Question

Reconsider Problem 8.8 and choosing ${\mathit{V}}_{\mathbf{b}\mathbf{c}}$as the reference, show that

${\mathit{V}}_{\mathbf{bc}\mathbf{,}\mathbf{0}}\mathbf{=}\mathbf{0}\mathbf{}\mathbf{;}\mathbf{}{\mathit{V}}_{\mathbf{bc}\mathbf{,}}\mathbf{=}\mathbf{-}\mathbf{-}\mathit{j}\sqrt{\mathbf{3}}{\mathit{V}}_{\mathbf{a}\mathbf{,}\mathbf{1}}\mathbf{}\mathbf{}\mathbf{;}\mathbf{}\mathbf{}{\mathit{V}}_{\mathbf{bc}\mathbf{,}\mathbf{2}}\mathbf{=}\mathit{j}\sqrt{\mathbf{3}}{\mathit{V}}_{\mathbf{a}\mathbf{,}\mathbf{2}}$

Short answer

Answer

The value of sequence components of the line voltages with $V_{bc}$as the reference are:

$$\begin{gathered} V_{bc,0}=0 \\ {V}_{bc,1}=-\sqrt{3}{V}_{a,1} \\ {V}_{bc,2}=j\sqrt{3}{V}_{a,2} \end{gathered}$$

Step-by-step solution

Write the given data from the question.

The $V_a,V_b$and $V_c$are phase voltages.

The $V_{ab},V_{bc}$and $V_{CB}$are line voltages.

The given line voltages are$V_{bc,0}=0;V_{bc,1}=-j\sqrt{3}{V}_{a,1}$and $V_{bc,2}=j\sqrt{3}{V}_{a,2}$

$a=1\angle {120}^{°}$and $a^2=1\angle {240}^{°}$.

Determine the formula of symmetrical components of the line voltages.

Write the formula of sequence component of line voltage of phase.

${\mathit{V}}_{\mathbf{b}\mathbf{c}\mathbf{,}\mathbf{0}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}}\left(V_{bc}+V_{CB}+V_{ab}\right)$ …… (1)

Here,${\mathit{V}}_{\mathbf{bc}}\mathbf{}\mathbf{}\mathbf{,}\mathbf{}\mathbf{}{\mathit{V}}_{\mathbf{ca}}$and${\mathit{V}}_{\mathbf{ab}}$are line voltages.

Write the formula of positive sequence component of line voltage of phase.

${\mathit{V}}_{\mathbf{bc}\mathbf{,}\mathbf{1}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}}\mathbf{(}{\mathbf{V}}_{\mathbf{bc}}\mathbf{+}{\mathbf{aV}}_{\mathbf{ca}}\mathbf{+}{\mathbf{a}}^{\mathbf{2}}{\mathbf{V}}_{\mathbf{ab}}\mathbf{)}$ …… (2)

Here,${\mathit{V}}_{\mathbf{bc}}\mathbf{}\mathbf{,}\mathbf{}{\mathit{V}}_{\mathbf{ca}}$and${\mathit{V}}_{\mathbf{ab}}$are line voltages.

${\mathit{V}}_{\mathbf{bc}\mathbf{,}\mathbf{2}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}}\mathbf{(}{\mathbf{V}}_{\mathbf{bc}}\mathbf{+}{\mathbf{a}}^{\mathbf{2}}{\mathbf{V}}_{\mathbf{ca}}\mathbf{+}{\mathbf{aV}}_{\mathbf{ab}}\mathbf{)}$ …… (3)

Here,${\mathit{V}}_{\mathbf{bc}}\mathbf{}\mathbf{}\mathbf{,}\mathbf{}{\mathit{V}}_{\mathbf{c}\mathbf{a}}$and${\mathit{V}}_{\mathbf{ab}}$are line voltages.

Determine the sequence components of the line voltages.

The relation between line voltages and phase voltages are:

$$\begin{gathered} V_{ab}=V_a-V_b \\ {V}_{bc}=V_b-V_c \\ {V}_{ca}=V_c-V_a \end{gathered}$$

Consider $V_{bc}$as reference from given sequence line voltages.

Determine the sequence components of line voltages $V_{bc,0}$of phase a.

Substitute $V_b-V_c$for $V_{bc},V_c-V_a$for $V_{ca}$and $V_a-V_b$for $V_{ab}$into equation (1).

role="math" localid="1656078107155" $$\begin{gathered} V_{ab,0}=\frac{1}{3}\left(V_b-V_c+V_c-V_a+V_a-V_b\right) \\ =0 \end{gathered}$$

Use this relation,

$$\begin{gathered} a^3=1\angle {\left(120\times 3\right)}^{°} \\ =1\angle {360}^{°} \\ =1\angle 0^{°} \end{gathered}$$

Determine the sequence components of line voltages $V_{bc,1}$of phase.

Substitutefor,forandforinto equation (2).$$\begin{gathered} V_{bc,1}=\frac{1}{3}\left(\left(V_b-V_c\right)+a\left(V_c-V_a\right)+a^2\left(V_a-V_b\right)\right) \\ =\frac{1}{3}\left(\left(a^2{V}_a+V_b+a{V}_c\right)-\left(a{V}_a+a^2{V}_b+V_c\right)\right) \\ =\frac{1}{3}\left(\left(a^2{V}_a+a^3{V}_b+a^3\times a{V}_c\right)-\left(a{V}_a+a^2{V}_b+a^3{V}_c\right)\right) \\ =\frac{1}{3}\left(a^2\left(V_a+a{V}_b+a^2{V}_a\right)-a\left(V_a+a{V}_b+a^2{V}_c\right)\right) \end{gathered}$$

Solve further as

$$\begin{gathered} V_{bc,1}=\frac{1}{3}\left(\left(a^2-a\right)V_{8\text{'}}\right) \\ =\left(\sqrt{3}\angle -{90}^{°}\right)V_{81} \\ =j\sqrt{3}{V}_{81} \end{gathered}$$

Determine the sequence components of line voltagesof phase.

Substitutefor,forandforinto equation (3).

Solve further as

Therefore, the sequence components of the line voltages withas the reference are$$\begin{gathered} V_{bc,0}=0 \\ {V}_{bc,1}=-j\sqrt{3}{V}_{a,1} \\ {V}_{bc,}=j\sqrt{3}{V}_{a,2} \end{gathered}$$.