Ece5318 ch4

1
ECE 5318/6352
Antenna Engineering
Chapter 4
Linear Wire Antennas

2
ƒƒINFINITESIMAL DIPOLE
(constant current)
l<<λ ; thin wire ;
G
(only electrical current present)
l ≤ λ/50
I
l / 2
l / 2
Io
θ
Impinging
Wave
z
o z I z I( ' ) = aˆ
I = 0 ⇒F =0
m
[4-1]

3
(CONT)
r = x2+y2+z2
l ≤ λ/50
Fig. 4.1(a) Geometrical arrangement
of an infinitesimal dipole

4
ƒƒ mixed coordinates in
expression - change to
spherical
for A << λ
R ≅ x 2 + y 2 + z 2
( ' , ' , ' ) '
μ
4
A
G G
d
R
(x,y,z) x y z e
jkR
c
e
o
−
A ≅ ∫ I
π
(x,y,z)
source
points
(x’,y’,z’)
l
[4-2]
(CONT)
l ≤ λ/50
( ) ( ) ( ) R = x − x' 2 + y − y' 2 + z − z' 2

5
ƒƒ mixed coordinates in expression
change to spherical
[4-4]
∫
−
−
(x,y,z) ≅
e
/ 2
/ 2
'
4
ˆ
A
A
G G
d z
r
jkr
o μ
A a I o z π
G GA a I z
μ
4
ˆ A
o o e jkr
r
(x,y,z) ≅ −
π
(x,y,z)
source
points
(x’,y’,z’)
l
(CONT)
l ≤ λ/50

6
R ≅ (x − x' )2 + (y − y' )2 + (z − z' )2
θ
μ
A = − A = I A −
sin o o jkr
z e
θ θ sin
4
r
π
θ
μ
A = − A = I A −
cos o o jkr
r z e
θ cos
4
r
π
∫c
d A' along source
= 0 φ A
(x,y,z)
source
points
(x’,y’,z’)
l
[4-6]
(CONT)
ƒ mixed coordinates in expression
need to change to spherical
l ≤ λ/50

7
ƒƒUsing Vector Potential A ,
calculate H & E fields
G G G
= 1 ∇× = ˆ 1 ∇×
[ ]φ φ μ μ
H A a A
) ( 1 A G
[ ] ⎡
⎤
⎥⎦
⎢⎣
∂
r A −
A
r ∂
∂
∂
∇× =
θ φ θ
r r
[4-7]
(CONT)
l ≤ λ/50

8
ƒƒUsing Vector Potential A ,
calculate H fields
[4-8]
⇒
G = 1
G
∇×
H A
μ
⎤
⎢⎣ ⎡
H = j k I A
sin 1+ 1
− ⎥⎦
o e jkr
r jkr
4
θ
π φ
= 0 r H
= 0 θ H
(CONT)
l ≤ λ/50

9
ƒƒUsing Maxwell’’s Eqns to
calculate E fields
G G
= ∇×
E I − ⎥⎦
⎤
⎡
E = j k I sin 1 + 1 − 1
− 2 2
⎥⎦
[4-10]
⇒
1
E H
jωε
η A
o jkr
r e
r jkr
⎤
⎢⎣ ⎡
= cos 1+ 1
2 2 θ
π
= 0 φ E
o e jkr
r jkr k r
⎢⎣
4
θ
π
η
θ
A
Fig. 4.1(b) Geometrical arrangement
of an infinitesimal dipole and its
associated electric-field components
on a spherical surface
(CONT)
l ≤ λ/50

10
ƒƒUsing Hφφ, Er, Eθθ, calculate the complex Poynting vector
= × ∗ = ( ∗ − ∗ ) θ φ θ φ E H E H r r W E H a aG G G G
( ) 1
2
ˆ
2
1
⎥⎦ ⎤
⎢⎣ ⎡
⎤
⎢⎣ ⎡
W η o A
θ
= − ⎥⎦
1 1 2
( )3
2 sin2
j
8 r
kr
I
r
λ
( )2 [4-12]
k Io cos sin = j ⎡ 2 3 1 + j
1 ⎤ 16 ⎢⎣ ( )
2 ⎥⎦
W η θ θ
θ π
r kr
A
(CONT)
l ≤ λ/50

11
ƒƒFind total outward flux through a closed sphere
(only contributions from Wr)
P W dsG G
2 2
0 ∫ ∫ = =
φ θ θ π
π
φ
= ∫∫ • [4-14]
s
⎤
⎥⎦
d W r d r sin
⎡
− ⎥⎦
1 1
θ
= ⎡ 3
⎢⎣
⎤
⎢⎣
2
3 kr
( )
I j oλ
ηπ A
0
=
(CONT)
l ≤ λ/50

12
(CONT)
ƒƒFind total outward flux through a closed sphere
Real P = total radiated power Prad
rad P I I 2 R
o r
ηπ A
o
2
1
2
= ⎡
3
⎤
= ⎥⎦
⎢⎣
λ
Radiation resistance
for free space where
λ
2
= π A r R
η = 120π 2
= = ⇒ = r λ R
[4-19]
[4-16]
0.02 0.316
50
A
2
80 2
λ
ƒƒExample [Ω]
l ≤ λ/50
(Impedance would also have a large capacitive term that is not calculated here.)

13
(CONT)
Imaginary part of P = reactive power in the radial direction
2 1
( )3
Io ⎤
⎥⎦
ηπ A
= − ⎡
⎢⎣
3 λ
kr
[4-17]
(Note: this → 0 as kr → ∞, so it is
essentially not present in far field;
only important in near field considerations)
l ≤ λ/50

14
ƒƒNear Field approximations
[ kr <<<< 1 ]
(field point very close or very low frequency case)
jkr
jkr
θ
E j I e
E ≅ − j I A
e
H I e
o
jkr
−
≅ A
o
−
o
φ sin
π 4 r2
Dominant terms ⇒
[4-20]
θ
π
η
cos
2 k r3
r
−
≅ − A
θ
π
η
θ sin
4 k r3
Like ‘quasistationary” fields
E near static electric dipole
H near static current element
(CONT)
l ≤ λ/50

15
ƒƒNear Field approximations
[ kr <<<< 1 ]
Biot – Savart Law :
infinitesimal ∧
current
element in az direction
(same as above when kr →0)
(note E and H are 90° out of
phase)
IG oA
H ≅ a
ˆ r2
NO RADIAL POWER FLOW --
REACTIVE FIELDS
θ
π φ sin
4
G G G
avg
W = 1 E×H∗
Re [ ]
2
= 0 avg W G
[4-21]
[4-22]
(CONT)
l ≤ λ/50

16
ƒƒIntermediate Fields
[ kr > 1]
(beginnings of radial power flow; still have radial fields)
E 1
r θ ∼ E 1
1
r 2 φ ∼
r E
r
∼
(CONT)
l ≤ λ/50

r = λ/2π (Radian Distance)
(Radius of Radian Sphere)
17
Energy
basically
imaginary
(stored)
Energy
basically
real
(radiated)
(CONT)
Fig. 4.2 Radiated field terms magnitude
variation versus radial distance
l ≤ λ/50

18
(CONT)
ƒƒFar Field [ kr >> 1 ]
Dominant terms ⇒
θ
E j k I e
jkr
−
≅ A
θ sin
θ [4-26]
o
4 π
r
η
H j k I e
π φ sin
4 r
jkr
o
−
≅ A
0 r r E E H H φ θ ≅ ≅ ≅ ≅
l ≤ λ/50

19
ƒƒFar Field [ kr >> 1 ]
η
E
θ =
H
φ
( both E and H are TEM to )
r aˆ
Similar to plane wave but propagation in direction
With 1
and sinθ
variations
r
[4-27]
(CONT)
r aˆ
l ≤ λ/50

20
ƒƒDirectivity (use Far Field approx.)
G G G
avg 2
W = 1 E×H∗
RADIATION INTENSITY
Re[ ]
2
2
2 sin
⎡
= a A
2 4
ˆ
r
k Io
r
θ
π
η
⎤
⎥ ⎥ ⎥
⎦
⎢ ⎢ ⎢
⎣
θ
⎤
⎡ = = k IoA
2 η 2 2
2 4 π
sin
⎥ ⎥⎦
⎢ ⎢⎣
avg U r W
2 2 sin
Io
= ⎡ A ⎤
Real ( W
r ) avg W θ
( Note: 2 as before for )
8 λ
r
η
⎥⎦
⎢⎣
[4-28]
[4-29]
(CONT)
l ≤ λ/50

3
8
⎤
⎡
A
I
D
(CONT)
(in θ = 90° direction) [4-31]
21
D = 4π Umax
o P
rad
2
⎤
⎡
η oA U k I
⎢⎣
=
max 2 4 π
⎥⎦
2
⎤
P I
η π oA
= ⎡
3 ⎥⎦
⎢⎣
λ
rad
1.5
2
3
2
2
= =
⎤
⎥⎦
⎡
⎢⎣
⎥⎦
⎢⎣
=
λ
ηπ
λ
η
A
o
o
o I
ƒƒDirectivity
l ≤ λ/50

22
ƒƒSMALL DIPOLE
Uniform current assumption - only valid for ideal case
( approximated by capacitor plate antenna)
value of fields compared
to constant current case
1_2
λ/50 < l < λ/10
λ/50 < l < λ/10
θ
E j k I e
π
η
θ sin
8 r
jkr
o
−
= A
θ
H j k I e
π φ sin
8 r
jkr
o
−
= A
[4-36]

23
ƒƒSMALL DIPOLE
(CONT)
For physical small dipole
triangular current distribution
value of case of
constant current
1_4
⎤
P I
ηπ oA
= ⎡
⎤
π A
= ⎡
same as constant
current case
λ/50 l λ/10
[4-37]
2
12 ⎥⎦
⎢⎣
λ
rad
2
20 2 ⎥⎦
⎢⎣
λ
r R
= 1.5 o D

24
ƒƒFINITE LENGTH DIPOLE
(length comparable to λ)
⎛ ′ 2
⎞
R r z z
' cos 1 sin2
= − + ⎜ ⎟ +
θ θ
2
r
⎝ ⎠
approx. error
(max error where θ = 90° ; 4th term = 0 there)

[4.41]
Fig. 4.5 Finite dipole geometry
and far-field approximations

25
ƒƒPhase and Magnitude considerations
ƒIn calculating phase assume
can tolerate phase error of π/8 (22°)
ƒMust choose r far enough
away so that ….
(CONT)

26
ƒƒPhase and Magnitude considerations
max z' = A
2
2
′ π
k z
r
≤
2 8
2 2 π
λ
2A2 ≤ ⇒ r
λ
ORIGIN OF
DEFINITION
OF FAR FIELD
8 8
π
r
A
1 R = r
ƒFor magnitude term ⇒ use
r
e− jkr
ƒFor phase term ⇒ use R = r − z' cosθ
[4-45]
(CONT)

27
(CONT)
ƒƒFinite dipole Current distribution
(“thin” wire, center fed, zero current at end points)
λ / 2 l λ
[4-56]
⎤
⎥⎦
⎡
a A
ˆ I sin k z z o
⎢⎣
⎞
⎟⎠
⎛ − '
2
⎜⎝
0 ≤ z' ≤ A
2
⎤
a A 0
⎥⎦
⎡
ˆ I sin k z z o
⎢⎣
⎞
⎟⎠
⎛ + '
2
⎜⎝
− A ≤ z' ≤
2
G
(x' = 0, y' = 0, z' ) = e I
(see Fig. 4.8)

28
ƒƒCurrent distribution for linear wire antenna
Fig. 4.8 Current distribution along
the length of a linear wire antenna
DIPOLE

29
ƒƒRadiated fields at (x, y, z)
of finite dipole
ƒFor infinitesimal dipole at z’ of length Δ z’
'
'
G
d E j k z e
( ) sin
d z
4
R
jkR
e θ
π
η θ
−
≅ I
ƒSince source
is only
along the z axis
( x' = 0, y' = 0 )
⇒ R = x 2 + y 2 + (z − z' )2
(CONT)

30
(CONT)
ƒƒRadiated fields of finite dipole at (x, y, z)
ƒIn far field region
in phase term
⇒ ( let R = r − z ' cos θ )
G
d E j k z e jkz
( ) e d z
cos '
'
' sin
4
r
jkr
e θ
θ θ
π
η
−
≅ I
[4-58]

31
ƒƒFar Field E H Radiating fields
ƒTotal Field
= / 2
∫−
/ 2
A
θ A θ E d E
−
E j k e jkz
/ 2 cos '
/ 2
η ∫−
≅ A
' ' sin ( )
4
I z e d z
r
e
jkr
θ
θ θ
π
A
[4-58a]
(CONT)

32
ƒƒFar Field E H Radiating fields
⎤
⎥ ⎥ ⎥ ⎥
⎦
⎡
⎢ ⎢ ⎢ ⎢
⎣
⎞
⎟⎠
− ⎛ ⎟⎠
⎜⎝
⎞
⎛
⎜⎝
≅
−
θ
θ
π
η
θ sin
2
cos cos
2
cos
2
kA kA
r
E j I e
jkr
o
ƒFor sinusoidal current distribution
[4-62]
H ≅ E
θ
η
φ
(CONT)

33
ƒƒ Power Density
2
⎡ ⎛ ⎞ ⎛ ⎞ ⎤ ⎢ ⎜ ⎟ − ⎜ ⎟ ⎥ = ⎢ ⎝ ⎠ ⎝ ⎠ ⎥
2
2 2
A A
cos cos cos
2 2
o
8 sin
r avg
k k
W I
r
θ
η
π θ
⎢ ⎥
⎢⎣ ⎥⎦
[4-63]
(CONT)

34
(CONT)
ƒƒ Radiation Intensity
2
2
2
2
− ⎛ ⎟⎠
θ
sin
2
cos cos
2
⎛
cos
η
8
⎤
⎥ ⎥ ⎥ ⎥
⎦
⎡
⎢ ⎢ ⎢ ⎢
⎣
⎞
⎟⎠
⎜⎝
⎞
⎜⎝
= =
θ
π
kA kA
U r W Io
avg [4-64]
l ≥ λ/2

35
(CONT)
ƒƒ 3-dB BEAMWIDTH
3-dB BEAMWIDTH
90°
87°
78°
64°
48°
A
.25 .5 .75 1 λ

36
ƒƒ 3-dB BEAMWIDTH
If allow A λ new
lobes begin to appear
Fig. 4.7(b) 2-D amplitude pattern for a thin dipole
l = 1.25 λ and sinusoidal current distribution
(CONT)

37
Elevation plane amplitude patterns for a thin dipole with sinusoidal current distribution
Fig. 4.6

38
(CONT)
ƒƒRadiated power
rad avg P dsG GW
= ∫∫ •
s
[4.66]
ƒResults of integration give terms involving Ci Si [4-68]

39
ƒƒRadiated power
ƒsin and cos integrals
(tabulated functions like trig. functions, but not as common)
ƒCan find Rr and Do in terms of Ci and Si
ƒDo, Rr, Rin plotted in fig. 4.9
[4-70] [4-75]
(CONT)

40
Radiation resistance, input resistance and directivity of a thin dipole with sinusoidal
current distribution
Fig. 4.9
FINITE LENGTH
DIPOLE

41
ƒƒInput Resistance
(note that Rr uses Imax in its derivation)
Z V in =
at input
I terminals
λ
2
for A ≥
in o I ≠ I
z’
Ie (z’)
max I I o =
(CONT)

42
ƒƒInput Resistance
So, even for lossless antenna ( RL = 0 )
⎡
R I
⎤
o
in R
[4-77a]
r
in
I
2
⎥⎦
⎢⎣
r in = R ≠ R ⇒
⎞
⎟⎠
sin2 ⎛
kA
⎜⎝
=
2
Rr
z’
Ie (z’)
max I I o =
(CONT)

43
(CONT)
ƒƒInput Resistance (cont)
Note: when A = n λ ; and →∞ in →0 R in I
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)

Ω in R
π A ⇒ G =
) 76 max ( ⎥⎦ ⎤
⎢⎣ ⎡
(CONT)
G = kA for dipole of length A
≤ G ≤ π ) 200 max ( ⎥⎦ ⎤
44
ƒƒEmpirical formula for Rin
Ω in R
) 12 max ( ⎥⎦ ⎤
⎢⎣ ⎡
4
0
λ
≤ A ≤
4
0
π
R 20G2 ≤ G ≤ in ≅
R 24.7G2.5 in ≅
R 11.14G4.17 in ≅
λ λ
≤ A ≤
4 2
λ
λ
0.64
2
≤ A ≤
π π
≤ G ≤
4 2
2
2
Ω in R
⎢⎣ ⎡
let 2
λ
[4-107] → [4-110]

45
ƒƒFor MONOPOLE
for wavelength monopole 1
4
Z 1 j in ≅ +
[73 42.5]
2
G = k A
R 1 in (monopole) = Rin (dipole)
2
Z ≅ 36.5 + j 21.2 [Ω] in
1
1
same current; voltage ⇒ impedance
2
2
[4-106]

46
ƒƒHALF WAVE DIPOLE
⎤
⎥ ⎥ ⎥ ⎥
⎦
⎡
⎢ ⎢ ⎢ ⎢
⎣
π θ
⎞
⎟⎠
⎛
cos
⎜⎝
E ≅
j I e
−
cos
2
θ
π
η
θ 2 r
sin
jkr
o
2
2
H j I e
jkr
2 2
⎛
cos
2
cos
2
cos
sin
⎛
⎡
cos
o
η
8
⎞
⎤
⎥ ⎥ ⎥ ⎥
⎦
⎡
⎢ ⎢ ⎢ ⎢
⎣
⎞
⎟⎠
⎜⎝
=
θ
θ
π
π
r
W Io
avg
⎤
⎥ ⎥ ⎥ ⎥
⎦
⎢ ⎢ ⎢ ⎢
⎣
⎟⎠
⎜⎝
≅
−
θ
θ
π
φ 2 π r
sin
l = λλ/2
0 20 40 60 80 100 120 140 160 180
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
θ (deg)
Normalized Power
sin 2θ
sin 3θ
[4-84]
θ
⎞
⎛
cos
2
η π P Io d
rad ∫
θ
θ
π
π
⎟⎠
⎜⎝
=
0
2
2
sin
cos
4
[4-85]
[4-86]
[4-88]

l = λλ/2
47
(CONT)
rad P = I C [4-89]
Slightly more
directive than
inf. dipole with
Do = 1.5
η
= 4 max ≅ 1.64
o P
rad
D π U
where in (2π ) ≅ 2.435 (2 ) C
2
8
π
π
in
o
[4-91]

l = λλ/2
48
(CONT)
η
R P
= = ≅ 73 [Ω]
r C
2 2 (2 ) π
4
π
in
rad
o
I
since (if lossless) in r R ≅ R in I = I max
X ≅ 42.5 [Ω] ⇒ Z ≅ 73 + j 42.5 [Ω] in in
[4-93]

49
ƒƒPRACTICAL DIPOLE
Usually choose l slightly less than 2 so that is totally real.
ƒFolded dipole
≅ 300 [Ω] in R
Useful for matching to two-wire
lines where
≅ 300 [Ω] o Z
l slightly λλ/2
λ
2
A ≅
λ
in in X →0 Z

50
ƒƒPRACTICAL DIPOLE
(CONT)
ƒResistance and Reactance Variations
(pure real for length slightly less than λ )
2
l slightly λλ/2
0.5 1.0 A λ
G , B
G
B

51
ƒƒIMAGE THEORY
ƒLinear antennas near an infinite ground plane
could approximate case of earth.
h1
Direct
Reflected
ƒCan calculate the fields in the UHP by removing the conductor
and finding the field due to the actual and image sources.
h2

52
ƒƒIMAGE THEORY
(CONT)
h
Observation
Point
μο, εο ⇒
→ →
h
h
ƒIn the Lower Half Plane, E = H = 0
Observation
Point
μο, εο
μο, εο
Image
σ = ∞
Actual Problem Equivalent Problem

53
ƒƒIMAGE THEORY
(CONT)
ƒFields due to image source are actually produced
by the induced currents in the ground plane
+
−
+
−
⇓ ⇓
⇓
actual
image
I
I
actual
image
I
I
actual
image
I
I

Electric dipoles above an infinite perfect electric conductor
54
VERTICAL DIPOLE HORIZONTAL DIPOLE
Fig. 4.12(a) Vertical electric dipole above an
Infinite, flat, perfect electric conductor
Fig. 4.24 Horizontal electric dipole, and its associated
image, above an infinite, flat, perfect electric conductor

55
Electric dipoles above ground plane
Fig. 4.14(a)
Fig. 4.25(a)

Electric dipoles above an infinite perfect electric conductor
56
Far Field
Fig. 4.14(b) Fig. 4.25(b)

57
ƒƒFAR FIELD RADIATING FIELDS
r1
h
h
r
r2
θ
h cos θ
x
y
z
h
h
r1
r
r
2
x
y
z
ψ
approx. in
phase terms
cosθ 1 r ≅ r − h
cosθ 2 r ≅ r + h
[4-97]
1 2 3 in magnitude terms r ≅ r ≅ r
[4-98]

58
E ≅ Ed +
ErSumming two contributions
θ θ1 θ 2 E Ed Er ψ ψ1 ψ 2 ≅ +
total = incident + reflected total = actual + imaginary
1
E j k I e
1
sin
4
1
θ
π
η
θ r
jkr
d o
−
≅ A
2
E j k I e
2
sin
4
2
θ
π
η
θ r
jkr
r o
−
≅ A
ψ
E j k I e
π
η
ψ sin
4 1
1
r
jkr
d o
−
≅ A
ψ
E j k I e
π
η
ψ sin
4 2
2
r
jkr
r o
−
≅ − A
[4-94]
[4-95]
[4-111]
[4-112]
(CONT)

cosψ = sinθ sinφ
sinψ = 1− sin2θ sin2φ
− A
E ≅ j k I e − −
η cos cos sin
4
59
VERTICAL DIPOLE
HORIZONTAL DIPOLE
[ θ θ ]
− A
E ≅ j k I e − +
η cos cos sin
θ θ
4
π
jkh jkh
jkr
o e e
r
[ θ θ ]
ψ ψ
π
jkh jkh
jkr
o e e
r
(CONT)

θ φ [ ( θ )]
60
−
η
E j k I o A
e kh
≅ ⎡⎣ ⎤⎦
r θ
[4-99]
[4-116]
sin 2 cos ( cos )
4
jkr
θ θ
π
Single source at origin array factor
− A
E j k I e
π
Single source at origin array factor
η
ψ 1 sin sin 2 sin cos
4
2 2 j kh
r
jkr
≅ o −
= 0 for θ E z 0
(CONT)

≅ 2 h [4-100] [4-117]
61
Amplitude patterns at different heights
Fig. 4.15
Fig. 4.26
Note minor lobes that are
formed for
Number of lobes
Note minor lobes that are
formed for
Number of lobes
h
λ
≅ 2 + 1
λ
4
h ≥
λ
2
h ≥
λ

62
Amplitude patterns at different heights
(CONT)
Fig. 4.16
Note max radiation is in θ = 90° direction
Fig. 4.28

63
ƒƒ RADIATION POWER
VERTICAL DIPOLE
HORIZONTAL DIPOLE
sin 2
P Io 1
cos 2
kh
rad λ
kh
= ⎡ 2 3
+ − ⎥⎦
[4-102]
[4-118]
sin 2
cos 2
P Io 1
sin 2
kh
rad λ
kh
kh
= ⎡ 2 3
+ − − ⎥⎦
R(kh)
( )
( )
( )
⎤
( ) ⎥⎦
⎡
⎢⎣
⎤
⎢⎣
2
2
2
3
kh
kh
ηπ A
( )
( )
( )
( )
( )
⎤
( ) ⎥⎦
⎡
⎢⎣
⎤
⎢⎣
2
2
2
2
3
kh
kh
kh
ηπ A

4sin2 π λ kh kh h
⎞
64
4 2
P kh
[4-104]
≤ ⎛ ≤
4 π λ
kh h
[4-123]
ƒƒDIRECTIVITY
Fig. 4.29 Radiation resistance and max. directivity
of a horizontal infinitesimal electric dipole as a
function of its height above an infinite perfect
electric conductor.
( )
( )
⎤
( )
( ) ⎥ ⎥ ⎥ ⎥ ⎥
⎦
⎡
⎢ ⎢ ⎢ ⎢ ⎢
⎣
⎤
⎥⎦
⎢⎣ ⎡
cos 2
− +
= =
sin 2
kh
2 3
max
2
2
1
3
kh
kh
D U
rad
o π
⎞
⎟⎠
≥ ⎛ ≥
⎜⎝
R(kh) 2 4
D 4π Umax
= =
o P
rad
( )
⎟⎠
⎜⎝
R(kh) 2 4
Fig. 4.18 Directivity and radiation resistance
Of a vertical infinitesimal electric dipole as a
function of its height above an infinite perfect
electric conductor.

65
ƒƒDIRECTIVITY
(CONT)
VERTICAL DIPOLE
HORIZONTAL DIPOLE
Limiting case of kh→ 0
Note:

x2 x4 x = − +
3! 5!
sin
x3 x5 x = x − +

2 4!
cos 1
cos 1 1 2
= − + x
x
2 2 ⋅ ⋅
sin 1 1 2
= − + x
x
3 2 ⋅ ⋅ ⋅
6 5 4 3 2
x x
2 4 3 2
x x
⇒ − x +
cos sin
2 3
1
3
x
x
x
1 2
2
3
⎡
1 1
≅ 1 + − = 4
=
6
1
6
1
2
3
⎤
⎥⎦
⎡
⎢⎣
⋅ ⋅ ⋅
⎤
1 1
+ − + ⎥⎦
⎢⎣
⋅ ⋅
≅ − − +
6 5 4 3 2
2 4 3 2
3
2
2
2
x
x
x
x
Note: direction of maximum radiation
changes as “h” is varied. Dg (θ=0)
Dg
(θ=0)
h/λ

66
2
Do
kh
2
Do
kh
kh h/λ Do
0 0 3
2.88 .458 6.57
∞ ∞ 6.0
⎡
kh
lim ( )
h/λ Do
0 7.5
slightly
6.0
.615+n/2
(n=1,2,3…)
∞ 6.0
6
3
1
lim
→ =
→∞
3
3
2
0
lim
→ =
→
( ) 2
0
⎤
7.5 sin ⎥⎦
⎢⎣
=
→
kh
Do
kh
[4-124]
ƒƒDIRECTIVITY
(CONT)

67
Input Impedance of a λλ/2 dipole above a
flat lossy electric conductive surface
VERTICAL DIPOLE
Fig. 4.20
in in in Z ≅ R + X Z ≅ 73 + j 42.5 [Ω] in

68
Input Impedance of a λλ/2 dipole above a
flat lossy electric conductive surface
HORIZONTAL DIPOLE
Fig. 4.30 in in in Z ≅ R + X Z ≅ 73 + j 42.5 [Ω] in

69
ƒƒGROUND EFFECTS
(“real” earth as ground plane)
ƒFinite conductivity σearth
h1
10 → 1 [S/m]
h2
Direct
Reflected
σearth
ƒAssume earth flat (ok. for Rearth λ)

70
ƒƒGROUND EFFECTS
(CONT)
(real earth as ground plane)
VERTICAL DIPOLE
Fig. 4.31 Elevation plane amplitude patterns
of an infinitesimal vertical dipole above a perfect
electric conductor σ=∞ and a flat earth σ= 0.01 [S/m]
HORIZONTAL DIPOLE
Fig. 4.32 Elevation plane ( φ = 90°)amplitude patterns
of an infinitesimal horizontal dipole above a perfect
electric conductor σ=∞ and a flat earth σ= 0.01 [S/m]

71
ƒƒGROUND EFFECTS
(CONT)
(real earth as ground plane)
σσ = ∞∞
σσearth
For low and medium frequency applications when
height is comparable to skin depth [ δ = 2/ωμσ ]
of the ground ⇒ increasing changes in input
impedance; less efficient; use of ground wires)

72
ƒƒGROUND EFFECTS
(CONT)
EARTH CURVATURE
Usually negligible effect
for observation angle ψ
greater than 3°.
Fig. 4.34 Geometry for reflections from a spherical surface

Divergence
factor
E
r
s
E
= rf
= ___________________
73
EARTH 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 of
grazing angle ψ.
reflected field from spherical surface
reflected field from flat surface
ƒƒGROUND EFFECTS
(CONT)

74
DIPOLE SUMMARY
(Resonant ⇒ XA=0; f = 100 MHz; σ = 5.7 x 107 S/m; Zc = 50; b = 3x10-4l)
l=λ/50 l=λ/10 l=λ/2 l=λ
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 ∞
0.9965
(-0.015 dB)
2.411
(3.822 dB)
2.4026
(3.807 dB)
0.9952
(-0.021 dB)
1.6409
(2.151 dB)
1.6331
(2.13 dB)
0.9339
(-0.296 dB)
1.5
(1.761 dB)
1.4009
(1.464 dB)
0.9188
(-0.368 dB)
1.5
(1.761 dB)
1.3782
(1.393 dB)
ecd
D0
G0
Γ -0.9863 -0.9189 0.18929 1
0
(-∞ Db)
0.9642
(-0.158 dB)
0.1556
(-8.08 dB)
0.0271
(-15.67 dB)
er
G0abs 0.0374 (-14.27 dB) 0.2181 (-6.613 dB) 1.5746 (1.972 dB) 0 (-∞ dB)

Ece5318 ch4

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