Question

A 500-km, 500-kV, 60-Hz, uncompensated three-phase line has a positive sequence series impedance${}^{\mathbf{z}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{03}\mathbf{+}\mathbf{j}\mathbf{0}\mathbf{.}\mathbf{35}\mathbf{}\mathbf{\Omega }\mathbf{/}\mathbf{km}}$ and a positive sequence shunt admittance${}^{\mathbf{Y}\mathbf{=}\mathbf{j}\mathbf{4}\mathbf{.}\mathbf{4}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{}\mathbf{S}\mathbf{/}\mathbf{Km}}\mathbf{}$. Calculate: (a)${}^{{\mathbf{Z}}_{\mathbf{c}}}$, (b)${}^{\left(\gamma l\right)}$ , and (c) the exact ABCD parameters for this line.

Short answer

(a)The value of Impedance${}^{{\mathbf{Z}}_{\mathbf{c}}}$ is${}^{\mathbf{282}\mathbf{.}\mathbf{55}\mathbf{\angle }\mathbf{-}\mathbf{2}\mathbf{.}{\mathbf{45}}^{\mathbf{°}}\mathbf{}\mathbf{\Omega }}$.

(b)The value of role="math" localid="1655364031829" ${}^{\mathit{\gamma }\mathit{l}}$is ${}^{\mathbf{0}\mathbf{.}\mathbf{0265}\mathbf{+}\mathbf{j}\mathbf{0}\mathbf{.}\mathbf{619}\mathbf{}\mathbf{pu}}$

(c)The value of ABCD parameters are${}^{\mathbf{A}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{815}\mathbf{\angle }\mathbf{1}\mathbf{.}{\mathbf{088}}^{\mathbf{°}}\mathbf{}\mathbf{pu}}$,${}^{\mathbf{B}\mathbf{=}\mathbf{163}\mathbf{.}\mathbf{88}\mathbf{\angle }\mathbf{85}\mathbf{.}{\mathbf{38}}^{\mathbf{°}}\mathbf{}\mathbf{\Omega }}$, ${}^{\mathbf{c}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{053}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{3}}\mathbf{}\mathbf{\angle }\mathbf{90}\mathbf{.}{\mathbf{28}}^{\mathbf{°}}}$and ${}^{\mathbf{D}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{815}\mathbf{\angle }\mathbf{1}\mathbf{.}{\mathbf{088}}^{\mathbf{°}}\mathbf{}\mathbf{pu}}$

Step-by-step solution

Given data.

A given Series impedance of uncompensated three-phase line${}^{\mathbf{z}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{03}\mathbf{+}\mathbf{j}\mathbf{0}\mathbf{.}\mathbf{35}\mathbf{}\mathbf{\Omega }\mathbf{/}\mathbf{km}}$ .

A given Shunt admittance of uncompensated three-phase line ${}^{\mathbf{Y}\mathbf{=}\mathbf{j}\mathbf{4}\mathbf{.}\mathbf{4}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{}\mathbf{S}\mathbf{/}\mathbf{Km}}\mathbf{}$.

Determine the required formula

Write the formula of${}^{{\mathbf{Z}}_{\mathbf{c}}}$.

${\mathbf{Z}}_{\mathbf{c}}\mathbf{=}\sqrt{\frac{\mathbf{z}}{\mathbf{y}}}$ …… (1)

Here, ${}^{{\mathbf{Z}}_{\mathbf{c}}}$is the characteristics impedance,${}^{\mathbf{z}}$ is series impedance and ${}^{\mathbf{y}}$is shunt admittance.

Write the formula of ${}^{\left(\gamma l\right)}$.

$\left(\gamma l\right)\mathbf{=}\sqrt{\mathbf{z}\mathbf{y}\mathbf{\times }\mathbf{l}}$ …… (2)

Here, ${}^{\mathbf{y}}$is the propagation constant, ${}^{\mathbf{l}}$is the length of the line, ${}^{\mathbf{z}}$is series impedance and ${}^{\mathbf{y}}$is shunt admittance.

Write the formula of ${}^{\mathbf{A}}$parameter.

$\mathit{A}\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\mathit{h}\left(\gamma l\right)$ …… (3)

Here, ${}^{\gamma l}$is dimensionless quantity.

Write the formula of ${}^{\mathbf{B}}$parameter.

$\mathit{B}\mathbf{=}{\mathit{Z}}_{\mathbf{c}}\mathit{s}\mathit{i}\mathit{n}\mathit{h}\left(\gamma l\right)$ …… (4)

Here, ${}^{{\mathbf{Z}}_{\mathbf{c}}}$is the characteristics impedance and ${}^{\mathit{\gamma }\mathit{l}}$is dimensionless quantity.

Write the formula of ${}^{\mathbf{C}}$parameter.

$\mathit{C}\mathbf{=}\frac{\mathbf{1}}{{\mathbf{Z}}_{\mathbf{c}}}\mathit{s}\mathit{i}\mathit{n}\mathit{h}\left(\gamma l\right)$ …… (5)

Here, ${}^{{\mathbf{Z}}_{\mathbf{c}}}$is the characteristics impedance and ${}^{\mathit{\gamma }\mathit{l}}$is dimensionless quantity.

Write the formula of ${}^{\mathbf{D}}$parameter.

$\mathit{D}\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\mathit{h}\left(\gamma l\right)$ …… (6)

Here, ${}^{\mathbf{\gamma }\mathbf{l}}$is dimensionless quantity

Determine the impedance.

(a)

Substitute ${}^{\mathbf{0}\mathbf{.}\mathbf{33}\mathbf{+}\mathbf{j}\mathbf{0}\mathbf{.}\mathbf{35}}$for ${}^{\mathbf{z}}$and ${}^{\mathbf{j}\mathbf{4}\mathbf{.}\mathbf{4}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}}$for${}^{\mathbf{y}}$into equation (1).

localid="1655369897030" $$\begin{gathered} {\mathbf{Z}}_{\mathbf{c}}\mathbf{=}\sqrt{\frac{\mathbf{0}\mathbf{.}\mathbf{33}\mathbf{+}\mathbf{j}\mathbf{0}\mathbf{.}\mathbf{35}\mathbf{}\mathbf{\Omega }\mathbf{}\mathbf{/}\mathbf{km}}{\mathbf{j}\mathbf{4}\mathbf{.}\mathbf{4}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{}\mathbf{S}\mathbf{/}\mathbf{km}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\sqrt{\mathbf{79}\mathbf{.}\mathbf{545}\mathbf{.}\mathbf{45}\mathbf{-}\mathbf{j}\mathbf{6818}\mathbf{.}\mathbf{18}\mathbf{}\mathbf{\Omega }} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\sqrt{\mathbf{79837}\mathbf{.}\mathbf{13}\mathbf{\angle }\mathbf{-}\mathbf{4}\mathbf{.}{\mathbf{9}}^{\mathbf{°}}}\mathbf{\Omega } \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{282}\mathbf{.}\mathbf{55}\mathbf{\angle }\mathbf{-}\mathbf{2}\mathbf{.}{\mathbf{45}}^{\mathbf{°}}\mathbf{\Omega } \end{gathered}$$

Determine the value of.

(b)

Substitute ${}^{\mathbf{0}\mathbf{.}\mathbf{03}\mathbf{+}\mathbf{j}\mathbf{0}\mathbf{.}\mathbf{35}}$for ${}^{\mathbf{z}}$and ${}^{\mathbf{j}\mathbf{4}\mathbf{.}\mathbf{4}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}}\mathbf{}$for${}^{\mathbf{y}}$ into equation (2).

Here, ${}^{\mathbf{z}}$is series impedance and ${}^{\mathbf{y}}$is shunt admittance.

$$\begin{gathered} \mathit{\gamma }\mathbf{=}\sqrt{\left(0.03+j0.35\right)\left(j4.4\times {10}^{-6}\right)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{=}\sqrt{\mathbf{-}\mathbf{1}\mathbf{.}\mathbf{54}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{+}\mathbf{j}\mathbf{1}\mathbf{.}\mathbf{32}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{7}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{=}\sqrt{\mathbf{1}\mathbf{.}\mathbf{54}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{3}}\mathbf{\angle }\mathbf{175}\mathbf{.}{\mathbf{1}}^{\mathbf{°}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{24}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{3}}\mathbf{\angle }\mathbf{87}\mathbf{.}{\mathbf{55}}^{\mathbf{°}} \end{gathered}$$

.

Now, determine the value of ${}^{\mathbf{\gamma }\mathbf{l}}$.

$$\begin{gathered} \mathit{\gamma }\mathit{l}\mathbf{=}\left(1.24\times {10}^{-3}\angle 87.{55}^{°}\right)\left(500\right) \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{62}\mathbf{\angle }\mathbf{87}\mathbf{.}{\mathbf{55}}^{\mathbf{°}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{0265}\mathbf{+}\mathbf{j}\mathbf{0}\mathbf{.}\mathbf{619}\mathbf{}\mathbf{pu} \end{gathered}$$

Determine the value ofparameter.

${}^{\mathbf{sinh}\mathbf{\left(}\mathbf{\gamma l}\mathbf{\right)}}$(c)

Substitute ${}^{\mathbf{0}\mathbf{.}\mathbf{0265}\mathbf{+}\mathbf{j}\mathbf{0}\mathbf{.}\mathbf{619}}$for ${}^{\mathit{\gamma }\mathit{l}}$into equation (3).

$$\begin{gathered} \mathit{A}\mathbf{}\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\mathit{h}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{0265}\mathbf{+}\mathbf{j}\mathbf{0}\mathbf{.}\mathbf{619}\mathbf{)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\mathit{h}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{0265}\mathbf{)}\mathit{c}\mathit{o}\mathit{s}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{619}\mathbf{}\mathbf{r}\mathbf{a}\mathbf{d}\mathbf{)}\mathbf{+}\mathit{s}\mathit{i}\mathit{n}\mathit{h}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{0265}\mathbf{)}\mathit{s}\mathit{i}\mathit{n}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{619}\mathbf{}\mathbf{r}\mathbf{a}\mathbf{d}\mathbf{)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{(}\mathbf{1}\mathbf{.}\mathbf{00035}\mathbf{)}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{8144}\mathbf{)}\mathbf{+}\mathit{j}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{0265}\mathbf{)}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{58}\mathbf{)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{815}\mathbf{+}\mathit{j}\mathbf{0}\mathbf{.}\mathbf{0154} \end{gathered}$$

Solve further as,

$$\begin{gathered} \mathit{A}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{815}\mathbf{\angle }\mathbf{0}\mathbf{.}{\mathbf{019}}^{\mathbf{°}}\mathbf{}\mathbf{pu} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{815}\mathbf{\angle }\mathbf{1}\mathbf{.}{\mathbf{088}}^{\mathbf{°}}\mathbf{}\mathbf{pu}\mathbf{} \end{gathered}$$

First we determine the value of${}^{\mathbf{sinh}\mathit{\left(}\mathit{\gamma }\mathit{l}\mathit{\right)}}$.

$\mathbf{sinh}\mathbf{\left(}\mathbf{\gamma }\mathbf{l}\mathbf{\right)}\mathbf{=}\mathbf{sinh}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{0265}\mathbf{+}\mathbf{j}\mathbf{0}\mathbf{.}\mathbf{619}\mathbf{)}$

Solve as,

$$\begin{gathered} \mathbf{sinh}\mathbf{\left(}\mathbf{\gamma l}\mathbf{\right)}\mathbf{=}\mathbf{sinh}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{0265}\mathbf{)}\mathbf{cos}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{619}\mathbf{}\mathbf{rad}\mathbf{)}\mathbf{+}\mathbf{}\mathbf{j}\mathbf{}\mathbf{cosh}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{0265}\mathbf{)}\mathbf{sin}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{619}\mathbf{}\mathbf{rad}\mathbf{)}\mathbf{}\mathbf{}\mathbf{}\mathbf{} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{0265}\mathbf{)}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{8144}\mathbf{)}\mathbf{+}\mathbf{j}\mathbf{(}\mathbf{1}\mathbf{.}\mathbf{00035}\mathbf{)}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{58}\mathbf{)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{0216}\mathbf{+}\mathbf{j}\mathbf{0}\mathbf{.}\mathbf{58} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{58}\mathbf{\angle }\mathbf{1}\mathbf{.}{\mathbf{533}}^{\mathbf{r}} \end{gathered}$$

Solve further as,

$\mathbf{sinh}\mathbf{\left(}\mathbf{\gamma }\mathbf{l}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{58}\mathbf{\angle }\mathbf{87}\mathbf{.}{\mathbf{83}}^{\mathbf{°}}$

Substitute ${}^{\mathbf{282}\mathbf{.}\mathbf{55}\mathbf{\angle }\mathbf{-}\mathbf{2}\mathbf{.}{\mathbf{45}}^{\mathbf{°}}}$for ${}^{{\mathbf{Z}}_{\mathbf{c}}}$and ${}^{\mathbf{0}\mathbf{.}\mathbf{58}\mathbf{\angle }\mathbf{87}\mathbf{.}{\mathbf{83}}^{\mathbf{°}}}$for into equation (4).

$$\begin{gathered} \mathit{B}\mathbf{=}\mathbf{(}\mathbf{282}\mathbf{.}\mathbf{55}\mathbf{\angle }\mathbf{-}\mathbf{2}\mathbf{.}{\mathbf{45}}^{\mathbf{°}}\mathbf{)}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{58}\mathbf{\angle }\mathbf{87}\mathbf{.}{\mathbf{83}}^{\mathbf{°}}\mathbf{)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{163}\mathbf{.}\mathbf{88}\mathbf{\angle }\mathbf{85}\mathbf{.}{\mathbf{38}}^{\mathbf{°}}\mathbf{}\mathit{\Omega } \end{gathered}$$

Substitute ${}^{\mathbf{282}\mathbf{.}\mathbf{55}\mathbf{\angle }\mathbf{-}\mathbf{2}\mathbf{.}{\mathbf{45}}^{\mathbf{°}}}$for ${}^{{\mathbf{Z}}_{\mathbf{c}}}$and ${}^{\mathbf{0}\mathbf{.}\mathbf{58}\mathbf{\angle }\mathbf{87}\mathbf{.}{\mathbf{83}}^{\mathbf{°}}}$for ${}^{\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{h}\left(\gamma l\right)}$into equation (5).

$$\begin{gathered} \mathit{c}\left(\frac{1}{282.55\angle -2.{45}^{°}}\right)\left(0.58\angle 87.{83}^{°}\right) \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{053}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{3}}\mathbf{\angle }\mathbf{90}\mathbf{.}{\mathbf{28}}^{\mathbf{°}}\mathbf{}\mathit{S} \end{gathered}$$

Substitute ${}^{\mathbf{0}\mathbf{.}\mathbf{0265}\mathbf{+}\mathbf{j}\mathbf{0}\mathbf{.}\mathbf{619}}$for ${}^{\mathit{\gamma }\mathit{l}}$into equation (6).

$$\begin{gathered} \mathbf{D}\mathbf{=}\mathbf{cosh}\left(0.0265+j0.619\right) \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{cosh}\left(0.0265\right)\mathbf{cos}\left(0.619\mathrm{rad}\right)\mathbf{+}\mathbf{}\mathbf{j}\mathbf{}\mathbf{sinh}\left(0.0265\right)\mathbf{}\mathbf{sinh}\left(0.619\mathrm{rad}\right) \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\left(1.00035\right)\left(0.8144\right)\mathbf{+}\mathbf{j}\mathbf{}\left(0.0265\right)\left(0.58\right) \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{815}\mathbf{+}\mathbf{j}\mathbf{0}\mathbf{.}\mathbf{0154} \end{gathered}$$

Solve further as,

$$\begin{gathered} \mathbf{D}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{815}\mathbf{\angle }\mathbf{0}\mathbf{.}{\mathbf{019}}^{\mathbf{°}}\mathbf{}\mathbf{pu} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{815}\mathbf{\angle }\mathbf{0}\mathbf{.}{\mathbf{088}}^{\mathbf{°}}\mathbf{}\mathbf{pu} \end{gathered}$$