Here are the basic equations of the three-phase symmetric components:

,

,

.

The vector has unit length and ° angular displacement.

The positive sequence is:

,

.

The negative sequence is:

,

.

The zero sequence is:

.

The phasor diagram comes from these equations. See [1], where these relations are given in terms of matrices. More details can be found in [2], where the matrix relation is also illustrated.

The sequence components can be used to create filters that identify the type of fault that occurs, depending on the magnitude and angle of the three components. Any kind of fault can be modeled and analyzed by symmetrical components, where each fault can be represented by an equivalent circuit. For example, in the case of a fault on phase A, the three phasors of sequence components are equal in magnitude and angle. All cases of unbalanced faults can be analyzed by symmetric components theory, like monophasic, biphasic, and biphasic with ground. References [3], [4], and [5] show the calculation methods and equivalent circuits for each type of fault, and [6] shows some phasor formats.

The phasor diagram illustrates the components of positive sequence, negative sequence, and zero sequence of a three-phase system, and the original vector of each phase. This Demonstration lets you see that the sums of the sequence components of each phase always result in the original vector.

The last two snapshots illustrate cases of failure: monophasic fault and biphasic without ground fault. Note that for the monophasic fault, the three symmetrical components have the same magnitude and angle. For biphasic without ground fault, the positive and negative components have the same length and the angle is ° out of phase, since the zero sequence is zero. These results are precisely what is expected when the equivalent circuits are solved.

Observations:

• The angles should be input using positive values (e.g., ).

• The green arrows are the positive sequence.

• The blue and yellow arrows are the negative sequence, which is shown twice to show (1) symmetry (yellow); and (2) that the sum of its components is the original vector (blue).

• The red arrows are the zero sequence.