Temperature dependence of a metallic resistor

What does this quantity describe?

For pure metals, resistivity rises approximately linearly with temperature. Given resistance R0 at reference temperature T0, resistance R1 at another temperature T1 follows from R1 = R0·[1 + α·(T1 − T0)], where α is the material's temperature coefficient.

This works similarly to the thermal expansion of a metal bar: both effects are a good linear approximation for small to moderate temperature changes and are described by a material-specific constant — except here it is electrical resistance rather than length that is affected, because stronger lattice vibrations at higher temperature impede electron motion more.

Formula and variables

R1 = R0 · [1 + α · (T1 − T0)]

• R1 = R0·[1 + α·(T1 − T0)]
• α = (R1/R0 − 1)/(T1 − T0)
• T1 = T0 + (R1/R0 − 1)/α

Choose the inputs correctly

R0, T0, T1 and temperature coefficient α determine R1. α is a material property referenced to a specific temperature, usually 20 °C, and is assumed constant over the temperature range considered.

How to use the calculator

Select the target quantity. For resistance at the target temperature, enter R0, T0, T1 and α. To find a measured temperature coefficient instead, choose α as the target and enter two measured resistances with their temperatures.

Worked example

A 100 Ω copper resistor at 20 °C with α = 0.00393/K, heated to 80 °C: R1 = 100 Ω · [1 + 0.00393 · 60] = 123.58 Ω, an increase of 23.58 Ω or +23.6%.

Understand the result and units

A 23.6% resistance increase for a 60 K rise shows why metallic winding resistances and heating elements have noticeably higher resistance at operating temperature than at room temperature — an effect resistance thermometers (Pt100) deliberately exploit for temperature measurement.

R0 and R1 are given in Ω, T0 and T1 in °C, and α in 1/K, equivalent to 1/°C for temperature differences.

Why doesn't this apply to NTC and PTC thermistors?

NTC thermistors are made from sintered polycrystalline mixed-oxide ceramic and show a pronounced but strongly nonlinear exponential decrease in resistance with rising temperature, typically 3 to 6% per kelvin. PTC thermistors, made from sintered metal oxides, show a step-like resistance increase of several orders of magnitude above a reference temperature. Both effects are nonlinear enough that the linear equation used here — valid for pure metals with a small, approximately constant α — cannot represent them; NTC and PTC components need their own characteristic-curve models from the specific component datasheet.

Typical applications

The relationship is used to correct cable and winding resistance to operating temperature, to design resistance thermometers, and to estimate motor and transformer winding heating from a resistance measurement taken before and after operation.

Assumptions, limits and common mistakes

The linear approximation works well for pure metals over moderate temperature ranges but increasingly deviates from the actual, slightly curved characteristic at very large temperature changes. For NTC and PTC thermistors — semiconductor ceramics with strongly nonlinear, exponential or step-like temperature-resistance behavior — this linear formula does not apply; they need their own material-specific characteristic models, such as the B-parameter equation for NTC thermistors.

Common mistake: A common mistake is applying this linear formula unchanged to NTC or PTC components, even though their resistance change is many times larger and nonlinear. It is also easy to overlook that α itself is slightly temperature-dependent, making the linear approximation less accurate for very large temperature swings well above 100 K.