Math Bridge: Coin-cell sag and link margin

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Math BridgeDesign MethodologyStruggle-friendly runway

Why can a comfortable link margin still brown out a coin cell?

Keep charge, pulse voltage, self-discharge, and the 915 MHz link budget in one reviewable topology ledger.

Blueprint Bina, the design guideBlueprint Bina guides
The one targetSeparate energy quantity from pulse delivery.
The chapter caseCR2032, 90 mA pulse, 915 MHz, 1500 m.
What it buys youA topology review that can spot brownout risk.

A field team faces an unresolved physical question: Why can a comfortable link margin still brown out a coin cell? They must answer it before changing tx 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 tx 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.

TX 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 link may have enough received power while the cell cannot hold the rail high enough to produce the transmit pulse.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for tx current is 90.

  2. 2

    Name the relationship. E=(225x3.00)/1000=0.675 Wh ΔV=0.090x20=1.80 V; Vload=1.20 V Five-year self-discharge≈11.0 mAh λ=0.328 m; FSPL=95.2 dB Pr=16-95.2=-79.2 dBm; margin=57.8 dB

  3. 3

    Substitute the chapter fixture. Set tx current to 90. 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 tx current

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

90
Chapter baseline
Nameplate energy

Observe The link may have enough received power while the cell cannot hold the rail high enough to produce the transmit pulse. Reset the control to 90 and compare nameplate energy.

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

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

1. Keep two battery stories apart

Capacity says how much charge exists. Internal resistance says how much voltage remains during a pulse. A large charge number cannot prevent a brownout when the pulse current is too high.

Blueprint Bina: Compare loaded voltage and RF margin on separate rows before choosing a topology.

2. Name every algebra move

1

Attach voltageE=CV/1000 for C in mAh.

2

Subtract internal dropVload=Voc−IRint.

3

Compound self-dischargeClost=C₀[1−(1−k)^t].

4

Find wavelengthλ=c/f.

5

Price free-space distanceFSPLdB=20log₁₀(4πd/λ).

6

Subtract the pathPr=EIRP−FSPL and margin=Pr−sensitivity.

3. Reproduce the yard-sensor case

E=(225×3.00)/1000=0.675 Wh
ΔV=0.090×20=1.80 V; Vload=1.20 V
Five-year self-discharge≈11.0 mAh
λ=0.328 m; FSPL=95.2 dB
Pr=16−95.2=−79.2 dBm; margin=57.8 dB

The ideal RF ledger has ample nominal margin while the battery pulse falls to a likely unusable voltage. A bigger mAh label does not repair the pulse path by itself.

4. Try the transmit current

TryMove the radio pulse current while the outdoor link assumptions stay fixed.

TX current
Nameplate energy
Pulse sag
Loaded voltage
Self-discharge loss
Wavelength
Free-space loss
Received power
Nominal margin

ObservePulse current changes terminal voltage but not the ideal RF margin; these are independent checks in this ledger.

ExplainThe link may have enough received power while the cell cannot hold the rail high enough to produce the transmit pulse.

Technical boundaries.

This combines two bounded checks, not a complete radio or cell model.

Cell
Internal resistance changes with lot, temperature, state of charge, and pulse history
Link
Free space omits terrain, antenna mismatch, polarization, fading, and interference
Buffer
A bulk capacitor needs ESR, leakage, recharge, and minimum-voltage checks

Measure the loaded rail and installed RF path before approving a coin-cell topology.

5. Test the fragile side first

Run the worst cold, aged-cell pulse with the intended bulk capacitor and regulator. Then measure installed packet delivery and margin instead of treating free-space loss as site evidence.

6. Record both acceptance gates

Store cell and lot, temperature, pulse shape, minimum rail, buffer, RF settings, antenna, distance, site loss, packet evidence, and topology decision.

7. Check yourself

Why is 0.675 Wh not enough to approve the radio?
Answer: It says nothing about terminal voltage during the 90 mA pulse.
Why can RF margin and brownout disagree?
Answer: Received-power arithmetic and battery internal resistance are separate constraints.
Does 57.8 dB qualify the yard link?
Answer: No. It is a free-space nominal result without site losses.
Honesty boundary.

The arithmetic reproduces the chapter's catalog-typical CR2032 and 915 MHz case.

1.20 V
Fixed-resistance pulse estimate, not a guaranteed rail trace
11.0 mAh
Compounded 1% annual self-discharge teaching estimate
57.8 dB
Free-space nominal margin, not installed coverage

Correct, not complete: this ledger does not qualify battery life or the yard link.

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