Notes #5, ECE594I, Fall 2009, E.R. Brown
93
FreeSpace Power Coupling for Two Special Cases: Radar and Radiometry
Friis' Transmission Formulation
Marconi was the pioneer for a new generation of electrical engineers working in the area of
“wireless”.
One of the truly brilliant amongst these was Friis working at Bell Laboratories in the
1920s and 30s.
Among other things, Friis was the first to take advantage of the inherent nature of
antennas as passive, reciprocal components, and treat the freespace propagation between a transmit
antenna and receive antenna as a twoport “link”.
This was done first and foremost for the wireless
communications “link”, which we review here first to set the stage for the following RF and THz
sensor (i.e., radar and radiometer) “link” formulation.
The first step in Friis’ formulation is the concept of an effective aperture A
eff
for the
receiving
antenna,
p
r
r
inc
eff
rec
S
A
P
ε
φ
θ
⋅
=
)
,
(
(1)
where P
rec
is the power available to the antenna for delivery to a load,
)
,
(
r
r
inc
S
is the average
Poynting vector for incoming radiation along the direction (
θ
r
.φ
r
) in the spherical coordinates
centered at the
receiving
antenna, and
ε
p
is the polarization coupling efficiency.
Note that this
expression applies only when
)
,
(
r
r
inc
S
is aligned with the direction of the beampattern
maximum
.
When there is misalignment, another factor is required which is the just the receive
beampattern,
p
r
r
inc
r
r
r
eff
rec
S
F
A
P
⋅
⋅
=
)
,
(
)
,
(
.
(2)
Next, we suppose that this
received
Poynting vector is generated by a second,
transmitting
antenna.
We can relate the received power to the properties of the transmitting antenna by:
)
(
4
)
,
(
)
(
4
)
,
(
)
,
,
(
)
,
(
2
2
r
r
F
G
P
r
r
F
D
P
r
S
S
t
t
t
t
in
t
t
t
t
rad
t
t
t
r
r
inc
τ
π
⋅
≡
⋅
=
≡
(3)
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View Full DocumentNotes #5, ECE594I, Fall 2009, E.R. Brown
94
where the subscript "t" is for transmitting, P
rad
is the total radiated power, P
in
is the power used to
drive the transmitting antenna (in the matched case, equal to P
rad
),
θ
t
and
φ
t
are the spherical angles
in the spherical coordinate system centered at the transmitting antenna,
τ
(r) is the power
transmission function including all attenuation effects, and r is the distance (i.e., the “range”
between transmitter and receiver.
In writing (3) it is understood that F
t
is taken in the direction
(
θ
t
,
φ
t
) pointing towards the receiver, which is not necessarily the direction of the maximum of F
t
.
Substitution of (3) into (2) yields the relationship
p
r
r
t
t
t
r
t
in
eff
rec
r
r
F
F
G
P
A
P
ε
τ
π
φ
θ
⋅
⋅
=
)
(
4
)
,
(
)
,
(
2
(4)
This can be simplified further in terms of the (ostensibly) known parameters of the receiving
antenna using the relationships,
2
eff
r
rec
r
r
rec
out
/
A
4
G
P
D
G
P
P
λ
π
≡
=
(5)
where P
out
is the power delivered to the load of the receiving antenna.
Substitution of (4) into (5)
yields
p
t
t
t
r
r
r
t
r
in
out
r
F
F
G
G
r
P
P
λ
⋅
⋅
⋅
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
)
(
)
,
(
)
,
(
4
2
1
(6)
the expression commonly known as Friis' formula after its originator.
It effectively treats the
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 Spring '09
 O
 Power, E.R. Brown, Radiometry Friis

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