Math Bridge: Coin-Cell Energy and Pulse Sag

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Math BridgeApplicationsStruggle-friendly runway

Can 225 mAh still start the radio after a year?

Separate stored charge from usable energy and pulse voltage.

Battery Bruno, the energy guideBattery Bruno guides
The one targetTurn a coin-cell label into energy, retention, and burst-voltage checks.
The chapter case225 mAh, 3.0 V, 12 mA BLE, 20 Ω fresh and 150 Ω aged.
What it buys youA battery-life claim that also checks whether the radio can start.

A field team faces an unresolved physical question: Can 225 mAh still start the radio after a year? They must answer it before changing radio 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 radio 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.

Radio 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. The same stored charge can support a low-current sleep state yet fail a short radio pulse because internal resistance turns current into voltage loss.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for radio current is 12.

  2. 2

    Name the relationship. E=(225/1000)(3.0)=0.675 Wh Vsag,fresh=(0.012)(20)=0.240 V; Vterm=2.760 V Vsag,aged=(0.012)(150)=1.800 V; Vterm=1.200 V Qyear=225(1-0.01)=222.75 mAh

  3. 3

    Substitute the chapter fixture. Set radio current to 12. The page ledger gives nameplate energy as 0.675 Wh.

  4. 4

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

Predict, then change radio current

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

12
Chapter baseline
Nameplate energy

Observe The same stored charge can support a low-current sleep state yet fail a short radio pulse because internal resistance turns current into voltage loss. Reset the control to 12 and compare nameplate energy.

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

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

1. Start with the physical story

Milliamp-hours count charge. A radio spends energy and needs terminal voltage while its current pulse is flowing. Internal resistance can therefore stop a pulse even when some charge remains.

Battery Bruno: Capacity says how much charge is stored; resistance says how hard it is to take that charge quickly.

2. Name every algebra move

1

Convert chargeDivide 225 mAh by 1,000 to get 0.225 Ah.

2

Find energyMultiply amp-hours by 3.0 V.

3

Convert currentDivide radio milliamps by 1,000 before using ohms.

4

Find sagMultiply current by internal resistance.

5

Find terminal voltageSubtract sag from open-circuit voltage.

6

Apply retentionMultiply charge by 0.99 for one year of 1% self-discharge.

3. Reproduce the chapter case

E=(225/1000)(3.0)=0.675 Wh
Vsag,fresh=(0.012)(20)=0.240 V; Vterm=2.760 V
Vsag,aged=(0.012)(150)=1.800 V; Vterm=1.200 V
Qyear=225(1−0.01)=222.75 mAh

The energy label looks healthy, but the aged-cell pulse falls below the example radio's 2.0 V operating floor.

4. Try one real input

TryMove radio current and predict fresh and aged terminal voltage.

Radio current
Nameplate energy
Fresh sag
Fresh terminal
Aged sag
Aged terminal
Charge after one year
Charge retained
Ideal continuous runtime
Aged cell clears 2.0 V

ObserveNameplate energy and one-year retention do not change with burst current, but sag grows directly with current and aged terminal voltage collapses first.

ExplainThe same stored charge can support a low-current sleep state yet fail a short radio pulse because internal resistance turns current into voltage loss.

Technical boundaries.

This is a constant-resistance pulse ledger, not a full electrochemical model.

Cell
Capacity and resistance vary with brand, pulse duration, temperature, state of charge, age, and recovery time.
Load
Regulator efficiency, brownout behavior, packet retries, receive current, sensing, and sleep current also spend the budget.
Life claim
Usable capacity needs measured duty cycle, derating, confidence bounds, and field validation.

Correct, not complete: this ledger does not prove one-year service or certify a CR2032 for a chosen radio.

5. Use the result in the design

Measure burst current and terminal voltage on fresh, cold, and aged cells. Set a brownout margin and retain the waveform, not only an average current.

6. Record the evidence state

Keep cell part and lot, age, temperature, state of charge, measured resistance, burst profile, regulator, brownout floor, packet outcome, sleep current, duty cycle, and derating.

7. Check yourself

Why is 225 mAh not already watt-hours?
Answer: Milliamp-hours count charge. Multiplying 0.225 Ah by voltage gives energy in watt-hours.
Why can an aged cell fail with charge remaining?
Answer: Higher internal resistance creates a larger I×R drop during the radio pulse.
Does 2.76 V guarantee a successful packet?
Answer: No. The radio floor, regulator, temperature, pulse duration, retries, antenna, and link conditions still matter.
Honesty boundary.

The arithmetic reproduces the chapter's 225 mAh, 3.0 V, 12 mA CR2032 example.

Cell
Capacity and resistance vary with brand, pulse duration, temperature, state of charge, age, and recovery time.
Load
Regulator efficiency, brownout behavior, packet retries, receive current, sensing, and sleep current also spend the budget.
Life claim
Usable capacity needs measured duty cycle, derating, confidence bounds, and field validation.

Correct, not complete: this ledger does not prove one-year service or certify a CR2032 for a chosen radio.

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