Math Bridge: Pulse Sag and Delivered Energy

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Math BridgeEnergy & PowerStruggle-friendly runway

When does a charge ledger stop being an energy ledger?

Weight each state by time, then let cell resistance change the voltage and joules delivered during the radio burst.

Battery Bruno, the energy and power guideBattery Bruno guides
The one targetKeep pulse voltage visible beside charge-weighted average current.
The chapter case599 s at 10 µA, 0.7 s at 20 mA, and 0.3 s at 120 mA.
What it buys youA source choice that passes both energy and brownout checks.

A field team faces an unresolved physical question: When does a charge ledger stop being an energy ledger? They must answer it before changing cycle time 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 cycle time. The middle card applies this page's relationship. The green card is cycle 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.

Cycle time changes cycle charge An input card leads through the page relationship to the cycle charge result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. A charge ledger can stay numerically unchanged while the source becomes unable to deliver that charge at a usable voltage.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for cycle time is 0.25.

  2. 2

    Name the relationship. Q=10 uAx599 s+20 mAx0.7 s+120 mAx0.3 s=55.99 mC Iavg=55.99 mC/600 s=93.3 uA Vsag=0.120x0.250=0.030 V Eradio=0.120x3.67x0.3=0.13212 J

  3. 3

    Substitute the chapter fixture. Set cycle time to 0.25. The page ledger gives cycle charge as 55.99.

  4. 4

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

Predict, then change cycle time

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

0.25
Chapter baseline
Cycle charge

Observe A charge ledger can stay numerically unchanged while the source becomes unable to deliver that charge at a usable voltage. Reset the control to 0.25 and compare cycle charge.

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

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

1. Start with the physical story

Current multiplied by time gives charge. That is enough for average current, but delivered energy also depends on terminal voltage. A real cell loses I×R inside itself, so the highest-current state can fail even when its share of cycle charge looks small.

Battery Bruno: Keep the charge ledger, voltage ledger, and cutoff gate in the same record.

2. Name every algebra move

1

Weight each stateMultiply every current by its duration.

2

Add charge and timeSum the state products and divide by the 600-second cycle.

3

Find radio sagMultiply 0.120 A by cell resistance.

4

Find terminal voltageSubtract sag from 3.70 V.

5

Find delivered energyMultiply radio current, terminal voltage, and 0.3 s.

6

Apply the service gateSubtract the 2.40 V cutoff from terminal voltage.

3. Reproduce the chapter case

Q=10 µA×599 s+20 mA×0.7 s+120 mA×0.3 s=55.99 mC
Iavg=55.99 mC/600 s=93.3 µA
Vsag=0.120×0.250=0.030 V
Eradio=0.120×3.67×0.3=0.13212 J

The open-circuit shortcut gives 0.13320 J, only 0.81% high for this cell. A 15 Ω source would request 1.80 V of sag and fail the 2.40 V cutoff.

4. Try one real input

TryRaise internal resistance and predict the first point where cutoff margin turns negative.

Cycle time
Cycle charge
Average current
Radio sag
Radio terminal
Cutoff margin
Delivered radio energy
Open-circuit estimate
Energy overestimate

ObserveAverage current and cycle charge stay fixed because the workload did not change. Sag, terminal voltage, delivered radio energy, and cutoff margin move together.

ExplainA charge ledger can stay numerically unchanged while the source becomes unable to deliver that charge at a usable voltage.

Technical boundaries.

This is a linear pulse ledger, not a full electrochemical model.

Resistance
Internal resistance changes with age, temperature, state of charge, and pulse duration.
Current request
A collapsing cell may never reach the requested 120 mA.
Other states
The simple energy output varies only the radio state's terminal voltage.

Correct, not complete: use measured pulse traces before qualifying the source.

5. Use the result in the design

Choose the cell against end-of-life pulse resistance, regulator cutoff, temperature, and the full radio waveform—not mAh alone.

6. Record the evidence state

Keep cell chemistry, age, temperature, open-circuit voltage, pulse current and duration, minimum terminal voltage, cutoff, and recovery.

7. Check yourself

Why does average current not move with resistance?
Answer: This control changes source voltage under load, not the requested current-time workload.
What makes charge a good energy proxy?
Answer: Terminal voltage must remain nearly constant across the states.
Which gate catches a brownout?
Answer: Terminal voltage minus cutoff voltage must remain positive with margin.
Honesty boundary.

The state times and currents come from the chapter; source resistance is an explicit scenario input.

Resistance
Internal resistance changes with age, temperature, state of charge, and pulse duration.
Current request
A collapsing cell may never reach the requested 120 mA.
Other states
The simple energy output varies only the radio state's terminal voltage.

Correct, not complete: use measured pulse traces before qualifying the source.

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