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Propagation of Electromagnetic Fields Over Flat Earth
ARL-TR-2352
Joseph R. Miletta
February 2001
Approved for public release; distribution unlimited.
The findings in this report are not to be construed as an official Department of the Army position unless so designated by other authorized documents.
Citation of manufacturer’s or trade names does not constitute an official endorsement or approval of the use thereof.
Destroy this report when it is no longer needed. Do not return it to the originator.
Army Research Laboratory
Adelphi, MD 20783-1197
ARL-TR-2352
February 2001
Propagation of Electromagnetic
Fields Over Flat Earth
Joseph R. Miletta
Sensors and Electron Devices Directorate
Approved for public release; distribution unlimited.
Abstract
This report looks at the interaction of radiated electromagnetic fields with earth ground in military or law-enforcement applications of high-power microwave (HPM) systems. For such systems to be effective, the microwave power density on target must be maximized. The destructive and constructive scattering of the fields as they propagate to the target will determine the power density at the target for a given source. The question of field polarization arises in designing an antenna for an HPM system. Should the transmitting antenna produce vertically, horizontally, or circularly polarized fields? Which polarization maximizes the power density on target? This report provides a partial answer to these questions. The problems of calculating the reflection of uniform plane wave fields from a homogeneous boundary and calculating the fields from a finite source local to a perfectly conducting boundary are relatively straightforward. However, when the source is local to a general homogeneous plane boundary, the solution cannot be expressed in closed form. An approximation usually of the form of an asymptotic expansion results. Calculations of the fields are provided for various source and target locations for the frequencies of interest. The conclusion is drawn that the resultant vertical field from an appropriately oriented source antenna located near and above the ground can be significantly larger than a horizontally polarized field radiated from the same location at a 1.3 GHz frequency at observer locations near and above the ground.
ii
Contents
Figures
1 Introduction
1
2 Problem Formulation
2
2.1 Vertical Dipole Over Earth . . . . . . . . . . . . . . . . . . . . 3
2.2 Horizontal Dipole Over Earth . . . . . . . . . . . . . . . . . . 4
2.3 Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3.1 Comparison of Field Components Near the Earth . . 6
3 Conclusion
14
References
15
Appendix. MATLAB m-Files
17
Distribution
27
Report Documentation Page
29
1 Geometry for vertical and horizontal dipole formulations . . . . 2
2 Comparison of main electric field components from a vertical (Ez) and from a horizontal (Eφ) broadside ideal dipole over a perfectly conducting ground . . . . . . . . . . . . . . . . . . . . . 7
3 Reflection coefficients for vertically and horizontally polarized plane waves of 1.3-GHz frequency incident on a flat ground for five earth-parameter cases listed in table 1 . . . . . . . . . . . . . 8
4 Fresnel reflection coefficient for a vertically polarized plane wave field compared to “reflection coefficient” as calculated by complete formulation for a vertical dipole over homogeneous earth as observed at 1-m height 1000 m down range at a frequency of 1.3 GHz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
5 Fresnel reflection coefficient for a horizontally polarized plane wave field compared to “reflection coefficient” as calculated by complete formulation for a horizontal dipole over homogeneous earth as observed at 1-m height, broadside 1000 m down range at a frequency of 1.3 GHz . . . . . . . . . . . . . . . . . . . . . . 10
iii
Tables
6 Comparison of principal fields from an ideal dipole oriented perpendicular and horizontal to a homogeneous flat earth . . . 11
7 Comparison of principal fields from an ideal dipole oriented perpendicular and horizontal to a homogeneous flat earth . . . 12
8 Effect of ground reflection on primary field components near ground for typical earth parameters . . . . . . . . . . . . . . . . 13
1 Earth parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
iv
1. Introduction
Effective military or law-enforcement applications of high-power microwave (HPM) systems in which the HPM system and the target system are on or near the ground or water require that the microwave power density on target be maximized. The power density at the target for a given source will depend on the destructive and constructive scattering of the fields as they propagate to the target. Antenna design for an HPM system includes addressing the following questions about field polarization: Should the fields the transmitting antenna produces be vertically, horizontally, or circularly polarized? Which polarization maximizes the power density on target? (The question of which polarization best couples to the target is beyond the scope of this report.) While this report does not completely answer these questions, it addresses the interaction of the radiated electromagnetic fields with earth ground. It is assumed that the transmitting antenna and the target (or receiver) are located above, but near the surface of a flat idealized earth (constant permittivity, ε, and conductivity, σ) ground. First an ideal vertical dipole (oriented along the z-axis perpendicular to the ground plane) is addressed. The horizontal dipole (parallel to the ground plane) follows.
1
2. Problem Formulation
The problems of calculating the reflection of uniform plane wave fields from a homogeneous boundary and calculating the fields from a finite source local to a perfectly conducting boundary are relatively straightforward. However, when the source is local to a general homogeneous plane boundary, it is found that the solution cannot be expressed in closed form. An approximation usually of the form of an asymptotic expansion results. The problem of an ideal dipole over a homogeneous half-space has been the topic of a number of studies starting at the turn of the last century with the solution provided by Sommerfeld (1949) and leading to the more contemporary work of Banos (1966) and King et al (1994, 1992). References such as Maclean and Wu (1993) address in detail the many approaches to solving the problem. Considerable controversy has surrounded these studies. We will not attempt to derive the solution or otherwise discuss the solution of the problem in this report. We will rely on the work of King et al for a complete and concise formulation of the problem. Figure 1 depicts the problem geometry. The King expressions have been encoded and solved in MATLAB (1984–1999). Comparisons of the field structures are provided for various source and target locations for frequencies of interest.
The expressions that follow are constrained by the magnitude of the wave numbers (k = ω(µ )1/2, ω = 2π × frequency) in each region:
|k1| ≥ 3 |k2| ,
(1)
where, after King, the subscript 2 denotes the upper half-space (air) and the subscript 1 denotes the lower half-space (earth). Also, µ is the permeability in henries per meter, ε is the permittivity in farads per meter, and σ is the conductivity in siemens per meter. The subscripts 0 and r represent free space and relative to free space, respectively.
Figure 1. Geometry for vertical and horizontal dipole formulations.
z θd
d θ
0
r1
r0 r2
Ψ
z′
Region 2 µ0, ε0, σ = 0
Region 1
ρ
µ0, ε2 = εr ε0,σ ≠ 0
y
φ
x
Looking down
2
2.1 Vertical Dipole Over Earth
The electromagnetic fields from a dipole with dipole moment I oriented perpendicular to and at a height z = d above the ground plane have three components in cylindrical coordinates. The magnetic field is symmetric about the z-axis, perpendicular to the direction of propagation. For that reason the fields will be termed transverse magnetic (TM) fields. (We will find that the horizontal dipole has TM and transverse electric (TE) components.) The formulation as provided in King et al (1994) is, referring to figure 1,
Hφ(ρ, z)
=
eik2r1 ρ
−
I 2π

2
−eik2r2 
r1
k23 k1
− + ik2 1
r1
r121
eik2 r2 2
ρ r2
π k2 r2
2 e−iP F (P )
ik2 r2
−
1 r22
 ,
(2)
Eρ(ρ, z)
=
− ωµ0I 2πk2
 
eik2r1 ρ
2
r1
+ eik2r2 ρ
2
r2
−
k2 k1
eik2
r2
z−d r1
ik2 r1
−
3 r12
−
3i k2 r13
z+d r2
ik2 r2
−
3 r22
−
3i k2 r23
ρ r2
ik2 r2
−
1 r22
−
k23 k1
π k2 r2
1
2 e−iP F (P )
  , and
(3)
Ez(ρ, z)
=
ωµ0I 2πk2
  
eik2 r1 2
ik2 r1
−
1 r12
−
i k2 r13
−
z−d 2 r1
+ eik2r2
2
ik2 r2
−
1 r22
−
i k2 r23
−
z+d r2
−eik2
r2
k23 k1
ρ r2
1
π k2 r2
2 e−iP F (P )
ik2 r1
−
3 r12
−
3i k2 r13
2
ik2 r2
−
3 r22
−
3i k2 r23
   ,
(4)
where
1
r1 = ρ2 + (z − d)2 2 ,
1
r2 = ρ2 + (z + d)2 2 ,
(5)
P
=
k23r2 2k12
k2r2 + k1 (z + d)
2
,
k2ρ
(6)
F (P ) =
∞ P
eit
(2πt)
1 2
dt
=
1 2
(1 + i) − C2 (P ) − iS2 (P )
,
(7)
and C2(P ), S2(P ) are the Fresnel integrals as defined in Abramowitz and Stegun (1970). Since
C2
(P )
+
iS2 (P )
=
1 2
(1
+
i)
−
i eiP w
√ iP
2
,
(8)
we have
F (P ) =
i eiP w
√ iP
,
2
(9)
where w(z), called the plasma dispersion function with complex argument, is a form of the error function defined in Abramowitz and Stegun (1970). A
3
convenient MATLAB m-file is available for solving the function wit√h complex argument (Chase) and is provided in the appendix. Note that i can
be
written
as
(i+1) 2
ing the exponential
and that F (P ) always appears with e−iP , thus cancelterm in equation (9). This is reflected in the MATLAB
m-files that calculate the fields (provided in the appendix).
The equations have been written such that the direct and reflected components appear first. The last term is referred to as the surface or lateral wave. Often it is called the Norton surface wave from the engineering models he developed in the mid-1930s. The equations reduce to the fields above a perfectly conducting ground when k1 → ∞; the surface or lateral wave term then goes to zero.
2.2 Horizontal Dipole Over Earth
The electromagnetic fields produced by an ideal dipole oriented at a height z = d above and parallel to the ground plane consist in general of both TM and TE components. The formulation as provided in King et al (1992) for the TM wave components in cylindrical coordinates, referring to figure 1 and the above definitions, is
Hφ(ρ, φ, z)
=
I 4π
cos
φ

eik2r1
z−d
r1 
+
2k2 k1
eik2r2

ik2 r1
−
1 r12
− eik2r2
ik2 r2
−
1 r22
−
i k2 r23
−
k23 k1
r2 ρ
π k2 r2
z+d r2
ik2 r2
−r122
1

2 e−iP F (P )
 ,
Eρ(ρ, φ, z)
=
ωµ0I 4πk2
cos
φ

 
eik2r1 −eik2
r2
2
r12  
+
2 r22
−
+
2i k2 r13
+
z−d 2 r1
ik2 r1
−
3 r12
−
3i k2 r13
+
2i k2 r23
+
z+d 2 r2
ik2 r2
−
3 r22
−
3i k2 r23
2k2 z+d
k1  r2
ik2 r2
−
1 r22
2k22 k12

ik2 r2
−
−
1 r22
k23
k1
−
i k2 r23
r2
π
ρ
k2 r2
1
2 e−iP F (P
)
 
  
 
, and
Ez(ρ, φ, z)
=
ωµ0I 4πk2
 −eik2r1
ρ r1
cos
φ

+eik2r2
ρ r2
−
2k2 k1
eik2
r2
z−d r1
ik2 r1
−
3 r12
−
3i k2 r13

z+d r2
ik2 r2
−
3 r22
−
3i k2 r23
ρ r2
ik2 r2
−
1 r22
−
k23 k1
π k2 r2
1
2 e−iP F (P )
 .
(10) (11) (12)
The
TM
fields
are
zero
broadside
to
the
dipole
orientation,
φ
=
π 2
.
The
TE
components are
4
Hρ(ρ, φ, z)
=
I 4π
sin φ  
eik2r1
z−d
r1 
+
2k2 k1
eik2r2

ik2 r1
−
1 r12
− eik2r2
2 r22
+
+z+kd22ri232
r2
+
ik22 k1ρ
ik2 r2
−
3 r22
z+d ik2
r2
r2
r22
π
ρ2 k2r2
−
3i k2 r23
−
1
1 r22
2 e−iP F (P )
 
 
,
(13)
Hz(ρ, φ, z)
=
I 4π
sin φ 
eik2r1
+2
ρ r2
ρ r1
irk12−
1 r12

k2 k1
eik2r2  −
− eik2r2
ρ r2
ik2 r2
−
1 r22
z+d
r2
ik2 r2
−
3 r22
−
3i k2 r23
k2 k1
2 
1 r22
+
z+d
r2
3i k22 r23
−
ik2
r2
3 k22 r24
+
−
6 r22
−
15i k2 r23
 
  

, and
(14)
Eφ(ρ, φ, z)
=
− ωµ0I 4πk2
sin
φ
 
eik2r1 −eik2r2
ik2
r1 

+
1 r12
−
i k2 r13
− eik2r2
−
2k2 k1
z+d
 r2
ik2 r2
−
1 r22
+
2k22 k12

2 r22
+
+z+kd22ri232
r2
ik2 r2
+
2ik24 k13 ρ
r2 2 π
ρ
k2 r2
ik2 r2
−
1 r22
−
−
3 r22
−
3i k2 r23
1
2 e−iP F (P )
i
k2r23
 
 
 
.
(15)
These are the predominant fields broadside to the dipole orientation. Here Eφ is often termed the horizontal electric field. Again, the equations reduce to the fields above a perfectly conducting ground when k1 → ∞; the surface or lateral wave term goes to zero.
2.3 Comparisons
The peak power radiated by a unit dipole is given in Collin and Zucker
(1969):
P = k22ζ0 , 12π
(16)
where ζ0 is the free-space impedance
µ0 ε0
. The fields and power density
comparisons that follow are normalized to one watt radiated peak power.
To obtain the unnormalized quantities, simply multiply the power results
1
by P and the field results by P 2 . The outward component of the complex
Poynting vector over a closed surface is
1/2 E × H ∗ · dS = −Pcomplex ,
(17)
S
where the negative sign indicates power flow away from the surface. Our interest is in the real part of the power density at the observer (or target) location. In our cylindrical coordinate system for the TM components, this becomes
1/2 Re (E × H∗) = 1/2 Re ρ0EzHφ ∗ + z0EρHφ ∗ ,
(18)
5
where ρ0 and z0 are the unit vectors in cylindrical coordinates. Near the ground, the power flow is predominantly radial with a small z component. And, for the TE components, we have
1/2 Re (E × H∗) = 1/2 Re ρ0EφHz ∗ − z0EφHρ ∗ .
(19)
For the vertical dipole, the power density at the observer (on target) will be
Pv = 1/2 (Re (EφHz ∗) + Re (EφHρ ∗)) .
(20)
The horizontal dipole in general will produce a target power density of
Ph = 1/2 ({Re (EφHz ∗) − Re (EzHφ ∗)} + {Re (EρHφ ∗) − Re (EφHρ ∗)}) .
(21)
For broadside calculations this becomes
Ph = 1/2 (Re (EφHz ∗) + Re (EφHρ ∗)) .
(22)
The calculations that follow are limited to a frequency of 1.3 GHz (wavelength, λ0, is 0.23 m) and dipole heights of 1 to 3 m. Observer (target) heights range from 0 to 5 m. The frequency of 1.3 GHz is chosen, since most of the HPM source and antenna design work at ARL is centered around that frequency. The height ranges are chosen to be consistent with ground vehicle source and target applications. Five classes of ground parameters that are representative of distinctly different terrain will be addressed. These five classes are those discussed in King et al (1994) and are given in table 1.
2.3.1 Comparison of Field Components Near the Earth
When thinking about the interaction of electromagnetic waves with the earth’s surface, we often view the earth as a perfect conductor. The horizontally polarized field is reflected with the opposite sign of the incident field, leading to a difference or destructively interfered-with field. The vertically polarized field is reflected with the same sign as the incident field, leading to a sum or constructively interfered-with field. Figure 2 compares the resultant primary field components produced from a horizontal and a vertical ideal dipole located at the same point in space over a perfectly conducting ground plane. From these calculations, one might conclude that
Table 1. Earth parameters.
Case
σ,* S/m εr*
1 Sea water
4
80
2 Wet earth
.4
12
3 Dry earth
.04
8
4 Lake water .004 80
5
Dry sand
.000
2
*The variable name representing σ in the MATLAB m-files is SIGMA and for εr it is EPSREL. These variable names appear on many of the figures.
6
Figure 2. Comparison of 3
main electric field
components from a
vertical (Ez) and from a 2.5
horizontal (Eφ)
broadside ideal dipole
over a perfectly
2
Observer height (m)
conducting ground. Ez
is a constructively
interfered-with field,
1.5
while Eφ is a
destructively
1
interfered-with field.
0.5
Dipole height = 2 m, frequency = 1.3 GHz, range = 1000 m
Horizontal E-field, horizontal dipole z-directed E-field, vertical dipole
0 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 Normalized field magnitude (V/m)
an antenna that radiated fields polarized such that the electric field was essentially perpendicular to the ground plane would produce the largest fields and, consequently, the greater power densities on a target or near the ground. This is generally not the case for real earth grounds. While the perfectly conducting ground plane model may provide some insight into the interaction with horizontally polarized fields, it clearly is an inadequate model for vertical polarization. The complex reflection coefficients for a plane wave polarized with the electric field in the plane of incidence (vertical polarization) and with a plane wave polarized with the electric field perpendicular to the plane of incidence (horizontal polarization) are given by the following Fresnel expressions. For vertical polarization, it is (Collin, 1985)
εr
−
i
σ ωε0
sin ψ −
εr
−
i
σ ωε0
− cos2 ψ
ρcomplex =
,
(23)
εr
−
i
σ ωε0
sin ψ +
εr
−
i
σ ωε0
− cos2 ψ
and for horizontal polarization, it is
sin ψ −
εr
−
i
σ ωε0
− cos2 ψ
ρcomplex =
,
(24)
sin ψ +
εr
−
i
σ ωε0
− cos2 ψ
where ψ is the angle of incidence as measured between the ray path and the surface of the earth (see fig. 1). These reflection coefficients for the five cases of table 1 are presented in figure 3. One can clearly see that for shallow grazing angles, a Brewster angle effect exists (often called a pseudoBrewster angle). The result is destructive interference of the vertically polarized field for incident angles that are less than this pseudo-Brewster
7
Figure 3. Reflection
(a) 1
coefficients for
Reflection coefficient
vertically and
0.8
horizontally polarized plane waves of 1.3-GHz 0.6
frequency incident on a 0.4
flat ground for five
earth-parameter cases
0.2
listed in table 1. Note
that case 5 provides a
0 0
true Brewster angle
(since conductivity is
zero) and case 4 is very (b) 1
near to producing a true 0.9 Brewster angle at
1.3-GHz frequency:
0.8
(a) vertical polarization
and (b) horizontal
0.7
polarization.
0.6
Reflection coefficient
0.5 0
Case 1 2
4
3
5
20
40
60
80
Incidence angle (°)
Case 1
42
3 5
20
40
60
80
Incidence angle (°)
Phase
Phase
0
–50
–100
–150
0
200 195 190 185 180 175 170
0
Case 1
2
5 4
3
20
40
60
80
Incidence angle (°)
4,5 3 Case 1 2
20
40
60
80
Incidence angle (°)
angle. This would lead one to conclude that horizontally and vertically polarized field levels above a ground would be comparable at shallow angles of incidence. The Fresnel equations (22) and (23) do not take into account the surface or lateral wave produced by finite sources above the ground. They are also not applicable to near-field problems. We can compare the reflection coefficients as determined from the Fresnel equations with the results of our complete solution. To do this with the complete equations (eqs (2) to (4) for the vertical dipole and eqs (10) to (15) for the horizontal dipole), we fix the radial (or range) distance to 1000 m. The dipole height is varied to provide a range of incident angles. The observer will be fixed at a 1-m height. The incident and ”reflected” fields are
Evtotal = Eρ2 + Ez2 ,
(25)
for the vertical dipole and
Ehtotal = Eφ2 + Eρ2 + Ez2 ,
(26)
for the horizontal dipole. For broadside calculations, this simply becomes Eφ. The terms involving r1 in equations (2) to (4) and (10) to (15) represent the incident field and the remaining terms represent the reflected and surface wave terms. The reflection coefficient will be calculated as the total incident electric field divided into the total reflected and bound (surface wave) fields. Figures 4 and 5 provide the comparison of the magnitude of the reflection coefficient for the five cases of table 1. For the most part, the Fresnel expressions are a good approximation for the calculation of horizontally polarized field values (fig. 5) above realistic earth ground at
8
Figure 4. Fresnel
(a) 1
reflection coefficient for 0.9
a vertically polarized
SIGMA = 4 S/m, EPSREL= 80 0.8
plane wave field
0.7
Fresnel Eq. Complete Eq.
Reflection coefficient
compared to “reflection 0.6
coefficient” as
0.5
calculated by complete 0.4
formulation for a
0.3
vertical dipole over
0.2
homogeneous earth as 0.1
observed at 1-m height 1000 m down range at a
0 0 10 20 30 40 50 60 70 80 90 Incidence angle (°)
frequency of 1.3 GHz: (a) sea water, (b) wet earth, (c) dry earth, (d) lake water, and (e) dry sand.
(c) 1 0.9 0.8 0.7 0.6
SIGMA = 0.04 S/m, EPSREL= 8
Fresnel Eq. Complete Eq.
Reflection coefficient
0.5
0.4
0.3
0.2
0.1
0 0 10 20 30 40 50 60 70 80 90 Incidence angle (°)
(b) 1 0.9
Fresnel Eq. Complete Eq.
0.8 SIGMA = 0.4 S/m, EPSREL= 12
Reflection coefficient
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0 0 10 20 30 40 50 60 70 80 90 Incidence angle (°)
(d) 1 0.9 0.8 SIGMA = 0.004 S/m, EPSREL= 80
Fresnel Eq. Complete Eq.
Reflection coefficient
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0 0 10 20 30 40 50 60 70 80 90 Incidence angle (°)
(e) 1 0.9 0.8
SIGMA = 0 S/m, EPSREL= 2
Fresnel Eq. Complete Eq.
Reflection coefficient
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0 0 10 20 30 40 50 60 70 80 90 Incidence angle (°)
1.3 GHz. The horizontally polarized fields diverge from the Fresnel expressions for low conductivities and relative dielectric constants. The resulting total fields in such cases predicted by the Fresnel expressions will be larger than the complete solution prediction. The results for vertical polarization (fig. 4) show significant divergence for shallow incident angles. In this case the resulting total fields predicted by the complete solution can be significantly higher than that predicted from employing the Fresnel expressions.
9
Reflection coefficient Reflection coefficient
Reflection coefficient Reflection coefficient
Figure 5. Fresnel
(a)
1
reflection coefficient for
0.9
a horizontally polarized
0.8
plane wave field
0.7
compared to “reflection 0.6
SIGMA = 4 S/m, EPSREL = 80
Fresnel eq. Complete eq.
coefficient” as
0.5
calculated by complete 0.4
formulation for a
0.3
horizontal dipole over
0.2
homogeneous earth as 0.1
observed at 1-m height, broadside 1000 m down
0 0 10 20 30 40 50 60 70 80 90 Incidence angle (°)
range at a frequency of
1.3 GHz: (a) sea water, (c) 1
(b) wet earth, (c) dry
0.9
earth, (d) lake water,
0.8
Fresnel eq. Complete eq.
and (e) dry sand.
0.7
0.6
0.5 SIGMA = 0.04 S/m, EPSREL = 8
0.4
0.3
0.2
0.1
0 0 10 20 30 40 50 60 70 80 90 Incidence angle (°)
(b) 1 0.9 0.8 0.7 0.6 SIGMA = 0.4 S/m, EPSREL = 12 0.5 0.4 0.3 0.2 0.1 0 0 10 20 30 40 50 60 Incidence angle (°)
Fresnel eq. Complete eq. 70 80 90
(d) 1 0.9
Fresnel eq. Complete eq.
0.8 SIGMA = 0.004 S/m, EPSREL = 80 0.7
0.6
0.5
0.4
0.3
0.2
0.1
0 0 10 20 30 40 50 60 70 80 90 Incidence angle (°)
Reflection coefficient
(e) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
SIGMA = 0 S/m, EPSREL = 2
Fresnel eq. Complete eq.
10 20 30 40 50 60 70 80 90 Incidence angle (°)
10
Figures 6 and 7 are plots of the fields produced by the two-dipole orientations; these plots compare the predominant field components for each dipole orientation at 1000 and 100 m, respectively. The plots for the 100-m case (fig. 7) show the beginning of the lobe effect produced by the phase difference between the incident and reflected wave. This is shown more dramatically in figure 8.
Observer height (m) Observer height (m)
Figure 6. Comparison of (a)
3
principal fields from an
SIGMA = 4 S/m, EPSREL = 80
dipole height = 2 m, frequency = 1.3 GHz
ideal dipole oriented
2.5 range = 1000 m
perpendicular and
horizontal to a
2
homogeneous flat earth. 1.5
In each case, dipole is
placed 2 m above
1
ground plane and observer or target is
0.5
Horizontal E-field, horizontal dipole
z-directed E-field, vertical dipole
1000 m down range: (a) sea water, (b) wet
0
0
1
2
3
4
5
6
7
Normalized field magnitude (V/m)
× 10–3
earth, (c) dry earth, (d) lake water, and (e) dry sand.
(c) 3
2.5
SIGMA = 0.04 S/m, EPSREL= 8 dipole height = 2 m, frequency = 1.3 GHz range = 1000 m
(b) 3 SIGMA = 0.4 S/m, EPSREL= 12
dipole height = 2 m, frequency = 1.3 GHz 2.5 range = 1000 m
2
1.5
1
0.5 0 0
Horizontal E-field, horizontal dipole z-directed E-field, vertical dipole
1
2
3
4
5
6
7
Normalized field magnitude (V/m)
× 10–3
(d) 3
2.5
SIGMA = 0.004 S/m, EPSREL= 80 dipole height = 2 m, frequency = 1.3 GHz range = 1000 m
Observer height (m) Observer height (m)
2
2
1.5
1.5
1
1
0.5
0 0
Horizontal E-field, horizontal dipole z-directed E-field, vertical dipole
1
2
3
4
5
6
7
Normalized field magnitude (V/m)
× 10–3
0.5
0 0
Horizontal E-field, horizontal dipole z-directed E-field, vertical dipole
1
2
3
4
5
6
7
Normalized field magnitude (V/m)
× 10–3
(e) 3 2.5
SIGMA = 0 S/m, EPSREL= 0.2 dipole height = 2 m, frequency = 1.3 GHz range = 1000 m
Observer height (m)
2
1.5
1
0.5
0 0
Horizontal E-field, horizontal dipole z-directed E-field, vertical dipole
1
2
3
4
5
Normalized field magnitude (V/m)
6 × 10–3
11
Observer height (m) Observer height (m)
Figure 7. Comparison of (a)
3 SIGMA = 4 S/m, EPSREL = 80
principal fields from an
dipole height = 2 m, frequency = 1.3 GHz
ideal dipole oriented
2.5 range = 100 m
perpendicular and
2
horizontal to a homogeneous flat earth. 1.5
In each case, dipole is
1
placed 2 m above
ground plane and observer or target is 100 m down range: (a) sea water, (b) wet
0.5
Horizontal E-field, horizontal dipole
z-directed E-field, vertical dipole
0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Normalized field magnitude (V/m)
earth, (c) dry earth, (d) lake water, and (e) dry sand.
(c) 3 SIGMA = 0.04 S/m, EPSREL = 8 dipole height = 2 m, frequency = 1.3 GHz
2.5 range = 100 m
(b) 3 SIGMA = 0.4 S/m, EPSREL = 12 dipole height = 2 m, frequency = 1.3 GHz
2.5 range = 100 m
2
1.5
1
0.5 00
Horizontal E-field, horizontal dipole z-directed E-field, vertical dipole
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Normalized field magnitude (V/m)
(d) 3 SIGMA = 0.004 S/m, EPSREL = 80 dipole height = 2 m, frequency = 1.3 GHz
2.5 range = 100 m
Observer height (m) Observer height (m)
2
2
1.5
1.5
1
1
0.5
Horizontal E-field, horizontal dipole
z-directed E-field, vertical dipole
0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Normalized field magnitude (V/m)
0.5
Horizontal E-field, horizontal dipole
z-directed E-field, vertical dipole
0 0
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Normalized field magnitude (V/m)
(e) 3 SIGMA = 0 S/m, EPSREL = 0.2 dipole height = 2 m, frequency = 1.3 GHz
2.5 range = 100 m
Observer height (m)
2
1.5
1
0.5
0 0
Horizontal E-field, horizontal dipole z-directed E-field, vertical dipole
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Normalized field magnitude (V/m)
12
Figure 8. Effect of ground reflection on primary field components near
(a) 30 SIGMA = 0.4 S/m, EPSREL = 12
dipole height = 2 m, frequency = 1.3 GHz
range = 100 m 25
ground for typical earth
parameters. Phase
Observer height (m)
difference between
20
incident and reflected
waves results in
development of field
15
lobes that are more
pronounced as radiating antenna is approached: (a) 100 m
10
Horizontal E-field, horizontal dipole
z-directed E-field, vertical dipole
down range and (b)
5
1000 m down range.
0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Normalized field magnitude (V/m)
(b) 30 25
SIGMA = 0.4 S/m, EPSREL = 12 dipole height = 2 m, frequency = 1.3 GHz range = 1000 m
Observer height (m)
20
15
10
5
Horizontal E-field, horizontal dipole
z-directed E-field, vertical dipole
0 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 Normalized field magnitude (V/m)
13
3. Conclusion
A suite of MATLAB m-files have been developed to calculate the electromagnetic fields produced by a vertical and a horizontal infinitesimal unit dipole over a homogeneous flat (ground) plane. Calculations for the fields above the ground plane have been made for various ground-plane conductivities and relative dielectric constants. The calculations bound the practical range of parameters representative of natural earth terrain. For a frequency of 1.3 GHz, where the dipole and the observer are close to the ground plane (<3 m), significant difference is seen in the magnitude of the fields from either dipole orientation. The power density on target will be much larger for vertical dipole orientation. The effects of rough terrain, foliage, or scattering from manmade or natural objects in the path from the dipole to target may alter this conclusion. How these other scatterers might affect the field structure at a target at 1.3 GHz is not known at this time. If the effects are random in nature, the present conclusion is most likely still valid. From the simple, smooth, flat ground model, then, one must conclude that vertical polarization (antenna radiating a vertically polarized field) will deliver the most energy to the target. Unless the target’s preference for field orientation for maximum pickup is known, a vertically polarized antenna may in fact be the best choice for a ground weapon system.
14
References
Abramowitz, M., and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Applied Mathematics Series–55, U.S. Department of Commerce (1970).
Banos, A., Dipole Radiation in the Presence of a Conducting Half-Space, Pergamon, Oxford (1966).
Chase, R., MATLAB m-file, Plasma Dispersion Function with Complex Argument (unpublished).
Collin, R. E., Antenna and Radiowave Propagation, McGraw-Hill, New York (1985).
Collin, R. E., and F. J. Zucker, Antenna Theory, Part 1, McGraw-Hill, New York (1969).
King, R.W.P., and S. S. Sandler, “The electromagnetic field of a vertical electric dipole over the earth or sea,” IEEE Trans. Antennas Propag. 42, No. 3 (March 1994), pp 382–389.
King, R.W.P., M. Owens, and T. T. Wu, Lateral Electromagnetic Waves, Springer-Verlag, New York (1992).
Maclean, T.S.M., and Z. Wu, Radiowave Propagation Over Ground, Chapman and Hall, London (1993).
MATLAB, release 5.3, The Math Works, Inc., Natick, MA (1984–1999). Sommerfeld, A., Partial Differential Equations in Physics, Academic Press,
New York (1949).
15
16
Appendix. MATLAB m-Files
WERF Function
This appendix documents the MATLAB m-files that implemented the field equations for the vertical and horizontal dipole and array cases. Slight modification may be required to obtain all the calculations presented.
function w = werf(z,N) % WERF(Z,N) Plasma Dispersion Function with complex argument.
% % Computes the function w(z)=exp(-z^2)*erfc(-iz) using a rational
% series with N terms. N should be a power of 2 or it gets SLOW.
% Default value of N is 64. z can be a matrix of values. % Taken from Siam Journal on Numerical Analysis, Oct 1994, V31,#5.
% Modified by R. Chase to work for all z (4/3/95).
%
%
N=32 gives approx 14 place accuracy, N=64 is better and
%
it seems as fast.
%
% See wfn.m -- Draws graph in Abramowitz & Stegun,
% Handbook of Mathematical Functions, p. 298
if nargin == 1, N=64 end M=2*N M2=2*M k= -M+1:1:M-1]' L=sqrt(N/sqrt(2)) theta=k*pi/M t=L*tan(theta/2) f=exp(-t.^2).*(L^2+t.^2) f= 0 f] a=real(fft(fftshift(f)))/M2 a=flipud(a(2:N+1)) nz=imag(z) <= 0 z(nz)=conj(z(nz)) Z=(L+i*z)./(L-i*z) p=polyval(a,Z) w=2*p./(L-i*z).^2+(1/sqrt(pi))./(L-i*z) w(nz)=2*exp(-(conj(z(nz)).^2)) - conj(w(nz)) %if all(imag(z)==0), w=real(w) end
% Default value for N % M2=no. of sampling points % Optimal choice of L % Variables thetaand t % function to be transformed % Coefficients of transform % Reorder coefficients % Find im(z) <=0 % Use conj for above % Polynomial evaluation % Evaluate w(z) % Handle im(z) <= 0 % Rtn real if real
17
Constants
function w,cv,epso,u0,zo,k0,k1,kappap,lo]=cnstdg(f) % This m-file contains the basic constants to be used by various % m-files and functions % % f is input in GHz % global EPSREL SIGMA w=2*pi*f*1e9 cv=2.99792458e8 epso=(1/(4*pi))*1e7*(1/cv^2) eps1=EPSREL u0=4*pi*1e-7 zo=sqrt(u0/epso) k0=w*sqrt(epso*u0) epsr=(eps1-j*SIGMA./(w*epso)) k1=k0*sqrt(epsr) kappap=-k0/sqrt(epsr+1) lo=cv/(f*1e9)
Vertical Dipole
function ez,er,hp]=dipg(f,d,rho,z)
% % Electromagnetic fields from a vertical z-directed current element
% at a height, d, over a conducting dielectric plane (ground) calculated
% by the King/Sandler model. This formulation is that developed
% as equations 6, 7, and 8 of "The Electromagnetic Field of a Vertical Electric
% Dipole over the Earth or Sea", IEEE Trans. on Antennas and Propagation,
% March 1994, Vol 42 No. 3, page 383. It is the same as the King/Owens/Wu
% formulation. This formulation is that developed as equations 4.2.30-32 of
% "Lateral Electromagnetic Waves", Springer-Verlag, 1992, pp 293-297.
%
%
> f- frequency rho- radial distance from the z-axis
%
< z- height of observer above ground plane (input as an array)
%
> ez,er,hp- field components at the observer, where the second letter
%
< designates the component -- z, r (rho), p (phi)
% % The formulation is subject to |k1(ground)|>3|k2(air)|
% w,c,eps0,u0,zo,k2,k1,kappap,lo]=cnstdg(f)
%
18
% King, et. al assume an exp(-i*omega*time)dependency thus we convert k1 % k1=conj(k1) %N2=(k1/k2)^2 k21=k2/k1 % % We assume a unit dipole % il=1 r1=(rho^2+(z-d).^2).^.5 r2=(rho^2+(z+d).^2).^.5 sd=rho./r1 sr=rho./r2 cd=(z-d)./r1 cr=(d+z)./r2 p=(r2*k2^3/(2*k1^2)).*((k2*r2+k1*(z+d))./(k2*rho)).^2 % % The attenuation function less the exp(i*p) term - related to the % error function is % Ferf=((1+i)/4)*werf(sqrt(i*p)) % % Develop the terms for the field expressions % t1=w*u0*il/(2*pi*k2) t11=-il/(2*pi) t2=exp(i*k2*r1)/2 t3=(i*k2./r1)-(1./r1.^2)-i./(k2*r1.^3) t31=(i*k2./r1)-(1./r1.^2) t4=(cd.^2).*((i*k2./r1)-(3./r1.^2)-(3*i)./(k2*r1.^3)) t41=((i*k2./r1)-(3./r1.^2)-(3*i)./(k2*r1.^3)) t5=exp(i*k2*r2)/2 t6=(i*k2./r2)-(1./r2.^2)-i./(k2*r2.^3) t61=(i*k2./r2)-(1./r2.^2) t7=(cr.^2).*((i*k2./r2)-(3./r2.^2)-(3*i)./(k2*r2.^3)) t71=((i*k2./r2)-(3./r2.^2)-(3*i)./(k2*r2.^3)) t8=(2*t5*(k2^3)/k1).*((pi./(k2*r2)).^.5).*sr.*Ferf t81=((k2^3)/k1)*((pi./(k2*r2)).^.5).*Ferf t82=(2*t5.*(k2^3)/k1).*((pi./(k2*r2)).^.5).*Ferf % % Calculate the fields % ez=t1*(t2.*(t3-t4)+t5.*(t6-t7)-t8) er=-t1*(t2.*sd.*cd.*t41+t5.*sr.*cr.*t71-k21*2*t5.*(sr.*t61-t81)) hp=t11*(t2.*sd.*t31+t5.*sr.*t61-t82)
19
Horizontal Dipole
function ez,ep,er,hz,hp,hr]=dipgh(f,d,phi,rho,z)
% % Electromagnetic fields from a horizontally directed (perpendicular to z)
% current element located in the phi=0 plane
% at a height, d, over a conducting dielectric plane (ground) calculated
% by the King/Owens/Wu formulation. This formulation is that developed
% as equations 7.10.75, 7.10.80, 7.10.84, 7.10.92, 7.10.93 and 7.10.94 of
% "Lateral Electromagnetic Waves", Springer-Verlag, 1992, pp 293-297.
%
%
> f- frequency(GHz) phi- angle about the z-axis, rho- radial distance from
%
< the z-axis, z- height of observer above ground plane (input as an array)
%
> ez,ep,er,hz,hp,hr- field components at the observer, where the second
%
< letter designates the component -- z, r (rho), p (phi)
% % % The formulation is subject to |k1(ground)|>3|k2(air)|
% w,c,eps0,u0,zo,k2,k1,kappap,lo]=cnstdg(f)
%
%
King, et. al assume an exp(-i*omega*time)dependency thus we convert k1
% k1=conj(k1)
%N2=(k1/k2)^2
k21=k2/k1
k2p=k2/rho
% % We assume a unit dipole
% il=1
cp=cos(phi)
sp=sin(phi)
r1=(rho^2+(z-d).^2).^.5
r2=(rho^2+(z+d).^2).^.5
sd=r1/rho
sdo=1./sd
sr=r2/rho
sro=1./sr
cd=(z-d)./r1
cr=(d+z)./r2
p=(r2*k2^3/(2*k1^2)).*((k2*r2+k1*(z+d))./(k2*rho)).^2
20
% % The attenuation function less the exp(i*p) term - related to the % error function is % Ferf=((1+i)/4)*werf(sqrt(i*p)) % % Develop the terms for the field expressions % t1=w*u0*il*cp/(4*pi*k2) t11=-w*u0*il*sp/(4*pi*k2) t12=il*sp/(4*pi) t13=il*cp/(4*pi) t2=exp(i*k2*r1) t32=(i*k2./r1)-(1./r1.^2)-i./(k2*r1.^3) t3=(2./r1.^2)+2*i./(k2*r1.^3) t31=(i*k2./r1)-(1./r1.^2) t4=(cd.^2).*((i*k2./r1)-(3./r1.^2)-(3*i)./(k2*r1.^3)) t41=(cd).*((i*k2./r1)-(3./r1.^2)-(3*i)./(k2*r1.^3)) t5=exp(i*k2*r2) t62=(i*k2./r2)-(1./r2.^2)-i./(k2*r2.^3) t6=(2./r2.^2)+2*i./(k2*r2.^3) t61=(i*k2./r2)-(1./r2.^2) t7=(cr.^2).*((i*k2./r2)-(3./r2.^2)-(3*i)./(k2*r2.^3)) t71=(cr).*((i*k2./r2)-(3./r2.^2)-(3*i)./(k2*r2.^3)) t72=((1./r2.^2)+(3*i)./(k2*r2.^3)-3./((k2^2)*r2.^4)) t73=(cr.^2).*((i*k2./r2)-(6./r2.^2)-(15*i)./(k2*r2.^3)) t8=((k2^3)/k1)*((pi./(k2*r2)).^.5).*sr.*Ferf t81=((k2^3)/k1)*((pi./(k2*r2)).^.5).*Ferf t82=(2*t5*(k2^3)/k1).*((pi./(k2*r2)).^.5).*Ferf t83=((pi./(k2*r2)).^.5).*Ferf % % Calculate the fields % ez=t1*(-t2.*sdo.*t41+t5.*sro.*t71-2*k21*t5.*(sro.*t61-t81)) er=t1*(t2.*(t3+t4)-t5.*(t6+t7-2*k21*t61+2*(k21^2)*(t62-t8))) ep=t11*(t2.*t32-t5.*t62-t5.*(-2*k21*cr.*t61+2*(k21^2)*(t6+t7)...
+2*i*k21^3*k2p*(sr.^2).*t83)) hr=t12*(t2.*cd.*t31-t5.*cr.*t61+2*k21*t5.*(t6+i*k21*k2p*(sr.^2).*t83+t7)) hp=t13*(t2.*cd.*t31-t5.*cr.*t61+2*k21*t5.*(t62-k21*(k2^2)*(sr).*t83)) hz=t12*(t2.*sdo.*t31-t5.*sro.*t61+2*sro.*t5.*(k21*t71(k21^2)*(t72+t73)))
21
Plot m-File for Fields
% % This m-file plots the fields over a conductive flat earth produced by an ideal
% dipole placed a distance d above the earth. It compares the results from
% a vertical and horizontal dipole.
% % % Establish the problem conditions
% % % EPSREL- Relative dielectric constant SIGMA- Earth conductivity (S/m)
% EPSREL=80 SIGMA=4 global EPSREL SIGMA
%EPSREL=12 SIGMA=.4 %EPSREL=8 SIGMA=.04 %EPSREL=80 SIGMA=.004 %EPSREL=2 SIGMA=.000 % % Location of dipole (m)
% d=2 % % Location of observer, rho (m) phi (radians) z (m) ---> an array
% rho=1000 phi=pi/2
z=.1* 1:1:30] % % f- frequency in GHz
% f=1.3
%
%
Field normalization factor - one Watt radiated
%
w,c,eps0,u0,zo,k2,k1,kappap,lo]=cnstdg(f)
Fn=sqrt(12*pi/((k2^2)*zo))
%
%
Horizontal dipole fields
% ez,ep,er,hz,hp,hr]=dipgh(f,d,phi,rho,z)
plot(Fn*abs(ep),z,'-b') hold
22
%
%
Vertical dipole fields
% ezv,erv,hpv]=dipg(f,d,rho,z)
plot(Fn*abs(ezv),z,'-.r')
ts=Fn*abs(erv(1))
title('Vertical and Horizontal Ideal Dipole fields over Ground')
text(ts,z(19), 'SIGMA = ',num2str(SIGMA),' S/m
EPSREL=
',num2str(EPSREL)])
text(ts,z(18), 'Dipole height = ',num2str(d),' meters Frequency =
',num2str(f),' GHz'])
text(ts,z(17), 'Range = ',num2str(rho),' meters'])
ylabel('Observer height - meters')
xlabel('Normalized field magnitude - V/m')
legend('Horizontal E-field, horizontal dipole','z-directed E-field,
vertical dipole')
hold
Fresnel Reflection Coefficients and Plots
function rcomv,rcomh]=Fresnel1(f,psi)
% % This routine calculates the Fresnel reflection coefficients for % vertical and horizontal incident fields % % after Collin, "Antenna and Radiowave Propagation", page 345 % % f-frequency in GHz psi- array of incident angles % % Code returns the arrays rcomv and rcomh, the vertical and horizontal % complex reflection coefficients, respectively. % global EPSREL SIGMA w=2*pi*f*1e9 cv=2.99792458e8 epso=(1/(4*pi))*1e7*(1/cv^2) epsr=(EPSREL-j*SIGMA./(w*epso)) rcomv=(epsr*sin(psi)-sqrt(epsr-cos(psi).^2))./(epsr*sin(psi)+sqrt(epsr-cos(psi).^2)) rcomh=(sin(psi)-sqrt(epsr-cos(psi).^2))./(sin(psi)+sqrt(epsr-cos(psi).^2))
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
%
23
24
% This m-file plots the reflection coefficient for vertical and horizontal % fields over a homogeneous ground plane % % % EPSREL- Relative dielectric constant SIGMA- Earth conductivity (S/m) % EPSREL=80 SIGMA=4 f=1.3 global EPSREL SIGMA psi= pi/1000:pi/1000:pi/2]
rcomv,rcomh]=Fresnel1(f,psi) EPSREL=12 SIGMA=.4
rcomv1,rcomh1]=Fresnel1(f,psi) EPSREL=8 SIGMA=.04
rcomv2,rcomh2]=Fresnel1(f,psi) EPSREL=80 SIGMA=.004
rcomv3,rcomh3]=Fresnel1(f,psi) EPSREL=2 SIGMA=.000
rcomv4,rcomh4]=Fresnel1(f,psi)
subplot(2,2,1),plot((180/pi)*psi,abs(rcomv),'k') hold axis( 0,90,0,1])
xlabel('Incidence angle (degrees)')
ylabel('Reflection Coefficient')
%text(5,.8, 'SIGMA = ',num2str(SIGMA),' S/m
EPSREL= ',num2str(EPSREL)])
subplot(2,2,1),plot((180/pi)*psi,abs(rcomv1),'r')
subplot(2,2,1),plot((180/pi)*psi,abs(rcomv2),'b')
subplot(2,2,1),plot((180/pi)*psi,abs(rcomv3),'g')
subplot(2,2,1),plot((180/pi)*psi,abs(rcomv4),'c')
hold
subplot(2,2,2),plot((180/pi)*psi,(180/pi)*angle(rcomv),'k') hold axis( 0,90,-190,5]) xlabel('Incidence angle (degrees)') ylabel('Phase') subplot(2,2,2),plot((180/pi)*psi,(180/pi)*angle(rcomv1),'r') subplot(2,2,2),plot((180/pi)*psi,(180/pi)*angle(rcomv2),'b') subplot(2,2,2),plot((180/pi)*psi,(180/pi)*angle(rcomv3),'g') subplot(2,2,2),plot((180/pi)*psi,-(180/pi)*angle(rcomv4),'c') hold
subplot(2,2,3),plot((180/pi)*psi,abs(rcomh),'k')
hold axis( 0,90,.5,1]) xlabel('Incidence angle (degrees)') ylabel('Reflection Coefficient') subplot(2,2,3),plot((180/pi)*psi,abs(rcomh1),'r') subplot(2,2,3),plot((180/pi)*psi,abs(rcomh2),'b') subplot(2,2,3),plot((180/pi)*psi,abs(rcomh3),'g') subplot(2,2,3),plot((180/pi)*psi,abs(rcomh4),'c') hold subplot(2,2,4),plot((180/pi)*psi,(180/pi)*angle(rcomh),'k') hold axis( 0,90,170,200]) xlabel('Incidence angle (degrees)') ylabel('Phase') subplot(2,2,4),plot((180/pi)*psi,(180/pi)*angle(rcomh1),'r') subplot(2,2,4),plot((180/pi)*psi,(180/pi)*angle(rcomh2),'b') subplot(2,2,4),plot((180/pi)*psi,(180/pi)*angle(rcomh3),'g') subplot(2,2,4),plot((180/pi)*psi,(180/pi)*angle(rcomh4),'c') hold
25
26
Distribution
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Distribution (cont’d)
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Mgmt ATTN AMSRL-CI-AP Techl Pub (2 copies) ATTN AMSRL-CI-LL Techl Lib (2 copies) ATTN AMSRL-SE-DP M Litz ATTN AMSRL-SE-DP R A Kehs (3 copies) ATTN AMSRL-SE-DS J Miletta (10 copies) ATTN AMSRL-SE-DS J Tatum (5 copies) ATTN AMSRL-SE-DS M Berry ATTN AMSRL-SE-DS W O Coburn Adelphi MD 20783-1197
28
REPORT DOCUMENTATION PAGE
Form Approved OMB No. 0704-0188
Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302, and to the Office of Management and Budget, Paperwork Reduction Project (0704-0188), Washington, DC 20503.
1. AGENCY USE ONLY (Leave blank)
2. REPORT DATE
February 2001
3. REPORT TYPE AND DATES COVERED
Summary, Oct 99 to Sept 00
4. TITLE AND SUBTITLE Propagation of Electromagnetic Fields Over Flat Earth 5. FUNDING NUMBERS
6. AUTHOR(S) Joseph R. Miletta
DA PR: AH94 PE: 62705A
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)
U.S. Army Research Laboratory
Attn: AMSRL-SE-DS
email: jmiletta@arl.army.mil
2800 Powder Mill Road
Adelphi, MD 20783-1197
9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)
U.S. Army Research Laboratory
2800 Powder Mill Road Adelphi, MD 20783-1197
8. PERFORMING ORGANIZATION REPORT NUMBER
ARL-TR-2352
10. SPONSORING/MONITORING AGENCY REPORT NUMBER
11. SUPPLEMENTARY NOTES
ARL PR: 0NE6YY AMS code: 622705.H94
12a. DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release; distribution 12b. DISTRIBUTION CODE unlimited.
13. ABSTRACT (Maximum 200 words)
This report looks at the interaction of radiated electromagnetic fields with earth ground in military or lawenforcement applications of high-power microwave (HPM) systems. For such systems to be effective, the microwave power density on target must be maximized. The destructive and constructive scattering of the fields as they propagate to the target will determine the power density at the target for a given source. The question of field polarization arises in designing an antenna for an HPM system. Should the transmitting antenna produce vertically, horizontally, or circularly polarized fields? Which polarization maximizes the power density on target? This report provides a partial answer to these questions. The problems of calculating the reflection of uniform plane wave fields from a homogeneous boundary and calculating the fields from a finite source local to a perfectly conducting boundary are relatively straightforward. However, when the source is local to a general homogeneous plane boundary, the solution cannot be expressed in closed form. An approximation usually of the form of an asymptotic expansion results. Calculations of the fields are provided for various source and target locations for the frequencies of interest. The conclusion is drawn that the resultant vertical field from an appropriately oriented source antenna located near and above the ground can be significantly larger than a horizontally polarized field radiated from the same location at a 1.3 GHz frequency at observer locations near and above the ground.
14. SUBJECT TERMS Antenna, ground interaction
15. NUMBER OF PAGES
33
16. PRICE CODE
17. SECURITY CLASSIFICATION OF REPORT
Unclassified
NSN 7540-01-280-5500
18. SECURITY CLASSIFICATION OF THIS PAGE
Unclassified
19. SECURITY CLASSIFICATION OF ABSTRACT
Unclassified
20. LIMITATION OF ABSTRACT
UL
Standard Form 298 (Rev. 2-89) Prescribed by ANSI Std. Z39-18 298-102
29