Three-Phase Rotating Field and Induction Torque

What is modelled

Three windings placed 120 degrees apart and driven by three currents 120 degrees apart in time add up to a field of constant size that rotates at the supply frequency divided by the number of pole pairs. A rotor in this field has currents induced in it only if it turns slower than the field, and this slip sets the torque. A weak phase adds a backward-rotating part that brakes the rotor.

Equations

• Synchronous speed ns = 60 f / p; slip s = (ns − n)/ns.
• Torque from the per-phase equivalent circuit T = 3 Vth² (R2/s) / (ωs [(Rth + R2/s)² + (Xth + X2)²]).
• Air-gap power Pag = T ωs, rotor copper loss s Pag, shaft power (1 − s) Pag.
• Unbalance: forward and backward parts from symmetrical components, net torque T(s) rf² − T(2 − s) rb².

Checks run in the page

• The field from summing three windings, its constant size and its rotation rate.
• The torque from a direct complex solution of the circuit against the Thevenin form.
• The air-gap, rotor-loss and shaft power split.
• The operating slip by bisection against a stepped start-up simulation.

Honest simplifications

The circuit has fixed winding resistances and reactances, no core loss, no friction and no saturation, the rotor is a single equivalent circuit without skin effect, and the start-up is a rigid inertia against a constant load torque. The default numbers describe a generic small motor, not the fictional machine of the 3D exhibit.

• Press Nearly no load to see the rotor run almost at the field speed with a tiny slip.
• Press Overload, stalls to see the load exceed the peak torque and the rotor stay near rest.
• Press One phase weak (20 %) to see the field become elliptical and the slip grow.