dq0 → abc (Inverse Park)

Inverse Park (dq0 → abc) transform. Maps the direct (d), quadrature (q), and zero-sequence (0) components of a synchronously rotating dq frame back to three instantaneous phase signals a, b, c, using the angle θ (radians, wired in — usually the same PLL / rotor-position signal that drives the forward transform). The d-axis is aligned with cos θ (q lags), so it is the exact inverse of the abc → dq0 block: a = C·(d·cosθ − q·sinθ) + C0·0, b = C·(d·cos(θ−2π/3) − q·sin(θ−2π/3)) + C0·0, c = C·(d·cos(θ+2π/3) − q·sin(θ+2π/3)) + C0·0. Choose amplitude-invariant scaling (C=1, C0=1) or power-invariant scaling (C=√(2/3), C0=√(1/3)); use the same convention on both blocks. Typical use is the modulation / reference-generation stage of an inverter or motor drive: a current/voltage controller produces d,q setpoints that this block turns into the per-phase references for the PWM stage.

Category: Math

Overview

The dq0 → abc block is the inverse Park transform: it takes the direct (d), quadrature (q), and zero-sequence (0) components of a frame spinning with the angle θ you wire in and reconstructs the three instantaneous phase signals $a,b,c$.

It is the output stage of a vector-control loop. A current or voltage regulator works in the rotating dq frame, where setpoints are DC and PI control is easy; this block turns those d,q commands back into the per-phase references that drive the modulator (PWM / SVPWM) of an inverter or motor drive.

It is the exact inverse of the abc → dq0 block (forward Park): feed one block's outputs through the other with the same angle and scaling and you recover the original signals.

The transform

With $\theta$ the frame angle (radians) and the d-axis aligned with $\cos \theta$ (q lags by 90°):

$$\begin{array}{rl}a& =C(d\cos \theta -q\sin \theta )+C_{0}0,\\ b& =C(d\cos (\theta -\frac{2\pi }{3})-q\sin (\theta -\frac{2\pi }{3}))+C_{0}0,\\ c& =C(d\cos (\theta +\frac{2\pi }{3})-q\sin (\theta +\frac{2\pi }{3}))+C_{0}0.\end{array}$$

The block is stateless — each output is an algebraic function of the present inputs only, evaluated every time step. No filtering or memory.

Scaling convention

The Scaling parameter sets the magnitude constants $(C,C_0)$ and must match the forward abc → dq0 block in the same loop:

Scaling	$C$	$C_0$	Pairs with
Amplitude (2/3)	$1$	$1$	forward $K=2/3,K_0=1/3$
Power (√2/3)	$\sqrt{2/3}$	$\sqrt{1/3}$	forward $K=\sqrt{2/3},K_0=\sqrt{1/3}$

• Amplitude-invariant reconstructs abc at the same amplitude as the d,q inputs — a d,q vector of magnitude $A$ yields phase waveforms of peak $A$. This is the usual motor / power-electronics choice.
• Power-invariant is the orthonormal inverse that preserves instantaneous power across the transform.

With matching scalings the forward → inverse round trip is the identity (see the abc → dq0 page for the forward constants).

Sign and alignment convention

The d-axis is aligned with $\cos \theta$ and q lags d by 90°, so this block is the exact inverse of the forward transform under the same convention. Driving it with the constant pair $d=A\cos \phi$, $q=A\sin \phi$ (amplitude scaling, $0=0$) produces the balanced set

$$a=A\cos (\theta +\phi ),b=A\cos (\theta +\phi -\frac{2\pi }{3}),c=A\cos (\theta +\phi +\frac{2\pi }{3}),$$

i.e. a vector of amplitude $A$ leading the frame angle by $\phi$. If your tooling uses the opposite (q-leading) convention, negate q.

Zero-sequence

The 0 input adds a common-mode term $C_0\cdot 0$ equally to all three phases and does not depend on θ. Leave zero unwired (treated as 0) for the usual balanced three-wire case; drive it to inject a deliberate common-mode component, for example third-harmonic / zero-sequence injection that extends the linear modulation range.

Wiring

• d, q ← the regulator outputs in the rotating frame.
• zero ← the zero-sequence command (optional; 0 if unwired).
• theta ← the frame angle in radians — use the same angle source that drives the forward abc → dq0 block (rotor position or PLL).
• a, b, c → the per-phase references, typically into the modulator (PWM / SVPWM) stage.

When to use something else

• abc → dq0 (forward Park) — the measurement-side map that takes sampled phase signals into the rotating frame for the regulators.

Ports

Name	Direction	Value type	Notes
d	input	double
q	input	double
zero	input	double
theta	input	double
a	output	double
b	output	double
c	output	double

Parameters

Name	Label	Type	Default	Units	Description
scaling	Scaling	enum (Amplitude (2/3) / Power (√2/3))	0	—	Magnitude-scaling convention; must match the forward abc → dq0 block in the same loop. Amplitude-invariant (C=1) reconstructs abc at the dq peak amplitude — the usual choice for motor / power-electronics control. Power-invariant (√(2/3)) is the orthonormal inverse that preserves instantaneous power across the transform.