ABC/DQ0 Transformation (power)

Description

ABC/DQ0 transformation in the power domain.

This component is an ideal reversible electrical multiport. The angle Theta is a control input in radians. With all electrical currents defined positive when entering the component, the component imposes:

$$v_{dq0} = T(\Theta)\,v_{abc}$$

$$i_{abc} = -T(\Theta)^T\,i_{dq0}$$

where:

$$v_{abc}=\begin{bmatrix} V_a\\ V_b\\ V_c \end{bmatrix},\quad v_{dq0}=\begin{bmatrix} V_d\\ V_q\\ V_0 \end{bmatrix},\quad i_{abc}=\begin{bmatrix} I_a\\ I_b\\ I_c \end{bmatrix},\quad i_{dq0}=\begin{bmatrix} I_d\\ I_q\\ I_0 \end{bmatrix}$$

The current equation uses the dual transpose relation. This makes the ideal multiport power-conserving:

$$v_{abc}^T i_{abc} + v_{dq0}^T i_{dq0} = 0$$

In "Power Invariant" mode, the Park matrix is:

$$T_p(\Theta)=\sqrt{\frac{2}{3}}\cdot \begin{bmatrix} \cos(\Theta) & \cos(\Theta-\frac{2\pi}{3}) & \cos(\Theta+\frac{2\pi}{3}) \\ -\sin(\Theta) & -\sin(\Theta-\frac{2\pi}{3}) & -\sin(\Theta+\frac{2\pi}{3}) \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{bmatrix}$$

This makes the transformation orthonormal: instantaneous power is directly conserved between the ABC and DQ0 electrical ports, and a balanced three-phase sinusoid with phase amplitude V gives an aligned DQ component with amplitude sqrt(3/2) * V.

In "Voltage Invariant" mode, the Park matrix is:

$$T_v(\Theta)=\frac{2}{3}\cdot \begin{bmatrix} \cos(\Theta) & \cos(\Theta-\frac{2\pi}{3}) & \cos(\Theta+\frac{2\pi}{3}) \\ -\sin(\Theta) & -\sin(\Theta-\frac{2\pi}{3}) & -\sin(\Theta+\frac{2\pi}{3}) \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \end{bmatrix}$$

In this mode, a balanced three-phase sinusoid with phase amplitude V gives an aligned DQ component with amplitude V. The current equations still use the dual transpose relation, so this electrical multiport remains power-conserving even though voltage and current components do not use the same normalization.

The inverse voltage relation is obtained from the selected matrix:

$$v_{abc}=T(\Theta)^{-1}v_{dq0}$$

The convention is receptor: positive current is entering the device.

Library

Electrical > Transformers

Parameters

Property	Display Name	Parameter Type	Description
Mode	Mode	ParkTransformModeParameter	Selects the Park transformation normalization.

Pins

Property	Pin Name	Type	Description
A	A	Electrical	Phase A voltage
B	B	Electrical	Phase B voltage
C	C	Electrical	Phase C voltage
D	D	Electrical	D-axis voltage
Q	Q	Electrical	Q-axis voltage
Zero	Zero	Electrical	Zero-sequence voltage
Theta	Theta	ControlIn	Angle input signal [rad]

Default Size

Width	Height
8	8