Antenna Characteristics
2.2 Beam Solid Angle, Directivity, and Gain
The total power radiated by an antenna is obtained by integrating the Poynt-ing vector over the entire area of the sphere of radius r.
Prad =
&
Ω
Srad(r, θ, φ)· ar dA W (2.21) where S(r, θ, φ) is the Poynting vector on the sphere and dA is the elemental area. In spherical coordinates the elemental area is given by
dA = r2sin θdθdφ (2.22)
Using Eqn (2.15), the total radiated power can also be written in terms of the radiation intensity as
Prad=
&
ΩU (θ, φ)dΩ W (2.23)
where the elemental solid angle, dΩ, is given by
dΩ = sin θdθdφ (2.24)
Using Eqn (2.17), U (θ, φ) = UmaxPn(θ, φ), the total radiated power is Prad = Umax
&
ΩPn(θ, φ)dΩ W (2.25)
Defining the beam solid angle, ΩA, as ΩA=
&
ΩPn(θ, φ)dΩ sr (2.26)
the radiated power can be expressed in terms of the beam solid angle and the maximum radiation intensity as
Prad = ΩAUmax W (2.27)
This equation suggests that if all the power is radiated uniformly within a solid angle ΩA, it would have an intensity equal to Umax. For a pencil beam pattern, the beam solid angle is approximately equal to the product of half-power beamwidths in the two principal planes
ΩA= Θ1HPΘ2HP sr (2.28)
where, Θ1HP(rad) and Θ2HP(rad) are the half power beamwidths in the two principal planes.
Since there are 4π steradians over a sphere, the average radiation intensity is
Uavg = Prad
4π W/sr (2.29)
The directivity of an antenna in a given direction, D(θ, φ), is defined as the ratio of the radiation intensity in that direction to the average radiation intensity
D(θ, φ) = U (θ, φ)
Uavg = 4πU (θ, φ)
Prad (2.30)
If the directivity of an antenna is specified without the associated direction, then the maximum value of D is assumed, i.e.
D = 4πUmax Prad = 4π
ΩA (2.31)
where, the second expression is obtained by using Eqn (2.27), which relates the radiated power to the beam solid angle. For a pencil-beam pattern, the maximum directivity can also be expressed in terms of the half-power beamwidths in the two principal planes
D = 4π
Θ1HPΘ2HP (2.32)
For an isotropic antenna, the radiation intensity, U0, is independent of the direction and the total radiated power is
Prad =
&
Ω
U0dΩ = 4πU0 W (2.33)
Thus, the radiation intensity of an isotropic antenna is given by U0 = Prad
4π (2.34)
This is the same as the Uavg of Eqn (2.29) and hence the directivity can also be looked upon as the ratio of radiation intensity in a given direction to that produced by an isotropic antenna, both radiating the same amount of power.
The directivity of an isotropic antenna is unity. The directivity of an antenna indicates how well it radiates in a particular direction in comparison with an isotropic antenna radiating the same amount of power.
The directivity is also expressed in decibels as
DdB = 10 log10D dB (2.35)
Sometimes, the reference to an isotropic antenna is explicitly indicated by the unit dBi instead of dB. Similarly, the power in decibel units referenced to 1 mW and 1 W are expressed in dBm and dBW units, respectively. The power is expressed in dBm units using
PdBm= 10 log10
P
1× 10−3
dBm (2.36)
where P is in watts. The same can be expressed in dBW units as PdBW = 10 log10
P 1
dBW (2.37)
The total power radiated by a Hertzian dipole is given by Prad=
&
Ω
U (θ, φ)dΩ =
&
Ω
U (θ, φ) sin θdθdφ (2.38)
Substituting U (θ, φ) from Eqn (2.16) and performing the integration over the sphere, the total radiated power is
Prad= ηπ 3
I0dl
λ
2
W (2.39)
The directivity of a Hertzian dipole is calculated by dividing the radiation intensity, U (θ, φ), by the average radiation intensity (Uavg = Prad/4π), which gives
D(θ, φ) = 1.5 sin2θ (2.40) The maximum value of the directivity of a Hertzian dipole is 1.5 (or 1.76 dBi) and occurs along θ = 90◦.
Any physical antenna has losses associated with it. Depending on the antenna structure, both ohmic and dielectric losses can be present in the antenna. Let Ploss be the power dissipated in the antenna due to the losses in the structure and Pin= Prad+ Ploss be the power input to the antenna.
Radiation efficiency, κ, of an antenna is the ratio of the total power radiated to the net power input to the antenna
κ = Prad Pin
(2.41) From the definition of the directivity, it is clear that directivity is a parameter dependent on the shape of the radiation pattern alone. To account for the losses in the antenna, we multiply the directivity by the efficiency and define this as the gain of the antenna.
G(θ, φ) = κD(θ, φ) = 4πU (θ, φ)
Pin (2.42)
Generally, when a single value is specified as the gain (or directivity) of an antenna, it is understood that it is the gain (or directivity) along the main beam peak or the maximum value. These are denoted by G and D without the argument (θ, φ).
EXAMPLE2.6
Calculate the radiation efficiency of an antenna if the input power is 100 W and the power dissipated in it is 1 W.
Solution: The input power, Pin= 100 W, and the power radiated is Prad = 100− 1 = 99 W. Substituting these values in Eqn (2.41), the radiation effi-ciency is κ = Prad/Pin= 99/100 = 0.99.
EXAMPLE2.7
Calculate the maximum power density at a distance of 1000 m from a Hertzian dipole radiating 10 W of power. Assume that the dipole has no losses. What are the electric and magnetic field intensities at the given point?
If the same power were to be radiated by an isotropic antenna, what will be the power density and the field intensities? How much power should the isotropic antenna radiate so that the field intensities of the isotropic antenna and the Hertzian dipole (along the maximum) are the same?
Solution: The maximum directivity of an antenna is given by Eqn (2.30) D = 4πUmax
Prad = 4πSmaxr2 Prad
where, Prad = 10 W is the total radiated power, r = 1000 m is the distance from the antenna to the field point, and Smax is the maximum radiated power density. Since the maximum directivity of a Hertzian dipole is 1.5, the maximum power density is
Smax= 1.5Prad
4πr2 = 1.5× 10
4π10002 = 1.1937× 10−6 W/m2
In the far-field region, the electric field intensity |E| and the power density are related by, S =|E|2/η, where η = 120π Ω, is the free-space impedance, thus the maximum electric field intensity is|Emax| =√
Smaxη = 0.0212 V/m.
The magnetic field intensity is given by,|H| = |E|/η = 5.627 × 10−5 A/m.
If Pradiis the power radiated and Smaxi is the radiation power density of an isotropic antenna, then
Smaxi = Pradi
4πr2 W/m2
For the maximum fields of the Hertzian dipole and the isotropic antenna to be identical, the radiated power density of the two antennas should be equal, i.e., Smaxi = 1.1937× 10−6 W/m2, which gives, Pradi= 4πr2Smaxi= 4π× 10002× 1.1937 × 10−6= 15 W.
EXAMPLE2.8
Calculate the directivity of an antenna the power pattern of which is given by U (θ, φ) =
sin θ sin φ 0≤ θ ≤ π; 0 ≤ φ ≤ π 0 0≤ θ ≤ π; π ≤ φ ≤ 2π
Solution: The total power radiated by the antenna is given by (Eqn (2.23)) Prad=
&
ΩU (θ, φ)dΩ W
where dΩ = sin θdθdφ. Substituting the value of the radiation intensity Prad= The directivity is given by
D(θ, φ) = 4πU (θ, φ)
The function of an antenna in a system is to radiate power into space or receive the electromagnetic energy from space. The antenna is generally connected to a transmitter or to a receiver via a transmission line or a waveguide with some characteristic impedance. The antenna acts as a trans-former between free space and the transmission line. In this section we shall treat the antenna as a transmitting antenna. The antenna as a receiver is treated separately later in this chapter. In a practical antenna connected to a transmitter via a transmission line, the applied radio frequency voltage establishes a current distribution on the antenna structure. This, in turn, radiates power into free space. A small part of the input power is dissipated due to ohmic/dielectric losses in the antenna. Further, the applied volt-age also establishes a reactive field in the vicinity of the antenna. One can think of the antenna as an equivalent complex impedance, Za, which draws exactly the same amount of complex power from the transmission line as