Reading — The Radar Range Equation

Reading — The Radar Range Equation#

By the end of this lesson you should be able to:

Write the radar range equation and explain the physical origin of each term.
Derive the maximum detection range \(R_\text{max}\) and show why it scales as the fourth root of every term.
Work a numerical example from transmit power and antenna gains to a detection range.
Quantify how an RCS reduction translates into a detection-range reduction.

Building the equation#

The range equation is just bookkeeping on power, applied twice — out to the target and back.

Step 1 — transmit toward the target. A transmitter radiates \(P_t\) watts. An antenna of gain \(G_t\) concentrates that into a beam, so the power density at range \(R\) is

\[ S_\text{inc} = \frac{P_t G_t}{4\pi R^2}. \]

The \(4\pi R^2\) is the surface area of a sphere of radius \(R\) — the one-way spreading loss from Lesson 2.

Step 2 — the target re-radiates. The target intercepts some power and scatters it back. Its radar cross section \(\sigma\) (units: m²) is defined exactly so that the power “captured and re-radiated isotropically” is \(S_\text{inc}\,\sigma\). The echo then spreads back over another sphere, giving a density at the radar of

\[ S_\text{echo} = \frac{P_t G_t}{4\pi R^2}\cdot\frac{\sigma}{4\pi R^2}. \]

Step 3 — the radar collects the echo. A receive antenna with gain \(G_r\) has effective aperture \(A_e = G_r \lambda^2 / 4\pi\). Multiplying the echo density by \(A_e\) gives the received power:

\[ \boxed{\,P_r = \frac{P_t\, G_t\, G_r\, \lambda^2\, \sigma}{(4\pi)^3\, R^4}\,} \]

The headline feature is the \(R^4\) in the denominator. The signal spreads out on the way to the target and again on the way back — two factors of \(1/R^2\) multiply into a fourth-power law.

Key Concept

The \(1/R^4\) in the radar range equation comes from two-way spreading: \(1/R^2\) out, \(1/R^2\) back. This single fact is why radar detection is so range-limited, and why it is the most leverage-rich link in the kill chain.

Maximum detection range#

A radar declares detection when received power meets or exceeds its minimum detectable signal \(S_\text{min}\). Setting \(P_r = S_\text{min}\) and solving for range:

\[ R_\text{max} = \left[\frac{P_t\, G_t\, G_r\, \lambda^2\, \sigma}{(4\pi)^3\, S_\text{min}}\right]^{1/4}. \]

Every term sits under a fourth root. Bundling everything except \(\sigma\) into a constant \(K\) recovers the Lesson 1 form,

\[ R_\text{max} = K\,\sigma^{1/4}. \]

The fourth root is brutal on the radar designer and merciful to the target. To double detection range you must raise \(P_t G_t G_r\) by a factor of 16. Conversely, the target gets the same leverage in reverse — a 16-fold RCS cut only halves the range, but the radar’s billion-watt options have run out long before the target’s shaping options do.

The cost of each term#

Want to double \(R_\text{max}\) by changing…	Required change	Practical?
Transmit power \(P_t\)	×16 (+12 dB)	Expensive; thermal and prime-power limits.
Antenna gain \(G_t\) or \(G_r\)	×16 each (+12 dB)	Huge apertures; physically bounded.
Minimum signal \(S_\text{min}\)	÷16	Limited by noise floor; integration helps a little.
Target RCS \(\sigma\)	÷16	This is the target’s lever — low-observable design.

The table is the whole survivability argument in one place: the radar pays \(\times 16\) for every range doubling, and there is no cheap term to push.

A worked example#

Take an S-band acquisition radar:

\(P_t = 1\) MW \(= 60\) dBW
\(G_t = G_r = 30\) dBi
\(\lambda = 0.1\) m (3 GHz, S-band)
\(S_\text{min} = -130\) dBm \(= -160\) dBW \(= 10^{-16}\) W (a very sensitive receiver — Lesson 8 shows how integration buys this)

Running the equation for a 1 m² target gives a maximum detection range of roughly

\[ R_\text{max} \approx 474\ \text{km}. \]

Now make the target stealthy. Drop the RCS to \(\sigma = -30\) dBsm (i.e. \(10^{-3}\ \text{m}^2\), a thousandfold reduction) — roughly a B-21-class target, and the figure every project in this course is built around. Scaling by the fourth root,

\[ R_\text{max,new} = 474\ \text{km}\times(10^{-3})^{1/4} = 474\times 0.178 \approx 84\ \text{km}. \]

A factor-of-1000 RCS reduction cuts the detection range from 474 km to about 84 km — roughly a 5.6× reduction in range for a 1000× reduction in RCS. That is the fourth root at work.

Type-along

Type this with me. One anonymous function, four one-line experiments off the worked example.

Rmax = @(Pt,Gt,Gr,lam,sig,Smin) ((Pt*Gt*Gr*lam^2*sig)/((4*pi)^3*Smin))^(1/4);
Pt = 1e6; Gt = 1e3; Gr = 1e3; lam = 0.1; sig = 1; Smin = 1e-16;
R0 = Rmax(Pt,Gt,Gr,lam,sig,Smin);
fprintf('Baseline: R_max = %.0f km\n\n', R0/1000);
name = {'RCS / 16','Power x 4','Gains x 2 each','S_min / 4'};  R = zeros(1,4);
R(1) = Rmax(Pt,Gt,Gr,lam,sig/16,Smin);
R(2) = Rmax(4*Pt,Gt,Gr,lam,sig,Smin);
R(3) = Rmax(Pt,2*Gt,2*Gr,lam,sig,Smin);
R(4) = Rmax(Pt,Gt,Gr,lam,sig,Smin/4);
dTerm = 10*log10([1/16 4 4 4]);
for k = 1:4
    fprintf('%-15s dTerm = %+6.2f dB   R_max = %4.0f km   dR_max = %+5.2f dB\n', ...
            name{k}, dTerm(k), R(k)/1000, 10*log10(R(k)/R0));
end

Three of these four rows are the radar spending real money. What do you notice about where they land?

Solution

Baseline: R_max = 474 km

RCS / 16        dTerm = -12.04 dB   R_max =  237 km   dR_max = -3.01 dB
Power x 4       dTerm =  +6.02 dB   R_max =  670 km   dR_max = +1.51 dB
Gains x 2 each  dTerm =  +6.02 dB   R_max =  670 km   dR_max = +1.51 dB
S_min / 4       dTerm =  +6.02 dB   R_max =  670 km   dR_max = +1.51 dB

Every dR_max is exactly one quarter of its dTerm. The fourth root does not care which term you moved — a dB is a dB.
The three radar upgrades are indistinguishable. Quadrupling the transmitter, doubling both antennas, and a fourfold-better receiver are all \(+6\) dB, and all three land on the same 670 km. The radar bought three different expensive things and got one answer.
The target’s single lever moves twice as far. One 16-fold RCS cut is \(-3\) dB of range against the radar’s \(+1.5\) dB — and the radar has to pay again for the next \(+1.5\) dB, while shaping is paid once at design time.

Wrap-Up#

The radar range equation, \(P_r = P_t G_t G_r \lambda^2 \sigma / [(4\pi)^3 R^4]\), is power bookkeeping over a two-way path, and its \(1/R^4\) forces \(R_\text{max} = K\,\sigma^{1/4}\). Every term sits under a fourth root, so doubling range costs a 16-fold change in power, gain, or sensitivity — while the target’s RCS lever, though equally fourth-rooted, is the one with room left to push.

So far the radar has been a continuous power budget. Real radars send pulses, and that changes everything about how range is measured — and introduces ambiguities the next lesson is built around.

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