What is the standard monostatic radar range equation for received power, including gains and losses?

What is the standard monostatic radar range equation for received power, including gains and losses?

• A. P_r = (P_t G_t G_r λ^2 σ) / ((4π)^3 R^4 L_t L_r)
• B. P_r = (P_t G_t G_r λ^2 σ) / ((4π)^2 R^4 L_t L_r)
• C. P_r = (P_t G_t G_r λ^2 σ) / ((4π)^3 R^4 L_t^2 L_r)
• D. P_r = (P_t G_t G_r λ^2 σ) / ((4π)^3 R^2 L_t L_r)

Answer: (A)

The key idea is that received power in a monostatic radar comes from a two‑way propagation, the antenna gains on both ends, the wavelength, the target’s radar cross section, and the system losses. Start with the power density that arrives at the target: the transmitter with gain G_t sends power P_t, and the signal spreads over a sphere of area 4πR^2, so the incident power density is P_t G_t /(4πR^2). The target then re-radiates power in proportion to its radar cross section σ, so the power scattered toward all directions is this incident density times σ. The receiver then sees the scattered energy as it travels back another distance R, so the power density at the receive location is this scattered power divided by the area of a sphere 4πR^2, giving a factor 1/(4πR^2) again. The receiving antenna with effective aperture A_e,r = G_r λ^2 /(4π) captures a portion equal to A_e,r times the power density, yielding a pre-loss received power proportional to P_t G_t G_r λ^2 σ /(4π)^3 R^4. Finally, the system losses L_t and L_r reduce this power, so you divide by L_t L_r. Putting it all together gives the standard monostatic radar range equation with gains and losses: P_r = P_t G_t G_r λ^2 σ / [ (4π)^3 R^4 L_t L_r ]. This form reflects the two‑way geometric spreading (the R^4 term) and the coupling through the antenna apertures (the (4π) factors and the λ^2 term). The other options would miscount the two‑way path, the antenna aperture contribution, or the way losses are applied.

The key idea is that received power in a monostatic radar comes from a two‑way propagation, the antenna gains on both ends, the wavelength, the target’s radar cross section, and the system losses. Start with the power density that arrives at the target: the transmitter with gain G_t sends power P_t, and the signal spreads over a sphere of area 4πR^2, so the incident power density is P_t G_t /(4πR^2). The target then re-radiates power in proportion to its radar cross section σ, so the power scattered toward all directions is this incident density times σ. The receiver then sees the scattered energy as it travels back another distance R, so the power density at the receive location is this scattered power divided by the area of a sphere 4πR^2, giving a factor 1/(4πR^2) again. The receiving antenna with effective aperture A_e,r = G_r λ^2 /(4π) captures a portion equal to A_e,r times the power density, yielding a pre-loss received power proportional to P_t G_t G_r λ^2 σ /(4π)^3 R^4. Finally, the system losses L_t and L_r reduce this power, so you divide by L_t L_r. Putting it all together gives the standard monostatic radar range equation with gains and losses:

P_r = P_t G_t G_r λ^2 σ / [ (4π)^3 R^4 L_t L_r ].

This form reflects the two‑way geometric spreading (the R^4 term) and the coupling through the antenna apertures (the (4π) factors and the λ^2 term). The other options would miscount the two‑way path, the antenna aperture contribution, or the way losses are applied.