How do bandwidth and range relate for different physical layer technologies? Let’s quantify the fundamental physics tradeoff.
Link budget equation constrains all wireless protocols: \(P_{RX} = P_{TX} - L_{path} + G_{TX} + G_{RX}\)
Where \(P_{RX}\) is received power, \(P_{TX}\) is transmit power, \(L_{path}\) is path loss, and \(G_{TX}\)/\(G_{RX}\) are transmit/receive antenna gains (all in dB).
Free-space path loss (Friis formula): \(L_{path}(\text{dB}) = 20\log_{10}(d) + 20\log_{10}(f) + 32.45\)
Where \(d\) = distance (km), \(f\) = frequency (MHz).
Key insight: To double range while maintaining the same received power, you must:
- Increase TX power by 6 dB (4x power), OR
- Reduce data rate to improve receiver sensitivity (halving rate gains ~3 dB), OR
- Reduce frequency to benefit from lower path loss
Quantifying protocol tradeoffs:
| Wi-Fi 2.4 GHz |
2400 MHz |
54 Mbps |
84 dB |
20 dBm |
~100 m |
| Zigbee |
2400 MHz |
250 kbps |
84 dB |
0 dBm |
~30 m (mesh extends) |
| LoRa SF7 |
915 MHz |
5.5 kbps |
148 dB |
14 dBm |
~1.5 km |
| LoRa SF12 |
915 MHz |
250 bps |
157 dB |
14 dBm |
~4 km |
Calculating LoRa’s range advantage over Wi-Fi:
Free-space path loss at 100 m (0.1 km), 2.4 GHz: \(20\log_{10}(0.1) + 20\log_{10}(2400) + 32.45 = -20 + 67.6 + 32.45 = 80\text{ dB}\)
Free-space path loss at 1 km, 915 MHz: \(20\log_{10}(1) + 20\log_{10}(915) + 32.45 = 0 + 59.2 + 32.45 = 91.7\text{ dB}\)
But LoRa SF12 uses chirp spread spectrum with 4096 chips per symbol, yielding processing gain: \(\text{Processing gain} = 10\log_{10}(4096) = 36.1\text{ dB}\)
Where LoRa’s advantage comes from:
- Lower frequency: ~8.4 dB less path loss at the same distance vs 2.4 GHz (\(20\log_{10}(2400/915) = 8.4\) dB)
- Lower data rate: improved receiver sensitivity from narrower noise bandwidth
- Spread spectrum: 36 dB processing gain from chirp spreading
- Combined link budget: 157 dB (LoRa SF12) vs 84 dB (Wi-Fi) = 73 dB advantage
Range extension: \(\text{Range ratio} = 10^{(73/20)} = 10^{3.65} \approx 4{,}467\text{x}\)
But that is theoretical maximum in free space. Real-world with obstacles, fading, and interference: ~40x range improvement (4 km vs 100 m).
Data rate cost: \(\frac{\text{54 Mbps (Wi-Fi)}}{\text{250 bps (LoRa SF12)}} = 216{,}000\text{x slower}\)
Key insight: There is no free lunch in physics. LoRa achieves 40x range by transmitting 216,000x slower and using spread spectrum. For a 50-byte packet: Wi-Fi takes ~7 microseconds, LoRa SF12 takes ~1.3 seconds. This is why you use Wi-Fi for video (needs Mbps) and LoRa for sensor readings (needs km range, not speed).
