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 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 usable fraction. 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 usable fraction An input card leads through the page relationship to the usable fraction result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. Changing resistance does not remove stored charge in this ledger; it changes whether the radio can draw that charge above its minimum voltage.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for nameplate energy 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 nameplate energy to 8. The page ledger gives usable fraction as 70.3%.

  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 usable fraction. Move one control, calculate, then check your prediction.

8
Chapter baseline
Usable fraction

Observe Changing resistance does not remove stored charge in this ledger; it changes whether the radio can draw that charge above its minimum voltage. Reset the control to 8 and compare usable fraction.

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

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.

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