Question

The total complex power delivered to a three-phase network equals (a) 1, (b) 2 , or (c) 3 times the total complex power delivered to the sequence networks.

Short answer

Answer

Therefore, the correct option is (c): 3

Step-by-step solution

Determine the equation to calculate the complex power delivered to three phase system.

The expression to deliver the total power to three phase network is given as follows.

${\mathit{S}}_{\mathbf{P}}\mathbf{=}{\mathit{V}}_{\mathbf{a}\mathbf{g}}{\mathit{I}}_{\mathbf{a}}^{\mathbf{*}}\mathbf{+}{\mathit{V}}_{\mathbf{b}\mathbf{g}}{\mathit{L}}_{\mathbf{b}}^{\mathbf{*}}\mathbf{+}{\mathit{V}}_{\mathbf{v}\mathbf{g}}{\mathit{I}}_{\mathbf{c}}^{\mathbf{*}}$ …… (1)

Here ${\mathit{V}}_{\mathbf{a}\mathbf{g}}\mathbf{,}{\mathit{V}}_{\mathbf{b}\mathbf{g}}\mathbf{,}{\mathit{V}}_{\mathbf{c}\mathbf{g}}$are the phase voltages and ${\mathit{I}}_{\mathbf{a}}\mathbf{,}{\mathit{I}}_{\mathbf{b}}\mathbf{,}{\mathit{I}}_{\mathbf{c}}$are the phase current.

The relationship between the sequence and phase voltage is given as follows.

${\mathit{V}}_{\mathbf{P}}\mathbf{=}\mathit{A}{\mathit{V}}_{\mathbf{S}}$ …… (2)

The relationship between the sequence and phase current is given as follows.

${\mathit{I}}_{\mathbf{P}}\mathbf{=}\mathit{A}{\mathit{I}}_{\mathbf{S}}$ ….. (3)

The identities of the operator,

$$\begin{gathered} \mathbf{1}\mathbf{+}\mathit{a}\mathbf{+}{\mathit{a}}^{\mathbf{2}}\mathbf{=}\mathbf{0} \\ {\mathit{a}}^{\mathbf{3}}\mathbf{=}\mathbf{1}\mathbf{\angle }{\mathbf{0}}^{\mathbf{°}} \\ {\mathit{a}}^{\mathbf{4}}\mathbf{=}\mathit{a} \end{gathered}$$

Calculate the complex power delivered to three phase system.

The phase power can be written as,

$$\begin{gathered} S_P=\left[V_{ag}{V}_{bg}{V}_{cg}\right]\left[\begin{array}{c}{I}_a^{*}\\ I_b^{*}\\ I_c^{*}\end{array}\right] \\ {S}_P={\left(V_P\right)}^T{\left(I_P\right)}^{*} \end{gathered}$$

Here T represent the transposed and * represents the conjugate of the current.

Substitute $A{V}_S$ for $V_P$ and $A{I}_S$for $I_P$into above equation.

$$\begin{gathered} S_P={\left(A{V}_S\right)}^T{\left(A{I}_S\right)}^{*} \\ {S}_P=V_S^T{A}^T{A}^{*}{I}_S^{*} \\ {S}_P=\left(V_S^T{I}_S^{*}\right)\left(A^T{A}^{*}\right) \end{gathered}$$

…… (4)

Calculate the term $A^T{A}^{*}$,

$$\begin{gathered} A^T{A}^{*}=\left[\begin{array}{ccc}1& 1& 1\\ 1& a^2& a\\ 1& a& a^2\end{array}\right]\left[\begin{array}{ccc}1& 1& 1\\ 1& a^2& a\\ 1& a& a^2\end{array}\right] \\ {A}^T{A}^{*}=\left[\begin{array}{ccc}1& 1& 1\\ 1& a^2& a\\ 1& a& a^2\end{array}\right]\left[\begin{array}{ccc}1& 1& 1\\ 1& a& a^2\\ 1& a^2& a\end{array}\right] \\ {A}^T{A}^{*}=\left[\begin{array}{ccc}1+1+1& 1+a+a^2& 1+a^2+a\\ 1+a^2+a& 1+a^3+a^3& 1+a^4+a^2\\ 1+a+a^2& 1+a^2+a^4& 1+a^3+a^3\end{array}\right] \end{gathered}$$

Substitute 1 for $a^{{}_3}$, a for $a^4$into above equation. 0

$$\begin{gathered} A^T{A}^{*}=\left[\begin{array}{ccc}1+1+1& 1+a+a^2& 1+a^2+a\\ 1+a^2+a& 1+1+1& 1+a+a^2\\ 1+a+a^2& 1+a^2+a& 1+1+1\end{array}\right] \\ {A}^T{A}^{*}=\left[\begin{array}{ccc}3& 1+a+a^2& 1+a^2+a\\ 1+a^2+a& 3& 1+a+a^2\\ 1+a+a^2& 1+a^2+a& 3\end{array}\right] \end{gathered}$$

Substitute 0 for $1+a+a^2$into above equation.

$$\begin{gathered} A^{{}_T}{A}^{*}=\left[\begin{array}{ccc}3& 0& 0\\ 0& 3& 0\\ 0& 0& 3\end{array}\right] \\ {A}^T{A}^{*}=3I \end{gathered}$$

Substitute $3I$for $A^T{A}^{*}$into equation (4).

$$\begin{gathered} S_P=\left(V_S^T{I}_S^{*}\right)\left(3I\right) \\ {S}_P=3\left(V_S^T{I}_S^{*}\right) \\ {S}_P=3\left[V_0{V}_1{V}_2\right]\left[\begin{array}{c}{I}_0^{*}\\ I_1^{*}\\ I_2^{*}\end{array}\right] \\ {S}_P=3S_S \end{gathered}$$

Hence, the total complex power delivered to a three-phase network equals 3 times the total complex power delivered to the sequence networks.