Download - Ece5318 ch4
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Chapter 4Chapter 4
Linear Wire AntennasLinear Wire Antennas
ECE 5318/6352ECE 5318/6352Antenna EngineeringAntenna Engineering
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INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE
(only electrical current present)
(constant current)
l ≤ λ/50
I
l / 2
l / 2
Io
θImpinging
Wave
z
oIz zaI ˆ)( ' =
; thin wire ;λ<<l
00 =⇒= FIm
[4-1]
3
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)
222 zyxr ++=
Fig. 4.1(a) Geometrical arrangementof an infinitesimal dipole
l ≤ λ/50
4
mixed coordinates in mixed coordinates in expression expression -- change to change to
sphericalspherical
222 zyxR ++≅
'''' ),,(4
dR
ezyx(x,y,z)jkR
ce
o−
∫≅ IAπ
μ
λ<<for
(x,y,z)
(x’,y’,z’)
source points
l
[4-2]
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)
l ≤ λ/50
( ) ( ) ( )2 2 2' ' 'R x x y y z z= − + − + −
5
mixed coordinates in expression mixed coordinates in expression change to sphericalchange to spherical
[4-4]
∫−
−
≅2/
2/
'
4ˆ zd
re(x,y,z)
jkroo IaA z π
μ
jkroo er
(x,y,z) −≅π
μ4
ˆ IaA z
(x,y,z)
(x’,y’,z’)
source points
l
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)
l ≤ λ/50
6
( ) ( ) ( ) 2'2'2' zzyyxxR −+−+−≅
θπ
μθθ sin4
sin jkrooz e
rIAA −=−=
θπ
μθ cos4
cos jkroozr e
rIAA −=−=
∫cd ' along source
0=φA
(x,y,z)
(x’,y’,z’)
source points
l
[4-6]
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)
mixed coordinates in expression mixed coordinates in expression need to change to sphericalneed to change to spherical
l ≤ λ/50
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Using Vector Potential Using Vector Potential A A , , calculate calculate HH & & EE fields fields
[ ] ⎥⎦⎤
⎢⎣⎡
∂∂
−∂∂
=×∇θθφ
rAArrr
)(1A
[ ]φφ μμAaAH ×∇=×∇=
1ˆ1
[4-7]
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)
l ≤ λ/50
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Using Vector Potential Using Vector Potential A A , , calculate calculate HH fields fields
[4-8]
⇒
AH ×∇=μ1
jkro ejkrr
IkjH −⎥⎦
⎤⎢⎣
⎡+=
11sin4
θπφ
0=rH
0=θH
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)
l ≤ λ/50
9
Using MaxwellUsing Maxwell’’s s EqnsEqns totocalculate calculate EE fields fields
[4-10]
⇒
HE ×∇=ωεj1
jkror e
jkrrIE −
⎥⎦
⎤⎢⎣
⎡+=
11cos2 2 θ
πη
0=φE
jkro erkjkrr
IkjE −⎥⎦
⎤⎢⎣
⎡−+= 22
111sin4
θπ
ηθ
Fig. 4.1(b) Geometrical arrangementof an infinitesimal dipole and its associated electric-field componentson a spherical surface
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)
l ≤ λ/50
10
Using Using HHφφ, , EErr, , EEθθ,, calculate the complex Poynting vectorcalculate the complex Poynting vector
( )∗∗∗ −=×= φθφθ HEHE rr aaHEW ˆ21)(
21
⎥⎦⎤
⎢⎣⎡= −⎥
⎦
⎤⎢⎣
⎡3)(
112
2sin2
8 krj
rI
roW θλ
η
[4-12]( )2cos sin 112 3 216 ( )
k Ioj j
r krW η θ θ
θ π+
⎡ ⎤= ⎢ ⎥⎣ ⎦
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)
l ≤ λ/50
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Find total outward flux through a closed sphereFind total outward flux through a closed sphere
(only contributions from Wr)
[4-14]∫∫ •=s
dP sW
⎥⎦
⎤⎢⎣
⎡−⎥⎦
⎤⎢⎣⎡= 3
2
)(11
3 krjIo
λπη
θθφπ
θ
π
φdrWd r sin
0
22
0 ∫∫ ===
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)
l ≤ λ/50
12
Find total outward flux through a closed sphereFind total outward flux through a closed sphere
roo
rad RIIP 22
21
3=⎥⎦
⎤⎢⎣⎡=
ληπ
316.002.050
=⇒== rRλλ
2
2280
λπ=rR
2120 πη =
Real P = total radiated power Prad
ExampleExample [Ω]
Radiation resistance
for free space where
[4-19]
[4-16]
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)
l ≤ λ/50
(Impedance would also have a large capacitive term that is not calculated here.)
13
( )3
2 13 kr
Io⎥⎦⎤
⎢⎣⎡−=
λπη
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 field considerations)
[4-17]
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)
l ≤ λ/50
14
Near Field approximations Near Field approximations [ [ krkr <<<< 1 ]1 ]
(field point very close or very low frequency case)
θπφ sin
4 2reIH
jkro
−
≅
Dominant terms ⇒
[4-20]
θπ
η cos2 3rk
eIjEjkr
or
−
−≅
θπ
ηθ sin
4 3rkeIjE
jkro
−
−≅
Like ‘quasistationary” fields
E near static electric dipole
H near static current element
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)
l ≤ λ/50
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Near Field approximations Near Field approximations [ [ krkr <<<< 1 ]1 ]
Biot – Savart Law : infinitesimal current element in directionaz
∧
(same as above when kr →0)
(note E and H are 90° out of phase)
NO RADIAL POWER FLOW --REACTIVE FIELDS
θπφ sin
4ˆ
2rIoaH ≅
][Re21 ∗×= HEWavg
0=avgW
[4-21]
[4-22]
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)
l ≤ λ/50
16
Intermediate FieldsIntermediate Fields[ [ krkr >> 1]1]
(beginnings of radial power flow; still have radial fields)
1Erθ ∼
1Erφ ∼2
1rE
r∼
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)
l ≤ λ/50
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r = λ/2π (Radian Distance)
(Radius of Radian Sphere)
Energy basically imaginary (stored)
Energybasically
real(radiated)
Fig. 4.2 Radiated field terms magnitude variation versus radial distance
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)
l ≤ λ/50
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Far Field Far Field [ [ krkr >>>> 1 ]1 ]
Dominant terms ⇒
[4-26]θπφ sin
4 reIkjH
jkro
−
≅
0r rE E H Hφ θ≅ ≅ ≅ ≅
θπ
ηθ sin
4 reIkjE
jkro
−
≅
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)
l ≤ λ/50
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Far Field Far Field [ [ krkr >>>> 1 ]1 ]
ηφ
θ =H
E( both E and H are TEM to )
ra
θsin
Similar to plane wave but propagation in direction
With and variationsr1
[4-27]
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)
ra
l ≤ λ/50
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DirectivityDirectivity (use Far Field approx.)
RADIATION INTENSITY
][Re21 ∗×= HEWavg 2
22 sin
42ˆ
rIk o
rθ
πη
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
= a
θπ
η 242 sin22
⎥⎥⎦
⎤
⎢⎢⎣
⎡== oIkavgWrU
( Note: as before for )2
22 sin8 r
oIavgW θ
λη
⎥⎦
⎤⎢⎣
⎡= )( Real rW
[4-28]
[4-29]
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)
l ≤ λ/50
21
(in θ = 90° direction) [4-31]
rado P
UD max4π=
2
max 42 ⎥⎦
⎤⎢⎣
⎡=
πη oIkU
2
3 ⎥⎦⎤
⎢⎣⎡=
λπη o
radIP
5.123
3
82
2
==
⎥⎦⎤
⎢⎣⎡
⎥⎦⎤
⎢⎣⎡
=
λπη
λη
o
o
oI
I
D
DirectivityDirectivity
INFINITESIMAL DIPOLEINFINITESIMAL DIPOLE(CONT)(CONT)
l ≤ λ/50
22
SMALL DIPOLESMALL 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
θπ
ηθ sin
8 reIkjE
jkro
−
=
θπφ sin
8 reIkjH
jkro
−
=
[4-36]
23
SMALL DIPOLESMALL DIPOLE(CONT)(CONT)
For physical small dipole triangular current distribution
value of case of constant current
1_4
same as constant current case
λ/50 < l < λ/10
[4-37]
2
12 ⎥⎦⎤
⎢⎣⎡=
ληπ o
radIP
2220 ⎥⎦
⎤⎢⎣⎡=λ
πrR
5.1=oD
24
FINITE LENGTH DIPOLEFINITE LENGTH DIPOLE(length comparable to λ)
(max error where θ = 90° ; 4th term = 0 there)
approx. error
2' 21cos sin
2zR r z
rθ θ
′⎛ ⎞= − + +⎜ ⎟
⎝ ⎠
[4.41]
Fig. 4.5 Finite dipole geometryand far-field approximations
25
Phase and Magnitude considerationsPhase and Magnitude considerations
In calculating phase assumecan tolerate phase error of π/8 (22°)
Must choose r far enough away so that ….
FINITE LENGTH DIPOLEFINITE LENGTH DIPOLE(CONT)(CONT)
26
Phase and Magnitude considerationsPhase and Magnitude considerations
2max ' =z
2
2 8k z
rπ′
≤
ORIGIN OF DEFINITION OF FAR FIELD
λ
22>r⇒≤
882 2 πλπ
r
jkre−For phase term ⇒ use θcos'zrR −=
For magnitude term ⇒ user1 rR =
[4-45]
FINITE LENGTH DIPOLEFINITE LENGTH DIPOLE(CONT)(CONT)
27
Finite dipole Current distributionFinite dipole Current distribution
(“thin” wire, center fed, zero current at end points)
λ / 2 < l < λ
[4-56]⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ − '
2sinˆ zkIoza
20 ' ≤≤ z
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ + '
2sinˆ zkIoza 0
2' ≤≤− z
=== ),0,0( ''' zyxeI
(see Fig. 4.8)
FINITE LENGTH DIPOLEFINITE LENGTH DIPOLE(CONT)(CONT)
28
Current distribution for linear wire antennaCurrent distribution for linear wire antenna
Fig. 4.8 Current distribution along the length of a linear wire antenna
DIPOLE
29
Radiated fields at (Radiated fields at (x, y, zx, y, z) ) of finite dipoleof finite dipole
''
sin4
)( zdRezkjEd
jkRe θ
πηθ
−
≅I
( ) 2'22 zzyxR −++=⇒
For infinitesimal dipole at z’ of length Δ z’
Since source is only along the z axis ( )0,0 '' == yx
FINITE LENGTH DIPOLEFINITE LENGTH DIPOLE(CONT)(CONT)
30
Radiated fields of finite dipole at (Radiated fields of finite dipole at (x, y, zx, y, z))
In far field regionin phase term
θcos'zrR −=( let )⇒
'cos'
'
sin4
)( zderezkjEd jkz
jkre θ
θ θπ
η−
≅I
[4-58]
FINITE LENGTH DIPOLEFINITE LENGTH DIPOLE(CONT)(CONT)
31
Far Field Far Field E & H E & H Radiating fields Radiating fields
∫−=
2/
2/ θθ EdE
'cos2/
2/
' '
)(sin4
zdezIr
ekjE jkze
jkrθ
θ θπ
η∫−
−
≅
Total Field
[4-58a]
FINITE LENGTH DIPOLEFINITE LENGTH DIPOLE(CONT)(CONT)
32
Far Field Far Field E & H E & H Radiating fields Radiating fields
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡⎟⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛
≅−
θ
θ
πη
θ sin2
coscos2
cos
2
kk
reIjE
jkro
For sinusoidal current distribution
[4-62]
ηθ
φEH ≅
FINITE LENGTH DIPOLEFINITE LENGTH DIPOLE(CONT)(CONT)
33
Power DensityPower Density
2
2
2 2
cos cos cos2 2
8 sino
r avg
k kIW
r
θηπ θ
⎡ ⎤⎛ ⎞ ⎛ ⎞−⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦
[4-63]
FINITE LENGTH DIPOLEFINITE LENGTH DIPOLE(CONT)(CONT)
34
Radiation IntensityRadiation Intensity
2
2
22
sin2
coscos2
cos
8⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡⎟⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛
==θ
θ
πη
kkIWrU o
avg [4-64]
FINITE LENGTH DIPOLEFINITE LENGTH DIPOLE(CONT)(CONT)
l ≥ λ/2
35
33--dB BEAMWIDTHdB BEAMWIDTH
3-dB
BE
AM
WID
TH
90°87°
78°64°
48°
.25 1.75.5 λ
FINITE LENGTH DIPOLEFINITE LENGTH DIPOLE(CONT)(CONT)
36
33--dB BEAMWIDTHdB BEAMWIDTH
λ>If allow new lobes begin to appear
Fig. 4.7(b) 2-D amplitude pattern for a thin dipolel = 1.25 λ and sinusoidal current distribution
FINITE LENGTH DIPOLEFINITE LENGTH DIPOLE(CONT)(CONT)
37
Elevation plane amplitude patterns for a thin dipole with sinusoElevation plane amplitude patterns for a thin dipole with sinusoidal current distributionidal current distribution
Fig. 4.6
38
Radiated power Radiated power
Results of integration give terms involving Ci & Si [4-68]
∫∫ •=s
avgrad dP sW [4.66]
FINITE LENGTH DIPOLEFINITE LENGTH DIPOLE(CONT)(CONT)
39
Radiated power Radiated power
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)
40
Radiation resistance, input resistance and directivity of a thinRadiation resistance, input resistance and directivity of a thin dipole with sinusoidal dipole with sinusoidal current distributioncurrent distribution
Fig. 4.9
FINITE LENGTH DIPOLE
41
Input ResistanceInput Resistance
(note that Rr uses Imax in its derivation)
2λ
≥for
oin II ≠
at input terminalsI
VZin =
z’
Ie (z’)
maxIIo =
FINITE LENGTH DIPOLEFINITE LENGTH DIPOLE(CONT)(CONT)
42
Input ResistanceInput Resistance
So, even for lossless antenna ( RL = 0 )
[4-77a]
rin
oin R
IIR
2
⎥⎦
⎤⎢⎣
⎡=inr RR ≠ ⇒
⎟⎠⎞
⎜⎝⎛
=
2sin2 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 ; andλn= ∞→inR0→inI
FINITE LENGTH DIPOLEFINITE LENGTH DIPOLE(CONT)(CONT)
44
Empirical formula for Empirical formula for RinRin
)12max( ⎥⎦⎤
⎢⎣⎡Ω<inR
40 λ
≤≤4
0 π≤≤ G220GRin ≅
17.414.11 GRin ≅
5.27.24 GRin ≅24λλ
≤≤
λλ 64.02
≤≤
24ππ
≤≤ G
22
≤≤ Gπ )200max( ⎥⎦⎤
⎢⎣⎡Ω<inR
)76max( ⎥⎦⎤
⎢⎣⎡Ω<inR
let 2kG = for dipole of length
λπ
=G⇒
[4-107] → [4-110]
FINITE LENGTH DIPOLEFINITE LENGTH DIPOLE(CONT)(CONT)
45
For MONOPOLEFor MONOPOLE
[ ]5.427321 jZin +≅
kG =
21Rin (monopole) = Rin (dipole)
[ ]Ω+≅ 2.215.36 jZin
for wavelength monopole14
same current; voltage ⇒ impedance21
21
[4-106]
46
HALF WAVE DIPOLEHALF WAVE DIPOLE
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡⎟⎠⎞
⎜⎝⎛
≅−
θ
θπ
πη
θ sin
cos2
cos
2 reIjE
jkro
2
22
2
sin
cos2
cos
8⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡⎟⎠⎞
⎜⎝⎛
=θ
θπ
πη
rIW o
avg
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡⎟⎠⎞
⎜⎝⎛
≅−
θ
θπ
πφ sin
cos2
cos
2 reIjH
jkro
ll = = λλ/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
θ (deg)
Nor
mal
ized
Pow
er
θ2sin
θ3sin
[4-84]
θθ
θπ
πη π
dIP orad ∫
⎟⎠⎞
⎜⎝⎛
=0
22
sin
cos2
cos
4[4-88]
[4-86]
[4-85]
47
ll = = λλ/2/2
HALF WAVE DIPOLE HALF WAVE DIPOLE (CONT)(CONT)
Slightly moredirective thaninf. dipole withDo = 1.5
64.14 max ≅=rad
o PUD π
where 435.2)2( ≅πinC)2(
2
8π
πη
ino
rad CIP = [4-89]
[4-91]
48
l l = = λλ/2/2
HALF WAVE DIPOLE HALF WAVE DIPOLE (CONT)(CONT)
since (if lossless)rin RR ≅ inII =max
[ ] [ ]Ω+≅⇒Ω≅ 5.42735.42 jZX inin
[ ]Ω≅== 734
2 )2(2 ππ
ηin
o
radr C
IPR
[4-93]
49
PRACTICAL DIPOLEPRACTICAL DIPOLE
[ ]Ω≅ 300inR
[ ]Ω≅ 300oZ
Folded dipole
Useful for matching to two-wire
lines where
l l slightly < slightly < λλ/2/2
2λ
≅
Usually choose ll slightly less than so that is totally real.2λ
ininX Z0 &→
50
PRACTICAL DIPOLEPRACTICAL DIPOLE(CONT)(CONT)
Resistance and Reactance Variations
2λ(pure real for length slightly less than )
l l slightly < slightly < λλ/2/2
0.5 1.0 λ
G , B
G
B
51
IMAGE THEORYIMAGE THEORY
Can calculate the fields in the UHP by removing the conductorand finding the field due to the actual and image sources.
Linear antennas near an infinite ground plane could approximate case of earth.
h1
Direct
Reflectedh2
52
IMAGE THEORYIMAGE THEORY(CONT)(CONT)
In the Lower Half Plane, E = H = 0→→
h
μο, εο
⇒
h
h
μο, εο
μο, εο
Image
σ = ∞
Actual Problem Equivalent Problem
Observation Point
Observation Point
53
IMAGE THEORY IMAGE THEORY (CONT)(CONT)
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
θ
h cos
θ
x
y
z
h
h
r1
r
r2
x
y
z
ψ
VERTICAL DIPOLEVERTICAL DIPOLE HORIZONTAL DIPOLEHORIZONTAL DIPOLE
approx. in phase terms
θcos1 hrr −≅θcos2 hrr +≅
in magnitude terms321 rrr ≅≅[4-97]
[4-98]
58
Summing two contributions
total = incident + reflected total = actual + imaginary
VERTICAL DIPOLEVERTICAL DIPOLE HORIZONTAL DIPOLEHORIZONTAL DIPOLE
rd EEE21 θθθ +≅
11
sin4
1
θπ
ηθ r
eIkjEjkr
od−
≅
22
sin4
2
θπ
ηθ r
eIkjEjkr
or−
≅
ψπ
ηψ sin
4 1
1
reIkjE
jkrod
−
≅
ψπ
ηψ sin
4 2
2
reIkjE
jkror
−
−≅
rd EEE21 ψψψ +≅
[4-94]
[4-95]
[4-111]
[4-112]
FAR FIELD RADIATING FIELDSFAR FIELD RADIATING FIELDS(CONT)(CONT)
59
VERTICAL DIPOLEVERTICAL DIPOLEHORIZONTAL DIPOLEHORIZONTAL DIPOLE
[ ]θθθ θ
πη coscossin
4jkhjkh
jkro ee
reIkjE +≅ −
−
[ ]θθψ ψ
πη coscossin
4jkhjkh
jkro ee
reIkjE −≅ −
−
φθψ 22 sinsin1sin −=
φθψ sinsincos =
FAR FIELD RADIATING FIELDSFAR FIELD RADIATING FIELDS(CONT)(CONT)
60
[4-99]
[4-116]
VERTICAL DIPOLEVERTICAL DIPOLE HORIZONTAL DIPOLEHORIZONTAL DIPOLE
Single source at origin array factor
( )sin 2 cos cos4
jkrok I eE j kh
rθη θ θ
π
−
≅ ⎡ ⎤⎣ ⎦
Single source at origin array factor
( )[ ]θφθπ
ηψ cossin2sinsin1
422 khj
reIkjE
jkro −≅
−
for 0=θE 0<z
FAR FIELD RADIATING FIELDSFAR FIELD RADIATING FIELDS(CONT)(CONT)
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
12+≅
λh
4λ
≥h2λ
≥h
λh2
≅[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 θ = 90° direction
Fig. 4.16Fig. 4.28
63
VERTICAL DIPOLEVERTICAL DIPOLE
HORIZONTAL DIPOLEHORIZONTAL DIPOLE[4-102]
[4-118]
R(kh)
RADIATION POWERRADIATION POWER
( )( )
( )( ) ⎥
⎦
⎤⎢⎣
⎡+−⎥⎦
⎤⎢⎣⎡= 32
2
22sin
22cos
31
khkh
khkhIP o
rad λπη
( )( )
( )( )
( )( ) ⎥
⎦
⎤⎢⎣
⎡+−−⎥⎦
⎤⎢⎣⎡= 32
2
22sin
22cos
22sin
31
khkh
khkh
khkhIP o
rad λπη
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.
( )( )
( )( ) ⎥
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
⎥⎦
⎤⎢⎣
⎡+−
==
32
max
22sin
22cos
31
24
khkh
khkhP
UDrad
o π
⎟⎠⎞
⎜⎝⎛ ≥≥
42R(kh)4 λπ hkh
==rad
o PUD max4π
( )⎟⎠⎞
⎜⎝⎛ ≤≤
42R(kh)sin4 2 λπ hkhkh
Fig. 4.18 Directivity and radiation resistanceOf a vertical infinitesimal electric dipole as afunction of its height above an infinite perfectelectric conductor.
65
DIRECTIVITYDIRECTIVITY(CONT)(CONT)
VERTICAL DIPOLEVERTICAL DIPOLEHORIZONTAL DIPOLEHORIZONTAL DIPOLE
Limiting case of kh→ 0
Note:
!5!3sin
53 xxxx +−=
!421cos
42 xxx +−=
2345611sin 2
23 ⋅⋅⋅+−=
xxx
x
234211cos 2
22 ⋅⋅+−=
xxx
x
32
sincos31
xx
xx
+−⇒
32
64
61
21
31
==−+≅
⎥⎦
⎤⎢⎣
⎡⋅⋅⋅
+−+⎥⎦
⎤⎢⎣
⎡⋅⋅
+−−≅23456
112342
1131 2
2
2
2
xx
xx
Note: direction of maximum radiationchanges as “h” is varied. Dg (θ=0)
Dg(θ=0)
h/λ
66
VERTICAL DIPOLEVERTICAL DIPOLE HORIZONTAL DIPOLEHORIZONTAL DIPOLE
6.0∞∞
6.57.4582.88
300
Doh/λkh
6.0∞
slightly
> 6.0.615+n/2(n=1,2,3…)
7.50
Doh/λ
63
12
lim
=→
∞→
oDkh
33
22
0lim
=→
→
oDkh
( ) 2
0lim )(
sin5.7 ⎥⎦
⎤⎢⎣
⎡=
→
khkh
oDkh
[4-124]
DIRECTIVITYDIRECTIVITY(CONT)(CONT)
67
VERTICAL DIPOLEVERTICAL DIPOLE
Input Impedance of a Input Impedance of a λλ/2 dipole above a /2 dipole above a flat lossy electric conductive surfaceflat lossy electric conductive surface
Fig. 4.20
ininin XRZ +≅ [ ]Ω+≅ 5.4273 jZin
68
HORIZONTAL DIPOLEHORIZONTAL DIPOLE
Input Impedance of a Input Impedance of a λλ/2 dipole above a /2 dipole above a flat lossy electric conductive surfaceflat lossy electric conductive surface
Fig. 4.30ininin XRZ +≅ [ ]Ω+≅ 5.4273 jZin
69
GROUND EFFECTSGROUND EFFECTS
Finite conductivity σearth
(“real” earth as ground plane)
h1
h2
Direct
Reflected
σearth
Assume earth flat (ok. for Rearth >> λ)
10 → 1 [S/m]
70
GROUND EFFECTSGROUND EFFECTS(CONT)(CONT)
(real earth as ground plane)
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]
VERTICAL DIPOLEVERTICAL DIPOLE
HORIZONTAL DIPOLEHORIZONTAL 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
(real earth as ground plane)
σσ = = ∞∞
σσearthearth
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)
GROUND EFFECTS GROUND EFFECTS (CONT)(CONT)
72
Usually negligible effect for observation angle ψgreater than 3°.
EARTH CURVATUREEARTH CURVATURE
Fig. 4.34 Geometry for reflections from a spherical surface
GROUND EFFECTS GROUND EFFECTS (CONT)(CONT)
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 ψ.
reflected field from spherical surface
reflected field from flat surface___________________=
DDivergence factor
= rf
rs
EE
GROUND EFFECTS GROUND EFFECTS (CONT)(CONT)
74
l=λl=λ/2l=λ/10l=λ/50
0 (-∞ dB)1.5746 (1.972 dB)0.2181 (-6.613 dB)0.0374 (-14.27 dB)G0abs
0 (-∞ Db)
0.9642 (-0.158 dB)
0.1556 (-8.08 dB)
0.0271(-15.67 dB)
er
10.18929-0.9189-0.9863Γ
2.4026(3.807 dB)
1.6331(2.13 dB)
1.4009(1.464 dB)
1.3782(1.393 dB)
G0
2.411 (3.822 dB)
1.6409 (2.151 dB)
1.5 (1.761 dB)
1.5 (1.761 dB)
D0
0.9965 (-0.015 dB)
0.9952 (-0.021 dB)
0.9339 (-0.296 dB)
0.9188(-0.368 dB)
ecd
∞731.97390.3158Rin
199731.97390.3158Rr
0.69810.3490.13960.0279RL
1.39620.6980.27920.0279Rhf
DIPOLE SUMMARYDIPOLE SUMMARY(Resonant ⇒ XA=0; f = 100 MHz; σ = 5.7 x 107 S/m; Zc = 50; b = 3x10-4l)