Temperature Coefficient of Resistance – Definition, Formula and Example

In this article, we will discuss the temperature coefficient of resistance, its definition, formula and numerical example.

As we know, the resistance of a material is affected by the change in its temperature. In the case of conductors, the resistance of the conductor increases with the increase in temperature. Therefore, the conductors (or metals) have a positive temperature coefficient of resistance.

In the case of semiconductors, electrolytes and insulators, the resistance decreases with the increase in temperature. Consequently, these materials have a negative temperature coefficient of resistance.

What is the Temperature Coefficient of Resistance?

The temperature coefficient of the resistance is the factor that gives information about changes in the resistance of a material with the variation in the temperature.

The temperature coefficient of resistance can be defined as the change in the resistance of a material with respect to the per unit change in the temperature.

Relation between Temperature and Resistance

In order to derive the relation between temperature and resistance, consider a metallic conductor having a resistance of R₀ at 0 °C, and it has a resistance of R_t at t °C.

Experimentally, it has been found that in the normal range of temperatures, the change in resistance, i.e.

`\ΔR=R_t-R_0`

(1). Is directly proportional to the initial resistance, i.e.

`\ΔR∝R_0`

(2). Is directly proportional to the change in the temperature, i.e.

`\ΔR∝t`

(3). Depends on the nature of the conductor material.

On combining the first two equations, we get,

`\ΔR∝R_0 t`

`\⟹R_t-R_0∝R_0 t`

`\⟹R_t-R_0=α_0 R_0 t`

Where, α₀­ is a constant of proportionality, and it is called the temperature coefficient of resistance at 0 °C. The value of the temperature coefficient depends on the nature of the material and the temperature of the conductor.

On rearranging the above equation, we get,

`\R_t=R_0 (1+α_0 t)`

Therefore, the temperature coefficient of resistance at 0 °C is given by,

`\α_0=(R_t-R_0)/(R_0 t)`

Thus, the unit of temperature coefficient of resistance is per degree Celsius (/°C).

Temperature Coefficient at Different Temperatures

From the above discussion, we can also calculate the temperature coefficient of resistance at different temperatures.

Consider α₀, α₁, and α₂ are the temperature coefficient at 0 °C, t₁ °C and t₂ °C respectively. Then, the value of these temperature coefficients can be given by the following expressions.

`\α_1=α_0/(1+α_0 t_1 )`

And,

`\α_2=α_0/(1+α_0 t_2 )`

In general,

`\α_T=α_0/(1+α_0 T)`

Special Case

Consider R_1­ and R₂ are the resistances of a conductor at T₁ °C and T₂ °C respectively. If the temperature coefficient of resistance at T₁ °C is α₁, then, the resistance R₂ is expressed as

`\R_2=R_1 [1+α_1 (T_2-T_1 )]`

Numerical Example – The armature winding of an electric motor has a resistance of 15 Ω at 20 °C and 18 Ω at 60 °C. If the temperature coefficient of resistance at 0 °C is 0.00426 /°C, then find (i) the resistance of the winding at 0 °C, (ii) the temperature coefficient at 20 °C.

Solution – Given data,

`\R_{20}=15 Ω`

`\R_{60}=18 Ω`

`\α_0=0.00426  ⁄°C`

(i). The resistance of winding at 0 °C:

`\R_{20}=R_0 (1+α_0×20)`

`\⟹R_0=R_{20}/((1+20α_0 ) )`

`\⟹R_0=15/((1+20×0.00426) )`

`\∴R_0=13.82 Ω`

(ii). The temperature coefficient at 20 °C:

`\α_{20}=α_0/(1+α_0×20)`

`\⟹α_{20}=0.00426/(1+(0.00426×20) )`

`\∴α_0=0.003925  ⁄°C`

Therefore, in this article, we discussed the temperature coefficient of resistance and its definition along with a solved numerical example for a better understanding of the concept.