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The dq model transforms the three-phase stator currents, voltages and fluxes of a permanent magnet motor into a rotating reference frame aligned with the rotor flux. This simplifies the dynamic equations and allows decoupled control of torque and flux.

Transformations

The dq0 Transformation converts phase quantities (a,b,c) to two stationary axes; the direct (subscript d) and quadrature (subscript q) axes. The third component, called the zero-sequence component is denoted with the subscript 0.

[SdSqS0]=23[cos(θ)cos(θ−120∘)cos(θ+120∘)−sin(θ)−sin(θ−120∘)−sin(θ+120∘)121212][SaSbSc] \begin{aligned} \left[\begin{array}{c} S_{d}\\ S_{q}\\ S_{0} \end{array}\right]=\frac{2}{3}\left[\begin{array}{ccc} cos(\theta) & cos(\theta-120^\circ) & cos(\theta+120^\circ)\\ -sin(\theta) & -sin(\theta-120^\circ) & -sin(\theta+120^\circ)\\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \end{array}\right]\left[\begin{array}{c} S_{a}\\ S_{b}\\ S_{c} \end{array}\right] \end{aligned}

Assumption

The following derivations assume balanced, sinusoidal quantities which lead the zero-sequence component to be zero.

The phase voltage equations are defined by:

va=Rsia+dψadtvb=Rsib+dψbdtvc=Rsic+dψcdt\begin{aligned} v_a &= R_s i_a + \frac{d\psi_a}{dt}\\ v_b &= R_s i_b + \frac{d\psi_b}{dt}\\ v_c &= R_s i_c + \frac{d\psi_c}{dt} \end{aligned}

where RsR_s is the stator winding resistance and ψ\psi the phase flux-linkage.

vd=Rsid−ωeψqvq=Rsiq+ωeψd\begin{aligned} v_d &= R_s i_d - \omega_e \psi_q\\ v_q &= R_s i_q + \omega_e \psi_d \end{aligned}

Here, idi_d controls the direct‑axis (flux‑producing) current and iqi_q controls the quadrature (torque‑producing) current. The electromagnetic torque for a three phase machine can be calculated in the following manner:

Te=32p[ψdiq−ψqid]T_e = \frac{3}{2} p \left[ \psi_d i_q - \psi_q i_d \right]

where pp is pole pair number.

Assumption

The derivations assume balanced, sinusoidal phase voltages and neglect saliency (i.e. we set Ld=LqL_d = L_q). The general case introduces coupling terms.

Takeaways

  1. Transforming to dq coordinates simplifies control and analysis.
  2. The d‑axis aligns with rotor flux; the q‑axis produces torque.
  3. Most modern motor controllers operate in the dq frame internally.