The HPBW is found by equating
sin
2
θ
with 1/2, yielding
θ
= 45
◦
, 135
◦
. The difference between these limits is the
HPBW, which is exactly 90
◦
in this case. Note the importance of correct normalization in this calculation. Discussing
the HPBW in the Hplane is not meaningful for a dipole.
Since the radiation pattern has nulls at
θ
= 0
◦
and 180
◦
, the BWFN is 180
◦
. Normally, the BWFN is nearly but not
exactly twice the HPBW. Note that there is no radiation off the ends of a dipole.
2.1.4
Total radiated power
We calculate the total power radiated by the antenna by integrating the Poynting flux over any closed surface surround
ing the antenna (a sphere or arbitrary radius being the natural choice). The computation frequently necessitates either
integral tables and variable transformations or numerical computation. The present case is elementary, however.
P
total
=
contintegraldisplay
s
P
r
dS
30
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Tx
tline
Z
line
Z
ant
= R
rad
+ jX
ant
Figure 2.3: Antenna viewed as circuit element with complex load impedance.
=
contintegraldisplay
s
P
r
r
2
d
Ω
=
integraldisplay
2
π
0
dφ
bracehtipupleft
bracehtipdownrightbracehtipdownleft
bracehtipupright
2
π
integraldisplay
π
0
dθ
Z
◦
8
parenleftbigg
Idl
rλ
parenrightbigg
2
r
2
sin
3
θ
=
2
πZ
◦
8
parenleftbigg
Idl
λ
parenrightbigg
2
integraldisplay
π
0
sin
3
θdθ
Notice that the result does not depend on
r
, as it must not. We frequently must calculate definite integrals of powers
of trigonometric functions. In this case,
integraltext
sin
3
θdθ
=
integraltext
sin
θ
(1
−
cos
2
θ
)
dθ
=
−
[cos
θ
−
cos
3
θ/
3]
π
0
= 4/3. Finally,
P
total
=
Z
◦
π
3
parenleftbigg
Idl
λ
parenrightbigg
2
(
W
)
2.1.5
Radiation resistance
Since an antenna dissipates power, it must have an associated resistance.
(Antennas are also reactive in general,
although one generally tries to design purely resistive antennas in practice so as to avoid impedance mismatches.)
We associate the
radiation resistance
R
of an antenna with the average flow of power into the antenna. For many
practical antennas, this resistance is typically tens or hundreds of Ohms. Radiation resistance is unrelated to the ohmic
resistance associated with the conductors from which the antenna is constructed. In most (but not all) circumstances,
ohmic losses can be neglected. This is tantamount to equating gain with directivity.
Equating the total radiated power for this antenna with the average power dissipated by an equivalent resistive
circuit element according to Ohm’s law yields:
P
total
=
I
2
rms
R
rad
=
1
2

I

2
R
rad
R
rad
=
80
π
2
(
dl/λ
)
2
(
Ohms
)
Whereas the leading coefficient in this expression is of the order of 1000, the ratio
(
dl/λ
)
is small by definition, and
the radiation resistance of an elemental dipole is apt to be very small. Consider an AM car radio antenna operating
at a frequency of
∼
1 MHz, for example. Here, the wavelength
λ
∼
300 m, whereas the antenna length
dl
∼
1m.
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 Spring '13
 HYSELL
 The Land, power density, Solid angle

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