Math Bridge: Battery Charge, Energy, and Self-Discharge

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Math BridgeOne learning thread

What 0.10 mAh/day does not tell you

Extend the chapter’s 2000 mAh beehive design into energy, voltage sag, and the self-discharge load its radio-only budget omits.

Phoebe guides this bridge
One targetSeparate charge, energy, and stored-energy loss.
Chapter case2000 mAh, 3.7 V, 0.102 mAh/day Zigbee.
What it buys youSize the solar path against the larger load.

See the relationship before changing it

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 daily device charge, so the numeric fixture does not switch without explanation.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 0.2 mAh/day.

  2. 2

    Name the relationship. life = 1,000 mAh / daily charge

  3. 3

    Substitute with units. 1,000 mAh / 0.20 mAh/day = 5,000 days

  4. 4

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

Predict, then change daily device charge

Try Predict the direction of life = 1,000 mAh / daily charge. Test another daily device charge, then compare ideal cell life.

0.2 mAh/day
Chapter baseline
Ideal cell life

Observe A larger daily charge ledger shortens ideal life before self-discharge. Reset daily device charge to 0.2 and compare ideal cell life.

Explain The cell contains a fixed usable charge, so spending more charge each day exhausts it sooner and reduces ideal operating days inversely.

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 daily device charge moves here. Field effects named in the technical boundary stay fixed.

Honesty note. This teaching model changes only daily device charge. It is not a field prediction.

1. Start with charge, then add voltage

Ampere-hours measure how much charge a cell can move. Watt-hours measure energy. Two cells can carry the same charge but deliver different energy when their voltages differ.

Phoebe: mAh answers “how much charge?” It does not finish “how much work?” until voltage joins it.

2. Turn the chapter’s capacity into energy

Ecell(Wh) = C(Ah) × V
1

Convert milliamp-hours to ampere-hours2000 mAh ÷ 1000 = 2.000 Ah.

2

Use the assumed nominal voltageE = 2.000 Ah × 3.7 V.

3

MultiplyE = 7.40 Wh.

The 3.7 V Li-ion chemistry is an explicit assumption because the wizard states capacity but not chemistry.

3. Test whether voltage sag matters

Vterminal = Voc − IRint
1

Convert current to amperes9.00 mA = 0.00900 A.

2

Use the assumed resistanceVsag = IR = 0.00900 × 0.100.

3

MultiplyVsag = 0.000900 V.

4

Convert volts to millivolts0.000900 × 1000 = 0.900 mV.

This is an honest negative finding: with the chapter’s 9.00 mA burst, sag is negligible in this simplified cell model.

4. Calculate self-discharge

The chapter uses a catalog-typical 2.00% per month assumption.

1

Convert percent to fraction2.00% = 0.0200.

2

Multiply by capacity2000 × 0.0200 = 40.0 mAh/month.

3

Convert month to daily average40.0/30.0 = 1.33 mAh/day.

4

Compare with the computed radio load1.33/0.102 = 13.1×.

5. Read the compounding model correctly

Eremaining(t) = Ecell(1 − k)t
1

Name the intervalt counts charge-to-charge periods in the same units used by k.

2

Keep the retained fractionWith k = 0.0200 per month, one month retains 1 − 0.0200 = 0.9800.

3

Apply repeated multiplicationAfter t months without recharge, multiply by 0.9800t.

Solar recharge interrupts this sealed-cell decay story. The useful decision is that the panel and controller must replace both radio use and roughly 1.33 mAh/day of assumed self-discharge.

6. Check yourself

1. How much nominal energy is 2000 mAh at 3.7 V?

2.000 Ah × 3.7 V = 7.40 Wh.

2. Is 0.900 mV sag important in this example?

No. Under the stated 9.00 mA and 0.100 Ω assumptions, it is negligible.

3. What dominates the simplified daily charge budget?

The assumed 1.33 mAh/day self-discharge, about 13.1× the 0.102 mAh/day active-plus-sleep load.

7. Honesty boundary

The simple lifetime division fixes a 1000 mAh capacity and leaves self-discharge out; the separate energy and sag examples use assumed lithium-ion voltage, resistance, and monthly loss. Those cell properties can change with temperature, age, charge state, cell quality, and protection circuitry, and radio current alone does not describe the whole device.

The wizard does not state chemistry
Battery chemistry absent from wizard inputs
3.7 V Li-ion
Assumed lithium-ion nominal voltage
0.100 Ω
Assumed battery internal resistance
2.00%/month
Assumed monthly capacity self-discharge
30 days/month are named assumptions
Assumed days used to convert monthly loss
Capacity
Usable cell charge capacity
self-discharge
Cell charge lost without external load
voltage
Cell voltage depends on operating conditions
resistance vary with temperature
Internal resistance depends on operating conditions
age
Cell ageing affects battery properties
state of charge
Remaining charge affects battery properties
cell quality
Cell quality affects battery properties
protection circuitry
Battery protection-circuit effects
Whole-device current is not only radio current
Total current includes loads beyond the radio

Go deeper in Put Numbers to the Decision and validate with a measured full-cycle current trace plus solar yield and storage tests.