Radar Equations

Chapter 2

IntroductionMaximum detection range depends on the SNR

of the received signal:Transmitted powerTarget range:As the target range is increased as the received

signal power is decreased, due to spreading over greater and greater area of the transmitted waves.

Antenna gain (directivity, efficient factor);(the gain can be expressed as the ratio of the

power radiated in the direction of max radiation, to the power that would have been radiated by a lossless isotropic antenna.

DG

IntroductionTarget radar cross section The targets intercepts a portion of the

transmitted signal and reflects it in various directions. How much of the signal is intercepted, how well the target reflects radar waves and how much of the reflected signal is actually directed back towards the radar, these affect and determine the size of the target as seen by the radar. The measure of this size is Radar Cross Section RCS ( in units area).

s Change from target to another and even from one orientation to another of the same target.

RCSThe conceptual definition of RCS includes the

fact that not all of the radiated energy falls on the target. A target’s RCS (F) is most easily visualized as the product of three factors:

= Projected cross section x Reflectivity x Directivity .Reflectivity: The percent of intercepted power

reradiated (scattered) by the target.Directivity: The ratio of the power scattered

back in the radar's direction to the power that would have been backscattered, the scattering been uniform in all directions (i.e. isotropically).

IntroductionAntenna effective areaThe reflected signal toward the radar is

intercepted by the receiving antenna, how much this area is important, how better the level of received signal at the receiver.

How much the Receiving Antenna area is big, the radar performance is better.

Derivation of radar equation (monostatic radar)

Case of no-loss path of the transmitted and reflected waves:

1. The transmitted wave from the TX antenna has as power Pt and it is propagated toward the target.

2. Calculate the Power density at distance R of the target

(in the case of isotropic antenna)(in the case of directive antenna) Power density

Where G is the maximun gain of the antenna (supposing that the target is situated at distance R inside the main beam )

24density power

R

Pt

24density power

R

GPt

Derivation of radar equation3. Calculate the Power reflected back to receiver

antenna, this is different to the power recieved by the antenna receiver

4. Power density of the reflected wave toward radar antenna receiver is equal to:

5. Calculate the intercepted power at the antenna receiver, this is related to the effective area of the antenna (area where the reflected signal is intercepted)

24 R

GPt

2

2

44

RR

GPt

et

r AR

GPP 224

Study of the Radar equationDependance of Range

Exp: 1. R2=2R1 then Pr2 =Pr1/162. How many the Range should be changed to

necessitate an increasing power of 3 dB Radar system where RCS, f, Ae, G constant values, gives.

et

r AR

GPP

424

then R if rP rr PdBP 10log10)(

Study of the Radar equationDependence on frequency

Ae is related to G,

restriction of this formula…..More representative formula

VHF : 30 MHZ 300 MHZ, increasing of 40 dBUHF: 300 MHZ 1GHZ, increasing of 20.89 dBLband: 1GHZ 2GHZ, increasing of 12 dB

4444

2arg fPfRc

AeAAPP r

eteTtr

fKP dBr log40

43

222

42 444 R

GPG

R

GPP ttr

Maximum detection rangeThe max detection Range Rmax is ultimately determined by

the minimum signal to noise ratio required by the receiver.For a given noise level at the input of the receiver, the

minimum signal to noise ratio depends on the minimum detectable signal power Smin, A signal weaker than Smin, would covered by noise and would probably not be detected.

Consequently for long ranges, the following parameters should be chosen accordingly: must be higherG must be higherSmin must be low (receiver ability to detect weak signal level)

Rmax in terms of transmitted frequency:

4/1

min2max

4

S

AGPR et

tP

fS

AeAAPR eteTt

4

min

2arg

max

4

min3

22

max4

S

GPR t

and

Radar equation with Losses introducedCase of monostatic radar with two

antennas

Suppose Pt is the output power of the transmitter, This power may be reduced by mismatch and losses in the microwave elements (duplexer, circulators, isolators, etc.) and transmission line (waveguide or coaxial line) that connects the transmitter to the antenna.

Radar equation with Losses introducedIf with Then the average peak power accepted at the receiver antenna is denoded as Pacc, where:

Lrt (radiation loss of the transmitting antenna) because some power is lost through heating effects in the structure of the antenna. This loss is denoted by and defined by:

is the efficiency of the transmitting antenna With these losses the average peak radiated power is :

Radar equation with Losses introduced If all the average peak radiated power occurred

from a nondirective (isotropic) antenna, the power density of the wave at distance R1 would be:

where Lch1 is the one path medium loss (due to all clear and unclear channel effects that may be present (atmospheric attenuations, effects of rain, snow, etc. )

If the antenna is directive with Gain in the direction of the target, then the power density toward the target direction is:

Radar equation with Losses introducedThe reflected power from the target in the

direction of the receiving antenna is given by:

The reflected power received by the RX antenna:

Radar equation with Losses introducedThe received power is then equal to:

Where Lrr is the antenna receiving loss :

And also,

Examplefind the maximum range R of a monostatic radar (with same antenna of trasmission and reception) that must provide an available received average peak signal power of 10- 12 W when frequency is 4.6 GHz, Pt = W, the antenna's aperture area is 2.0 m2 , aperture efficiency is 0.64, radar cross section is 1.4 m2 , loss of transmitting antenna is 1.2, loss of antenna’s radiation = 1.04, and loss of path = 1.43.