Symmetrical components

The three-phase case

Symmetrical components are most commonly used for analysis of three-phase electrical power systems. The voltage or current of a three-phase system at some point can be indicated by three phasors, called the three components of the voltage or the current.

This article discusses voltage; however, the same considerations also apply to current. In a perfectly balanced three-phase power system, the voltage phasor components have equal magnitudes but are 120 degrees apart. In an unbalanced system, the magnitudes and phases of the voltage phasor components are different.

Decomposing the voltage phasor components into a set of symmetrical components helps analyze the system as well as visualize any imbalances. If the three voltage components are expressed as phasors (which are complex numbers), a complex vector can be formed in which the three phase components are the components of the vector. A vector for three phase voltage components can be written as

${\displaystyle {\mathbf{v}}_{abc}=\left[\begin{array}{c}{V}_a\\ V_b\\ V_c\end{array}\right]}$

and decomposing the vector into three symmetrical components gives

${\displaystyle \left[\begin{array}{c}{V}_a\\ V_b\\ V_c\end{array}\right]=\left[\begin{array}{c}{V}_{a,0}\\ V_{b,0}\\ V_{c,0}\end{array}\right]+\left[\begin{array}{c}{V}_{a,1}\\ V_{b,1}\\ V_{c,1}\end{array}\right]+\left[\begin{array}{c}{V}_{a,2}\\ V_{b,2}\\ V_{c,2}\end{array}\right]}$

where the subscripts 0, 1, and 2 refer respectively to the zero, positive, and negative sequence components. The sequence components differ only by their phase angles, which are symmetrical and so are ${\displaystyle \frac{2}{3}\pi }$ radians or 120°.

A matrix

Define a phasor rotation operator ${\displaystyle \alpha }$, which rotates a phasor vector counterclockwise by 120 degrees when multiplied by it:

${\displaystyle \alpha \equiv e^{\frac{2}{3}\pi i}}$.

Note that ${\displaystyle {\alpha }^3=1}$ so that ${\displaystyle {\alpha }^{-1}={\alpha }^2}$.

The zero sequence components have equal magnitude and are in phase with each other, therefore:

${\displaystyle V_0\equiv V_{a,0}=V_{b,0}=V_{c,0}}$,

and the other sequence components have the same magnitude, but their phase angles differ by 120°. If the original unbalanced set of voltage phasors have positive or abc phase sequence, then:

${\displaystyle \begin{array}{cc}{V}_1& \equiv V_{a,1}=\alpha V_{b,1}={\alpha }^2{V}_{c,1}\end{array}}$,

${\displaystyle \begin{array}{cc}{V}_2& \equiv V_{a,2}={\alpha }^2{V}_{b,2}=\alpha V_{c,2}\end{array}}$,

meaning that

${\displaystyle \begin{array}{c}{V}_{b,1}={\alpha }^2{V}_1\end{array}}$,

${\displaystyle \begin{array}{c}{V}_{c,1}=\alpha V_1\end{array}}$,

${\displaystyle \begin{array}{c}{V}_{b,2}=\alpha V_2\end{array}}$,

${\displaystyle \begin{array}{c}{V}_{c,2}={\alpha }^2{V}_2\end{array}}$.

Thus,

${\displaystyle \begin{array}{cc}{\mathbf{v}}_{abc}& =\left[\begin{array}{c}{V}_0\\ V_0\\ V_0\end{array}\right]+\left[\begin{array}{c}{V}_1\\ {\alpha }^2{V}_1\\ \alpha V_1\end{array}\right]+\left[\begin{array}{c}{V}_2\\ \alpha V_2\\ {\alpha }^2{V}_2\end{array}\right]\\ & =\left[\begin{array}{ccc}1& 1& 1\\ 1& {\alpha }^2& \alpha \\ 1& \alpha & {\alpha }^2\end{array}\right]\left[\begin{array}{c}{V}_0\\ V_1\\ V_2\end{array}\right]\\ & =\mathbf{\text{A}}{\mathbf{v}}_{012}\end{array}}$

where

${\displaystyle {\mathbf{v}}_{012}=\left[\begin{array}{c}{V}_0\\ V_1\\ V_2\end{array}\right],\mathbf{\text{A}}=\left[\begin{array}{ccc}1& 1& 1\\ 1& {\alpha }^2& \alpha \\ 1& \alpha & {\alpha }^2\end{array}\right]}$

If instead the original unbalanced set of voltage phasors have negative or acb phase sequence, the following matrix can be similarly derived:

${\displaystyle {\mathbf{\text{A}}}_{acb}=\left[\begin{array}{ccc}1& 1& 1\\ 1& \alpha & {\alpha }^2\\ 1& {\alpha }^2& \alpha \end{array}\right]}$

Decomposition

The sequence components are derived from the analysis equation

${\displaystyle {\mathbf{v}}_{012}={\mathbf{\text{A}}}^{-1}{\mathbf{v}}_{abc}}$

where

${\displaystyle {\mathbf{\text{A}}}^{-1}=\frac{1}{3}\left[\begin{array}{ccc}1& 1& 1\\ 1& \alpha & {\alpha }^2\\ 1& {\alpha }^2& \alpha \end{array}\right]}$

The above two equations tell how to derive symmetrical components corresponding to an asymmetrical set of three phasors:

• Sequence 0 is one-third the sum of the original three phasors.
• Sequence 1 is one-third the sum of the original three phasors rotated counterclockwise 0°, 120°, and 240°.
• Sequence 2 is one-third the sum of the original three phasors rotated counterclockwise 0°, 240°, and 120°.

Visually, if the original components are symmetrical, sequences 0 and 2 will each form a triangle, summing to zero, and sequence 1 components will sum to a straight line.

Intuition

The phasors ${\displaystyle V_{(ab)}=V_{(a)}-V_{(b)};V_{(bc)}=V_{(b)}-V_{(c)};V_{(ca)}=V_{(c)}-V_{(a)}}$ form a closed triangle (e.g., outer voltages or line to line voltages). To find the synchronous and inverse components of the phases, take any side of the outer triangle and draw the two possible equilateral triangles sharing the selected side as base. These two equilateral triangles represent a synchronous and an inverse system.

If the phasors V were a perfectly synchronous system, the vertex of the outer triangle not on the base line would be at the same position as the corresponding vertex of the equilateral triangle representing the synchronous system. Any amount of inverse component would mean a deviation from this position. The deviation is exactly 3 times the inverse phase component.

The synchronous component is in the same manner 3 times the deviation from the "inverse equilateral triangle". The directions of these components are correct for the relevant phase. It seems counter intuitive that this works for all three phases regardless of the side chosen but that is the beauty of this illustration. The graphic is from Napoleon's Theorem, which matches a graphical calculation technique that sometimes appears in older references books.

Poly-phase case

It can be seen that the transformation matrix A above is a DFT matrix, and as such, symmetrical components can be calculated for any poly-phase system.

Contribution of harmonics to symmetrical components in 3-phase power systems

Harmonics often occur in power systems as a consequence of non-linear loads. Each order of harmonics contributes to different sequence components. The fundamental and harmonics of order ${\displaystyle 3n+1}$ will contribute to the positive sequence component. Harmonics of order ${\displaystyle 3n-1}$ will contribute to the negative sequence. Harmonics of order ${\displaystyle 3n}$ contribute to the zero sequence.

Note that the rules above are only applicable if the phase values (or distortion) in each phase are exactly the same. Please further note that even harmonics are not common in power systems.

Consequence of the zero sequence component in power systems

The zero sequence represents the component of the unbalanced phasors that is equal in magnitude and phase. Because they are in phase, zero sequence currents flowing through an n-phase network will sum to n times the magnitude of the individual zero sequence currents components. Under normal operating conditions this sum is small enough to be negligible. However, during large zero sequence events such as lightning strikes, this nonzero sum of currents can lead to a larger current flowing through the neutral conductor than the individual phase conductors. Because neutral conductors are typically not larger than individual phase conductors, and are often smaller than these conductors, a large zero sequence component can lead to overheating of neutral conductors and to fires.

One way to prevent large zero sequence currents is to use a delta connection, which appears as an open circuit to zero sequence currents. For this reason, most transmission, and much sub-transmission is implemented using delta. Much distribution is also implemented using delta, although "old work" distribution systems have occasionally been "wyed-up" (converted from delta to wye) so as to increase the line's capacity at a low converted cost, but at the expense of a higher central station protective relay cost.