Question

As shown in Figure 13.33, two identical, single-phase, two-wire, lossless lines are connected in parallel at both the sending and receiving ends. Each line has a $\mathbf{400}-\mathbf{\Omega }$ characteristic impedance,$\mathbf{3}\times {\mathbf{10}}^8\mathbf{m}/\mathbf{s}$ velocity of propagation, and$\mathbf{100}-\mathbf{km}$ line length. The source voltage at the sending end is a$\mathbf{100}-\mathbf{kV}$ step with source impedance ${\mathbf{Z}}_G=\mathbf{100}\mathbf{\Omega }$. The receiving end is shorted $({\mathbf{Z}}_R=\mathbf{0})$. Both lines are initially unenergized. (a) Determine the first forward traveling voltage waves that start at time$\mathbf{t}=\mathbf{0}$ and travel on each line toward the receiving end. (b) Determine the sending- and receiving-end voltage reflection coefficients in per-unit,

(c) Draw the Bewley lattice diagram for $\mathbf{0}\le \mathbf{t}\le \mathbf{2}.\mathbf{0}\mathbf{ms}$.

(d) Plot the voltage at the center of one line versus time$\mathbf{t}$ for $\mathbf{0}\le \mathbf{t}\le \mathbf{2}.\mathbf{0}\mathbf{ms}$.

Short answer

(a) The value of first forward travelling wave ${\mathrm{V}}_1\left(\mathrm{t}\right)$is $66.67u_{-1}\left(t\right)\text{\hspace{0.17em}kV}$.

(b) The value of sending end and receiving end voltage regulation coefficient in per unit$I_S^{-1}$ and$I_{\text{R}}^{-1}$ is$-0.33\text{\hspace{0.17em}per\hspace{0.17em}unit}$ and $-1\text{\hspace{0.17em}per\hspace{0.17em}unit}$.

(c) Draw the Bewley lattice diagram for$\text{0}\le \text{t}\le \text{2.0ms}$.

(d) Plot the voltage at the center of one line versus time$\mathrm{t}$ for $\text{0}\le \text{t}\le \text{2.0ms}$.

Step-by-step solution

Write the given data from the question.

Consider the following data:

Length of the cable line, $l=100\text{\hspace{0.17em}km}$.

Source voltage at sending end,$e_{\text{G}}\left(t\right)=\text{E}{u}_{-1}\left(t\right)\text{\hspace{0.17em}kV}$

$\begin{array}{l}\text{E}=100\\ {\text{E}}_G\left(s\right)=\frac{100}{s}\end{array}$

Source impedance, ${\text{Z}}_{\text{G}}=100\Omega$

Characteristic impedance, ${\text{Z}}_{\text{c}}=400\Omega$

Receiving end impedance, $Z_{\text{R}}=0$

Velocity of propagation, $\nu =3\times {10}^8\frac{\text{m}}{\text{s}}$

Determine the formula of of first forward travelling wave, sending end and receiving end voltage regulation coefficient in per unit.

Write the formula of first forward travelling wave.

${\mathbf{V}}_1\left(\mathbf{t}\right)=\mathrm{E}\mathbf{\left[}\frac{\frac{{\mathbf{Z}}_{\mathbf{c}}}{\mathbf{2}}}{{\mathbf{Z}}_{\mathbf{G}}\mathbf{+}\frac{{\mathbf{Z}}_{\mathbf{c}}}{\mathbf{2}}}\mathbf{\right]}{\mathbf{u}}_{-\mathbf{1}}\left(\mathbf{t}\right)\mathbf{kV}$ …… (1)

Here, ${\mathbf{Z}}_c$is characteristics impedance, ${\mathbf{Z}}_G$is source impedance and ${\mathbf{Eu}}_{-\mathbf{1}}\left(\mathbf{t}\right)\mathbf{kV}$is source voltage at sending end.

Write the formula ofsending end voltage regulation coefficient.

${\mathbf{I}}_S^{-\mathbf{1}}=\frac{\frac{\frac{{\mathbf{Z}}_G}{{\mathbf{Z}}_c}}{2}-\mathbf{1}}{\frac{\frac{{\mathbf{Z}}_G}{{\mathbf{Z}}_c}+\mathbf{1}}{2}}$ …… (2)

Here, ${\mathbf{Z}}_c$is characteristics impedance, ${\mathbf{Z}}_G$is source impedance.

Write the formula ofreceiving end voltage regulation coefficient.

${\mathbf{I}}_R^{-\mathbf{1}}=\frac{\frac{\frac{{\mathbf{Z}}_R}{{\mathbf{Z}}_c}}{2}-\mathbf{1}}{\frac{\frac{{\mathbf{Z}}_R}{{\mathbf{Z}}_c}+\mathbf{1}}{2}}$ …… (3)

Here,${\mathbf{Z}}_R$ is receiving end impedance and${\mathbf{Z}}_c$ is characteristics impedance.

(a) Determine the value of first forward travelling wave.

Draw the circuit diagram of single-phase, two-wire, lossless line.

Figure 1

Determine the transit time,$\tau$ .

Draw the simplified circuit diagram of the system.

Figure 2

Determine the first forward travelling wave.

Substitute $100$for $E$, $400$for $Z_c$, $100$for $Z_G$and $400$for $Z_C$into equation (2).

$\begin{array}{c}{\text{V}}_{\text{1}}\left(\text{t}\right)=100\left[\frac{200}{200+100}\right]{\mathrm{u}}_{-1}\left(\mathrm{t}\right)\text{\hspace{0.17em}kV}\\ ={\text{66.67u}}_{-1}\left(\mathrm{t}\right)\text{\hspace{0.17em}kV}\end{array}$

Therefore, the fast forward travelling wave$V_1\left(t\right)$ is$66.67u_{-1}\left(t\right)\text{\hspace{0.17em}kV}$ .

(b) Determine the value of sending end and receiving end voltage regulation coefficient.

Determine the sending end voltage reflection coefficient in per unit, ${I^{\text{'}}}_S$.

Substitute $100$for $Z_G$and $400$for $Z_C$into equation (2).

$\begin{array}{c}{I}_S^{-1}=\frac{\frac{100}{200}-1}{\frac{100}{200}+1}\\ =-0.33\end{array}$

Determine the receiving end voltage reflection coefficient in per unit,${I^{\text{'}}}_{\text{R}}$.

Substitute $0$for $Z_R$and $400$ for $Z_C$into equation (3).

$\begin{array}{c}{I}_{\text{R}}^{-1}=\frac{0-1}{0+1}\\ =-1\end{array}$

Therefore, the value of sending end and receiving end voltage regulation coefficient in per unit $I_S^{-1}$and$I_{\text{R}}^{-1}$ is$-0.33\text{\hspace{0.17em}per\hspace{0.17em}unit}$ and $-1\text{\hspace{0.17em}per\hspace{0.17em}unit}$.

(c) Draw the Bewley lattice diagram for 0≤t≤2.0ms.

Determine the${\text{i}}^{\text{th}}$ travelling wave,${\text{V}}_{\text{i}}\left(\text{s}\right)$ .

For$(i=1,2,3,\mathrm{...})$ at $t=0$;

Determine the${\text{i}}^{\text{th}}$ travelling wave for $\text{i}=1$

Determine the ${\text{i}}^{\text{th}}$travelling wave for $\text{i}=2$.

$\begin{array}{c}{V}_2\left(s\right)=I_{\text{R}}^{-1}{V}_1\left(s\right)\\ =-0.1\times \frac{66.67}{s}\\ =-\frac{66.67}{s}\end{array}$

Determine the${\text{i}}^{\text{th}}$ travelling wave for $\text{i}=3$.

Determine the subsequent waves in the same manner.

Draw the lattice diagram is shown below.

Figure 3

(d) Plot the voltage at the center of one line versus time t for  0≤t≤2.0ms

Draw a vertical dotted line at$x=\frac{1}{2}$ on the lattice diagram to represent the voltage at $x=\frac{1}{2}$that point.

As you descend, add each voltage wave as it crosses the dotted line.

The graph plot$\nu (1/2,t)$ versus$t(\text{m}/\text{s})$ is shown below.

Figure 4