Symmetrical Components

The method of symmetrical components is used to simplify fault analysis by converting a three-phase unbalanced system into two sets of three phase balanced phasors (=> same magnitude) and one set of three phase parallel phasors called symmetrical components.

Make use of a rotational operator, denoted by a as the ∠120°and given formally by :

a	=	=	= 1∠+120°	Negative Sequence one time
a2	= 1∠+120° x  1∠+120°	= 1∠+240° or	= 1∠-120°	Negative Sequence two time or Positive Sequence one time
a3	= 1∠+120° x  1∠+120° x  1∠+120°	= 1∠+360° or	= 1	One round circle

Note  :  1 + a + a²  =  1 + 1∠+120° + 1∠-120°  =  0

 

By using the operator a, any unbalance three phase system I_L1, I_L2, I_L3 in actual world can be broken down into three theoretical balanced system, I(1) system (positive sequence system), I(2) system (negative sequence system), I(0 )system (zero sequence system).

 

In terms of operator a, in positive sequence system, their relationship is :

I_{L1(1)   =}                                                  I_L1(1)

I_{L2(1)   =}                                                 I_{L1(1) x} a²

I_L3(1)   = I_{L1(1) x a² x a²}  or  I_{L1(1) x} a        _∵ a³ _{= 1  ∴} a⁴ ₌ a³_x a _{= 1 x} a ₌ a

In terms of operator a, in negative sequence system, their relationship is :

I_L1(2)   =                                                 I_L1(2)

I_L2(2)   =                                                 I_{L1(2) x} a

I_L3(2)   = I_{L1(2) x} a _x a   or  I_{L1(2) x} a²

In zero sequence system, as there is no rotation, their relationship is :

I_L1(0)  =   I_L2(0)   =   I_L3(0)

 

Since  I = I _{(Pos) +} I _{(Neg) +} I _(Zero)

I_L1 = I_{L1(1)   +}    I_{L1(2)   +}  I_L1(0)

I_L2 = I_{L2(1) +} I_{L2(2) +} I_L2(0)  ∵ I_L2(1) = a² I_{L1(1)  and}  I_L2(2) = aI_{L1(2)  and}  I_L2(0) = I_L1(0)

I_L2 = a² I_{L1(1)  +}  aI_{L1(2)   +}  I_L1(0)

I_L3 = I_{L3(1) +} I_{L3(2) +} I_L3(0)  ∵ I_L3(1) = a I_{L1(1)  and}  I_L3(2) = a²I_{L1(2)  and}  I_L2(0) = I_L1(0)

I_L3 = a I_{L1(1)   +}  a²I_{L1(2)   +}   I_L1(0)

Now we have

      Pos      Neg     Zero

I_L1 =  I_{L1(1)   +}   I_{L1(2)   +}  I_L1(0)

I_L2 = a² I_{L1(1)  +}  aI_{L1(2)   +}  I_L1(0)

I_L3 = a I_{L1(1)   +}  a²I_{L1(2)   +}   I_L1(0)

 

For 3 phase 4 wire system, I_N = I_L1 + I_L2 + I_{L3 .}

I_N = I_{L1(1) +} I_{L1(2) +} I_{L1(0)  +}  I_{L2(1) +} I_{L2(2) +} I_{L2(0)  +}  I_{L3(1) +} I_{L3(2) +} I_L3(0)

= I_{L1(1) +} I_{L2(1) +} I_{L3(1)  +}  I_{L1(2) +} I_{L2(2) +} I_{L3(2)  +}  I_L1(0)+ I_{L2(0) +} I_L3(0)

_{∵ Positive and negative sequence system are 3 phase balanced systems,}

_{their vector sum is zero.}

= 0 ₊ 0 ₊ I_L1(0)+ I_{L2(0) +} I_L3(0) 

= 3 I_L1(0)

or

I_N = I_{L1(1) +} I_{L1(2) +} I_{L1(0)    +}

a² I_{L1(1) +} aI_{L1(2) +}I_{L1(0)  +}

a I_{L1(1) +} a²I_{L1(2) +} I_L1(0)

= I_{L1(1) x [1+} a²₊ a_{]  +}  I_{L1(2) x [1+} a²₊ a_{]  +}  I_{L1(0) x 3}

= I_{L1(1) x [ 0 ]  +}  I_{L1(2) x [ 0 ]  +}  I_{L1(0) x 3}     ∵ 1 + a + a² = 0

= 0 ₊ 0 ₊ I_{L1(0) x 3}

= 3 I_L1(0)

 

∴ For 3 phase 4 wire un-balanced system, I_N is  3 I_{(0)  and may be with an angle if angle not = 0.}