Detailed Notes: Symmetrical Components | Power Systems - Electrical Engineering (EE) PDF Download

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CHAPTER  3:  SYMMETRICAL COMPONENTS
[CONTENTS: Introduction, The a operator, Power in terms of symmetrical components, Phase shift in Y-
? transformer banks, Unsymmetrical series impedances, Sequence impedances, Sequence
networks, Sequence networks of an unloaded generator, Sequence networks of elements,
Sequence networks of power system]
3.1    INTRODUCTION
Power systems are large and complex three-phase systems. In the normal operating
conditions, these systems are in balanced condition and hence can be represented as an
equivalent single phase system.  However, a fault can cause the system to become
unbalanced. Specifically, the unsymmetrical faults: open circuit, LG, LL, and LLG faults
cause the system to become unsymmetrical. The single-phase equivalent system method
of analysis (using SLD and the reactance diagram) cannot be applied to such
unsymmetrical systems. Now the question is how to analyze power systems under
unsymmetrical conditions? There are two methods available for such an analysis:
Kirchhoff’s laws method and Symmetrical components method.
The method of symmetrical components developed by C.L. Fortescue in 1918 is a
powerful technique for analyzing unbalanced three phase systems. Fortescue defined a
linear transformation from phase components to a new set of components called
symmetrical components. This transformation represents an unbalanced three-phase
system by a set of three balanced three-phase systems. The symmetrical component
method is a modeling technique that permits systematic analysis and design of three-
phase systems. Decoupling a complex three-phase network into three simpler networks
reveals complicated phenomena in more simplistic terms.
Consider a set of three-phase unbalanced voltages designated as V
a
, V
b
, and V
c
.
According to Fortescue theorem, these phase voltages can be resolved into following
three sets of components.
1. Positive-sequence components, consisting of three phasors equal in magnitude,
displaced from each other by 120
0
in phase, and having the same phase sequence as
the  original phasors, designated as V
a1
, V
b1
, and V
c1
2. Negative-sequence components, consisting of three phasors equal in magnitude,
displaced from each other by 120
0
in phase, and having the phase sequence opposite
to that of the original phasors, designated as V
a2
, V
b2
, and V
c2
3. Zero-sequence components, consisting of three phasors equal in magnitude, and with
zero phase displacement from each other, designated as V
a0
, V
b0
, and V
c0
Since each of the original unbalanced phasors is the sum of its components, the
original phasors expressed in terns of their components are
V
a
= V
a1
+ V
a2
+ V
a0
V
b
= V
b1
+ V
b2
+ V
b0
V
c
= V
c1
+ V
c2
+ V
c0
(3.1)