Analysis and Design of Conical Spiral Antennas in Free

Analysis and Design of Conical Spiral

Antennas in Free Space and over Ground

A Thesis

Presented to

The Academic Faculty

by

Thorsten W. Hertel

In Partial Fulfillment

of the Requirements for the Degree of

Doctor of Philosophy in Electrical and Computer Engineering

Georgia Institute of Technology

November 2001

Analysis and Design of Conical Spiral Antennas in

Free Space and over Ground

Approved:

Glenn S. Smith, Chairman

Waymond R. Scott, Jr.

Andrew F. Peterson

John A. Buck

Karsten Schwan

Date Approved

ii

ACKNOWLEDGMENTS

This dissertation is the result of my graduate studies at the Georgia Institute of

Technology. Obviously, this work would not have been accomplished without the

support of many people; I would like to express my gratitude to those who have

contributed to this work in one way or another.

Dr. Glenn S. Smith deserves my deepest gratitude for reaching this goal. His

immense knowledge and uncompromising quest for excellence provided an optimum

working environment at Georgia Tech. Dr. Smith has been instrumental in ensuring

my academic and professional well-being over the last five years.

Also, I would like to thank the members of my dissertation committee, Dr.

Waymond Scott, Jr., Dr. Andrew F. Peterson, Dr. John A. Buck, and Dr. Karsten

Schwan. Especially, I would like to express my gratitude to Dr. Scott for giving useful

advice for the measurements and for his courtesy of letting us use his measurement

equipment and the Beowulf Cluster to perform important computer simulations.

In addition, I would like to thank Dr. James G. Maloney and the Georgia Tech

Research Institute for letting us use their calibration kit. Throughout my first few

years at Georgia Tech, Dr. Maloney provided insightful comments on my work.

The discussions and cooperations with all of my colleagues have contributed

substantially to this work: L.E. Rickard Petersson was very helpful in backing up

theoretical issues of my work and Benny Venkatesan assembled and maintained the

Beowulf Cluster and was of valuable help in computer related issues. In particular,

I am very grateful to Dr. Christoph T. Schroder who has been a long-time friend

during college, both in Braunschweig, Germany, and in Atlanta, USA. His academic

iii

ACKNOWLEDGMENTS

excellence and his way of life have always been a great source of motivation.

I am very grateful to many of my friends for their assistance, support, and

extracurricular activities: Dr. Sumeer K. Bhola, Dr. Fabian E. Bustamante, Pawan

Hegde, Holger Junker, Tony V. Mule, Dr. Christoph T. Schroder, Aparna Pappu,

Benny Venkatesan, and Ram Voorakaranam. In particular, I owe special gratitude to

Fabian for many hours of mind-opening discussions and for his culinary treats.

Finally, I would like thank my parents, Jurgen and Elke Hertel, and my brother,

Arne Hertel, for their continuous and unconditional support of all my undertakings,

scholastic and otherwise.

This work was supported in part by the Army Research Office under Contract

DAAG55-98-1-0403.

iv

Contents

ACKNOWLEDGMENTS ii

LIST OF TABLES viii

LIST OF FIGURES ix

SUMMARY xvii

1 Introduction 1

1.1 Historical Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Contribution of Research . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 The Two-Arm Conical Spiral Antenna in Free Space 10

2.1 Description of the CSA . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Characteristics of the CSA in Free Space . . . . . . . . . . . . . . . . 16

3 Parametric Study and Design Graphs 29

4 Resistive Terminations for the CSA 42

4.1 Methods for Terminating the CSA . . . . . . . . . . . . . . . . . . . 42

4.2 Characteristics of the Loaded CSAs . . . . . . . . . . . . . . . . . . . 44

5 The Two-Arm Conical Spiral Antenna over the Ground 57

5.1 General Characteristics of the CSA over the Ground . . . . . . . . . 58

v

CONTENTS

5.2 Numerical Analysis of the CSA over the Ground . . . . . . . . . . . . 66

5.3 GPR Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.3.1 GPR: Detection of Buried Rods . . . . . . . . . . . . . . . . . . 76

5.3.2 GPR: Detection of Buried Mines . . . . . . . . . . . . . . . . . 82

5.4 Removing the Dispersion in the CSA . . . . . . . . . . . . . . . . . . 86

6 FDTD Analysis 98

6.1 FDTD Update Equations . . . . . . . . . . . . . . . . . . . . . . . . 99

6.1.1 Update Equations in Continuous Space . . . . . . . . . . . . . . 100

6.1.2 Update Equations in Discretized Space . . . . . . . . . . . . . . 105

6.1.3 Medium Properties in the FDTD Update Equations . . . . . . . 110

6.1.4 FDTD Update Equations for the Resistive Elements . . . . . . . 115

6.1.5 FDTD Update Equations for a Dipole . . . . . . . . . . . . . . 117

6.2 Validation of PML . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.3 FDTD Modeling of the CSA . . . . . . . . . . . . . . . . . . . . . . . 122

6.4 FDTD Antenna Feed . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.5 Near-Field to Far-Field Transformer . . . . . . . . . . . . . . . . . . 132

6.6 Program/Code Parallelization . . . . . . . . . . . . . . . . . . . . . . 133

7 Validation of FDTD Results 138

7.1 Validation using Published Results . . . . . . . . . . . . . . . . . . . 138

7.1.1 CSA in Free Space . . . . . . . . . . . . . . . . . . . . . . . . . 138

7.1.2 Horizontal Dipole over Lossy Media . . . . . . . . . . . . . . . . 142

7.2 Validation using Georgia Tech Measurements . . . . . . . . . . . . . 144

7.2.1 Manufacturing of the CSA . . . . . . . . . . . . . . . . . . . . . 145

7.2.2 Feed System of the CSA . . . . . . . . . . . . . . . . . . . . . . 149

7.2.3 Termination of the CSA . . . . . . . . . . . . . . . . . . . . . . 155

7.2.4 Impedance Measurements . . . . . . . . . . . . . . . . . . . . . 159

7.2.5 Realized Gain Measurements . . . . . . . . . . . . . . . . . . . . 165

7.2.6 Dipole Measurements . . . . . . . . . . . . . . . . . . . . . . . . 169

vi

CONTENTS

8 Conclusions 175

APPENDIX A Analysis of the Resistive Sheet Termination 178

APPENDIX B Convergence Study for FDTD Feed Models for An-

tennas 183

BIBLIOGRAPHY 197

VITA 201

vii

List of Tables

3.1 Hybrid bandwidth results as a function of D/d for a CSA with a)θ = 15, α = 80, δ = 90, b) θ = 7.5, α = 75, δ = 90, and c)θ = 5, α = 60, δ = 90. . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.1 Electrical parameters for different soil conditions at f = 1.8GHz. . . 63

6.1 Short-hand notation and staggered positions in time and space for allelectric and magnetic fields within the Yee cell (i, j, k). . . . . . . . . 107

6.2 Averaging of the electrical properties in the FDTD update equations. 1126.3 Electrical parameters for different soil conditions at f = 0.7GHz. . . 1226.4 Relative errors for the electric field at various observation points in the

grid for three different soil conditions. . . . . . . . . . . . . . . . . . . 1236.5 Normalization of the parameters in the transmission-line equations. . 128

B.1 Parameters of the discretized linear dipole antenna of square crosssection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

B.2 Ratio of numerical phase velocity to the speed of light for differentdiscretizations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

B.3 Resonant frequencies and the conductances at anti-resonance for thehard source (best case for convergence). . . . . . . . . . . . . . . . . . 196

viii

List of Figures

1.1 Geometry for the practical two-arm CSA (top) and side view of thebasic cone (bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Schematic drawings for two models of the CSA: a) using thin wires, b)antenna arms with expanding arm width. . . . . . . . . . . . . . . . . 4

1.3 a) “Infinite balun” introduced by Dyson [9], b) balun and impedancetransformer introduced by Wills [16]. . . . . . . . . . . . . . . . . . . 6

1.4 The two-arm CSA placed in air directly over the ground. . . . . . . . 7

2.1 a) Geometry for the practical two-arm CSA, b) side view of the basiccone, and c) top view of the highlighted planar cut. . . . . . . . . . . 11

2.2 a) Schematic model for the a) left-handed CSA and b) right-handedCSA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 a) Schematic drawing for pulses of charge at three instances of time(top view of CSA), b) polarization ellipse for the electric field for anobserver beyond the small end looking in the direction of the negative zaxis, c) polarization ellipse for the electric field for an observer beyondthe large end looking in the direction of the positive z axis, and d)illustration for the position and orientation of the observers. . . . . . 15

2.4 a) Differentiated Gaussian pulse as a function of normalized time, t/τp,b) spectrum of the differentiated Gaussian pulse as a function of nor-malized frequency, ωτp, and c) far-zone electric field on axis. . . . . . 17

2.5 Theoretical results for the reflected voltage at the terminals of theunloaded CSA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.6 Comparison of theoretical (FDTD) and measured terminal quantities:a) magnitude of the reflection coefficient, b) input resistance, and c)input reactance for the unloaded CSA with θ = 7.5, α = 75, δ = 90,D/d = 8 (d = 1.9 cm). . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.7 Theoretical results for the input impedance plotted on a Smith chartfor a) 0.3GHz ≤ f ≤ 3.5GHz and b) 0.5GHz ≤ f ≤ 3.5GHz . . . . . 21

2.8 Comparison of theoretical (FDTD) and measured input reactances. Asmall capacitance, C = 0.12 pF, has been added in parallel with theterminals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

ix

LIST OF FIGURES

2.9 Theoretical results for the far-zone pattern for the right-handed andleft-handed circularly polarized components of the electric field at thefrequencies: a) f = 0.75GHz, b) f = 1.50GHz, and c) f = 2.25GHz. 23

2.10 Theoretical results for the front-to-back ratio referred to a) the to-tal far-zone electric field and b) the left-handed circularly polarizedcomponent of the far-zone electric field. . . . . . . . . . . . . . . . . . 25

2.11 a) Schematic drawing for the polarization ellipse, b) theoretical resultsfor the axial ratio for the electric field in the far zone of the CSA. . . 26

2.12 a) Theoretical and measured results for the realized gain and b) the-oretical results for the gain. The results are for the direction of maxi-mum radiation (−z). . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.1 Frequencies that define various bandwidths: a) the bandwidth forVSWR, b) the bandwidth for directivity, and c) the hybrid bandwidth. 30

3.2 Design graphs for the unloaded CSA based on the results for the VSWR(Zc = R∞) showing a) the bandwidth, b) the maximum wavelength,λmax, and c) the minimum wavelength, λmin, as functions of the anglesθ and α, with δ = 90. . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3 Design graphs for the unloaded CSA based on the results for the direc-tivity showing a) the hybrid bandwidth, b) the maximum wavelength,λmax, and c) the minimum wavelength, λmin, as functions of the anglesθ and α, with δ = 90. . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4 Design graphs for the unloaded CSA showing a) the hybrid bandwidth,b) the maximum wavelength, λmax, and c) the minimum wavelength,λmin, as functions of the angles θ and α, with δ = 90. . . . . . . . . 34

3.5 Design graph for the unloaded CSA showing the average input impedanceof the infinitely-long CSA as a function of the angles θ and α, withδ = 90. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.6 Design graphs for the unloaded CSA showing a) the average directivity,b) the average half-power beamwidth, and c) the average front-to-backratio as functions of angles θ and α, with δ = 90. . . . . . . . . . . 37

3.7 Design graph for the CSA showing the average axial ratio as a functionof the angles θ and α, with δ = 90. . . . . . . . . . . . . . . . . . . 38

3.8 Theoretical results for the VSWR (a,c,e) and the directivity (b,d,f)as a function of frequency with D/d as parameter. The results arefor a CSA with a) and b) θ = 15, α = 80, δ = 90, c) and d)θ = 7.5, α = 75, δ = 90, and e) and f) θ = 5, α = 60, δ = 90. . . 39

4.1 a) The loaded CSA including the feed system, b) simple transmission-line model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2 Schematic drawings for the two resistive terminations: a) two lumpedresistors connecting the ends of the spiral arms, b) a resistive sheetcovering the entire end of the cone. . . . . . . . . . . . . . . . . . . . 44

x

LIST OF FIGURES

4.3 a) Theoretical results for the reflected voltage at the terminals of theantenna that contains only the reflection from the drive point, b) the-oretical results for the input impedance of the infinitely long CSA. . . 46

4.4 Theoretical results for the reflected voltage at the terminals of theantenna for the unloaded CSA (solid line), the CSA terminated withlumped resistors (dashed line), and the CSA terminated with a resistivesheet (dash-dotted line). . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.5 Theoretical results for the CSA with and without the resistive termina-tions: a) the magnitude of the reflection coefficient, b) input resistance,and c) input reactance. . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.6 Smith charts for a) the unloaded CSA, b) the CSA terminated withlumped resistors, and c) the CSA terminated with a resistive sheet. . 49

4.7 Theoretical results for the CSA terminated with the lumped resistors:the far-zone patterns for the right-handed and left-handed circularlypolarized components of the electric field at the frequencies: a) f =0.75GHz, b) f = 1.50GHz, and c) f = 2.25GHz. . . . . . . . . . . . 50

4.8 Theoretical results for the CSA terminated with a resistive sheet: thefar-zone patterns for the right-handed and left-handed circularly po-larized components of the electric field at the frequencies: a) f =0.75GHz, b) f = 1.50GHz, and c) f = 2.25GHz. . . . . . . . . . . . 51

4.9 Theoretical results for a) the front-to-back ratio referred to the totalfar-zone electric field, b) front-to-back ratio referred to the left-handedcircularly polarized component of the far-zone electric field, and c)axial ratio for the total far-zone electric field. . . . . . . . . . . . . . . 53

4.10 a) Schematic drawing for determining the dissipated power for theCSA terminated with lumped resistors (left) and the CSA terminatedwith a resistive sheet (right). Theoretical results for the b) dissipatedpower and c) efficiency for the CSA terminated with lumped resistors(solid) and for the CSA terminated with a resistive sheet (dashed). . 54

4.11 Theoretical results for the a) realized gain, b) gain, and c) directivityin the direction of maximum radiation (−z). . . . . . . . . . . . . . . 56

5.1 CSA over the ground with the virtual apex of the antenna placed atthe air/ground interface. . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.2 Schematic drawing to illustrate the scaling of the electric field of theCSA in the ground. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.3 Schematic drawings of the polarization ellipses for the electric field:for transmission, maximum reception, and null reception. The ellipsesin a) are for the most general case (elliptical polarization) and in b)for the special case of circular polarization; c) position and orientationof the observers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.4 Idealized polarization characteristics of the left-handed CSA for trans-mission and reception. . . . . . . . . . . . . . . . . . . . . . . . . . . 63

xi

LIST OF FIGURES

5.5 a) Geometry of the plane-wave analysis. Analytical results for themagnitude of the reflection coefficient for the electric field b) parallelto the plane of incidence and c) perpendicular to the plane of incidence. 64

5.6 Analytical results for the ratio |ErefLHCP/E

refRHCP|2 as a function of angle

of incidence αinc for three different types of soil. Polarization ellipsesfor the incident and reflected electric fields are shown for the medium-moist soil. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.7 Schematic drawing to illustrate that the use of circular polarizationreduces the effect of the air/ground interface on the antenna response. 67

5.8 Comparison of the input impedances for the CSA in free space andover medium-moist soil. . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.9 Theoretical results for a) the reflected voltage of the CSA over groundand b) the voltage of the signal scattered from the air/ground interface. 69

5.10 a) Theoretical results for the normalized electric field on axis as afunction of depth: in free space (solid line), in the lossy, medium-moistsoil (dashed line), and in the lossless medium-moist soil (dash-dottedline). b) Theoretical results for the axial ratio on axis for the CSA infree space at the apex (solid line), for the CSA over the medium-moistsoil observed 4 cm (dashed line), and 1m (dash-dotted line) below theair/ground interface. . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.11 a) The normalized electric field distribution 4 cm below the air/groundinterface as a function of frequency and distance from the axis. b)Results for the normalized 3 dB width as a function of frequency forthe antenna in free space observed at the apex (solid line) and for theantenna over the medium-moist ground observed at 4 cm below theapex (dashed line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.12 The axial ratio at a depth of 4 cm below the air/ground interface as afunction of frequency and distance from the axis. . . . . . . . . . . . 74

5.13 Theoretical results for a) the electric field of the CSA in free space inthe near field of the antenna and b) the electric field scattered fromthe air/ground interface. . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.14 Schematic drawing for a monostatic GPR that uses a single CSA todetect thin metallic rods buried in the ground. . . . . . . . . . . . . . 76

5.15 Illustration for the detection approach using a CSA in a monostaticGPR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.16 Drawing detailing the parametric study for four differently sized rodsburied in the ground using a) a single CSA and b) four resonant dipolesas the GPR antennas. . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.17 Theoretical results for the parametric study for a) the monostatic GPRusing a single CSA and b) a series of resonant dipoles to detect thinmetallic rods buried in the ground. . . . . . . . . . . . . . . . . . . . 80

5.18 Theoretical results for the electric field scattered from the rod buriedin the ground. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

xii

LIST OF FIGURES

5.19 Results for the monostatic GPR with the thin rods buried in the drysoil. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.20 a) Schematic drawing for the monostatic GPR used to detect buriedplastic mines, b) parameters that describe the mine and the relativepositioning of the antenna and the mine. . . . . . . . . . . . . . . . . 83

5.21 Theoretical results for the parametric study for the monostatic GPRusing a single CSA to detect buried plastic mines: a) DM = 18 cm, b)DM = 13.5 cm, and c) DM = 9 cm. . . . . . . . . . . . . . . . . . . . 85

5.22 Theoretical results for the electric field scattered from the buried mine:a) on axis and b) off axis. . . . . . . . . . . . . . . . . . . . . . . . . 87

5.23 Schematic model for a monostatic radar to detect an infinitely longpipe in free space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.24 Theoretical results for the reflected voltage of the CSA a) in free spaceand b) the CSA in free space with the target; c) results for the voltageof the signal scattered from the pipe in free space. . . . . . . . . . . . 90

5.25 Examples for scatterers of co polarization using a) an anisotropic mediumand b) a grid of closely spaced wires. . . . . . . . . . . . . . . . . . . 91

5.26 Theoretical results for the circularly polarized components of the elec-tric field scattered from the grid, normalized by the incident, LHCPcomponent of the electric field. . . . . . . . . . . . . . . . . . . . . . . 92

5.27 Theoretical results for the received voltage of the signal scattered fromthe pipe in free space with the dispersion removed: a) signal in thefrequency domain, b) signal in the time domain for −0.5 ≤ t/τL ≤ 4,and c) signal in the time domain for 0.5 ≤ t/τL ≤ 1.5. . . . . . . . . . 94

5.28 Schematic model for a monostatic radar to detect the infinitely longpipe buried in the ground. . . . . . . . . . . . . . . . . . . . . . . . . 95

5.29 Theoretical results for a) the reflected voltage of the CSA above the dryground with the pipe present and b) the voltage of the signal scatteredfrom the pipe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.30 Theoretical results for the received voltage of the signal scattered fromthe pipe buried in dry ground with the dispersion removed: a) is for−0.5 ≤ t/τL ≤ 4, and b) for 0 ≤ t/τL ≤ 1. . . . . . . . . . . . . . . . 97

6.1 Schematic drawing for the open-region problem truncated by the PMLabsorbing boundary condition (CSA is omitted for simplicity). . . . . 101

6.2 Yee cell in three dimensions. . . . . . . . . . . . . . . . . . . . . . . . 1066.3 Details about the correct averaging at an interface of the electrical

properties for a) the electric field Ex and b) the magnetic field Hz. . . 1116.4 a) Spatial variation of the conductivity σx a) along its longitudinal

direction x and b) along its transverse direction z with the air/groundinterface present. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.5 Implementing the lumped resistor in the FDTD method. . . . . . . . 115

xiii

LIST OF FIGURES

6.6 Implementing the resistive sheet in the FDTD method: a) the three-dimensional Yee cell containing the resistive sheet and b) slice of theFDTD grid in the vicinity of the sheet. . . . . . . . . . . . . . . . . . 117

6.7 a) Schematic model of the dipole antenna, b) drawing of a Yee cellthat contains the antenna conductor. . . . . . . . . . . . . . . . . . . 118

6.8 Cross-sectional view of a) a single Yee cell that contains the antennaconductor, b) Yee cells in the vicinity of the drive-point gap, and c)Yee cell near the antenna end. . . . . . . . . . . . . . . . . . . . . . . 120

6.9 Schematic drawing for the approach to validate the PML (only theregion surrounding the antenna is shown, the PML is omitted for sim-plicity). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.10 Schematic drawing detailing the FDTD modeling of the antenna armsin a two-step process. . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.11 FDTD model of the conical spiral antenna. Only 10% of the antennaat the feed end is shown. . . . . . . . . . . . . . . . . . . . . . . . . 124

6.12 Discretized version of the feeding disc. . . . . . . . . . . . . . . . . . 1256.13 Schematic model of a) the one-dimensional transmission line and b)

the feed region including the virtual transmission line. . . . . . . . . . 1276.14 Model of the surface used for the near-field to far-field transformation. 1346.15 Division of the solution space into subspaces. . . . . . . . . . . . . . 1356.16 Illustration for the necessary message passing of interface data (fields). 137

7.1 Comparison of theoretical (FDTD) and measured reflection coefficientsplotted on a Smith chart for a) the unloaded CSA and b) the CSAterminated with lumped resistors. The results are for Ramsdale’s CSA[17] with θ = 15, α = 60, δ = 90, and D/d = 12.77 with D = 15.2 cm.140

7.2 Comparison of theoretical (FDTD) and measured far zone patternsfor the circularly polarized components of the electric field for a) theunloaded CSA and b) the CSA terminated with lumped resistors (RL =300Ω) at f = 0.5GHz. The results are for Ramsdale’s CSA [17] withθ = 15, α = 60, δ = 90, and D/d = 12.77 with D = 15.2 cm. . . . . 141

7.3 Schematic model for the horizontal dipole placed over the ground. . . 1427.4 Comparison of the analytical and FDTD proximity losses for the hor-

izontal dipole over the ground. The analytical results are from [48]. . 1437.5 Comparison of the theoretical (FDTD) and measured results for the in-

put admittance of the horizontal dipole antenna over tap water arounda) the first resonance and b) the second resonance. The experimentalresults are from [49]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

7.6 Illustration on how the conical surface becomes a circular sector andvice versa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

7.7 Schematic drawings for the unwrapped surface of the cone for a) θ ≤14.5 and b) 14.5 < θ ≤ 30. . . . . . . . . . . . . . . . . . . . . . . 147

xiv

LIST OF FIGURES

7.8 a) Template for the copper etching process, b) photograph of the man-ufactured flexible circuit board. . . . . . . . . . . . . . . . . . . . . . 148

7.9 a) Photograph of the planar circuit board before it takes on the shapeof the conical surface, b) photograph of the assembled CSA. . . . . . 150

7.10 Details of the method used to feed the CSAs in the measurements. . 1517.11 a) Photograph of the rigid circuit board used to connect the feed cable

with the antenna arms, b) photograph of the feed region. . . . . . . . 1527.12 Photograph of a) the vector network analyzer and b) the balun used

in the measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1537.13 Schematic drawings for the two-port measurements of the balun: a) is

for determining S11, S12, S21, and S22, b) for S11, S13, S31, and S33, andc) for S22, S23, S32, and S33. . . . . . . . . . . . . . . . . . . . . . . . . 154

7.14 Measured results for the a) insertion loss, b) return loss, and c) phasedifference between the output ports. . . . . . . . . . . . . . . . . . . . 156

7.15 Photograph of a) the lumped resistor termination (chip resistor sol-dered to the wires) and b) the big end of the CSA when terminatedwith lumped resistors. . . . . . . . . . . . . . . . . . . . . . . . . . . 157

7.16 Schematic drawing for the measurement of the lumped resistor. . . . 1577.17 Measured results for the impedance of the lumped resistor with RL =

200Ω. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1587.18 Photograph of a) the resistive sheet, modified to properly attach at

the antenna end and b) the big end of the CSA when terminated withthe resistive sheet.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

7.19 a) Schematic model for the resistive sheet used in the measurement,b) comparison of the analytical and experimental results for the DCresistance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

7.20 Photograph of the measurement setup to determine the input impedance.1617.21 Comparison of theoretical (FDTD) and measured terminal quantities

for the unloaded antenna: a) magnitude of the reflection coefficient, b)input resistance, and c) input reactance for the CSA with θ = 7.5,α = 75, δ = 90, D/d = 8 (d = 1.9 cm). . . . . . . . . . . . . . . . . 162

7.22 Comparison of theoretical (FDTD) and measured terminal quantitiesfor the antenna terminated with lumped resistors: a) magnitude of thereflection coefficient, b) input resistance, and c) input reactance for theCSA with θ = 7.5, α = 75, δ = 90, D/d = 8 (d = 1.9 cm). . . . . . 163

7.23 Comparison of theoretical (FDTD) and measured terminal quantitiesfor the antenna terminated with a resistive sheet: a) magnitude of thereflection coefficient, b) input resistance, and c) input reactance for theCSA with θ = 7.5, α = 75, δ = 90, D/d = 8 (d = 1.9 cm). . . . . . 164

7.24 Comparison of theoretical (FDTD) and measured input reactances forthe antenna terminated with a resistive sheet. A small capacitance,C = 0.12 pF, has been added in parallel with the terminals. . . . . . . 165

xv

LIST OF FIGURES

7.25 a) Photograph of the measurement setup to determine the realizedgain, b) schematic model for the two-antenna method. . . . . . . . . 166

7.26 Drawing detailing the wavelength-dependent distance R(λ) betweenthe active regions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

7.27 Comparison of theoretical (FDTD) and measured realized gains (lin-ear) for the CSA with θ = 7.5, α = 75, δ = 90, D/d = 8(d = 1.9 cm): a) the unloaded antenna, b) the antenna terminatedwith lumped resistors, and c) the antenna terminated with a resistivesheet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

7.28 Schematic model of the dipole used in the measurement. . . . . . . . 1697.29 Measurement setup for the dipole antenna. . . . . . . . . . . . . . . . 1707.30 Schematic FDTD model of a) the dipole with square cross section

including the details of the feed region and b) the simple straight dipole.1707.31 Comparison of theoretical (FDTD) and measured results for the dipole

with the square cross section including the details of the antenna feed:a) input impedance, b) realized gain, and c) gain. . . . . . . . . . . . 172

7.32 Comparison of theoretical (FDTD) and measured results for the dipolewith the straight cross section omitting the details of the antenna feed:a) input impedance, b) realized gain, and c) gain. . . . . . . . . . . . 174

A.1 Drawing detailing the conformal mappings. . . . . . . . . . . . . . . . 179A.2 Drawing detailing the Schwarz-Christoffel transformation. . . . . . . 181A.3 Normalized DC sheet resistance as a function of the angular electrode

width δ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

B.1 Schematic drawings showing the feed region of the antenna. . . . . . 184B.2 a) Model of the linear dipole with square cross section, b) discretization

of the model (L/w = 30.5). . . . . . . . . . . . . . . . . . . . . . . . . 186B.3 Feeding techniques shown for the first three discretizations (worst case

for convergence). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189B.4 Results for the input admittance in the worst case for convergence: a)

hard-source model and b) simple feed. . . . . . . . . . . . . . . . . . 190B.5 Schematic illustration of the change of the susceptance with varying

length of the gap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190B.6 Schematic drawing for the fringing of the electric field in the vicinity

of the drive-point gap. . . . . . . . . . . . . . . . . . . . . . . . . . . 191B.7 Feeding techniques shown for the first three discretizations (best case

for convergence). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192B.8 a) Proper scaling of the reference planes in the transmission line, b)

correct virtual connection of the transmission line at the feed point. . 193B.9 Results for the input admittance in the best case for convergence: a)

hard-source model and b) simple feed. . . . . . . . . . . . . . . . . . 194

xvi

LIST OF FIGURES

B.10 Comparison of the results for the admittance: a) worst case and b)best case for convergence. . . . . . . . . . . . . . . . . . . . . . . . . 195

xvii

SUMMARY

In this research, the two-arm, conical spiral antenna (CSA) is investigated for its

use as a transmitting and receiving antenna for all applications in free-space, e.g.,

telecommunications and electromagnetic compatibility and for its use when placed

over the ground, e.g., in a ground-penetrating radar (GPR). This antenna belongs

to the class of frequency-independent antennas, and has uniform performance char-

acteristics, such as input impedance, gain, and circular polarization over a broad

frequency range. These are desirable features for the above applications. The CSA

is analyzed using the finite-difference time-domain (FDTD) numerical method and

validated by comparison with measurements of the input impedance and the realized

gain. A parametric study is performed using the FDTD analysis, and the results

from the study are used to produce new design graphs for this antenna. The graphs

supplement and extend the existing, mainly empirical, design base for this antenna.

Two resistive terminations, intended to improve the low-frequency performance, are

examined. One is a termination formed from two lumped resistors, and the other is

a new termination formed from a thin disc of resistive material. When the CSA is

placed directly over the ground with its axis normal to the surface of the ground, it

may be useful for applications in which signals must be transmitted into the ground

and/or received from within the ground. Qualitative arguments based on the geo-

metrical and electrical properties of the CSA isolated in free space show that the

beneficial properties of the CSA in free space are preserved when placed over the

ground. Results from a complete analysis of the CSA over the ground are used to

quantitatively verify these arguments. Illustrative examples are presented in which

xviii

SUMMARY

the CSA is used in a monostatic GPR.

xix

CHAPTER 1

Introduction

In this work, the two-arm, conical spiral antenna (CSA) is investigated when isolated

in free space, e.g., for its use in telecommunications and electromagnetic compatibility

[1]-[4], and when it is placed over the ground, e.g., for its use in a ground-penetrating

radar (GPR) [5]. This antenna belongs to the class of frequency-independent anten-

nas, i.e., it has uniform performance characteristics, such as input impedance, gain,

and circular polarization over a broad range of frequencies, which makes it beneficial

for the above applications.

1.1 Historical Overview

The genesis of the CSA was in the work of Rumsey during the 1950s [6, 7]. Rumsey

introduced two necessary conditions for practical, frequency-independent antennas of

this type: the angle principle and the truncation principle. Briefly stated, the angle

principle says that the performance of an antenna that is defined entirely by angles

will be frequency independent. Evidently, this implies an infinite antenna geometry

and thus an infinite frequency range of operation, so an additional consideration is

needed for practical antennas. The truncation principle says that the antenna must

have an “active region” of finite size. An active region is that part of the antenna that

contributes the most to the radiation at one frequency. As the frequency is changed,

it moves on the antenna in such a way that the dimensions of this region, expressed

1


in terms of wavelength, remain constant; hence, the performance remains constant.

The antenna performance is then practically frequency independent over the range

of frequencies for which the active region is completely contained within the finite

structure of the antenna.

In 1959, Dyson first introduced the equiangular planar spiral antenna as a prac-

tical frequency-independent antenna that is defined mainly by angles [8]. Based on

measurements, he created design curves for some important antenna characteristics,

such as the beamwidth and the axial ratio. The planar spiral antenna radiates equally

in two directions simultaneously, and for many applications an antenna is needed with

a beam in only one direction. The CSA described below satisfies this requirement.

The two-arm CSA that satisfies Rumsey’s principles for practical frequency in-

dependence is shown in Fig. 1.1. Three angles define the geometry of the infinite CSA:

the half angle of the cone θ, the wrap angle α, and the angular width of the antenna

arms δ. The practical CSA is bounded in its minimum and maximum diameters of

the truncated cone, d and D, respectively. The active region, as shown by the gray

areas at both ends of the CSA, moves along the cone towards increasing diameter

with decreasing frequency. Consequently, the highest frequency is radiated from the

small end where the CSA is usually fed and the lowest frequency is radiated from the

large end. Dyson first described this antenna in 1959 [9]-[11], and later in a seminal

paper he provided comprehensive information for its design [12]. This paper contains

an extensive, experimentally based survey of two-arm CSAs. Dyson performed a se-

ries of measurements on these antennas and used these results to construct design

graphs. An important element in this procedure was the measurement of the current

distribution on the arms of the CSA (actually the magnetic field close to the arms).

The current was then used to determine the active region, and the movement in terms

of wavelength of the active region along the arms was used to establish the bandwidth

of operation.

Dyson showed that the active region is that part of the antenna that contributes

the most to the radiation at a particular frequency. This is because the current outside

2


r1 r1

o

d

D

x

y

z

Apex

Spiral ArmSpiral Arm

Active Regionfor min d

Active Regionfor max D

Range ofActiveRegionfor FiniteAntenna

Feed Region

Figure 1.1: Geometry for the practical two-arm CSA (top) and side view of thebasic cone (bottom).

of the active region is very small at that frequency. In the simplest approximation,

the active region at a given frequency occurs where the cross section of the CSA is

roughly one wavelength in circumference: kρ = 2π(ρ/λ) ≈ 1 with d/2 ≤ ρ ≤ D/2.

Thus, the active region moves from the small end to the large end as the wavelength of

the radiation goes from λmin ≈ πd to λmax ≈ πD. The bandwidth of operation, BW,

is then approximately equal to the ratio of the maximum diameter to the minimum

diameter of the cone: BW ≈ fmax/fmin = λmax/λmin = D/d. Dyson showed that the

actual bandwidth of the CSA is somewhat less than that stated above.

3


(b)(a)

SpiralArm #1

SpiralArm #1

SpiralArm #2

SpiralArm #2

Figure 1.2: Schematic drawings for two models of the CSA: a) using thin wires,b) antenna arms with expanding arm width.

Yeh and Mei [13, 14] applied Hallen’s integral equation to the CSA to solve for

the currents on the antenna arms using the Method of Moments. In their models, the

method is often based on a thin-wire model for the CSA, see Fig. 1.2a, in which the

antenna arms are modeled with an effective wire radius. Since these model does not

scale with operating wavelength, it is not truly frequency independent. For frequency

independence, the arms must expand in width as they move away from the apex of

the cone, as shown in Fig. 1.2b.

In 1977, Miller first investigated CSAs in the time domain with a time-domain

integral equation method [15]. The main objective was to optimize the antenna for

pulsed excitation. Ideally, the incident signal should cover a wide range of frequencies,

but often, the low-frequency content of a pulsed signal causes long settling times of

the antenna response and consequently requires long execution times. Miller studied

various pulse forms that are commonly used in numerical time-domain methods and

that represent a trade-off between broadband incident signals and small settling times.

He proposed the modulated Gaussian pulse; this is a sinusoidal signal modulated with

a Gaussian envelope.

Wills proposed a two-arm CSA for submarine communication [16]. While most

researchers before him investigated the thin-wire model, Fig. 1.2a, he studied the

antenna with thin metallic, expanding antenna arms, Fig. 1.2b, empirically. In the

4

1.2 Contribution of Research

proposed submarine antenna system, the feed system was integrated within the cone.

This design was meant to operate under extreme conditions in the sea. His new ap-

proach to feed the antenna at the small end (diameter d) included a balun and an

impedance transformer. Due to their symmetry, spiral antennas are balanced struc-

tures, while the coaxial lines that are generally used to feed the antenna are inherently

unbalanced. To reduce the mismatch between the interface of both structures, a balun

must be used. The first balun implementation for CSAs was based on Dyson’s “in-

finite balun” [9]. Both Dyson’s and Wills’ balun implementations are illustrated in

Fig. 1.3. Figure 1.3a shows a photograph of one of Dyson’s experimental CSAs, in

which the feeding coaxial line is carried along the antenna arm. In order to guarantee

symmetry, an unused (dummy) coaxial line must be carried along the other antenna

arm. This technique reduces the unbalanced mode on the line significantly. Wills’

technique to suppress the unbalanced modes is based on a gradual change of the

transmission-line geometry: from a coaxial line (unbalanced structure) to a two-wire

line (balanced structure due to symmetry), as illustrated in the schematic drawing in

Fig. 1.3b.

Ramsdale also investigated the CSA with expanding arms [17]. He proposed a

new approach to improve the low-frequency performance of the antenna by termi-

nating the big end (diameter D) with lumped resistors. The objective of matching

this open end is to minimize the reflection from this end, which mainly degrades

the performance of the antenna at the lowest frequencies within the operational fre-

quency range. He showed that the termination significantly improved the impedance

mismatch and the pattern at low frequencies.


Due to the nature of this thesis, i.e., the analysis and design of the CSA using the

FDTDmethod, the thesis will mainly make contributions to the CSA with less empha-

sis on the FDTD method, which has been discussed extensively in the literature. The

5


main contributions of this research are three fold, viz, providing new design graphs

for the CSA, investigating resistive terminations for CSAs, and analyzing CSAs in

free space when placed directly over the ground. In this research, the frequency-

independent model of the CSA, i.e., with expanding arm width, is examined both

numerically and experimentally. To date, only the wire model has been studied

numerically that often does not satisfy the conditions for practical frequency inde-

pendence and that is clearly outperformed by the full model of the CSA (with the

expanding arm width).

(a)

(b)

FeedingCoaxialLine

DummyCoaxialLine

Section ThroughConductors

N-Type 50

Coaxial InputConnector

100 alanced

Line Output

DielectricInnerConductor

OuterConductor

Figure 1.3: a) “Infinite balun” introduced by Dyson [9], b) balun andimpedance transformer introduced by Wills [16].

6


A parametric study for the CSA is performed using the FDTD analysis, and the

results from this study are used to produce new design graphs. These graphs supple-

ment and extend the design information provided by Dyson [12]. The bandwidths for

the impedance and the pattern of the antenna are calculated directly from the results

for these quantities, unlike Dyson’s study, where the bandwidth is inferred from an

examination of the active region for the antenna.

The CSA has desirable features for applications in which antennas are placed

in air over the ground, see Fig. 1.4, and used to transmit signals into the ground or

to receive signals coming from within the ground. Simple physical arguments show

that the frequency-independent performance of the CSA isolated in free space can

be preserved when placed over the ground, and that the use of circular polarization

can minimize the effect of the air/ground interface on the antenna response. Thus,

parameters such as the input impedance should be the same as those when the antenna

is isolated in free space. Using simple scaling, it can be shown that the near field also

exhibits frequency independence. To date, no one has investigated the properties of

this antenna over the ground in such detail. This knowledge about the characteristics

of the CSA over the ground is used to study the CSA in a monostatic GPR.

Earth

Figure 1.4: The two-arm CSA placed in air directly over the ground.

7

1.3 Outline

In earlier work, based on limited experimental results, the resistive termination

at the open end of the antenna was shown to improve the low-frequency performance

for just two properties [17]: the impedance and the pattern. In this thesis, the effect

of two different resistive terminations on the performance of the antenna is examined

in more detail. In the first configuration, lumped resistors are connected in parallel

with the antenna arms. In the second and new configuration, a circular disc cut from

a thin sheet of resistive material connects the two arms.

1.3 Outline

The outline of this thesis is as follows. In Chapter 2, the two-arm CSA is inves-

tigated when isolated in free space. Main performance characteristics such as the

impedance, the gain, and the pattern of this antenna are presented, and the accu-

racy of the FDTD model is verified using measurements of the input impedance and

the realized gain. In Chapter 3, a parametric study is performed using the FDTD

analysis, and the results from this study are used to produce new design graphs for

this antenna. These graphs supplement and extend the empirical results of Dyson.

Two resistive terminations, intended to improve the low-frequency performance of

the CSA, are studied in Chapter 4: lumped resistors connected between the arms at

the open end, and a new configuration, a thin disc of resistive material connected to

the arms at the open end. The effect of the resistive terminations on the performance

of the CSA is examined, and some guidelines are offered for their use. In Chapter

5, qualitative arguments based on the geometrical and electrical properties of the

isolated CSA are used to determine the performance when the CSA is placed over

the ground. Results from a complete analysis of the CSA over the ground, performed

with the FDTD method, are used to quantitatively verify these arguments. Illus-

trative examples are given in which the CSA is used in a monostatic GPR. Chapter

6 is concerned with the finite-difference time-domain (FDTD) method used to ana-

lyze this antenna numerically. The typical FDTD update equations are derived for

8

1.3 Outline

the perfectly matched layer (PML) absorbing boundary condition that truncates the

open region surrounding the antenna and absorbs outgoing electromagnetic energy

with negligible reflection. Other key elements in this analysis are the near-field to

far-field transformer (for the results in free space) to obtain the radiated or far-zone

field of the antenna, and the one-dimensional transmission-line feed to conveniently

study the voltage reflected from the antenna drive point. In order to greatly reduce

the execution time of the simulation, the computer code can be parallelized to run

the simulation on a multiple-processor architecture. In Chapter 7, the FDTD method

is validated with measured results from the published literature and with new mea-

surements performed at Georgia Tech. The published results are for a CSA in free

space both unloaded and loaded (terminated using lumped resistors) and for hori-

zontal dipoles placed directly over the ground. The Georgia Tech measurements are

for a CSA in free space unloaded and loaded (terminated using lumped resistors and

a resistive sheet). For these measurements, two identical CSAs were manufactured

using flexible printed circuit board material with the metallic antenna arms on this

sheet formed by a wet chemical etch. Measured results for the input impedance and

the realized gain are used to validate the FDTD analysis.

There are two appendices to this thesis. In Appendix A, the DC resistance of

the circular resistive sheet measured between circular electrodes of angular width δ is

determined. The solution is accomplished by means of two conformal mappings. In

Appendix B, two common FDTD feed models for antennas are studied for convergence

with increasing discretization. The objective of this convergence study is to provide

guidelines for the proper positioning and scaling of elements in the feed model.

9

CHAPTER 2

The Two-Arm Conical Spiral Antenna in

Free Space

This chapter is concerned with the analysis and design of the two-arm, conical spiral

antenna (CSA) in free space using the finite-difference time-domain (FDTD) method.

First, the antenna geometry and characteristics of the radiated electric field are de-

scribed in Section 2.1. In Section 2.2, several antenna performance characteristics are

presented for one antenna geometry. To validate the numerical analysis, results for

the input impedance and the realized gain are compared with measurements.

2.1 Description of the CSA

The geometry for the two-arm CSA is shown in Fig. 2.1. The antenna is constructed

by winding two metallic strips around the surface of a truncated cone. Three angles

define the geometry of the frequency-independent CSA. These are the half angle of

the cone, θ, the wrap angle, α, and the angular width of the arms, δ. The half angle

of the cone is measured between the symmetry axis and the side of the cone. When

θ = 90, the CSA becomes a planar spiral, which radiates equally in two directions

(±z) [8]. When θ is small, the radiation from the CSA is predominantly along the

axis of the cone in the direction of the apex (−z).1 The rate of wrap of the arms

1The main radiation direction (−z) for the CSA might be unexpected; one might expect the

main beam of the antenna to be end fire, i.e., in the propagation direction of the guided wave that

10


o

d

D

y

x

z

x

y

z

Apex

Arm 2Arm 1

rd

r1 r1

R

(a)

(b) (c)

z

Figure 2.1: a) Geometry for the practical two-arm CSA, b) side view of thebasic cone, and c) top view of the highlighted planar cut.

around the conical surface is defined by the angle α. This is the angle between the

spiral arm and the radial line from the apex of the cone, as shown in Fig. 2.1a. The

third angle, δ, defines the constant angular width of the arms everywhere along the

cone and is illustrated in Fig. 2.1c, which is the top view of the highlighted plane in

Fig. 2.1a. The results in this thesis are limited to the most common configuration,

is launched at the small end and traveling towards the big end (towards +z direction). Dyson

compared the CSA with a bifilar helix and showed using a Brillouin diagram, also referred to as

k−β diagram [18, 19], that the radiation from the CSA is in fact backfire (−z) within the frequency

range of operation [12].

11


δ = 90, which is sometimes referred to as “self-complementary design.” In this case,

the metallic arms are identical in size and shape to the open regions on the conical

surface. Generally, this value gives the most desirable characteristics for radiation.

For this antenna to be practical, it must be of finite size. As shown in Fig. 2.1b,

the extent of the antenna is limited by the minimum and maximum diameters of the

cone, d and D, respectively.

The boundaries of the arms for the two-arm CSA can be described mathemati-

cally with expressions for the radial distance, r, from the apex of the cone to a point

on the conical surface [12]. As illustrated in Fig. 2.1a, r1 describes the distance from

the apex of the cone to one boundary of the first arm, and r1δ describes the distance

from the apex to the other boundary of the same arm, i.e.,

r1(φ) = rd eb|φ|, |φ| ≥ 0 (2.1)

and

r1δ(φ) = rd eb(|φ|−δ), |φ| ≥ δ. (2.2)

Here, b is defined as

b =sin θtanα

, (2.3)

and rd = d/(2 sin θ) is the radial distance from the apex of the cone to the smaller

end of the cone (diameter d). Note that the magnitude of φ is not limited to 2π.

Values of |φ| > 2π are permitted, because the exponential functions in (2.1) and (2.2)

are not periodic. The two arms are symmetric to the z axis (diametrically opposite),

so the boundaries of the second arm can be obtained by rotating the boundaries of

the first arm by the angle π, i.e.,

r2(φ) = rd eb(|φ|−π), |φ| ≥ π (2.4)

and

r2δ(φ) = rd eb(|φ|−π−δ), |φ| ≥ π + δ. (2.5)

Simple geometry can be used to determine other characteristic parameters of the

antenna, that will frequently be used throughout the analysis and the design of CSAs:

12


the height of the cone (measured from the apex of the cone to the big end)

hc =D

2 tan θ, (2.6)

the height of the CSA (measured from the small end to the big end)

hs =D − d

2 tan θ, (2.7)

and the total length of the spiral arms (measured along the surface of the cone)

Ls =D − d

2 sin θ cosα. (2.8)

The radiation from a well-designed CSA is maximum in the direction of the −z

axis, and in this direction, the electric field is predominantly circularly polarized. The

sense of the circular polarization, viz, right-handed or left-handed, is determined by

the direction in which the arms are wound around the cone. For the antenna shown

in Fig. 2.1a, the polarization is left-handed circular, and the antenna is referred to

as a left-handed CSA. There is an easy way to remember the relationship between

the state of polarization for the radiation and the direction of the winding: For an

antenna that radiates left-handed circular polarization, with the thumb of the left

hand pointing in the direction of maximum radiation (−z), the fingers, when curled

to form a fist, point in the direction the arms are traced out in going from the small

end to the large end of the cone,2 see Fig. 2.2a. The radii that describe the antenna

arms of the left-handed CSA (r1, r1δ, r2, and r2δ) are given by (2.1)-(2.5) with φ ≥ 0

and z > 0. As shown in Fig. 2.2b, for a right-handed CSA, the winding sense for the

arms is reversed; therefore, φ ≤ 0 and z > 0.

2Notice that the arms of the left-handed CSA in Fig. 2.1a are traced out with the sense of a

right-handed screw: The arms spiral around the +z axis in clockwise sense as z is increased. So

another way of remembering the relationship between the state of polarization for the radiation and

the direction of the winding is that a left-handed CSA (one that produces a left-handed circularly

polarized field in the direction for maximum radiation) has the arms wound with the sense of a

right-handed screw.

13


(b)(a)

SpiralArm #2

SpiralArm #2

SpiralArm #1

SpiralArm #1

Left-Handed CSA Right-Handed CSA

Figure 2.2: a) Schematic model for the a) left-handed CSA and b) right-handedCSA.

The characteristics of the electric field radiated from a left-handed CSA are

illustrated schematically in Fig. 2.3. Figure 2.3a is the top view looking from the big

end of the CSA down towards the small end. For simplicity, the thin-wire model of

the antenna is used for this simple argument. Since the antenna is fed at the small

end, pulses of charge propagate along each antenna arm towards the open end of the

CSA. Due to the out-of-phase excitation, one antenna arm carries a pulse of positive

charge and the other arm a pulse of negative charge, between which an electric field

can be measured. The positions of the pulses of charge and the corresponding electric

fields are shown for three instances in time, t = t1, t2, and t3, with t1 < t2 < t3. The

circular rotation of the electric field with time is evident and illustrated schematically

using polarization ellipses in Figs. 2.3b and c. Notice that the ellipse in Fig. 2.3b is

traced out counterclockwise for an observer located beyond the small end looking in

the direction of the negative z axis, while the ellipse in Fig. 2.3c is traced out clockwise

for an observer located beyond the big end looking in the direction of the positive z

axis. Both directions are directions of propagation for the corresponding waves. The

positions of the observers and their orientation are illustrated in Fig. 2.3d. Following

the convention used by the IEEE [20], the polarization ellipse in Fig. 2.3b corresponds

to left-handed circular polarization (LHCP), while the polarization ellipse in Fig. 2.3c

corresponds to right-handed circular polarization (RHCP).

14


-

-

-

+

+

+

E( )t2

E( )t1

E( )t3

Pulse ofNegativeCharge

Pulse ofPostitiveCharge

y

ExEx

x

EyEy

z

z-z

RHCPLHCP

(a)

(b) (c)

(d)

AntennaArms

xy

z

Observer (b)

Observer (c)Direction ofPropagation(- )z

Sense ofRotationfor ( )E t

Direction ofPropagation( )z

Figure 2.3: a) Schematic drawing for pulses of charge at three instances oftime (top view of CSA), b) polarization ellipse for the electric fieldfor an observer beyond the small end looking in the direction ofthe negative z axis, c) polarization ellipse for the electric field foran observer beyond the large end looking in the direction of thepositive z axis, and d) illustration for the position and orientationof the observers.

15

2.2 Characteristics of the CSA in Free Space


In this section, important antenna parameters are presented to demonstrate the ben-

eficial properties of CSAs, such as uniform impedance, unidirectional radiation, and

circular polarization over a broad range of frequencies. The results presented are for

the left-handed CSA with the following parameters: θ = 7.5, α = 75, δ = 90,

D/d = 8 with d = 1.9 cm, and Zc = 100Ω. A pair of identical CSAs with above pa-

rameters was constructed, and measurements of the input impedance and the realized

gain of these antennas were used to validate the FDTD analysis. The details of the

measurements (construction, feeding, results) will be discussed in detail in Chapter

7, while only some results are presented here.

In this research, the antenna is excited in the feeding transmission line by an

incident differentiated Gaussian voltage pulse of the form

Vinc(t) = −V

(t

τp

)e0.5−0.5(t/τp)

2

, (2.9)

where τp is the characteristic time for the pulse. This pulse is shown in Fig. 2.4a as

a function of normalized time, t/τp, and its spectrum3 in terms of ωτp is shown in

Fig. 2.4b. From the spectrum, it can be seen that this pulse does not contain signifi-

cant low-frequency components that can cause long settling times in the simulation.

The ratio of characteristic times between the pulse (τp) and the antenna (τL = Ls/c)

is chosen to be τp/τL = 1.6 · 10−2, i.e., approximately 14 pulse widths fit along theantenna arm length.

All of the results from the FDTD analysis are time-varying signals. As an exam-

ple, the θ component of the far-zone electric field, Erθ , in the direction for maximum

radiation (−z) is shown in Fig. 2.4c. Notice that this is a chirp signal with the higher

frequencies arriving before the lower frequencies. This is what one would expect: The

3The analytical expression for the spectrum of (2.9) is

Vinc(ω) = jV√2π τp(ωτp) e0.5−0.5(ωτp)2

16


-4 -2 0 2 4

-1.0

-0.5

0.0

0.5

1.0

t / p

Vin

c(

)/

tV

o

(a)

t / L

0.0 0.5 1.0 1.5 2.0

-1.0

-0.5

0.0

0.5

1.0

E,

no

rm

()t

r

(c)

-4 -2 0 2 4

-1.0

-0.5

0.0

0.5

1.0

p

(b)

j VV

inc(

)/ (

)

o

Figure 2.4: a) Differentiated Gaussian pulse as a function of normalized time,t/τp, b) spectrum of the differentiated Gaussian pulse as a functionof normalized frequency, ωτp, and c) far-zone electric field on axis.

higher frequencies are radiated near the small end of the cone so they travel a shorter

distance to reach the far zone than the lower frequencies that are radiated near the

large end of the cone. The frequency-domain results for the CSA are obtained by

Fourier transforming signals such as those shown in Fig. 2.4a and c.

The first result for the CSA is the reflected voltage as a function of normalized

17


-1 0 1 2 3 4 5 6

-0.2

-0.1

0.0

0.1

0.2

0.3

t / L

Vre

fo

/V

Drive PointReflection

Second EndReflection

First EndReflection

Figure 2.5: Theoretical results for the reflected voltage at the terminals of theunloaded CSA.

time, t/τL, see Fig. 2.5. The reflection at t/τL ≈ 0 is caused by the drive point, i.e.,

the interface between the feeding transmission line and the antenna terminals. A

pulse with an amplitude of roughly 80% of the incident pulse is transmitted through

this interface. The reflection from the antenna end enters the antenna response at

integer multiples of the round-trip time along the antenna arms, i.e., multiples of 2τL.

Notice that the two end reflections are small and thus suggest that most of the signal

is radiated by the CSA.

In Fig. 2.6, the FDTD results (solid line) and measured results (dashed line)

for the magnitude of the reflection coefficient and the input impedance (resistance

and reactance) of the CSA with θ = 7.5, α = 75, δ = 90, D/d = 8 (d =

1.9 cm) are graphed as a function of frequency (logarithmic scale). The resistance

is about 150Ω and the reactance is small over a large range of frequencies, i.e.,

0.5GHz ≤ f ≤ 3.5GHz.4 Notice that the FDTD results match the measured results

“ripple for ripple,” although there is some offset of the two curves, which will be

discussed later. At frequencies below 0.5GHz, the large oscillations in the resistance

4Most frequency domain results in this section are presented over the frequency range 0.3GHz ≤f ≤ 3.5GHz. For comparison, the range determined from the active region, in its simplest approxi-mation, extends from fmin = c/(πD) = 0.6GHz to fmax = c/(πd) = 5.0GHz.

18


0.3 0.5 1 1.5 2 2.5 3 3.5

0.00

0.25

0.50

0.75

1.00

f [GHz]

||

0.3 0.5 1 1.5 2 2.5 3 3.5

0

50

100

150

200

f [GHz]

R[

]

0.3 0.5 1 1.5 2 2.5 3 3.5

-100

-50

0

50

100

f [GHz]

X[

]

(a)

(b)

(c)

FDTD

Measurement

Figure 2.6: Comparison of theoretical (FDTD) and measured terminal quan-tities: a) magnitude of the reflection coefficient, b) input resis-tance, and c) input reactance for the unloaded CSA with θ = 7.5,α = 75, δ = 90, D/d = 8 (d = 1.9 cm).

19


and reactance, Figs. 2.6b and c, are caused by reflections from the open end. At

frequencies greater than about 2.0GHz, the differences in the numerical and measured

results are probably caused by small differences in the geometry of the feed region in

the FDTD and experimental models. The good match of the CSA to the transmission

line is illustrated further in the Smith chart in Fig. 2.7. The impedance is plotted over

the frequency range 0.3GHz ≤ f ≤ 3.5GHz on the complete Smith chart in Fig. 2.7a

and plotted over the frequency range 0.5GHz ≤ f ≤ 3.5GHz on the expanded Smith

chart in Fig. 2.7b. Two points are marked on the graphs, the impedance for the

frequencies 0.5GHz and 1.0GHz. Notice that the CSA is already fairly well matched

for f ≥ 0.5GHz.

To give an idea of how sensitive the input impedance is to small changes in

the feed region, the reactance was computed with a small capacitance, C = 0.12 pF,

added in parallel with the terminals. Results for this case are shown in Fig. 2.8, and

they should be compared with those in Fig. 2.6c. Notice that the capacitance has

shifted the FDTD results for the reactance downward so that they are now in better

agreement with the measurements. To put this amount of capacitance in perspective,

it is roughly equivalent to the capacitance of a 1mm length of one of the semi-rigid

coaxial lines (each about 3m long) used in the feeding network in the experimental

studies, which will be described in Chapter 7.

In Figs. 2.9–2.12, features of the pattern are shown, e.g., the far-zone electric

field, the front-to-back ratio (FTBR), and the gain. First, an overview of the different

electric fields are given that are used to describe and characterize the radiation from

the antenna [21]. The electric field in the far zone of the antenna is usually expressed

using the spherical components: Erφ = Er

φ ejφφ and Er

θ = Erθ e

jφθ . In these expressions,

the superscript r denotes the far-zone or radiated electric field component, and the

tilde is used for a complex phasor. Assuming an ejωt time dependence, the field vector

of the radiated electric field propagating in the r direction can thus be written as

&Er =(Er

θ θ + Erφ φ

)e−jkr, (2.10)

20


j

j

j

2

1

1

j

j

j

j

j

j

0.5

0.5

0.2

0.2

0.2

0.2

0

-

-

-

-

-

j

j

0.5

0.5

-j 2

1

1

2

2

50.50.2

f = 0.5 GHz

f = 1.0 GHz

(a)

(b)

Figure 2.7: Theoretical results for the input impedance plotted on a Smithchart for a) 0.3GHz ≤ f ≤ 3.5GHz and b) 0.5GHz ≤ f ≤ 3.5GHz

where k = ω/c = 2π/λ is the wave number and r the radial distance to the observation

point in the far zone of the antenna. This expression can always be rewritten as a

combination of right-handed and left-handed circularly polarized components, i.e.,

&Er = ErL (θ + φ ejπ/2) e−jkr + Er

R (θ + φ e−jπ/2) e−jkr (2.11)

= &ErLHCP e−jkr + &Er

RHCP e−jkr, (2.12)

with

ErR = Er

R ejφR =Er

θ + jErφ

2(2.13)

21


0.3 0.5 1 1.5 2 2.5 3 3.5

-100

-50

0

50

100

f [GHz]

X[

]

FDTD

Measurement

Figure 2.8: Comparison of theoretical (FDTD) and measured input reactances.A small capacitance, C = 0.12 pF, has been added in parallel withthe terminals.

and

ErL = Er

L ejφL =Er

θ − jErφ

2. (2.14)

It is important to note that the magnitudes of the circularly polarized field com-

ponents, used exclusively in this research, are ErRHCP = | &Er

RHCP| =√2Er

R and

ErLHCP = | &Er

LHCP| =√2Er

L, respectively, which are due to the linear combination

of the orthogonal unit vectors θ and φ in (2.11).

In Fig. 2.9, vertical-plane (φ = 0), far-zone patterns are shown for three dif-

ferent frequencies (f = 0.75GHz, 1.50GHz, and 2.25GHz). The patterns are given

for both of the circularly polarized components of the electric field, viz, left-handed

(ErLHCP, solid line) and right-handed (E

rRHCP, dashed line). For this left-handed CSA,

the LHCP field is clearly dominant, and the radiation is concentrated near the −z

direction.

The results for the front-to-back ratio are presented in Fig. 2.10. It is important

to note that this ratio is referred to two different radiated electric fields.5 In the first

5The front direction is assumed to be the direction of maximum radiation (−z or equivalently θ =

180).

22


0-10-20

0

-10

-20

0 -10 -20

0

-10

-20

0

30

6090

120

150

180

210

240270

300

330

0-10-20

0

-10

-20

0 -10 -20

0

-10

-20

0

30

6090

120

150

180

210

240270

300

330

0-10-20

0

-10

-20

0 -10 -20

0

-10

-20

0

30

6090

120

150

180

210

240270

300

330

LHCPRHCP

dB

dB

(a)

(b)

(c)

dB

Figure 2.9: Theoretical results for the far-zone pattern for the right-handedand left-handed circularly polarized components of the electric fieldat the frequencies: a) f = 0.75GHz, b) f = 1.50GHz, and c)f = 2.25GHz.

23


interpretation, shown in Fig. 2.10a, the front-to-back ratio is defined by

FTBR = 20 dB log

∣∣∣∣∣Er(θ = 180)Er(θ = 0)

∣∣∣∣∣ (2.15)

and is referred to the total electric field in the far zone of the antenna, Er = | &Er|,see (2.10), while in the second interpretation, shown in Fig. 2.10b, the front-to-back

ratio is defined by

FTBRLHCP = 20 dB log

∣∣∣∣∣ErLHCP(θ = 180

)ErLHCP(θ = 0

)

∣∣∣∣∣ (2.16)

and is referred to the main circularly polarized component of the radiated electric field,

viz, the left-handed circularly component, ErLHCP = | &Er

LHCP|, see (2.11). Note thatthe different interpretations of the front-to-back ratio are graphed on a logarithmic

scale. As seen previously in the far-zone plots for the electric field, the radiation in

the main radiation direction (−z) is orders of magnitude larger than that in the back

direction (z). Also, the front-to-back ratio is significantly higher when it is referred

to the main circularly polarized component instead of the total field.

The axial ratio, AR, for the electric field, is defined as the ratio of the major

axis OA to minor axis OB of the polarization ellipse [21], see Fig. 2.11a:

AR =OA

OB=

1

tan[0.5 sin−1(sin δE sin 2γE)], (2.17)

where

δE = Eφ(ω)

Eθ(ω)

, (2.18)

and

tan γE =

∣∣∣∣∣Eφ(ω)

Eθ(ω)

∣∣∣∣∣ . (2.19)

For circular polarization (γE = π/4, δE = ±π/2), the axial ratio is AR = 1.0. The

theoretical results for the axial ratio are shown as a function of frequency in Fig. 2.11b.

Sketches for the polarization ellipse in the top of Fig. 2.11b further illustrate the

polarization of the CSA. The graphs in Figs. 2.10 and 2.11 confirm what was observed

earlier: the CSA radiates circular polarization in predominantly one direction, i.e., in

the direction of the apex of the cone.

24


0.3 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0

10

20

30

40

(a)

FT

BR

[dB

]

f [GHz]

FT

BR

[dB

]L

HC

P

f [GHz](b)

0.3 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0

10

20

30

40

50

60

70

Figure 2.10: Theoretical results for the front-to-back ratio referred to a) the totalfar-zone electric field and b) the left-handed circularly polarizedcomponent of the far-zone electric field.

The last quantities to be examined are the gain, realized gain, and directivity.

The gain, G, is defined by the IEEE [22] as “the ratio of the radiation intensity,

in a given direction, to the radiation intensity that would be obtained if the power

accepted by the antenna were radiated isotropically.” It therefore does not include

the mismatch of the feeding transmission line; often however, the realized gain, Gr,

is of interest, which includes the mismatch. For the CSA isolated in free space, the

realized gain in the direction of maximum radiation is written as

Gr =(2π/η) |rEr(θ = 180)|2

Pavail

, (2.20)

25


(b)

AR

f [GHz]

0.3 0.5 1.0 1.5 2.0 2.5 3.0 3.5

1.00

1.25

1.50

1.75

2.00CircularPolarization

O

AB

Ex( )t

Ey( )t

z

PolarizationEllipse

(a)

Figure 2.11: a) Schematic drawing for the polarization ellipse, b) theoreticalresults for the axial ratio for the electric field in the far zone of theCSA.

where η =√µ/ε is the wave impedance of free-space, Pavail = Pinc = |Vinc|2/(2Zc)

the available power supplied by the generator to the feeding line, and Er = | &Er| themagnitude of the total electric field in the far zone of the antenna. Similarly, the gain

is then written as

G =(2π/η) |rEr(θ = 180)|2

Pant

, (2.21)

where Pant = (1−|Γ|2)Pavail is the power accepted by the antenna (incident minus re-

flected power). The directivity, defined by the IEEE [22] as “the ratio of the radiation

intensity in a given direction from the antenna to the radiation intensity averaged over

all directions,” is determined solely from the pattern of the antenna and is written as

D =(2π/η) |rEr(θ = 180)|2

Prad

=G

η, (2.22)

26


where Prad is the total power radiated by the antenna. Notice, that the directivity

can be related to the gain using the efficiency of the antenna:

η =G

D=

Prad

Pant

=Pant − Pdiss

Pant

, (2.23)

where Pdiss is the power dissipated in the antenna. In this research, the dissipation is

assumed to be solely caused by the conduction loss in the resistive termination, see

Chapter 4. Since the CSA in this section is studied without any termination at the

big end (η = 1), the directivity for this unloaded CSA equals the gain.

In Fig. 2.12a, the FDTD results (solid line) and measured results (dashed line)

for the realized gain in the direction for maximum radiation (−z) are graphed as

a function of the frequency, and the theoretical results for the gain are graphed in

Fig. 2.12b. Notice that the different forms of the gain are displayed on a linear scale.

The agreement between the numerical and the measured results is seen to be good

(within about 1 dB). For frequencies f ≥ 0.7GHz, the difference between the gain and

the realized gain is small since the antenna is fairly well matched to the transmission

line, see Fig. 2.6. However, for lower frequencies, f ≤ 0.7GHz, the results for the

gain and realized gain differ due to the mismatch. While the realized gain decays

smoothly, the result for the gain has ripples superimposed, that are similar to those

seen in the results for the impedance, Figs. 2.6b and c.

Several important performance characteristics of the CSA with θ = 7.5, α =

75, δ = 90, D/d = 8 (d = 1.9 cm) and Zc = 100Ω were studied. When all perfor-

mance characteristics are considered, the frequency range of acceptable performance is

seen to be limited at the low end by the impedance mismatch, see Fig. 2.6, and limited

at the high end by the gain, see Fig. 2.12. Thus, the frequency range of operation with

acceptable performance for this CSA approximately extends from 0.5GHz to 3.5GHz,

which corresponds to a bandwidth of BW = 7. Notice that this bandwidth, which

is determined from actual antenna performance characteristics, is significantly larger

than that suggested by Dyson, which is determined from the movement of the active

region with frequency, i.e., BW = 3.6 with fmin = 0.7GHz and fmax = 2.4GHz.,

27


0.3 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0

2

4

6

8

0.3 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0

2

4

6

8

G

f [GHz](b)

f [GHz]

Gr

(a)

FDTD

Measurement

Figure 2.12: a) Theoretical and measured results for the realized gain and b)theoretical results for the gain. The results are for the direction ofmaximum radiation (−z).

but smaller than that predicted by the simple approximation (BW = D/d = 8 with

fmin = 0.6GHz and fmax = 5.0GHz).

28

CHAPTER 3

Parametric Study and Design Graphs

In this chapter, a parametric study for the unloaded CSA is performed using the

FDTD analysis. The results from this study are then used to produce design graphs,

intended to supplement and extend the graphs presented by Dyson [12]. An important

difference between the approach used here and Dyson’s is the method for determin-

ing the bandwidth of the antenna. Dyson measured the current distribution on the

antenna and determined the bandwidth from an examination of the movement of the

active region with frequency. Here, the voltage standing wave ratio (VSWR) and the

directivity are calculated, and the variation of these quantities with frequency is used

to establish the overall bandwidth.

In the parametric study, the angles that define the geometry of the antenna cover

the following ranges: 5 ≤ θ ≤ 15 and 60 ≤ α ≤ 80, with δ = 90. The remaining

parameter, the ratio of the largest diameter to the smallest diameter, D/d, is chosen

in the following manner. A simple analysis, which is presented at the end of this

chapter, shows that with the angles held fixed, the scaled bandwidth, BW/(D/d), for

a particular parameter, e.g., VSWR, is independent of the ratio D/d for sufficiently

large D/d. Thus, there is some flexibility in the choice of D/d used in the study.

Values in the range 5 ≤ D/d ≤ 7 are selected, because they make the best use of the

available computational resources.

For frequency-independent antennas, the bandwidth is often defined as the ratio

of the maximum to minimum frequencies, at which the performance of the antenna

29


f

Directivity

Dmax

0.0

fminDir fmax

Dir

Dmax

2

f1.0

1.5

fminVSWR fmax

VSWR

(a)

(c)

(b)

f

fminfmax

maxmin

fminfmax

maxmin

Dir Dir

Dir Dir

VSWR VSWR

VSWR VSWRRange for Hybrid Bandwidth

VSWR ( = )Z Rc

Figure 3.1: Frequencies that define various bandwidths: a) the bandwidth forVSWR, b) the bandwidth for directivity, and c) the hybrid band-width.

meets some minimum requirement: BW = fmax/fmin. Two bandwidths were exam-

ined in this study, one based on the voltage standing wave ratio, VSWR, and the

other one based on the directivity, D, in the direction for maximum radiation.1 The

bandwidth for the VSWR is obtained in the following manner. The input impedance

is determined with the antenna fed from a transmission line with Zc = R∞; here, the

overbar indicates an average over the operational bandwidth of the antenna. That is,

the characteristic impedance of the line is set equal to the average input resistance

of the infinitely-long CSA, which will discussed in more detail in Section 4.1. Briefly

stated, R∞ is the optimum resistance for the feeding transmission line to reduce the

mismatch between the line and the antenna over a wide range of frequencies. This

is also the optimum resistance to terminate the big end of the antenna in order to

1The directivity is calculated from the power density of the total field, viz, the sum of the left-

and right-handed circularly polarized components. Of course, for the left-handed CSA, the field is

predominantly left-handed circularly polarized.

30


reduce the end reflections. The VSWR (for Zc = R∞) is then calculated from the

input impedance,2 and, as shown in Fig. 3.1a, the bandwidth is determined from the

frequencies at which the VSWR exceeds 1.5: BWVSWR = fVSWRmax /fVSWR

min . The band-

width for the directivity is determined, as shown in Fig. 3.1b, from the frequencies

at which the directivity dropped to one half (−3 dB) of its maximum value, Dmax:

BWDir = fDirmax/fDirmin.

For the purpose of design, a single bandwidth is desirable. This was chosen

to be the region where the bandwidths for the VSWR and the directivity over-

lapped; that is, where the requirements on the VSWR and on the directivity are

met simultaneously, see Fig. 3.1c. This region is referred to as the “hybrid band-

width,” BWhybrid = fhybridmax /fhybridmin , and its limits are fhybridmin = maxfVSWRmin , fDirmin,

and fhybridmax = minfVSWRmax , fDirmax. For all cases examined in this study, it can be found

fhybridmin = fVSWRmin and fhybridmax = fDirmax. For convenience, the superscript “hybrid” will

be omitted, and the bandwidth and limiting wavelengths will be normalized in the

following manner: BW/(D/d), λmin/(πd), and λmax/(πD).

The normalized bandwidth and the limiting wavelengths are shown as functions

of the two angles θ and α based both on the VSWR, Fig. 3.2, and on the directivity,

Fig. 3.3. Figure 3.4 is for the combination of the two, i.e., for the hybrid results. In

these design graphs, the light gray areas bounded by solid lines are the results from

this study, and the dark gray areas bounded by dashed lines are Dyson’s results. For

clarity, only the curves for α = 60 and 80 are shown; linear interpolation can be used

between these two values. When the shortest and largest limiting wavelengths for the

results based on the VSWR and the directivity are compared, Figs. 3.2 and 3.3, they

2It is not necessary to perform a separate FDTD calculation just to determine the VSWR for a

particular characteristic impedance when the VSWR for the same antenna is already known for a

different characteristic impedance. In both cases, the input impedance must be the same since it is

independent of the characteristic impedance of the feeding line when the simple FDTD feed model

is used. Simple transmission-line equations can then be used to renormalize the existing VSWR to

any characteristic impedance.

31


5.0 7.5 10.0 12.5 15.0

0.00

0.25

0.50

0.75

1.00

1.25

= 80o

= 60oThis Study

Dyson = 60

o

= 80o

5.0 7.5 10.0 12.5 15.0

0.0

1.0

2.0

3.0

4.0

o

o

BW

/(/

)V

SW

RD

d

m

in/

d

= 80o

= 80o

= 60o

= 60 o

5.0 7.5 10.0 12.5 15.0

0.0

0.5

1.0

1.5

o

max

/D

= 60o

= 80 o

= 80o

(a)

(b)

(c)

= 60o

VS

WR

VS

WR

VSWR

VSWR

VSWR

Figure 3.2: Design graphs for the unloaded CSA based on the results for theVSWR (Zc = R∞) showing a) the bandwidth, b) the maximumwavelength, λmax, and c) the minimum wavelength, λmin, as func-tions of the angles θ and α, with δ = 90.

32


5.0 7.5 10.0 12.5 15.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

= 80o

= 60o

= 60o

= 80o

5.0 7.5 10.0 12.5 15.0

0.00

0.25

0.50

0.75

1.00

1.25

1.50

Dyson = 80

o

= 60o

= 80o

= 60o This Study

o

o

BW

/(/

)D

irD

d

m

in/

d

o

max/

D

(a)

(b)

(c)

5.0 7.5 10.0 12.5 15.0

0.0

1.0

2.0

3.0

4.0

= 80o

= 80o

= 60 o

= 60o

Dir

Dir

Directivity

Directivity

Directivity

Figure 3.3: Design graphs for the unloaded CSA based on the results for the di-rectivity showing a) the hybrid bandwidth, b) the maximum wave-length, λmax, and c) the minimum wavelength, λmin, as functions ofthe angles θ and α, with δ = 90.

33


5.0 7.5 10.0 12.5 15.0

0.0

1.0

2.0

3.0

4.0

5.0 7.5 10.0 12.5 15.0

0.00

0.25

0.50

0.75

1.00

o

o

BW

/(/

)D

d

m

in/

d

= 80o

= 80o

= 80o

= 80o

= 60o

= 60o

= 60o

= 60 o

5.0 7.5 10.0 12.5 15.0

0.0

0.5

1.0

1.5

o

max/

D

= 60o

= 80 o

= 80o

= 60o

This Study

Dyson

(a)

(b)

(c)

Hybrid

Hybrid

Hybrid

Figure 3.4: Design graphs for the unloaded CSA showing a) the hybrid band-width, b) the maximum wavelength, λmax, and c) the minimumwavelength, λmin, as functions of the angles θ and α, with δ = 90.

34


o

5.0 7.5 10.0 12.5 15.0

0

25

50

75

100

125

150

175

= 80o

Dyson

= 80o

= 60o

This Study

|X |

= 60o

R|X

|,

[]

R R

Figure 3.5: Design graph for the unloaded CSA showing the average inputimpedance of the infinitely-long CSA as a function of the anglesθ and α, with δ = 90.

are seen to extend over different ranges. As illustrated qualitatively in Fig. 3.1, the

operational range of the antenna based on the directivity is shifted towards larger

wavelengths/lower frequencies relative to the range based on the VSWR. In both

these cases, the simplest approximation for the bandwidth, discussed in Section 2.1,

describes the performance of the CSA pretty well, i.e., BWVSWR ≈ BWDir ≈ D/d.

Also notice, that the range of acceptable performance is significantly wider than that

suggested by Dyson. The hybrid results from this study, Fig. 3.4, indicate a useful

bandwidth that is at least 50% greater than that proposed by Dyson. In this case,

the simple approximation, BW = D/d, clearly overestimates the bandwidth.

In Figs. 3.5-3.7, each of the parameters displayed was obtained by averaging

over the hybrid bandwidth. Figure 3.5 shows the average input impedance, R∞ and

|X∞|, of the infinitely long CSA as a function of the angle θ with the angle α as a

parameter. For the ranges displayed, both of these angles are seen to have very little

effect on the input impedance. The input resistance is R∞ ≈ 150Ω, and the reactance

is small, |X∞| < 25Ω. It is important to mention that the input resistance of the CSA

of finite length, averaged over the hybrid bandwidth, that is R, is essentially the same

as the average resistance for the infinitely long CSA, R∞. Hence, for the purposes of

35


design, Fig. 3.5 can be used as a graph for the input impedance of the CSA of finite

length: Z = R + jX ≈ R∞, since |X| R. The three values of measured input

resistance from Dyson’s work (solid dots) are seen to agree with the values from this

study.

Figure 3.6 shows three characteristics of the pattern of the CSA: the average

directivity in the direction for maximum radiation (−z), D, the average half-power

beamwidth, HPBW, averaged over two orthogonal planes (φ = 0 and φ = 90), and

the front-to-back ratio, FTBR.3 These quantities are plotted as functions of the angle

θ with the angle α as parameter. Note that the directivity is plotted on a linear scale,

and the front-to-back ratio is plotted on a logarithmic scale. Again, the solid lines

are the results from this study, and the dashed lines are Dyson’s results. The average

directivity for this study exceeds that from Dyson’s study; the same is true for the

half-power beamwidth, which agree better with Dyson’s results if the results for this

study are averaged over Dyson’s bandwidth.

The state of polarization is given by the axial ratio, AR. The average axial ratio

for the electric field in the direction of maximum radiation (−z) is shown in Fig. 3.7.

For circular polarization AR = 1.0. Clearly, as the sketches for the polarization ellipse

at the two extremes show, the radiation is very nearly circularly polarized over the

whole range of angles displayed.

All of the design graphs, Figs. 3.2-3.7, show that the best performance is ob-

tained from the CSA when the angle of the cone, θ, is small, and the arms are tightly

wrapped, viz, the angle α is close to 90.

The following analysis shows that the results for the normalized bandwidth,

BW/(D/d), and limiting wavelengths, λmax/(πD) and λmin/(πd), are independent of

the ratio of biggest to smallest cone diameter, D/d. To show that these results hold

for any D/d, if chosen sufficiently large, the performance characteristics VSWR and

directivity are examined for three different CSA geometries and for three different

3All of these quantities are calculated from the power density of the total field, viz, the sum of

the left-handed and right-handed circularly polarized components.

36


5.0 7.5 10.0 12.5 15.0

0

2

4

6

8

10

= 60o

= 80 o

= 60 o

= 80 o

5.0 7.5 10.0 12.5 15.0

1

1

0

20

30

40

o

FT

BR

[dB

]

= 80 o

D

o

This Study

Dyson

5.0 7.5 10.0 12.5 15

=

60 o

.0

0

45

90

135

180

o

= 60o = 60

o

= 80o = 80

o

HP

BW

[Deg

.]

(a)

(b)

(c)

Figure 3.6: Design graphs for the unloaded CSA showing a) the average di-rectivity, b) the average half-power beamwidth, and c) the averagefront-to-back ratio as functions of angles θ and α, with δ = 90.

37


o

AR

5.0 7.5 10.0 12.5 15.0

1.0

1.1

1.2

1.3

1.4

1.5

= 80o

= 60o

CircularPolarization

Figure 3.7: Design graph for the CSA showing the average axial ratio as afunction of the angles θ and α, with δ = 90.

D/d, i.e., D/d = 5, 7, and 9. The CSA geometries are chosen to cover the range of

values studied for θ and α, i.e., the studied cases are CSAs with i) θ = 15, α = 80,

ii) θ = 7.5, α = 75, and iii) θ = 5, α = 60. The angular width is δ = 90 and

for the results plotted in Fig. 3.8, the smallest diameter of the cone is d = 2 cm. For

the above CSA geometries, the results for the voltage standing wave ratio and the

directivity are presented in Fig. 3.8 as a function of frequency with the ratio D/d as

parameter. Notice that by keeping the small end of the CSA constant (d = 2 cm),

the antenna performance at the high end of the operational frequency range is almost

the same for all D/d, while the range of acceptable performance extends to lower

frequencies with increasing D/d according to the scaling behavior of the CSA. Key

results from the parametric study presented in this work, i.e., the hybrid bandwidth

and its limiting normalized wavelengths, are listed in Table 3.1 for the studied cases

and clearly show that the results of the parametric study are fairly independent of

the parameter D/d.

To illustrate the use of these design graphs, the CSA will be considered that

was built and measured, as discussed in Section 7. The parameters for this antenna

are θ = 7.5, α = 75, δ = 90, D/d = 8 with d = 1.9 cm, and Zc = 100Ω. From

Fig. 3.4, the bandwidth is determined to be BW/(D/d) ≈ 0.79 or BW ≈ 6.3, with the

38


f [GHz]

f [GHz]

f [GHz]

f [GHz]

f [GHz]

f [GHz]

0.4 1.0 2.0 3.0 4.0

0

1

2

3

4

5

0.4 1.0 2.0 3.0 4.0

0

2

4

6

8

0.3 1.0 2.0 3.0 4.0

0

1

2

3

4

5

0.3 1.0 2.0 3.0 4.0

0

2

4

6

8

0.2 1.0 2.0 3.0 4.0

0

1

2

3

4

5

0.2 1.0 2.0 3.0 4.0

0

2

4

6

D dD dD d

/ = 5/ = 7/ = 9

D dD dD d

/ = 5/ = 7/ = 9

VS

WR

VS

WR

VS

WR

DD

D

(a) (b)

(c) (d)

(e) (f)

Figure 3.8: Theoretical results for the VSWR (a,c,e) and the directivity (b,d,f)as a function of frequency with D/d as parameter. The results arefor a CSA with a) and b) θ = 15, α = 80, δ = 90, c) and d)θ = 7.5, α = 75, δ = 90, and e) and f) θ = 5, α = 60, δ = 90.

39


Table 3.1: Hybrid bandwidth results as a function of D/d for a CSA with a)θ = 15, α = 80, δ = 90, b) θ = 7.5, α = 75, δ = 90, and c)θ = 5, α = 60, δ = 90.

θ = 15, α = 80

D/d = 5 D/d = 7 D/d = 9

λmin/πd 1.41 1.42 1.42

λmax/πD 0.88 0.94 0.95

BW/(D/d) 0.62 0.66 0.67

(a)

θ = 7.5, α = 75

D/d = 5 D/d = 7 D/d = 9

λmin/πd 1.49 1.50 1.46

λmax/πD 1.15 1.14 1.14

BW/(D/d) 0.77 0.76 0.78

(b)

θ = 5, α = 60

D/d = 5 D/d = 7 D/d = 9

λmin/πd 2.13 2.13 2.15

λmax/πD 1.41 1.36 1.35

BW/(D/d) 0.66 0.64 0.63

(c)

following limits for the useful frequency range λmax/(πD) ≈ 1.21 or fmin ≈ 0.52GHz,

and λmin/(πd) ≈ 1.54 or fmax ≈ 3.3GHz. From Fig. 3.5, the input impedance is

Z ≈ R ≈ 150Ω, and from Fig. 3.6a, the average directivity should be D ≈ 6.2. The

average realized gain is easily calculated from these results:

Gr = D(1− |Γ|2) = D

1−

∣∣∣∣∣R− Zc

R + Zc

∣∣∣∣∣2 ≈ 6.0. (3.1)

40


Now these predictions can be compared with the measured results in Figs. 7.21 and

2.12a. Both the prediction for the resistance and the prediction for the realized gain

are seen to be close to the measured values when the latter are averaged over the

specified frequency range: 0.52GHz ≤ f ≤ 3.3GHz.

41

CHAPTER 4

Resistive Terminations for the CSA

In this chapter, two resistive terminations for the open end of the CSA are examined,

and the impact of these terminations on the antenna performance is studied. In

Section 4.1, the two resistive terminations are examined. A simple argument suggests

that when the open end of the antenna is terminated with an optimal load resistance,

the low-frequency performance can be improved. In Section 4.2, several performance

characteristics for the resistively terminated CSAs are studied in detail and compared

to those obtained for the unloaded CSA, presented previously in Section 2.2

4.1 Methods for Terminating the CSA

When a signal propagating along the arms of the CSA reaches the open end (diameter

D), it is reflected back towards the small end where the CSA is fed. Since the

largest wavelengths within the bandwidth of operation are radiated near the open

end, this reflection mainly degrades the low-frequency performance of the antenna.

The reflection can be reduced by terminating the antenna with a load resistor Rload,

see Fig. 4.1a. A practical choice for Rload is obtained using the following simple

argument. Consider the model for the CSA shown in Fig. 4.1b. In this model, the

antenna is a uniform transmission line with characteristic impedance Zc,CSA. An

estimate for Zc,CSA is determined from the input impedance of the infinitely long

CSA: Z∞ = R∞ + jX∞ ≈ R∞. We choose Zc,CSA = R∞, where the overbar indicates

42

4.1 Methods for Terminating the CSA

Zc

Rload

Zc

Rload

(a) (b)

FeedSystem

RZc,CSA

Figure 4.1: a) The loaded CSA including the feed system, b) simpletransmission-line model.

an average over the operational bandwidth of the CSA. To reduce the end reflection,

the load resistance is set equal to Zc,CSA: Rload = R∞. The value of R∞ is easily

obtained from the FDTD simulation. The antenna is excited by an incident pulse in

the feeding transmission line. The width of the pulse in time and the duration of the

calculation are chosen so the end reflection does not appear in the reflected voltage

at the terminals of the antenna (it is “windowed out”). The Fourier transform of the

reflected voltage is used to obtain R∞ and R∞.

The two configurations used to implement the resistive termination are shown

in Fig. 4.2. In the first configuration, Fig. 4.2a, lumped resistors of value RL connect

the closest edges of the two arms. Since there are two resistors in parallel, we must

choose RL = 2R∞. In the second configuration, Fig. 4.2b, a disc cut from a thin

sheet of resistive material connects the two arms. The sheet is characterized by its

resistance per square Rs = 1/σsts, where σs is the electrical conductivity, and ts is the

thickness. The DC resistance of this termination, as measured between the two arms

(between the perfectly conducting arcs of angle δ), is set equal to R∞. An analysis

based on conformal mappings, which is summarized in Appendix A, shows that the

43

4.2 Characteristics of the Loaded CSAs

Resistive Sheet

D

Rs sts

1=

(b)

LumpedResistor

y

x

z

Spiral Arms

RL

RL

(a)

Figure 4.2: Schematic drawings for the two resistive terminations: a) twolumped resistors connecting the ends of the spiral arms, b) a re-sistive sheet covering the entire end of the cone.

resistance per square should be

Rs = R∞K

(k =

√1− u42

)2K(k = u22)

, (4.1)

where K(k) is the complete elliptic integral of first kind with the modulus k, and

u2 =cos(δ/2)

1 + sin(δ/2). (4.2)

For the case of interest in this work, δ = 90, (4.1) simplifies to become Rs = R∞.


In this section, several performance characteristics are presented for the CSA ter-

minated resistively using lumped resistors and a resistive sheet. The objective is

to study the effect of these terminations on the performance of the loaded CSAs

and to compare the performance to that of the unloaded CSA. As in Section 2.2,

the results presented are for the left-handed CSA with the following parameters:

θ = 7.5, α = 75, δ = 90, D/d = 8 with d = 1.9 cm, and Zc = 100Ω.

In order to properly study the performance characteristics of the CSA with

resistive terminations, it is necessary to determine the most practical load resistance,

Rload = R∞, first. The following analysis outlines the technique introduced in Section

44


4.1, i.e., to obtain R∞ with a FDTD simulation. For the drive-point response of

the practical CSA to resemble that of the infinite CSA, the voltage measured at the

drive-point terminals must only contain the drive-point reflection. It is therefore

necessary to choose an incident signal with finite and sufficiently narrow temporal

extent to guarantee that drive-point reflection for the unloaded CSA vanishes well

before the first end reflection arrives at t/τL = 2. The differentiated Gaussian voltage

pulse, introduced in Section 2.2, with τp/τL = 1.6 · 10−2 is a good choice, because itsufficiently satisfies above criterion, see Fig. 2.5. The result for the reflected voltage

of such a FDTD simulation is shown in Fig. 4.3a as a function of normalized time,

t/τL, which is almost identical to the reflected voltage in Fig. 2.5 but with all signals

arriving later than the drive-point reflection windowed out, i.e., setting the voltage

to zero for t/τL ≥ 1. Ideally, this signal is the voltage response of the infinitely

long CSA and thus sufficient to determine the input impedance of the infinite CSA,

Z∞ = R∞ + jX∞, which is shown as a function of frequency in Fig. 4.3b. Evidently,

the resistance is almost constant and the reactance is small over the frequency range

displayed, i.e., |X∞| R∞. The load resistance minimizing the reflection from the

big end of the antenna is now approximated by the average resistance over the hybrid

frequency range, i.e., Rload = R∞ ≈ 150Ω.1 Thus, the CSA terminated with lumped

resistors, see Fig. 4.2a, should have RL ≈ 300Ω, while the CSA terminated with a

resistive sheet, see Fig. 4.2b, should have Rs ≈ 150Ω.

The first result for the CSA with and without resistive terminations is for the

reflected voltage as a function of normalized time, t/τL, with τp/τL = 1.6 · 10−2.The solid line in Fig. 4.4 is for the unloaded antenna, the dashed line is for the

CSA terminated with two lumped resistors, and the dash-dotted line is for the CSA

terminated with a resistive sheet. As expected, the drive-point reflection at t/τL ≈ 0

is the same for all cases. It can be seen that the terminations significantly reduce the

reflections from the antenna end. The amplitude of the reflected pulse at t = 2τL for

1Recall that the results for this resistance, R∞, are presented for CSAs with 5 ≤ θ ≤ 15, 60 ≤α ≤ 80 and δ = 90 in Fig. 3.5.

45


-0.25 0.00 0.25 0.50 0.75 1.00

-0.2

-0.1

0.0

0.1

0.2

0.3

0.3 0.5 1.0 1.5 2.0 2.5 3.0 3.5

-50

0

50

100

150

200

t / L

Vre

fo

/V

f [GHz]

X

RR

(a)

(b)

Hybrid Bandwidth

RX

,[

]

Figure 4.3: a) Theoretical results for the reflected voltage at the terminals ofthe antenna that contains only the reflection from the drive point,b) theoretical results for the input impedance of the infinitely longCSA.

the resistively terminated antennas is about one tenth that of the unloaded antenna.

The magnitude of the reflection coefficient and the input impedance (resistance

and reactance separately) are shown in Fig. 4.5 as a function of frequency on a

logarithmic scale for the antenna with and without the resistive terminations. As for

the previous results and those that follow, the solid line is for the unloaded antenna,

46


-1 0 1 2 3 4 5 6

-0.2

-0.1

0.0

0.1

0.2

0.3

t / L

Vre

fo

/V

Drive PointReflection

Second EndReflection

First EndReflection

UnloadedLumped ResistorsResistive Sheet

Figure 4.4: Theoretical results for the reflected voltage at the terminals of theantenna for the unloaded CSA (solid line), the CSA terminatedwith lumped resistors (dashed line), and the CSA terminated witha resistive sheet (dash-dotted line).

the dashed line is for the CSA terminated with two lumped resistors, and the dash-

dotted line is for the CSA terminated with a resistive sheet. The loading of the open

end only affects the low frequencies f < 0.8GHz; the curves overlap for mid-range

and high frequencies. The resistive terminations clearly improve the match of the

antenna to the transmission line at low frequencies. This is further illustrated in the

Smith charts in Fig. 4.6. The impedance is plotted separately for each CSA over the

frequency range 0.3GHz ≤ f ≤ 3.5GHz.

For the resistively terminated CSAs, vertical-plane (φ = 0), far-zone patterns

are shown in Fig. 4.7 (lumped resistors) and Fig. 4.8 (resistive sheet) for three different

frequencies (f = 0.75GHz, 1.50GHz, and 2.25GHz). As for the unloaded CSA, see

Fig. 2.9, these patterns are given for both of the circularly polarized components

of the electric field, viz, left-handed (LHCP, solid line) and right-handed (RHCP,

dashed line). Especially at low frequencies, both terminations are seen to improve

the front-to-back ratio for the LHCP component by several orders of magnitude:

compare the solid lines at θ = 180 with those at θ = 0. Both terminations are

also seen to significantly improve the axial ratio: compare the LHCP (solid line) and

47


f [GHz]

||

(a)

0.3 0.5 1 1.5 2 2.5 3 3.5

0.00

0.25

0.50

0.75

1.00

f [GHz]

R[

]

(b)

0.3 0.5 1 1.5 2 2.5 3 3.5

0

50

100

150

200

f [GHz]

X[

]

(c)

0.3 0.5 1 1.5 2 2.5 3 3.5

-100

-50

0

50

100


Figure 4.5: Theoretical results for the CSA with and without the resistive ter-minations: a) the magnitude of the reflection coefficient, b) inputresistance, and c) input reactance.

48


j 2

-j 2

j 1

-j 1

j 0.2

-j 0.2

j 0.5

-j 0.5

j 2

-j 2

j 1

-j 1

j 0.2

-j 0.2

j 0.5

-j 0.5

j 2

-j 2

j 1

-j 1

j 0.2

-j 0.2

j 0.5

-j 0.5

(a)

(b)

(c)

0 1 2 50.50.2

0 1 2 50.50.2

0 1 2 50.50.2

Figure 4.6: Smith charts for a) the unloaded CSA, b) the CSA terminated withlumped resistors, and c) the CSA terminated with a resistive sheet.

49


0-10-20

0

-10

-20

0 -10 -20

0

-10

-20

0

30

6090

120

150

180

210

240270

300

330

0-10-20

0

-10

-20

0 -10 -20

0

-10

-20

0

30

6090

120

150

180

210

240270

300

330

LHCPRHCP

dB

(a)

0-10-20

0

-10

-20

0 -10 -20

0

-10

-20

0

30

6090

120

150

180

210

240270

300

330

dB

(b)

(c)

dB

Figure 4.7: Theoretical results for the CSA terminated with the lumped re-sistors: the far-zone patterns for the right-handed and left-handedcircularly polarized components of the electric field at the frequen-cies: a) f = 0.75GHz, b) f = 1.50GHz, and c) f = 2.25GHz.

50


0-10-20

0

-10

-20

0 -10 -20

0

-10

-20

0

30

6090

120

150

180

210

240270

300

330

0-10-20

0

-10

-20

0 -10 -20

0

-10

-20

0

30

6090

120

150

180

210

240270

300

330

LHCPRHCP

dB

dB

(a)

(b)

0-10-20

0

-10

-20

0 -10 -20

0

-10

-20

0

30

6090

120

150

180

210

240270

300

330

(c)

dB

Figure 4.8: Theoretical results for the CSA terminated with a resistive sheet:the far-zone patterns for the right-handed and left-handed circularlypolarized components of the electric field at the frequencies: a)f = 0.75GHz, b) f = 1.50GHz, and c) f = 2.25GHz.

51


RHCP (dashed line) components at the angle θ = 180. The resistive disc is better at

reducing the RHCP component in the backward direction (θ = 0) than the lumped

resistors.

The results for the front-to-back ratio and the axial ratio are shown in Fig. 4.9.

Recall, two different interpretations of this ratio were introduced in Section 2.2: the

front-to-back ratio, FTBR, referred to the total radiated electric field, is shown in

Fig. 4.9a, and the front-to-back ratio, FTBRLHCP, referred to the LHCP component

of the radiated electric field, is shown in Fig. 4.9b. These graphs confirm what was

observed earlier: The resistive terminations improve both the front-to-back ratio and

the axial ratio for low frequencies.

The last quantities to be examined are the gain, realized gain, and directivity.

These features of the far-field are defined in Section 2.2. Recall that the power

dissipated in the antenna must be calculated first in order to determine the directivity.

Figure 4.10a illustrates the approaches used for determining the power dissipated in

each termination. For the termination with the lumped resistors, the power dissipated

is obtained using the current, IL, passing through each lumped element, i.e., the total

dissipated power in both lumped resistors is Pdiss = 2(|IL|2/2RL). For the termination

with a resistive sheet, the power dissipated is obtained using an approach that is

based on the conservation of energy. The total power passing through two planes of

square cross section, that contain the resistive sheet, is determined: Pbot right below

the sheet and Ptop right above the sheet. The power passing through the vertical

planes that connect the top and bottom horizontal planes is neglected in this simple

analysis, i.e., the power passing through these planes is assumed to be negligible. The

power dissipated in the termination is then approximately the difference between the

power entering the bottom plane and the power leaving through the top plane, i.e.,

Pdiss = Pbot − Ptop. The dissipated powers, normalized by the power supplied to

the antenna, Pinc = |Vinc|2/(2Zc), are shown in Fig. 4.10b for the CSA terminated

with lumped resistors (solid line) and for the CSA terminated with a resistive sheet

(dashed line). It can be seen that most of the available power is absorbed in the

52


(a)

FT

BR

[dB

]

f [GHz]

0.3 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0

10

20

30

40

50

FT

BR

LH

CP

[dB

]

f [GHz](b)

0.3 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0

10

20

30

40

50

60

70


AR

f [GHz](c)

0.3 0.5 1.0 1.5 2.0 2.5 3.0 3.5

1.00

1.25

1.50

1.75

2.00CircularPolarization

Figure 4.9: Theoretical results for a) the front-to-back ratio referred to thetotal far-zone electric field, b) front-to-back ratio referred to theleft-handed circularly polarized component of the far-zone electricfield, and c) axial ratio for the total far-zone electric field.

53


IL

ILPbot

PtopSide View:

Top View:

RL

RLCircular ResistiveSheet

(a)

(c)

f [GHz]

0.3 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0.00

0.25

0.50

0.75

1.00

1.25

(b)

f [GHz]

PP

dis

sin

c/

0.3 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0.00

0.25

0.50

0.75

1.00

Lumped ResistorsResistive Sheet

Figure 4.10: a) Schematic drawing for determining the dissipated power for theCSA terminated with lumped resistors (left) and the CSA termi-nated with a resistive sheet (right). Theoretical results for theb) dissipated power and c) efficiency for the CSA terminated withlumped resistors (solid) and for the CSA terminated with a resistivesheet (dashed).

54


resistive terminations for frequencies f ≤ 0.5GHz, while the dissipated power for

f > 0.8GHz is insignificant. The shape of the normalized powers is roughly the same

for both terminations; however, the lumped resistors absorb slightly more than the

resistive sheet. Similar conclusions can be drawn from the results for the efficiency,

η, shown in Fig. 4.10c. For frequencies f ≤ 0.8GHz, the efficiency deviates from the

ideal value of 1, and for frequencies f ≤ 0.5GHz, the efficiency for the resistive sheet

clearly exceeds that for the lumped resistors.

The results for the realized gain (with mismatch), gain (without mismatch), and

directivity for the unloaded and resistively terminated CSAs are shown in Fig. 4.11;

they are plotted on a linear scale. The results for the realized gain, Fig. 4.11a, and the

gain, Fig. 4.11b, clearly show that the terminations have no beneficial effect as far as

the radiation from the antenna is concerned. When the mismatch of the transmission

line is included, the results for the realized gain are almost identical for all of the

CSAs, i.e., for the terminated CSAs, the reduced reflection from the antenna end

balances out the conduction loss in the termination. The conduction loss in the

terminations cause the gain for the resistively terminated CSAs to drop compared to

the gain for the unloaded CSA. The directivity, shown in Fig. 4.11c, shows that the

terminations can, as expected, improve the directive nature of the antenna.

In conclusion, several performance characteristics of the CSA with θ = 7.5, α =

75, δ = 90, D/d = 8 and Zc = 100Ω have been studied for the unloaded CSA, for the

CSA terminated with lumped resistors, and for the CSA terminated with a resistive

sheet. The terminations can be seen to improve the impedance match for the antenna

and several features of the pattern at low frequencies. However, the terminations have

a negligible effect on the realized gain.

55


0.3 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0

2

4

6

8

G

f [GHz](b)

0.3 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0

2

4

6

8

D

f [GHz](c)

0.3 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0

2

4

6

8

f [GHz]

Gr

(a)


Figure 4.11: Theoretical results for the a) realized gain, b) gain, and c) directiv-ity in the direction of maximum radiation (−z).

56

CHAPTER 5

The Two-Arm Conical Spiral Antenna

over the Ground

In certain applications, antennas are placed in air over the ground and used to trans-

mit signals into the ground or to receive signals coming from within the ground.

Examples are antennas used in communication links to underground tunnels and

antennas used in ground-penetrating radars (GPRs). When the signals to be radi-

ated/received have a wide bandwidth, antennas with uniform properties over this

bandwidth are most suitable. The two-arm, conical spiral antenna (CSA), when iso-

lated in free space, has broadband performance characteristics, such as, uniform input

impedance, gain, and circular polarization, see Chapter 2. When the CSA is placed

over the ground, several interesting performance characteristics result. In Section

5.1, qualitative arguments for these characteristics are presented, e.g., the frequency

independence of the CSA isolated in free space can be preserved when placed over the

ground and circular polarization can be used to reduce the effect of the air/ground

interface on the antenna performance. These characteristics are verified and quanti-

fied with an accurate numerical analysis (FDTD) in Section 5.2. In Section 5.3, a few

applications for the CSA in a monostatic GPR are described. A practical approach

to improve and clarify the time-domain signature of targets buried in the ground by

removing the dispersion of the pulse upon transmission and reception is outlined in

Section 5.4.

57

5.1 General Characteristics of the CSA over the Ground

VirtualApex at theAir/GroundInterface

Earth rg g,

h

Figure 5.1: CSA over the ground with the virtual apex of the antenna placedat the air/ground interface.

5.1 General Characteristics of the CSA over the

Ground

A schematic drawing for the CSA over the ground is shown in Fig. 5.1. The ground

is modeled as a half space of material with the following electrical properties: relative

permittivity εrg, conductivity σg, and relative permeability µrg = 1. The axis of

the CSA is normal to the air/ground interface, with the small end of the antenna

(diameter d) at the height h above the interface. Notice, when the virtual apex of the

CSA is on the interface, that is, when h = hc − hs = d/(2 tan θ) as in Fig. 5.1, no

additional angles or lengths are needed to describe the CSA. Stated differently, the

geometry of the CSA over the half space is completely described by the same angles

and lengths used to describe the isolated CSA in free space. From this observation, we

expect the performance of the CSA over the ground to exhibit frequency independence

similar to that for the CSA in free space, provided any dispersion introduced by the

electrical properties of the ground is ignored.

The active-region concept implies that there is certain scaling of the electro-

magnetic field within the ground. The following results qualitatively demonstrate the

uniform performance with wavelength of the CSA over the ground. Therefore, the

58


scaling relations for Maxwell’s equations must be summarized first. In normalized

form, the Ampere-Maxwell law in lossless media can be written as

∇λ × λ &Hη = j(ωλ/c)εr λ&E (5.1)

and Faraday’s law as

∇λ × λ&E = −j(ωλ/c)µr λ &Hη, (5.2)

where λ is the wavelength,∇λ = λ∇ the normalized differential operator, η =√µ/ε

the free-space wave impedance. When the total current in the antenna arm is assumed

to be the same in each active region, these equations can be used to infer the frequency

independence of the field radiated by the CSA. Consider the schematic drawing shown

in Fig. 5.2. The active region of the CSA for wavelength λ1 produces the electric field

&E1 on the horizontal plane at depth s1 in the ground. Similarly, the active region for

wavelength λ2 produces the electric field &E2 on the horizontal plane at depth s2. Now,

when the dissipation in the ground is ignored, Maxwell’s equations and the theory of

scale models [23] show that at depths that satisfy

s2λ2=

s1λ1

(5.3)

the electric fields are related by

λ2 &E2[ρ2 = (λ2/λ1)ρ1] = λ1 &E1(ρ1). (5.4)

Notice, in Fig. 5.2 the relative sizes of the two sketches for the electric field indicate

this scaling. It is interesting to note that the scaling of the electric field is satisfied

for all wavelengths/frequencies at just one observation point, that is, at the apex

(s = 0). It will be shown that this condition can be relaxed, such that even for points

in the vicinity of the apex with s = const, frequency independence is obtained over

the frequency range of operation.

In the following, the use of circular polarization for the CSA over the ground is

discussed. Therefore, both the transmission and reception characteristics of the CSA

are of great importance. Recall that the transmission characteristics are discussed in

59


2

1

Active Regionfor 1

Active Regionfor 2

Air

Earth

s2

s1

1 o/ 2 tan

2 o/ 2 tan

|E2|

|E1|

Figure 5.2: Schematic drawing to illustrate the scaling of the electric field ofthe CSA in the ground.

Section 2.2. The general reception characteristics of antennas are discussed in detail

in [21] using plane waves, repeated here just briefly, and then applied to the ideal

CSA. It is necessary to characterize the electric fields transmitted and received by

the antenna based on its polarization. In general, the electric field of a plane wave

propagating in the r direction (ejωt time dependence) can be written using spherical

coordinates

&E(r, t) = Eθ(r, t) θ + Eφ(r, t) φ (5.5)

= Aθ cos[ω(t− r/c) + φθ] θ + Aφ cos[ω(t− r/c) + φφ] φ (5.6)

= Aθcos[s(r, t)] θ + tan(γ) cos[s(r, t) + δ] θ, (5.7)

where s(r, t) = ω(t− r/c) + φθ. The relative amplitude and the relative phase of the

electric field are expressed with the parameters γ = tan(Aφ/Aθ) and δ = φφ−φθ. The

polarization of the electric field can thus be characterized using the set of parameters

(γ, δ), e.g., a left-handed circularly polarized wave has (γ = π/4, δ = π/2) and a

60


right-handed circularly polarized wave has (γ = π/4, δ = −π/2). A straightforward

analysis shows that the ability of an antenna to receive a certain state of polarization

can be described using the polarization mismatch factor, pr, i.e.,

pr =1 + tan2(γtrans) tan

2(γrec)− 2 tan(γtrans) tan(γrec) cos(δtrans + δrec)

sec2(γtrans) sec2(γrec), (5.8)

where (γtrans, δtrans) characterizes the polarization of the electric field transmitted

by the antenna and (γrec, δrec) characterizes the polarization of the electric field being

received by the antenna. The mismatch factor varies between pr = 0 for null reception,

i.e., the electric field incident upon the antenna is not received at all, and pr = 1 for

maximum reception, i.e., the electric field incident upon the antenna is completely

received. The shape and orientation of the polarization ellipse, and the sense in which

the polarization ellipse is traced out (handedness) for null and maximum reception

are illustrated in Fig. 5.3 relative to the shape, orientation, and handedness of the

polarization ellipse for transmission. Figure 5.3a is for the most general case in which

the antenna transmits an elliptically polarized wave, while Fig. 5.3b is for a special

case in which the antenna transmits a left-handed circularly polarized wave. The

polarization ellipses on the left are for the electric field transmitted by the antenna

and plotted for an observer located beyond to antenna looking in the main radiation

direction. The polarization ellipses in the center (on the right) are for the electric field

with maximum (null) reception and plotted for an observer located at the antenna

looking in the direction of the wave incident upon the antenna; see Fig. 5.3c for the

position and orientation of the observers. Notice that for this choice of the coordinate

systems, the shape and handedness of the polarization ellipses do not change for

maximum reception, while the axis of the ellipse (orientation) is the mirror image

with respect to the φ axis relative to that for transmission. For null reception, the

shape remains the same, while the orientation of the ellipse changes by 90 relative

to that for maximum reception and the sense in which the polarization ellipse is

traced out reverses. The results presented in Fig. 5.3b can be directly applied to the

ideal left-handed CSA that radiates predominantly left-handed circular polarization

61


Transmission Maximum Reception Null Reception

(a)

(b)

E

E

RHCP

E

E

-rE

E

LHCP

E

E

-r

-rE

E

LHCP

E

E

r

r

(c)

r

^

^

^

^

^

^

^

^

^Antenna

Direction ofPropagation( )r

Direction ofPropagation(- )r

Transmission Reception

-r

Figure 5.3: Schematic drawings of the polarization ellipses for the electric field:for transmission, maximum reception, and null reception. The el-lipses in a) are for the most general case (elliptical polarization)and in b) for the special case of circular polarization; c) positionand orientation of the observers.

in its main radiation direction, i.e., r is equivalent to −z, compare Fig. 5.3b with

Fig. 2.3b. Following the analysis outlined above, when the left-handed CSA is used

as a receiving antenna, the left-handed circularly polarized component of the electric

field incident upon the antenna from the forward direction is completely received,

while the right-handed circularly component of the electric field passes by the CSA

undetected. These idealized transmission and reception characteristics for the left-

handed CSA are illustrated schematically in Fig. 5.4.

The circularly polarized field of the CSA can be used to reduce the effect of the

air/ground interface on the antenna response. To demonstrate this point qualitatively,

62


LHCP RHCP

Transmission Maximum Reception Null Reception

LHCP

Left-Handed CSA

RHCP

Figure 5.4: Idealized polarization characteristics of the left-handed CSA fortransmission and reception.

a simple plane-wave analysis based on the geometry shown in Fig. 5.5a is used. A

plane wave in free space with its electric field components parallel and perpendicular

to the plane of incidence, Einc‖ and Einc

⊥ , respectively, is incident upon the air/ground

interface with the angle αinc. The field is partially reflected into the free-space region

(µr = 1, εr = 1, σ = 0) and partially transmitted into the lossy medium (µrg =

1, εrg, σg). The reflected field components, Eref⊥ and Eref

‖ , can be written in terms of

the incident field components using the reflection coefficients for the corresponding

orientation (parallel and perpendicular to the plane of incidence), i.e., Eref⊥ = Γ⊥Einc

⊥

and Eref‖ = Γ‖Einc

‖ [24]. The results for the magnitude of the reflection coefficients

are graphed as a function of angle of incidence αinc in Figs. 5.5b and c for three soils:

dry, medium moist, and wet. The electrical parameters at the frequency f = 1.8GHz

(midband frequency of the CSA investigated in the next section) are given in Table

5.1 [25].

Table 5.1: Electrical parameters for different soil conditions at f = 1.8GHz.

εrg σg [S/m]

dry 4.0 0.02

medium 8.0 0.07

wet 15.0 0.15

63


Earth rg g,

Free Space

kt

Et

Et

inc inc

Eref

Eref

krefEinc

Einc

kinc

(a)

(b)

(c)

0 10 20 30 40

0.00

0.25

0.50

0.75

1.00

0 10 20 30 40

0.00

0.25

0.50

0.75

1.00

inc

inc

DryMediumWet

Figure 5.5: a) Geometry of the plane-wave analysis. Analytical results for themagnitude of the reflection coefficient for the electric field b) par-allel to the plane of incidence and c) perpendicular to the plane ofincidence.

64


A similar plane-wave analysis based on these reflection coefficients shows that

the use of circular polarization reduces the effect of the air/ground interface on the

antenna response. The schematic drawing for this analysis is shown in the inset of

Fig. 5.6. Here, a left-handed circularly polarized plane wave is incident upon the

air/ground interface with the angle αinc. The reflected wave is decomposed into two

waves: a left-handed circularly polarized wave (copolarized component) and a right-

handed circularly polarized wave (cross-polarized component). The ratio of these two

field components, |ErefLHCP/E

refRHCP|2, can be expressed in the most general form for an

arbitrarily polarized incident electric field (γinc, δinc) as∣∣∣∣∣ErefLHCP

ErefRHCP

∣∣∣∣∣2

=

∣∣∣∣∣1− j(Γ⊥/Γ‖) tan(γinc) exp(jδinc)1 + j(Γ⊥/Γ‖) tan(γinc) exp(jδinc)

∣∣∣∣∣2

. (5.9)

When the incident signal is assumed to be left-handed circularly polarized, i.e., γinc =

π/4 and δinc = π/2, (5.9) simplifies to

∣∣∣∣∣ErefLHCP

ErefRHCP

∣∣∣∣∣2

=

∣∣∣∣∣1 + (Γ⊥/Γ‖)1− (Γ⊥/Γ‖)

∣∣∣∣∣2

. (5.10)

This ratio is graphed as a function of the angle of incidence in Fig. 5.6. Results

are shown again for three soils: dry, medium moist, and wet, with the electrical

parameters at the frequency f = 1.8GHz given in Table 5.1. From these results,

it is clear that the electric field reflected from the surface is predominantly cross

polarized to that of the incident wave, viz, RHCP. This point is illustrated further by

the polarization ellipses for the incident electric field (dashed line) and the reflected

electric field (solid line) shown at the top of Fig. 5.6. The ellipses are for the medium-

moist soil and for three different angles of incidence: αinc = 5, 20, and 35. In all

three cases, the ellipse for the reflected field is seen to be almost a circle and to have

a sense of orientation opposite to that for the incident field.

The results for this plane-wave analysis are summarized in Fig. 5.7 for its use in

a GPR application that employs a CSA with the transmission and reception charac-

teristics shown in Fig. 5.4. Based on the argument presented above, which can easily

be extended to multiple parallel layers [26], an ideal CSA that radiates a left-handed

65

5.2 Numerical Analysis of the CSA over the Ground

inc = 35oinc = 5

o inc = 20o

0 10 20 30 400.000

0.025

0.050

0.075

inc

Earth

Free Space

inc inc

ELHCPELHCP

ERHCPinc

ref

ref

EL

HC

P

ref

ER

HC

P

ref

/2

DryMediumWet

ReflectedIncident

Figure 5.6: Analytical results for the ratio |ErefLHCP/E

refRHCP|2 as a function of

angle of incidence αinc for three different types of soil. Polarizationellipses for the incident and reflected electric fields are shown forthe medium-moist soil.

circularly polarized wave toward the air/ground interface does not receive the wave

reflected from any media interface, because it is predominantly right-handed circu-

larly polarized.1 This feature can be useful in applications, such as GPRs, where

a signal from within the ground is to be measured, not a signal reflected from the

surface of the ground, which is considered to be “clutter.”


In this section, an accurate analysis of the CSA over the ground is performed using

the FDTD method, and results obtained from this analysis are used to verify and

1Precisely, this is correct only for media interfaces of two parallel, isotropic media infinite in ex-

tent. It will be shown in Section 5.4 that an anisotropic medium can scatter a significant copolarized

signal.

66


RHCP

LHCP

Earth

LHCP

RHCP

Left-handed CSA

Layer

Figure 5.7: Schematic drawing to illustrate that the use of circular polariza-tion reduces the effect of the air/ground interface on the antennaresponse.

quantify the characteristics of the CSA that are discussed qualitatively in the last

section. The FDTD technique has been applied to the CSA in free space in Chapter 2,

and numerical results for the input impedance and realized gain are in good agreement

with measurements, see Section 2.2. Thus, the same techniques can be applied to a

similar CSA over the ground with confidence.

The two-arm CSA has the geometry shown in Fig. 2.1 with the following pa-

rameters: θ = 7.5, α = 75, δ = 90, and D/d = 8 (d = 2 cm). The antenna

is fed from a transmission line with the characteristic impedance Zc = 100Ω. To

improve the low-frequency performance, the CSA is terminated with a resistive sheet

to reduce the reflections from the open end, see Chapter 4. The electrical and ge-

ometrical parameters of the sheet are chosen to make the DC resistance, measured

between the two arms, 150Ω.2 When this antenna is isolated in free space, it has an

operational (hybrid) bandwidth of about 6.3, extending from 0.49GHz to 3.1GHz,

see Chapter 3. For all of the results to be presented, the virtual apex of the CSA

is placed at the air/ground interface, so the height h of the drive point above the

2Recall, this is the most practical choice for the characteristic impedance of the line and for the

load resistance, see Section 4.1 and Fig. 3.5.

67


RX

,[

]

f [GHz]

0.25 0.50 1.00 1.50 2.00 2.50 3.00 3.50

-25

0

25

50

75

100

125

150

175

R

X

j2

j1

j0.5

j0.2

0

- 0.2j

- 0.5j- 1j

- 2j

1 2 50.50.2

R Z/ c

X Z/ c

Free SpaceOver Ground

Operational Bandwidth

Figure 5.8: Comparison of the input impedances for the CSA in free space andover medium-moist soil.

ground is h = 7.6 cm.

The input impedances of the CSA when isolated in free space (solid line) and

when placed over the medium-moist soil (dashed line) are compared in Fig. 5.8. The

presence of the ground is seen to have almost no effect on the input impedance, and

both the resistance and reactance are seen to be nearly frequency independent over the

operational bandwidth of the antenna. Notice that the reactance is small compared

to the resistance, and that the resistance is approximately R ≈ 140Ω. Thus, over

the operational bandwidth, the antenna is well matched to the feeding transmission

line (Zc = 100Ω). This is clearly shown on the Smith chart that is inset in Fig. 5.8.

Similar results are obtained for the other two soil conditions: dry and wet. So for this

arrangement, the input impedance of the CSA is essentially independent of frequency

and independent of the electrical properties of the ground.

The effect of the ground on the time-domain results for the antenna response

is qualitatively shown in Fig. 5.9. In Fig. 5.9a, the reflected voltage for the CSA

over the ground, Vref,g, is graphed as a function of normalized time, t/τL, with thecharacteristic time of the incident, differentiated Gaussian pulse, τp/τL = 1.3 · 10−2.Other than the drive-point reflection at t/τL ≈ 0 and the antenna-end reflection at

68


-0.5 0.0 1.0 2.0 3.0 4.0

-0.2

-0.1

0.0

0.1

0.2

t / L

Vre

f,g

/V

o

(a)

-0.5 0.0 1.0 2.0 3.0 4.0

-0.010

-0.005

0.000

0.005

0.010

t / L

Vsc

att

,g

/V

o

(b)

Figure 5.9: Theoretical results for a) the reflected voltage of the CSA overground and b) the voltage of the signal scattered from theair/ground interface.

t/τL ≈ 2, no additional reflected signals, e.g., from the ground, are obvious. The

effect of the air/ground interface can be quantified when the reflected voltage of the

CSA over the ground, Vref,g, is subtracted from the voltage of the CSA in free space,

Vref,fs, i.e., the voltage of the signal scattered from the air/ground interface can be

written Vscatt,g = Vref,fs − Vref,g. This voltage is shown in Fig. 5.9b; as expected, thisvoltage is very small.

Two factors contribute to the observed behavior for the input impedance and

the reflected voltage of the CSA over the ground. The lack of a major effect due to

the presence of the ground is a result of the CSA not receiving the signal reflected

from the ground. The frequency independence is a result of the angle principle and

69


the truncation principle being satisfied to the same degree for the CSA isolated in

free space and for the CSA with its virtual apex at the air/ground interface.

Figure 5.10 shows the characteristics of the electric field along the z axis (axis

of the CSA) in the ground. The graph in Fig. 5.10a verifies the scaling of the electric

field given by (5.4). It shows the normalized electric field as a function of the depth

in the ground. For the plot, depths in the range 0.2m ≤ s ≤ 1m are used, and

at each depth, the free-space wavelength at which the electric field is evaluated is

set equal to 0.6 times the depth; that is, s/λ = const = 1.67, which satisfies (5.3).

Results are shown for free space (εrg = 1.0, σg = 0, solid line), for the medium-moist

soil with loss (εrg = 8.0, σg = 0.07 S/m, dashed line) and for the medium-moist soil

without loss (εrg = 8.0, σg = 0, dash-dotted line). Notice that the exponential factor

exp(αs), where α ≈ σgη/2√εrg is the attenuation constant in the soil, is included in

the normalization to remove the effect of the attenuation in the lossy ground. The

curves are seen to be nearly the same and to be almost uniform. This verifies the

scaling of the electric field described by (5.4).

The graph in Fig. 5.10b shows the axial ratio for the polarization ellipse of the

electric field versus frequency for two very different depths in the medium-moist soil

(s = 4 cm, dashed line; and s = 1m, dash-dotted line). Here, the axial ratio, usually

a far-field quantity, is determined in the near zone of the CSA using the electric field

components normal to the axis of the antenna, Ex and Ey. At the depths used, the

component of the electric field parallel to this axis, Ez, is orders of magnitude smaller

than those normal to the axis. As a point of reference, the axial ratio for the electric

field at the apex of the antenna when isolated in free space is also shown (solid line).

Recall, for circular polarization, the axial ratio is AR = 1.0. The electric field in

the ground is seen to be nearly circularly polarized over the operational bandwidth

of the antenna (AR ≤ 1.25, for 0.49GHz ≤ f ≤ 3.1GHz) and to be similar to that

for the isolated antenna. This point is further illustrated by plots of the polarization

ellipse at the top of the figure. They are for the electric field at the depth s = 1m

(solid line) in the medium-moist soil and for circular polarization (dotted line) at the

70


(a)

()

|| /

||

exp(

)E

Vs

inc

s [m]

0.2 0.4 0.6 0.8 1.0

0.00

0.25

0.50

0.75

1.00

1.25

Free SpaceLossy GroundLossless Ground

(b)

0.25 0.50 1.00 1.50 2.00 2.50 3.00 3.50

1.00

1.25

1.50

1.75

2.00 Free Space (at Apex)Ground ( =4cm)Ground (Circ. Polarization

ss =1m)

f [GHz]

AR

Figure 5.10: a) Theoretical results for the normalized electric field on axis as afunction of depth: in free space (solid line), in the lossy, medium-moist soil (dashed line), and in the lossless medium-moist soil (dash-dotted line). b) Theoretical results for the axial ratio on axis forthe CSA in free space at the apex (solid line), for the CSA overthe medium-moist soil observed 4 cm (dashed line), and 1m (dash-dotted line) below the air/ground interface.

frequencies f = 0.50, 1.75, and 3.00GHz.

In some applications, such as GPRs used to detect landmines buried in the

ground or reinforcing bars embedded in concrete, the characteristics of the electric

field at a shallow depth in the ground are of interest. At such depths, the distance

from the buried object to the virtual apex of the CSA is small. So the characteristics

71


of the incident field are very similar to those at the apex of the CSA, i.e., at the

surface of the ground. The apex is a special point, because the scaling relation (5.4)

is satisfied for all frequencies when s = 0. This can be seen by setting s1 = s2 = 0 in

(5.3).

In Fig. 5.11 results are shown for the normalized electric field at the shallow

depth s = 4 cm in the medium-moist soil. Figure 5.11a is a surface plot of the nor-

malized electric field, λ| &E|/|Vinc|, versus the normalized position, ρ/λ, and frequency,f . The distribution for the normalized electric field is seen to be essentially the same

for frequencies within the operational bandwidth. To qualitatively evaluate the off-

axis behavior, the points on the horizontal plane are determined at which the electric

field has dropped by 3 dB from its value on axis, see the inset of Fig. 5.11a. Ac-

cording to the scaling described in Section 5.1, this width when normalized to the

wavelength, i.e., w3dB/λ, should be independent of the wavelength/frequency. Results

for the normalized width are given in Fig. 5.11b, and, as expected, it is seen to be

fairly independent of frequency.

Figure 5.12 is a surface plot of the axial ratio at the depth s = 4 cm in the

medium-moist soil versus the normalized position, ρ/d, and frequency, f . Here as in

the previous example, the axial ratio is determined on a plane of constant z in the

near zone of the CSA using the components of the electric field tangential to this

plane, Ex and Ey. The field component normal to this plane, Ez, is generally small

compared to Ex and Ey, particularly for points near the axis of the CSA. The electric

field is seen to be predominantly circularly polarized for ρ/d ≤ 10 (ρ/D ≤ 1.25) and

for frequencies within the operational bandwidth. Thus, it can be concluded that

directly below the surface of the ground over an area centered on the antenna at least

as large as 2D × 2D, the electric field scales as given in (5.4) and is predominantlycircularly polarized. The results for the near field along with those for the impedance

clearly show that the frequency-independent performance of the CSA is preserved

when it is placed over the ground.

The polarization of the radiation from the CSA in the near field of the an-

72


f [GHz]

w3

dB

/

0.3 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0.0

0.5

1.0

1.5

2.0

Free Space (at Apex)Ground ( =4cm)s

|

| /|

| [d

B]

EV

inc

/f [GHz]

(a)

(b)

|E |norm

w3dB

0.51.0

1.52.0

2.53.0

3.5

Figure 5.11: a) The normalized electric field distribution 4 cm below theair/ground interface as a function of frequency and distance fromthe axis. b) Results for the normalized 3 dB width as a function offrequency for the antenna in free space observed at the apex (solidline) and for the antenna over the medium-moist ground observedat 4 cm below the apex (dashed line).

tenna and the scattering characteristics from the air/ground interface are studied

qualitatively using the electric field in the vicinity of the antenna. In Fig. 5.13a, the

normalized, left-handed and right-handed circularly polarized field components of the

incident electric field, Einc,n = |Einc/(VincLs)|, are plotted as a function of frequencyfor the CSA in free space. For clarity, these fields, ELHCP

inc,n and ERHCPinc,n , are normalized

by the maximum value of the LHCP component, ELHCP,maxinc,n . The electric field is

observed on axis and 0.5 cm above the virtual apex of the antenna. Similar to the

73


/df [GHz]

AR

2.0

1.8

1.6

1.4

1.2

1.0

0.51.0

1.52.0

2.53.0

3.5

Figure 5.12: The axial ratio at a depth of 4 cm below the air/ground interfaceas a function of frequency and distance from the axis.

electric field in the far field of the antenna, the polarization of the electric field in the

near field of the antenna is predominantly left handed. Recall that the air/ground in-

terface predominantly scatters the cross-polarized component of the incident electric

field. For normal incidence, the reflection coefficients for the electric field perpendicu-

lar and parallel to the plane of incidence, are the same, i.e., Γ = Γ⊥ = Γ‖ for αinc = 0.

For this case, the ratio of the cross-polarized component of the electric field scattered

from the air/ground interface to the corresponding component of the incident electric

field must equal the reflection coefficient, i.e., ERHCPref /ELHCP

inc = ELHCPref /ERHCP

inc = Γ.

This behavior is verified in Fig. 5.13b for the ratio ERHCPref /ELHCP

inc as a function of

frequency. The reflected/scattered electric field (on axis 0.5 cm above the apex) is

obtained by subtracting the electric field for the CSA over the ground, &Eground, from

the electric field component of the CSA in free space, &Einc, i.e., Eref = | &Eground− &Einc|.This ratio is graphed for three different soils: dry (solid line), medium moist (dashed

line), and wet (dash-dotted line), and the corresponding reflection coefficients are

graphed with dotted lines. Since the ratio of the electric fields matches the value for

the reflection coefficient well over the entire frequency range, the scattering from the

air/ground interface can be correctly described by the plane-wave analyses in Section

74

5.3 GPR Applications

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0.00

0.25

0.50

0.75

1.00

1.25

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0.00

0.25

0.50

0.75

1.00

f [GHz]

(a)

||

EE

/in

c,

inc,

nn

LH

CP,

max

f [GHz]

(b)

LH

CP

|Ere

f/

Ein

c|

RH

CP

DryMediumWet

LHCPRHCP

wet

medium

dry

Figure 5.13: Theoretical results for a) the electric field of the CSA in free spacein the near field of the antenna and b) the electric field scatteredfrom the air/ground interface.

5.1. Similar agreement can be obtained for the results for the ratio of ELHCPref /ERHCP

inc ,

which are omitted since this reflected component of the electric field is very small. In

conclusion, this simple analysis confirms and quantifies that the air/ground interface

predominantly reflects the cross-polarized component of the incident electric field.


In the following, monostatic GPRs are investigated that use a single two-arm CSA

for transmission and reception. Different objects buried in the ground are used to

75


RHCPLHCP

Earth

LHCP

LPLHCP

Thin Rod

+

Left-handed CSA

Lr

sr

h

Figure 5.14: Schematic drawing for a monostatic GPR that uses a single CSAto detect thin metallic rods buried in the ground.

illustrate the sensitivity of the CSA to the polarization of the scattered electric field

of the target. In the first case, the targets are thin metallic rods that predominantly

scatter linear polarization, while in the second case, the targets are buried plastic

mines.

5.3.1 GPR: Detection of Buried Rods

Here, a monostatic GPR is investigated for detecting thin metallic rods buried in the

ground. Figure 5.14 is a schematic drawing for this GPR that uses a left-handed,

two-arm CSA. A rod of length Lr buried at the depth sr in the ground predomi-

nantly scatters linear polarization, which can be decomposed into equal left-handed

and right-handed circularly polarized components. Ideally, only the copolarized com-

ponent (LHCP) can be received by this CSA and used to detect the rod buried in

the ground. It will be shown that the rod buried in the ground with electrical pa-

rameters µrg = 1, εrg, and σg can be detected at the resonant frequencies of the

76


Earth

Target

Vref,t

Earth

Vref,g

V V Vtarget ref, ref,= -t g

Figure 5.15: Illustration for the detection approach using a CSA in a monostaticGPR.

rod: fres ≈ 0.85c/(√εrg2Lr) (half-wave resonance) and 3fres. In this study, the first

resonant frequency, fres, is of main interest.

The detection process used throughout this section is based on the voltage for

the signal scattered from the target, Vtarget = Vref,t − Vref,g, which is the difference

between the voltages received by the GPR antenna with (Vref,t) and without (Vref,g)

the target present, as illustrated in Fig. 5.15. Ideally, this voltage only contains the

response from the target (solid line), while other responses, e.g., from the ground and

the antenna end (dashed lines), are cancelled out.

In the following, a parametric study is performed to show that throughout the

entire frequency range of operation for the CSA differently sized rods buried at shallow

depths in the ground can be detected. For the rods to be detected equally well, they

must scale in length as well as in depth in the ground, i.e., small rods at small

depths and large rods at large depths. The CSA in this study has the parameters

θ = 7.5, α = 75, δ = 90, D/d = 5 (d = 2 cm), and is at the practical height of h =

7.6 cm above the ground, which places the apex of the CSA on the interface. In order

to reduce clutter, the antenna end is terminated with a resistive sheet that has Rs =

150Ω. For simplicity, the ground is chosen to be lossless with εrg = 8. The parametric

study is performed for four differently sized rods buried at different depths in the

ground. The dimensions of the rods are chosen to address the entire range of active

77


regions along the antenna, as illustrated in Fig. 5.16a. Notice that four active regions

are indicated on the CSA (numbered 1, . . . , 4), and that each active region is for

the resonant frequency of the corresponding rod. In the simplest approximation, the

resonant frequency (half-wave resonance) of the buried rod matches the characteristic

frequency of the active region. The range of frequencies over which the antenna can

be operated ranges approximately from 0.78GHz to 3.1GHz. Thus, the lengths of

the rods are chosen to illustrate the uniform performance over this bandwidth. The

longest rod has Lr1 = 5.0 cm and fres1 ≈ 0.9GHz; the others have Lri = Lr1/i and

fresi = ifres1, where i = 1, . . . , 4. The burial depth in the ground is chosen to be

sr ≈ 0.7Lr, and thus scales with the length of the rod.

The results from the parametric study for the CSA are shown in Fig. 5.17a;

the normalized voltage for the signal scattered from the rods, |V rodtarget/Vinc| is shown

as a function of frequency. From these results, it can be seen that the large rods

are detected at low frequencies and the small rods at high frequencies. As mentioned

earlier, the level of detection should ideally be the same for rods when properly scaled

in length and depth; however, the received signals scattered from the smaller rods

(detection at high frequencies) are somewhat smaller than those for the larger rods.

This is probably due to reduced radiation at higher frequencies; in the far field this

is evident in the realized gain, see Fig. 4.11a, which trails off at the high end of the

frequency range.

To demonstrate the efficiency of the CSA in a GPR for detecting buried rods, a

similar study was performed for a GPR that uses a series of resonant dipole antennas

to detect the buried rods, Fig. 5.16b. To minimize the polarization loss, the dipole

antennas are aligned with the rods. The length of each of the dipoles is chosen so

that the dipole is resonant: Ldip ≈ 3.3Lr ≈ √εrgLr. The height of each dipole above

the ground is hdip ≈ 2.2Ldip and thus is approximately the same as the height of the

active region of the CSA. Results from this study are shown in Fig. 5.17b. Notice

that the signals received by each dipole peaks at roughly the resonant frequency of

the rod and that the level of detection is almost the same for all of the dipoles, as

78


Earth rg g,

Earth rg g,

L Lr r4 1= /4

L Lr r3 1= /3

L Lr r2 1= /2

Lr1

Active Regionsfor the ResonantFrequencies ofthe Buried Rods

Dipole Antennasthat Resonant at theResonant Frequenciesof the Buried Rods

1

1

2

2

3

3

4

4

(a)

(b)

Figure 5.16: Drawing detailing the parametric study for four differently sizedrods buried in the ground using a) a single CSA and b) four resonantdipoles as the GPR antennas.

it should be when the rods and dipoles are properly scaled. Comparing Figs. 5.17a

and 5.17b, the CSA is seen to detect the buried rods significantly better than the

resonant dipoles.

The scattering characteristics for the longest rod (Lr1 = 5.0 cm, sr1 = 3.2 cm,

and fres1 = 0.9GHz) buried in the lossless medium-moist ground are now examined

using the approach outlined in Section 5.2. The objective is to show that the rod

scatters a significant copolarized signal. Therefore, the electric field (on axis 0.5 cm

79


1 2 3 4

1 2 3 4

1.0

1.0

0.5

0.5

2.0

2.0

3.0

3.0

4.0

4.0

0.000

0.004

0.008

0.012

0.000

0.004

0.008

0.012

f [GHz]

f [GHz]

(a)

(b)

|/

|V

Vta

rget

inc

rod

|/

|V

Vta

rget

inc

rod

Figure 5.17: Theoretical results for the parametric study for a) the monostaticGPR using a single CSA and b) a series of resonant dipoles to detectthin metallic rods buried in the ground.

above the apex) scattered from the rod is obtained by subtracting the electric field for

the CSA over the ground with the rod present, &Erod, from the electric field component

of the CSA over the ground without the rod, &Eground, i.e., Eref, rod = | &Erod − &Eground|.This electric field, normalized by the LHCP component of the incident electric field

ELHCPinc (obtained from the CSA in free space) is graphed in Fig. 5.18 as a function of

frequency. The rod is seen to scatter linear polarization because the right-handed and

left-handed circularly polarized components of the scattered electric field are equal.

As noted earlier, the rod scatters predominantly at the resonant frequencies, fres and

3fres.

80


0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0.0

0.1

0.2

0.3

f [GHz]

LH

CP

||

Ere

f,ro

d/

Ein

c

LHCPRHCP

LHCP RHCP

Figure 5.18: Theoretical results for the electric field scattered from the rodburied in the ground.

To conclude the discussion of the GPR application to detect buried rods in the

ground, another parametric study is performed using rods buried in lossy ground

and at larger depths. In order to achieve good penetration of the electric field into

the ground, the operational frequency range for the monostatic radar is chosen to be

0.16GHz ≤ f ≤ 1.0GHz [25, 27]. The ground is dry soil, which has the following

electrical properties within this bandwidth: εrg ≈ 4 and σg ≈ 0.007 S/m.3 The left-

handed CSA has the dimensions θ = 7.5, α = 75, δ = 90, D/d = 8 (d = 6 cm)

and is placed at a height of h = 22.8 cm over the ground. Notice that this is a scaled

version of the CSA discussed earlier in Section 5.2: all of the dimensions are three

times larger and the operational bandwidth is lower in frequency by a factor of three.

As in the previous case, the CSA is terminated with a resistive sheet (Rs = 150Ω).

The lengths of the five rods used in the parametric study are chosen to illustrate

the uniform performance of the radar over the operational bandwidth. For the longest

rod Lr1 = 32 cm, and fres1 ≈ 0.2GHz; and for the other rods Lri = Lr1/i and fres i ≈ifres1, i = 2, . . . , 5. The rods are buried at depths such that sri/Lri = const = 3.1,

3Notice that this conductivity is lower than the value given for dry soil in Table 5.1. This is caused

by the difference in the frequency: The value in Table 5.1 is for the frequency f = 1.8GHz, while

the value used in the present calculation is for the center frequency of the operational bandwidth,

f = 0.6GHz.

81


0.15 0.30 0.45 0.60 0.75 0.90 1.05 1.200.000

0.002

0.004

0.006

0.008

0.010

f [GHz]

1 2 3 4 5

|VV

sta

rget

inc

/| e

xp

(2)

r

rod

Figure 5.19: Results for the monostatic GPR with the thin rods buried in thedry soil.

so for the longest rod, the depth is sr1 = 1m, and for the shortest rod, the depth is

sr5 = 0.2m.

In Fig. 5.19, the voltage |V rodtarget/Vinc| exp(2αsr) is plotted versus frequency. The

factor exp(2αsr) accounts for the attenuation of the waves propagating the round-

trip distance 2sr (to and from the rod) in the ground. Notice that there is a peak

at the resonant frequency of each rod, and that all of the peaks have comparable

amplitudes. These results show that the monostatic radar based on the CSA is

equally good for detecting small targets at shallow depths and large targets at large

depths. The overall conclusion that can be drawn from these parametric studies is

that a monostatic radar that uses a single CSA can detect buried rods, because the

rods scatter a significant copolarized signal.

5.3.2 GPR: Detection of Buried Mines

In this section, a monostatic GPR is investigated for detecting plastic landmines

buried in the ground. The model for this GPR is shown schematically in Fig. 5.20a.

Recall for an ideal CSA, the detection of buried mines must be based on the reception

of the copolarized components of the field scattered from the mine. The ideal CSA

should not detect a mine that is concentric with the antenna (the axis of the mine

82


RHCPLHCPLHCP

EarthLHCP

+

Left-handed CSA

Mine

(a)

Mine

DM

sMdM

h

hM

Earth rg g,

rM

(b)

Figure 5.20: a) Schematic drawing for the monostatic GPR used to detect buriedplastic mines, b) parameters that describe the mine and the relativepositioning of the antenna and the mine.

aligned with the axis of the CSA). The detection of the mine must therefore be based

on the copolarized component of the scattered field produced when the antenna and

mine are not concentric, i.e., for detection the scattering from the edges of the mine

should produce a sufficient copolarized signal, see Fig. 5.20a

A parametric study was performed to study the performance of the radar to

detect differently sized mines buried at different depths in the ground. The antenna

parameters were chosen to be θ = 7.5, α = 75, δ = 90, D/d = 5 (d = 2 cm), and

83


h = 7.6 cm. The CSA was terminated at the big end with a resistive sheet that has

Rs = 150Ω. The ground with a relative permittivity εrg = 8 was modeled both as a

lossless medium (σg = 0) and as a lossy medium, in which case the conductivity at

the midband frequency (f = 1.8GHz) was σg = 0.07 S/m. In this parametric study,

three differently sized mines were used, the details of the mine are shown in Fig. 5.20b.

The mines were lossless with relative permittivity εrM = 3, which corresponds to the

electrical properties of commonly used explosives, and their diameters were DM =

9.0 cm, 13.5 cm, and 18.0 cm. The sizes of these mines range from antipersonnel to

antitank mines. The height of the mine hM and the depth in the ground dM (measured

between the top of the mine and the air/ground interface) were scaled to the diameter

of the mine, i.e., hM ≈ DM/2 and dM ≈ DM/2. Two lateral positions of the axis

of the CSA relative to the mine were considered: the CSA centered over the mine

(sM = 0) and the CSA centered over the edge of the mine (sM = DM/2).

The results from the parametric study are shown in Fig. 5.21. The voltage for

the signal scattered from the mine, V minetarget, is plotted as a function of frequency for

the biggest mine (DM = 18 cm) in Fig. 5.21a, for the medium-sized mine (DM =

13.5 cm) in Fig. 5.21b, and for the smallest mine (DM = 9 cm) in Fig. 5.21c. For

clarity, the results from this study are plotted after post processing. A straightforward

(smoothing) filter was applied to the raw data in order to reduce high-frequency

ripples on the results. In these plots, the black curves correspond to the lossless ground

while the gray curves correspond to the lossy ground. The two shifts considered in this

study are shown with different linestyles, the solid curve is for the CSA centered over

the mine and the dashed curve is for the CSA centered over the edge of the mine. The

scattering from the biggest mine can be seen throughout the frequency range, while

scattering from the smallest mine can only be seen at the higher frequencies. When

the ground is lossy, the attenuation causes the signal to decrease over the entire range

of frequencies (compare black and gray lines). Contrary to the previous expectations,

the effect of the mine, i.e., the scattered signal received by the CSA, is biggest when

the antenna is concentric with the mine, and not when it is centered over the edge of

84


Lossless Ground

Lossy Ground

Centered over MineCentered over Edge

0.000

0.000

0.000

0.004

0.004

0.004

0.008

0.008

0.008

0.012

0.012

0.012

1 2 3 4

f [GHz]

(a)

0.5

1 2 3 4

f [GHz]

(c)

0.5

1 2 3 4

f [GHz]

(b)

0.5

||

VV

targ

et

inc

/m

ine

||

VV

targ

et

inc

/m

ine

||

VV

targ

et

inc

/m

ine

Figure 5.21: Theoretical results for the parametric study for the monostaticGPR using a single CSA to detect buried plastic mines: a) DM =18 cm, b) DM = 13.5 cm, and c) DM = 9 cm.

85

5.4 Removing the Dispersion in the CSA

the mine. This result must be caused by the imperfections in the antenna that allow

it to receive the cross-polarized component of the scattered signal. The conclusion

that can be drawn from this parametric study is that a monostatic radar that uses

a single CSA can only marginally detect buried plastic mines. The reason for this is

that the plastic mines do not scatter a sufficient amount of copolarized signal. This

is true even when the CSA is centered over the edge of the mine.

The scattering characteristics of the mine in the lossless ground are examined in

the same manner as in the previous GPR application, Section 5.3.1, to confirm that

the mine does not scatter a significant copolarized signal. The electric field (on axis

0.5 cm above the apex) scattered from the smallest mine (DM = 9 cm) is obtained

by subtracting the electric field for the CSA over the ground with the mine present,

&Emine, from the electric field component of the CSA over the ground without the mine,

&Eground, i.e., Eref,mine = | &Emine− &Eground|. This electric field, normalized by the LHCPcomponent of the incident electric field ELHCP

inc (obtained from the CSA in free space)

is graphed for the mine on axis in Fig. 5.22a, and for the mine off axis in Fig. 5.22b.

From these graphs, it is clear that the copolarized signal scattered from the mine is

too small for the monostatic GPR to clearly detect the mine buried in the ground.


This section is concerned with CSAs that are used in radars that radiate and receive

short pulses. Specifically, a practical approach will be described for removing the

dispersion introduced by the CSA. Upon transmission and reception from the CSA,

signals disperse due to the movement of the active region with frequency. This dis-

persion is undesired, e.g., when CSAs are used in radars, since the received signals

are spread out in time and not clearly distinguishable from other reflections, e.g., the

antenna end reflection and the reflection from the target. A straightforward and also

practical approach is introduced which removes the dispersion from the response; this

increases the chances for successful detection.

86


(b)

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0.0

0.1

0.2

0.3

0.4

f [GHz]

LH

CP

Ere

f,m

ine

/E

inc

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0.0

0.1

0.2

0.3

0.4

f [GHz]

LH

CP

Ere

f,m

ine

/E

inc

(a)

LHCPRHCP

Figure 5.22: Theoretical results for the electric field scattered from the buriedmine: a) on axis and b) off axis.

The radar investigated uses a CSA in free space with the geometry θ = 7.5,

α = 75, δ = 90, and D/d = 8 (d = 2 cm). To reduce clutter, the CSA is terminated

at the big end using a resistive sheet (Rs = 150Ω). A perfectly conducting pipe of

square cross section with equal height and width of w = 15 cm and of infinite length4

is displaced from the apex of the cone by s = 1m (measured from the apex to the top

of the pipe), see Fig. 5.23. The conventional algorithm used to obtain the signature

of the target is outlined in Fig. 5.15 and based on the spectrum of the signal scattered

from the target. This voltage is the basis for another approach; however, instead of

using the voltage in the frequency domain, the main interest is now the voltage in

4In the FDTD, this infinitely long pipe is modeled by extending the pipe into the PML.

87


s

w

w

…

…

Figure 5.23: Schematic model for a monostatic radar to detect an infinitely longpipe in free space

the time domain. Figure 5.24a shows the time-domain reflected voltage of the CSA

in free space without the target, Vref,fs, and Fig. 5.24b shows the reflected voltage ofthe CSA in free space with the target, Vref,t, as functions of normalized time t/τL.

The time τs = s/c is the time for an electromagnetic wave to propagate the distance

s in free space. For this example, this time, normalized by the characteristic time of

the antenna, is τs/τL = s/Ls ≈ 0.5. The two reflections in Fig. 5.24a are the drive-

point reflection at t/τL ≈ 0 and the reflection from the open end of the antenna at

t/τL ≈ 2. Additional, very small reflections due to the target can be seen in Fig. 5.24b

starting at t/τL ≈ 1.1. The signal scattered from this target, Vpipetarget(t) = Vref,t−Vref,fs,

shown expanded in Fig. 5.24c, is a chirp signal with the highest frequencies arriving

first at t/τL ≈ 1.0 and the lowest frequencies arriving last at t/τL ≈ 3.5. The

arrival times approximately match the theoretical arrival times for the CSA. The

highest frequency is radiated and received at the small end of the antenna, so it

propagates the round-trip distance 2(hc − hs) + 2s ≈ 2s = 2.0m and arrives at

t/τL ≈ 2τs/τL = 2s/Ls = 1. The lowest frequency is radiated and received at the big

end of the antenna, so it propagates the round-trip distance 2Ls+2hc+2s = 7.4m and

88


arrives at t/τL ≈ 3.6. Hence, the dispersion of the pulse is caused by the movement

with a change in radiated wavelength of the active region, or, in other words, different

wavelengths are transmitted and received at different times.

The approach to remove the dispersion must eliminate or compensate for the

delay times of the spectral components of the received voltage according to the move-

ment of the active region. This can be accomplished by eliminating the terms in

the frequency domain that cause the dispersion of the pulse. Consider the following

decomposition of the signal scattered from the target

Vtarget(ω) = Vref,t − Vref,fs = Vinc FLHCPtrans F LHCP

target FLHCPrec = Vinc FCSA F LHCP

target , (5.11)

where F LHCPtrans and F LHCP

rec denote the transfer functions for transmission and reception

of the LHCP components. Here, a left-handed CSA is used, so ideally the transmission

and reception are solely based on the LHCP components of the electric field. The

transfer function of the CSA, FCSA(ω) = F LHCPtrans F LHCP

rec , contains the dispersion of

the pulse upon transmission and reception, i.e., this is the part of the response that

needs to be eliminated in the received signal to remove the dispersion. The transfer

function for the target, F LHCPtarget , includes the scattering of the LHCP component from

the target and the propagation to and from the target.

The main objective of this analysis is to determine the transfer function of the

CSA, FCSA, using a calibration target, and then to apply this transfer function in the

detection process for any target. Ideally, for the calibration, a simple and practical

target is desirable that only scatters the copolarized component of the incident signal

(LHCP) at normal incidence, see Fig. 5.25a, such that the transfer function of the

target is F LHCPtarget = −1. However, such scattering characteristics can only be realized

with an anisotropic material, since any isotropic material predominantly reflects the

cross-polarized component of the incident signal, see Section 5.1. A practical imple-

mentation for this material is illustrated in Fig. 5.25b. Here a grid of closely spaced,

thin metallic, and infinitely long wires is placed at the height of the apex.5 Recall from

5In a practical approach, the dimensions of the grid must exceed those of the big end of the CSA.

89


-0.5 0.0 1.0 2.0 3.0 4.0

-0.2

-0.1

0.0

0.1

0.2

t / L

Vre

f,fs

/V

o

(a)

-0.5 0.0 1.0 2.0 3.0 4.0

-0.2

-0.1

0.0

0.1

0.2

t / L

Vre

f,t/

Vo

(b)

-0.5 0.0 1.0 2.0 3.0 4.0

-5.0x10-3

-2.5x10-3

0.0

2.5x10-3

5.0x10-3

Vta

rget

/V

o

t / L

(c)

pip

e

Figure 5.24: Theoretical results for the reflected voltage of the CSA a) in freespace and b) the CSA in free space with the target; c) results forthe voltage of the signal scattered from the pipe in free space.

90


AnisotropicMaterial

…

…

…

LHCP

LHCP

Left-handed CSA

Grid

……

…

…LHCP

LHCP

Left-handed CSA

RHCP

(a)

(b)

12

12

Figure 5.25: Examples for scatterers of co polarization using a) an anisotropicmedium and b) a grid of closely spaced wires.

Section 5.3.1 that a thin metallic rod scatters predominantly linear polarization that

can be decomposed equally into left-handed and right-handed circular components.

The scattering characteristics of the grid are quantified in Fig. 5.26. The circularly

polarized components of the scattered electric field, Eref,grid, i.e., the difference of the

electric field with and without the grid present, are shown as a function of frequency.

Again, the scattered electric field is normalized by the LHCP component of the in-

cident electric field for the CSA in free space, ELHCPinc , and observed on axis 0.5 cm

91


0.5 1.0 2.0 3.0 4.0

0.00

0.25

0.50

0.75

1.00

f [GHz]

|EE

ref,

gri

din

c/

|L

HC

P

LHCPRHCP

Figure 5.26: Theoretical results for the circularly polarized components of theelectric field scattered from the grid, normalized by the incident,LHCP component of the electric field.

above the apex. It can be seen that the grid scatters linear polarization since the two

circularly polarized components are almost equal. The relative amplitudes of these

components confirm the simple argument in Fig. 5.25b: The incident LHCP signal

is reflected equally into LHCP and RHCP signals, with the magnitude of each scat-

tered signal being half that of the incident signal, i.e., ELHCPref,grid = ERHCP

ref,grid = 0.5ELHCPinc .

Consequently, the transfer function of the target can be written as F LHCPtarget,grid = −0.5,

neglecting the propagation to and from the grid. The transfer function for the CSA

can thus be determined from (5.11) to be

FCSA(ω) =Vtarget

Vinc F LHCPtarget,grid

= −2 V gridref,t − Vref,fs

Vinc. (5.12)

To determine FCSA, it is only necessary to measure the reflected voltages of the CSA

with and without the grid present. Once FCSA is determined, it can be used to remove

the dispersion from the voltage of a signal scattered from any other target, e.g., that

shown in Fig. 5.24c. The spectrum of the dispersion-less voltage of the signal scattered

from any target is

V disp.lesstarget (ω) =

VtargetFCSA

= −12

Vref,t − Vref,fs

V gridref,t − Vref,fs

Vinc. (5.13)

The results for this voltage when transformed into the time domain are unsatisfactory,

92


because spurious reflections are superimposed on the signal scattered by the target.

To eliminate these spurious reflections, it is necessary for the spectrum of this voltage

to be smoothly tapered to zero at the limits of the operational frequency range, which

is illustrated in Fig. 5.27a. The dashed line is for the raw results for this voltage,

while the solid line is for the smoothly tapered voltage.

The result for the time-domain voltage scattered from the pipe in free space is

then shown in Figs. 5.27b and c as a function of normalized time t/τL. Notice that

the reflected signal is, as expected, centered around t/τL ≈ 1. By placing the grid

at the apex of the CSA, the spectral components of the modified signals, (5.13), are

delayed such that each frequency of this signal is received at t/τL = 0 when reflected

from the apex. Since the pipe is 1m below the apex of the CSA, the scattered signal

must enter the antenna response at t/τL ≈ 2τs/τL ≈ 1. Clearly, the reflected signal

is not a chirp signal any more, compare with Fig. 5.24c. Instead, it resembles the

differentiated version of the incident pulse, which is due to the transfer function of this

particular target, i.e., the infinite pipe of square cross section scatters approximately

the derivative of the incident signal.

The remainder of this section is concerned with a practical approach for the

pulsed CSA in a monostatic GPR to improve and clarify the time-domain signature

of targets buried in the ground, e.g., for its use in detecting long, perfectly conducting

pipes buried at large depths in the ground. This GPR uses a CSA with the geometry

θ = 7.5, α = 75, δ = 90, and D/d = 8 (d = 6 cm). Notice that this is a

scaled version of the CSA used in the previous example, which was for free space.

All of the dimensions are three times larger and the operational bandwidth is lower

in frequency by a factor of three. The ground is dry soil, which has the following

electrical properties within this bandwidth: εrg ≈ 4 and σg ≈ 0.007 S/m. The same

pipe as in the previous example is used, buried at the depth of 1m in the ground. A

schematic drawing of this radar is shown in Fig. 5.28. Notice that the apex of the

CSA is again located at the air/ground interface, by setting h = 22.8 cm.

The procedure to remove the dispersion from the voltage and thus to determine

93


0.50 0.75 1.00 1.25 1.50

-15x10-3

-10x10-3

-5x10-3

0.0

5x10-3

10x10-3

15x10-3

t / L

(c)

Vta

rget

/V

o

dis

p.le

ss

0.5 1.0 1.5 2.0 2.5 3.0 3.5

0.00

0.01

0.02

0.03

0.04

0.05

-0.5 0.0 1.0 2.0 3.0 4.0

-15x10-3

-10x10-3

-5x10-3

0.0

5x10-3

10x10-3

15x10-3

t / L

(b)

Vta

rget

/V

o

dis

p.le

ss

|VV

targ

et/

inc|

dis

p.le

ss

f [GHz](a)

Tapered VoltageRaw Voltage

Figure 5.27: Theoretical results for the received voltage of the signal scatteredfrom the pipe in free space with the dispersion removed: a) signalin the frequency domain, b) signal in the time domain for −0.5 ≤t/τL ≤ 4, and c) signal in the time domain for 0.5 ≤ t/τL ≤ 1.5.

94


Earth

s

w

w

h

…

…

Figure 5.28: Schematic model for a monostatic radar to detect the infinitely longpipe buried in the ground.

the signature of this buried target is similar to the one outlined above. First, the

following reflected voltages must be determined separately: the reflected voltage of

the CSA in free space over the ground, first with the target in the ground, Vref,t,and then without the target in the ground, Vref,g. In Fig. 5.29a, the theoretical

result for the reflected voltage of the CSA over the ground with the pipe buried

in the ground is shown as a function of normalized time t/τL. Again, the drive-

point reflection clearly dominates the reflected voltage, since the reflections from the

air/ground interface and the target are small compared with this initial reflection

at t/τL ≈ 0. In Fig. 5.29b, the voltage for the signal scattered from the target,

Vtarget(t) = Vref,t−Vref,g, is shown. Due to the presence of the ground, the time for anelectromagnetic wave to propagate the distance s from the apex to the target is now

τs =√εrgs/c, or, in this example, in normalized form τs/τL =

√εrgs/Ls = 0.3. The

95


-0.5 0.0 1.0 2.0 3.0 4.0

-1x10-3

0.5x10-3

0

0.5x10-3

1x10-3

-0.5 0.0 1.0 2.0 3.0 4.0

-0.2

-0.1

0.0

0.1

0.2

t / L

Vre

f,t/

Vo

(a)

Vta

rget

/V

o

t / L

(b)

rod

Figure 5.29: Theoretical results for a) the reflected voltage of the CSA above thedry ground with the pipe present and b) the voltage of the signalscattered from the pipe.

scattered signal in Fig. 5.29b is a chirp signal with the highest frequencies arriving

first at t/τL ≈ [2(hc − hs) + 2√εrgs]/Ls = 0.7 and the lowest frequencies arriving

last at t/τL ≈ [2Ls + 2hc + 2√εrgs]/Ls = 3.2. Similar to the previous example, the

dispersion can be removed using the transfer function of the CSA, FCSA, obtained

by subtracting the reflected voltages of the CSA in free space with and without the

grid in free space, see (5.12). The spectrum of the dispersion-less voltage can thus be

written as

V disp.lesstarget (ω) =

VtargetFCSA

= −12

Vref,t − Vref,g

V gridref,t − Vref,fs

Vinc. (5.14)

This voltage, when tapered at the limits of the operational frequency range and

then Fourier transformed, is graphed in Fig. 5.30 and shows a pulse centered at

96


0.00 0.25 0.50 0.75 1.00

-5.0x10-3

-2.5x10-3

0.0

2.5x10-3

5.0x10-3

-0.5 0.0 1.0 2.0 3.0 4.0

-5.0x10-3

-2.5x10-3

0.0

2.5x10-3

5.0x10-3

t / L

(a)

Vta

rget

/V

o

dis

p.le

ss

t / L

(b)

Vta

rget

/V

o

dis

p.le

ss

Figure 5.30: Theoretical results for the received voltage of the signal scatteredfrom the pipe buried in dry ground with the dispersion removed:a) is for −0.5 ≤ t/τL ≤ 4, and b) for 0 ≤ t/τL ≤ 1.

t/τL ≈ 2τs/τL = 2√εrgs/Ls = 0.6. The signature of the target shown in Fig. 5.30b

(with the dispersion removed) is significantly better than that in Fig. 5.29b (without

the dispersion removed).

97

CHAPTER 6

FDTD Analysis

This chapter is concerned with the finite-difference time-domain (FDTD) method in

three-dimensions for its use in the analysis and design of antennas. Since then, the

FDTD method is a time-domain method in which transient fields are computed as

a function of time, and which allows the accurate characterization of complex in-

homogeneous structures for which analytical methods are ill-suited. The genesis of

this numerical method in electromagnetics was in the work of Yee in 1966 [28]. In

his seminal paper, he proposed a three-dimensional central difference approximation

for Maxwell’s curl equations, both in space and time. The FDTD method has been

applied to many areas in electromagnetics and documented extensively in the litera-

ture [29]-[31]. With increasing computational power,1 the FDTD method has evolved

as one of the most popular and powerful numerical methods over the last decade,

particularly for its use in analyzing and designing antennas [33]-[35]. This numeri-

cal method, inherently a time-domain technique, is especially beneficial in analyzing

broadband antennas such as the conical spiral antenna (CSA), because antenna char-

acteristics can be obtained over a broad range of operating frequencies with just a

single simulation; compare this with a series of simulations that would be necessary

with a frequency-domain method, e.g. Finite Elements or Method of Moments.

Section 6.1 is mainly concerned with the derivation of a single set of update

1Following Moore’s Law [32], the computational power (precisely, the number of transistors per

integrated circuit) approximately doubles every 18 months.

98

6.1 FDTD Update Equations

equations that applies to all media used in the problem, i.e., free space, conductive

ground, and the conductive, uniaxial, anisotropic medium used to implement the

perfectly matched layer (PML) absorbing boundary condition. Section 6.2 verifies

that this absorbing boundary condition absorbs outgoing electromagnetic energy with

negligible reflection. The two-step discretization scheme of CSAs is outlined in Section

6.3, while the discretization of the feed region and the feeding technique with a virtual

transmission line are described in Section 6.4. To obtain the electric fields in the far

zone of the antenna, a near-field to far-field transformation must be employed; the

governing convolution is discussed rather briefly in Section 6.5 for the CSA isolated

in free space. Section 6.6 outlines the savings in execution time by parallelizing the

computer code and running the simulations on multiple-processor architectures.


The objective of this section is to derive a single set of FDTD update equations

that is general enough to be applied to all media present in the problem studied. In

Section 6.1.1, the differential forms of Faraday’s Law and the Ampere-Maxwell Law

are presented for conductive, uniaxial, anisotropic media, used to implement the per-

fectly matched layer (PML) absorbing boundary condition. The electric properties

of this medium are sufficient to describe the media of interest in this work. This

layer truncates the open region surrounding the antenna and effectively extends this

region to infinity by absorbing all impinging waves. In Section 6.1.2, Maxwell’s curl

equations in continuous space are discretized with the approach suggested by Yee in

1966: central differences in time and space. Furthermore, an efficient implementa-

tion of these update equations in the computer program is introduced. Section 6.1.3

is concerned with the correct averaging of the electrical properties (µr, εr, σ) at me-

dia interfaces and the spatial distribution of the electrical parameters of PML. The

CSA terminations (lumped resistor and resistive sheet) require modification of the

electromagnetic fields in the vicinity of the resistive elements, which is described in

99


Section 6.1.4. Section 6.1.5 outlines the implementation of dipole antennas with finite

thickness in the FDTD method.

6.1.1 Update Equations in Continuous Space

In this research, the CSA is investigated when isolated in free space or in free

space placed directly over the ground. This problem, as almost all antenna prob-

lems/applications, is an open-region problem in which the media surrounding the

antenna extend to infinity. Obviously, the infinite space cannot be modeled numer-

ically, so the region surrounding the antenna must be truncated by an absorbing

boundary condition, which should ideally absorb any outgoing electromagnetic radi-

ation and thus simulate the extension of the studied domain to infinity. Hence, the

reception/transmission characteristics or the near field of the antenna can be deter-

mined explicitly, while the field in the far zone can only be determined using near-field

to far-field transformations, discussed in Section 6.5. Over the last decade, absorbing

boundary conditions have been developed that absorb electromagnetic waves with

negligible reflection independent of the frequency, polarization, and angle incident

upon the interface. The genesis of these PML absorbing boundary conditions was

in the work of Berenger [36]. He proposed to replace the medium surrounding the

studied geometries with a highly effective absorbing material of finite thickness, dPML,

shown schematically in Fig. 6.1. The computational domain of this open-region prob-

lem is truncated by the PML and is thus bounded in length (X), width (Y ), and

height (Z). In continuous space, the electrical properties of the PML can be perfectly

matched to those of the ambient medium to result in a vanishing reflection coeffi-

cient. Upon discretization, this absorbing material must be adjusted appropriately to

absorb the outgoing electromagnetic energy with negligible reflection. The uniaxial

material method [37, 38] is more demonstrative and more efficient than Berenger’s

original formulation that involved the splitting of fields, and thus causes computa-

tionally expensive modifications of Maxwell’s equations. The complex coordinate

100


x

y

z

X

dPML

Z

YPML

Ground

Air

Figure 6.1: Schematic drawing for the open-region problem truncated by thePML absorbing boundary condition (CSA is omitted for simplicity).

stretching method is another straightforward approach to formulate the PML, which

can easily be applied to spherical and cylindrical coordinate systems [39]. Although

the PML absorbing boundary condition is computationally more expensive than for-

merly used absorbing boundary conditions, the reflections from the truncated grid are

reduced significantly and almost independent of frequency, polarization, and angle of

incidence.

The PML formulation used in this research is based on Gedney’s uniaxial PML

[40]. The electrical properties of the conductive, uniaxial, anisotropic material are

described by tensors with zero off-diagonal entries. To achieve a perfect match be-

tween the open region surrounding the antenna and the PML, the tensors for the

permittivity, ε, and the permittivity, µ, have to satisfy

ε

εεr=

µ

µµr

=

syszsx

0 0

0sxszsy

0

0 0sxsysz

. (6.1)

101


The tensor elements can be written as

si = κi +σi

jωεwith i = x, y, z (6.2)

and correspond to loss terms in the longitudinal direction i. The purely real term

κi ≥ 1 is intended to suppress evanescent modes,2 while the anisotropic conductivity

σi is intended to suppress the propagating modes within the thin layer. To match

the ambient medium to the PML, it must extend through the PML with the same

µr, εr, and σ, i.e., the PML is defined electrically by the set of anisotropic conductiv-

ities (σx, σy, σz) and the set of isotropic properties (µr, εr, σ) of the ambient medium.3

When using the phasor notation,4 the relative permittivity εr and the conductivity σ

are often combined in the complex permittivity

εr = εr(1− jp) = εr

(1− j

σ

ωεεr

), (6.3)

where p is the loss tangent p = σ/(ωεεr).

The governing FDTD update equations are based on Maxwell’s curl equations;

therefore, the transient electromagnetic fields within conductive, uniaxial, anisotropic

media will eventually be computed from Faraday’s Law

∇× &E = −jωµ &H (6.4)

and the Ampere-Maxwell Law

∇× &H = jωε &E. (6.5)

2In this research, κi = 1 is chosen.3Notice the difference between the conductivities σ and σi: While σ describes the isotropic

conduction loss, e.g., in the ground, σi describes the anisotropic conduction loss along its longitudinal

direction.4The vector phasor F (ω) is defined with

F(t) = Re

F (ω)ejωt

,

where F(t) is the time-harmonic function. Notice that throughout this work, the time-domainvariables are shown with a calligraphic font, while phasors are shown with a roman font.

102


Writing out the tensors and arranging terms to form dimensionless quantities, (6.4)

can be rewritten as

(Ls∇)× &E ′ = −jωτLµr

syszsx

0 0

0sxszsy

0

0 0sxsysz

&H ′ (6.6)

and (6.5) as

(Ls∇)× &H ′ = (ησLs + jωτLεr)

syszsx

0 0

0sxszsy

0

0 0sxsysz

&E ′. (6.7)

In the above equations, each of the terms is a dimensionless quantity. The parameters

used for normalization are the length of the spiral arms, Ls, the characteristic time

of the antenna, τL = Ls/c, and the free-space wave impedance, η. Notice that the

primed fields, &E ′ and &H ′, are normalized as well, i.e., the normalized electric field is

&E ′ =&E

(V/Ls)(6.8)

and the normalized magnetic field is

&H ′ =η &H

(V/Ls), (6.9)

where V is the peak of the incident voltage used to excite the antenna, see (2.9).

In the remainder of this chapter, just the x components of Maxwell’s curl equa-

tions will be investigated for simplicity and then used to derive the FDTD update

equations for the electric field, E ′x, and the magnetic field, H′

x. The y and z com-

ponents of these fields can be derived similarly and are omitted here. Consequently,

Faraday’s Law (eventually used to determine H′x) is

∂E ′z

∂(y/Ls)− ∂E ′

y

∂(z/Ls)= −jωτLµr

syszsx

H ′x, (6.10)

103


and the Ampere-Maxwell Law (eventually used to determine E ′x) is

∂H ′z

∂(y/Ls)− ∂H ′

y

∂(z/Ls)= (ησLs + jωτLεr)

syszsx

E ′x. (6.11)

These equations contain several mixed jω terms (both in the numerator and de-

nominator) that yield convolutions when transformed to the time domain. Numer-

ically, such convolutions are computationally expensive and can be avoided by de-

coupling (6.10) and (6.11) [37]. Consequently, multiple-step update equations will

result. For the central finite-difference scheme of second order, each decoupled up-

date equation must have at most one frequency-dependent term of the form a+ jωb,

e.g., ησyLs + jωτLsy = jωτL, in the numerator. Introducing the “dummy” field

Bx = sz/sx H′x or equivalently jωτLsx Bx = jωτLsz H

′x, Faraday’s Law (6.10) can

be decoupled and the following two-step update equation for the magnetic field H ′x

results: step 1 is

−µr(ησyLs + jωτL)Bx =∂E ′

z

∂(y/Ls)− ∂E ′

y

∂(z/Ls)(6.12)

and step 2

(ησzLs + jωτL)H′x = (ησxLs + jωτL)Bx. (6.13)

Obviously, this dummy field Bx causes an increase of memory requirements in the

computer simulation but is computationally less expensive than convolutions. Note

that this field does not correspond to the magnetic flux, &B = µ &H; this is just the no-

tation commonly used in the literature. Introducing the dummy fields Dx = sz/sx E′x

and Fx = sy Dx, the Ampere-Maxwell Law (6.11) can be decoupled and the following

three-step5 update equation for the electric field E ′x results: step 1 is

(ησLs + jωτLεr)Fx =∂H ′

z

∂(y/Ls)− ∂H ′

y

∂(z/Ls), (6.14)

step 2

(ησyLs + jωτL)Dx = jωτLFx, (6.15)

5Here, the frequency-dependent term introduced by the isotropic conductivity of the medium, σ,

causes an additional step compared to the two steps required to solve for the magnetic field.

104


and step 3

(ησzLs + jωτL)E′x = (ησxLs + jωτL)Dx. (6.16)

Since the frequency-dependent terms in Maxwell’s curl equations are decoupled,

(6.12)-(6.16) can be conveniently transformed to the time domain by replacing the

jωτL terms with time derivatives∂

∂(t/τL). Therefore, the time-domain form of Fara-

day’s Law in continuous space can be written in step 1

−ησyLsµr Bx − µr∂Bx

∂(t/τL)=

∂E ′z

∂(y/Ls)− ∂E ′

y

∂(z/Ls)(6.17)

and in step 2

ησzLs H′x +

∂H′x

∂(t/τL)= ησxLs Bx +

∂Bx

∂(t/τL)(6.18)

Similarly, the time-domain form of the Ampere-Maxwell Law in continuous space can

be written in step 1

ησLs Fx + εr∂Fx

∂(t/τL)=

∂H′z

∂(y/Ls)− ∂H′

y

∂(z/Ls), (6.19)

in step 2

ησyLs Dx +∂Dx

∂(t/τL)=

∂Fx

∂(t/τL), (6.20)

and in step 3

ησzLs E ′x +

∂E ′x

∂(t/τL)= ησxLs Dx +

∂Dx

∂(t/τL). (6.21)

In the next section, these equations will be discretized following Yee’s approach to

obtain typical FDTD update equations.

6.1.2 Update Equations in Discretized Space

The spatial discretization of the update equations for the electric and magnetic fields,

derived in the previous section, is based on the Yee cell [28], shown in Fig. 6.2. The

dimensions of the cube are denoted by ∆x,∆y, and ∆z for its length, width, and

height. The Yee cell is the basic building block of the computational domain and

used to discretize and model arbitrary structures of complex composition. Notice

105


that the magnetic fields are normal to the sides of the cell and determined at the

center of the sides, while the electric fields are tangential to the edges of the cell

and determined at the center of the edges. The discretization in time is such that

the magnetic fields are evaluated at half time steps, i.e., t = (n + 0.5)∆t, while the

electric fields are evaluated at full time steps, i.e., t = (n + 1)∆t. Throughout this

work, a short-hand notation for the field components is used, e.g., the normalized

x component of the electric field is referred to as E ′x|n+1i,j,k = E ′n+1

x (i, j, k) = E ′x(x =

(i−0.5)∆x, y = (j−1)∆y, z = (k−1)∆z, t = (n+1)∆t). A list of staggered positions

in time and space for all electric and magnetic fields within the Yee cell (i, j, k) is

presented in Table 6.1. It is important to note that the spatial indices (i, j, k) refer to

the cell in which the field components are located rather than to the actual positions

of the field components in the grid. Notice also that the fields that are referred to

cell (i, j, k) are shown with black symbols in the Yee cell, Fig. 6.2, while the gray field

components belong to neighboring cells.

In the following, Maxwell’s Equations (6.17)-(6.21) will be discretized to obtain

the typical FDTD update equations, so that the transient fields can be computed using

the well-known “leapfrog” scheme of the FDTD, i.e., the update of the magnetic field

x

y

z

Hz

Ex

Ey

Ez

Hx

Hy

y

x

z

Yee Cell ( , , ):i j k

[ ]( -1) , ( -1) , ( -1) )i x j y k z

Figure 6.2: Yee cell in three dimensions.

106


Table 6.1: Short-hand notation and staggered positions in time and space for all

electric and magnetic fields within the Yee cell (i, j, k).

Time Coordinates

Field t x y z

H′x|n+0.5i,j,k (n+ 0.5)∆t (i− 1)∆x (j − 0.5)∆y (k − 0.5)∆z

H′y|n+0.5i,j,k (n+ 0.5)∆t (i− 0.5)∆x (j − 1)∆y (k − 0.5)∆z

H′z|n+0.5i,j,k (n+ 0.5)∆t (i− 0.5)∆x (j − 0.5)∆y (k − 1)∆z

E ′x|n+1i,j,k (n+ 1)∆t (i− 0.5)∆x (j − 1)∆y (k − 1)∆z

E ′y|n+1i,j,k (n+ 1)∆t (i− 1)∆x (j − 0.5)∆y (k − 1)∆z

E ′z|n+1i,j,k (n+ 1)∆t (i− 1)∆x (j − 1)∆y (k − 0.5)∆z

is based on the previously calculated electric field and vice versa. Throughout this

work, the temporal and spatial derivatives are discretized as suggested by Yee using

central-difference approximations of second order [28]. Thus, the spatial derivatives

become, e.g., for E ′y,

∂E ′y

∂(z/Ls)=

E ′y(i, j, k + 1)− E ′

y(i, j, k)

(∆z/Ls)(6.22)

the temporal derivatives become, e.g., for E ′x,

∂E ′x

∂(t/τL)=

E ′n+1x (i, j, k)− E ′n

x (i, j, k)

(∆t/τL). (6.23)

The update equations for the electric and magnetic fields can thus be expressed

in discretized space and time: The two-step update equations to solve for the x

component of the magnetic field become in step 1

Bx|n+0.5i,j,k =1− 0.5 ησyτL (∆t/τL)

1 + 0.5 ησyτL (∆t/τL)Bx|n−0.5i,j,k − (∆t/τL)

µr[1 + 0.5 ησyτL (∆t/τL)]

×[ E ′

z|ni,j+1,k − E ′z|ni,j,k

(∆y/Ls)− E ′

z|ni,j,k+1 − E ′z|ni,j,k

(∆z/Ls)

](6.24)

107


and in step 2

H′x|n+0.5i,j,k =

1− 0.5 ησzτL (∆t/τL)

1 + 0.5 ησzτL (∆t/τL)H′

x|n−0.5i,j,k +1

1 + 0.5 ησzτL (∆t/τL)

×[1 + 0.5 ησxτL (∆t/τL)] Bx|n+0.5i,j,k

+ [−1 + 0.5 ησxτL (∆t/τL)] Bx|n−0.5i,j,k

. (6.25)

Similarly, the three-step update equations to solve for the x component of the electric

field become in step 1

Fx|n+1i,j,k =εr − 0.5 ηστL (∆t/τL)

εr + 0.5 ηστL (∆t/τL)Fx|ni,j,k +

(∆t/τL)

εr + 0.5 ηστL (∆t/τL)

×H′

z|n+0.5i,j+1,k − H′z|n+0.5i,j,k

(∆y/Ls)− H′

z|n+0.5i,j,k+1 − H′z|n+0.5i,j,k

(∆z/Ls)

, (6.26)

in step 2

Dx|n+1i,j,k =1− 0.5 ησyτL (∆t/τL)

1 + 0.5 ησyτL (∆t/τL)Dx|ni,j,k

+1

1 + 0.5 ησyτL (∆t/τL)

(Fx|n+1i,j,k − Fx|ni,j,k

), (6.27)

and in step 3:

E ′x|n+1i,j,k =

1− 0.5 ησzτL (∆t/τL)

1 + 0.5 ησzτL (∆t/τL)E ′x|ni,j,k +

1

1 + 0.5 ησzτL (∆t/τL)

×[1 + 0.5 ησxτL (∆t/τL)] Dx|n+1i,j,k

+ [−1 + 0.5 ησxτL (∆t/τL)] Dx|ni,j,k. (6.28)

In (6.24) and (6.25), the fields Bx and H′x must be interpolated at times t = n∆t since

they do not exist at the full time steps, e.g., Bnx = 0.5(Bn+0.5

x + Bn−0.5x ), similarly in

(6.26)-(6.28) the fields Fx,Dx, and E ′x must be interpolated at times t = (n+0.5)∆t.

Obviously, the above equations are rather complex, so the following technique is used

108


in the computer program to simplify their structure and to optimize the updating

procedure. Equations (6.24)-(6.28) are therefore expressed using “update variables”

that contain the terms in front of the field components. For the magnetic field, step

1 becomes

Bx|n+0.5i,j,k = upd Bx1(j)Bx|n−0.5i,j,k

+ upd Bx2(j)

[E ′z|ni,j+1,k − E ′

z|ni,j,k(∆y/Ls)

− E ′z|ni,j,k+1 − E ′

z|ni,j,k(∆z/Ls)

](6.29)

and step 2

H′x|n+0.5i,j,k = upd Hx1(k)H′

x|n−0.5i,j,k + upd Hx2(i, k)Bx|n+0.5i,j,k

+ upd Hx3(i, k)Bx|n−0.5i,j,k . (6.30)

For the electric field, step 1 becomes

Fx|n+1i,j,k = upd Fx1(i, j, k)Fx|ni,j,k

+ upd Fx2(i, j, k)

[H′z|n+0.5i,j+1,k −H′

z|n+0.5i,j,k

(∆y/Ls)− H′

z|n+0.5i,j,k+1 −H′z|n+0.5i,j,k

(∆z/Ls)

],(6.31)

step 2

Dx|n+1i,j,k = upd Dx1(j)Dx|ni,j,k

+ upd Dx2(j)(Fx|n+1i,j,k −Fx|ni,j,k

), (6.32)

and step 3:

E ′x|n+1i,j,k = upd Ex1(k) E ′

x|ni,j,k + upd Ex2(i, k)Dx|n+1i,j,k

+ upd Ex3(i, k)Dx|ni,j,k. (6.33)

Notice that the update variables in the above equations are one, two or three-

dimensional. This is mainly because the terms in the update variables contain a

109


combination of PML conductivities σi. While the anisotropic conductivities, σi, can

be constant in continuous space to satisfy the matching condition, (6.1), the σi must

be tapered from zero (at the PML interface) to σi,max (at the outer grid boundary) in

discretized space along their longitudinal direction to effectively absorb the outgoing

waves,6 i.e., σx = σx(i), σy = σy(j), and σz = σz(k). In the above equations, it was

furthermore assumed that the isotropic conductivity is piecewise constant along the

z direction, i.e., σ = σ(k), and that the permittivity can be piecewise constant in any

direction, i.e., εr = εr(i, j, k). The advantage of this implementation of the FDTD

equations is that the update variables can be calculated once in the initialization

stage of the computer program and then used throughout the rest of the program,

i.e., for the update of &E and &H. It is therefore not necessary to perform the costly

operations of calculating these terms every time the fields are updated. The draw-

back of this approach is that especially the 3D variables, e.g., upd Fx1 and upd Fx2,

require a significant amount of memory for the computer simulation to run.7 For the

program to run most time efficient on common computer architectures, it is often

most important to reduce the number of floating-point operations instead of reducing

the memory consumption.

6.1.3 Medium Properties in the FDTD Update Equations

Various isotropic media are used in problems studied, e.g., that in which the CSA in

free space is placed directly over the ground to detect mines buried in the ground.

This open region is divided into the free-space region that can be described with

the electrical properties (µr = 1, εr = 1, and σ = 0), the soil with the electrical

properties (µrg = 1, εrg > 1, and σg > 0), and the mine with the electrical properties

6The anisotropic conductivities must be tapered within the PML because the discontinuity would

otherwise cause significant artificial reflections.7Some versions of the computer code are based on a hybrid approach, in which the 1D and 2D

update variables are implemented as shown, while the fields F are updated explicitly without the

3D update variables.

110


r1 1,

( , -1, )i j k

r1

( , , )i j k

r2 2,

( , , )i j k

r3 3,

( , -1, -1)i j k

r2

( , , -1)i j k

r4 4,

( , , -1)i j k

(a)

(b)

x

y

z

x

y

z

Figure 6.3: Details about the correct averaging at an interface of the electricalproperties for a) the electric field Ex and b) the magnetic field Hz.

(µrM = 1, εrM , and σM = 0). Obviously, the anisotropic conductivities σi must be

zero throughout these media.

Since the media interfaces run along the boundaries of Yee cells, it is sufficient

to define the medium properties (µr, εr, σ) for each cell instead of for each node.

This approach requires the averaging of the corresponding electrical properties right

at media interfaces, which is illustrated for one electric field and magnetic field in

Fig. 6.3. Figure 6.3a details how the permittivity and conductivity must be averaged

for the electric field Ex and Fig. 6.3b how the permeability must be averaged for

the magnetic field Hz. Notice that the node for the electric field, Ex, is surrounded

by four cells, hence, the effective permittivity, εr, and effective conductivity, σ, for

111


Table 6.2: Averaging of the electrical properties in the FDTD update equa-tions.

Updated Field Averaged Medium Property

H′x|n+0.5i,j,k µr =

µr(i,j,k)+µr(i−1,j,k)2

H′y|n+0.5i,j,k µr =

µr(i,j,k)+µr(i,j−1,k)2

H′z|n+0.5i,j,k µr =

µr(i,j,k)+µr(i,j,k−1)2

E ′x|n+1i,j,k εr =

εr(i,j,k)+εr(i,j−1,k)+εr(i,j,k−1)+εr(i,j−1,k−1)4

E ′y|n+1i,j,k εr =

εr(i,j,k)+εr(i−1,j,k)+εr(i,j,k−1)+εr(i−1,j,k−1)4

E ′z|n+1i,j,k εr =

εr(i,j,k)+εr(i−1,j,k)+εr(i,j−1,k)+εr(i−1,j−1,k)4

this node are the average of the electrical properties εr and σ of the surrounding

cells. Similarly, the node for the magnetic field is surrounded by two cells, hence,

the effective permeability, µ, for this node is the average of the permeabilities of the

surrounding cells. The proper averaging of the electrical properties is outlined in

Table 6.2 for the electric8 and magnetic fields.

The PML, truncating the region surrounding the antenna can be subdivided in

three general regions: corners, edges, and sides. Within these regions, the anisotropic

conductivities σi are nonzero when the PML extends in the longitudinal direction

and zero when it extends in any transverse direction, e.g., σx = σx(i) = 0 for the

PML extending along x, and σx = 0 for the PML extending along y and z. Hence,

in the corners all conductivities σi are nonzero, in edges only two conductivities σi

are nonzero, e.g., σx = σx(i) = 0 and σy = σy(j) = 0 for the edge region with PMLinterfaces normal to x and y direction, and in the sides only one σi is nonzero, e.g.,

σx = 0 for the side region with a PML interface normal to the x direction.

In Fig. 6.4a, the variation of the conductivity σx along the longitudinal direc-

tion x is illustrated. The conductivity is zero within the open region surrounding

8The averaging technique for the conductivity σ is the same as for the permittivity εr (replace

εr with σ).

112


x,max

x

dPML X-dPML X

( -1)i x ( -0.5)i x

Cell ( , , )i j k

x

(a)

(b)

Air ( = 1)r

Ground ( , ) rg g

PML

z

x

m sx,air

m sx,ground

PEC

Figure 6.4: a) Spatial variation of the conductivity σx a) along its longitudi-nal direction x and b) along its transverse direction z with theair/ground interface present.

the antenna and tapered from zero at the interface between the open region and the

PML to the maximum value σx,max at the outermost boundaries of the computational

domain.9 Commonly, a polynomial taper is chosen to mathematically express the con-

ductivities σi along its longitudinal direction [40]. The expression for the normalized

9The optimum value for σi,max must be a trade off between two different types of reflection. When

the value for σi,max is chosen too small, waves are barely attenuated within the PML and reflect

from the PEC interface that is behind the PML. When the value for σi,max is too big, significant

reflections occur from within the PML due to large medium discontinuities between cells.

113


conductivity ησxLs, as it appears in the update equations, can thus be written

ησxLs =

ησx,maxLs

(dPML−xdPML

)mfor x ≤ dPML

0 for dPML < x < X − dPML

ησx,maxLs

[x−(X−dPML)

dPML

]mfor x ≥ X − dPML

(6.34)

In this research, the exponent of the polynomial taper is chosen to be m = 2.1

[41]. The following rule of thumb for evaluating the maximum value of this taper is

proposed given the thickness of the PML is 10 cells. The condition

ησx,maxLs =τL∆t

(6.35)

should be satisfied to obtain good results for the reflection coefficient from the in-

terface. Notice that this expression can be remembered/derived easily from the

discretized form of the update equations, (6.29)-(6.33). Terms of the form 1 +

0.5ησiLs(∆t/τL) will be become 1 + 0.5 at the outermost boundaries of the com-

putational domain.

For the best PML performance, the anisotropic conductivities must be deter-

mined with respect to the actual position of the node instead of the position of the

cell the field is located in. In the presented case, shown in the inset of Fig. 6.4a,

the update equations for the fields E ′y|i,j,k, E ′

z|i,j,k, and H′x|i,j,k require the calculation

of the conductivity σx at positions x = (i − 1)∆x, while the update equations for

the fields E ′z|i,j,k,H′

y|i,j,k, and H′z|i,j,k require the calculation of the conductivity at

positions x = (i− 0.5)∆x.

When the open region surrounding the antenna is a homogeneous dielectric (εr),

the tensor elements si are often expressed as si = κi+σi/(jωεεr) and thus dependent

on the relative permittivity of the ambient medium. However, when the open region

has media interfaces with piecewise constant εr, e.g., the air/ground interface, the

above expression for si causes an accumulation of charges at these discontinuities and

thus leads to instabilities [40]. To satisfy Gauss’ Law, the si terms can only vary

along its longitudinal direction. Therefore, the imaginary part of si, Imsi must be

114


…

…

…

…

RL

Ex ( )i , j ,kL L L

n+1

x

yz

Figure 6.5: Implementing the lumped resistor in the FDTD method.

continuous at interfaces along its transverse directions, as illustrated in Fig. 6.4b for

Imsx along its transverse direction z near the air/ground interface. By expressing

the terms si in (6.2) with εr = 1, the PML is optimized for the free space region.

Since the antenna is located in air, this region has the strongest fields and requires the

best performance. The ground is most often lossy, so that the natural attenuation in

the soil could make up for the increase in the reflection from the PML in the ground.

6.1.4 FDTD Update Equations for the Resistive Elements

This section is concerned with the FDTD update equations for the lumped resistor

elements and the resistive sheet. More detailed discussions on the lumped resistors

can be found in [30] and on the resistive sheet in [42, 43].

To model the lumped resistor, RL, connected at two terminals that extend in

the x direction, consider the schematic drawing in Fig. 6.5. In discretized space, the

implementation of this resistor involves the modification of the tangential electric

fields, Ex, between the terminals. In the presented case, the modification of a single

electric field En+1x (iL, jL, kL) is discussed; however, this analysis can easily be applied

to a multiple-cell gap. When the resistance of the node within this gap is RL, the

115


current (measured positive in +x direction) through this node can be expressed as

In+0.5L (iL, jL, kL) =

∆x[En+1x (iL, jL, kL) + En

x (iL, jL, kL)]

2RL

, (6.36)

which is Ohm’s Law. This conduction current must be considered in the derivation

of the update equations for the electric field at this node, i.e., the Ampere-Maxwell

Law must then be

∇× &H = εεr∂ &E∂t+ &JL = εεr

∂ &E∂t+

JL,x

0

0

. (6.37)

Upon discretization of this equation, the new update equation for the electric field

within the gap is

Ex|n+1iL,jL,kL=

1− ∆t∆x2RLεεr∆z∆y

1 + ∆t∆x2RLεεr∆z∆y

Ex|niL,jL,kL+

∆tεεr

1 + ∆t∆x2RLεεr∆z∆y

×(Hz|n+0.5iL,jL+1,kL

−Hz|n+0.5iL,jL,kL

∆y− Hz|n+0.5iL,jL,kL+1

−Hz|n+0.5iL,jL,kL

∆z

).(6.38)

Notice that this update equation is written in its original and, hence, un-normalized

form.

The resistive sheet, commonly characterized by the resistance per square Rs =

1/(σsts) (conductivity σs and thickness ts), covers the cross-sectional area of the big

end of the CSA and thus spreads over a significant number of cells in the computa-

tional grid. However, the thickness of the sheet is just a fraction of the height of the

Yee cell, as shown in Fig. 6.6a; hence, a subcell model must be introduced to include

this thin resistive sheet in the FDTD method. This subcell approach is illustrated in

Fig. 6.6b, which is a cross-sectional view of the modified Yee cell in the xz plane. The

top row of cells contains the resistive sheet, shown in gray, and within these cells a

subcell is introduced. These special cells contain two electric field components Ez: the

electric field in the center of the subcell (resistive sheet), Ez,s, and the regular electric

field in the center of the Yee cell, Ez,r. Reference [42] outlines, how this subcell model

116


x

y

z

Hz

Ex

Ey

Ez

Hx

Hy

ts

(a)

(b)

x

z

Ex

Ex

Ex

Ex

Ex

Ex

Ex

Ex

Ex

Ez,r Ez,r

Ez EzEz Ez

Ez,s Ez,sEz,s Ez,s

Ez,r Ez,rHy

Hy

Hy

Hy

Hy

Hy

Subcell

Figure 6.6: Implementing the resistive sheet in the FDTDmethod: a) the three-dimensional Yee cell containing the resistive sheet and b) slice ofthe FDTD grid in the vicinity of the sheet.

modifies the electric and magnetic fields within those special cells, i.e., incorporat-

ing the additional electrical properties and the additional electric field in the FDTD

update equations.

6.1.5 FDTD Update Equations for a Dipole

This section outlines the modeling of linear dipole antennas in the FDTD method.

The schematic model for the dipole of length L and conductor radius a is shown

in Fig. 6.7a. The case investigated in this section is that in which the radius of the

conductor, a, is smaller than the dimensions of the cell size, in particular a < ∆y/2 ≈

117


xyz

(a)

(b)

L

2a

x

y

z

Hz

Ex

Ey

Ez

Hx

Hy

y

x

z

Figure 6.7: a) Schematic model of the dipole antenna, b) drawing of a Yee cellthat contains the antenna conductor.

∆z/2, see Fig. 6.7b. Obviously, the regular FDTD scheme is not able to model the

cross section of this conductor, therefore, an approximate equivalence for a round

wire based on a quasi-static approach is introduced here [44]. For electrically thin

dipoles, i.e., a/λ 1, usually just the tangential electric field components, Ex, along

the conductor are set to zero.10 To incorporate the finite radius of the conductor in

the FDTD method, modified FDTD update equations for the magnetic fields in the

immediate vicinity are derived. Consider the cross-sectional view of the Yee cell in

the xz plane that contains the antenna conductor, Fig. 6.8a. In the following quasi-

static solution approach, the electric and magnetic field components, Ez and Hy, in

10It is interesting to note that the dipoles implemented in this simplest approach have an effective

radius of about a ≈ ∆y/7 ≈ ∆z/7.

118


the immediate vicinity of the conductor are assumed to be zero within z ≤ a and to

have a 1/ρ dependence (ρ is the radial distance measured from the center of the wire)

within a < z ≤ ∆z, i.e.,

Ez(z) =∆z

2zEz(∆z/2) (6.39)

and

Hy(z) =∆z

2zHy(∆z/2). (6.40)

The integral form of Faraday’s Law

∮&E · &dl = −µ

∫∫ ∂ &H∂t

· &dS (6.41)

together with (6.39) and (6.40) can now be used to determine the modified update

equation for the magnetic fields Hy along the rod

Hy|n+0.5i,j,k = Hy|n−0.5i,j,k − ∆t

µ∆z

2

ln(∆z/a)(Ex|ni,j,k+1 − Ex|ni,j,k) (6.42)

+∆t

µ∆x(Ez|ni+1,j,k − Ez|ni,j,k)

Similarly, the update equation for the magnetic field Hy in the immediate vicinity

of the conductor can be derived and is omitted here for simplicity.

Within the gap, the tangential electric field, Ex, is related to the voltage that

different source models insert across the gap, which is explained in more detail in

Section 6.4 and in Appendix B. A simple argument shows that the above set of

update equations applies also for the magnetic fields around the drive-point gap, see

Fig. 6.8b for a one-cell gap. Notice that the electric fields, Ez, along sides A and B are

assumed to vary by 1/ρ in the update equations for the magnetic field surrounding

the conductor. For consistency, they must therefore be treated the same way when

updating the magnetic field components surrounding the gap. A similar argument

must be applied at the ends of the antenna conductors, see Fig. 6.8c. Here, the

electric field Ez along side C is still assumed to vary by 1/ρ, while the electric field

along side D and the magnetic field in between must be treated as being constant

119


x

z

x

x

z

z

Ex = 0

Ex = 0

Ex = 0

Ex = 0Ex = ( )f V

Ex = ( )f V

Ex

Ex

Ex

Ex

Ex

Ex

Ez

Ez

Ez

Ez

Ez

Ez

Ez

Ez

Ez

Hy

Hy

Hy

Hy

Hy

Hy

az

x

(a)

(b)

(c)

A

C

B

D1

1

1

1

1

1

1

Figure 6.8: Cross-sectional view of a) a single Yee cell that contains the antennaconductor, b) Yee cells in the vicinity of the drive-point gap, andc) Yee cell near the antenna end.

along z within the cell. It is therefore necessary to derive a separate update equation

for the magnetic field one cell beyond the antenna end, i.e.,

Hy|n+0.5i,j,k = Hy|n−0.5i,j,k − ∆t

µ∆z(Ex|ni,j,k+1 − Ex|ni,j,k) (6.43)

+∆t

µ∆x

[Ez|ni+1,j,k − 0.5 ln(∆z/a) Ez|ni,j,k

]

120

6.2 Validation of PML

x

z

y

A

B

CD

E

F

G

HI

J

KL

oL

pL

qL

Figure 6.9: Schematic drawing for the approach to validate the PML (only theregion surrounding the antenna is shown, the PML is omitted forsimplicity).

6.2 Validation of PML

The performance of the PML has been investigated in many ways. Just one approach

is outlined here; a schematic drawing for this approach is shown in Fig. 6.9. A linear

dipole with total length of L = 20.4 cm is placed horizontally in free space at a height

of h = 2 cm above the ground. The electric field at points marked A−K in Fig. 6.9

are recorded for three soils: dry, medium moist, and wet. The electrical parameters

for these soils at the half-wave resonant frequency of the dipole (f = 0.7GHz) are

given in Table 6.3. Notice that the observation points are in the immediate vicinity

of the PML corners and edges and also near the air/ground interface. In order to

validate the PML, the electric fields are recorded for two different grid sizes. In the

first case, the PML interfaces are displaced by 0.3L from the conductors of the dipole

(o ≈ 0.6, p ≈ 0.6, q ≈ 1.6), and in the second case the PML interface is moved an

additional distance of 3.3L further out in each direction (o ≈ 7.2, p ≈ 7.2, q ≈ 8.2).

In both cases, the distances from the observation points for the electric field to the

antenna are identical and the thickness of the PML is chosen to be 10 cells = 0.3L.

121

6.3 FDTD Modeling of the CSA

Table 6.3: Electrical parameters for different soil conditions at f = 0.7GHz.

εrg σg [S/m]

dry 4.0 0.008

medium 8.0 0.03

wet 15.0 0.07

The relative error is then calculated for the total electric field with

Rel. Error =max |&Efar − &Eclose|

max |&Efar|, (6.44)

where &Eclose is the recorded electric field when the PML interface is displaced by 0.3Lfrom the conductors and &Efar the electric field when the PML interface is displaced by3.6L from the conductors. The largest errors observed over time are listed in Table 6.4

for each observation point and for each soil condition. The errors for the electric fields

are seen to be small, largest for observation points closest to the antenna conductor

(as expected) and increasing for increasing conductivity of the ground. It should be

noted, that the errors in the reflected voltage in the feeding transmission line, which

is often of greater interest, are smaller than those for the electric fields surrounding

the antenna by several orders of magnitude.


In the FDTD model, space is divided into cubical cells (∆x = ∆y = ∆z), and

the perfectly conducting arms of the CSA are approximated by a staircased surface.

This surface is constructed using the two-step procedure outlined below and shown

in Fig. 6.10. Figure 6.10a is for a plane of constant height z. The solid circle in

this figure is the intersection of the conical surface of the CSA with this plane, and

the dashed circle is the intersection of the conical surface with the plane one cell

below (z−∆z). In the first step, the circular boundary of the CSA is represented by

122


Table 6.4: Relative errors for the electric field at various observation points inthe grid for three different soil conditions.

Rel. Error [dB]

Pos. dry medium wet

A -78 -67 -57

B -78 -69 -59

C -79 -73 -68

D -81 -68 -59

E -78 -67 -60

F -78 -71 -67

G -83 -80 -65

H -85 -80 -68

I -86 -84 -80

J -75 -64 -56

K -85 -73 -66

Step 1:

(a)

Step 2:

(b)

Figure 6.10: Schematic drawing detailing the FDTD modeling of the antennaarms in a two-step process.

vertical PEC faces. In the figure, these faces are dashed areas. They project above

the plane at the solid circle and below the plane at the dashed circle. On the plane

(z), these vertical faces are connected by a horizontal PEC surface that is gray in

123


……

10% of Antennaat Feed End

Figure 6.11: FDTD model of the conical spiral antenna. Only 10% of the an-tenna at the feed end is shown.

the figure. In the second step, shown in Fig. 6.10b, the angular sectors of width δ

are determined (dashed lines), and the horizontal and vertical faces that are within

these sectors are extracted to form the arms on this plane. This process is repeated

on successive planes until the complete antenna is obtained. Figure 6.11 shows the

staircased surface that models a CSA with θ = 7.5, α = 75, δ = 90, and D/d = 8.

For clarity, only a small portion (about 10%) of the antenna near the drive point is

shown. In the FDTD model, the tangential component of the electric field is set to

zero on the faces of the staircased surface.

124

6.4 FDTD Antenna Feed

xy

z

Hy

Hz

Ex

Feed Points

Figure 6.12: Discretized version of the feeding disc.


As outlined in Chapter 7, the CSAs investigated experimentally and numerically in

this research are fed through a circular disc whose circular sectors make electrical

contact with the spiral arms, see Fig. 7.10 for a schematic drawing or Fig. 7.11b

for a photograph of the feed region in the experiments. However, in the numerical

model, the fine details of the feed region cannot be reproduced; hence, a simplified

model for the antenna feed must be used. Figure 6.12 shows the discretized version

of the feeding disc used in the validation of the measurements for the CSA, which

can also be seen in the bottom of Fig. 6.11 attached to the small end of the CSA.

In this model, 24 cells were placed along the smallest diameter d, i.e., cell sizes with

∆x = ∆y = ∆z = d/24 ≈ 0.8mm are used to model the circular, perfectly conducting

sectors. The feed points are indicated in the drawing using the black dots; notice that

these positions differ from those used in the measurements, see Fig. 7.11.

The “simple-feed” model [34, 45] is used to excite the CSA with the incident

pulse, chosen to be the differentiated Gaussian pulse, (2.9), throughout this work.

The spectral width of this pulse is sufficient to obtain accurate, broadband results in

the frequency domain whenever the time-domain results are Fourier transformed. The

main aspects of this numerical feed model will be discussed to provide the background

125


for the convergence analysis of basic FDTD antenna feeds in Appendix B. This

non-physical antenna feed, used exclusively for the analysis of CSAs in this work,

involves a one-dimensional transmission line that virtually attaches to the drive point

of the antenna and that is modeled concurrently with the transient fields in the three-

dimensional FDTD grid. The schematic model of this line is shown in Fig. 6.13a.

The spatial step in the line, ∆s, is usually chosen to be same as one of the Yee cell

parameters (∆x, ∆y, or ∆z), and the time step, ∆t, is identical for both FDTD grids.

The variables in the transmission line are the current, I, and the voltage, V . The unitcell in Fig. 6.13a shows the discretized components in the one-dimensional FDTD

grid. Both components are staggered in space and in time, i.e., Vn+1(l) = V(s =(l− 1)∆s, t = (n+1)∆t) and In+0.5(l) = I(s = (l− 0.5)∆s, t = (n+0.5)∆t). Notice

that this notation is similar to the notation used for the three-dimensional FDTD

update equations; the spatial index l refers to the cell in which the transmission-line

components are located instead of the actual position in the grid.

The equations that couple the voltage and the current in the line can be ex-

pressed with the following differential equations

∂I∂t= − c

Zc

∂V∂s

(6.45)

and∂V∂t

= −cZc∂I∂s

. (6.46)

In these equations, c is the speed of light in the line, and Zc is the characteristic

impedance of the transmission line. The update equation In+0.5(l) is obtained by

discretizing (6.45) at t = n∆t and s = (l − 0.5)∆s, and the update equation for

Vn+1(l) is obtained by discretizing (6.46) at t = (n+0.5)∆t and s = (l−1)∆s. Then,

the update equation for the current is

In+0.5(l) = In−0.5(l)− 1

Zc

c∆t

∆s

[Vn(l + 1)− Vn(l)

](6.47)

and the update equation for the voltage is

Vn+1(l) = Vn(l)− Zcc∆t

∆s

[In+0.5(l)− In+0.5(l − 1)

]. (6.48)

126


Antenna DrivePoint

…

…

xyz

Hy

Hz

Termination(ABC)

Source

Observation

V ( )n+1

l

V ( +1)n+1

l

I ( )n+0.5

ls

Drive Point

Reflected T.L.Components

Total T.L.Components

l l= max

l l= src

l l= obs

l=1

I ( ) = ( , )l fmax H Hx y

1D Transmisssion Line:

Simple Feed:

Ex = f l( ( ))V max

(a)

(b)

dp

Figure 6.13: Schematic model of a) the one-dimensional transmission line andb) the feed region including the virtual transmission line.

Just like the update equations of the electromagnetic field in the three-dimensional

grid, these equations can be implemented in a normalized fashion using the same

parameters for normalization; these are the peak value of the incident pulse, V,

the characteristic length of the antenna, Ls, the characteristic time of the antenna,

τL = Ls/c, and the wave impedance of s free space, η =√µ/ε. Table 6.5 lists each

term in the transmission-line update equations in the normalized form; however, for

simplicity, the original and hence un-normalized form of the update equations is used

127


Table 6.5: Normalization of the parameters in the transmission-line equations.

Original Parameter Normalized Parameter

V V/V

I I/(V/η)

c∆t/∆s (∆t/τL)/(∆s/Ls)

Zc Zc/η

throughout the remainder of this section.

The source excitation is established at the source plane within this transmission

line, see Fig. 6.13a, using the “one-way injector” [34, 45]. This non-physical source

implementation makes it possible to study the reflected voltage from the antenna

below the source plane without the superposition of the incident signal and is com-

monly referred to as one-way injector since the incident signal is launched at the

source plane in only one direction—towards the virtual drive point. Consequently,

the transmission-line components above the source plane at s = (lsrc− 1)∆s are total

voltages and total currents in the transmission-line equations, i.e., they contain the

incident and reflected components. The computational domain below and right at

this plane only contains the reflected transmission line components. For simplicity,

the voltages and currents in the line will be referenced either as total or reflected

components. The notation for the current is then

I =

Itot for l ≥ lsrc

Iref for l < lsrc

and for the voltage

V =

Vtot for l > lsrc

Vref for l ≤ lsrc.

For the transition between these two domains, one update equation for each compo-

nent (V and I) must be modified near the source plane by adding/subtracting the

128


time-domain expression of the excitation. Slightly modified updates for V and I re-sult for those update equations that contain mixed (total and reflected) components

in its original form, (6.47)-(6.48). The update for the current In+0.5(lsrc), right above

the source plane and therefore a total current is

In+0.5tot (lsrc) = In−0.5

tot (lsrc)− 1

Zc

c∆t

∆s

[Vntot(lsrc + 1)− Vn

tot(lsrc)]. (6.49)

However, the voltage Vntot(lsrc) observed right at the source plane is not a total voltage

but a reflected voltage in the transmission line. Since the total voltage is simply the

sum of incident and reflected voltage, i.e.,

Vntot(lsrc) = Vn

inc + Vnref(lsrc), (6.50)

(6.49) can be rewritten as

In+0.5tot (lsrc) = In−0.5

tot (lsrc)− 1

Zc

c∆t

∆s

[Vntot(lsrc + 1)− Vn

ref(lsrc)]

+1

Zc

c∆t

∆sVninc. (6.51)

Similarly, the modified update equation for the voltage right at the source plane can

be obtained, which must contain purely reflected transmission-line components, such

that (6.48) must be of the form

Vn+1ref (lsrc) = Vn

ref(lsrc)− Zcc∆t

∆s

[In+0.5ref (lsrc)− In+0.5

ref (lsrc − 1)]. (6.52)

However, the current I(lsrc) is a total current and must therefore be substituted bythe reflected component of the current

In+0.5ref (lsrc) = In+0.5

tot (lsrc)− In+0.5inc (lsrc)

= In+0.5tot (lsrc)− Vn+0.5

inc (s = (lsrc − 0.5)∆s)

Zc

. (6.53)

Note that the spatial offset of the incident voltage, determined ∆s/2 above the

source plane, must be converted into a temporal shift (delay), i.e., Vn+0.5inc (s = (lsrc −

129


0.5)∆s)) = Vinc(t = (n+0.5)∆t− 0.5∆s/c). Consequently, the final update equation

for the voltage at the source plane, can be written as

Vn+1ref (lsrc) = Vn

ref(lsrc)− Zcc∆t

∆s

[In+0.5tot (lsrc)− In+0.5

ref (lsrc − 1)]

+c∆t

∆sVn+0.5−0.5∆s/(c∆t)inc . (6.54)

A simple absorbing boundary condition is placed at the bottom end of the

transmission line. Note that for the Courant-Friedrichs-Levy (CFL) condition ∆t =

0.5∆s/c, the pulse propagates with velocity c one spatial step ∆s within two time

increments ∆t. For negligible reflection from the grid boundary, the first voltage in

this grid, V(1) must therefore be updated by the second voltage in this grid, V(2),observed two time steps earlier, i.e., Vn+1(1) = Vn−1(2).

To couple the transmission line to the three-dimensional FDTD grid and vice

versa, the line virtually attaches to the drive-point gap. Therefore, the voltage and

the current in the last cell are related to electric and magnetic fields in the vicinity of

the gap as shown schematically in Fig. 6.13b. To couple the line in the drive point,

the final voltage in the line is related to the tangential electric fields, that spread

across the gap between the antenna arms. In general, the voltage in the line and the

electric fields in the gap can be related using the line integral of the field across the

drive point, e.g., in the x direction,

V(lmax, t) = −∫gap

Ex dx. (6.55)

In the FDTD method, this integral can be broken up into summations to derive the

new update equations for the electric fields within the gap. As shown in Fig. 6.12 for

the drive point of the CSA, each electric field within the two-cell gap, that is marked

with a light gray line, is updated with

En+1x

∣∣∣dp= −Vn+1(lmax)

2∆x. (6.56)

To couple the drive point in the transmission line, the conduction current into

the antenna arm is related to the final current at the end of the transmission line,

130


In+0.5(lmax). This current can be expressed with a closed-path integral around the

conductor

I(lmax, t) =∮C&H · &ds. (6.57)

Ideally, the contour C should not include any displacement current but only a con-duction current. Therefore, the current contour should be shifted away from the gap

not to include any fringing of the field, which is explained in more detail in Appendix

B. As shown for the CSA in Fig. 6.12, the current contour consists of the magnetic

fields surrounding one sector of the feeding disc, that are marked with black lines and

thus determines the conduction current.

The apparent drive-point impedance [23] is determined using an approach that

involves both the incident and reflected voltage. Recall that the reflected voltage,

Vref , is observed at the plane with s = (lobs − 1)∆s, while the incident voltage Vincis injected at the source plane with s = (lsrc − 1)∆s, located above the observation

plane, shown qualitatively in Fig. 6.13a. Following basic transmission-line theory, the

input impedance can be determined using the reflection coefficient at the drive point

and the characteristic impedance of the line, i.e.,

Z(ω) = Zc1 + Γdp1− Γdp . (6.58)

This reflection coefficient, Γdp, can be written as the ratio of the reflected voltage and

the incident voltage provided both are observed at the same plane (drive point), i.e.,

Γdp(ω) =V dpref (ω)

V dpinc (ω)

=FVdp

ref(t)FVdp

inc(t). (6.59)

Notice, that the incident and reflected voltages are referenced to different planes well

beyond the drive point. In a straightforward approach, these voltages can easily be

transformed to the drive point. The reflected voltage, observed at the drive point,

Vdpref , is simply the time-advanced version of the reflected voltage at the observation

plane, Vref , that is

Vdpref(t) = Vref(t+ τobs) = Vref(t+ (lmax − lobs)∆s/c). (6.60)

131

6.5 Near-Field to Far-Field Transformer

Similarly, the observed incident voltage at the drive point, Vdpinc, is a time-delayed

version of the incident voltage at the source plane, Vinc, that is

Vdpinc(t) = Vinc(t− τsrc) = Vinc(t− (lmax − lsrc)∆s/c). (6.61)

These voltages can then be used to determine the reflection coefficient in (6.58)

Γdp(ω) =V dpref (ω)

V dpinc (ω)

=Vref(ω) e

−jωτobs

Vinc(ω) ejωτsrc=

Vref(ω)

Vinc(ω) ejω(τsrc+τobs). (6.62)

Instead of Fourier-transforming the original voltages, Vref(t) and Vinc(t), and subse-quently applying a phase shift, a slightly different approach is implemented in the

FDTD method that was shown to be more exact [41]. Notice that the denominator

term in (6.62), i.e., Vinc(ω) exp(jω(τsrc+ τobs)) or equivalently Vinc(t− (τsrc+ τobs)) in

the time domain, can be viewed as the (observed) incident voltage that propagated

the distance [(lmax − lobs) + (lmax − lsrc)]∆s in the transmission line, which is the

distance from the source plane to the drive point and back to the observation plane.

To avoid the shifts in the frequency domain and to account for spurious effects in

the transmission line, e.g., dispersion, this time-delayed voltage is determined in the

FDTD calculation using the following straightforward approach. By applying a short

circuit at the drive point with V(lmax) = 0, the reflected voltage at the observation

plane is

Vref |V(lmax)=0= −Vinc(t− (τsrc + τobs)), (6.63)

which, apart from the minus sign corresponds to the voltage in the denominator of

(6.62). This voltage can be determined concurrently within the same program that

is used to determine the antenna response and hence does not require additional

computational efforts.

6.5 Near-Field to Far-Field Transformer

Since the PML is truncating the FDTD lattice close to the antenna, only the field

in the space surrounding the antenna (near zone) is computed. However, often the

132

6.6 Program/Code Parallelization

field behavior at a large radial distance from the antenna (far zone) is of greater

interest. The electric field in the far zone of the CSA isolated in free space is obtained

with a near-field to far-field transformation based on Huygens’ Principle [31]. The

electric and magnetic field components tangential to the closed surface surrounding

the antenna, see Fig. 6.14, can be used to determine the radiated electric field in the

far zone of the antenna using a convolution integral:

&Er(&r, t) =µ4πr

∫∫S

r × r × ∂

∂t′[n× &H(&r ′, t′)

]

− 1

ηr × ∂

∂t′[n× &E(&r ′, t′)

]dS|t′=t−|%r−%r ′|/c. (6.64)

The primed variables, &r ′ and t′, denote the position and time for the fields observed

on the surface, while the unprimed variables, &r and t, refer to the position and time

for the fields observed in the far zone. The unit vector n is the outward normal to the

surface. The radiated field at the angle θ and at time t depends on the summation

of all tangential fields on the surface at the retarded times t − τi. The retardation

times τi = |&r − &r ′|/c are the times for a wave to propagate from the points on the

transforming box Qi to the far-field point P , as illustrated in Fig. 6.14.


The solution space of a finite-difference model, especially for three-dimensional prob-

lems, can become quite large. Thus, in order to reduce the execution time of the simu-

lation, large parallel computers are commonly used. This section briefly lists common

parallel-computer architectures and outlines the code parallelization intended to take

full advantage of these architectures.

With Single-Process Multiple-Data (SPMD) multi processing (MP), multiple

CPUs execute a single program simultaneously. This approach can significantly in-

crease the performance11 and has been employed with great success on Unix supercom-

11Ideally, the execution time can be reduced by almost a factor of Np provided a parallel computer

133


TransformingBox

x

y

z

P

Q2

Q1

r2 2 2 f ( , )

r1 1 1 f ( , )

Figure 6.14: Model of the surface used for the near-field to far-field transforma-tion.

puter systems for several years. Improvements in integrated circuit (IC) and network-

ing/communication technology have made a transition from these large and costly

systems towards practical MP technologies on inexpensive microprocessor-based per-

sonal computers (PCs) possible. There are two types of MP solutions typically im-

plemented: shared-memory processing (SMP), and distributed-memory processing

(DMP). A shared-memory parallel system is a single computer system utilizing mul-

tiple CPUs that share access to a common set of memory addresses. The OpenMP

standard is commonly used for programming this shared-memory architecture on

various platforms. A distributed-memory processing system is a multiple individual

computer system connected over a high-speed network to execute a single program.

with Np processors is employed. In reality, however, hardware latencies and parts of the computer

code that cannot be parallelized efficiently cause the program to run somewhat slower. The actual

improvement is often quantified with the “speed up,” defined as the ratio of the execution time for

the simulation spread over Np processors to that run on a single processor.

134


Subspace

Entire Solution Space Solution Space Dividedinto Subspaces

Figure 6.15: Division of the solution space into subspaces.

This approach has proven to be very successful dealing with extremely large problems

and has become popular within many research areas. The typical hardware config-

uration is a group of commodity computers connected via high-speed interconnects,

e.g., 100 Mbit/s Ethernet or Gigabit Ethernet. This configuration, dubbed as cluster

based computing or often referred to as “Beowulf cluster,” passes instructions and

data between systems via Message Passing Interface (MPI).

Figure 6.15 shows the basic parallelization scheme. In the conventional (serial)

three-dimensional finite-difference model, the transient fields are calculated through-

out the entire solution space by a single processor. To parallelize the finite-difference

model, the entire solution space is divided into several equal-sized subspaces. Each

subspace is assigned to one processor of the multi-processor architecture, which then

calculates the transient fields throughout its individual subspace.

At each time step, interface data must be exchanged between neighboring sub-

spaces, which is outlined in Fig. 6.16. Recall that one electric (magnetic) field com-

ponent at a discretized location in the grid depends on the difference of magnetic

(electric) field components in the immediate surrounding, i.e., four components in to-

tal. The fields located well inside the subspaces are independent of the fields in other

subspaces since the nearest neighbors are located in the same subspace. However,

the electric field located on the interface between two subspaces depends on at least

one magnetic field component from the neighboring subspaces, and the magnetic field

135


in the immediate vicinity of such an interface depends on at least one electric field

component from neighboring subspaces. In the following, two examples are discussed

that require the exchange of information between two subspaces. In the first example,

illustrated in the top left corner in Fig. 6.16 for an interface with x = const, the elec-

tric field tangential to the interface between subspaces 1 and 2, Ex, depends on four

magnetic field components: Three of those are located entirely within the subspace 1,

while the fourth magnetic field, Hz, is located in subspace 2. In the second example,

illustrated in the top right corner of Fig. 6.16 for an interface with x = const, the

magnetic field component, Hz, in subspace 2 right next to the interface depends on

four electric field components: Three of those are located within subspace 2, while the

remaining electric field, Ex, belongs to subspace 1 (even though it is located on the

shared interface). Therefore, it is necessary to pass the interface data of the electric

field right at the interface and that of the magnetic field right next to the interface

from one subspace to its immediate neighbor, as shown in the bottom of Fig. 6.16.

As part of the research for this dissertation, a Fortran90 computer code was par-

allelized using both the shared-memory approach (using OpenMP) and the message-

passing approach (using MPI). Initially, the OpenMP program was executed on a

SMP architecture, i.e., a Dell Poweredge Quad Pentium III-Xeon (500 MHz) server

with 2 GB RAM. Later, the program was rewritten using the MPI approach to run

simulations on both SMP and DMP architectures. The latter was a Beowulf cluster

with 50 Pentium III (500 MHz) processors and a total of 6 GB RAM, interconnected

via FastEthernet (100 Mbit/s).

To parallelize an already existing serial version of the Fortran90 program, al-

most all subroutines needed to be modified to operate on the corresponding subspace

instead of the entire domain. Therefore, the following MPI commands were essen-

tial [46], such as MPI INIT, MPI COMM RANK, MPI COMM SIZE, MPI SEND, MPI RECEIVE,

MPI BARRIER, and MPI FINALIZE.12

12Within an MPI environment enclosed by MPI INIT and MPI FINALIZE, a user defined number

of processes are created. In the current implementation, each process updates the electromagnetic

136


Subspace 1 Subspace 2

Subspace 1 Subspace 2

Hz

Ex

Ey

Hy

Figure 6.16: Illustration for the necessary message passing of interface data(fields).

fields in the corresponding subspace; the rank of the process is determined with MPI COMM RANK and

the total number of processes, equal to the number of subspaces, is obtained with MPI COMM SIZE.

The essential commands are MPI SEND and MPI RECEIVE to send and receive data in order to pass

interface data between adjacent subspaces. For code synchronization purposes, MPI BARRIER is used.

137

CHAPTER 7

Validation of FDTD Results

This chapter is concerned with the validation of the FDTD method used to model the

CSA in free space and over ground. In Section 7.1, numerical results are compared

with results published in the literature. First, results obtained for an unloaded CSA

and a CSA terminated with lumped resistors are compared with measurements made

elsewhere. Then, results obtained for linear dipole antennas placed horizontally over

lossy media are compared with analytical and experimental results of others. In

Section 7.2, measurements performed at Georgia Tech are described. The fabrication

of the CSA, the feed system, and the measurement techniques for the input impedance

and the realized gain are explained.

7.1 Validation using Published Results

7.1.1 CSA in Free Space

In this section, measurements performed by Ramsdale [17] for the input impedance

and the electric field in the far zone of the CSA are compared with numerical results

obtained using the FDTD method. The results presented here are for a right-handed

CSA with the following parameters: θ = 15, α = 60, δ = 90, and D/d = 12.77

with D = 15.2 cm. The antenna is fed from a transmission line with the characteristic

impedance Zc = 100Ω. The results are for the unloaded CSA and the CSA termi-

nated with lumped resistors. Each of the lumped resistors, connected in parallel at

138


the closest edges of the two arms, has a resistance of RL = 300Ω. For the FDTD cal-

culations, the dimensions of the cells are ∆x = ∆y = ∆z ≈ 1.2mm; this corresponds

to a discretization of 125 cells per wavelength at the highest frequency studied.

The reflection coefficient is plotted on a Smith chart in Fig. 7.1a for the unloaded

antenna and in Fig. 7.1b for the loaded antenna. The FDTD results are plotted with

a black line, the measurements are plotted with a gray line. Points are marked that

correspond to a reflection coefficient at particular frequencies. For low frequencies,

the reflection coefficient for the unloaded antenna agrees well with the measurements,

while some deviations can be seen for higher frequencies. Since high frequencies/short

wavelengths are radiated from the small end of the CSA, these deviations probably

result from the differences in the feed system. The modeling of the feed in the exper-

imental model and in the numerical model are indeed different. In the experimental

model, two rigid coaxial lines are attached to the feed point at the disc, while in

the FDTD model a two-wire transmission line is connected to these feed points, as

outlined in Section 6.4. For the loaded case, both curves show that the reflections are

centered in a small region about the point R/Zc = 1.5. Although the shapes of the

reflection coefficient curves are different, both curves circle around the same point,

which corresponds to an impedance of 150Ω. As stated earlier, the end termination

should not affect the high frequencies significantly. This behavior can be seen in the

numerical results for the reflection coefficient, since both curves look similar in the

high-frequency range.

In Fig. 7.2, vertical-plane (φ = 90), far-zone patterns are shown for the fre-

quency f = 0.5GHz. The patterns, graphed on a logarithmic scale, are given for

both of the circularly polarized components of the electric field for the unloaded CSA

in Fig. 7.2a and for the loaded CSA in Fig. 7.2b. The numerical results for the

left-handed circularly polarized component of the electric field, ErLHCP, are shown

with a dotted line using hollow dots, and those for the right-handed circularly polar-

ized component of the field, ErRHCP, are shown with a dotted line using solid dots.

The corresponding experimental results are shown with a solid line. The agreement

139


0.2 GHz

2GHz

0.6GHz

0.2 GHz

2 GHz

(a)

(b)

FDTDMeasurement

1

1

Figure 7.1: Comparison of theoretical (FDTD) and measured reflection coeffi-cients plotted on a Smith chart for a) the unloaded CSA and b) theCSA terminated with lumped resistors. The results are for Rams-dale’s CSA [17] with θ = 15, α = 60, δ = 90, and D/d = 12.77with D = 15.2 cm.

140


-30 -20 -10 0

-30

-20

-10

0

-30-20-100

-30

-20

-10

0

0

30

60

90

120

150

180

210

240

270

300

330

-30 -20 -10 0

-30

-20

-10

0

-30-20-100

-30

-20

-10

0

0

30

60

90

120

150

180

210

240

270

300

330

(a)

(b)

E

ER

L

(FDTD)

(FDTD)

Measurement

dB

dB

Figure 7.2: Comparison of theoretical (FDTD) and measured far zone patternsfor the circularly polarized components of the electric field for a) theunloaded CSA and b) the CSA terminated with lumped resistors(RL = 300Ω) at f = 0.5GHz. The results are for Ramsdale’sCSA [17] with θ = 15, α = 60, δ = 90, and D/d = 12.77 withD = 15.2 cm.

141


r ,

h

L

Lossy Medium

Figure 7.3: Schematic model for the horizontal dipole placed over the ground.

between the FDTD results and the measurements is seen to be fair in both cases.

Differences can be noted for the unloaded case in the broadside direction, that is

when θ ≈ 90. For the loaded antenna, the radiated fields deviate for the positive z

direction (θ = 0).

7.1.2 Horizontal Dipole over Lossy Media

The objective of this section is to validate results for the input impedance obtained

for dipole antennas placed horizontally in free space over a lossy half space. The

FDTD results are compared with analytical results [47, 48] and experimental results

[49]. The model for the two cases studied is shown in Fig. 7.3. A dipole antenna of

length L is placed horizontally at height h above the lossy medium with electrical

parameters µr = 1, εr, and σ.

In the first case, the proximity loss of a thin, short dipole over the ground is

investigated, which is defined as the decibel ratio of the resistance of the antenna over

the ground, Rg, to the resistance of the antenna in free space, Rfs, that is

PL = 10 dB log(Rg/Rfs). (7.1)

This equation should be independent of frequency as long as the dipole antenna

is electrically short, i.e., λ/L ≥ 50 [50]. The dipole investigated has a length of

L = 20.4 cm and the frequency at which the resistances Rfs and Rg are determined is

f = 24MHz, which satisfies the above condition for an electrically short antenna, i.e.

142


10-3 10-2 10-1 100 101 102 103

-10

-5

0

5

10

15

20

25

30

S

Pro

xm

ity

Loss

[dB

]

= 0.01

= 0.02

= 0.05

TheoryFDTD

Figure 7.4: Comparison of the analytical and FDTD proximity losses for thehorizontal dipole over the ground. The analytical results are from[48].

λ/L = 62.5 ≥ 50 with λ = c/f = 12.8m the wavelength of operation. The results

for the proximity loss are shown in Fig. 7.4 as a function of normalized conductivity

of the ground S = εrp = ησλ/2π (loss tangent p = σ/ωεεr) for three different

normalized heights of the antenna above the ground, α = 4πh/λ. The theoretical

results [48] are shown with solid lines and the numerical results are marked using

dots. The agreement between these results is good.

In the second case, the input admittance of a horizontal dipole above tap wa-

ter is studied around its first and second resonant frequencies and compared with

measurements [49]. The length of the antenna is L = 51.2 cm and the height above

the lossy half space is h = 2.54 cm. This dipole has a finite thickness, i.e., the ra-

dius of the conductor is a = 8.3mm. The electrical properties of the tap water are

εr = 80 and σ = 0.004 S/m around the first resonance (fres ≈ c/2L = 293MHz) and

σ = 0.04 S/m around the second resonance. The input admittance of this antenna is

shown in Fig. 7.5 as a function of frequency: around the first resonant frequency in

Fig. 7.5a and around the second resonant frequency in Fig. 7.5b. Again, the results

are seen to be in good agreement and show that the FDTD correctly models antennas

placed directly above lossy media.

143

7.2 Validation using Georgia Tech Measurements

240 260 280 300 320 340 360

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

f [MHz]

f [MHz]

GB

,[m

S]

GB

,[m

S]

B

G FDTDMeasurement

780 800 820 840 860 880 900

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

B

G

(a)

(b)

Figure 7.5: Comparison of the theoretical (FDTD) and measured results for theinput admittance of the horizontal dipole antenna over tap wateraround a) the first resonance and b) the second resonance. Theexperimental results are from [49].


This section is mainly concerned with the validation of numerical results obtained for

the CSA with θ = 7.5, α = 75, δ = 90, and D/d = 8 (d = 1.9 cm) using measure-

ments performed at Georgia Tech. A pair of identical CSAs was constructed, and

measurements of the input impedance and realized gain of these antennas were used

to validate the FDTD analysis. Several aspects of the measurements will be discussed

in some detail. The approach to manufacture the CSA using a flexible circuit board

144


material is explained in Section 7.2.1. The antenna feeding technique together with

the experimental validation of the balun are presented in Section 7.2.2. In Section

7.2.3, the experimental implementations of the resistive terminations (lumped resis-

tors and the resistive sheet) are discussed, and each termination is examined with

separate measurements. The measurement for the input impedance is explained in

Section 7.2.4, while the two-port measurement for the realized gain is explained in

Section 7.2.5. In Section 7.2.6, the same two-port measurement technique is used to

determine the input impedance, realized gain, and gain for a dipole antenna. These

results are then compared to numerical results obtained with the FDTD method.

7.2.1 Manufacturing of the CSA

The objective for the CSA design was to build and measure a CSA with performance

characteristics such as uniform impedance, unidirectional radiation, and circular po-

larization over a broad range of frequencies, that are beneficial for applications when

isolated in free space or placed over the ground. Therefore, CSAs with small half-cone

angles, θ, and large wrap angles, α, were of main interest; thus, the characteristic

angles of the antenna were chosen to be θ = 7.5, α = 75, and δ = 90. The re-

maining parameters d = 1.9 cm and D/d = 8 were chosen for practical reasons. The

small end is where the CSA is fed, the smallest diameter must therefore be sufficiently

big to fit the feed cables and the connectors that attach to the feeding end of the

antenna. The ratio of biggest to smallest diameter, D/d = 8, guaranteed that the

CSA would just fit on the biggest available flexible circuit-board sheet that measured

43.2 cm× 58.4 cm (17′′ × 23′′).The general approach for fabricating the CSA is to form the metallic spiral arms

on a flexible circuit board by etching away the unwanted portion of the coating.

The circuit board is then wrapped around a solid conical mandrel, and the arms

are soldered together along the seam. This manufacturing of the CSA requires the

proper design of the planar sheet that when wrapped on a cone becomes the CSA.

145


Figure 7.6: Illustration on how the conical surface becomes a circular sectorand vice versa.

Figure 7.6 illustrates that by unwrapping the conical surface on a plane a sector is

obtained that has circular arcs at the small and the big end. Schematic drawings for

this planar surface are shown in Fig. 7.7. Simple geometry can be used to show that

the maximum dimensions of the sector are its height

Ys =

D sin(2π sin θ)2 sin θ for θ ≤ 14.5

2D for 14.5 < θ ≤ 30(7.2)

and its width

Xs =

D2 sin θ −

d cos(2π sin θ)2 sin θ for θ ≤ 14.5

2D[1− cos(2π sin θ)] for 14.5 < θ ≤ 30. (7.3)

Note that these equations must be determined separately for different ranges of θ, i.e.,

each time the angular width of the sector, β = 2π sin θ, exceeds an integer multiple

of 90. Here, just the results for the most common ranges of θ are presented, as

illustrated in Fig. 7.7.

In the next step, the boundaries of the metallic arms must be transferred on

the planar surface using simple geometrical expressions. The template for the copper

etching process of the unwrapped CSA with θ = 7.5, α = 75, δ = 90, and D/d = 8

is shown in Fig. 7.8a. Here, the two antenna arms are shown in two different shades

of gray. When this model is wrapped around a conical mandrel, the arms properly

connect at the seam, as illustrated in the inset of Fig. 7.8a. For the actual manu-

facturing process, the boundary information of the arms was stored in a Gerber1 file

format, and the arms were then formed on a flexible circuit board material through a

1The Gerber format is used primarily by designers who need to make format checkplots of PC

146


d

D

Ys

Xs

d

Ys

Xs

D

(a)

(b)

Figure 7.7: Schematic drawings for the unwrapped surface of the cone for a)θ ≤ 14.5 and b) 14.5 < θ ≤ 30.

wet chemical etch provided by Innovative Circuits. The substrate of this circuit board

was Kapton with a thickness of tK = 0.051mm, and the coating was copper with a

thickness of tc = 0.071mm. At the lowest frequency of operation (f = 0.52GHz), the

thickness of the copper layer normalized by its skin depth δc is tc/δc = 24 and at the

highest frequency of operation (f = 3.3GHz), this ratio is tc/δc = 60. The manufac-

boards and hybrid circuits. Gerber data is a simple, generic means of transferring printed circuit

board information to a wide variety of devices that convert the electronic printed circuit board

(PCB) data to artwork produced by a photoplotter. It is a software structure consisting of x, y

coordinates supplemented by commands that define where the PCB image starts, what shape it will

take, and where it ends. In addition to the coordinates, Gerber data contains aperture information,

which defines the shapes and sizes of lines, holes, and other features.

147


Kapton(Substrate)

Copper (Coating),Metallic Arms

(a)

(b)

Figure 7.8: a) Template for the copper etching process, b) photograph of themanufactured flexible circuit board.

tured flexible circuit board with the spiral arms formed on the surface is shown in a

photograph in Fig. 7.8b.

Next, the planar surface must take on the shape of the conical surface. First,

small patches of solder were placed on the metallic (copper) regions along the sides

of the planar circuit board, as shown in Fig. 7.9a. Then, the surface was wrapped

148


around a conical mandrel and the spiral arms were soldered together at the seam.

This was done by soldering small patches of brass sheet on each previously prepared

end of the spiral arm. The assembled CSA is shown in Fig. 7.9b.

7.2.2 Feed System of the CSA

The following section is concerned with the antenna feed system used for the CSA

measurements. The feeding technique consists of three main components: the feeding

disc at the small end of the CSA, the feed cable running along the axis of the antenna,

and the balun that is placed in between the feed cable and the vector network analyzer.

These components are illustrated schematically in Fig. 7.10.

The details of the feed point of the antenna are shown at the top of Fig. 7.10.

The small disc of rigid circuit board contained two symmetric angular sectors of

conductor. The disc just fits into the open end of the CSA. Holes were drilled into

each metallic sector, into which metallic pins were soldered. The pins protruded into

the cone. The feeding disc used in the measurements is shown in Fig. 7.11a with

and without the pins attached. The metallic sectors of the feeding disc were put in

electrical contact with the two arms of the CSA by soldering both metallic regions

together using brass sheets, see Fig. 7.11b.

A pair of semi-rigid coaxial lines (Zc = 50Ω) ran along the axis of the cone.

After leaving the balun, these lines were electrically bonded together to suppress

unbalanced modes, i.e., currents on the surface of the coaxial lines. The lines did not

separate again until they reached the feed point. Female connectors were soldered to

the ends of these lines, and the pins on the arms of the antenna were inserted into the

center conductors of these connectors to temporarily attach the antenna to the feed

cables. The network analyzer was calibrated by connecting standard terminations

(matched load, short circuit, and open circuit) simultaneously to both of the female

connectors. Thus, the plane for calibration of the measurement was right at the

terminals of the antenna.

149


(a)

(b)

Brass Sheet

Seam

Solder

Figure 7.9: a) Photograph of the planar circuit board before it takes on theshape of the conical surface, b) photograph of the assembled CSA.

150


CalibrationPlane

CalibrationPlane

Conductor

Dielectric Disk

to NetworkAnalyzer(HP8720)

FeedCable

PicosecondPulse LabsModel 5315Balun

PulseSplitter

Phas

eIn

ver

ter

FeedingPins

…

FemaleConnectors

Lineselectricallybondedtogether

Figure 7.10: Details of the method used to feed the CSAs in the measurements.

151


(a)

(b)

Figure 7.11: a) Photograph of the rigid circuit board used to connect the feedcable with the antenna arms, b) photograph of the feed region.

The CSA is a balanced structure, and the measurements to be described were

made using a vector network analyzer (HP8720) with coaxial ports (unbalanced), see

Fig. 7.12a for a photograph. Therefore, a balun had to be used with each antenna.

The balun selected was the Picosecond Pulse Labs Model 5315, see the bottom of

Fig. 7.10 and Fig. 7.12b for a photograph. All ports of the balun had the characteristic

impedance Zc = 50Ω, so the balun can be used to effectively connect an unbalanced

line with Zc = 50Ω to a balanced line with Zc = 100Ω. The balun was located well

outside the antenna so as to not affect the performance of the antenna.

The balun was examined quantitatively using the measurement setup shown in

Fig. 7.13. Three separate two-port measurements were performed to determine the

insertion loss of each combination of ports and the return loss of the three ports.

These measurements were also used to verify that the two output ports are 180 out

of phase. In each of the experiments, two ports i and j of the balun were connected

152


(a)

(b)

Port 1(Coaxial)

Port 2(Coaxial)

OutputPort

OutputPort

InputPort

Figure 7.12: Photograph of a) the vector network analyzer and b) the balun usedin the measurement.

with the network analyzer, while port k was terminated with a 50Ω load. With this

measurement setup, the insertion loss (power loss through the device from port i to

port j and vice versa), Sij = Sji, and the return losses (power reflected from the

ports i and j), Sii and Sjj, can be determined. The results for this measurement are

presented in Fig. 7.14 as a function of frequency with 50MHz ≤ f ≤ 16GHz. Notice

that the operational range of the CSA (0.52GHz ≤ f ≤ 3.3GHz) is just a small part

of that range, as marked on the graph. The results for the insertion loss, shown in

Fig. 7.14a, clearly indicate that the power loss through the device is independent of

the combination of input and output ports chosen. The results for the return loss in

Fig. 7.14b are similar, the power reflected from the port is almost independent of the

port. The presented results compare well with results provided by the manufacturer,

omitted here. The last result, shown in Fig. 7.14c, is for the phase difference between

153


50

`

1

2

3

50

To VNA(Port #2)

To VNA(Port #1)

(a)

1

2

3

50

To VNA(Port #2)

To VNA(Port #1)

(b)

1

2

3

To VNA(Port #2)

To VNA(Port #1)

(c)

Figure 7.13: Schematic drawings for the two-port measurements of the balun:a) is for determining S11, S12, S21, and S22, b) for S11, S13, S31, andS33, and c) for S22, S23, S32, and S33.

154


the two output ports, S12 − S13. It can be seen that the phase difference has an

almost constant value of 180, which shows that the output ports are out of phase,

as illustrated in the inset.

7.2.3 Termination of the CSA

In the following, the implementation of the two resistive terminations (lumped resis-

tors and resistive sheet) is described. The lumped resistors used were chip resistors

(type 0805) that measured 2.1mm × 1.3mm × 0.5mm. In order to obtain a total

value of RL = 300Ω, two chip resistors with 200Ω and 100Ω were connected in se-

ries. To connect them to the antenna arms, two thin wires were carefully soldered

to the metallic wraparound of the chip resistor, which is shown in a photograph in

Fig. 7.15a. Figure 7.15b shows a photograph of the big end of the CSA terminated

with the lumped resistors. The wires were bent at the end to properly align them

with the surface of the cone, and then soldered to the antenna arms. Notice, the

terminations are far enough away from the axis of the CSA, so that the terminations

are not in the way of the feed cable. In a separate measurement, a lumped resis-

tor was studied experimentally. Small leads were soldered to both ends of a lumped

resistor with a value of RL = 200Ω, see Fig. 7.16. These thin wires were inserted

into the center conductors of the female connectors of the feeding transmission lines.

The measured impedance of this lumped resistor is shown in Fig. 7.17. As expected,

the resistance has a constant value of R ≈ 200Ω. The metallic leads introduce an

inductance which makes the reactance increase almost linearly with frequency. At

the low end of the operational bandwidth for the CSA, 0.52GHz ≤ f ≤ 3.2GHz, the

reactance is small compared to the resistance.

For the configuration using the resistive sheet, the disc was formed from a sheet

of carbon-black filled, conductive, polycarbonate film, with a resistance per square

of Rs = 146Ω. To properly terminate the big end of the CSA, the sheet required

modifications for the feeding cables to run along the axis of the cone. A slit was cut

155


0.05 2 4 6 8 10 12 14 16

0.0

0.2

0.4

0.6

0.8

1.0

0.05 2 4 6 8 10 12 14 16

0.0

0.2

0.4

0.6

0.8

1.0

0.05 2 4 6 8 10 12 14 16

-225

-180

-135

-90

-45

0

Inse

rtio

nL

oss

Ret

urn

Lo

ss

f [GHz]

f [GHz]

f [GHz]

(a)

(b)

(c)

| |

| |

| |

S

S

S

12

13

23

| |

| |

| |

S

S

S

11

22

33

SS

12

13

-

1

2

3

Frequency Range for CSA

Figure 7.14: Measured results for the a) insertion loss, b) return loss, and c)phase difference between the output ports.

156


(a)

(b)

Chip Resistor

Figure 7.15: Photograph of a) the lumped resistor termination (chip resistor sol-dered to the wires) and b) the big end of the CSA when terminatedwith lumped resistors.

FeedCable

…

RL = 200

Figure 7.16: Schematic drawing for the measurement of the lumped resistor.

157


0.3 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0

50

100

150

200

250

f [GHz]

R,X

[]

R

X

Figure 7.17: Measured results for the impedance of the lumped resistor withRL = 200Ω.

from the side into the center of the sheet where resistive material equal in size and

shape to the cross section of the feed cable was removed, see Fig. 7.18a. Then, the

resistive sheet was wrapped around the transmission line and attached to the arms

of the antenna with metallic tape, as shown in Fig. 7.18b. Notice that the sheet

was attached such that the slit is tangential to the direction of current flow, which is

indicated schematically on the disc.

A separate measurement was performed to validate the analytical results for the

DC resistance, R, of the circular disc between two electrodes of angle δ, derived in

Appendix A. For the measurements, conductive paper with a resistance per square

of Rs ≈ 5000Ω was cut into circular discs with a diameter of D = 14.7 cm. The

electrodes were marked by sectors with an angular width δ and a radial width of

1.3 cm. These sectors were covered with silver paint to make them conductive. The

schematic model for the discs used in the measurement is shown in Fig. 7.19a. In the

measurements, a total of 7 discs were used, i.e., the angular widths were δ = i 22.5

with i = 1, . . . , 7. A digital multimeter was connected to each electrode to measure

the DC resistance of the disc. Since the resistance per square of the sheets was

not known exactly, the DC resistance of the sheet with the electrode’s angular width

δ = 90 was set to the resistance per square, Rs. Recall, that the analytical results for

158


(a)

(b)

Hole Cut outfor Feed Cable

Slit

I

Figure 7.18: Photograph of a) the resistive sheet, modified to properly attach atthe antenna end and b) the big end of the CSA when terminatedwith the resistive sheet..

the resistive sheet obtained R = Rs for δ = 90. The results measured for other values

of δ were normalized to this value. As it can be seen in Fig. 7.19b, the agreement

between the analytical and experimental results is good.

7.2.4 Impedance Measurements

The measurement setup for the input impedance is shown in Fig. 7.20: The CSA was

mounted vertically with the feed cable running along its axis until it almost reached

159


14.7 cm

1.3cm

ConductivePaper

Electrode(Silver Paint)

0 20 40 60 80 100 120 140 160 180

0.0

0.5

1.0

1.5

2.0

2.5

RR

/s

(a)

(b)

Theory, /

Measurement,

R Rs

R R/ ( = 90 )o

……

Figure 7.19: a) Schematic model for the resistive sheet used in the measurement,b) comparison of the analytical and experimental results for the DCresistance.

the ground. A wooden stick was used to support the feed cable. Another set of

semi-rigid coaxial lines, bent to form the shape of a helix, was used to make the

transition between the set of feed cables and the balun. A high precision, semi-rigid

coaxial line then connected the input port of the balun with one port of the vector

network analyzer. The floor and all of the measurement equipment (feed cables,

balun, network analyzer) were covered with absorber. The ceiling, which was in

the direction of maximum radiation of the CSA, was covered with high-performance,

broadband absorber.

In Figs. 7.21–7.23, the FDTD results (solid line) and measured results (dashed

line) for the magnitude of the reflection coefficient and input impedance (resistance

and reactance) of the antennas are graphed as a function of frequency (logarithmic

160


1

1

2

2

Absorber

FeedCable

NetworkAnalyzer

BalunWooden Stick

Figure 7.20: Photograph of the measurement setup to determine the inputimpedance.

scale). Figure 7.21 is for the unloaded antenna, Fig. 7.22 is for the antenna terminated

with the lumped resistors (RL = 300Ω), and Fig. 7.23 is for the antenna terminated

with the resistive sheet (Rs = 146Ω). For the FDTD calculations, the dimensions of

the cells are ∆x = ∆y = ∆z ≈ 0.8mm; this corresponds to 107 cells per wavelength

at the highest frequency investigated. For all cases, the resistance is about 150Ω and

the reactance is small over the operational (hybrid) bandwidth of the unloaded CSA,

0.52GHz ≤ f ≤ 3.3GHz. Notice that the FDTD results match the measured results

“ripple for ripple,” although there is some offset of the two curves, which will be

discussed later. At frequencies below 0.5GHz, the large oscillations in the terminal

quantities of the unloaded antenna, Fig. 7.21, are caused by reflections from the open

end. These reflections are clearly reduced by addition of the resistive terminations,

Figs. 7.22b and 7.23. At frequencies greater than about 2.0GHz, the differences in

the numerical and measured results are probably caused by small differences in the

geometry of the feed region in the FDTD and experimental models.

161


0.3 0.5 1 1.5 2 2.5 3 3.5

0.00

0.25

0.50

0.75

1.00

f [GHz]|

|

0.3 0.5 1 1.5 2 2.5 3 3.5

0

50

100

150

200

f [GHz]

R[

]

0.3 0.5 1 1.5 2 2.5 3 3.5

-100

-50

0

50

100

f [GHz]

X[

]

(a)

(b)

(c)

FDTD

Measurement

Operational Bandwidth

Figure 7.21: Comparison of theoretical (FDTD) and measured terminal quanti-ties for the unloaded antenna: a) magnitude of the reflection coeffi-cient, b) input resistance, and c) input reactance for the CSA withθ = 7.5, α = 75, δ = 90, D/d = 8 (d = 1.9 cm).

162


f [GHz]|

|

f [GHz]

R[

]

f [GHz]

X[

]

(a)

(b)

(c)

0.3 0.5 1 1.5 2 2.5 3 3.5

0.00

0.25

0.50

0.75

1.00

0.3 0.5 1 1.5 2 2.5 3 3.5

0

50

100

150

200

0.3 0.5 1 1.5 2 2.5 3 3.5

-100

-50

0

50

100

FDTD

Measurement

Figure 7.22: Comparison of theoretical (FDTD) and measured terminal quanti-ties for the antenna terminated with lumped resistors: a) magni-tude of the reflection coefficient, b) input resistance, and c) inputreactance for the CSA with θ = 7.5, α = 75, δ = 90, D/d = 8(d = 1.9 cm).

163


f [GHz]|

|

f [GHz]

R[

]

f [GHz]

X[

]

(a)

(b)

(c)

0.3 0.5 1 1.5 2 2.5 3 3.5

0.00

0.25

0.50

0.75

1.00

0.3 0.5 1 1.5 2 2.5 3 3.5

0

50

100

150

200

0.3 0.5 1 1.5 2 2.5 3 3.5

-100

-50

0

50

100

FDTD

Measurement

Figure 7.23: Comparison of theoretical (FDTD) and measured terminal quanti-ties for the antenna terminated with a resistive sheet: a) magnitudeof the reflection coefficient, b) input resistance, and c) input reac-tance for the CSA with θ = 7.5, α = 75, δ = 90, D/d = 8(d = 1.9 cm).

164


0.3 0.5 1 1.5 2 2.5 3 3.5

-100

-50

0

50

100

f [GHz]

X[

]

FDTD

Measurement

Figure 7.24: Comparison of theoretical (FDTD) and measured input reactancesfor the antenna terminated with a resistive sheet. A small capaci-tance, C = 0.12 pF, has been added in parallel with the terminals.

To give an idea of how sensitive the input impedance is to small changes in

the feed region, the reactance of the antenna terminated with the resistive sheet was

computed with a small capacitance, C = 0.12 pF, added in parallel with the terminals.

Results for this case are shown in Fig. 7.24, and they should be compared with those

in Fig. 7.23c. Notice that the capacitance has shifted the FDTD results downward so

that they are now in better agreement with the measurements. To put this amount

of capacitance in perspective, it is roughly equivalent to the capacitance of a 1mm

length of one of the semi-rigid coaxial lines (each about 3m long, measured between

the feeding disc and the balun) used in the feeding network shown in Fig. 7.10.

7.2.5 Realized Gain Measurements

The realized gain (gain including mismatch) of the antennas was determined using

the two-antenna method [51]. The measurement setup is shown in the photograph in

Fig. 7.25a. The two identical CSAs are hung from the ceiling at equal height, with

their small ends facing each other and 1.2m apart. In order to attach the CSA to

the strings coming from the ceiling, small hooks were soldered to the antenna arms.

Styrofoam discs were inserted into the CSAs to keep the circular cross section of the

antenna and to keep the feed cable on the axis of the CSA. As for the impedance mea-

165


(b)

(a)

S21

R

Figure 7.25: a) Photograph of the measurement setup to determine the realizedgain, b) schematic model for the two-antenna method.

surement, absorbers were placed in the vicinity of the antennas and the measurement

equipment to reduce unwanted reflections.

With the two-antenna method, the realized gain can be calculated from Friis’

transmission formula:

Gr =4πR

λ

√Pr

Pt

, (7.4)

where the ratio of the power received by one antenna to the power transmitted by the

166


Ro

R( )

hs

Active Regionfor Wavelength

Figure 7.26: Drawing detailing the wavelength-dependent distance R(λ) be-tween the active regions.

other antenna, i.e., Pr/Pt, is determined from a two-port measurement of the scatter-

ing parameter S21 =√Pr/Pt with the network analyzer, as illustrated schematically

in Fig. 7.25b. Usually, the distance between the two antennas, R in the above for-

mula, is so large that the movement with a change of frequency of the phase center for

radiation is inconsequential for determining the realized gain. In these measurements,

however, the distance between the feed points of the two antennas, R, was only about

twice their length, i.e., R/hs ≈ 2, so this effect had to be taken into account, see

Fig. 7.26. At a given frequency, the point at which radiation occurs on the antenna

(active region) was assumed to be where the circumference is one wavelength. The

distance between the phase centers of the two antennas is then

R(λ) = R +d

tan θ

(λ

πd− 1

). (7.5)

This wavelength-dependent distance was used in (7.4) to determine the measured

realized gain.

In Fig. 7.27, the FDTD results (solid line) and measured results (dashed line)

for the realized gain in the direction for maximum radiation (−z direction in Fig. 2.1)

are graphed as a function of the frequency. Note that the gain is displayed on a linear

scale. The agreement between the numerical and the measured results is seen to be

good (within about 1 dB) for the three cases: Fig. 7.27a for the unloaded antenna,

Fig. 7.27b for the antenna terminated with the lumped resistors, and Fig. 7.27c for

the antenna terminated with the resistive sheet. The resistive terminations are seen

to have almost no effect on the realized gain.

167


1.0 2.0 3.0 4.0

0

2

4

6

8

1.0 2.0 3.0 4.0

0

2

4

6

8

1.0 2.0 3.0 4.0

0

2

4

6

8

Gr

f [GHz]

0.3G

r

f [GHz]

0.3

Gr

f [GHz]

0.3

(a)

(b)

(c)

FDTD

Measurement

Figure 7.27: Comparison of theoretical (FDTD) and measured realized gains(linear) for the CSA with θ = 7.5, α = 75, δ = 90, D/d = 8 (d =1.9 cm): a) the unloaded antenna, b) the antenna terminated withlumped resistors, and c) the antenna terminated with a resistivesheet.

168


FeedCable

…

h

lghg

2a

Figure 7.28: Schematic model of the dipole used in the measurement.

7.2.6 Dipole Measurements

This section presents an additional check on the measurement technique used in the

previous section for the CSA. Here, for a simple dipole antenna, the input impedance

is determined (using S11), and the two-antenna method is used to determine the

realized gain (using S21) and the gain (using a combination of S11 and S21). A

schematic drawing for the dipole antenna is in Fig. 7.28. The antenna, inherently

a balanced structure due to its symmetry, was fed through the same feed system

(balun and feeding transmission line) as the CSA. Each arm of the dipole was of

length h = 71.4mm and radius a = 0.5mm. Pins at the ends of the arms were

inserted into the center conductors of the feeding transmission-line connectors. The

vertical pins had a height of hg = 3.2mm and were separated by the drive-point gap

of length lg = 8mm. A photograph for this measurement setup is shown in Fig. 7.29:

Each of the two, identical dipole antennas was connected to one port of the vector

network analyzer. The dipole antennas, separated by the distance R = 1.19m, were

aligned to be parallel to minimize the polarization loss.

The measurements will be compared with results for two different FDTD mod-

els, which are shown schematically in Fig. 7.30. In the first model, see Fig. 7.30a, the

dipole has a square cross section. The side of the square conductor, ae, was chosen

to obtain the approximate equivalence for the circular conductor used in the mea-

surement: ae = 1.69a. In this model, details of the antenna near the feed region are

169


Dipole

FeedingTransmissionLine

Figure 7.29: Measurement setup for the dipole antenna.

Hy

Hy

Hy

Hz

Ex = f ( )V( )lmax

Ex = f ( )V( )lmax

Ex =0

Ex =0

xyz

xyz

…

…

…

…

(a)

(b)

x

a=^ a = ae 1.69

Figure 7.30: Schematic FDTD model of a) the dipole with square cross sectionincluding the details of the feed region and b) the simple straightdipole.

170


preserved by implementing the bent in the antenna conductors. However, since the

details of the experimental feed cannot be modeled with the FDTDmethod, the dipole

was fed with the “simple feed” across the pins. In the present case, 2 cells were chosen

across the height and width of the conductor, i.e., ∆x = ∆y = ∆z = ae/2 ≈ 0.43mm.

Consequently, the gap was modeled with 18 cells, the height of the pins with 8 cells

and the horizontal length of the antenna arm with 168 cells. In Fig. 7.30a, those

tangential electric field components, Ex, are marked with black lines that relate to

the voltage in the transmission line, V(lmax), (6.55). Following the guidelines in Ap-pendix B, the current contour C is chosen to obtain the conduction current in thearms. Therefore, the magnetic field components, Hx and Hy, surrounding the verti-

cal pins are chosen to form the contour C. The numerical results obtained for this

dipole are compared with the measurements in Fig. 7.31. The results for the input

impedance, Fig. 7.31a, for the realized gain, Fig. 7.31b, and the gain, Fig. 7.31c, are

seen to be in reasonable agreement.

The second FDTD model investigated, shown in Fig. 7.30b, neglects the fine

details of the feed region and just models the straight, linear antenna. Here, the

length of the antenna arm, h, is discretized with 48 cells (the electric field components,

Ex = 0, that mark the conductors of the thin dipole are shown with thick black

lines), i.e., ∆x ≈ 1.49mm, and the gap is discretized with 5 cells (the electric field

components that relate to the voltage in the transmission line are shown with thin

black lines). As described in Section 6.1.5, the metallic antenna arms are marked

by setting the tangential electric field components, Ex, to zero and modifying the

magnetic field components surrounding the antenna conductor as well as the gap

with the modified FDTD update equations to incorporate the finite thickness of the

conductors. Since this approximate equivalence for a round wire requires the cell

size to be at least twice the antenna radius, i.e., ∆x > 2a, this discretization is

much coarser than that used for the previous model. The tangential electric field

components, Ex, within the gap that relate to the voltage in the transmission line,

V(lmax), are shown in Fig. 7.30b, as well as the magnetic field components,Hy and Hz,

171


0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

-1000

-500

0

500

1000

1500

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0.0

0.5

1.0

1.5

2.0

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0

1

2

3

4

R,X

[]

f [GHz]

f [GHz]

f [GHz]

Gr

G

(a)

(b)

(c)

FDTD

Measurement

R

X

Figure 7.31: Comparison of theoretical (FDTD) and measured results for thedipole with the square cross section including the details of theantenna feed: a) input impedance, b) realized gain, and c) gain.

172


that mark the current contour C. Notice the shift of the contour away from the drive-point gap in order to determine the conduction current. The numerical results for the

input impedance, the realized gain, and the gain are compared with the measured

results in Fig. 7.32. Again, the numerical results are in reasonable agreement with

the measurements.

Notice that for both models the results for the gain is in good agreement with

the measurements. This is because the mismatch between the antenna and the trans-

mission line is not included in this form of the gain. However, the realized gain that

by definition includes the mismatch, is seen to differ for both feed models and to de-

viate from the measured results for frequencies greater than the resonant frequency,

fres ≈ 0.9GHz due to the differences in the reflection coefficient/input impedance.

173


0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

-1000

-500

0

500

1000

1500

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0.0

0.5

1.0

1.5

2.0

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0

1

2

3

4

R,X

[]

f [GHz]

f [GHz]

Gr

G

(b)

(c)

FDTD

Measurement

R

X

f [GHz](a)

Figure 7.32: Comparison of theoretical (FDTD) and measured results for thedipole with the straight cross section omitting the details of theantenna feed: a) input impedance, b) realized gain, and c) gain.

174

CHAPTER 8

Conclusions

The two-arm, conical spiral antenna in free space was shown to have many beneficial

and desirable characteristics over a wide range of frequencies, such as uniform input

impedance, unidirectional radiation and circular polarization. The numerical results

obtained using the FDTD method were validated using accurate measurements per-

formed at Georgia Tech.

A parametric study was performed with the FDTD analysis, and the results from

the study were used to produce new design graphs for this antenna. One graph gives

the useful bandwidth of operation for the antenna, i.e., the bandwidth over which

simultaneously the VSWR is less than 1.5 and the directivity is within −3 dB of itsmaximum value. Other graphs give the parameters that describe the performance of

the antenna (impedance, directivity, half-power beamwidth, front-to-back ratio, and

axial ratio) averaged over this bandwidth. These graphs supplement and extend the

empirical results presented in Dyson’s seminal paper. In general, the results from this

study agree with Dyson’s; a notable difference, however, is that the results from this

study predict a larger useful bandwidth.

Two resistive terminations intended to improve the low-frequency performance

of the CSA were examined: a pair of lumped resistors connected between the arms at

the open end and a thin disc of resistive material connected to the arms at the open

end. The latter is a new configuration that was analyzed to determine the resistance

using conformal mappings. Both terminations were shown to improve the impedance

175

Conclusions

match, front-to-back ratio, and axial ratio of the CSA at low frequencies but to

have a negligible effect on the realized gain. Compared to the lumped resistors, the

resistive sheet was seen to have less conduction loss and slightly better performance

characteristics in some features of the pattern.

The two-arm CSA was investigated when placed directly over the ground with

its axis normal to the surface of the ground. Both qualitative arguments and a

complete FDTD analysis were used to examine how the well-known properties of the

CSA in free space are modified when the antenna is placed over the ground. When

the virtual apex of the antenna is on the air/ground interface, the input impedance

of the antenna was shown to be approximately resistive and nearly constant over

the operational bandwidth of the antenna. The electric field within the ground was

shown to scale with position and frequency, as one would expect from the active-region

concept for the CSA. Also, the electric field in the ground directly below the CSA

was shown to be predominantly circularly polarized. Calculations for a monostatic

GPR that uses a CSA showed how the broadband properties of the CSA could be

used to detect buried objects that scatter an electric field with a component that is

copolarized to the circularly polarized electric field transmitted by the CSA.

The FDTD method was shown to be a powerful tool to analyze and design CSAs

in free space over the ground. The numerical analysis was validated in many ways.

Results for the impedance and the radiated electric field for the unloaded CSA and

the loaded CSA (terminated with lumped resistors) agree well with results published

in the literature. The implementation of the conductive medium was validated with

published analytical and experimental input impedance results for horizontal dipole

antennas placed directly above ground or in tap water. With measurements performed

at Georgia Tech, the FDTD input impedance and realized gain were validated for the

unloaded CSA and the loaded CSAs (terminated using lumped resistors and the

resistive sheet).

Two common FDTD antenna feed models were analyzed for convergence using

the linear dipole antenna with square cross section. It was found that elements of

176

Conclusions

the antenna feed must be properly scaled and positioned for convergence; these are

the physical length of the gap, the size and position of the current contour, as well

as the reference planes within the transmission line. This analysis showed that the

discretization must be chosen finer that one expects in order to reach convergence.

177

APPENDIX A

Analysis of the Resistive Sheet

Termination

The objective of this analysis is to determine the DC resistance between the two

electrodes of angular width δ on the circular disc of diameter D with resistance

per square Rs, see Fig. 4.2b. This is equivalent to solving the electromagnetostatic

problem described in Fig. A.1a: Laplace’s equation must be solved for the potential,

Φ, on the circular disc subject to the boundary conditions Φ = V on one electrode,

Φ = 0 on the other electrode (Dirichlet’s boundary condition) and ∂Φ/∂φ = 0 on

the remainder of the boundary (Niemann’s boundary condition). The solution of this

mixed-boundary problem is accomplished by means of conformal mappings that are

based on analytic functions [52]. Such transformations from one plane to another

preserve the magnitude and sense of angles, and the real and imaginary parts of such

mappings automatically satisfy the two-dimensional Laplace equation. The purpose of

this analysis is to transform the original mixed-boundary problem to a straightforward

geometry with known potential/known DC resistance between the two electrodes.

Thus, two conformal mappings will be used to transform the circular electrodes to a

pair of parallel electrodes with equal DC resistance.

The end points of the circular electrodes, Pmi with i = 1, . . . , 4, can be expressed

mathematically as a function of the diameter of the disc D and the angular width

of the electrodes δ, e.g., Pm1 = (D/2) exp(j(π/2 − δ/2)). Notice that the enclosed

178

Analysis of the Resistive Sheet Termination

= V

= 0

k

l

= 0P1

m

P4

mP3

m

P2

m

m

P1

wP2

wP3

wP4

w

u

vw

-u1 u2-u2 u1

P1

z

P2

zP3

z

P4

z

x

yz

lz

hz

(a)

(b)

(c)

Figure A.1: Drawing detailing the conformal mappings.

179


resistive material lies to the left when the end points are traversed in counterclock

direction (going from Pm1 to Pm

4 ).

The first mapping is a bilinear transformation that transforms the circular disc

in the complex plane m = k + jl into the upper half space in the complex plane

w = u+ jv:

w(m) = j

(2m/D + 1

1− 2m/D

). (A.1)

The result of this transformation is shown in Fig. A.1b. Notice that the electrodes

are now a pair of symmetric strips on the real axis with the end points given by

u2 =1

u1=

cos(δ/2)

1 + sin(δ/2). (A.2)

The resistive material is now located on the positive imaginary half plane and is still

on the left when the points are traversed in the same order as before (going from Pw1

to Pw4 ).

The second mapping is based on the Schwarz-Christoffel transformation, which

transforms a line of constant v into the polygon with N corner points in the z = x+jy

plane. This mapping is illustrated for N = 4 in Fig. A.2 for the general case (left)

and for the special case considered here (right). The general expression for this

transformation can be written as

z = A∫ w

−∞(w′ − w1)

β1π−1(w′ − w2)

β2π−1 . . . (w′ − wN)

βNπ

−1dw′ +B, (A.3)

where the points wi are points along the infinite line with constant v in the w = u+jv

plane that become the corners of the polygon in the z plane. The angle βi is measured

counterclockwise between adjacent segments at the ith corner of the polygon in the

z plane (traversing the polygon from corner 1 to N). In the special case considered

here, the upper half space in the w = u+ jv plane is transformed into the rectangular

region in the z = x+ jy plane with the transformation

z(w) =∫ w

−∞dw′√

w′2 − u22√w′2 − 1/u22

= u2

∫ w/u2

−∞dξ√

(1− ξ2)(1− u42 ξ2). (A.4)

180


w1

-u1

v v

u u

w2

-u2

w3

u2

w4

u1

x x

y y

1

3

2

4

3

2

4

1

Schwarz-Christoffel Transformation (General Case)

Schwarz-Christoffel Transformation (Special Case)

General Case ( = 4):N Special Case:

2 2 2 2

1 2

wdw

w u w u

z

31 2 41 1 1 1

1 2 3 4( ) ( ) ( ) ( )

w

A w w w w w w w w dwz B

1

1

2

23 3

4

4

Figure A.2: Drawing detailing the Schwarz-Christoffel transformation.

The result of this transformation is shown in Fig. A.1c. Notice that the electrodes

are now a pair of parallel strips of width hz separated by the distance lz, and that the

resistive material is again enclosed to the left when the points along the boundary

are traversed from P z1 to P z

4 . Both hz and lz can be written as the distance between

two points, P zi . After using (A.4), we have

lz = |P z3 − P z

2 | = |z(u2)− z(−u2)| = 2u2∫ 1

0

dξ√(1− ξ2)(1− u42 ξ

2)(A.5)

181


= 2u2K(k = u22), (A.6)

and

hz = |P z4 − P z

3 | = |z(u1)− z(u2)| = u2

∫ 1

0

dξ√(1− ξ2)[1− (1− u42) ξ

2](A.7)

= u2K(k =

√1− u42

), (A.8)

where K is the complete elliptic integral of the first kind and modulus k [53]. The

resistance of the disc is now simply determined from the geometry in Fig. A.1c:

R =lzhz

Rs =2K(k = u22)

K(k =

√1− u42

) Rs. (A.9)

with u2 given by (A.2). The normalized resistance, R/Rs, is shown in Fig. A.3 as a

function of the angular width of the electrodes δ. Notice that for δ = 90, the DC

resistance of the sheet reduces to R = Rs.

RR

/s

0 45 90 135 180

0

1

2

3

Figure A.3: Normalized DC sheet resistance as a function of the angular elec-trode width δ.

182

APPENDIX B

Convergence Study for FDTD Feed

Models for Antennas

The properties of antennas, such as the input impedance and the field pattern, are

now routinely determined using numerical techniques for solving Maxwell’s equa-

tions, such as the finite-difference time-domain (FDTD) method. A critical region in

these simulations, particularly for determining the input impedance/admittance, is

the drive point. Often the geometry of the feed point for the actual antenna is quite

complex, so a simplified geometry for the feed point is used in the numerical model.

Differences then exist between the calculated and measured impedances, so a com-

parison of the calculated impedances with the measurements cannot strictly be used

to establish the accuracy of the simulation. In particular, such comparisons cannot

be used to determine if the numerical calculation has converged, which in the case of

the FDTD method generally means that the size of the Yee cell is sufficiently small.

In the following, the convergence of FDTD calculations of the input admittance of a

simple dipole antenna are examined for two different feed models.

Two feed models commonly used for analyzing and designing antennas with the

FDTD method are studied in this work and referred to as the “hard source” and

the “simple feed.” Common properties of both antenna feed models are illustrated

in Fig. B.1. Near the feed region, the antenna is a metallic (PEC) structure which

is separated at the drive-point by a gap of width lg. Outside the feed region, the

183

Convergence Study for FDTD Feed Models for Antennas

Antenna DrivePoint

Perfectly-ConductingWire

…

xyz

Hy

Hz

Ex = f ( )V

I = ( , )f H Hx y

s

lg…

Figure B.1: Schematic drawings showing the feed region of the antenna.

antenna can have any arbitrary shape. The electric field in the gap is related to the

voltage between the drive-point terminals using the relation

V = −∫

&E · &dl. (B.1)

In the FDTD method, this integral is converted into summations and can thus easily

be rewritten to obtain the modified update equations for the electric field in the gap as

a function of the terminal voltage V . In the hard-source model, this voltage is simplythe total impressed voltage, Vt = Vinc(t), see (2.9), while in the simple-feed modelthis voltage is generally the last voltage in the virtual transmission line, as outlined

in Section 6.4. In both feed models, the current at the drive point, I, is determinedusing the magnetic field surrounding the drive-point region with the relation

I =∮C&H · &ds, (B.2)

This contour C is shifted away from the center of the gap in the transverse direction

by the distance s. In the hard-source model, the Fourier transform of this cur-

rent, together with that of the impressed voltage, are used to determine the input

impedance of the antenna. In the simple-feed model, this current is used to couple

the three-dimensional FDTD grid into the transmission line, and the apparent drive-

point impedance is determined within the transmission line using both the incident

and reflected voltages, as outlined in Section 6.4.

The convergence study is based on the input admittance and is performed for

the perfectly conducting, linear dipole antenna of length L with square cross section

184


of equal height and width, w, as shown in Fig. B.2a. This structure is very easy to

discretize with the FDTD method using cubic Yee cells. Here, approximations that

are often introduced in the FDTD analysis of antennas that can affect the convergence

are avoided, such as an approximate equivalence for a round wire, and staircasing of

the geometry for the conductors. For simplicity, at each step in the convergence

study, the next finer discretization is achieved by decreasing the size of the cells by

a factor of two in each direction, i.e., doubling the number of cells along the antenna

in each direction. In the convergence study, the dipole’s length to width ratio is held

fixed at L/w = 30.5 with L = 21.2 cm. Figure B.2b shows the coarsest discretization

used for the antenna: 61 cells along its total length, 2 cells along its width, and 1 cell

across the gap. The shaded areas in this picture are the faces of the FDTD cells along

which the tangential electric field components are set to zero, so they delineate the

boundaries of the perfectly conducting structure. The excitation at the drive point

for the hard-source model (Vt) and within the transmission line (Vinc) for the simplefeed is a differentiated Gaussian voltage pulse, (2.9), with a ratio of characteristic

times between the pulse (τp) and the antenna (τL = L/c) of τp/τL = 1/π ≈ 0.32.

Generally, four different discretizations are studied for convergence of the ad-

mittance results: coarse, medium, fine, and superfine; however, in many results, just

the first three discretizations are shown for clarity. The number of cells per antenna

length, L/∆x, and the number of cells per width, w/∆x, are listed for each discretiza-

tion in Table B.1. In numerical analyses, the number of cells per wavelength at the

highest frequency is of great interest and is often used as an indicator for the accuracy

of the results. A rule of thumb for the FDTD analysis is that at least 10 cells per

shortest wavelength should be used in the simulation to obtain acceptable results, i.e.,

to reduce dispersion to a reasonable level. As shown in Table B.1, even the coars-

est discretization should give negligible dispersion with approximately 30 cells per

shortest wavelength, λmin. In this work, the maximum frequency (minimum wave-

length) is defined to be fmax ≈ 3.5fpeak = 3.5/(2πτp) (λmin = c/fmax ≈ 1.80 τp/τL L),

which is the frequency at which the spectrum decayed to 180to that of the peak

185


Hy

Hz

Exw

L

Drive Point

xy

z

Schematic Model:

w

FDTD Model:

Faces of CubicFDTD Cells

(a)

(b)

Figure B.2: a) Model of the linear dipole with square cross section, b) discretiza-tion of the model (L/w = 30.5).

value at f = fpeak. A straightforward analysis shows that dispersion is negligible in

this convergence study. Recall, numerical dispersion describes the deviation of the

phase velocity of the numerical wave in the FDTD grid from the speed of light [30].

This effect strongly depends on the propagation direction in the grid and the grid

discretization. Since the pulse propagates in just the x direction (along the metal

rods) in the present case, the simplified numerical dispersion relation for the FDTD

algorithm can be written as

1

c∆tsin

(ω∆t

2

)=

1

∆xsin

(kx∆x

2

), (B.3)

where kx is the numerical wavevector and is related to the numerical phase velocity,

vp, by kx = ω/vp. The dispersion relation, (B.3), can then be solved for the ratio of

186


Table B.1: Parameters of the discretized linear dipole antenna of square cross section.

coarse medium fine superfine

L/∆ x 61 122 244 488

w/∆x 2 4 8 16

λmin/∆x 35 70 139 279

numerical phase velocity to the speed of light, i.e.,

vp

c=

π

Nλ sin−1 [ 1

CFLsin

(π CFLNλ

)] . (B.4)

Note that this equation contains only normalized parameters such as the number of

cells per wavelength, Nλ = λ/∆x, or the Courant-Friedrichs-Levy (CFL) number,1

CFL = c∆t/∆x, that links the spatial time increment to the dimensions of the Yee

cell. In three-dimensional, cubic FDTD grids, this number is commonly CFL = 0.5.

The results for this ratio vp/c are listed in Table B.2 for various numbers of cells per

shortest wavelength; most of these numbers correspond to the discretizations studied,

see Table B.1. Recall that λ/∆x ≈ 10 is commonly used as a minimum requirement

in FDTD analyses to obtain accurate results at that frequency/wavelength. These

results show that the dispersion will be insignificant for the cases considered.

In the following, the motivation for this work is addressed by presenting the

results for the admittance when the simplest feeding technique is employed for each

discretization. The feeding technique for this “worst case” for convergence is illus-

1The Courant-Friedrichs-Levy condition in a three-dimensional FDTD grid

c∆t

√1∆x2

+1∆y2

+1∆z2

≤ 1

must be satisfied for stability for the finite-difference scheme of second order in time and space.

When cubic cells are chosen, this condition simplifies to

c∆t

∆x= CFL ≤ 1√

3.

187


Table B.2: Ratio of numerical phase velocity to the speed of light for differentdiscretizations.

λmin/∆x 10 35 70 139 279(coarse) (medium) (fine) (superfine)

vp/c 0.98726 0.99899 0.99975 0.99994 0.99998

trated in Fig. B.3 for the first three discretizations (coarse, medium, and fine). The

left side is the “bird’s eye” view of the feed region, the center shows the top view

of the feed, and the right side shows the cross-sectional view of the drive point. In

each discretization, a one-cell gap is employed for each discretization, i.e., the phys-

ical length of the gap differs for each discretization, see the center of Fig. B.3, while

the total length of the antenna remains the same. Here, just a single electric field in

the center of the gap is updated with the modified update equation, based on (B.1).

The current is determined by integrating directly around the drive-point gap, i.e., the

contour is not shifted in the transverse direction and has the smallest cross-sectional

area to just enclose the surface of the conductor. The magnetic fields, Hy and Hz,

that mark the rectangular contour C are seen to move closer to the antenna conduc-tor the finer the discretization. The distance between the surface of the conductor

and the magnetic field component is exactly a half spatial step of the corresponding

grid. The results for this convergence study are shown in Fig. B.4 for the first three

discretizations. Figure B.4a is for the hard-source model and Fig. B.4b is for the

simple feed. Obviously, the results for either feed are not converging with increasing

discretization. Recall that these feed elements are chosen in the simplest and usually

most common way for each discretization; thus, some of these choices can be expected

to cause the results not to converge.

In the remainder of this section, it will be shown that it is crucial to properly po-

sition and scale the elements in the feed model for convergence. Figure B.5 illustrates

that different geometrical lengths of the gap introduce a change in the susceptance.

As shown schematically, the decrease in length increases the capacitive effect at the

188


xyz

xy

z

y

Hy

Ex

Ex

Hy

Hz

Hz

Coarse:

Medium:

Fine:

C

Figure B.3: Feeding techniques shown for the first three discretizations (worstcase for convergence).

drive point. To avoid this source of error, the length of the drive point-gap must be

the same for each discretization. Another important effect must be considered in this

study: the fringing of the electric field in the vicinity of the gap, shown schemati-

cally in Fig. B.6. Fringing occurs since charges accumulate at the terminals of the

antenna conductors that represent a discontinuity in the antenna geometry. Obvi-

ously, by varying the radial extent of the contour when placed in the vicinity of the

gap, different portions of the fringing field are included in the current contour. The

contour surrounding the gap thus yields a displacement current; however, since the

calculation of the input impedance for the hard-source model and the components of

the transmission line require the total current into the antenna terminals, an offset

contour with s > lg/2 should be chosen instead to yield the conduction current.

The above criteria are met in the following feeding technique, referred to as “best

case” for convergence. Schematic drawings of the feed region are shown in Fig. B.7.

189


GB

,[m

S]

0.0 0.5 1.0 1.5 2.0 2.5

-4

0

4

8

12

16

f [GHz]

G

B

GB

,[m

S]

0.0 0.5 1.0 1.5 2.0 2.5

-4

0

4

8

12

16

f [GHz]

G

B

CoarseMediumFine

(a)

(b)

Figure B.4: Results for the input admittance in the worst case for convergence:a) hard-source model and b) simple feed.

Ccoarse Cmedium Cfine< <

Figure B.5: Schematic illustration of the change of the susceptance with varyinglength of the gap.

Here, the drive-point gap properly scales for each discretization by keeping the length

of the gap constant with lg = w/2. For discretizations finer than the coarsest one, this

requires multiple-cell gaps. In order to apply the voltage in the gap more uniformly, all

190


Fringing Field

Figure B.6: Schematic drawing for the fringing of the electric field in the vicinityof the drive-point gap.

tangential electric field components within the gap are updated based on (B.1). The

current contour in this new approach yields a conduction current, i.e., the transverse

shift of the contour is s = w to avoid the fringing of the electric field. Furthermore, the

radial extent of the contour is chosen to be same for every discretization. Notice that

in order to properly scale and position the current contour, averages in the transverse

direction and the radial extent are necessary for the discretizations finer than the

coarsest one. This can clearly be seen in the schematic drawings in the center (top

view) and on the right (cross-sectional view) of Fig. B.7.

For the simple-feed model, it is crucial to properly position the reference planes

for those voltages and currents in the transmission line that virtually couple both the

transmission line to the drive point, as illustrated in B.8. In the present approach,

the reference plane for the current in the last cell2 is chosen to be the same for all

discretizations, as shown in Fig. B.8a with the solid line. The reference plane for the

voltage that couples the one-dimensional transmission line to the three-dimensional

FDTD grid must be placed at the same location for all discretizations.3 This plane

is set by the position of the voltage in the last cell of the coarsest grid, shown in

Fig. B.8a with a dashed line. Notice that for discretizations finer than the coarsest

one, averages are necessary to determine the corresponding voltage at this plane. In

2Recall, that I(lmax) couples the three-dimensional FDTD grid in the one-dimensional transmis-

sion line.3Recall, that in the regular approach V(lmax) is used for the coupling.

191


xyz

xy

z

y

Hy

Ex

ExHy

Hz

Hz

Coarse:

Medium:

Fine:

Average

over the TwoContours

H

Average

over the TwoContours

H

Figure B.7: Feeding techniques shown for the first three discretizations (bestcase for convergence).

Fig. B.8b, the virtual connections between the transmission line and the drive-point

of the antenna are shown for the first three discretizations, and the voltages are listed

that are required to couple the line to the antenna domain at the drive point.

When the above criteria are met, the results for the input admittance as a

function of frequency are as shown in Fig. B.9 for the first three discretizations.

The results can be seen to converge for both the hard-source feed in Fig. B.9a and

the simple feed in Fig. B.9b. To better examine the convergence of the results, the

admittances (both hard-source feed and simple feed) are shown for the worst case

in Fig. B.10a, and for the best case in Fig. B.10b over the limited frequency range,

1.9GHz ≤ f ≤ 2.0GHz. This range is where the results for the admittance deviate

the most, see Figs. B.4 and B.9. The three discretizations (coarse, medium, and

fine) are seen not to converge for either the conductance G or the susceptance B for

the worst case. However, these components are clearly seen to converge for the four

discretizations (course, medium, fine, and superfine) for the best case.

To analyze the results for the best case for convergence quantitatively, the reso-

192


…

…

…

…

…

…

xyz

Hy

Hy

Hy

Hz

Hz

Hz

Vn+1

( )lmaxE x

n+1

dp

E x

n+1

dp

E x

n+1

dp

Vn+1

( )+lmax Vn+1

( -1)lmax

Vn+1

( -1)+lmax Vn+1

( -2)lmax

Coarse:

Medium:

Fine:

Coarse Medium Fine1D Line:

(a)

(b)

Plane for I (3D 1D)

Plane for V (3D 1D)

Figure B.8: a) Proper scaling of the reference planes in the transmission line,b) correct virtual connection of the transmission line at the feedpoint.

nant frequency and the conductance at anti-resonance (the point where B is zero for

the second time, λres ≈ L or equivalently fres = c/L) are determined for the hard-

source feed, see Fig. B.9. The results for this analysis are listed in Table B.3. Clearly,

the two parameters fres and Gres are seen to converge with increasing discretization;

the deviation in the resonant frequency between the coarsest and the finest grid is

approximately 2% and the deviation in the conductance is 4%. It should be noted

that for errors in these parameters less than about 2% the number of cells at that

193


GB

,[m

S]

f [GHz]

GB

,[m

S]

f [GHz](a)

(b)

0.0 0.5 1.0 1.5 2.0 2.5

-4

0

4

8

12

16

0.0 0.5 1.0 1.5 2.0 2.5

-4

0

4

8

12

16

CoarseMediumFine

Figure B.9: Results for the input admittance in the best case for convergence:a) hard-source model and b) simple feed.

particular wavelength, λ must be at least λ/∆x ≈ 100.

In conclusion, for the FDTD feed models studied (hard source and simple feed)

to converge the elements of the feed must be properly scaled and positioned. These

elements are the length of the drive-point gap and the size and position of the cur-

rent contour C. For the simple feed, this also includes the proper positioning of thereference plane for the voltage in the transmission line that couples the line into the

drive point of the antenna. It was found that the discretization for convergence has

to be much finer than one typically expects based on the negligible dispersion alone.

194


1.90 1.95 2.00

10

12

14

1.90 1.95 2.00

10

12

14

1.90 1.95 2.00

8

10

12

1.90 1.95 2.00

8

10

12

G[m

S]

f [GHz]

1.90 1.95 2.00

0

2

4

6

8

10

B[m

S]

f [GHz]1.90 1.95 2.00

0

2

4

6

8

10

B[m

S]

f [GHz]

1.90 1.95 2.00

-2

0

2

4

6

B[m

S]

f [GHz]1.90 1.95 2.00

0

2

4

6

8

B[m

S]

f [GHz]

G[m

S]

f [GHz]G

[mS

]

f [GHz]

G[m

S]

f [GHz]

Hard Source

Hard Source

Simple Feed

Simple Feed

Hard Source

Hard Source

Simple Feed

(a)

(b)

CoarseMediumFineSuperfine

Figure B.10: Comparison of the results for the admittance: a) worst case and b)best case for convergence.

195


Table B.3: Resonant frequencies and the conductances at anti-resonance forthe hard source (best case for convergence).

coarse medium fine superfine

fres [GHz] 1.17 1.18 1.19 1.19

Gres [mS] 1.91 1.86 1.83 1.83

λres/∆x 74 146 291 581

196

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201

VITA

Thorsten W. Hertel was born in Holzminden, Germany, on April 19, 1974. He at-

tended the Campe Gymnasium in Holzminden and graduated from there in 1993.

He received the Vordiplom in Electrical Engineering from the Technische Universitat

Braunschweig, Germany, in 1996. The same year he entered the Graduate Program in

Electrical and Computer Engineering at the Georgia Institute of Technology, Atlanta,

GA (USA), where he was a graduate teaching assistant, and a graduate research as-

sistant. He received the M.S. degree in Electrical and Computer Engineering from

the Georgia Institute of Technology in 1998 with a thesis entitled: “Pulse Radiation

from an Insulated Antenna: An Analogue of Cherenkov Radiation from a Moving

Charge.”

His special interests include numerical modeling with the Finite-Difference Time-

Domain (FDTD) Method and antenna analysis and design.

202

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Analysis and Design of Conical Spiral Antennas in Free