. Since the radiation is uniform in the Hplane here, the HPBW is not defined
in that plane. Another measure is the beamwidth between first nulls (BWFN), which usually works out to be about
twice the HPFB. In this case, it is 180
◦
(exactly twice).
Directivity
A more objective measure of the degree of confinement is the directivity. The definition of directivity,
D
, can be
expressed as
D
≡
power density at center of main beam at range
R
avg
.
power density at range
R
=
power density at center of main beam at range
R
total power transmitted
/
4
πR
2
where the average means an average over all directions. This definition is slightly clumsy. If we interpret power
density as power per unit area, then we have to specify some range
R
where this can be evaluated. It’s more natural to
calculate power densities per unit solid angle, since that quantity does not vary with range. The latter definition will
have to wait until we have discussed solid angles, however.
Gain
Gain
G
(sometimes called directive gain) and directivity differ only for antennas with losses. Whereas the directivity
is defined in terms of the total power transmitted, the gain is defined in terms of the power delivered to the antenna by
the transmitter. Not all the power delivered is transmitted if the antenna has significant ohmic losses.
G
≡
power density at center of main beam at range
R
power delivered to antenna
/
4
πR
2
Note that
G
≤
D
. In many applications, antenna losses are negligible, and the gain and directivity are used inter
changeably. Both directivity and gain are dimensionless quantities, and it is often useful to represent them in decibels
(dB).
D
≈
G
=
10 log
10
P
max
P
avg
11
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Note also that a high gain implies a small HPBW.
Beam solid angle
We can characterize the main beam of an antenna pattern by the solid angle it occupies. Solid angle is the generalization
of an angle to a radial shape in three dimensions. Just as the arc length swept out on a circle by a differential angle
dθ
is
Rdθ
, the differential area swept out on a sphere by a differential solid angle is
dA
=
R
2
d
Ω
. Thus, we define
d
Ω
=
dA/R
2
. In spherical coordinates, we can readily see that
d
Ω
=
sin
θdθdφ
. The units of solid angle are steradians (Str.)
The total solid angle enclosed by any closed surface is
4
π
Str.
Tx
R
d
Ω
θ
φ
dA
x
y
z
Figure 1.4: Antenna radiation pattern characterized by its solid angle
d
Ω
.
As mentioned earlier, the definitions of directivity and gain given above are somewhat clumsy since they are stated
in terms of power densities which vary with range even though the directivity and gain do not. We can now redefine
the directivity more naturally in terms of solid angle with the knowledge that a perunitarea quantity multiplied by
R
2
becomes a perunitsolidangle quantity.
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 Spring '13
 HYSELL
 The Land, power density, Solid angle

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