Math Bridge: WSN Battery Ledger

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Math BridgeWSNStruggle-friendly runway

Why is 2000 mAh not yet a lifetime?

Keep charge, energy, voltage sag, usable capacity, and average current in their own rows.

Packet Pete, the guidePacket Pete guides
The one targetBuild a bounded leaf-node battery ledger.
The chapter case3.6 V, 2000 mAh, 30 mA pulse, 50 µA average.
What it buys youA lifetime number with its assumptions visible.

A field team has a real problem to settle: Why is 2000 mAh not yet a lifetime? They must decide what happens before they change average current on the device. Predict the direction first.

See the relationship first

The figure reads from left to right. The blue card is average current. The middle card uses this page's rule. The green card is nameplate energy. Follow the arrows: set the input, use the rule, then read the result and its unit.

The audit later on checks more than one number. Here, the added model uses the baseline named below and holds every other chapter value fixed. That sentence bridges the fixtures, so the numbers do not change without a reason.

Average current changes nameplate energy An input card leads through the page rule to the nameplate energy result. SET INPUT ONE CONTROL USE RULE predict calculate check units READ RESULT
Follow the arrows. The control changes the capacity/current division. The pulse uses a separate 30 mA input, so the two battery questions remain separate.

Derive the baseline in four moves

  1. 1

    Name the input. The chapter baseline for average current is 50.

  2. 2

    Name the rule. Enameplate=2.000x3.6=7.20 Wh ΔV=0.030x3=0.0900 V; Vterm=3.51 V Qusable=0.80x2000=1600 mAh; Eusable=5.76 Wh t=1600/0.050=32,000 h=3.65 years

  3. 3

    Put in the chapter value. Set average current to 50. The page rule gives nameplate energy as 7.20 Wh.

  4. 4

    Read the result. Keep Wh next to the value. Use it only within the limits on this page.

Predict, then change average current

Try Predict what happens to nameplate energy. Move one control, calculate, then check your idea.

50
Chapter baseline
Nameplate energy

Observe The control changes the capacity/current division. The pulse uses a separate 30 mA input, so the two battery questions remain separate. Reset to 50 and compare nameplate energy.

Explain Only average current moves here. The other chapter values stay fixed.

Check yourself

What should you do before you trust the result?
Answer: Predict its direction, use the shown rule, keep the units, and reset to the worked baseline.
What does this small model leave out?
Answer: Only average current moves. Field effects named in the page limits stay fixed.

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

1. Charge and energy are related, not identical

Milliamp-hours count charge. Watt-hours include voltage. A real cell's terminal voltage falls during a current pulse because current also crosses the cell's internal resistance.

Packet Pete: Never jump from the label to years without a usable-capacity and current step.

2. Name every algebra move

1

Convert mAh to AhQAh=QmAh/1000.

2

Approximate energyEWh=QAhV.

3

Use Ohm's law inside the cellΔV=IRint; Vterm=Voc−ΔV.

4

Apply design deratingQusable=fQnominal.

5

Divide capacity by average currentth=Qusable/Iavg.

3. Reproduce the leaf-node case

Enameplate=2.000×3.6=7.20 Wh
ΔV=0.030×3=0.0900 V; Vterm=3.51 V
Qusable=0.80×2000=1600 mAh; Eusable=5.76 Wh
t=1600/0.050=32,000 h=3.65 years

These numbers form one bounded comparison, not a discharge curve.

4. Try average current

TryMove the duty-cycle average while cell and pulse assumptions stay fixed.

Average current
Nameplate energy
Pulse sag
Pulse terminal voltage
Usable capacity
Usable energy
Bounded runtime
Bounded years

ObserveAt 50 µA the bounded runtime is 32,000 h. Doubling average current halves this simple runtime while pulse sag stays fixed.

ExplainThe control changes the capacity/current division. The pulse uses a separate 30 mA input, so the two battery questions remain separate.

Technical boundaries.

The ledger uses fixed voltage, resistance, derating, and average current.

Cell
Temperature, age, self-discharge, rate effects, and capacity curves are omitted
Load
Startup, sensing, processing, receive windows, retries, and regulators need a state trace
Cutoff
Usable life ends at the system threshold, not necessarily zero stored charge

Measure pulse voltage and integrate the complete deployed load schedule.

5. Keep sag and life separate

A cell may have charge left and still reset the node during a pulse. Check pulse voltage against cutoff as well as the average-current lifetime.

6. Build the node record

Record cell chemistry and lot, voltage, capacity test conditions, internal resistance, cutoff, all state currents and durations, temperature, retries, margin, owner, and retest trigger.

7. Check yourself

Why is the nameplate energy 7.20 Wh?
Answer: Convert 2000 mAh to 2.000 Ah, then multiply by the 3.6 V approximation.
Why does the pulse voltage fall by 0.0900 V?
Answer: Ohm's law gives 0.030 A×3 Ω=0.0900 V inside the cell.
Does 3.65 years predict a deployed node?
Answer: No. It is a constant-average, derated ledger that omits real load and cell behaviour.
Honesty boundary.

The 3.6 V, 2000 mAh, resistance, derating, and current values are the chapter's explicit typical example.

7.20 Wh
Nameplate constant-voltage approximation
5.76 Wh
Simple 80% design ledger
3.65 years
Bounded result, not a field guarantee

Go deeper in energy duty cycling and replace typical inputs with measurements.

Return to Wireless Sensor Networks Introduction with the maths unpacked →