Question

Consider three ideal single-phase transformers (with a voltage gain of $\mathbf{\eta }$)put together as a $\mathbf{∆}\mathbf{-}\mathbf{\Omega }$three-phase bank as shown in Figure 3.35. Assuming positive-sequence voltages for${\mathit{V}}_{\mathbf{a}\mathbf{n}}\mathbf{}\mathbf{,}\mathbf{}{\mathit{V}}_{\mathbf{b}\mathbf{n}}\mathbf{}\mathbf{,}\mathbf{}\mathbf{and}\mathbf{}{\mathit{V}}_{\mathbf{c}\mathbf{n}}\mathbf{}\mathbf{,}$find${\mathit{V}}_{\mathbf{a}\mathbf{\text{'}}\mathbf{n}\mathbf{\text{'}}}\mathbf{}\mathbf{,}\mathbf{}{\mathit{V}}_{\mathbf{b}\mathbf{\text{'}}\mathbf{n}\mathbf{\text{'}}}\mathbf{}\mathbf{,}\mathbf{}\mathbf{and}\mathbf{}{\mathit{V}}_{\mathbf{c}\mathbf{\text{'}}\mathbf{n}\mathbf{\text{'}}}$in terms of${\mathit{V}}_{\mathbf{an}}\mathbf{}\mathbf{,}\mathbf{}{\mathit{V}}_{\mathbf{bn}}\mathbf{}\mathbf{,}\mathbf{}\mathbf{and}\mathbf{}{\mathit{V}}_{\mathbf{cn}}\mathbf{}\mathbf{,}$respectively.

(a) Would such relationships hold for the line voltages as well?

(b) Looking into the current relationships, express${\mathit{I}}_{\mathbf{a}}\mathbf{\text{'}}\mathbf{,}\mathbf{}{\mathit{I}}_{\mathbf{b}}\mathbf{\text{'}}\mathbf{}\mathbf{,}\mathbf{}\mathbf{and}\mathbf{}{\mathit{I}}_{\mathbf{c}}\mathbf{\text{'}}$in terms

of${\mathit{I}}_{\mathbf{a}}\mathbf{}\mathbf{,}\mathbf{}{\mathit{I}}_{\mathbf{b}}\mathbf{}\mathbf{}\mathbf{,}\mathbf{}\mathbf{and}\mathbf{}{\mathit{I}}_{\mathbf{c}}$,respectively.

(c) Let $\mathit{S}\mathit{\text{'}}$and $\mathit{S}$be the per-phase complex power output and input,

respectively. Find $\mathit{S}\mathbf{\text{'}}$in terms of $\mathit{S}$.

Short answer

(a) Expressing line voltages in terms of phasevoltages is possible.

(b) The current relations are $I_a\text{'}=\frac{I_a}{C},I_b\text{'}=\frac{I_b}{C}and{I}_c\text{'}=\frac{I_c}{C}$.

(c) The complex power relation is $S\text{'}=S$.

Step-by-step solution

Determine the formulas of phase and line voltages

Write the formula of phase voltage at star connection with respect to line voltages at delta connection.

${\mathbf{V}}_{\mathbf{a}\mathbf{\text{'}}\mathbf{n}\mathbf{\text{'}}}\mathbf{=}{\mathbf{\eta V}}_{\mathbf{ab}}$ ……. (1)

Write the line voltage in terms of phase voltages.

${\mathbf{V}}_{\mathbf{ab}}\mathbf{=}{\mathbf{V}}_{\mathbf{an}}\mathbf{-}{\mathbf{V}}_{\mathbf{bn}}$ ……. (2)

Substitute equation (2) into equation (1).

$$\begin{gathered} {\mathbf{V}}_{\mathbf{a}\mathbf{\text{'}}\mathbf{n}\mathbf{\text{'}}}\mathbf{=}\mathbf{\eta }\left(V_{\mathrm{an}}-V_{\mathrm{bn}}\right) \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\sqrt{\mathbf{3}}{\mathbf{\eta e}}^{\mathbf{j}\mathbf{30}}{\mathbf{V}}_{\mathbf{an}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}{\mathbf{CV}}_{\mathbf{an}} \end{gathered}$$ …… (3)

Similarly, the phase voltages for ‘b’ and ‘c’ phases are $V_{b\text{'}n\text{'}}=C{V}_{bn}$and $V_{c\text{'}n\text{'}}=C{V}_{cn}$.

Determine the relationship between line and phase voltages

(a)

Write the formula of line voltage at star connection with respect to phase voltages at delta connection.

${\mathrm{V}}_{\mathrm{a}\text{'}\mathrm{b}\text{'}}=\frac{{\mathrm{V}}_{\mathrm{an}}}{\mathrm{\eta }}$

Therefore, the relation shown by equation (3) is true for line voltages also.

Determine the relationship between line and phase currents

(b)

Write the formula of line current ${\mathrm{I}}_{\mathrm{a}}$.

$$\begin{gathered} I_a=\eta \left(I_{ab}-I_{ca}\right) \\ =\eta \left(I_a\text{'}-I_c\text{'}\right) \\ =\sqrt{3}\eta e^{-j30}{I}_a\text{'} \\ =C{I}_a\text{'} \end{gathered}$$

Solving further

$I_a\text{'}=\frac{I_a}{C}$ …… (4)

Similarly, the line currents for ‘b’ and ‘c’ phases are role="math" localid="1655289082110" $I_b\text{'}=\frac{I_b}{C}\mathrm{and}{I}_c\text{'}=\frac{I_c}{C}$.

Determine the relationship between output and input power

(c)

Write the formula of output complex power.

$S\text{'}=V_{a\text{'}n\text{'}}{\left(I_a\right)}^{*}$ …… (5)

Write the formula of input complex power.

$S=V_{an}{\left(I_a\right)}^{*}$ …… (6)

Substitute role="math" localid="1655289337007" $C{V}_{an}\mathrm{for}\hspace{0.17em}{V}_{a\text{'}n\text{'}}\mathrm{and}\frac{I_a}{C}\mathrm{for}{I}_a\text{'}$in equation (5).

$$\begin{gathered} S\text{'}=C{V}_{an}{\left(\frac{I_a}{C}\right)}^{*} \\ =V_{an}{\left(I_a\right)}^{*} \\ S\text{'}=S \end{gathered}$$

Therefore, the relation between the complex input and the power per phase is $S\text{'}=S$.