TEMPERATURE CO-EFFICIENT OF RESISTANCE

Consider a metallic resistor having a resistance of R₀ and 0°C and R_t at t°C. The increase in resistance (R_t − R₀) is directly proportional to its initial resistance, that is,

 

(R_t − R₀) ∝ R₀

(R_t − R₀) is directly proportional to rise in temperature, that is

 

(R_t − R₀) ∝ t

 (R_t − R₀) depends on the nature of its material.

Thus,

 

(R_t − R₀) ∝ R₀t

 or

 

(R_t − R₀) = α₀R₀t        (1.1)

 

where α₀ is a constant called temperature coefficient of resistance at 0°C. Its value depends upon the nature of resistor material.

By rearranging the equation (1.1), we get

 

R_t = R₀ (1 + α₀t)        (1.2)

 

and

        (1.3)

If R₀ = 1 ohm; t = 1°C; then, α₀ = (R_t − R₀)

Hence, temperature coefficient of resistance at 0°C may be defined as the change in resistance per ohm original resistance per °C change in temperature.

Unit: We know that, α₀ = (R_t − R₀)/R₀ t

Substituting the units of various quantities, we get,

Unit of 

Hence, the unit of temperature coefficient is per °C.

1.20  TEMPERATURE CO-EFFICIENT OF COPPER AT 0°C

It has been seen that R_t = R₀ (1 + α₀t)

The above equation holds good for both rise and fall in temperature. The temperature verses resistance graph of copper material is a straight line as shown in Figure 1.9.

Fig. 1.9  Graph between temperature and resistance

If this line is extended in the backward direction, it would cut the temperature axis at point A where temperature is −234.5°C.

Putting the value of R_t = 0 and t = −234.5°C in the above equation, we get,

 

0 = R₀ [1 + α₀ (−234.5)]

or

 

234.5 α₀ = 1

or

where α₀ is the temperature coefficient of resistance of copper at 0°C.

However, in practice, the curve departs (at point B) from a straight line at very low temperature and the resistance never becomes zero.

1.21  EFFECT OF TEMPERATURE ON α

The value of temperature coefficient of resistance (α ) is not constant. Its value depends upon the initial temperature on which the increment in resistance is based. If the initial temperature is 0°C, the value of α is α₀. Similarly, if the initial temperature is t₁°C, the value of α is α₁.

Relation between α₀ and α₁.

Consider a conductor of resistance R₀ at 0°C. When its temperature is raised to t₁°C, its resistance increases to say R₁.

∴

 

(R₁ − R₀) = R₀α₀t₁        (1.4)

or

 

R₁ = R₀ (1 + α₀t₁)        (1.5)

Let us suppose that the conductor of resistance R₁ at t₁°C be now cooled down to 0°C to give a resistance of final value R₀.

Then,

 

R₀ = R₁ (1 + α₁ (−t₁)) = R₁ − t₁R₁α₁

or

Substituting the value of (R₁ − R₀) from equation (1.4) and the value of R₁ from equation (1.5), we get,

or

Relation between α₁ and α₂.

Rearranging the equation (1.6), we get,

Similarly, if the initial temperature is t₂°C and, the value of α is α₂, then

Subtracting equation (1.7) from (1.8), we get,

or

The following conclusions were drawn from the discussion

1. From equation (1.10), it is clear that with the rise in temperature, the value of α decreases. Thus, α₀ has the maximum value.
2. If the initial resistance at 0°C is R₀ and the final resistance at t₁ °C is R₁, then R₁ = R₀ (1 + α₀t₁) 
3. If the initial resistance at t₁ °C is R₁ and the final resistance at t₂ °C is R₂, then R₂ = R₁ (1 + α₁ (t₂ − t₁)) 
4. If the value of α at t₁°C is α₁, then its value at t₂°C will be