Math Bridge: Ten-Year Battery Requirement

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Math BridgeApplicationsStruggle-friendly runway

What current budget hides inside a ten-year requirement?

Derate the cell, divide by service hours, and compare the resulting microamp budget with one wake-and-transmit window.

Bex, the requirements guideBex guides
The one targetTranslate service years into a whole-device current ceiling.
The chapter case2400 mAh, 99% yearly retention, 15% reserve, and a ten-minute report.
What it buys youA requirement that asks for measurable evidence.

A field team faces an unresolved physical question: What current budget hides inside a ten-year requirement? They must answer it before changing active window 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 active window. The middle card applies this page's relationship. The green card is usable capacity. 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.

Active window changes usable capacity An input card leads through the page relationship to the usable capacity result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. A short 45 mA state is thousands of times larger than sleep current, so tenths of a second materially change a microamp-scale lifetime budget.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for active window is 300.

  2. 2

    Name the relationship. Cusable=2400x0.99^10x0.85=1850 mAh Ibudget=1850/87,600=21.1 uA Iavg(300 ms)=0.3/600x45+599.7/600x0.0015=24.0 uA ΔV=0.045x15=0.675 V Vload=3.6-0.675=2.93 V

  3. 3

    Substitute the chapter fixture. Set active window to 300. The page ledger gives usable capacity as 1845 mAh.

  4. 4

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

Predict, then change active window

Try Predict the direction of usable capacity. Move one control, calculate, then check your prediction.

300
Chapter baseline
Usable capacity

Observe A short 45 mA state is thousands of times larger than sleep current, so tenths of a second materially change a microamp-scale lifetime budget. Reset the control to 300 and compare usable capacity.

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

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

1. Start with the physical story

A ten-year claim allocates a small amount of charge to every hour. Self-discharge and reserve reduce that allocation. Each wake, sample, and radio state must fit inside it, while pulse voltage must remain above the device cutoff.

Bex: Treat years as a budget, not a label on a cell.

2. Name every algebra move

1

Retain chargeMultiply 2400 mAh by 0.99 ten times.

2

Keep reserveMultiply by 0.85.

3

Make an hourly ceilingDivide usable charge by 87,600 hours.

4

Average statesWeight 45 mA active and 0.0015 mA sleep by their time fractions.

5

Estimate serviceDivide usable mAh by average mA.

6

Check pulse sagMultiply 0.045 A by 15 ohms and subtract from 3.6 V.

3. Reproduce the chapter case

Cusable=2400×0.99^10×0.85=1850 mAh
Ibudget=1850/87,600=21.1 µA
Iavg(300 ms)=0.3/600×45+599.7/600×0.0015=24.0 µA
ΔV=0.045×15=0.675 V
Vload=3.6−0.675=2.93 V

The 300 ms example already exceeds the average-current budget; at 600 ms, ideal service falls to about 5.17 years.

4. Try one real input

TryLengthen the active window and predict the average-current overrun and achievable life.

Active window
Usable capacity
Ten-year budget
Average current
Budget overrun
Achievable life
Pulse sag
Loaded voltage

ObserveAt 300 ms the average is about 24.0 microamps, roughly 14% above the ten-year allocation.

ExplainA short 45 mA state is thousands of times larger than sleep current, so tenths of a second materially change a microamp-scale lifetime budget.

Technical boundaries.

This is a constant-state duty-cycle estimate.

Radio
Join, retries, acknowledgements, spreading factor, and temperature change active time.
Cell
Capacity, retention, internal resistance, and cutoff vary over life.
Estate
Fleet TCO also depends on failures, maintenance, network coverage, and replacement policy.

Correct, not complete: meeting average current does not prove ten years of reliable warehouse service.

5. Use the result in the requirement

Replace “ten-year battery” with a measured-state budget, pulse-voltage floor, environmental range, retry case, reserve, and acceptance test.

6. Record the evidence state

Keep chemistry, lot, capacity test, retention, temperature, sleep current, wake trace, sensor time, radio settings, packet result, internal resistance, cutoff, firmware, and confidence bound.

7. Check yourself

Why is usable capacity below 2400 mAh?
Answer: Ten years of self-discharge and the 15% reserve are removed before allocating charge.
Why does 300 ms matter inside a ten-minute period?
Answer: The active current is 30,000 times the 1.5 microamp sleep current.
Does a 21.1 microamp average ceiling guarantee pulse survival?
Answer: No. Loaded voltage must be checked separately during the 45 mA burst.
Honesty boundary.

The calculation uses the chapter's illustrative warehouse interval and catalog-style cell constants.

Radio
Join, retries, acknowledgements, spreading factor, and temperature change active time.
Cell
Capacity, retention, internal resistance, and cutoff vary over life.
Estate
Fleet TCO also depends on failures, maintenance, network coverage, and replacement policy.

Correct, not complete: meeting average current does not prove ten years of reliable warehouse service.

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