**By Robert D. Castro, University of Southern California**

Engineers typically view symmetrical components as a mathematical machination to transform unbalanced vectors into sets of balanced vectors. This article relates symmetrical components to physical concepts encountered in power system operations.

In a balanced system, the system impedances in each phase are identical, and the three-phase voltages and currents are completely balanced. In a balanced system, analysis can proceed on a single-phase basis because the knowledge of the voltage and current in one phase is sufficient to completely determine the voltages and currents in the other phases.

Because most faults are not balanced, however–only 5 percent of faults are three-phase–this is not applicable to most faults.

The method of symmetrical components is the only practical method of determining the voltages and currents in all parts of a power system under conditions of single-phase load or unbalanced short-circuit. Symmetrical components can be viewed as one of those methods by which a complicated problem can be transformed into a set of simpler ones.

One way to look at symmetrical components is to view them as a transformation of unbalanced vectors into an arena where they are balanced and form what is called (in abstract algebra) a group, where the a operator can be used to solve them. Once the symmetrical components are used to solve for the value, the unbalanced values can be derived from the symmetrical components (kind of an inverse transformation back into reality).

Symmetrical components typically are used as an interim device to go from system fault conditions to calculated fault values. Viewed in this light, symmetrical components are an abstract tool having little relevance in the real world. This turns out not to be the case; we can use the sequence components of symmetrical components in the real world to determine if the system is operating in an unbalanced fashion.

## Electromagnetic Negative Sequence Relays

Modern relays rely on numerical techniques that calculate sequence values from standard inputs of voltage and currents. Before these solid-state technologies were adopted by the protection sector, however, engineers had to use electromechanical methods for developing their relays. Let’s see how they developed their negative sequence relays using the properties of symmetrical components. This will demonstrate that although phase sequence components are mathematical concepts, we can separate out actual quantities from three-phase values that are directly proportional to any of the phase sequence components, in effect, making sequence components physical quantities.

It is sometimes useful to operate protective relaying equipment from a specific sequence component of the three-phase system currents. Consider the negative sequence filter from an electromechanical relay shown in Figure 1 (left). We’ll see that the mutual reactor coils are connected in such a fashion that any negative sequence voltage is detected.

Without getting into too much detail of the filter design, it might be appropriate to point out that the filter operates by:

- 1. Shifting I
_{b}240 degrees counterclockwise. - 2. Shifting I
_{c}120 degrees counterclockwise. - 3. Adding I
_{a}to the vector sum of the shifted I_{b}and the shifted I_{c}. - 4. The inductors are wound on the same core, with phase C wound with the opposite polarity.

We can then choose X_{m} such that V_{Ref} becomes zero for positive sequence components and positive for zero sequence components.

It is evident that forming a counterclockwise Kirchoff loop around the closed circuit yields the values in Figure 2 (page 48).

## Analyzing for Positive Sequence Currents

Let’s analyze what happens to this equation when we have positive sequence currents entering it (assume phase sequence a-b-c).

We know that for phase sequence abc, the positive phase currents I_{B} and I_{C} are related to I_{A} by I_{B} = a^{2} I_{A} and I_{C} = a I_{A}

Substituting these into Figure 2 leads to the values in Figure 3 (right).

So we see that positive sequence currents do not make it through this filter. That is, under normal operation the reference voltage, V_{Ref}, will remain at 0 volts.

## Analyzing for Negative Sequence Currents

Let’s analyze what happens to this equation when we have negative sequence currents entering it. We know that for phase sequence abc, the negative phase currents IB and IC are related to IA by IB = a IA and IC = a2 IA

Substituting these into Figure 2 leads to the values in Figure 4 (right).

So we see that negative sequence currents do make it through this filter and will register a positive voltage amount at VRef. We see that negative sequence currents do make it through this filter and can be used to indicate unbalanced operation or an unbalanced fault (e.g., line-to-line, single line to ground) on the circuit. For ungrounded systems, however, the negative sequence current is too small to be used for ground fault detection; for that we will rely on zero sequence currents.

We’ve introduced a negative sequence filter and analyzed it to demonstrate how negative sequence components can be viewed in a physical context. In this way we’ve demonstrated that sequence components need not be viewed as a strictly mathematical construct and can be measured to indicate a problem in operation.

Robert Castro teaches at the University of Southern California and develops wind generation for a local utility. Reach him at robert.castro@alumni.usc.edu.