USAFA DFEC

ECE 495 — Electromagnetic Warfare and Survivability

Block 1: Radar Fundamentals
Lesson 3: The Radar Range Equation
Juan Jurado, Col, USAF, Ph.D.
Permanent Professor and Head
Electrical and Computer Engineering
Fall 2026
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Block 1 · Lesson 3 · Where We Left Off

Where We Left Off

L1 hinted: $R_{\max} \propto \sigma^{1/4}$.

L2 gave us one-way FSPL: $L_{\text{fs}} \propto (R/\lambda)^{2}$.

Today
The full radar range equation that ties them together — the central equation of the course.

Block 1 · Lesson 3 · Objectives

Lesson Objectives

By the end of this lesson, you will be able to:

  • Derive the radar range equation from one-way propagation and a passive target
  • Identify each term and what it represents physically
  • Compute $R_{\max}$ for a notional engagement
  • Connect L1's fourth-power law back to the full equation
One equation, and every term in it sits under a fourth root.

Block 1 · Lesson 3 · Part One

The Range Equation

Power bookkeeping over a two-way path. Nothing in the derivation is new — it is L2's spreading loss, applied twice.

Block 1 · Lesson 3 · Two-Way Path

From One-Way to Two-Way

L2 was a transmitter and a receiver. Radar adds a passive target in between.

  • One-way path: one factor of $1/R^{2}$ from spherical spreading
  • Radar path: two factors of $1/R^{2}$ — out and back
  • The target itself acts as a passive antenna with effective area $\sigma$
Two trips through free space, one bounce off the target.

Block 1 · Lesson 3 · Derivation

Deriving the Radar Range Equation

1
Tx radiates $P_t$ with gain $G_t$. Power density at the target:
\[ S_t = \frac{P_t \, G_t}{4 \pi R^{2}} \]
2
Target with RCS $\sigma$ re-radiates:
\[ P_{\text{ref}} = S_t \, \sigma \]
3
Power density back at the radar (second $1/R^{2}$):
\[ S_r = \frac{P_{\text{ref}}}{4 \pi R^{2}} = \frac{P_t \, G_t \, \sigma}{(4 \pi)^{2} \, R^{4}} \]
4
Receiver effective aperture $A_e = G_r \lambda^{2} / (4\pi)$ captures:
\[ P_r = S_r \, A_e \]

Block 1 · Lesson 3 · The Equation

The Radar Range Equation

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

Detection threshold: $P_r \geq S_{\min}$. Solving at equality:

Maximum detection range
\[ R_{\max} = \left[\frac{P_t \, G_t \, G_r \, \lambda^{2} \, \sigma}{(4\pi)^{3} \, S_{\min}}\right]^{1/4} \]
  • Every lever the radar designer pulls lives in the numerator
  • Every term the target controls (just $\sigma$) lives in the numerator too
  • $R_{\max} \propto (\text{everything})^{1/4}$ — the fourth-power penalty

Block 1 · Lesson 3 · Part Two

The Fourth Root

Everything in the equation is fourth-rooted before it reaches range. That is brutal on the radar designer and merciful to the target.

Block 1 · Lesson 3 · Cost of Each Term

Each Term Has a Cost

TermWho controls itEngineering cost
$P_t$Radar designerPrime power, cooling, hardware
$G_t, G_r$Radar designerAntenna aperture, mechanical size
$\lambda$Mission plannerBand selection (atmosphere, hardware)
$\sigma$Aircraft designerShape, materials, EMCON
$S_{\min}$Radar designerReceiver design, BW, integration
  • Every parameter pays its dB at $\tfrac{1}{4}$ efficiency in $R_{\max}$
  • $\sigma$ is the only lever the aircraft fights for directly

Block 1 · Lesson 3 · Fourth-Power Law

The Fourth-Power Law Revisited

L1's law, unhidden
\[ R_{\max} = K \cdot \sigma^{1/4}, \qquad K \equiv \left[\frac{P_t \, G_t \, G_r \, \lambda^{2}}{(4\pi)^{3} \, S_{\min}}\right]^{1/4} \]
  • $K$ is everything except $\sigma$ — fixed once the threat is chosen
  • Halving $R_{\max}$ takes $\sigma$ down by 12 dB — the cost of LO
  • L1's notional $K = 100$ km / m$^{1/2}$ was the radar range equation in disguise
L1's fourth-power law was the radar equation hiding in one constant.

Block 1 · Lesson 3 · Worked Example

S-band Acquisition vs a 1 m$^{2}$ Fighter

$P_t = 1$ MW $\Rightarrow$ 60 dBW
$\sigma = 1$ m$^{2}$ $\Rightarrow$ 0 dBsm
$G_t = G_r =$ 30 dBi
$S_{\min} = -130$ dBm $\Rightarrow$ $-160$ dBW
$\lambda = 0.1$ m $\Rightarrow$ $-20$ dB-m
$(4\pi)^{3} \Rightarrow$ 33 dB
Working in dB
\[ 10\log_{10}\!\left(R_{\max}^{4}\right) = P_t + G_t + G_r + 20\log\lambda + \sigma - 33 - S_{\min} \] \[ = 60 + 30 + 30 - 20 + 0 - 33 + 160 = 227 \text{ dB} \;\Rightarrow\; 10\log_{10} R_{\max} = 56.75 \text{ dB-m} \]
474km
$\sigma = 1$ m$^{2}$ (0 dBsm) — legacy fighter

Block 1 · Lesson 3 · Worked Example

Now Swap In the B-21

Same radar, same band, same receiver. Only $\sigma$ changes — from the fighter's 1 m$^{2}$ to roughly $-30$ dBsm.

Scaling by the fourth root
\[ R_{\max,\text{new}} = 474 \text{ km} \times \left(10^{-3}\right)^{1/4} = 474 \times 0.178 \approx 84 \text{ km} \]
84km
$\sigma \approx -30$ dBsm — B-21 · a 1000× RCS cut buys a 5.6× range cut

Today's Interactive Demo: Range Equation Explorer

Companion script: L3_RadarRangeEquation.m
In L4 we add the time dimension: pulse width, PRF, and ambiguity
Live demo

Range Equation Explorer

usafa-ece.github.io/ece-495-ew

Block 1 · Lesson 3 · Type-along

Your Turn

Laptops open, MATLAB up. One anonymous function and four one-line experiments.

Block 1 · Lesson 3 · Quick Exercise

Quick Exercise / Type-along: Four Levers

Type this with me. One anonymous function, four one-line experiments.

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?

Which lever would you rather own — and which one do you have to keep buying?

Block 1 · Lesson 3 · Wrap-Up

Wrap-Up

Key ideas
  • $P_r \propto P_t G_t G_r \lambda^{2} \sigma / R^{4}$ — the central equation of B1
  • $R_{\max} \propto (\text{everything})^{1/4}$ — fourth-power penalty for everyone
  • $\sigma$ is the airframe's lever; $S_{\min}$ is the receiver designer's
Next lesson
  • L4: Pulse radars — PW, PRF, duty cycle, and range ambiguity
  • Work the L4 pre-flight before class — you could be the one up front