Math Bridge: Smartwatch Charge Ledger

A technician must decide whether total daily charge is safe before changing always-on display time on the real device. The result is unresolved until the rule and units are checked. Predict the direction first.

See the relationship before changing it

The figure reads from left to right. The blue card is always-on display time. The middle card applies this page's rule. The green card is total daily 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 model keeps those stated values fixed and changes only always-on display time, so the numeric fixture does not switch without explanation.

Walk the arrows. Longer dim-display time adds charge every day.

Derive the baseline in four named moves

1
Name the input. The chapter baseline is 21 h/day.
2
Name the relationship. charge = 115.36 mAh/day + 3 mA x display hours
3
Substitute with units. 115.36 + 3 x 21 = 178.36 mAh/day
4
Read the result. Keep the unit beside the value. Use it only inside the technical boundary on this page.

Predict, then change always-on display time

Try Predict the direction of charge = 115.36 mAh/day + 3 mA x display hours. Test another always-on display time, then compare total daily charge.

Observe Longer dim-display time adds charge every day. Reset always-on display time to 21 and compare total daily charge.

Explain Longer dim-display time adds charge every day.

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 always-on display time moves here. Field effects named in the technical boundary stay fixed.

1. Start with the physical story

Current is charge flow. Multiplying each roughly constant current by its active hours gives mAh; adding states gives daily charge. A modest always-on current can dominate because it runs for most of the day.

Battery Bruno: A small current paid for many hours can beat a large current paid briefly.

2. Name every algebra move

Price each stateMultiply milliamps by hours.

Add the daySum MCU active and sleep, display, PPG, and BLE charge.

Add always-onMultiply 3 mA by its enabled hours.

Find runtimeDivide 300 mAh by mAh/day.

Add GPSPrice 25 mA for one hour.

Check the railAdd simultaneous currents and multiply by 0.2 ohm.

3. Reproduce the chapter case

QMCU=5×6+0.02×18=30.36 mAh/day
Qbase=30.36+15×3+1.5×24+8×0.5=115.36 mAh/day
Tbase=300/115.36=2.60 days
QAOD=3×21=63; T=300/178.36=1.68 days
QGPS=25×1; T=300/203.36=1.48 days
ΔV=(54.5 mA)(0.2 Ω)=10.9 mV

The state ledger reproduces the chapter totals; the small Li-poly sag means this example is charge-limited rather than pulse-delivery-limited.

4. Try one real input

TryChange always-on display hours and predict daily charge and runtime.

ObserveTwenty-one hours at 3 mA adds 63 mAh/day—more than the active display's 45 mAh/day.

ExplainCharge depends on both current and duration. Always-on features repeatedly win the duration term.

Technical boundaries.

This is a fixed-state nameplate-charge ledger.

Loads: Real current varies with brightness, sample rate, radio traffic, processing, temperature, and firmware.
Overlap: State durations and simultaneous peaks must come from a measured trace, not independent table rows.
Battery: Usable capacity, conversion loss, cutoff, ageing, temperature, self-discharge, and resistance vary.

Correct, not complete: nameplate mAh division does not prove wearable service life.

5. Use the result in the design

Measure a complete representative current trace, assign energy to features, then optimise display duty, PPG schedule, radio batching, processing, and GPS against the user outcome.

6. Record the evidence state

Keep firmware, brightness, display schedule, sensor rate, BLE interval and traffic, GPS conditions, CPU state, rail efficiency, current trace, cell voltage, temperature, age, cutoff, and user workflow.

7. Check yourself

Why is mA times hours a charge unit?

Answer: Current is charge per time, so multiplying by time leaves charge.

Why can 3 mA dominate 15 mA?

Answer: The 3 mA state runs 21 hours, while the 15 mA state runs only three.

Does 10.9 mV sag prove the rail is safe?

Answer: No. It uses one assumed resistance and load combination; conversion, ageing, temperature, and timing still matter.

Honesty boundary.

The arithmetic reproduces the chapter's illustrative 300 mAh watch table.

Loads: Real current varies with brightness, sample rate, radio traffic, processing, temperature, and firmware.
Overlap: State durations and simultaneous peaks must come from a measured trace, not independent table rows.
Battery: Usable capacity, conversion loss, cutoff, ageing, temperature, self-discharge, and resistance vary.

Correct, not complete: nameplate mAh division does not prove wearable service life.