Math Bridge: CoAP API Charge and Pulse Survival

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Math BridgeCoAPStruggle-friendly runway

Can NON last longer and still brown out?

Keep yearly message charge and instant pulse voltage separate, then use both to review a CoAP API cadence.

Eddie, the electronics guideEddie guides
The one targetJoin a yearly CON/NON charge comparison to a separate IR pulse-survival check.
The chapter case525,600 messages/year, a 220 mAh CR2032, and 10 mA TX pulses.
What it buys youA design review that does not mistake remaining mAh for guaranteed voltage.

A field team faces an unresolved physical question: Can NON last longer and still brown out? 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 con annual 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.

Internal resistance changes con annual charge An input card leads through the page relationship to the con annual charge result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. A milliamp-hour integral does not depend on voltage; pulse survival does. Both facts can be true at once.

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. CON = 525,600(0.278 uAh)/1000 + 43.8 = 189.92 mAh/year NON = 525,600(0.139 uAh)/1000 + 43.8 = 116.86 mAh/year life = 220/annual charge = 1.16 and 1.88 years fresh TX sag = 0.010(15) = 0.150 V fresh loaded voltage = 3.00-0.150 = 2.85 V

  3. 3

    Substitute the chapter fixture. Set internal resistance to 15. The page ledger gives con annual charge as 189.92 mAh/year.

  4. 4

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

Predict, then change internal resistance

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

15
Chapter baseline
CON annual charge

Observe A milliamp-hour integral does not depend on voltage; pulse survival does. Both facts can be true at once. Reset the control to 15 and compare con annual charge.

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

Charge tells us how much current flowed over time. Terminal voltage tells us whether the radio can remain powered during a pulse. A message strategy can save charge and still fail if an old or cold coin cell cannot hold the rail up.

Eddie: Use two ledgers: yearly mAh for endurance and IR sag for instant survival.

2. Name every algebra move

1

Accumulate message chargeMultiply per-message µAh by 525,600 and divide by 1,000 for mAh/year.

2

Add background useAdd the chapter's 43.8 mAh/year to both strategies.

3

Find lifeDivide 220 mAh by each annual total.

4

Find sagMultiply the 10 mA transmit current by internal resistance.

5

Test marginSubtract sag and then subtract the 2.0 V cutoff.

3. Reproduce the chapter case

CON = 525,600(0.278 µAh)/1000 + 43.8 = 189.92 mAh/year
NON = 525,600(0.139 µAh)/1000 + 43.8 = 116.86 mAh/year
life = 220/annual charge = 1.16 and 1.88 years
fresh TX sag = 0.010(15) = 0.150 V
fresh loaded voltage = 3.00−0.150 = 2.85 V

NON's longer life comes from charge. The fresh-cell voltage check does not alter that mAh result; it answers the different question of whether one pulse stays above cutoff.

4. Try one real input

TryMove cell resistance from 15 ohm toward 110 ohm. Annual charge stays fixed while transmit and ACK voltage margins collapse.

Internal resistance
CON annual charge
NON annual charge
CON lifetime
NON lifetime
NON advantage
TX sag
TX terminal voltage
TX cutoff margin
TX internal loss
ACK sag
ACK terminal voltage

ObserveChanging resistance moves six voltage outputs but not either annual charge or lifetime result.

ExplainA milliamp-hour integral does not depend on voltage; pulse survival does. Both facts can be true at once.

Technical boundaries.

This ledger reproduces the chapter's annual message charges and a lumped-resistance pulse check.

Traffic
One fixed yearly count, fixed message charges, and fixed background use are assumed.
Cell
Resistance is a single illustrative value; chemistry, recovery, temperature, state of charge, and capacitance are not simulated.
Rail
Regulator dropout, wiring, capacitor ESR, and pulse waveform are omitted.

Correct, not complete: this ledger does not qualify a CR2032 or prove one CoAP API design best.

5. Use the result in the design

Choose the message type from delivery semantics first. Then measure annual charge and worst-case pulse voltage on the real cell, temperature range, hardware rail, and retry pattern.

6. Record the evidence state

Record API cadence, method and message type, payload, retry count, annual background load, cell lot, temperature, state of charge, pulse current, terminal minimum, and reset threshold.

7. Check yourself

Why does resistance not change the mAh totals?
Answer: Those totals integrate current over time. Resistance changes voltage and energy loss, not the stated current-time charge.
At 100 ohm, what happens to a 10 mA pulse?
Answer: Sag is 1.00 V, so a 3.00 V open-circuit cell reaches 2.00 V, right at the example cutoff.
Does NON's longer arithmetic life make it correct for every resource?
Answer: No. Delivery importance, replacement data, congestion, retries, security, and application semantics decide the message contract.
Honesty boundary.

This ledger reproduces the chapter's annual message charges and a lumped-resistance pulse check.

Traffic
One fixed yearly count, fixed message charges, and fixed background use are assumed.
Cell
Resistance is a single illustrative value; chemistry, recovery, temperature, state of charge, and capacitance are not simulated.
Rail
Regulator dropout, wiring, capacitor ESR, and pulse waveform are omitted.

Correct, not complete: this ledger does not qualify a CR2032 or prove one CoAP API design best.

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