### Radar Range Equation

R_{max} = ( P_{peak} G^{2} λ^{2} σ ) ^{(1/4)}
( _______________________ )
(m) ( (4 π)^{3} k T B F L SNR_{min} )

The radar range equation (Oftentimes simply called the radar equation) is used to attempt to calculate the maximum range at which a radar can detect a target.

The radar range equation is rather simple to derive, and doing so is a valid question on the trade qualification board for Combat Systems Engineers in the Canadian Navy. I got sonar theory instead.

Radars work by transmitting electromagnetic radiation into the atmosphere, and then receiving that energy back after it has bounced off a target. The time lapse between transmission and reception is used to calculate the range to the target, since the energy travels at a constant speed.

So, to start the derivation, we assume that we have an omnidirectional antenna transmitting out into space, at a certain power level. As the energy spreads out, the power density decreases. You have the same amount of energy total, but it is spread thinner, over the surface of an imaginary bubble.

The power density of the surface of that bubble will be the peak power of the transmitter, divided by the surface area of the bubble.

Power Density = P_{peak}
_______
(W/m^{2}) 4 π R^{2}

Now, radars don't work by transmitting in all directions at once. They concentrate the majority of the energy into a particular direction. The degree to which this is done is primarily determined by the function of the radar. Search radars will have a relatively wide beam pattern, in order to be able to cover more volume, whereas track radars typically have very tight beam patterns, concentrating most of its energy in a beam about a degree wide. We introduce into the equation a parameter called the Gain of the antenna, which is a ratio of the power in main lobe of the beam, to what the power would be if it was an omnidirectional antenna.

Power Density = P_{peak} G
________
(W/m^{2}) 4 π R^{2}

This power travels out until it hits a target. The energy will bounce off the target. How much energy bounces off the target is dependant upon a parameter called the radar cross section. It is basically a measurement of how "big" something looks to a radar. This can vary greatly for any particular object, depending upon the angle which you're looking at the object from, the frequency of the radar, or the type of paint used.

The radar cross section used in the equation should be based on what you are looking for. If you are attempting to pick up an incoming anti-ship missile, you should use an RCS of about 0.1 m^{2}. For a ship, the displacement tonnage of the ship, in m^{2} is a good estimate.

So, the total energy bouncing off the target is equal to:

Reradiated Power = P_{peak} G σ
_________
(W) 4 π R^{2}

This experiences the same spherical spreading losses as it did when it was first transmitted, until it reaches back to the radar antenna. The power density at that point will be:

Power density = P_{peak} G σ
____________
(W/m^{2}) (4 π)^{2} R^{4}

The antenna will then receive this power. The amount of power that it receives will be proportional to the size of the antenna. Obviously, a larger antenna will receive more energy, because it has more energy hitting it. The effective antenna aperture size is denoted by A_{e}. Thus, the signal received by the radar will be equal to:

Signal = P_{peak} G σ A_{e}
____________
(W) (4 π)^{2} R^{4}

Assuming that we are using the same antenna for transmission and reception, as the vast majority of radar systems do, we can simplify the above equation because of a relationship between the antenna aperture, and the gain of the antenna.

G = 4 π A_{e}
________
λ^{2}
λ = the wavelength of the EM radiation being emitted (m)

Replacing A_{e} with this relationship in the above equation results in:

Signal = P_{peak} G^{2} λ^{2} σ A_{e}
______________
(W) (4 π)^{3} R^{4}

Every real system, however, will also have some losses associated with it. This will be due to things such as imperfections in the waveguide, causing you to lose energy between the antenna and the radar receiver. To account for this, we introduce a loss factor, L.

Signal = P_{peak} G^{2} λ^{2} σ A_{e}
______________
(W) (4 π)^{3} R^{4} L

As well as the valid signal being received by the radar, there will also be noise. This noise is caused by the random movement of electrons in the receiver, and increases as the temperature does.

Receiver noise power = k T B F
Where:
k = Boltzmann's constant 1.38 × 10^{-23} m^{2} kg s^{-2}^{-1}
T = receiver temperature (K)
B = signal bandwidth (s^{-1})
F = noise figure (This accounts for non-idealities in the receiver)

This allows us to calculate the signal to noise ratio.

SNR = P_{peak} G^{2} λ^{2} σ A_{e}
_______________
(4 π)^{3} R^{4} k T B F L

But, we're not trying to find the SNR here! It's the radar *range* equation! We want to find the maximum range that our radar will be able to pick up a target at. So, we replace SNR with SNR_{min}, the minimum SNR needed by the radar's receiver to actually determine a valid signal. And then we solve for R.

R_{max} = ( P_{peak} G^{2} λ^{2} σ ) ^{(1/4)}
( _______________________ )
(m) ( (4 π)^{3} k T B F L SNR_{min} )

So, there you have it. This equation helps reveal some of the less intuitive relationships between various parameters that affect a radar's performance, such as the fact that in order to double a radar's range, you would need to increase the power output sixteen-fold! It is invaluable when designing a new radar system, as well as to gain an understanding of an existing system.

**Sources:**

Canadian Forces Engineering School (Halifax). "Naval Combat Systems Engineering Applications Course notes."

Payne, Craig. "Principles of Naval Weapon Systems," *Naval Institute Press.* 2006. Pages 37 - 41.

Croci, Renato. "Radar Basics." *Radar's Corner.* 06 Oct 02. <www.alphalpha.org/radar/intro_e.html> (12 Nov 2008.)