Math Bridge: Offloading Energy and Service Days

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Math BridgeEnergy & PowerStruggle-friendly runway

How does per-event energy become days of service?

Convert cell charge to usable energy, then price the chapter's local and Wi-Fi paths at a chosen event rate.

Battery Bruno, the energy and power guideBattery Bruno guides
The one targetTurn placement energy into a service interval.
The chapter case900 mJ local, 261 mJ remote, and 1,440 events/day.
What it buys youA placement comparison tied to battery reserve and cadence.

A field team faces an unresolved physical question: How does per-event energy become days of service? They must answer it before changing events/day 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 events/day. The middle card applies this page's relationship. The green card is nameplate 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 added model holds every other chapter fixture fixed, so the numeric fixture does not switch without explanation.

Events/day changes nameplate energy An input card leads through the page relationship to the nameplate energy result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. Cadence multiplies both daily ledgers equally; it does not change the energy ratio of the two event paths.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for events/day is 1440.

  2. 2

    Name the relationship. Ecell=0.150 Ahx3.7 V=0.555 Wh=1,998,000 mJ Eusable=1,998,000x0.98x0.85=1,664,334 mJ Elocal/day=900x1440=1,296,000 mJ Eremote/day=261x1440=375,840 mJ service=1.28 days local or 4.43 days remote

  3. 3

    Substitute the chapter fixture. Set events/day to 1440. The page ledger gives nameplate energy as 1998 J.

  4. 4

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

Predict, then change events/day

Try Predict the direction of nameplate energy. Move one control, calculate, then check your prediction.

1440
Chapter baseline
Nameplate energy

Observe Cadence multiplies both daily ledgers equally; it does not change the energy ratio of the two event paths. Reset the control to 1440 and compare nameplate energy.

Explain Only events/day 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 events/day moves. Field effects named in the page's technical boundary stay fixed.

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

1. Start with the physical story

Power multiplied by time gives event energy. Battery charge becomes energy only after multiplying by voltage, and some of that energy must be removed for self-discharge and reserve.

Battery Bruno: Per-event winners keep the same ratio at a fixed cadence, but cadence sets how quickly both budgets disappear.

2. Name every algebra move

1

Convert chargeMultiply 0.150 Ah by 3.7 V.

2

Convert unitsMultiply Wh by 3.6 million to get mJ.

3

DerateApply 98% retention and keep a 15% reserve.

4

Price a dayMultiply each event energy by events per day.

5

Find serviceDivide usable energy by daily energy.

3. Reproduce the chapter case

Ecell=0.150 Ah×3.7 V=0.555 Wh=1,998,000 mJ
Eusable=1,998,000×0.98×0.85=1,664,334 mJ
Elocal/day=900×1440=1,296,000 mJ
Eremote/day=261×1440=375,840 mJ
service=1.28 days local or 4.43 days remote

The 3.45× service advantage matches 900/261 because both paths share the same cadence and usable battery.

4. Try one real input

TryChange event cadence and predict both service intervals before reading them.

Events/day
Nameplate energy
Usable energy
Local daily energy
Remote daily energy
Local service
Remote service
Service advantage

ObserveDoubling cadence halves both service intervals but leaves their 3.45× ratio unchanged.

ExplainCadence multiplies both daily ledgers equally; it does not change the energy ratio of the two event paths.

Technical boundaries.

This compares two fixed event ledgers, not whole-device battery life.

Radio state
Retries, reconnects, and tail energy must be measured.
Battery
Voltage, retention, and reserve are illustrative.
Service
Latency, privacy, availability, and result quality still gate offloading.

Correct, not complete: an energy win does not by itself approve remote placement.

5. Use the result in the design

Measure local compute, upload, wait, download, transition, retry, and preprocessing energy under the same workload and service boundary.

6. Record the evidence state

Keep event cadence, payload, radio state, network condition, local model, remote response, energy trace, reserve, latency, privacy, and fallback rule.

7. Check yourself

Why multiply Ah by volts?
Answer: Ampere-hours count charge; voltage turns that charge into energy.
Why does cadence not change the 3.45× ratio?
Answer: The same event count multiplies both path energies.
Does 4.43 days prove Wi-Fi offload is correct?
Answer: No. Service, privacy, retry, and network evidence remain necessary.
Honesty boundary.

The arithmetic extends the chapter's fixed event energies through an illustrative wearable cell.

Radio state
Retries, reconnects, and tail energy must be measured.
Battery
Voltage, retention, and reserve are illustrative.
Service
Latency, privacy, availability, and result quality still gate offloading.

Correct, not complete: an energy win does not by itself approve remote placement.

Return to Context Energy Offloading with the maths unpacked →