Math Bridge: Source and Load Resistance

← Back to Electricity Fundamentals
Math BridgeElectronicsStruggle-friendly runway

Why is maximum power not maximum battery life?

Separate maximum delivered power from battery efficiency using a source internal-resistance model.

Eddie, the electronics guideEddie guides
The one targetCompute current, load power, source heating, and efficiency together.
The chapter case1.50 V alkaline cell, 0.150 ohm internal resistance, and a 330 ohm node load.
What it buys youA load choice that does not confuse a power theorem with efficient operation.

A field team faces an unresolved physical question: Why is maximum power not maximum battery life? They must answer it before changing load resistance 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 load resistance. The middle card applies this page's relationship. The green card is load/internal ratio. 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.

Load resistance changes load/internal ratio An input card leads through the page relationship to the load/internal ratio result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. Maximum power occurs at equal resistances because current and load share balance there; maximum efficiency pushes load resistance much higher.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for load resistance is 330.

  2. 2

    Name the relationship. I=1.50/(330+0.150)=4.543 mA Vload=1.499 V Pload=6.81 mW Pinternal=0.00310 mW efficiency=330/(330.150)=99.955%

  3. 3

    Substitute the chapter fixture. Set load resistance to 330. The page ledger gives load/internal ratio as 2200.0 times.

  4. 4

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

Predict, then change load resistance

Try Predict the direction of load/internal ratio. Move one control, calculate, then check your prediction.

330
Chapter baseline
Load/internal ratio

Observe Maximum power occurs at equal resistances because current and load share balance there; maximum efficiency pushes load resistance much higher. Reset the control to 330 and compare load/internal ratio.

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

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

1. Start with the physical story

A real cell hides resistance inside itself. The same current that powers the load also heats that resistance, so delivered power and efficiency are separate questions.

Eddie: The matched-load theorem maximises power at 50% efficiency; an IoT node usually wants far less current and far less internal heat.

2. Name every algebra move

1

Add series resistanceRtotal=Rint+Rload.

2

Find currentI=Voc/Rtotal.

3

Find load voltageVload=IRload.

4

Split powerPload=I^2Rload and Pint=I^2Rint.

5

Form efficiencyeta=Pload/(Pload+Pint).

3. Reproduce the chapter case

I=1.50/(330+0.150)=4.543 mA
Vload=1.499 V
Pload=6.81 mW
Pinternal=0.00310 mW
efficiency=330/(330.150)=99.955%

At Rload=Rint=0.150 ohm the load power is maximal but half the source power becomes heat.

4. Try one real input

TryMove the control, predict the direction, then compare every output.

Load resistance
Load/internal ratio
Current
Load voltage
Load power
Internal heat
Efficiency

ObserveIncreasing load resistance cuts current and source heating while efficiency rises, although delivered load power eventually falls.

ExplainMaximum power occurs at equal resistances because current and load share balance there; maximum efficiency pushes load resistance much higher.

Technical boundaries.

This is a transparent first-order teaching ledger tied to the chapter constants.

Cell model
Internal resistance is treated as fixed and purely ohmic.
Dynamics
Pulse sag, chemistry, capacity, recovery, ageing, and temperature are omitted.
Load
A real node changes resistance across sleep, radio, and actuator states.

Correct, not complete: this ledger does not qualify a battery, regulator, pulsed load, or runtime claim.

5. Use the result in the design

Evaluate every load state and pulse, compare sag with brownout limits, then use a cell model or measured profile for lifetime.

6. Record the evidence state

Record open-circuit voltage, pulse current, loaded voltage, inferred internal resistance, temperature, state of charge, pulse duration, and recovery.

7. Check yourself

Why is matched load only 50% efficient?
Answer: Equal source and load resistances dissipate equal I-squared-R power.
Why not choose infinite load resistance?
Answer: Efficiency approaches 100%, but current and useful delivered power approach zero.
Does 330 ohms predict battery life?
Answer: No. Capacity, duty cycle, regulator loss, chemistry, temperature, and ageing are still needed.
Honesty boundary.

The arithmetic reproduces the named chapter case; it is an inspectable model, not a component approval.

Cell model
Internal resistance is treated as fixed and purely ohmic.
Dynamics
Pulse sag, chemistry, capacity, recovery, ageing, and temperature are omitted.
Load
A real node changes resistance across sleep, radio, and actuator states.

Correct, not complete: this ledger does not qualify a battery, regulator, pulsed load, or runtime claim.

Return to Electricity Fundamentals with the maths unpacked →