3.4 Temperature effects on resistance

Temperature affects resistance in all materials, changing how they conduct electricity. Understanding this relationship helps you predict how circuits behave when conditions change, whether that's a wire heating up under load or a sensor responding to a temperature shift.

The core idea: resistance can increase or decrease with temperature depending on the material. Metals typically show increased resistance when heated, while semiconductors often show decreased resistance. Knowing which way a material goes (and by how much) is essential for designing and troubleshooting circuits.

Temperature Coefficients

Impact of Temperature on Resistance

The temperature coefficient of resistance quantifies how much a material's resistance changes per degree of temperature change. It's represented by the Greek letter alpha ($\alpha$) and typically expressed in units of $\frac{1}{°C}$.

Why does resistance change at all? In metals, higher temperatures cause atoms to vibrate more, which makes it harder for electrons to flow through the lattice. That increased "scattering" of electrons raises resistance. In semiconductors, higher temperatures free up more charge carriers, which actually makes it easier for current to flow, lowering resistance.

Types of Temperature Coefficients

• Positive Temperature Coefficient (PTC): Resistance increases as temperature rises. Most metals behave this way, including copper and aluminum.
• Negative Temperature Coefficient (NTC): Resistance decreases as temperature rises. Semiconductors and certain ceramics fall into this category.

The sign of $\alpha$ tells you which type you're dealing with. A positive $\alpha$ means PTC; a negative $\alpha$ means NTC. The magnitude of $\alpha$ tells you how sensitive the material is to temperature changes.

Calculating Resistance Change

You can calculate the new resistance at a given temperature using this formula:

$R_T = R_0[1 + \alpha(T - T_0)]$

• $R_T$ = resistance at the new temperature $T$
• $R_0$ = resistance at the reference temperature $T_0$
• $\alpha$ = temperature coefficient of resistance for that material

Example walkthrough: Find the resistance of a copper wire at $50°C$ if it has $R_0 = 10\,\Omega$ at $T_0 = 20°C$ and $\alpha = 0.00393\,\frac{1}{°C}$.

1. Find the temperature change: $T - T_0 = 50 - 20 = 30°C$

2. Multiply by $\alpha$: $0.00393 \times 30 = 0.1179$

3. Add 1: $1 + 0.1179 = 1.1179$

4. Multiply by $R_0$: $10 \times 1.1179 = 11.18\,\Omega$

So the resistance increases by about 12% over that 30°C rise. That's a meaningful change if you're designing a circuit that needs to stay accurate across temperatures.

Temperature-Sensitive Devices

Thermistors

Thermistors are resistors specifically designed to have a large, predictable resistance change with temperature. Most common thermistors are NTC types, meaning their resistance drops significantly as temperature rises.

This makes them useful as temperature sensors. You measure the thermistor's resistance, and because the resistance-temperature relationship is well-characterized, you can determine the temperature. Common applications include:

• Temperature sensors in home appliances (ovens, refrigerators)
• Automotive engine temperature monitoring
• Industrial process control
• Thermal protection circuits that shut down equipment if it overheats

Linear Approximation

Over small temperature ranges, the resistance-temperature relationship of a thermistor can be treated as roughly linear. This simplifies the math considerably and makes thermistors easier to integrate into control systems.

The linear approximation formula looks just like the general resistance formula:

$R_T \approx R_0[1 + \beta(T - T_0)]$

Here, $\beta$ is the thermistor's temperature coefficient (in $\frac{1}{°C}$). For NTC thermistors, $\beta$ is negative. This approximation works well for temperature changes of about $\pm 10°C$ to $\pm 20°C$ from the reference point. Beyond that range, the actual relationship is nonlinear and this formula becomes less accurate.

Reference Temperature and Resistance

Thermistor datasheets specify a reference temperature ($T_0$) and the resistance at that temperature ($R_0$). Common reference temperatures are $25°C$ and $20°C$. These values serve as your starting point for any resistance calculation.

Example walkthrough: A thermistor has $\beta = -0.04\,\frac{1}{°C}$, $R_0 = 1000\,\Omega$ at $T_0 = 25°C$. Find its approximate resistance at $T = 30°C$.

1. Find the temperature change: $30 - 25 = 5°C$

2. Multiply by $\beta$: $-0.04 \times 5 = -0.20$

3. Add 1: $1 + (-0.20) = 0.80$

4. Multiply by $R_0$: $1000 \times 0.80 = 800\,\Omega$

The resistance dropped by 20% with just a 5°C increase. That large sensitivity is exactly what makes thermistors useful as temperature sensors, but it also shows why the linear approximation breaks down over wider temperature ranges.