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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

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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

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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

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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

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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

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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

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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.

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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

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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

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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.

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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

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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.

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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)

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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’.

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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

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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

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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

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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)

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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.

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−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

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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

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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)

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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

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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

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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

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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.

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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)

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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]

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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.

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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.

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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]

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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]

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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

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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

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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

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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.

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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

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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

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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.

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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.

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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

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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.

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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.

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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

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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,

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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.

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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

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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.

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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

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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

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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

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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

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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

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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

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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.

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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.

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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

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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

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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:

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∇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.

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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

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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.

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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.

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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 .

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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

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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.

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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.

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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)

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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).

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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

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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

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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.

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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.

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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.

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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

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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

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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.

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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

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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

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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

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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.

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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

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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

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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.

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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

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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].

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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

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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:

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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.

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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

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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.

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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

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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

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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.