Page 37 - Antennas for Base Stations in Wireless Communications

10 Chapter One

radiation and reduces efficiency. Mathematically, efficiency is calculated
as radiation resistance divided by total resistance.

1.1.6 Directivity and Gain
The directivity of an antenna has been defined as “the ratio of the radia-
tion intensity in a given direction from the antenna to the radiation
intensity averaged over all directions.” In other words, the directivity
of a nonisotropic source is equal to the ratio of its radiation intensity in
a given direction, over that of an isotropic source 1–4 :

U 4π U
D = = (1.14)
U i P r
where D is the directivity of the antenna; U is the radiation intensity
of the antenna; U is the radiation intensity of an isotropic source; and
i
P is the total power radiated.
r
Sometimes, the direction of the directivity is not specified. In this
case, the direction of the maximum radiation intensity is implied and
the maximum directivity is given as

U 4π U
D max = U i = P r max (1.15)
max
where D max is the maximum directivity and U max is the maximum radia-
tion intensity.
A more general expression of directivity includes sources with radia-
tion patterns as functions of spherical coordinate angles q and f:
4π
D = (1.16)
Ω A

where Ω A is the beam solid angle and is defined as the solid angle in
which, if the antenna radiation intensity is constant (and maximum
value), all power would flow through it. Directivity is a dimensionless
quantity because it is the ratio of two radiation intensities. Therefore,
it is generally expressed in dBi. The directivity of an antenna can be
easily estimated from the radiation pattern of the antenna. An antenna
that has a narrow main lobe would have better directivity than the one
that has a broad main lobe; hence, this antenna is more directive. In the
case of antennas with one narrow major lobe and very negligible minor
lobes, the beam solid angle can be approximated as the product of the
half-power beamwidths in two perpendicular planes:

Ω = Θ Θ 2 r (1.17)
r
1
A