SYMMETRICAL COMPONENTS

 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 120^o 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 120^o 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 120^o

In circuit analysis, ‘j’ causes a 90^o rotation

In complex number notation ‘i’ causes a 90^o rotation

In numerical analysis and number notation and representation ‘-1’ causes a 180^o rotation

A rotation of 90o + 90o  is achieved by multiplying ‘ j x j ’ which causes a 180^o rotation j² = -1.

With this understanding,

a = 1/120 = 1e^j2π/3 = -0.5 + j0.866

apply ‘a’ twice to achieve a 240^o rotation

when ‘a’ is applied three times the phasor is rotated 360o

Hence

a² = 1/240 = 1e^j4π/3 = -0.5 - j0.866

a³ = 1/360 = 1e^j2π = 1/0^o = 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  =  V^TA^T

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)

 

                                       A^t  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.