Math Bridge: Self-Discharge Sensitivity

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Math BridgeEnergy & PowerStruggle-friendly runway

When does one missing input halve a battery-life estimate?

Add self-discharge as an equivalent current, then see how it changes both lifetime and the value of saving one more microamp.

Battery Bruno, the energy and power guideBattery Bruno guides
The one targetMake self-discharge a first-class sensitivity input.
The chapter case1000 mAh, 26.9 µA device current, and 2% self-discharge/month.
What it buys youA lifetime result whose largest hidden assumption can be swept and challenged.

See the relationship before changing it

The figure reads from left to right. The blue card is cell self-discharge rate. The middle card applies this page's rule. The green card is self-discharge leak current. 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 model keeps those stated values fixed and changes only cell self-discharge rate, so the numeric fixture does not switch without explanation.

Cell self-discharge rate changes self-discharge leak current An input card leads through the rule leak current = 1,000 mAh x rate / 720 h to the self-discharge leak current result. INPUT PAGE INPUT APPLY THE RULE predict calculate check units OUTPUT RESULT
Walk the arrows. A higher cell leak raises a current floor firmware cannot sleep away.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 2 %/month.

  2. 2

    Name the relationship. leak current = 1,000 mAh x rate / 720 h

  3. 3

    Substitute with units. 1,000 x 0.02 / 720 = 27.8 uA

  4. 4

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

Predict, then change cell self-discharge rate

Try Predict the direction of leak current = 1,000 mAh x rate / 720 h. Test another cell self-discharge rate, then compare self-discharge leak current.

2 %/month
Chapter baseline
Self-discharge leak current

Observe A higher cell leak raises a current floor firmware cannot sleep away. Reset cell self-discharge rate to 2 and compare self-discharge leak current.

Explain A higher cell leak raises a current floor firmware cannot sleep away.

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 cell self-discharge rate moves here. Field effects named in the technical boundary stay fixed.

Honesty note. This teaching model changes only cell self-discharge rate. It is not a field prediction.

1. Start with the physical story

The device is not the only path that empties a cell. Chemistry also consumes stored charge. A calculator that divides capacity by device current alone silently assigns zero to that path, so its answer and sensitivity can both be badly wrong.

Battery Bruno: Put every drain on one average-current ledger before comparing optimizations.

2. Name every algebra move

1

Find monthly lossMultiply the self-discharge fraction by 1000 mAh.

2

Convert to currentDivide monthly mAh by 720 hours.

3

Add drainsAdd equivalent self-discharge current to 26.9 µA.

4

Find corrected lifeDivide capacity by total current.

5

Compare headlinesMeasure the percentage loss against device-only life.

6

Measure sensitivityThe hours gained per saved microamp follow C/I².

3. Reproduce the chapter case

Isd=(0.02×1000 mAh)/720 h=27.8 µA
Itotal=26.9+27.8=54.7 µA
Lifecorrected=1000 mAh/0.0547 mA=2.09 years
|∂Life/∂I|=1000×1000/54.7²=334 h per saved µA

Without shelf loss, the same model says 4.24 years and about 1,382 hours per saved microamp. The missing input changes the optimization ranking, not just the last decimal.

4. Try one real input

TrySweep monthly self-discharge and predict when it overtakes device current.

Equivalent self-discharge
Total average current
Device-only life
Corrected life
Life reduction
Hours per saved µA

ObserveThe device-only headline stays fixed while corrected life and marginal value fall as the omitted drain rises.

ExplainLifetime is inversely proportional to total current. Its slope is even more sensitive: doubling total current cuts the value of another one-microamp saving by four.

Technical boundaries.

This is a constant equivalent-current sensitivity model.

Rate
Real self-discharge depends on chemistry, temperature, age, and state of charge.
Capacity
Nameplate mAh still needs derating and cutoff checks.
Time model
A fixed monthly percentage actually compounds; this input is converted to a comparison current.

Correct, not complete: use measured cell retention and pulse evidence for release.

5. Use the result in the design

Sweep the largest uncertain drains first. If self-discharge dominates, changing chemistry or storage conditions may beat another firmware sleep optimization.

6. Record the evidence state

Keep chemistry, capacity test, storage temperature, observed retention, device current trace, cutoff, lifetime result, and sensitivity ranking.

7. Check yourself

When does self-discharge exceed the device load here?
Answer: At about 1.94% per month, its equivalent current exceeds 26.9 µA.
Why does the naive result stay fixed?
Answer: It deliberately divides by device current only.
Why does sensitivity use I squared?
Answer: Differentiating C/I gives a magnitude of C/I².
Honesty boundary.

The arithmetic exposes the chapter's omitted self-discharge input rather than claiming one universal rate.

Rate
Real self-discharge depends on chemistry, temperature, age, and state of charge.
Capacity
Nameplate mAh still needs derating and cutoff checks.
Time model
A fixed monthly percentage actually compounds; this input is converted to a comparison current.

Correct, not complete: use measured cell retention and pulse evidence for release.

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