Clarke transform (abc-αβ0)#

The Clarke transform (abc-αβ0) block performs a forward or reverse Clarke Transform. The block has a choice between standard and power invariant transformation, the type of location of axis \(\beta\) relative to the phase A axis.

Direct Clarke transform allows switching from a three-phase coordinate system abc to a stator-related electronic motor fixed two-phase system \(\alpha \beta\)0 with zero component

Note

For symmetric system, this component will be 0

The reverse Clarke Transform allows making the opposite transition: from a two-phase coordinate system \(\alpha \beta\)0 to a three-phase coordinate system abc.

Note

The transformation type is determined by the selected value of property Transformation Direction.

Direct transformation is performed using the formula (1), reverse – using the formula (2) [1] [2]:

(1)#\[\begin{split} \begin{pmatrix}\alpha \\ \beta \\ 0 \\ \end{pmatrix}=\frac{2}{3} \cdot \begin{pmatrix} 1 & -\frac{1}{2} & -\frac{1}{2}\\ 0 & \frac{\sqrt{3}}{2\cdot cos\Theta_{\beta}} & -\frac{\sqrt{3}}{2\cdot cos\Theta_{\beta}} \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ \end{pmatrix}\cdot \begin{pmatrix}A \\ B \\ C \\ \end{pmatrix} \end{split}\]

(2)#\[\begin{split}\begin{pmatrix}A \\ B \\ C \\ \end{pmatrix}=\begin{pmatrix} 1 & 0 & 1\\ -\frac{1}{2} & \frac{\sqrt{3}}{2\cdot cos\Theta_{\beta}} & 1 \\ -\frac{1}{2} & - \frac{\sqrt{3}}{2\cdot cos\Theta_{\beta}} & 1 \\ \end{pmatrix}\cdot \begin{pmatrix}\alpha \\ \beta \\ 0 \\ \end{pmatrix} \end{split}\]

where:

A, B, C – components of the three-phase coordinate system abc;

\(\alpha\),\( \beta\),0 – components of the two-phase coordinate system \(\alpha \beta\)0;

\(\Theta_{\beta}\) – angle between the \(\beta\) axis and the vertical axis.

Note

\(\Theta_{\beta}\) value is set in the block in the property Location of axis β. Depending on the selected value, axis \(\beta\) can be ahead of phase A axis by 90⁰ (see Figure 1 below) or be 90⁰ behind it (see Figure 2 below) [3]. In the first case, the angle \(\Theta_{\beta}\) = 0, in the second \(\Theta_{\beta}\) = 180⁰.

Location_of_coordinate_system_axes_when_axis_is_advanced — Fig. 1 Location of axes of coordinate systems at advance of axis \(\beta\)#

Location_of_axes_of_coordinate_systems_when_axis_lags — Fig. 2 Location of axes of coordinate systems at lagging of axis \(\beta\)#

Expressions given in the formulas (1) and (2) represent the standard Clarke Transform.

Important

If we calculate the active and reactive powers of the motor using these transformations, they will not coincide with similar values in a three-phase calculation system.

Hint

In order to preserve the power values after the transformation, the Clarke Transform is used, invariant in power, which is calculated using the formulas (3) and (4) [4]:

(3)#\[\begin{split}\begin{pmatrix}u_{\alpha} \\ u_{\beta} \\ u_0 \\ \end{pmatrix}=\sqrt{\frac{2}{3}} \cdot \begin{pmatrix} 1 & -\frac{1}{2} & -\frac{1}{2}\\ 0 & \frac{\sqrt{3}}{2\cdot cos\Theta_{\beta}} & -\frac{\sqrt{3}}{2\cdot cos\Theta_{\beta}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \end{pmatrix}\cdot \begin{pmatrix}u_{A} \\ u_{B} \\ u_{C} \\ \end{pmatrix}\end{split}\]

(4)#\[\begin{split}\begin{pmatrix}u_{A} \\ u_{B} \\ u_{C} \\ \end{pmatrix}=\sqrt{\frac{2}{3}} \cdot \begin{pmatrix} 1 & 0 & \frac{1}{\sqrt{2}}\\ -\frac{1}{2} & \frac{\sqrt{3}}{2\cdot cos\Theta_{\beta}} & \frac{1}{\sqrt{2}} \\ -\frac{1}{2} & -\frac{\sqrt{3}}{2\cdot cos\Theta_{\beta}} & \frac{1}{\sqrt{2}} \\ \end{pmatrix}\cdot \begin{pmatrix}u_{\alpha} \\ u_{\beta} \\ u_0 \\ \end{pmatrix}\end{split}\]

One can select this transformation type in the property Transformation type.

Fig. 1 and 2 show vector F - a generalized or depicting vector of a three-phase system rotating at a speed of ωt.

Properties

Property	Default Value	Description	Units of measurement
Transform direction: Direct; Inverse	Direct	-	-
Transform type	Standard	-	-
\(\beta\) axis position	90\(^0\) before A axis	-	-

Inputs and outputs

Note

The name of the input and output parameters depends on the value of property Transformation direction

Parameters	Input/Output	Description	Units of measurement	Port type
A (“Direct”) / \(\alpha\) (“Reverse”)	Input	-	-	Automation
B (“Direct”) / \(\beta\) (“Reverse”)	Input	-	-	Automation
C (“Direct”) / 0 (“Reverse”)	Input	-	-	Automation
\(\alpha\) (“Direct”) / A (“Reverse”)	Output	-	-	Automation
\(\beta\) (“Direct”) / B (“Reverse”)	Output	-	-	Automation
0 (“Direct”) / C (“Reverse”)	Output	-	-	Automation

In order to see the connection capabilities of the block, you need to go to page Description of block ports

1. Code generation#

Important

Block supports C source code generation

Clarke transform (abc-αβ0)#

Properties

Inputs and outputs

1. Code generation#

2. Sources#