Temperature Rise & Hot Resistance

Every watt in a field coil becomes heat, and heat raises the copper's resistance. On a fixed voltage that reduces the current and the flux. This plate finds the equilibrium temperature, the warm-up curve, and lets you take a coil's temperature with an ohmmeter.

Parameters

Drive

Applied voltage

V

Resistance at 20 °C

R_{20}

Coil outer diameter

D_o

Coil length

L_c

Heat-transfer coefficient

h

≈ 8–14 still air incl. radiation; 4–7 inside a potW/m²K

Ambient temperature

T_a

°C

Thermal mass (copper-equivalent)

m

Fig. 13Coil at thermal equilibrium. Insulation class limits (A 105 °C, B 130 °C, F 155 °C, H 180 °C) marked on the scale.

Thermal equilibrium

Final temperature

T_\infty

53 °C

Temperature rise

\Delta T

28 K

Hot resistance

R_\infty

1.69 kΩ

Power cold → hot6.54 → 5.9 W

Surface area

A

176 cm²

Surface loading0.034 W/cm²

Thermal time const.

\tau_{th}

12.2 min

Thermal resistance

R_{th}

4.74 K/W

☞ Settles after ≈ 49 min. Field weakens 9.73 % as it warms.

Warm-up curve

Resistance-method thermometer

Cold resistance

R_c

Cold temperature

T_c

°C

Hot resistance (measured)

R_h

Winding temperature

Hot temperature

T_h

62.4 °C

Rise42.4 K

Resistance change16.7 %

How it works

Theory · assumptions · formulas

Heat balance

In equilibrium the electrical power equals the heat shed from the coil surface. Lumping convection and radiation into one coefficient $h$ :

P(T) = h\,A\,(T - T_a), \qquad A \approx \pi D_o L_c + \tfrac{\pi}{2} D_o^2

The power depends on temperature through copper's resistance, $R(T) = R_{20}[1 + \alpha(T-20)]$ :

P_V(T) = \frac{V^2}{R(T)} \quad\text{(voltage drive)}, \qquad P_I(T) = I^2 R(T) \quad\text{(current drive)}

Voltage drive is self-limiting: hotter coil, less power. Current drive has positive feedback and can run away when $\alpha I^2 R_{20} > hA$ . The calculator solves the balance by damped fixed-point iteration.

Warm-up

With thermal capacity $C_{th} = m\,c_{Cu}$ (copper, 385 J/kg·K; enlarge $m$ to account for iron in contact), the temperature obeys

C_{th}\,\frac{dT}{dt} = P(T) - hA\,(T - T_a), \qquad \tau_{th} = \frac{C_{th}}{hA}

integrated numerically over five thermal time constants. Large field coils take an hour or more to stabilise.

Measuring temperature by resistance

The inside of a winding cannot be reached with a thermometer, but its resistance can. Copper's resistance is proportional to $234.5 + T$ , giving the standard formula (IEC 60034, IEEE 112):

T_h = \frac{R_h}{R_c}\,(234.5 + T_c) - 234.5

PracticeMeasure the cold resistance after the coil has stood overnight at a known room temperature; measure the hot value immediately after switching off. Each 1 % rise in resistance means about 2.5 K of heating.