Clarke Transform

Implement abc to αβ0 transform

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Libraries:
Simscape / Electrical / Control / Mathematical Transforms

Description

The Clarke Transform block converts the time-domain components of a three-phase system in an abc reference frame to components in a stationary ɑβ0 reference frame. The block can preserve the active and reactive powers with the powers of the system in the abc reference frame by implementing a power invariant version of the Clarke transform. For a balanced system, the zero component is equal to zero.

If you have a license for Motor Control Blockset™ toolbox, you can use this block for embedded implementation. Use the Number of inputs parameter to specify either two or three inputs. When you set the Number of inputs parameter to Two inputs, the block accepts two signals out of the three phases (abc), automatically calculates the third signal, and outputs the corresponding components in the αβ reference frame. (since R2025a)

The figures show:

• The direction of the magnetic axes of the stator windings in the abc reference frame and the stationary ɑβ0 reference frame

  

• Equivalent ɑ, β, and zero components in the stationary reference frame

  

• The time-response of the individual components of equivalent balanced abc and ɑβ0 systems

  

Equations

The block implements the Clarke transform as

$\left[\begin{array}{c}\alpha \\ \beta \\ 0\end{array}\right]=\frac{2}{3}\left[\begin{array}{ccc}1& -\frac{1}{2}& -\frac{1}{2}\\ 0& \frac{\sqrt{3}}{2}& -\frac{\sqrt{3}}{2}\\ \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\end{array}\right]\left[\begin{array}{c}a\\ b\\ c\end{array}\right],$

where:

• a, b, and c are the components of the three-phase system in the abc reference frame.

• α and β are the components of the two-axis system in the stationary reference frame.

• 0 is the zero component of the two-axis system in the stationary reference frame.

The block implements the power invariant version of the Clarke transform as

$\left[\begin{array}{c}\alpha \\ \beta \\ 0\end{array}\right]=\sqrt{\frac{2}{3}}\left[\begin{array}{ccc}1& -\frac{1}{2}& -\frac{1}{2}\\ 0& \frac{\sqrt{3}}{2}& -\frac{\sqrt{3}}{2}\\ \sqrt{\frac{1}{2}}& \sqrt{\frac{1}{2}}& \sqrt{\frac{1}{2}}\end{array}\right]\left[\begin{array}{c}a\\ b\\ c\end{array}\right].$

Examples

Three-Phase Matrix Converter

A three-phase matrix converter that drives a static load and draws unity power factor at the source. The Scopes subsystem contains scopes that allow you to see the simulation results.

• Open Model

Ports

Input

abc — a-, b-, and c-phase components

vector

Components of the three-phase system in the abc reference frame.

Dependencies

To enable this port, either clear the Use block for embedded implementation (requires Motor Control Blockset™) parameter or select the Use block for embedded implementation (requires Motor Control Blockset™) parameter and set Number of inputs to Three inputs. (since R2025a)

Data Types: single | double

a — a-phase component

scalar

Since R2025a

a-phase component of the three-phase system in the abc reference frame.

Dependencies

To enable this port, select the Use block for embedded implementation (requires Motor Control Blockset™) parameter and set Number of inputs to Two inputs.

Data Types: single | double

b — b-phase component

scalar

Since R2025a

b-phase component of the three-phase system in the abc reference frame.

Dependencies

To enable this port, select the Use block for embedded implementation (requires Motor Control Blockset™) parameter and set Number of inputs to Two inputs.

Data Types: single | double

Output

αβ0 — α-β axis and zero components

vector

Alpha-axis component,α, beta-axis component β, and zero component in the stationary reference frame.

Dependencies

To enable this port, either clear the Use block for embedded implementation (requires Motor Control Blockset™) parameter or select the Use block for embedded implementation (requires Motor Control Blockset™) parameter and set Number of inputs to Three inputs. (since R2025a)

Data Types: single | double

α — Alpha-axis component

scalar

Since R2025a

Alpha-axis component,α, in the stationary reference frame.

Dependencies

To enable this port, select the Use block for embedded implementation (requires Motor Control Blockset™) parameter and set Number of inputs to Two inputs.

Data Types: single | double

β — Beta-axis component

scalar

Since R2025a

Beta-axis component,β, in the stationary reference frame.

Dependencies

To enable this port, select the Use block for embedded implementation (requires Motor Control Blockset™) parameter and set Number of inputs to Two inputs.

Data Types: single | double

Parameters

Power Invariant — Enable power invariance

off (default) | on

Option to preserve the active and reactive power of the system in the abc reference frame.

Use block for embedded implementation (requires Motor Control Blockset™) — Option to use block for embedded implementation

off (default) | on

Since R2025a

Option to use the block for embedded implementation. This option requires the Motor Control Blockset toolbox.

Number of inputs — Number of block inputs

Two inputs (default) | Three inputs

Since R2025a

Select the number of inputs that you can specify:

• Two inputs — Configure the block to accept two separate input signals a and b. The block generates two separate output signals α and β.

• Three inputs — Configure the block to accept a multiplexed input containing a, b, and c signals. The block generates a multiplexed output containing α, β, and 0 signals.

Dependencies

To enable this parameter, select the Use block for embedded implementation (requires Motor Control Blockset™) parameter.

References

[1] Krause, P., O. Wasynczuk, S. D. Sudhoff, and S. Pekarek. Analysis of Electric Machinery and Drive Systems. Piscatawy, NJ: Wiley-IEEE Press, 2013.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2017b

R2025a: Optimize block for embedded implementation

If you have a license for Motor Control Blockset toolbox, you can optimize the Clarke Transform block for embedded implementation. To optimize this block for embedded implementation, select the Use block for embedded implementation (requires Motor Control Blockset™) parameter.