This page is a section of Ultimate Electronics: Practical Circuit Design and Analysis, a free, online, interactive electronics textbook by Michael F. Robbins.

## Practical Resistors: Temperature Coefficient

### Microscopic Origins of Resistor Temperature Dependence

As discussed in Resistance and Ohm's Law, it is the thermal velocity which dominates setting the mean time betwen electron-atom collisions.

As temperature goes up, resistance goes up in most resistive materials.

At higher temperatures, there's greater thermal energy, leading to higher electron velocity. This leads to more frequent collisions (reducing $t_{\text{collision}}$), which reduces the peak drift velocity reached before a collision occurs. This causes reduced current flow for the same external electric field, which manifests itself as higher resistance.

### Temperature Dependence of Resistance

The linearized model of resistivity has a material-dependent temperature coefficient $\alpha_{\text{material}}$:

$$\rho_T = \rho_{ref} + \alpha_{\text{material}} * (T - T_{ref})$$

While this is a linearized model, the real behavior is highly nonlinear. This is just an approximation over a small range.

Most materials have a positive temperature coefficient (PTC), $\alpha > 0$.

Some materials, however, have a negative temperature coefficient (NTC).

In many consumer, industrial, and other situations, we often want a zero temperature coefficient, so that our circuit's performance doesn't change over our operating temperature range.

Since resistance changes with temperature, we may be able to use a resistor as a temperature sensor.

### Self-Heating

A resistor, in normal operation, turns electrical energy into heat. The mass of the resistor heats up.

For a positive-temperature-coefficient resistor, the resistance also rises as the temperature goes up.

Normally, a new equilibrium is found. This equilibrium is at a temperature higher than ambient, and (for a PTC resistor) at a resistance higher than the resistance at ambient.

If a PTC resistor is driven by a constant current source, then when temperature goes up, so does the power dissipation (since $P = i^2 R$ and $R$ is increasing). This can lead to destructive runaway.

If a PTC resistor is driven by a constant voltage source, then when temperature goes up, the power dissipation goes down (since $P = \frac {v^2} {R}$).

In any case, using any resistor as a temperature sensor always involves running a current through it, so there will always be self-heating, which can bias the measurement.

### Fuses

As discussed in the Practial Resistors: Power Rating (Wattage) section, it is possible that no equilibrium is reached before the resistor material heats up to a point where the material changes dramatically and possibly fails permanently.

### Resistor as Temperature Sensor

There are three common types of resistors used as temperature sensors:

• PTC Thermistor. Highly nonlinear resistance versus temperature curve, but an overall positive relationship.
• NTC Thermistor. Highly nonlinear resistance versus temperature curve, but an overall negative relationship.
• RTD, such as Pt100. Almost linear, but not quite, over an extraordinarly wide temperature range.

Thermistors are often used intentionally with self-heating effects for current limiting or as self-resetting fuses.

RTDs are often used for precision temperature measurement.

### What's Next

In the next section, Practical Resistors: Potentiometers, we'll discuss mechanically adjustable resistors, also known as potentiometers.

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