RH.
All tutorials

Direct Vector Control of Induction Machines · Part 1 of 4

Transformations & the decoupled model

How Concordia and Park turn three-phase AC into two DC quantities: flux and torque.

01

Why decoupling

In a DC motor, field flux and armature torque are naturally orthogonal: one winding sets the flux, the other the torque. In a three-phase induction machine the same stator currents create both at once. To control an AC machine like a DC machine, we project the currents onto a frame that rotates with the rotor flux.

(a,b,c)  3 windings @120°  ──Concordia──▶  (α,β) 2 axes @90°  ──Park──▶  (d,q) DC
   sinusoidal AC                            sinusoidal AC               d = flux, q = torque

02

Concordia: 3 → 2

The first step replaces the three windings at 120° with two equivalent orthogonal axes α and β fixed to the stator. The power-invariant Concordia matrix is:

(1)
[isαisβ]=23[1−12−12032−32][isaisbisc]\begin{bmatrix} i_{s\alpha} \\ i_{s\beta} \end{bmatrix} = \sqrt{\tfrac{2}{3}}\begin{bmatrix} 1 & -\frac{1}{2} & -\frac{1}{2} \\ 0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2} \end{bmatrix} \begin{bmatrix} i_{sa} \\ i_{sb} \\ i_{sc} \end{bmatrix}

For a balanced system the αβ currents are still sinusoidal, simply described with two signals instead of three.

03

Park: the rotating frame

We then rotate the axes by the angle ρ of the rotor flux. Viewed from a frame turning at the same speed as the current vector, the vector stands still: the d and q components become constant.

(2)
[isdisq]=[cos⁡ρsin⁡ρ−sin⁡ρcos⁡ρ][isαisβ]\begin{bmatrix} i_{sd} \\ i_{sq} \end{bmatrix} = \begin{bmatrix} \cos\rho & \sin\rho \\ -\sin\rho & \cos\rho \end{bmatrix} \begin{bmatrix} i_{s\alpha} \\ i_{s\beta} \end{bmatrix}

04

Live canvas

Change the frame offset to see the vector split differently between d and q. With zero offset, all current sits on the d axis.

abcdq
-4Phase currents
abc
-4Stationary αβ
αβ
-4Rotating dq
d = 1.22q = 0.00
Left: space vector with the fixed a, b, c axes and the rotating d-q frame. Right: phase currents, αβ components and dq components over the last two periods.

05

Field orientation

Choosing ρ so that the rotor flux lies entirely on the d axis (Φrq = 0) decouples the machine into two independent linear channels: i_sd sets the flux through a first-order lag, i_sq sets the torque directly.

(3)
Φr=M1+Trs isd,Tr=LrRr\Phi_r = \frac{M}{1 + T_r s}\, i_{sd}, \qquad T_r = \frac{L_r}{R_r}
(4)
Cem=npMLr Φr isqC_{em} = n_p \frac{M}{L_r}\, \Phi_r\, i_{sq}

This is exactly the DC-machine behaviour we wanted: flux and torque are controlled by two separate DC currents. Φrq=0⇒Φrd=Φr\Phi_{rq}=0 \Rightarrow \Phi_{rd}=\Phi_r