Radar Equation and detection of stealth Fighters

There are so many arguments and debates related to the detection of stealth fighters are happening around. Here we are presenting a simple method to find the detection range of radars against stealth fighters.
Below you can see the radar equation, various parameters that affect the range of radar and how these variables individually affect the detection range. 

R max = 4th root of ((Pt * G2*lamda 2 * RCS) / ((4 Pie) 3 * P Min))
The variables in the above equation are constant and radar dependent except target RCS.

G – Antenna Gain
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An important characteristic of radar is its antenna gain. Some antennas are highly directional; that is, more energy is propagated in certain directions than in others. The ratio between the amount of energy propagated in these directions compared to the energy that would be propagated if the antenna were not directional (Isotropic Radiation) is known as its gain.
In simple terms, Ratio of how much energy is transmitted in or received from certain direction by the radar antennas when compared to a hypothetical antenna which transmits the same energy evenly in all directions are called antenna gain.
 
Lamda – Wavelength

Wavelength measured in meters, which is inverse of frequency, the higher the frequency the shorter the wave length. To detecting further away a longer wave length is better.

Pt – Transmit power – (in watts ie; emitted by the radar)
 
The more energy the radar system transmits, the greater the target detection range will be.
 
RCS

 It is also important to note that the target RCS of an aircraft depends upon the wave length used.

P min – Minimum detectable signal

Minimum detectable signal (Pmin) depends on receiver bandwidth (B), noise flore (F), temperature (T), and required signal-to-noise ratio (S/N). A narrow bandwidth receiver will be more sensitive than a wider bandwidth receiver. Noise figure is a measure of how much noise a device (the receiver) contributes to a signal: the smaller the noise figure, the less noise the device contributes. Increasing temperature affects receiver sensitivity by increasing input noise.
 
Noise flore is a ver important characteristic of a Radar. There is always some kind of noise man made/ natural. To work properly radar must reject signals which are weaker than the noise flore. Noise flore of the radar can be reduced by using more advanced algorithms & processors to filter out noise & look for patterns or by using more advanced radar tech to reduce the amount of noise & interference produced by the radar itself. 

Det Range = 4 root of ((Transmit power * Antenna gain 2 * wavelength 2 * RCS) / (4Pie) 3 * Noise flore))

Certain features has a greater effect on detection range. Have a look on each of the variables in the radar equation.

Det Range = 4th root of Power.

Assume all other variables are constant; the relationship between Detection Range & Power is to the 4TH root. This is due to the inverse square law the radar experiences this law twice on the way to the target and the return trip. The energy twice as far from the source is spread over 4 times the area hence ¼ th the intensity.  From this it’s clear that if you want to increase the detection range you need to increase the power significantly.

For eg if you want to increase the Detection Range twice without adjusting any other variables

2 * Det Range = 4th root of 16* power
4 * Det Range = 4th root of 256* power

In real situation increasing the power has its own disadvantages, such as increase in noise flore. This means we would have to increase power even further to account for detection range loses to noise. 

Look at the next variable Antenna Gain

Det Range = 4 root of gain 2

Gain has a square root relationship with Det Range. So this means we can increase the detection range by twice by quadruply the gain

2 * Det Range = root of 4* gain
4 * Det Range = root of 16 * gain 

 
Wavelength also has a square root relation

2 * Det Range = root of 4* WL
4 * Det Range = root of 16 * WL 

Noise flore has an inverse relationship

Det Range = 4th  root (1/Noise flore)

To improve ddetection rnge we need to significantly improve noise flore of radar .

2 * Det Range = NF / 16
4 * Det Range = NF / 256

And the next one is RCS. This is a key capability of the stealth fighter

Det Range = 4th root RCS
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2 * Det Range = 16 * RCS
4 * Det Range = 256 * RCS

To reduce detection range aircrafts need to reduce RCS significantly.

1/2 * Det Range = RCS/16
1/4 * Det Range = RCS/256

Find out the different methods of reducing RCS

Calculating the detection range using known RCS and Range

Stealth fighter det Range = Known radar det range /4 root (Known radar det Range RCS/ stealth fighter rcs)

Calculating the detection range of Irbis – E

According to the official website Irbis E can detect a fighter with RCS of 3m2 at 350 Km. We can use these figures to calculate its detection range against F22. F22 RCS is 0.0001m2.

Det range = 350 Km / (4th root of 3/ 4th root of 0.0001) = 27Km
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So irbis E can detect a fighter aircraft with 0.0001 m2 RCS at 27 km approximately. 

Calculating the detection range of AN APG-77

AN APG can detect 1M2 target at 400Km, substituting this values to the equation will give

Det Range = 400Km/ (4th root of 1/4th root of 0.0001) = 40 Km

AN APG 77 can detect a fighter aircraft with 0.0001 m2 RCS at 40 km approximately. 


Calculate other Radars detection range and comment below if you found our article informative . 

Picture Courtesy - Youtube
Info Courtesy - Radartutorial.eu