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zsu~2x_Ff4>JM))vKwG}8Smvs#C9cL_J^56v-hd`dinkgj^^BPQhc`7NHVTObotGDb Lfk8%97UTZ`E4wl5 diff --git a/differential-equations/electrical.tex b/differential-equations/electrical.tex index 68b7dfc..0278658 100755 --- a/differential-equations/electrical.tex +++ b/differential-equations/electrical.tex @@ -45,11 +45,10 @@ \title{Dynamics in Electrical Systems} -\author{Jakub~Hanak, - Peter~Babic\\% <-this % stops a space +\author{Peter~Babic\\% <-this % stops a space Dept. of Computers and Informatics, FEI TU of Kosice\\% Slovak Republic\\% - jakub.hanak2@gmail.com, babicpet@gmail.com% + babicpet@gmail.com% } % The paper headers @@ -80,7 +79,7 @@ circuits, cycle, differential, dynamics, electrical, equation, limit, modeling, \IEEEPARstart{T}{his} this paper is intended to sum up the research done in order to understand the Dynamics in electrical systems and their underlying differential equations. - + \hfill June 03, 2015 \section{Dynamical Systems} @@ -89,13 +88,13 @@ circuits, cycle, differential, dynamics, electrical, equation, limit, modeling, For the most part, applications fall into three broad categories: predictive (also referred to as generative), in which the objective is to predict future states of the system from observations of the past and present states of the system, diagnostic, in which the objective is to infer what possible past states of the system might have led to the present state of the system (or observations leading up to the present state), and, finally, applications in which the objective is neither to predict the future nor explain the past but rather to provide a theory for the physical phenomena. These three categories correspond roughly to the need to predict, explain, and understand physical phenomena. \subsection{Differential Equations} -A \textbf{differential equation} is any equation which contains derivatives, either ordinary derivatives or partial derivatives. Almost every physical situation that occurs in nature can be \textit{described} with an appropriate differential equation. +A \textbf{differential equation} is any equation which contains derivatives, either ordinary derivatives or partial derivatives. Almost every physical situation that occurs in nature can be \textit{described} with an appropriate differential equation. The process of describing a physical situation with a differential equation is called \textbf{modeling}. Differential equations are generally concerned about three questions: \begin{enumerate} -\item Given a differential equation will a solution exist? +\item Given a differential equation will a solution exist? \item If a differential equation does have a solution how many solutions are there? \item If a differential equation does have a solution can we find it? \end{enumerate} @@ -108,7 +107,7 @@ Differential equations fall to two groups - \textit{ordinary differential equati \subsection{Direction Field} -Understanding \textbf{direction fields} (or \textbf{slope fields)} and what they tell us about a differential equation and its solution is important and can be introduced without any knowledge of how to solve a differential equation and so can be done before the getting to actually solving them. +Understanding \textbf{direction fields} (or \textbf{slope fields)} and what they tell us about a differential equation and its solution is important and can be introduced without any knowledge of how to solve a differential equation and so can be done before the getting to actually solving them. The direction fields are important because they can provide a \textit{sketch of solution}, if exist, and a \textit{long term behavior} - most of the time we are interested in general picture about what is happening, as the time passes. @@ -118,7 +117,7 @@ Example direction field, embedded in phase portrait is shown in \cref{f:vdp_m}. \subsection{Laplace Transform} The \textbf{Laplace transform} is an integral transform perhaps second only to the Fourier transform in its utility in solving physical problems. The Laplace transform \eqref{eq:lpl} is particularly useful in solving linear ordinary differential equations such as those arising in the analysis of electronic circuits. The Laplace transform $\mathcal{L}$ \begin{equation} -\label{eq:lpl} +\label{eq:lpl} \mathcal{L}[f(t)](s)=\int{0}{\infty} f(t)e^{-st}dt \end{equation} where $f(t)$ is defined for $t\le 0$ - this is it's most common form and is called \textit{unilateral}. @@ -127,7 +126,7 @@ Most important properties of Laplace transform is that differentiation and integ \section{Periodic Orbits} -A periodic orbit corresponds to a special type of solution for a dynamical system, namely one which repeats itself in time. A dynamical system exhibiting a stable periodic orbit is often called an \textit{oscillator}. +A periodic orbit corresponds to a special type of solution for a dynamical system, namely one which repeats itself in time. A dynamical system exhibiting a stable periodic orbit is often called an \textit{oscillator}. \subsection{Limit Cycle} A \textbf{limit cycle} is an isolated closed trajectory. \textit{Isolated} means that neighboring trajectories are not closed - they spiral either towards or away from the limit cycle. The particle on the limit cycle, appears after one period on the exact same spot. Limit cycle appears on on a plane, opposed to a periodic orbit, that happens to be a vector. @@ -139,7 +138,7 @@ A \textbf{limit cycle} is an isolated closed trajectory. \textit{Isolated} means \label{f:lc_st} \end{figure} -If all neighboring trajectories approach the limit cycle, we say the limit cycle is \textbf{stable} or \textit{attracting}, as shown on \cref{f:lc_st}. Otherwise the limit cycle is \textbf{unstable}, or in exceptional cases, \textbf{half-stable}. Stable limit cycles are very important scientifically as they model systems that exhibit self-sustained oscillations. In other words, these systems oscillate even in the absence of external periodic forcing. +If all neighboring trajectories approach the limit cycle, we say the limit cycle is \textbf{stable} or \textit{attracting}, as shown on \cref{f:lc_st}. Otherwise the limit cycle is \textbf{unstable}, or in exceptional cases, \textbf{half-stable}. Stable limit cycles are very important scientifically as they model systems that exhibit self-sustained oscillations. In other words, these systems oscillate even in the absence of external periodic forcing. \begin{figure}[ht!] \centering @@ -276,7 +275,7 @@ The RL circuit shown on \cref{f:rl} has a resistor and an inductor connected in Applying the KVL, we obtain the algebraic sums of all the voltage drops as an ODE with respect to time $$Ri+L\,\frac{di}{dt}=V(t)$$ -solving which we obtain +solving which we obtain $$i=\frac{V}{R}\left(1-e^{-(R/L)t}\right)$$ The solving process is quite lenghty and is not a point of this work. For more details see \cite{bird2014electrical}. @@ -352,7 +351,7 @@ The main goal was to get some general idea about the connections between the ter %\appendices %\section{Proof of the First Zonklar Equation} -%Appendix one text goes here. +%Appendix one text goes here. % %% you can choose not to have a title for an appendix %% if you want by leaving the argument blank @@ -397,7 +396,7 @@ The authors would like to thank professor Carlos Par\'{e}s for having patience w %\begin{figure}[!t] %\centering %\includegraphics[width=2.5in]{myfigure} -% where an .eps filename suffix will be assumed under latex, +% where an .eps filename suffix will be assumed under latex, % and a .pdf suffix will be assumed for pdflatex; or what has been declared % via \DeclareGraphicsExtensions. %\caption{Simulation Results.} @@ -413,7 +412,7 @@ The authors would like to thank professor Carlos Par\'{e}s for having patience w % The subfigure \label commands are set within each subfloat command, % and the \label for the overall figure must come after \caption. % \hfil is used as a separator to get equal spacing. -% Watch out that the combined width of all the subfigures on a +% Watch out that the combined width of all the subfigures on a % line do not exceed the text width or a line break will occur. % %\begin{figure*}[!t] @@ -432,7 +431,7 @@ The authors would like to thank professor Carlos Par\'{e}s for having patience w % reference/describe all of them (a), (b), etc., within the main caption. -% An example of a floating table. Note that, for IEEE style tables, the +% An example of a floating table. Note that, for IEEE style tables, the % \caption command should come BEFORE the table. Table text will default to % \footnotesize as IEEE normally uses this smaller font for tables. % The \label must come after \caption as always. @@ -657,8 +656,8 @@ The authors would like to thank professor Carlos Par\'{e}s for having patience w % \let\MYoriglatexcaption\caption % \renewcommand{\caption}[2][\relax]{\MYoriglatexcaption[#2]{#2}} %\fi -% endfloat.sty was written by James Darrell McCauley, Jeff Goldberg and -% Axel Sommerfeldt. This package may be useful when used in conjunction with +% endfloat.sty was written by James Darrell McCauley, Jeff Goldberg and +% Axel Sommerfeldt. This package may be useful when used in conjunction with % IEEEtran.cls' captionsoff option. Some IEEE journals/societies require that % submissions have lists of figures/tables at the end of the paper and that % figures/tables without any captions are placed on a page by themselves at @@ -687,7 +686,7 @@ The authors would like to thank professor Carlos Par\'{e}s for having patience w % http://www.ctan.org/tex-archive/macros/latex/contrib/endfloat/ % % The IEEEtran \ifCLASSOPTIONcaptionsoff conditional can also be used -% later in the document, say, to conditionally put the References on a +% later in the document, say, to conditionally put the References on a % page by themselves. @@ -740,7 +739,7 @@ The authors would like to thank professor Carlos Par\'{e}s for having patience w %\end{thebibliography} % biography section -% +% % If you have an EPS/PDF photo (graphicx package needed) extra braces are % needed around the contents of the optional argument to biography to prevent % the LaTeX parser from getting confused when it sees the complicated