Radar Equation for Point Targets Consider an isotropic radiator transmitting a pulse with power P t t and a target that is at distance R. The target has an area Aσ. At a distance R from the transmitter the power density is
=
S
P t
4πR 2
Next add an an antenna antenna with gain G that redirects the power towards the target. The power intercepted by the target with area Aσ is P σ
=
GP A t σ
4πR 2
Now make the assumption assumption that that (a) there there are no losses losses at the target, target, and (b) the power interce intercepted pted by the target target is reradiated reradiated isotropically. isotropically. Some of the the reradiated reradiated power reaches reaches the the transmitter. transmitter. This amount is
=
P r
P Ae σ
4πR 2
where Ae is the effective area of the transmitter antenna. Substitute the expression for P σ and we have P r
=
GP A t σ
4π R
2
Ae 2
4π R
=
GP A Ae t σ
(4π )2 R 4
The effective area of the antenna is by definition Ae
=
2
Gλ
4π
Thus the received power is
P r
=
GP A t σ
2
2
Gλ
=
(4π )2 R 4 4π
2
G λ P A t σ
64π
3
R
4
The physical target area Aσ is related to, but different from what appears to the radar. Call the (RCS) area σ, and the received power is radar cross sectional (RCS) 2
P r
=
2
G λ P σ t
64π 3 R 4
Caveat: there are several variations of the radar equation in use. Some use energy rather than power, some some include include the radar radar pulse width, and and different different symbols are used. used. This does does not change change the basic relationships between transmitted power, distance from the radar, gain, etc. This equation is called the
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Example. The received power from a point target located at 110 km from a radar is 10 14 W. What is the received power of two identical point targets at 180 km?
-
Use the radar equation for point targets to determine the target area P ( R1 ) r
=
2
2
64π
3
G λ P σ t R1
4
=> σ = P ( R1 ) r
64π 3 R1 2
2
4
G λ P t
m2
where R1 = 110 km. Use the radar equation for point targets to determine the received power at R2 = 180 km: 4 2 2 2 2 G λ P G λ P 64π 3 R1 σ t t P ( R2 ) = = ⋅ P r ( R1 ) 2 2 r 4 4 G λ P 64π 3 R2 64π 3 R2 t R1
4
R2
4
= P ( R1 ) r
4
= 10
−14
110 180
This is the received power from one target. The received power from two point targets is -15 twice this, or 2.78×10 W.
Radar Cross Section (RCS) 2
RCS and measured in m . It depends on three factors σ
= (Projected Cross Section) × (Reflectiv ity) × (Directivi ty)
Target Geometric Cross Section (Projected Cross Section)
This is the geometric area/cross section A presented to the radar, i.e., the projection of the target. For example, the geometric area of a ship is larger broadside compared to head 2 on. The units are m . Target Reflectivity
Not all the intercepted power is reflected. For example, aircraft can be coated with absorbent materials, and there are inherent ohmic losses in all scatterers. Reflectivity is expressed as a ratio so it is dimensionless. Target Directivity
This refers to the power scattered back in the direction of the transmitting radar. Simple, idealized targets scatter isotropically, but real targets don’t. They scatter more in some
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directions than others. Target directivity is defined as the ratio of backscattered power to backscatter from a target that scatters isostropically. Directivity is dimensionless.
Sphere, radius = a
πa
Cylinder, radius = a, height = h
2πah2
2
λ
Flat plate, width = a, height =
b
4πa 2b 2 2
λ
Square trihedral corner reflector, width = length = height = a
12πa 4 2
λ
Some radar cross sections
The table above shows (the maximum) RCS for some simple geometric scatterers, under the assumption that the wavelength is much smaller that the largest dime nsion of the object, and that the range is much larger than the wavelength, and the reflectivity is 1. Note that for a sphere there is no dependence on wavelength. Example. What is the maximum RCS for a flat plate 0.094 m × 0.094 m at 1 GHz and 10 GHz? Solution
At 1 GHz, the wavelength is 0.1 m, so the RCS is RCS 1 GHz
=
4πa 2b2 2
λ
=
4π
× 0.0942 × 0.0942 2
0.1
= 1 m2
At 10 GHz, the wavelength is 10 times smaller, and the RCS is 100 times smaller, so RCS 10 GHz
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= 0.01 m 2
3
