Maximum Torque⚠ unverified
Electrical / Motors · Compute the pull-out (breakdown) torque of an induction motor
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| V | V | V | 1.0 | Per-phase stator voltage |
| R1 | R1 | — | 1.0 | Stator winding resistance |
| R2 | R2 | — | 1.0 | Rotor resistance referred to the stator, in ohms (Ohm). Present for interface symmetry; the breakdown-torque expression does not depend on it |
| X1 | X1 | — | 1.0 | Stator leakage reactance |
| X2 | X2 | — | 1.0 | Rotor leakage reactance referred to the stator |
| f | f | Hz | 60.0 | Supply line frequency, in hertz (Hz). Default is 60.0 |
| poles | poles | — | 4.0 | Number of magnetic poles of the machine. Default is 4 |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | Tmax | N.m | Maximum (breakdown) torque, in newton-metres (N.m) |
The science & history
Understanding the Parameters
-
$V$ — breakdown torque scales as $V^2$, so it drops sharply with sagging supply voltage (a common cause of stalls).
-
$R_1$, $X_1+X_2$ — the stator resistance and total leakage reactance limit the peak.
- $R_2$ — sets where (which slip) the peak occurs, not how high it is.
Derivation (Approaching a Proof)
Take the torque–slip expression $T(s) = \dfrac{3}{\omega_s}\dfrac{V^2 (R_2/s)}{(R_1 + R_2/s)^2 + (X_1+X_2)^2}$ (see Starting Torque). Treating $R_2/s$ as the variable and setting $dT/d(R_2/s) = 0$ gives the condition for the peak:
$$\frac{R_2}{s_{max}} = \sqrt{R_1^2 + (X_1 + X_2)^2},$$
i.e. the peak occurs at slip $s_{max} = R_2/\sqrt{R_1^2+(X_1+X_2)^2}$. Substituting this back and simplifying, the $R_2$ cancels and leaves
$$T_{max} = \frac{3\,V^2}{2\,\omega_s\big(R_1 + \sqrt{R_1^2 + (X_1 + X_2)^2}\big)}.$$
So rotor resistance shifts the peak along the slip axis (used to get high torque at start) but cannot raise the breakdown torque itself — a key design insight.
History
This "maximum torque" result and the slip at which it occurs are classic outcomes of Steinmetz's equivalent‑circuit analysis. Breakdown torque (typically 200–300 % of rated) is a defining spec for motor selection and appears directly in NEMA torque‑class definitions.
Related Concepts: Starting Torque, Slip, Synchronous Speed, Motor Torque
Notes: Registry calculator maximum-torque (unverified; $R$/$X$ mislabelled dimensionless — should
be ohms; $R_2$ input is accepted but unused). Uses $V$/$R_1$ as Thévenin approximations.