Math Bridge: From 79 Ideal Years to a Derated Battery Ledger

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Math BridgeApplication ProtocolsStruggle-friendly runway

Why does a 79-year estimate become about 31 years?

Turn charge into energy, model radio-burst sag, reserve unusable charge, and add self-discharge to the chapter protocol comparison.

Bex, the guideBex guides
The one targetBuild a derated battery-life denominator.
The chapter case1000 mAh, 30% reserve, 1%/year self-discharge.
What it buys youKeep protocol savings in proportion to cell physics.

A field team faces an unresolved physical question: Why does a 79-year estimate become about 31 years? They must answer it before changing reserved charge 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 reserved charge. The middle card applies this page's relationship. The green card is burst sag. 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.

Reserved charge changes burst sag An input card leads through the page relationship to the burst sag result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. Reserve changes every lifetime in proportion, while the common 10.0 mAh/year self-discharge term compresses the protocols’ relative separation.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for reserved charge is 30.

  2. 2

    Name the relationship. t=Qnameplate(1-δ)/(365·Qactive/day+Qnameplate·s)

  3. 3

    Substitute the chapter fixture. Set reserved charge to 30. The page ledger gives burst sag as 0.670 V.

  4. 4

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

Predict, then change reserved charge

Try Predict the direction of burst sag. Move one control, calculate, then check your prediction.

30
Chapter baseline
Burst sag

Observe Reserve changes every lifetime in proportion, while the common 10.0 mAh/year self-discharge term compresses the protocols’ relative separation. Reset the control to 30 and compare burst sag.

Explain Only reserved charge 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 reserved charge moves. Field effects named in the page's technical boundary stay fixed.

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

1. mAh is charge, not a complete battery promise

Charge becomes energy only after multiplying by voltage. The terminal voltage falls during a radio burst, and some nameplate charge must remain unusable at cutoff or cold conditions. Self-discharge spends charge even while firmware sleeps.

Bex: Keep the units and the assumptions beside every number.

2. Name every algebra move

1

Convert charge to energyEusable=(Qnameplate·(1−δ)/1000)·Voc.

2

Check burst voltageVterm=Voc−IRint.

3

Add every annual drainlife=Qusable/(Qactive/year+Qself/year).

3. The denominator changes the protocol story

t=Qnameplate(1−δ)/(365·Qactive/day+Qnameplate·s)

The fixed self-discharge term is the same tax on CoAP, MQTT, and HTTP. Adding it shrinks the relative benefit of reducing protocol activity, even though CoAP still leads this chapter’s illustrative ledger.

4. Try one controlled change

t=Qnameplate(1−δ)/(365·Qactive/day+Qnameplate·s)

TryChange the reserved fraction while protocol activity and self-discharge stay fixed.

Reserve
Burst sag
Burst terminal
Usable charge
Usable energy
Self-discharge
CoAP life
MQTT life
HTTP life
CoAP advantage

ObserveAt 30%, 700 mAh or 2.52 Wh remains. The model gives about 30.9 years for CoAP, 26.9 for MQTT, and 26.7 for HTTP.

ExplainReserve changes every lifetime in proportion, while the common 10.0 mAh/year self-discharge term compresses the protocols’ relative separation.

Technical boundaries.

The Li-SOCl2 voltage, 15 Ω internal resistance, 1%/year self-discharge, and 30% reserve are catalog-typical illustrations.

pulse capability and passivation
Cell transient evidence
temperature and converter efficiency
Operating-condition evidence
sleep current and voltage curve
Measured load and discharge evidence
capacity-rate effect, aging, and cutoff
Lifetime-boundary evidence

These require a selected cell and measured load profile.

5. Reproduce the chapter values

The chapter’s 0.00267 mAh over 0.215 s implies 44.7 mA. Through 15 Ω that sags 0.671 V, leaving 2.93 V from 3.6 V. A 30% reserve leaves 700 mAh=2.52 Wh. CoAP uses 12.7 mAh/year; adding 10.0 gives 700/22.7=30.9 years. MQTT gives 26.9 and HTTP 26.7 years.

6. Carry the evidence forward

Choose a real cell, then log its lot, temperature, pulse profile, voltage sag, converter/cutoff behavior, sleep current, radio retries, protocol bytes, self-discharge assumption, reserve policy, and periodic field capacity checks.

7. Check yourself

Why is 1000 mAh not 3.6 Wh usable here?
Answer: The 30% reserve leaves 0.700 Ah; at 3.6 V that is 2.52 Wh before conversion losses.
Why does protocol advantage shrink?
Answer: A fixed self-discharge drain is added to every protocol’s annual activity.
Does 30.9 years predict a deployment?
Answer: No. It is a bounded illustrative ledger; the chosen cell and measured load must replace the typical inputs.
Honesty boundary.

These are the worked values and named assumptions for this bridge.

1000 mAh
Nameplate charge
3.6 V
Open-circuit voltage
44.7 mA
Worked active current
15 Ω
Illustrative internal resistance
0.671 V
Worked burst sag
2.93 V
Worked terminal voltage
700 mAh
Usable charge
2.52 Wh
Usable energy
10.0 mAh/year
Self-discharge ledger
30.9 years
Illustrative CoAP life
26.9 years
Illustrative MQTT life
26.7 years
Illustrative HTTP life

The Li-SOCl2 voltage, 15 Ω internal resistance, 1%/year self-discharge, and 30% reserve are catalog-typical illustrations. Pulse capability, passivation, temperature, converter efficiency, sleep current, voltage curve, capacity-rate effect, aging, and cutoff require a selected cell and measured load profile.

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