Implement *+-0* to *abc*
transform

**Library:**Simscape / Electrical / Control / Mathematical Transforms

The Inverse Symmetrical-Components Transform block implements an
inverse symmetrical transform of a positive, negative, and zero phasor. The transform
splits a symmetrical set of three phasors into the equivalent unbalanced set of
*a*, *b*, and *c* phasors.

Use this transform to regenerate a three-phase signal from a system that was decoupled using the Symmetrical-Components Transform block.

Use the `Power invariant`

property to choose between the Fortescue
transform, and the alternative, power-invariant version.

The inverse symmetrical-components transform regenerates an unbalanced three-phase
signal
[*V _{a},V_{b},V_{c}*]
from the

$$\left[\begin{array}{c}{V}_{a}\\ {V}_{b}\\ {V}_{c}\end{array}\right]=\frac{1}{K}\left[\begin{array}{ccc}1& 1& 1\\ {a}^{2}& a& 1\\ a& {a}^{2}& 1\end{array}\right]\left[\begin{array}{c}{V}_{a+}\\ {V}_{a-}\\ {V}_{a0}\end{array}\right].$$

where, *a* is the complex rotation operator

$$a={e}^{2\pi i/3},$$

and *K* is the constant that determines the type
of transform:

$$\{\begin{array}{cc}K=1& \text{Fortescuetransform}\\ K=\sqrt{3}& \text{Power-invarianttransform}\end{array}$$

If the transform was performed using the power-invariant option,
enable the `Power invariant`

property to select the
power-invariant inverse transform and regenerate the correct *abc*
signal.

The symmetrical-components transform separates an unbalanced three-phase signal given in phasor quantities into three balanced sets of phasors:

$$\left[\begin{array}{c}{v}_{a}\\ {v}_{b}\\ {v}_{c}\end{array}\right]=\left[\begin{array}{c}{v}_{a+}\\ {v}_{b+}\\ {v}_{c+}\end{array}\right]+\left[\begin{array}{c}{v}_{a-}\\ {v}_{b-}\\ {v}_{c-}\end{array}\right]+\left[\begin{array}{c}{v}_{a0}\\ {v}_{b0}\\ {v}_{c0}\end{array}\right],$$

where:

*v*,_{a}*v*, and_{b}*v*make up the original, unbalanced set of phasors._{c}*v*,_{a+}*v*, and_{b+}*v*make up the balanced, positive set of phasors._{c+}*v*,_{a-}*v*, and_{b-}*v*make up the balanced, negative set of phasors._{c-}*v*,_{a0}*v*, and_{b0}*v*make up the balanced, zero set of phasors._{c0}

The symmetrical-components transform calculates the symmetric
*a*-phase as:

$$\left[\begin{array}{c}{V}_{a+}\\ {V}_{a-}\\ {V}_{a0}\end{array}\right]=\frac{K}{3}\left[\begin{array}{ccc}1& a& {a}^{2}\\ 1& {a}^{2}& a\\ 1& 1& 1\end{array}\right]\left[\begin{array}{c}{V}_{a}\\ {V}_{b}\\ {V}_{c}\end{array}\right].$$

Because the remaining two sets of symmetrical phasors are not often used in
calculation, the transformation only generates the first set. However, you can
calculate the *b*- and *c*-sets in terms of simple
rotations of the first:

$$\left[\begin{array}{c}{V}_{b+}\\ {V}_{b-}\\ {V}_{b0}\end{array}\right]=\left[\begin{array}{ccc}{a}^{2}& 0& 0\\ 0& a& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{V}_{a+}\\ {V}_{a-}\\ {V}_{a0}\end{array}\right],$$

and

$$\left[\begin{array}{c}{V}_{c+}\\ {V}_{c-}\\ {V}_{c0}\end{array}\right]=\left[\begin{array}{ccc}a& 0& 0\\ 0& {a}^{2}& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{V}_{a+}\\ {V}_{a-}\\ {V}_{a0}\end{array}\right].$$

The three sets of balanced phasors generated by the symmetrical-components transform have the following properties:

The positive set has the same order as the unbalanced set of phasors

*a-b-c*.The negative set has the opposite order as the unbalanced set of phasors

*a-c-b*.The zero set has no order because all three phasor angles are equal.

This diagram visualizes the separation performed by the transform.

In the diagram, the top axis shows an unbalanced three-phase signal
with components *a*, *b*, and
*c*. The bottom set of axes separates the three-phase signal into
symmetrical positive, negative, and zero phasors.

Observe that in each case, the *a*, *b*, and
*c* components are symmetrical and are separated by:

+120 degrees for the positive set.

*-*120 degrees for the negative set.0 degrees for the zero set.

[1] Anderson, P. M. *Analysis of Faulted Power Systems.*
Hoboken, NJ: Wiley-IEEE Press, 1995.