chapter 4 linear wire antennas 1 ece 5318/6352 antenna engineering dr. stuart long
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Chapter 4Chapter 4
Linear Wire AntennasLinear Wire Antennas
1
ECE 5318/6352Antenna Engineering
Dr. Stuart Long
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE
oIz zaI ˆ)( '
2
(only electrical current present)
(constant current)
l /50
I
l / 2
l / 2
Io
Impinging Wave
z
; thin wire ;l
00 FI
m
[4-1]
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)
3
222 zyxr
Fig. 4.1(a) Geometrical arrangementof an infinitesimal dipole
l /50
mixed coordinates in mixed coordinates in expression - change to sphericalexpression - change to spherical
222 zyxR
'''' ),,(4
dR
ezyx(x,y,z)
jkR
c
eo
IA
2 2 2' ' 'R x x y y z z
4
for
(x,y,z)
(x’,y’,z’)
source points
l
[4-2]
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)
l /50
mixed coordinates in expression mixed coordinates in expression change to sphericalchange to spherical
5
[4-4]
2/
2/
'
4ˆ
zd
r
e(x,y,z)
jkroo I
aA z
jkroo er
(x,y,z)
4
ˆ
IaA z
(x,y,z)
(x’,y’,z’)
source points
l
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)
l /50
mixed coordinates in expression mixed coordinates in expression need to change to sphericalneed to change to spherical
2'2'2' zzyyxxR
sin4
sin jkrooz e
r
IAA
6
cos cos4
jkro or z
IA A e
r
cd' along source
0A
(x,y,z)
(x’,y’,z’)
source points
l
[4-6]
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)
l /50
7
Using Vector Potential Using Vector Potential A A , , calculate calculate HH & & EE fields fields
rAAr
rr)(
1A
AaAH
1
ˆ1
[4-7]
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)
l /50
AH
1
jkro ejkrr
IkjH
11sin
4
0rH
0H
8
Using Vector Potential Using Vector Potential A A , , calculate calculate HH fields fields
[4-8]
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)
l /50
9
Using Maxwell’s Eqns toUsing Maxwell’s Eqns to calculate calculate EE fields fields
[4-10]
HE
j
1
jkror e
jkrr
IE
11cos
2 2
0E
jkro erkjkrr
IkjE
22
111sin
4
Fig. 4.1(b) Geometrical arrangementof an infinitesimal dipole and its
associated electric-field components on a spherical surface
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)
l /50
Using Using HH, , EErr, , EE,, calculate the complex Poynting vectorcalculate the complex Poynting vector
1 1ˆ( )
2 2
2 2sin 11
2 38 ( )
2cos sin 1
12 3 216 ( )
E H E Hr r
IoW j
r r kr
k IoW j j
r kr
W E H a a
10
[4-12]
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)
l /50
s
dP sW
3
2
)(
11
3 krj
Io
drWd r sin
0
22
0
11
Find total outward flux through a closed sphereFind total outward flux through a closed sphere
(only contributions from Wr)
[4-14]
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)
l /50
12
Find total outward flux through a closed sphereFind total outward flux through a closed sphere
0.02 0.31650
Example: rR
22
22
2
1
3 2
Radiation resistance 80
for free-space where 120
orad o r
r
IP I R
R
Real P total radiated power Prad
[4-19]
[4-16]
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)
l /50
(Impedance would also have a large capacitive term that is not calculated here.)
13
32
1
3 kr
Io
Imaginary part of P reactive power in the radial direction
(Note: this 0 as kr , so it is essentially not present in far field;
only important in near and intermediate field considerations)
[4-17]
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)
l /50
Near Field approximations Near Field approximations [ kr [ kr 1 ] 1 ]
(field point very close or very low frequency case)
sin
4 2r
eIH
jkro
14
Dominant terms
[4-20]
cos
2 3rk
eIjE
jkro
r
sin
4 3rk
eIjE
jkro
Like ‘quasistationary” fields E near static electric dipole
H near static current element
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)
l /50
Near Field approximations Near Field approximations [ kr [ kr 1 ] 1 ]
sin
4ˆ
2r
IoaH
15
Biot – Savart Law : infinitesimal current element in direction az
(same as above when kr 0)
(note E and H are 90° out of phase)
NO RADIAL POWER FLOW -- REACTIVE FIELDS
][Re2
1 HEW
avg
0avgW
[4-21]
[4-22]
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)
l /50
Intermediate FieldsIntermediate Fields[ kr [ kr >> 1] 1]
(induction zone; still have radial fields)
• E 1/r • H 1/r
16
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)
l /50
• Er 1/r
17
r = /2 (Radian Distance)
(Radius of Radian Sphere)
Energy basically imaginary
(stored)Near field
Energybasically
Real(radiating)
Far field
Fig. 4.2 Radiated field terms magnitude variation versus radial distance
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)
l /50
IntermediateRegion
Induction Zone
Far Field Far Field [ kr [ kr >>>> 1 ] 1 ]
18
Dominant terms
[4-26] sin
4 r
eIkjH
jkro
0r rE E H H
sin
4 r
eIkjE
jkro
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)
l /50
Far Field Far Field [ kr [ kr >>>> 1 ] 1 ]
H
E
ˆSimilar to plane wave, but propagation in direction
1With and sin variations
r
r
a
19
( both E and H are TEM to )[4-27]
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)
ra
l /50
DirectivityDirectivity (use Far Field approx.)
21 sin2ˆRe[ ]22 2 4
22 2sin
2 4
k Io
avg r r
k IoU r W
avg
W E H a
20
RADIATION INTENSITY
2 2sinas before for RNote ea: l ( )
28 r
IoW W
avg r
[4-28]
[4-29]
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)
l /50
DirectivityDirectivity
21
[4-31]
2
2
in =90 direct
38max4
io
4 1.52
3
2
max 2n
4
2
3
o
o
o
IU
D Do P Irad
k IoU
IoP
rad
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)
l /50
SMALL DIPOLESMALL DIPOLE
Uniform current assumption - only valid for ideal case( approximated by capacitor plate antenna)
sin
8 r
eIkjE
jkro
sin
8 r
eIkjH
jkro
22
½ value of fields compared to constant current case
/50 < l < /10
/50 < l < /10
[4-36]
SMALL DIPOLESMALL DIPOLE(CONT)(CONT)
23
For physical small dipole triangular current distribution
value of case of constant current
14
same as constant current case
/50 < l < /10
[4-37]
2
12
orad
IP
2220
rR
5.1oD
FINITE LENGTH DIPOLEFINITE LENGTH DIPOLE
2' 21cos sin
2
zR r z
r
24
(length comparable to )
(max error where = 90° ; 4th term = 0 there)
approx. error
[4.41]
Fig. 4.5 Finite dipole geometryand far-field approximations
Phase and Magnitude considerationsPhase and Magnitude considerations
25
In calculating phase assume can tolerate phase error of /8 (22°)
Must choose r far enough away so that ….
FINITE LENGTH DIPOLEFINITE LENGTH DIPOLE(CONT)(CONT)
Phase and Magnitude considerationsPhase and Magnitude considerations
for magnitude term use
for phase term us
1
- coe sjkr
R rr
e R r z
2 22 2 2' max 2 8 2 8 8
k zz r
r r
26
ORIGIN OF DEFINITION
OF FAR FIELD
[4-45]
FINITE LENGTH DIPOLEFINITE LENGTH DIPOLE(CONT)(CONT)
Finite dipole Current distributionFinite dipole Current distribution
' 'ˆ sin 02 2' ' '( 0, 0, )
' 'ˆ sin 02 2
I k z zz o
x y ze
I k z zz o
a
I
a
27
(“thin” wire, center fed, zero current at end points)
/ 2 < l <
[4-56]
(see Fig. 4.8)
FINITE LENGTH DIPOLEFINITE LENGTH DIPOLE(CONT)(CONT)
Current distribution for linear wire antennaCurrent distribution for linear wire antenna
28
Fig. 4.8 Current distribution along the length of a linear wire antenna
DIPOLE
Radiated fields at (Radiated fields at (x, y, zx, y, z) ) of finite dipoleof finite dipole
''
sin4
)(zd
R
ezkjEd
jkRe
I
2'22 zzyxR
29
For infinitesimal dipole at z’ of length z’
Since source is only along the z axis ( )
0,0 '' yx
FINITE LENGTH DIPOLEFINITE LENGTH DIPOLE(CONT)(CONT)
Radiated fields of finite dipole at (Radiated fields of finite dipole at (x, y, zx, y, z))
'cos'
'
sin4
)(zde
r
ezkjEd jkz
jkre
I
30
In far field region in phase term
cos'zrR ( let )
[4-58]
FINITE LENGTH DIPOLEFINITE LENGTH DIPOLE(CONT)(CONT)
Far Field Far Field E & H E & H Radiating fields Radiating fields
2/
2/
EdE
'cos2/
2/
' '
)(sin4
zdezIr
ekjE jkz
e
jkr
31
Total Field
[4-58a]
FINITE LENGTH DIPOLEFINITE LENGTH DIPOLE(CONT)(CONT)
Far Field Far Field E & H E & H Radiating fields Radiating fields
sin2
coscos2
cos
2
kk
r
eIjE
jkro
E
H
32
For sinusoidal current distribution
[4-62]
FINITE LENGTH DIPOLEFINITE LENGTH DIPOLE(CONT)(CONT)
Power DensityPower Density
2
2
2 2
cos cos cos2 2
8 sino
r avg
k kI
Wr
33
[4-63]
FINITE LENGTH DIPOLEFINITE LENGTH DIPOLE(CONT)(CONT)
2
2
22
sin2
coscos2
cos
8
kkI
WrU oavg
34
Radiation IntensityRadiation Intensity
[4-64]
FINITE LENGTH DIPOLEFINITE LENGTH DIPOLE(CONT)(CONT)
l ≥ /2
3-dB BEAMWIDTH3-dB BEAMWIDTH
35
3-dB
BE
AM
WID
TH
90° 87°
78°
64°48°
.25 0.5 0.75 1
FINITE LENGTH DIPOLEFINITE LENGTH DIPOLE(CONT)(CONT)
3-dB BEAMWIDTH3-dB BEAMWIDTH
36
If allow new lobes begin to appear
Fig. 4.7(b) 2-D amplitude pattern for a thin dipole l = 1.25 and sinusoidal current distribution
FINITE LENGTH DIPOLEFINITE LENGTH DIPOLE(CONT)(CONT)
Elevation plane amplitude patterns for a thin dipole with sinusoidal current distributionElevation plane amplitude patterns for a thin dipole with sinusoidal current distribution
37
Fig. 4.6
Radiated power Radiated power
s
avgrad dP s
W
38
Results of integration give terms involving Ci & Si [4-68]
FINITE LENGTH DIPOLEFINITE LENGTH DIPOLE(CONT)(CONT)
Radiated power Radiated power
39
sin and cos integrals (tabulated functions like trig. functions, but not as common)
Can find Rr and Do in terms of Ci and Si
Do, Rr, Rin plotted in fig. 4.9
[4-75][4-70]
FINITE LENGTH DIPOLEFINITE LENGTH DIPOLE(CONT)(CONT)
Radiation resistance, input resistance and directivity of a thin dipole with sinusoidal current Radiation resistance, input resistance and directivity of a thin dipole with sinusoidal current distributiondistribution
40
Fig. 4.9
FINITE LENGTH DIPOLE
Input ResistanceInput Resistance
2
oin II
41
(note that Rr uses Imax in its derivation)
for
at input terminalsI
VZ in
z’
Ie (z’)
maxIIo
FINITE LENGTH DIPOLEFINITE LENGTH DIPOLE(CONT)(CONT)
Input ResistanceInput Resistance
42
So, even for lossless antenna ( RL = 0 )
[4-77a]
rin
oin R
I
IR
2
inr RR
2sin 2 k
Rr
z’
Ie (z’)
maxIIo
FINITE LENGTH DIPOLEFINITE LENGTH DIPOLE(CONT)(CONT)
43
Input Resistance (cont)Input Resistance (cont)
Not true in practical case, current not exactly sinusoidal at the feed point(due to non-zero radius of wire and finite feed gap at terminals)
Numerous ways to account for more exact current distribution, result in currents that are both in and out of phase, and in Rin + j Xin
(subject of extensive research, numerical and analytical)
Note: when ; andn inR0inI
FINITE LENGTH DIPOLEFINITE LENGTH DIPOLE(CONT)(CONT)
Empirical formula for Empirical formula for RinRin
)200max(
inR )12max(
inR
44
40
40
G220GRin
17.414.11 GRin
5.27.24 GRin 24
64.0
2
24
G
22
G
)76max(
inR
let2
kG for dipole of length
G
[4-107] [4-110]
FINITE LENGTH DIPOLEFINITE LENGTH DIPOLE(CONT)(CONT)
For MONOPOLEFor MONOPOLE
5.42732
1jZ in
kG
2.215.36 jZ in
45
2
1Rin (monopole) Rin (dipole)
for wavelength monopole1
4
same current; voltage impedance2
1
2
1
[4-106]
HALF WAVE DIPOLEHALF WAVE DIPOLE
sin
cos2
cos
2 r
eIjE
jkro
2
22
2
sin
cos2
cos
8
r
IW o
avg
sin
cos2
cos
2 r
eIjH
jkro
dI
P orad
0
22
sin
cos2
cos
4
46
ll = = /2/2
0 20 40 60 80 100 120 140 160 1800
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(deg)
No
rma
lize
d P
ow
er
2sin
3sin
[4-84]
[4-88]
[4-86]
[4-85]
64.14 max rad
o P
UD
47
ll = = /2/2
HALF WAVE DIPOLE HALF WAVE DIPOLE (CONT)(CONT)
Slightly moredirective thaninf. dipole with
Do = 1.5
2
(2 ) where (2 ) 2.4358
IoP C C
rad in in
[4-89]
[4-91]
max
2 (2 ) 732 4
since if lossless
42.5 73 42.5
r
in r in
in in
PradR C
inIo
R R I I
X Z j
48
l l = = /2/2
HALF WAVE DIPOLE HALF WAVE DIPOLE (CONT)(CONT)
[4-93]
PRACTICAL DIPOLEPRACTICAL DIPOLE
300
Useful for matching to wire lines where
300
in
o
R
Z
49
Folded dipole
l l slightly < slightly < /2/2
2
Usually choose slightly less than 2
so that 0 & Z is totally realXin in
PRACTICAL DIPOLEPRACTICAL DIPOLE(CONT)(CONT)
50
Resistance and Reactance Variations
2
(pure real for length slightly less than )
l l slightly < slightly < /2/2
0.5 1.0
G , B
G
B
IMAGE THEORYIMAGE THEORY
51
Can calculate the fields in the UHP by removing the conductor and finding the field due to the actual and image sources.
Linear antennas near an infinite ground plane could approximate case of earth.
h1
Direct
Reflected
h2
IMAGE THEORYIMAGE THEORY(CONT)(CONT)
52
In the Lower Half Plane, E = H = 0
h
h
h
Image
Actual Problem Equivalent Problem
Observation Point
Observation Point
IMAGE THEORY IMAGE THEORY (CONT)(CONT)
53
Fields due to image source are actually produced by the induced currents in the ground plane
actual
image
I
I
image
actual
I
Iactual
image
I
I
54
VERTICAL DIPOLEVERTICAL DIPOLE HORIZONTAL DIPOLEHORIZONTAL DIPOLE
Electric dipoles above an infinite perfect electric conductorElectric dipoles above an infinite perfect electric conductor
Fig. 4.12(a) Vertical electric dipole above anInfinite, flat, perfect electric conductor
Fig. 4.24 Horizontal electric dipole, and its associated image, above an infinite, flat, perfect electric conductor
55
VERTICAL DIPOLEVERTICAL DIPOLE HORIZONTAL DIPOLEHORIZONTAL DIPOLE
Electric dipoles above ground planeElectric dipoles above ground plane
Fig. 4.14(a) Fig. 4.25(a)
56
Far FieldFar Field
Electric dipoles above an infinite perfect electric conductorElectric dipoles above an infinite perfect electric conductor
Fig. 4.14(b) Fig. 4.25(b)
57
FAR FIELD RADIATING FIELDSFAR FIELD RADIATING FIELDS
r1
h
h
r
r2
h co
s
x
y
z
h
h
r1
r
r2
x
y
z
VERTICAL DIPOLEVERTICAL DIPOLE HORIZONTAL DIPOLEHORIZONTAL DIPOLE
approx. in phase terms
cos1 hrr cos2 hrr
in magnitude terms321 rrr [4-97]
[4-98]
rd EEE21
rd EEE21
58
Summing two contributions
total = incident + reflected total = actual + imaginary
VERTICAL DIPOLEVERTICAL DIPOLE HORIZONTAL DIPOLEHORIZONTAL DIPOLE
11
sin4
1
r
eIkjE
jkrod
22
sin4
2
r
eIkjE
jkror
sin
4 1
1
r
eIkjE
jkrod
sin
4 2
2
r
eIkjE
jkror
[4-94]
[4-95]
[4-111]
[4-112]
FAR FIELD RADIATING FIELDSFAR FIELD RADIATING FIELDS(CONT)(CONT)
coscossin
4jkhjkh
jkro ee
r
eIkjE
59
VERTICAL DIPOLEVERTICAL DIPOLEHORIZONTAL DIPOLEHORIZONTAL DIPOLE
coscossin
4jkhjkh
jkro ee
r
eIkjE
22 sinsin1sin
sinsincos
FAR FIELD RADIATING FIELDSFAR FIELD RADIATING FIELDS(CONT)(CONT)
sin 2 cos cos4
jkrok I e
E j khr
60
[4-99]
[4-116]
VERTICAL DIPOLEVERTICAL DIPOLE HORIZONTAL DIPOLEHORIZONTAL DIPOLE
Single source at origin array factor
Single source at originarray factor
cossin2sinsin1
422 khj
r
eIkjE
jkro
for 0E 0z
FAR FIELD RADIATING FIELDSFAR FIELD RADIATING FIELDS(CONT)(CONT)
12
h
4
h
2
h
61
Amplitude patterns at different heightsAmplitude patterns at different heights
Fig. 4.15Fig. 4.26
Number of lobes
Note minor lobes that are
formed for
HORIZONTAL DIPOLEHORIZONTAL DIPOLEVERTICAL DIPOLEVERTICAL DIPOLE
Number of lobes
Note minor lobes that are
formed for
h2
[4-100] [4-117]
62
Amplitude patterns at different heightsAmplitude patterns at different heights
(CONT)(CONT)
VERTICAL DIPOLEVERTICAL DIPOLE HORIZONTAL DIPOLEHORIZONTAL DIPOLE
Note max radiation is in = 90° direction
Fig. 4.16Fig. 4.28
32
2
2
2sin
2
2cos
3
1
kh
kh
kh
khIP o
rad
32
2
2
2sin
2
2cos
2
2sin
3
1
kh
kh
kh
kh
kh
khIP o
rad
63
VERTICAL DIPOLEVERTICAL DIPOLE
HORIZONTAL DIPOLEHORIZONTAL DIPOLE
[4-102]
[4-118]
R(kh)
RADIATION POWERRADIATION POWER
32
max
2
2sin
2
2cos31
24
kh
kh
kh
khP
UD
rado
42R(kh)
4 hkh
rad
o P
UD max4
42R(kh)
sin4 2 hkh
kh
64
[4-104]
[4-123]
DIRECTIVITYDIRECTIVITYVERTICAL DIPOLEVERTICAL DIPOLE HORIZONTAL DIPOLEHORIZONTAL DIPOLE
Fig. 4.29 Radiation resistance and max. directivityof a horizontal infinitesimal electric dipole as afunction of its height above an infinite perfectelectric conductor.
Fig. 4.18 Directivity and radiation resistanceOf a vertical infinitesimal electric dipole as afunction of its height above an infinite perfectelectric conductor.
!5!3
sin53 xx
xx
!42
1cos42 xx
x
23456
11sin 2
23
x
xx
x
2342
11cos 2
22
x
xx
x
65
DIRECTIVITYDIRECTIVITY (CONT)(CONT)
VERTICAL DIPOLEVERTICAL DIPOLEHORIZONTAL DIPOLEHORIZONTAL DIPOLE
Limiting case of kh
Note:
32
sincos
3
1
x
x
x
x
3
2
6
4
6
1
2
1
3
1
23456
11
2342
11
3
1 2
2
2
2
x
x
x
x
Note: direction of maximum radiationchanges as “h” is varied. Dg (=0)
Dg
(=0)
h/
kh h/ Do
0 0 3
2.88 .458 6.57
6.0
h/ Do
0 7.5
.615+n/2(n=1,2,3…)
slightly
6.0
6.0
66
VERTICAL DIPOLEVERTICAL DIPOLE HORIZONTAL DIPOLEHORIZONTAL DIPOLE
6
312
lim
oDkh
3
322
0lim
oDkh
2
0lim
)(
sin5.7
kh
khoD
kh
[4-124]
DIRECTIVITYDIRECTIVITY (CONT)(CONT)
in in inZ R jX
5.4273 jZ in
67
VERTICAL DIPOLEVERTICAL DIPOLE
Input Impedance of a Input Impedance of a /2 dipole above a /2 dipole above a flat lossy electric conductive surfaceflat lossy electric conductive surface
Fig. 4.20
ininin XRZ
5.4273 jZ in
68
HORIZONTAL DIPOLEHORIZONTAL DIPOLE
Input Impedance of a Input Impedance of a /2 dipole above a /2 dipole above a flat lossy electric conductive surfaceflat lossy electric conductive surface
Fig. 4.30
GROUND EFFECTSGROUND EFFECTS
69
Finite conductivity earth
(“real” earth as ground plane)
h1
h2
Direct
Reflected
earth
Assume earth flat (ok. for Rearth )
10 1 [S/m]
GROUND EFFECTSGROUND EFFECTS (CONT)(CONT)
70
(real earth as ground plane)
Fig. 4.31 Elevation plane amplitude patterns of an infinitesimal vertical dipole above a perfect
electric conductor = and a flat earth = 0.01 [S/m]
VERTICAL DIPOLEVERTICAL DIPOLE HORIZONTAL DIPOLEHORIZONTAL DIPOLE
Fig. 4.32 Elevation plane ( = 90°)amplitude patterns of an infinitesimal horizontal dipole above a perfect
electric conductor = and a flat earth = 0.01 [S/m]
GROUND EFFECTS GROUND EFFECTS (CONT)(CONT)
71
(real earth as ground plane)
For low and medium frequency applications when height is comparable to skin depth [ ]
of the ground increasing changes in input impedance; less efficient; use of ground wires)
GROUND EFFECTS GROUND EFFECTS (CONT)(CONT)
72
Usually negligible effect for observation angle greater than 3°.
EARTH CURVATUREEARTH CURVATURE
Fig. 4.34 Geometry for reflections from a spherical surface
73
EARTH CURVATUREEARTH CURVATURE
Curved surfaces spreads out radiation (divergent) that is reflected more than from flat surface. (can introduce a divergence factor)
Fig. 4.35 Divergence factor for a 4/3 radius earth(ae = 5,280 mi = 8,497.3 km) as a function ofgrazing angle .
reflected field from spherical surface
reflected field from flat surface
DDivergence factor
rf
rs
E
E
GROUND EFFECTS GROUND EFFECTS (CONT)(CONT)
74
l=/50 l=/10 l=/2 l=
Rhf 0.0279 0.2792 0.698 1.3962
RL 0.0279 0.1396 0.349 0.6981
Rr 0.3158 1.9739 73 199
Rin 0.3158 1.9739 73
ecd 0.9188 (-0.368 dB)
0.9339 (-0.296 dB)
0.9952 (-0.021 dB)
0.9965 (-0.015 dB)
D0 1.5 (1.761 dB)
1.5 (1.761 dB)
1.6409 (2.151 dB)
2.411 (3.822 dB)
G0 1.3782 (1.393 dB)
1.4009 (1.464 dB)
1.6331(2.13 dB)
2.4026 (3.807 dB)
-0.9863 -0.9189 0.18929 1
er 0.0271 (-15.67 dB)
0.1556 (-8.08 dB)
0.9642 (-0.158 dB)
0 (- Db)
G0abs 0.0374 (-14.27 dB) 0.2181 (-6.613 dB) 1.5746 (1.972 dB) 0 (- dB)
DIPOLE SUMMARYDIPOLE SUMMARY(Resonant XA=0; f = 100 MHz; = 5.7 x 107 S/m; Zc = 50; b = 3x10-4l)
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