Free-Space Propagation Loss - Understanding GPS: Principles and Applications - page 671

2 2

A GGc
f

λ
π

=

R

=

R

(C.3)

R

4

2

4 π

Observe that the effective area of an antenna having a given gain is inversely
proportional to the square of the frequency. For the same antenna gain with increas-
ing frequency, the antenna's area must become smaller.
Returning to the earlier discussion of an electromagnetic wave emanating out-
ward from a transmitter, the power spatial density (having units of W/m 2 ) at a point
on a sphere with radius d from the transmit antenna is

PG
d

Φ=

TT
4

(C.4)

2

π

The power spatial density is also known as the power flux density (PFD).
Observe that the PFD decreases with the square of the distance from the transmitter,
so that the PFD (the received power per unit area) is independent of frequency and
depends only on the distance from the transmitter.
The power at the receive antenna's terminals is given by the product of the PFD
at the receive antenna and the effective area of the receive antenna

P A
R

=Φ (C.5)

R

Substituting (C.3) and (C.4) into (C.5) yields

2

PG
d

G λ

π

P =

TT R

R

2

4 π 4

(C.6)

2

λ
π

= PGG

TTR

4 d

Expression (C.6), often called the Friis equation [2], allows calculation of the
received power, given the EIRP ( P T G T ) and the receive antenna gain ( G R ). When
(C.6) is calculated for an isotropic receive antenna, for which G R =

1, the result is the

RIP.
Sometimes the free-space propagation model is generalized to account for an
excess propagation loss beyond the free-space loss. This excess propagation loss
could be caused by attenuation due to the atmosphere, foliage penetration, building
penetration, or polarization mismatch. The effect of this excess power loss is mod-
eled by a dimensionless multiplicative factor L that takes on values between unity
and infinity, with unity indicating no excess loss and infinity indicating complete
blockage. As in the definition of propagation loss, L is defined to match common
terminology (e.g., “an excess loss of 2 dB”). The resulting expression for received
power is

2

PGG
L

λ
π

P =

TTR

(C.7)

R

4 d