Math Bridge: The Fixed Leak in a Duty-Cycle Budget

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Math BridgeBluetooth and BLEStruggle-friendly runway

Why does the same leak matter 15× more to BLE?

Add self-discharge to the chapter’s SPP and BLE charge ledgers and show why the same leak matters more to the lower-power profile.

Radio Remi, the guideRadio Remi guides
The one targetAdd a fixed leakage current.
The chapter case2%/month: 27.8 µA and 0.667 mAh/day.
What it buys youKeep long battery estimates honest without erasing BLE’s win.

See the relationship before changing it

The figure reads from left to right. The blue card is monthly self-discharge. The middle card applies this page's rule. The green card is equivalent leak current. 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 monthly self-discharge, so the numeric fixture does not switch without explanation.

Monthly self-discharge changes equivalent leak current An input card leads through the rule leak = 1,000 mAh x monthly percent / 720 h to the equivalent leak current result. INPUT PAGE INPUT APPLY THE RULE predict calculate check units OUTPUT RESULT
Walk the arrows. The fixed cell leak can rival a low-duty Bluetooth profile current.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 2 %/month.

  2. 2

    Name the relationship. leak = 1,000 mAh x monthly percent / 720 h

  3. 3

    Substitute with units. 1,000 x 0.02 / 720 = 27.8 uA

  4. 4

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

Predict, then change monthly self-discharge

Try Predict the direction of leak = 1,000 mAh x monthly percent / 720 h. Test another monthly self-discharge, then compare equivalent leak current.

2 %/month
Chapter baseline
Equivalent leak current

Observe The fixed cell leak can rival a low-duty Bluetooth profile current. Reset monthly self-discharge to 2 and compare equivalent leak current.

Explain The fixed cell leak can rival a low-duty Bluetooth profile current.

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 monthly self-discharge moves here. Field effects named in the technical boundary stay fixed.

1. A fixed leak belongs in the denominator

A profile’s daily charge comes from its radio states. Self-discharge is different: it runs whether the radio is busy or asleep. Convert the monthly fraction to charge per day, then add it to each profile’s daily draw.

Radio Remi: Keep units and assumptions beside every number.

2. Name every algebra move

1

Convert the fractionQleak/day=C0 fmonth/30.

2

Add the daily drainsQtotal/day=Qprofile/day+Qleak/day.

3

Divide capacitylife=C0/Qtotal/day.

3. One leak, two relative effects

leak share=100(Q_leak/day)/(Q_profile/day)

The numerator is identical for both profiles. Dividing it by BLE’s smaller profile draw makes its relative share larger, even though BLE’s corrected life remains much longer.

4. Try one controlled change

leak share=100(Q_leak/day)/(Q_profile/day)

TryChange only the cell’s monthly self-discharge; both profile duty-cycle ledgers stay fixed.

Self-discharge
Leak current
Leak charge/day
SPP leak share
BLE leak share
SPP life
BLE life
BLE days lost
BLE shortfall
Corrected advantage

ObserveAt 2%, the 0.667 mAh/day leak is 0.208% of SPP draw but 3.12% of BLE draw. Corrected lives are about 3.119 and 45.4 days.

ExplainA fixed tax is most visible when the designed load is small. The correction lowers BLE’s advantage from 15.0× to about 14.6× without reversing the decision.

Technical boundaries.

The 2% per month self-discharge rate is catalog-typical.

Chemistry and age
Change the storage loss
Temperature and state
Change both leakage and capacity
Pulse load and cutoff
Change the usable charge

Measure the selected cell and electronics across the intended storage and load profile.

5. Reproduce the chapter values

For the chapter’s 1000 mAh cell at 2%/month, Qleak/day=1000×0.02/30=0.667 mAh/day and Ileak=0.667/24=0.0278 mA=27.8 µA. SPP life is 1000/(320+0.667)=3.119 days. BLE life is 1000/(21.36+0.667)=45.4 days, 1.4 days or about 3.0% below its 46.8-day charge-only result. The corrected ratio is about 14.6×.

6. Carry the evidence forward

Record chemistry, lot, age, storage and operating temperature, measured quiescent current, profile state timing, reconnect costs, capacity test, cutoff, and uncertainty beside the life claim.

7. Check yourself

Why is the leak not 2% of each profile draw?
Answer: It is defined from cell capacity, so it adds the same charge per day to both ledgers.
Why does BLE lose more days?
Answer: Its ideal life is longer, leaving more calendar time for the fixed leak to act.
Does the correction erase BLE’s advantage?
Answer: No. The chapter comparison remains about 14.6× under these illustrative assumptions.
Honesty boundary.

The comparison isolates profile current and one stated self-discharge rate.

1,000 mAh
Illustrative nominal capacity
2% per month
Illustrative storage loss
SPP and BLE life
Idealized current-only results

The ratio is useful for reasoning, not a warranty of field life.