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 *R _{0}* at 0 °C, and it has a
resistance of

*R*at

_{t}*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.

(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, *α*_{0}* *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 *α*_{0}*, **α** _{1}*,
and

*α*

*are the temperature coefficient at 0 °C, t*

_{2}_{1}°C and t

_{2}°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

*R*are the resistances of a conductor at

_{2}*T*°C and

_{1}*T*°C respectively. If the temperature coefficient of resistance at

_{2}*T*°C is

_{1}*α*

*, then, the resistance*

_{1}*R*is expressed as

_{2}`\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.

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