Characterization of Alternative Mounting of Vehicular ...
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Characterization of Alternative Mounting of Vehicular Antenna Systems Near Real Grounds By Timothy W. Samson BSEE, Mercer University, 2012 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfilments Of the requirements for the degree of Masters of Science Department of Electrical and Computer Engineering 2014
Transcript of Characterization of Alternative Mounting of Vehicular ...
Characterization of Alternative Mounting of Vehicular Antenna Systems
Near Real Grounds
By
Timothy W. Samson
BSEE, Mercer University, 2012
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfilments
Of the requirements for the degree of
Masters of Science
Department of Electrical and Computer Engineering
2014
The thesis entitled:
Characterization of Alternative Mounting of Vehicular Antenna Systems Near Real Grounds
Written by Timothy W. Samson
Has been approved for the Department of Electrical and Computer Engineering
___________________________________________
Dejan S. Filipovic
___________________________________________
Neill Kefauver
___________________________________________
Maxim Ignatenko
Date:___________
The final copy of this thesis has been examined by the signatories, and we find that both the
content and the form meet acceptable presentation standards of scholarly work in the above
mentioned discipline.
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Samson, Timothy W.
Characterization of Alternative Mounting of Vehicular Antenna Systems Near Real Grounds
Thesis directed by Dejan S. Filipovic
Vehicular antenna systems for jamming and communications are frequently mounted on either the
roof or rear bumper of the vehicle. While this facilitates easier mounting and maintenance of the
antenna systems, it reduces the maneuverability and increases the visual profile of the vehicle. This
thesis researches alternative mounting locations such as underneath the vehicle or on the sides of
the vehicle.
For either mounting location, the presence of the ground electrically close to the antenna system
may drastically modify its performance and characteristics. Therefore, it is vital to understand how
the presence of the ground affects the antenna’s patterns, impedance, efficiency, and axial ratio
when the antennas are designed to be circularly polarized. Once the effects of the grounds on the
antenna systems are understood, steps can be taken to mitigate potential adverse effects and
capitalize on any benefits. To better understand propagation close to the ground, both with and
without the vehicle model, ideal sources were initially studied. More conductive soils offer higher
efficiencies, but cause a greater amount of depolarization for circularly polarized sources.
For sources under the vehicle, the source is not just having a stronger coupling with the ground,
but the vehicle body redirects energy back towards the ground. To mitigate this, the parallel plate
source is proposed for HF and VHF frequencies. Small modifications to the geometry can achieve
significantly higher efficiencies for sources under the vehicle. Other antennas are considered for
operation in the UHF region.
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For antenna systems mounted on the side of the vehicle, two mounting approaches (for spiral
antennas) are considered. First, a cavity-backed aperture is mounted flush with the vehicle’s body.
The second configuration has the spiral antenna aperture offset from the vehicle thus requiring
minimum changes to the vehicle’s body. While the first approach is more commonly used, the
latter provides greater gain and efficiency at the lower end and comparable gain and efficiency at
higher frequencies while maintaining good axial ratio.
For side mounted sources, the four arm spiral provides a lower and more stable axial ratio when
compared to the two arm spiral. Both antennas offer similar efficiency, but the four arm spiral
maintains higher quality performance with the introduction of a cavity backing. The performance
of side mounted sources can be further improved with the use of the spiral-helix that improves
both the low and high frequency performance.
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Acknowledgements
I would like to acknowledge the support and contributions from the following people:
Committee Chair and Academic Advisor Professor Dejan S. Filipovic, for all of his support,
guidance, and advice for the work found in this thesis.
Committee Member Dr. Maxim Ignatenko for evaluating this thesis and all of his advice and
contributions while we worked together.
Committee Member Dr. Neill Kefauver for his work in evaluating this thesis.
Dr. Jim Lovejoy and Tom Cencich at Lockheed Martin for their advice and suggestions.
My father Jerry Samson for all his advice and help.
And lastly, all of my friends and family for their love and support.
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Contents
Acknowledgements ......................................................................................................................... v
Chapter 1 Introduction .................................................................................................................... 1
1.1 Background ...................................................................................................................... 1
1.1.1. Improvised Explosive Devices ................................................................................. 1
1.1.2. Jamming .................................................................................................................... 3
1.1.3. Common Vehicle Mounted Antennas ....................................................................... 5
1.2 Requirements .................................................................................................................... 6
1.3 Thesis Outline .................................................................................................................. 6
Chapter 2 Computational Electromagnetics: Solvers and Modeling .............................................. 8
2.1 Method of Moments ......................................................................................................... 8
2.1.1 Theoretical Basics of the Method of Moments ......................................................... 9
2.1.2 Green’s Function ..................................................................................................... 11
2.1.3 Source Models ........................................................................................................ 12
2.1.4 Basis and Weighting Functions .............................................................................. 13
2.1.5 FEKO Implementation of MoM ............................................................................. 15
2.2 Physical Optics ............................................................................................................... 17
2.3 Hybrid MoM and PO...................................................................................................... 20
2.4 Sources over Grounds .................................................................................................... 23
2.4.1 Boundary Conditions .............................................................................................. 24
2.4.2 Image Theory .......................................................................................................... 26
2.4.3 Fresnel Reflection Coefficients ............................................................................... 27
2.4.4 Computational Ground Models in FEKO ............................................................... 29
2.5 Humvee Model ............................................................................................................... 30
2.5.1 Complex Humvee Model ........................................................................................ 31
2.5.2 Basic Humvee Model .............................................................................................. 31
2.5.3 Rough Humvee Model ............................................................................................ 32
2.5.4 Approximations with PEC Surfaces ....................................................................... 33
2.6 Summary and Conclusions ............................................................................................. 36
Chapter 3 Characterization of Fields from Ideal Sources ............................................................. 37
3.1 Ideal Sources .................................................................................................................. 37
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3.1.1 Vertical Electric Dipole .......................................................................................... 38
3.1.2 Horizontal Magnetic Dipole ................................................................................... 38
3.1.3 Crossed Electric Dipoles ......................................................................................... 38
3.2 Ideal Sources over Grounds without Vehicle Model ..................................................... 39
3.2.1 Efficiency of VED .................................................................................................. 39
3.2.2 Efficiency of Horizontal Magnetic Dipole ............................................................. 41
3.2.3 Efficiency of Crossed Electric Dipoles ................................................................... 42
3.3 Vehicle Mounted Ideal Sources over Flat Grounds ....................................................... 43
3.3.1 Azimuthal Pattern Uniformity Measured as the Wobble of the Wave (WoW) ...... 45
3.3.2 Attenuation .............................................................................................................. 47
3.3.3 Depolarization ......................................................................................................... 48
3.3.4 Efficiency ................................................................................................................ 49
3.4 Summary and Conclusions ............................................................................................. 51
Chapter 4 Antenna Mounting on the Vehicle Bottom ................................................................. 53
4.1 Stand-Alone HF-VHF Parallel Plate Antenna ............................................................... 54
4.1.1 Understanding Operation of Parallel Plate Source ................................................. 56
4.1.2 Parametric Studies .................................................................................................. 61
4.1.3 Modifications of the Parallel Plate Source .............................................................. 66
4.2 Integration of Parallel Plate Source with Vehicle Model ............................................... 68
4.3 Effect of Wheels in HF and VHF Bands ........................................................................ 71
4.4 Parallel Plate Source with Scaled Humvee Model ......................................................... 73
4.5 Standalone UHF Antennas ............................................................................................. 75
4.5.1 Mode-2 Spiral Antenna ........................................................................................... 76
4.5.2 Mode-2 Spiral–Helix Antenna ................................................................................ 79
4.5.3 Modified Monocone Antenna ................................................................................. 81
4.5.4 Comparison of Antennas Performance ................................................................... 82
4.6 Vehicle Mounted UHF Sources over Grounds .............................................................. 84
4.6.1 Numerical Model .................................................................................................... 84
4.6.2 Size of PEC Plate .................................................................................................... 87
4.6.3 Efficiency Comparison ........................................................................................... 88
4.6.4 Near-Field Coverage ............................................................................................... 89
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4.6.5 Role of Wheels at UHF ........................................................................................... 90
4.7 Summary ........................................................................................................................ 92
Chapter 5 Side Mounting of Spiral Antennas ............................................................................... 94
5.1 Introduction to Spiral Antennas ..................................................................................... 94
5.2 Standalone Sources ...................................................................................................... 102
5.2.1 Antennas Parameters ............................................................................................. 102
5.2.2 Numerical Model .................................................................................................. 103
5.2.3 Free Space Antenna Performance ......................................................................... 104
5.3 Mounting Antennas on Vehicle Side ........................................................................... 105
5.3.1 Model Description ................................................................................................ 105
5.3.2 Choice of Mounting Approach ............................................................................. 107
5.3.3 Choice of the Number of Arms ............................................................................. 110
5.3.4 Effect of Grounds on Spiral Antenna Performance .............................................. 114
5.4 Variations and Reductions to Vehicle Model .............................................................. 122
5.5 Spiral-Helix Antenna.................................................................................................... 123
5.5.1 Antenna Parameters .............................................................................................. 123
5.5.2 Model Description and Validation ........................................................................ 125
5.5.3 Antenna Performance............................................................................................ 127
5.6 Spiral-Helix Two-Element Antenna Array .................................................................. 130
5.6.1 Array performance ................................................................................................ 131
5.7 Summary ...................................................................................................................... 134
Chapter 6 Summary, Conclusions and Future Research ............................................................. 135
6.1 Summary and Conclusions ........................................................................................... 135
6.2 Future Research ............................................................................................................ 140
References ................................................................................................................................... 142
Appendix: Modeling Validation ................................................................................................. 146
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Tables
Table 2.1. Dielectric properties of ground types used in this thesis ............................................. 30
Table 2.2. Percent reduction in memory and CPU time at 400MHz over PEC ground ............... 35
Table 4.1 Zeroes of Bessel functions of the first J0 and second Y0 kinds and their corresponding
frequencies. Disc radius R = 2m. ........................................................................................... 58
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Figures
Figure 1.1. Humvee with several large antennas mounted on the rear of the vehicle. [5] .............. 5
Figure 2.1. Two triangular surfaces (Tn+ and Tn-) with areas An that share an edge ln ............. 14
Figure 2.2. Incident field on a conductive scatterer producing both an illuminated and shadowed
region. The surface current is nonzero in the illuminated region. ......................................... 19
Figure 2.3. Ideal vertical electric sources (a), horizontal electric sources (b), vertical magnetic
sources (c), and horizontal magnetic sources (d) and their images at a height h over an infinite
PEC ground plane. ................................................................................................................. 27
Figure 2.4. Incident, transmitted, and reflected electric fields at a dielectric interface between two
media. .................................................................................................................................... 28
Figure 2.5. Dimensions of the complex Humvee model. The vehicle body is modeled as PEC with
rubber wheels with a εr of 3 and conductivity of 10-15S/m. .................................................. 31
Figure 2.6. Dimensions the basic Humvee model. The vehicle body is modeled as PEC with rubber
wheels with a εr of 3 and conductivity of 10-15S/m. .............................................................. 32
Figure 2.7. Dimensions of the rough Humvee model. The entire model is modeled as PEC. ..... 33
Figure 2.8. View of rough Humvee model with side mounted antenna modeled only with MoM.
The reduced rough Humvee is modeled using either MoM or PO. ....................................... 34
Figure 2.9. Comparison of the three models. (a) Comparison of azimuthal near field at 5m and (b)
far field elevation patterns in the yz plane so that 90 degress corresponds to the boresight
direction. ................................................................................................................................ 35
Figure 3.1. Efficiencies for the VED at both (a.) 0.3m and (b.) 1.78m above various grounds for
different radii. ........................................................................................................................ 40
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Figure 3.2. Efficiencies for the HMD at both (a.) 0.3m and (b.) 1.78m above various grounds for
different radii. ........................................................................................................................ 41
Figure 3.3. Efficiencies for the crossed electric dipoles at both (a.) 0.3m and (b.) 1.78m above
various grounds for different radii. ........................................................................................ 42
Figure 3.4. Comparison of the efficiencies for the three ideal sources at 0.3m above a dry sand
ground. ................................................................................................................................... 43
Figure 3.5. Location of the antennas’ mounting for the rough Humvee model. The top source is in
the middle of the roof, the bottom source is at the middle of the bottom, and the rear is 0.1m
from the rear corner. .............................................................................................................. 44
Figure 3.6. Electric field strength 10cm above dry sand for under the vehicle model at 100MHz
for each mounting position. ................................................................................................... 45
Figure 3.7. Wobble of Wave (WoW) for the three antenna positions on the vehicle model over a
PEC ground............................................................................................................................ 46
Figure 3.8. Attenuation of the electric field due to losses in the ground propagating near the ground
for the vertical electric dipole for both dry sand (a) and wet soil (b). ................................... 47
Figure 3.9. Depolarization of a circularly polarized source above lossy grounds at 0.3m (a) and
1.78m (b). .............................................................................................................................. 49
Figure 3.10. Efficiency of the vertical electric dipole above dry sand for the three different vehicle
positions as well as the source without the vehicle model. ................................................... 50
Figure 3.11. Efficiency of the horizontal magnetic dipole over both dry sand (a) and wet soil (b)
both with and without the vehicle model. .............................................................................. 51
Figure 3.12. Efficiency of the crossed electric dipoles for the three antenna positions over dry sand
with and without the vehicle model. ...................................................................................... 51
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Figure 4.1. Efficiency of the ideal vertical electric dipole above a dry sand ground both with and
without the vehicle model for the top and bottom mounting positions. The inclusion of the
vehicle model enhances the efficiency of the top mounted source, but greatly diminishes the
efficiency of the bottom source. ............................................................................................ 53
Figure 4.2 (a) View and (b) geometry of the parallel plate source. The top disc simulates the bottom
floor of the vehicle and the bottom disc reflects and directs the wave away from the lossy
grounds and to potentially increase the efficiency. The source is an ideal vertical electric
dipole in the center. ............................................................................................................... 55
Figure 4.3. (a) Efficiency of an ideal vertical dipole and (b) the basic parallel plate source over dry
sand, asphalt, and wet soil ground types. Heights of the sources are 0.3m. Parallel plates
improve the efficiency at most frequency points and introduce oscillatory behavior. .......... 56
Figure 4.4. (a) Efficiency of the parallel plate source over the minimally lossy ground modeled
with the Sommerfeld ground model. There is a correlation between the standing wave
frequencies and the minima and maxima in the efficiency. (b) Electric field 0.5m directly
under the parallel plate source in free space. The local minima and maxima in the electric field
strength correlate with the standing waves in the efficiency. ................................................ 59
Figure 4.5 Comparison of parallel discs source 0.3m above dry sand and 2,4, and 16 horizontal
Hertzian magnetic dipoles to approximate a ring of magnetic current for efficiency (a) and
elevation cut at 300MHz (b). ................................................................................................. 60
Figure 4.6. Geometry of the parallel plate source over dry sand with the separation between the
plates varied as the other dimensions remain constant. The height of 0.3m is at the midpoint
between the plates. ................................................................................................................. 61
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Figure 4.7. Elevation patterns of the parallel plate source over dry sand for various separation
distances of the plates. ........................................................................................................... 62
Figure 4.8. Efficiency of the parallel plate source 0.3m above dry sand with the separation between
the top and bottom plate h varied. There is little change in the efficiency due to the very small
changes in separation in terms of wavelengths. However, some differences begin to become
apparent at higher frequencies. .............................................................................................. 62
Figure 4.9. Geometry of the parallel plate source over dry sand with the height of the source varied
as the other dimensions remain constant. .............................................................................. 63
Figure 4.10. Efficiency of the 2m radius parallel plate source at various heights h above a dry sand
ground. As the height of the source increaes, the effiency increases to oscillate just over 50%
............................................................................................................................................... 64
Figure 4.11. Geometry of the parallel plate source over the dry sand ground as the radius of the
plate is changed. The separation and height above the ground remain constant ................... 65
Figure 4.12. Efficiency of the parallel plate source 0.3m above dry sand with three different radii.
............................................................................................................................................... 65
Figure 4.13. Geometry of the parallel plate source with the fins. The fins are either 7cm or 14cm
long and extend out from the bottom plate at 45 degrees. ..................................................... 66
Figure 4.14. Elevation cut of the electric near field measured 2.5m from the center of the parallel
plate source in free space. ...................................................................................................... 67
Figure 4.15. Efficiency of the parallel plate source with and without fins over the dry sand ground.
Even with the addition of electrically very small fins, there is a significant increase in
efficiency above 50MHz ....................................................................................................... 67
Figure 4.16. Geometry and dimensions of a rough vehicle model. .............................................. 69
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Figure 4.17. Geometry of plate with fins under the vehicle model. The fins extend 5cm out from
the side and 5cm up. The top plate is the bottom of the vehicle. .......................................... 69
Figure 4.18 (a) Efficiency of the VED under the vehicle compared to the parallel plate source with
and without the vehicle model over dry sand. (b) Model of the parallel plate source under the
vehicle and the parallel plate source without the vehicle. ..................................................... 70
Figure 4.19. Normalized electric field strength in dB at several frequencies 32cm above a PEC
around the basic Humvee model with rubber, PEC, and no wheels. ..................................... 72
Figure 4.20. Scaled basic Humvee model without (a) and with (b) the scaled parallel plate source
under the vehicle. ................................................................................................................... 74
Figure 4.21. S11 measurements of the scaled parallel plate source over a conductive ground plane
both without (a) and with (b) the scaled Humvee model. ..................................................... 75
Figure 4.22. (a) Co-polarized far-field patterns of mode-1 and mode-2 of a four-arm spiral in free
space without any backing (only top half is shown). (b) Phase progressions in feeding of the
four-arm spiral antenna to achieve mode-1 and mode-2. The excitation magnitudes at all four
arms are the same. ................................................................................................................. 76
Figure 4.23. Geometry of the mode-2 four-arm spiral antenna and close up of feed that will be
used under the vehicle. .......................................................................................................... 77
Figure 4.24. VSWR of the 25.4cm diameter spiral for both mode-1 and mode-2 operation in free
space. VSWR is given with respect to nominal impedance. ................................................. 78
Figure 4.25. Real and imaginary impedance of the mode-2 spiral antenna (a) and the far-field total
gain elevation cuts over the UHF band (b). Elevation cuts are calculated from 700 MHz to 3
GHz with step 250 MHz. ....................................................................................................... 78
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Figure 4.26. General geometry of the spiral-helix antenna. It has a metal to slot ratio of 50% and
is mounted on the bottom of a PEC plate. When ground is modeled, the plate is 37cm above
the ground. ............................................................................................................................. 79
Figure 4.27. Real and imaginary parts of impedance of the mode-2 spiral-helix antenna under the
PEC plate (a) and the far-field gain elevation cuts over the UHF band (b). Elevation cuts are
calculated from 700 MHz to 3 GHz with step 250 MHz....................................................... 80
Figure 4.28. Geometry of the modified monocone antenna over a ground plane (a) and general
view of numerical model (b). ................................................................................................ 81
Figure 4.29. Real and imaginary parts of impedance of the modified bicone antenna (a) and the
far-field gain elevation cuts (b) over the UHF band. Elevation cuts are calculated from 700
MHz to 3 GHz with step 250 MHz. Free space. .................................................................... 82
Figure 4.30. VSWR of the three considered UHF sources in free space with respect to their nominal
impedances. ........................................................................................................................... 83
Figure 4.31. (a) Efficiency of the three considered UHF sources 0.32m above dry sand without the
vehicle model. (b) Percentage of power dissipated in the loads of the spiral and spiral-helix.
............................................................................................................................................... 84
Figure 4.32. Model of the vehicle bottom with PEC wheels. ....................................................... 85
Figure 4.33. Model of PEC plate for modeling the bottom of the vehicle and the height of the
source and plate above the ground. ....................................................................................... 86
Figure 4.34 Efficiency of monocone antenna mounted on the 2m x 2m PEC plate and vehicle
bottom model over dry sand. ................................................................................................. 86
Figure 4.35. A geometry used to evaluate the effects of size variation of the square PEC plate with
the mode-2 four-arm spiral over dry sand. ............................................................................ 87
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Figure 4.36. Efficiency of the mode 2 four arm spiral antenna placed between dry sand and square
PEC plates of various sizes. The 2m x 2m plate has very similar efficiency to the vehicle
bottom model. ........................................................................................................................ 87
Figure 4.37. (a) Efficiency of the mode-2 spiral, mode-2 spiral-helix and monocone antennas under
the 2m x 2m PEC plate over dry sand. (b) Efficiency of the spiral-helix antenna for dry sand,
asphalt, and wet soil............................................................................................................... 88
Figure 4.38. (a) WoW for the modified monocone under the 2m x 2m PEC plate over various
grounds. The wet soil ground has the lowest WoW, while the more lossy asphalt and dry sand
have very similar WoWs. (b) Comparison of the WoW for the monocone and spiral-helix over
dry sand ground. .................................................................................................................... 89
Figure 4.39. Azimuthal electric near field patterns in dB 5m and 10m from the source and 0.32m
above the dry sand ground under the 2m x 2m PEC plate. There is low variability for the
modified monocone source compared to the other sources. The modified monocone source
also has a much stronger electric field near the source. ........................................................ 90
Figure 4.40. WoW for the monocone and spiral helix source under the vehicle bottom model for
both PEC and dry sand ground. Note the decreased WoW for the dry sand case compared to
the PEC ground and the significantly higher WoW with the inclusion of the PEC wheels. . 91
Figure 5.1. Two-arm Archimedean spiral depicting the currents in the feed and active region.
Points A, B, C, A’, B’, C’ are on the active region of the antenna when the circumference of
points A, B, A’, B’ and C, C’ radiate constructively in broadside direction. ........................ 97
Figure 5.2. Geometry and parameters for a two arm Archimedean spiral. ................................... 98
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Figure 5.3. Geometry of the two (left) and four-arm (right) spiral antennas both with and without
the 5.08cm cavity. The spirals without the cavity have a ring load around the arms. ......... 103
Figure 5.4. Detail of feeding structure for the two (a) and four-arm (b) spirals as well as the location
of the excitation sources for the spiral arms. ....................................................................... 104
Figure 5.5. (a) VSWR and (b) boresight axial ratio of the two- and four-arm spiral antennas with
and without the cavity in free space. The VSWR performances of the antennas are similar, but
the two-arm spiral has higher axial ratio. ............................................................................ 105
Figure 5.6. (a) Geometry of the vehicle side and location of antenna source and (b) possible
mounting strategies. View of reduced model with flush mounted spiral (c) and side views of
both the flush and offset mounted spirals with the reduced model (d). ............................... 106
Figure 5.7. VSWR for the four-arm spiral in either the offset or flush mounting position with
respect to the nominal impedance for the four-arm spiral, 130Ω. ....................................... 107
Figure 5.8. (a) Efficiency for the flush and offset mounting of the four-arm spiral over dry sand
ground and (b) the percentage of power dissipated in the loads and at the end of the spiral
arms rather than radiated ..................................................................................................... 108
Figure 5.9. (a) Axial ratio for the offset and flush mounting over dry sand for the four-arm spiral.
While there is more variation found in the flush mounting, the axial ratios for each position
are very similar. (b) Elevation cut of the axial ratio over dry sand for the offset and flush
mounting. ............................................................................................................................. 109
Figure 5.10. Far field elevation gain patterns for the offset and flush mounting over dry sand at
various frequencies for the four arm spiral. The gain of the offset mounting is greater than or
equal to the flush mounting. ................................................................................................ 110
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Figure 5.11. VWSR for the 2 and 4 arm spiral with respect to their nominal impedances over the
frequencies of interest. ......................................................................................................... 111
Figure 5.12. (a) Efficiency of the two and four-arm spiral antennas over a dry sand ground. (b) The
power dissipated in the loads. .............................................................................................. 112
Figure 5.13. Gain pattern of the two and four-arm spirals offset mounted over dry sand. ......... 112
Figure 5.14. Axial ratio for the two and four arm antennas offset mounted on the vehicle over dry
sand. ..................................................................................................................................... 113
Figure 5.15. Theta and phi polarizations of the far field gain patterns of the two and four arm spirals
in free space. The four arm spiral patterns are the same at boresight and corresponds to its
lower axial ratio. .................................................................................................................. 114
Figure 5.16. VSWR of the four-arm spiral offset mounted over the dry sand, asphalt, and wet soil.
............................................................................................................................................. 115
Figure 5.17. Efficiency of the offset mounted four arm spiral antenna source over the various
ground types. ....................................................................................................................... 116
Figure 5.18. Azimuth and elevation far field gain patterns of the offset four-arm spiral over dry
sand, asphalt, and wet soil at 1GHz. The far field patterns are constant over the different
grounds over the frequencies of interest. ............................................................................. 117
Figure 5.19. Elevation gain patterns of the co and cross polarization of the four-arm spiral offset
mounted over a dry sand ground. ........................................................................................ 117
Figure 5.20. Azimuthal gain patterns of the offset four-arm spiral over dry sand. The azimuthal
pattern was computed at 5 degrees from the ground. .......................................................... 118
Figure 5.21. Axial ratio of the offset mounted four arm spiral antenna over the different grounds.
............................................................................................................................................. 119
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Figure 5.22. Elevation (a) and azimuthal (b) cuts of the axial ratio of the four arm spiral over dry
sand, asphalt, and wet soil. The elevation cuts are along the boresight axis while the azimuthal
cuts are from 5 degrees above the ground. .......................................................................... 120
Figure 5.23. Depolarization of circularly polarized signal over both dry sand (a) and wet soil (b)
over a variety of frequencies. .............................................................................................. 121
Figure 5.24. Geometry of the vehicle side with the windows. Both windows are the same size.
............................................................................................................................................. 122
Figure 5.25. Near-field and far-field patterns of both geometries. Both have very good agreement
in the direction of propagation with differences in the backlobe. ....................................... 123
Figure 5.26. Model geometry of the spiral-helix antenna mounted on the PEC plate over the
ground. ................................................................................................................................. 124
Figure 5.27 Original dimensions of reduced vehicle model (a) and the antenna source mounted on
the PEC plate (b).................................................................................................................. 125
Figure 5.28 Elevation and azimuth cuts of the far field of both the vehicle side and plate geometries
over the dry sand ground. There is good agreement between the two with discrepancies in the
backlobes. ............................................................................................................................ 126
Figure 5.29. VSWR of the spiral helix antenna source mounted on the plate with respect to its
nominal impedance of 150 Ω. ............................................................................................. 127
Figure 5.30. (a) Efficiency of a single spiral helix mounted on a 0.83m x 0.83m PEC plate over
various grounds and (b) the percentage of power dissipated in the loads of the spiral-helix
antenna. A significant portion of the power below 1GHz is absorbed by the loads. .......... 128
Figure 5.31. Boresight gain of the mounted spiral-helix in free space. ...................................... 129
Figure 5.32. Axial ratio of the spiral helix antenna above dry sand, asphalt, and wet soil. ....... 129
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Figure 5.33. Geometry of the two element spiral helix array mounted on the PEC plate. ......... 130
Figure 5.34. Elevation cut of the gain of a single spiral-helix and two element spiral-helix antenna
array in free space. ............................................................................................................... 131
Figure 5.36. (a) Efficiency of the spiral helix and spiral helix array over both dry sand and wet soil
grounds. The efficiency of the array is slightly higher than with a single element. (b) Axial
ratio for the spiral-helix and spiral-helix array over both the dry sand and wet soil grounds.
The spiral helix array has greater degradation of the axial ratio on the lower end and is less
stable on the higher end. ...................................................................................................... 132
Figure 5.37 Elevation cuts of the RHCP and LHCP fields of the dual polarized, two element spiral
helix system. ........................................................................................................................ 133
Figure 5.38. Comparison of the spiral helix antenna as a single element, in a two element array,
and as a two element system with opposite polarizations over dry sand. ........................... 133
Figure A.1 (a) Dimensions of the rough Humvee model. The entire model is modeled as PEC. (b)
Magnitude of the electric field for different radii around the rough Humvee model with a top
mounted Hertzian source over a PEC ground for both MoM and FEM at 100MHz. ......... 146
Figure A.2 Reduced rough Humvee model geometries. (a) Full rough Humvee model solved with
either MoM or PO, (b) Partial Humvee with just sides modeled as PO, and (c) side of the
Humvee modeled as PO. ..................................................................................................... 147
Figure A.3 Near field 0.765m above the PEC ground for the alternative geometries. There is good
agreement between the full rough Humvee with MoM and the Humvee side with PO. ..... 148
1
Chapter 1
Introduction
Since the first radio antennas were mounted on automobiles in the early 20th
century, the number of applications for antenna systems on vehicles has skyrocketed. Since
1923, antennas have been mounted on cars to receive AM radio transmissions. Today,
automobiles have a plethora of antennas for a wide number of applications, from
communications purposes to sensors to aid in driving. For consumers, vehicle mounted
antennas are used to communicate with nearby cell towers to provide better cell service
and internet access for passengers in the vehicle. As well, antennas receive GPS signals to
provide the driver with directions and navigation.
Vehicle mounted antennas also provide a wide range of military applications.
Humvees require secure and reliable communications from a variety of distances, both
nearby and across the globe. As well, there has been a growing need for Electronic Warfare
and Countermeasures applications. These applications utilize the RF spectrum from HF
(3MHz-30MHz) to the UHF (300MHz-3GHz) and beyond.
1.1 Background
1.1.1. Improvised Explosive Devices
Improvised Explosive Devices (IEDs) are cheap, easy to manufacture bombs
utilizing lethal, incendiary materials to kill personnel and destroy vehicles. They are
typically built from readily available consumer goods and parts. This makes them very
cheap and simple to produce and deploy. Thus, they are a common weapon used by
2
insurgents, terrorists, and guerrillas since the Second World War. Today, they represent
the most effective threat to US and Coalition Forces in Iraq and Afghanistan.
Because IEDs are built from whatever materials are available, they are diverse in
both design and use. Typically, any explosive device that was not manufactured in an
industrial plant can be classified as an IED. IEDs fall into one of three categories: Victim
Operated Devices, Timed Devices, or Command Operated Devices. [1] Victim operated
devices are triggered by the victims of the devices, typically by use of pressure plates.
Timed devices are triggered by a set timer. Command operated devices are detonated by
an adversary, typically by removing a safety pin or sending an electromagnetic signal from
a distance.
Remote Controlled Improvised Explosive Devices (RCIEDs) fall under the
command operated devices and are triggered by a radio signal. The trigger is typically a
modified commercially available consumer good. Typical items used range from remote
key fobs, RC toys, cell phones, satellite phones, garage door openers, to HF/VHF/UHF
radio transceivers. This represents a wide range of frequencies from as low as 1MHz to
3GHz and beyond. [1] [2]
The trigger can be prevented from detonating by broadcasting a jamming signal at
a power level that overwhelms the intended signal. However, it is impractical, if not
impossible, to effectively jam that entire frequency range. While jamming all of those
signals would also prevent adversaries from utilizing communication systems, it also
3
prevents friendly communications. This can be at least partially mitigated by allowing
small ‘windows’ in the spectrum for friendly communications. [1]
1.1.2. Jamming
The effectiveness of a jamming system is measured by the jamming to signal ratio
(JSR). When the jamming signal power is sufficiently larger than the transmitter signal,
the trigger is blocked from detonating. The JSR is determined by
𝐽𝑆𝑅𝑑𝐵 = 𝐸𝑅𝑃𝐽 − 𝐸𝑅𝑃𝑆 − 𝐿𝐽 − 𝐿𝑆 + 𝐺𝑅𝐽 + 𝐺𝑅 (1)
Where
ERPJ – Effective radiated power of the jamming system in dB
ERPS – Effective radiated power of the signal in dB
LJ – Propagation loss between jamming system and RCIED in dB
LS - Propagation loss between signal and RCIED in dB
GRJ – Gain of receiver antenna in direction of jamming system in dB
GR – Gain of receiver antenna in direction of signal in dB
The effective radiated power of the signal, its propagation loss, and the gain of the
receiver in the direction of the signal source are unknown. However, because they are
typically built with off the shelf consumer electronics, some assumptions can be made for
worst case scenarios. Similarly, the location, and therefore propagation loss between the
jammer and RCIED and the gain are also unknown. [1]
RCIEDs are typically placed very low to the ground and may be placed in any
orientation. The receiving antenna will then be of an unknown polarization and orientation.
This alone can result in tens of dB of loss from polarization loss factor. The polarization
4
loss factor is the measure of how well aligned the incident jamming signal’s polarization
is to the receiving antenna. [3]
As well, because the location of the RCIED is unknown, the jammer should be
uniform in azimuth to provide a range of protection around the personnel or convoy. The
radius of the zone of protection can be estimated [1] by
𝑅 = 𝑅𝑆√
𝑃𝐽
𝑃𝑆 𝐾(𝐻𝑆𝐻𝐽)2
𝑛
(2)
Where:
PS – Effective power of adversary signal
HJ – Relative height of jamming antenna
HS – Relative height of adversary antenna
RJ – Distance between jamming antenna and receiver (km)
RS – Distance between adversary antenna and receiver (km)
K – Tuning accuracy constant (2 for FM, 3 for AM/CW in VHF)
n – Ground conductivity and terrain quality factor
The ground conductivity and terrain quality factor varies between 2 for a very level and
conductive ground and 5 for a very rough and lossy ground. [1]
There are a variety of properties by which one can judge the effectiveness of an
RCIED jamming system. Those factors fall into three broad categories of the
receiver/detector capabilities, the jamming systems capabilities, and the effects of the
vehicle and terrain on propagation. This work will be primarily focused on the third
category in seeking to better understand how mounting and ground types can affect the
propagation of electromagnetic waves near a vehicle.
5
1.1.3. Common Vehicle Mounted Antennas
Vertical whip antennas are widely used as they are simple, inexpensive, and easy
to mount to the vehicle. The antenna is typically 2m-5m long and may have a wide
bandwidth. Also, because they are frequently used for HF communications, they have a
very low radiation efficiency without a tuning circuit to match the whip to the desired
frequency. Another common vehicle mounted antenna is a loop. Typically, a half loop
is mounted with the vehicle body acting like a ground plane. Like the vertical whip, the
loop has a narrow impedance bandwidth and requires a tuning circuit to match to the
desired frequency. [4]
Figure 1.1. Humvee with several large antennas mounted on the rear of the vehicle. [5]
Both of these antennas are quite large and are typically mounted on either the roof
or rear bumper of the vehicle. The size of the antennas hinder the maneuverability of the
vehicle as well as having a large visual profile (Figure 1.1). This can disclose the purpose
of the vehicle and can make it a target for adversaries. Additionally, the roof and rear
6
bumper are attractive locations for mounting antennas. This can result in coupling between
the antennas that can interfere with the performance of the different systems.
To combat this, alternative mounting locations are explored, including both the side
of the vehicle as well as underneath the vehicle. For both mounting locations, the antennas
are electrically close to the ground. Therefore, it is vital to understand how the ground
affects the performance of the antenna’s patterns, impedance, efficiency, and axial ratio.
Once the effects of the different grounds are known, steps can be taken to mitigate potential
adverse effects and capitalize on any benefits. As well, it is important to understand how
the vehicle body itself affects the performance.
1.2 Requirements
The main objective of this research is to explore potential alternative mounting
locations for antenna systems, primarily the bottom and sides of the vehicle. In order for
there to be efficient jamming or communications, the overall efficiency of the source
should be as high as possible. For sources under the vehicle, this is especially important
and one of the most important aspects to overcome. As well, the effects of wheels on the
performance should be considered. For circularly polarized systems, it is vital to
understand how the presence of the vehicle and ground affect the polarization.
1.3 Thesis Outline
This thesis is organized as follows: Chapter 2 discusses the basis for computational
electromagnetics and modeling used throughout this thesis. Chapter 2 also discusses how
the vehicle models and various grounds are modeled to give accurate results and
7
computational efficiency. Chapter 3 considers three different idealized sources over ground
models both with and without a vehicle. These ideal sources are considered in multiple
locations on the vehicle models to explore the potential for alternative mounting locations.
Chapter 4 interrogates potential sources under the vehicle using a parallel plate source.
Chapter 5 investigates the potential for sectorial sources mounted on the sides of a vehicle.
The sources considered are spiral antennas for their wide band performance and circular
polarization to explore the effects of both mounting approaches and soils on antenna
performance and propagation. Finally, Chapter 6 summarizes the conclusions and potential
future work.
8
Chapter 2
Computational Electromagnetics: Solvers and Modeling
While performing experiments in the field will provide the most definitive results
for a given environment, performing such experiments is both expensive and time
consuming. Rather, numerical models of the problems permit inexpensive and quick results
for a wide variety of situations and environments. Understanding the strengths and
limitations of numerical modeling for electromagnetics comes from a comprehensive
knowledge of the basic principles on which it is based. This chapter will start with a
discussion of the basics behind how FEKO solves electromagnetic problems. Then, there
will be a discussion on the vehicle graphical models and how they can be altered to provide
fast, efficient, and accurate results.
2.1 Method of Moments
There exist a variety of computational methods for modeling antennas. Some of the
most common methods include the method of moments (MoM), the finite element method
(FEM), and the finite difference time domain (FDTD). The finite element method meshes
an entire volume. Each meshed volume can have its own electrical properties. Therefore,
the FEM excels at solving for more complex, inhomogenous geometries. However, it
requires meshing of the entirety of the solved space. Similarly, FDTD requires the entire
volume to be meshed. As the name would suggest, the FDTD solves in the time domain.
This allows for rapid solving over a wide range of frequencies for inhomogenous media.
However, for problems that may have a narrow Q, the FDTD will require a long
9
computation time. [6] Note that the FEM can be formulated in the time domain [7], and
that MoM can mesh volumes [8]; however, both formulations are not commonly utilized.
The method of moments is a numerical technique that often applies the weighted
residual method to reduce integral equations into a system of linear equations that can be
expressed as a matrix equation. The solution of this matrix equation can be then determined
through elimination, inversion, or iterative methods. What makes the method of moments
so ubiquitous is that it can be a1pplied to both closed problems, such as waveguide or
resonance cavities, as well as open problems, such as radiation and scattering. Because it
only meshes the surface, it is especially adept at modeling planar surface as well as PEC
or homogeneous dielectric objects. [6]
2.1.1 Theoretical Basics of the Method of Moments
Most electromagnetic problems can be described as an inhomogeneous equation.
The general procedure for the method of moments is to first derive the appropriate integral
equation. This integral equation is then converted into a matrix equation using the basis
and weighting functions. This process is also called discretization. Then, the matrix
elements are evaluated and used to solve the matrix equation. [6] This allows for
determining the parameters of interest. The following is a basic description of those steps.
𝐿𝑓 = 𝑔 (3)
In equation (3), L is a linear operator and is typically an integro-differential type
but could also be a differential or integral operator only. The unknown function, f, is the
charges or currents. The excitation is represented by g and is a known function.
10
The unknown function is expanded into a sum of N basis functions.
𝑓 = ∑𝛼𝑛𝑓𝑛
𝑁
𝑛=1
(4)
Where an are the unknown weighting coefficients and fn is a basis function associated with
an nth element. This is then substituted into the equation (3) so that
∑𝑎𝑛𝐿(𝑓𝑛)
𝑁
𝑛=1
≈ 𝑔 (5)
The residual is then formed as:
𝑅 = ∑𝑎𝑛𝐿(𝑓𝑛)
𝑁
𝑛=1
− 𝑔
(6)
The inner product of the basis function and the weighting function can be expressed as
equation (7) and is referred to as the moment, thus the name method of moments.
< 𝑓𝑚, 𝑓𝑛 >= ∫ 𝑓𝑚(𝑟)𝑓𝑚
°∫ 𝑓𝑛(𝑟′)𝑓𝑛
𝑑𝑟𝑑𝑟′ (7)
Then, the inner product of the residual and the weighting functions is found to get equation
(8).
∑𝑎𝑛 < 𝑓𝑚,
𝑁
𝑛=1
𝐿(𝑓𝑛) > = < 𝑓𝑚, 𝑔 > (8)
This results in an N x N matrix equation (9)
11
𝑍𝑎 = 𝑏 (9)
Where Zmn=<fm,L(fn)> and bm=<fm,g>
This can then be solved numerically for the unknown coefficients in a. For the
method of moments, each basis function interacts with all other basis functions through the
Green’s function. This results in a full system matrix, as opposed to other methods, such
as the finite element method which typically has very sparse system matrices. [6]
2.1.2 Green’s Function
Determining the integral equation from the partial differential equation is done with
an auxiliary equation known as the Green’s function. The Green’s function is a kernel
function determined from the linear boundary value problem and forms the relationship
between differential and integral formulations. It also reduces the inhomogeneous equation
to a homogenous one instead of requiring a series expansion to solve the inhomogeneous
boundary value problem. [6]
The Green’s function finds the effects of each point and sums up all of the effects.
G(𝑟, 𝑟’) is the field at some observation point r that is due to a point source at point r’. The
field at 𝑟 can then be determined by integrating the source (g(𝑟’)) and the Green’s function
over the region of the source. Green’s function can be thought of as the potential at some
point 𝑟 due to a point charge at a point 𝑟’. The Green’s function represents the response of
a linear system to an impulse applied at the point 𝑟=𝑟’.
12
There are four main properties of the Green’s function. The first is that Green’s
function satisfies the equation LG=0 except at 𝑟’, which is the source point. The Green’s
function is also symmetric so that G(𝑟, 𝑟’)=G(𝑟’, 𝑟). This is also known as the principle of
reciprocity and says that the observation point and source point can be interchanged
without affecting the Green’s function. And finally, the derivative will have a discontinuity
at 𝑟’. The Green’s function can be expressed as
lim𝜌→0
∮𝜕𝐺
𝜕𝑛𝑑𝑠 = 1
(10)
Where n is the normal to a sphere with a radius ρ so that ρ2=|𝑟-𝑟’|.
2.1.3 Source Models
Methods of moments is most efficient for modeling metallic wires and surfaces in
free space. For modeling a source on a wire, there are two common methods: the delta gap
and the magnetic frill. The delta gap models a small gap between two wire segments with
a voltage difference excited between the gap and converted into a current source by
applying the Norton equivalent. This current will be constant over the length of the gap
and zero everywhere else. While this method is easy to implement, its accuracy is
dependent on the size of the gap; larger gaps decrease the accuracy. [6]
The second method is the magnetic frill and is based off a coaxial line going through
a ground plane. Rather than a gap between two segments, an annulus of magnetic current
density is impressed in the area between the inner and outer conductor of the coax. The
13
magnetic current will induce an electric field on the surface of the wire resulting in a fully
populared excitation vector in (g). [6]
2.1.4 Basis and Weighting Functions
The accuracy of the method of moments is affected by the choice of basis and
weighting functions. Simpler problems may be solved using linear, triangular, or
sinusoidal basis functions. But, such simple basis functions may be impractical for more
complex problems. To solve more complex problems, a tradeoff between the accuracy of
the solution and the computation time should be taken into account.
The simplest method is only testing the function at discrete points. This is known
as point matching and effectively makes the testing functions a series of delta functions.
[9] Therefore, only the source function needs to be integrated. While this is a simple
approach to implement, the accuracy is affected because the boundary conditions are only
enforced at specific points [6]. A more complicated but more accurate method is the
Galerkin’s method. The Galerkin method is where the weighting functions are the same as
the basis functions. This will enforce the boundary conditions over the entire domain. [6]
When choosing the basis function, it must satisfy the boundary conditions.
However, when solving an integral equation, the boundary conditions are built into the
equation and are not necessary for the basis functions. If the operator involves
differentiation, then the basis functions must be continuous. Additionally, the basis
functions determine the complexity of the integrals for the matrix elements. If sufficiently
simple basis functions are used, an analytical solution can be found. Finally, the
14
smoothness of the basis functions will determine the convergence and accuracy of the
numerical solution. [9]
FEKO, a commercially available tool that is frequently used in this thesis, uses
piecewise linear triangular basis functions within the Galerkin method for either wires or
surfaces. [10] It meshes surfaces as two dimensional triangles and uses the Rao-Wilton-
Glisson (RWG) [11] vector basis function (Figure 2.1) which is described as
𝑓𝑛(𝑟) =
𝑙𝑛2𝐴𝑛
+ (𝑟 − 𝑎3,𝑛) 𝑓𝑜𝑟 𝑟 ∈ 𝑇𝑛
+
−𝑙𝑛2𝐴𝑛−
(𝑟 − 𝑎3,𝑛) 𝑓𝑜𝑟 𝑟 ∈ 𝑇𝑛−
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
(11)
Where:
Tn± designates two triangles sharing an edge
ln length of the shared edge
a3,n± two vertices of the triangles opposite to the shared edge
An± areas of the two triangles
Figure 2.1. Two triangular surfaces (Tn+ and Tn-) with areas An that share an edge ln
15
2.1.5 FEKO Implementation of MoM
The source is considered to be impressed electric and magnetic fields. These fields
could be the result of a uniform plane wave or from a voltage source on a wire segment.
These imposed fields create electric currents (IMM) along the wire segments and surface
current densities (JMM) on the surface. These currents will then radiate in an isotropic,
homogenous, and linear medium which will produce scattered fields designated Es and Hs.
The total electromagnetic fields will be the superposition of both the impressed and
scattered fields.
𝐸𝑠 = 𝐽𝐸𝐽𝑀𝑀 + 𝐼
𝐸𝐼𝑀𝑀 (12)
𝐻𝑠 = 𝐽𝐻𝐽𝑀𝑀 + 𝐼
𝐻𝐼𝑀𝑀 (13)
Where the integro-differential operators are defined as
𝐽𝐸𝐽𝑀𝑀 = −
𝑗
4𝜋휀𝜔∇∬(∇𝐴
′ ∙ 𝐽(𝑟′)) ∙ 𝐺(𝑟, 𝑟 ′)𝑑𝐴′ − 𝑗𝜔𝜇
4𝜋∬𝐽(𝑟′) ∙ 𝐺(𝑟, 𝑟 ′)𝑑𝐴′
(14)
𝐼𝐸𝐼𝑀𝑀 = −
𝑗
4𝜋휀𝜔∇∫
𝜕𝐼(𝑟′)
𝑑𝑙′∙ 𝐺(𝑟, 𝑟 ′)𝑑𝑙′ − 𝑗𝜔
𝜇
4𝜋∫ 𝐼(𝑟′) ∙ 𝑙′ ∙ 𝐺(𝑟, 𝑟 ′)𝑑𝑙′
(15)
𝐽𝐻𝐽𝑀𝑀 =
1
4𝜋∇ ×∬𝐽(𝑟′) ∙ 𝐺(𝑟, 𝑟 ′)𝑑𝐴′
(16)
𝐼𝐻𝐼𝑀𝑀 =
1
4𝜋∇ × ∫ 𝐼(𝑟′) ∙ 𝑙′ ∙ 𝐺(𝑟, 𝑟 ′)𝑑𝑙′
(17)
𝐺(𝑟, 𝑟 ′) =𝑒−𝑗𝑘|𝑟−𝑟′|
|𝑟 − 𝑟′|
(18)
16
The direction of the current flow along the wire is represented by the unit vector 𝑙
and is the wavenumber k=2π/λ. As previously mentioned, the unknown currents are
described as a series of the products of the unknown coefficients and the basis functions.
The RWG vector basis function, fn, is applied to triangular surfaces (Figure 2.1) and
triangular basis functions, gn, for the wire segments. [12]
𝐽𝑀𝑀 = ∑ 𝛼𝑛𝑓𝑛
𝑁𝐽𝑀𝑀
𝑛=1
(19)
𝐼𝑀𝑀 = ∑ 𝛽𝑛𝑔𝑛
𝑁𝐼𝑀𝑀
𝑛=1
(20)
From here, constraining the boundary conditions allows for the determination of
the unknown coefficients. For example, the tangential electric field on a PEC surface or
along a wire segment will be enforced to be zero.
𝑡𝑎𝑛 = 0 (21)
The electric field integral equation can then be found by combining equations (12),
(14), and (15) and then including equations (19) and (20) to describe the currents as a series
of unknowns and basis functions will result in equation (22). FEKO then applies the
Galerkin method to convert the integral equation into a system of linear equations. This
system of equations then forms a matrix equation that can then be solved for numerically
to determine the unknown coefficients. These coefficients can be used to determine a
variety of parameters about the problem.
17
−𝑖 𝑡𝑎𝑛 = ∑ 𝛼𝑛
𝑁𝐽𝑀𝑀
𝑛=1
∙ (𝐽𝐸𝑓𝑛)𝑡𝑎𝑛
+ ∑ 𝛽𝑛 ∙ (𝐼𝐸𝑛)𝑡𝑎𝑛
𝑁𝐼𝑀𝑀
𝑛=1
(22)
The method of moments produces a square matrix of rank N and will require
memory on the order of O(N2) and a computation time on the order of O(N2) to O(N3).
[12] To properly mesh the geometry, the mesh should be approximately λ/10 in length for
the basic method of moments. If using higher order basis functions, the length of the mesh
element can be increased. Increase in frequency will increase the number of mesh elements
and basis functions. For wire segments, the number of basis functions will be proportional
to frequency. For a 2D triangular segment with N basis functions will be proportional to f2.
If the geometry contains both surfaces and wire segments, the required memory will be
proportional to f4 with a computation time proportional to f4 to f6. This means that increased
frequency can quickly surpass computational resources. [10] This is especially important
to consider as the vehicle models considered become very large electrically at higher
frequencies.
2.2 Physical Optics
Because of the large increase in memory and computation time requirements, the
method of moments loses its utility as the frequency increases. This can be mitigated by
employing high frequency methods for arriving at a solution. These approaches, such as
Geometric Optics, Uniform Theory of Diffraction, and Physical Optics, are applicable
when the wavelength is small compared to the scattering object. This makes these
approaches especially adept at solving for electrically large structures where other codes
18
like the method of moments or finite element analysis have poor convergence and require
immense computational resources.
The basis for these codes is the idea of ray theory. Ray theory is applicable when
the wavelength of the wave is much smaller than the scatterer. Energy flows in straight line
paths known as rays. The rays are typically required to be in an isotropic and lossless media,
but do not necessarily need to be in a homogeneous media. These rays will have a surface
normal to their direction known as the eikonal. For a plane wave, the eikonal is a plane.
For a point source, the eikonal will be a spherical shell. Knowledge of either the rays or
the eikonal is necessary. [13]
Because the energy only flows along these rays, rays that are incident on a
scattering surface will necessarily create illuminated and shadowed regions (Figure 2.2).
Physical Optics considers each point on the illumination region of the scatterer as if it was
scattering off an infinite plane tangent to that point. In the shadowed regions, the incident
field is considered to be zero. Illuminated regions will have a surface current proportional
to the incident field, while shadowed regions will not have a surface current. [13]
𝐽𝑃𝑂 = × 𝑡𝑎𝑛 𝑖𝑙𝑙𝑢𝑚𝑖𝑛𝑎𝑡𝑒𝑑0 𝑠ℎ𝑎𝑑𝑜𝑤𝑒𝑑
(23)
Image theory dictates that the tangential components of the magnetic field at a PEC
surface are twice that of those from the same source in when the PEC surface is removed.
Therefore, the surface currents in the illuminated region will be
19
𝐽𝑃𝑂 = 2( × 𝑖) (24)
Figure 2.2. Incident field on a conductive scatterer producing both an illuminated and
shadowed region. The surface current is nonzero in the illuminated region.
The scattered electric field is simply (25) and the scattered magnetic field can be
found by taking the curl of (25).
𝑠 = −𝑗𝑤𝜇∬𝐽𝑃𝑂 𝐺(𝑟, 𝑟′)𝑑𝑠′
(25)
𝑠 = ∇ ×∬𝐽𝑃𝑂 𝐺(𝑟, 𝑟′)𝑑𝑠′
(26)
This can be rewritten as
𝑠 =∬(∇ 𝐺(𝑟, 𝑟 ′) × 𝐽𝑃𝑂)𝑑𝑠′ (27)
When the Green’s function is considered in the far field, the expression for ∇𝐺(𝑟, 𝑟 ′)
becomes
𝐺(𝑟, 𝑟 ′) = −𝑟 1 + 𝑗𝛽𝑟
4𝜋𝑟2𝑒−𝑗𝛽𝑟𝑒𝑗𝛽𝑟 ∙𝑟′
(28)
20
This results in a scattered magnetic field of
𝑠 = 𝑒−𝑗𝛽𝑟∬(𝐽𝑃𝑂 × 𝑟 )1 + 𝑗𝛽𝑟
4𝜋𝑟2𝑒𝑗𝛽𝑟 ∙𝑟′ 𝑑𝑠′
(29)
A ray incident on a surface will produce a current that will generate a scattered field
in all directions. This provides an accurate solution to the approximate scattered field as
well as providing a nonzero approximation for the scattered radiation pattern. The accuracy
of physical optics is dependent on how well the physical optics surface current
approximates the actually physical currents. A close approximation can be used for the
illuminated regions that are sufficiently far from the edge of the scatterer. As will be shown,
a hybridization of the method of moments with physical optics will still generate accurate
results with some discrepancies far from boresight. [13]
2.3 Hybrid MoM and PO
Because physical optics, like the method of moments, is a current based method,
the current can be continuous across the divide between a method of moments and physical
optics region. This means that there is an overlap of the basis functions where the method
of moments and physical optics regions meet. Therefore, there will not be a discontinuity
in the first order of the boundary conditions.
Physical optics decreases the number of basis functions required for surfaces. It
does not work for thin wires and they should instead be modeled with the method of
moments. Regions that are designated to be solved using physical optics should be chosen
based off illuminated and shadowed regions of the model, with illuminated regions with
21
PO and shadowed regions with MoM. Like the method of moments, the surface current
densities are expressed as a series of basis functions and unknown coefficients.
𝐽𝑃𝑂 = ∑ 𝛾𝑛𝑓𝑛
𝑁𝐽𝑀𝑀+𝑁𝐽
𝑃𝑂
𝑛=1+𝑁𝐽𝑀𝑀
(30)
However, unlike the method of moments, the unknown coefficients in physical
optics are not determined by a system of equations but through the applications of the
principles of physical optics. The surface current density in the physical optics region will
be described as
𝐽𝑃𝑂(𝑟) = 2𝛿𝑖 ∙ × 𝑖(𝑟) + ∑ 2𝛼𝑛𝛿𝐽,𝑛 ∙ ×
𝑁𝐽𝑀𝑀
𝑛=1
𝐽𝐻𝑓𝑛 + ∑ 2𝛽𝑛𝛿𝐼,𝑛 ∙ ×
𝑁𝐼𝑀𝑀
𝑛=1
𝐼𝐻𝑔𝑛
(31)
The first term is the surface current density due to the incident impressed magnetic
field and where δ is constant due to shadowing and is a unit vector normal to surface.
The rest of the equation is due to the presence of radiation and scattering sources in the
method of moments region. The wire and surface basis functions for the method of
moments will be acted upon by the operators as described in the method of moments
section. This will yield the magnetic fields. The normal unit vector and the shadowing
coefficient will determine the induced current from the MoM region. The unknown
physical optics coefficients can then be determined by introducing two unit vectors normal
to the edge between the two triangles located at the midpoint of the edge. Multiplying both
22
sides of equation (30) with this unit vector and adding the two equations will produce an
equation for the unknown. [12]
𝛾𝑘 =1
2(𝑘+ + 𝑘
−) ∙ 𝐽𝑃𝑂(𝑟𝑘) (32)
𝑓𝑜𝑟 𝑘 = 𝑁𝐽𝑀𝑀 + 1 +⋯𝑁𝐽
𝑀𝑀 + 𝑁𝐽𝑃𝑂
Then by combining equation (31) with equation (32), one can describe the unknown
coefficient as
𝛾𝑘 = (𝑘+ + 𝑘
−) ∙ 𝛿𝑖 ∙ × 𝑖(𝑟) + ∑ 𝛼𝑛 ∙ (𝑘+ + 𝑘
−) ∙ 𝛿𝐽,𝑛 ∙ ×
𝑁𝐽𝑀𝑀
𝑛=1
𝐽𝐻𝑓𝑛
+ ∑ 𝛽𝑛 ∙ (𝑘+ + 𝑘
−) ∙ 𝛿𝐼,𝑛 ∙ ×
𝑁𝐼𝑀𝑀
𝑛=1
𝐼𝐻𝑔𝑛
(33)
The coupling between the method of moments and physical optics regions will arise
in two different ways. The currents in the method of moments region will illuminate
portions of the physical optics region to induce currents. Additionally, the currents in the
physical optics region will radiate and the fields will need to be taken into account in
method of moments region when constraining the boundary conditions. By taking the
electric field integral equation and the boundary conditions for both regions, the resulting
equation is (34).
From here, the weighting functions can be applied just like in the original method
of moments. The unknown coefficients for the method of moments regions, α and β, can
23
then be determined by solving the system of equations. Once those have been determined,
the unknown coefficient for the physical optics region, γ, can be found using equation (33).
[12]
−𝑖 𝑡𝑎𝑛 − ∑ (𝑘+ + 𝑘
−) ∙ 𝛿𝑖 ∙ × 𝑖(𝑟)
𝑁𝐽𝑀𝑀+𝑁𝐽
𝑃𝑂
𝑘=1+𝑁𝐽𝑀𝑀
∙ (𝐽𝐸𝑓𝑘)𝑡𝑎𝑛
= ∑ 𝛼𝑛
𝑁𝐽𝑀𝑀
𝑛=1
∙ [(𝐽𝐸𝑓𝑛)𝑡𝑎𝑛
+ ∑ (𝑘+ + 𝑘
−) ∙ 𝛿𝐽,𝑘 ∙ ×
𝑁𝐽𝑀𝑀+𝑁𝐽
𝑃𝑂
𝑘=1+𝑁𝐽𝑀𝑀
𝐽𝐻𝑓𝑘 ∙ (𝐿𝐽
𝐸𝑓𝑘)]
+ ∑ 𝛽𝑛 ∙ [(𝐼𝐸𝑘)𝑡𝑎𝑛 ∑ (𝑘
+ + 𝑘−) ∙ 𝛿𝐼,𝑘 ∙ ×
𝑁𝐽𝑀𝑀+𝑁𝐽
𝑃𝑂
𝑘=1+𝑁𝐽𝑀𝑀
𝐼𝐻𝑓𝑘 ∙ (𝐽
𝐸𝑘)𝑡𝑎𝑛]
𝑁𝐼𝑀𝑀
𝑛=1
(34)
2.4 Sources over Grounds
When electromagnetic sources are in close proximity to the ground, a variety of
interactions with the ground will modify the source’s performance. The ground can be
modeled as an interface between two dielectric media. Reflections off the ground result in
modifications of the radiation pattern, while power transmitted into the ground will be
dissipated and lost for communications and jamming purposes.
24
2.4.1 Boundary Conditions
The physical boundary between two different materials will cause discontinuities
in the electric and magnetic fields at that boundary. One can consider a point at the
boundary between two different materials and a normal unit vector from the first region
into the second. The tangential fields can be computed from the cross product from the
incident fields and the normal vector. As long as either surface electric or magnetic currents
exist on the boundary, the fields can be discontinuous at the interface between the two
materials. [14] These surface currents are determined by
𝐽𝑠 = × (2 − 1) (35)
𝑠 = − × (2 − 1) (36)
The different dielectric and magnetic properties of the materials and the subsequent
induced electric and magnetic charges alter the normal components of the fields.
𝜌𝑠 = ∙ (𝜖22 − 𝜖11) (37)
𝜏𝑠 = ∙ (𝜇22 − 𝜇11) (38)
Where ρs is the electric surface charge density and τs is the magnetic surface charge density.
For lossless dielectic materials, the currents will be zero and the equations reduce to
0 = × (2 − 1) (39)
0 = × (2 − 1) (40)
25
This means that the tangential fields will be continuous across the boundary of the
materials.
Two special cases for the boundary conditions are for a perfect electric conductor
(PEC) and a perfect magnetic conductor (PMC). Both of these special cases can be used to
simplify calculations and analysis. For PEC materials, the fields inside the material go to
zero and only electric currents are induced on it. [15] As well, the tangential electric field
will go to zero.
0 = × 2 (41)
𝐽𝑠 = × 2 (42)
For PMC materials, the interior fields also go to zero, but only magnetic currents are
allowed on the surface. Similarly, the tangential magnetic field goes to zero.
0 = × 2 (43)
𝑠 = × 2 (44)
While many metals are close approximations to the PEC material at radio
frequencies, there is no real world analogue to the PMC. Over narrow frequency bands,
PMC can be approximated with ferrite [16] or metamaterials [17] [18]. The concept of
PMC materials is incredibly useful for calculations and simplifying problems. Both of these
materials can be used with the concept of images to greatly simplify problems. [15]
26
2.4.2 Image Theory
Image theory considers a source at a height above an infinite PEC ground plane.
This ground plane can be removed with an additional source added to produce the
“reflected” wave. Imagine a source over a ground plane at a height d and the source is
producing a plane wave directed towards the ground plane. When the wave reaches the
ground plane at z=0, it will be reflected off the ground and travel in the positive Z direction.
This will result in a standing wave between the source and the ground plane.
Image theory says that this ground plane can be removed and a source with a current
in the opposite direction can be placed at a height of –d (Figure 2.3). The fields above the
now removed ground plane (z>0) can now be found by the superposition of the two
sources. This method will only solve for fields above the ground plane as there will be no
fields below the PEC ground.
For a PEC surface, the tangential filed components will be forced to go to zero to
satisfy the boundary conditions. The imaginary source that is added must satisfy these
conditions. For a vertically polarized electric source, the image source will have the same
polarity. For a horizontally polarized source, the imaged source will have the opposite
polarity of the original source. In the case of vertical or horizontal magnetic sources, their
polarity is the opposite as the electric case. The vertical source will have an image with the
opposite polarity while the horizontal magnetic source image will have the same polarity.
27
Figure 2.3. Ideal vertical electric sources (a), horizontal electric sources (b), vertical
magnetic sources (c), and horizontal magnetic sources (d) and their images at a height h
over an infinite PEC ground plane.
2.4.3 Fresnel Reflection Coefficients
When a uniform plane wave is incident on an interface between two dielectric
media (Figure 2.4), a percentage of the power will be reflected off while some will be
transmitted into the new media. How much is reflected and transmitted is based off the
electrical properties of the two media as well as the polarization of the incident wave. There
are two canonical cases for an incident uniform plane wave: the electric filed oriented in
the xz plane (parallel polarization) and the electric field oriented normal to the xz plane
(perpendicular polarization). [19] Any given uniform plane wave can be constructed as a
linear combination of these two canonical cases.
28
Figure 2.4. Incident, transmitted, and reflected electric fields at a dielectric interface
between two media.
Parallel Polarization
The reflection and transmission coefficients for the parallel polarization are:
𝛤 =𝜂2 cos(𝜃𝑡) − 𝜂1 cos(𝜃𝑖)
𝜂2 cos(𝜃𝑡) + 𝜂1 cos(𝜃𝑖)
(45)
𝑇 =2𝜂2 cos(𝜃𝑖)
𝜂2 cos(𝜃𝑡) + 𝜂1 cos(𝜃𝑖)
(46)
For the parallel polarization, there exists an angle θB, known as the Brewster angle,
where the reflection coefficient goes to zero. When the angle of incidence is equal to the
Brewster angle, the numerator for the reflection coefficient for parallel polarization goes
to zero making all reflected waves perpendicularly polarized. This makes for a straight
forward method of determining the Brewster angle for a given interface of two media
neglecting permeability. [19]
29
sin(𝜃𝐵) =1
√1 +𝜖1𝜖2⁄
(47)
Perpendicular Polarization
The reflection and transmission coefficients for the perpendicular polarization are:
𝛤 =𝜂2 cos(𝜃𝑖) − 𝜂1 cos(𝜃𝑡)
𝜂2 cos(𝜃𝑖) + 𝜂1 cos(𝜃𝑡)
(48)
𝑇 =2𝜂2 cos(𝜃𝑖)
𝜂2 cos(𝜃𝑖) + 𝜂1 cos(𝜃𝑡)
(49)
Unlike the parallel polarization case, there is no Brewster angle for the
perpendicular polarization. This can be seen as there is no possible solution for the
numerator of the reflection coefficient that can equal zero. [19]
2.4.4 Computational Ground Models in FEKO
FEKO offers the choice of infinite ground planes. These can be modeled as a perfect
electric conductor or as a dielectric medium. For dielectric mediums, they model a
homogenous half space using either the reflection coefficient approximation or with the
exact Sommerfeld integrals. Ensuring that the ground is modeled properly and accurately
is crucial for our studies. One part of this is using parameters that correlate to real world
values. Three ground conditions are considered: dry sand, asphalt, and wet soil so to
provide a wide range of soil characteristics. The dielectric properties of these ground types
can be found in Table 2.1. [20]
30
Table 2.1. Dielectric properties of ground types used in this thesis
εr σ [S/m]
Wet Soil 35 0.700
Asphalt 3.4 0.040
Dry Sand 2.5 0.002
Reflective Ground Model
This sets up an infinite ground plate in the z=0 plane with the defined dielectric
properties. For the reflective approximation, each reflected component of the fields are
added to the fields. While this method is faster than using the exact Sommerfeld integrals,
it is less accurate. Structures must be more than a tenth of a wavelength away from the
gorund to guarantee good results. [21]
Sommerfeld Integrals
Like the reflective ground model, the properties of the ground are based off the
predefined dielectric properties. The Sommerfeld integral [22] uses an approximate
Green’s function [21] to determine the exact boundary conditions. Unlike the reflective
ground model, structures can be arbitrarily close and even embedded within the ground.
2.5 Humvee Model
The vehicle model that is used plays a large role in the required computational
resources for this study. Electrically larger and more complex models will require more
memory and longer computation times. Ostensibly, the more complex model will have
more accurate results baring no convergence issues. It is imperative that the vehicle model
31
is simple enough to be run with low memory and time requirements while maintaining a
sufficient level of detail to provide accuracy.
2.5.1 Complex Humvee Model
The complex Humvee model dimensions can be seen in Figure 2.5. The body of
the vehicle is PEC with rubber wheels with a relative permittivity of 3 and conductivity of
10-15S/m. The model contains a variety of features such as doors, mirrors, and front grill.
All of these features require fine meshing to properly model and thus increase the required
memory and computation time. The inclusion of the dielectric wheels even further
increases the computational cost.
Figure 2.5. Dimensions of the complex Humvee model. The vehicle body is modeled as
PEC with rubber wheels with a εr of 3 and conductivity of 10-15S/m.
2.5.2 Basic Humvee Model
The basic Humvee model is an alternative model to the complex Humvee model.
Its geometry, seen in Figure 2.6, is based on other common Humvee geometries, but has a
32
reduced number of fine features when compared to the complex Humvee model. For
example, this model does not have the side mirrors or front grill of the complex model. The
exclusion of these fine features significantly reduces the amount of fine mesh elements,
and therefore the computational cost associated with simulating the model.
Figure 2.6. Dimensions the basic Humvee model. The vehicle body is modeled as PEC
with rubber wheels with a εr of 3 and conductivity of 10-15S/m.
2.5.3 Rough Humvee Model
The rough Humvee model is an even further reduction of the complex Humvee
model. The finer details such as the front grill, wheels, and mirrors are removed. The rough
model is comprised of two rectangular PEC blocks to represent the main body and cab of
the vehicle (Figure 2.7). Like the complex model, the rough model is 4.64m long and
2.31m wide. The body is 0.35m above the ground with the top of the body 1.18m above
the ground. The top of the cabin is 1.73m above the ground and is slightly forward of the
33
body’s center. This model is primarily for lower frequency calculations where the finer,
electrically small details are less critical for accurate determinations of the near and far
fields. For this reason, the rough Humvee model is used for frequencies between 3MHz
and 300MHz.
Figure 2.7. Dimensions of the rough Humvee model. The entire model is modeled as PEC.
2.5.4 Approximations with PEC Surfaces
Despite simplifications of the Humvee model geometry, this is not good enough for
efficient and accurate calculations at higher frequencies. At 2GHz, even a very coarse mesh
of λ/5 requires 553 GBytes of memory and a more reasonable coarse mesh of λ/8 requires
over 1 TB. This by far exceeds reasonable computational resources and an alternative will
be needed for higher frequencies.
34
The solution to this takes a two pronged approach. First, the Humvee model is even
further reduced. In the case of side mounted antennas (Chapter 5), only the side of the
rough Humvee model is modeled (Figure 2.8). In the case of bottom mounted antennas
(Chapter 4), only the bottom of the Humvee is modeled. Next, the Humvee side is modeled
using physical optics to use a hybrid MoM/PO code available in FEKO.
Figure 2.8. View of rough Humvee model with side mounted antenna modeled only with
MoM. The reduced rough Humvee is modeled using either MoM or PO.
To verify that this is a viable method for obtaining efficient and accurate results, a
comparison between the rough Humvee model modeled entirely with method of moments,
the side of the rough Humvee model modeled with method of moments, and the side of the
rough Humvee model modeled using physical optics was performed. All the models were
over a PEC ground and operated at 400MHz. Both the near field electric field and far field
patterns were compared.
As can be seen in Figure 2.9, all three models have very good agreement for the
near field electric field in the azimuth. The main lobes of each models also have very good
35
agreement. The only location with poor agreement is the back lobes for the reduced rough
Humvee model for both method of moments and physical optics. This results in a loss of
detail to the backlobes. However, for higher frequency studies, the backlobes are less of a
concern and such models are acceptable for these studies. The gains in required memory
and computation time can be seen in Table 2.2.
Table 2.2. Percent reduction in memory and CPU time at 400MHz over PEC ground
Models Memory CPU Time
Humvee (MoM) 100% 100%
Side (MoM) 1.99% 2.27%
Side (PO) 0.19% 0.86%
Figure 2.9. Comparison of the three models. (a) Comparison of azimuthal near field at 5m
and (b) far field elevation patterns in the yz plane so that 90 degress corresponds to the
boresight direction.
Modeling only the side of the Humvee model already greatly reduces the required
memory and computation time. Reducing the model in conjunction with the physical optics
36
modeling significantly reduces the computational cost. Both the near field and far field
performance have very good agreement between the different methods. This allows for
efficient and accurate calculations at higher frequencies and will be primarily used for
studies between 300MHz and 3GHz.
2.6 Summary and Conclusions
While prototypes and measurements can provide the most accurate results for field
performance, such methods are time intensive and expensive. Computational simulations
provide relatively rapid results at a significantly lower cost. FEKO is, in its core, a method
of moments based code that allows for modeling of a vehicle body over a variety of real
ground models. By reducing the complexity of the vehicle model, the computational
resources required for solving these models can be greatly reduced. The rough Humvee
model allows for faster, efficient, and accurate measurements for antennas at HF and VHF
frequencies. For even higher frequency bands, further reduction techniques should
probably be employed. By reducing the models to shaped PEC surfaces modeled using a
hybrid method of moments and physical optics code, the computational cost can be greatly
reduced. This is especially important when studying frequencies in the UHF band.
37
Chapter 3
Characterization of Fields from Ideal Sources
Before studying practical antenna sources, it is useful to understand the general
effects of the grounds and vehicle model on propagation. To do this, ideal sources are used
to allow the focus to be placed on the waves and to remove effects from specific antenna
designs. First, we will consider these ideal sources over a variety of lossy grounds to
characterize the propagation over those grounds. Then, the vehicle model is included with
the grounds to determine its effects on the performance. The primary parameter that will
be considered is the efficiency of the source above the lossy ground types. Unless otherwise
mentioned, the efficiency in this thesis is defined as the percentage of power that is radiated
in the far field compared to the known amount of power radiated by the source. For lossy
grounds, a percentage of power will be dissipated by the ground. Other important
parameters included the wobble of the wave, attenuation, and the depolarization of
circularly polarized sources.
3.1 Ideal Sources Ideal sources are based off either the Hertzian electric or magnetic dipole. These
sources are infinitesimally small and thin with ideal dipole radiation patterns and have a
directivity of 1.76dB. [23] By using ideal sources, the focus can be placed on the radiated
fields and how they are scattered and absorbed off the ground rather than on the
performance of a specific antenna.
38
3.1.1 Vertical Electric Dipole
The vertical electric dipole (VED) is a vertically oriented Hertzian electric dipole
antenna. This idealized source is an infinitesimally small and thin dipole with a length
much less than a wavelength and with a constant current magnitude and phase over that
length. Practical realizations of such a source can be difficult due to any physical
representation of the excitation method. However, this current distribution on a very short
dipole can be approximated by using capacitive loading hats at each end. [23] The
understanding of how this ideal dipole operates is the basis for the understanding of larger
and more complicated antennas. Larger antennas can be decomposed into a series of these
ideal antennas. [13] Therefore, understanding the performance of this ideal antenna can
provide insight on the performance of other antennas. FEKO permits choosing the ideal
source’s location, orientation, magnitude, and phase.
3.1.2 Horizontal Magnetic Dipole
The horizontal magnetic dipole is modeled as either a loop of electric current or as
a line of magnetic current. The loop of electric current will perform like an electrically
small loop antenna. When practically realized, a small loop should have constant current
with no phase change along its circumference. Both the electric current loop and magnetic
line current produce the same near fields and far fields. For this study, the electric current
loop model is used. [23]
3.1.3 Crossed Electric Dipoles
The crossed electric dipoles consist of two Hertzian electric dipoles fed 90 degree
out of phase with the other and oriented orthogonal to each other to produce right hand
circular polarization. This configuration creates a circularly polarized ideal source. This
39
source allows for studies on how different polarizations are influenced by scattering from
both the vehicle body as well as scattering and attenuation from lossy grounds. [13]
3.2 Ideal Sources over Grounds without Vehicle Model The following studies are performed by placing the ideal sources above the three
different flat ground models for dry sand, asphalt, and wet soil. The dielectric parameters
of the grounds can be found in Chapter 2, Table 1. Two different heights are considered
for this study, 0.3m and 1.78m. These heights correspond to either 5cm above or below the
vehicle model. These positions will be used for the ideal sources mounted on the vehicle
body.
The efficiency of the sources is defined as the percent power radiated above the
ground. This is determined by creating hemispherical shells that encapsulate the source.
The source then radiates with a known power. The power radiated through the shell is
considered the percent power radiated compared to the known amount at the source. Both
near field and far field shells are used to determine the amount of power dissipated into the
ground at different distances from the source.
3.2.1 Efficiency of VED
For the first case, the vertical electric dipole is placed at either 0.3m or 1.78m above
one of the following, dry sand, asphalt, or wet soil. The source radiates between 3MHz and
300MHz. This study is primarily concerned with the near field performances of the sources
and will compute the efficiencies from 5m and 25m radius hemispherical shells. The
efficiencies for both heights and distances can be found in Figure 3.1.
40
For a height of 0.3m, a greater efficiency than 15% is not achieved until 75MHz.
For both heights, the efficiency significantly decreases for very low frequencies. This is
due in large part to coupling of the source with the ground. This results in having a large
amount of power dissipated in the grounds. For lower frequencies, as the electrical height
of the source is increased, the efficiency will increase.
Figure 3.1. Efficiencies for the VED at both (a.) 0.3m and (b.) 1.78m above various grounds
for different radii.
It is also apparent that the 1.78m location has greater efficiency than for the 0.3m
height for the same reasons. For the 1.78m height, the efficiency will level off as the
electrical height above the ground increases. As well, the efficiency is higher closer to the
source. This is due to the presence of the non-radiating, reactive near field. At a height of
0.3m, the maximum efficiency is approximately 50% for the wet soil. However, the more
lossy dry sand and asphalt decrease the overall efficiency.
41
There is a peak in the efficiency at 50MHz for the 0.3m height. The electrical height
of the source at that frequency is 0.05 wavelengths. This correlates with the peak near
8MHz for the 1.78m high source. The roll off in efficiency observed for the 0.3m source
for the asphalt and wet soil correlates to the roll off in efficiency for the 1.78m height
between 10MHz and 50MHz.
3.2.2 Efficiency of Horizontal Magnetic Dipole
Like the vertical electric dipole, the horizontal magnetic dipole is placed at either
0.3m or 1.78m above the three different ground types. The source operates from 3MHz to
300MHz and the fields are measured on hemispherical shells in the relative near field at
5m and 25m. The efficiencies for the different cases for the horizontal magnetic dipole can
be found in Figure 3.2.
Figure 3.2. Efficiencies for the HMD at both (a.) 0.3m and (b.) 1.78m above various
grounds for different radii.
Like the vertical electric dipole, the efficiency goes to zero at very low frequencies.
For the 0.3m height, the source achieves 30% efficiency at 50MHz over the wet soil. As
42
the electrical height of the source increases, the efficiency increases. For the 1.78m height,
the increase in efficiency from the increase in electrical height eventually levels off
between 50% and 80%. The maximum efficiency for the 0.3m height is at 100MHz with
an efficiency of about 70%. Like the vertical electric dipole, the efficiency is higher for the
more conductive wet soil.
3.2.3 Efficiency of Crossed Electric Dipoles
The crossed electric dipoles are two electric dipoles at the same height, but
orthogonal to each other and 90 degrees out of phase. Like the previous ideal sources, they
are placed at either 0.3m or 1.78m above the various ground types and operate between
3MHz and 300MHz.
Figure 3.3. Efficiencies for the crossed electric dipoles at both (a.) 0.3m and (b.) 1.78m
above various grounds for different radii.
Like the previous sources, the efficiency (Figure 3.3) goes to zero for very low
frequencies but increases as the electrical height of the source increases. For the 0.3m case,
43
the efficiency for all grounds is below 20% at < 75MHz. Unlike the previous sources, the
crossed electric dipoles exhibit a leveling off of the efficiency over the wet soil ground for
the 0.3m height. This leveling off corresponds to the lack of roll off observed at the 1.78m
height for the same electrical height or 1/6th the frequency.
Figure 3.4. Comparison of the efficiencies for the three ideal sources at 0.3m above a dry
sand ground.
Comparing all three sources above a dry sand ground at 0.3m shows that the
horizontal magnetic dipole offers the best efficiency below 200MHz. The crossed electric
dipoles have the best efficiency above 200MHz. This behavior bodes well for circularly
polarized sources near grounds at higher frequencies in terms of their efficiency.
3.3 Vehicle Mounted Ideal Sources over Flat Grounds While understanding the performance of the sources over flat, lossy grounds is
important, it is also vital to understand how the sources interact with the vehicle body.
44
Where the source is mounted on the vehicle can have profound effects on its overall
performance. [24] The ideal sources are mounted in three different locations on the rough
Humvee model. These three positions are the top, bottom, and rear. The top position is 5cm
above the center of the vehicle cab. The bottom position is 5cm below the center of the
vehicle body. The rear position is 10cm from the left corner and 10cm from the rear of the
vehicle. These positions can be seen in Figure 3.5. Like the previous study, the sources
will operate between 3MHz and 300MHz.
Figure 3.5. Location of the antennas’ mounting for the rough Humvee model. The top
source is in the middle of the roof, the bottom source is at the middle of the bottom, and
the rear is 0.1m from the rear corner.
This study in interested in finding a source with strong, uniform fields around the
vehicle body. The electric field strength was computed 0.1m above the ground directly
45
under the vehicle and up to 5m away from its center. All three mounting positions were
considered over the dry sand ground at 100MHz. While the top position offers uniform
fields, the bottom position has greater field strength while maintaining good field
uniformity (Figure 3.6). The rear position has stronger fields near the rear portion of the
vehicle, but has poor uniform field coverage.
Figure 3.6. Electric field strength 10cm above dry sand for under the vehicle model at
100MHz for each mounting position.
3.3.1 Azimuthal Pattern Uniformity Measured as the Wobble of the Wave
(WoW)
Uniform coverage around the vehicle is due to the source-vehicle system having a
uniform azimuthal pattern. This is characterized with a parameter known as the Wobble of
the Wave (WoW). The WoW characterizes the variation of the electric field by comparing
the maximum in the magnitude of the electric field pattern and the minimum of the
magnitude of the electric field pattern as seen in equation (50). Each measurement is taken
at the same elevation angle. For our studies, we are interested in the pattern uniformity very
close to the ground and will be looking at an elevation angle at or very close to 90 degrees.
46
This can be done over a range of frequencies to see the variation in pattern uniformity over
the entire frequency band of interest.
𝑊𝑜𝑊 = 10 log10|𝐸𝜃𝑚𝑎𝑥|
|𝐸𝜃𝑚𝑖𝑛| (50)
Because the WoW is a comparison between the minimum and maximum electric
field magnitudes, the smaller the value, the more azimuthally isotropic the pattern. A
pattern with a large WoW of tens of dB will have at least one large null in the azimuthal
pattern and a non-uniform field. Figure 3.7 shows the WoW for the three different antenna
positions over an infinite PEC ground.
Figure 3.7. Wobble of Wave (WoW) for the three antenna positions on the vehicle model
over a PEC ground.
The top position has the lowest overall WoW over the frequency band. Except near
140MHz and 160MHz, the bottom position has a lower WoW than the rear position. The
47
peak in WoW for the rear position at 68MHz indicates a deep null in the azimuthal pattern,
while the relatively low WoW for the top position is indicative of an omnidirectional
azimuthal pattern. The WoW for the bottom position at this frequency is between that of
the WoW for the top and rear position. As a result, there will be some variation in the
azimuthal field.
3.3.2 Attenuation
For more conductive ground types, such as the wet soil, the attenuation is
decreased. The magnitude of the electric field 10cm above the ground is calculated. In
Figure 3.8, the product of the electric field and the distances from the source is plotted
with respect to the distance. A completely straight line would represent free space
propagation loss.
Figure 3.8. Attenuation of the electric field due to losses in the ground propagating near the
ground for the vertical electric dipole for both dry sand (a) and wet soil (b).
It is apparent from Figure 3.8 that propagation over the wet soil has much lower
attenuation than the much more lossy dry sand. As well, the higher the frequency, the
48
greater the attenuation of the electric field. The height corresponding to the bottom position
experiences greater attenuation than the top position. While the antenna position does play
a role in determining the attenuation, the soil composition is the major factor in determining
the attenuation.
3.3.3 Depolarization
The depolarization is measured as the ratio of the magnitude of the horizontal
electric field to the vertical electric field. For a circularly polarized wave in free space, this
ratio will be unity. If the source is not perfectly circularly polarized or propagating in an
inhomogeneous media, the ratio will deviate from unity. Using two ideal electric dipole
sources that are both orthogonal to each other and 90 degrees out of phase will result in
this perfect circular polarization.
These crossed electric dipoles are then placed over the various ground types over a
range of frequencies (Figure 3.9). By measuring this depolarization ratio, the purity of the
circular polarization at different distances can be inferred. When the source is very close
electrically to the ground, the depolarization ratio becomes less than one. This is due to a
more rapid attenuation of the horizontal component of the electric field when propagating
very close to the ground. The depolarization is even greater for the more conductive wet
soil than for the more lossy dry sand. The boundary conditions of the conductive surfaces
do not support tangential polarizations and are forced to zero for a PEC surface.
When the source is raised to 1.78m, the ratio becomes more frequency dependent.
At 30MHz, the source is much closer electrically to the ground than at 300MHz. This
height change results in greater attenuation of the horizontal component at 30MHz than at
49
300MHz. Close to the source, the horizontal component experiences less attenuation than
the vertical component. Because of this, it is better for a circularly polarized source to
propagate higher above the ground. While the physical dimensions may prevent the source
from being moved much higher, the same effect can be achieved by operating at higher
frequencies and thus increasing the electrical height above the ground.
Figure 3.9. Depolarization of a circularly polarized source above lossy grounds at 0.3m (a)
and 1.78m (b).
3.3.4 Efficiency
With the introduction of the vehicle body, the most significant change in
performance is observed in the efficiency. The vehicle acts as a scatterer that scatters the
fields away from the vehicle body. In the case of the top position, a larger percentage of
the fields are scattered up and away from the lossy ground. This results in a significant
increase in efficiency seen in Figure 3.10. Likewise, the fields for the bottom position are
reflected down, into the lossy medium and a larger percentage of the power is dissipated
50
in the ground. For the dry sand, the efficiency for the bottom position is under 10%. The
rear position is between the efficiency of the bottom and top positions.
Figure 3.10. Efficiency of the vertical electric dipole above dry sand for the three different
vehicle positions as well as the source without the vehicle model.
This trend holds for the other sources, the horizontal magnetic dipole and the
crossed electric dipoles. Figure 3.11 shows the effect of the ground type on the efficiency
with the vehicle model. Like the previous case with the vertical electric dipole, the
inclusion of the vehicle model increases the top mounting efficiency and decreases the
efficiency for the bottom position. However, while the efficiency is only about 10% for the
bottom position over dry sand, it maxes out at 40% and remains over 25% over the majority
of the frequency band for the wet soil. Therefore, one of the prime predictors of the
efficiency of the bottom source will be the conductivity of the ground. As seen in Figure
3.12, the low efficiency for the bottom position holds true for the crossed electric dipoles
51
as well. Increasing the efficiency will be the most important part of improving the utility
of the bottom as a mounting location.
Figure 3.11. Efficiency of the horizontal magnetic dipole over both dry sand (a) and wet soil (b)
both with and without the vehicle model.
Figure 3.12. Efficiency of the crossed electric dipoles for the three antenna positions over dry
sand with and without the vehicle model.
3.4 Summary and Conclusions For sources that are close to the ground, there will be significant effects on their
performance characteristics. For lossier grounds, like the dry sand, there is a large decrease
52
in efficiency when compared to more conductive ground types like the wet soil. However,
the more conductive the soil, the greater are the effects of depolarization of circularly
polarized sources. To mitigate this, circularly polarized sources that are propagating along
the ground should be mounted as far from the ground as possible.
Ideally, an antenna system for jamming applications would have azimuthal pattern
uniformity with a low attenuation up to 50m from the vehicle and a high efficiency. A
source mounted on the roof of the vehicle offers a small WoW and good azimuthal pattern
uniformity. It also offers the greatest amount of efficiency. The bottom source potentially
offers a better WoW than the rear position, but comes at the cost of a poor efficiency. If
the efficiency of the bottom position could be improved while maintaining a low WoW, it
would become a promising mounting position. Two potential ways of doing this is by either
preventing coupling with the grounds to increase the efficiency or by using sectorial
sources. Both of these approaches will be explored in the following chapters.
53
Chapter 4
Antenna Mounting on the Vehicle Bottom
This chapter considers antennas mounted underneath vehicles as an alternative
mounting location to top and rear mounting locations with a variety of antennas. By
mounting antenna sources under the vehicle body, the visual profile of the antenna system
is considerably reduced. An antenna mounted on the bottom can be used as a source for
either jamming or possibly short-range communication applications.
As discussed in Chapter 3, the inclusion of the vehicle body greatly reduces the
efficiency of the bottom mounted ideal source (Figure 4.1). This section is mainly focused
on the ways to improve the efficiency of bottom (belly) practical antennas. Results for
omni-directionality of the considered sources in azimuthal plane are also presented.
Figure 4.1. Efficiency of the ideal vertical electric dipole above a dry sand ground both
with and without the vehicle model for the top and bottom mounting positions. The
inclusion of the vehicle model enhances the efficiency of the top mounted source, but
greatly diminishes the efficiency of the bottom source.
54
Two different frequency ranges are considered. The first is from 3MHz to 300MHz (HF-
VHF) and the second - from 300MHz to 3GHz (UHF). In the first case, the source is a
parallel plate antenna. Goal of the plates is twofold. Firstly, they “shield” the source from
the grounds and thus may help to increase the antenna efficiency. Secondly, they
capacitively load the antenna, and thus improve its performance. In the case of UHF
antennas, the initial hypothesis is that antenna pattern nulls to the direction of ground and
vehicle reduce the interaction of the antenna with them. Mode-2 spiral, mode-2 spiral-helix
antennas and a modified monocone antenna are explored. Since spiral antennas are
wideband and flash mountable, they are good candidates for the antennas concealed on the
vehicle side, and next chapter considers the electrical performance of spiral antennas on
vehicle side. All calculations are performed by means of FEKO (MoM) software [21].
4.1 Stand-Alone HF-VHF Parallel Plate Antenna
The motivation for considering a metallic (PEC) plate parallel to the vehicle’s
bottom with the source embedded in-between comes from the need to mitigate interactions
between the source and the ground. A conductive plate can provide both some shielding
and redirect the radiated power to prevent it from coupling with the ground while also being
redirected away from it.
Before studying the role of the modified vehicle bottom source in a more realistic
and possibly operational environment, the stand-alone parallel-plate source is studied in
greater depth. The parallel plate source is comprised of two PEC discs with radii of 2m,
separated by 10cm. An ideal vertical electric dipole source is sandwiched between the two
plates. This is then placed at the height 0.3m above the ground (Figure 4.2). Because the
55
top plate is intended to emulate the bottom of the vehicle model, its height corresponds to
the bottom of the vehicle model - 0.35m above the ground. The second plate is 10cm
directly below it at 0.25m above the ground. The main objective is to determine if there are
ways to theoretically (through validated numerical simulations) demonstrate that the
inherently very low efficiencies associated with the near-ground antennas may be improved
using means that may be practically realizable.
Figure 4.2 (a) View and (b) geometry of the parallel plate source. The top disc simulates
the bottom floor of the vehicle and the bottom disc reflects and directs the wave away from
the lossy grounds and to potentially increase the efficiency. The source is an ideal vertical
electric dipole in the center.
Compared to the case of an ideal source at the same height above the ground, the
introduction of parallel plates significantly improves the efficiency at most frequency
points and exhibits an oscillatory behavior (Figure 4.3). This is especially apparent at the
low frequency end. The oscillatory behavior of the efficiency is associated with the
excitation of standing waves within the parallel plates. Also, sources over the more
56
conductive wet soil typically have a higher efficiency than the less conductive dry sand
and asphalt grounds. The efficiencies over dry sand and asphalt are similar.
In this study, the dry sand is considered to be the worst case scenario in terms of efficiency
and it is used as the baseline performance.
Figure 4.3. (a) Efficiency of an ideal vertical dipole and (b) the basic parallel plate source
over dry sand, asphalt, and wet soil ground types. Heights of the sources are 0.3m. Parallel
plates improve the efficiency at most frequency points and introduce oscillatory behavior.
4.1.1 Understanding Operation of Parallel Plate Source
Minimally Lossy Ground
As previously noted, the efficiency exhibits oscillatory behavior with periodic
maxima and minima. There is a correlation between the locations of the minima and
maxima and the analytical solution for the magnetic field inside the source. The magnetic
field distribution is governed by the Helmholtz wave equation:
57
∇2 + 𝑘2 = 0, (51)
where k is the wavenumber in free space. Neglecting fringing field and taking into account
cylindrical symmetry of the problem, the magnetic field becomes a one-dimensional
function given by
𝐻𝜑 = 𝐴 𝐽0(𝑘𝑟) + 𝐵 𝑌0(𝑘𝑟), (52)
where J0 and Y0 are Bessel functions of the first and second kinds, respectively; unknown
coefficients A and B are determined by the boundary conditions.
The first boundary condition states that the magnetic field is zero at the edge of the
discs since there is no radial current |𝐽r| = |n x , where n is the unit vector perpendicular
to the disc surface. The second boundary condition forces the magnetic field to either be a
finite value at the disc’s center or infinite. In former case, ~ J0 (kR), while in latter
~ Y0 (kR), here R is the disc radius.
This solution describes a set of standing waves. The frequencies for which these
waves occur are determined by the zeroes of the Bessel functions. The first four zeroes of
the Bessel functions are shown in Table 4.1. To verify that there is correlation between the
standing waves and oscillatory behavior of the efficiency, the parallel plate source is solved
above a ‘lossy air’ where the relative permittivity is unity and the loss tangent is very small.
In this case, it is chosen to be 0.0001.
58
Table 4.1 Zeroes of Bessel functions of the first J0 and second Y0 kinds and their
corresponding frequencies. Disc radius R = 2m.
𝐽0(𝑘𝑅) = 𝟎 𝑌0(𝑘𝑅) = 𝟎
kR f, MHz kR f, MHz
1 2.40 57.3 0.90 21.5
2 5.52 131.8 3.95 94.3
3 8.65 206.5 7.05 168.3
4 11.79 281.5 10.2 243.5
This represents a lossy medium for dissipation while minimizing the loading effects
of the ground on the parallel plate source. The efficiency of the parallel plate over the lossy
air was then computed in the same manner as the other ground types.
As seen in Figure 4.4 (a), the efficiency exhibits maxima and minima that correlate
well with the frequencies of the standing waves in the analytical model. Maxima in the
efficiency correspond to the zeroes of the Bessel function of the first kind, minima
correspond to zeroes in the second kind. The oscillations are centered at 50%. This is what
one would expect as waves propagating in the upper half do not experience any attenuation
or loss, while waves propagating in the downward direction will not be reflected and are
dissipated in the lossy medium.
The electric field 0.5m directly under the parallel plate source was measured in free
space (Figure 4.4 (b)). Far-field pattern of the source has null at this direction, and therefore
this near-field electric field is comprised entirely of the radial component of the electric
near field and does not propagate. However, there is still power stored in these fields and
59
it is dissipated when close to a lossy ground. The local maxima in the electric field correlate
to the local minima in the efficiency for the parallel plate source over the lossy air.
Similarly, the local minima in the electric field correlates to the maxima in the efficiency.
Figure 4.4. (a) Efficiency of the parallel plate source over the minimally lossy ground
modeled with the Sommerfeld ground model. There is a correlation between the standing
wave frequencies and the minima and maxima in the efficiency. (b) Electric field 0.5m
directly under the parallel plate source in free space. The local minima and maxima in the
electric field strength correlate with the standing waves in the efficiency.
Magnetic Current Loop
An alternative way to describe the operation of the parallel plate source is to
consider magnetic current excited along the outermost boundary of the parallel plates.
Specifically, the edges of the parallel discs can be thought of as a ring of magnetic current
computed as a vector product of the electric fields between the plates and the outward
normal to the side. This magnetic current ring runs along the circumference of the parallel
plates, and it can be approximated as a series of ideal horizontal magnetic dipoles arranged
tangentially to the circumference of the discs.
60
The simplest model would be two horizontal magnetic dipoles facing opposite
directions four meters apart to simulate the parallel discs and excited circumferential
constant magnetic current. As the number of magnetic current elements is increased, the
results converge to the fields and efficiencies for the parallel discs (Figure 4.5).
Figure 4.5 Comparison of parallel discs source 0.3m above dry sand and 2,4, and 16
horizontal Hertzian magnetic dipoles to approximate a ring of magnetic current for
efficiency (a) and elevation cut at 300MHz (b).
When the number of horizontal magnetic dipoles is 16, there is a good agreement with
parallel plate results in efficiency (Figure 4.5). While the model of a constant magnetic
current loop does compare well with the parallel discs, there are some discrepancies in the
computed performances. This is due to the fixed height of the parallel discs resulting in a
magnetic current loop that is 10 cm tall. For the magnetic current loop source model
presented above, it was comprised of ideal sources with an infinitesimal thickness.
61
4.1.2 Parametric Studies
Separation of Plates
For practical considerations, the bottom mounted source should be as thin as
possible. If a large separation is required to achieve adequate performance, then it would
not be a viable option as a source under the vehicle due to physical clearance.
This study varies the separation of the plates from 10 cm to 30 cm (Figure 4.6).
Both the efficiency and far-field elevation patterns are considered to determine the
associated effects. The maximum separation is limited to 30 cm because of the limited
space between the ground and the hypothetical bottom of the vehicle.
Figure 4.6. Geometry of the parallel plate source over dry sand with the separation between
the plates varied as the other dimensions remain constant. The height of 0.3m is at the
midpoint between the plates.
The analysis has shown there is little appreciable difference in the elevation far-field
patterns as the separation between the plates is varied (Figure 4.7). Similarly, there is little
difference in the efficiency (Figure 4.8). As the separation of the discs is increased, there
is a slight shift in the peaks to lower frequencies and a slight decrease in the maximum
efficiency of those peaks when compared to the smaller separation. The maximum
difference in efficiency is about five percent. This effect becomes especially noticeable at
frequencies above 200MHz as the separation is no long negligible compared to a
wavelength .
62
Figure 4.7. Elevation patterns of the parallel plate source over dry sand for various
separation distances of the plates.
Figure 4.8. Efficiency of the parallel plate source 0.3m above dry sand with the separation
between the top and bottom plate h varied. There is little change in the efficiency due to the
very small changes in separation in terms of wavelengths. However, some differences begin
to become apparent at higher frequencies.
Overall, there is a very weak sensitivity to the change in separation between the
discs. This is due to the insignificant electrical widths associated with the changes in the
separation. At the higher frequencies, the decrease in efficiency for the larger separation is
due to fringing fields being closer to the ground. The distance between the local minima
63
and maxima in the efficiency is due to the source being loaded by the ground. As this
loading becomes more pronounced, there is a greater shift.
Height of Source above Ground
To better understand the effects of the ground, the height of the source above the
ground is varied (Figure 4.9). In all cases, the separation between the discs and the radius
of the discs remain the same at 10cm and 2m respectively. Note that the height above
ground is measured from the middle of the source.
Figure 4.9. Geometry of the parallel plate source over dry sand with the height of the source
varied as the other dimensions remain constant.
The height of the source affects efficiency at the lower frequency end along with
the magnitude and mean level of oscillations (Figure 4.10). One can speculate that the
decrease in efficiency for lower frequencies can be explained either by the source at a
height being comparable to a wavelength (Figure 4.10, h = 5 and 10m) or by noting that
the electrically small aperture is closer to the ground (Figure 4.10, h = 0.3 and 1m). In the
latter case, the interactions of the reactive near field with the ground can be reduced or
perhaps even eliminated by modifying the geometry of the aperture.
64
Figure 4.10. Efficiency of the 2m radius parallel plate source at various heights h above a
dry sand ground. As the height of the source increases, the efficiency increases to oscillate
just over 50%
When the antenna is moved farther away from the ground, the effect of the ground
on antenna performance is reduced, and efficiency level is at about 50% for dry sand. For
other grounds, this level will be different.
Radius of Discs
The final parameter to consider is the radius of the disc. The radius is varied from
1m to 2.5m. The separation of the discs is kept constant at 10cm and the height of the
source is kept at 0.3m (Figure 4.11). As the radius of the disc is increased, the number of
minima and maxima increases over the frequency range (Figure 4.12). As well, the
oscillatory behavior in the efficiency starts at lower frequencies as the radius is increased.
Therefore, a larger radius contributes to the increased efficiency at lower frequencies.
65
Figure 4.11. Geometry of the parallel plate source over the dry sand ground as the radius
of the plate is changed. The separation and height above the ground remain constant
Figure 4.12. Efficiency of the parallel plate source 0.3m above dry sand with three different
radii.
As previously mentioned, the locations of the first maxima in the efficiency are
based off the radius of the disc. Three markers are located at the first maximum for each
disc radius. That first peak occurs when the frequency is:
𝑓 =𝑐02𝑟
(53)
66
This corresponds to 66MHz, 94MHz, and 110MHz, for parallel discs sources with a radius
of 2.5m, 1.75m, and 1m respectively. Clearly, the size of the parallel discs changes the
performance at different frequencies. Later, when this is implemented with the vehicle
model, it will allow for an understanding of how other geometries for the plates will
operate. This is discussed further in later sections.
4.1.3 Modifications of the Parallel Plate Source
While the bottom antenna design and optimization are not one of the objectives of
this thesis, it is useful to consider if some simple modifications to the geometry of the
parallel plate radiator may induce favorable performance improvements. Rather than just
having two parallel discs, short fins are added to the bottom plate. These fins are 7 cm long
and extend 5 cm out from the disc and 5 cm up above the lower disc (Figure 4.13).
Figure 4.13. Geometry of the parallel plate source with the fins. The fins are either 7cm or
14cm long and extend out from the bottom plate at 45 degrees.
The addition of the fins for the stand-alone antenna configuration in free-space
(with no ground present) shows that the near-field elevation patterns are bent up by the
fins, even at low frequencies (Figure 4.14). This indicates that the fields themselves are
being directed away from the ground and may account for the increase in efficiency. Note
that some front-to-back ratio is now observable (in free space).
67
Figure 4.14. Elevation cut of the electric near field measured 2.5m from the center of the
parallel plate source in free space.
Figure 4.15. Efficiency of the parallel plate source with and without fins over the dry sand
ground. Even with the addition of electrically very small fins, there is a significant increase
in efficiency above 50MHz
68
Despite being electrically small, the addition of these fins greatly increases the
efficiency of the parallel discs source. As can be seen in Figure 4.15, the fins increase the
efficiency of the parallel discs source by about twenty percent when compared to the discs
without the fins when over the dry sand ground.
The efficiency can be even further increased by increasing the length of the fins.
By making the fins extend 10cm out and 10cm up (fins length of 14cm), the efficiency can
be increased by nearly an additional twenty percent than the source with the shorter fins
(Figure 4.15). These results are all over the dry sand, which represents the worst case
scenario for the efficiency. Note that our objective was not to find an optimal topology, but
rather to show that the extremely low values of efficiency over different grounds for the
bottom antenna may be mitigated during the antenna design process. Even very small
changes to the parallel disc geometry can have drastic improvements on the efficiency and
an optimization of this geometry could prove to offer a very efficient source under a
vehicle.
4.2 Integration of Parallel Plate Source with Vehicle Model
The above discussed parallel plate antenna is integrated with the basic Humvee
vehicle model. Geometry of the used vehicle model is shown again in Figure 4.16. As
depicted in Figure 4.17, a plate that is 2.31m wide and 4.64m long; corresponding to the
dimensions of the vehicle; is placed 10cm below the vehicle’s bottom. For this rough
model, the wheels are excluded to allow for more rapid calculations. The effects of the
wheels will be discussed in more detail in Section 4.3. Fins are added to the edges of the
69
plate to extend 5cm out from the vehicle and 5cm up. Like previous simulations, the source
is the vertical Hertzian electric dipole placed in the center of the plates.
Figure 4.17. Geometry of plate with fins under the vehicle model. The fins extend 5cm out
from the side and 5cm up. The top plate is the bottom of the vehicle.
Figure 4.16. Geometry and dimensions of a rough vehicle model.
70
Figure 4.18 (a) Efficiency of the VED under the vehicle compared to the parallel plate
source with and without the vehicle model over dry sand. (b) Model of the parallel plate
source under the vehicle and the parallel plate source without the vehicle.
This modification drastically increases the efficiency of the source when compared
to the vertical Hertzian electric dipole under the vehicle without the plate (Figure 4.18).
Note, this source was not optimized in terms of geometry or performance, thus even further
improvements are possible. Nevertheless, these results clearly indicate that the bottom side
of a vehicle may be a viable destination for an antenna. Two configurations of the vehicle
were modeled. The first was the entire rough vehicle model with the parallel plate source
with fins. The second configuration was the same parallel plate with fins geometry, but the
vehicle model was removed. Only the parallel plate source in the shape of the vehicle was
modeled in this case. For this geometry, there is a similar increase in efficiency.
71
4.3 Effect of Wheels in HF and VHF Bands
For lower frequencies, such as the HF band, the wavelengths vary from 300m to
10m. On the lower end, when the wavelength is hundreds of meters, the wavelength is
significantly larger than both the antenna and vehicle dimensions. Therefore, the entire
antenna and vehicle system can be considered as an electrically small antenna. When the
vehicle is over the ground, it will have an image underneath the ground. This can be
modeled as a flat plate representing the undercarriage of the vehicle. Its image will result
in a lumped capacitance and the tires can be modeled as a lumped impedance.
However, for sources that are under the vehicle, the wheels have a potential for an
even larger effect than just on the impedance of the source. They will also absorb and
reflect energy that can change the radiation pattern and the field strength in close proximity
to the vehicle. To determine how much of an effect there is on the field strength and
patterns, the Basic Humvee model is used over a PEC ground.
An ideal Hertzian electric dipole source is placed under the vehicle 5cm from the
bottom from 3MHz to 300MHz. Then the basic Humvee model is modeled with no wheels,
rubber wheels, and PEC wheels. The electric field is calculated 32cm above the ground
around the vehicle at 3, 5, and 25m. At 100MHz, the PEC wheels clearly contribute to deep
nulls in the normalized electric field strength. The presence of the rubber wheels also result
in nulls, however they are not as deep as the PEC wheels. As well, the fields around the
vehicle with the rubber wheels are much closer to the case without wheels than with the
PEC wheels.
72
Figure 4.19. Normalized electric field strength in dB at several frequencies 32cm above a
PEC around the basic Humvee model with rubber, PEC, and no wheels.
At 30MHz, the effect of the rubber wheels is nearly negligible when compared to
the model without the wheels (Figure 4.19). At 3m, there is a 3% difference between the
two cases. When measured further at 25m, the percent difference is less than 6%. However,
as the frequency increases, the effects of the wheels begin to have a large impact on the
distribution of the electric field around the vehicle. As the frequency is increased even
further, the resulting field with the rubber wheels begins to converge to that of the PEC
wheels. This is especially apparent for fields further from the vehicle. For fields further
73
from the vehicle and at frequencies well above 300MHz, the wheels can be modeled as
PEC to reduce computation time and costs while preserving good results.
4.4 Parallel Plate Source with Scaled Humvee Model
Considering the effects of the wheels on sources under the vehicle, the parallel plate
source is then integrated with the basic Humvee model with the rubber wheels. The parallel
plate source is smaller than the one used for the rough Humvee model. Rather than using a
plate the size of the undercarriage of the model, the parallel plate source is only directly
under the cabin of the vehicle to avoid the wheels. This is a rectangular plate that is 2.22m
by 2m and 10cm from the bottom of the Humvee model.
In order to allow for practical measurements of the parallel plate source under the
Humvee, a basic Humvee model is scaled to be reduced in size by a factor of 20. This
Humvee model was 3D printed and then plated in copper. The model was printed with a
Makerbot Replicator 5th Generation [25] which uses a fused deposition modeling process.
This process uses a plastic filament that is unwound as it passes through an extrusion nozzle
that controls the flow of the filament. The extrusion nozzle heats and melts the material
and moves horizontally to deposit the material in the required areas. This thin strand of
plastic rapidly cools and hardens to bond with the adjacent strands. Once that layer is
complete, the nozzle moves up to produce the next layer. [25]
Structures on the model that stick out, like “T” shapes, require scaffolding to be
printed. [25] This scaffolding is then removed after the printing is finished. This means that
the inside of the model cannot be hollow, but needs some material to support the rest of the
74
model. Once finished, the excess scaffolding is removed and the model is sanded down to
polish it. Then, the 3D printed model is metalized. The entire process is relatively quick
and inexpensive compared to machining the model out of metal.
A scaled model for the parallel plate source was made by using a coaxial cable and
removing the outer conductor and dielectric from the last 1.2cm. The inner conductor was
then bent downwards and soldered to an 11cm by 10cm copper plate. This model was then
taped under the scaled model so that the bottom of the model would act as the top of the
parallel plate (Figure 4.20). To improve the mechanical stability, a 10cm by 11cm by 1cm
Styrofoam block was added to the model. This Styrofoam has a relative permittivity of
1.03 and a loss tangent of 0.0003. The model and antenna were then placed on a 2.22m by
2.22m ground plane for measurements.
Figure 4.20. Scaled basic Humvee model without (a) and with (b) the scaled parallel plate
source under the vehicle.
75
The scaled model both with and without the Humvee shows a good match (Figure
4.21) at 11.1GHz with the Humvee model and at 11.6GHz without the model. For a
potential use, a tuning circuit can be used to more closely match to the desired frequency.
However, this source does not exhibit wide band behavior, which is advantageous for
jamming applications.
Figure 4.21. S11 measurements of the scaled parallel plate source over a conductive ground plane
both without (a) and with (b) the scaled Humvee model.
4.5 Standalone UHF Antennas
In this section, the performances of a few UHF antennas mounted on the bottom of
vehicle are considered. The antennas studied are the mode-2 spiral and mode-2 spiral-helix
antennas [26], along with the modified monocone antenna [27]. All antennas have antenna
patterns with nulls in the far-field at the direction of the ground. The initial hypothesis is
that the antenna – ground interactions can be reduced by pointing the null toward the
76
ground. Note that the ground is still in the near-field of the considered antennas where the
field structure is different than that in the far-field.
4.5.1 Mode-2 Spiral Antenna
It is well known that the far-field radiation of a spiral antenna can be decomposed
into a series of modes [4]. Typically, the antenna is designed with two arms to operate with
a single, dominant mode. However, higher order modes can be excited resulting in pattern
distortions in terms of axial ratio and/or wobble-of-the-wave (WoW). The mode-1 of the
spiral antenna has a maximum at boresight, while all the higher order modes have null at
boresight [Figure 4.22 (a)]. To operate at a specific mode m, the spiral arms need to be fed
in a m*360 phase progression [Figure 4.22 (b)] [4].
Figure 4.22. (a) Co-polarized far-field patterns of mode-1 and mode-2 of a four-arm spiral
in free space without any backing (only top half is shown). (b) Phase progressions in
feeding of the four-arm spiral antenna to achieve mode-1 and mode-2. The excitation
magnitudes at all four arms are the same.
Cross-polarized modes have indices –m and their magnitudes are the same in shape
as their co-polarized counterparts. In the case of spiral antennas, the cross polarized modes
77
are suppressed by the wrapping of the spiral arms. A more detailed description of the
operation of spiral antennas is given in Chapter 5.
To evaluate the effects of ground proximity and viability of mounting a spiral
antenna on the bottom of a vehicle, the mode-2 four-arm spiral antenna with a 25.4 cm
diameter was selected to be used. The configuration is a self-complementary Archimedean
spiral with the layout shown in Figure 4.23.
Each arm has two turns and is loaded with a 130Ω resistor at the ends to assist with
the dissipation of the non-radiated travelling current wave. These loads may also help in
mitigating the excitation of the cross-polarized mode -2 due to different bounces. A
metallic ring is included to improve lower frequency performance and thus decrease its
turn on frequency [28]. It also allows for easier (practical) soldering of arm-terminating
resistors.
Figure 4.23. Geometry of the mode-2 four-arm spiral antenna and close up of feed that will
be used under the vehicle.
r1 = 0.88cm
r2 = 14.2cm
w = 0.76cm
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Figure 4.25. Real and imaginary impedance of the mode-2 spiral antenna (a) and the far-field
total gain elevation cuts over the UHF band (b). Elevation cuts are calculated from 700 MHz
to 3 GHz with step 250 MHz.
Based on the largest circumference of the spiral, the turn on frequency for mode 1
is 375MHz. However, the turn on frequency for the mode 2 operation will be about twice
Figure 4.24. VSWR of the 25.4cm diameter spiral for both mode-1 and mode-2 operation
in free space. VSWR is given with respect to nominal impedance.
500 1000 1500 2000 2500 30001
1.5
2
2.5
3
3.5
4
4.5
5
Frequency [MHz]
VS
WR
Mode 1
Mode 2
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that of mode 1. This can be observed in Figure 4.24, where the VSWR of mode 1 is below
2 at 375MHz and below 2 at 750MHz for mode 2 operation. The impedance of the 25.4
cm mode-2 four-arm spiral is very stable above 900MHz, but has typical ringing behavior
at lower frequencies (Figure 4.25). The far-field elevation cuts are very stable over the
UHF band.
4.5.2 Mode-2 Spiral–Helix Antenna
The mode-2 spiral-helix antenna operates in a similar way as the mode-2 four-arm
spiral [26]. The helical terminations of the spiral arms increase the lower bandwidth of the
antenna while maintaining a small aperture diameter. The spiral-helix aperture has a
diameter of 15.24cm and a height of 5.08cm (Figure 4.26).
Figure 4.26. General geometry of the spiral-helix antenna. It has a metal to slot ratio of
50% and is mounted on the bottom of a PEC plate. When ground is modeled, the plate is
37cm above the ground.
The self-complementary Archimedean helix arms have ¾ turns. The back of the
spiral-helix cavity has a PEC cylinder with a diameter of 14.23cm and a height of 1.27cm
80
to decrease destructive interference at higher frequencies. Like the four-arm spiral, it is
operated in mode-2 so that there is a null present at boresight in the direction of the ground.
The height of the plate is 37cm.
The impedance of the stand-alone spiral-helix antenna in free space, seen in Figure
4.27, experiences some variation, but has a nominal value of about 100Ω. The inclusion of
different ground types does not affect the impedance of the antenna (not shown). There is
some variation in the far-field elevation cuts of the spiral-helix on the lower end below
1GHz, but is stable above this frequency. Note that spiral-helix antenna directs more power
in the front direction. For the given geometry, this corresponds to the direction towards the
ground.
Figure 4.27. Real and imaginary parts of impedance of the mode-2 spiral-helix antenna under
the PEC plate (a) and the far-field gain elevation cuts over the UHF band (b). Elevation cuts
are calculated from 700 MHz to 3 GHz with step 250 MHz.
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4.5.3 Modified Monocone Antenna
The modified monocone antenna is based on the design by J. McDonald for a
modified monocone over a PEC ground plane [27]. The modified monocone starts as a
classical conical antenna with a cone angle of 80 degrees. The cone is combined with a
circular cylinder that increases its height. This modification increases its bandwidth
performance [27]. The 2D cross section of the monocone antenna is presented in Figure
4.28. The chosen dimensions are scaled for the frequencies of interest.
The monocone requires ground plane for the operation. In order to see the
performance of the stand-alone antenna above ground, the modified monocone antenna is
mirrored to create a modified bicone antenna. In free space, the impedance and far-field
Figure 4.28. Geometry of the modified monocone antenna over a ground plane (a) and
general view of numerical model (b).
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elevation cuts are stable over the majority of the UHF band (Figure 4.29). Because it is a
bicone, its impedance is twice that of the monocone.
4.5.4 Comparison of Antennas Performance
The voltage standing wave ratio (VSWR) characterizes the matching of the
transmission line to the antenna with unity being a perfect match and maximum power
transfer to/from the antenna. As seen in Figure 4.30, all antennas have VSWR less than 2
at frequencies higher than 1.1 GHz in free space. Nevertheless, the antennas have different
turn on frequencies due to their physical size and more importantly, their physics of
operation. The spiral-helix has the lowest VSWR at most frequency points at expense of
lower efficiency (Figure 4.31). Turn on frequencies of the spiral and monocone antennas
are 600 MHz and 750 MHz, respectively.
Figure 4.29. Real and imaginary parts of impedance of the modified bicone antenna (a) and
the far-field gain elevation cuts (b) over the UHF band. Elevation cuts are calculated from
700 MHz to 3 GHz with step 250 MHz. Free space.
83
Figure 4.30. VSWR of the three considered UHF sources in free space with respect to their
nominal impedances.
In order to model the effect of ground on stand-alone antennas, the spiral, spiral-
helix and bicone antennas are placed at the same height where they would be mounted on
a vehicle (when mounted, the bicone is replaced with an equivalent monocone
configuration). The efficiencies of the three sources above dry sand without the vehicle
model are compared in Figure 4.31. Both the monocone and planar spiral antennas achieve
efficiency above 50%, while the spiral-helix antenna’s efficiency remains relatively low
throughout the considered frequency range. This lower efficiency of the spiral-helix is due
to the geometry and environmental induced losses. Specifically, the increased power
dissipated in the loads reduces the efficiency on the lower end, while at the higher end the
spiral-helix more readily couples with the ground. Nevertheless, the computed efficiencies
84
are good and may be even further improved with a more dedicated design and optimization
process.
4.6 Vehicle Mounted UHF Sources over Grounds
4.6.1 Numerical Model
In this section, the performance of vehicle mounted antennas is considered. To
obtain reasonable data at higher frequencies, the full-wave simulations in MoM require
smaller mesh triangles. The number of mesh elements and therefore the number of
unknowns needed to be solved greatly increases the computational cost of the calculations.
In order to reduce the required memory and computation time, the geometric models must
be as small as possible while still giving reasonably good and reliable results. A significant
amount of effort and time was placed in developing computational models that have a good
balance between the computational cost (memory and time) and achieved accuracy. The
presented results are theoretically validated using these computational studies.
Figure 4.31. (a) Efficiency of the three considered UHF sources 0.32m above dry sand
without the vehicle model. (b) Percentage of power dissipated in the loads of the spiral and
spiral-helix.
85
For reference calculations, the bottom vehicle profile is modeled as shown in
Figure 4.32. This model includes some details of the vehicle geometry around the wheels
and axles of the vehicle since their size is comparable to the wavelengths of interest. The
PEC wheels are also included in the model. Modeling the wheels as dielectric would
significantly increase the running time and memory and most of these studies will require
more extensive computational resources than what we had. However, as previously shown,
the use of PEC wheels for higher frequencies can still give good results. The entire model
is simulated using MoM. To further reduce the complexity of the models and keep
computational time reasonable, maximum frequency is set to 750 MHz.
Figure 4.32. Model of the vehicle bottom with PEC wheels.
The reduced model of the vehicle bottom is a PEC square plate of 2m x 2m size
with the sources placed at the center of the plate. With this model, the calculations can be
done up to 3 GHz on available desktop computer and FEKO license.
86
Figure 4.33. Model of PEC plate for modeling the bottom of the vehicle and the height of
the source and plate above the ground.
Figure 4.34 Efficiency of monocone antenna mounted on the 2m x 2m PEC plate and vehicle
bottom model over dry sand.
To verify that the calculated efficiency obtained with a 2m x 2m plate is valid, the
modified monocone antenna is placed under both vehicle bottom models over dry sand and
computed results are plotted in Figure 4.34. As seen, the obtained efficiencies are nearly
identical; therefore validating that the 2m x 2m PEC is a good reduced order complexity
model that can be used to accurately determine the efficiencies of mounted antennas up to
750MHz.
87
4.6.2 Size of PEC Plate
To start the studies, the sensitivity of the sources’ performances to the size of the
PEC plate is determined. To do so, the size of the square PEC plate is varied from 2m edge
lengths to 0.5m as shown in Figure 4.35. The selected antenna is the mode-2 planar four-
arm spiral placed 32cm above ground and facing the ground. The height of the plate above
the ground is set at 37cm.
Figure 4.36. Efficiency of the mode 2 four arm spiral antenna placed between dry sand and
square PEC plates of various sizes. The 2m x 2m plate has very similar efficiency to the
vehicle bottom model.
Figure 4.35. A geometry used to evaluate the effects of size variation of the square PEC
plate with the mode-2 four-arm spiral over dry sand.
88
There is clear dependence of computed efficiency on the size of utilized PEC plate.
Efficiencies for the smaller PEC plates increase as their size decreases (Figure 4.36).
However, none of the smaller plates have similar efficiencies to the vehicle bottom model
other than the 2m size. For accurate calculations of the efficiency of the bottom source, the
plate should be 2m by 2m.
4.6.3 Efficiency Comparison
Computed efficiencies for studied antennas are seen in Figure 4.37. Antennas are
placed 5cm below the 2m x 2m PEC plate facing dry sand. The plate is 37cm above the
ground.
Figure 4.37. (a) Efficiency of the mode-2 spiral, mode-2 spiral-helix and monocone antennas
under the 2m x 2m PEC plate over dry sand. (b) Efficiency of the spiral-helix antenna for dry
sand, asphalt, and wet soil.
As seen, the modified monocone has the highest efficiency. However, its efficiency
does not exceed 30% and indicates that the losses in the ground are still high. The efficiency
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of the mode-2 spiral and mode-2 spiral-helix is lower than 10% since these antennas have
additional loss mechanism – the resistive terminations. As expected, dry sand is the worst
case scenario for efficiency and the wet soil has the highest efficiency.
4.6.4 Near-Field Coverage
Since the PEC plate used is flat and symmetric, the patterns are also symmetric with
relatively small WoW as shown in Figure 4.38. The electric field in the near-field is
computed 5m and 10m away from the sources, at 0.32m above the ground. This height
corresponds to the height of the sources. As expected, the strength of the electric field
follows the trends of efficiency (Figure 4.39). The modified monocone has the strongest
electric field at all frequencies and distances.
Figure 4.38. (a) WoW for the modified monocone under the 2m x 2m PEC plate over
various grounds. The wet soil ground has the lowest WoW, while the more lossy asphalt
and dry sand have very similar WoWs. (b) Comparison of the WoW for the monocone and
spiral-helix over dry sand ground.
90
Figure 4.39. Azimuthal electric near field patterns in dB 5m and 10m from the source and
0.32m above the dry sand ground under the 2m x 2m PEC plate. There is low variability
for the modified monocone source compared to the other sources. The modified monocone
source also has a much stronger electric field near the source.
4.6.5 Role of Wheels at UHF
The impact of the wheels on omni-directionality of the sources is studied with a
non-flat and non-symmetric vehicle floor model used before for reference calculations
(Figure 4.32). Herein considered sources are monocone and mode-2 spiral-helix antennas.
As expected, the inclusion of PEC wheels greatly reduces the quality of the azimuthal
uniformity in the UHF band. As previously discussed, the wheels can be modeled as PEC
at higher frequencies for sources under the vehicle.
91
Over a PEC ground, both the modified monocone and mode-2 spiral-helix sources
have WoWs larger than 20dB as shown in Figure 4.40. Dry sand reduces the value for
WoW, however, the overall level is still very high. Considering the high value of WoW,
mounting a single UHF band antenna under the vehicle becomes much more problematic
and needs to be heavily taken into account when designing a source for under the vehicle
in these frequency ranges.
As seen, the source realized as a modified monocone antenna offers the best
performance in terms of near-field strength and efficiency. Thus, for sources mounted
under the vehicle in the UHF region, the modified monocone source is preferred. However,
due to the generally poor WoW performance and the low efficiency for sources under the
Figure 4.40. WoW for the monocone and spiral helix source under the vehicle bottom
model for both PEC and dry sand ground. Note the decreased WoW for the dry sand case
compared to the PEC ground and the significantly higher WoW with the inclusion of the
PEC wheels.
92
vehicle in the UHF band, mounting antenna sources under the vehicle becomes much more
difficult. One way this range of frequencies can be utilized is by mounting the sources on
other portions of the vehicle model. This will be further explored in Chapter 5 where UHF
sources are mounted on the sides of the vehicle body.
4.7 Summary
When antenna sources are very close to the ground, the fields will couple with the
ground and significantly reduce the radiation efficiency. If these sources are not just close
to the ground, but mounted underneath a vehicle, the reduction in efficiency is substantial.
In order for the bottom of the vehicle to be a viable alternative mounting location, the
coupling of the source with the ground must be prevented, or at least mitigated.
For antennas in the HF and VHF bands, the parallel plate source can be integrated
with the vehicle to greatly increase the efficiency of a source mounted on the bottom
position. Most of the variation in the efficiency is the result of standing waves produced
within the parallel plates. Loading from the ground will exacerbate the variations in
efficiency. The efficiency of the source is increased as the separation of the plates is
reduced.
At UHF frequencies, the practicality of omni-directional bottom mounted sources
is decreased. The modified monocone, which is a practical realization of the VED, offers
the highest efficiency of the bottom mounted sources in UHF. It also has the strongest
electric field in proximity to the vehicle while offering the lowest WoW. However, the
93
inclusion of PEC wheels greatly increases the WoW. For the UHF band, a small, vertically
polarized source offers the best performance for underneath the vehicle.
94
Chapter 5
Side Mounting of Spiral Antennas
This chapter considers mounting wideband, circularly polarized antennas on the
sides of vehicles as an alternative mounting location to top and rear locations. By mounting
a source on the side of the vehicle body, the visual profile of the antenna system is reduced
and it can also be used as a sectorial source for either communications or jamming
applications. Two different mounting configurations are considered: flush and offset
mounting. With flush mounting, the antenna face is flush with the side of the vehicle body.
Offset mounting has the face of the antenna displaced from the vehicle body.
This chapter also considers how the antennas can be efficiently modeled on the
vehicle side and which mounting configuration offers the best performance. As well, the
role of the grounds and mounting on antenna performance will be explored.
5.1 Introduction to Spiral Antennas
The spiral antenna belongs to a family of frequency independent antennas. [13]
Truly frequency independent antennas maintain a constant impedance and radiation
characteristics that are completely independent of frequency. Such an antenna would
require an infinitely large aperture with an infinitely small feed to eliminate the lower and
upper bounds on frequency. As well, the feed for the antenna would also need to be
frequency independent [29]. Frequency independent antennas eschew length in favor of
angles to define their geometry. This avoids geometries with fixed lengths and allow for
95
wide bandwidth. There is also a focus on self-complementary structures in frequency
independent antenna geometry.
The frequency independence relates to the concept of active regions, or bands, with
a current that radiates. For antennas with circular geometries like the spiral antenna, the
active regions are found along a ring with a circumference equal to an integer multiple of
the wavelength. The currents between the active region and the feed operate in a
transmission line mode and do not radiate. Beyond the active region, the currents will have
a low magnitude as most of the power would have been lost due to radiation. Because of
the spiral antenna’s smooth shape, there is a continuous active region as the frequency is
changed. This allows for the same electrical performance over a wide frequency range [13].
In this study, we consider the performance of both the equiangular and Archimedean spiral.
The equiangular spiral has an exponential growth rate. The Archimedean spiral antenna
has a slow flare rate that is linearly proportional to the angle. With tightly wound
Archimedean spirals, their far field performance is indistinguishable from the equiangular
spiral with low growth rate [4].
Of course, no antenna can be truly frequency independent. Rather, antennas are
considered to be frequency independent if they maintain a consistent radiation pattern and
impedance over an extremely wide bandwidth. Typically, this bandwidth is no more than
a few octaves. Common frequency independent antennas include the spiral, log-periodic
[30], sinuous [31] [32], and modulated arm width (MAW) spirals [33] [34]. When these
antennas are planar, they are bidirectional. But, can be made unidirectional by introducing
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a cavity. This cavity is typically filled with absorbing material that significantly degrades
the efficiency. The cavity also introduces a fixed length that will deteriorate the frequency
independence.
For a two-arm spiral, the two arms are fed 180 degrees out of phase at the center
(Figure 5.1, points F1 and F2). This feeding results in currents on one arm to be opposite to
those on the other. The arms are the same length and therefore have equal phase shifts from
the feed to the ends of each arm This maintains the current directions along the length of
the spiral arms. When the circumference is one wavelength, the currents will be within the
active region. At the points labeled A, B, and C in Figure 5.1, the magnitudes of the
currents will be close to equal. There will be a phase shift of 180 degrees from point A and
A’ due to the half wavelength distance between them. The other adjacent points (A, B’ and
A’, B) on the other arms are also in phase due to the 180 degree phase shift from the half
turn. This results in the electric fields adding in phase in the broadside direction. [13]
Between the active region and the feed, the circumference is electrically small. This
results in very low radiation from the region and can be considered a transmission line
mode. Beyond the active region, the currents should be very low as power is radiating away
in the active region. However, there can still be significant currents beyond the active
region. These currents can be reflected back from the ends of the spiral arms and will
degrade the performance. Resistive loads are commonly added to the ends of the spiral
arms to absorb these currents and reduce their reflections [13].
97
A consequence of both this feeding and geometry is that the resulting radiation is
circularly polarized. A quarter turn around the spiral results in a 90 degree phase shift. The
currents at C will be 90 degrees behind the currents at A. As well, these currents are
orthogonal to each other. These currents will also have equal magnitudes. These three
conditions are sufficient for circularly polarized radiation.
The growth rate of the Archimedean spiral is linearly proportional to the angle and
the geometry of the spiral is defined by
𝑟 = 𝑟0𝜃 + 𝑟1, (54)
Figure 5.1. Two-arm Archimedean spiral depicting the currents in the feed and active
region. Points A, B, C, A’, B’, C’ are on the active region of the antenna when the
from the currents at points A, B, A’, B’ and C, C’ radiate constructively in broadside
direction.
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where the inner radius of the spiral is r1, and r2 is the maximum radius (Figure 5.2). The
proportionality constant ro is determined based on the width of the spiral arms (w) and the
spacing between the arms (s). For a self-complementary spiral, the spacing between the
turns is equal to the width of the arms. The proportionality constant is determined by
𝑟0 =𝑠 + 𝑤
𝜋.
(55)
Figure 5.2. Geometry and parameters for a two arm Archimedean spiral.
The spacing between the arms for the two-arm spiral is determined by
𝑠 =𝑟2 − 𝑟12𝑁
− 𝑤, (56)
or by
99
𝑠 = 𝑤 =𝑟2 − 𝑟14𝑁
, (57)
for self-complementary spirals. In this case, N is the number of turns for each arm.
In this report, both two and four-arm spiral antennas are considered. For four-arm
spirals, the proportionality constant is given by
𝑟0,4 𝑎𝑟𝑚 =4𝑤
𝜋, (58)
and for a self-complementary spiral, the width of the spiral arms is determined as
𝑤4 𝑎𝑟𝑚 =𝑟2 − 𝑟18𝑁
. (59)
The growth rate is one of the design parameters determining performance of an
antenna. The smaller the growth rate, the more turns required for a given maximum radius.
A smaller growth rate helps maintain the frequency independence of the spiral antenna
[35]. The growth rate also influences the axial ratio off the broadside of the antenna. This
will affect the field of view of the spiral antenna while larger growth rates will deteriorate
the field of view of the spiral antenna [35].
The turn on and cutoff frequencies of the spiral antenna are based on the maximum
radius r2 and the inner radius r1, respectively. Because the active region exists where the
circumference is a wavelength, the larger the radius is, the lower the turn on frequency.
Similarly, the inner radius of the center of the spiral and its feeding structure determine the
smallest circumference and highest frequency:
100
𝑓𝑙𝑜𝑤 =𝑐
2𝜋𝑟2, (60)
𝑓ℎ𝑖𝑔ℎ =𝑐
2𝜋𝑟1. (61)
Equiangular antennas operate on the same principles as the Archimedean spirals,
but have an exponentially growing arm that is completely described by angles. The arm of
the equiangular spiral is defined as [15]
𝑟 = 𝑟0𝑒𝑎𝜃 (62)
Where r0 is the inner radius and a is a constant defined as
𝑎 =1
𝑡𝑎𝑛𝛼 (63)
The wrap angle, α, is determined by the expansion factor which is the ratio of the
increase in radius in a single turn of the spiral arm. The expansion factor (EF) is then
determined by the number of turns and the inner and outer radius.
𝐸𝐹 = (𝑟0
𝑟𝑜𝑢𝑡𝑒𝑟)1/𝑡𝑢𝑟𝑛𝑠 (64)
Once the expansion factor is determined, a can be found by
𝑎 =ln (𝐸𝐹)
2𝜋 (65)
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Like the Archimedean spiral, the region between the feed and active region act like
a transmission line. In order to maximize the amount of power radiated, a minimum of
power should be lost to resistive losses in this transmission line region. [15]This can be
accomplished by ensuring the arm length is not too long. The loss can be estimated by
modeling the arm as a coplanar transmission line. The length of the arm is determined by
the expansion factor [15] as
𝐿 = (𝑟𝑜𝑢𝑡𝑒𝑟 − 𝑟0)√1 +1
𝑎2 (66)
Typical values for a are around 0.22 to achieve a close enough wrap of the spiral
arms while keeping the arm length relatively short. The second edge of the spiral arm is
simply the arm rotated by an angle δ. This is determined based off the number of arms and
the gap to arm ratio. [15]
𝛿 =2𝜋
𝑁(1 +𝑔𝑎𝑝𝑎𝑟𝑚)
(67)
N is the number of arms and the arm to gap ratio determines the thickness of the metal arm
compared to the space between the arms. For a self-complementary spiral, the gap to arm
ratio is one.
The planar spiral antenna produces a bidirectional beam that is orthogonal to the
plane of the spiral. However, it is frequently desirable to have a unidirectional antenna, for
example, to integrate the antenna on a vehicle. Cavity backing is used to allow radiation in
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only a single direction. However, this adds a fixed physical dimension to the spiral antenna
and will degrade the frequency independent behavior of the antenna. This can be mitigated
by filling the cavity with absorber, but comes at the cost of efficiency [36]. For these
studies, the cavities are air filled and absorber is not used.
5.2 Standalone Sources
For now, two different geometries are considered: Archimedean two arm spiral and
four-arm spiral antennas. Both of these spiral antennas are considered with and without a
cavity backing (Figure 5.3).
5.2.1 Antennas Parameters
Two-Arm Spiral Antenna
The two arm spiral has two identical spiral arms that are fed 180 degrees out of
phase with each other. The two-arm spiral antenna used for this study is a self-
complementary Archimedean spiral with a diameter of 25.4 cm and an inner radius of
0.8cm (Figure 5.3). This gives a turn on frequency of 375MHz and an upper cutoff of
5.9GHz. There is a ring loading the spiral to improve the low frequency performance [28].
It is designed to operate within the UHF region from 300MHz to 3GHz. It has a 5.08cm
cavity backing. Each arm has five turns and is terminated with 200Ω loads to the cavity.
The resistance of the loads matches the nominal mode impedance of the spiral.
Four-Arm Spiral Antenna
The four-arm spiral undergoes a similar process, but each arm is fed 90 degrees out
of phase with the adjacent arm to achieve the same results [3]. Like the two-arm spiral, the
four-arm spiral is a self-complementary Archimedean spiral that operates in the UHF
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region. It also has a 25.4cm diameter and an inner radius of 0.8cm (Figure 5.3). The arms
have two turns each. It also has a 5.08cm cavity backing and ring loading. Theoretical turn-
on and upper cutoff frequencies are the same as for a two-arm spiral antenna (375 MHz
and 5.9 GHz, respectively). The arms have two turns and are terminated in 130Ω loads to
the cavity. The loads are matched to the impedance of the four-arm spiral.
Figure 5.3. Geometry of the two (left) and four-arm (right) spiral antennas both with and
without the 5.08cm cavity. The spirals without the cavity have a ring load around the arms.
5.2.2 Numerical Model
The spiral antenna is modeled using MoM. The model is fed in the center with
either two or four edge ports, for the two and four-arm spiral respectively (Figure 5.4). For
the two-arm spiral, there is an edge port at the center of the spiral feeding the two arms.
For the four-arm spiral, there are four edge ports using star feeding of the spiral arms.
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Figure 5.4. Detail of feeding structure for the two (a) and four-arm (b) spirals as well as
the location of the excitation sources for the spiral arms.
5.2.3 Free Space Antenna Performance
The two and four-arm spiral antennas are considered both with and without the
cavity. The spirals without the cavity have a ring along the spiral.
Without the cavity, the four-arm spiral has a clear turn on at 500MHz and 400MHz for the
two-arm spiral, as seen in Figure 5.5 (a). The inclusion of the cavity backing degrades the
VSWR performance at the lower end. For both the two and four arm antennas, the VSWR
is consistently less than 2 after 750MHz with the cavity.
The use of four arms instead of two significantly improves the boresight axial ratio.
The axial ratio of the four-arm spiral is close to 0 dB both with and without the cavity over
the entire frequency range, while the boresight axial ratio for the two-arm spiral is typically
greater than that of the four-arm spiral, seen in Figure 5.5 (b). The introduction of the
reflective cavity degrades the two-arm spiral’s axial ratio, especially at lower frequencies.
105
Considering the degradation in the axial ratio of the two-arm spiral compared to the
four-arm spiral, the four-arm spiral is initially a more promising source. A more thorough
comparison on the role of the number of spiral arms will be discussed later in this chapter.
5.3 Mounting Antennas on Vehicle Side
5.3.1 Model Description
Two methods of mounting the spiral antennas are considered: flush and offset
mounting (Figure 5.6). The flush mounting has the plane of the spiral antenna flush with
the vehicle body. A 5.08cm cavity is behind the antenna and embedded in the vehicle body.
Loads are placed at the end of the spiral arms connecting the arms to the cavity walls. The
resistance of the loads is the same as in free space configuration. The offset mounting has
the face of the antenna 5.08cm from the vehicle body. It does not use a cavity, but uses the
face of the vehicle body in much the same way as the back of a cavity. There is a 28.5cm
Figure 5.5. (a) VSWR and (b) boresight axial ratio of the two- and four-arm spiral antennas
with and without the cavity in free space. The VSWR performances of the antennas are
similar, but the two-arm spiral has higher axial ratio.
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diameter ring around the offset spiral that is connected to the ends of the spiral arms with
loads. Both the vehicle model and antenna sources are modeled as PEC.
Figure 5.6. (a) Geometry of the vehicle side and location of antenna source and (b) possible
mounting strategies. View of reduced model with flush mounted spiral (c) and side views of
both the flush and offset mounted spirals with the reduced model (d).
Because this study is performed in the UHF region from 300MHz up to 3GHz, the
traditional method of moments is prohibitively computationally expensive. As discussed
in Chapter 2, the vehicle model is reduced to only the side of the vehicle body. This side
is then modeled using physical optics while the antenna itself is modeled with method of
moments. This significantly reduces the memory and time requirements while maintaining
good agreement.
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5.3.2 Choice of Mounting Approach
Mounting the four-arm spiral antenna in both the flush and offset configurations is
considered. In all cases, the vehicle model is placed above dry sand because it is the worst
case scenario in terms of the efficiency.
VSWR
The nominal impedance for the four-arm spiral is 130 Ω. It is used to measure the
VSWR for both the offset and flush mounting (Figure 5.7). The offset mounting exhibits
a sharp turn on frequency seen with the four-arm spiral in free space without the resonate
cavity. This is due to the presence of the ring around the spiral in the offset position. The
VSWR between the two mountings is then the same with some deviation near 950 MHz.
Figure 5.7. VSWR for the four-arm spiral in either the offset or flush mounting position with
respect to the nominal impedance for the four-arm spiral, 130Ω.
Efficiency
The efficiency of the offset mounted antenna significantly outperforms that of the
flush mounted spiral at frequencies below 700 MHz, while at higher frequencies their
500 1000 1500 2000 2500 30001
1.5
2
2.5
3
3.5
4
4.5
5
Frequency [MHz]
VS
WR
Offset
Flush
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efficiencies are comparable (Figure 5.8). Notice that a large fraction of power is dissipated
in the loads at the low frequency end. Close to the turn on frequency, the active region is
located on the outer spiral edge, making currents at the loads relatively strong. The offset
mounted spiral has a ring, and therefore more effectively radiates the currents at lower
frequencies. The flush mounted antenna does not have the ring, and currents are dissipated
by the loads.
Figure 5.8. (a) Efficiency for the flush and offset mounting of the four-arm spiral over dry sand
ground and (b) the percentage of power dissipated in the loads and at the end of the spiral arms
rather than radiated
Axial Ratio
Figure 5.9 (a) shows comparable axial ratios between the flush and offset mounting
in the direction of broadside above the dry sand ground. This performance is similar for
other ground types. An elevation cut of the axial ratio along the boresight plane is also
considered. This measures the axial ratio in the elevation plane and shows the region near
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boresight that still exhibits a low axial ratio. Between the offset and flush mounting, Figure
5.9 (b) shows both have the same axial ratio up to 30 degrees from the ground plane.
Figure 5.9. (a) Axial ratio for the offset and flush mounting over dry sand for the four-arm
spiral. While there is more variation found in the flush mounting, the axial ratios for each
position are very similar. (b) Elevation cut of the axial ratio over dry sand for the offset
and flush mounting.
Gain
Mounting the antenna in the offset position lowers backlobes and increases the gain
at lower frequencies and offers similar gain with lower backlobes as the frequency is
increased, as seen in Figure 5.10. This holds true for both polarizations and similar
performances were observed at other frequencies.
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Figure 5.10. Far field elevation gain patterns for the offset and flush mounting over dry
sand at various frequencies for the four arm spiral. The gain of the offset mounting is
greater than or equal to the flush mounting.
Conclusion
Both the offset and flush mount offer good performance in VSWR and axial ratio.
The offset mount has slightly higher VSWR near 950MHz, but the flush mounting has a
slightly less stable axial ratio. The efficiency of the offset mount is greater on the lower
end but is slightly lower at the higher end. The gain of the offset is higher at the lower end,
and equal at higher frequencies. Due to the slight advantage of the offset mounting in gain
and efficiency and comparable performance in axial ratio and VSWR, it will be explored
further.
5.3.3 Choice of the Number of Arms
The two and four-arm self-complementary Archimedean spiral antennas offer very
wide bandwidth and circular polarization. This facilitates the study by including the effects
of different polarizations over a wide frequency range. In this way, the results from this
study can be applied to other side mounted antenna sources even if they are not spiral
antennas or circularly polarized sources. This section will primarily compare the axial ratio
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and efficiency of the two and four-arm spiral antennas. Unless otherwise noted, the
antennas are offset mounted and the vehicle model is placed above dry sand ground.
VSWR
The VSWR of the two and four-arm spiral offset mounted configuration are
considered (Figure 5.11). Both offer similar performance with a very flat VSWR beyond
1.1GHz. After the turn on frequency, the VSWR does not exceed 2 when considered with
respect to the nominal impedance for the two and four-arm spirals.
Figure 5.11. VWSR for the 2 and 4 arm spiral with respect to their nominal impedances
over the frequencies of interest.
Efficiency
Efficiencies of both antenna configurations are very similar to each other (Figure
5.12). However, the efficiency of the four arm-spiral is slightly lower than the two arm
spiral. This is due to the increased power dissipated in the four arm spiral’s loads. For both
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antennas, the efficiency over dry sand is above 50% for frequencies above 1GHz and it
approaches 60%.
Gain
The gain pattern of the two and four-arm spirals offset mounted over dry sand are
compared (Figure 5.13). Results are very similar. Some relative variation over frequency
range is expected due to variations in the efficiency.
Figure 5.13. Gain pattern of the two and four-arm spirals offset mounted over dry sand.
Figure 5.12. (a) Efficiency of the two and four-arm spiral antennas over a dry sand ground.
(b) The power dissipated in the loads.
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Axial Ratio
The axial ratio at boresight for the two antennas is compared in Figure 5.14. As
expected, the four-arm spiral offers a much lower axial ratio over the entire frequency range
of interest. [35] This behavior is maintained when the antennas are placed over lossy
grounds, however there is some degradation of the axial ratio especially at the lower ends.
Generally, the axial ratio is consistent when offset mounted to the side of the vehicle with
the exception of a peak in the axial ratio at 2.95GHz.
Figure 5.14. Axial ratio for the two and four arm antennas offset mounted on the vehicle
over dry sand.
This is also observed in the far-field patterns of the two and four arm spirals in free
space (Figure 5.15). The two-arm spiral has a greater amount of pattern rotation that is
observed as the theta and phi polarizations change dominance as the frequency changes.
This correlates with its poorer axial ratio performance compared to the four-arm spiral. The
four-arm spiral has both polarizations nearly equal in magnitude at boresight over the
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majority of the frequency band. This is desirable for jamming applications, as the
polarization and orientation of the jammed antenna is unknown.
Figure 5.15. Theta and phi polarizations of the far field gain patterns of the two and four arm
spirals in free space. The four arm spiral patterns are the same at boresight and corresponds
to its lower axial ratio.
The four-arm spiral antenna has a consistently lower axial ratio over the entirety of
the UHF band compared to the two arm spiral. The two arm spiral’s poorer axial ratio
performance below 1.5GHz is worsened by the presence of grounds. While the inclusion
of the grounds does deteriorate the four arm spiral performance, it is much less susceptible
to variations in the ground parameters than the two arm spiral. Both the two and four arm
spirals achieve a good level of efficiency approaching 60%. For these reasons, the four arm
spiral antenna is preferred over the two arm spiral.
5.3.4 Effect of Grounds on Spiral Antenna Performance
Now that the baseline performance of the two and four-arm spiral antennas is
understood, a more comprehensive study of the role of the grounds on the antenna
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performance can be undertaken. The four-arm spiral antenna is considered for these studies
because of its superior performance compared to the two-arm spiral. In all cases, the
antenna is offset mounted.
VSWR
The impedance of the four-arm spiral does not change with the change in grounds
(Figure 5.16). This results in a VSWR that will be consistent despite changes in the
grounds and their electrical parameters. Insensitivity of the antenna impedance to grounds
can be explained by the fact that the antenna height is electrically large at frequencies above
the turn on frequency.
Figure 5.16. VSWR of the four-arm spiral offset mounted over the dry sand, asphalt, and
wet soil.
Efficiency
As discussed in Chapter 3, the more conductive ground type causes the higher
propagation efficiency for the given electrical height (Figure 5.17). This is a small peak in
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efficiency near 330MHz due to the presence of the ring around the four-arm spiral that
improves its low frequency performance. All ground types have efficiencies greater than
50% above 1GHz and quickly rise to greater than 60%.
Figure 5.17. Efficiency of the offset mounted four arm spiral antenna source over the various
ground types.
Gain
The side mounted source is intended to be used as a sectorial source where each
source covers a different quadrant. Having the coverage be consistent over different ground
types is desirable. As can be seen in Figure 5.18, the elevation pattern is nearly identical
over the three different grounds below 30 degrees. The azimuthal pattern was measured at
5 degrees above the grounds. Like the elevation pattern, there is very little difference
between the patterns over the different grounds. The azimuthal pattern has a 3dB
beamwidth of 68 degrees at 1GHz and the elevation 3dB beamwidth is 5.5 degrees.
Frequency dependence of the elevation pattern along vehicle is presented in Figure 5.19,
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azimuthal pattern at 5 degree above ground is shown in Figure 5.20. In all cases the ground
is dry sand.
Figure 5.18. Azimuth and elevation far field gain patterns of the offset four-arm spiral over
dry sand, asphalt, and wet soil at 1GHz. The far field patterns are constant over the different
grounds over the frequencies of interest.
Figure 5.19. Elevation gain patterns of the co and cross polarization of the four-arm spiral
offset mounted over a dry sand ground.
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Figure 5.20. Azimuthal gain patterns of the offset four-arm spiral over dry sand. The
azimuthal pattern was computed at 5 degrees from the ground.
One thing that becomes apparent as the frequency is increased, is that the azimuthal
beam is not symmetric. This is due to the asymmetry of the vehicle model where the
‘engine’ of the model is shorter than the ‘trunk.’ Also, as the frequency increases, the
strength of the cross polarization increases. The reflection of the circularly polarized source
of the PEC backing results in a reflection polarized with the opposing sense. This
destructive interference is what drives the deterioration of the axial ratio discussed earlier.
Axial Ratio
The axial ratio is measured over the dry sand, asphalt, and wet soil ground types
for the offset mounted four-arm spiral antenna source (Figure 5.21). The more conductive
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the ground type, the higher the axial ratio. The wet soil ground type has a significantly
higher axial ratio than the other ground types, since it supports a vertical polarized signal
much better than a horizontally polarized one. There is a peak in the axial ratio at 2.9GHz
for all ground types. It appears when the distance between the antenna and the vehicle wall
is half of a wavelength. This results in destructive interference between original wave and
the wave reflected from the vehicle.
Figure 5.21. Axial ratio of the offset mounted four arm spiral antenna over the different
grounds.
As expected, axial ratio deteriorates further from the boresight (Figure 5.22). The
more conductive grounds have smaller angle with small axial ratio. Generally, the axial
ratios are low up to about 15 degrees above the ground. As the frequency increases, the
region of small axial ratio decreases from about 25 degrees at 500MHz to about 15 degrees
at 1GHz. Similarly, the axial ratio is measured in the azimuthal plane. In this case, it is
measured 5 degrees up from the ground. As expected, the axial ratio for the more
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conductive wet soil is higher than for the dry sand and asphalt. There is also a limited range
to the relatively low axial ratio. There is about 30 degrees before the axial ratio rapidly
increases. This illustrates the limitations of a side mounted circularly polarized source.
Outside of its range from boresight, the circular polarization breaks down. In order to
provide uniform coverage around a vehicle, multiple sources would need to be placed
around the vehicle body and each antenna operated as a sectorial source to provide the
required coverage.
Figure 5.22. Elevation (a) and azimuthal (b) cuts of the axial ratio of the four arm spiral over
dry sand, asphalt, and wet soil. The elevation cuts are along the boresight axis while the
azimuthal cuts are from 5 degrees above the ground.
Depolarization
The depolarization is the ratio of the magnitude of the horizontal electric field
component to the vertical component. Unlike the axial ratio, the depolarization is measured
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along specific distances from the source to determine how the ratio of the fields changes
with distance. For a circularly polarized wave, this ratio is unity. If the source is not
perfectly circularly polarized or propagating in an inhomogeneous media, the ratio will
diverge from unity. The electric field is measured at 0.765m above the ground, the same
height as the center of the source.
Figure 5.23 shows the depolarization of an offset mounted four-arm spiral antenna
over both dry sand and wet soil. The more conductive wet soil will have stronger
depolarization than the less conductive dry sand.
Figure 5.23. Depolarization of circularly polarized signal over both dry sand (a) and wet soil
(b) over a variety of frequencies.
There is also an increase in the distance before the wave exhibits a constant
decaying behavior due to interactions between the source and the vehicle. For the dry sand,
this constant decaying behavior occurs by 20m from the source for all frequencies. The
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source vehicle interactions are even stronger over the wet soil and do not die down to a
constant decaying behavior until beyond 80m.
Overall, the four-arm spiral antenna offers a lower and more stable axial ratio and
radiation patterns compared to the two-arm spiral over the entirety of the UHF band. Both
the two and four arm spirals have comparable efficiency. Propagation over more
conductive soils will increase the efficiency at the cost of a deteriorated axial ratio. The
offset mounting offers lower backlobes and higher or comparable gain while maintaining
comparable axial ratios.
5.4 Variations and Reductions to Vehicle Model
The previously discussed sources were mounted on the side of the vehicle model.
This model was greatly simplified to allow for efficient calculations. One of the
simplifications was the removal of windows from the vehicle model. The windows were
added back to the side vehicle model to determine their potential effects on propagation
(Figure 5.24).
Figure 5.24. Geometry of the vehicle side with the windows. Both windows are the same
size.
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The four-arm spiral is then offset mounted onto this vehicle model over dry sand.
However, there is little change in the main direction of the antenna and only some small
differences with the backlobes (Figure 5.25).
Figure 5.25. Near-field and far-field patterns of both geometries. Both have very good
agreement in the direction of propagation with differences in the backlobe.
5.5 Spiral-Helix Antenna
5.5.1 Antenna Parameters
The spiral-helix antenna is an equiangular four-arm spiral with a helix structure at
the end of the spiral arms. The inclusion of the helix terminations for the spiral arms
increases the lower bandwidth of the antenna without increasing the diameter of the spiral.
The design is based on the work of Dr. Radway and James Bargeron [35] [26]. The helix
is 5.08cm long and makes ¾ turns and the spiral aperture has a diameter of 15.24cm
(Figure 5.26). This would have a theoretical turn on of 627 MHz, but the presence of the
helix decreases the turn on frequency to allow for performance across the entirety of the
UHF band. The spiral is also self-complementary with four arms making two turns. The
helix arms are terminated with 150Ω loads into a ground plane.
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The spiral aperture of the spiral-helix antenna is 5.08cm from the ground plane. At
lower frequencies, this is electrically close to the PEC backing (≈λ/20 at 300MHz). This
results in strong coupling with the backing that will degrade the performance. However,
the helix structure radiates in the boresight direction. This reduces the coupling between
the spiral and the ground plane [35]. Like the previous antennas, the cavity is air filled, but,
unlike the previous antennas, the spiral helix cavity has a 1.27cm tall cylinder with a
diameter of 14.23cm at the bottom of the cavity. The presence of this cylinder increases
the frequency for which destructive interference from the cavity occurs. This decreases the
cavity depth from 5.08cm to 3.81cm. This improves the higher frequency performance by
removing the half wavelength interference that would occur for a 5.08cm cavity near 3GHz
[29].
The considered system is a spiral helix antenna operating as either a single element
or with multiple elements in an array. The spiral helix antenna is mounted onto a square
Figure 5.26. Model geometry of the spiral-helix antenna mounted on the PEC plate over the
ground.
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0.83m x 0.83m PEC plate to simulate the side of the vehicle body. This is then considered
both in free space as well as over dry sand, asphalt, and wet soil ground types.
5.5.2 Model Description and Validation
As the complexity of the source increases, the computational cost of the simulations
increases; thus the vehicle model needs to be further reduced. This is especially important
for more complex sources like the spiral-helix antenna. Considering the only effect is a
minor change in the backlobes with the inclusion of the windows to the vehicle model,
there is a prospect for further reduction in the complexity of the model.
In this section, the vehicle is modeled as a square plate 0.83m by 083m (Figure
5.27). This dimension corresponds to the smallest length of the rear of the rough Humvee
model discussed in previous chapters. Like the vehicle side, the PEC plate is modeled using
physical optics while the antenna is modeled with the method of moments. The fields
around these sources for these two different model geometries are then compared.
Figure 5.27 Original dimensions of reduced vehicle model (a) and the antenna
source mounted on the PEC plate (b).
The reductions to the model are verified that they maintain good agreement with
the previous models (Figure 5.28). Both the elevation and azimuthal patterns of the two
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geometries have very good agreement. There are discrepancies between the two
geometries’ backlobes. This is acceptable for this study, as it is concerned with the use of
these antennas as sectorial sources.
The good agreement between the square PEC plate model and vehicle side model show
that the antenna performance is determined by the model geometry in the immediate
vicinity of the antenna. Details farther from the source play a role in the smaller features
and backlobes of the results. As well, it verifies that the results for the vehicle side can be
applied to other side locations of the vehicle such as the front or rear face.
Figure 5.28 Elevation and azimuth cuts of the far field of both the vehicle side and plate
geometries over the dry sand ground. There is good agreement between the two with
discrepancies in the backlobes.
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5.5.3 Antenna Performance
A variety of parameters are considered, including the axial ratio, efficiency, and
VSWR. Using the spiral helix antenna for a two element array and as a dual polarized
system is also considered.
VSWR
The impedance of the spiral helix antenna experiences some variation, but has a
nominal impedance of 150Ω. The inclusion of different ground types does not affect the
impedance of the antenna. In contrast to the results for two-arm and four-arm spirals, the
VSWR of the spiral-helix antenna does not have a sharp turn on and remains below 2 over
the frequency range of interest.
Figure 5.29. VSWR of the spiral helix antenna source mounted on the plate with respect to
its nominal impedance of 150 Ω.
Efficiency
The efficiency of the spiral-helix antenna increases from nearly zero to reach a local
maximum around 1GHz (Figure 5.30). Note that at frequencies below 1 GHz, the
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efficiency of the helix-spiral antenna is much less than that of the offset mounted spiral
antenna shown before. At these frequencies, the main loss mechanism is the total loss in
loads. The main features of the efficiency curves are that they are not ground dependent.
For example, the dip in efficiency at 1.5GHz correlates to the increased power dissipated
in the loads. Like previous sources, the wet soil has greater efficiency than the more lossy
dry sand and asphalt ground types. The dry sand reaches a maximum efficiency of about
65% while the wet soil reaches a maximum of about 75%.
Figure 5.30. (a) Efficiency of a single spiral helix mounted on a 0.83m x 0.83m PEC plate
over various grounds and (b) the percentage of power dissipated in the loads of the spiral-
helix antenna. A significant portion of the power below 1GHz is absorbed by the loads.
Gain
The spiral-helix antenna offers a fairly stable gain pattern over the UHF band, but
does experience a dip in gain near 1.5GHz associated with the increased power dissipated
in the loads (Figure 5.31). The realized gain of the source is greater than zero above
500MHz.
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Figure 5.31. Boresight gain of the mounted spiral-helix in free space.
Axial Ratio
Initially, the spiral-helix antenna is modeled in free space to determine a baseline
performance. As seen in Figure 5.32, the axial ratio for this source in free space is very
low.
Figure 5.32. Axial ratio of the spiral helix antenna above dry sand, asphalt, and wet soil.
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As with the two and four-arm spirals, the inclusion of the ground increases the axial
ratio. The presence of the ground mostly increases the axial ratio at lower frequencies
below 1GHz. The more conductive wet soil exhibits a higher axial ratio than the more lossy
dry sand and asphalt grounds. This performance tracks well with the four-arm spiral
antenna performance.
5.6 Spiral-Helix Two-Element Antenna Array
The spiral-helix antenna array is a two element array of the spiral-helix antennas
with their centers separated by 16.24cm in the vertical axis (Figure 5.33). The midpoint
between them is at 0.765m above the ground.
Figure 5.33. Geometry of the two element spiral helix array mounted on the PEC plate.
The array configuration decreases the beam width at 1 GHz and 3 GHz compared to single
element (Figure 5.34). This may decrease the amount of power directed towards the
ground and increase the amount of power directed along or above the ground and therefore
increase the efficiency of the source.
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Figure 5.34. Elevation cut of the gain of a single spiral-helix and two element spiral-helix
antenna array in free space.
5.6.1 Array performance
Efficiency
As expected, efficiency of array configuration is higher than that of a single element
(Figure 4.35 (a)). The difference is about 10% in the case of the dry sand, but it is nearly
identical for the wet soil. As with previous studies, the increased conductivity of the wet
soil increases the efficiency performance.
Axial Ratio
The axial ratio performance of the array is degraded from the single element
performance (Figure 5.35 (b)). At the lower end, the axial ratio is increased. Above
1.25GHz, the axial ratio is less stable and varies rapidly. This is exacerbated for more
conductive soils. This slight increase in efficiency comes at the cost of a more complex
system and a degraded axial ratio.
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Figure 5.35. (a) Efficiency of the spiral helix and spiral helix array over both dry sand and
wet soil grounds. The efficiency of the array is slightly higher than with a single element.
(b) Axial ratio for the spiral-helix and spiral-helix array over both the dry sand and wet soil
grounds. The spiral helix array has greater degradation of the axial ratio on the lower end
and is less stable on the higher end.
Dual Polarized System
The two spiral-helix elements do not have to be used as an array. Instead, each
element can be used for each polarization sense; one can be RHCP and the other LHCP.
This allows for the transmission and reception of both polarizations. Like the array, there
is some coupling and scattering between the two antennas. This can be used to an advantage
as the beam of the top source is squinted up and away from the ground. The bottom source
is squinted towards the ground (Figure 5.36).
The efficiencies of all considered systems are very similar (Figure 5.37). The use
of the spiral helix as an array slightly increases the efficiency. The dual polarized system
has a slightly lower efficiency than the single element near 1GHz. This is due to the power
radiated by the system being split between the two sources. Because the bottom source is
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bent downwards toward the ground, more of its power is dissipated than from a source with
a more even pattern.
Figure 5.37. Comparison of the spiral helix antenna as a single element, in a two element
array, and as a two element system with opposite polarizations over dry sand.
This can allow for the sources to be used for different applications. For example,
one may need the higher efficiency and propagation away from the ground for
Figure 5.36 Elevation cuts of the RHCP and LHCP fields of the dual polarized, two element
spiral helix system.
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communication applications and while also requiring sources that have strong fields near
the ground for jamming applications.
5.7 Summary
The offset modeling can easily utilize the hybrid MoM/PO code to significantly
reduce the computational cost of the simulations. This offset mounting offers higher or
comparable gain while having lower backlobes compared to the flush mounting. The offset
mounting of the four-arm spiral also has higher efficiency while maintaining similar axial
ratios.
The four-arm spiral offers a lower and more stable axial ratio when compared to
the two arm spiral antenna, all while maintaining a comparable efficiency. While the
performance of the two-arm spiral is good without a cavity, the introduction of the metallic
backing greatly decreases its axial ratio performance compared to the four-arm spiral.
The spiral-helix antenna offers improved performance over the basic four arm
spiral. The helix structure increases the low frequency performance of the antenna. This
allows for a smaller aperture diameter compared to the original four-arm spiral. The spiral-
helix also offers increased efficiency and axial ratio for all ground types. However, soils
with higher conductivities offer higher efficiencies at the cost of a deteriorated axial ratio.
For slightly improved efficiency performance, the spiral-helix can be used as an array
element, but this comes at the cost of a deteriorated axial ratio and higher complexity.
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Chapter 6
Summary, Conclusions and Future Research
In this thesis, antenna systems mounted on vehicles while close to the ground were
studied and the effects on these systems were characterized. Based off this, alternative
mounting locations for antenna systems were examined. For sources under the vehicle, the
parallel plate source was explored. Ultra wideband spiral antennas were studied as a
potential for side mounted, sectorial sources. Section 6.1 of this chapter summarizes this
thesis and Section 6.2 discusses potential avenues for future work and research.
6.1 Summary and Conclusions
As we move forward into the 21st century, the need for electronic warfare and
electronic countermeasures is only increasing. Current antenna systems for such
applications as well as for communication applications frequently use large antennas
mounted on the roof or rear bumper of vehicles. While these can be attractive mounting
locations for ease of mounting and their radiation propagation properties, they contribute
to a large visual profile that can hinder the maneuverability and increase the vulnerability
of the vehicle.
Chapter 2 considered the theoretical basis for the computational tools used to
perform the studies found in this thesis; specifically, it discussed the method of moments,
physical optics, and a hybridization of the method of moments with physical optics. As
well, the ground and vehicle models used in this thesis were discussed.
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Computational simulations allow for accurate and relatively rapid results at a
significantly lower cost when compared to designing, building, and measuring prototypes.
FEKO is a commercially available code that allows for modeling of a variety of vehicles
over a different of real ground models. The rough Humvee model allows for faster and
efficient calculations while still maintaining accurate results for antennas at HF and VHF
frequencies compared to the complex Humvee model. At higher frequencies, even the
rough Humvee model becomes computationally expensive. By reducing the models to PEC
surfaces and using a hybrid method of moments and physical optics code, the
computational cost can be significantly reduced. This is especially important when
studying frequencies in the UHF band.
Chapter 3 discusses and characterizes three different ideal sources with the ground
models, both with and without the vehicle model. The three ideal sources considered are
the vertical electric Hertzian dipole, the horizontal magnetic Hertzian dipole, and the
vertically crossed electric Hertzian dipoles. For sources that are electrically close to the
ground, there will be significant changes in the source’s characteristics and performance.
One of the biggest differences in performance is in the overall efficiency of the source over
different ground types. For soils with more loss, like dry sand, the efficiency is much lower
than for more conductive soils, like wet soil. However, the more conductive soils will cause
a greater amount of depolarization for circularly polarized sources propagating over the
ground. This depolarization effect can be mitigated by increasing the electrical height of
the source above the ground.
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The introduction of the vehicle model also has a profound impact on the
characteristics of these sources. Three different mounting locations are considered: the top,
rear, and bottom. One that is largely dictated by the geometry of the vehicle model is the
azimuthal pattern around the vehicle model. One of the ways this is characterized is through
the parameter known as the wobble of the wave, or WoW. For sources mounted on the roof
of the vehicle, a small WoW and good azimuthal pattern uniformity can be achieved.
Sources on the top of the vehicle also are both electrically farther from the ground and also
have power reflected off the body of the vehicle and away from the ground. This increases
the efficiency of the source and results in the top mounting position having largest
efficiency of the three mounting locations. The bottom source offers a better WoW than
the rear position, but has a significantly poorer efficiency as the power is reflected off the
bottom of the vehicle model towards the ground. Improving the efficiency of sources
mounted on the bottom of the vehicle while maintaining relatively low WoW would make
this an attractive alternative mounting location.
Chapter 4 discusses sources mounted underneath the vehicle and very close to the
ground. When antenna sources are very close electrically to the ground, the fields will
couple strongly with the ground. This coupling significantly reduces the radiation
efficiency. If these sources are not just close to the ground, but also mounted underneath a
vehicle, there is a substantial reduction in efficiency. In order for the bottom of the vehicle
to be a viable alternative mounting location, the coupling of the source with the ground
must be prevented, or at least mitigated. This is the idea behind the parallel plate source.
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The parallel plate source is considered as a method for improving the efficiency for
HF and VHF sources mounted under the vehicle. Initially, the parallel plate source is
studied on its own to better understand its performance. There is a variation in the
efficiency that is the result of standing waves produced within the parallel plates. These
standing waves are determined by the geometry of the parallel plate source. These
variations in the efficiency are exasperated when the source is loaded by the ground.
Decreasing the separation of the plates may slightly improve the efficiency of the source.
A version of the parallel plate source was integrated with the vehicle model to show
improved efficiency performance for a source mounted underneath the vehicle. The
addition of small fins to the parallel plate source further improved the efficiency
performance.
Finally, three alternative sources were considered for use under the vehicle in the
UHF region. These antennas include the mode 2 spiral, mode 2 spiral-helix, and a modified
monocone antenna. The modified monocone, which is a practical realization of the VED,
has the highest efficiency for sources under the vehicle in the UHF region. It also has the
strongest electric field in proximity to the vehicle while offering the lowest WoW.
However, the inclusion of PEC wheels greatly increases the WoW. These wheels are a
reasonable approximation for the performance with rubber wheels at higher frequencies.
For the UHF band, a small, vertically polarized source offers the best performance for
sources under the vehicle. However, for higher frequencies, sectorial sources may be a
better option.
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Chapter 5 explores ultra wideband, circularly polarized sources mounted on the
sides of the vehicle models that operate as sectorial jammers. In order to understand the
performance of side mounted sources, the mounting style, ground effects, and antenna
design are considered. For the mounting, two mounting strategies are compared, flush
mounting and offset mounting. Flush mounting has the face of the antenna flush with the
vehicle body while offset mounting has the antenna face mounted away from the vehicle
body.
The offset mounting offers higher or comparable gain while having lower
backlobes compared to the flush mounting. The offset modeling can also more easily utilize
the hybrid MoM/PO code to significantly reduce the computational cost of the simulations.
The offset mounting of the four-arm spiral has higher efficiency at the lower end and
comparable efficiency for the higher end while maintaining similar axial ratios.
The four-arm spiral has a lower and more stable axial ratio when compared to the
two arm spiral antenna. Both the two and four arm spirals have similar efficiencies.
However, the introduction of the cavity backing greatly decreases the axial ratio
performance of the two arm spiral when compared to the four-arm spiral.
The four-arm spiral can be improved upon by using the helix loading. The helix
structure improves the low frequency performance of the antenna, allowing for a smaller
aperture diameter. The spiral-helix also has increased efficiency and axial ratio
performance for all ground types when compared to the Archimedean four arm spiral. Soils
with higher conductivities offer higher efficiencies, but come at the cost of a deteriorated
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axial ratio. For slightly improved efficiency performance, the spiral-helix can be used as
an array element, but this increases the complexity of the system and deteriorates the axial
ratio performance.
6.2 Future Research
There are three main avenues for future work and research: layered ground models,
optimization of the parallel plate source, and alternate ultra wideband sources for sectorial
use. This thesis considered three different uniform grounds. To better understand the
effects of grounds on propagation and antenna performance, layered ground types could be
explored. These layered ground types could be as simple as combining the already used
ground parameters, such as asphalt of variable thicknesses over dry sand or wet soil.
The addition of the fins to the parallel plate source greatly improved its
performance. If such small modifications to the geometry can yield improved performance,
there is still a lot of potential for even further improvements. The current method of exciting
the antenna yields a narrow bandwidth. Exploring potential modifications to improve the
efficiency and bandwidth of the antenna would move it forward from a concept to a
potential reality. As well, a tuning circuit could be designed and implemented to improve
its performance and allow operation across a wider range of frequencies, even if they
cannot all be covered simultaneously.
Finally, alternative frequency independent sources could be explored for mounting
on the side of the vehicle for sectorial applications. Such antennas would include the MAW
spiral, log periodic, and sinuous antenna. These antennas can offer dual polarization which
141
can be advantageous. As well, there can be opportunities for these sources or modifications
to the spiral-helix to be better suited for jamming applications by improving the efficiency
and power handling. As well, understanding how mounting and ground characteristics
might potentially affect power handling beyond mismatch loss and the power dissipated in
the loads would be useful.
142
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Appendix: Modeling Validation
In order to verify the accuracy of the developed models and obtained results, the
same computational model is simulated with different computational methods. The
simulations were performed using FEKO with either the method of moments or a hybrid
method of moments / physical optics. Ansys HFSS based on the finite element method is
also utilized to further demonstrate the validity of the herein presented results.
Figure A.1 (a) Dimensions of the rough Humvee model. The entire model is modeled as
PEC. (b) Magnitude of the electric field for different radii around the rough Humvee model
with a top mounted Hertzian source over a PEC ground for both MoM and FEM at
100MHz.
Because the applications for this study emphasize the fields near the vehicle, the
first validation test involves calculation of electric field magnitude around the rough
Humvee model (reproduced in Error! Reference source not found. (a)). The vehicle is
placed over a PEC ground and it is modeled at 100MHz with MoM and FEM. The fields
147
are calculated 10cm above the ground and as seen in Figure A.1 (b) a very good agreement
between the two different methods is obtained.
Note on high frequency methods: FEKO offers a large variety of high frequency
methods that can be hybridized with the method of moments. For electrically larger
problems, alternative solving methods include the Multi-level Fast Multipole Method
(MLFMM), Geometrical Optics (GO), Physical Optics (PO), and the Uniform Theory of
Diffraction (UTD). Unfortunately, all but the physical optics method are compatible with
an infinite ground plane and having sources and calculated fields in close proximity to that
ground. For this reason, this was the primary high frequency method that was considered
for the studies.
Figure A.2 Reduced rough Humvee model geometries. (a) Full rough Humvee model
solved with either MoM or PO, (b) Partial Humvee with just sides modeled as PO, and (c)
side of the Humvee modeled as PO.
In order to reduce the computational cost and time of modeling the rough Humvee
model, the complexity of the model is further reduced as shown in Figure A.2. These
148
models are solved at 400MHz over a PEC ground (Figure A.3). Excellent agreement
validates the computational results throughout this thesis.
Figure A.3 Near field 0.765m above the PEC ground for the alternative geometries. There
is good agreement between the full rough Humvee with MoM and the Humvee side with
PO.
