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-%$Header: /cvsroot/latex-beamer/latex-beamer/solutions/generic-talks/generic-ornate-15min-45min.en.tex,v 1.4 2004/10/07 20:53:08 tantau Exp $
-\documentclass{beamer}
-\mode<presentation>
-{
-  \usecolortheme{seahorse}
-  \usefonttheme{professionalfonts}
-  \useinnertheme{rounded}
-  \useoutertheme{shadow}
-%  \useoutertheme{smoothbars}
-}
-%\setbeamertemplate{background canvas}[vertical shading][bottom=white!10,top=blue!5]
-\usepackage{verbatim} 
-\usepackage[english]{babel}
-\usepackage[latin1]{inputenc}
-\usepackage{pgf,pgfarrows,pgfnodes,pgfautomata,pgfheaps,pgfshade}
-\usepackage{amsmath,amsfonts,amsthm,amssymb}
-\usepackage{times}
-\usepackage[T1]{fontenc}
-\usepackage{graphics}
-\usepackage{graphicx}
-%\usepackage{psfig}
-\usepackage{algorithmic}
-
-\title
-{Scilab based Mini Circuit Simulator}
-
-\author[]
-{Yogesh Dilip Save}
-\institute
-{
-  Department of Electrical Engineering\\
-  Indian Institute of Technology, Bombay
-}
-%\pgfdeclareimage[height=0.7cm]{university-logo}{iitblogo.eps}
-%\logo{\pgfuseimage{university-logo}}
-
-
-\date[seminar] % (optional)
-{Sept., 2011 / \small{Software Freedom Day}}
-
-
-\begin{document}
-%***************************************************************************************
-\begin{frame}
-  \titlepage
-\end{frame}
-%***************************************************************************************
-%\begin{frame}
-%  \frametitle{Presentation Outline}
-%  \setcounter{tocdepth}{1}
-%  \tableofcontents
-%\end{frame}
-%***************************************************************************************
-
-\section{Introduction}
-\begin{frame}
- \frametitle{Motivation}
-\begin{block}{Objective}
-To assist students in improving their knowledge in field of circuit simulation.
-\end{block}
-\begin{block}{Problem with commercial simulators}
-\begin{itemize}
-\item Generally software codes are not available.
-\item Software codes are written in higher level language (C Programming and Fortran....).
-\item Complex due to implementation of many features and complex modelling.
-\end{itemize}
-\end{block}
-\end{frame}
-
-\begin{frame}
- \frametitle{Motivation}
-\begin{block}{Objective}
-To assist students in improving their knowledge in field of circuit simulation.
-\end{block}
-\begin{block}{Mini simulator}
-\begin{itemize}
-\item used Scilab for coding.
-\item integrated least number of component.  
-\item different versions for add-on features. 
-\end{itemize} 
-\end{block}
-\end{frame}
-
-\section{Features}
-\begin{frame}
- \frametitle{Features}
-\begin{itemize}
-  \item {\color{red} Various Analysis options.}
-    \begin{itemize}
-      \item Operating Point Analysis
-      \item DC Analysis 
-      \item Transient Analysis  
-      \item AC Analysis 
-    \end{itemize}
-  \item Facility to define a new component.
-  \item Provides circuit equations for debugging as well as learning circuit simulator.
-  \item Easy to integrate and test a new method such as convergence technique, integration method etc. 
-\end{itemize} 
-\end{frame}
-
-\begin{frame}
-\frametitle{Full Wave Bridge Rectifier with Filter}
-\begin{minipage}[!b]{0.47\linewidth} % A minipage that covers half the page
-  \begin{small}  {\bf Circuit Diagram and Netlist} \end{small}
-\vspace{-0.5cm}
-\begin{figure}[h]
-\centering
-\includegraphics[scale=0.47]{../figures/bridgeFilter.eps}
-\end{figure}
-\vspace{-0.5cm}
-\begin{tiny}
-* Full Wave Bridge Rectifier 
-\newline
-\vspace{-0.1cm}
-V1 1 2 sine (5 50) 
-\newline
-\vspace{-0.1cm}
-D1 1 3 mymodel (1e-8 0.026) 
-\newline
-\vspace{-0.1cm}
-D2 2 3 mymodel (1e-8 0.026) 
-\newline
-\vspace{-0.1cm}
-D3 0 1 mymodel (1e-8 0.026) 
-\newline
-\vspace{-0.1cm}
-D4 0 2 mymodel (1e-8 0.026) 
-\newline
-\vspace{-0.1cm}
-R1 3 0 10000 
-\newline
-\vspace{-0.1cm}
-C1 3 0 1e-2 
-\newline
-\vspace{-0.1cm}
-.tran 0 100 0.5 
-\newline
-\vspace{-0.1cm}
-.plot v(1)-v(2) v(3) 
-\newline
-\vspace{-0.1cm}
-.end 
-\end{tiny}
-\end{minipage}
-\hspace{0.1cm} % To get a little bit of space between the figures
-\begin{minipage}[!b]{0.47\linewidth} % A minipage that covers half the page
-\begin{figure}[h]
-\centering
-\includegraphics[scale=0.3]{../figures/bridgeFilterOutput.eps}
-\caption{Input-Output Waveform} 
-\end{figure}
-\end{minipage}
-\end{frame}
-
-\begin{frame}
- \frametitle{Features}
-\begin{itemize}
-  \item Various Analysis options.
-    \begin{itemize}
-      \item Operating Point Analysis
-      \item DC Analysis 
-      \item Transient Analysis  
-      \item AC Analysis 
-    \end{itemize}
-  \item {\color{red} Facility to define a new component.}
-  \item Provides circuit equations for debugging as well as learning circuit simulator.
-  \item Easy to integrate and test a new method such as convergence technique, integration method etc. 
-\end{itemize} 
-\end{frame}
-
-\begin{frame}
-\frametitle{User defined Components}
-Consider, a non-linear resistance,
-$$I=\frac{1}{R}V^3$$
-
-\begin{itemize}
-\item Create  a file \$CompName.sci
-\item Define
-\begin{itemize}
-\item Function in the $i=g(v)$ form
-\item Jacobian of the function
-\end{itemize}
-\end{itemize}
-
-%{\bf Syntax:-}
-%\newline
-%function I=\$CompName\_func(voltage,parameter)
-%\$par\_2=parameter(2)
-%\$par\_3=parameter(3)
-\end{frame}
-
-\begin{frame}
-\frametitle{Non-linear Resistance}
-\begin{minipage}[!b]{0.43\linewidth} % A minipage that covers half the page
-\begin{figure}[h]
-\centering
-\includegraphics[scale=0.7]{../figures/myR.eps}
-\end{figure}
-\begin{tiny}
-function I=myR\_func(voltage,param)\newline
-\hspace*{1cm}R=param(2); \newline
-\hspace*{1cm}I=1/R*(voltage$^3$);\newline
-endfunction 
-
-function Gj=myR\_Jacobian(voltage,param)\newline 
-\hspace*{1cm}R=param(2); \newline
-\hspace*{1cm}Gj=3/R*(voltage$^2$);\newline
-endfunction
-\end{tiny}
-\end{minipage}
-\hspace{0.5cm} % To get a little bit of space between the figures
-\begin{minipage}[!b]{0.5\linewidth} % A minipage that covers half the page
-\begin{figure}[h]
-\centering
-\includegraphics[scale=0.3]{../figures/myROutput.eps}
-\end{figure}
-\end{minipage}
-\end{frame}
-
-\begin{frame}
- \frametitle{Features}
-\begin{itemize}
-  \item Various Analysis options.
-    \begin{itemize}
-      \item Operating Point Analysis
-      \item DC Analysis 
-      \item Transient Analysis  
-      \item AC Analysis 
-    \end{itemize}
-  \item Facility to define a new component.
-  \item {\color{red} Provides circuit equations for debugging as well as learning circuit simulator.}
-  \item Easy to integrate and test a new method such as convergence technique, integration method etc. 
-\end{itemize} 
-\end{frame}
-
-\begin{frame}
-\begin{block}{Example}
-%\begin{minipage}[!b]{0.4\linewidth} % A minipage that covers half the page
-\begin{figure}[!ht]
-\begin{center}
-\includegraphics[scale=0.35]{../figures/modified_figure.eps}
-\caption{ Example for MNA } \label{modifiedfig}
-\end{center}
-\end{figure}
-%\end{minipage}
-%\begin{minipage}[!b]{0.55\linewidth} % A minipage that covers half the page
-\begin{tiny}
-$$\left[
-\begin{array}{cccccc}
-G_{1}+G_{4} & -G_{1} & -G_{4} & 1 & 0 \\
--G_{1} & G_{1}+G_{2}+G_{3} & -G_{3} & 0 & 0 \\
--G_{4} & -G_{3} & G_{3}+G_{4} & 0 & 1  \\
-1 & 0 & 0 & 0 & 0 \\
-0 & 0 & 1 & 0 & 0
-\end{array}
-\right] \left[
-\begin{array}{c}
-v_{1}\\
-v_{2}\\
-v_{3}\\
-i_{V_1}\\
-i_{V_2}\\
-\end{array}
-\right]= \left[
-\begin{array}{c}
-0\\
-0\\
-0\\
-V_{1}\\
-V_{2}
-\end{array}
-\right]$$
-\end{tiny}
-%\end{minipage}
-\end{block}
-\end{frame}
-
-\begin{frame}
- \frametitle{Features}
-\begin{itemize}
-  \item Various Analysis options.
-    \begin{itemize}
-      \item Operating Point Analysis
-      \item DC Analysis 
-      \item Transient Analysis  
-      \item AC Analysis 
-    \end{itemize}
-  \item Facility to define a new component.
-  \item Provides circuit equations for debugging as well as learning circuit simulator.
-  \item {\color{red} Easy to integrate and test a new method such as convergence technique, integration method etc.} 
-\end{itemize} 
-\end{frame}
-
-\begin{frame}
-	\begin{center}
-	{\Huge	Thank You}
-\end{center}
-%	\smiley
-\end{frame}
-% 
-% \section{Operating Point Analysis}
-% \begin{frame}
-% \begin{block}{Operating Point (OP) Analysis}
-% \begin{itemize}
-% \item OP Analysis is the central part of a circuit simulator.
-% \item The equations that describe the electrical system are nonlinear and algebraic and their solution gives operating point.
-% \item Systems of nonlinear equations are solved by iteratively formulating and solving systems of linear algebraic equations. 
-% \item The overall efficiency of a circuit simulator is dependent upon the performance of the linear DC analyzer.
-% %\item Thus, our work is towards improving the performance of linear DC Analyzers and handling convergence issues related to large size nonlinear circuits.
-% \end{itemize}
-% \end{block}
-% \end{frame}
-% 
-% \begin{frame}
-% \begin{block}{\small Nodal Analysis}
-% \begin{itemize}
-% \begin{small}
-% \item Applicable when the network has only current sources and conductances type devices i.e., $i=g(v)$.
-% \item Let, $\mathbf{A}_r$ be the reduced incidence matrix of $\cal{G}$ which is a representative matrix of $V_v(\cal{G})$. \\
-% \end{small}
-% \begin{tiny}
-% The KCL constraints are
-% $$\mathbf{A_ri}=\mathbf{0}$$
-% $$\left[\begin{array}{cc}
-%   \mathbf{A}_{rG} & \mathbf{A}_{rJ}
-% \end{array}\right]
-% \left[\begin{array}{c}
-%   \mathbf{i}_{G} \\
-%   \mathbf{i}_{J}
-% \end{array}\right]
-% =\mathbf{0}$$
-% $$\mathbf{A}_{rG}\mathbf{i}_{G}=-\mathbf{A}_{rJ}\mathbf{i}_{J}$$
-% 
-% $$\mathbf{A}_{rG}\mathbf{G}\mathbf{v}_{G}=-\mathbf{A}_{rJ}\mathbf{i}_{J}\ \ \ \ \ \ \ \ (As, \mathbf{i}_{G}=\mathbf{G}\mathbf{v}_{G})$$
-% 
-% The KVE constraints are
-% $$\left[\begin{array}{c}
-%   \mathbf{v}_{G} \\
-%   \mathbf{v}_{J}
-% \end{array}\right]
-% =
-% \left[\begin{array}{c}
-%   \mathbf{A}_{rG}^T \\
-%   \mathbf{A}_{rJ}^T
-% \end{array}\right]
-% \mathbf{v}_n$$
-% 
-% \begin{equation}
-% \mathbf{A}_{rG}\mathbf{G}\mathbf{A}_{rG}^{T}\mathbf{v}_{n}=-\mathbf{A}_{rJ}\mathbf{i}_{J}
-% \label{nodal_equation}
-% \end{equation}
-% \end{tiny}
-% \end{itemize}
-% \end{block}
-% \end{frame}
-% 
-% \begin{frame}
-% \begin{block}{Matrix Formulation}
-% \begin{itemize}
-% \item The diagonal entries of the matrix are the sum of conductances incident on the corresponding nodes.
-% \item The off diagonal entries $(i,j)^{th}$ of the matrix is the negative of conductances between node $i$ and $j$.
-% \item The $\mathbf{A}_{rJ}\mathbf{i}_{J}$ is the sum of current sources leaving the nodes.
-% \end{itemize}
-% \end{block}
-% \begin{block}{Example}
-% \end{block}
-% \begin{minipage}[!b]{0.4\linewidth} % A minipage that covers half the page
-% \begin{figure}[h]
-% \centering
-% \includegraphics[scale=0.35]{../figures/nodal_figure.eps}
-% \end{figure}
-% \end{minipage}
-% \begin{minipage}[!b]{0.55\linewidth} % A minipage that covers half the page
-% \begin{tiny}
-% $$\left[
-% \begin{array}{ccc}
-% G_{1}+G_{2} & -G_{2} & 0\\
-% -G_{2} & G_{2}+G_{3}+G_{4} & -G_{4}\\
-% 0 & -G_{4} & G_{4}+G_{5}
-% \end{array}
-% \right] \left[
-% \begin{array}{c}
-% v_{1}\\
-% v_{2}\\
-% v_{3}
-% \end{array}
-% \right]= \left[
-% \begin{array}{c}
-% I_{1}\\
-% 0\\
-% I_{2}
-% \end{array}
-% \right]$$
-% \end{tiny}
-% \end{minipage}
-% \end{frame}
-% 
-% 
-% \begin{frame}
-% \begin{block}{Modified Nodal Analysis}
-% \begin{small}
-% \begin{itemize}
-% \item applicable to all kinds of networks.
-% \item Let $\mathbf{A}_{r}$ be the reduced incidence matrix of ${\cal{G}}$
-% By Tellegan's theorem,
-% \begin{tiny}
-% $$\mathbf{A_ri}=\mathbf{0}$$
-% $$\left[\begin{array}{ccc}
-%   \mathbf{A}_{rG} & \mathbf{A}_{rT} & \mathbf{A}_{rJ}
-% \end{array}\right]
-% \left[\begin{array}{c}
-%   \mathbf{i}_{G} \\
-%   \mathbf{i}_{T} \\
-%   \mathbf{i}_{J}
-% \end{array}\right]
-% =\mathbf{0}$$
-% 
-% $$\left[\begin{array}{cc}
-%   \mathbf{A}_{rG}\mathbf{G} & \mathbf{A}_{rT}
-% \end{array}\right]
-% \left[\begin{array}{c}
-%   \mathbf{v}_{G} \\
-%   \mathbf{i}_{T}
-% \end{array}\right]
-% =-\mathbf{A}_{rJ}\mathbf{i}_{J}$$
-% 
-% \begin{equation}
-% \label{mna_eq1}
-% \left[\begin{array}{cc}
-%   \mathbf{A}_{rG}\mathbf{G}\mathbf{A}_{rG}^{T} & \mathbf{A}_{rT}
-% \end{array}\right]
-% \left[\begin{array}{c}
-%   \mathbf{v}_{n} \\
-%   \mathbf{i}_{T}
-% \end{array}\right]
-% =-\mathbf{A}_{rJ}\mathbf{i}_{J}
-% \end{equation}
-% 
-% Device characteristics of the branches in $T$ be
-% $$\left[\begin{array}{cc}
-%   \mathbf{M} & \mathbf{N}
-% \end{array}\right]
-% \left[\begin{array}{c}
-%   \mathbf{i}_{T} \\
-%   \mathbf{v}_{T}
-% \end{array}\right]
-% =\mathbf{S}_{T}$$
-% 
-% \begin{equation}
-% \label{mna_eq2}
-% \left[\begin{array}{cc}
-%   \mathbf{NA}_{rT}^{T} & \mathbf{M}
-% \end{array}\right]
-% \left[\begin{array}{c}
-%   \mathbf{v}_{n} \\
-%   \mathbf{i}_{T}
-% \end{array}\right]
-% =\mathbf{S}_{T}
-% \end{equation}
-% \end{tiny}
-% \end{itemize}
-% \end{small}
-% \end{block}
-% \end{frame}
-% 
-% \begin{frame}
-% \begin{block}{Example}
-% %\begin{minipage}[!b]{0.4\linewidth} % A minipage that covers half the page
-% \begin{figure}[!ht]
-% \begin{center}
-% \includegraphics[scale=0.35]{../figures/modified_figure.eps}
-% \caption{ Example for MNA } \label{modifiedfig}
-% \end{center}
-% \end{figure}
-% %\end{minipage}
-% %\begin{minipage}[!b]{0.55\linewidth} % A minipage that covers half the page
-% \begin{tiny}
-% $$\left[
-% \begin{array}{cccccc}
-% G_{1}+G_{4} & -G_{1} & -G_{4} & 1 & 0 \\
-% -G_{1} & G_{1}+G_{2}+G_{3} & -G_{3} & 0 & 0 \\
-% -G_{4} & -G_{3} & G_{3}+G_{4} & 0 & 1  \\
-% 1 & 0 & 0 & 0 & 0 \\
-% 0 & 0 & 1 & 0 & 0
-% \end{array}
-% \right] \left[
-% \begin{array}{c}
-% v_{1}\\
-% v_{2}\\
-% v_{3}\\
-% i_{V_1}\\
-% i_{V_2}\\
-% \end{array}
-% \right]= \left[
-% \begin{array}{c}
-% 0\\
-% 0\\
-% 0\\
-% V_{1}\\
-% V_{2}
-% \end{array}
-% \right]$$
-% \end{tiny}
-% %\end{minipage}
-% \end{block}
-% \end{frame}
-% 
-% \begin{frame}
-% \frametitle{Controlled Sources}
-% \begin{minipage}[!b]{0.47\linewidth} % A minipage that covers half the page
-%   \begin{figure}[!ht]
-%       \centering
-%       \includegraphics[scale=0.6]{../figures/VCCS.eps}
-%       \caption{Voltage Controlled Current Source (VCCS)}
-%       \label{vccs}
-%   \end{figure}
-% \end{minipage}
-% %\hspace{0.5cm} % To get a little bit of space between the figures
-% \begin{minipage}[!b]{0.47\linewidth}
-%   \begin{figure}[!ht]
-%      \centering
-%       \includegraphics[scale=0.6]{../figures/VCVS.eps}
-%       \caption{Voltage Controlled Voltage Source (VCVS) }
-%       \label{vcvs}
-%   \end{figure}
-%     \end{minipage}
-% \begin{minipage}[!b]{0.47\linewidth} % A minipage that covers half the page
-%   \begin{figure}[!ht]
-%       \centering
-%       \includegraphics[scale=0.6]{../figures/CCCS.eps}
-%       \caption{Current Controlled Current Source (CCCS)}
-%       \label{cccs}
-%   \end{figure}
-% \end{minipage}
-% %\hspace{0.5cm} % To get a little bit of space between the figures
-% \begin{minipage}[!b]{0.47\linewidth}
-%   \begin{figure}[!ht]
-%      \centering
-%       \includegraphics[scale=0.6]{../figures/CCVS.eps}
-%       \caption{Current Controlled Voltage Source (CCVS) }
-%       \label{ccvs}
-%   \end{figure}
-%     \end{minipage}
-% \begin{small}
-% \begin{itemize}
-% \item In voltage controlled devices, we have added a $0A$ current source as controlling branch
-% %without disturbing the incidence relationship of existing edges (i.e., the addition is 'soldering type') and its voltage is used for calculating the value of the devices.
-% \item In current controlled devices, we have added a $0V$ voltage source as controlling branch
-% %by splitting a node (i.e., plier type entry) and the current through it is used for calculating the value of the devices.
-% \end{itemize}
-% \end{small}
-% \end{frame}
-% 
-% \begin{frame}
-% \frametitle{Linearization of Nonlinear Elements}
-% \begin{minipage}[!b]{0.5\linewidth}
-% Diode characteristics,
-% $$I_D=I_S(e^{qV/kT}-1)$$
-% $$I_D=I_D|_{V=V_0} + (V-V_0)\frac{I_D}{V}|_{V=V_0}$$
-% $$I_D=I_{D0}+(V-V_0)G_{D0}$$
-% \begin{figure}[h]
-% \begin{center}
-% \includegraphics[scale=0.4]{../figures/diodeI.eps}
-% \begin{small}Modeling of Diode\end{small}
-% \label{diodeI}
-% \end{center}
-% \end{figure}
-% \end{minipage}
-% \begin{minipage}[!b]{0.4\linewidth}
-% \begin{figure}[h]
-% \begin{center}
-% \includegraphics[scale=0.3]{../figures/diodechar1.eps}
-% \begin{small}Linearized approximation of diode model\end{small}
-% \begin{tiny}$$I_{DN0}=I_{D0}-V_0G_{D0}$$\end{tiny}
-% \end{center}
-% \end{figure}
-% \end{minipage}
-% \end{frame}
-% 
-% 
-% \begin{frame}
-% {\bf Procedure:}{Operating Point Analysis}
-% \small
-% \begin{algorithmic}[1]
-% \STATE Find Node Potential and Current through devices whose device characteristic can not be expressed in terms of voltage.
-% \STATE Find branch voltage and node potentail.
-% \STATE Find branch current from branch voltage using device characteristics.
-% \IF{Non-linear component}
-% \STATE {\bf NR:} Check  device characteristics of non-linear devices.
-% \IF {Device characteristics is not satisfied}
-% \STATE Call Newton Raphson procedure
-% \STATE Find Node Potential and Current through devices whose device characteristic can not be expressed in terms of voltage.
-% \STATE Find branch current from branch voltage using device characteristics.
-% \STATE Go to {\bf NR}
-% \ENDIF
-% \STATE Check for KCL
-% \ENDIF
-% \end{algorithmic}
-% \normalsize
-% \end{frame}
-% 
-% \begin{frame}
-% \frametitle{Full Wave Bridge Rectifier}
-% \begin{minipage}[!b]{0.4\linewidth} % A minipage that covers half the page
-% \begin{figure}[h]
-% \centering
-% \includegraphics[scale=0.5]{../figures/bridge.eps}
-% \end{figure}
-% \end{minipage}
-% \hspace{0.5cm} % To get a little bit of space between the figures
-% \begin{minipage}[!b]{0.5\linewidth} % A minipage that covers half the page
-% \begin{figure}[h]
-% \centering
-% \includegraphics[scale=0.3]{../figures/bridgeOutput.eps}
-% \end{figure}
-% \end{minipage}
-% \end{frame}
-% 
-% \section{DC Analysis}
-% \begin{frame}
-% \frametitle{DC Analysis}
-% {\bf Procedure:}{DC Analysis}
-% \small
-% \begin{algorithmic}[1]
-% \STATE Modify the value of the sweep source and update Modified Nodal matrix.
-% \STATE Do Operating Point Analysis.
-% \end{algorithmic}
-% \normalsize
-% \end{frame}
-% 
-% \begin{frame}
-% \frametitle{Voltage Sweep}
-% \begin{minipage}[!b]{0.4\linewidth} % A minipage that covers half the page
-% \begin{figure}[h]
-% \centering
-% \includegraphics[scale=0.8]{../figures/V_Sweep.eps}
-% \caption{Example of DC Analysis (Vsweep.ckt)}
-% \end{figure}
-% \end{minipage}
-% \hspace{0.5cm} % To get a little bit of space between the figures
-% \begin{minipage}[!b]{0.5\linewidth} % A minipage that covers half the page
-% \begin{figure}[h]
-% \centering
-% \includegraphics[scale=0.3]{../figures/V_SweepOutput.eps}
-% \end{figure}
-% \end{minipage}
-% \end{frame}
-% 
-% \section{Transient Analysis}
-% \begin{frame}
-%   \begin{block}{What is Transient Analysis?}
-%      \begin{itemize}
-%        \item Computes the response of a circuit as function of time.
-%        \item Time is discretized and the solution is computed piecewise.
-%      \end{itemize}
-%    \end{block}
-%   \begin{block}{Important factors}
-%     \begin{itemize}
-%       \item Proper time Stepping.
-%       \item Integration methods.
-%     \end{itemize}
-%   \end{block}
-% \end{frame}
-% 
-% \begin{frame}
-% \frametitle{Discreatization}
-% Consider, a capacitor
-% \begin{tiny}
-% $$I_C(t_n)=C\frac{\partial{V}_C(t_n)}{\partial{t}}$$
-% Using Backward Euler's method,
-% $$I_C(t_n)=C\frac{V(t_n)-V(t_{n-1})}{t_n-t_{n-1}}$$
-% $$I_C(t_n)=\frac{C}{h}V(t_n)-\frac{C}{h}V(t_{n-1})$$
-% $$I_C(t_n)=G_C^{(k)}V(t_n)-I_C^{(k)}$$
-% \end{tiny}
-% \begin{figure}[h]
-% \centering
-% \includegraphics[scale=0.8]{../figures/Ceq.eps}
-% \end{figure}
-% \end{frame}
-% 
-% \begin{frame}
-% \frametitle{RC Circuit}
-% \begin{minipage}[!b]{0.4\linewidth} % A minipage that covers half the page
-% \begin{figure}[h]
-% \centering
-% \includegraphics[scale=0.8]{../figures/RC.eps}
-% \end{figure}
-% \end{minipage}
-% \hspace{0.5cm} % To get a little bit of space between the figures
-% \begin{minipage}[!b]{0.5\linewidth} % A minipage that covers half the page
-% \begin{figure}[h]
-% \centering
-% \includegraphics[scale=0.3]{../figures/RCOutput.eps}
-% \end{figure}
-% \end{minipage}
-% \end{frame}
-% 
-% 
-% \begin{frame}
-% \frametitle{PseudoCode}
-% {\bf Procedure:}{Transient Analysis}
-% \small
-% \begin{algorithmic}[1]
-% \STATE Discretize time dependent Component and Update Modified Nodal matrix.
-% \STATE Do Operating Point Analysis.
-% \end{algorithmic}
-% \normalsize
-% 
-% {\bf Procedure:}{Discretization}
-% \small
-% \begin{algorithmic}[1]
-% \STATE Compute time dependent source value at time t.
-% \STATE Compute the values of static model of dynamic component at time t.
-% \STATE Update Modified Nodal matrix.
-% \end{algorithmic}
-% \normalsize
-% \end{frame}
-% 
-% %\begin{frame}
-% %\frametitle{CMOS Inverter}
-% %\begin{minipage}[!b]{0.4\linewidth} % A minipage that covers half the page
-% %\begin{figure}[h]
-% %\centering
-% %\includegraphics[scale=0.4]{../figures/inverter.eps}
-% %\end{figure}
-% %\end{minipage}
-% %\hspace{0.5cm} % To get a little bit of space between the figures
-% %\begin{minipage}[!b]{0.5\linewidth} % A minipage that covers half the page
-% %\begin{figure}[h]
-% %\centering
-% %\includegraphics[scale=0.3]{../figures/inverterOutput.eps}
-% %\end{figure}
-% %\end{minipage}
-% %\end{frame}
-% 
-\end{document}
-