Question

As shown in Figure 13.34, an ideal current source consisting of a $\mathbf{10}\mathbf{-}\mathit{k}\mathit{A}$pulse with $\mathbf{50}\mathbf{-}\mathit{\mu }\mathit{s}$duration is applied to the junction of a single-phase, lossless cable and a single-phase, lossless overhead line. The cable has a $\mathbf{200}\mathbf{-}\mathit{\Omega }$characteristic impedance, $\mathbf{2}\mathbf{\times }{\mathbf{10}}^{\mathbf{8}}\mathbf{}\mathit{m}\mathbf{/}\mathit{s}$velocity of propagation, and $\mathbf{20}\mathbf{-}\mathit{k}\mathit{m}$length. The overhead line has a $\mathbf{300}\mathbf{-}\mathit{\Omega }$characteristic impedance, $\mathbf{3}\mathbf{\times }{\mathbf{10}}^{\mathbf{8}}\mathbf{}\mathit{m}\mathbf{/}\mathit{s}$velocity of propagation, and $\mathbf{60}\mathbf{-}\mathit{k}\mathit{m}$length. The sending end of the cable is terminated by $\mathbf{400}\mathbf{-}\mathit{\Omega }$resistor, and the receiving end of the overhead line is terminated by a $\mathbf{100}\mathbf{-}\mathit{\Omega }$resistor. Both the line and cable are initially unenergized. (a) Determine the voltage reflection coefficients ${\mathit{T}}_{\mathbf{S}}$, ${\mathit{T}}_{\mathbf{R}}$, ${\mathit{T}}_{\mathbf{AA}}$, ${\mathit{T}}_{\mathbf{AB}}$, ${\mathit{T}}_{\mathbf{BA}}$, and ${\mathit{T}}_{\mathbf{BB}}$(b) Draw the Bewley lattice diagram for $\mathbf{0}\mathbf{\le }\mathit{t}\mathbf{\le }\mathbf{0}\mathbf{.}\mathbf{8}\mathbf{}\mathit{m}\mathit{s}$. (c) Determine and plot the voltage $\mathit{\upsilon }\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{t}\mathbf{)}$at $\mathit{x}\mathbf{=}\mathbf{0}$versus time $\mathit{t}$for $\mathbf{0}\mathbf{\le }\mathit{t}\mathbf{\le }\mathbf{0}\mathbf{.}\mathbf{8}\mathbf{}\mathit{m}\mathit{s}$.

Figure 13.34

Short answer

Answer

(a) The value of voltage reflection coefficient $T_S,T_R,T_{AA},T_{AB},T_{BA},$and $T_{BB}$are $\frac{1}{3},-\frac{1}{2},\frac{1}{5},\frac{4}{5},\frac{6}{5}$and $\frac{1}{5}$.

(b) Draw the Bewley lattice diagram for$0\le t\le 0.8ms$.

(c) Plot the graph of $\upsilon \left(0,t\right)$versus time $\left(t\right)$.

Step-by-step solution

Write the given data from the question.

Consider the following data:

Sending end impedance, $Z_G=400\Omega$

Receiving end impedance, $Z_R=100\Omega$

Length of the cable line, $I_A=20km$

Velocity of propagation, $v_A=2\times {10}^6\frac{m}{s}$

Characteristic impedance, $Z_A=200\Omega$

Length of the overhead line, $I_8=60km$

Velocity of propagation, $v3=3\times {10}^8\frac{m}{s}$

Characteristics impedance, $Z_B=300\Omega$

Determine the formula of voltage reflection coefficient.

Write the formula of sending end voltage reflection coefficient in per unit, $T_S$.

${\mathit{T}}_{\mathbf{s}}\mathbf{=}\frac{{\displaystyle \frac{{\mathbf{Z}}_{\mathbf{G}}}{{\mathbf{Z}}_{\mathbf{A}}}}\mathbf{-}\mathbf{1}}{{\displaystyle \frac{{\mathbf{Z}}_{\mathbf{G}}}{{\mathbf{Z}}_{\mathbf{A}}}}\mathbf{+}\mathbf{1}}$ …… (1)

Here, ${\mathit{Z}}_{\mathbf{G}}$is source impedance and ${\mathit{Z}}_{\mathbf{A}}$is Characteristic impedance.

Write the formula of receiving end voltage reflection coefficient in per unit, localid="1656764426061" $T_R$.

localid="1656764421830" ${\mathit{T}}_{\mathbf{R}}\mathbf{=}\frac{{\displaystyle \frac{{\mathbf{Z}}_{\mathbf{R}}}{{\mathbf{Z}}_{\mathbf{B}}}\mathbf{-}\mathbf{1}}}{{\displaystyle \frac{{\mathbf{Z}}_{\mathbf{R}}}{{\mathbf{Z}}_{\mathbf{B}}}\mathbf{+}\mathbf{1}}}$ …… (2)

Here, localid="1656764431763" ${\mathit{Z}}_{\mathbf{R}}$is receiving end impedance and localid="1656764435958" ${\mathit{Z}}_{\mathbf{B}}$is characteristics impedance.

Write the formula of reflection coefficient for the waves arriving at the junction from A in per unit.

${\mathit{T}}_{\mathbf{A}\mathbf{A}}\mathbf{=}\frac{{\displaystyle \frac{{\mathbf{Z}}_{\mathbf{B}}}{{\mathbf{Z}}_{\mathbf{A}}}}\mathbf{-}\mathbf{1}}{{\displaystyle \frac{{\mathbf{Z}}_{\mathbf{B}}}{{\mathbf{Z}}_{\mathbf{A}}}}\mathbf{+}\mathbf{1}}$ …… (3)

Here, ${\mathit{Z}}_{\mathbf{B}}$is characteristics impedance and ${\mathit{Z}}_{\mathbf{A}}$is Characteristic impedance.

Write the formula of refraction coefficient for the waves returning to the junction from B in per unit.

${\mathit{T}}_{\mathbf{AB}}\mathbf{=}\frac{\mathbf{2}\mathbf{\left(}{\displaystyle \frac{Z_A}{Z_B}}\mathbf{\right)}}{{\displaystyle \frac{{\mathbf{Z}}_{\mathbf{A}}}{{\mathbf{Z}}_{\mathbf{B}}}\mathbf{+}\mathbf{1}}}$ …… (4)

Here, ${\mathit{Z}}_{\mathbf{A}}$is Characteristic impedance and ${\mathit{Z}}_{\mathbf{B}}$is characteristics impedance.

Write the formula of refraction coefficient for the waves arriving at the junction from A in per unit.

${\mathit{T}}_{\mathbf{BA}}\mathbf{=}\frac{\mathbf{2}\mathbf{\left(}{\displaystyle \frac{Z_B}{Z_A}}\mathbf{\right)}}{{\displaystyle \frac{{\mathbf{Z}}_{\mathbf{B}}}{{\mathbf{Z}}_{\mathbf{A}}}\mathbf{+}\mathbf{1}}}$ …… (5)

Here, ${\mathit{Z}}_{\mathbf{B}}$is characteristics impedance and ${\mathit{Z}}_{\mathbf{A}}$is Characteristic impedance.

Write the formula of reflection coefficient for the waves returning to the junction from Bin per unit.

${\mathit{T}}_{\mathbf{BB}}\mathbf{=}\frac{{\displaystyle \frac{{\mathbf{Z}}_{\mathbf{A}}}{{\mathbf{Z}}_{\mathbf{B}}}\mathbf{-}\mathbf{1}}}{{\displaystyle \frac{{\mathbf{Z}}_{\mathbf{A}}}{{\mathbf{Z}}_{\mathbf{B}}}\mathbf{+}\mathbf{1}}}$ …… (6)

Here, ${\mathit{Z}}_{\mathbf{A}}$is Characteristic impedance and ${\mathit{Z}}_{\mathbf{B}}$is characteristics impedance.

(a) Determine the value of voltage reflection coefficient.

Draw the circuit diagram of single-phase lossless overhead line.

Determine the transit time, ${\zeta }_A$.

$$\begin{gathered} {\zeta }_A=\frac{I_A}{v_A} \\ =\left(\frac{20\times {10}^3}{2\times {10}^5}\right)s \\ =\left({10}^{-4}\right)S \\ =0.1ms \end{gathered}$$

Determine the transit time, ${\zeta }_B$.

$$\begin{gathered} {\zeta }_B=\frac{I_B}{V_R} \\ =\frac{60\times {10}^3}{3\times {18}^8}s \\ =2\times {10}^{-4}s \\ =0.2ms \end{gathered}$$

Determine the sending end voltage reflection coefficient in per unit, $T_s$.

Substitute 400 for $Z_G$and 200 for $Z_A$into equation (1).

$$\begin{gathered} T_s=\frac{{\displaystyle \frac{400}{200}}-1}{{\displaystyle \frac{400}{200}}+1} \\ =\frac{1}{3} \end{gathered}$$

Determine the receiving end voltage reflection coefficient in per unit, $T_R$.

Substitute 100 for $Z_R$and 300 for $Z_B$into equation (2).

$$\begin{gathered} T_R=\frac{{\displaystyle \frac{100}{300}}-1}{{\displaystyle \frac{100}{300}}+1} \\ =-\frac{1}{2} \end{gathered}$$

Determine the reflection coefficient for the waves arriving at the junction from A in per unit.

Substitute 300 for $Z_B$and 200 for $Z_A$into equation (3).

$$\begin{gathered} T_{AA}=\frac{{\displaystyle \frac{300}{200}}-1}{{\displaystyle \frac{300}{200}}+1} \\ =\frac{1}{5} \end{gathered}$$

Determine the refraction coefficient for the waves returning to the junction from B in per unit.

Substitute 200 for $Z_A$and 300 for $Z_B$into equation (4).

$$\begin{gathered} T_{AB}=\frac{2\left({\displaystyle \frac{200}{300}}\right)}{{\displaystyle \frac{200}{300}}+1} \\ =\frac{4}{5} \end{gathered}$$

Therefore, the value of voltage reflection coefficient $T_s,T_R,T_{AA},T_{AB}$, $,T_{BA},$and $T_{BB}$are $\frac{1}{3},\frac{1}{2},\frac{1}{5},\frac{4}{5},\frac{6}{5}$and $-\frac{1}{5}$.

(b) Draw the Bewley lattice diagram for.

Assume the following data:

$$\begin{gathered} i\left(t\right)=10u_{-1}\left(t-\zeta \right)kA \\ \zeta =50\mu s \end{gathered}$$

At $t=0$, the parallel combination of $200\Omega$and $300-\Omega$impedance is encountered by the ideal current source.

$$\begin{gathered} 200\Omega \boxed{\xcancel{}}300\Omega =\frac{\left(200\right)\left(300\right)}{200+300} \\ =120\Omega \end{gathered}$$

Determine the magnitude of voltage.

$$\begin{gathered} E=\left(120\Omega \right)\left(10kA\right) \\ =1200kV \end{gathered}$$

Determine the $e_G$.

$$\begin{gathered} e_G\left(t\right)=1200u_{-1}\left(t-\zeta \right)kV \\ {E}_G\left(s\right)=\frac{1200e^{-st}}{s} \end{gathered}$$

The first voltage waves which travel on the cable as well as on the overhead line are:

$V_1\left(s\right)=\frac{E}{s}\left(1-e^{-st}\right)$

Determine the remaining voltage waves similarly using localid="1656764784087" $T_s,T_R,T_{AA},$$T_{AB},T_{BA},$and $T_{BB}$

The lattice diagram is shown below:

(c) Plot the graph of υ(0,t) versus time (t).

A vertical dotted line should be drawn at x = 0 on the lattice diagram to indicate the voltage x = 0. When each voltage wave meets the dotted line, go downward and add it.

The graph of $\upsilon \left(0,t\right)$versus time $t$is shown below:

Figure 3