Thermal Time Constant Formulas
Heating and cooling curve formulas for a cable, motor or transformer: the temperature rise after a time, the time to reach a rise and the cooling curve.
The formulas
| To find | Formula |
|---|---|
| Rise after time t (heating) | θ = θ∞ × (1 − e^(−t / τ)) |
| Time to reach a rise θ | t = −τ × ln(1 − θ / θ∞) |
| Rise after time t (cooling) | θ = θ0 × e^(−t / τ) |
θ∞ is the final steady rise over the ambient in kelvin, which a constant load gives after a long time, τ the thermal time constant in the same unit of time as t and θ0 the rise at the start of the cooling. The model treats the equipment as one lump with one time constant. Real machines have several, such as a quick winding and a slow core, so use the maker's curves for a close answer. After one time constant the rise has reached 63.2% of its final value, and after three it has reached 95%. The values here are examples.
Worked example: a final rise of 60 K and τ = 3 h
- After 2 h, θ = 60 × (1 − e^(−2 / 3)) = 29.19 K
- Time to reach 40 K: t = −3 × ln(1 − 40 / 60) = 3.296 h
- Cooling from 60 K, after 2 h θ = 60 × e^(−2 / 3) = 30.81 K
Share of the final rise reached while heating
| Time | Share of the final rise |
|---|---|
| 1 × τ | 63.21% |
| 2 × τ | 86.47% |
| 3 × τ | 95.02% |
| 5 × τ | 99.33% |
Frequently asked questions
What is the time constant?
The time in which the rise covers 63.2% of the distance to its final value. It depends on the mass and the cooling of the equipment, and not on the load.
How long until the temperature is steady?
About five time constants reach 99%, so a machine with a long time constant takes hours to settle.
Why can a short overload be allowed?
The temperature takes time to rise, so a load above the rating for a time much shorter than τ gives only a small rise. Use the maker's overload curve for the limit.
Is the cooling time constant the same?
Often not. A motor that stands still without its fan cools more slowly than it heats, so check the maker's data.
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