Temperature-Dependent Resistance — Theory
Electrical resistance often varies with temperature because carrier mobility and scattering change with \(T\).
Metals typically show an approximately linear increase over moderate temperature ranges, while many semiconductors and
thermistors follow stronger, nonlinear (often exponential) behavior.
1) Metals (linear approximation)
For many metals near a reference temperature \(T_0\), a common approximation is:
This linear model is an approximation; at very high temperatures \(\alpha\) may not be constant, and radiation/geometry changes can matter.
2) NTC thermistors (Beta model)
A widely-used engineering model for many NTC thermistors is the Beta equation:
3) PTC devices (simple exponential preview)
Real PTC thermistors can have complex behavior, but a simple exponential can capture “rapid increase with temperature” qualitatively:
4) Power and self-heating (idea)
If a voltage is applied across a resistor, the dissipated power is:
\[
P=\frac{V^2}{R(T)}.
\]
Self-heating can increase temperature, which can further change \(R\). The calculator includes an optional “toy” warm-up demo for the metal model
using a first-order thermal system.
5) University note: superconductivity
In some materials at cryogenic temperatures, resistance can drop dramatically and may become effectively zero below a critical temperature.
That behavior is not captured by the linear or simple exponential models here.
6) Plot interaction note
The plot shows \(R\) versus \(T\). The cursor slider only moves a dot along the curve (useful for reading off values); it does not modify inputs.
Use mouse wheel to zoom and drag to pan; double-click resets.
