Library / Sensors & detection / SEN-03
Sensors & detection
Radar range from echo delay
Range to a target from the round-trip time of the pulse.
- Ref
- SEN-03
- Rev
- v1
- Field
- Sensors & detection
- Result
- R [km]
τ is the time between sending a pulse and receiving its echo, entered in microseconds, and c the speed of light. The pulse covers the distance twice, out and back, so the one-way range is cτ/2. The answer is in kilometres.
Range to a target from the round-trip time of the pulse. The factor of two accounts for the out-and-back path.
Calculator
- Range in nautical miles16.19 nmi
- Echo delay per kilometre of range6.671 µs/km
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Radar echo takes 6.67 µs for each kilometre of range (12.4 µs per nautical mile). A 200 µs delay means a target about 30 km away.
- The next pulse should not be sent until the echo from the furthest target of interest has returned; otherwise a close echo from the new pulse is confused with a distant one from the old pulse.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (km) | Local step | Change in result |
|---|---|---|---|---|---|---|
| τ Echo delay | 1 to 10,000 µs | 0.1499 | 1,499 | 1,499 | +10% | +10 % |
Sweep of Echo delay
| Echo delay (µs) | Range (km) |
|---|---|
| 1 | 0.1499 |
| 3.162 | 0.474 |
| 10 | 1.499 |
| 31.62 | 4.74 |
| 100 | 14.99 |
| 316.2 | 47.4 |
| 1,000 | 149.9 |
| 3,162 | 474 |
| 10,000 | 1,499 |
Sweep one variable
Variables and how each one contributes
Range
What the equation gives you. Everything else feeds it.
Echo delay
The echo delay: the time between transmitting the pulse and receiving the echo. Directly proportional: each extra microsecond of delay adds 150 m of range.
Worked example with the values above
- Convert the delay: τ = 200 µs = 2×10−4 s.
- Round-trip distance: c τ = 2.998×108 × 2×10−4 = 59,960 m.
- Halve it: R = 29,980 m = 29.98 km.
Assumptions and limits
- The pulse travels at the speed of light in free space; air slows it by only about 0.03 %.
- A single echo from a target that is effectively stationary during the pulse.
Common mistakes
- Forgetting the factor of 2 for the round trip.
- Mixing µs and ms. A delay of 200 ms would be 30 000 km.
Statistics and uncertainty
The range error equals c/2 times the timing error: 1 µs of timing error is 150 m, regardless of range. Atmospheric refraction slows the pulse by about 0.03 % in air, a small effect (30 m at 100 km), but timing jitter and the width of the pulse usually dominate.
Where it is used
- Converting measured echo delay to range.
- Setting the range gate and the pulse repetition interval of a radar.
Maths you need
- Speed, distance and time: d = vt
- Standard form: c = 3 × 10⁸ m/s