Symmetrical components are applied to
calculations of unbalanced fault currents and voltages and in rotating machine
analysis. Theory of symmetrical components can be briefly stated thus: a
coplanar vector is defined by the position of its terminal and length and has 2
degrees of freedom.
A three-phase balanced system has 2 degrees
of freedom because the current or voltage vectors (phasors) are displaced from
each other by equal angles of separation of 120◦ and are of equal length. A
three-phase unbalanced system of currents or voltages has 6 degrees of freedom
because the vectors are of varying length at varying displacement angles from
each other.
Such an unbalanced system can be resolved
into three symmetrical systems, each system having three vectors with 2 degrees
of freedom. Positive-sequence system is a set of balanced three-phase
components of the same phase sequence as the original unbalanced set.
Negative-sequence system is a set of
three-phase components of opposite phase sequence to the positive sequence
system but vectors (phasors) of the same magnitude. Zero-sequence system
consists of three single-phase components of the same magnitude and cophasial.
and
Where Va, Vb, and Vc are the original
unbalanced voltages; a is a unit vector operator that rotates 120◦ in the
counterclockwise direction; and V+a, V −a, and V0a are the positive, negative,
and zero sequence components of the original unbalanced set.
Characteristics of Sequence Components
In a three-phase wye connected and
ungrounded system, no zero sequence current flows. If the wye point is
grounded, neutral carries the out-of-balance current. In a delta connection, no
zero sequence currents can appear in the line currents.
In a balanced three-phase system with
balanced loads, only positive sequence currents can flow. Negative sequence
currents are set up in circuits of unbalanced impedances and voltages.
In symmetrical circuits, currents and
voltages of different sequence do not affect each other (i.e., the positive
sequence currents produce only positive sequence voltage drops and the theorem
of superposition applies). Sequence impedance networks must be constructed for
unbalanced fault current calculations and data input to digital computers.
As an example, the single-line-to-ground
fault is given by the expression:
Where Ig is the single-line-to-ground fault
current; E is the line-to-neutral voltage; and Z+, Z−, and Z0 are the positive,
negative, and zero sequence impedances to the fault point.
No comments:
Post a Comment