Question

For the unbalanced three-phase system described by${\mathit{l}}_{\mathbf{a}}\mathbf{=}\mathbf{10}\mathbf{\angle }\mathbf{0}\mathbf{°}\mathbf{,}\mathbf{}{\mathit{l}}_{\mathbf{b}}\mathbf{=}\mathbf{8}\mathbf{\angle }\mathbf{-}\mathbf{90}\mathbf{°}\mathit{A}$ and ${\mathit{l}}_{\mathbf{c}}\mathbf{=}\mathbf{6}\mathbf{\angle }\mathbf{150}\mathbf{°}\mathit{A}$

Compute the symmetrical components${\mathit{l}}_{\mathbf{0}}$, ${\mathit{l}}_{\mathbf{1}}$ and ${\mathit{l}}_{\mathbf{2}}$.

Short answer

The positive sequence current is$l_1=7.74\angle 17.56°A$.

The negative sequence current is$l_2=1.22\angle -33.07°A$.

The zero sequence current is$l_0=2.31\angle -46.15°A$.

Step-by-step solution

Write the given data from the question.

Assume that a line current of an un-balanced three-phase network are$l_a=10\angle 0°,l_b=8\angle -90°A,l_c=6\angle 150°A$

Determine the formula of phase sequence current.

Write the formula of positive sequence current.

${\mathit{l}}_{\mathbf{1}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}}\mathbf{(}{\mathbf{l}}_{\mathbf{a}}\mathbf{+}{\mathbf{al}}_{\mathbf{b}}\mathbf{+}{\mathbf{a}}^{\mathbf{2}}{\mathbf{l}}_{\mathbf{c}}\mathbf{)}$ …… (1)

Here,$l_a$, $l_b$and $l_c$are line current of three phase network.

Write the formula of negative sequence current.

${\mathit{l}}_{\mathbf{2}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}}\left(l_a+a^2{l}_b+a{l}_c\right)$ …… (2)

Here,$l_a$, $l_b$and $l_c$are line current of three phase network.

Write the formula of zero sequence current.

${\mathit{l}}_{\mathbf{0}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}}\left(l_a+l_b+l_c\right)$ …… (3)

Here,role="math" localid="1654692795853" $l_{a,}{l}_b$and $l_c$are line current of three phase network.

Determine the line current of an un-balanced three-phase network.

First we convert the line voltage in polar form into rectangular form for further solving.

Write the value of line current$l_a$of three phase network.

$$\begin{gathered} l_a=10\angle 0° \\ =10A \end{gathered}$$

Write the value of line current$l_b$of three phase network.

$$\begin{gathered} l_b=8\angle -90° \\ =-j8A \end{gathered}$$

Write the value of line current$l_c$ of three phase network.

Determine the positive sequence current.

Substitute the value of,andinto equation (1).

As further solve as

Determine the negative sequence current.

Substitute the value of,andinto equation (2).

Determine the zero sequence current.

Substitute the value of,andinto equation (3).

Therefore, the symmetrical components of line currents are given as,and.