Math Bridge: CR2032 Pulse Sag and Shelf Life

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How can a 77-year message budget fail in one pulse?

Place nominal stored energy, instant terminal voltage, delivered message energy, cutoff, and shelf life in one transparent ledger.

Eddie, the electronics guideEddie guides
The one targetTest a nominal coin-cell message budget against the voltage available during a radio pulse.
The chapter case2,430 J, 3.00 V, 20 mA, 0.0864 mJ/message, and 1,000 messages/day.
What it buys youA visible reason why unused chemical energy can remain behind an electrical cutoff.

A field team faces an unresolved physical question: How can a 77-year message budget fail in one pulse? They must answer it before changing internal resistance 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 internal resistance. The middle card applies this page's relationship. The green card is pulse sag. 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.

Internal resistance changes pulse sag An input card leads through the page relationship to the pulse sag result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. The capacity ledger does not know whether the terminal voltage stays high enough to release that stored energy through the radio pulse.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for internal resistance is 15.

  2. 2

    Name the relationship. fresh sag = 0.020(15) = 0.300 V fresh loaded voltage = 3.00-0.300 = 2.70 V pulse time = 0.0000864/(0.020x3.00) = 1.44 ms fresh delivered energy = 0.020(2.70)(0.00144) = 0.07776 mJ naive days = 2,430/(0.0000864x1,000) = 28,125

  3. 3

    Substitute the chapter fixture. Set internal resistance to 15. The page ledger gives pulse sag as 300 mV.

  4. 4

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

Predict, then change internal resistance

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

15
Chapter baseline
Pulse sag

Observe The capacity ledger does not know whether the terminal voltage stays high enough to release that stored energy through the radio pulse. Reset the control to 15 and compare pulse sag.

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

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

1. Start with the physical story

A datasheet capacity says how much charge or nominal energy a fresh cell stores under specified conditions. A transmitter asks for current now. Internal resistance turns part of the open-circuit voltage into heat inside the cell before the radio sees it.

Eddie: Stored joules answer “how much”; loaded voltage answers “can the next pulse run?”

2. Name every algebra move

1

Find sagMultiply 0.020 A by internal resistance.

2

Find loaded voltageSubtract sag from 3.00 V.

3

Recover pulse timeDivide the nominal 0.0000864 J by 0.020 A × 3.00 V.

4

Find delivered energyMultiply current, loaded voltage, and pulse time.

5

Compare ceilingsDivide 2,430 J by daily nominal message energy, then compare years with the stated ten-year shelf ceiling.

3. Reproduce the chapter case

fresh sag = 0.020(15) = 0.300 V
fresh loaded voltage = 3.00−0.300 = 2.70 V
pulse time = 0.0000864/(0.020×3.00) = 1.44 ms
fresh delivered energy = 0.020(2.70)(0.00144) = 0.07776 mJ
naive days = 2,430/(0.0000864×1,000) = 28,125

The nominal ledger reaches about 77.1 years, but the same cell's electrical pulse margin and roughly ten-year shelf ceiling can stop useful service much earlier.

4. Try one real input

TryMove cell resistance from 15 ohm toward 100 ohm. Watch loaded voltage, delivered pulse energy, cutoff margin, and internal loss change while the naive message count stays fixed.

Internal resistance
Pulse sag
Loaded voltage
Cutoff margin
Pulse survives
Pulse time
Delivered message energy
Internal voltage loss
Naive message-only days
Naive years
Years / shelf ceiling

ObserveAt 15 ohm, the radio receives 2.70 V and 90% of nominal pulse energy. At 100 ohm, the rail reaches 1.00 V and fails the 2.00 V example cutoff.

ExplainThe capacity ledger does not know whether the terminal voltage stays high enough to release that stored energy through the radio pulse.

Technical boundaries.

This is a one-resistor Thevenin teaching model around the chapter's message-energy figures.

Cell
Resistance is not constant with chemistry, temperature, state of charge, pulse history, or recovery.
Rail
Regulator behaviour, wiring, decoupling capacitance, ESR, and radio brownout dynamics are omitted.
Lifetime
The ten-year shelf figure is a bounding comparison, not a guaranteed service life.

Correct, not complete: this ledger does not qualify a coin cell, power rail, or CoAP traffic plan.

5. Use the result in the design

Test the radio burst on the oldest, coldest allowed cell with real wiring and decoupling. Keep the message-energy budget, shelf ceiling, and pulse-voltage qualification as separate evidence lines.

6. Record the evidence state

Record cell part and lot, temperature, state of charge, rest time, pulse current and duration, internal resistance estimate, rail minimum, cutoff, capacitor state, reset log, and message cadence.

7. Check yourself

Where does the missing voltage go?
Answer: The IR drop appears across the cell's internal resistance and produces internal heat.
Why does the naive 77.1-year result stay fixed as resistance moves?
Answer: That result uses the nominal message energy and stored joules only; it deliberately does not model deliverability.
Does a fresh 2.70 V pulse guarantee ten years?
Answer: No. Resistance, capacity, self-discharge, temperature, cutoff, and the rest of the product load change over time.
Honesty boundary.

This is a one-resistor Thevenin teaching model around the chapter's message-energy figures.

Cell
Resistance is not constant with chemistry, temperature, state of charge, pulse history, or recovery.
Rail
Regulator behaviour, wiring, decoupling capacitance, ESR, and radio brownout dynamics are omitted.
Lifetime
The ten-year shelf figure is a bounding comparison, not a guaranteed service life.

Correct, not complete: this ledger does not qualify a coin cell, power rail, or CoAP traffic plan.

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