latex-papers-slides/differential-equations/electrical.tex

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\documentclass[journal]{IEEEtran}


\usepackage[style=numeric-comp,backend=biber,url=false]{biblatex}
\addbibresource{electrical.bib}

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\begin{document}
\boldmath

\title{Dynamics in Electrical Systems}

\author{Jakub~Hanak,
        Peter~Babic\\% <-this % stops a space
        Dept. of Computers and Informatics, FEI TU of Kosice\\%
        Slovak Republic\\%
        jakub.hanak2@gmail.com, babicpet@gmail.com%
}

% The paper headers
\markboth{Paper for Modeling course, June~2015}%
{}


% make the title area
\maketitle


\begin{abstract}
The abstract goes here.
\end{abstract}

% Note that keywords are not normally used for peerreview papers.
\begin{IEEEkeywords}
differential, dynamics, electrical, equation, modeling, ordinary, system
\end{IEEEkeywords}



\IEEEpeerreviewmaketitle



\section{Introduction}

\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}
\textbf{Dynamical systems} are mathematical objects used to model physical phenomena whose state (or instantaneous description) changes over time. These models are used in financial and economic forecasting, environmental modeling, medical diagnosis, industrial equipment diagnosis, and a host of other applications.

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. 

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 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}

There are two types of differential equations. \textit{Ordinary differential equations} (PDE) and \textit{Partial differential equations}. Our study won't go into further detail about PDE and will stay focused mainly on ODE.


\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. 



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.

\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}. 

\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.

\begin{figure}[ht!]
	\centering
	\includegraphics[width=.6\linewidth]{lcycle_stable}
	\caption{Stable limit cycle. Trajectories spiral towards it.}
	\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 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
	\includegraphics[width=.55\linewidth]{lcycle_unstable}
	\caption{Unstable limit cycle. Trajectories spiral away from it.}
	\label{f:lc_unst}
\end{figure}

Of the countless examples that could be given, we mention only a few: the beating of a heart; the periodic ring of a pace maker neuron; daily rhythms in human body temperature and hormone secretion; chemical reactions that oscillate spontaneously; and dangerous self-excited vibrations in bridges and airplane wings. In each case, there is a standard oscillation of some preferred period, waveform, and amplitude. Oscillations are important part of electronics \cite{oscillations}, too.

If the system is perturbed slightly, it always returns to the standard cycle. Limit cycles are inherently nonlinear phenomena; they cant occur in linear systems \cite{strogatz2008nonlinear}.

\begin{figure}[ht!]
	\centering
	\includegraphics[width=.5\linewidth]{lcycle_hstable}
	\caption{Half-stable (or semi-stable) limit cycle. Attract trajectories from one side and repel them from other side.}
	\label{f:lc_hst}
\end{figure}

\subsection{Damping}
Mentioning damping is important mainly because, in a real world, oscillations eventually stop, due to Newton's law of Thermodynamics (the frictional force). In electronics, there is no ideal oscillator, too - small amount of energy is lost every cycle, due to electric resistance.

Generally, the damping is linear either linear or nonlinear. As a rule of thumb, the linear one is easily modeled mathematically, obeying known rules, while the nonlinear one is not \cite{institute1989estimation}. There are some use cases, where nonlinear damping is advantageous, but the research is still ongoing about this topic.

\section{Conducted Studies}
sadfa

\subsection{Li\'{e}nard Equation}
A nonlinear second-order ordinary differential equation
\begin{equation}
\label{eq:lnrd}
y''+f(x)x'+x=0
\end{equation}

This equation describes the dynamics of a system with one degree of freedom in the presence of a linear restoring force and nonlinear damping. The function $f$ has properties
\begin{align*}
f(x)&<0\quad for\,small\,|x| \\
f(x)&>0\quad for\,large\,|x|
\end{align*}
that is, if for small amplitudes the system absorbs energy and for large amplitudes dissipation occurs, then in the system one can expect self-exciting oscillations.

Li\'{e}nard equation was intensely studied as it can be used to model oscillating circuits. Under certain additional assumptions Li\'{e}nard's theorem guarantees the uniqueness and existence of a limit cycle for such a system.

\subsection{Van der Pol Equation}
One of the most well-known oscillator model in dynamics is \textbf{Van der Pol oscillator}, which is a special case of Li\'{e}nard's equation \eqref{eq:lnrd} and is described by a differential equation
\begin{equation}
\label{eq:vdp}
y''-\mu\left(1-y^2\right)y'+y=0
\end{equation}
where $y$ is the dynamical variable and $\mu>0$ is a parameter. If $\mu=0$, then the equation reduces to the equation of simple harmonic motion
$$y''+y=0$$
The $\mu$ parameter determines the shape of the limit cycle. As it approaches 0, it gets the shape of a circle. On the other hand, increasing the paramter, involves sharpening of the curves.

The Van der Pol equation \eqref{eq:vdp} arises in the study of circuits containing vacuum tubes (triode) and is derived from earlier, Rayleigh equation \cite{nahin2001science}, known also as Rayleigh-Plesset equation - an ordinary differential equation explaining the dynamics of a spherical bubble in an infinite body of liquid.

Van der Pol oscillator is \textbf{self-sustainable}, \textbf{relaxation} oscillator. Self-sustainability in this context means, that the energy is fed into small oscillations and removed from large oscillations. Relaxation means, that the energy is gradually accumulating over time and then quickly released (relaxed). In electronics jargon, the relaxation oscillator is also called a \textit{free-running} oscillator. As already explained, it does not require neither one (monostable), nor two (bistable) inputs for transitioning between states, it "runs" by itself, thus free-running.

\subsection{Periodicity in Van der Pol's Oscillator}
Li\'{e}nard's theorem can be used to prove that the system described by  Van der Pol equation \eqref{eq:vdp} has a limit cycle \cite{sternberg2014dynamical}. If we want to visualize it, the one-dimensional form of equation must be first \textit{transformed} to the two-dimensional form. Applying the Li\'{e}nard transformation $$y=x-\frac{x^3}{3}-\frac{\dot x}{\mu}$$ where dot indicates the time derivative, the system can be written in it's two-dimensional form \cite{kaplan2012understanding}:
\begin{align*}
\dot x &= \mu \left(x-\frac13 x^3 -y\right) \\
\dot y &= \frac{1}{\mu} x
\end{align*}

However, this form is not well-known. Far common form uses the transformation $y=\dot x$, that yields
\begin{align*}
\dot x &= y \\
\dot y &= \mu\left(1-x^2\right)y-x
\end{align*}
which can be plotted onto direction field, as shown on \cref{f:vdp_m}. It is possible to see the stable limit cycle as well as trajectories from both sides attracted towards it.

The Van der Pol oscillator can be forced too, however, this work does not aim to investigate further in this direction.

\begin{figure}[ht!]
	\centering
	\includegraphics[width=.85\linewidth]{vdp_maxima}
	\caption{Phase portrait of the unforced Van der Pol oscillator, showing a limit cycle and the direction field Parameter $\mu=1$. The wxMaxima computing software was used for this purpose. }
	\label{f:vdp_m}
\end{figure}


\section{Applications in Electronics}
%The modeling met some more advanced applications in electronics field, too. We will briefly summarize some of them, in this section.

\subsection{Josephson Junctions}
\textbf{Josephson junctions} are superconducting devices that are capable of generating voltage oscillations of extraordinary high frequency, typically 10\textsuperscript{10} - 10\textsuperscript{11} cycles per second \cite{van1981principles}. They consist of two superconducting layers, separated by a very thin insulator that weakly couples them, as shown on \cref{f:jjunc}.

\begin{figure}[ht!]
	\centering
	\includegraphics[width=.4\linewidth]{jjunc}
	\caption{The physical structure of a Josephson Junction. Shown for ilustration purposes.}
	\label{f:jjunc}
\end{figure}

Although quantum mechanics is required to explain the origin of the Josephson effect, we can nevertheless dive into dynamics of Josephson junctions in classical terms. They have been particularly useful for \textit{experimental} studies of nonlinear dynamics, because the equation governing a single junction resembles the one of a pendulum \cite{strogatz1994nonlinear}.

Josephson junctions are used to detect extremely low electric potentials and are used for instance, to detect far-infrared radiation from distant galaxies. They are also formed to arrays, because there is a great potential seen in this configuration, however, all the effects are yet to be fully understood.

\section{Conclusion}
The conclusion goes here.


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The authors would like to thank...


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% \includegraphics command within an optional argument. (You could create
% your own custom macro containing the \includegraphics command to make things
% simpler here.)
%\begin{IEEEbiography}[{\includegraphics[width=1in,height=1.25in,clip,keepaspectratio]{mshell}}]{Michael Shell}
% or if you just want to reserve a space for a photo:

%\begin{IEEEbiography}{Michael Shell}
%Biography text here.
%\end{IEEEbiography}

% if you will not have a photo at all:
%\begin{IEEEbiographynophoto}{John Doe}
%Biography text here.
%\end{IEEEbiographynophoto}

% insert where needed to balance the two columns on the last page with
% biographies
%\newpage

%\begin{IEEEbiographynophoto}{Jane Doe}
%Biography text here.
%\end{IEEEbiographynophoto}

% You can push biographies down or up by placing
% a \vfill before or after them. The appropriate
% use of \vfill depends on what kind of text is
% on the last page and whether or not the columns
% are being equalized.

%\vfill

% Can be used to pull up biographies so that the bottom of the last one
% is flush with the other column.
%\enlargethispage{-5in}