Question

Reconsider Problem 3.29. If${\mathbf{V}}_{\mathbf{an}}\mathbf{,}\mathbf{}{\mathbf{V}}_{\mathbf{bn}}\mathbf{,}$and${\mathbf{V}}_{\mathbf{cn}}$are a negative-sequence set, how would the voltage and current relationships change?

(a) If${\mathbf{C}}_{\mathbf{1}}$is the complex positive-sequence voltage gain in Problem 3.29

and (b) if ${\mathbf{C}}_{\mathbf{2}}$is the negative sequence complex voltage gain, express the

relationship between${\mathbf{C}}_{\mathbf{1}}$and${\mathbf{C}}_{\mathbf{2}}$.

Short answer

The relationship between voltage gain is$C_1^{*}=C_2.$

Step-by-step solution

Determine the formulas of phase voltage and line current

Draw the given diagram.

Using problem 3.29, write the formula of the phase voltage at star connection with respect to line voltages at delta connection for negative sequence set.

$$\begin{gathered} {\mathbf{V}}_{\mathbf{a}\mathbf{\text{'}}\mathbf{n}\mathbf{\text{'}}}\mathbf{=}\sqrt{\mathbf{3}}{\mathbf{ne}}^{\mathbf{-}\mathbf{j}\mathbf{}\mathbf{30}}{\mathbf{V}}_{\mathbf{an}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}{\mathbf{C}}^{\mathbf{*}}{\mathbf{V}}_{\mathbf{ana}} \end{gathered}$$ ……. (1)

Also, write the formula of line current in delta connection with respect to star connection for negative sequence set.

$$\begin{gathered} {\mathbf{I}}_{\mathbf{a}}\mathbf{=}\sqrt{\mathbf{3}}{\mathbf{ne}}^{\mathbf{j}\mathbf{30}}{\mathbf{I}}_{\mathbf{a}}\mathbf{\text{'}} \\ {\mathbf{I}}_{\mathbf{a}}\mathbf{=}{\mathbf{C}}^{\mathbf{*}}{\mathbf{I}}_{\mathbf{a}}\mathbf{\text{'}} \end{gathered}$$ ……. (2)

Determine the relationship between voltage gain for the positive and negative sequence set.

Assume that $C_1$is the complex voltage gain for the positive sequence set and $C_2$is the complex voltage gain for the negative sequence set.

From problem 3.29., for positive sequence set, the relation in phase voltage and line voltage is as follows:

${\mathrm{V}}_{\mathrm{a}\text{'}\mathrm{n}\text{'}}={\mathrm{C}}_1{\mathrm{V}}_{\mathrm{an}}......\left(3\right)$

Now, compairing equations (1) and (3),

$C_1=C_{1.}^{*}$

From the given, replace $C_1^{*}$with$C_2$; then $C_1=C_2$or $C_1^{*}=C_2$.