Fortescue's Theorem for a Three-Phase Unbalanced System

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In 1918, C. L. Fortescue stated his theorem: unbalanced phasors can be represented by systems of balanced phasors. Sequence components were created to facilitate calculations in unbalanced circuits and systems. Using the theory of symmetrical components, it is easier to analyze the problems of unbalanced systems (e.g. unbalanced load) and cases of faults in electric power systems (e.g. monophasic faults). According to [1], "Fortescue defined a linear transformation from three-phase components to a new set of components called symmetrical components. The advantage of this transformation is that for balanced three-phase networks the equivalent circuit obtained for the symmetrical components, called sequence networks, are separated into three uncoupled networks. Furthermore, for unbalanced three-phase systems, the three sequence networks are connected only at the points of unbalance. As a result, sequence networks for many cases of unbalanced three-phase systems are relatively easy to analyze. The symmetrical component method is basically a modeling technique that permits systematic analysis and design of three-phase systems. Decoupling a detailed three-phase network into three simpler sequence networks reveals complicated phenomena in more simplistic terms. Sequence network results can then be superimposed to obtain three-phase results. The application of symmetrical components to unsymmetrical fault studies is indispensable." The positive sequence consists of three phasors, one for each phase, with degrees of angular displacement and equal magnitude. Thus, phase B is degrees from phase A, and phase C is degrees from phase B. A negative sequence is similar to the positive component, but instead of degrees, the displacement is degrees. The zero sequence is represented by three vectors of equal magnitude and equal angular displacement.


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.


• 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.


Contributed by: Daniel Motter (August 2011)
Based on a program by: Harley H. Hartman

Open content licensed under CC BY-NC-SA


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