Induction Machines & Speed Control

Three-phase stator currents create a rotating magnetic field. For $p$ pole pairs at electrical frequency $f$,

$$n_s=\frac{60f}{p}\quad\text{rpm}.$$

Rotor conductors develop induced current only when their mechanical speed differs from the field. Slip is

$$s=\frac{n_s-n}{n_s}.$$

At motoring load $0<s<1$; rotor electrical frequency is $sf$. Zero slip means no rotor induced frequency and, in the ideal induction mechanism, no torque.

Torque production and equivalent circuit #

The air gap behaves like a transformer whose secondary frequency changes with slip. Stator resistance/leakage, magnetizing reactance, and rotor resistance/leakage referred to the stator predict current, power factor, torque, and losses. Rotor copper loss is approximately slip times air-gap power, which explains why high slip means heat.

Starting #

At standstill $s=1$, rotor frequency equals stator frequency and current can be large. Direct-on-line starting must fit supply voltage dip, contactor duty, motor starting capability, driven-load acceleration torque, and thermal limit. Star-delta and autotransformer methods reduce voltage and therefore reduce induction torque roughly with voltage squared; they are not free current reduction.

Scalar $V/f$ control #

Back EMF is approximately proportional to frequency times flux. Holding $V/f$ roughly constant maintains flux below base speed, with a low-speed voltage boost to overcome resistance. Above base frequency, inverter voltage is limited, flux weakens, and available torque falls while approximate constant-power operation may continue.

Open-loop $V/f$ does not directly regulate torque-producing current or compensate slip. It is adequate for many fans and pumps, not precise low-speed torque.

Field-oriented control #

FOC transforms measured phase currents into a rotating $d$–$q$ frame aligned with estimated rotor flux. One current component controls flux and the orthogonal component controls torque, recovering DC-machine-like control. The transformations are simple; accurate flux angle, motor parameters, current sampling, dead-time compensation, and voltage limits are the hard parts.

Diagnostic map #

Commission in layers: phase order and insulation; current offsets and polarity; open-loop rotation; motor parameter/flux estimate; torque-current response; then speed and position loops.