4: Introduction to symmetrical components

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James Kirtley
Massachusetts Institute of Technology via MIT OpenCourseWare

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Introduction

Installment 3 of these notes dealt primarily with networks that are balanced, in which the three voltages (and three currents) are identical but for exact 120^o phase shifts. Unbalanced conditions may arise from unequal voltage sources or loads. It is possible to analyze some simple types of unbalanced networks using straightforward solution techniques and wye-delta transformations. However, power networks can be come quite complex and many situations would be very difficult to handle using ordinary network analysis. For this reason, a technique which has come to be called symmetrical components has been developed.

Symmetrical components, in addition to being a powerful analytical tool, is also conceptually useful. The symmetrical components themselves, which are obtained from a transformation of the ordinary line voltages and currents, are useful in their own right. Symmetrical components have become accepted as one way of describing the properties of many types of network elements such as transmission lines, motors and generators.

4.1: The Symmetrical Component Transformation
The basis for this symmetrical component transformation technique is a transformation of the three voltages and three currents into a second set of voltages and currents. This second set is known as the symmetrical components.
4.2: Sequence Impedances
Many different types of network elements exhibit different behavior to the different symmetrical components. For example, as we will see shortly, transmission lines have one impedance for positive and negative sequence, but an entirely different impedance to zero sequence. Rotating machines have different impedances to all three sequences.
4.3: Unbalanced Sources
Consider the network shown in Figure 4. A balanced three-phase resistor is fed by a balanced line (with mutual coupling between phases). Assume that only one phase of the voltage source is working, so that:
4.4: Rotating Machines
Some network elements are more readily represented by sequence networks than by ordinary phase networks. This is the case, for example, for synchronous machines. synchronous motors and generators produce a positive sequence internal voltage and have terminal impedance.
4.5: Transformers
Transformers provide some interesting features in setting up sequence networks. The first of these arises from the fact that wye-delta or delta-wye transformer connections produce phase shifts from primary to secondary. Depending on connection, this phase shift may be either plus or minus 30o from primary to secondary for positive sequence voltages and currents.
4.6: Unbalanced Faults
A very common application of symmetrical components is in calculating currents arising from unblanced short circuits. For three-phase systems, the possible unbalanced faults are: