### 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 **R _{1}** at temperature

**q**and

_{1}**R**at temperature

_{2}**q**, then:

_{2}**R**

_{1}/ (q_{1}– q_{0}) = R_{2}/ (q_{2}– q_{0}) = (R_{2}– R_{1}) / (q_{2}– q_{1})where

**q**is the extrapolated temperature for zero resistance.

_{0}The ratio of resistances **R _{2}** and

**R**is:

_{1}**R**

_{2}/ R_{1}= (q_{2}– q_{0}) / (q_{1}– q_{0})The average temperature rise **Dq** of a winding under load may be estimated from measured values of the cold winding resistance **R _{1}** at temperature

**q**(usually ambient temperature) and the hot winding resistance

_{1}**R**at temperature

_{2}**q**, using:

_{2}**Dq = q**

_{2}– q_{1}= (q_{1}– q_{0}) (R_{2}– R_{1}) / R_{1}Rearranging for per-unit change in resistance **DR _{pu}** relative to

**R**:

_{1}**DR**

_{pu}= (R_{2}– R_{1}) / R_{1}= (q_{2}– q_{1}) / (q_{1}– q_{0}) = Dq / (q_{1}– q_{0})Note that the resistance values are measured using a small direct current to avoid thermal and inductive effects.

*Copper Windings*

The value of **q _{0}** for copper is

**– 234.5 °C**, so that:

**Dq = q**

_{2}– q_{1}= (q_{1}+ 234.5) (R_{2}– R_{1}) / R_{1}If **q _{1}** is

**20 °C**and

**Dq**is

**1 degC**:

**DR**

_{pu}= (R_{2}– R_{1}) / R_{1}= Dq / (q_{1}– q_{0}) = 1 / 254.5 = 0.00393The temperature coefficient of resistance of copper at 20 °C is

**0.00393**per degC.

*Aluminium Windings*

The value of **q _{0}** for aluminium is

**– 228 °C**, so that:

**Dq = q**

_{2}– q_{1}= (q_{1}+ 228) (R_{2}– R_{1}) / R_{1}If **q _{1}** is

**20 °C**and

**Dq**is

**1 degC**:

**DR**

_{pu}= (R_{2}– R_{1}) / R_{1}= Dq / (q_{1}– q_{0}) = 1 / 248 = 0.00403The 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.

NOTATION | ||||||

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BE f G I j k m N n P p R | susceptance induced voltage frequency conductance current j-operator coefficient number of phases number of turns rotational speed power pole pairs resistance | [siemens, S] [volts, V] [hertz, Hz] [siemens, S] [amps, A] [1Ð90°] [number] [number] [number] [revs/min] [watts, W] [number] [ohms, W] | SsTVXYZdFfhqw | voltamperes slip torque terminal voltage reactance admittance impedance loss angle magnetic flux phase angle efficiency temperature angular speed | [volt-amps, VA] [per-unit] [newton-metres, Nm] [volts, V] [ohms, W] [siemens, S] [ohms, W] [degrees, °] [webers, Wb] [degrees, °] [per-unit] [centigrade, °C] [radians/sec] |