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?
Best Answer
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: $$ \mathrm{TCR}_2 = \frac{\mathrm{TCR}_1}{1+\mathrm{TCR}_1(T_{\mathrm{ref},2}-T_{\mathrm{ref},1})}. $$