Math Bridge: Why is a 4000 mAh phone not a 4000 mAh gateway?

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Why is a 4000 mAh phone not a 4000 mAh gateway?

Connect charge, energy, derating, shift current, self-discharge, and burst voltage sag for a mobile gateway.

Gateway Gus, the guideGateway Gus guides
The one targetTurn a phone label into a bounded gateway power budget.
The chapter case4000 mAh at 3.85 V, 20% reserve, eight hours, 1.5 A burst.
What it buys youA runtime and brownout guardrail that names its assumptions.

A field team faces an unresolved physical question: Why is a 4000 mAh phone not a 4000 mAh gateway? They must answer it before changing peak current 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 peak current. 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.

Peak current 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. The reserve limits average gateway work, while internal resistance creates a separate instantaneous voltage limit during radio and screen bursts.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for peak current is 1.5.

  2. 2

    Name the relationship. E=15.40 Wh; Cusable=3200 mAh; Eusable=12.32 Wh; Isustainable=400 mA Vsag=IRint; at 1.5 A: 0.225 V and Vterminal=3.625 V

  3. 3

    Substitute the chapter fixture. Set peak current to 1.5. The page ledger gives nameplate energy as 15.40 Wh.

  4. 4

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

Predict, then change peak current

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

1.5
Chapter baseline
Nameplate energy

Observe The reserve limits average gateway work, while internal resistance creates a separate instantaneous voltage limit during radio and screen bursts. Reset the control to 1.5 and compare nameplate energy.

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

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

1. Start with the physical question

Turn a phone label into a bounded gateway power budget. A runtime and brownout guardrail that names its assumptions.

Gateway Gus: Keep the units and the model boundary visible from the first line.

2. Name every algebra move

1

Convert mAh to AhDivide by 1000 before multiplying by volts.

2

Hold the reserveMultiply charge and energy by 0.80.

3

Spread charge across the shiftIsustainable=Cusable/t.

4

Apply Ohm’s law to the burstVsag=IRint.

5

Subtract the sagVterminal=Voc−Vsag.

3. Reproduce the chapter case

E=15.40 Wh; Cusable=3200 mAh; Eusable=12.32 Wh; Isustainable=400 mA
Vsag=IRint; at 1.5 A: 0.225 V and Vterminal=3.625 V

The arithmetic reproduces the chapter case while keeping its assumptions explicit.

4. Try the controlling input

TryMove the control and watch every displayed result come from the shown formula.

Peak current
Nameplate energy
Usable capacity
Usable energy
Shift current
Shift self-discharge
Voltage sag
Terminal voltage

ObserveAt 1.50 A, nameplate energy is 15.40 Wh, usable capacity is 3200 mAh, shift current is 400.000 mA, and terminal voltage is 3.625 V.

ExplainThe reserve limits average gateway work, while internal resistance creates a separate instantaneous voltage limit during radio and screen bursts.

Technical boundaries.

This compact engine isolates one relationship; it is not a deployment certificate.

Phone load
The 400 mA budget excludes calls, display, navigation, and other apps
Battery
Temperature, ageing, cutoff curve, and power-conversion loss are omitted
Burst
One resistance value cannot qualify every state of charge

Measure the real system and reopen the decision when its inputs change.

5. Budget average and burst separately

Measure the gateway duty-cycle average with ordinary user load, then capture worst radio, GPS, encryption, and display bursts against the real shutdown threshold.

6. Keep the mobile power record

Record phone model, battery health, usable reserve, shift target, user workload, radio policy, average and peak current, voltage floor, temperature, result, owner, and retest trigger.

7. Check yourself

Why is nameplate energy 15.40 Wh?
Answer: 4.000 Ah multiplied by 3.85 V equals 15.40 Wh.
Why is the eight-hour current 400 mA?
Answer: The 20% reserve leaves 3200 mAh; 3200/8=400 mA.
Does a 3.625 V result prove the phone will stay on?
Answer: No. The actual cutoff, state of charge, ageing, temperature, converter behavior, and simultaneous load must be measured.
Honesty boundary.

The worked values are traceable chapter examples or explicitly labelled teaching assumptions.

4000 mAh and 3.85 V
Representative modern phone pack
20%
Explicit operating reserve
0.150 Ω
Catalog-typical teaching resistance

Correct, not complete: field evidence still decides acceptance.

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