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\vspace{3cm}

\title{ 
%\logo{
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Circuits and Transforms
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}
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%	\logo{Matrix Analysis through Octave}{\begin{center}\includegraphics[scale=.24]{tlc}\end{center}}{}{HAMDSP}
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\author{ G V V Sharma$^{*}$ %<-this  stops a space
\thanks{*The author is with the Department
of Electrical Engineering, Indian Institute of Technology, Hyderabad
502285 India e-mail:  gadepall@iith.ac.in.  All content in the manuscript is 
released under GNU GPL.  Free to use for anything. }% <-this % stops a space
%\thanks{J. Doe and J. Doe are with Anonymous University.}% <-this % stops a space
%\thanks{Manuscript received April 19, 2005; revised January 11, 2007.}}
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\bigskip

\begin{abstract}
This manual provides a simple introduction to Transforms
\end{abstract}


%% Copyright (C) 2020 Saurabh Joshi
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 \section{Definitions}
\begin{enumerate}[label=\arabic*.,ref=\thesection.\theenumi]
\numberwithin{equation}{section}
\numberwithin{figure}{section}
\item The unit step function is 
\begin{align}
u(t) =
\begin{cases}
1 & t > 0
\\
	\frac{1}{2} & t = 0
\\
0 & t < 0
\end{cases}
\end{align}
\item The Laplace transform of $g(t)$ is defined as 
\begin{align}
	G(s) = \int_{-\infty}^{\infty} g(t) e^{-st}\, dt
\end{align}
 \end{enumerate}

 \section{Laplace Transform}
\begin{enumerate}[label=\arabic*.,ref=\thesection.\theenumi]
\numberwithin{equation}{section}
\item In the circuit, the switch S is connected to position P for a long time so that the charge on the capacitor
	becomes $q_1 \, \mu C$. Then S is switched to position Q.  After a long time, the charge on the capacitor is
		$q_2 \, \mu C$.
		\begin{figure}[!ht]
			\centering
			\includegraphics[width=\columnwidth]{figs/ckt.jpg}
			\caption{}
			\label{fig:ckt}
\end{figure}
\item Draw the circuit using latex-tikz.
\item Find $q_1$.
	\item Show that the Laplace transform of $u(t)$ is $\frac{1}{s}$ and find the ROC.
	\item Show that 
		\begin{align}
			e^{-at}u(t) \system{L} \frac{1}{s+a}, \quad a > 0
		\end{align}
		and find the ROC.
	\item Now consider the following resistive circuit transformed from 
			Fig. \ref{fig:ckt}
		\begin{figure}[!ht]
			\centering
			\includegraphics[width=\columnwidth]{figs/lap-ckt.jpg}
			\caption{}
			\label{fig:lap-ckt}
\end{figure}
		where 
		\begin{align}
			u(t) \system{L} V_1(s)
			\\
			2u(t) \system{L} V_2(s)
		\end{align}
		Find the voltage across the capacitor $V_{C_0}(s)$.
	\item Find $v_{C_0}(t)$.  Plot using python.
	\item Verify your result using ngspice.
	\item Obtain Fig. 
			\ref{fig:lap-ckt}
			using the equivalent differential equation.
\end{enumerate}

 \section{Initial Conditions}
\begin{enumerate}[label=\arabic*.,ref=\thesection.\theenumi]
\numberwithin{equation}{section}
\item Find $q_2$ in Fig. 
			\ref{fig:ckt}.
\item Draw the equivalent $s$-domain resistive circuit when S is switched to position Q.  Use variables $R_1, R_2, C_0$ for the passive elements.
Use latex-tikz.
		\label{prob:init}
	
		\item $V_{C_0}(s)$ = ?  
	\item $v_{C_0}(t)$ = ? Plot using python.
	\item Verify your result using ngspice.
	\item Find $v_{C_0}(0-), v_{C_0}(0+)$ and  $v_{C_0}(\infty) $. 
	\item Obtain the Fig.  in problem 
		\ref{prob:init}
			using the equivalent differential equation.


	\end{enumerate}
 \section{Bilinear Transform}
\begin{enumerate}[label=\arabic*.,ref=\thesection.\theenumi]
\numberwithin{equation}{section}
\item In Fig. 
			\ref{fig:ckt},
			consider the case when $S$ is switched to $Q$ right in the beginning. Formulate the differential equation.
		\item 			Find $H(s)$ considering the ouput voltage at the capacitor.
		\item Plot $H(s)$.  What kind of filter is it?
		\item Using trapezoidal rule for integration, formulate the difference equation
			by considering 
		\begin{align}
			y(n) = y(t)\vert_{t=n}
		\end{align}
	\item Find $H(z)$.
	\item How can you obtain $H(z)$ from $H(s)$?
	\end{enumerate}

\end{document}