Math Bridge: Tiny Endpoint Life Budget

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Math BridgeApplicationsStruggle-friendly runway

What does a ten-year coin-cell claim leave for each report?

Join antenna scale and retained charge without hiding the sleep floor.

Battery Bruno, the power guideBattery Bruno guides
The one targetTurn ten-year retained charge into a reporting interval.
The chapter case2.4 GHz, 220 mAh, 1% annual loss, 20% reserve.
What it buys youA physically consistent tiny, cheap, sleepy endpoint.

A field team faces an unresolved physical question: What does a ten-year coin-cell claim leave for each report? They must answer it before changing report interval 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 report interval. The middle card applies this page's relationship. The green card is wavelength. 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.

Report interval changes wavelength An input card leads through the page relationship to the wavelength result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. The radio's small wavelength helps board cost; only the wake schedule helps the current budget.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for report interval is 20.4.

  2. 2

    Name the relationship. λ=3x10⁸/2.4x10⁹=0.125 m; λ/4=3.125 cm Qusable=220x0.99¹⁰x0.8=159.2 mAh Ibudget=159.2/(10x365x24)=1.82 uA Iwake budget=1.82-1.00=0.82 uA 70.6 reports/dayx0.05 s=3.53 s/day

  3. 3

    Substitute the chapter fixture. Set report interval to 20.4. The page ledger gives wavelength as 12.50 cm.

  4. 4

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

Predict, then change report interval

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

20.4
Chapter baseline
Wavelength

Observe The radio's small wavelength helps board cost; only the wake schedule helps the current budget. Reset the control to 20.4 and compare wavelength.

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

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

1. Start with the physical story

A short 2.4 GHz wavelength lets the antenna live on the PCB. The battery claim is harder: after calendar loss and reserve, sleep current consumes most of the allowed average before the radio wakes.

Battery Bruno: Cheap hardware only becomes long-lived when its seconds-awake ledger fits the microamp budget.

2. Name every algebra move

1

Find wavelengthDivide light speed by 2.4 GHz, then divide by four.

2

Retain chargeCompound 99% annual retention for ten years.

3

Reserve chargeKeep only 80% after derating.

4

Set current budgetDivide usable mAh by ten years of hours.

5

Count reportsDivide one day by the interval.

6

Price wakesConvert active seconds per day into average current, then add sleep.

3. Reproduce the chapter case

λ=3×10⁸/2.4×10⁹=0.125 m; λ/4=3.125 cm
Qusable=220×0.99¹⁰×0.8=159.2 mAh
Ibudget=159.2/(10×365×24)=1.82 µA
Iwake budget=1.82−1.00=0.82 µA
70.6 reports/day×0.05 s=3.53 s/day

About one short report every 20.4 minutes spends the remaining wake allowance after the 1 microamp sleep floor.

4. Try one real input

TryMove report interval and watch awake current compete with the fixed sleep floor.

Report interval
Wavelength
Quarter-wave
Usable charge
Ten-year budget
Wake allowance
Reports per day
Active seconds/day
Wake average
Total average
Derated ideal life

ObserveThe antenna numbers stay fixed while shorter intervals raise active average current and cut life.

ExplainThe radio's small wavelength helps board cost; only the wake schedule helps the current budget.

Technical boundaries.

This is an ideal quarter-wave scale and fixed-current report model.

Antenna
PCB dielectric, ground plane, enclosure, matching, efficiency, certification, and hand or soil detuning change the design.
Workload
Sensor warm-up, processing, joining, retries, receive windows, OTA, and fault states add awake time.
Battery
Pulse resistance, cutoff, temperature, leakage, capacity spread, and annual retention vary.

Correct, not complete: this screen does not prove a sub-dollar BOM or ten-year field life.

5. Use the result in the design

Measure sleep and a complete report, then choose an interval that preserves reserve for retries, maintenance traffic, and worst-case cells.

6. Record the evidence state

Keep antenna geometry and match, enclosure, firmware, report payload and interval, current trace, cell part and lot, retention assumption, temperature, cutoff, retries, and OTA policy.

7. Check yourself

Why is the quarter-wave only 3.12 cm?
Answer: The 2.4 GHz wavelength is 12.5 cm; one quarter is 3.125 cm.
Where did the 1.82 microamp budget come from?
Answer: Divide 159.2 retained and derated mAh by all 87,600 hours in ten years.
Does one report every 20.4 minutes guarantee ten years?
Answer: No. It uses fixed currents and omits protocol, environment, cutoff, pulse, and maintenance behavior.
Honesty boundary.

The arithmetic reproduces the chapter's 2.4 GHz, 220 mAh, ten-year sample-and-sleep screen.

Antenna
PCB dielectric, ground plane, enclosure, matching, efficiency, certification, and hand or soil detuning change the design.
Workload
Sensor warm-up, processing, joining, retries, receive windows, OTA, and fault states add awake time.
Battery
Pulse resistance, cutoff, temperature, leakage, capacity spread, and annual retention vary.

Correct, not complete: this screen does not prove a sub-dollar BOM or ten-year field life.

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