Math Bridge: LPWAN Battery and Airtime Coupling

← Back to Choosing the Right Network Type
Math BridgeCore NetworkingStruggle-friendly runway

How can a stronger LPWAN setting cost 32 times the airtime?

Connect cell energy and derating to LoRa spreading factor, symbol time, and relative transmit energy.

Eddie, the electronics guideEddie guides
The one targetJoin usable battery charge to the spreading-factor airtime multiplier.
The chapter caseA 2400 mAh, 3.6 V soil-sensor cell over five years at SF7 to SF12.
What it buys youA reach-rate-energy classification record instead of a range-only label.

A field team faces an unresolved physical question: How can a stronger LPWAN setting cost 32 times the airtime? They must answer it before changing spreading factor on the real device. Predict the direction first.

See the relationship before changing it

The figure reads from left to right. The blue card is spreading factor. The middle card applies this page's relationship. The green card is nameplate energy. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

The retained audit below checks several chapter fixtures. This added model holds every other chapter fixture fixed, so the numeric fixture does not switch without explanation.

Spreading factor changes nameplate energy An input card leads through the page relationship to the nameplate energy result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. Robustness can recover link margin, but it consumes the same finite usable battery budget for longer on every matching transmission.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for spreading factor is 12.

  2. 2

    Name the relationship. nameplate=2.400 Ah(3.6 V)=8.64 Wh usable=2400(0.993)^5(0.85)=1970 mAh Tsymbol,SF12=2^12/125000=32.768 ms airtime multiplier=2^(12-7)=32

  3. 3

    Substitute the chapter fixture. Set spreading factor to 12. The page ledger gives nameplate energy as 8.64 Wh.

  4. 4

    Read the result. Keep Wh beside the value. Use it only inside the technical boundary on this page.

Predict, then change spreading factor

Try Predict the direction of nameplate energy. Move one control, calculate, then check your prediction.

12
Chapter baseline
Nameplate energy

Observe Robustness can recover link margin, but it consumes the same finite usable battery budget for longer on every matching transmission. Reset the control to 12 and compare nameplate energy.

Explain Only spreading factor moves here. The other chapter fixtures remain fixed.

Check yourself

What should you do before trusting a moved-control result?
Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.
What does this small model leave out?
Answer: Only spreading factor moves. Field effects named in the page's technical boundary stay fixed.

Honesty note. This teaching model changes only spreading factor. It replays the chapter fixture; it is not a field prediction.

1. Start with the physical story

A higher LoRa spreading factor makes each symbol last longer so a weak signal is easier to detect. The radio pays for that robustness with extra on-air time.

Eddie: Follow one quantity at a time; every displayed result comes from the same ledger.

2. Name every algebra move

1

Turn charge into nameplate energyMultiply amp-hours by volts.

2

Apply the slow leakKeep 99.3% of charge once for each year.

3

Hold a design reserveKeep only the remaining 85% for planned use.

4

Find symbol durationDivide 2^SF by bandwidth.

5

Compare with SF7The time multiplier is 2^(SF−7).

3. Reproduce the chapter case

nameplate=2.400 Ah(3.6 V)=8.64 Wh
usable=2400(0.993)^5(0.85)=1970 mAh
Tsymbol,SF12=2^12/125000=32.768 ms
airtime multiplier=2^(12−7)=32

The arithmetic stays visible so that units and assumptions can be checked before the result is used.

4. Try one real input

TryMove the spreading factor. Predict how many doublings separate SF7 and SF12.

Spreading factor
Nameplate energy
Usable charge
Usable share
Symbol duration
Symbol rate
Airtime vs SF7
TX energy term vs SF7

ObserveEach step doubles symbol duration. SF12 is five doublings above SF7, so its same-symbol-count airtime term is 32 times larger.

ExplainRobustness can recover link margin, but it consumes the same finite usable battery budget for longer on every matching transmission.

Technical boundaries.

This transparent ledger reproduces the named chapter case.

Battery
Catalog capacity, self-discharge, and reserve are illustrative; pulse sag and regulator loss are separate.
Radio
The comparison holds bandwidth and symbol count fixed.
Network
Duty-cycle law, ADR, retries, collisions, gateway diversity, and receive windows are outside it.

Correct, not complete: this ledger does not select a battery, prove coverage, or predict field lifetime.

5. Use the result in the design

Classify reach, data rate, airtime, duty-cycle limits, and energy together; improve antenna and gateway placement before assuming maximum spreading factor.

6. Record the evidence state

Record cell chemistry, temperature, pulse capability, reserve, SF, bandwidth, coding settings, payload, retries, measured link margin, and airtime.

7. Check yourself

Why is 2400 mAh not 8.64 Wh by itself?
Answer: Because charge becomes energy only after multiplying by the cell voltage.
Why does SF12 give a 32-fold term over SF7?
Answer: There are five spreading-factor steps, and 2^5 equals 32.
Does the 32-fold term prove battery life falls by 32?
Answer: No. Sleep, sensing, payload, overhead, retries, regulation, and transmit frequency also shape total life.
Honesty boundary.

This transparent ledger reproduces the named chapter case.

Battery
Catalog capacity, self-discharge, and reserve are illustrative; pulse sag and regulator loss are separate.
Radio
The comparison holds bandwidth and symbol count fixed.
Network
Duty-cycle law, ADR, retries, collisions, gateway diversity, and receive windows are outside it.

Correct, not complete: this ledger does not select a battery, prove coverage, or predict field lifetime.

Return to Choosing the Right Network Type with the maths unpacked →