S.673) and Karimullah

** (IEEE MTT-28 S.1218)

** Parameters for segmentation

IP 0.0277 0.33 0.70

** Geometric Structure

** lower half of the dipole antenna

DP A 0 0 -2.77

DP B 0 0 -0.34

DP C 0 0 0.34

BL A B

** lower half of the dielectric solid

DP E -0.1 0.2 -0.85

DP F 0.1 0.2 -0.85



DP G 0.1 0.4 -0.85

DP H -0.1 0.4 -0.85

DP I -0.1 0.2 0

DP J 0.1 0.2 0

DP K 0.1 0.4 0

DP L -0.1 0.4 0

ME 1 0

BP E H G F

BP E F J I

BP J F G K

BP L K G H

BP I L H E

** Mirroring around the plane z=0 (xy-plane) -- ideal electric conducting plane

SY 1 0 0 2

** Create excitation segment with label 1

ME 0

LA 1

BL B C

** End of the geometric input

EG 1 0 0 0 0

** Dielectric data

DI 1 113 1 0.62


FR 1 0 27.0E6

** Excitation by means of a voltage source (E-field) on dipole

A1 0 1 1

** Calculate the electric near field (along the y axis)

FE 1 1 100 1 0 0 -0.395 0 0 0.01 0

** Far field calculation

FF 1 1 181 0 90 0 0 2

** End

EN

Here we use an ME card to define the dielectric volume and a second one to switch backto conductors in free space in order to construct the feed segment (which must be definedafter specifying the symmetry).

Below we list some extracts from the output file example_05.out





Current in A 1.1076E-02 -3.2407E-03 1.1540E-02 -16.31

Admitt. in A/V 1.1076E-02 -3.2407E-03 1.1540E-02 -16.31










0 unknown 0.00000E+00 0.00000E+00 5.54361E-03 unknown

1 1.64229E-04 0.00000E+00 0.00000E+00 0.00000E+00 1.64229E-04



LOCATION EX ...


0 0.00000E+00 -3.95000E-01 0.00000E+00 0.00000E+00 0.00

0 0.00000E+00 -3.85000E-01 0.00000E+00 0.00000E+00 0.00

...

1 0.00000E+00 2.05000E-01 0.00000E+00 0.00000E+00 0.00

1 0.00000E+00 2.15000E-01 0.00000E+00 0.00000E+00 0.00

EY EZ


0.00000E+00 0.00 3.37382E-01 173.86

0.00000E+00 0.00 3.39730E-01 174.37

0.00000E+00 0.00 1.83275E-01 -122.12

0.00000E+00 0.00 1.70781E-01 -124.80





90.00 0.00 7.295E-01 67.46 0.000E+00 0.00 2.1694 -999.9999 2.1694

90.00 2.00 7.296E-01 67.46 0.000E+00 0.00 2.1705 -999.9999 2.1705

90.00 4.00 7.297E-01 67.47 0.000E+00 0.00 2.1716 -999.9999 2.1716

POLARISATION


0.0000 180.00 LINEAR

0.0000 180.00 LINEAR

0.0000 180.00 LINEAR



3.6 Example 6: Magnetic field probe

DC

A

B


The current in a circular loop wire can be used to measure the magnitude of a magneticfield. The conducting wire is shielded against electric fields by a conducting sleeve. Thestructure is shown in figure 3-12. The input file is listed below.

** A magnetic field probe in the form of a frame antenna with

** shielding against electric fields. Wave length approximately 10 m.


IP 0.005 0.25 0.15

** Create a quarter of the torus in the area y<0 and z<0

DP A 0 0 0

DP B 0 0 1

DP C -1 0 0

DP D -1.1 0 0

TO A B C D 175 180 0.25 0.12 0

** Mirroring around the plane z=0 (xy-plane only geometrically symmetric)

SY 1 0 0 1

** Create the inner conductor in the form of a half circle

CL A B C 180 0.15

** mirror around the plane y=0 (xz-plane, ideal conducting electric plane)

SY 1 0 2 0

** End of the geometric input -- no geometry data in output file

EG 1 0 0 0 0




FR 1 0 30.0E6

** Examine multiple incident fields. Step the angle of incidence THETA

** from 0 to 90 in steps of 10 degrees. Polarisation is such that for

** THETA=90 the magnetic field is perpendicular to the plane of the

** circle. For any angle of THETA the electric field is always in the

** y direction.

A0 0 10 1 1 0 0 0 90 10

** Start the calculation with the OS card and output the segment currents

OS 3

** End

EN

Some extracts from the output file example_06.out are







Direction of propag.: BETA0X = 0.00000E+00

BETA0Y = 0.00000E+00

BETA0Z = -6.28754E-01

Field strength in V/m: |E0X| = 0.00000E+00 ARG(E0X) = 0.00

(Phase in deg.) |E0Y| = 1.00000E+00 ARG(E0Y) = 0.00

|E0Z| = 0.00000E+00 ARG(E0Z) = 0.00

VALUES OF THE CURRENT IN THE SEGMENTS in A

Segment centre IX IY ...

number x/m y/m z/m magn. phase magn. phase

1 -9.94415E-01 -7.45211E-02 0.00000E+00 5.204E-04 -91.14 6.944E-03 88.86

2 -9.72202E-01 -2.21899E-01 0.00000E+00 1.536E-03 -91.14 6.729E-03 88.86

3 -9.28271E-01 -3.64319E-01 0.00000E+00 2.477E-03 -91.14 6.312E-03 88.86

4 -8.63604E-01 -4.98602E-01 0.00000E+00 3.300E-03 -91.14 5.715E-03 88.86

5 -7.79645E-01 -6.21746E-01 0.00000E+00 3.965E-03 -91.14 4.972E-03 88.86

IZ

magn. phase

0.000E+00 0.00

0.000E+00 0.00

0.000E+00 0.00

0.000E+00 0.00

0.000E+00 0.00










BETA0Y = 0.00000E+00

BETA0Z = -6.19201E-01



|E0Z| = 0.00000E+00 ARG(E0Z) = 0.00




1 -9.94415E-01 -7.45211E-02 0.00000E+00 5.260E-04 -94.88 7.019E-03 85.12

2 -9.72202E-01 -2.21899E-01 0.00000E+00 1.553E-03 -94.88 6.802E-03 85.12

3 -9.28271E-01 -3.64319E-01 0.00000E+00 2.504E-03 -94.88 6.380E-03 85.12

4 -8.63604E-01 -4.98602E-01 0.00000E+00 3.335E-03 -94.88 5.777E-03 85.12

5 -7.79645E-01 -6.21746E-01 0.00000E+00 4.008E-03 -94.88 5.026E-03 85.12

IZ

magn. phase

0.000E+00 0.00

0.000E+00 0.00

0.000E+00 0.00

0.000E+00 0.00

0.000E+00 0.00



3.7 Example 7: Monopole on a finite circular ground plane

AE

B

D


A monopole antenna of length λ4 and radius 10−5 λ is located in the middle of a circular

ground plane with a radius R such that 2 π Rλ = 3. The structure is shown in figure 3-13

and the input file is

** A lambda/4 monopole antenna, on a finite circular ground plane with

** radius R, where 2*PI*R/lambda = 3, i.e. R=1.91m.

** lambda=4m, and the wire radius is 10^-5*lambda.

** Comparisons to the results obtained can be found in the book "Monopole

** Elements on Circular Ground Planes" from Melvin M. Weiner et.al.

** PP. 66, 175, 230

** Parameters for segmentation

#lam = 4

#tri_len = #lam/10

#segl = #lam/40

IP 1e-5*#lam #tri_len #segl

** Radius of the ground plane

#rad = 3 * #lam / ( 2 * #pi )


DP A 0 0 0

DP B 0 0 1

DP D #rad 0 0

DP E 0 0 0.6*#segl

** Create a quarter of the circular ground plane, that will be treated

** with the MoM.

KR A B D 90 #tri_len

** Mirror this to form a full circle (ideal magnetic wall)

SY 1 3 3 0



** Create monopole, and the feed segment (label 1)

LA 0

BL E B

LA 1

BL A E

** End of geometric input

EG 1 0 0 0 0


FR 1 0 #c0/#lam

** Excitation be means of a voltage gap at the first segment of the

** monopole -- radiating exactly 2 watt

A1 0 1 1 0

PW 1 2

** Write the surface currents to the output file

OS 1 1

** To display the full pattern in GraphFEKO, we need a calculation

** where theta goes from 0 to 360 while phi remains constant

FF 1 181 1 0 0 0 2

** Calculate the far field and integrate this to obtain the radiated

** power. The pattern should be omni-directional in the phi-direction

** such that we can use large increments for this angle.

FF 3 90 12 0 1 15 2 30

** For FEKO LITE replace the line above with

** FF 3 90 6 0 1 90 2 60

** End

EN

Note that we switch off averaging of the currents at the OS card. Averaging may requirea significant amount of run time and is no longer required for the post processor. Someof extracts from the output file example_07.out are given below.



Current in A 2.4577E-01 -1.9206E-01 3.1191E-01 -38.01

Admitt. in A/V 1.5101E-02 -1.1801E-02 1.9165E-02 -38.01







0.00 0.00 1.683E-16 107.91 2.454E-15 169.09 -336.2662 -312.9904 -312.9701

2.00 0.00 1.020E+00 103.82 0.000E+00 0.00 -20.6209 -999.9999 -20.6209

4.00 0.00 2.033E+00 103.78 0.000E+00 0.00 -14.6280 -999.9999 -14.6280



POLARISATION


0.0600 -91.90 LEFT

0.0000 180.00 LINEAR

0.0000 180.00 LINEAR

Gain is a factor of 1.00000E+00 ( 0.00 dB) larger than directivity

The directivity/gain is based on an active power of 2.00000E+00 W

and on a power loss of 0.00000E+00 W






0.00 .. 180.00 deg. 0.00 .. 360.00 deg. 1.92627E+00 Watt

1.00 .. 179.00 deg. 15.00 .. 345.00 deg. 1.76578E+00 Watt

The power supplied by the voltage source is 2 W, but the power calculated by integratingin the far field is 1.926 W — a difference of 3.7%. A difference this large implies that afiner segmentation may be required. The radiation pattern is shown in figure 3-14.

Electric far field |E_Theta|

-5

-10

-15

-20

345º330º

315º

300º

285º

270º

255º

240º

225º

210º195º180º165º

150º

135º

120º

105º

90º

75º

60º

45º

30º15º 0º

Figure 3-14: Radiation pattern of the monopole as a function of ϑ



3.8 Example 8: A horn antenna

CZDZ

CY

C

BZ

BY

B

AZ

AY

A


A horn antenna structure, as shown in figure 3-15, is examined next. The input file is

** A pyramidal horn antenna for the frequency 1.645 GHz

** Set a scaling factor so that all dimensions below are in cm

#scal = 0.01

SF 1 #scal

** Parameters of the structure (all in cm)

#freq = 1.645e9 ** frequency

#lam = #c0/#freq/#scal ** wavelength (scale so that also in cm)

#wa = 12.96 ** waveguide width

#wb = 6.48 ** waveguide height

#ha = 55.00 ** horn width

#hb = 42.80 ** horn height

#wl = 30.20 ** length of the waveguide section

#fl = #wl - #lam/4 ** position of the feed wire in the waveguide

#hl = 46.00 ** length of the horn section


#edgelen = #lam / 6 ** note that a mesh of lam / 6 is rather coarse

#seglen = #lam / 15

#segrho = #seglen / 12

IP #segrho #edgelen #seglen



** Define the corner points for a quarter of the horn in the

** quadrants y>0 and z>0 including the feed structure.

**

** Wave guide end

DP C -#wl #wa/2 #wb/2

DP CZ -#wl 0 #wb/2

DP CY -#wl #wa/2 0

DP C0 -#wl 0 0

** Transition from wave guide to horn

DP B 0 #wa/2 #wb/2

DP BZ 0 0 #wb/2

DP BY 0 #wa/2 0

** Horn opening

DP A #hl #ha/2 #hb/2

DP AZ #hl 0 #hb/2

DP AY #hl #ha/2 0

** Feed

#z = 0.45*#seglen

DP DU -#fl 0 -#z

DP DO -#fl 0 #z

DP DZ -#fl 0 #wb/2

** Create the surfaces in the quadrants y>0 and z>0

** Wave guide end

BP C CZ C0 CY

** Wave guide top

BT C CZ DZ

BQ C DZ BZ B

** Wave guide walls

BP C B BY CY

** Horn walls

BQ B A AY BY

** Horn top

BQ B A AZ BZ


SY 1 0 3 0

** Create half of the feed wire

BL DO DZ


SY 1 0 0 2

** Create the feed segment with the label 1

LA 1

BL DU DO


EG 1 0 0 0 0

** The excitation

FR 1 0 #freq

A1 0 1 1 0



** Calculate the horizontal radiation pattern

FF 1 1 361 0 90 -90 0 1

** Calculate the vertical radiation pattern

FF 1 361 1 0 0 0 1 0

** Integration of the full 3-D pattern over a sphere to get the radiated power

** accurately (use symmetry, only 1/4 sphere, multiply power by 4).

** A test using a finer angular stepping has shown that a stepping of 5 deg.

** is fully sufficient for the dimensions under consideration (must be adjusted

** if a horn antenna with higher gain is modelled, since then more sidelobes occur)

#stepping = 5

#nthe = 90/#stepping + 1

#nphi = 180/#stepping + 1

FF 3 #nthe#nphi0 0 0 #stepping #stepping

** End

EN

Note the use of a triangle on the upper waveguide wall to ensure an attachment point forthe feed wire which runs from top to bottom. Symmetry will ensure that there is also anattachment point on the bottom wall.

Some extracts from the output file example_08.out are given below.



Current in A 1.0511E-03 -1.8467E-04 1.0672E-03 -9.96

Admitt. in A/V 1.0511E-03 -1.8467E-04 1.0672E-03 -9.96








...

90.00 -4.00 1.293E+00 102.13 0.000E+00 0.00 16.6567 -999.9999 16.6567

90.00 -3.00 1.316E+00 102.70 0.000E+00 0.00 16.7980 -999.9999 16.7980

90.00 -2.00 1.333E+00 103.11 0.000E+00 0.00 16.8991 -999.9999 16.8991

90.00 -1.00 1.343E+00 103.36 0.000E+00 0.00 16.9597 -999.9999 16.9597

POLARISATION


0.0000 180.00 LINEAR

0.0000 180.00 LINEAR

0.0000 180.00 LINEAR



0.0000 180.00 LINEAR





0.00 0.00 1.212E-01 -151.71 0.000E+00 0.00 -3.3176 -999.9999 -3.3176

1.00 0.00 1.245E-01 -135.52 0.000E+00 0.00 -3.0824 -999.9999 -3.0824

2.00 0.00 1.272E-01 -119.72 0.000E+00 0.00 -2.8949 -999.9999 -2.8949

3.00 0.00 1.290E-01 -104.17 0.000E+00 0.00 -2.7700 -999.9999 -2.7700

4.00 0.00 1.299E-01 -88.71 0.000E+00 0.00 -2.7127 -999.9999 -2.7127

POLARISATION


0.0000 180.00 LINEAR

0.0000 180.00 LINEAR

0.0000 0.00 LINEAR

0.0000 0.00 LINEAR

0.0000 0.00 LINEAR






-2.50 .. 92.50 deg. -2.50 .. 182.50 deg. 1.74560E-04 Watt

0.00 .. 90.00 deg. 0.00 .. 180.00 deg. 1.26874E-04 Watt

Note that the source power is calculated from a single current and in some cases thismight not be very accurate. In this example this is 0.525569 mW. We can obtain a moreaccurate calculation of the radiated power by integrating the far field. (Note that thisgives the radiated power which is not the same as the source power for lossy antennas.)In this example, the integrated radiated power is 4 times 0.12687 mW (we only integratedover a quarter of the far field region). Thus the actual power is 0.152 dB higher than thecalculated source power. Since the power is used to normalise the directivity calculation,the actual directivity is 0.152 dB lower than the calculated values. The directivity mighttherefore be compensated by specifying a -0.152 dB offset in POSTFEKO. These resultsare shown in figures 3-16 and 3-17 for the horizontal and vertical radiation patternsrespectively.



2702402101801501209060300-30-60-90

20

15

10

5

0

-5

-10

-15

-20

-25

-30

Phi (degrees)

Directivity

(dB

)


3603303002702402101801501209060300

20

18

16

14

12

10

8

6

4

2

0

-2

-4

-6

-8

-10

Theta (degrees)

Dire

ctivity

(dB

)




3.9 Example 9: Dielectric cube


In Example 4 a dielectric sphere was examined using the equivalent surface currentmethod. Here a dielectric cube is to be examined with the volume current method.

The cube edge length is a = 2 m and the dielectric constant is εr = 4. The cube issituated at the origin of the coordinate system. As in Example 4, the excitation is anx polarised (electric field) incident wave propagating in the z direction (see figure 3-8).The free space wavelength in this case is 10 m. The cube is shown in figure 3-18.

** A dielectric cube consisting of volume elements.

** The side length is 2 meters.

** Calculation of the near field and RCS when excited by a plane wave

** Set segmentation parameters

#freq = 30.0e6

#epsr = 4

#lambda = #c0/#freq / sqrt(#epsr)

IP #lambda/10

** Define the corner points

DP A 0 0 0

DP B 1 1 1

** Set the medium

ME Cube

** Create an eighth of the cube

QU A B 1

** Mirror the eight to form the whole cube

SY 1 2 3 1




EG 1 0 0 0 0

** Set the material parameters

DI Cube #epsr 1

** Plane wave excitation

FR 1 0 #freq

A0 0 1 0 180 0 0

** Calculate the electric near field along the z axis

FE 1 1 1 40 0 0 0 -5 0 0 0.1

FE 7

FE 1 1 1 40 0 0 0 1.1 0 0 0.1

** Calculate the far field (RCS)

FF 1 91 1 0 0 0 2

** End

EN

Some of the results extracted from the output file example_09.out are


in free space

LOCATION EX EY EZ


0.0E+00 0.0E+00 -5.0000 1.060E+00 178.30 0.000E+00 0.00 0.000E+00 0.00

0.0E+00 0.0E+00 -4.9000 1.065E+00 175.07 0.000E+00 0.00 0.000E+00 0.00

0.0E+00 0.0E+00 -4.8000 1.069E+00 171.87 0.000E+00 0.00 0.000E+00 0.00

0.0E+00 0.0E+00 -4.7000 1.072E+00 168.70 0.000E+00 0.00 0.000E+00 0.00


inside the dielectric cuboids

LOCATION EX EY EZ SAR cuboid no.

X/m Y/m Z/m magn. phase magn. phase magn. phase in W/kg

0.167 0.167 0.167 7.646E-01 -18.06 7.336E-03 -14.69 3.474E-02 74.50 0.000E+00 1

0.167 0.167 0.500 7.978E-01 -32.85 7.392E-03 -19.50 3.086E-02 43.47 0.000E+00 2

0.167 0.167 0.833 8.203E-01 -47.49 7.348E-03 -22.31 2.712E-02 7.58 0.000E+00 3





0.00 0.00 5.036E-01 -8.97 0.000E+00 0.00 3.187E+00

2.00 0.00 5.033E-01 -8.97 0.000E+00 0.00 3.183E+00

4.00 0.00 5.024E-01 -8.97 0.000E+00 0.00 3.171E+00



POLARISATION


0.0000 0.00 LINEAR

0.0000 0.00 LINEAR

0.0000 0.00 LINEAR

Figure 3-19 shows the distribution of the near field along the z axis. (The second FEcard calculates the fields at the centres of the cuboids. Due to the piecewise constantbasis functions, the field on the z axis is equal to that inside the adjacent cuboids. Thesevalues are used in the plot.) Figure 3-20 shows the RCS in the vertical plane ϕ = 0.

Figure 3-19: Near field along the z axis



Figure 3-20: Bistatic radar cross section in the plane ϕ = 0



3.10 Example 10: Yagi-Uda antenna over a real ground

E2

E1

D2

D1

C2

C1

B2

B3

B1

A2

A1


In this example we consider the radiation of a horizontally polarised Yagi-Uda antennaconsisting of a dipole, a reflector and three directors. The frequency is 400 MHz. Theantenna is located 3 m above a real ground which is modelled with the reflection coef-ficient approximation as well as the Green’s function formulation. For comparison, thecalculation is also done without a ground plane. The geometry is shown in figure 3-21and the input file is

** A horizontally polarised Yagi-Uda antenna 3m above a ground plane, at 400 MHz

** The antenna consists of a reflector, a dipole and 3 directors

** Set some parameters

#freq = 400.0e6 ** Frequency

#lambda = #c0 / #freq ** Wave length

#h = 3 ** Height of the antenna

#d = 0.25*#lambda ** Distance between the elements

#lr = 0.477*#lambda ** Length of the reflector

#li = 0.451*#lambda ** Length of the dipole

#ld = 0.442*#lambda ** Length of the directors

#rho = 0.0025*#lambda ** The wire radius

** Parameter of the ground

#epsr = 10 ** Relative permittivity

#mur = 1 ** Relative permeability

#sigma = 1.0e-3 ** Conductivity


#segl = #lambda / 15

IP #rho #segl

** Create half of the reflector

DP A1 -#d 0 #h

DP A2 -#d #lr/2 #h

BL A1 A2



** Create half the dipole (without the feed segment)

DP B1 0 0.4*#segl #h

DP B2 0 #li/2 #h

BL B1 B2

** Create half of each of the 3 directors

DP C1 #d 0 #h

DP C2 #d #ld/2 #h

BL C1 C2

DP D1 2*#d 0 #h

DP D2 2*#d #ld/2 #h

BL D1 D2

DP E1 3*#d 0 #h

DP E2 3*#d #ld/2 #h

BL E1 E2

** Mirror around the plane y=0 (electric wall)

SY 1 0 2 0


DP B3 0 -0.4*#segl #h

LA 1

BL B3 B1

** End of geometry

EG 1 0 0 0 0

** Set frequency and excitation (1 watt)

FR 1 0 #freq

A1 0 1 1 0

PW 1 1

** --------------

** Firstly calculate the horizontal and the vertical radiation patterns

** without the ground plane

FF 1 1 181 0 90 0 0 2

FF 1 181 1 0 0 0 2 0

** --------------

** Repeat with a real ground (reflexion coefficient approximation)

BO 1 #epsr #sigma #mur

** Far field calculations now only in upper hemisphere (z>0)

** Avoid calculating the far field exactly at the interface

FF 1 1 181 1 85 0 0 2

FF 1 360 1 1 -89.75 0 0.5 0

** --------------

** Repeat for real ground plane using the exact Sommerfeld formulation

BO 0

GF 11 1 0 1 1 0

#epsr 1 #sigma

FF 1 1 181 1 85 0 0 2

FF 1 360 1 1 -89.75 0 0.5 0



** End

EN

The following are some extracts from the output file example_10.out


free space



Current in A 2.0866E-01 1.3553E-01 2.4881E-01 33.00

Admitt. in A/V 2.1770E-02 1.4140E-02 2.5959E-02 33.00

Impedance in Ohm 3.2306E+01 -2.0983E+01 3.8523E+01 -33.00






90.00 0.00 0.000E+00 0.00 2.732E+01 -76.56 -999.9999 10.952297 10.952297

90.00 2.00 0.000E+00 0.00 2.726E+01 -76.61 -999.9999 10.932851 10.932851

90.00 4.00 0.000E+00 0.00 2.708E+01 -76.77 -999.9999 10.874376 10.874376

POLARISATION


0.0000 90.00 LINEAR

0.0000 90.00 LINEAR

0.0000 90.00 LINEAR

GROUND PLANE IS PRESENT (reflection coefficient approximation)

relative permittivity: 10.0000

relative permeability: 1.0000

conductivity in S/m: 1.0000E-03

electric loss factor: 4.4938E-03

magnetic loss factor: 0.0000E+00



Current in A 2.0840E-01 1.3452E-01 2.4804E-01 32.84

Admitt. in A/V 2.1715E-02 1.4017E-02 2.5846E-02 32.84







LOCATION ETHETA EPHI gain in dB ...


85.00 0.00 0.000E+00 0.00 4.287E+01 14.26 -999.9999 14.865021 14.865021

85.00 2.00 8.017E-02 80.94 4.278E+01 14.21 -39.69811 14.845511 14.845527

85.00 4.00 1.595E-01 80.78 4.249E+01 14.06 -33.72558 14.786843 14.786904

POLARISATION


0.0000 90.00 LINEAR

0.0017 89.96 RIGHT

0.0034 89.92 RIGHT




ground plane present top GPLANE_TOP = No

bottom GPLANE_BOT = No

Data for the single layers

no. height z_max z_min relative relative conductivity ...

in m in m in m permitt. permeab. in S/m

0 infinity +infinity 0.00000E+00 1.00000 1.00000 0.00000E+00

1 1.50000E-03 0.00000E+00 -infinity 10.00000 1.00000 1.00000E-03

tan(delta) tan(delta)

(electric) (magnetic)

0.00000E+00 0.00000E+00

4.49378E-03 0.00000E+00



Current in A 2.0850E-01 1.3022E-01 2.4583E-01 31.99

Admitt. in A/V 2.1737E-02 1.3576E-02 2.5628E-02 31.99







85.00 0.00 0.000E+00 0.00 4.290E+01 14.22 -999.9999 14.8697 14.8697

85.00 2.00 8.016E-02 80.91 4.280E+01 14.17 -39.6990 14.8502 14.8503

85.00 4.00 1.594E-01 80.75 4.251E+01 14.01 -33.7265 14.7916 14.7916

POLARISATION


0.0000 90.00 LINEAR

0.0017 89.96 RIGHT

0.0034 89.92 RIGHT



Figure 3-22: Radiation pattern in the horizontal plane ϑ = 90 (E plane) in the absenceof the ground plane

Figure 3-23: Radiation pattern in the vertical plane ϕ = 0 (H plane) in the absence ofthe ground plane



Figure 3-24: Radiation pattern in the plane ϑ = 85 with a ground plane (reflectioncoefficient method)

Figure 3-25: Radiation pattern in the vertical plane ϕ = 0 (H plane) with a groundplane (reflection coefficient method)



Figure 3-26: Radiation pattern in the plane ϑ = 85 with a ground plane (Green’sfunction)

Figure 3-27: Radiation pattern in the vertical plane ϕ = 0 (H plane) with a groundplane (Green’s function)



3.11 Example 11: A dipole antenna in front of a metallic platewith PO and edge currents

As in Example 3, a dipole in front of a metallic plate is treated with the physical opticsapproximation. Here additional “correction” currents are added to the edges of the metalplate. This significantly improves the accuracy compared to the standard PO implemen-tation. The geometry is shown in figure 3-28.

A

D

B

EF

C

G


The input file is given below. The two KA cards specify the edges for which edge correctionmust be taken into account — note that they are effected by symmetry.

** A dipole antenna, of length lambda/2, in front of a metallic plate

** of dimension 3*lambda by 5*lambda. The dipole is d=3/4*lambda away from the plate.

** The plate region is solved by means of the physical optics approximation

** and the effects of the bordering edges have been taken into

** account using edge currents

** Variables

#lambda = 1 ** Wave length

#h = #lambda/4 ** Dipole height

#a = 2.5*#lambda ** Half edge length of plate

#b = 1.5*#lambda ** Half edge width of plate

#d = 3/4*#lambda ** Distance between dipole and plate

#tri_len = #lambda/5 ** Maximum edge length of the triangles

#seglen = #lambda/28 ** Maximum segment length



** Set the segmentation parameters

IP 0.002 #tri_len #seglen

** Create a quarter of the plate with the label 2

DP A -#a 0 0

DP B 0 0 0

DP C 0 0 #b

DP D -#a 0 #b

LA 2

BP A B C D

** Edges for the edge currents

KA C D 2

KA D A 2

** Mirror around the plane x=0 (magnetic wall)

SY 1 3 0 0 0

** Create upper half of the dipole with the label 0

#temp = 0.45*#seglen

DP E 0 -#d -#temp

DP F 0 -#d #temp

DP G 0 -#d #h

LA 0

BL F G

** mirror around the plane z=0 (electrical wall)

SY 1 0 0 2 0

** create the feed segment with the label 1

LA 1

BL E F

** Use PO approximation on the plate

PO 2 1 1 0 0


EG 1 0 0 0 0

** The excitation


FR 1 0 #freq

** Choose voltage so that exactly 1 Watt is radiated

** (This illustrates scaling the volatage, one could also use a PW card.)

A1 0 1 15.2007 0

** Current distribution on the plate

OS 2 1

** Far field

FF 1 1 361 0 90 0 0 1

** End

EN

Some of the results extracted from the output file example_11.out are



Current in A 1.3048E-01 -7.5950E-02 1.5098E-01 -30.20

Admitt. in A/V 8.5841E-03 -4.9965E-03 9.9324E-03 -30.20





VALUES OF THE CURRENT DENSITY VECTOR ON TRIANGLES in A/m

Triangle centre JX JY ...


1 -2.43590E+00 0.00000E+00 6.25000E-02 2.592E-04 118.60 0.000E+00 0.00

2 -2.37179E+00 0.00000E+00 1.25000E-01 8.446E-04 98.74 0.000E+00 0.00

3 -2.43590E+00 0.00000E+00 2.50000E-01 5.931E-04 136.07 0.000E+00 0.00

JZ Current magnitude in the

magn. phase 3 corner points

7.563E-03 36.91 7.531E-03 8.055E-03 7.001E-03

6.935E-03 35.51 7.562E-03 7.001E-03 8.055E-03

6.542E-03 40.91 7.001E-03 7.562E-03 5.427E-03





90.00 0.00 8.852E+00 87.86 0.000E+00 0.00 1.1982932 -999.9999 1.1982932

90.00 1.00 8.432E+00 83.37 0.000E+00 0.00 0.7760847 -999.9999 0.7760847

90.00 2.00 8.026E+00 78.72 0.000E+00 0.00 0.3473062 -999.9999 0.3473062

POLARISATION


0.0000 -180.00 LINEAR

0.0000 -180.00 LINEAR

0.0000 -180.00 LINEAR

Figure 3-29: Radiation pattern of the horizontal plane ϑ = 90



3.12 Example 12: A metallic sphere coated with a dielectric

This example considers a dielectric coated metallic sphere, i.e. the triangular patches onthe metallic sphere also represent the surface of the dielectric. The inside of the sphere isfree space. It is also possible to solve a metallic sphere embedded in a spherical dielectricGreen’s function region, but we will not consider it in this example.

A cut through the geometry is shown in figure 3-30.


The input file is

** Metallic sphere coated with a dielectric sphere (FEKO results can

** be compared to the exact solution, see figures in Examples Guide)

** Variables

#b = 1 ** Radius of the dielectric sphere

#a = 0.25 ** Radius of the metallic sphere

#eps = 4 ** Relative dielectric constant

#lambda = 2.0944 ** Free space wavelength


#tri_len = #lambda / sqrt(#eps) / 5

IP #tri_len



** Define points

DP A 0 0 0

DP B 0 -#a 0

DP C 0 -#b 0

DP D 0 0 #a

DP E 0 0 #b

** Create an eighth of the dielectric sphere in the quadrants x>0,y<0,z>0

ME 1 0

KU A E C 0 0 0 90 90

** Create an eighth of the metallic sphere, that forms the inner edge of the

** dielectric sphere (we assume that the interiour of the metallic sphere is

** filled with air, one could also fill it with the dielectric material)

ME 1 0 1

#len = 0.7 * #tri_len

KU A D B 1 0 0 90 90 #len

** Mirroring of the eighth of the sphere using electric and magnetic symmetry

SY 1 2 3 1


EG 1 0 0 0 0

** Program control

** Parameters of the dielectric medium

DI 1 #eps 1



FR 1 0 #freq

A0 0 1 1 1 0 -180 0 0

** Calculate the far field (bistatic RCS)

FF 1 181 1 0 0 0 1

** Near field along the z axis (avoid the surfaces of the sphere and coating)

FE 1 1 1 80 0 0 0 -1.975 0 0 0.05

** End

EN






0.00 0.00 2.590E+00 -106.96 0.000E+00 0.00 8.43144E+01

1.00 0.00 2.589E+00 -106.98 0.000E+00 0.00 8.42144E+01

2.00 0.00 2.584E+00 -107.04 0.000E+00 0.00 8.39151E+01



POLARISATION


0.0000 0.00 LINEAR

0.0000 0.00 LINEAR

0.0000 0.00 LINEAR



LOCATION EX ...


0 0.00000E+00 0.00000E+00 -1.97500E+00 1.32744E+00 -36.03

0 0.00000E+00 0.00000E+00 -1.92500E+00 1.39808E+00 -40.39

0 0.00000E+00 0.00000E+00 -1.87500E+00 1.44917E+00 -44.39

...

1 0.00000E+00 0.00000E+00 -9.75000E-01 7.17665E-01 106.88

1 0.00000E+00 0.00000E+00 -9.25000E-01 1.01341E+00 97.48

1 0.00000E+00 0.00000E+00 -8.75000E-01 1.29607E+00 90.61

EY EZ


0.00000E+00 0.00 0.00000E+00 0.00

0.00000E+00 0.00 0.00000E+00 0.00

0.00000E+00 0.00 0.00000E+00 0.00

0.00000E+00 0.00 0.00000E+00 0.00

0.00000E+00 0.00 0.00000E+00 0.00

0.00000E+00 0.00 0.00000E+00 0.00

Figure 3-31: Variation of the Ex field along the z axis compared to the exact solution



3.13 Example 13: Sphere with dielectric and magnetic propertiessolved with the volume equivalent current method

This example examines a homogeneous dielectric and magnetic sphere with εr = µr = 4,excited by an incident plane wave. The calculations are done with the volume currentmethod.

The geometry is shown in figure 3-32.


The input file is as follows

** A dielectric and magnetic sphere in the field of an incident plane

** wave (volume current method)

** Set some parameters as input

#r = 1.02 ** Sphere radius

#betrad = 0.3 ** Product of propagation constant times sphere radius

#mur = 4 ** The relative permeability

#epsr = 4 ** The relative dielectric constant

** Compute some derived parameters

#beta = #betrad / #r ** Propagation constant

#lambda = 2*#pi / #beta ** Wavelength in free space

#lambda_di = #lambda / sqrt(#mur*#epsr) ** Wavelength in the medium



#freq = #c0 / #lambda ** The frequency


** We can use here a rather fine mesh since the problem is electrically

** small and we don’t get many elements

#cube_len = #lambda_di / 20

IP #cube_len

** Define corner points

DP A 0 0 0

DP B #r 0 0

DP C 0 #r 0

DP D 0 0 #r

** Set the medium for the sphere

ME Sph

** Create an eighth of the sphere

DK A B C D 3 #cube_len

** Mirror around all there coordinate planes

** yz-plane: ideal electric conducting plane

** xz-plane: ideal magnetic conducting plane

** xy-plane: only geometric symmetry

SY 1 2 3 1


EG 1 0 0 0 0

** Set the material parameters

DI Sph #epsr #mur


FR 1 0 #freq

A0 0 1 1 1 0 -180

** Near field calculation along the z axis

** Note that the near fields cannot be calculated on the surface of a cuboid,

** thus a small offset is required

#offs = #r/1000

FE 3 1 1 201 0 #offs #offs -5+#offs 0 0 0.05

** Output fields inside the cuboids (forming the sphere)

FE 7

** Radar cross section in the vertical plane Phi=0

FF 1 181 1 0 0 0 1

** End

EN





in free space

LOCATION EX EY EZ


0.00100 0.00100 -4.9990 1.003E+00 84.29 6.476E-10 -3.25 3.011E-06 141.55

0.00100 0.00100 -4.9490 1.004E+00 83.45 6.757E-10 -3.03 3.120E-06 142.13

0.00100 0.00100 -4.8990 1.004E+00 82.62 7.054E-10 -2.81 3.233E-06 142.70

0.00100 0.00100 -4.8490 1.004E+00 81.78 7.369E-10 -2.59 3.352E-06 143.27


in free space

LOCATION HX HY HZ


0.00100 0.00100 -4.9990 1.719E-12 -3.25 2.664E-03 84.29 7.994E-09 141.55

0.00100 0.00100 -4.9490 1.794E-12 -3.03 2.664E-03 83.45 8.281E-09 142.13

0.00100 0.00100 -4.8990 1.873E-12 -2.81 2.664E-03 82.62 8.583E-09 142.70

0.00100 0.00100 -4.8490 1.956E-12 -2.59 2.664E-03 81.78 8.899E-09 143.27


inside the dielectric cuboids

LOCATION EX EY EZ SAR cuboid no.

X/m Y/m Z/m magn. phase magn. phase magn. phase in W/kg

0.128 0.128 -0.128 6.130E-01 4.80 3.415E-04 -0.70 3.623E-02 91.27 0.000E+00 1

0.128 0.128 -0.383 6.045E-01 15.20 3.619E-04 -21.50 3.649E-02 93.92 0.000E+00 2

0.128 0.128 -0.637 5.847E-01 25.63 1.932E-03 -157.66 3.652E-02 99.97 0.000E+00 3

0.128 0.128 -0.860 5.202E-01 33.46 7.309E-03 20.56 3.698E-02 132.30 0.000E+00 4


inside the magnetic cuboids

LOCATION HX HY HZ cuboid no.


0.128 0.128 -0.128 9.064E-07 -0.70 1.627E-03 4.80 9.618E-05 91.27 289

0.128 0.128 -0.383 9.607E-07 -21.50 1.605E-03 15.20 9.685E-05 93.92 290

0.128 0.128 -0.637 5.129E-06 -157.66 1.552E-03 25.63 9.694E-05 99.97 291

0.128 0.128 -0.860 1.940E-05 20.56 1.381E-03 33.46 9.816E-05 132.30 292







0.00 0.00 7.924E-02 -0.44 0.000E+00 0.00 7.890E-02

1.00 0.00 7.923E-02 -0.44 0.000E+00 0.00 7.889E-02

2.00 0.00 7.921E-02 -0.44 0.000E+00 0.00 7.885E-02

3.00 0.00 7.918E-02 -0.44 0.000E+00 0.00 7.879E-02

4.00 0.00 7.914E-02 -0.44 0.000E+00 0.00 7.871E-02

POLARISATION


0.0000 0.00 LINEAR

0.0000 0.00 LINEAR

0.0000 0.00 LINEAR

0.0000 0.00 LINEAR

0.0000 0.00 LINEAR

Figure 3-33 shows the distribution of the near field. The exact solution and the solutionobtained with equivalent surface currents (similar to Example 4) are also given. For thevolume current method inside the dielectric, two sets of calculations are requested by thetwo commands

** Near field along the z axis

FE 3 1 1 201 0 #offs #offs -5+#offs 0 0 0.05

** Fields inside the sphere

FE 7

The first card calculates the field on regular intervals and is the one used for figure 3-33.Note that the first parameter of this card specifies Both fields in free space rather thaninside a dielectric. The option Both fields inside dielectric is used to select the interiorproblem in the surface equivalence problem and should not be used with volume currents.Also note the small offset which is used to avoid sampling the near field on the surface ofthe cuboids.

The second card gives the field at the centre of each cuboid. This data is not on a regulargrid/line and is not simple to plot with POSTFEKO.



Figure 3-33: Distribution of the electric near field just off the z axis



3.14 Example 14: Perfectly conducting cube modelled with POand wedge correction

Figure 3-34 shows an ideal conducting cube with an incident plane wave as excitation.The surface currents will be calculated with the physical optics approximation. Wedgecorrection terms are also taken into account.

BB

DD

B

D

GG

HH

A

E

AAF

EE

FF



** Ideal conducting cube with the edge length a.

** Excitation by means of an incident plane wave with wave length 1 m.

** Calculation with the PO using correction currents on the wedges.

** Variables

#lam = 1

#a = 2*#lam ** Cube edge length

** Parameters for Segmentation

IP #lam/5.5

** Define points

DP A 0 0 #a/2

DP B 0 -#a/2 #a/2

DP C 0 -#a/2 0

DP D #a/2 -#a/2 0

DP E #a/2 0 0

DP F #a/2 0 #a/2

DP G #a/2 -#a/2 #a/2



** Create an eighth of the cube

LA 3

BP A B G F

LA 2

BP B C D G

LA 1

BP F G D E

** Mirror three times

** Only geometric symmetry due to rounding errors when searching for

** shaded opposite sides.

** One face of the cube is always illuminated and the other in the shade

SY 1 1 0 0 3

CB 6 3

CB 5 2

SY 1 0 1 0 4

CB 5 1

CB 7 3

CB 8 4

SY 1 0 0 1 6

CB 7 1

CB 8 2

CB 10 4

CB 12 6

** Apply PO to all surfaces. (For the cube all normals point

** outward and the surface is closed. Thus, as no triangles

** may be illuminated form behind, use the option NOSHADE=2.)

PO 1 2 0 0 0 9

** Create the four wedges on the front side

DP AA -#a/2 #a/2 #a/2

DP BB -#a/2 -#a/2 #a/2

DP CC -#a/2 -#a/2 -#a/2

DP DD #a/2 -#a/2 -#a/2

DP EE #a/2 #a/2 -#a/2

DP FF #a/2 #a/2 #a/2

DP GG #a/2 -#a/2 #a/2

DP HH -#a/2 #a/2 -#a/2

KL GG BB DD FF 2 3

KL BB CC GG AA 2 4

KL CC DD BB HH 2 9

KL DD GG CC EE 2 1

KL GG FF BB DD 3 1

KL FF AA GG EE 3 6

KL AA BB FF HH 3 4

** Create another two that are just illuminated, PHI<270 degrees

KL AA HH BB FF 4 6

KL HH CC AA EE 4 9

** End of geometric input

EG 1 0 0 0 0



** Set the frequency for a 1m wavelength

FR 1 0 #c0/#lam

** Excitation by means of an incident plane wave. Not quite perpendicular,

** to ensure that the one side of the cube is illuminated and the other is

** shaded. Determine the monostatic radar cross section.

A0 0 90 1 1 0 0.5 269.99 0 1

FF 2

** End

EN

It is sometimes required to relabel structures after applying cards that increment labels(such as SY and TG) — this is done with the CB card.






0.50 269.99 3.907E+00 91.43 3.533E-05 155.76 1.918E+02

...

1.50 269.99 3.813E+00 91.61 3.866E-05 164.50 1.827E+02

...

2.50 269.99 3.642E+00 91.16 3.571E-05 176.16 1.667E+02

...

3.50 269.99 3.557E+00 91.05 4.229E-05 150.52 1.590E+02

...

4.50 269.99 3.196E+00 91.98 5.233E-05 -176.93 1.284E+02

POLARISATION


0.0000 -180.00 LINEAR

0.0000 -180.00 LINEAR

0.0000 -180.00 LINEAR

0.0000 -180.00 LEFT

0.0000 180.00 LEFT

Figure 3-35 shows the calculated monostatic radar cross section.



Figure 3-35: The monostatic radar cross section of the cube



3.15 Example 15: Dipole antenna in front of a dielectric sphere

Figure 3-36 shows a dipole antenna in front of a dielectric sphere that will be examinedin this example. The sphere can be modelled with either the equivalent surface current orthe volume current. Here a third method is used namely a special Green’s function. Thissaves a large amount of memory, but is only applicable to spherical or planar dielectrics.

Figure 3-36: The geometry of Example 15


** A Dipole Antenna in front of a dielectric sphere

** Set parameters

#lambda = 1 ** Wave length

#a = #lambda ** Distance between the dipole and the sphere

#r = 0.4*#lambda ** Radius of the sphere

#epsr = 4 ** Relative dielectric constant


#segr = #lambda/1000

#segl = #lambda/20

#cube_len = #lambda/sqrt(#epsr)/6

IP #segr #segl #cube_len

** Define corner points

DP P1 #r+#a 0 -#lambda/4

DP P2 #r+#a 0 -0.4*#segl

DP P3 #r+#a 0 0.4*#segl



** Create the lower half of the antenna

BL P1 P2

**

** Mirror around the plane z=0

SY 1 0 0 2

**

** Create the feed segment

LA 1

BL P2 P3

**


EG 1 0 0 0 0

**

** Use special Green’s functions for the sphere

GF 1 0 #r #epsr 1

**

** Excitation

#freq = #c0/#lambda

FR 1 0 #freq

A1 0 1 1 0

**

** The Antenna radiates 1 Watt of power

PW 1 1

**

** Far field in the horizontal plane

FF 1 1 73 1 90 0 5

**

** End

EN

Note that only the dipole is created in the geometry section. The sphere is added as aGreen’s function with the GF card. In this case a single dielectric sphere is used, but itmay be up to three layers thick.




Current in A 1.5096E-01 -5.4399E-02 1.6046E-01 -19.82

Admitt. in A/V 1.1395E-02 -4.1061E-03 1.2112E-02 -19.82









90.00 0.00 7.152E+00 -148.25 0.000E+00 0.00 -0.689351 -999.9999 -0.689351

90.00 5.00 7.188E+00 -150.72 0.000E+00 0.00 -0.645993 -999.9999 -0.645993

90.00 10.00 7.305E+00 -157.97 0.000E+00 0.00 -0.506468 -999.9999 -0.506468

90.00 15.00 7.519E+00 -169.56 0.000E+00 0.00 -0.254768 -999.9999 -0.254768

POLARISATION


0.0000 180.00 LINEAR

0.0000 180.00 LINEAR

0.0000 180.00 LINEAR

0.0000 180.00 LINEAR

Figure 3-37: Radiation pattern in the horizontal plane



3.16 Example 16: Dipole antenna in front of a metallic cylinder(Fock theory)

Figure 3-38 shows a dipole antenna, placed in front of a metallic cylinder. In this examplethe current distribution on the surface of the cylinder is approximated using Fock theory.

D

A

B

P1

P2

P3C



** Metallic cylinder with the dipole antenna in front of it.

** Calculations are done by means of Fock-theory

**

** Set the parameters

#lambda = 1 ** the wave length

#r = 0.5*#lambda ** radius of the cylinder

#h = 1.0*#lambda ** half the height of the cylinder

#l = 0.5*#lambda ** length of the dipole antenna

#d = 0.8*#lambda ** Distance of the dipole from the cylinder



** Parameters for the segmentation

#segr = #lambda/1000

#segl = #lambda/15

#tri_len = #lambda/6.5

IP #segr #tri_len #segl

**


DP A 0 0 0

DP B 0 0 #h

DP C #r 0 0

DP D 0 0 -#h

#temp = 0.4*#segl

DP P1 0 -#r-#d -#temp

DP P2 0 -#r-#d #temp

DP P3 0 -#r-#d #l/2

**

** Create an eighth of the cylinder

LA 2

ZY A B C 90 #tri_len

**

** Mirroring

SY 1 3 1 0 0

**

** Create half of the dipole

LA 0

BL P2 P3

**

** Mirroring

SY 1 0 0 2 0

** Feed segment

LA 1

BL P1 P2

**

** Define the cylinder surface as a Fock region

FO 1 2 D B 0 0

**


EG 1 0 0 0 0



FR 1 #freq

** Excitation

A1 0 1 1 0

** Radiation Pattern in the horizontal plane

FF 1 1 181 0 90 0 0 2

** End

EN

Some extracts from the output file example_16.out are given on the next page.





Current in A 1.0285E-02 -3.9435E-03 1.1015E-02 -20.98

Admitt. in A/V 1.0285E-02 -3.9435E-03 1.1015E-02 -20.98







90.00 0.00 8.965E-01 47.59 0.000E+00 0.00 4.1607962 -999.9999 4.1607962

90.00 2.00 9.352E-01 34.25 0.000E+00 0.00 4.5278265 -999.9999 4.5278265

90.00 4.00 9.674E-01 21.07 0.000E+00 0.00 4.8220364 -999.9999 4.8220364

POLARISATION


0.0000 -180.00 LINEAR

0.0000 0.00 LINEAR

0.0000 0.00 LINEAR

Figure 3-39 shows the radiation pattern in the horizontal plane.




3.17 Example 17: Hertzian dipole in front of a parabolic reflector

Figure 3-40 shows a meshed parabolic reflector. A Hertzian dipole is placed at the focalpoint of the parabolic reflector.



** Parabolic reflector with a Hertzian dipole at its focal point

**

** Parameters

#lam = 1 ** Wavelength

#r = 4*#lam ** Radius of the parabolic reflector

#h = 2*#lam ** Height of the parabolic reflector

#f = (#r^2) / (4*#h) ** Focal distance


#tri_len = #lam/5

IP #tri_len



** Create a quarter of the reflector

DP A 0 0 0

DP B 1 0 0

DP C 0 0 #r

DP D #h 0 #r

PB A B C D 90 #tri_len

** Using symmetry

SY 1 0 3 2

** Treat the reflector with PO

PO 0 1 1 0 0

** End of the geometrical input

EG 1 0 1 0 0

** Excitation

#freq = #c0 / #lam

FR 1 0 #freq

A5 0 1 0 #f 0 0 0 0

** Far field calculation

FF 1 361 1 0 0 0 1

** End

EN






...

88.00 0.00 2.210E+03 1.98 0.000E+00 0.00 23.139655 -999.9999 23.139655

89.00 0.00 2.299E+03 2.05 0.000E+00 0.00 23.482399 -999.9999 23.482399

90.00 0.00 2.329E+03 2.08 0.000E+00 0.00 23.596207 -999.9999 23.596207

POLARISATION


0.0000 0.00 LINEAR

0.0000 0.00 LINEAR

0.0000 0.00 LINEAR

Figure 3-41 shows the radiation pattern in the vertical plane.



Figure 3-41: Radiation pattern in the vertical plane



3.18 Example 18: UHF antenna array

Figure 3-42 shows a UHF antenna array consisting of 32 UHF antenna elements — anindividual element is shown in figure 3-43.




Figure 3-43: A single UHF antenna “element”

The input file is as follows** UHF Antenna Array

** Parameters for the Problem (everything in millimetres)

#freq = 102.1e6 ** Frequency (Hertz)

#br = 2450 ** Width of the reflector

#hr = 1700 ** Height of the reflector

#xd = 737 ** Distance between the dipole and the reflector

#yd = 784 ** sideways displacement of the dipole (referenced to the middle)

#h = 690 ** Height (half of the length) of the dipole

#xiver = 3000 ** Distance of the inner fields (in the direction of the mast)

#xaver = 4000 ** Distance of the outer fields (in the direction of the mast)

#zver = 3200 ** Height displacement of the individual fields

#pha = 30 ** Phase (in degrees) in the first plane (bottom)

#phb = 10 ** Phase in the second plane

#phc =-60 ** Phase in the third plane

#phd =-40 ** Phase in the fourth plane

#phe =-40 ** Phase in the fifth plane

#phf =-40 ** Phase in the sixth plane

#phg = 50 ** Phase in the seventh plane

#phh = 80 ** Phase in the eighth plane (top)


#lambda = #c0 / #freq * 1000 ** Wavelength in mm



** (without the wire that lie in the plane of symmetry)

#dya = 325

#dyb = 625

#dyc = 925

#dyd = #br/2 ** 1225mm

#dz = #hr /2

DP A 0 0 0

DP B 0 #dya 0

DP C 0 #dyb 0



DP D 0 #dyc 0

DP E 0 #dyd 0

DP F 0 0 #dz

DP G 0 #dya #dz

DP H 0 #dyb #dz

DP I 0 #dyc #dz

DP J 0 #dyd #dz

DP M 0 0 -#dz

IP 20 #segl

LA 10

BL F G

BL G H

BL H I

BL I J

BL J E

IP 6

BL B G

BL H C

BL I D

** Create a quarter of the dipole antenna

IP 20

DP K #xd #yd 0

#z = 0.7*#segl

DP N #xd #yd #z

DP L #xd #yd #h

LA 1

BL K N

LA 10

BL N L

** Mirror the whole geometry in the plane z=0

SY 1 0 0 1 1

CB 11 10

** Create the wires in the symmetry plane z=0

** Reflector

IP 20

BL A B

BL B C

BL C D

BL D E

** Dipole mount

IP 20

BL A K

** Mirror the whole geometry in the plane y=0

SY 1 0 1 0 2

CB 12 10

** Create the wire in the symmetry plane y=0

IP 20

BL M A

BL A F



** Shift the previous antenna section up and forwards by half

TG 0 0 10 0 0 #xaver/2 #zver/2

** Create the section above, and increase label by 10

TG 1 1 10 10 0 #zver

** Create another section above, an increase label by another 10

#dx = (#xiver-#xaver)/2

TG 1 11 20 10 0 #dx #zver

** Create another section above, an increase label by another 10

TG 1 21 30 10 0 #zver

** Rotate the four sections that have been generated by 45 degrees

TG 0 0 40 0 0 45

** Mirror the geometry, to create the total geometry

SY 1 0 0 1 40

SY 1 3 0 0 80

SY 1 0 3 0 160

** Scale the millimetres to metres

SF 0.001


EG 1 0 0 0 0

** Program control (save solution to a *.str file)

PS 0 1 1 0

** Frequency specification

FR 1 0 #freq

** Excitation of the 32 dipoles

** First plane (bottom)

A1 0 72 1 #pha

A1 1 152 1 #pha

A1 1 154 1 #pha

A1 1 314 1 #pha

A1 1 312 1 #pha

A1 1 232 1 #pha

A1 1 234 1 #pha

A1 1 74 1 #pha

A1 1 71 1 #pha+180

A1 1 151 1 #pha+180

A1 1 153 1 #pha+180

A1 1 313 1 #pha+180

A1 1 311 1 #pha+180

A1 1 231 1 #pha+180

A1 1 233 1 #pha+180

A1 1 73 1 #pha+180

** Second Plane

A1 1 62 1 #phb

A1 1 142 1 #phb



A1 1 144 1 #phb

A1 1 304 1 #phb

A1 1 302 1 #phb

A1 1 222 1 #phb

A1 1 224 1 #phb

A1 1 64 1 #phb

A1 1 61 1 #phb+180

A1 1 141 1 #phb+180

A1 1 143 1 #phb+180

A1 1 303 1 #phb+180

A1 1 301 1 #phb+180

A1 1 221 1 #phb+180

A1 1 223 1 #phb+180

A1 1 63 1 #phb+180

** Third plane

A1 1 52 1 #phc

A1 1 132 1 #phc

A1 1 134 1 #phc

A1 1 294 1 #phc

A1 1 292 1 #phc

A1 1 212 1 #phc

A1 1 214 1 #phc

A1 1 54 1 #phc

A1 1 51 1 #phc+180

A1 1 131 1 #phc+180

A1 1 133 1 #phc+180

A1 1 293 1 #phc+180

A1 1 291 1 #phc+180

A1 1 211 1 #phc+180

A1 1 213 1 #phc+180

A1 1 53 1 #phc+180

** Fourth plane

A1 1 42 1 #phd

A1 1 122 1 #phd

A1 1 124 1 #phd

A1 1 284 1 #phd

A1 1 282 1 #phd

A1 1 202 1 #phd

A1 1 204 1 #phd

A1 1 44 1 #phd

A1 1 41 1 #phd+180

A1 1 121 1 #phd+180

A1 1 123 1 #phd+180

A1 1 283 1 #phd+180

A1 1 281 1 #phd+180

A1 1 201 1 #phd+180

A1 1 203 1 #phd+180

A1 1 43 1 #phd+180

** Fifth plane

A1 1 1 1 #phe

A1 1 81 1 #phe

A1 1 83 1 #phe

A1 1 243 1 #phe

A1 1 241 1 #phe



A1 1 161 1 #phe

A1 1 163 1 #phe

A1 1 3 1 #phe

A1 1 2 1 #phe+180

A1 1 82 1 #phe+180

A1 1 84 1 #phe+180

A1 1 244 1 #phe+180

A1 1 242 1 #phe+180

A1 1 162 1 #phe+180

A1 1 164 1 #phe+180

A1 1 4 1 #phe+180

** Sixth plane

A1 1 11 1 #phf

A1 1 91 1 #phf

A1 1 93 1 #phf

A1 1 253 1 #phf

A1 1 251 1 #phf

A1 1 171 1 #phf

A1 1 173 1 #phf

A1 1 13 1 #phf

A1 1 12 1 #phf+180

A1 1 92 1 #phf+180

A1 1 94 1 #phf+180

A1 1 254 1 #phf+180

A1 1 252 1 #phf+180

A1 1 172 1 #phf+180

A1 1 174 1 #phf+180

A1 1 14 1 #phf+180

** Seventh plane

A1 1 21 1 #phg

A1 1 101 1 #phg

A1 1 103 1 #phg

A1 1 263 1 #phg

A1 1 261 1 #phg

A1 1 181 1 #phg

A1 1 183 1 #phg

A1 1 23 1 #phg

A1 1 22 1 #phg+180

A1 1 102 1 #phg+180

A1 1 104 1 #phg+180

A1 1 264 1 #phg+180

A1 1 262 1 #phg+180

A1 1 182 1 #phg+180

A1 1 184 1 #phg+180

A1 1 24 1 #phg+180

** Eighth plane (top)

A1 1 31 1 #phh

A1 1 111 1 #phh

A1 1 113 1 #phh

A1 1 273 1 #phh

A1 1 271 1 #phh

A1 1 191 1 #phh

A1 1 193 1 #phh

A1 1 33 1 #phh



A1 1 32 1 #phh+180

A1 1 112 1 #phh+180

A1 1 114 1 #phh+180

A1 1 274 1 #phh+180

A1 1 272 1 #phh+180

A1 1 192 1 #phh+180

A1 1 194 1 #phh+180

A1 1 34 1 #phh+180

** Calculate Radiation Patterns

FF 1 1 361 0 90 0 1

FF 1 721 1 0 0 0 0.5

FF 1 721 1 0 0 45 0.5

** High resolution in the vertical plane

FF 1 201 1 0 90 0 0.1

FF 1 201 1 0 90 45 0.1

** Determine the power in the far field (1/8 sphere)

FF 3 180 18 0 0.5 1.25 1 2.5

** End

EN




Current in A 1.7444E-02 1.4629E-02 2.2766E-02 39.99

Admitt. in A/V 2.2421E-02 3.9475E-03 2.2766E-02 9.99







90.00 0.00 3.446E+01 -31.05 0.000E+00 0.00 11.351448 -999.9999 11.351448

90.00 1.00 3.442E+01 -30.92 1.101E-02 -91.15 11.341705 -58.56179 11.341705

90.00 2.00 3.431E+01 -30.55 2.199E-02 -91.37 11.312743 -52.55015 11.312745

POLARISATION


0.0000 0.00 LINEAR

0.0003 0.01 RIGHT

0.0006 0.02 RIGHT

Figures 3-44 and 3-45 shows the radiation patterns in the horizontal and vertical planes.



3.19 Example 19: Dipole antenna in front of a UTD plate

This example considers a dipole antenna in front of a metallic plate which is used as areflector, similar to Examples 2 and 3. In Example 2 the whole structure is treated withthe moment method, whereas in Example 3, the currents on the surface on the reflectorare approximated by means of physical optics(PO). In this example the plate is treatedby means of diffraction theory (UTD). The geometry is shown in figure 3-46.



** A half lambda dipole in front of a metallic plate with

** an edge length 3*lambda d=3/4 lambda apart. The dipole

** is treated with MoM and the plate with UTD

#lambda = 3 ** Wavelength

#h = #lambda/4 ** Half the dipole height

#a = 1.5*#lambda ** Half edge length of the plate

#d = 3/4*#lambda ** Distance between dipole and plate

#seglen = #lambda/28 ** Maximum segment length

#rho = 0.002*#lambda ** Segment radius

**

IP #rho #seglen



** The upper half of the dipole is assigned the label 0

#temp = 0.45*#seglen

DP E #d 0 -#temp

DP F #d 0 #temp

DP G #d 0 #h

LA 0

BL F G

** Mirroring around the plane z=0 (electric wall)

SY 1 0 0 2 0

** Excitation segment is assignment the label 1

LA 1

BL E F

** Create the plate (polygon in the UTD region)

DP A 0 -#a -#a

DP B 0 #a -#a

DP C 0 #a #a

DP D 0 -#a #a

PY A B C D

** Parameters for the UTD:

** GO and diffraction, no double diffraction and no edge diffraction

UT 1 1 0 0 3 0


EG 1 0 0 0 0

** Frequency


FR 1 0 #freq

** Excitation by means of a voltage source with the power of 1 Watt

A1 0 1 1 0

PW 1 1

** Far Field (horizontal pattern)

FF 1 1 181 0 90 0 0 2

** Near field

FE 1 70 1 1 0 -1.97 0 0 0.1 0 0

** End

EN

Some extracts from the output file example_19.out are shown on the next page.





Current in A 1.3111E-01 -7.6455E-02 1.5177E-01 -30.25

Admitt. in A/V 8.5945E-03 -5.0119E-03 9.9491E-03 -30.25




Factor e^(-j*BETA*R)/R not considered (R= 1.0800E+03 m)



90.00 0.00 1.757E+01 -32.18 0.000E+00 0.00 7.1191372 -999.9999 7.1191372

90.00 2.00 1.761E+01 -32.30 0.000E+00 0.00 7.1362156 -999.9999 7.1362156

90.00 4.00 1.771E+01 -32.64 0.000E+00 0.00 7.1855920 -999.9999 7.1855920

90.00 6.00 1.787E+01 -33.15 0.000E+00 0.00 7.2617670 -999.9999 7.2617670

90.00 8.00 1.806E+01 -33.77 0.000E+00 0.00 7.3558859 -999.9999 7.3558859

POLARISATION


0.0000 0.00 LINEAR

0.0000 0.00 LINEAR

0.0000 0.00 LINEAR

0.0000 0.00 LINEAR

0.0000 0.00 LINEAR


in free space (determined with UTD)

LOCATION EX EY EZ


-1.9700 0.0E+00 0.0E+00 0.000E+00 0.00 0.000E+00 0.00 7.063E-02 79.63

-1.8700 0.0E+00 0.0E+00 0.000E+00 0.00 0.000E+00 0.00 6.778E-02 84.33

-1.7700 0.0E+00 0.0E+00 0.000E+00 0.00 0.000E+00 0.00 6.484E-02 88.83

-1.6700 0.0E+00 0.0E+00 0.000E+00 0.00 0.000E+00 0.00 6.179E-02 93.11

...

4.63000 0.0E+00 0.0E+00 0.000E+00 0.00 0.000E+00 0.00 5.409E+00 -62.21

4.73000 0.0E+00 0.0E+00 0.000E+00 0.00 0.000E+00 0.00 5.251E+00 -73.75

4.83000 0.0E+00 0.0E+00 0.000E+00 0.00 0.000E+00 0.00 5.102E+00 -85.31

4.93000 0.0E+00 0.0E+00 0.000E+00 0.00 0.000E+00 0.00 4.963E+00 -96.90

Figure 3-47 shows the near field values along the x axis and figure 3-48 the radiationpattern in the horizontal plane.



Figure 3-47: Electric near field




3.20 Example 20: Monopole antenna on a metallic UTD plate

This example considers the UTD solution of a monopole antenna on a square plate.Figure 3-49 shows the geometry.


In the past the attachment point between a wire and a UTD polygonal plate was specifiedby defining a very short stub on the opposite side of the plate. This is no longer required— in fact it will result in an error — as FEKO now automatically determines contactpoints between wires and polygonal plates (and the ground plane specified by the BO andGF cards).


** Monopole antenna on a square ground plate of finite size using the

** MoM/UTD hybrid method (coupling between MoM and UTD for the plate)

** Parameters for the geometry

#lam = 1 ** Wave length

#h = #lam/4 ** Height of the monopole antenna

#a = 5*#lam ** Edge length of the plate

** Segmentation parameters for the wire antenna

#seglen = #lam/20

#segrad = #lam/1000

IP #segrad #seglen

** Defining the edge points of the plate

DP P1 #a/2 #a/2 0

DP P2 -#a/2 #a/2 0

DP P3 -#a/2 -#a/2 0

DP P4 #a/2 -#a/2 0

** Generate the plate

PY P1 P2 P3 P4



** Points for the monopole antenna

DP A 0 0 0

#temp = 0.9*#seglen

DP B 0 0 #temp

DP C 0 0 #h

** Create monopole antenna (Excitation segment with label 1)

LA 1

BL A B

LA 0

BL B C

** Parameters for the UTD (Edge and corner diffraction is taken into account)

UT 1 2 0 0 7 0


EG 0 0 0 0 0

** Excitation by a voltage source (1 Watt)

#freq = #c0 / #lam

FR 1 0 #freq

A1 0 1 1 0

PW 1 1

** Calculated the far field in 2 vertical planes

FF 1 181 1 0 0 0 2 0

FF 1 181 1 0 0 45 2 0

** End

EN


At node 1 the segments 1 and 2 are located.

Only one half of the basis function gn over the segment 1 is considered.

The segment 2 is regarded as not existant and will be ignored.



Current in A 2.0472E-01 -8.5277E-02 2.2178E-01 -22.61

Admitt. in A/V 2.0956E-02 -8.7292E-03 2.2701E-02 -22.61









0.00 0.00 1.077E-18 103.30 2.160E-18 78.69 -377.1356 -371.0914 -370.1270

2.00 0.00 1.253E+00 99.18 0.000E+00 0.00 -15.8175 -999.9999 -15.8175

4.00 0.00 2.229E+00 98.05 0.000E+00 0.00 -10.8148 -999.9999 -10.8148

6.00 0.00 2.738E+00 95.66 0.000E+00 0.00 -9.0289 -999.9999 -9.0289

POLARISATION


0.1711 -115.18 RIGHT

0.0000 180.00 LINEAR

0.0000 180.00 LINEAR

0.0000 180.00 LINEAR





0.00 45.00 2.242E-18 86.82 8.929E-19 57.89 -370.7665 -378.7625 -370.1270

2.00 45.00 1.283E+00 99.18 0.000E+00 0.00 -15.6133 -999.9999 -15.6133

4.00 45.00 2.450E+00 98.21 0.000E+00 0.00 -9.9944 -999.9999 -9.9944

6.00 45.00 3.390E+00 96.79 0.000E+00 0.00 -7.1739 -999.9999 -7.1739

POLARISATION


0.1711 -160.18 RIGHT

0.0000 180.00 LINEAR

0.0000 180.00 LINEAR

0.0000 180.00 LINEAR

Figures 3-50 and 3-51 presents the radiation patterns in the vertical plane.




0.8

0.6

0.4

0.2

345º330º

315º

300º

285º

270º

255º

240º

225º

210º195º180º165º

150º

135º

120º

105º

90º

75º

60º

45º

30º15º 0º

Figure 3-50: Radiation pattern in the plane ϕ = 0

345º330º

315º

300º

285º

270º

255º

240º

225º

210º195º180º165º

150º

135º

120º

105º

90º

75º

60º

45º

30º15º 0º


0.8

0.6

0.4

0.2

Figure 3-51: Radiation pattern in the plane ϕ = 45



3.21 Example 21: Mobile communications antenna on the roofof a building

A mobile communications antenna which transmits at 900 MHz, has been placed on top ofa building. The resulting radiation pattern must be calculated. To simplify the modellingthe antenna is replaced with a Hertzian dipole. Of course more complex antennas canbe considered at the expense of solution time. The building is assumed to be a perfectlyconducting body. Figure 3-52 shows the geometry under consideration.

A4

B4C4

A2

B2

C1B1

D1

D4

D2

C2

C3

D3

B3

A3A1

Figure 3-52: Geometry of Example 21


** A mobile communications antenna (in this case replaced by a Hertzian dipole)

** on the roof of a building

** Frequency

#freq = 900.0e6

** Dimensions

#a = 30

#b = 20

#c = 20

#d = 26

#e = 16

#f = 3



** Declare points

DP A1 #b/2 -#a/2 0

DP A2 #b/2 #a/2 0

DP A3 -#b/2 #a/2 0

DP A4 -#b/2 -#a/2 0

DP B1 #b/2 -#a/2 #c

DP B2 #b/2 #a/2 #c

DP B3 -#b/2 #a/2 #c

DP B4 -#b/2 -#a/2 #c

DP C1 #e/2 -#d/2 #c

DP C2 #e/2 #d/2 #c

DP C3 -#e/2 #d/2 #c

DP C4 -#e/2 -#d/2 #c

DP D1 #e/2 -#d/2 #c+#f

DP D2 #e/2 #d/2 #c+#f

DP D3 -#e/2 #d/2 #c+#f

DP D4 -#e/2 -#d/2 #c+#f

** Construct building

PY A1 A2 A3 A4

PY A1 A2 B2 B1

PY A2 A3 B3 B2

PY A3 A4 B4 B3

PY A4 A1 B1 B4

PY B1 B2 C2 C1

PY B2 B3 C3 C2

PY B3 B4 C4 C3

PY B4 B1 C1 C4

PY C1 C2 D2 D1

PY C2 C3 D3 D2

PY C3 C4 D4 D3

PY C4 C1 D1 D4

PY D1 D2 D3 D4

** Parameters for the UTD

UT 1 2 0 0 15 0

** End of the Geometry

EG 1 0 0 0 0

** Excitation by means of a Hertzian dipole

FR 1 0 #freq

#x = #b/2 - 1.75

#y = 0

#z = 21.5

A5 0 1 #x #y #z

** Far field calculations

FF 1 1 720 0 89.99 0.25 0.5

FF 1 720 1 0 0.25 0.01 0.5

FF 1 1440 1 0 0.125 90.01 0.25

** End

EN




DATA FOR GO/UTD

Consideration of:

geometrical optics (direct and refl.rays): yes

edge and wedge diffraction (not multiple): yes

corner diffraction: yes

double diffraction: yes

creeping waves: no

tip diffraction: no

Maximum number of reflections/diffractions: 2

Using UTD according to Kouyoumjian

Coupling MoM-UTD is taken into account





WARNING: Largeness parameter K= 0.627466636 is less than 1

WARNING 1010: Assumptions of UTD not fulfilled

...

89.99 0.25 1.098E+03 7.67 4.838E-01 138.39 7.5221168 -59.59419 7.5221176

89.99 0.75 1.109E+03 6.59 9.479E-01 141.86 7.6113774 -53.75230 7.6113806

89.99 1.25 1.101E+03 5.53 1.630E+00 138.88 7.5446845 -49.04441 7.5446941

89.99 1.75 1.107E+03 3.44 2.338E+00 136.42 7.5985177 -45.91079 7.5985371

POLARISATION


0.0003 -0.02 LEFT

0.0006 -0.03 LEFT

0.0011 -0.06 LEFT

0.0015 -0.08 LEFT





0.25 0.01 4.451E+01 0.14 7.102E-03 -151.46 -20.31874 -96.26041 -20.31874

0.75 0.01 4.835E+01 85.62 1.005E-02 -26.20 -19.59954 -93.24160 -19.59954

1.25 0.01 4.943E+01 168.50 1.425E-02 81.05 -19.40837 -90.21248 -19.40837

1.75 0.01 4.897E+01 -112.03 1.940E-02 -179.19 -19.48928 -87.53100 -19.48928

2.25 0.01 4.535E+01 -35.71 2.629E-02 -83.34 -20.15654 -84.89069 -20.15654

POLARISATION


0.0001 -0.01 RIGHT

0.0002 180.00 RIGHT

0.0003 -180.00 RIGHT

0.0004 0.01 RIGHT

0.0004 0.02 RIGHT







0.13 90.01 6.106E+00 -72.02 4.434E+01 144.44 -37.57305 -20.35080 -20.26924

0.38 90.01 1.225E+01 -67.70 4.441E+01 144.08 -31.52408 -20.33881 -20.02022

0.63 90.01 7.767E+00 -41.69 4.445E+01 143.36 -35.48249 -20.33007 -20.19946

0.88 90.01 7.313E+00 15.89 4.442E+01 142.25 -36.00593 -20.33594 -20.21980

1.13 90.01 9.597E+00 -12.28 4.435E+01 140.77 -33.64459 -20.34883 -20.15011

POLARISATION


0.0808 -83.64 RIGHT

0.1376 -76.54 RIGHT

0.0149 -80.12 RIGHT

0.1313 -84.33 LEFT

0.0945 -78.98 LEFT

Note the warning 1010 which states that some ray paths do not have the required min-imum length. These paths starts to violate the far field assumption of the UTD. Thisdoes not imply that the results will be incorrect, but does indicate that the user shouldverify it in some way. Figures 3-53, 3-54 and 3-55 presents the radiation patterns in thehorizontal and the two vertical planes.




Figure 3-54: Radiation pattern in the vertical plane with ϕ = 0.01

Figure 3-55: Radiation pattern in the vertical plane with ϕ = 90



3.22 Example 22: Planar dipole antenna (modelled using wiresegments) on a substrate

xy

z

~l = 144 mm

w= 1.25 mm

h = 1.5 mm

r= 4.5

Figure 3-56: Figure of a planar dipole antenna on a substrate

Figure 3-56 shows a planar dipole antenna with length l = 144 mm and width w =1.25 mm located on a planar substrate with thickness h = 1.5 mm. The substrate doesnot have a ground plane.

The input impedance of the antenna as a function of frequency in the band 700 MHz to900 MHz, as well as the vertical radiation pattern at 830 MHz must be calculated.

Relatively narrow planar antennas may be modelled as a wire and this approach will beused in this example. (Example 24 shows the use of surface triangles.) The input file is

** Example of a printed dipole antenna on a

** dielectric substrate

** Parameters

#w = 1.25 ** Width of the metal strips (all dimensions in mm)

#rad = 0.25*#w ** Equivalent radius of the segments

#len = 144 ** Length of the antenna

** Maximum segmentation length

#lambda = 1000*#c0/900E6 ** Free space wavelength (in mm) at 900 MHz


** Set the segmentation parameters

IP #rad #segl



** Generate the dipole antenna

#x = 0.45*#segl

DP A -#x 0 0

DP B #x 0 0

DP C #len/2 0 0

BL B C

SY 1 2 0 0

LA 1

BL A B

** Scaling factor (dimensions in mm)

SF 1.0e-3

** End of geometry input

EG 1 0 0 0 0

** Substrate (no ground plane)

#epsr = 4.5 ** Relative permittivity

#h = 1.5 ** Height (in mm)

GF 11 2 1 1 0

#h #epsr 1 0

0 1 1 0

** Voltage source as excitation

A1 0 1 1 0

** Frequency loop 700 .. 900 MHz

FR 9 0 700e6 900e6

** Initiate the computation

** (we are interested in the input impedance only)

OS 0

** Now we consider the frequency 830 MHz and compute the

** radiation patterns in the xz- and yz-planes

FR 1 0 830e6

FF 1 179 1 1 -89 0 1

FF 1 179 1 1 -89 90 1

** End

EN


EXCITATION BY VOLTAGE SOURCE AT SEGMENT






Source at segment w. label: ULA = 1

Absolute number of segment: UNR = 9






ground plane present? GPLANE = No





1 1.50000E-03 0.00000E+00 -1.50000E-03 4.50000 1.00000 0.00000E+00

2 infinity -1.50000E-03 -infinity 1.00000 1.00000 0.00000E+00



0.00000E+00 0.00000E+00

0.00000E+00 0.00000E+00

0.00000E+00 0.00000E+00



Current in A 3.1627E-03 8.9264E-03 9.4702E-03 70.49

Admitt. in A/V 3.1627E-03 8.9264E-03 9.4702E-03 70.49









Source at segment w. label: ULA = 1

Absolute number of segment: UNR = 9



Current in A 5.4795E-03 1.0562E-02 1.1899E-02 62.58

Admitt. in A/V 5.4795E-03 1.0562E-02 1.1899E-02 62.58









-89.00 0.00 1.614E-02 -97.09 0.000E+00 0.00 -32.3024 -999.9999 -32.3024

-88.00 0.00 2.704E-02 -102.32 0.000E+00 0.00 -27.8209 -999.9999 -27.8209

-87.00 0.00 3.844E-02 -106.12 0.000E+00 0.00 -24.7643 -999.9999 -24.7643

POLARISATION


0.0000 0.00 LINEAR

0.0000 0.00 LINEAR

0.0000 0.00 LINEAR





-89.00 90.00 0.000E+00 0.00 2.924E-01 -5.55 -999.9999 -7.1396 -7.1396

-88.00 90.00 0.000E+00 0.00 4.973E-01 10.93 -999.9999 -2.5270 -2.5270

-87.00 90.00 0.000E+00 0.00 6.171E-01 22.43 -999.9999 -0.6530 -0.6530

POLARISATION


0.0000 90.00 LINEAR

0.0000 90.00 LINEAR

0.0000 90.00 LINEAR

The input impedance as a function of frequency (extracted with POSTFEKO), is

Freq (Hz) Real (ohm) Imag (ohm) Magnitude Phase

7.0000E+02 3.5265E+01 -9.9532E+01 1.0560E+02 -70.490

7.2500E+02 3.8702E+01 -7.4600E+01 8.4042E+01 -62.580

7.5000E+02 4.2418E+01 -5.0203E+01 6.5724E+01 -49.805

7.7500E+02 4.6437E+01 -2.6265E+01 5.3350E+01 -29.493

8.0000E+02 5.0788E+01 -2.6996E+00 5.0860E+01 -3.043

8.2500E+02 5.5500E+01 2.0575E+01 5.9191E+01 20.341

8.5000E+02 6.0611E+01 4.3645E+01 7.4690E+01 35.757

8.7500E+02 6.6158E+01 6.6559E+01 9.3846E+01 45.173

9.0000E+02 7.2187E+01 8.9396E+01 1.1490E+02 51.079

The input impedance as a function of frequency and the radiation pattern at 830 MHzare presented in figures 3-57 and 3-58 respectively.



Figure 3-57: Smith chart of the input impedance in the frequency range from 700 MHzto 900 MHz. The curve runs clockwise on the Smith chart.

xz plane

yz plane

Far field gain

0º 345º

330º

315º

300º

285º

270º

255º

240º

225º

210º

195º180º165º

150º

135º

120º

105º

90º

75º

60º

45º

30º

15º 0º

0

-5

-10

-15

-20

0º

0

-5

-10

-15

-20

O

O

O

O

OO

O

O

O

O

O

O

O

OO

O

O

O

O

O

O

O

O

O

Figure 3-58: Vertical radiation pattern (Gain in dB) of the antenna at 830 MHz



3.23 Example 23: Dielectric cone on top of a metallic cylinder

Figure 3-59: Meshed geometry of Example 23

This example is also considered in the description of the ME card in the User’s Manual.The geometry is shown in figure 3-59. The metal cylinder can be considered to be filledwith air. In this case the input file is as follows (example_23a.pre)

** Computation of the RCS of a dielectric cone on top of a

** metallic cylinder. See Fig. 3 in IEEE Trans. on Antennas

** and Propagation, vol. 39, no. 7, July 1991, p. 1036.

** Here: Inside of the metallic cylinder is assumed to be filled

** with air (medium 0)


#a = 0.3*#lambda ** Radius of the cylinder

#h = 0.6*#lambda ** Height of the cylinder and the cone




IP #tri_len




DP A 0 0 0

DP AO 0 0 #h

DP AU 0 0 -#h

DP C #a 0 0

DP CU #a 0 -#h

** Generate the geometry (only one quarter)

** Cylinder side and bottom - triangles are in air (medium 0)

ME 0

KR AU A CU 90

ZY AU A CU 90

** Cone - triangles on the surface of a dielectric

ME 1 0

KK A AO C 90 0

** Cylinder top/cone bottom - metallic triangles on a dielectric surface

ME 0 1 1

KR A AO C 90

** Symmetry

SY 1 2 3 0

** End of geometry

EG 1 0 0 0 0

** Dielectric properties

DI 1 #epsr 1 0

** Frequency


FR 1 0 #freq

** Incident plane wave

A0 0 1 1 1 0 0 0 0

** Bistatic RCS computation

FF 1 181 0 0 0 0 1

** End

EN

The model can also be constructed such that the cylinder is filled with the same dielectricas the cone. (Such an approach will decouple the internal and external problems and canbe very useful for certain applications with a high shielding factor. One must, however,be aware of the fact that the number of basis functions will be larger, which will influencethe run time and memory.) In this case the ME cards in the geometry block of the inputfile (example_23b.pre) will be slightly different and the block will be as follows.


** Cylinder side and bottom - metallic triangles on a dielectric surface

** Note that the normal directions require two ME cards

ME 0 1 1

KR AU A CU 90

ME 1 0 1



ZY AU A CU 90

** Cone - triangles on the surface of a dielectric

ME 1 0

KK A AO C 90 0

** Cylinder top/cone bottom - metallic triangles inside a dielectric

ME 1

KR A AO C 90

One may also fill the cylinder with a different dielectric medium. (This will require evenmore basis functions than filling the cylinder with the cone dielectric.) Then the geometryblock of the input file (example_23c.pre) will then be as follows.


** Cylinder side and bottom - metallic triangles on a dielectric surface

** The surface between medium 2 and air (medium 0)

ME 0 2 1

KR AU A CU 90

ME 2 0 1

ZY AU A CU 90

** Cone - triangles on the surface of a dielectric (medium 1)

ME 1 0

KK A AO C 90 0

** Cylinder top/cone bottom - metallic triangles on a dielectric surface

** The surface between medium 1 and medium 2

ME 2 1 1

KR A AO C 90

Some extracts from the output file example_23a.out are given below.





0.00 0.00 6.226E-01 45.43 0.000E+00 0.00 4.189E+00

1.00 0.00 6.223E-01 45.43 0.000E+00 0.00 4.186E+00

2.00 0.00 6.215E-01 45.45 0.000E+00 0.00 4.176E+00

3.00 0.00 6.203E-01 45.47 0.000E+00 0.00 4.160E+00

4.00 0.00 6.185E-01 45.50 0.000E+00 0.00 4.137E+00

POLARISATION


0.0000 180.00 LINEAR

0.0000 180.00 LINEAR

0.0000 180.00 LINEAR

0.0000 180.00 LINEAR

0.0000 180.00 LINEAR

Figure 3-60 shows the radar cross section pattern.



Figure 3-60: The bistatic radar cross section as a function of the angle ϑ



3.24 Example 24: Planar dipole antenna on a substrate (usingtriangles)

The example of a planar dipole antenna in figure 3-56 is examined again, this time as aplanar metallic strip modelled by triangular elements. (Note that the dimensions differfrom those in figure 3-56.) The model of the dipole, without substrate, is shown infigure 3-61.

Figure 3-61: Discretised strip dipole antenna


** Example of a printed dipole antenna on a substrate

** Infinite substrate with Green’s function

** Dipole parameters

#w = 8 ** width of the metallisation (all dimensions in mm)

#len = 2*100 ** length of the dipole antenna

** Substrate parameters

#h = 6 ** height

#epsr = 4 ** rel. permittivity

** Maximum segmentation length

#freq = 1.0e9

#lambda = #c0 / #freq

#tri_len = #lambda / 25 * 1000 ** in mm

** Generate the dipole antenna

IP #tri_len

DP A 0 -#w/2 0

DP B #len/2 -#w/2 0

DP C #len/2 #w/2 0

DP D 0 #w/2 0

LA 1

BP A B C D

SY 1 2 0 0 1



** Scaling factor (dimensions in mm)

SF 1.0e-3


EG 1 0 0 0 0

** Green’s function

GF 11 2 0 1 1 0

#h #epsr 1 0

1 1 0

** Voltage source across an edge as excitation

AE 0 1 2 1

** Frequency

FR 1 0 #freq

** Far-field pattern

FF 1 90 1 1 -89 0 2

FF 1 90 1 1 -89 90 2

** End

EN

Note the use of the AE card rather than the A7 card to excite the dipole. The AE cardis much simpler to use, especially for structures where there is more than one edge inparallel.

The extracts from the output file example_24.out are similar to those of Example 22,but they cannot be compared directly since the dimensions of the dipoles differ.









1 6.00000E-03 0.00000E+00 -6.00000E-03 4.00000 1.00000 0.00000E+00

2 infinity -6.00000E-03 -infinity 1.00000 1.00000 0.00000E+00



0.00000E+00 0.00000E+00

0.00000E+00 0.00000E+00

0.00000E+00 0.00000E+00





Current in A 1.4973E-03 2.5676E-03 2.9723E-03 59.75

Admitt. in A/V 1.4973E-03 2.5676E-03 2.9723E-03 59.75


Capacitance in F 5.4761E-13






...

1.00 0.00 2.517E-01 13.52 2.606E-05 85.44 1.7301741 -77.96789 1.7301741

3.00 0.00 2.510E-01 13.56 7.802E-05 85.43 1.7057965 -68.44361 1.7057970

5.00 0.00 2.496E-01 13.63 1.295E-04 85.40 1.6570512 -64.04288 1.6570524

POLARISATION


0.0001 0.00 LEFT

0.0003 0.01 LEFT

0.0005 0.01 LEFT



3.26 Example 26: Input impedance of a two wire transmissionline

A two wire transmission line with length l = 0.6 λ and characteristic impedance Zo =360 Ω is terminated with a real load Zl = 50 Ω. The input impedance of the transmissionline may be found from transmission line theory

Zi = (75.6 + j253.9)Ω

This structure is modelled in FEKO. The wire radius is 2mm and the separation iscalculated to give a characteristic impedance of 360 Ω. The remaining parameters maybe determined from the input file

** This is an example to compute the input impedance

** of a loaded two wire transmission line

** Parameters

#a = 0.002 ** wire radius

#Z_0 = 360 ** the desired characteristic impedance

#Z_l = 50 ** the load

#freq = 20.0e6 ** the frequency

#ll = 0.6 ** length of the transmission line in wavelengths

** compute the spacing of the two wires (approximate

** formula for D>>a)

#D = #a * exp(#Z_0 / 120)

** compute the wavelength

#lam = #c0 / #freq

** the length of the transmission line

#l = #ll*#lam


** (Note that the segment length should be short as compared

** to the wavelength but also not too long as compared to the

** separation distance between the two parallel wires.)

#segl = min ( #lam/10, 5*#D )

IP #a #segl

** definition of the structure

#z = min ( 0.45*#segl, #D/6 )

DP A 0 0 -#z

DP B 0 0 #z

DP C 0 0 #D/2

DP D #l 0 #D/2

DP E #l 0 #z

DP F #l 0 -#z

BL B C

BL C D

BL D E



SY 1 0 0 2

LA 1

BL A B

LA 2

BL E F

** end of geometry

EG 1 0 0 0 0

** excitation

FR 1 0 #freq

A1 0 1 1 0

** load

LZ 2 #Z_l 0

** Just compute the input impedance

OS 0

** end

EN

The FEKO result for the input impedance is:



Current in A 1.0466E-03 -3.5339E-03 3.6856E-03 -73.50

Admitt. in A/V 1.0466E-03 -3.5339E-03 3.6856E-03 -73.50



This result shows good agreement with the result from transmission line theory. Notethat FEKO also models the radiation from the transmission line correctly while this isnot included in the transmission line theory.

In a similar manner it is possible, for example for EMC purposes, to investigate thecoupling of electromagnetic fields into cables.



3.27 Example 27: Yagi-Uda antenna in front of a cylindrical(UTD) mast

With this example the use of UTD for a cylinder is shown. A four element Yagi-Uda an-tenna is placed in front of a cylindrical mast. To reduce computational time, the couplingbetween the MoM region (antenna) and the UTD region (cylinder) is neglected. Fig-ure 3-62 shows the geometry of the model. Note that the cylinder is considered infinitelylong for computational purposes.


The exact data for the geometry may be determined from the input file

** A four element Yagi-Uda antenna in front of a cylindrical mast, which

** is considered as infinitely long and treated with UTD.

** Only the influence of the mast on the radiation pattern is investigated,

** the influence on the input impedance of the antenna is not considered.

** For acceleration of the computation the coupling is neglected.

** Define some necessary variables


#lambda = #c0 / #freq ** Wavelength

#d = 0.30 * #lambda ** Distance between individual elements

#lr = 0.475 * #lambda ** Length of reflector

#l = 0.453 * #lambda ** Length of the dipole

#ld = 0.446 * #lambda ** Length of the director



#h = 0 ** Height of the element

#a = 1.5 ** Longitudinal shift of element from origin

#b = 1 ** Radius of mast

#c = 2 ** Half of mast length (only for input purposes,

** The UZ card actually uses an infinite mast)


#rho = 0.0025*#lambda

#segl = #lambda/20

IP #rho #segl

** Generation of the Yagi-Uda Antenna

DP A 0 0 -#c

DP B 0 0 #c

DP C #b 0 -#c

#x = #a

DP A1 #x 0 #h

DP A2 #x #lr/2 #h

#x = #x + #d

DP B1 #x -0.4*#segl #h

DP B2 #x 0.4*#segl #h

DP B3 #x #l/2 #h

#x = #x + #d

DP C1 #x 0 #h

DP C2 #x #ld/2 #h

#x = #x + #d

DP D1 #x 0 #h

DP D2 #x #ld/2 #h

** Define half of the antenna

LA 0

BL A1 A2

BL B2 B3

BL C1 C2

BL D1 D2

** Mirror it in the plane y=0

SY 1 0 2 0

** Feed segment

LA 1

BL B1 B2

** Define the mast as an infinite UTD cylinder

UZ A B C 360 -1 -1

UT 1 2 0 0 31 1

** End of geometry

EG 1 0 0 0 0

** Excitation

FR 1 0 #freq

A1 0 1 1 0



** Radiation diagram

FF 1 1 181 0 90 0 0 2

FF 1 181 1 0 0 0 2 0

** End

EN




Current in A 3.5051E-02 9.5055E-03 3.6317E-02 15.17

Admitt. in A/V 3.5051E-02 9.5055E-03 3.6317E-02 15.17







90.00 0.00 0.000E+00 0.00 3.332E+00 -164.63 -999.9999 10.239742 10.239742

90.00 2.00 0.000E+00 0.00 3.320E+00 -165.29 -999.9999 10.206376 10.206376

90.00 4.00 0.000E+00 0.00 3.281E+00 -167.27 -999.9999 10.105962 10.105962

POLARISATION


0.0000 -90.00 LINEAR

0.0000 -90.00 LINEAR

0.0000 -90.00 LINEAR





0.00 0.00 0.000E+00 0.00 0.000E+00 0.00 -999.9999 -999.9999 -999.9999

2.00 0.00 0.000E+00 0.00 5.514E-01 61.40 -999.9999 -5.385792 -5.385792

4.00 0.00 0.000E+00 0.00 9.118E-01 111.32 -999.9999 -1.017019 -1.017019

POLARISATION


0.0000 0.00 UNDEF.

0.0000 -90.00 LINEAR

0.0000 -90.00 LINEAR

In horizontal and vertical radiation patterns of the Yagi-Uda antenna in front of a cylinderare shown in figures 3-63 and 3-64.



Figure 3-63: Radiation pattern in the horizontal plane (ϑ = 90)

Figure 3-64: Radiation pattern in the vertical plane (ϕ = 0)



3.28 Example 28: Resonant dipole antenna in front of a dielectriccylinder treated with physical optics (PO)

With this example we demonstrate the use of the PO for a dielectric body. A resonantdipole antenna (i.e. the input impedance of the antenna on its own in free space is purelyreal) is placed in front of a dielectric cylinder. The impedance is investigated as a functionof the distance to the cylinder.

Since the PO is applied to the dielectric cylinder, the equivalent currents are zero in theregion shadowed from the sources. For a metallic cylinder one could simply leave out theback of the cylinder and the top and bottom surfaces. However, in the case of a dielectriccylinder we need to define the complete cylinder in order to uniquely define the regionsof the different media (i.e. air and dielectric). The geometry is shown in figure 3-65.Since this is a closed body, we must ensure that all normals point outwards and thenselect the option “Full ray tracing, illumination only from outside” at the PO card. Thisavoids doing ray tracing to determine if the back triangles are illuminated and leads to asignificant saving in computation time.

Figure 3-65: Dipole antenna in front of a dielectric cylinder



For this example OPTFEKO could be used to vary the distance — see OPTFEKO in theUser’s Manual. The input file below is constructed for a fixed distance using the “defined”function such that it may be used as is with OPTFEKO. (Without the “defined” function,the values OPTFEKO writes to the top of the file will merely be overwritten.)

** Resonant dipole in front of a dielectric cylinder

** Treated with PO

** Variable that may be varied by OPTFEKO:

!!if not(defined(#d)) then

#d = 0.4 ** Distance from antenna to cylinder

!!endif

** Other variables


#lambda = #c0 / #freq ** Wavelength

#h = 0.25*#lambda * 0.9627087 ** Dipole length (chosen such that the free

** space input impedance is purely real)

#b = #lambda ** Radius of cylinder

#c = 2*#lambda ** Half the cylinder height

#epsr = 4

#muer = 1

#sigma = 0.05



#segr = #lambda / 1000



** Create one an eighth of a cylinder with label 5 (make sure

** that the normal vector points outwards, can be used at the

** PO card to accelerate ray tracing)

LA 5

ME 1 0

DP A 0 0 0

DP B 0 0 #c

DP C #b 0 0

ZY A B C 0 90 #tri_len

DP F 0 0 2*#c

DP E #b 0 #c

KR B F E 90 #tri_len

** Mirror the structure

SY 1 1 0 3

** Half of dipole antenna (in free space, with label 0)

DP AD #d+#b -0.4*#segl 0

DP BD #d+#b 0.4*#segl 0

DP CD #d+#b #h 0

ME 0

LA 0

BL BD CD



** Mirroring

SY 1 2

** Define the feed segment with label 1

LA 1

BL AD BD

** Use the PO approximation on dielectric cylinder (use NOSHADE=2

** as no triangles may be illuminated from behind/inside)

PO 5 2 1 0 0

** End of geometry

EG 1 0 1 0 0

** Excitation

FR 1 0 #freq

A1 0 1 1 0

** Set the dielectric parameters

DI 1 #epsr #muer #sigma

** Only impedance of antenna as output

OS 0

** End

EN

The calculated impedance is:



Current in A 1.4195E-02 -6.4119E-04 1.4209E-02 -2.59

Admitt. in A/V 1.4195E-02 -6.4119E-04 1.4209E-02 -2.59


When, as indicated above, OPTFEKO is used to vary the distance #d, one can obtain, forexample, the imaginary part of the input impedance as a function of distance as shownin figure 3-66. The symbols represent the method of moments result, which is bothcomputationally and storage wise very intensive. The smooth graph shows the result ofthe much more efficient PO method.



Figure 3-66: Variation of the imaginary part of the input impedance with distance



3.29 Example 29: Pin fed patch antenna on a finite dielectricsubstrate

Feed pin

Ground plane

Figure 3-67: Pin fed patch antenna on a finite dielectric substrate. The geometry hasbeen cut away to show the feed pin.

The structure in figure 3-67 is a rectangular patch (31.1807 mm × 46.7480 mm) on afinite dielectric substrate (50 mm × 80 mm) operating at 3 GHz. The patch is excitedwith a feed pin 8.9 mm from the centre of the long edge. The input file is as follows

** Pin-fed rectangular patch antenna on a finite dielectric substrate

** Scaling factor since all dimensions below in mm

SF 1 0.001

** Dimensions of the patch

#len_x = 31.1807

#len_y = 46.7480

** Dimensions of the substrate

#gnd_x = 50

#gnd_y = 80

** Feed location and wire diameter

#feed_x = 8.9

#diam = 1.3


#h = 2.87 ** Height


** Frequency (for the discretisation)

#freq = 3.0e9

#lam = 1000 * #c0 / #freq / sqrt(#epsr) ** Wavelength in mm


#tri_len = #lam / 12

#fine_tri = #lam / 16

#segl = #lam / 15

#segr = #diam/2




** Generate one half of the structure


#x1 = #len_x - #feed_x

#x2 = #len_x/2 - #feed_x - #gnd_x/2

#x3 = #len_x/2 - #feed_x + #gnd_x/2

DP A -#feed_x 0 0

DP B #x1 0 0

DP C #x1 #len_y/2 0

DP D 0 0 0

DP E -#feed_x #len_y/2 0

DP F #x3 0 0

DP G #x3 #gnd_y/2 0

DP H #x2 #gnd_y/2 0

DP I #x2 0 0

DP J #x3 0 -#h

DP K #x3 #gnd_y/2 -#h

DP L #x2 #gnd_y/2 -#h

DP M #x2 0 -#h

DP N 0 0 -#h

** Dielectric substrate

ME 1 0

BQ B F G C #fine_tri

BQ C G H E #fine_tri

BQ E H I A #fine_tri

BP F J K G

BP G K L H

BP H L M I

** Metallic patch

ME 1 0 1

BT D B C #fine_tri #fine_tri #fine_tri

BQ D C E A #fine_tri #fine_tri #fine_tri #fine_tri

** Metallic ground plane

BT N K J

BQ N M L K

** Symmetry to create the full structure

SY 1 0 3 0

** Feed wire (will be a single segment) with label 1

LA 1

BL N D


EG 1 0 0 0 0

** Dielectric properties

DI 1 #epsr 1

** Frequency

FR 1 0 #freq



** Voltage source at the wire centre with impressed power

PW 1 1

A1 0 1 1 0

** Far-field pattern

FF 1 73 1 1 0 0 5

FF 1 73 1 1 0 90 5

** Compute the radiated power in the far-field (only one half)

FF 3 37 37 0 0 0 5 5

** End

EN

Some extracts from the output file are



Current in A 2.5527E-01 -1.8584E-01 3.1575E-01 -36.06

Admitt. in A/V 3.2581E-02 -2.3720E-02 4.0301E-02 -36.06










0 unknown 0.00000E+00 0.00000E+00 0.00000E+00 unknown

1 -1.29485E+00 0.00000E+00 0.00000E+00 1.00000E+00 -2.94849E-01






-2.50 .. 182.50 deg. -2.50 .. 182.50 deg. 4.96085E-01 Watt

0.00 .. 180.00 deg. 0.00 .. 180.00 deg. 4.81045E-01 Watt

Note that the power flowing into the dielectric is negative as the source (which has beenscaled to 1 W) is located inside it. We would, however, have expected 1 W to be flowingout of the dielectric surface. The integral of the power in the far field is also a little lessthan the expected 0.5 W. This indicates that the power is not calculated very accuratelyand the mesh should be refined if this is a critical parameter. The radiation patternson the other hand are not that sensitive to the mesh density. Figure 3-68 compares the



pattern in the plane ϕ = 90 to the pattern (calculated in the next example) for an infiniteground plane and Green’s function dielectric.

Figure 3-68: Far field gain of the patch antenna on a finite substrate in the plane ϕ = 90.Note that the pattern obtained with the Green’s function has to be 0 in the region90 ≤ ϕ ≤ 270 as the ground plane is assumed to be infinite.



3.30 Example 30: Patch antenna on a dielectric substrate

Y

Z

X

Substrate height: 2.87mm

Rel dielectric constant: 2.2Metallic ground plane

Figure 3-69: Patch antenna on a dielectric substrate

The structure in figure 3-69 is a patch antenna on a dielectric substrate with a groundplane. The patch is 31.1807 mm by 46.7480 mm and is to be fed 8.9 mm inward form thecentre of the long side (at the origin of the coordinate system in the figure). The structureis excited with an A2 card which applies a voltage between the bottom of a vertical pinand the ground plane. The patch has been created similar to the one in Example 29such that there is a node at the pin position. The input impedance is calculated as afunction of frequency and the radiation patterns only at the centre frequency. The inputfile example_30a.pre is as follows

** A rectangular patch antenna on a dielectric substrate with

** a metallic ground plane (wire pin feed)


SF 1 0.001


#len_x = 31.1807

#len_y = 46.7480


#feed_x = 8.9

#diam = 1.3


#h = 2.87 ** Height





#freq = 3.0e9



IP #diam/2 #lam/15 #lam/15

** Generate one quarter of the structure


#x = #len_x - #feed_x

DP A -#feed_x 0 0

DP B #x 0 0

DP C #x #len_y/2 0

DP D 0 0 0


DP N 0 0 -#h

** Patch

BT D B C

BQ D C E A

** Symmetry to create the full structure

SY 1 0 3 0

** Feed wire (will be a single segment) with label 1

LA 1

BL N D

** End of geometry

EG 1 0 0 0 0

** Substrate (with groundplane)

GF 10 1 0 1 1

#h #epsr 1

** Voltage source at the wire centre with impressed power

A2 0 -1 1 0 0 0 -#h

** Frequency loop in order to compute the impedance

FR 17 0 2.8e9 3.2e9

** Change the line above as shown below to run with FEKO LITE

** FR 10 0 2.8e9 3.2e9

** Just compute the impedance, no output of surface currents

OS 0

** Far-field pattern at centre frequency

FR 1 0 3.0e9

FF 1 73 1 1 0 0 5

FF 1 73 1 1 0 90 5

** End

EN



Some extracts from the output file example_30a.out are

EXCITATION BY VOLTAGE SOURCE AT NODE






Source at segment w. label: ULA = -1

Absolute number of node: UNR = 1

Location of the excit. in m: X = 0.00000E+00

Y = 0.00000E+00

Z = -2.87000E-03

Positive feed direction: X = 0.00000E+00

Y = 0.00000E+00

Z = -1.00000E+00






1 2.87000E-03 0.00000E+00 -2.87000E-03 2.20000 1.00000 0.00000E+00



0.00000E+00 0.00000E+00

0.00000E+00 0.00000E+00

metallic ground plane at z= -0.00287 m

Allocated for the interpolation table (potentials): 0.055 MByte



Current in A 3.4713E-03 -1.5844E-02 1.6220E-02 -77.64

Admitt. in A/V 3.4713E-03 -1.5844E-02 1.6220E-02 -77.64




The structure could also be excited with a coaxial probe approximation. When the A4card is used, the source (a surface charge effectively representing a current source) isplaced at the centre of the nearest triangle. Thus we create the complete patch with asingle BP card — shown in figure 3-70 — such that there will be a triangle whose centreis reasonably close to the feed position. (Note the positions of the specified and actualfeed points in the figure.)



Y

Z

X

Substrate height: 2.87mm

Rel dielectric constant: 2.2

Specified excitation point

Actual probe position

Metallic ground plane

Figure 3-70: Patch antenna on a dielectric substrate

The input file, example_30b.pre, is as follows** A rectangular patch antenna on a dielectric substrate with

** a metallic ground plane (coaxial probe feed model)


SF 1 0.001


#len_x = 31.1807

#len_y = 46.7480


#feed_x = 8.9

#diam = 1.3


#h = 2.87 ** Height



#freq = 3.0e9



IP #lam/15

** Points for the patch

#x = #len_x - #feed_x

DP C #x #len_y/2 0

DP CY #x -#len_y/2 0


DP EY -#feed_x -#len_y/2 0



** The whole patch (we use no symmetry so that there might be a

** triangle with its centroid close to the desired feed location)

BP EY CY C E

** End of geometry

EG 1 0 0 0 0

** Substrate (with groundplane)

GF 10 1 0 1 1

#h #epsr 1

** Excitation by coaxial probe

A4 0 -1 1 1 0 0 0 0 #diam/2

** Frequency loop in order to compute the impedance

FR 17 0 2.8e9 3.2e9

** Change the line above as shown below to run with FEKO LITE

** FR 10 0 2.8e9 3.2e9

** Just compute the impedance, no output of surface currents

OS 0

** Far-field pattern at centre frequency

FR 1 0 3.0e9

FF 1 73 1 1 0 0 5

FF 1 73 1 1 0 90 5

** End

EN

The output file is very similar to the one for the A2 feed above, differing mainly in thedescription of the sources. Some extracts from the output file example_30b.out are

EXCITATION BY CURRENT SOURCE AT TRIANGLE

Number of current source: N = 1



Feed current in A: |I0| = 1.00000E+00

Phase in deg.: ARG(I0) = 0.00

Source at triangle w. label: ULA = -1

Absolute number of triangle: UNR = 34

Location of the excit. in m: X = -1.47602E-03

Y = 7.08303E-04

Z = 0.00000E+00

Radius feed pin in m: RAD = 6.50000E-04


...

Allocated for the interpolation table (potentials): 0.005 MByte

DATA OF THE CURRENT SOURCE NO. 1




Voltage in V 1.0550E+01 6.4159E+01 6.5021E+01 80.66

Admitt. in A/V 2.4954E-03 -1.5176E-02 1.5380E-02 -80.66



Reference plane for impedance: z = -0.00287 m


The S11 results for the two models are presented on a Smith chart in figure 3-71. Theradiation patterns of the two models — as shown in figure 3-68 — are virtually the same.The pin model requires more time for the calculation of the Green’s function interpolationtables, but for subsequent runs the solution time is about the same for the two models.Also, with the A4 approximation, the interpolation tables requires only 5 kByte of storageversus the 48 kByte required for the pin model and the associated z directed currents.This will, however, only be significant for very large problems.

The difference in the impedance is caused by the inaccuracy in the position of the A4probe as well as the approximations used in the probe model. The probe model decreasesin accuracy as the wire radius and/or the dielectric thickness increase. In most cases wherethe input impedance is of significance, the wire pin model will be worth the additionalcomputational requirements.

Figure 3-71: Input impedance of the patch antennas with different feed models as afunction of frequency



3.31 Example 31: Wire antenna penetrating a real ground

The structure for this example, shown in figure 3-72, is a dipole antenna, constructedfrom wire segments, partly buried in a real ground.

Lossless dielectric ground with = 16r

Free space

Wire radius0.25 mm

62.5m

m

132.4m

m

~

117.6

mm



** Wire antenna partly buried in the earth

**

** See also: K.A. Michalski and D. Zheng, "Electromagnetic Scattering and

** Radiation by Surfaces of Arbitrary Shape in Layered Media, Part II:

** Implementation and Results for Contiguous Half-Spaces",

** IEEE Trans. on Antennas and Propagation, vol. 38, pp. 345-352,

** Mar. 1990

** There is a comparison with NEC-4 data

** Some parameters

#alpha = 45 ** Tilt angle of the wire


#epsr = 16 ** Earth parameters

** Positions along the strip

#minus_l = -0.0625 ** Length in earth

#plus_l = 0.25 ** Length in air

#feed_l = 0.1176 ** Position of the voltage source


#lambda = #c0 / #freq / sqrt(#epsr)


#segrad = 0.00025

IP #segrad #seglen



** Points for the wire antenna

#x = #minus_l * sin(rad(#alpha))

#z = #minus_l * cos(rad(#alpha))

DP A #x 0 #z

DP B 0 0 0

#x_feed = #feed_l * sin(rad(#alpha))

#z_feed = #feed_l * cos(rad(#alpha))

DP C #x_feed 0 #z_feed

#x = #plus_l * sin(rad(#alpha))

#z = #plus_l * cos(rad(#alpha))

DP D #x 0 #z

** Create the wire

BL A B

BL B C

BL C D

** End of geometry

EG 1 0 0 0 0

** Excitation

FR 1 0 #freq

A2 0 -1 1 0 #x_feed 0 #z_feed

** Earth (dielectric half space)

GF 11 1 0 1 1

#epsr 1

** Print the currents along the wire

OS 1 1

** End

EN

The current distribution along the wire as well as the input impedance is calculated.Some extracts from the output file follows

EXCITATION BY VOLTAGE SOURCE AT NODE






Source at segment w. label: ULA = -1

Absolute number of node: UNR = 17

Location of the excit. in m: X = 8.31558E-02

Y = 0.00000E+00

Z = 8.31558E-02

Positive feed direction: X = 7.07107E-01

Y = 0.00000E+00

Z = 7.07107E-01











1 infinity 0.00000E+00 -infinity 16.00000 1.00000 0.00000E+00



0.00000E+00 0.00000E+00

0.00000E+00 0.00000E+00



Current in A 9.1472E-03 -2.7838E-03 9.5614E-03 -16.93

Admitt. in A/V 9.1472E-03 -2.7838E-03 9.5614E-03 -16.93






1 -4.05113E-02 0.00000E+00 -4.05113E-02 1.504E-03 -27.47 0.000E+00 0.00

2 -3.31456E-02 0.00000E+00 -3.31456E-02 4.199E-03 -26.96 0.000E+00 0.00

3 -2.57799E-02 0.00000E+00 -2.57799E-02 6.337E-03 -26.19 0.000E+00 0.00

4 -1.84142E-02 0.00000E+00 -1.84142E-02 7.950E-03 -25.30 0.000E+00 0.00

IZ

magn. phase

1.504E-03 -27.47

4.199E-03 -26.96

6.337E-03 -26.19

7.950E-03 -25.30



Figure 3-73 compares the current calculated with FEKO with published NEC-4 results.

Real part, NEC resultImag part, NEC resultReal part, FEKO resultImag part, FEKO result

Figure 3-73: Current distribution along the partly buried dipole antenna



3.32 Example 32: RCS of a thin dielectric sheet

z

y

x

a = 2 m

b = 1 m

i

i

iEi

Si

Hi

n

Figure 3-74: Geometry of Example 32 with the incident plane wave

The geometry for this example is shown in figure 3-74 — a thin dielectric plate. Thesize, thickness and material parameters can be determined from the input file below. Theplate is illuminated by an incident plane wave such that the bistatic radar cross sectionmay be calculated.

As indicated in the section “Dielectric solids” in the “General comments” chapter of theUser’s manual, there are a number of ways with which such a thin dielectric plate may betreated in FEKO. In principle we may use the volume equivalence principle, discretisingthe dielectric into small cuboids (as was done for the cube in Example 9). However, ituses substantially less memory to realise the sheet with the SK card. The input file is asfollows

** RCS (radar cross section) computation of a thin dielectric plate

** Customisable parameters

#a = 2 ** Length of the plate

#b = 1 ** Width of the plate

#d = 0.004 ** Thickness of the plate


#tand = 0.03 ** Loss tangent


#thetai = 20 ** Angle of incidence

#phii = 50 ** - " -

#etai = 60 ** Polarisation angle


#lambda0 = #c0 / #freq

#lambda = #lambda0 / sqrt(#epsr)


IP #tri_len



** quarter plate

DP A 0 0 0

DP B #a/2 0 0

DP C #a/2 #b/2 0

DP D 0 #b/2 0

BP A B C D

** Symmetry (Geometrical only due to arbitrary incidence direction)

SY 1 1 1 0

** End of geometry

EG 1 0 0 0 0

** Excitation

FR 1 1 #freq

A0 0 1 1 1 0 #thetai #phii #etai

** Define the thin dielectric sheet

SK 0 4 #d 1 0 #tand #epsr

** Bistatic RCS (vertical cut)

FF 1 181 1 0 0 0 2 0

** End

EN

The geometry is discretised into triangular elements, similar to conducting plates. Thethin dielectric sheet formulation is then applied to all triangles with the given label. Weare interested in the calculated RCS. Some extracts from the output file follows








BETA0Y = -5.49117E-01

BETA0Z = -1.96945E+00

Field strength in V/m: |E0X| = 9.65425E-01 ARG(E0X) = 180.00

(Phase in deg.) |E0Y| = 1.96747E-01 ARG(E0Y) = 0.00

|E0Z| = 1.71010E-01 ARG(E0Z) = 0.00

DATA OF LABELS



Eps_r = 7.000E+00 Sigma = 1.168E-03 S/m tan(delta) = 3.000E-02

All segments and triangles without a listed label are perfectly conducting






0| 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 | 4.3373E-06

total| 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 | 4.3373E-06








0.00 0.00 1.489E-02 177.12 2.862E-03 -2.65 2.889E-03

2.00 0.00 1.472E-02 177.11 2.828E-03 -2.65 2.822E-03

4.00 0.00 1.450E-02 177.10 2.789E-03 -2.65 2.742E-03

POLARISATION


0.0008 169.12 RIGHT

0.0008 169.12 RIGHT

0.0008 169.12 RIGHT

Figure 3-75 shows the bistatic RCS as a function of the angle ϑ in the plane ϕ = 0.

Figure 3-75: Bistatic RCS of a thin dielectric sheet



3.33 Example 33: Shielding effectiveness of a thin hollow sphere

Figure 3-76: The meshed geometry of the thin-walled hollow sphere used in Example 33

This example determines the shielding effectiveness of a hollow sphere. The sphere radiusis 1 m and the silver walls have a thickness of only 2.5 nm. The geometry is shown infigure 3-76.

In the input file listed below, the hollow sphere is excited by an incident plane wave andthe electric and magnetic fields are calculated at the centre thereof.

** Computation of the electric and magnetic shielding factor of

** a thin hollow silver sphere.

** Customisable parameters

#r0 = 1 ** Sphere radius

#fmax = 50.0e6 ** Maximum frequency (for the segmentation)

#d = 2.5e-9 ** Thickness of the shell

#sigma = 6.1e7 ** Conductivity (silver)


#lambda = #c0 / #fmax

#tri_len = min (#lambda/8, #r0/4)

IP #tri_len

** Quarter sphere

DP A 0 0 0

DP B #r0 0 0

DP C 0 0 #r0

KU A B C 0 0 0 90 90 #tri_len

** Mirroring using symmetry

SY 1 1 3 2




EG 1 0 0 0 0

** Define the losses

SK 0 3 #d 1 #sigma

** excitation by a plane incident wave

A0 0 1 1 1 0 90 180 0

** Here we consider just one single frequency (can be extended to a loop)

FR 1 0 #fmax

** Electric and magnetic near-field at the sphere centre

FE 3 1 1 1 0 0 0 0

** End

EN

Some extracts from the output file follows







Direction of propag.: BETA0X = 1.04792E+00

BETA0Y = 0.00000E+00

BETA0Z = 0.00000E+00



|E0Z| = 1.00000E+00 ARG(E0Z) = 0.00

DATA OF LABELS



Sigma = 6.100E+07 S/m Mue_r = 1.000E+00 tan(delta_mu) = 0.000E+00

Penetration depth of the skin effect: 9.11319E-06 m




0| 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 | 6.9674E-04

total| 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 | 6.9674E-04







in free space

LOCATION EX EY EZ


0.0E+00 0.0E+00 0.0E+00 0.000E+00 0.00 0.000E+00 0.00 3.2075E-02 53.42


in free space

LOCATION HX HY HZ


0.0E+00 0.0E+00 0.0E+00 0.000E+00 0.00 1.0249E-04 105.82 0.000E+00 0.00

From the two field strength values E = 3.206 · 10−2 Vm and H = 1.025 · 10−4 A

m , and theincident field strength Ei = 1 V

m and Hi = 2.6544 · 10−3 Am (derived from Ei and the free

space wave impedance), we can determine the shielding factor

ae = −20 logE

Ei= 29.881 dB

am = −20 logH

Hi= 28.265 dB

These values may be compared to those in the book by Kaden: ae = 29.698 dB for electricshielding factor and am = 28.310 dB for magnetic screen absorption. These results are inexcellent agreement with the FEKO results.



3.34 Example 34: Coaxial cable (modelled with surface triangles)

Figure 3-77: Coaxial cable showing the 8 segments used for the loading

This example discusses modelling a coaxial cable in FEKO. For typical coaxial cables (upto a few hundred Ohm characteristic impedance), the inner conductor is thick relativeto the rest of the geometry. Thus it cannot be modelled with a single wire and bothconductors are constructed with meshed cylinders.

To get an accurate calculation of the input impedance, it is important that the excitationshould give the best possible approximation of a proper TEM mode. Therefore we use 8segments radially connecting the inner conductor to the outer conductor as can be seenin figure 3-77. The example also demonstrates how to use a shorted stub to enclose thefeed segments to prevent them from causing external radiation.

The input file (example_34.pre) is as follows.** Model of a coaxial cable using cylindrical surface elements.

** This example illustrates how to

** - excite a proper TEM mode (by using more feed elements)

** - match the open wire end (adding a lambda/4 long shorted stub)

** The coaxial cable is loaded at the end:

** load resistor: 40 Ohm

** wave impedance: 70 Ohm

** length of the cable: 80 mm

** wavelength: 120 mm

**

** The theoretical input impedance (assuming ideal transmission lines) is as follows:

** reflection coefficient at the load

** r_2 = (40 Ohm - 70 Ohm) / (40 Ohm + 70 Ohm)

** = -0.2727

**



** reflection coefficient at the source

** r_1 = r_2 * exp(-j * 2*pi * 2 * 80 mm / 120 mm)

** = 0.1364 + j*0.23621

**

** input impedance

** Z_in = 70 Ohm * (1+r_1) / (1-r_1)

** = (80.8247 + j*41.2478) Ohm

**

** We will use 8 voltage sources to excite a proper TEM mode in the cable

** (see note at the A1 cards below). Thus the "per source" input impedance is

** Z_source = 8 * (80.8247 + j*41.2478)

** = (646.6 + j*330.0)

** User defined variables

#scaling = 1.0e-3 ** unit for the geometrical dimensions (here in mm)

#r_inner = 2 ** inner radius in mm

#Zc = 70 ** wave impedance of the coaxial cable in Ohm

#Rload = 40 ** load resistor at the end

#len = 80 ** length of the cable in mm

#lambda = 120 ** wavelength in mm

** Compute outer radius of the cable

#r_outer = #r_inner * exp(#Zc/60)

** We have to adjust the maximum number of connections

** (otherwise FEKO will print an error message that we have to do so)

#maxnv = 2*16

** Select the maximum edge length of the triangles for the curved inner and

** outer sections so that the geometry of the arc can be represented accurately

#curved_len_i = 2*#pi*#r_inner / 8

#curved_len_o = 2*#pi*#r_outer / 8

** Segmentation parameters and wire radius

#edge_len = min (#lambda/10, 4*#curved_len_i)

#seg_len = #edge_len

#seg_rad = #seg_len / 50

IP #seg_rad #edge_len #seg_len

** Some point definitions for the actual cable

DP A 0 0 0

DP B #len 0 0

DP Ci 0 #r_inner 0

DP Co 0 #r_outer 0

** If we just want to use the coaxial cable to excite, for example a horn

** antenna, we need not be concerned about radiation to the open side. But

** if we want to prevent this, or want to get more accurate input impedance

** values, a shorted lambda/4 long stub can be added to the feed side. This

** transforms to an open at the feed position. As stated above, the stub

** can generally be omitted, but we include it to demonstrate the principle.

#stub_len = #lambda / 4

DP Bs -#stub_len 0 0

DP Csi -#stub_len #r_inner 0

DP Cso -#stub_len #r_outer 0



** We create a small angular section of the cylinders to ensure that we have

** nodes at the future connection points. We also create the short in this

** section to ensure connection.

#angle = 360 / 16

#maxlen_i = min (#edge_len, #curved_len_i)

#maxlen_o = min (#edge_len, #curved_len_o)

ZY A B Ci 0 #angle #maxlen_i ** Main inner conductor

ZY Bs A Csi 0 #angle #maxlen_i ** Stub inner conductor

ZY A B Co 0 #angle #maxlen_o ** Main outer conductor

ZY Bs A Cso 0 #angle #maxlen_o ** Stub outer conductor

KR Bs A Cso Csi #angle #maxlen_o #maxlen_i ** Shorting plate

** Make copies while rotating around the x-axis to create a quarter structure

TG 3 0 0 0 1 #angle

** Add wires to be used later as feed and load. If some feed and load wires

** lie in the principle planes, we cannot make optimal use of symmetry. Thus

** we rotate the wires out of this plane.

LA 1

BL Ci Co

TG 1 1 2 #angle

TG 1 1 1 1 2 2*#angle

TG 1 1 2 9 2 #len

** Adjust the labels of all load segments so that they are all 10

CB 11 10

** Create the full model using symmetry (again adjust the load labels)

SY 1 0 3 0 2

CB 12 10

SY 1 0 0 3 4

CB 14 10

** Scale the whole geometry (units are in mm)

SF 1 #scaling

** End of geometry definition

EG 1 0 0 0 0


#freq = #c0 / (#lambda*#scaling)

FR 1 0 #freq

** Excite the eight wires in phase to get a proper TEM mode. One could also

** use a single wire and feed unsymmetrically. This, however, will cause the

** excitation of higher order modes which will corrupt the input impedance

** (even though these modes decay as the wave propagates away from the feed).

A1 0 1 1 0

A1 1 2 1 0

A1 1 3 1 0

A1 1 4 1 0

A1 1 5 1 0

A1 1 6 1 0

A1 1 7 1 0



A1 1 8 1 0

** Load the end of the coaxial cable (note that the eight wires are in

** parallel which gives an additional factor 8 for the resistance of each

** wire). The length of the wires is #scaling*(#r_outer-#r_inner) and the

** required distributed resistance is:

#Rdist = 8 * #Rload / (#r_outer - #r_inner) / #scaling

LD 10 #Rdist

** Impedance computation and output of surface currents for the WinFEKO

** display (e.g. graphical check for the TEM mode)

OS 1 1

** Compute the near-field in a cross section across the cable close to the

** feed, can be used in order to check the proper TEM mode

#x = min(#len/5, #r_outer)

#n = 40

#start = -1.2*#r_outer

#width = 2.4*#r_outer

#delta = #width / (#n-1)

FE 3 1 #n #n 0 #x #start #start #delta #delta

** Compute the near-fields inside of the coaxial in a plane parallel to the

** axis, this allows to display e.g. the standing wave pattern

#nx = 60

#nz = 20

#deltax = (#len + #stub_len) / #nx

#deltaz = (#r_outer - #r_inner) / #nz

#startx = -#stub_len + #deltax/2

#startz = #r_inner + #deltaz/2

FE 3 #nx 1 #nz 0 #startx 0 #startz #deltax #deltaz

** End

EN

Note that the TG as SY cards use the option to increment the label. This ensures that thefeed segments all have unique labels (labels 1 to 8). We have to use separate excitation(A1 cards) for each of these segments. Also, the CB card is used a number of times toensure that all the load segments have the same label (label 10 in this case).

The output file contains the following output for the sources (each of the other six sourcesare symmetrical to one of these two)



Current in A 1.1173E-03 -6.2787E-04 1.2816E-03 -29.33

Admitt. in A/V 1.1173E-03 -6.2787E-04 1.2816E-03 -29.33








Current in A 1.1209E-03 -6.2993E-04 1.2858E-03 -29.34

Admitt. in A/V 1.1209E-03 -6.2993E-04 1.2858E-03 -29.34




The calculated impedances (680.22 + j 382.2)Ω and (678.0 + j 381.0)Ω are, as expected,nearly identical and are quite close to the theoretically predicted (646.6 + j 330.0)Ω —see the derivation at the start of the *.pre file). Note that the model includes radiationfrom the open load as well as the fact that there will be some higher order modes.

A simple test with 4 radial wires instead of 8 (the segment loading is adjusted to retaina 40Ω load) results in an input impedance of (399.2 + j 178.6)Ω which is significantlydifferent from the expected (323.3 + j 165.0)Ω — it should be half of value for the 8 wirecase). In this case there is possibly also a larger radiation from the open end.

The electric near field on an orthogonal cross section near the feed segments is shown infigure 3-78. This shows that the field distribution is close to that of the TEM mode, butnot exactly so. Note that the near field is calculated on a rectangular grid which explainsthe stepped behaviour of the contours near the conductors.

Figure 3-78: Electric near field orthogonal to the coaxial cable near the feed



3.35 Example 35: Horn antenna in front of a reflector replacedwith equivalent surface currents or far field pattern

Figure 3-79: Horn in front of a parabolic reflector

This example considers a horn antenna in front of a parabolic reflector. The reflector istreated with the PO and the horn with the MoM. Even if the MoM is decoupled fromthe PO, the calculation of the interaction between the MoM basis functions and the POtriangles can be quite time consuming. More so if the solution has to be repeated anumber of times, for example to optimise the shape of the reflector.

The antenna can be removed by using the equivalence theorem, in particular the aperturescreated with the AP card. This requires two model files. The first (example_35a.pre)calculates the near fields on six planes surrounding the antenna and writes it to *.efeand *.hfe files for later use. (It also calculates the far field pattern, but that will bediscussed later.)

** Example_35 considers a horn antenna in front of a parabolic reflector

** The example is split into four parts

** -This first part calculates the near fields radiated by the horn in

** the absence of the reflector. The fields are written to *.efe and

** *.hfe files to use for the aperture excitation. It also calculates

** the far fields and write this to a *.ffe file for use as a point

** source with a specified pattern.

** -The second part uses these near fields to define an equivalent

** aperture to replace the horn. The aperture is then used to excite

** the parabolic reflector.

** -The third part models the horn and reflector together for verification.

** -The forth part use the far fields as a point source.



** We include the constants as well as the scaling from a file

** such that the dimensions need only be modified in one place

IN 0 "example_35.inc"

** Define the corner points for a quarter horn in the quadrant y>0 and z>0

** Points on the waveguide back wall

DP C #xback #wg_w/2 #wg_h/2

DP CZ #xback 0 #wg_h/2

DP CY #xback #wg_w/2 0

DP C0 #xback 0 0

** Points on the transition from wave guide to horn

DP B -#horn_l #wg_w/2 #wg_h/2

DP BZ -#horn_l 0 #wg_h/2

DP BY -#horn_l #wg_w/2 0

** Points on the horn opening

DP A 0 #horn_w/2 #horn_h/2

DP AZ 0 0 #horn_h/2

DP AY 0 #horn_w/2 0

** Points along the feed wire

DP DU #xfeed 0 -#seg_l/2

DP DO #xfeed 0 #seg_l/2

DP DZ #xfeed 0 #wg_h/2

** Create the surfaces in the quadrant y>0 and z>0

** Wave guide end

BP C CZ C0 CY

** Wave guide top

BT C CZ DZ

BQ C DZ BZ B

** Wave guide walls

BP C B BY CY

** Horn walls

BQ B A AY BY

** Horn top

BQ B A AZ BZ


SY 1 0 3 0

** Create half of the feed wire

BL DO DZ


SY 1 0 0 2


LA 1

BL DU DO


EG 1 0 0 0 0



** The excitation

FR 1 0 #freq

A1 0 1 1 0

** We will write the near and far fields to file

DA 1 1 1 0 0

** Now calculate the near fields on a closed aperture around the horn

** Planes of constant x

FE 3 1 #Ny #Nz 0 #xneg #ystart #zstart #xskip #yskip #zskip

FE 3 1 #Ny #Nz 0 #xpos #ystart #zstart #xskip #yskip #zskip

** Planes of constant y

FE 3 #Nx 1 #Nz 0 #xstart -#ypos #zstart #xskip #yskip #zskip

FE 3 #Nx 1 #Nz 0 #xstart #ypos #zstart #xskip #yskip #zskip

** Planes of constant z

FE 3 #Nx #Ny 1 0 #xstart #ystart -#zpos #xskip #yskip #zskip

FE 3 #Nx #Ny 1 0 #xstart #ystart #zpos #xskip #yskip #zskip

** To determine the phase centre, we calculate the near field in front of the horn

** (In the far field region, 20m to 30m, or about about 100 to 150 wavelengths)

** FE 1 101 1 1 0 20/#sf 0 0 0.1/#sf 0 0

** Finally we calculate the far field pattern referenced to the phase centre

** (See the example guide on how to determine the phase centre)

OF 1 0 -0.215/#sf 0 0

FF 1 37 73 0 0 0 5 5

** End

EN

The second (example_35b.pre) uses the AP card with the field data in the *.efe and*.hfe files to replace the horn with an equivalent aperture:



** -The first part calculates the near fields radiated by the horn in





** -This second part uses these near fields to define an equivalent




** -The forth part use the far fields as a point source.







** First set the reflector meshing (keep the segment settings)

IP #seg_rad #ref_tri #seg_l

** Use label 2 in order to specify the reflector for PO

LA 2

** Define the points -- the reflector face in the negative x-direction

DP R1 #focal 0 0

DP R2 #focal/2 0 0

DP R3 #focal 0 #ref_rad

DP R4 #ref_rim 0 #ref_rad

PB R1 R2 R3 R4 90 #ref_tri

** Mirror the quarter in the plane y=0 (xz-plane, ideal magnetic wall)

** and the plane z=0 (xy-plane, ideal electric wall) at the same time.

SY 1 0 3 2

** Apply PO to the reflector

PO 2 1 0 0 0

** Define the corner points for the six apertures

DP A1 #xneg -#ypos -#zpos

DP A2 #xpos -#ypos -#zpos

DP A3 #xneg #ypos -#zpos

DP A4 #xpos #ypos -#zpos

DP A5 #xneg -#ypos #zpos

DP A6 #xpos -#ypos #zpos

DP A7 #xneg #ypos #zpos


EG 1 0 0 0 0

** The excitation

FR 1 0 #freq

** Define the apertures which replace the horn

** - Note that the two aperture axes must be in the order x, y, z (any two of these,

** depending on the aperture orientation); and pointing in the direction of the

** positive axis. (This is the order of the data samples in the near field files.)

** On some surfaces this leads to normals pointing inward whereas the surface

** equivalence formulation requires normals pointing outward. This is rectified

** by adding a 180 degree phase to these apertures. (Effectively reversing the

** normal vector.)

** - Note also that all data points are in the same file. To keep track of where

** to start the data of a specific aperture, the variable #start is incremented

** with the size of each aperture to indicate the start of the next one’s data.

** Plane x = #xneg (inward pointing normal)

#start = 1

AP 0 -5 A1 A3 A5 #start #Ny #Nz 1 180 ...

"example_35a.efe" "example_35a.hfe"



** Plane x = #xpos

#start = #start + #Ny*#Nz

AP 1 -5 A2 A4 A6 #start #Ny #Nz 1 0 ...


** Plane y = -#ypos

#start = #start + #Ny*#Nz

AP 1 -5 A1 A2 A5 #start #Nx #Nz 1 0 ...


** Plane y = #ypos (inward pointing normal)

#start = #start + #Nx*#Nz

AP 1 -5 A3 A4 A7 #start #Nx #Nz 1 180 ...


** Plane z = -#zpos (inward pointing normal)

#start = #start + #Nx*#Nz

AP 1 -5 A1 A2 A3 #start #Nx #Ny 1 180 ...


** Plane z = #zpos

#start = #start + #Nx*#Ny

AP 1 -5 A5 A6 A7 #start #Nx #Ny 1 0 ...


** Calculate the far-fields in the principle planes

** Vertical cut

FF 1 181 1 0 0 180 1

** Horizontal cut

FF 1 1 361 0 90 0 1

** End

EN

Both of these read the crucial dimensions from the file example_35.inc such that themodel parameters need only be changed in one location:

** Common definitions for example_35

** Scaling. All coordinates are entered in mm and scaled with #sf to metre

** The field positions are also scaled.

#sf = 0.001 ** Multiple dimensions with this factor to get metre

** Implement scaling

SF 1 #sf

** Frequency, wavelength and segmentation

#freq = 1.645E9 ** Frequency in Hertz

#lam = (#c0/#freq)/#sf ** Wavelength in mm (coordinates before scaling)

#seg_l = #lam/15 ** Maximum wire segment length

#tri_l = #lam/7 ** Maximum triangle edge length

#seg_rad = 1 ** Segment radius



** Implement segmentation

IP #seg_rad #tri_l #seg_l

** Waveguide and horn parameters -- dimensions in mm

#horn_w = 550 ** Width at the horn opening (parallel to y-axis)

#horn_h = 428 ** Height at the horn opening (parallel to z-axis)

#horn_l = 460 ** Length (along x-axis) of horn flare

#wg_w = 129.6 ** Waveguide width

#wg_h = 64.8 ** Waveguide height

#wg_l = 302 ** Length of the section of straight waveguide.

#feedsep = 46 ** Distance between back wall and feed pin

** Derived parameters

#xback = -#wg_l - #horn_l ** x-coordinate at the back wall

#xfeed = #xback + #feedsep ** x-coordinate at the feed point

** Aperture parameters

#xpos = 30 ** x-coordinate of the plane at constant positive x

#xneg = -800 ** x-coordinate of the plane at constant negative x

#ypos = 300 ** y-coordinate of the plane at constant positive y

#zpos = 280 ** z-coordinate of the plane at constant positive z

#sample = #lam/2.5 ** Target sample density on aperture


#Nx = CEIL((#xpos-#xneg)/#sample)

#xskip = (#xpos-#xneg)/#Nx

#xstart = #xneg + #xskip/2

#Ny = CEIL(2*#ypos/#sample)

#yskip = 2*#ypos/#Ny

#ystart = -#ypos + #yskip/2

#Nz = CEIL(2*#zpos/#sample)

#zskip = 2*#zpos/#Nz

#zstart = -#zpos + #zskip/2

** Reflector parameters

#ref_rad = 6*#lam ** Radius of the parabolic reflector

#ref_h = #lam ** Height of the parabolic reflector

#ref_tri = #lam/4 ** Allow coarser meshing on the reflector


#focal = (#ref_rad^2) / (4*#ref_h) ** Focal distance

#ref_rim = #focal - #ref_h ** x-coordinate of the front rim of the reflector

A third model (example_35c.pre) is constructed for comparison. This model containsboth the MoM horn and the PO reflector. The coupling between the MoM and PO isnot taken into account during the solution. Using the aperture replaces the 4072 basisfunctions of the horn with 2128 point dipoles. This considerably reduces the time requiredto calculate the PO currents on the reflector (from 72.5 to 36.5 seconds on a 1.2 GHzAMD). For more complex feed models the effect will be even larger. Figures 3-80 and 3-81compare the far field patterns calculated with the aperture replacement to those calculatedwith the MoM PO hybrid method.



One may also calculate the near fields on a spherical surface around the horn (using theOF card to specify a local origin) and use a single spherical aperture. This may requiremore dipoles (612 more dipoles if one requires the same maximum spacing and the samenearest point to the horn — due to the fact that the separation will decrease towards thepoles of the sphere), but is much simpler to set up.It is also possible to replace the horn with its far field pattern. The pattern is alsocalculated in example_35a.pre and then used with an AR card in example_35d.pre:



** -The first part calculates the near fields radiated by the horn in





** -The second part uses these near fields to define an equivalent




** -This forth part use the far fields as a point source.





** First set the reflector meshing (keep the segment settings)

IP #seg_rad #ref_tri #seg_l

** Use label 2 in order to specify the reflector for PO

LA 2

** Define the points -- the reflector face in the negative x-direction

DP R1 #focal 0 0

DP R2 #focal/2 0 0

DP R3 #focal 0 #ref_rad

DP R4 #ref_rim 0 #ref_rad

PB R1 R2 R3 R4 90 #ref_tri

** Mirror the quarter in the plane y=0 (xz-plane, ideal magnetic wall)

** and the plane z=0 (xy-plane, ideal electric wall) at the same time.

SY 1 0 3 2

** Apply PO to the reflector

PO 2 1 0 0 0


EG 1 0 0 0 0

** The excitation

FR 1 0 #freq



** Use the calculated radiation pattern excitation

** Note that the point source must be placed at the phase origin used to

** calculate the far field. In this case this is 176mm inside the horn mouth.

AR 0 1 1 37 73 1 0 -0.176/#sf 0 0 ...

0 0 0 "example_35a.ffe"

** Calculate the far-fields in the principle planes

** Vertical cut

FF 1 181 1 0 0 180 1

** Horizontal cut

FF 1 1 361 0 90 0 1

** End

EN

Note that when using the AR card, an antenna is replaced by a point source with thespecified pattern. It is very important that this point source is located at the phase centreof the antenna it represents and that the far field pattern is calculated relative to the samephase centre.

One may determine the phase centre of an antenna by using the fact that the electric farfield of the antenna should decay at 1/r and by assuming that the origin of this decaywill be the phase centre of the antenna. Thus if one inverts the near field, extending thelinear limit of this line to its intersection with the distance axis should give the phasecentre of the antenna in the coordinate system where the near field was calculated. Forthe horn antenna, symmetry dictates that the phase centre must be on the x axis.

To determine the phase centre of the horn, we calculate the electric near field from 20 mto 30 m on the x axis in front of the antenna. (This is about 100 to 150 wavelengths— if we do not obtain a consistent result, we must increase the distance — see below.)Next we plot this as a function of distance (x) in POSTFEKO and obtain the inverse byselecting the Perform calculations from series button. The expression to use is #series1 -1 to obtain a result that should be proportional to x barring some offset. Switch tothe series 2 tab. From the table on this panel, it can be seen that the inverse at 20m is15.8947 and at 30m it is 23.7571. The slope of this line is determined from the differencebetween these numbers divided by the extent (here 7.8624/10 = 0.78624). The next step isto determine the inverted near field value at x = 0. From elementary linear mathematics,it is known that the axis crossing (c) can be determined from c = y − mx where m isthe gradient of the line. This is used to calculate the crossing point at the start of theline, and the end. These two axis crossing values are 0.169262 and 0.198943 respectively.These two numbers should be relatively close together. If this difference is too great, theresult is not consistent and we must increase the distance from the antenna. This is theleft axis value where x is 0 and dividing this with the slope results in the distance fromthe origin to the phase centre. Here it is 0.215 m.

The result is almost the same as for the more exact techniques (see figure 3-81 — thereflector is well inside the 2D2/λ far field criterion for the horn, such that one should



expect approximate results when using the far field source approximation), but the runtime decreased from 36.5 seconds to less than 0.1 seconds.

MoM/PO

AP/PO

Theta (degrees)1801501209060300

Dire

ctivity

(dB

)

30

25

20

15

10

5

0

-5

-10

-15

-20

Figure 3-80: Far field pattern in the vertical plane

MoM/PO

AP/PO

Phi (degrees)3603303002702402101801501209060300

Dire

ctivity

(dB

)

30

25

20

15

10

5

0

-5

-10

-15

-20

-25

-30

Figure 3-81: Far field pattern in the horizontal plane



MoM/PO

AP/PO

AR/PO

Theta (degrees)1051009590858075

Directivity

(dB

)

28

26

24

22

20

18

16

14

12

10

8

6

4

2

Figure 3-82: Far field pattern around the main beam in the vertical plane



3.36 Example 36: Example of S-parameter calculation above aground plane

0

1

2

3

4

Figure 3-83: Strip dipole and grounded wire above an infinite ground plane

This example considers a strip dipole next to a grounded wire above a ground plane asshown in figure 3-83. The centre feed of the dipole and the two feed segments connectingthe wire to the ground plane are considered to be a three port system. The S-parametersare calculated with the SP card.

The input file (example_36.pre) is as follows.** Example file using the SP card to calculate S-parameters

** A strip dipole next to a wire attached to a ground plane

** Parameters

#lambda = 1

#hgt_w = 0.05 * #lambda ** Height of the wire above ground

#len_w = 0.5*#lambda ** Length of half the wire

#hgt_s = 0.05*#lambda ** Height of the strip dipole above ground

#len_s = 0.75*#lambda ** Length of half the strip dipole

#wid_s1 = 0.03*#lambda ** Distance from wire to near end of strip

#wid_s2 = 0.08*#lambda ** Distance from wire to near end of strip

** Segmentation

#edgelen = #lambda/15

#seglen = #lambda/20

#segrho = #seglen/100

IP #segrho #edgelen #seglen

** Define the points in space

DP A 0 #len_w 0

DP B 0 #len_w #hgt_w

DP C 0 0 #hgt_w

DP B1 -#wid_s1 0 #hgt_s

DP B2 -#wid_s1 #len_s #hgt_s

DP B3 -#wid_s2 #len_s #hgt_s

DP B4 -#wid_s2 0 #hgt_s



** Create the geometry

** Horizontal wire

BL C B

** Strip dipole arm

LA 1

BP B1 B2 B3 B4

** Use symmetry in the y=0 plane

SY 1 0 1 0 1

** We must add the port segment AFTER symmetry, since below different

** load impedances are used, and thus geometrical symmetry cannot be

** used (FEKO gives warning 536)

LA 3

BL A B

** Create the symmetrical element

TG 1 3 3 1 2 -2*#len_w

** End of geometry

EG 1 0 0 0 0

** Specify an infinite ground plane

BO 2

** Excitations of all 3 ports

** Only specify a port impedance for the second one, the rest will use the SP card value

AE 0 1 2 1 0

A1 1 4 1 0 100

A3 1 3 1 0 #segrho 2*#segrho

** Export the S-parameters to a Touchstone *.SnP file

DA 0 0 0 0 0 1

** Frequency loop


FR 10 0 0.9*#freq 1.1*#freq

** Initiate the S-parameter computation

SP 50

** End

EN

Note that the port wires are created after geometrical symmetry has been defined. Sym-metry — even geometrical symmetry — implies that the loading will also be symmetrical.In this example, different port impedances are specified and thus these segments are loadedunsymmetrically.

Some extracts of the S-parameters as listed in the output file are given on the next page.



SCATTERING PARAMETERS

ports magnitude phase

sink source real part imag. part linear in dB in deg.

S 1 1 4.40767E-01 -8.05506E-01 9.18213E-01 -0.7411 -61.31

S 2 1 2.27276E-01 -5.91538E-02 2.34848E-01 -12.5843 -14.59

S 3 1 -1.59453E-01 4.80535E-02 1.66536E-01 -15.5698 163.23

Sum |S|^2 of these S-parameters: 9.26003E-01 -0.3339

The S-parameters are also listed in a Touchstone format file (example 36.s3p) requestedby the DA card. They are displayed in figure 3-84 — note that S21 ≈ S12 and S23 ≈ S32

which is expected for a passive device. (The S-parameters are calculated from a smallnumber of port current values and may therefore be sensitive to the mesh density —the maximum segment and edge lengths used in this example are therefore smaller thannormal. Note that the small separation between the strip and the wire also influences themesh requirements.)

Figure 3-84: S-parameters of a three port system above a BO card ground



3.37 Example 37: Proximity coupled circular patch antenna withmicrostrip feed

Figure 3-85: Proximity coupled circular patch antenna. The lighter triangles are on alower level (closer to the ground plane). The dielectric layers are hidden to show thegeometry of the triangular elements.

This example considers a proximity coupled circular patch antenna. The geometry ofthe triangles is shown in figure 3-85 and the parameters of the dielectric layers can beobtained from the GF card in the listing of the input file (example_37.pre) below. Themesh size is related to the width of the strip to avoid having triangles with a large aspectratio. Note that magnetic symmetry is used to reduce the number of unknowns. The AEcard is used to define a line between points as the strip line feed port — this line mustextend to both sides of the symmetry plane.

** Analysis of a circular patch antenna proximity coupled to microstrip line.

** Compare with Fig.’s 7 and 8 in the paper "Green’s Functions Analysis of

** Planar Circuits in a Two-Layer Grounded Medium", F Alonso-Monferrer,

** A A Kishk and A W Glisson, IEEE Trans. on Antennas and Propagation,

** vol. 40, no.6, pp. 690-696, June 1992

** Everything in mm, set correct scaling factor

SF 1 0.001

** Variables:

#freq = 2.8e9 ** the starting frequency

#d = 1.590 ** half of the dielectric thickness

#er = 2.62 ** relative permittivity

#ur = 1.00 ** relative permeability

#w = 4.373 ** width of feedline

#r = 17.5 ** radius of the circular patch

#l = 79 ** length of the microstrip line


#lam = 1000*#c0 / #freq / sqrt(#er) ** the wavelength in the dielectric

#edgelen = min(#lam/12,0.8*#w) ** mesh size on patch and feed strip

IP #edgelen



** Define the points required by the geometry

** Circular patch

DP s1 0 0 0

DP s2 0 0 #r

DP s3 #r 0 0

** Rectangular feed-line

DP p1 0 0 -#d

DP p2 #w/2 0 -#d

DP p3 #w/2 #l -#d

DP p4 0 #l -#d

DP p3m -#w/2 #l -#d

** Build the circular patch

LA 0

KR s1 s2 s3 90

** Symmetry plane y=0

SY 1 0 1 0

** Feedline of the antenna

LA 1

BP p3 p4 p1 p2

** Magnetic symmetry plane x=0

SY 1 3 0 0

** Geometry end

EG 1 0 0 0 0

** The dielectric layers

GF 10 1 0 1 1

2*#d #er #ur

** Excitation of the microstrip line

AE 0 p3 p3m 3 1 0

** Frequency loop:

FR 8 0 #freq 0.05e9

** Just calculate the impedance

OS 0

** The end

EN

Some extracts of the S-parameters as listed in the output file are given on the next page.Figure 3-86 shows the input impedance on the Smith chart. There is a small frequencyshift which can be reduced by using a finer mesh.











431 432




ground plane present? GPLANE = Yes



Current in A 5.2283E-02 -3.7638E-02 6.4422E-02 -35.75

Admitt. in A/V 5.2283E-02 -3.7638E-02 6.4422E-02 -35.75




Figure 3-86: Reflection coefficient of the proximity coupled patch.



3.38 Example 38: Microstrip filter

Port 1 Port 2

Figure 3-87: Simple microstrip filter

In this example we consider a simple two port microstrip filter. A single stub is usedto block transmission at the centre frequency. The SP card is used to determine theS-parameters. The two ports can be fed by making a physical connection to ground andfeeding the edge between the line and the vertical strip. (The “voltage” in a microstripline is between the line and ground. It yields better results to feed the edge at the linethan the one connected to ground.) The input file, example_38a.pre, is as follows

** Microstrip filter example, see G. V. Eleftheriades and J. R. Mosig,

** "On the Network Characterization of Planar Passive Circuits Using

** the Method of Moments", IEEE Trans. MTT, vol. 44, no. 3, March 1996,

** pp. 438-445, Fig. 9.

**

** The analysis here is just for the microstrip circuit, without the

** shielding cavity. This can of course be added as well in FEKO.

** Here we model the microstrip lines with vertical connections to

** ground (see options used at the AE-cards).

** All dimensions in mm

SF 1 0.001

** Substrate height

#t = 1.57

** Meshsize

#fmax = 4.0e9

#lam = #c0 / #fmax * 1000 ** in mm

#l1 = #lam / 10 ** along lines

#l2 = #lam / 20 ** across lines

IP #l1

** Define the points for the filter structure

#x = 0

#y = 23

DP P1 #x #y 0

DP Q1 #x #y -#t

#x = #x + 41.4

DP P2 #x #y 0

#x = #x + 4.6

DP P3 #x #y 0



#x = 92

DP P4 #x #y 0

DP Q4 #x #y -#t

#y = #y + 4.6

DP P5 #x #y 0

DP Q5 #x #y -#t

#x = 41.4 + 4.6

DP P6 #x #y 0

#y = #y + 18.4

DP P7 #x #y 0

#x = #x - 4.6

DP P8 #x #y 0

#y = #y - 18.4

DP P9 #x #y 0

#x = #x - 41.4

DP P10 #x #y 0

DP Q10 #x #y -#t

** Create the structure

LA 1

BP P1 P2 P9 P10 #l2

LA 3

BP P2 P3 P6 P9 #l2 #l2

BP P9 P6 P7 P8 #l2

LA 2

BP P3 P4 P5 P6 #l2

** Vertical elements to ground

LA 5

BP P1 P10 Q10 Q1 #l2

BP P4 P5 Q5 Q4 #l2

** End of geometry

EG 1 0 0 0 0

** Green’s function

GF 10 1 0 1 1

#t 2.33 1

** Excitation of the two ports

** (The edges at the top of the vertical connections to ground)

AE 0 1 5 0 1

AE 1 2 5 0 1

** Frequency loop and S-parameter computations (Use the first FR card with FEKO LITE)

** FR 10 0 2.0e9 3.8e9

FR 51 0 1.0e9 4.0e9

SP

** End

EN



Some extracts from the output file example_38a.out are









2 6




S 1 1 -6.09731E-02 2.68205E-01 2.75049E-01 -11.2118 102.81

S 2 1 -9.57949E-01 -6.99244E-02 9.60498E-01 -0.3501 -175.83










90 92




S 1 2 -9.58381E-01 -6.99558E-02 9.60931E-01 -0.3462 -175.83

S 2 2 2.06992E-02 2.74190E-01 2.74970E-01 -11.2143 85.68


CPU-time runtime

...

Initialisation of the Greens function 1465.330 2017.974

...

Calcul. of matrix A 62.960 94.624

...

total times: 1529.310 2114.128



One may also — see the file example_38b.pre — use the AE card to excite the end ofthe microstrip line without a connection to ground. The two vertical strips and unusedpoints are then omitted; and the two AE cards modified to specify the line where theport is located:

AE 0 P1 P10 3 1 0

AE 1 P4 P5 3 1 0

The comparative extracts from the output file example_38b.out are









89 90




S 1 1 -7.26299E-02 2.33305E-01 2.44349E-01 -12.2398 107.29

S 2 1 -9.59998E-01 -1.27969E-01 9.68489E-01 -0.2781 -172.41










91 92




S 1 2 -9.59980E-01 -1.27966E-01 9.68472E-01 -0.2783 -172.41

S 2 2 9.63821E-03 2.44251E-01 2.44441E-01 -12.2365 87.74


CPU-time runtime

...

Initialisation of the Greens function 25.130 53.800

...

Calcul. of matrix A 49.030 100.063

...

total times: 75.080 155.595



The results agree very well as can also be seen in figure 3-88, but there is a significantdifference in the run time. The model using the vertical connections has vertical currentsand thus requires a 3D interpolation table for the Green’s functions while the modelwithout it needs only a 2D interpolation table. The difference in run time reflects thecalculation time required for these interpolation tables.

Figure 3-88: S-parameters of the single stub filter



3.39 Example 39: Log periodic antenna

0

1

2

3

4

5

6

7

8

9

10

11

12

Labels

Figure 3-89: Log periodic antenna. Note the unique labels of the centre segments.

This example uses the non-radiating transmission lines to form the connections for a logperiodic antenna. The location of the transmission lines can be shown with POSTFEKO.Note that the lines on the figure cross to show crossed transmission lines as required for alog periodic antenna. This depends on the orientation of the segment as well as whethera crossed transmission line is specified. The example also demonstrates how one may use!!FOR . . . !!NEXT loops to create repetitive geometry. The last transmission line is alsoused to specify a termination load. The input file, example_39.pre, is as follows

** Analysis of a 12-element logarithmic periodic antenna

** (corresponds to the NEC example 5).

** Some definitions for the geometry

#sigma = 0.70 ** initial spacing

#tau = 0.93 ** scaling factor for elements

#len = 2 ** length of first element (the shortest element)

#rad = 0.00667 ** radius of first element

#Zline = 50 ** transmission line impedance

#Zload = 50 ** load impedance at the last element

** Frequency specification and segmentation

#freq = 46.29e6 ** frequency

#lambda = #c0 / #freq


IP #seglen

** We can either use a !!FOR ... !!NEXT loop to construct the 12 elements,

** or we can use the TG-card. With the TG card one would create one

** element and duplicate it 11 times using the scaling factor 1/#tau.



** However, then the number of segments used for each dipole would be

** the same. Therefore use here rather a !!FOR ... !!NEXT loop construction.

** Initial values for the loop

!!for #i = 1 to 12

!!if (#i = 1) then

** This is the first element to be created, at origin

#x = 0

!!else

** Other elements with a certain distance from previous element

#x = #x - #sigma

!!endif

** Create the wire with the correct radius, use a unique

** label #i for the centre segment

#y = 0.4*#seglen ** ensure that just one segment at the centre

DP P1 #x -#len/2 0

DP P2 #x -#y 0

DP P3 #x #y 0

DP P4 #x #len/2 0

LA 0

BL P1 P2 #rad

LA #i

BL P2 P3 #rad

LA 0

BL P3 P4 #rad

** Apply scaling

#sigma = #sigma/#tau

#len = #len/#tau

#rad = #rad/#tau

!!next


EG 1 0 0 0 0

** Create all the transmission lines (again a loop is very useful)

!!for #i = 1 to 11

** Extra shunt admittance at the last element 12

!!if (#i=11) then

#YS = 1 / #Zload

!!else

#YS = 0

!!endif

** Define the transmission line from label #i to label #i+1 (crossed)

TL 1 #i #i+1 1 -1 #Zline #YS

!!next

** Excitation by a voltage source

FR 1 0 #freq

A1 0 1 1 0

** Vertical radiation pattern

FF 1 73 1 0 90 0 -5

EN




DATA FOR TRANSMISSION LINES (TL cards)

No. Type Segments Length Transm. line impedance Shunt adm. port 1 in S ...

numbers in m real part imag. part real part imag. part

1 2 3 -8 7.52688E-01 5.00000E+01 0.00000E+00 0.00000E+00 0.00000E+00

2 1 8 -13 8.09342E-01 5.00000E+01 0.00000E+00 0.00000E+00 0.00000E+00

3 1 13 -18 8.70260E-01 5.00000E+01 0.00000E+00 0.00000E+00 0.00000E+00

...

11 1 60 -68 1.55520E+00 5.00000E+01 0.00000E+00 0.00000E+00 0.00000E+00

Shunt adm. port 2 in S

real part imag. part

0.00000E+00 0.00000E+00

0.00000E+00 0.00000E+00

0.00000E+00 0.00000E+00

2.00000E-02 0.00000E+00

NETWORK DATA FOR TRANSMISSION LINES (TL cards)

No. Port Port voltage in V Port current in A Port impedance in Ohm ...

real part imag. part real part imag. part real part imag. part

1 1 1.0000E+00 0.0000E+00 2.1868E-02 -4.5764E-04 4.5710E+01 9.5661E-01

1 2 -7.2976E-01 7.2933E-01 1.6292E-02 -1.3682E-02 -4.8314E+01 4.1928E+00

2 1 -7.2976E-01 7.2933E-01 -1.5311E-02 1.4273E-02 4.9259E+01 -1.7146E+00

2 2 1.1594E-02 -1.0570E+00 -5.1672E-04 2.0413E-02 -5.1765E+01 7.4237E-01

3 1 1.1594E-02 -1.0570E+00 -2.9827E-03 -2.2954E-02 4.5221E+01 6.3813E+00

3 2 8.5021E-01 5.9067E-01 -1.7784E-02 -1.5421E-02 -4.3728E+01 4.7038E+00

Power in W

1.0934E-02

-1.0934E-02

1.0792E-02

-1.0792E-02

1.2114E-02

-1.2114E-02

SUMMARY OF LOSSES

Metallic elements: 0.0000E+00 W

Dielectric (surface equiv. princ.): 0.0000E+00 W

Dielectric (volume equiv. princ.): 0.0000E+00 W

Mismatch at feed: 0.0000E+00 W

Non-radiating networks: 1.6650E-03 W

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

Sum of all losses: 1.6650E-03 W

Efficiency of the antenna: 85.7209 %

(based on a total active power: 1.1660E-02 W)







90.00 0.00 0.000E+00 0.00 2.559E+00 -27.83 -999.9999 10.3854 10.3854

85.00 0.00 0.000E+00 0.00 2.542E+00 -26.85 -999.9999 10.3267 10.3267

80.00 0.00 0.000E+00 0.00 2.490E+00 -23.92 -999.9999 10.1475 10.1475

POLARISATION


0.0000 90.00 LINEAR

0.0000 90.00 LINEAR

0.0000 90.00 LINEAR

The vertical pattern is shown in figure 3-90.

345º330º

315º

300º

285º

270º

255º

240º

225º

210º195º180º165º

150º

135º

120º

105º

90º

75º

60º

45º

30º15º 0º

10

50

-5

-10

-15

Figure 3-90: Directivity of a log periodic antenna in the vertical plane



3.40 Example 40: Coupling between impedance matched dipoles

Port 1, label 1

Port 2, label 2

Figure 3-91: Two half wavelength dipole antennas, each fed with an active impedancematching network.

This example shows how to use impedance loading in the presence of an SP card. Thetwo dipoles are fed with active impedance matching networks. This is modelled by addinga load to cancel the imaginary part of the impedance and calculating the S-parametersfor a source impedance equal to the real part of the input impedance. This results inmaximum power transfer to the second dipole at all frequencies. (The two dipoles areidentical, thus we will use the same load on both ports.)

The input file (example_40.pre) which contains two frequency loops, is as follows

** Coupling between two half wavelength dipoles fed with matching networks

** Set the frequency and wavelength

#f_min = 400e6 ** Start frequency

#f_o = 470e6 ** Centre frequency

#f_max = 490e6 ** End frequency

#Nf = 10 ** Number of frequencies

#lam_o = #c0/#f_o ** Wavelength at the centre frequency (defines the geometry)

#lam_m = #c0/#f_max ** Wavelength at the highest frequency (defines the mesh size)


#seg_len = #lam_m / 20

#seg_rad = 0.001

IP #seg_rad #seg_len



** Geometry parameters

#h = #lam_o/4 ** Half the dipole length

#d = 2*#lam_o ** Separation between dipoles

#l = 0.4*#seg_len ** Half the length of the feed segment


DP A 0 0 -#h

DP B 0 0 -#l

DP C 0 0 #l

** Define the lower half of the first dipole (without feed)

BL A B

** Mirror the lower half of the dipole upwards

SY 1 0 0 2

** Create feed segment with the label 1

LA 1

BL B C

** Create the second dipole by copying the first

** Note that this does not destroy the symmetry

** Note also that label 0 is not incremented

TG 1 0 1 1 0 #d

** End of the geometric input -- write complete geometry to output file

EG 1 0 0 0 0

** Define the frequency

FR #Nf 0 #f_min #f_max

** Calculate the S-parameters in a 50 ohm system and write them to file

** Excite both dipoles as ports for S-parameter calculations

DA 0 0 0 0 0 1

A1 0 1 1 0

A1 1 2 1 0

SP 50

** Now we create a FOR loop frequency loop to allow loading the dipoles.

** We load the dipoles in order to cancel the imaginary part of the input

** impedance and then calculate the S-parameters in a system impedance

** that match the real part of the input impedance.

** Note that loading ports change the interaction matrix such that it is

** not a severe penalty to create a second frequency loop here.

** #Nf = -1 ** For the first run we need to skip this loop

!!for #n = 1 to #Nf

** Read the frequency and antenna input impedance from the prepared file

#freq = fileread("example_40.dat",#n+1,1) * 1.0E6

#Zr = fileread("example_40.dat",#n+1,2)

#Zi = fileread("example_40.dat",#n+1,3)




FR 1 #freq

** Now determine the load to cancel the imaginary part of the impedance

** Note that we cannot use the LZ card as this impedance is overwritten

** by the SP card

!!if #Zi < 0 then

** Negative impedance, add an inductive load

#L = -#Zi/(2*#pi*#freq)

LS 1 #L

LS 2 #L

!!elseif

** Positive impedance, add a capacitive load

#C = 1/(#Zi*2*#pi*#freq)

LS 1 #C

LS 2 #C

!!endif

** Finally calculate the S-parameters (we won’t write them to file)

** Now add the sources, specifying the system impedance

DA 0 0 0 0 0 0

A1 0 1 1 0 #Zr

A1 1 2 1 0 #Zr

** Note that we have specified the system impedance at the ports

** (This could allow using different system impedances for the various ports)

SP

** End of frequency loop

!!next

** End

EN

The first frequency loop calculates the S-parameters in a 50 Ω system. (Note that #Nf isset to -1 to skip the second loop during this phase as the second loop tries to read fromthe file example_40.dat which is calculated from the result of the first run.) The resultsare written to a Touchstone format file (example_40.s2p). From this we can determinethe input impedance at one port if the second port is terminated in a conjugate matchedload. (These calculations were done in Mathematica and involved transforming fromS-parameters to Z-parameters and solving a complex matrix equation for the optimuminput impedance given that the second port is loaded with the complex conjugate of thisinput impedance.) Note that, since the coupling between the two dipoles is very small,the input impedance of one dipole is not very dependent on the load at the second dipole.Thus one will get a very similar result by just plotting the input impedance — with 50 Ωloading subtracted — in POSTFEKO and writing this to a data file.



The calculated input impedance is

"Frequency [MHz]" "ReZ_in matched load" "ImZ_in matched load"

400 51.35897294752684 -98.24440217970796

410 54.78298951057599 -78.49672704243486

420 58.55367334913904 -59.10669618038157

430 62.65276893582022 -40.11841191634703

440 67.01399757492561 -21.55044496478654

450 71.56254189197807 -3.38173332666909

460 76.22303420171083 14.46290879409742

470 80.95653423584226 32.08598498017869

480 85.75940433192570 49.62447813617270

490 90.70622025526940 67.19024349043431

For comparison, the input impedance if the second dipole is loaded with 50 Ω is

"Frequency [MHz]" "ReZ_in" "ImZ_in"

400 51.69 -98.47

410 54.88 -78.84

420 58.42 -59.36

430 62.44 -40.16

440 66.92 -21.44

450 71.64 -3.29

460 76.36 14.39

470 80.97 31.87

480 85.57 49.42

490 90.36 67.15

The second loop reads the frequency and impedance from this file. This loop is done witha FOR–NEXT loop rather than a FR card loop to allow reading the impedance valuesfrom file. Both ports are then loaded to cancel the imaginary part of the input impedance.It would be quite simple to do this with an LZ card, but the SP card overwrites the LZcard. Thus we use an LS card with an inductor to cancel negative imaginary values anda capacitor to cancel positive values.

Note that one may pay a significant penalty in performance if you construct two frequencyloops like this for, for example, two sets of sources. However, changing the loading atports changes the interaction matrix which requires a new LU decomposition such thatin this case there is not a severe penalty for using a second loop. (We do calculate theelement interaction twice, but this is usually a very small part of the total run time.)

Some extracts from the second frequency loop in the output file example_40.out are

LOAD IMPEDANCES AT PORTS

port impedance in Ohm

1 7.15625E+01

2 7.15625E+01








DATA OF LABELS


Rss= 0.000E+00 Ohm/m Lss= 0.000E+00 H/m Css= 0.000E+00 F

Rs = 0.000E+00 Ohm Ls = 1.196E-09 H Cs = 0.000E+00 F

Rp = 0.000E+00 Ohm Lp = 0.000E+00 H Cp = 0.000E+00 F

Zs = ( 7.156E+01 +j 0.000E+00) Ohm (indep. of freq.)

Zlasts=( 0.000E+00 +j 0.000E+00) Ohm/m (freq. dep.)

Zlast =( 7.156E+01 +j 3.382E+00) Ohm (freq. dep.)



Current in A 6.9869E-03 -1.0335E-07 6.9869E-03 0.00

Admitt. in A/V 6.9869E-03 -1.0335E-07 6.9869E-03 0.00

Impedance in Ohm 1.4313E+02 2.1170E-03 1.4313E+02 0.00




S 1 1 2.47111E-06 1.47915E-05 1.49965E-05 -96.4802 80.52

S 2 1 -2.12123E-02 6.34272E-02 6.68803E-02 -23.4940 108.49

Figure 3-92 shows the input impedance at the voltage source ports of either dipole. Thefirst two lines (solid red and green with + symbols) shows the input impedance in the 50 Ωsystem of the first frequency loop. Note that the real part includes the 50 Ω load addedby the SP card. The next two lines (yellow with square markers and blue with circularmarkers) shows the input impedance at the voltage sources for the matched system. Asexpected, the imaginary part is zero while the real part includes the loading (which isapproximately equal to the real part of the 50 Ω system less the 50 Ω load) which explainsthe increased slope.

Finally figure 3-93 shows the S-parameters. The reflection coefficient is drastically reduced— it is less than -80 dB for the matched system — which shows in the increased couplingto the second dipole away from the resonance frequency. (Using the input impedance fora single dipole when the second dipole is loaded with 50 Ω instead of a matched load,increases S11 to about -60 dB.)



ReZ_s 50 ImZ_s 50 ReZ_s matched ImZ_s matched

Frequency [MHz]490480470460450440430420410400

Impedance

[Ohm

]200

150

100

50

0

-50

-100

Figure 3-92: The impedance at the source port of either dipole in a 50 Ω system and amatched system. Note that the added loads contribute to these impedances.

|S11| 50 system |S21| 50 system |S21| matched system

Frequency [MHz]490480470460450440430420410400

Spa

ram

ete

rs[d

B]

0

-5

-10

-15

-20

-25

-30

Figure 3-93: S-parameters for the two dipole antennas in a 50 Ω system and a matchedsystem. Note that S11 for the matched system is less than -80 dB and is not included inthe figure.



3.41 Example 41: Using the MLFMM

Y

X

Z

Figure 3-94: Plane wave incident on an electrically large trihedral

In this example we consider a single plane wave incident (from ϑ = 60 and ϕ = 0) ona trihedral. The size of the trihedral (13.5λ2 surface area) was chosen such that we canstill solve it incore on a PC with 768 MByte of RAM. This is on the small side for theMLFMM, but enough to demonstrate the advantage thereof.

The file example_41.pre is listed below. Note the use of the FM card at the end of thegeometry section in the input file and the wrapped EG card which includes the Singleprecision field (the 1 in column 101).

** RCS computation of a tetrahedral using the

** Multilevel Fast Multipole Method (MLFMM)

** Some general parameter settings


#a = 3*#lambda ** Length of an edge



IP #tri_len

** Define the points for the plate



#b = sqrt(0.5)*#a ** Projected length

DP O 0 0 0

DP L #b -#b 0

DP R #b #b 0

DP T 0 0 #a

** Create trihedral

BT O L R

BT O R T

BT O T L

** Use the MLFMM instead of the standard MoM

FM

** End of the geometry (we switch here to single precision storage

** to reduce the MLFMM memory requirement)

EG 1 0 0 0 ...

1



FR 0 #freq


A0 0 1 1 1 0 60 0 0

** Bistatic radar cross section in the vertical plane Phi=0

FF 1 91 1 0 0 0 2

** End

EN


DATA FOR MEMORY USAGE

Number of metallic triangles: 3915 ...

FAST MULTIPOLE METHOD (FMM)

Multilevel FMM is used

Storage of some elements with single precision (saves memory)

DATA FOR THE FMM

Level of the MLFMM: 5

Finest box size DELTA/LAMBDA: 0.2300

Number of nearfield matrix elements: 1437056 ( 4.2601 % of full MoM)

SUMMARY OF MEMORY REQUIREMENT FMM (in MByte)



Near-field matrix: 16.4458

Far-field matrix

Direction vectors Fourier trans: 0.0304

Fourier trans. basis functions: 8.5078

Transfer function: 3.0772

Interpolation and Filtering: 0.1733

Matrix-vector-multiplication: 2.8754

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

total: 31.1098 MByte

for comparison classical MoM: 514.7227 MByte

PRE-CONDITIONING OF THE LINEAR SET OF EQUATIONS

...

Memory requirement for preconditioner: 54.443 MByte



The solution required 47 seconds on a Pentium 4 2.4 GHz PC. For comparison the MoMresult required about 515 MByte of RAM and 240 seconds solution time. Even if onesets up the model to exploit the single plane of symmetry, the MoM requires 262 MByteor RAM and 66 seconds solution time. As the problem size increase, the difference willbecome more and more significant. Figure 3-95 compares the results obtained with theMLFMM with those obtained with the MoM.

Theta (degrees)

1801651501351201059075604530150

RC

S(d

Bs

m)

30

25

20

15

10

5

0

MoM MLFMM

Figure 3-95: Bistatic RCS of a trihedral. Comparison of the MLFMM and MoM results.


IndexA0 card, 3-14, 3-21, 3-32A1 card, 3-1, 3-4A2 card, 3-114, 3-120A4 card, 3-114A5 card, 3-64, 3-83AE card, 2-12, 3-148, 3-151AP card, 3-135aperture, 3-135AR card, 3-135attachment to UTD plate, 3-79

biological tissue, 2-18BL card, 3-1, 3-4BO card, 3-36BP card, 3-4, 3-10BQ card, 3-27BT card, 3-27

CB card, 3-54, 3-67circular section, 3-24CL card, 3-21coaxial cable, 3-130coaxial probe feed, 3-114cone section, 3-93copy geometry, 3-67coupling, 3-160cuboid sphere, 3-49currents

output, 3-21, 3-24, 3-43cylinder, 3-61, 3-93, 3-106

UTD, 3-102

DI card, 3-14, 3-18, 3-46dielectric

coated sphere, 3-46fields inside, 3-52finite substrate, 3-110Green’s function, 3-36, 3-58parameter setting, 3-14, 3-18physical optics, 3-106sphere, 3-58substrate, 3-88, 3-97, 3-110, 3-114surface equivalence, 3-14, 3-18, 3-93, 3-110volume equivalence, 3-32, 3-49

DK card, 3-49

DP card, 3-1

edge corrected PO, 3-43edge excitation, 3-97elements, 2-1, 2-5, 3-4, 3-10, 3-27excitation, see sourcesexposure analysis, 2-18

far fields, 2-1, 3-4, 3-6, 3-10as source, 3-135

FE card, 2-1, 3-4, 3-10, 3-52feed, see sourcesFEM/MoM hybrid, 2-18FF card, 2-1, 3-1, 3-4, 3-10fields in dielectrics, 3-52FM card, 2-15, 3-166FO card, 3-61Fock theory, 3-61for loops, 3-156FR card, 3-1, 3-4

gainpower compensation, 2-5, 3-27

geometrycopy and transform, 3-67

GF card, 3-36, 3-58, 3-88, 3-97, 3-114Green’s function

coaxial probe feed, 3-114planar substrate, 3-88, 3-97, 3-151real ground, 3-36sphere, 3-58

groundperfect conducting, 3-145real, 3-36reflection, 3-36Sommerfeld, 3-36wires inside, 3-120

Hertzian dipole, 3-83

impedanceloading, 3-100, 3-160transformation, 3-130

infinite substrate, 3-88input impedance, 3-1

I-1

IP card, 3-1, 3-4

KA card, 3-43KK card, 3-93KL card, 3-54KR card, 3-24KU card, 3-14, 3-46

LA card, 3-1, 3-4labels, 3-2, 3-6, 3-54large models, 2-15, 3-166loading

and SP card, 3-160wire segments, 3-100

loopsFOR–NEXT, 3-156

losses on plates, 3-6LS card, 3-160LZ card, 3-100, 3-160

magnetic media, 3-49ME card, 3-14, 3-18, 3-46, 3-93, 3-110microstrip

feed line, 2-12, 3-148filter, 3-151wire approximation, 3-88

MLFMM, 2-15, 3-166

near fields, 2-1, 3-4, 3-10non-radiating network, 3-156

OPTFEKO, 1-1optimising parameters, 1-1OS card, 3-21, 3-24, 3-43

parabolic reflector, 3-64, 3-135patch antenna, 2-12, 3-88, 3-97, 3-110, 3-114,

3-148pattern source, 3-135PB card, 3-64physical optics, 2-1, 3-10

dielectric media, 3-106edge correction, 3-43Fock currents, 3-61wedge correction, 3-54

plane wave incidence, 3-14, 3-21, 3-32PO card, 2-1, 3-10, 3-106, 3-145points, 3-5

polygons, 3-75, 3-79power

gain compensation, 2-5, 3-27output setting, 3-24, 3-36

proximity coupling, 2-12, 3-148PS card, 3-4PW card, 3-24, 3-36PY card, 3-75, 3-79

QU card, 3-32

radiation pattern, 3-1real (reflection) ground, 3-36resistive loading, see loading

S-parameters, 3-145, 3-151scaling, 2-5, 3-14, 3-27segments, 3-1, 3-2, 3-4SF card, 2-5, 3-14, 3-27SK card, 2-9, 3-4, 3-124, 3-127skin effect losses, 3-6sources

far field pattern, 3-135feed pin, 3-110, 3-114Hertzian dipole, 3-64, 3-83microstrip line, 2-12, 3-148, 3-151on coaxial cable, 3-130plane wave, 3-14, 3-21, 3-32transmission line, 3-156voltage at a node, 3-120voltage on a segment, 3-1, 3-4voltage on an edge, 2-12, 3-97, 3-148, 3-151

SP card, 3-145, 3-151, 3-160spherical section, 3-14, 3-46stub

shorted, 3-130substrate

finite, 3-110infinite, 2-12, 3-88, 3-148, 3-151

surface equivalence, 3-14, 3-18as source, 3-135

SY card, 3-1, 3-4symmetry, 3-2, 3-4

loading requirements, 3-146

TG card, 3-67, 3-160time domain, 1-1TIMEFEKO, 1-1

I-2

TL card, 3-156TO card, 3-21torus section, 3-21transform geometry, 3-67transmission lines, 3-100, 3-156

UT card, 3-75, 3-79UTD, 3-75, 3-83

current attachment, 3-79cylinder, 3-102

UZ card, 3-102

volume equivalence, 3-32, 3-49

wedge corrected PO, 3-54

ZY card, 3-61, 3-93, 3-106

I-3

feko exampleguide

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