# Symmetrical Components

**Introduction to symmetrical components:**

In power systems, it is a common occurrence that there are unbalanced voltages and currents on the distribution side of the grid. But these voltages and currents might be unbalanced anywhere through the grid in the case of a Fault, most commonly a short circuit fault. The simplest method to calculate the before mentioned quantities is a method termed “symmetrical per phase” technique given by Fortescue’s Theorem. There are certain other name conventions and details assigned to this specific method as well.

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**Importance of symmetrical components in power systems:**

Calculating important phase angles and magnitudes of various voltages and currents during a fault state can be quite a tedious and difficult task as the system is quite large and unbalanced values also tend to make the calculations complex.

The method of symmetrical components simplified the problem of three-phase unbalanced systems. Symmetrical components, also, to be a powerful analytical tool, is conceptually useful.

**Balanced set of phasors:**

So what is the balanced set of phasors? Well, it’s like a spinning wheel with three arrows. Let’s name the three arrows A, B, and C. For 3-phase balanced set the angle between the phasors is 120° and has the same magnitude. The three arrows (phases) are 120° apart. The wheel can rotate in two possible directions counter-clockwise and clockwise. Each dimensional angle and magnitude represents an important working aspect in the overall function of a set of phasors.

*The difference in balanced and unbalanced Phasors:*

There is a major difference in balanced and unbalanced phasors. The balanced or symmetrical set of phasors are those which have the same magnitude and are all 120° apart. Whereas the unbalanced set of phasors host a different concept overall.

*Breakdown of the Symmetrical Components:*

As the name three-component method suggests that the symmetrical components comprise a set of components that are balanced in nature.

- Positive sequence component
- Negative sequence component
- Zero sequence component

For a symmetrical system, the source(generator) voltages are equal in magnitude and are in phase, with their three phases 120° apart.

*Zero Sequence Component:*

The zero phase sequence components are the set of three phasors that are equal in magnitude and at zero phase displacement from each other. The zero-phase sequence component is as shown in the figure here.

*Positive Sequence Component:*

The positive sequence has Three balanced phasors with the same magnitude and phase difference. They also have the same phase sequence as that of the original set of phasors.

*Negative Sequence Component:*

The Negative sequence component only slightly differs from the Positive sequence in terms of Phase frequency.

*Operator ‘a’*

As we have seen that in symmetrical component theory the concept of 120º displacement is quite important therefore, we require some parameter or operator that might cause this 120° rotation. The operator ‘*a*’ is thus used. It is defined as under:

The ‘a’ operator is a unit vector at an angle of 120°. It can be written as

*a=1∠120° and a²=1∠240°*

*Mathematical Equations:*

The unbalanced phase currents in a 3-phase system can be expressed in terms of symmetrical components as illustrated.\

**Positive sequence Components:**

We will now be using the ‘a’ operator to relate the positive sequence components of each phase.

*Iₐ₁ = Iₐ₁*

*Iᵦ₁ = (1∠240°) Iₐ₁ = a²Iₐ₁*

*I𝒸₁ = (1∠120°) Iₐ₁ = aIₐ₁*

So, it can be seen here that positive sequence components of phase ‘B’ and phase ‘C’ are ‘** a²**’ and ‘

**’ times the positive sequence component of phase ‘A’. Hence a set of balanced positive sequence currents is achieved.**

*a***Negative sequence Components:**

We will now be using the ‘a’ operator to relate the negative sequence components of each phase.

*Iₐ₂ = Iₐ₂Iᵦ₂ = (1∠120°)Iₐ₂ = aIₐ₂I𝒸₂ = (1∠240°)Iₐ₂ = a² Iₐ₂*

It can also be seen here that negative sequence components of phase ‘B’ and phase ‘C’ are ‘** a**’ and ‘

**’ times the negative sequence component of phase ‘A’. Hence a set of balanced negative sequence currents is achieved.**

*a²*Zero sequence Components:

It should be noted that zero sequence components for each phase remain the same.

*Iₐ₀ = Iₐ₀Iᵦ₀ = Iₐ₀I𝒸₀= Iₐ₀*

Or

*Iₐ₀ = Iᵦ₀ = I𝒸₀*

A balanced set of zero sequence components is achieved here.

**Simplified equations:**

The simplified set of equations consisting of symmetrical components of phase A, balanced set of 3 systems and Voltage equations for the 3 components are illustrated further.

**Importance of symmetrical components**:

We can see those symmetrical components are a way of analyzing unbalanced phasors which arise due to faults. Converting a set of unbalanced phasors into sets of balanced components greatly simplifies our analysis and is helpful in performing fault calculations, power flow studies, and stability studies.

Along with phasor values of current and voltage, the zero-sequence and negative-sequence components are also an important input to protective relays that sense these values to decide the nature and type of the fault.

Moreover, we can calculate sequence impedances using symmetrical components, from which we can derive sequence circuits of transmission lines and transformers and ultimately sequence networks of a complete power system. This helps can positively help in a number of ways.

Let us know if you have any queries regarding this topic and do provide us with your feedback in the comments.

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# About The Author

**Abdur Rehman** is a professional electrical engineer with more than eight years of experience working with equipment from 208V to 115kV in both the Utility and Industrial & Commercial space. He has a particular focus on Power Systems Protection & Engineering Studies.

Abdur Rehman is the CEO and co-founder of **allumiax.com** and creator of **GeneralPAC by AllumiaX**. He has been actively involved in various roles in the **IEEE Seattle Section**, IEEE PES Seattle, IEEE Region 6, and IEEE MGA.