Question

The sequence components${\mathit{V}}_{\mathbf{0}}\mathbf{}\mathbf{,}\mathbf{}{\mathit{V}}_{\mathbf{1}}$, and ${\mathit{V}}_{\mathbf{2}}$can be expressed in terms of phase components ${\mathit{V}}_{\mathbf{a}}\mathbf{}\mathbf{}\mathbf{,}\mathbf{}\mathbf{}{\mathit{V}}_{\mathbf{b}}$, and ${\mathit{V}}_{\mathbf{c}}$ .

${\mathit{V}}_{\mathbf{0}}\mathbf{=}$_______________;${\mathit{V}}_{\mathbf{1}}\mathbf{=}$_______________; ${\mathit{V}}_{\mathbf{2}}\mathbf{=}$_______________

Short answer

Answer

The zero $\left(V_0\right)$, positive $\left(V_1\right)$and negative $\left(V_2\right)$sequence component in terms of phase component are $\frac{1}{3}\left(V_a+V_b+V_c\right),\frac{1}{3}\left(V_a+a{V}_b+a^2{V}_c\right)$and $\frac{1}{3}\left(V_a+a^2{V}_{b}a{V}_c\right)$respectively.

Step-by-step solution

Determine the equation to find the sequence component in terms of phase components.

The relationship between the sequence and phase component is given as,

$$\begin{gathered} {\mathit{V}}_{\mathbf{P}}\mathbf{=}\mathit{A}{\mathit{V}}_{\mathbf{s}} \\ {\mathit{V}}_{\mathbf{S}}\mathbf{=}{\mathit{A}}^{\mathbf{-}\mathbf{1}}{\mathit{V}}_{\mathbf{P}} \end{gathered}$$ …… (1)

The matrix that given the $A^{-1}$is given as follows.

${\mathit{A}}^{\mathbf{-}\mathbf{1}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}}\left[\begin{array}{ccc}1& 1& 1\\ 1& a& a^2\\ 1& a^2& a\end{array}\right]$

Determine the sequence component in terms of phase components.

Let assume $V_a,V_b,V_c$are the phase component and $V_0,V_1,V_2$are the sequence component.

Then matrix of the phase components,

$V_P=\left[\begin{array}{c}{V}_a\\ V_b\\ V_c\end{array}\right]$

The matrix of sequence components,

$V_P=\left[\begin{array}{c}{V}_a\\ V_2\\ V_2\end{array}\right]$

Derive the sequence component.

Substitute $\left[\begin{array}{c}{V}_a\\ V_b\\ V_c\end{array}\right]$for $V_P,\left[\begin{array}{c}{V}_a\\ V_2\\ V_2\end{array}\right]$for $V_s$and $\frac{1}{3}\left[\begin{array}{ccc}1& 1& 1\\ 1& a& a^2\\ 1& a^2& a\end{array}\right]$for $A^{-1}$into equation (1).

$\left[\begin{array}{c}{V}_0\\ V_2\\ V_2\end{array}\right]=\frac{1}{3}\left[\begin{array}{ccc}1& 1& 1\\ 1& a& a^2\\ 1& a^2& a\end{array}\right]\left[\begin{array}{c}{V}_a\\ V_b\\ V_c\end{array}\right]$

The zero-sequence component,

$V_1=\frac{1}{3}\left(V_a+a{V}_b+a^2{V}_c\right)$

The positive sequence component,

$V_1=\frac{1}{3}\left(V_a+a{V}_b+a^2{V}_c\right)$

The negative sequence component,

$V_2=\frac{1}{3}\left(V_a+a{V}_b+a^2{V}_c\right)$

Hence, the zero$\left(V_0\right)$, positive$\left(V_1\right)$and negative$\left(V_2\right)$sequence component in terms of phase component are $\frac{1}{3}\left(V_a+V_b+V_c\right)$,$\frac{1}{3}\left(V_a+a{V}_b+a^2{V}_c\right)$and $\frac{1}{3}\left(V_a+a^2{V}_b+a{V}_c\right)$respectively.