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Field and Wave Electromagnetic
Chapter11
Antennas and Radiating Systems
Radio Technology Laboratory 2
Introduction (1)- Potential functions and
1-
for time harmonic cases
- The potential functions and are solutions of
non-homogeneous wave equations.
- For harmonic time dependence, the ph
A V
H A
E V j A
A V
μ
ω
= ∇×
= −∇ −
'
'
asor retarded potentials.
'4
1 2'
4
jkR
v
jkR
v
JA dv
R
V dv kR
e
e
μπ
ρ πω μεπε λ
−
−
=
= = =
∫
∫
Radio Technology Laboratory 3
Introduction (2)
0 Lorentz Gauge cf)
. 0
continuity equation
.
No need to evaluate the both integrals
VA
t
i e A j V
Jt
i e J j
με
ωμε
ρ
ωρ
∂⎛ ∇• + = →⎜ ∂⎜⎜ ∇• + =⎝
∂⎛ ∇• = − →⎜ ∂⎜⎜ ∇• = −⎝
∴
Radio Technology Laboratory 4
Introduction (3)-
1
- Three steps to determine electromagnetic fields from
a current distribution.
1. Determine from
2. Find from
3. Find from
E and H
E Hj
A J
H A
E H
ωε= ∇×
Radio Technology Laboratory 5
Radiation Fields of Elemental Dipoles* Elemental electric dipoles
Elemental oscillating electric dipole.
Short conducting wire of length terminated in two small conductive
spheres disks (capacitive loading)assume the
dl
Q
⇒current in the wire to be
uniform and to vary sinusoidally with time ( ) cos [ ]j ti t t e Ie ωω= = ℜ
Radio Technology Laboratory 6
- The current vanishes at the ends of the wire,
charge must be deposited there.
( )( ) ( ) Re[ ]
+ sign : for the charge on the upper end
j tdq ti t q t Qe
dtI j Q
Ior Q
j
ω
ω
ω
= ± =
= ±
= ± ⇒
sign : for the charge on the lower end
The pair of equal and opposite charges separated by
a short distance effectively form an electric dipole.
( )
This kind of oscillating di
p zQdl C m
−⇒
= ⋅⇒ pole is called a Hertzian dipole
The Elemental Electric Dipole (1)
Radio Technology Laboratory 7
0
0
- Electromagnetic fields of a Hertzian dipole
( )4
2 where = k =
c
cos - sin
j RIdl eA z
R
z R
βμπ
ω πβλ
θ θ θ
−
=
=
=
The Elemental Electric Dipole (2)
Radio Technology Laboratory 8
The Elemental Electric Dipole (3)
0
0
0 0
22
cos ( ) cos4
- sin ( )sin4
0
1 1[ ( ) ]
1 1 - sin [ ]
4 ( )
R
j R
R z
j R
z
R
j R
A RA A A
Idl eA A
R
Idl eA A
RA
AH A RA
R R
Idle
j R j R
θ φ
β
β
θ
φ
θ
β
θ φ
μθ θπμθ θπ
φμ μ θ
φ β θπ β β
−
−
−
= + +
⎡= =⎢
⎢⎢
= = −⎢⎢
=⎢⎢⎣
∂∂= ∇× = −
∂ ∂
= +
Radio Technology Laboratory 9
The Elemental Electric Dipole (4)
0
0
20 2 3
20 2 3
00
0
1
1 1 1 [ ( sin ) ( )]
sin
1 1 2cos [ ]
4 ( ) ( )
1 1 1 sin [ ]
4 ( ) ( )
0
where 120
j RR
j R
E Hj
R H RHj R R R
IdlE e
j R j R
IdlE e
j R j R j R
E
φ φ
β
βθ
φ
ωε
θ θωε θ θ
η β θπ β β
η β θπ β β β
μη πε
−
−
= ∇×
∂ ∂= −
∂ ∂
= − +
= − + +
=
= =
Radio Technology Laboratory 10
The Elemental Electric Dipole - Near Field (1)
2
2
- Near field
2 In the near zone, 1,
Consider only leading term.
( )sin , where 1- - 1
4 2j R
RR
Idl RH e j R
Rβ
φ
πβλ
βθ βπ
−
=
= +
Magnetic field intensity due to a current elementIdl is obtained using the Biot-Savart law in magnetostatics
'
' '0 0
2) ( ) ,
4 4 C
I Idl R dlcf dB A
R R
μ μπ π
×= = ∫
Radio Technology Laboratory 11
The Elemental Electric Dipole - Near Field (2)
30
30
30
2cos4
sin4
) , , ( 2cos sin )4
Note : the near field of an oscillating time-varying dipole
are the
R
R
E RE E
pE
R
pE
R
I pcf p zQdl Q E R
j R
θ
θ
θ
θπε
θπε
θ θ θω πε
= +
=
=
= = ± = +
quasi-static field
Electric field due to an elemental electric dipole of a moment in the e-direction
ρ
Radio Technology Laboratory 12
The Elemental Electric Dipole - Far Field (1)
0
- Far field
2 RIn the far zone, = 1,
Far zone leading terms of E and H fields are
( ) sin4
( ) sin , ( 0)4
j R
j R
R
R
Idl eH j
R
Idl eE j E
R
β
φ
β
θ
πβλ
β θπ
η β θπ
−
−
=
= =
Radio Technology Laboratory 13
The Elemental Electric Dipole - Far Field (2)
0
1. and are in space quardrature and in time phase.
2. : constant equal to the intrinsic impedance of
the medium.
3. The same propert
note E H
E
θ φ
θ
φ
η=Η
ies as those of a plane wave
(at very large distances from the dipole a spherical
wavefront closely resembles a plane wavefront)
4. The magnitude of the far-zone fields varies inversely
with the distance from the source.
5. The phase is a periodic function of R.
2 ( )
2
cR
f
π λλβ π
= =
Radio Technology Laboratory 14
The Elemental Magnetic Dipole (1)
* Elemental magnetic dipole
- Small filamentary loop of
radius b.
2
- Uniform time harmonic current
( ) cos
:
vector phasor magnetic moment
i t I t
m zI b zm
ω
π
=
= =
Radio Technology Laboratory 15
The Elemental Magnetic Dipole (2)
1
1 1
'0
1
( )
1
''0
1
' '
( )4
[1 ( )]
[(1 ) - ]4
) (- sin
j R
j R j R Rj R
j R
j R
I dlA e
R
e e e
e j R R
I dlA e j R j dl
R
cf dl x
β
β ββ
β
β
μπ
β
μ β βπ
φ
−
− − −−
−
−
=
=
− −
∴ = +
= +
∫
∫ ∫
' 'cos )y bdφ φ
0
Radio Technology Laboratory 16
The Elemental Magnetic Dipole (3)
'For every , there is another symmetrically
located differential current element on the
other side of the y-axis an equal amount
ˆof contribution to in the - .
But an equal amount of con
Idl
A x
⇒
tribution to
ˆin the opposite direction of
A
y
Radio Technology Laboratory 17
The Elemental Magnetic Dipole (4)
''0 2
12
2 2 21
'
12' 2 22
21
1' 2
sin (1 )
2
2 cos
cos sin sin
1 1 2 (1 sin sin ) (R )
1 2 (1 sin sin )
j RIbA j R e d
R
where R R b bR
R R
b bb
R R R R
b
R R
πβ
πμ φφ β φπ
ψ
ψ θ φ
θ φ
θ φ
−
−
−
−
= +
= + −
=
∴ = + −
−
∫
'1 (1 sin sin )
b
R Rθ φ+
Radio Technology Laboratory 18
The Elemental Magnetic Dipole (5)
20
2
02
202
202 3
0
202 3
0
(1 ) sin4
(1 ) sin4
1 1 sin [ ]
4 ( )
1 1 2cos [ ]
4 ( ) ( )
1 1 1 sin [ ]
4 ( ) ( )
j R
j R
j R
j RR
IbA j R e
Rm
j R eRj m
E ej R j R
j mH e
j R j R
j mH e
j R j R j R
β
β
βφ
β
θ
μφ β θ
μφ β θπωμ β θπ β βωμ β θπη β βωμ β θπη β β β
−
−
−
−
−
= +
= +
∴ = +
= − +
= − + + j Rβ
Radio Technology Laboratory 19
The Elemental Magnetic Dipole (6)
0
0
0
- Far field
( ) sin4
( ) sin4
j R
j R
m eE
R
m eH
R
β
φ
β
θ
ωμ β θπωμ β θπη
−
−
=
= −
Radio Technology Laboratory 20
The Elemental Magnetic Dipole (7)
0
H.W 11-1, 11-2, 11-3, 11-4, 11-5
) let ( , ) : Electric and magnetic field of the electric dipole.
( , ) : Electric and magnetic field of the magnetic dipole.
e e
m m
e
cf E H
E H
E Hη= 00
00 0
0
, ( )
i.e Hertzian electric dipole and elemental
magnetic dipole are dual devices.
mm e
EEH
H
if Idl j m where
φ
θ
ηη
ωμβ β ω μ εη
= − =
= = =
Radio Technology Laboratory 21
Antenna Patterns and Antenna Parameters (1)
* Antenna patterns and antenna parameters
- Antenna pattern (radiation pattern) : the relative far-zone field
strength versus direction at a fixed distance from an antenna.
- E-plane pattern = the magnitude of the normalized field strength
(with respect to the peak value) versus for a constant
- H-plane pattern = the magnitude of the normalized field strength
versus
θ φ
for πφ θ =2
Radio Technology Laboratory 22
Antenna Patterns and Antenna Parameters (2)
e.g. Hertzian dipole
θ φ
φ θ π= /2
Radio Technology Laboratory 23
Antenna Patterns and Antenna Parameters (3)
- Antenna parameters
(1) Width of main beam = beamwidth (3dB beamwidth)
(2) Sidelobe levels unwanted radiation (first sidelobe : -40dB)
(3) Directivity
- Directive gain in terms of radiation
⇒
2
intensity.
- Radiation intensity = the time-average power per unit solid angle
: (W/sr) cf) sr : steradian (solid angle)
Radiation intensity (W/sr)
- Total time-average pavU R∴ = P
ower radiated
(W)
differential solid angle sin
r avP ds Ud
d d dθ θ φ
= = Ω
Ω = =∫ ∫iP
Radio Technology Laboratory 24
Antenna Patterns and Antenna Parameters (4)
- Directive gain
( , ) = the ratio of the radiation intensity
in the direction ( , ) to the
average radiation intensity
( , ) 4 (
/ 4
D
r
G
U U
P
θ φθ φ
θ φ π θπ
= =
max max
2
max2 2
0 0
, )
- Directivity = the maximum directive gain of an antenna
4 (Dimensionless)
4
( , ) sin
av r
Ud
U UD
U P
E
E d dπ π
φ
π
π
θ φ θ θ φ
Ω
= =
=
∫
∫ ∫
Radio Technology Laboratory 25
Antenna Patterns and Antenna Parameters (5)
22 2
02 2
22 2 2
02
22
2 2
0 0
. : Hertzian dipole
1 1 ( ) sin
2 2 32
( ) U sin
32
4 ( , ) 4 sin 3 ( , ) sin
2(sin )sin
av
av
D
Ex
Idle E H E H
R
IdlR
UG
Ud d d
D
θ φ
π π
η β θπ
η β θπ
π θ φ π θθ φ θθ θ θ φ
∗
−
= ℜ × = =
= =
∴ = = =Ω
=
∫ ∫ ∫
P
P
( , ) 1.5 1.762DG dBπ φ = →
Radio Technology Laboratory 26
Antenna Patterns and Antenna Parameters (6)
p- Power gain G = the ratio of its maximum radiation intensity
to the radiation intensity of a
lossless isotropic source with
i
max
the same input power P
total input power ( : ohmic loss power)
4
- Radiation efficiency
- Radiation resistan
i r l l
pi
p rr
i
P P P P
UG
P
G P
D P
π
η
= +
=
= =
22
ce = the value of a hypothetical resistance
21
2
rr m r r
m
PP I R R
I= ∴ =
Radio Technology Laboratory 27
Thin Linear Antennas (1)* Thin linear antennas
( ) sin ( ),
sin ( ), 0
sin ( ), 0.
m
m
m
I z I h z
I h z z
I h z z
β
ββ
= −
− >⎧= ⎨ + <⎩
Radio Technology Laboratory 28
Thin Linear Antennas (2)
'
0 0'
1' 2 2 2
'
-00
- The far-field contribution from the differential current element is
( ) sin .4
( 2 cos ) cos
1 1
sin sin ( )
4
j R
j R j zm
Idz
Idz edE dH j
R
R R z Rz R z
R RI
E H j e h z eR
β
θ φ
β βθ φ
η η β θπ
θ θ
η β θη βπ
−
= =
= + − ≅ −
≅
= = − cos
cos
sin ( ) : even function
cos( cos ) sin( cos )
h
h
j z
dz
h z
e z j z
θ
β θ
β
β θ β θ
−
−
= +
∫
Odd functioneven function
Radio Technology Laboratory 29
Thin Linear Antennas (3)-0
0 0
-
' '
sin sin ( )cos( cos )
260
( ),
cos( cos ) cos where ( )
sin
) sin( ) ?
1, , sin( ),
hj Rm
j Rm
Ax
Ax Ax
IE H j e h z z dz
Rj I
e FR
h hF
cf e Bx C dx
u e u e v Bx C vA
βθ φ
β
η β θη β β θπ
θ
β θ βθθ
= = −
=
−=
+ =
= = = +
∫
∫cos( )
sin( ) Ax
B Bx C
I e Bx C dx
= +
= +∫
H.W
Radio Technology Laboratory 30
Thin Linear Antennas (4)
2
2
2 2
2 2
1sin( ) cos( )
1sin( ) [ cos( ) sin( )
1sin( ) cos( )
[ sin( ) cos( )]
cos , , -
Ax Ax
Ax Ax Ax
Ax Ax
Ax
Be Bx C e Bx C dx
A AB
e Bx C e Bx C B e Bx C dxA A
B Be Bx C e Bx C I
A A A
eI A Bx C B Bx C
A Blet A jk B k C khθ
= + − +
= + − + + +
= + − + −
∴ = + − ++= = =
∫
∫
Radio Technology Laboratory 31
Thin Linear Antennas (5)
- Half wave dipole
2cos( cos ) cos( ) cos( cos )
4 4 2 ( ) sin sin
cos( cos )60 2 sin
cos( cos )2
2 sin
j Rm
j Rm
F
j IE e
R
jIH e
R
βθ
βφ
λ π λ πβ θ θλθ
θ θπ θ
θ
π θ
π θ
−
−
−= =
⎧ ⎫⎪ ⎪
∴ = ⎨ ⎬⎪ ⎪⎩ ⎭⎧ ⎫⎪ ⎪
= ⎨ ⎬⎪ ⎪⎩ ⎭
Radio Technology Laboratory 32
Thin Linear Antennas (6)2
2
2
2
2 2 2
0 0 0
2
0
cos( cos )151 22 sin
cos ( cos )2sin 30
sin
cos ( cos )2 1.218 obtained by numerical solution
sin
mav
r av m
IE H
R
P R d d I d
d
θ φ
π π π
π
π θ
π θ
π θθ θ φ θ
θπ θ
θθ
∗
⎧ ⎫⎪ ⎪
= = ⎨ ⎬⎪ ⎪⎩ ⎭
= =
= ⇒
∫ ∫ ∫
∫
P
P
using Simpson's law
Radio Technology Laboratory 33
Thin Linear Antennas (7)
2
2 0 2max
max
2 73.1
15 (90 )
4 60 1.64
36.54
rr
m
av m
r
PR
I
U R I
UD
P
ππ
∴ = = Ω
= =
= = =
P
Radio Technology Laboratory 34
Thin Linear Antennas (8)
cos0
- Effective antenna length
Assume a center-feed linear dipole
and a general phasor current distribution ( )
30 {sin ( ) }
(0) : the input curr
hj R j z
h
I z
jE H e I z e dz
RI
β β θθ φη β θ−
−= = ∫
0
cos
ent at the feed point of the antenna
30 (0) ( )
sin where ( ) ( )
(0)
the effective length of the transmitting antenna
j Re
h j ze h
j IE H e l
R
l I z e dz
βθ φ
β θ
η β θ
θθ
−
−
⇒ = =
=Ι
⇒
∫
Radio Technology Laboratory 35
Thin Linear Antennas (9)
1 ( ) ( )
2 (0)
The length of an equivalent linear antenna with
a uniform current (0) such that it radiates the
same
h
e hl I z dz
I
I
πθ−
= =
⇒
∫
far-zone field in the plane2
πθ =
Radio Technology Laboratory 36
Thin Linear Antennas (10)
Ex 11-6) Assume a sinusoidal current distribution on a center-fed, thin
straight half-wave dipole. Find its effective lenth.
What is its maximum value?
cos4
4
( ) sin sin ( )4
cos( cos )2 2 ( )sin
j ze
e
l z e dz
l
λβ θ
λλθ θ β
π θθ
β θ
−= −
⎡ ⎤⎢ ⎥
= ⎢ ⎥⎢ ⎥⎣ ⎦
∫
Radio Technology Laboratory 37
Thin Linear Antennas (11)
2 maximum value of ( ) ( )
2 2
note
1 ( )
(0)
Valid only for relatively short antennas having
a current maximum at the fee
e e
h
e h
l l
l I z dzI
π λ λθβ π
−
= = = <
=
⇒
∫
d point
Radio Technology Laboratory 38
Thin Linear Antennas (12)
( )
Negative sign is to conform with the convention that the
electric potential increases in a direction opposite to
that of the electric field
oce
i
Vl
Eθ = −
⇒
The effective length of an antenna
for receiving is the same as that
for transmitting.
⇒
Receiving Antennas and Reciprocity
39
Circuit relation for Radiation into Free Space
40
Effective Area of Receiving Antenna
41
Effective Area for Hertzian Dipole
42
z
Δ z
-h
h
0
r'
For matched termination:
⎟⎟⎠
⎞⎜⎜⎝
⎛=⋅= 2
2
1EAAaPP eerR η
)(θθ fr
eZIaE
jkr−
=
θθ sin23)( =f
322
ηλl
jZ =
θθ 2sin)23()( =g
2
6
4⎟⎠⎞
⎜⎝⎛=λ
πη lRr
)6/4()/(8
)sin(
8 2
22
πληθ
l
El
R
VP
r
OCR ==
πλθθλ
π 4)()sin(
8
3 22 gAe ==
cf) Normalization g( , )d =4θ φ πΩ∫