Math Bridge: Mixed-Rail Duty-Cycle Power

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Math BridgeElectronicsStruggle-friendly runway

How does heater duty cycle become battery runtime?

Turn currents on two rails into one energy ledger.

Eddie, the electronics guideEddie guides
The one targetCompare mixed-voltage loads with power, not raw current.
The chapter case92.4 mA at 3.3 V, a 150 mA heater at 5 V, 300 s cycles, and 7.40 Wh.
What it buys youAn honest first-pass runtime and a visible heater-duty trade-off.

A field team faces an unresolved physical question: How does heater duty cycle become battery runtime? They must answer it before changing heater on-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 heater on-time. The middle card applies this page's relationship. The green card is heater duty. 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.

Heater on-time changes heater duty An input card leads through the page relationship to the heater duty result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. Duty fraction scales the heater row before power rows are added. Battery energy is fixed, so more average watts consume it faster.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for heater on-time is 30.

  2. 2

    Name the relationship. duty=30/300=10.0% Iheater,avg=150 mAx0.10=15.0 mA Pbase=(3.3 V)(92.4 mA)=305 mW Pheater=(5 V)(15.0 mA)=75.0 mW t=7.40 Wh/0.380 W=19.5 h

  3. 3

    Substitute the chapter fixture. Set heater on-time to 30. The page ledger gives heater duty as 10.0%.

  4. 4

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

Predict, then change heater on-time

Try Predict the direction of heater duty. Move one control, calculate, then check your prediction.

30
Chapter baseline
Heater duty

Observe Duty fraction scales the heater row before power rows are added. Battery energy is fixed, so more average watts consume it faster. Reset the control to 30 and compare heater duty.

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

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

1. Start with the physical story

A milliamp on 5 V consumes more power than a milliamp on 3.3 V. Add each rail as voltage times time-weighted current, then compare total average power with battery energy. Duty-cycling the heater changes only its time share.

Eddie: Current is charge flow. A mixed-rail battery budget needs the energy rate, VI, before the rows can be added.

2. Name every algebra move

1

Find duty fractionDivide heater-on seconds by the 300 s cycle.

2

Average heater currentMultiply 150 mA by that fraction.

3

Convert each rail to powerUse P=VI for the 3.3 V base and 5 V heater.

4

Find stored energy2.000 Ah×3.7 V=7.40 Wh.

5

Estimate runtimeDivide watt-hours by average watts.

3. Reproduce the chapter case

duty=30/300=10.0%
Iheater,avg=150 mA×0.10=15.0 mA
Pbase=(3.3 V)(92.4 mA)=305 mW
Pheater=(5 V)(15.0 mA)=75.0 mW
t=7.40 Wh/0.380 W=19.5 h

Keeping the non-heater loads continuously active makes the ideal duty-cycled result close to the chapter's 19.5-hour estimate.

4. Try one real input

TryMove heater-on time and predict total power and ideal runtime.

Heater on-time
Heater duty
Average heater current
3.3 V rail current
Base power
Heater power
Total power
Battery energy
Ideal runtime

ObserveHeater current and power rise linearly with on-time; total power rises and ideal runtime falls nonlinearly because the fixed base load remains.

ExplainDuty fraction scales the heater row before power rows are added. Battery energy is fixed, so more average watts consume it faster.

Technical boundaries.

This is an ideal energy budget with fixed rail currents and lossless voltage conversion.

Battery
Usable energy changes with load, temperature, ageing, cutoff voltage, and cell chemistry.
Converters
Regulator efficiency and quiescent current must be included per rail.
Loads
Boot, radio retries, sensor warm-up, heater regulation, and sleep states vary.

Correct, not complete: this ledger does not predict field battery life or qualify the heater cycle.

5. Use the result in the design

Measure every rail in each state, multiply by real state durations and voltages, include conversion loss and reserve, then test the heater duty needed for valid sensing.

6. Record the evidence state

Record rail voltages/currents, state timing, regulator efficiency, heater warm-up requirement, battery chemistry/capacity/cutoff, temperature, radio retries, and measured cycle energy.

7. Check yourself

Why not add the 3.3 V and 5 V currents directly?
Answer: Equal currents at different voltages consume different power, so convert each rail with P=VI first.
Why does 30 seconds mean 15 mA average heater current?
Answer: Thirty of 300 seconds is 10%, and 10% of 150 mA is 15 mA.
Does 19.5 hours predict field runtime?
Answer: No. It omits conversion loss, battery derating, varying states, retries, warm-up needs, and reserve.
Honesty boundary.

The arithmetic reproduces the chapter's 92.4 mA base rail, 150 mA heater, 300 s cycle, and 7.40 Wh battery case.

Battery
Usable energy changes with load, temperature, ageing, cutoff voltage, and cell chemistry.
Converters
Regulator efficiency and quiescent current must be included per rail.
Loads
Boot, radio retries, sensor warm-up, heater regulation, and sleep states vary.

Correct, not complete: this ledger does not predict field battery life or qualify the heater cycle.

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