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By temperature coefficient of resistance of a material about a reference point $T_0$, I mean $${1\over R(T_0)} \left.{dR\over dT}\right|_{T_0}.$$

All the sources (that I’ve seen) that quote this value for different materials (for a given temperature range where it can be assumed constant) don’t specify the materials’ geometry. But that would imply that in the given temperature range, the resistances of all the geometries are linearly proportional to each other.

Is that really so?

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  • $\begingroup$ What do you mean by the material's geometry? $\endgroup$ – Marco Ocram Oct 13 at 17:50
  • $\begingroup$ @MarcoOcram Its 3d shape. $\endgroup$ – Atom Oct 13 at 17:52
  • $\begingroup$ Some materials are anisotropic, have different resistivities and temperature coefficients in different crystal directions. Is that what you are asking about? $\endgroup$ – Pieter Oct 13 at 23:35
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The temperature coefficient of a resistor depends on several factors in a complicated way. If we consider a resistor in the shape of an unconstrained right prism of length $L$ and cross-sectional area $A$, composed of a homogeneous and isotropic material with resistivity $\rho$, assuming uniform longitudinal current distribution, we can write its resistance with the well-known formula

$$R = \rho\frac{L}{A}\tag{1}$$

From here, taking the logarithmic derivative, we can determine the temperature coefficient of the resistance as

$$\frac{1}{R}\frac{\mathrm{d} R}{\mathrm{d} T} = \frac{1}{\rho}\frac{\mathrm{d} \rho}{\mathrm{d} T}+\frac{1}{L}\frac{\mathrm{d} L}{\mathrm{d} T}-\frac{1}{A}\frac{\mathrm{d} A}{\mathrm{d} T}\tag{2}.$$

For an isotropic prism, the thermal expansion is the same in all directions, such that

$$\frac{1}{A}\frac{\mathrm{d} A}{\mathrm{d} T} = \frac{2}{L}\frac{\mathrm{d} L}{\mathrm{d} T}.$$

Substituting the above in (2) yields

$$\frac{1}{R}\frac{\mathrm{d} R}{\mathrm{d} T} = \frac{1}{\rho}\frac{\mathrm{d} \rho}{\mathrm{d} T}-\frac{1}{L}\frac{\mathrm{d} L}{\mathrm{d} T},$$

or

$$\alpha_R = \alpha_\rho-\alpha_L,\tag{3}$$

where with $\alpha_X$ I denoted the temperature coefficient of the quantity $X$.

Remark 1. Equation (1) is valid only with the assumption listed at the beginning. For a general geometry that equation no longer holds, but probably scaling arguments lead to the same equation (3). So, the answer to your question, if I interpret it correctly, is that under the stated assumptions the temperature coefficient of resistance does not depend (too much) on the shape and size of the resistive element.
Remark 2. Equation (3) tells you that in the temperature coefficient of resistance, there are (at least) two components, one associated to the material, actually the most significant, one associated to the geometry.

This is not the end of the story, though. In a real resistor, the resistive element is usually constrained. For example, a film resistor is usually composed of a resistive film deposited on a ceramic substrate. When the temperature changes, if the thermal expansion coefficients of the two materials don't match, the substrate strains the resistive element which, in turn, responds to the strain with an additional change of resistance. This is due to a phenomenon called piezoresistivity (I wrote a bit about it in this answer). In this common case, (3) is no longer valid and should be modified as follows (I shall simplify a number of details). Let $\alpha_S$ be the thermal expansion coefficient of the substrate. The relative longitudinal expansion of the constrained resistive element is then $\alpha_S-\alpha_L$ and due do the piezoresistive effect the resistance changes by $F(\alpha_S-\alpha_L)$, where $F$ is the effective sensitivity coefficient due to the piezoresistive effect. This effective coefficient embeds also the transverse strain and sensitivity, which may depend on the shape of the resistive element. As a result, we can write the overall temperature coefficient as

$$\alpha_R = \alpha_\rho-\alpha_L+F(\alpha_S-\alpha_L).\tag{4}$$

Therefore, for most real resistors the resistance temperature coefficient not only does it depend on the resistive material, but also on the substrate material and on the shape of the whole structure.

Remark 3. When you need a very predictable and stable temperature coefficient you therefore want to leave the resistive element as unconstrained as possible. This is done, for instance, in so-called standard platinum resistor thermometers (SPRTs), which are used in a wide range of temperatures to interpolate between known thermodynamic temperatures (and which may cost up to several thousand euros).
Remark 4. Equation (4) is exploited by a couple of resistor manufacturers to design resistors with a very low temperature coefficient, of the order of $10^{-6}/\mathrm{K}$ or less. This is achieved by carefully combining the thermal properties of the resistive element and those of the substrate.

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But that would imply that in the given temperature range, the resistances of all the geometries are linearly proportional to each other.

Unless I am misunderstanding what you are saying, I don't believe that is true.

The temperature coefficient of resistance is a material property. The electrical resistance of a conductor depends on its geometry. I don't believe you can make a general statement about the electrical resistance of all geometries being linearly proportional to each other.

For example, how would you compare the relationship between the resistance of circular conductor in the top figure below, to the resistance of conductor in the figure below it? Obviously for the same material (same coefficient of resistance), the resistance of the conductor in the top figure is linearly proportional to its length. But the resistance of the conductor in the bottom figure is not linearly proportional to its length. Given that, you can't say the electrical resistance of these two conductors are "linearly proportional to each other".

Hope this helps.

enter image description here

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The temperature coefficient of resistance is simply a measure of how resistivity varies with temperature. It is analogous to the coefficient of expansion, which is a measure of how the density of a material changes with temperature, which is independent of the shape the material forms. It doesn't imply that the resistance of all geometries are proportional, but it does imply that the percentage change of resistance with temperature is the same regardless of geometry.

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