Temperature Rise

Formulas

The resistance of copper and aluminium windings increases with temperature, and the relationship is quite linear over the normal range of operating temperatures. For a linear relationship, if the winding resistance is R₁ at temperature q₁ and R₂ at temperature q₂, then:
R₁ / (q₁ – q₀) = R₂ / (q₂ – q₀) = (R₂ – R₁) / (q₂ – q₁)
where q₀ is the extrapolated temperature for zero resistance.

The ratio of resistances R₂ and R₁ is:
R₂ / R₁ = (q₂ – q₀) / (q₁ – q₀)

The average temperature rise Dq of a winding under load may be estimated from measured values of the cold winding resistance R₁ at temperature q₁ (usually ambient temperature) and the hot winding resistance R₂ at temperature q₂, using:
Dq = q₂ – q₁ = (q₁ – q₀) (R₂ – R₁) / R₁

Rearranging for per-unit change in resistance DR_pu relative to R₁:
DR_pu = (R₂ – R₁) / R₁ = (q₂ – q₁) / (q₁ – q₀) = Dq / (q₁ – q₀)

Note that the resistance values are measured using a small direct current to avoid thermal and inductive effects.

Copper Windings
The value of q₀ for copper is – 234.5 °C, so that:
Dq = q₂ – q₁ = (q₁ + 234.5) (R₂ – R₁) / R₁

If q₁ is 20 °C and Dq is 1 degC:
DR_pu = (R₂ – R₁) / R₁ = Dq / (q₁ – q₀) = 1 / 254.5 = 0.00393
The temperature coefficient of resistance of copper at 20 °C is 0.00393 per degC.

Aluminium Windings
The value of q₀ for aluminium is – 228 °C, so that:
Dq = q₂ – q₁ = (q₁ + 228) (R₂ – R₁) / R₁

If q₁ is 20 °C and Dq is 1 degC:
DR_pu = (R₂ – R₁) / R₁ = Dq / (q₁ – q₀) = 1 / 248 = 0.00403

The temperature coefficient of resistance of aluminium at 20 °C is 0.00403 per degC.

Note that aluminium has 61% of the conductivity and 30% of the density of copper, therefore for the same conductance (and same resistance) an aluminium conductor has 164% of the cross-sectional area, 128% of the diameter and 49% of the mass of a copper conductor.