2. Radian and Steradian
• Radian
– A measure of a plane angle is a radian.
– One radian is defined as” the plane angle with its
vertex at the centre of a circle of radius r that is
subtended by an are whose length is r.
– Since the circumference of a circle of radius r is C = 2πr
there are 2πr rad ( 2πr r ) in a full circle.
3.
4. Radian and Steradian
• Steradian
– The measure of a solid angle is a steradian.
– One steradian is defined as “ the solid angle with its vertex
at the centre of a sphere of radius r that is subtended by a
spherical surface area equal to that of a square with each
side of length r.
– Since the area of a sphere of radius r is A = 4πr there are
2
( 4πr 2 2 ) in a closed sphere.
r
– The infinitesimal area dA on the surface of a sphere of
radius r is dA = r 2 sin θdθ m 2
– Therefore the element of solid angle dΩ of a sphere can
be written as
dΩ = sin θdθdφ sr
7. Radiation Power Density
The quantity used to describe power associated with
an electromagnetic wave is the instantaneous
Poynting vector defined as:-
W = E × H (W/m2)
W = instantaneous Poynting vector W/m2
E = instantaneous electric field intensity V/m
H= instantaneous magnetic field intensity A/m
2
8. Radiation Intensity
Power radiated by an antenna per unit solid angle
Far field parameter
U = r2 Wrad
where
U = radiation intensity (W/unit solid angle)
Wrad = radiation density (W/m2)
or U = r2 Prad/A= Prad/A/ r2 = Prad/ Ω
The total power is obtained by integrating the
radiation intensity over the entire solid angle of 4π
Prad = ∫∫ U dΩ = ∫∫ U Sin(θ) dθdφ
9. Directivity
Ratio of radiation intensity in a given direction to the radiation
Intensity averaged over all directions.
D = U/Uo = U / Prad / 4π
=4πU / Prad
If direction not specified – Direction of max radiation intensity Do
Dmax = Do = Umax / Uo =4π Umax / Prad
D = directivity (dimensionless quantity)
Do = maximum directivity
U = radiation intensity (W/unit solid angle)
Umax=maximum radiation intensity(W/unit solid angle)
Uo=radiation intensity of isotope (W/unit solid angle)
10. Partial Directivities: For orthogonal polarization
components
“ That part of radiation intensity corresponding
to a given polarization divided by total radiation
intensity “
Do = Dθ + Dφ
Do = 4π Uθ /Prad + 4π Uφ /Prad
Implies how well a radiator directs em energy in
a certain direction
11. Antenna Gain
Another useful measure describing the performance of an
antenna is the gain. Although the gain of the antenna is closely
related to the directivity.
It is a measures that takes into account the efficiency of the
antenna as well as its directional capabilities.
Absolute gain of an antenna (in a given direction) is defined as “
the ratio of the intensity in a given direction to the radiation
intensity that would be obtained if the power accepted by the
antenna were radiated isotropically.
Mathematically represented as:-
Gain = 4π radiation intensity = 4π U (θ,φ)
total input (accepted) power Pin
12. Antenna Gain
An alternate way to define antenna gain is :-
G = Power radiated by an ant
Power radiated by ref ant
The i/p power to both the antenna is the same and the reference
ant generally chosen is an isotrope.
13. Antenna Efficiency (eo)
eo is to take into account losses in antenna
– Reflection and mismatch losses
– Conduction losses (I2R)
eo = er ec ed (overall efficiency)
eo = total efficiency
er = reflection (mismatch) efficiency = (1-|Γ|2)
ed = dielectric efficiency
Γ= voltage reflection coefficient at the input
terminals of antenna
14. Beam Efficiency
To judge the quality of transmission/reception
BE = Power transmitted (received) within cone angle θ1
power transmitted (received) by the antenna
15. Bandwidth
“Range of frequencies within which performance
of an antenna with respect to some characteristic
conforms to a specified standard”
Characteristics within acceptable values of centre
frequency (Gain, beam direction, side lobe level,
Polarization).
Broadband antenna bandwidth described in ratio
of upper to lower freq. (e.g. 10:1)
Narrow band antenna described in %age of B.W.
Antenna chars. don’t vary in the same manner
Pattern Bandwidth, Impedance Bandwidth
16. Polarization
Polarization is defined as “that property of the
electromagnetic wave describing the time varying
direction and relative magnitude of the electric field
vector; specially the figure traced out as a function of
time by the extremity of the vector at a fixed location in
space and the sense in which it is traced as observed
along the direction of propagation.
Polarization is the curve traced out by the end point of
the arrow representing the instantaneous electric field.
The field must be observed along the direction of
propagation.
Polarization can be classified as linear, circular or
elliptical. If the vector that describes the electric field at a
point in space as a function of time is always directed
along a line, the field is said to be linearly polarized.
17. Polarization (contd)
In general however, the figure that the electric field
traces is an ellipse and the field is said to be elliptically
polarized.
Linear and circular polarizations are special cases of
elliptical and they can be obtained when the ellipse
becomes a straight line or a circle respectively.
24. Radiation Resistance
• An important property of a transmitting ant is its radiation
resistance which is associated with the power radiated by the
ant. If
I = rms ant current
Rr = antenna radiation resistance
Then power radiated is I2 Rr watts where Rr is a fictitious
resistance which accounts for the radiated power somewhat
like a acct resistance which dissipates heat.
• The radiation resistance should be large as the greater Rr is,
the greater the power radiated by ant.
• In contrast, for a receiving antenna its i/p impedance is
important. The i/p impedance is defined as the ratio of voltage
to correct at its i/p and it should be matched to connecting lines
or cables.
• The i/p impedance may or may not equal to its radiation
resistance, though very often it does.
25. Effective Length
• An antenna with a non-uniform distribution of current
over its length l can be considered as having a shorter
effective length le over which the current is assumed to
be uniform and equal to its peak value. The relationship
b/w le and l is given by:-
le = area under non – uniform current distribution
l area under uniform peak current distribution
26. Effective Aperture
• The power received by an antenna can be associated with a
collecting area. Every antenna may be considered to have
such a collecting area which is called its effective aperture A.
• If Pd is the power density at the antenna and Pr is the received
power then.
• Pr = Pd A watts
or Pr 2
A= m
Pd
For an antenna with power gain G, the effective aperture A at
the operating wavelength λ is given by
Gλ2
A=
4π
