Clarke Transformation

The Clarke transformation , named after Edith Clarke and also referred to as α, β transformation , is used to convert multi-phase variables, as in a three-phase machine with the axes U, V, W, ... into a simpler two-axis coordinate system with the axes α, β convict. The Clarke transformation, together with the d / q transformation, is one of the mathematical principles for vector control of three-phase machines and describes one of several possible space vector representations .

General

Arrangement of the coils mounted in the stator in the so-called stator-fixed αβ coordinate system.

In the Clarke transformation, the underlying right-angled coordinate system is chosen to be the same as the stator at rest and is mapped in the complex plane with the real part α and the imaginary part β, with the sum of the three external conductor currents always being zero. In the three-phase system, the three coils of the stator of an induction machine are each offset by an angle of 120 °, whereby by definition the axis U coincides with the real axis α, as shown in the adjacent figure.

The Clarke transformation converts the three phase currents , and into two equivalent currents and . ${\displaystyle I_{U}}$ ${\displaystyle I_{V}}$ ${\displaystyle I_{W}}$ ${\displaystyle I_{\alpha }}$ ${\displaystyle I_{\beta }}$

${\displaystyle I_{\alpha }=\left(\cos(0^{\circ })\cdot I_{U}+\cos(120^{\circ })\cdot I_{V}+\cos(240^{\circ })\cdot I_{W}\right){\frac {2}{3}}}$

${\displaystyle I_{\beta }=\left(\sin(0^{\circ })\cdot I_{U}+\sin(120^{\circ })\cdot I_{V}+\sin(240^{\circ })\cdot I_{W}\right){\frac {2}{3}}}$

In element-wise matrix notation , it is:

${\displaystyle {\begin{bmatrix}I_{\alpha }\\I_{\beta }\end{bmatrix}}={\frac {2}{3}}{\begin{bmatrix}1&-{\frac {1}{2}}&-{\frac {1}{2}}\\0&{\frac {\sqrt {3}}{2}}&-{\frac {\sqrt {3}}{2}}\\\end{bmatrix}}{\begin{bmatrix}I_{U}\\I_{V}\\I_{W}\end{bmatrix}}}$

Due to the requirement that the sum of the three external conductor currents is always zero at all times, this equation can be simplified to:

${\displaystyle {\begin{bmatrix}I_{\alpha }\\I_{\beta }\end{bmatrix}}={\begin{bmatrix}1&0\\{\frac {1}{\sqrt {3}}}&{\frac {2}{\sqrt {3}}}\\\end{bmatrix}}{\begin{bmatrix}I_{U}\\I_{V}\end{bmatrix}}}$

In practice, the simplification means that the current only actually has to be measured with two and not three strings, for example by a current transformer .

The inverse Clarke transform is:

${\displaystyle {\begin{bmatrix}I_{U}\\I_{V}\\I_{W}\end{bmatrix}}={\begin{bmatrix}1&0\\-{\frac {1}{2}}&{\frac {\sqrt {3}}{2}}\\-{\frac {1}{2}}&-{\frac {\sqrt {3}}{2}}\end{bmatrix}}{\begin{bmatrix}I_{\alpha }\\I_{\beta }\end{bmatrix}}}$

The transformation is not only limited to the electrical currents, but can also be used analogously for all other electrical quantities such as the electrical voltages that occur or the magnetic flux densities .

extension

In the case of a three-phase system that is not in equilibrium, the α, β transformation can be extended to the α, β, γ transformation by adding a third parameter within the framework of the theory of symmetrical components . is the sum of the three phase currents: ${\displaystyle I_{\gamma }}$ ${\displaystyle I_{\gamma }}$

${\displaystyle I_{\gamma }={\frac {1}{3}}(I_{U}+I_{V}+I_{W})}$

with the α, β, γ transformation then also valid for three-phase systems that are not in equilibrium:

${\displaystyle {\begin{bmatrix}I_{\alpha }\\I_{\beta }\\I_{\gamma }\end{bmatrix}}={\frac {2}{3}}{\begin{bmatrix}1&-{\frac {1}{2}}&-{\frac {1}{2}}\\0&{\frac {\sqrt {3}}{2}}&-{\frac {\sqrt {3}}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}&{\frac {1}{2}}\\\end{bmatrix}}{\begin{bmatrix}I_{U}\\I_{V}\\I_{W}\end{bmatrix}}}$

and the inverse α, β, γ transformation:

${\displaystyle {\begin{bmatrix}I_{U}\\I_{V}\\I_{W}\end{bmatrix}}={\begin{bmatrix}1&0&1\\-{\frac {1}{2}}&{\frac {\sqrt {3}}{2}}&1\\-{\frac {1}{2}}&-{\frac {\sqrt {3}}{2}}&1\end{bmatrix}}{\begin{bmatrix}I_{\alpha }\\I_{\beta }\\I_{\gamma }\end{bmatrix}}}$

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• Edith Clarke: Circuit Analysis of AC Power Systems. Vol. I . J. Wiley & sons, New York 1943.
• Field Orientated Control of 3-Phase AC Motors (PDF; 99 kB), application note BPRA073, Texas Instruments Europe, 1998