Math Bridge: Self-Discharge, Sag, and Cutoff

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Math BridgeEnergy & PowerStruggle-friendly runway

How can a battery hold charge yet fail during one radio pulse?

Use one equation for slow shelf loss and another for instantaneous voltage under load.

Battery Bruno, the energy and power guideBattery Bruno guides
The one targetKeep capacity derating and pulse capability as separate review questions.
The chapter case1% annual shelf loss, a 35-point derating gap, and a 15 mA pulse.
What it buys youA source decision that checks both remaining charge and voltage above cutoff.

A field team faces an unresolved physical question: How can a battery hold charge yet fail during one radio pulse? They must answer it before changing internal resistance 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 internal resistance. The middle card applies this page's relationship. The green card is charge retained. 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.

Internal resistance changes charge retained An input card leads through the page relationship to the charge retained result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. Capacity and pulse capability can fail independently. A single usable-capacity percentage is useful for planning, but it must not hide the pulse-voltage check.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for internal resistance is 25.

  2. 2

    Name the relationship. Retention after two years=0.99²=98.01% Shelf loss=1.99%≈2.00 points Share of 35-point gap=1.99/35=5.69% Fresh pulse: 3.0-0.015x25=2.625 V End-of-life linear check: 2.6-0.015x200=-0.400 V

  3. 3

    Substitute the chapter fixture. Set internal resistance to 25. The page ledger gives charge retained as 98.01%.

  4. 4

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

Predict, then change internal resistance

Try Predict the direction of charge retained. Move one control, calculate, then check your prediction.

25
Chapter baseline
Charge retained

Observe Capacity and pulse capability can fail independently. A single usable-capacity percentage is useful for planning, but it must not hide the pulse-voltage check. Reset the control to 25 and compare charge retained.

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

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

1. Start with the physical story

Self-discharge removes stored charge slowly even when the node is off. Internal resistance causes a voltage drop only while current flows. A cell can therefore show plenty of remaining charge and still fall below the device cutoff during a radio burst.

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

Compound shelf retentionQ(t)=Q0(1−r)^t.

2

Find shelf lossSubtract retained percent from 100%.

3

Compare with the derating gapDivide shelf-loss points by the 35-point gap.

4

Find pulse dropΔV=I×Rint.

5

Find terminal voltageVterm=Voc−ΔV.

6

Test the cutoffMargin=Vterm−Vcutoff.

3. Reproduce the chapter case

Retention after two years=0.99²=98.01%
Shelf loss=1.99%≈2.00 points
Share of 35-point gap=1.99/35=5.69%
Fresh pulse: 3.0−0.015×25=2.625 V
End-of-life linear check: 2.6−0.015×200=−0.400 V

A negative linear-model voltage is not a physical cell output. It is a warning that the requested pulse is outside this simple model and the cell will current-limit or collapse before reaching it.

4. Try one real input

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

Internal resistance
Charge retained
Shelf loss
Share of derating gap
Fresh terminal voltage
Cutoff margin
End-of-life linear result

ObserveRaising internal resistance lowers pulse terminal voltage and cutoff margin. Retained charge does not move because shelf loss is a different mechanism.

ExplainCapacity and pulse capability can fail independently. A single usable-capacity percentage is useful for planning, but it must not hide the pulse-voltage check.

Technical boundaries.

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

Shelf rate
One percent per year is an illustrative primary-cell value.
Thevenin model
A single resistance omits nonlinear polarisation and current limiting.
Cutoff
The actual regulator and load waveform decide usable pulse margin.

Correct, not complete: the ledger separates two failure mechanisms; it does not qualify a cell without end-of-life pulse evidence.

5. Use the result in the design

Record self-discharge separately from cutoff, temperature, aging, and pulse resistance. Test the real current waveform at the lowest expected open-circuit voltage.

6. Record the evidence state

Keep chemistry, age, temperature, storage time, open-circuit voltage, internal resistance method, pulse current and duration, regulator cutoff, and recovery voltage.

7. Check yourself

Why does retained charge stay fixed when resistance moves?
Answer: Self-discharge and pulse resistance are separate inputs in this model.
What does the negative end-of-life result mean?
Answer: The linear model has left its valid range; the requested pulse cannot be sustained.
Does 2% shelf loss explain a 35% derating?
Answer: No. It explains only about 5.7% of that gap; other losses need separate evidence.
Honesty boundary.

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

Shelf rate
One percent per year is an illustrative primary-cell value.
Thevenin model
A single resistance omits nonlinear polarisation and current limiting.
Cutoff
The actual regulator and load waveform decide usable pulse margin.

Correct, not complete: the ledger separates two failure mechanisms; it does not qualify a cell without end-of-life pulse evidence.

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