Math Bridge: From Message Energy to a Battery Bound

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Math BridgeApplication ProtocolsStruggle-friendly runway

Why do 11.4 J and 25.9 J not predict battery life?

Turn the chapter energy-per-message values into annual radio energy, voltage sag, and an honest upper bound on service life.

Bex, the guideBex guides
The one targetBuild the radio-only battery ledger.
The chapter case525,600 messages: 11.4 J CoAP, 25.9 J MQTT.
What it buys youSee why protocol energy alone cannot predict battery life.

A technician must decide whether annual coap radio energy is safe before changing messages per minute on the real device. The result is unresolved until the rule and units are checked. Predict the direction first.

See the relationship before changing it

The figure reads from left to right. The blue card is messages per minute. The middle card applies this page's rule. The green card is annual coap radio energy. 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 messages per minute, so the numeric fixture does not switch without explanation.

Messages per minute changes annual coap radio energy An input card leads through the rule energy = rate x 525,600 x 0.00002176 J to the annual coap radio energy result. INPUT PAGE INPUT APPLY THE RULE predict calculate check units OUTPUT RESULT
Walk the arrows. More messages repeat the same bounded radio cost more often.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 1 messages/min.

  2. 2

    Name the relationship. energy = rate x 525,600 x 0.00002176 J

  3. 3

    Substitute with units. 1 x 525,600 x 0.00002176 = 11.4 J

  4. 4

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

Predict, then change messages per minute

Try Predict the direction of energy = rate x 525,600 x 0.00002176 J. Test another messages per minute, then compare annual coap radio energy.

1 messages/min
Chapter baseline
Annual CoAP radio energy

Observe More messages repeat the same bounded radio cost more often. Reset messages per minute to 1 and compare annual coap radio energy.

Explain More messages repeat the same bounded radio cost more often.

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 messages per minute moves here. Field effects named in the technical boundary stay fixed.

Honesty note. This teaching model changes only messages per minute. It is not a field prediction.

1. Energy per event becomes energy per year

Multiplication is enough only when the event count is known. One message each minute means 60 each hour, 1,440 each day, and 525,600 each year. The units cancel in a chain: joules per message times messages per year gives joules per year.

Bex: Keep units and assumptions beside every number.

2. Name every algebra move

1

Count the eventsNyear=r×60×24×365.

2

Spend radio energyEyear=Nyear×Emessage.

3

Bound the yearstradio=Eusable/Eyear; this includes no other load.

3. The quotient is an upper bound

t_radio=E_usable/(N_year E_message); V_term=V_oc−(P/V_oc)R_int

The first formula asks how many identical radio events fit in a derated energy budget. The second checks whether the nominal voltage stays available during the pulse. Neither formula includes the rest of the device.

4. Try one controlled change

t_radio=E_usable/(N_year E_message); V_term=V_oc−(P/V_oc)R_int

TryIncrease only the message rate. The cell, derating, and per-message radio costs stay fixed.

Rate
Messages/year
Usable energy
Pulse sag
Pulse terminal
CoAP/year
MQTT/year
CoAP radio bound
MQTT radio bound

ObserveAt one message per minute, the widget reproduces 525,600 messages, 11.4 J for CoAP, 25.9 J for MQTT, and radio-only bounds near 166 and 73.4 years.

ExplainDoubling the message rate doubles annual radio energy and halves each radio-only bound. It still says nothing about sleep current or shelf life.

Technical boundaries.

The arithmetic uses an illustrative CR2032 model.

Capacity and derating
Need cell and temperature evidence
Internal resistance
Changes with pulse, age, and temperature
Radio energy
Excludes sensing, processing, listening, and retries

Measure pulse capability, cutoff, self-discharge, and converter loss before predicting service life.

5. Reproduce the chapter values

One message per minute gives 525,600 per year. CoAP uses 525,600×21.76 µJ=11.437 J/year; MQTT uses 525,600×49.28 µJ=25.902 J/year. The derated budget is 0.220 Ah×3.0 V×0.8=0.528 Wh=1900.8 J, so 1900.8/11.437=166 years and 1900.8/25.902=73.4 years. A 10 mW pulse draws 3.33 mA and sags about 0.050 V through 15 Ω.

6. Carry the evidence forward

Record payload size, headers, acknowledgements, retries, measured current trace, message rate, listening windows, sleep current, sensor and processor duty, cell lot, temperature, cutoff, and field packet results.

7. Check yourself

Why is 166 years not a battery forecast?
Answer: Its denominator contains only CoAP transmit energy. Every other drain and shelf limit is missing.
What happens at two messages per minute?
Answer: Annual radio energy doubles and each radio-only lifetime bound halves.
Why retain the sag check?
Answer: A cell can have charge remaining yet fail when a current pulse pulls its terminal below the device cutoff.
Honesty boundary.

The comparison reproduces the chapter's radio-only teaching case.

220 mAh and 80%
Illustrative usable capacity
15 Ω and 3.0 V
Illustrative cell and voltage
CoAP and MQTT totals
Computed radio energy, not device energy

Correct arithmetic is not a complete battery-life claim.

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