Math Bridge: Coin-cell charge and voltage sag

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Math BridgeUX DesignStruggle-friendly runway

How can a 19-year charge budget still brown out?

Separate the slow milliamp-hour ledger from the radio's fast voltage test.

UX Uma, the guideUX Uma guides
The one targetCalculate both average-current life and burst voltage sag.
The chapter case220 mAh; 80% derating; 8 mA for 3 ms every 15 min.
What it buys youA device review that tests brownout before trusting battery life.

A field team faces an unresolved physical question: How can a 19-year charge budget still brown out? They must answer it before changing cell 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 cell resistance. The middle card applies this page's relationship. The green card is usable charge. 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.

Cell resistance changes usable charge An input card leads through the page relationship to the usable charge result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. That is why installed antenna loss, retries, and peak current can end service long before the mAh estimate.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for cell resistance is 200.

  2. 2

    Name the relationship. Qusable = 0.8x220 = 176 mAh; E = 0.528 Wh d = 0.003/900 = 3.33x10⁻⁶ Iavg = 1.03 uA; charge-only life ≈ 19.6 years at 200 ohm: sag = 0.008x200 = 1.60 V; Vterm = 1.40 V

  3. 3

    Substitute the chapter fixture. Set cell resistance to 200. The page ledger gives usable charge as 176 mAh.

  4. 4

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

Predict, then change cell resistance

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

200
Chapter baseline
Usable charge

Observe That is why installed antenna loss, retries, and peak current can end service long before the mAh estimate. Reset the control to 200 and compare usable charge.

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

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

1. Separate two ways a battery fails

A charge budget asks how long average current can continue. A voltage-sag test asks whether the source can supply one short radio burst now. A worn coin cell can pass the first calculation and fail the second.

UX Uma: A long lifetime number is not proof that the device survives its main action.

2. Name every algebra move

1

Derate chargeQusable=fQnominal.

2

Find duty cycled=Tactive/Tinterval.

3

Weight currentsIavg=Isleep(1−d)+Iactive d.

4

Subtract burst dropVterm=Voc−IRint.

3. Reproduce the beacon ledger

Qusable = 0.8×220 = 176 mAh; E = 0.528 Wh
d = 0.003/900 = 3.33×10⁻⁶
Iavg = 1.03 µA; charge-only life ≈ 19.6 years
at 200 Ω: sag = 0.008×200 = 1.60 V; Vterm = 1.40 V

The two-year average-current budget is 176 mAh ÷ 17,520 h = 10.0 µA, so charge alone looks comfortable.

4. Try the cell resistance

TryAge the cell by increasing internal resistance while leaving its charge ledger unchanged.

Cell resistance
Usable charge
Usable energy
2-year budget
Duty cycle
Average current
Charge-only life
Burst sag
Radio voltage

ObserveLifetime stays fixed because average current is fixed, while burst voltage collapses as resistance rises.

ExplainThat is why installed antenna loss, retries, and peak current can end service long before the mAh estimate.

Technical boundaries.

The cell is represented by a fixed open-circuit voltage and one resistance.

Cell
Real voltage and resistance vary with temperature, age, and pulse history
Radio
Startup, receive windows, retries, and updates add current
Life
Self-discharge and cutoff voltage reduce usable service

Measure the complete installed current trace and minimum radio voltage.

5. Test the worst moment

Repeat the longest radio action at cold temperature, low state of charge, poor link quality, and after aging. Watch supply voltage and reset state, not only an average-current meter.

6. Record the device state

Store cell type, lot, age, temperature, resistance, current trace, antenna installation, retries, brownout threshold, recovery behavior, firmware, and the change condition for replacement.

7. Check yourself

Why does charge predict 19.6 years?
Answer: The 3 ms burst every 900 s keeps average current near 1.03 µA.
Why can the same cell brown out?
Answer: An 8 mA burst through 200 Ω drops 1.60 V immediately.
Does adding mAh fix voltage sag?
Answer: Not by itself. The source must also keep resistance low enough at peak current.
Honesty boundary.

The 15-minute and two-year product claim comes from the chapter; the CR2032 and radio values are stated typical assumptions.

220 mAh at 3.0 V
Coin-cell teaching case
8 mA for 3 ms
BLE burst assumption
10 Ω to 200 Ω
Fresh-to-aged resistance illustration

Correct, not complete: this two-state ledger does not qualify a battery-powered connected product.

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