What are symmetrical components? It’s simply a tool used in many areas of the power systems. One application is to calculate fault current magnitude and angle – notably, for unbalanced faults.

There are many different types of faults that occur in power systems – only perfect three phase faults are considered balanced or symmetrical fault – this occurs when all three phases touch each other at exactly the same time.

When a tree contacts a single phase and makes a connection to ground (Line-to-ground fault), or lightning bolt strikes a conductor, or two conductors slap each other due to strong winds, these type of faults produce unbalanced fault current and voltage – these type of faults are considered as ** unbalanced faults.**

**Balanced power system**: When all three phase currents and voltages are in normal state or running in normal conditions, this simply means that the phase current and voltages have equal magnitude and exactly 120 degrees phase displacement.

**When a line-to-ground, or line-to-line fault occurs – the system three phase voltages and currents phasors become unbalanced the moment the fault occurs.** Or in other words, during a faulted condition, the system 3 phase voltage and current have unequal magnitudes and unequal phase displacement.

The most powerful tool for dealing with these “unbalanced set of phasors” is to use the *method of symmetrical components* introduced by C. L. Fortsecue.

Through symmetrical components, we are able to simplify the power system and calculate fault currents, clearly discriminate different types of faults, increase the security of correct operation of system protection, etc – symmetrical components have many applications.

**NOTE:**The method of symmetrical components and the symmetrical components system is to help us aid in fault calculations – among many other things. However, the various sequence components do not exist in physical entities in power systems.

According to Fortescue’s theorm, any three unbalanced set of phasors of a three-phase system (such as Phase A, B, and C of voltage and current) can be expressed as three balanced set of phasors – which are called components.

# ABC and ACB Phase Sequence / Rotation

There are two types of phase sequence used in the United States and much of the world – ABC and ACB phase sequence. Suppose we have a balanced power system with equal phase magnitudes and 120 degrees phase displacement, the ABC and ACB phase sequence will look something like below:

#### ABC Phase Sequence

#### ACB Phase Sequence

### How do we determine ABC vs ACB phase sequence?

It’s very important to understand the system phase sequence for the purpose of intuitively understanding symmetrical components.

**Note**: phase sequence is also commonly referred to as

*phase rotation*.

# Definition of Symmetrical Components

## Positive Sequence Symmetrical Component

**Positive sequence symmetrical component** – consists of three phasors equal in magnitude with a 120 degrees phase displacement between each phase, and having the same phase sequence as the original phasor.

#### Positive Sequence Component Characteristics

- Equal phase magnitude
- 120 degrees phase displacement
- Same phase sequence as the system phasors
- Same phase rotation as the system phasors

## Negative Sequence Symmetrical Component

**Negative symmetrical components** – consists of three phasors equal in magnitude with a 120 degrees phase displacement between each phase, and having the opposite phase sequence of the original phasors

#### Negative Sequence Component Characteristics

- Equal phase magnitude
- 120 degrees phase displacement
- Opposite phase sequence as the system phasors
- Same phase rotation as the system phasors

## Zero Sequence Symmetrical Component

**Zero symmetrical components** – consists of three phasors equal in magnitude with no phase displacement – all three phasors sit on top of each other

#### Zero Sequence Component Characteristics

Representing the symmetrical components mathematically, we get the following equations

It’s very clear that the symmetrical components for each phase are superimposed to create the system phase voltages or currents. Which simply means that ${V}_{a}^{(0)}+{V}_{a}^{(1)}+{V}_{a}^{(2)}$ adds up vectorically to get ${V}_{a}$ – the system phase voltage.

Before we go too deep, let’s look at how the system phase sequence or phase rotation play a role in symmetrical components.

# How do we represent Symmetrical Components of Phase B and C with respect to Phase A?

It’s very clear from our mathematical representation of symmetrical components for a three phase system that we’ll have positive, negative, and zero sequence components for each phase – that’s a total of 9 symmetrical components that we must keep track.

Luckily for us, there is a trick to represent Phase B and C symmetrical components with respect to phase A. Let’s see how that works.

**NOTE:**The increment of 120 degrees phase rotation depends on the phase sequence. From our earlier discussions on the characteristics of positive sequence, we know that the positive and negative symmetrical component is the same phase sequence as the system phase sequence. However, negative sequence component is an

*opposite*phase sequence.

### Positive Sequence Rotation (ABC system)

Because positive symmetrical components by definition are equal magnitude and equal phase displacement, we can use ${V}_{a}^{(1)}$ to refer to ${V}_{b}^{(1)}$ and ${V}_{c}^{(1)}$ by simply shifting ${V}_{a}^{(1)}$ by increments of 120 degrees.

**Suppose our system was an ABC phase rotation – rotating counter-clockwise direction.** If we shift
${V}_{a}^{(1)}$
by 240 degrees in CCW direction, we get
${V}_{b}^{(1)}$. Similarly, if we shift
${V}_{a}^{(1)}$
by 120 degrees in CCW direction, we get
${V}_{c}^{(1)}$
.

#### Mathematical Representation (ABC system)

$${V}_{b}^{(1)}=\left(1\angle 240\xb0\right)\cdot {V}_{a}^{(1)}$$ $${V}_{c}^{(1)}=\left(1\angle 120\xb0\right)\cdot {V}_{a}^{(1)}$$### Negative Sequence Rotation (ABC system)

Because negative symmetrical components by definition are equal magnitude and equal phase displacement, we can use ${V}_{a}^{(2)}$ to refer to ${V}_{b}^{(2)}$ and ${V}_{c}^{(2)}$ by simply shifting ${V}_{a}^{(2)}$ by increments of 120 degrees.

**Suppose our system was an ABC phase rotation – rotating counter-clockwise direction**. If we shift
${V}_{a}^{(2)}$
by 120 degrees in CCW direction, we get
${V}_{b}^{(2)}$. Similarly, if we shift
${V}_{a}^{(2)}$
by 240 degrees in CCW direction, we get
${V}_{c}^{(2)}$
.

#### Mathematical Representation

$${V}_{b}^{(2)}=\left(1\angle 120\xb0\right)\cdot {V}_{a}^{(2)}$$ $${V}_{c}^{(2)}=\left(1\angle 240\xb0\right)\cdot {V}_{a}^{(2)}$$### Zero Sequence Rotation (ABC system)

Zero sequence components $({V}_{a}^{\left(0\right)},{V}_{b}^{\left(0\right)},{V}_{c}^{\left(0\right)})$ by definition are all rotating in the same direction as the system phase rotation (CCW in our example) – they have equal magnitude but no phase displacement since the all three zero sequence components are are in phase and overlapping each other. Phase shifting is not required.

#### Mathematical Representation

$${V}_{b}^{(0)}={V}_{a}^{(0)}$$ $${V}_{c}^{(0)}={V}_{a}^{(0)}$$# Where does the “a” operating quantity of symmetrical components come from?

In the previous section, we talked about representing phase B and C symmetrical components with respect to phase A. The ubiquitous operating quantity “a” simply does exactly that.

Mathematical definition of the “a” quantity:

$$a=1\angle 120\xb0$$ $${a}^{2}=1\angle 240\xb0$$When we multiply any rotating phasor with $a$ or ${a}^{2}$ we’re essentially shifting the rotating phasor by a factor of 120 degrees with respect to the original phase rotation. Therefore, we can represent phase B and C symmetrical components with respect to phase A. Why do we this? To simply our problem of course!

Once we determine the symmetrical components with respect to phase A, we can get phase B and C symmetrical components by shifting counter clock-wise (for ABC Phase sequence) by a factor of 120 degrees.

Remember our fundamental definition of symmetrical components? Symmetrical components of each phase are superimposed to create the system three phase voltage or current. By superposition, we simply mean that ${V}_{a}^{\left(0\right)}+{V}_{a}^{\left(1\right)}+{V}_{a}^{\left(2\right)}$ adds up vectorically to get ${V}_{a}$

Here are the equations again -

$${V}_{a}={V}_{a}^{(0)}+{V}_{a}^{(1)}+{V}_{a}^{(2)}$$ $${V}_{b}={V}_{b}^{\left(0\right)}+{V}_{b}^{\left(1\right)}+{V}_{b}^{\left(2\right)}$$ $${V}_{c}={V}_{c}^{(0)}+{V}_{c}^{(1)}+{V}_{c}^{(2)}$$But now that wan represent ${V}_{b}^{\left(0\right)},{V}_{b}^{\left(1\right)},{V}_{b}^{\left(2\right)},{V}_{c}^{(0)},{V}_{c}^{(1)},{V}_{c}^{(2)}$ with respect to ${V}_{a}^{(0)},{V}_{a}^{(1)},{V}_{a}^{(2)}$ – our equation simplifies to the following –

$${V}_{a}={V}_{a}^{(0)}+{V}_{a}^{(1)}+{V}_{a}^{(2)}$$ $${V}_{b}={V}_{a}^{(0)}+{a}^{2}\cdot {V}_{a}^{(1)}+a\cdot {V}_{a}^{(2)}$$ $${V}_{c}={V}_{a}^{(0)}+a\cdot {V}_{a}^{(1)}+{a}^{2}\cdot {V}_{a}^{(2)}$$If we look at the above equation in matrix form, we get the following –

$$\left[\begin{array}{c}{V}_{a}\\ {V}_{b}\\ {V}_{c}\end{array}\right]=\left[\begin{array}{ccc}1& 1& 1\\ 1& {a}^{2}& a\\ 1& a& {a}^{2}\end{array}\right]\cdot \left[\begin{array}{c}{V}_{a}^{\left(0\right)}\\ {V}_{a}^{\left(1\right)}\\ {V}_{a}^{\left(2\right)}\end{array}\right]$$Another common way to represent the above matrix is as follows –

$$\left[\begin{array}{c}{V}_{a}\\ {V}_{b}\\ {V}_{c}\end{array}\right]=A\cdot \left[\begin{array}{c}{V}_{a}^{\left(0\right)}\\ {V}_{a}^{\left(1\right)}\\ {V}_{a}^{\left(2\right)}\end{array}\right]$$Where $A=\left[\begin{array}{ccc}1& 1& 1\\ 1& {a}^{2}& a\\ 1& a& {a}^{2}\end{array}\right]$

**NOTE:**the above matrix multiplication is specifically for ABC phase rotation. If we have an ACB phase rotation, we’ll need to shift the symmetrical components in the opposite direction.

Watch an excellent video tutorial on matrix multiplication here for those of us who need a quick refresher.

What if we wanted to solve for the symmetrical components matrix, $\left[\begin{array}{c}{V}_{a}^{\left(0\right)}\\ {V}_{a}^{\left(1\right)}\\ {V}_{a}^{\left(2\right)}\end{array}\right]$ instead?

How would we accomplish this? We simply multiply both sides of the equation by the inverse of the “A” matrix. For example:

$$\left[\begin{array}{c}{V}_{a}\\ {V}_{b}\\ {V}_{c}\end{array}\right]\cdot {A}^{-1}=A\cdot {A}^{-1}\cdot \left[\begin{array}{c}{V}_{a}^{\left(0\right)}\\ {V}_{a}^{\left(1\right)}\\ {V}_{a}^{\left(2\right)}\end{array}\right]$$ where $${A}^{-1}=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\cdot \left[\begin{array}{ccc}1& 1& 1\\ 1& a& {a}^{2}\\ 1& {a}^{2}& a\end{array}\right]$$ $$\left[\begin{array}{c}{V}_{a}^{\left(0\right)}\\ {V}_{a}^{\left(1\right)}\\ {V}_{a}^{\left(2\right)}\end{array}\right]={A}^{-1}\cdot \left[\begin{array}{c}{V}_{a}\\ {V}_{b}\\ {V}_{c}\end{array}\right]$$The $A\cdot {A}^{-1}$ gives us the identity matrix which is essentially equivalent to multiplying by 1. Learn more about the inverse matrix and how to solve them here.

Solving for the symmetrical component gives us the following -

$$\left[\begin{array}{c}{V}_{a}^{\left(0\right)}\\ {V}_{a}^{\left(1\right)}\\ {V}_{a}^{\left(2\right)}\end{array}\right]={A}^{-1}\cdot \left[\begin{array}{c}{V}_{a}\\ {V}_{b}\\ {V}_{c}\end{array}\right]$$or

$$\left[\begin{array}{c}{V}_{a}^{\left(0\right)}\\ {V}_{a}^{\left(1\right)}\\ {V}_{a}^{\left(2\right)}\end{array}\right]=\frac{1}{3}\cdot \left[\begin{array}{ccc}1& 1& 1\\ 1& a& {a}^{2}\\ 1& {a}^{2}& a\end{array}\right]\cdot \left[\begin{array}{c}{V}_{a}\\ {V}_{b}\\ {V}_{c}\end{array}\right]$$### Summary of the “a” operator

If we were to summarize this entire section with only two equations, it would be the following two. They are the fundamental building blocks of symmetrical components and all power engineers should understand where they come from.

For ABC System Phase Sequence

For ABC System Phase Sequence

# What are the symmetrical components for a balanced three phase system with ABC phase rotation?

The balanced three phase characteristic of symmetrical component is very important and should be understood by all of us.

From our earlier discussion, we said that balanced three phase systems have equal magnitude and equal phase displacement. A balanced three phase (ABC Phase Rotation) voltage looks like the following:

Where

$${V}_{a}=100V\angle 0\xb0$$ $${V}_{b}=100V\angle 240\xb0$$ $${V}_{c}=100V\angle 120\xb0$$It is clear that when all three voltage phasors above rotate in a counter-clockwise direction, it forms an ABC phase rotation.

The positive, negative, and zero sequence components for the above three balanced set of phasors are –

For a balanced three phase system, the positive, negative, and zero symmetrical components are -

$${V}_{a}^{\left(0\right)}=0$$ $${V}_{a}^{\left(1\right)}=100V\angle 0\xb0$$ $${V}_{a}^{\left(2\right)}=0$$It’s very important for us to understand the above symmetrical components are only for phase A – for phase B and C, we simply follow what we’ve learned in the above sections.

It’s very clear that for a balanced three phase system, phase A, B, and C positive sequence components are equal to phase A, B, and C of the original three phase system. However, zero and negative sequence components are all zero.

### Summary

For balanced three phase systems – only positive sequence components exists. Zero and Negative sequence components are zero.