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 = e

^{i2π/3 }^{}

= Cos(2π/3) + jSin(2π/3)

= -0.5 + j0.866

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

λ

^{2}= e^{i4π/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

λ

^{3}= e^{i6π/3}= e^{i2π}^{}

= Cos(2π) + jSin(2π)

= 1

⇒
λ

^{3}– 1 = 0
⇒
(λ
+ 1)(1 + λ

^{2}+ λ) = 0
As (λ + 1) cannot be zero,
therefore

1 + λ

^{2}+ λ = 0
Thus to summarize the
properties of operator λ,

**λ**

^{3}= 1

**λ**

^{4}= λ^{3}. λ = λ

**1 + λ**

^{2}+ λ = 0

Consider the figure below
where a three phase unbalanced voltages V

_{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
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