Math Bridge: Usable Source and Pulse Sag

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Math BridgeEnergy & PowerStruggle-friendly runway

Where did the missing 30% of usable capacity go?

Multiply explicit retention factors, then keep pulse sag separate from the mAh result.

Battery Bruno, the energy and power guideBattery Bruno guides
The one targetRebuild the chapter's 70% source assumption and test its radio headroom.
The chapter case2400 mAh at 3.6 V, three retention factors, and a 42 mA pulse.
What it buys youA budget that separates stored charge from deliverable voltage.

A field team faces an unresolved physical question: Where did the missing 30% of usable capacity go? They must answer it before changing cell internal resistance (ohm) on the real device. Predict the direction first.

See the relationship before changing it

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.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for cell internal resistance (ohm) is 8.

  2. 2

    Name the relationship. Enameplate=2.4x3.6=8.64 Wh fusable=0.95x0.87x0.85=70.25% Qusable=2400x0.702525=1686.1 mAh Vsag=0.042x8=0.336 V; margin=3.6-0.336-2.4=0.864 V

  3. 3

    Substitute the chapter fixture. Set cell internal resistance (ohm) to 8. The page ledger gives terminal voltage as 3.26 V.

  4. 4

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

Predict, then change cell internal resistance (ohm)

Try Predict the direction of terminal voltage. Move one control, calculate, then check your prediction.

8
Chapter baseline
Terminal voltage

Observe Predict a decrease in terminal voltage as the input increases over this range. Reset to the chapter baseline and compare.

Explain Only cell internal resistance (ohm) moves here. The other chapter fixtures remain fixed.

Source: content/energy-power/energy-aware-introduction.qmd — Phoebe's Field Notes: Where This Chapter's "70% Usable" Actually Comes From. Use the chapter equation and fixed ledger above; only the named input varies.

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 internal resistance (ohm) moves. Field effects named in the page's technical boundary stay fixed.

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

1. Start with the physical story

Nameplate capacity is measured under stated conditions. Shelf loss, cold or pulse-rate effects, and cutoff each remove a fraction of usable charge. Even after that derating, internal resistance can pull terminal voltage below brownout during a short radio burst.

Battery Bruno: Derating answers “how much charge?” Pulse sag answers “at what usable voltage?”

2. Name every algebra move

1

Convert charge to energyMultiply 2.4 Ah by 3.6 V.

2

Stack retentionMultiply 0.95, 0.87, and 0.85.

3

Find usable chargeMultiply 2400 mAh by the combined fraction.

4

Find pulse sagMultiply 0.042 A by internal resistance.

5

Find terminal voltageSubtract sag from 3.6 V.

6

Apply brownoutSubtract the 2.4 V limit.

3. Reproduce the chapter case

Enameplate=2.4×3.6=8.64 Wh
fusable=0.95×0.87×0.85=70.25%
Qusable=2400×0.702525=1686.1 mAh
Vsag=0.042×8=0.336 V; margin=3.6−0.336−2.4=0.864 V

At 40 Ω, sag becomes 1.680 V and terminal voltage falls to 1.920 V, so the same charge budget fails its service voltage.

4. Try one real input

TryRaise internal resistance and predict the brownout crossing.

Nameplate energy
Usable fraction
Usable capacity
Pulse sag
Terminal voltage
Brownout margin

ObserveNameplate energy and usable mAh stay fixed while sag grows. Brownout margin reaches zero near 28.6 Ω.

ExplainChanging resistance does not remove stored charge in this ledger; it changes whether the radio can draw that charge above its minimum voltage.

Technical boundaries.

This is a factor ledger plus a linear pulse model.

Factors
The 5%, 13%, and 15% losses are labelled scenario assumptions.
Resistance
Cell impedance varies with chemistry, age, temperature, and pulse duration.
Regulation
Converter efficiency and dropout are not modelled.

Correct, not complete: qualify the source with measured end-of-life pulses and the real power path.

5. Use the result in the design

Replace each retention factor with evidence, then test the worst radio pulse at low temperature and end-of-life impedance.

6. Record the evidence state

Keep cell part and lot, retention assumptions, capacity test conditions, cutoff, pulse waveform, impedance, terminal minimum, and regulator state.

7. Check yourself

Why multiply the three retention factors?
Answer: Each factor acts on the charge remaining after the previous factor.
Does 1686 mAh prove the radio will work?
Answer: No. The pulse must also stay above brownout.
What does a negative margin mean?
Answer: The linear model predicts terminal voltage below the service limit.
Honesty boundary.

The arithmetic reconstructs the chapter's approximate 70% case with explicit factors.

Factors
The 5%, 13%, and 15% losses are labelled scenario assumptions.
Resistance
Cell impedance varies with chemistry, age, temperature, and pulse duration.
Regulation
Converter efficiency and dropout are not modelled.

Correct, not complete: qualify the source with measured end-of-life pulses and the real power path.

Return to Energy-Aware Design with the maths unpacked →