THE RADAR EQUATION
ELC 451
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RADAR EQUATION: DEFINITION
1.
The radar equation represents the physical
dependences of the transmit power, that is the
wave propagation up to the receiving of the
echo-signals.
2.
The power PR returning to the receiving
antenna is given by the radar equation,
depending on the transmitted power Pt, the
slant range R, and the reflecting
characteristics of the target (described as the
the radar scattering cross section).
3.
At known sensibility of the radar receiver, the
radar equation determines what can be
achieved by a given radar set theoretically
maximum range.
4.
In general, the performance of the radar
system can be assessed with the radar
equation.
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DEFINITION
• The Radar Equation
represents the
fundamental relation
between the
characteristics of the
radar, the target, and
the received signal.
Scatterer
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RADAR EQUATION: DERIVATION (1)
1. If high-frequency energy is
emitted by an isotropic
transmitter, the energy
propagate uniformly in all
directions.
2. The same amount of energy
spreads out on an incremental
basis on spherical surface
with an incremented spherical
radius.
3. That means: the power density
on the surface of a sphere is
inversely proportional to the
square of the radius of the
sphere and can be
represented as:
4. where Ss is the power density
at the scatterer.
Scatterer
4
RADAR EQUATION: DERIVATION (2)
5. To obtain the total power intercepted by the
scatterer, the power density must be
multiplied by the effective receiving area of
the scatterer:
6. The effective area Ars is not the actual area
of the incident beam intercepted by the
scatterer, but rather is the area of the
incident beam from which all power would
be removed if one assumed that the power
going through all the rest of the beam
continued uninterrupted.
7. The actual value of Ars depends on the
effectiveness of the scatterer as a
receiving antenna.
5
RADAR EQUATION: DERIVATION (3)
8. Some of the power
received by the scatterer
is absorbed in losses in
the scatterer.
9. The rest is reradiated in
various directions.
10. If the fraction absorbed is
fa, then fraction
reradiated is 1- fa, and
the total reradiated power
is:
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RADAR EQUATION: DERIVATION (4)
11. The conduction and displacement
currents that flow in the scatterer
result in re-radiation that has a
pattern (like an antenna pattern).
12. The effective receiving area of the
scatterer is a function of its
orientation relative to the
incoming beam, so that
Ars applies only for the direction of
the incoming beam.
13. The re-radiation pattern may not
be the same as the pattern of Ars, 14. Where Pts is the total reradiated
power, Gts is the gain of the scatterer
and the gain in the direction of the
in the direction of the receiver and
receiver is the relevant value in
the re-radiation pattern. Hence:
15. Si is called the spreading factor for
the re-radiation.
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RADAR EQUATION: DERIVATION (5)
14. The power entering the receiver
is
Pr  S r A r
15. where the area Ar is the effective
aperture of the receiving
antenna, not its actual area.
16. Not only is Ar function of
direction, but it is also a function
of the load impedance the
receiver provides to the antenna.
17. For example, Pr would have to be
zero if the load were a short
circuit or an open circuit.
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RADAR EQUATION: DERIVATION (6)
18.
Combining equations discussed in 1-17
above yields:
19.
The factors associated with the scatterer
are combined in the square brackets.
20.
These factors are difficult to measure
individually, and their relative
contributions are not of interest to one
wishing to know the size of the received
radar signal. Hence they are normally
combined into one factor, the radar
scattering cross section:
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RADAR EQUATION: DERIVATION (7)
• The final form of the radar
equation is:
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RADAR EQUATION: DERIVATION (8)
•
The most common situation is that
for which receiving and
transmitting locations are the
same, so that the transmitter and
receiver distances are the same
and the same antenna is used for
transmitting and receiving, so the
gains and effective apertures are
the same, that is:
11
RADAR EQUATION: DERIVATION (9)
• Since the effective area of an
antenna is related to its gain by
•
The radar equation can be
written as:
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END
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