An Isotropic Antenna is a (theoretical) antenna that radiates
energy uniformly in all directions.
Thus if the power radiated by an isotropic antenna is P_{t} then the
power flux density (PFD) at a distance s metres from the antenna (in free space) is:
PFD = P_{t }/ (4*pi*s^{2}) W/m^{2}
Note that (4*pi*s^{2}) which is the area of the sphere of radius s is called the
"spreading area".
A real receiving antenna will collect power in an effective area Ae m^{2} and if it is
at a distance s metres from the transmitting antenna then the power received (P_{r})
is:
P_{r }= Ae . PFD = Ae . P_{t} / (4*pi*s^{2}) W
The relationship between the gain G of an antenna and its effective area (where l is the
wavelength in metres) is:
Ae = (G* λ^{2}) / (4*pi) m^{2}
The effective area A_{i} of an isotropic antenna, which by definition has unit gain is
therefore:
A_{i} = λ^{2} / (4*pi) m^{2}
Free Space Attenuation:
The power P_{r }received by an isotropic antenna as a distance s from an isotropic
transmitter is therefore:
P_{r} = A_{i} . PFD = A_{i} . P_{t }/ (4*pi*s^{2})
W
= P_{t} / L W
Where: L = (4*pi*s^{2}) / A_{i} = (4*pi*s^{2}) / (λ^{2} /
4*pi)
i.e. L = (4*pi*s / λ^{2})
So, L, which is the ratio of the spreading area to the area of an isotropic antenna is the
freespace attenuation between isotropic antennas and is often called the "path
loss".
Received Power Between Antennas:
Now, if the transmitting and receiving antennas have gains G_{t} and G_{r}
respectively then the power C received is:
C = (Pt . Gt) . Gr / L = EIRP . (Gr / L) W
Note two ways of finding the power received:
 Using the PFD and effective area of the receiving antenna
 Using the EIRP and gain of the receiving antenna
The effective area A_{e} of an antenna is related to the physical area
of its aperture A_{a} by the expression:
A_{e} = h . A_{a}
where h is the efficiency of the antenna.
The efficiency is less than 100% because the antenna is not perfect and the main factors are:

Spillover past the subreflector and main reflector (rays A and C in the
figure)

Blockage of the antenna aperture by the subreflector (B) and supports (not
shown)

Losses due to profile and other manufacturing errors

Ohmic losses

Nonuniform amplitude and phase distribution in the aperture
We have already seen that:
A_{e} = (G*λ^{2}) / (4*pi*m^{2})
So: G = (4*pi*A_{e}) / λ^{2}
= (4*pi*h*A_{a}) / λ^{2}
i.e. G = h * (pi*D / λ)^{2
}where D is the antenna diameter in metres.
As gain increases, the beamwidth decreases.
The halfpower (3 dB) beamwidth is given by:
HPBW = (N*λ) / D degrees
Where N is a constant dependant on the aperture illumination.
For an ideally illuminated aperture (i.e. each point in the aperture is illuminated with an RF
signal of the same amplitude and phase)
N = 58 but this is not achievable in practice.
For an efficient real antenna, N is approximately 65.
