Math Bridge: Bluetooth Burst Sag

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Math BridgeBluetooth and BLEStruggle-friendly runway

How can a 3.0 V coin cell collapse to 1.50 V during radio work?

Separate charge from energy, convert Bluetooth dBm classes, and calculate voltage sag as a cell’s internal resistance grows.

Radio Remi, the guideRadio Remi guides
The one targetConnect Bluetooth power class to battery sag.
The chapter case7.5 mA burst from a 3.0 V, 220 mAh coin cell.
What it buys youExplain brownouts that an open-circuit meter misses.

A technician must decide whether radio terminal voltage is safe before changing cell internal resistance on the real device. The result is unresolved until the rule and units are checked. Predict the direction first.

See the relationship before changing it

The figure reads from left to right. The blue card is cell internal resistance. The middle card applies this page's rule. The green card is radio terminal voltage. 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 model keeps those stated values fixed and changes only cell internal resistance, so the numeric fixture does not switch without explanation.

Cell internal resistance changes radio terminal voltage An input card leads through the rule terminal = 3.000 V - 0.0075 A x resistance to the radio terminal voltage result. INPUT PAGE INPUT APPLY THE RULE predict calculate check units OUTPUT RESULT
Walk the arrows. Aged-cell resistance creates more voltage sag during the same burst.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 10 ohm.

  2. 2

    Name the relationship. terminal = 3.000 V - 0.0075 A x resistance

  3. 3

    Substitute with units. 3.000 - 0.0075 x 10 = 2.925 V

  4. 4

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

Predict, then change cell internal resistance

Try Predict the direction of terminal = 3.000 V - 0.0075 A x resistance. Test another cell internal resistance, then compare radio terminal voltage.

10 ohm
Chapter baseline
Radio terminal voltage

Observe Aged-cell resistance creates more voltage sag during the same burst. Reset cell internal resistance to 10 and compare radio terminal voltage.

Explain Aged-cell resistance creates more voltage sag during the same burst.

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 cell internal resistance moves here. Field effects named in the technical boundary stay fixed.

Honesty note. This teaching model changes only cell internal resistance. It is not a field prediction.

1. Charge is not yet energy

A 220 mAh label counts charge. Multiply 0.220 Ah by 3.0 V to obtain 0.660 Wh. Neither number says whether the terminal voltage remains high enough during a transmit burst.

Radio Remi: Test a cell under the same pulse that the radio demands.

2. Name each conversion

1

Charge to energyE=V(QmAh/1000).

2

dBm to milliwattsPmW=10^(PdBm/10).

3

Resistance to sagVterm=Voc−IRint.

3. Age changes the pulse result

ΔV=IRint; Q(t)=Q0(1−r)^t

At 7.5 mA and 10 Ω, sag is 0.075 V. At the same current and 200 Ω, it is 1.50 V. Separately, 1% annual self-discharge leaves about 209 mAh after five years.

4. Try one controlled change

Vterm=Voc−ItxRint

TryIncrease only cell internal resistance from fresh toward aged. Open-circuit voltage, burst current, nameplate charge, power classes, and self-discharge stay fixed.

Pulse sag
Terminal voltage
Nameplate energy
Class 1 (+20 dBm)
Class 2 (+4 dBm)
Charge after 5 years
Shelf loss
Shelf loss fraction

ObserveAt 10 Ω, sag is 0.075 V and terminal voltage is 2.925 V. At 200 Ω, the same control yields 1.50 V sag and only 1.50 V at the radio.

ExplainChanging resistance does not change the open-circuit meter reading in this model. It changes the voltage lost only when current flows, which is why a transmit pulse reveals ageing.

Technical boundaries.

This is a static Thevenin snapshot of a dynamic electrochemical cell.

Internal resistance
Varies with state of charge, temperature, age, chemistry, and pulse duration
Transmit current
Depends on SoC, power setting, supply voltage, and board losses
Energy budget
Nameplate Wh does not include cutoff, recovery, passivation, or regulator limits

Capture loaded voltage and current on the chosen cell, board, and radio setting.

5. Reproduce the chapter values

0.220 Ah×3.0 V=0.660 Wh. +20 dBm is 100 mW and +4 dBm is 2.51 mW. At 10 Ω, 0.0075×10=0.075 V; at 200 Ω it is 1.50 V. Self-discharge gives 220×0.99⁵=209 mAh, a 10.8 mAh or 4.90% shelf loss.

6. Carry the evidence forward

Record cell chemistry and lot, open-circuit and loaded voltage, impedance versus state of charge and temperature, pulse shape, power class, board current, regulator dropout, brownout threshold, shelf time, and recovery behaviour.

7. Check yourself

Why is 220 mAh not the same as 0.660 Wh?
Answer: mAh is charge; multiplying 0.220 Ah by 3.0 V gives energy.
Why can an open-circuit 3.0 V reading be misleading?
Answer: No load means almost no IR sag; the radio pulse exposes the resistance loss.
Does +4 dBm mean 4 mW?
Answer: No. 10^(4/10)=2.51 mW.
Honesty boundary.

The page joins four chapter equations without claiming a complete battery model.

1.50 V terminal
Illustration at 7.5 mA and fixed 200 Ω
209 mAh after five years
Constant 1% annual self-discharge model
Power classes
RF output conversions, not battery-input power

A real brownout decision belongs to measured loaded-voltage evidence.

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