Question

For the circuit given in Problem 13.8, replace the circuit elements by their discrete-time equivalent circuits. Use $\mathbf{∆}\mathit{t}\mathbf{10}\mathbf{=}\mathbf{50}\mathit{s}\mathbf{}\mathit{\mu }\mathit{s}\mathbf{=}{\mathbf{5}}^{\mathbf{-}\mathbf{5}}$and $\mathit{E}\mathbf{=}\mathbf{100}\mathbf{}\mathit{k}\mathit{V}$. Determine and show all resistance values on the discrete-time circuit. Write nodal equations for the discrete-time circuit, giving equations for all dependent sources. Then solve the nodal equations and determine the sending-and receiving-end voltages at the following times: $\mathit{t}\mathbf{=}\mathbf{50}\mathbf{,}\mathbf{100}\mathbf{,}\mathbf{}\mathbf{150}\mathbf{,}\mathbf{}\mathbf{200}\mathbf{,}\mathbf{}\mathbf{250}$and $\mathbf{300}\mathbf{}\mathit{m}\mathit{s}$.

Short answer

Answer

For $\mathit{t}\mathbf{=}\mathbf{50}\mathbf{}\mathit{\mu }\mathit{s}$

The value for role="math" localid="1656760999733" ${\mathit{V}}_{\mathbf{k}}\left(50\times {10}^{-6}\right)\mathbf{,}\mathbf{}\mathbf{}{\mathit{V}}_{\mathbf{m}}\left(50\times {10}^{-6}\right)\mathbf{,}\mathbf{}{\mathit{I}}_{\mathbf{k}}\left(50\times {10}^{-6}\right)\mathbf{,}\mathbf{}{\mathit{I}}_{\mathbf{m}}\left(50\times {10}^{-6}\right)\mathbf{,}$and ${\mathit{I}}_{\mathbf{C}}\left(50\times {10}^{-6}\right)\mathbf{,}\mathbf{}$ are $\mathbf{0}\mathbf{.}\mathbf{0025}\mathbf{}\mathit{k}\mathit{V}\mathbf{}\mathbf{,}\mathbf{}\mathbf{0}\mathit{V}\mathbf{}\mathbf{,}\mathbf{}\mathbf{0}\mathit{k}\mathit{A}\mathbf{}\mathbf{,}\mathbf{}\mathbf{1}\mathbf{.}\mathbf{675}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{50}}\mathbf{}\mathit{k}\mathit{A}$and $\mathbf{0}\mathbf{}\mathit{k}\mathit{A}$respectively.

For $\mathit{t}\mathbf{=}\mathbf{100}\mathbf{}\mathit{\mu }\mathit{s}$

The value for ${\mathit{V}}_{\mathbf{k}}\mathbf{(}\mathbf{100}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{,}\mathbf{}{\mathit{V}}_{\mathbf{m}}\mathbf{(}\mathbf{100}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{,}\mathbf{}{\mathit{I}}_{\mathbf{k}}\mathbf{(}\mathbf{100}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{,}\mathbf{}{\mathit{I}}_{\mathbf{m}}\mathbf{(}\mathbf{100}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{,}\mathbf{}$and ${\mathit{I}}_{\mathbf{C}}\mathbf{(}\mathbf{100}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{,}\mathbf{}$ are $\mathbf{0}\mathbf{.}\mathbf{005}\mathbf{}\mathit{k}\mathit{V}\mathbf{}\mathbf{,}\mathbf{}\mathbf{0}\mathit{V}\mathbf{}\mathbf{,}\mathbf{}\mathbf{0}\mathbf{}\mathit{k}\mathit{A}\mathbf{}\mathbf{,}\mathbf{}\mathbf{3}\mathbf{.}\mathbf{35}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathbf{}\mathit{k}\mathit{A}$and $\mathbf{0}\mathbf{}\mathit{k}\mathit{A}$respectively.

For $\mathit{t}\mathbf{=}\mathbf{100}\mathbf{}\mathit{\mu }\mathit{s}$

The value for ${\mathit{V}}_{\mathbf{k}}\mathbf{(}\mathbf{150}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{,}\mathbf{}{\mathit{V}}_{\mathbf{m}}\mathbf{(}\mathbf{150}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{,}\mathbf{}{\mathit{I}}_{\mathbf{k}}\mathbf{(}\mathbf{150}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{,}\mathbf{}{\mathit{I}}_{\mathbf{m}}\mathbf{(}\mathbf{150}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{,}\mathbf{}$and ${\mathit{I}}_{\mathbf{C}}\mathbf{(}\mathbf{150}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{,}\mathbf{}$ are $\mathbf{}\mathbf{0}\mathbf{.}\mathbf{0075}\mathbf{}\mathit{k}\mathit{V}\mathbf{}\mathbf{,}\mathbf{}\mathbf{0}\mathit{V}\mathbf{}\mathbf{,}\mathbf{}\mathbf{0}\mathbf{}\mathit{k}\mathit{A}\mathbf{}\mathbf{,}\mathbf{}\mathbf{5}\mathbf{.}\mathbf{025}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathbf{}\mathit{k}\mathit{A}$and$\mathbf{}\mathbf{0}\mathbf{}\mathit{k}\mathit{A}$respectively.

For $\mathit{t}\mathbf{=}\mathbf{200}\mathbf{}\mathit{\mu }\mathit{s}$

The value for ${\mathit{V}}_{\mathbf{k}}\mathbf{(}\mathbf{200}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{,}\mathbf{}{\mathit{V}}_{\mathbf{m}}\mathbf{(}\mathbf{200}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{,}\mathbf{}{\mathit{I}}_{\mathbf{k}}\mathbf{(}\mathbf{200}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{,}\mathbf{}{\mathit{I}}_{\mathbf{m}}\mathbf{(}\mathbf{200}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{,}\mathbf{}$and ${\mathit{I}}_{\mathbf{C}}\mathbf{(}\mathbf{200}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{,}\mathbf{}$ are$\mathbf{}\mathbf{0}\mathbf{.}\mathbf{01}\mathbf{}\mathit{k}\mathit{V}\mathbf{}\mathbf{,}\mathbf{}\mathbf{0}\mathit{V}\mathbf{}\mathbf{,}\mathbf{}\mathbf{0}\mathbf{}\mathit{k}\mathit{A}\mathbf{}\mathbf{,}\mathbf{}\mathbf{6}\mathbf{.}\mathbf{7}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathbf{}\mathit{k}\mathit{A}$and$\mathbf{}\mathbf{0}\mathbf{}\mathit{k}\mathit{A}$respectively.

For $\mathit{t}\mathbf{=}\mathbf{250}\mathbf{}\mathit{\mu }\mathit{s}$

The value for ${\mathit{V}}_{\mathbf{k}}\mathbf{(}\mathbf{250}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{,}\mathbf{}{\mathit{V}}_{\mathbf{m}}\mathbf{(}\mathbf{250}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{,}\mathbf{}{\mathit{I}}_{\mathbf{k}}\mathbf{(}\mathbf{250}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{,}\mathbf{}{\mathit{I}}_{\mathbf{m}}\mathbf{(}\mathbf{250}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{,}\mathbf{}$and ${\mathit{I}}_{\mathbf{C}}\mathbf{(}\mathbf{250}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{,}$arerole="math" localid="1656762238787" $0.0125kV,0V,3.24\times {10}^{-4}V,1.446\times {10}^{-5}kA,8.375\times {10}^{-5}kA$and $\mathbf{2}\mathbf{.}\mathbf{592}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathbf{}\mathit{k}\mathit{A}$respectively.

For $\mathit{t}\mathbf{=}\mathbf{300}\mathbf{}\mathit{\mu }\mathit{s}$

The value for ${\mathit{V}}_{\mathbf{k}}\mathbf{(}\mathbf{300}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{,}\mathbf{}{\mathit{V}}_{\mathbf{m}}\mathbf{(}\mathbf{300}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{,}\mathbf{}{\mathit{I}}_{\mathbf{k}}\mathbf{(}\mathbf{300}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{,}\mathbf{}{\mathit{I}}_{\mathbf{m}}\mathbf{(}\mathbf{300}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{,}\mathbf{}$and ${\mathit{I}}_{\mathbf{C}}\left(300\times {10}^{-8}\right)\mathbf{,}$ are $0.015kV,1.816\times {10}^{-4}kV,3.229\times {10}^{-5}kA,1.005\times {10}^{-4}kA$and $\mathbf{1}\mathbf{.}\mathbf{1392}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathbf{}\mathit{k}\mathit{A}$respectively.

Step-by-step solution

Write the given data from the from the question.

The voltage source,$E=100kV$

Time,$∆t=5\times {10}^{-5}s$

The series inductance,$L=0.999\times {10}^{-6}\frac{H}{m}$

The shunt capacitance,$C=1.112\times {10}^{-11}\frac{F}{m}$

The length of the line,$I=60kM$

The source impedance,$Z_G\left(s\right)=Z_c$

The source voltage,

$$\begin{gathered} e_G\left(t\right)=Et{u}_{-1}\left(t\right)kV \\ {e}_G\left(t\right)=Et{u}_{-2}\left(t\right)kV \end{gathered}$$

Determine the equation to calculate the sending-end receiving end voltages.

The equation to calculate characteristics impedance is given as follows.

${\mathit{Z}}_{\mathbf{c}}\mathbf{=}\sqrt{\frac{\mathbf{L}}{\mathbf{C}}}$ …… (1)

The equation to calculate the value of velocity is given as follows.

$\mathit{v}\mathbf{=}\frac{\mathbf{1}}{\sqrt{\mathbf{LC}}}$ …… (2)

The equation to calculate the transit time is given as follows.

$\mathit{T}\mathbf{=}\frac{\mathbf{I}}{\mathbf{v}}$ …… (3)

Calculate the sending-end receiving end voltages.

Consider the diagram using the terminal variables in the time domain.

Redraw the circuit diagram by using the equivalent and also convert the voltage and current sources with the help of source transformation.

Calculate the characteristics impedance.

Substitute $\mathbf{0}\mathbf{.}\mathbf{999}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\frac{\mathbf{H}}{\mathbf{m}}$for L, and $\mathbf{1}\mathbf{.}\mathbf{112}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{11}}\frac{\mathbf{F}}{\mathbf{m}}$for C into equation (1).

$$\begin{gathered} {\mathit{Z}}_{\mathbf{c}}\mathbf{=}\sqrt{\frac{\mathbf{0}\mathbf{.}\mathbf{999}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}}{\mathbf{1}\mathbf{.}\mathbf{112}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{11}}}} \\ {\mathit{Z}}_{\mathbf{C}}\mathbf{=}\sqrt{\mathbf{8}\mathbf{.}\mathbf{983}\mathbf{\times }{\mathbf{10}}^{\mathbf{4}}} \\ {\mathit{Z}}_{\mathbf{C}}\mathbf{=}\mathbf{299}\mathbf{.}\mathbf{73}\mathbf{}\mathit{\Omega } \end{gathered}$$

Calculate the value of the velocity.

Substitute $\mathbf{0}\mathbf{.}\mathbf{999}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\frac{\mathbf{H}}{\mathbf{m}}$for L, and $\mathbf{1}\mathbf{.}\mathbf{112}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{11}}\frac{\mathbf{F}}{\mathbf{m}}$for C into equation (2).

$$\begin{gathered} \mathit{v}\mathbf{=}\frac{\mathbf{1}}{\sqrt{\mathbf{0}\mathbf{.}\mathbf{999}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{\times }\mathbf{1}\mathbf{.}\mathbf{112}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{11}}}} \\ \mathit{v}\mathbf{=}\sqrt{\mathbf{9}\mathbf{\times }{\mathbf{10}}^{\mathbf{16}}} \\ \mathit{v}\mathbf{=}\mathbf{3}\mathbf{\times }{\mathbf{10}}^{\mathbf{8}}\frac{\mathbf{m}}{\mathbf{s}} \end{gathered}$$

Calculate the value of the transit time.

Substitute 60 km for I and $\mathbf{3}\mathbf{\times }{\mathbf{10}}^{\mathbf{8}}\frac{\mathbf{m}}{\mathbf{s}}$for v into equation (3).

$$\begin{gathered} \mathit{\zeta }\mathbf{=}\frac{\mathbf{60}\mathbf{\times }{\mathbf{10}}^{\mathbf{3}}}{\mathbf{3}\mathbf{\times }{\mathbf{10}}^{\mathbf{S}}} \\ \mathit{\zeta }\mathbf{=}\mathbf{200}\mathbf{}\mathit{\mu }\mathit{s} \end{gathered}$$

Apply the nodal analysis at node 1.

role="math" localid="1656766636362" $$\begin{gathered} \frac{{\mathbf{V}}_{\mathbf{k}}\left(t\right)}{\left(Z_c\boxed{\xcancel{}}{Z}_G\right)}\mathbf{-}{\mathit{i}}_{\mathbf{G}}\left(t\right)\mathbf{+}{\mathit{I}}_{\mathbf{k}}\left(t-\zeta \right)\mathbf{=}\mathbf{0} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\frac{{\mathbf{V}}_{\mathbf{k}}\left(t\right)}{\left(Z_c\boxed{\xcancel{}}{Z}_G\right)}\mathbf{=}{\mathit{i}}_{\mathbf{G}}\left(t\right)\mathbf{-}{\mathit{I}}_{\mathbf{k}}\left(t-\zeta \right) \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}{\mathit{V}}_{\mathbf{k}}\left(t\right)\mathbf{=}\left[i_{Gk}\left(t-\zeta \right)\left(t\right)-I\right]\left(Z_c\boxed{\xcancel{}}{Z}_G\right) \end{gathered}$$

Substitute role="math" localid="1656766648626" $\frac{{\mathbf{e}}_{\mathbf{G}}\left(t\right)}{{\mathbf{Z}}_{\mathbf{G}}}$for ${\mathit{i}}_{\mathbf{G}}$into above equation.

${\mathit{V}}_{\mathbf{k}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{[}\frac{{\mathbf{e}}_{\mathbf{G}}\left(t\right)}{{\mathbf{Z}}_{\mathbf{G}}}\mathbf{-}{\mathbf{I}}_{\mathbf{K}}\left(t-\zeta \right)\mathbf{]}\mathbf{(}{\mathbf{Z}}_{\mathbf{c}}\mathbf{}\boxed{\xcancel{}}\mathbf{}{\mathbf{Z}}_{\mathbf{G}}\mathbf{)}$

Substitute $\mathbf{200}\mathbf{}\mathit{\mu }\mathit{s}$for $\mathit{\zeta }\mathbf{}\mathbf{,}\mathbf{}\mathbf{100}\mathit{t}{\mathit{u}}_{\mathbf{-}\mathbf{1}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathit{k}\mathit{V}$for ${\mathit{e}}_{\mathbf{G}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{}\mathbf{,}\mathbf{}\mathbf{299}\mathbf{.}\mathbf{73}\mathbf{}\mathit{\Omega }$for ${\mathit{Z}}_{\mathbf{G}}$into above equation.

$$\begin{gathered} {\mathit{V}}_{\mathbf{k}}\left(t\right)\mathbf{=}\left[\frac{100t}{299.73}-I_k\left(t-200\times {10}^{-6}\right)\right]\left(299.73\boxed{\xcancel{}}299.73\right) \\ {\mathit{V}}_{\mathbf{k}}\left(t\right)\mathbf{=}\left[0.334t-I_k\left(t-200\times {10}^{-6}\right)\right]\mathbf{149}\mathbf{.}\mathbf{87}\mathbf{}\mathit{k}\mathit{V} \end{gathered}$$ …… (4)

Therefore, the expression for the voltage ${\mathit{V}}_{\mathbf{k}}\left(t\right)$is $\mathbf{[}\mathbf{0}\mathbf{.}\mathbf{334}\mathbf{t}\mathbf{-}{\mathbf{I}}_{\mathbf{k}}\left(t-200\times {10}^{-6}\right)\mathbf{]}\mathbf{149}\mathbf{.}\mathbf{87}\mathbf{}\mathit{V}$.

Apply the nodal analysis at node 2,

$$\begin{gathered} \frac{{\mathbf{V}}_{\mathbf{m}}\left(t\right)}{\left(Z_c\boxed{\xcancel{}}R\boxed{\xcancel{}}{\displaystyle \frac{∆t}{2C}}\right)}\mathbf{-}{\mathit{I}}_{\mathbf{m}}\left(t-\zeta \right)\mathbf{-}{\mathit{I}}_{\mathbf{C}}\left(t-∆t\right)\mathbf{=}\mathbf{0} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}{\mathit{V}}_{\mathbf{m}}\left(t\right)\mathbf{=}\left[I_m\left(t-\zeta \right)-I_c\left(t-∆t\right)\right]\left(Z_C\boxed{\xcancel{}}R\boxed{\xcancel{}}\frac{∆t}{2C}\right) \end{gathered}$$

Substitute $\mathbf{200}\mathbf{}\mathit{\mu }\mathit{s}$for $\mathit{\zeta }\mathbf{}\mathbf{,}\mathbf{}\mathbf{50}\mathit{m}\mathit{s}$for $\mathbf{∆}\mathit{t}\mathbf{}\mathbf{,}\mathbf{}\mathbf{299}\mathbf{.}\mathbf{73}\mathit{\Omega }$for ${\mathit{Z}}_{\mathbf{c}}\mathbf{}\mathbf{,}\mathbf{}\mathbf{150}\mathbf{}\mathit{\Omega }$and $\mathbf{1}\mathit{\mu }\mathit{F}$for C into above equation.

$$\begin{gathered} {\mathit{V}}_{\mathbf{m}}\left(t\right)\mathbf{=}\left[I_m\left(t-200\times {10}^{-6}\right)-I_c\left(t-50\times {10}^8\right)\right]\left(299.73\boxed{\xcancel{}}150\boxed{\xcancel{}}\frac{50\times {10}^{-6}}{2\times 1\times {10}^{-6}}\right) \\ {\mathit{V}}_{\mathbf{m}}\left(t\right)\mathbf{=}\left[I_m\left(t-200\times 10\right)-I_c\left(t-50\times {10}^{-6}\right)\right]\left(100\boxed{\xcancel{}}25\right) \end{gathered}$$

${\mathit{V}}_{\mathbf{m}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{[}{\mathbf{I}}_{\mathbf{m}}\left(t-200\times 10\right)\mathbf{-}{\mathbf{I}}_{\mathbf{c}}\left(t-50\times {10}^{-6}\right)\mathbf{]}\mathbf{\times }\mathbf{20}\mathbf{}\mathit{k}\mathit{V}$ …… (5)

Hence the expression for the voltage ${\mathit{V}}_{\mathbf{m}}\left(t\right)$is $\left[I_m\left(t-200\times {10}^{-6}\right)-I_c\left(t-50\times {10}^{-6}\right)\right]\mathbf{\times }\mathbf{20}\mathbf{}\mathit{k}\mathit{V}$.

The expression for the current ${\mathit{I}}_{\mathbf{k}}\mathbf{(}\mathbf{t}\mathbf{-}\mathbf{\zeta }\mathbf{)}$.

${\mathit{I}}_{\mathbf{k}}\mathbf{(}\mathbf{t}\mathbf{-}\mathbf{\zeta }\mathbf{)}\mathbf{=}{\mathit{I}}_{\mathbf{m}}\mathbf{(}\mathbf{t}\mathbf{-}\mathbf{\zeta }\mathbf{)}\mathbf{-}\frac{\mathbf{2}}{{\mathbf{Z}}_{\mathbf{c}}}{\mathit{v}}_{\mathbf{m}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$

Replace $\mathit{t}\mathbf{-}\mathit{\zeta }$with t into above equation.

${\mathit{I}}_{\mathbf{k}}\mathbf{=}{\mathit{I}}_{\mathbf{m}}\mathbf{(}\mathbf{t}\mathbf{-}\mathbf{\zeta }\mathbf{)}\mathbf{-}\frac{\mathbf{2}}{{\mathbf{Z}}_{\mathbf{C}}}{\mathit{v}}_{\mathbf{m}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$

Substitute $\mathbf{200}\mathbf{}\mathit{\mu }\mathit{s}$for $\mathit{\zeta }$and $\mathbf{299}\mathbf{.}\mathbf{73}\mathbf{}\mathit{\Omega }$for ${\mathit{Z}}_{\mathbf{c}}$into above equation.

${\mathit{I}}_{\mathbf{k}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathit{I}}_{\mathbf{m}}\mathbf{(}\mathbf{t}\mathbf{-}\mathbf{200}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{-}\frac{\mathbf{2}}{\mathbf{299}\mathbf{.}\mathbf{73}}{\mathit{v}}_{\mathbf{m}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$

${\mathit{I}}_{\mathbf{k}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathit{I}}_{\mathbf{m}}\mathbf{(}\mathbf{t}\mathbf{-}\mathbf{200}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{0067}{\mathit{v}}_{\mathbf{m}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathit{k}\mathit{A}$ …… (6)

Hence the expression for the currentis.

The expression for the current.

$I_m\left(t-\zeta \right)=I_k\left(t-2\zeta \right)+\frac{2}{Z_0}{v}_k\left(t-\zeta \right)$

Replacewithinto above equation.

${\mathit{I}}_{\mathbf{m}}\left(+t\right)\mathbf{=}{\mathit{I}}_{\mathbf{k}}\left(t-\zeta \right)\mathbf{+}\frac{\mathbf{2}}{{\mathbf{Z}}_{\mathbf{0}}}{\mathit{v}}_{\mathbf{k}}\left(t\right)$

Substituteforandforinto above equation.

${\mathit{I}}_{\mathbf{m}}\left(t\right)\mathbf{=}{\mathit{I}}_{\mathbf{k}}\left(t-200\times {10}^6\right)\mathbf{+}\frac{\mathbf{2}}{\mathbf{299}\mathbf{.}\mathbf{73}}{\mathit{v}}_{\mathbf{k}}\left(t\right)$

${\mathit{I}}_{\mathbf{m}}\left(t\right)\mathbf{=}{\mathit{I}}_{\mathbf{k}}\left(t-200\times {10}^{-6}\right)\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{0067}{\mathit{v}}_{\mathbf{k}}\left(t\right)\mathit{k}\mathit{A}$ …… (7)

Hence the expression current ${\mathit{I}}_{\mathbf{m}}\left(t\right)\mathbf{}\mathit{i}\mathit{s}\mathbf{}{\mathit{I}}_{\mathbf{k}}\left(t-200\times {10}^{-6}\right)\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{0067}{\mathit{v}}_{\mathbf{k}}\left(t\right)\mathit{k}\mathit{A}$.

The expression for the current ${\mathit{I}}_{\mathbf{c}}\left(t-∆t\right)$.

${\mathit{I}}_{\mathbf{c}}\left(t-∆t\right)\mathbf{=}\mathbf{-}{\mathit{I}}_{\mathbf{C}}\left(t-∆t\right)\mathbf{+}\frac{{\mathbf{v}}_{\mathbf{m}}\left(t\right)}{\mathbf{∆}\mathbf{t}\mathbf{/}\mathbf{4}\mathbf{C}}$

Replacewithinto above equation.

Substituteforandforinto above equation.

…… (8)

Hence the expression foris.

For

Recall the equation (4).

Substituteforinto above equation.

The value of the termis zero since input is composed of unit function.

Recall equation (5),

Substituteforinto above equation.

The value of the termis zero since input is composed of unit function.

Recall the equation (6),

Substituteforinto above equation.

The value of the termis zero since input is composed of unit function.

Substituteforinto above equation.

Recall equation (7).

Substituteforinto above equation.

The value of the termis zero since input is composed of unit function.

Substituteforinto above equation.

Recall the equation (8),

Substituteforinto above equation.

Substituteforandforinto above equation.

Hence the value forand are,,,andrespectively.

For

Recall the equation (4).

Substituteforinto above equation.

The value of the termis zero since input is composed of unit function.

Recall equation (5),

Substituteforinto above equation.

The value of the termis zero since input is composed of unit function.

Substituteforinto above equation.

Recall the equation (6),

Substituteforinto above equation.

The value of the termis zero since input is composed of unit function.

Substituteforinto above equation.

Recall equation (7).

Substituteforinto above equation.

The value of the termis zero since input is composed of unit function.

Substituteforinto above equation.

Recall the equation (8),

Substituteforinto above equation.

Substituteforandforinto above equation.

Hence the value forand are,,,andrespectively.

For

Recall the equation (4).

Substituteforinto above equation.

The value of the termis zero since input is composed of unit function.

Recall equation (5),

Substituteforinto above equation.

The value of the termis zero since input is composed of unit function.

Substituteforinto above equation.

Recall the equation (6),

Substituteforinto above equation.

The value of the termis zero since input is composed of unit function.

Substituteforinto above equation.

Recall equation (7).

Substituteforinto above equation.

The value of the termis zero since input is composed of unit function.

Substituteforinto above equation.

Recall the equation (8),

Substituteforinto above equation.

Substituteforandforinto above equation.

Hence the value forand are,,,andrespectively.

For

Recall the equation (4).

Substituteforinto above equation.

Recall equation (5),

Substituteforinto above equation.

Substituteforinto above equation.

Recall the equation (6),

Substituteforinto above equation.

Substituteforinto above equation.

Recall equation (7).

Substituteforinto above equation.

Substituteforinto above equation.

Recall the equation (8),

${\mathit{I}}_{\mathbf{C}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathit{I}}_{\mathbf{c}}\left(t-50\times {10}^{-6}\right)\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{08}{\mathit{v}}_{\mathbf{m}}\left(t\right)\mathit{k}\mathit{A}$

Substituteforinto above equation.

$$\begin{gathered} {\mathit{I}}_C\mathbf{(}\mathbf{200}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{=}\mathbf{-}{\mathit{I}}_C\mathbf{(}\mathbf{200}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{-}\mathbf{50}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathbf{)}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{08}{\mathit{v}}_m\mathbf{}\mathbf{(}\mathbf{200}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)} \\ {\mathit{I}}_{\mathbf{m}}\mathbf{(}\mathbf{250}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{=}{\mathit{I}}_c\mathbf{(}\mathbf{150}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{08}{\mathit{v}}_m\mathbf{}\mathbf{(}\mathbf{200}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)} \end{gathered}$$

Substituteforandforinto above equation.

$$\begin{gathered} {\mathit{I}}_{\mathbf{m}}\mathbf{(}\mathbf{200}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{=}\mathbf{0}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{8}\left(0\right) \\ {\mathit{I}}_{\mathbf{m}}\mathbf{(}\mathbf{200}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{=}\mathbf{0}\mathbf{}\mathit{k}\mathit{A} \end{gathered}$$

Hence the value for $V_k\left(200\times {10}^{-6}\right),V_m\left(200\times {10}^{-6}\right),I_k\left(200\times {10}^{-6}\right),I_m\left(200\times {10}^{-6}\right)$and $I_C\left(200\times {10}^{-6}\right),$ are $0.01kV,0V,6.7\times {10}^{-6}kA$and $0kA$respectively.

For $\mathit{t}\mathbf{=}\mathbf{250}\mathbf{}\mathit{\mu }\mathit{s}$

Recall the equation (4).

$V_k\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{[}\mathbf{0}\mathbf{.}\mathbf{334}\mathbf{t}\mathbf{-}{\mathbf{I}}_{\mathbf{k}}\mathbf{(}\mathbf{t}\mathbf{-}\mathbf{200}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{]}\mathbf{149}\mathbf{.}\mathbf{87}\mathbf{}\mathit{k}\mathit{A}$

Substitutefor t into above equation.

$$\begin{gathered} {\mathit{I}}_{\mathbf{m}}\mathbf{(}\mathbf{250}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{=}\left[0.334\times 250\times {10}^{-6}-I_k\left(250\times {10}^{-6}-200\times {10}^{-6}\right)\right]\mathbf{149}\mathbf{.}\mathbf{87} \\ {\mathit{I}}_{\mathbf{m}}\mathbf{(}\mathbf{250}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{=}\left[83.5\times {10}^{-6}-I_k\left(50\times {10}^{-6}\right)\right]\mathbf{149}\mathbf{.}\mathbf{87} \end{gathered}$$

The value of the termis zero since input is composed of unit function.

$$\begin{gathered} {\mathit{I}}_{\mathbf{m}}\mathbf{(}\mathbf{250}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{=}\left[83.5\times {10}^{-6}-0\right]\mathbf{149}\mathbf{.}\mathbf{87} \\ {\mathit{I}}_{\mathbf{m}}\mathbf{(}\mathbf{250}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{0125}\mathbf{}\mathit{k}\mathit{V} \end{gathered}$$

Recall equation (5),

$V_{\mathbf{m}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\left[I_m(t-200\times {10}^{-8})-I_c(t-50\times {10}^{-6})\right]\mathbf{\times }\mathbf{20}\mathbf{}\mathit{k}\mathit{A}$

Substituteforinto above equation.

$$\begin{gathered} V_{\mathbf{m}}\mathbf{(}\mathbf{250}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{=}\left[I_m\left(250\times {10}^{-6}-200\times {10}^{-6}\right)-I_c\left(250\times {10}^{-6}-50\times {10}^{-6}\right)\right]\mathbf{\times }\mathbf{20} \\ {V}_{\mathbf{m}}\mathbf{(}\mathbf{250}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{=}\left[I_m\left(50\times {10}^{-6}\right)-I_c\left(200\times {10}^{-6}\right)\right]\mathbf{\times }\mathbf{20} \end{gathered}$$

Substituteforandforinto above equation.

$$\begin{gathered} V_{\mathbf{m}}\mathbf{(}\mathbf{250}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{=}\left[1.675\times {10}^{-5}-0\right]\mathbf{\times }\mathbf{20} \\ {V}_{\mathbf{m}}\mathbf{(}\mathbf{250}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{24}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{4}}\mathbf{}\mathit{k}\mathit{V} \end{gathered}$$

Recall the equation (6),

${\mathit{I}}_{\mathbf{m}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathit{I}}_m\mathbf{(}\mathbf{t}\mathbf{-}\mathbf{200}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{0067}{\mathbf{v}}_{\mathbf{k}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathit{k}\mathit{A}$

Substituteforinto above equation.

$$\begin{gathered} {\mathit{I}}_{\mathbf{m}}\mathbf{(}\mathbf{250}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{=}{\mathit{I}}_m\mathbf{(}\mathbf{250}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{-}\mathbf{200}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{0067}{\mathit{v}}_{\mathbf{k}}\mathbf{(}\mathbf{250}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)} \\ {\mathit{I}}_{\mathbf{m}}\mathbf{(}\mathbf{250}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{=}{\mathit{I}}_{\mathbf{k}}\mathbf{(}\mathbf{50}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{0067}{\mathit{v}}_{\mathbf{k}}\mathbf{(}\mathbf{250}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)} \end{gathered}$$

Substituteforandinto above equation.

$$\begin{gathered} {\mathit{I}}_{\mathbf{m}}\mathbf{(}\mathbf{250}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{675}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{0067}\mathbf{(}\mathbf{3}\mathbf{.}\mathbf{24}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{4}}\mathbf{)} \\ {\mathit{I}}_{\mathbf{m}}\mathbf{(}\mathbf{250}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{675}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathbf{-}\mathbf{2}\mathbf{.}\mathbf{2378}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{4}}\mathit{k}\mathit{A} \\ {\mathit{I}}_{\mathbf{k}}\left(250\times {10}^{-6}\right)\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{446}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathit{k}\mathit{A} \end{gathered}$$

Recall equation (7).

${\mathit{I}}_{\mathbf{m}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathit{I}}_{\mathbf{k}}\left(t-200\times {10}^{-8}\right)\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{0067}{\mathbf{v}}_{\mathbf{k}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathit{k}\mathit{A}$

Substituteforinto above equation.

$$\begin{gathered} {\mathit{I}}_{\mathbf{m}}\mathbf{(}\mathbf{250}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{=}{\mathit{I}}_{\mathbf{k}}\mathbf{(}\mathbf{250}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{-}\mathbf{200}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{0067}{\mathit{v}}_{\mathbf{k}}\mathbf{(}\mathbf{250}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)} \\ {\mathit{I}}_{\mathbf{m}}\mathbf{(}\mathbf{250}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{=}{\mathit{I}}_{\mathbf{k}}\mathbf{(}\mathbf{50}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{0067}{\mathit{v}}_{\mathbf{k}}\mathbf{(}\mathbf{250}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)} \end{gathered}$$

Substituteforandforinto above equation.

$$\begin{gathered} {\mathit{I}}_{\mathbf{m}}\left(250\times {10}^{-6}\right)\mathbf{=}\mathbf{0}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{0067}\left(0.0125\right) \\ {\mathit{I}}_{\mathbf{m}}\left(250\times {10}^{-5}\right)\mathbf{=}\mathbf{8}\mathbf{.}\mathbf{375}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathit{k}\mathit{A} \end{gathered}$$

Recall the equation (8),

${\mathit{I}}_{\mathbf{c}}\left(t\right)\mathbf{=}\mathbf{-}{\mathit{I}}_{\mathbf{c}}\left(t-50\times {10}^{-6}\right)\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{08}{\mathit{v}}_{\mathbf{m}}\left(t\right)\mathit{k}\mathit{A}$

Substituteforinto above equation.

$$\begin{gathered} {\mathit{I}}_{\mathbf{c}}\left(250\times {10}^{-6}\right)\mathbf{=}\mathbf{-}{\mathit{I}}_{\mathbf{c}}\left(250\times {10}^{-5}\right)\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{08}{\mathit{v}}_{\mathbf{m}}\left(250\times {10}^{-6}\right) \\ {\mathit{I}}_{\mathbf{C}}\left(250\times {10}^{-6}\right)\mathbf{=}\mathbf{-}{\mathit{I}}_{\mathbf{C}}\left(200\times {10}^{-6}\right)\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{08}{\mathit{v}}_{\mathbf{m}}\left(250\times {10}^{-6}\right) \end{gathered}$$

Substituteforandforinto above equation.

$$\begin{gathered} {\mathit{I}}_{\mathbf{c}}\left(250\times {10}^{-6}\right)\mathbf{=}\mathbf{0}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{08}\left(3.24\times {10}^{-4}\right) \\ {\mathit{I}}_{\mathbf{C}}\left(250\times {10}^{-6}\right)\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{592}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathit{k}\mathit{A} \end{gathered}$$

Hence the value for $V_k\left(250\times {10}^{-6}\right),V_m\left(250\times {10}^{-6}\right),I_k\left(250\times {10}^{-6}\right),I_m\left(250\times {10}^{-6}\right)$and $I_C\left(250\times {10}^{-6}\right),$ are $0.0125kV,3.24\times {10}^{-4}V,1.446\times {10}^{-5}kA,8.375\times {10}^{-5}kA$and $2.592\times {10}^{-5}kA$respectively.

For $\mathit{t}\mathbf{=}\mathbf{300}\mathbf{}\mathit{\mu }\mathit{s}$

Recall the equation (4).

${\mathit{V}}_{\mathbf{k}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{[}\mathbf{0}\mathbf{.}\mathbf{334}\mathbf{t}\mathbf{-}{\mathbf{I}}_{\mathbf{k}}\mathbf{(}\mathbf{t}\mathbf{-}\mathbf{200}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{]}\mathbf{149}\mathbf{.}\mathbf{87}\mathit{k}\mathit{V}$

Substitute $\mathbf{300}\mathbf{}\mathit{\mu }\mathit{s}$for into above equation.

$$\begin{gathered} {\mathit{V}}_{\mathbf{k}}\left(300\times {10}^{-6}\right)\mathbf{=}\left[0.334\times 300\times {10}^{-6}-l_k\left(300\times {10}^{-6}-200\times {10}^{-6}\right)\right]\mathbf{149}\mathbf{.}\mathbf{87} \\ {\mathit{V}}_{\mathbf{k}}\left(300\times {10}^{-6}\right)\mathbf{=}\left[100.2\times {10}^{-6}-I_k\left(100\times {10}^{-6}\right)\right]\mathbf{149}\mathbf{.}\mathbf{87} \end{gathered}$$

The value of the term ${\mathit{V}}_{\mathbf{k}}\mathbf{(}\mathbf{300}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}$is zero since input is composed of unit function.

$$\begin{gathered} {\mathit{V}}_{\mathbf{k}}\mathbf{(}\mathbf{300}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{=}\mathbf{[}\mathbf{100}\mathbf{.}\mathbf{2}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{-}\mathbf{0}\mathbf{]}\mathbf{149}\mathbf{.}\mathbf{87} \\ {\mathit{V}}_{\mathbf{k}}\mathbf{(}\mathbf{300}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{015}\mathbf{}\mathit{k}\mathit{V} \end{gathered}$$

Recall equation (5),

${\mathit{V}}_{\mathbf{m}}\left(t\right)\mathbf{=}\left[I_m\left(t-200\times {10}^{-6}\right)-I_C\left(t-50\times {10}^{-6}\right)\right]\mathbf{\times }\mathbf{20}\mathbf{}\mathit{k}\mathit{V}$

Substitute $\mathbf{300}\mathbf{}\mathit{\mu }\mathit{s}$for t into above equation.

$$\begin{gathered} {\mathit{V}}_{\mathbf{m}}\left(300\times {10}^{-6}\right)\mathbf{=}\left[I_m\left(300\times {10}^{-6}-200\times {10}^{-6}\right)-I_C\left(300\times {10}^{-6}-50\times {10}^{-6}\right)\right]\mathbf{\times }\mathbf{20} \\ {\mathit{V}}_{\mathbf{m}}\left(300\times {10}^{-6}\right)\mathbf{=}\left[I_m\left(100\times {10}^{-6}\right)-I_c\left(250\times 10-6\right)\right]\mathbf{\times }\mathbf{20} \end{gathered}$$

Substitute $\mathbf{2}\mathbf{.}\mathbf{592}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathit{k}\mathit{A}$for ${\mathit{I}}_{\mathbf{c}}\mathbf{(}\mathbf{250}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}$and $\mathbf{3}\mathbf{.}\mathbf{35}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathit{k}\mathit{A}$for ${\mathit{I}}_{\mathbf{m}}\mathbf{(}\mathbf{100}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}$into above equation.

$$\begin{gathered} {\mathit{V}}_{\mathbf{m}}\left(300\times {10}^{-6}\right)\mathbf{=}\left[3.35\times {10}^{-5}-2.592\right]\mathbf{\times }\mathbf{20} \\ {\mathit{V}}_{\mathbf{m}}\left(300\times {10}^{-6}\right)\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{816}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{4}}\mathit{k}\mathit{V} \end{gathered}$$

Recall the equation (6),

${\mathit{I}}_{\mathbf{K}}\left(t\right)\mathbf{=}{\mathit{I}}_{\mathbf{m}}\left(t-200\times {10}^{-6}\right)\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{0067}{\mathit{v}}_{\mathbf{m}}\left(t\right)\mathit{k}\mathit{A}$

Substituteforinto above equation.

$$\begin{gathered} {\mathit{I}}_{\mathbf{k}}\mathbf{(}\mathbf{300}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{=}{\mathit{I}}_{\mathbf{m}}\mathbf{(}\mathbf{300}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{-}\mathbf{200}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{-}\mathbf{0}\mathbf{.}{\mathbf{0067}}_{\mathbf{m}}\mathbf{(}\mathbf{300}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)} \\ {\mathit{I}}_{\mathbf{k}}\mathbf{(}\mathbf{300}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{=}{\mathit{I}}_{\mathbf{m}}\mathbf{(}\mathbf{100}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{0067}{\mathit{v}}_{\mathbf{m}}\mathbf{(}\mathbf{300}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)} \end{gathered}$$

Substitute $\mathbf{1}\mathbf{.}\mathbf{816}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{4}}\mathit{k}\mathit{V}$for ${\mathit{V}}_{\mathbf{m}}\mathbf{(}\mathbf{300}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}$and $\mathbf{3}\mathbf{.}\mathbf{35}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathit{k}\mathit{A}$for ${\mathit{I}}_{\mathbf{m}}\mathbf{(}\mathbf{100}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}$ into above equation.

$$\begin{gathered} {\mathit{I}}_{\mathbf{k}}\left(300\times {10}^{-6}\right)\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{35}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{0067}\left(1.816\times {10}^{-4}\right) \\ {\mathit{I}}_{\mathbf{k}}\left(300\times {10}^{-6}\right)\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{35}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{121}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}} \\ {\mathit{I}}_{\mathbf{K}}\left(300\times {10}^{-6}\right)\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{229}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathit{k}\mathit{A} \end{gathered}$$

Recall equation (7).

${\mathit{I}}_{\mathbf{m}}\left(t\right)\mathbf{=}{\mathit{I}}_{\mathbf{k}}\left(t-200\times {10}^{-6}\right)\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{0067}{\mathit{v}}_{\mathbf{k}}\left(t\right)\mathit{k}\mathit{A}$

Substitute $\mathbf{300}\mathbf{}\mathit{\mu }\mathit{s}$for t into above equation.

$$\begin{gathered} {\mathit{I}}_{\mathbf{m}}\left(300\times {10}^{-6}\right)\mathbf{=}{\mathit{I}}_{\mathbf{K}}\left(300\times {10}^{-6}-200\times {10}^{-6}\right)\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{0067}{\mathit{v}}_{\mathbf{k}}\left(300\times {10}^{-6}\right) \\ {\mathit{I}}_{\mathbf{m}}\left(300\times {10}^{-6}\right)\mathbf{=}{\mathit{I}}_{\mathbf{K}}\left(300\times {10}^{-6}\right)\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{0067}{\mathit{v}}_{\mathbf{k}}\left(300\times {10}^{-6}\right) \end{gathered}$$

Substitute 0.015 kV for ${\mathit{V}}_{\mathbf{k}}\left(300\times {10}^{-6}\right)$and 0 A for ${\mathit{I}}_{\mathbf{k}}\mathbf{(}\mathbf{100}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}$into above equation.

$$\begin{gathered} {\mathit{I}}_{\mathbf{m}}\left(300\times {10}^{-6}\right)\mathbf{=}\mathbf{0}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{0067}\left(0.015\right) \\ {\mathit{I}}_{\mathbf{m}}\left(300\times {10}^{-6}\right)\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{005}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{4}}\mathit{k}\mathit{A} \end{gathered}$$

Recall the equation (8),

${\mathit{I}}_{\mathbf{C}}\left(t\right)\mathbf{=}\mathbf{-}{\mathit{I}}_{\mathbf{C}}\left(t-50\times {10}^{-6}\right)\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{08}{\mathit{v}}_{\mathbf{m}}\left(t\right)\mathit{k}\mathit{A}$

Substitute $\mathbf{300}\mathbf{}\mathit{\mu }\mathit{s}$for t into above equation.

$$\begin{gathered} {\mathit{I}}_{\mathbf{C}}\mathbf{(}\mathbf{300}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{=}\mathbf{-}{\mathit{I}}_{\mathbf{C}}\left(300\times {10}^{-6}-50\times {10}^{-6}\right)\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{08}{\mathit{v}}_{\mathbf{m}}\left(300\times {10}^{-6}\right) \\ {\mathit{I}}_{\mathbf{C}}\mathbf{(}\mathbf{300}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}\mathbf{=}\mathbf{-}{\mathit{I}}_{\mathbf{C}}\left(250\times {10}^{-6}\right)\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{08}{\mathit{v}}_{\mathbf{m}}\left(300\times {10}^{-6}\right) \end{gathered}$$

Substitute $\mathbf{1}\mathbf{.}\mathbf{816}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{4}}\mathit{k}\mathit{V}$for ${\mathit{V}}_{\mathbf{m}}\mathbf{(}\mathbf{300}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}$and $\mathbf{2}\mathbf{.}\mathbf{592}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathit{k}\mathit{V}$for ${\mathit{I}}_{\mathbf{C}}\mathbf{(}\mathbf{250}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{)}$into above equation.

$$\begin{gathered} {\mathit{I}}_{\mathbf{C}}\left(300\times {10}^{-6}\right)\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{592}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathbf{}\mathit{k}\mathit{A}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{08}\left(1.816\times {10}^{-4}kV\right) \\ {\mathit{I}}_{\mathbf{C}}\left(300\times {10}^{-6}\right)\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{1392}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathbf{}\mathit{k}\mathit{A} \end{gathered}$$

Hence the value for${\mathit{V}}_{\mathbf{k}}\left(300\times {10}^{-6}\right)\mathbf{,}\mathbf{}{\mathit{V}}_{\mathbf{m}}\left(300\times {10}^{-6}\right)\mathbf{,}\mathbf{}{\mathit{I}}_{\mathbf{k}}\left(300\times {10}^{-6}\right)\mathbf{,}\mathbf{}{\mathit{V}}_{\mathbf{m}}\left(300\times {10}^{-6}\right)\mathbf{,}\mathbf{}$and ${\mathit{I}}_{\mathbf{c}}\left(300\times {10}^{-6}\right)\mathbf{,}\mathbf{}$ are $\mathbf{0}\mathbf{.}\mathbf{015}\mathbf{}\mathit{k}\mathit{V}\mathbf{}\mathbf{,}$$\mathbf{1}\mathbf{.}\mathbf{816}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{4}}\mathbf{}\mathit{k}\mathit{V}$,$\mathbf{}\mathbf{3}\mathbf{.}\mathbf{229}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathbf{}\mathit{k}\mathit{V}$,$\mathbf{1}\mathbf{.}\mathbf{005}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{4}}\mathbf{}\mathit{k}\mathit{A}$and $\mathbf{1}\mathbf{.}\mathbf{005}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{4}}\mathbf{}\mathit{k}\mathit{A}$respectively.