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.

^{i2π/3 }

^{ }

^{2}= e

^{i4π/3}

^{ }

^{ }= Cos(4π/3) + jSin(4π/3)

^{3}= e

^{i6π/3}= e

^{i2π}

^{ }

^{3}– 1 = 0

^{2}+ λ) = 0

^{2}+ λ = 0

**λ**

^{3}= 1

**λ**

^{4}= λ^{3}. λ = λ

**1 + λ**

^{2}+ λ = 0

_{a}, V

_{b}and V

_{c}are resolved into three set of balanced voltages.

According to the Concept of Symmetrical components,

V_{a} = V_{a1}+ V_{a2} + V_{a0} …………………(1)

V_{b} = V_{b1}+ V_{b2} + V_{b0} ………………….(2)

V_{c} = V_{c1}+ V_{c2} + V_{c0} …………………..(3)

But taking V_{a1} reference and applying the concept of operator λ,

V_{b1} = λ^{2}V_{a1}

V_{c1} = λV_{a1}

Similarly for Negative Sequence we can write as

V_{b2} = λV_{a2}

V_{c2} = λ^{2}V_{a2}

Fortunately for Zero Sequence,

V_{a0} = V_{b0}= V_{c0}

Thus from equation (2) and (3),

V_{b} = λ^{2}V_{a1}+ λV_{a2} + V_{b0} ………………(4)

V_{c} = λV_{a1}+ λ^{2}V_{a2} + V_{c0} ……………….(5)

Now, multiplying equation (4) by λ and (5) by λ^{2} and adding them to equation (1), we get

V_{a} + λV_{b}+ λ^{2}V_{c }

= V_{a1}(1+ λ^{3}+ λ^{3}) + V_{a2}(1+ λ^{2}+ λ^{4}) + V_{a0}(1+ λ + λ^{2})

= 3V_{a1} + V_{a2}(1+ λ + λ^{2})

= 3V_{a1}

⇒ **V _{a1} = (V_{a}+ λV_{b} + λ^{2}V_{c} ) / 3 …………………(6)**

For getting negative sequence component, multiply equation (4) by λ^{2} and (5) by λ & add them to equation (1),

V_{a} + λ^{2}V_{b}+ λV_{c}

= V_{a1}(1+ λ^{4}+ λ^{2}) + V_{a2}(1+ λ^{3}+ λ^{3}) + V_{a0}(1+ λ + λ^{2})

= 3V_{a2}

⇒ **V _{a2} = (V_{a}+ λ^{2}V_{b }+ λV_{c}) / 3 ……………………(7)**

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

V_{a }+ V_{b}+ V_{c}

= V_{a1}(1+ λ+ λ^{2}) + V_{a2}(1+ λ+ λ^{2}) + 3V_{a0}

= 3V_{a0}

⇒ **V _{a0} = (V_{a}+ V_{b} + V_{c}) / 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.

**I _{a1}= (I_{a} + λI_{b} + λ^{2}I_{c} ) / 3 **

**I _{a2}= (I_{a} + λ^{2}I_{b }+ λI_{c}) / 3 **

**I _{a0}= (I_{a} + I_{b} + I_{c}) / 3 **