Question

For a balanced-Y impedance load with per-phase impedance of $X_y$and a neutral impedance $Z_n$connected between the load neutral and the close space ground, the $3\times 3$phase-impedance matrix consists of equal diagonal elements given by ___________ and equal nondiagonal elements given by ___________.

Short answer

For a balanced-Y impedance load with per-phase impedance of $Z_y$and a neutral impedance $Z_n$connected between the load neutral and the close space ground, the $3\times 3$phase-impedance matrix consists of equal diagonal elements given by $Z_n$and equal non-diagonal elements given by $Z_y+Z_n$.

Step-by-step solution

Determine the voltage equation.

Consider the balanced -Y impedance load.

Here, $1_a,1_b$and $1_c$are the line currents, $1_n$is the neutral current, $Z_y$is the star connected impedances and $Z_n$is the neutral impedance.

Apply the Kirchhoff’s law at the node N, the neutral current is equal to sum of the all three currents.

$1_n=1_a+1_b+1_c$

Calculate the line to ground voltage from the above figure.

$V_{ag}=Z_y{1}_a+Z_n{1}_n$

Substitute $1_a+1_b+1_c$for $1_n$into above equation.

$$\begin{gathered} V_{ag}=Z_y{1}_a+Z_n\left(1_a+1_b+1_c\right) \\ {V}_{ag}=\left(Z_y+Z_n\right)1_a+Z_n{1}_b+Z_n{1}_c \end{gathered}$$ …… (1)

Similarly, for the phase b, the equation can be calculated as,

$V_{bg}=Z_n{1}_a+\left(Z_y+Z_n\right)1_b+Z_n{1}_c$ …… (2)

Similarly, for the phase c, the equation can be calculated as,

$V_{cg}=Z_n{1}_a+Z_n{1}_b+\left(Z_y+Z_n\right)1_c$ …… (3)

Determine the diagonal and non-diagonal elements of the impedance matrix.

Write the equations (1), (2) and (3) into matrix form.

$\left[\begin{array}{c}{V}_{ag}\\ V_{bg}\\ V_{cg}\end{array}\right]=\left[\begin{array}{ccc}{Z}_y+& Z_n& Z_n\\ Z_n& Z_y+Z_n& Z_n\\ Z_n& Z_n& Z_y+Z_n\end{array}\right]\left[\begin{array}{c}{1}_a\\ 1_b\\ 1_c\end{array}\right]$

From the above matrix, the impedance matrix is $\left[\begin{array}{ccc}{Z}_y+Z_n& Z_n& Z_n\\ Z_n& Z_y+Z_n& Z_n\\ Z_n& Z_n& Z_y+Z_n\end{array}\right]$consists of diagonal element given by $Z_y+Z_n$and non-diagonal elements$Z_n$.