Tuesday, 27 September 2016

Calculation of Symmetrical Components


We are now aware of the concept of Sequence components of current / voltage. If you have miss this concept, please read Concept of Symmetrical Components.

Now we are at a stage to calculate the zero, positive and negative sequence components of current / voltage. As already discussed any three phase unbalanced voltage / current can be resolved into three set of balanced vectors. Thus we will use this concept to calculate the positive, negative and zero sequence components of voltages. Mind that the same philosophy is applicable for current also.



Before going into the calculation part, let us introduce ourselves with an operator λ. λ is an operator which when multiplied to any vector quantity, rotates the vector by an angle of 120° in anticlock wise direction without changing the magnitude of the vector. This means that λ must have a magnitude unity. From this definition we can write λ as below.

λ = ei2π/3

   = Cos(2π/3) + jSin(2π/3)

   = -0.5 + j0.866

Why not to explore more properties of λ? Sure, we must…

λ2 = ei4π/3

   = Cos(4π/3) + jSin(4π/3)

   = Cos(2π - 2π/3) + jSin(2π - 2π/3)

   = Cos(2π/3) - jSin(2π/3)

   = -0.5 - j0.866

and

λ3 = ei6π/3 = ei2π

    = Cos(2π) + jSin(2π)

    = 1

λ3 – 1 = 0

(λ + 1)(1 + λ2 + λ) = 0

As (λ + 1) cannot be zero, therefore

1 + λ2 + λ = 0

Thus to summarize the properties of operator λ,

λ3 = 1

λ4 = λ3. λ = λ

1 + λ2 + λ = 0

Consider the figure below where a three phase unbalanced voltages Va, Vb and Vc are resolved into three set of balanced voltages.




Va = Va1 + Va2 + Va0  …………………(1)

Vb = Vb1 + Vb2 + Vb0 ………………….(2)

Vc = Vc1 + Vc2 + Vc0 …………………..(3)

But taking Va1 reference and applying the concept of operator λ,

Vb1 = λ2Va1

Vc1 = λVa1

Similarly for Negative Sequence we can write as

Vb2 = λVa2

Vc2 = λ2Va2

Fortunately for Zero Sequence,

Va0 = Vb0 = Vc0

Thus from equation (2) and (3),

Vb = λ2Va1 + λVa2 + Vb0  ………………(4)

Vc = λVa1 + λ2Va2 + Vc0  ……………….(5)

Now, multiplying equation (4) by λ and (5) by λ2 and adding them to equation (1), we get

Va + λVb + λ2Vc

= Va1(1+ λ3+ λ3) + Va2(1+ λ2+ λ4) + Va0(1+ λ + λ2)

= 3Va1 + Va2(1+ λ + λ2)

= 3Va1

Va1 = (Va + λVb + λ2Vc ) / 3  …………………(6)

For getting negative sequence component, multiply equation (4) by λ2 and (5) by λ & add them to equation (1),

Va + λ2Vb + λVc

= Va1(1+ λ4+ λ2) + Va2(1+ λ3+ λ3) + Va0(1+ λ + λ2)

= 3Va2

Va2 = (Va + λ2Vb + λVc) / 3  ……………………(7)

For Zero Sequence component, add equation (1), (4) and (5),

Va + Vb + Vc

= Va1(1+ λ+ λ2) + Va2(1+ λ+ λ2) + 3Va0

= 3Va0

Va0 = (Va + Vb + Vc) / 3  ……………………(8)

Therefore from equation (6), (7) and (8), we have completely calculated the positive, negative and zero sequence voltages.

In the same way, we can calculate the three components of currents. For currents we can write as below.

Ia1 = (Ia + λIb + λ2Ic ) / 3 

Ia2 = (Ia + λ2Ib + λIc) / 3 


Ia0 = (Ia + Ib + Ic) / 3  

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