The radar equation is the most fundamental mathematical relationship in radar theory. It quantifies the power received by a radar system from a target, relating it to the system's transmitted power, antenna characteristics, the target's reflective properties, and its distance. This equation is indispensable for radar system design, performance prediction, and understanding operational limitations.

Let's systematically derive the radar equation, assuming a monostatic radar configuration (transmitter and receiver at the same location):

1. Power Density from Isotropic Radiator: Imagine a point source radiating power (Pt) uniformly in all directions. At a distance R from this source, the power spreads over the surface of a sphere with radius R. The power density (power per unit area) at this distance is:
   Pdensity,isotropic = 4πR²Pt
   Here, 4πR² is the surface area of a sphere of radius R.

Radar antennas are designed to be directional, meaning they concentrate the transmitted power in a specific direction. This concentration is quantified by the Transmitting Antenna Gain (Gt). An antenna with gain Gt will effectively multiply the isotropic power density by Gt in its main beam direction.
Pdensity,directed = 4πR²Pt Gt
This is the power density incident on the target located at range R.

When the radar wave reaches the target, a portion of this incident power is intercepted and re-radiated. The target's ability to intercept and scatter radar energy is characterized by its Radar Cross-Section (RCS), denoted by σ (sigma). RCS has units of area (e.g., square meters). It represents an effective area that the target presents to the radar.
Pintercepted = Pdensity,directed × σ = 4πR²Pt Gt σ

The intercepted power (Pintercepted) is then scattered by the target. For the purpose of the basic radar equation, we assume this scattered power is re-radiated isotropically (uniformly in all directions) from the target. The power density of this scattered wave, back at the radar receiver (which is also at distance R from the target), is:
Pscattered_density = 4πR²Pintercepted = 4πR² (4πR²Pt Gt σ) = (4π)²R⁴Pt Gt σ

The radar's receiving antenna captures a portion of this scattered power. The amount of power captured depends on the Effective Aperture Area (Ae) of the receiving antenna. The effective aperture area is related to the antenna's gain and the radar signal's wavelength (λ).
For a receiving antenna, its gain (Gr) is related to its effective aperture area (Ae) and the wavelength (λ) by the formula:
Ae = 4πGrλ²
For a monostatic radar (where the same antenna is used for transmitting and receiving, or identical antennas are used with Gt = Gr = G), the received power (Pr) is:
Pr = Pscattered_density × Ae = (4π)²R⁴Pt Gt σ × 4πGrλ²
Substituting Gr = G:
Pr = (4π)³R⁴Pt G²λ²σ
This is the fundamental form of the monostatic radar equation.

Let's break down each parameter and its significance:

• Pr (Received Power): The power measured at the input of the radar receiver, typically in Watts. This is the ultimate signal that the radar processing chain must detect and analyze.
• Pt (Transmitted Power): The peak power generated by the radar transmitter, also in Watts. Higher transmitted power means more energy is radiated, leading to potentially longer detection ranges.
• G (Antenna Gain): A dimensionless ratio (often expressed in dB) representing how well an antenna concentrates power in a particular direction compared to an isotropic radiator. A higher gain means a narrower beam and more focused energy.
• λ (Wavelength): The spatial period of the electromagnetic wave, in meters. It is inversely related to the operating frequency (f) by the speed of light (c): λ = c/f.
• σ (Radar Cross-Section - RCS): The effective area of the target as seen by the radar, in square meters. This measures the target's ability to intercept and scatter radar energy back to the receiver. A larger σ means a stronger echo.
• R (Range): The distance from the radar to the target, in meters. The R⁴ dependency is crucial. It means that if the range to a target doubles, the received power drops by a factor of 24 = 16. This rapid decrease highlights the challenge of long-range detection.

For a target to be detected, the received power (Pr) must exceed a certain threshold, which is typically determined by the noise present in the radar receiver. This threshold is known as the Minimum Detectable Signal (Smin). Smin is the smallest signal power at the receiver input that can be reliably detected above the noise floor.
Smin is fundamentally linked to the receiver's thermal noise and the required Signal-to-Noise Ratio (SNR) for a given probability of detection. The noise power (N) in a receiver is given by:
N = kT0 BF
Where:
- k is Boltzmann's constant (1.38 × 10−23 Joules/Kelvin)
- T0 is the standard noise temperature (usually taken as 290 Kelvin, representing room temperature)
- B is the receiver's noise bandwidth in Hertz
- F is the receiver's Noise Figure (a dimensionless value greater than or equal to 1, indicating how much the receiver degrades the SNR of the signal).
To achieve a desired detection performance, a minimum SNR (SNRmin) is required at the receiver output. Therefore, Smin is often expressed as:
Smin = N × SNRmin = kT0 BF(SNRmin)
By substituting Smin for Pr in the radar equation, we can solve for the Maximum Detectable Range (Rmax), which is the greatest distance at which a target can be reliably detected:
Smin = (4π)³Rmax⁴ Pt G²λ²σ
Rearranging for Rmax:
Rmax = ((4π)³Smin Pt G²λ²σ)^(1/4)
This formula is critical for radar system design, as it directly specifies the operational range given system parameters.

Let's work through a detailed example:
A ground-based air surveillance radar has the following characteristics:
- Peak Transmitted Power (Pt) = 250 kW (2.5 × 10^5 W)
- Antenna Gain (G) = 35 dB
- Operating Frequency (f) = 3 GHz
- Minimum Detectable Signal (Smin) = −120 dBm (decibels relative to 1 milliwatt)
- Target Radar Cross-Section (σ) = 5 m²
Step 1: Convert all units to linear (non-dB) scale.
- Antenna Gain G: 35 dB = 10^(35/10)=10^3.5 ≈ 3162.28
- Minimum Detectable Signal Smin: −120 dBm means 10^(−120/10) milliwatts = 10^(−12) milliwatts.
Since 1 milliwatt = 10^−3 Watts, Smin = 10^−12 × 10^−3 W = 10^−15 W
Step 2: Calculate the Wavelength (λ).
- f = 3 GHz = 3 × 10^9 Hz
- λ = c/f = (3 × 10^8 m/s)/(3 × 10^9 Hz) = 0.1 m
Step 3: Substitute values into the Rmax equation.
Rmax = ((4π)³Smin Pt G²λ²σ)^(1/4)
Rmax = ((4π)³ × (10^−15) × (2.5 × 10^5) × (3162.28)² × (0.1)² × 5)^(1/4)
Rmax ≈ 500 km
This calculation shows that under these conditions, the radar could detect a target with a 5 m² RCS at a maximum range of approximately 500 kilometers.