 ### Symmetrical Components in Electrical Engineering

Introduction

Symmetrical components method was discovered by C. L. Fortescue. The knowledge of symmetrical components is very useful for the study of unsymmetrical faults in three phase power networks. The concept  is also useful for studying the three phase machine behavior under unbalanced condition.

Symmetrical Components

Why symmetrical components? Symmetrical component technique is used for analyzing unbalanced three phase systems. When the system is balanced, analysis is very simple. We do not analyze for all the three phases instead we analyze it as single phase system. So the three phase system is reduced to simpler single phase system. Symmetrical component method helps to apply the single phase analysis tools also to the unbalanced three phase system. How?

In Symmetrical component method, any unbalanced three phase system can be resolved into three sets of symmetrical components. These three sets are positive sequence, negative sequence and zero sequence.

Considering counterclockwise abc sequence as positive sequence, then acb will be negative sequence (See Fig-A). Both the positive and negative sequence components are balanced. It means that the three phasors have the same magnitude and the phase angle between any two phasors is 120 degrees. The three phasors of zero sequence are of same magnitude and aligned in the same direction. So, in case of zero sequence the angle between any two phasors is zero. All these phasors rotate counterclockwise with frequency of the system. So the relative position between the phasors remains the same. .

For identification purposes we have used the symbols +, - and 0 for positive, negative and zero sequence components respectively.

Every three phase unbalanced system can be decomposed to three balanced systems as in Fig-A.

A three phase unbalanced system is shown in Fig-B.
The unbalnced system in Fig-B can be resolved to symmetrical components like Fig-A. In Figure-C just see  how each unbalanced component is made up of +ve, -ve and 0 sequence components.

From the diagram above it is easy to verify the below equations.

Now is the time to apply the phasor operator  a  that we learned previously.
(Phasor operator when applied to a phasor rotates the phasor anticlockwise by 120 degrees).

So the above equation can be written as below

In the above equation we have eliminated both b and c phase positive, negative and zero sequence components.

When the unbalanced system is known. we know Va, Vb and Vc.
Of course we also know the value of phasor operator a which is constant.

So the above three equations has three unknown Va+, Va- and Va0.
We can solve the equations and find the three unknowns by using school maths.

Now from Va+, Va- and Va0 that we calculated we can construct the full symmetrical components as in Fig-A. It will simplify for per phase analysis.