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_a and s_α are both equal to 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:

• Clarke followed by a rotation of -θ
• Concordia followed by a rotation of -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_q become:

Properties of the Transformation

• As expected, s_d and s_q are time-invariant since they do not depend on ωt;
• HYPERSIM uses this transformation;

Since s_d vanishes, it is hard to check whether s_q leads or lags s_d. For that, consider a three-phase system with phase delay δ as follows:

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_q become: