Starting Torque⚠ unverified
Electrical / Motors · Compute the approximate starting torque of an induction motor (slip = 1)
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 |
| 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 | T | N.m | Starting torque, in newton-metres (N.m) |
The science & history
Understanding the Parameters
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$V$ drives torque as $V^2$ — halving the voltage quarters the starting torque (why reduced‑voltage starters limit both inrush and torque).
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$R_2$ appears in the numerator: more rotor resistance gives more starting torque (up to a point), the principle behind wound‑rotor motors and deep‑bar/double‑cage designs.
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$\omega_s = 4\pi f/p$ — the synchronous angular speed (from $f$ and $p$).
Derivation (Approaching a Proof)
From the Thévenin‑reduced per‑phase equivalent circuit of the induction motor, the developed torque at slip $s$ is
$$T(s) = \frac{3}{\omega_s}\cdot\frac{V^2\,(R_2/s)}{(R_1 + R_2/s)^2 + (X_1 + X_2)^2},$$
(using $V$, $R_1$ for the Thévenin source and impedance). Starting torque is this at $s = 1$, where $R_2/s = R_2$:
$$T_{start} = \frac{3\,V^2 R_2}{\omega_s\big((R_1 + R_2)^2 + (X_1 + X_2)^2\big)}.$$
The factor $3$ is the three phases; $\omega_s$ converts air‑gap power to torque. Because $T_{start}\propto R_2$ for small $R_2$, adding external rotor resistance (wound‑rotor) or shaping the rotor bars raises the break‑away torque at the cost of running efficiency.
History
The torque–slip relation comes from Steinmetz's induction‑motor equivalent circuit (1890s). Managing starting torque versus inrush current drove a century of designs — NEMA design classes A–D, wound‑rotor starters, and today's variable‑frequency drives that start motors softly at controlled slip.
Related Concepts: Maximum Torque, Slip, Synchronous Speed, Motor Torque
Notes: Registry calculator starting-torque (unverified; $R_1$, $R_2$, $X_1$, $X_2$ mislabelled
dimensionless — should be ohms). Uses $V$/$R_1$ as the Thévenin approximations.