Math Bridge: Coin-Cell Radio Brownout

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Math BridgeApplicationsStruggle-friendly runway

Why can BLE start when Wi-Fi cannot?

Convert radio power to current before trusting coin-cell energy.

Radio Remi, the radio guideRadio Remi guides
The one targetTurn radio power into loaded coin-cell voltage.
The chapter case3.0 V, 220 mAh, 15 Ω; BLE 10 mW, Wi-Fi 300–500 mW.
What it buys youA radio choice that can physically start.

A field team faces an unresolved physical question: Why can BLE start when Wi-Fi cannot? They must answer it before changing radio power 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 radio power. The middle card applies this page's relationship. The green card is active current. 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.

Radio power changes active current An input card leads through the page relationship to the active current result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. Energy capacity and pulse capability answer different questions. The wearable must satisfy both before clinical or wellness logic matters.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for radio power is 10.

  2. 2

    Name the relationship. IBLE=10 mW/3.0 V=3.33 mA ΔVBLE=0.00333x15=0.050 V; Vterm=2.95 V IWiFi=300/3 to 500/3=100–167 mA ΔVWiFi=1.50–2.50 V; Vterm=1.50–0.50 V Enameplate=0.220x3.0=0.660 Wh

  3. 3

    Substitute the chapter fixture. Set radio power to 10. The page ledger gives active current as 3.3 mA.

  4. 4

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

Predict, then change radio power

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

10
Chapter baseline
Active current

Observe Energy capacity and pulse capability answer different questions. The wearable must satisfy both before clinical or wellness logic matters. Reset the control to 10 and compare active current.

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

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

1. Start with the physical story

Nameplate charge describes how much can flow over time. A radio pulse asks how fast it can flow now, and the coin cell loses current times internal resistance from its terminal voltage.

Radio Remi: A charged cell can still be the wrong source for a high-power radio.

2. Name every algebra move

1

Find currentDivide radio milliwatts by cell volts.

2

Convert unitsTurn milliamps into amps for the resistance equation.

3

Find sagMultiply amps by 15 ohms.

4

Load the railSubtract sag from 3.0 V.

5

Check floorSubtract the 2.0 V brownout voltage.

6

Keep energy separateMultiply 0.220 Ah by 3.0 V.

3. Reproduce the chapter case

IBLE=10 mW/3.0 V=3.33 mA
ΔVBLE=0.00333×15=0.050 V; Vterm=2.95 V
IWiFi=300/3 to 500/3=100–167 mA
ΔVWiFi=1.50–2.50 V; Vterm=1.50–0.50 V
Enameplate=0.220×3.0=0.660 Wh

BLE leaves nearly a volt above the illustrative radio floor. Wi-Fi collapses this source below that floor despite the same remaining charge.

4. Try one real input

TryMove radio power from BLE toward Wi-Fi and watch brownout arrive.

Radio power
Active current
Cell sag
Terminal voltage
Brownout headroom
Nameplate energy
Ideal continuous hours
Power versus BLE
Above radio floor

ObserveNameplate energy never changes, but terminal voltage falls directly with requested power.

ExplainEnergy capacity and pulse capability answer different questions. The wearable must satisfy both before clinical or wellness logic matters.

Technical boundaries.

This is a constant-power radio and fixed-resistance source screen.

Cell
Resistance changes with part, pulse duration, state of charge, temperature, recovery time, and age.
Load
LEDs, analogue front end, MCU, regulation, startup, and radio waveform overlap on the real rail.
Measurement
Beer-Lambert optics and calibrated SpO2 estimation are separate from the radio-power result.

Correct, not complete: this ledger does not prove wearable runtime, signal quality, safety, or clinical performance.

5. Use the result in the design

Measure terminal voltage during the complete optical-sample and radio event on fresh, aged, warm, and cold cells; add capacitance or choose another source if needed.

6. Record the evidence state

Keep cell part and lot, state of charge, temperature, age, pulse profile, rail trace, regulator, PPG LED current, MCU state, radio mode, packet outcome, and firmware.

7. Check yourself

Why does 10 mW become 3.33 mA?
Answer: Current is power divided by voltage: 10 mW divided by 3 V is 3.33 mA.
Why can Wi-Fi fail with charge remaining?
Answer: Its high current creates enough internal-resistance sag to pull terminal voltage below the radio floor.
Does BLE rail headroom prove an SpO2 monitor works?
Answer: No. Optical coupling, calibration, motion, skin, algorithms, alarms, safety, and workflow remain separate evidence.
Honesty boundary.

The arithmetic reproduces the chapter's 10 mW BLE, 300–500 mW Wi-Fi, and catalog-typical CR2032 screen.

Cell
Resistance changes with part, pulse duration, state of charge, temperature, recovery time, and age.
Load
LEDs, analogue front end, MCU, regulation, startup, and radio waveform overlap on the real rail.
Measurement
Beer-Lambert optics and calibrated SpO2 estimation are separate from the radio-power result.

Correct, not complete: this ledger does not prove wearable runtime, signal quality, safety, or clinical performance.

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