# Temperature coefficient of resistivity (resistance) in function of temperature

If I'm correct, the definition of the temperature coefficient of resistivity at a certain reference temperature is the derivative of the resistivity in function of the temperature, divided by the reference resistivity.

Are there materials where the temperature coefficient is constant over different reference temperatures? I only seem to find its value at reference temperature 20 degree Celsius... If it would be constant you'd get: $\alpha=\frac{\partial R(T)}{R(T)\partial T} \iff \alpha R(T)=R'(T)$ Which is an easy to solve differential equation and gives an exponential function for $R(T)$.

Could it be that simple or am I missing something?

The definition of the thermal coefficient of resistance (TCR) is the change in resistance per change in temperature divided by the resistance at a specified, fixed reference temperature: $$\mathrm{TCR} = \frac{1}{R(T_\mathrm{ref})} \left.{\frac{\mathrm{d}R}{\mathrm{d}T}}\right|_{T=T_{ref}}.$$ Note that $R(T_\mathrm{ref})$ is a fixed value. It is the resistance given at a single temperature $T_\mathrm{ref}$. It does not depend on $T$. Typically $T_\mathrm{ref}$ is specified to be $20^\circ\mathrm{C}$ or $25^\circ\mathrm{C}$ ~ room temperature. Less often $0^\circ\mathrm{C}$ is used for $T_\mathrm{ref}$.

With the above definition the TCR is a constant and the resistance can be written as a linear function of temperature around $T_{ref}$: \begin{gather} \mathrm{TCR} = \frac{1}{R(T_\mathrm{ref})} \frac{R(T)-R(T_\mathrm{ref})}{(T-T_\mathrm{ref})}; \\ R(T) = R(T_\mathrm{ref}) \big(1 + \mathrm{TCR} (T-T_\mathrm{ref})\big). \end{gather}

The TCR is going to be different for different reference temperatures. You have to provide the value used for $T_\mathrm{ref}$ along with the TCR! For example, suppose that I measure a resistor and find that its resistance is a linear function of temperature over a given temperature range (intercept $b$ and slope $m$): $$R(T) = b + mT.$$ Then the TCR for this resistor can be found using the definition of $\mathrm{TCR}$ and a given $T_\mathrm{ref}$: \begin{gather} \left.{\frac{\mathrm{d}R}{\mathrm{d}T}}\right|_{T=T_{ref}} = m; \\ R(T_\mathrm{ref}) = b + m T_\mathrm{ref}; \\ \mathrm{TCR} = \frac{1}{R(T_\mathrm{ref})} \left.{\frac{\mathrm{d}R}{\mathrm{d}T}}\right|_{T=T_{ref}} = \frac{m}{b+mT_\mathrm{ref}}. \end{gather}

Note that the TCR depends on the $T_\mathrm{ref}$ you specify. The TCR will be lower if you choose a larger $T_\mathrm{ref}$. The $T_\mathrm{ref}$ should be specified along with values given in material property tables.

In the end, the value of a TCR is that it allows you to estimate the resistance of a resistor as a function of temperature given only a measure of its resistance at $T_{ref}$.

One more thing. If you know the value $\mathrm{TCR}_1$ at reference temperature $T_{\mathrm{ref},1}$, then you can find $\mathrm{TCR}_2$ at a different $T_{\mathrm{ref},2}$ using this equation: $$\mathrm{TCR}_2 = \frac{\mathrm{TCR}_1}{1+\mathrm{TCR}_1(T_{\mathrm{ref},2}-T_{\mathrm{ref},1})}.$$

• I suggest to learn the latex markup used here, it is really simple with wonderful results. – peterh - Reinstate Monica Nov 27 '14 at 6:53
• Hi Mike Johnson, and welcome to Physics Stackexchange! In case you're not familiar with latex-style formatting, a basic intro can be found here. Far more thorough coverage is given here. – user10851 Nov 27 '14 at 8:16
• It was late and my brain did not want to remember latex for the equations. It has been while since I have had to use it. Thanks for the editing! – Mike Johnson Nov 29 '14 at 4:52
• And as noted below. Yes, using a TCR assumes that the resistivity is linear with temperature. So a TCR value is only valid for a limited temperature range around $T_{ref}$. I like the example in the NIST article (by R. A Matula) at the link in the answer below. It has a nice example for copper (Figures 5 and 6). – Mike Johnson Nov 29 '14 at 5:02

In addition to Mike's answer, there are a few things that need to be noted.

First, I would say that the existence of TCR is a consequence of the assumption that the resistivity is linear. It is not because the TCR is constant that the resistivity is linear. A given TCR is valid only for a specific range of temperature. Basically that is the slope of the tangent near the "middle" of the valid temperature range.

Second, the resistivity is not linear, expecially at really low temperatures, but also at higher temperatures. Many other laws do exist.

Sources and examples : http://www.nist.gov/data/PDFfiles/jpcrd155.pdf

http://www.memsnet.org/material/glasssio2bulk/