SYMMETRICAL
COMPONENTS
An unbalanced system of n related phasors can be resolved
into n systems of balanced phasors called symmetrical components of the
original phasors.
These symmetrical components are balanced. They are equal in
length and the angle between them are also equal.
In the Electrical Power System, The Symmetrical Components
used, has the Positive Sequence component, the negative Sequence component and the
zero Sequence components.
The characteristic of these components are as follows:
The Positive Sequence component,
The Positive Sequence components consist of three phasors
equal in magnitude displaced from each other by 120o in phase and
having the same phase sequence as the original Phasors. ,
The negative Sequence component,
The negative Sequence components consist of three phasors
equal in magnitude displaced from each other by 120o in phase and
having a phase sequence opposite that of the original Phasors. ,
The Zero Sequence component,
The Zero Sequence Components
consist of three phasors equal in magnitude with zero phase displacement from
each other.
V(a) = V(a1)+V(a2)+V(a0) (i)
V(b) = V(b1)+V(b2)+V(b0) (ii)
V(c) = V(c1)+V(c2)+V(c0) (iii)
Positive Sequence Component
abc(phase sequence of original phasors) ------- abc (phase sequence of
symmetrical components)
Negative Sequence Component
abc(phase sequence of original phasors) -------- acb(phase sequence of symmetrical components)
1=> +ve
2=> -ve
0=> zero
In symmetrical Component Analysis, ‘a’ causes a rotation of
120o
In circuit analysis, ‘j’ causes a 90o rotation
In complex number notation ‘i’ causes a 90o
rotation
In numerical analysis and number notation and representation
‘-1’ causes a 180o rotation
A rotation of 90o + 90o
is achieved by multiplying ‘ j x j ’ which causes a 180o
rotation j2 = -1.
With this understanding,
a = 1/120 = 1ej2π/3
= -0.5 + j0.866
apply ‘a’ twice to achieve a 240o rotation
when ‘a’ is applied three times the phasor is rotated 360o
Hence
a2 = 1/240 = 1ej4π/3
= -0.5 - j0.866
a3 = 1/360 = 1ej2π
= 1/0o = 1
V(b1) = a2V(a1)
V(b2) = aV(a2)
V(b0) = V(a0)
V(c1) = aV(a1)
V(c2) = a2V(a2)
V(c0) = V(a0)
V(a) = V(a1) + V(a2) + V(a0)
V(b) = a2V(a1) + aV(a2) + V(a0)
V(c) =aV(a1) + a2V(a2) + V(a0)
 |
|||
 |
|||
A =
 |
A-1 =
 |
|||
 |
|||
A-1 = 1/3
A-1 = 1/3
V(a0) = 1/3 (V(a) + V(b) + V(c))
V(a10) = 1/3 (V(a) + aV(b) + a2V(c))
V(a20) = 1/3 (V(a) + a2V(b)
+ aV(c))
Let us take a look at the effect of Symmetrical Components
on the Impedance with respect to the Electrical Power System.
V(abc) = Z(abc)I(abc)
Where
Z(abc) 3x3 matrix
giving the self and mutual impedances in and between phases
However
V(abc) = [A]V(012)
I(abc) = [A]I(012)
[A]V(012) = Z(abc)[A]I(012)
Or
V(012) = [A]-1Z(abc)[A]I(012)
Let us define
Z(012) = [A]-1Z(abc)[A]
V(012) = Z(012)I(012)
Power Considerations
We shall apply the Reversal Rule of Matrix Algebra.
The Reversal Rule of Matrix Algebra.
The Reversal Rule Of Matrix Algebra states that the
transpose of the Product of two matrices is equal to the Product of the
transpose of the matrices in reverse order hence :
[AV]T
= VTAT
Complex Power can be expressed as follows in the Electrical
Power System:
S(3φ) = V(a)I*(a) + V(b)I*(b)+ V(c)I*(c)
Let ‘t’ => transpose (interchange of rows and columns
in a matrix v*(i,j) = v(j,i)
)
Let ’*’ =>
conjugate (the mirror image along the x axis of the complex plane of a
complex number or phasor )
In Matrix Notation, This can be written as:
S(3φ) = Vt(abc)I*(abc)
=
{[A]V(012)}t{[A]I(012)} *
=
V(012)tAtA*I*(012)
 |
At A* =
 |
=
Therefore
S(3φ) = 3V(012)t I* (012)
S(3φ) = 3(V(0)I*(0)V(1)I*(1)V(2)I*(2))
Notice that there are no cross terms or mixed quantities in
the expression for S(3φ). This is useful when equivalent circuits are being
considered.
Symmetrical Components are very important in the Analysis of
the Electrical Power System.
It is particularly very useful in the analysis of unbalanced
faults as a result of the unbalanced phasors associated with them.
No comments:
Post a Comment