Math Bridge: Battery test energy and sag

← Back to End-to-End Test Strategy
Math BridgeDesign MethodologyStruggle-friendly runway

Why can a battery test pass and the field unit still run short?

Carry nameplate charge through voltage, pulse sag, derating, average current, and installed-antenna power.

Blueprint Bina, the design guideBlueprint Bina guides
The one targetTurn a power-row claim into a reproducible ledger.
The chapter case3.6 V, 2400 mAh, 3 Ω, 80% usable, 80 µA.
What it buys youA test that uses the real load and antenna.

A field team faces an unresolved physical question: Why can a battery test pass and the field unit still run short? They must answer it before changing average current 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 average current. The middle card applies this page's relationship. The green card is nameplate energy. 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.

Average current changes nameplate energy An input card leads through the page relationship to the nameplate energy result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. A long-run budget and a pulse-voltage check can fail independently; both must use the installed radio path.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for average current is 80.

  2. 2

    Name the relationship. Enameplate=2.400x3.6=8.64 Wh ΔV=0.120x3=0.360 V; Vterm=3.24 V Qusable=0.8x2400=1920 mAh; Eusable=6.91 Wh Life=1920/0.080=24,000 h=2.74 years Pt=20-6=14 dBm; RF-stage ratio=3.98x

  3. 3

    Substitute the chapter fixture. Set average current to 80. The page ledger gives nameplate energy as 8.64 Wh.

  4. 4

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

Predict, then change average current

Try Predict the direction of nameplate energy. Move one control, calculate, then check your prediction.

80
Chapter baseline
Nameplate energy

Observe A long-run budget and a pulse-voltage check can fail independently; both must use the installed radio path. Reset the control to 80 and compare nameplate energy.

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

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

1. Separate charge from energy

Milliamp-hours count charge. Watt-hours include the voltage that pushes that charge. A current pulse then changes terminal voltage through the cell's internal resistance.

Blueprint Bina: Keep nameplate, pulse, and long-run budgets on separate rows.

2. Name every algebra move

1

Attach voltageE(Wh)=Q(Ah)×V.

2

Subtract pulse sagVterm=Voc−IRint.

3

Apply the derateQusable=fderateQnominal.

4

Divide by currentLife=Qusable/Iaverage.

5

Account for antenna gainPt=EIRP−G.

3. Reproduce the test-rig case

Enameplate=2.400×3.6=8.64 Wh
ΔV=0.120×3=0.360 V; Vterm=3.24 V
Qusable=0.8×2400=1920 mAh; Eusable=6.91 Wh
Life=1920/0.080=24,000 h=2.74 years
Pt=20−6=14 dBm; RF-stage ratio=3.98×

The service-life result belongs to the current trace and installed antenna used by the test, not merely to the cell label.

4. Try the sleep-average current

TryMove the measured average current while the cell and pulse case stay fixed.

Average current
Nameplate energy
Pulse sag
Terminal voltage
Usable charge
Usable energy
Life
Life
Installed RF power
RF-stage ratio

ObserveLife changes inversely with average current, while the separate 120 mA pulse still sets sag.

ExplainA long-run budget and a pulse-voltage check can fail independently; both must use the installed radio path.

Technical boundaries.

This is a bounded battery test ledger, not a cell-discharge model.

Derate
80% is a stated design allowance, not a guaranteed capacity curve
Resistance
3 Ω changes with temperature, age, state of charge, and pulse duration
RF power
Conducted-power ratio is not the same as total radio-current ratio

Use the intended cell, antenna, enclosure, temperature, firmware, and current trace in the release test.

5. Recreate field stress

Run cold, aged, weak-link, retry, and update cases. Verify minimum terminal voltage and average current from the same firmware and antenna configuration.

6. Preserve the test state

Store cell lot, temperature, resistance method, firmware, current trace, RF configuration, derate, cutoff voltage, and the requirement ID.

7. Check yourself

Why is 2400 mAh not yet an energy figure?
Answer: Energy also needs voltage.
Why can 6.91 Wh still fail a transmit pulse?
Answer: Total energy does not show the instantaneous 0.360 V sag.
Does 6 dBi guarantee 3.98× lower battery current?
Answer: No. It gives an RF output-power ratio, not the radio's full electrical-current curve.
Honesty boundary.

The arithmetic reproduces the chapter's catalog-typical test-rig values.

2.74 years
Charge-only result at 80 µA after an explicit 80% derate
3.24 V
One pulse estimate using a fixed 3 Ω resistance
3.98×
RF-stage output ratio, not a service-life multiplier

Correct, not complete: this ledger does not qualify field battery life.

Return to End-to-End Test Strategy with the maths unpacked →