Deriving the Link Budget and LoRa Sensitivity
Ada derives the three link-budget equations and works one real LoRa link end to end, showing why the same link closes at SF12 and fails at SF7
ADA · CALCULATION AUDIT
Deriving the Link Budget and LoRa Sensitivity
Three short equations explain every number in a link budget — and the same arithmetic is what ADR re-runs every time it changes a setting.
The chapter’s review example puts a device at 14 dBm EIRP with 1 dB of enclosure and connector loss, a path survey estimating 126 dB of loss, and 2 dB of gateway antenna gain — an expected received level near −111 dBm. If the chosen data rate needs about −123 dBm and the project demands 10 dB of margin, only about 2 dB is left above target. This audit derives the three short equations behind those numbers and works one real link end to end, asking why the same physical link can close at SF12 and fail at SF7.
Companion to the chapter LoRaWAN Link Budget and ADR — every number here comes from that chapter.
Why decibels
1. Why decibels. Radio power spans a factor of about 1016 between transmitter and receiver floor, so we work with logarithms: a ratio expressed as 10·log10(P1/P2) dB. Logarithms turn multiplication of gains and losses into addition, so an entire radio link becomes one line of bookkeeping. Two anchors worth memorising: +3 dB ≈ ×2 power, +10 dB = ×10 power.
What distance costs
2. What distance costs. A wave spreading over a sphere thins with the square of distance; converting that square law into decibels gives the 20·log terms of free-space path loss (d in km, f in MHz; the 32.45 packs the unit conversions and 4π factors):
The square law means every doubling of distance costs exactly 6 dB. Real environments obstruct and reflect, so engineers replace the exponent 2 with a measured path-loss exponent n (≈2.7–3.5 urban) relative to a 1 km reference:
What the receiver can still hear
3. What the receiver can still hear. Thermal physics sets a noise floor of −174 dBm per Hz at room temperature. Widen the bandwidth and you collect more noise; imperfect electronics add a noise figure NF; and the modulation needs some signal-to-noise ratio to demodulate. LoRa’s defining trick is that its slowest spreading factor demodulates at SNRmin = −20 dB — twenty decibels below the noise floor:
For BW = 125 kHz and NF = 6 dB: SF12 gives −174 + 51.0 + 6 − 20 = −137.0 dBm; SF7 (SNRmin = −7.5 dB) gives −124.5 dBm.
One real link, end to end
4. One real link, end to end. A basement meter 5 km from the gateway, EU868, 14 dBm transmit, unity antennas, urban exponent n = 3.2, 20 dB building penetration, 10 dB fade allowance:
| Step | Calculation | Result |
| Reference loss at 1 km | 32.45 + 20·log10(868) | 91.2 dB |
| Urban loss at 5 km | 91.2 + 10·3.2·log10(5) | 113.6 dB |
| Received power | 14 − 113.6 | −99.6 dBm |
| After building + fade | −99.6 − 20 − 10 | −129.6 dBm |
| Margin at SF12 (−137.0) | −129.6 − (−137.0) | +7.4 dB — link closes |
| Margin at SF7 (−124.5) | −129.6 − (−124.5) | −5.1 dB — link fails |
What the mathematics buys you: the same physical link works at SF12 and fails at SF7 — that single comparison is ADR. Every ADR decision this chapter reviews is the network re-running this arithmetic against fresh SNR evidence. Now scroll back to the link-budget calculator in the Overview and reproduce the table: set 5 km, urban, 14 dBm, and watch the margin flip sign between SF7 and SF12.
Every number above is derived from the chapter’s own three link-budget equations and its one worked link, step by step.