Question

Express the sequence impedance matrix $Z_s$in terms of the phase impedance matrix $Z_p$, and the transformation matrix A which relates $V_p=A{V}_s$and $1_p=A{I}_s$. $Z_s=$___________.

Short answer

The sequence matrix is $Z_s=A^{-1}{Z}_{p}A$.

Step-by-step solution

determine the relationship between the voltage and current.

Consider the balanced -impedance load.

Here $I_a,I_b$and $I_c$are the line currents, $I_n$is the neutral current, $Z_y$is the star connected impedances and 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.

$I_n=I_a+I_b+I_c$

Calculate the line to ground voltage from the above figure,

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

Substitute $I_a+I_b+I_c$for $I_n$into above equation.

$V_{ag}=Z_y{I}_a+Z_n\left(I_a+I_b+I_c\right)$ …… (1)

$V_{ag}=\left(Z_y+Z_n\right)I_a+Z_n{I}_b+Z_n{I}_c$

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

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

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

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

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

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

Determine the sequence impedances.

The phase voltage matrix is given as follows.

$V_p=Z_p{I}_p$

Substitute $A{I}_s$for $I_p$and $A{V}_s$for $V_p$into above equation.

$A{V}_s=Z_{p}A{I}_s$

Multiply by $A^1$in both the sides of the above equation.$$\begin{gathered} A^{-1}A{V}_s=A^{-1}{Z}_{p}A{I}_s \\ {V}_s=\left(A^1{Z}_{p}A\right)I_s \end{gathered}$$$$\begin{gathered} A^{-1}A{V}_s=A^{-1}{Z}_{p}A{I}_s \\ {V}_s=\left(A^1{Z}_{p}A\right)I_s \end{gathered}$$

Substitute $Z_s$for $A^{-1}{Z}_{p}A$into above equation.

$V_s=Z_s{I}_s$

Hence, the sequence matrix is $Z_s=A^{-1}{Z}_{p}A$.