Math Bridge: Peukert Relay Lifetime

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Math BridgeWSNStruggle-friendly runway

Why can the relay lose more life than current alone predicts?

Turn one chemistry exponent into an honest correction to the chapter's relay ledger.

Packet Pete, the guidePacket Pete guides
The one targetUse Peukert's law without treating it as a battery oracle.
The chapter case50 µA leaf reference and 300 µA relay.
What it buys youA visible chemistry penalty beside linear life.

See the relationship before changing it

The figure reads from left to right. The blue card is relay average current. The middle card applies this page's rule. The green card is linear battery runtime. 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 relay average current, so the numeric fixture does not switch without explanation.

Relay average current changes linear battery runtime An input card leads through the rule linear runtime = 1,600 mAh / relay current to the linear battery runtime result. INPUT PAGE INPUT APPLY THE RULE predict calculate check units OUTPUT RESULT
Walk the arrows. More relay current shortens reference life before the Peukert penalty.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 0.3 mA.

  2. 2

    Name the relationship. linear runtime = 1,600 mAh / relay current

  3. 3

    Substitute with units. 1,600 / 0.300 = 5,333 hours

  4. 4

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

Predict, then change relay average current

Try Predict the direction of linear runtime = 1,600 mAh / relay current. Test another relay average current, then compare linear battery runtime.

0.3 mA
Chapter baseline
Linear battery runtime

Observe More relay current shortens reference life before the Peukert penalty. Reset relay average current to 0.3 and compare linear battery runtime.

Explain More relay current shortens reference life before the Peukert penalty.

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 relay average current moves here. Field effects named in the technical boundary stay fixed.

Honesty note. This teaching model changes only relay average current. It is not a field prediction.

1. Linear division is the starting line

Capacity divided by average current assumes usable capacity stays fixed as current changes. Peukert's model adds a chemistry-sensitive exponent that makes higher sustained current cost extra runtime.

Packet Pete: The exponent corrects one assumption; it does not model the whole field battery.

2. Name every algebra move

1

Set the reference lifetref=Cusable/Iref.

2

Write Peukert's invariantCp=I^k t.

3

Substitute the referenceCp=Iref^k tref.

4

Divide by relay current raised to ktrelay=tref(Iref/Irelay)^k.

5

Compare with linear lifepenalty=100(1−tPeukert/tlinear).

3. Reproduce both chapter examples

tlinear=1600/0.300=5,333 h
k=1.05: 32,000(0.050/0.300)^1.05=4,876 h=0.557 years
penalty=8.6%
k=1.30: runtime=3,116 h

The alkaline-style exponent makes a much larger correction than the low-rate lithium-style exponent.

4. Try the exponent

TryMove k from the ideal linear case toward a stronger rate penalty.

Exponent
Linear relay life
Corrected relay life
Corrected years
Penalty from linear
Peukert capacity

ObserveAt k=1.05 the corrected life is 4,876 h, 8.6% below linear. Increasing k makes the penalty grow.

ExplainThe current ratio is below one. Raising it to a larger exponent makes that factor smaller, so predicted runtime falls.

Technical boundaries.

This is a sustained-current Peukert comparison around one reference point.

Load
A pulsed radio schedule is not identical to one sustained current
Chemistry
Exponent varies with cell type, temperature, age, and discharge regime
Electronics
Voltage cutoff, regulator efficiency, pulse sag, and self-discharge are omitted

Use manufacturer curves and measured state traces for the selected cell.

5. Keep the baseline visible

Report both linear and corrected life. Hiding the baseline makes the size of the chemistry correction hard to audit.

6. Build the battery evidence record

Record chemistry, capacity test point, exponent source, temperature, current waveform, relay burden, cutoff, pulse voltage, age, margin, owner, and retest trigger.

7. Check yourself

What does k=1 mean here?
Answer: The rate correction disappears and the current ratio gives the linear estimate.
Why is 4,876 h below 5,333 h?
Answer: With k above one, the sixfold current increase carries an extra rate penalty.
Does k=1.05 certify this relay battery?
Answer: No. It is a catalog-typical teaching input, not a measured cell result.
Honesty boundary.

The 50 µA, 300 µA, 1600 mAh, and worked results come from the chapter's bounded examples.

1.05/1.30
Catalog-typical teaching exponents
4,876 h
Low-rate example, not a warranty
3,116 h
High-penalty comparison

Correct, not complete: validate the actual cell and current waveform.

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