Math Bridge: Four-state battery life

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Math BridgeUX DesignStruggle-friendly runway

How can one timing change cut battery life below two years?

Weight every current state by time, then keep cell aging as a separate loss.

UX Uma, the guideUX Uma guides
The one targetCalculate average current and life from four states.
The chapter case10 µA sleep; 5/15/120 mA actions; 300 s and 60 s reports.
What it buys youA service-interval claim tied to real firmware timing.

A field team faces an unresolved physical question: How can one timing change cut battery life below two years? They must answer it before changing report interval 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 report interval. The middle card applies this page's relationship. The green card is sleep time. 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.

Report interval changes sleep time An input card leads through the page relationship to the sleep time result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. The numerator loses sleep time and keeps almost the same active charge while the denominator shrinks.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for report interval is 300.

  2. 2

    Name the relationship. At 300 s: Iavg=(0.010x299.83+5x0.05+15x0.02+120x0.10)/300=51.8 uA Qusable=2400x0.99²=2352 mAh; 2-year budget=134 uA; margin=2.59x At 60 s: Iavg=219 uA; life=2352/0.219/8760≈1.23 years

  3. 3

    Substitute the chapter fixture. Set report interval to 300. The page ledger gives sleep time as 299.83 s.

  4. 4

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

Predict, then change report interval

Try Predict the direction of sleep time. Move one control, calculate, then check your prediction.

300
Chapter baseline
Sleep time

Observe The numerator loses sleep time and keeps almost the same active charge while the denominator shrinks. Reset the control to 300 and compare sleep time.

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

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

1. Start with current multiplied by time

Current tells how quickly charge moves. Multiplying current by the seconds spent in a state gives that state's share of one cycle. A short 120 mA transmit burst can matter more than a longer 5 mA sensing step.

UX Uma: Do not average the four current labels; average the charge they spend.

2. Name every algebra move

1

Find sleep timetsleep=T−tsense−tcompute−ttx.

2

Multiply each stateQi=Ii ti.

3

Add and divideIavg=ΣIi ti/T.

4

Derate the cellQusable=Qnominal(1−r)y.

5

Convert charge to lifelife=Qusable/Iavg.

3. Reproduce both report intervals

At 300 s: Iavg=(0.010×299.83+5×0.05+15×0.02+120×0.10)/300=51.8 µA
Qusable=2400×0.99²=2352 mAh; 2-year budget=134 µA; margin=2.59×
At 60 s: Iavg=219 µA; life=2352/0.219/8760≈1.23 years

Only the interval changes. The shorter cycle repeats the sensing, computing, and radio charge five times as often.

4. Try the report interval

TryMove the interval from five minutes toward one minute without changing any current or active duration.

Report interval
Sleep time
Average current
Usable charge
2-year budget
Budget margin
Modelled life

ObserveShorter intervals barely change sleep current, but repeat all three active charges more often.

ExplainThe numerator loses sleep time and keeps almost the same active charge while the denominator shrinks.

Technical boundaries.

This is a repeating current ledger, not an electrochemical cell simulation.

States
Real firmware adds startup, retry, update, and fault states
Cell
Temperature, pulse load, cutoff voltage, and aging change usable charge
Timing
Measure installed durations instead of trusting catalog examples

Use current traces and worst-case timing before signing a service interval.

5. Test the timing claim

Capture a complete cycle at normal, weak-link, update, and recovery conditions. Repeat it at the coldest supported temperature and at low cell voltage.

6. Record the power state

Store firmware, interval, every current and duration, radio retries, cell lot, temperature, cutoff, measured trace, and the changes that force a rerun.

7. Check yourself

Why not average 10 µA, 5 mA, 15 mA, and 120 mA directly?
Answer: The states last very different times, so each current must be time-weighted.
Why does 60 seconds fail the two-year budget?
Answer: Its 219 µA average exceeds the aged cell's roughly 134 µA two-year budget.
Does the 5.18-year arithmetic qualify the product?
Answer: No. It omits voltage sag, temperature, retries, cutoff, and measured variability.
Honesty boundary.

The four currents, durations, intervals, and cell-aging case reproduce the chapter's worked region.

2,400 mAh
Catalog-typical Li-SOCl2 teaching cell
1% per year
Simplified compounding self-discharge assumption
Four states
Useful ledger, not the complete installed waveform

Correct, not complete: a four-state average does not qualify battery life.

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