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# Clarke, Concordia and Park Transformations

# Three-Phase System

Consider a direct three-phase system given by:

# Transformations from (a,b,c) to (α, β)

Knowing that for a three-phase system the summation of signals is zero, it makes sense to get rid of the summation equation and to move to a two-signals system in which those signals are orthogonal, therefore, non-dependent. For that, two transformations are suggested: Clarke and Concordia.

## Clarke Transformation

Clarke transformation applied to the three-phase system leads to:

Therefore,

From which the inverse of Clarke transformation is computed as:

### Properties of Clarke Transformation

- This transformation keeps the amplitude invariant, hence the amplitudes of
*s*and_{a}*s*are both equal to_{α}*S√2*. However, the power is variant - The determinant of this transformation is
*2/(3√3)*, hence this is not a rotation - SPS solver uses this transformation

## Concordia Transformation

Therefore,

### Properties of Concordia Transformation

- This transformation changes the amplitude but keeps the power invariant
- The determinant of this transformation is 1. Additionally, columns are orthogonal and have unity-norm. Consequently, this is a rotation and the inverse is nothing but the transpose
- SSN solver uses this transformation

# Transformations from (α, β) to (d,q)

## Rotation of Angle -θ

This transformation is used to move from (*α, β*) stationary frame to (*d, q*) rotating frame. The goal behind is to remove the *ωt *component and to end-up with time-invariant signals *s** _{d}* and

*s*

*.*

_{q}Knowing that the three-phase system depicted in Fig. 2 is undergoing counterclockwise rotation at *ω *rd*/*s, a rotation of an angle *θ *= -*ωt *is to be applied in order to get time-invariant signals in the (*d, q*) frame, leading to:

Obviously, this transformation corresponds to a rotation matrix Q with det = 1. The calculation of the speed matrix goes as follows:

It is noteworthy that the speed matrix Ω is skew-symmetric, which is a necessary condition for the speed term.

# Transformations from (a,b,c) to (d,q)

Obviously there are four possible combinations to bring the three-phase system (*a, b, c*) to a (*d, q*) one, namely:

*θ**θ*- Clarke followed by a rotation of -
*θ*+*pi/*2 - Concordia followed by a rotation of -
*θ*+*pi/*2

Notice that these transformations are known as Park transformations in the literature.

## Combination of Clarke and Rotation of -θ

For the three-phase system defined as in the equations at the top of this page, *s _{d} *and

*s*become:

_{q}### Properties of the Transformation

- As expected,
*s*and_{d}*s*are time-invariant since they do not depend on_{q}*ωt*; **HYPERSIM uses this transformation;**

Since *s _{d }*vanishes, it is hard to check whether

*s*leads or lags

_{q}*s*. For that, consider a three-phase system with phase delay δ as follows:

_{d}Applying Clarke transformation leads to:

Applying rotation of -θ leads to:

Figures 3 and 4 show the spatial representation of this system for *δ = pi/6* and *ωt = 0*, then *ωt = pi/3*. It's worth noticing that *s _{q}* lags

*s*.

_{d}## Combination of Concordia and Rotation of -θ

For the three-phase system defined as in the equations at the top of this page, *s _{d}* and

*s*become:

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