9.2 Three-phase to Two-phase Transformations

For a deeper analysis of electric drives with filters, knowledge of filter models is required.

The natural frame of co-ordinates for a three-phase system is a three-axis co-ordinate with a 120-degree shift, but for a better analysis of the system, the use of the orthogonal co-ordinates is more useful, as well as easier to understand. Therefore, for the next analysis, proper co-ordinate transformations are presented. Two kinds of transformations are used, one for constant vectors magnitudes and another for constant power of the three-phase and two-phase frames of reference.

The transformation matrix for the conversion from a three-phase ABC co-ordinate to a two-phase αβ0 co-ordinate for constant vector magnitude is described as

and the transformation matrix for constant power of the system is

Using equations (9.1) or (9.2), each of the model variables x can be transformed from ABC to two-phase αβ0 co-ordinates according to

(9.3)

where the variables retain the same magnitude in each co-ordinate.

The next relation is used when the constant power of both systems is fulfilled:

(9.4)

The parameter transformations are required for ...

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