Question

Consider a three-phase balanced Y-connected load with self and mutual impedances as shown in Figure 8.23. Let the load neutral be grounded through impedance$Z_n$. Using Kirchhoff’s laws, develop the equations for line-to-neutral voltages, and then determine the elements of the phase impedance matrix. Also find the elements of the corresponding sequence impedance matrix.

Short answer

The elements of the phase impedance matrix$\left[Z_P\right]$is$\left[\begin{array}{ccc}{Z}_s+Z_n& Z_m+Z_n& Z_m+Z_N\\ Z_m+Z_N& Z_s+Z_n& Z_m+Z_N\\ Z_m+Z_N& Z_m+Z_N& Z_s+Z_n\end{array}\right]$.

The elements of the corresponding sequence impedance matrix.$\left[\begin{array}{ccc}{Z}_s+3Z_n+2Z_m& 0& 0\\ 0& Z_s-Z_m& 0\\ 0& 0& Z_s-Z_m\end{array}\right]$

Step-by-step solution

Write the given data from the question.

Consider a self-impedance load.$Z_s$

Consider a mutual impedance load$Z_m$

Consider neutral be grounded through impedance load$Z_n$.

Determine the formula of unbalance phase voltage.

Write the formula of phase impedance matrix.

$\left[{\mathbf{Z}}_P\right]=\left[\begin{array}{ccc}{\mathbf{Z}}_s\mathbf{+}{\mathbf{Z}}_n& {\mathbf{Z}}_m\mathbf{+}{\mathbf{Z}}_n& {\mathbf{Z}}_m\mathbf{+}{\mathbf{Z}}_n\\ {\mathbf{Z}}_m\mathbf{+}{\mathbf{Z}}_n& {\mathbf{Z}}_s\mathbf{+}{\mathbf{Z}}_n& {\mathbf{Z}}_m\mathbf{+}{\mathbf{Z}}_n\\ {\mathbf{Z}}_m\mathbf{+}{\mathbf{Z}}_n& {\mathbf{Z}}_m\mathbf{+}{\mathbf{Z}}_n& {\mathbf{Z}}_s\mathbf{+}{\mathbf{Z}}_n\end{array}\right]$…… (1)

Here,${\mathbf{Z}}_s$is self-impedance load,$Z_m$is mutual impedance load and${\mathbf{Z}}_n$is neutral be groundedimpedance load.

Write the formula of sequence impedance matrix.

${\mathbf{Z}}_S\mathbf{=}{\mathbf{A}}^{-1}{\mathbf{Z}}_P\mathbf{A}$ …… (2)

Here,${\mathbf{Z}}_P$ is phase impedance matrix,$A^{-1}$ is inverse of matrix and is$\mathbf{A}$ adjoin matrix.

Step 3:Determine the elements of the phase impedance matrix and sequence impedance matrix.

Draw the following circuit diagram.

Apply Kirchhoff’s voltage law for the loop $V_a$to neutral.

$V_a=I_a{Z}_s+I_b{Z}_m+I_c{Z}_m+I_n{Z}_n$ …… (3)

Here, $I_a$is phase current$Z_s$,is self-impedance load,$I_b$is phase current,$Z_m$is mutual impedance load,$I_c$is phase current,$I_n$is neutral current and$Z_n$is neutral be groundedimpedance load.

Apply Kirchhoff’s voltage law for the loop $V_b$to neutral.

$V_b=I_c{Z}_s+I_b{Z}_m+I_a{Z}_m+I_a{Z}_n$ …… (4)

Here, localid="1656778214214" $I_a$is phase current$Z_s$,is self-impedance load,$I_b$is phase current,$Z_m$is mutual impedance load, $I_c$is phase current, $I_n$is neutral current and$Z_n$is neutral be groundedimpedance load.

Apply Kirchhoff’s voltage law for the loop $V_c$to neutral.

$V_c=I_c{Z}_s+I_b{Z}_m+I_a{Z}_m+I_a{Z}_n$ …… (5)

Here,$I_a$is phase current$Z_s$,is self-impedance load,$I_b$is phase current,$Z_m$is mutual impedance load, $I_c$is phase current, $I_n$is neutral current and $Z_n$is neutral be groundedimpedance load.

From the figure 1, the neutral current is,

localid="1656781286533" $I_n=I_a+I_b+I_c$ …… (6)

Here localid="1656781478060" $I_a$localid="1656781485047" $I_b$,and localid="1656781495196" $I_c$are phase current.

Substitute localid="1656781298602" $I_a+I_b+I_c$for localid="1656781503368" $I_n$into equation (3).

localid="1656781304038" $\begin{array}{c}{V}_a=I_a{Z}_s+I_b{Z}_m+I_c{Z}_m+\left(I_a+I_b+I_c\right)Z_n\\ =I_a\left(Z_s+Z_n\right)+I_b\left(Z_m+Z_n\right)+I_c\left(Z_m+Z_n\right)\end{array}$ …… (7)

Substitute localid="1656781308376" $I_a+I_b+I_c$for localid="1656781510337" $I_n$into equation (4).

localid="1656781291673" $\begin{array}{c}{V}_b=I_b{Z}_s+I_a{Z}_m+I_c{Z}_m+\left(I_a+I_b+I_c\right)Z_n\\ =I_a\left(Z_m+Z_n\right)+I_b\left(Z_s+Z_n\right)+I_c\left(Z_m+Z_n\right)\end{array}$ …… (8)

Substitute localid="1656781312847" $I_a+I_b+I_c$for localid="1656781521448" $I_n$into equation (5).

localid="1656781318073" $\begin{array}{c}{V}_c=I_c{Z}_s+I_b{Z}_m+I_a{Z}_m+\left(I_a+I_b+I_c\right)Z_n\\ =I_a\left(Z_m+Z_n\right)+I_b\left(Z_m+Z_n\right)+I_c\left(Z_s+Z_n\right)\end{array}$ …… (9)

Write the equations (7), (8) and (9) in matrix format.

localid="1656781323129" $\left[\begin{array}{c}{V}_a\\ V_b\\ V_c\end{array}\right]=\left[\begin{array}{ccc}{Z}_s+Z_n& Z_m+Z_n& Z_m+Z_n\\ Z_m+Z_n& Z_s+Z_n& Z_m+Z_n\\ Z_m+Z_n& Z_m+Z_n& Z_s+Z_n\end{array}\right]\left[\begin{array}{c}{I}_a\\ I_b\\ I_c\end{array}\right]$ …… (10)

Determine the phase impedance matrix is,

From the equation (10), the phase impedance matrix is,

localid="1656781328691" $\left[Z_P\right]\text{=}\left[\begin{array}{ccc}{Z}_s+Z_n& Z_m+Z_n& Z_m+Z_n\\ Z_m+Z_n& Z_s+Z_n& Z_m+Z_n\\ Z_m+Z_n& Z_m+Z_n& Z_s+Z_n\end{array}\right]$

Determine the sequence impedance matrix.

We know the relation,

localid="1656781334914" $1+a+a^2=0$

Substitute localid="1656781340320" $\frac{1}{3}\left[\begin{array}{ccc}1& 1& 1\\ 1& a& a^2\\ 1& a^2& a\end{array}\right]$forlocalid="1656781379486" $A^{-1}$, localid="1656781347133" $Z_P$for localid="1656781385611" $\left[\begin{array}{ccc}{Z}_s+Z_n& Z_m+Z_n& Z_m+Z_n\\ Z_m+Z_n& Z_s+Z_n& Z_m+Z_n\\ Z_m+Z_n& Z_m+Z_n& Z_s+Z_n\end{array}\right]$and localid="1656781402944" $\left[\begin{array}{ccc}1& 1& 1\\ 1& a^2& a\\ 1& a& a^2\end{array}\right]$for into equation (2).

localid="1656781395715" $\begin{array}{c}{Z}_s=\frac{1}{3}\left[\begin{array}{ccc}1& 1& 1\\ 1& a& a^2\\ 1& a^2& a\end{array}\right]\left[\begin{array}{ccc}{Z}_s+Z_n& Z_m+Z_n& Z_m+Z_n\\ Z_m+Z_n& Z_s+Z_n& Z_m+Z_n\\ Z_m+Z_n& Z_m+Z_n& Z_s+Z_n\end{array}\right]\left[\begin{array}{ccc}1& 1& 1\\ 1& a^2& a\\ 1& a& a^2\end{array}\right]\\ =\frac{1}{3}\left[\begin{array}{ccc}{Z}_s+3Z_n+2Z_m& Z_s+3Z_n+2Z_m& Z_s+3Z_n+2Z_m\\ Z_s+\left(a+a^2\right)Z_m& a{Z}_s+\left(1+a^2\right)Z_m& a^2{Z}_s+\left(1+a\right)Z_m\\ Z_s+\left(a+a^2\right)Z_m& a^2{Z}_s+\left(1+a\right)Z_m& a{Z}_s+\left(1+a^2\right)Z_m\end{array}\right]\left[\begin{array}{ccc}1& 1& 1\\ 1& a^2& a\\ 1& a& a^2\end{array}\right]\\ =\frac{1}{3}\left[\begin{array}{ccc}3\left(Z_s+3Z_n+2Z_m\right)& 0& 0\\ 0& 3\left(Z_s-Z_m\right)& 0\\ 0& 0& 3\left(Z_s-Z_m\right)\end{array}\right]\\ =\left[\begin{array}{ccc}{Z}_s+3Z_n+2Z_m& 0& 0\\ 0& Z_s-Z_m& 0\\ 0& 0& Z_s-Z_m\end{array}\right]\end{array}$

Therefore the phase impedance and sequence impedance matrix is given by.

localid="1656781351793" $\left[Z_P\right]\text{=}\left[\begin{array}{ccc}{Z}_s+Z_n& Z_m+Z_n& Z_m+Z_n\\ Z_m+Z_n& Z_s+Z_n& Z_m+Z_n\\ Z_m+Z_n& Z_m+Z_n& Z_s+Z_n\end{array}\right]$and .localid="1656781364168" $Z_s=\left[\begin{array}{ccc}{Z}_s+3Z_n+2Z_m& 0& 0\\ 0& Z_s-Z_m& 0\\ 0& 0& Z_s-Z_m\end{array}\right]$