This post gives a brief about power in symmetrical components. Let us take a look at symmetrical component transformation in time-dependent form used in electric-machine theory, with the help of few mathematical equations, and vector diagrams.
Charles Legeyt Fortescue proved that any set of unbalanced phasors or polyphase signal can be represented as the sum of N symmetrical sets of balanced phasors where N is a prime number. These phasors represent only a single frequency component.
One of the sets of phasors has the same phase as the underlying system, second set has a reverse phase and in the last set all individual phasors are in phase with each other.This forms three independent sources which makes the analysis of asymmetric faults easy and tractable.This technique can also be extended to higher order phase systems.This analytical technique has been accepted and progressed by engineers at General Electric and Westinghouse.
In a 3 phase winding, a positive sequence set of currents produce normal rotating field, a negative sequence set produces a field with the opposite rotation, and the zero sequence set with all phasors in phase with each other produces an oscillating field and not a rotating one. These effects can be detected physically. This mathematical tool has become the basis for the design of protection relays. In such cases negative-sequence voltages and currents are used as a reliable indicator of fault conditions.
Symmetrical component transformation application in the time-dependent form is used in electric-machine theory. In the field of power systems this transformation is applied to steady state sinusoidal phasors for fault corrections. Although they were formulated for 3-phase phasors, they are fundamental to the transformation of random instantaneous variables. So, now let us try to understand symmetrical components power systems with the help of some mathematical equations.
The aiee paper aircraft symmetrical components are also based on the following equations. The transformation matrix used in the first application is as given below:
This transformation was extended to m-n winding machines and the unitary form was used, which is advantageous since power and torque need no back transformation, since unitary transformation is power invariant. In this unitary form the symmetrical transformation matrix is as follows:
These equations can also be used to update simpler version of wagner evans symmetrical components.
The above matrix can also be applied to both current and voltages. Application of this transformation to 3-phase and m-phase systems has shown its property of decomposing higher space harmonics into special groups.
The new variables are related to old variables as:
Let be a general asymmetrical three-phase voltage:
where u- the instantaneous value
– the rms value of the phase voltage.
Transformation of Power into Symmetrical Components
Before transformation of power into symmetrical components, the result can be written as the sum of 2 complex conjugated terms:
Transformation of this power into symmetrical components, can be further simplified by substituting:
where,
From the above symmetrical components solutions, conclusion can be drawn that the use of time-dependent symmetrical components in network calculations is advantageous in several ways.
Firstly Network-component data are available in these coordinates mostly and the simple relation with their steady-state phasors supports the interpretation of results that are calculated by the other popularly used steady-state phasor theories like in case of asymmetric faults.
An asymmetric or unbalanced fault is a fault, which does not affect each of the three phases equally. Hence, it is not possible to use one-line diagram. Since power systems are linear, the voltages and currents are considered as superposition of the aforementioned power in symmetrical components on which, 3 phase analysis can be effectively performed.
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