Math Bridge: Airtime, Battery Sag, and Runtime

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Math BridgeWSNStruggle-friendly runway

Can the battery survive the burst behind the average?

Follow one 160-bit message through airtime, current, internal resistance, derating, and runtime.

Packet Pete, the guidePacket Pete guides
The one targetCheck the weakest supply moment as well as average draw.
The chapter case160 bits, 4 kb/s, 24 mA, 40 ms.
What it buys youA battery claim with sag, derating, and airtime visible.

A field team faces an unresolved physical question: Can the battery survive the burst behind the average? They must answer it before changing radio bit rate in bits per second 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 bit rate in bits per second. The middle card applies this page's relationship. The green card is burst 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.

Radio bit rate in bits per second changes burst charge An input card leads through the page relationship to the burst charge result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. Bit rate changes airtime. Pulse current changes sag. The page keeps them separate so one improvement is not credited for another.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for radio bit rate in bits per second is 4000.

  2. 2

    Name the relationship. ttx=B/R; Qtx=Itx·ttx; Vterm=Voc-ItxRint; trun=Qusable/Iavg

  3. 3

    Substitute the chapter fixture. Set radio bit rate in bits per second to 4000. The page ledger gives burst charge as 960.

  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 radio bit rate in bits per second

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

4000
Chapter baseline
Burst charge

Observe Bit rate changes airtime. Pulse current changes sag. The page keeps them separate so one improvement is not credited for another. Reset the control to 4000 and compare burst charge.

Explain Only radio bit rate in bits per second 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 bit rate in bits per second moves. Field effects named in the page's technical boundary stay fixed.

Honesty note. This teaching model changes only radio bit rate in bits per second. It replays the chapter fixture; it is not a field prediction.

1. Charge is not yet energy

Current multiplied by time counts charge. Energy also needs voltage. A real cell loses some terminal voltage during a pulse because current crosses its internal resistance. The average can look safe while the radio browns out during its largest burst.

Packet Pete: Keep both ledgers: how much charge leaves and how low voltage falls.

2. Name every algebra move

1

Divide bits by bits per secondttx = payload/bit rate.

2

Multiply current by timeThe burst contribution is Itx × ttx.

3

Multiply current by resistanceVoltage sag is ΔV = IR.

4

Stack derating factorsMultiply nameplate charge by each remaining fraction.

5

Divide charge by average currentRuntime hours = usable mAh / average mA.

3. Work the chapter burst

160/4000=0.040 s; 24×40=960 mA·ms; 3.0−0.024(20)=2.52 V

The illustrative CR2032 keeps 225×0.98×0.85×0.84 = 157 mAh. Against the chapter's 0.213 mA average, that is about 739 h or 30.8 days.

4. Try one controlled change

ttx=B/R; Qtx=Itx·ttx; Vterm=Voc−ItxRint; trun=Qusable/Iavg

TryMove only bit rate. Payload, transmit current, cell, derating factors, and average-current baseline stay fixed.

Transmit time
Burst charge mA·ms
Voltage sag
Terminal voltage
Brownout margin
Usable charge
Runtime
Runtime days

ObserveAt 4,000 bit/s the message lasts 40.0 ms and spends 960 mA·ms. Raising bit rate shortens that term; voltage sag stays at 0.480 V because current and resistance did not change.

ExplainBit rate changes airtime. Pulse current changes sag. The page keeps them separate so one improvement is not credited for another.

Technical boundaries.

This compact model uses fixed currents and derating factors.

Radio
Headers, coding, acknowledgements, retries, startup, and receive windows add airtime
Cell
Internal resistance changes with age, temperature, state of charge, and pulse history
Runtime
A constant average omits load variation and regulator efficiency

Measure the chosen PHY and cell over the real temperature and lifetime range.

5. Do not trade range for airtime silently

A faster PHY may shorten airtime, but band, modulation, coding, sensitivity, and propagation change link range and reliability. A clean ledger names those choices instead of treating bit rate as a free slider.

6. Carry a release-ready battery record

Record payload and overhead bits, bit rate, output power, current trace, retries, open-circuit and pulse voltage, cell age, temperature, cutoff, regulator efficiency, measured average, and the brownout result.

7. Check yourself

Why is 960 mA·ms not a complete energy result?
Answer: It counts charge; energy also needs voltage.
Why does the cell fall to 2.52 V?
Answer: The 24 mA pulse loses IR = 0.024×20 = 0.480 V.
Does a faster bit rate guarantee longer field life?
Answer: No. It shortens this burst term, but real PHY overhead, retries, range, current, sleep, and cell behaviour remain.
Honesty boundary.

The 24 mA, 40 ms, and 0.213 mA values come from the chapter; the CR2032 condition and derating factors are illustrative.

40 ms
Chapter transmit state
157 mAh
Illustrative usable charge
30.8 days
Bounded constant-average estimate

Go deeper in the chapter, then replace catalog assumptions with measured pulse evidence.

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