Induction machines

The steady-state induction-machine model includes the equivalent circuit, scheduled power, slip, load-torque laws, saturation, and locked-rotor behavior.

Equivalent circuit

An induction machine is the standard double-cage equivalent circuit, per unit on the machine base MBASE and rated voltage RATEKV. The terminal voltage in machine per unit is the bus voltage scaled by the ratio of the bus base voltage to the rated voltage. With \(v\) the machine terminal voltage and \(s\) the slip,

\[ Z_s = R_A + jX_A, \qquad Z_k = \frac{R_k}{s} + jX_k \quad (k = 1, 2), \]

\[ Z_{rot} = jX_3 + (Z_1 \parallel Z_2), \qquad I = \frac{v}{Z_s + (jX_M \parallel Z_{rot})} , \]

and the machine draws \(P + jQ = v\,\bar{I}\) times MBASE from its bus, in MW and Mvar, positive when operating as a motor.

Scheduled power

PSET schedules the machine power in MW: a positive value draws power as a motor, a negative value delivers power as a generator. PCODE selects the scheduled quantity: 1 schedules the mechanical output power, 2 the electrical power drawn.

Slip solution

At the solved terminal voltage, the electromagnetic torque equals the load torque:

\[ T_e(v,s) = T_L(1-s). \]

The result reports the slip, electrical power, reactive power, and stall state.

Load-torque laws

TCODE selects the load-torque law. TCODE = 1 is the power law \(\tau(\omega) = \omega^D\). TCODE = 2 is the polynomial form

\[ \tau(\omega) = A\,\omega^2 + B\,\omega + D\,\omega^E + C_0 , \]

with \(C_0\) chosen to normalize the curve at rated speed.

Saturation

The equivalent circuit can include leakage and main-flux saturation. Leakage saturation changes stator reactance with current. Main-flux saturation changes the magnetizing reactance with internal voltage.

Stall (locked rotor)

The locked-rotor state is the standard slip-one equivalent circuit. The result identifies a machine that operates in this state.