Motor Thermal Time Constant⚠ unverified
Electrical / Motors · Compute the thermal time constant of a motor
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| Rth | Rth | K/W | 1.0 | Thermal resistance |
| Cth | Cth | J/K | 1.0 | Thermal capacitance (heat capacity) |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | τ | s | Thermal time constant, in seconds (s) |
The science & history
Understanding the Parameters
- $R_{th}$ — sets the final temperature rise ($\Delta T = P_{loss}R_{th}$, see Temperature Rise).
- $C_{th}$ — the thermal mass ($mc_p$); a heavy motor heats slowly and can tolerate longer overloads.
- $\tau$ — their product, in seconds (often minutes to tens of minutes for real machines).
Derivation (Approaching a Proof)
Model the motor as one thermal node: heat is generated at rate $P_{loss}$, stored in capacitance $C_{th}$, and leaks to ambient through $R_{th}$. Energy balance gives a first‑order differential equation identical in form to a charging capacitor:
$$C_{th}\frac{d(\Delta T)}{dt} + \frac{\Delta T}{R_{th}} = P_{loss},$$
whose solution is $\Delta T(t) = P_{loss}R_{th}\big(1 - e^{-t/\tau}\big)$ with
$$\tau = R_{th}\,C_{th}.$$
The mathematics is exactly the RC charging curve (see RC Time Constant), with the thermal–electrical analogy $\Delta T\!\leftrightarrow\!V$, $P\!\leftrightarrow\!I$, $R_{th}\!\leftrightarrow\!R$, $C_{th}\!\leftrightarrow\!C$. The long thermal $\tau$ is why motors survive brief overloads that would be impossible thermally at steady state.
History
Lumped RC thermal models were adapted from circuit theory to machines and power electronics in the mid‑20th century; the thermal time constant underlies motor overload ("$I^2t$") protection, which allows short overloads but trips on sustained ones.
Related Concepts: Temperature Rise, RC Time Constant, Motor Efficiency
Notes: Registry calculator motor-thermal-time-constant (unverified). Single lumped node; detailed
models use multiple nodes with separate winding and frame time constants.