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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 / (q1q0) = R2 / (q2q0) = (R2 – R1) / (q2q1)
where q0 is the extrapolated temperature for zero resistance.

The ratio of resistances R2 and R1 is:
R2 / R1 = (q2q0) / (q1q0)

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 = q2q1 = (q1q0) (R2 – R1) / R1

Rearranging for per-unit change in resistance DRpu relative to R1:
DRpu = (R2 – R1) / R1 = (q2q1) / (q1q0) = Dq / (q1q0)

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 = q2q1 = (q1 + 234.5) (R2 – R1) / R1

If q1 is 20 °C and Dq is 1 degC:
DRpu = (R2 – R1) / R1 = Dq / (q1q0) = 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 = q2q1 = (q1 + 228) (R2 – R1) / R1

If q1 is 20 °C and Dq is 1 degC:
DRpu = (R2 – R1) / R1 = Dq / (q1q0) = 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.

NOTATION
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B
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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]
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f
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q
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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]

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Page edited by E.C. (Google).

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