Antennas
Antennas
Transmitter
Transmitting
Antenna
I
Electromagnetic
Wave
Transmission Line
I
Receiver
I
Receiving
Antenna
Electromagnetic
Wave
Transmission Line
I
Antennas are transducers that transfer electromagnetic energy
between a transmission line and free space.
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1
Antennas
Here are a few examples of common antennas:
Linear monopole
fed by a single wire
over a ground plane
Linear dipole fed
by a two-wire line
Ground plane
Coaxial ground
plane antenna
Linear elements connected
to outer conductor of the
coaxial cable simulate the
ground plane
Multiple loop antenna wound
around a ferrite core
Loop antenna
Uda-Yagi dipole array
Loop dipole
Parabolic (dish) antenna
Log−periodic array
Passive elements
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2
Antennas
From a circuit point of view, a transmitting antenna behaves like an
equivalent impedance that dissipates the power transmitted
Transmitter
Transmitting
Antenna
I
Electromagnetic
Wave
Transmission Line
I
1
P( t ) = Req I 2
2
Transmitter
Zg
Vg
I
Zeq
Transmission Line
= Req + jXeq
I
The transmitter is equivalent to a generator.
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3
Antennas
A receiving antenna behaves like a generator with an internal
impedance corresponding to the antenna equivalent impedance.
Receiver
I
Receiving
Antenna
Electromagnetic
Wave
Transmission Line
I
I
ZR
Zeq
Zin
Transmission Line
1
P( t ) = Rin I 2
2
Veq
I
The receiver represents the load impedance that dissipates the time
average power generated by the receiving antenna.
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4
Antennas
Antennas are in general reciprocal devices, which can be used both
as transmitting and as receiving elements. This is how the
antennas on cellular phones and walkie−talkies operate.
The basic principle of operation of an antenna is easily understood
starting from a two−wire transmission line, terminated by an open
circuit.
|I|
Zg
ZR → ∞
Vg
Open circuit
|I|
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Note: This is the return
current on the second wire,
not the reflected current
already included in the
standing wave pattern.
5
Antennas
Imagine to bend the end of the transmission line, forming a dipole
antenna. Because of the change in geometry, there is now an
abrupt change in the characteristic impedance at the transition
point, where the current is still continuous. The dipole leaks
electromagnetic energy into the surrounding space, therefore it
reflects less power than the original open circuit ⇒ the standing
wave pattern on the transmission line is modified
| I0 |
Zg
|I|
Vg
Z0
|I|
| I0 |
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6
Antennas
In the space surrounding the dipole we have an electric field. At
zero frequency (d.c. bias), fixed electrostatic field lines connect the
metal elements of the antenna, with circular symmetry.
E
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E
7
Antennas
At higher frequency, the current oscillates in the wires and the field
emanating from the dipole changes periodically. The field lines
propagate away from the dipole and form closed loops.
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8
Antennas
The electromagnetic field emitted by an antenna obeys Maxwell’s
equations
∇ × E = − jω µ H
∇ × H = J + jω ε E
Under the assumption of uniform isotropic medium we have the
wave equation:
∇ × ∇ × E = − j ω µ ∇ × H = − j ω µ J + ω 2µ ε E
∇ × ∇ × H = ∇ × J + jω ε ∇ × E
= ∇ × J + ω 2µ ε H
Note that in the regions with electrical charges ρ
∇ × ∇ × E = ∇∇ ⋅ E − ∇ 2 E = ∇ ( ρ ε ) − ∇ 2 E
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9
Antennas
In general, these wave equations are difficult to solve, because of
the presence of the terms with current and charge. It is easier to
use the magnetic vector potential and the electric scalar potential.
The definition of the magnetic vector potential is
B=∇×A
Note that since the divergence of the curl of a vector is equal to
zero we always satisfy the zero divergence condition
∇ ⋅ B = ∇ ⋅ ( ∇ × A) = 0
We have also
∇ × E = − jω µ H = − jω ∇ × A ⇒ ∇ × ( E + jω A) = 0
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10
Antennas
We define the scalar potential φ first noticing that
∇ × ( ±∇φ ) = 0
and then choosing (with sign convention as in electrostatics)
∇ × ( E + jω A) = ∇ × ( −∇φ ) ⇒ E = − jω A − ∇φ
Note that the magnetic vector potential is not uniquely defined,
since for any arbitrary scalar field ψ
B = ∇ × A = ∇ × ( A + ∇ψ )
In order to uniquely define the magnetic vector potential, the
standard approach is to use the Lorenz gauge
∇ ⋅ A + jω µ ε φ = 0
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11
Antennas
From Maxwell’s equations
∇×H=
1
µ
∇ × B = J + jω ε E
∇ × B = µ J + jω µ ε E
⇒ ∇ × ( ∇ × A) = µ J + jω µ ε ( − j ω A − ∇φ )
From vector calculus
∇ × ( ∇ × …) = ∇ ( ∇ ⋅ …) − ∇ 2 …
∇ × ( ∇ × A) = ∇ ( ∇ ⋅ A) − ∇ 2 A = µ J + ω 2 µ ε A − jω µ ε ∇φ
Lorenz Gauge
∇ ⋅ A = − jω µ ε φ
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⇒ ∇ ( ∇ ⋅ A) = − jω µ ε ∇φ
12
Antennas
Finally, the wave equation for the magnetic vector potential is
∇2A + ω 2 µ ε A = ∇2A + β 2 A = − µ J
For the electric field we have
ρ
∇ ⋅ D = ρ ⇒ ∇ ⋅ E = ∇ ⋅ ( − j ω A − ∇φ ) =
ε
ρ
2
2
∇ φ + jω ∇ ⋅ A = ∇ φ + jω ( − jω µ ε φ ) = −
ε
The wave equation for the electric scalar potential is
ρ
∇ φ +ω µε φ = ∇ φ + β φ = −
ε
2
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2
2
2
13
Antennas
The wave equations are inhomogenoeous Helmholtz equations,
which apply to regions where currents and charges are not zero.
We use the following system of coordinates for an antenna body
J( r ')
ρ ( r ')
z
Observation point
r − r'
dV '
r'
r
y
x
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Radiating antenna body
14
Antennas
The generals solutions for the wave equations are
− jβ r − r '
J ( r ') e
µ
A( r ) =
4π ∫∫∫V
r − r'
φ ( r) =
1
4πε ∫∫∫V
ρ ( r ') e− jβ r − r '
r − r'
dV '
dV '
The integrals are extended to all points over the antenna body
where the sources (current density, charge) are not zero. The effect
of each volume element of the antenna is to radiate a radial wave
e
− jβ r − r '
r − r'
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15
Antennas
Infinitesimal Antenna
z
Observation point
J(0)
ρ (0)
r = r − r'
∆S
r' = 0
∆z
y
dV '
x
Infinitesimal antenna body
I = constant
phasor
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∆ z << λ
Dielectric medium (ε , µ)
16
Antennas
The current flowing in the infinitesimal antenna is assumed to be
constant and oriented along the z−axis
I = ∆ S ⋅ J ( r ') = ∆ S ⋅ J ( 0 )
∆V ' = ∆ S ⋅ ∆ z
∆V 'J ( r ') = I ∆ z iz
The solution of the wave equation for the magnetic vector potential
simply becomes the evaluation of the integrand at the origin
µ I ∆ z e− j β r
A=
iz
4π r
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1

 H= µ∇×A

⇒ 
 E= 1 ∇×H

jω ε
17
Antennas
There is still a major mathematical step left. The curl operations
must be expressed in terms of spherical coordinates
z
iϕ
r
ir
iθ
Polar angle
θ
x
ϕ
Azimuthal angle
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y
iϕ
18
Antennas
In spherical coordinates
ir
r iθ
r sin θ i ϕ
∂
∇×A= 2
r sin θ ∂ r
∂
∂θ
∂
∂ϕ
Ar
rAθ
r sin θ Aϕ
1
1  ∂
∂

=
sin θ Aϕ −
( Aθ )  i r

∂ϕ
r sin θ  ∂ θ

(
)
∂
1 1 ∂

+ 
( Ar ) − r Aϕ  iθ
r  sin θ ∂ ϕ
∂r

1∂
∂

+  ( r Aθ ) −
( Ar )  i ϕ
r ∂ r
∂θ

(
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)
19
Antennas
We had
µ I ∆ z e− j β r
A=
iz
4π r
⇒
with
i z = i r cos θ − i θ sin θ
j µ β I ∆ z e− j β r 
1 
1+
sin θ
∇ × A = iϕ



4π r
jβ r 
For the fields we have
− jβ r
I
j
β
∆
z
e

1
1 
H = ∇ × A = iϕ
1+
sin θ



4π r
jβ r 
µ
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20
Antennas
− jβ r
µ jβ I ∆ z e
E=
∇×H=
jω ε
ε
4π r
1

 1
1 
×  2 cos θ 
+
 ir
2

 jβ r ( jβ r ) 

1
1  
+ sin θ  1 +
+
 iθ 
2
jβ r ( jβ r )  


The general field expressions can be simplified for observation
point at large distance from the infinitesimal antenna
1 >>
1
jβ r
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>>
1
( jβ r )
2
⇒ βr=
2π
λ
r >> 1
21
Antennas
At large distance we have the expressions for the Far Field
jβ I ∆ z e− j β r
H ≈ iϕ
sin θ
4π r
E ≈ iθ
− jβ r
µ jβ I ∆ z e
sin θ
4π r
ε
2π r >> λ
• At sufficient distance from the antenna, the radiated fields are
perpendicular to each other and to the direction of propagation.
• The magnetic field and electric field are in phase and
µ
E =
H =η H
ε
These are also properties of uniform plane waves.
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22
Antennas
However, there are significant differences with respect to a uniform
plane wave:
• The surfaces of constant phase are spherical instead of planar,
and the wave travels in the radial direction.
• The intensities of the fields are inversely proportional to the
distance, therefore the field intensities decay while they are
constant for a uniform plane wave.
• The field intensities are not constant on a given surface of
constant phase. The intensity depends on the sine of the polar
angle θ.
The radiated power density is
{
}
2
1
1 µ
*
P( t ) = Re E × H = i r
Hϕ
2
2 ε
2
η  β I ∆ z
2
= ir 
sin
θ

2  4π r 
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23
Antennas
z
P( t )
Hϕ
J
θ
r
Eθ
ϕ
The spherical wave resembles a plane wave locally in a small
neighborhood of the point ( r, θ, ϕ ).
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24
Antennas
Radiation Patterns
Electric Field and Magnetic Field
Eθ or Hϕ
z
y
θ
x
x
Fixed r
Plane containing the antenna
Plane perpendicular to the antenna
proportional to sinθ
omnidirectional or isotropic
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25
Antennas
Time−average Power Flow (Poynting Vector)
y
P( t )
z
θ
x
x
Fixed r
Plane containing the antenna
Plane perpendicular to the antenna
proportional to sin2θ
omnidirectional or isotropic
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26
Antennas
Total Radiated Power
The time−average power flow is not uniform on the spherical wave
front. In order to obtain the total power radiated by the infinitesimal
antenna, it is necessary to integrate over the sphere
2π
Ptot =
dϕ
0
= 2π
∫
π
∫0
dθ r 2 sin θ P (t )
2
ηβ I ∆z
2 π
3
= 
2
π
θ
sin
θ
r
d

∫
0
2  4π r 
=4 3
4π η  β I ∆ z 
=


3  4π 
2
Note: the total radiated power is independent of distance. Although
the power density decreases with distance, the integral of the
power over concentric spherical wave fronts remains constant.
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27
Antennas
Ptot1 = Ptot 2
Ptot 2
Ptot1
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28
Antennas
The total radiated power is also the power delivered by the
transmission line to the real part of the equivalent impedance seen
at the input of the antenna
2
1 2
4πη  2π I ∆ z  1 2  2πη  ∆ z  2 
Ptot = I Req =
= I 





2
3  λ 4π  2  3
λ 
Req
The equivalent resistance of the antenna is usually called radiation
resistance. In free space
2
∆
z
µo
 
η = ηo =
= 120π [ Ω ] ⇒ Req = 80π 2
Ω]
[
 λ 
εo
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29
Antennas
The total radiated power is also used to define the average power
density emitted by the antenna. The average power density
corresponds to the radiation of a hypothetical omnidirectional
(isotropic) antenna, which is used as a reference to understand the
directive properties of any antenna.
Power radiation pattern of an
omnidirectional average antenna
Power radiation pattern
of the actual antenna
z
θ
x
Pave
P( t, θ )
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30
Antennas
The time−average power density is given by
Pave =
Total Radiated Power
Surface of wave front
=
η
η  β I ∆ z
2 1
β I ∆ z)
=
=
= 
2 12π (
2
3  4π r 
4π r
4π r
Ptot
2
The directive gain of the infinitesimal antenna is defined as
D(θ , ϕ ) =
P( t , r, θ )
Pave
2  −1
 η  β I ∆ z
η  β I ∆ z
2
= 
si
n
θ 



2  4π r 
 3  4π r  
2
3 2
= sin θ
2
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31
Antennas
The maximum value of the directive gain is called directivity of the
antenna. For the infinitesimal antenna, the maximum of the
directive gain occurs when the polar angle θ is 90°
3 2π
= 1.5
Directivity = max { D(θ , ϕ )} = sin
 2
2
The directivity gives a measure of how the actual antenna performs
in the direction of maximum radiation, with respect to the ideal
isotropic antenna which emits the average power in all directions.
z
Pave
P
max
90°
x
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32
Antennas
The infinitesimal antenna is a suitable model to study the behavior
of the elementary radiating element called Hertzian dipole.
Consider two small charge reservoirs, separated by a distance ∆z,
which exchange mobile charge in the form of an oscillatory curent
I( t)
t
−
+
−
+
−
+
+
−
+
−
+
−
∆z
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33
Antennas
The Hertzian dipole can be used as an elementary model for many
natural charge oscillation phenomena. The radiated fields can be
described by using the results of the infinitesimal antenna.
Assuming a sinusoidally varying charge flow between the
reservoirs, the oscillating current is
I( t)
current flowing
out of reservoir
d
d
= q( t) = qo cos(ω t)
dt
dt
charge on
reference reservoir
⇒ I o = jω qo
phasor
Radiation
pattern
Io
qo
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34
Antennas
A short wire antenna has a triangular current distribution, since the
current itself has to reach a null at the end the wires. The current
can be made approximately uniform by adding capacitor plates.
Imax
∆z
Io
Io
The small capacitor plate antenna is equivalent to a Hertzian dipole
and the radiated fields can also be described by using the results of
the infinitesimal antenna. The short wire antenna can be described
by the same results, if one uses an average current value giving the
same integral of the current
Io = Imax 2
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35
Antennas
Example − A Hertzian dipole is 1.0 mm long and it operates at the
frequency of 1.0 GHz, with feeding current Io = 1.0 Ampéres. Find
the total radiated power.
λ = c f ≈ 3 × 10 8 109 = 0.3 m = 300 mm
∆ z = 1 mm λ ⇒ Hertzian dipole
4π η  2π Io ∆ z  2
1
2π 2
=
120π (
) ( 1 ⋅ 10 −3 )2
Ptot =
3  λ 4π 
12π η
0.3
o
λ
Io
∆z
= 4.39 mW
For a short dipole with triangular current distribution and maximum
current Imax = 1.0 Ampére
Io = Imax 2 ⇒
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Ptot = 4.39 / 4 ≈ 1.09 mW
36
Antennas
Time−dependent fields - Consider the far−field approximation
{
}
 jβ I ∆ z sin θ j (ω t − β r ) 
H ( t ) = Re H e
e
≈ i ϕ Re 

4π r


 β I ∆ z sin θ

2
≈ i ϕ Re 
jcos(ω t − β r )+ j sin(ω t − β r ) 
4π r


β I ∆ z sin θ
≈ −iϕ
sin(ω t − β r )
4π r
jω t
(
{
E ( t ) = Re E e jω t
)
}
β I ∆ z sin θ
≈ − iθ η
sin(ω t − β r )
4π r
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37
Antennas
Linear Antennas
Consider a dipole with wires of length comparable to the
wavelength.
z
L2
θ'
∆z
z'
r'
iθ '
iθ
r
θ
− L1
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38
Antennas
Because of its length, the current flowing in the antenna wire is a
function of the coordinate z. To evaluate the far−field at an
observation point, we divide the antenna into segments which can
be considered as elementary infinitesimal antennas.
The electric field radiated by each element , in the far−field
approximation, is
− jβ r '
µ jβ I ∆ z e
sin θ '
4π r '
ε
∆ E' = iθ
In far−field conditions we can use these additional approximations
θ ≈θ'
r ' ≈ r − z 'cos θ
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39
Antennas
The lines r and r’ are nearly parallel under these assumptions.
z
L2
θ'≈θ
r'
r
∆z
z'
θ
z 'cos θ
© Amanogawa, 2006 – Digital Maestro Series
This length is neglected if
r ' ≈ r − z 'cos θ
40
Antennas
The electric field contributions due to each infinitesimal segment
becomes
you cannot
neglect here
∆ E' = iθ
− jβ r jβ z 'cosθ
j
β
∆
z
e
e
I
µ
sin θ
ε
4π r − 4π z 'cos θ
you can
neglect here
The total fields are obtained by integration of all the contributions
E = iθ
L2
µ jβ e− j β r
sin θ ⋅ ∫
I( z) e jβ z cosθ dz
− L1
ε 4π r
L2
jβ e− j β r
H = iϕ
sin θ ⋅ ∫
I( z) e jβ z cosθ dz
− L1
4π r
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41
Antennas
Short Dipole
Consider a short symmetric dipole comprising two wires, each of
length L << λ . Assume a triangular distribution of the phasor
current on the wires
 Imax ( 1 − z L )
I( z) = 
 Imax ( 1 + z L )
z≥0
z<0
The integral in the field expressions becomes
∫
L
L
2L
jβ z cosθ
I( z) e
I( z) dz =
dz ≈
Imax
−L
−L
2
≈1
since max
∫
βz = β ⋅ L =
2π
λ
L
1 for a short dipole
⇒ e jβ z cosθ ≈ 1
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42
Antennas
The final expression for far−fields of the short dipole are similar to
the expressions for the Hertzian dipole where the average of the
triangular current distribution is used
E = iθ
− jβ r
µ jβ e
ε 4π r
∆z
Imax
sin θ ⋅ 2 L ⋅
2
average
current
= iθ
µ jβ Imax L e− jβ r
sin θ
ε
4π r
jβ Imax L e− jβ r
H = iϕ
sin θ
4π r
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43
Antennas
Half−wavelength dipole
Consider a symmetric linear antenna with total length λ/2 and
assume a current phasor distribution on the wires which is
approximately sinusoidal
I( z) = Imax cos(β z)
The integral in the field expressions is
λ 4
∫
−λ 4
Imax cos ( β z) e
© Amanogawa, 2006 – Digital Maestro Series
jβ z cos θ
 π cos θ 
dz =
cos
2
 2 
β sin θ
2 Imax
44
Antennas
We obtain the far−field expressions
E = iθ
µ j e− jβ r Imax
 π cos θ 
cos
 2 
ε 2π r sin θ
j e− jβ r Imax
 π cos θ 
H = iϕ
cos
 2 
2π r sin θ
and the time−average Poynting vector
P( t ) = i r
2
µ
Imax
2  π cos θ 
cos
2
2
2
 2 
ε 8π r sin θ
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45
Antennas
The total radiated power is obtained after integration of the
time−average Poynting vector
1 2
Ptot = Imax
2
µ 1 2π  1 − cos ( u) 
du


∫
0


u
ε 4π
≈ 2.4376
µ
1 2
= Imax
⋅ 0.193978
2
ε
Req
The integral above cannot be solved analytically, but the value is
found numerically or from published tables.
The equivalent
resistance of the half−wave dipole antenna in air is then
µ
Req ( λ 2) =
⋅ 0.193978 ≈ 73.07 Ω
ε
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46
Antennas
The direction of maximum radiation strength is obtained again for
polar angle θ =90° ande we obtain the directivity
D=
P( t, r, 90°)
Ptot 4π r
2
=
2
µ Imax
ε 8π 2 r 2
1
2
I
2 2 max
8π r
µ
⋅ 2.4376
ε
≈ 1.641
The directivity of the half−wavelength dipole is marginally better
than the directivity for a Hertzian dipole (D = 1.5).
The real improvement is in the much larger radiation resistance,
which is now comparable to the characteristic impedance of typical
transmission line.
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47
Antennas
From the linear antenna applet
Radiation Pattern for E and H
© Amanogawa, 2006 – Digital Maestro Series
Power Radiation Pattern
48
Antennas
For short dipoles of length 0.0005 λ to 0.05 λ
Radiation Pattern for E and H
© Amanogawa, 2006 – Digital Maestro Series
Power Radiation Pattern
49
Antennas
Radiation Pattern for E and H
© Amanogawa, 2006 – Digital Maestro Series
Power Radiation Pattern
50
Antennas
Radiation Pattern for E and H
© Amanogawa, 2006 – Digital Maestro Series
Power Radiation Pattern
51
Antennas
For general symmetric linear antennas with two wires of length L, it
is convenient to express the current distribution on the wires as
I ( z) = Imax sin { β ( L − z ) }
The integral in the field expressions is now
L
∫
−L
=
Imax sin  β ( L − z )  e jβ z cosθ dz =
2 Imax
2
β sin θ
{ cos ( β L cosθ ) − cos ( β L) }
© Amanogawa, 2006 – Digital Maestro Series
52
Antennas
The field expressions become
E = iθ
= iθ
L2
µ j β e− jβ r
sin θ ⋅ ∫
I( z) e jβ z cosθ dz
− L1
ε 4π r
µ j Imax e− jβ r
cos ( β L cos θ ) − cos ( β L ) }
{
ε 2π r sin θ
L2
j β e− jβ r
H = iϕ
sin θ ⋅ ∫
I( z) e jβ z cosθ dz
− L1
4π r
j Imax e− jβ r
= iϕ
cos ( β L cos θ ) − cos ( β L ) }
{
2π r sin θ
© Amanogawa, 2006 – Digital Maestro Series
53
Antennas
Examples of long wire antennas
Radiation Pattern for E and H
© Amanogawa, 2006 – Digital Maestro Series
Power Radiation Pattern
54
Antennas
Radiation Pattern for E and H
© Amanogawa, 2006 – Digital Maestro Series
Power Radiation Pattern
55
Antennas
Radiation Pattern for E and H
© Amanogawa, 2006 – Digital Maestro Series
Power Radiation Pattern
56
Antennas
Radiation Pattern for E and H
© Amanogawa, 2006 – Digital Maestro Series
Power Radiation Pattern
57
Antennas
Radiation Pattern for E and H
© Amanogawa, 2006 – Digital Maestro Series
Power Radiation Pattern
58
Antennas
Radiation Pattern for E and H
© Amanogawa, 2006 – Digital Maestro Series
Power Radiation Pattern
59
Antennas
Radiation Pattern for E and H
© Amanogawa, 2006 – Digital Maestro Series
Power Radiation Pattern
60
Antennas
Radiation Pattern for E and H
© Amanogawa, 2006 – Digital Maestro Series
Power Radiation Pattern
61
Antennas
Radiation Pattern for E and H
© Amanogawa, 2006 – Digital Maestro Series
Power Radiation Pattern
62
Antennas
Radiation Pattern for E and H
© Amanogawa, 2006 – Digital Maestro Series
Power Radiation Pattern
63