Math Bridge: Hourly battery ledger

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How does a ten-second radio burst become a battery-life result?

Turn device states into one explicit charge ledger.

Data Dora, the guideData Dora guides
The one targetCompute lifetime from timed current states.
The chapter case30 s active; 10 s transmit; 2,500 mAh cell.
What it buys youA reviewable power claim.

A field team faces an unresolved physical question: How does a ten-second radio burst become a battery-life result? They must answer it before changing transmit time 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 transmit time. The middle card applies this page's relationship. The green card is active 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.

Transmit time changes active charge An input card leads through the page relationship to the active charge result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. Usable lifetime is a measured charge ledger, not the printed cell capacity divided by a guessed current.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for transmit time is 10.

  2. 2

    Name the relationship. 25(30)/3600 = 0.208 mAh; 120(10)/3600 = 0.333 mAh 0.01(3,560)/3600 = 0.00989 mAh; Iavg = 0.552 mA 2,500/0.552 = 4,530 h = 189 days

  3. 3

    Substitute the chapter fixture. Set transmit time to 10. The page ledger gives active charge as 0.208.

  4. 4

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

Predict, then change transmit time

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

10
Chapter baseline
Active charge

Observe Usable lifetime is a measured charge ledger, not the printed cell capacity divided by a guessed current. Reset the control to 10 and compare active charge.

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

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

1. Count charge, not labels

Current is charge flow. Multiply each state's current by its time, sum the hourly charge, and divide usable capacity by that average current.

Data Dora: “Sleep most of the time” is not a budget until every state has a current and duration.

2. Name every algebra move

1

Convert seconds to hoursQi=Iiti/3600.

2

Add statesIavg=ΣQi for a one-hour ledger.

3

Derate capacityCusable=C(1−d).

4

Dividetlife=Cusable/Iavg.

5

Subtract sagVload=Voc−IR.

3. Reproduce the chapter

25(30)/3600 = 0.208 mAh; 120(10)/3600 = 0.333 mAh
0.01(3,560)/3600 = 0.00989 mAh; Iavg = 0.552 mA
2,500/0.552 = 4,530 h = 189 days

The chapter's 4,545-hour result uses its rounded 0.55 mA display; the unrounded ledger gives 4,530 hours.

4. Try the radio time

TryShorten the burst while keeping the 30-second sensing state.

Transmit time
Active charge
Transmit charge
Sleep time
Sleep charge
Average current
Usable capacity
Lifetime
Lifetime days
Lifetime years
Loaded voltage

ObserveThe radio dominates this baseline ledger; shortening it moves lifetime directly.

ExplainUsable lifetime is a measured charge ledger, not the printed cell capacity divided by a guessed current.

Technical boundaries.

States are constant-current rectangles.

Cell
No nonlinear discharge curve, ageing, or temperature response
Radio
No startup, retry, or network-search variation
Voltage
One resistance term, not a transient cell model

Profile the complete device across temperature and network conditions.

5. Add real losses

Repeat the ledger with cold derating, self-discharge, converter efficiency, retries, and the measured cutoff voltage.

6. Keep evidence

Version firmware state timings, current traces, cell lot, temperature, network condition, and calculation sheet.

7. Check yourself

Why divide seconds by 3,600?
Answer: It converts each state duration to hours for mAh.
Why do 4,530 and 4,545 hours differ?
Answer: The latter divides by the rounded 0.55 mA display.
Does nominal mAh guarantee field life?
Answer: No; usable capacity depends on load, cutoff, temperature, ageing, and leakage.
Honesty boundary.

The baseline timings and currents come from the chapter; sag terms are stated checks.

30 s, 10 s, 3,560 s
Chapter states
2,500 mAh
Chapter baseline cell
2 Ω
Interactive teaching assumption

Correct, not complete: a compact ledger does not qualify a battery-powered product.

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