Math Bridge: Software Energy and Battery Aging

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Math BridgeEnergy & PowerStruggle-friendly runway

Why is an 8,000-day current budget not an 8,000-day battery?

Separate charge, energy, calendar loss, and pulse voltage before calling a software saving a field lifetime.

Battery Bruno, the energy and power guideBattery Bruno guides
The one targetTranslate a tiny average current into an honest long-life ceiling.
The chapter caseA 2 Ah, 3.7 V pack, 2% monthly self-discharge, and an 8 mA event.
What it buys youA software comparison that names calendar aging and correct pulse arithmetic.

A field team faces an unresolved physical question: Why is an 8,000-day current budget not an 8,000-day battery? They must answer it before changing nameplate energy 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 nameplate energy. The middle card applies this page's relationship. The green card is after five years. 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.

Nameplate energy changes after five years An input card leads through the page relationship to the after five years result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. The 8,000-day figure is a load-only ceiling. Calendar chemistry can end the budget sooner even when software makes average current extremely small.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for nameplate energy is 2.

  2. 2

    Name the relationship. E=2.000x3.7=7.40 Wh Five years: 0.98^60=29.8% remains Ten years: 0.98^120=8.85% remains Fresh sag: 0.008x0.15=1.20 mV Aged sag: 0.008x1.0=8.00 mV

  3. 3

    Substitute the chapter fixture. Set nameplate energy to 2. The page ledger gives after five years as 29.8%.

  4. 4

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

Predict, then change nameplate energy

Try Predict the direction of after five years. Move one control, calculate, then check your prediction.

2
Chapter baseline
After five years

Observe The 8,000-day figure is a load-only ceiling. Calendar chemistry can end the budget sooner even when software makes average current extremely small. Reset the control to 2 and compare after five years.

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

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

1. Start with the physical story

Ampere-hours measure charge. Watt-hours measure energy, so voltage must be included. A perfect sleeping workload can still lose stored charge through chemistry each month. During a short event, internal resistance also lowers terminal voltage. These are different limits and need different equations.

Battery Bruno: Keep the units beside every number. They show whether a result is charge, energy, voltage, time, or a ratio.

2. Name every algebra move

1

Convert charge to energyEWh=QAh×V.

2

Turn a percent into retentionMonthly retention is 1−r.

3

Compound through timeAfter n months, remaining fraction=(1−r)^n.

4

Convert the ideal ceilingDivide 8,000 days by 365 days per year.

5

Check pulse sagΔV=I×Rint.

6

Check arithmetic units0.008 A×0.15 Ω=0.0012 V=1.20 mV.

3. Reproduce the chapter case

E=2.000×3.7=7.40 Wh
Five years: 0.98^60=29.8% remains
Ten years: 0.98^120=8.85% remains
Fresh sag: 0.008×0.15=1.20 mV
Aged sag: 0.008×1.0=8.00 mV

The legacy box multiplied both pulse-sag results by fifteen. The corrected values are 1.20 mV and 8.00 mV, not 18 mV and 120 mV. The correction strengthens the main lesson: for this small handler current, calendar loss dominates long before pulse sag does.

4. Try one real input

TryMove the control and predict which outputs should change before reading them.

Nameplate energy
After five years
After ten years
Ideal draw ceiling
Fresh pulse sag
Aged pulse sag
One event energy

ObserveA small change in monthly self-discharge compounds into a large calendar-life change. Pulse sag and one-event energy stay fixed because their electrical inputs did not move.

ExplainThe 8,000-day figure is a load-only ceiling. Calendar chemistry can end the budget sooner even when software makes average current extremely small.

Technical boundaries.

This is a small formula ledger, not a complete source qualification.

Self-discharge rate
Real retention depends on chemistry, temperature, age, and state of charge.
Sag model
One resistance omits transient electrochemistry and recovery.
Software scope
The handler is only one contributor to system energy.

Correct, not complete: this bridge corrects the pulse arithmetic and bounds the ideal software ceiling; it does not predict a particular pack.

5. Use the result in the design

Keep the polling-versus-event comparison, but add cell retention, end-of-life resistance, cutoff voltage, sensor current, radio current, and regulator loss before promising years.

6. Record the evidence state

Keep capacity basis, nominal voltage, chemistry, storage temperature, self-discharge assumption, event waveform, internal resistance, cutoff, and the measured whole-device average.

7. Check yourself

Why multiply Ah by volts?
Answer: That converts charge capacity into energy capacity in Wh.
What was wrong in the legacy sag line?
Answer: 8 mA×0.15 Ω is 1.20 mV, not 18 mV; 8 mA×1 Ω is 8 mV, not 120 mV.
Does 29.8% after five years predict every Li-ion cell?
Answer: No. It is the result of the stated illustrative 2% monthly rate.
Honesty boundary.

The arithmetic uses the chapter's named or clearly labelled catalog-typical inputs.

Self-discharge rate
Real retention depends on chemistry, temperature, age, and state of charge.
Sag model
One resistance omits transient electrochemistry and recovery.
Software scope
The handler is only one contributor to system energy.

Correct, not complete: this bridge corrects the pulse arithmetic and bounds the ideal software ceiling; it does not predict a particular pack.

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