Induction machine dynamics & direct vector control

01

Machine block diagram

The machine is a chain of four physical stages: stator voltages drive currents through an RL dynamic, currents build the rotor flux, flux and current produce torque, and torque accelerates the shaft. Hover or focus a block to highlight its equation.

Induction machine signal flow in the αβ frame.

02

State equations

With the total leakage factor σ = 1 − M²/(LsLr), the model in the stationary αβ frame is:

1. Stator RL dynamics → isαβ

(5)

v_{s\alpha} = R_s i_{s\alpha} + \frac{d\Phi_{s\alpha}}{dt}, \quad v_{s\beta} = R_s i_{s\beta} + \frac{d\Phi_{s\beta}}{dt}

2. Rotor flux & inductances → Φrαβ

(6)

\Phi_{r\alpha} = \frac{L_r}{M}\left(\Phi_{s\alpha} - \sigma L_s i_{s\alpha}\right), \quad \Phi_{r\beta} = \frac{L_r}{M}\left(\Phi_{s\beta} - \sigma L_s i_{s\beta}\right)

3. Torque equation → Cem

(7)

C_{em} = n_p \frac{M}{L_r}\left(\Phi_{r\alpha} i_{s\beta} - \Phi_{r\beta} i_{s\alpha}\right)

4. Mechanical acceleration → Ω

(8)

J \frac{d\Omega}{dt} = C_{em} - C_r - f_0\, \Omega

03

Direct vector control

Outer PI loops turn flux and speed references into dq current references. An inverse Park rotation by the estimated angle ρ brings them back to αβ, where fast current loops produce the voltage references sent to the PWM inverter. The estimator closes the loop by reconstructing Φr and ρ from measured currents and speed.

Direct vector control in the αβ frame. Dashed lines are measurements and estimates fed back.