Math Bridge: Resistor Temperature Drift

A field team faces an unresolved physical question: Why does a 260 ohm resistor change when it gets hot? They must answer it before changing temperature 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 temperature. The middle card applies this page's relationship. The green card is temperature change. 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.

Walk the arrows. The coefficient changes R first; current and power then recompute from that same R instead of from a memorised nominal value.

Derive the baseline in four named moves

1
Name the input. The chapter baseline for temperature is 85.
2
Name the relationship. ΔT=85-25=60°C Rfilm=260[1+(100x10⁻⁶)(60)]=261.56 ohm driftfilm=(261.56/260-1)100=0.600% Rcopper=260[1+(0.00393)(60)]=321.31 ohm Iled=(3.3-2.0)/261.56=4.97 mA
3
Substitute the chapter fixture. Set temperature to 85. The page ledger gives temperature change as 60.0 degrees C.
4
Read the result. Keep degrees C beside the value. Use it only inside the technical boundary on this page.

Predict, then change temperature

Try Predict the direction of temperature change. Move one control, calculate, then check your prediction.

Observe The coefficient changes R first; current and power then recompute from that same R instead of from a memorised nominal value. Reset the control to 85 and compare temperature change.

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

1. Start with the physical story

A resistor is not a perfect fixed number. As a metal warms, its lattice vibrates more and scatters charge carriers more often. A purpose-built metal-film resistor is chosen so this effect is small; copper changes much more.

Eddie: Temperature does not break Ohm's law. It changes the resistance that Ohm's law must use.

2. Name every algebra move

Find the temperature changeSubtract 25°C from the operating temperature.

Turn ppm into a fraction100 ppm/°C is 0.000100 per °C.

Scale the resistanceMultiply 260 Ω by 1+αΔT.

Find resistor voltageSubtract the 2.0 V LED drop from 3.3 V.

Check current and heatUse I=V/R, then P=I²R.

3. Reproduce the chapter case

ΔT=85−25=60°C
Rfilm=260[1+(100×10⁻⁶)(60)]=261.56 Ω
driftfilm=(261.56/260−1)100=0.600%
Rcopper=260[1+(0.00393)(60)]=321.31 Ω
Iled=(3.3−2.0)/261.56=4.97 mA

The metal-film part moves only 1.56 Ω at 85°C. The equal-value copper path moves 61.31 Ω, showing why material belongs in the assumption.

4. Try one real input

TryMove temperature and predict resistance, LED current, and resistor heat.

ObserveWarming raises both resistances, but copper changes 39.3 times faster. The higher resistor value slightly lowers current and heat.

ExplainThe coefficient changes R first; current and power then recompute from that same R instead of from a memorised nominal value.

Technical boundaries.

This is a linear local temperature model with fixed LED forward voltage and GPIO voltage.

Resistor: Part tolerance, self-heating, ageing, humidity, and nonlinear TCR can add error.
LED: Forward voltage and brightness change with current, temperature, colour, and part spread.
GPIO: The loaded pin voltage and source limit must be measured on the real controller.

Correct, not complete: this ledger does not qualify a resistor, LED, or GPIO over temperature.

5. Use the result in the design

Choose the next higher standard resistor, check current at voltage and temperature limits, and compare calculated dissipation with a derated package rating.

6. Record the evidence state

Record resistor technology, value, tolerance, TCR, temperature range, LED part and voltage, GPIO level and limit, measured current, and measured hot resistance.

7. Check yourself

What does 100 ppm/°C mean?

Answer: A fractional change of 100/1,000,000, or 0.000100, for each degree Celsius.

Why does LED current fall as this resistor warms?

Answer: The model keeps resistor voltage fixed while R rises, so I=V/R falls.

Does a 0.600% calculation certify the part?

Answer: No. It omits tolerance, self-heating, nonlinear drift, package limits, and measured circuit conditions.

Honesty boundary.

The arithmetic reproduces the chapter's 260 Ω and 100 ppm/°C temperature case and carries it into the chapter's 3.3 V LED loop.

Resistor: Part tolerance, self-heating, ageing, humidity, and nonlinear TCR can add error.
LED: Forward voltage and brightness change with current, temperature, colour, and part spread.
GPIO: The loaded pin voltage and source limit must be measured on the real controller.

Correct, not complete: this ledger does not qualify a resistor, LED, or GPIO over temperature.