Math Bridge: Cold-Chain Tag Battery

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Math BridgeRFID tagBattery screen

Will the tag still wake when the coin cell is cold?

Turn the chapter's 24-month BAP lifecycle into a pulse-voltage and average-charge screen.

Eddie, the electronics guideEddie guides
The one targetConnect a service charge budget to cold pulse headroom.
The chapter caseA 225 mAh CR2032-class cell, 15 mA pulse, and 24-month interval.
What it buys youA bounded bench hypothesis for warm and cold pulse tests.

A field team faces an unresolved physical question: Will the tag still wake when the coin cell is cold? They must answer it before changing pulse current 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 pulse current. The middle card applies this page's relationship. The green card is nameplate energy. 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.

Pulse current changes nameplate energy An input card leads through the page relationship to the nameplate energy result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. The same pulse current produces three times the sag when internal resistance triples.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for pulse current is 15.

  2. 2

    Name the relationship. E = 0.225 x 3 = 0.675 Wh Qusable = 225 x 0.99² x 0.85 = 187.4 mAh Iavg = 187.4 Ah / 17,520 h = 10.7 uA Vcold = 3 - 0.015 x 60 = 2.10 V

  3. 3

    Substitute the chapter fixture. Set pulse current to 15. The page ledger gives nameplate energy as 0.675 Wh.

  4. 4

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

Predict, then change pulse current

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

15
Chapter baseline
Nameplate energy

Observe The same pulse current produces three times the sag when internal resistance triples. Reset the control to 15 and compare nameplate energy.

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

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

1. Start with the physical story

A BAP tag can have charge left yet reset during a brief cold pulse because internal resistance turns current into voltage sag.

Eddie: Nameplate energy, average current, and pulse voltage answer different questions; keep all three visible.

2. Name every algebra move

1

Convert chargeUse E = QV for nameplate watt-hours.

2

Derate timeCompound self-discharge, then reserve cutoff headroom.

3

AverageDivide usable mAh by the 24-month hours.

4

Load the cellUse Vterm = Voc − IR for warm and cold resistance.

3. Reproduce the chapter case

E = 0.225 × 3 = 0.675 Wh
Qusable = 225 × 0.99² × 0.85 = 187.4 mAh
Iavg = 187.4 Ah / 17,520 h = 10.7 µA
Vcold = 3 − 0.015 × 60 = 2.10 V

The arithmetic explains why a warm bench read cannot stand in for the refrigerated pulse test.

4. Try one real input

TryMove pulse current while capacity, service interval, and the two resistance screens stay fixed.

Pulse current
Nameplate energy
Charge after self-discharge
Usable charge
Service hours
Average budget
Warm sag
Warm terminal
Warm cutoff headroom
Cold sag
Cold terminal
Cold cutoff headroom
Fleet replacements/month

ObserveAt 15 mA the warm model keeps 0.70 V above cutoff, while the cold model keeps only 0.10 V.

ExplainThe same pulse current produces three times the sag when internal resistance triples.

Technical boundaries.

This is a constant-resistance pulse screen, not a cell discharge simulation.

Cell data
Resistance, capacity, and cutoff vary with part, temperature, state of charge, pulse length, and ageing.
Load
Sensing, logging, leakage, regulator loss, and reader-field effects need measured duty traces.
Fleet
The replacement rate is a planning average, not a failure-time distribution.

Correct, not complete: this ledger does not approve a cell or predict tag life.

5. Use the result in the lab

Replay the measured pulse at warm and cold limits, log the cell terminals at the tag, and check reset and data retention.

6. Record the evidence state

Keep cell lot, age, temperature, pulse trace, cutoff, tag firmware, load profile, failures, and retest trigger.

7. Check yourself

Does 187 mAh prove two years of service?
Answer: No. It is a derated charge ceiling before the real load trace and cell curve.
Why can a cell with charge left still reset?
Answer: Pulse current through internal resistance can pull terminal voltage below logic cutoff.
Does the monthly replacement average predict each failure?
Answer: No. It only spreads a planned interval across the population.
Honesty boundary.

The ledger converts explicit chapter constants into a benchable cold-pulse hypothesis.

Computed
Charge derating, average budget, sag, terminal voltage, and cutoff headroom are reproducible.
Specified
The selected cell's temperature curves and tag cutoff replace the illustrative constants.
Observed
Oscilloscope traces, resets, read success, and retained logs decide the outcome.

Correct, not complete: test the real cell and tag across the claimed cold interval.

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