Math Bridge: MQTT Load and Battery Self-Discharge

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Math BridgeMQTTStruggle-friendly runway

What does an 833-day MQTT estimate leave out?

Put daily load, elapsed months, compounding self-discharge, a deeper Wi-Fi estimate, and pulse sag on the same review page.

Eddie, the electronics guideEddie guides
The one targetShow how a long C/I lifetime creates time for self-discharge to matter.
The chapter case2,000 mAh, 2.4 or 76.8 mAh/day, 2%/month, and a 120 mA pulse.
What it buys youA reason to challenge an attractive always-on QoS0 battery estimate.

A field team faces an unresolved physical question: What does an 833-day MQTT estimate leave out? They must answer it before changing daily charge 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 daily charge. The middle card applies this page's relationship. The green card is naive lifetime. 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.

Daily charge changes naive lifetime An input card leads through the page relationship to the naive lifetime result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. A very low daily-load claim creates a long time horizon; long horizons expose losses that a simple capacity/load division cannot represent.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for daily charge is 2.4.

  2. 2

    Name the relationship. naive days = 2,000/2.4 = 833.3 days months = 833.3/30.4375 = 27.38 months remaining shelf charge = 2,000(0.98)^27.38 = 1,150 mAh self-discharge loss = 42.5% of nameplate fresh pulse sag = 0.120(0.10) = 0.012 V

  3. 3

    Substitute the chapter fixture. Set daily charge to 2.4. The page ledger gives naive lifetime as 833.3 days.

  4. 4

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

Predict, then change daily charge

Try Predict the direction of naive lifetime. Move one control, calculate, then check your prediction.

2.4
Chapter baseline
Naive lifetime

Observe A very low daily-load claim creates a long time horizon; long horizons expose losses that a simple capacity/load division cannot represent. Reset the control to 2.4 and compare naive lifetime.

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

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

1. Start with the physical story

Dividing capacity by daily load treats the pack like a bucket emptied only by the device. A lithium pack also loses stored charge with time. A low load predicts more elapsed months, so self-discharge gets more time to compete.

Eddie: First calculate the naive days. Then use those same days to ask how much shelf charge remains.

2. Name every algebra move

1

Find naive daysDivide 2,000 mAh by mAh/day.

2

Convert days to monthsDivide by the average 30.4375 days per month.

3

Compound remaining chargeMultiply 2,000 by 0.98 raised to the number of elapsed months.

4

Find lossSubtract remaining charge from nameplate capacity.

5

Check pulse voltageUse IR separately for the 120 mA radio burst.

3. Reproduce the chapter case

naive days = 2,000/2.4 = 833.3 days
months = 833.3/30.4375 = 27.38 months
remaining shelf charge = 2,000(0.98)^27.38 = 1,150 mAh
self-discharge loss = 42.5% of nameplate
fresh pulse sag = 0.120(0.10) = 0.012 V

The compounding result is not an improved lifetime prediction by itself. It demonstrates that the naive load-only claim spans enough time for a second loss mechanism to become material.

4. Try one real input

TryMove daily load from the attractive 2.4 mAh/day estimate toward the chapter's 76.8 mAh/day deeper model. Watch naive days and self-discharge exposure fall together.

Daily charge
Naive lifetime
Elapsed months
Shelf-model remaining charge
Shelf-model remaining
Self-discharge loss
Self-discharge loss share
Deep-sleep estimate
Fresh pulse sag
Fresh loaded voltage

ObserveAt 2.4 mAh/day, the load-only result spans 27.4 months and the separate shelf model loses about 42.5%. At 76.8 mAh/day, the 26-day load dominates and shelf loss is small.

ExplainA very low daily-load claim creates a long time horizon; long horizons expose losses that a simple capacity/load division cannot represent.

Technical boundaries.

This ledger compares two chapter load estimates with one simple compounding shelf model.

Load
Daily charge is treated as constant; reconnects, signal quality, TLS, retries, sensors, regulator loss, and firmware states are not simulated.
Cell
Two percent per month is an illustrative constant rate, not a qualified LiPo ageing or temperature model.
Combination
Load draw and self-discharge compete for the same changing state of charge; the two independent ledgers must not be subtracted as a finished lifetime solver.

Correct, not complete: this ledger does not predict ESP32 field life or prove a QoS choice.

5. Use the result in the design

Measure a full connect-publish-ack-sleep cycle at representative signal strength, include reconnection and broker security, then fit a battery model that combines load, ageing, temperature, cutoff, and usable capacity.

6. Record the evidence state

Record firmware, Wi-Fi association time, MQTT session state, QoS, TLS, publish interval, current trace, daily charge, pulse voltage, pack chemistry, temperature, age, self-discharge source, and cutoff.

7. Check yourself

Why does lower device load increase the self-discharge percentage shown?
Answer: Lower load produces more elapsed months, so the separate monthly shelf-loss process has longer to compound.
Can we subtract the displayed shelf loss from 2,000 mAh and divide again for the true life?
Answer: Not reliably. Load and self-discharge act together on a changing cell, so a coupled battery model or measurement is needed.
Does 12 mV fresh pulse sag settle the power design?
Answer: No. Resistance rises with age and temperature, and wiring, regulator response, capacitance, and real pulse shape also matter.
Honesty boundary.

This ledger compares two chapter load estimates with one simple compounding shelf model.

Load
Daily charge is treated as constant; reconnects, signal quality, TLS, retries, sensors, regulator loss, and firmware states are not simulated.
Cell
Two percent per month is an illustrative constant rate, not a qualified LiPo ageing or temperature model.
Combination
Load draw and self-discharge compete for the same changing state of charge; the two independent ledgers must not be subtracted as a finished lifetime solver.

Correct, not complete: this ledger does not predict ESP32 field life or prove a QoS choice.

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