Motor temperature
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_{1}
and R_{2} at temperature q_{2},
then:
R_{1} / (q_{1}
 q_{0})
= R_{2} / (q_{2}
 q_{0})
= (R_{2}  R_{1}) / (q_{2}
 q_{1})
where q_{0}
is the extrapolated temperature for zero resistance.
The ratio of resistances R_{2}
and R_{1} is:
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_{1}
(usually ambient temperature) and the hot winding resistance R_{2}
at temperature
q_{2},
using:
Dq =
q_{2}

q_{1}
= (q_{1}

q_{0})
(R_{2}  R_{1}) / R_{1}
Rearranging for perunit 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 = 0.00393
The 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 = 0.00403
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 crosssectional area, 128% of the diameter and 49% of the mass of a copper conductor.
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_{1}
and R_{2} at temperature q_{2},
then:
R_{1} / (q_{1}
 q_{0})
= R_{2} / (q_{2}
 q_{0})
= (R_{2}  R_{1}) / (q_{2}
 q_{1})
where q_{0}
is the extrapolated temperature for zero resistance.
The ratio of resistances R_{2}
and R_{1} is:
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_{1}
(usually ambient temperature) and the hot winding resistance R_{2}
at temperature
q_{2},
using:
Dq =
q_{2}

q_{1}
= (q_{1}

q_{0})
(R_{2}  R_{1}) / R_{1}
Rearranging for perunit 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 = 0.00393
The 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 = 0.00403
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 crosssectional area, 128% of the diameter and 49%
of the mass of a copper conductor.