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| 1 | +\sectionframe{Original Model} |
| 2 | +\section{OG} |
| 3 | + |
| 4 | +\begin{frame}{Model Origin} |
| 5 | + \begin{itemize} |
| 6 | + \item DC/AC power converter |
| 7 | + \item The converter switches $\Rightarrow$ models are piecewise-smooth and discontinuous |
| 8 | + \pause \vspace{1em} |
| 9 | + \item We focus on the time-discrete model |
| 10 | + \item It maps phase of the last switch $\theta_n$ to the phase of the next switch $\theta_{n+1}$ |
| 11 | + \end{itemize} |
| 12 | +\end{frame} |
| 13 | + |
| 14 | +\begin{frame}{Model Definition (1/2)} |
| 15 | + \vspace{-2.0em} |
| 16 | + \begin{align} |
| 17 | + \theta_{n+1} & = F(\theta_n) \mod 2 \pi |
| 18 | + \\ |
| 19 | + F(\theta) & = \begin{cases} |
| 20 | + F_1(\theta) & \text{if } q \cdot \cos(\theta) > 0 \\ |
| 21 | + F_2(\theta) & \text{if } q \cdot \cos(\theta) < 0 |
| 22 | + \end{cases} |
| 23 | + \\ |
| 24 | + F_1(\theta) & = \begin{cases} |
| 25 | + \theta + z_{L_+} + z_1 & \text{if } z_{L_+} < z_{L_0} \\ |
| 26 | + \theta + z_{L_0} + z_2 & \text{if } z_{L_+} > z_{L_0} |
| 27 | + \end{cases} |
| 28 | + \\ |
| 29 | + F_2(\theta) & = \begin{cases} |
| 30 | + \theta + z_{R_+} + z_3 & \text{if } z_{R_+} < z_{R_0} \\ |
| 31 | + \theta + z_{R_0} + z_4 & \text{if } z_{R_+} > z_{R_0} |
| 32 | + \end{cases} |
| 33 | + \end{align} |
| 34 | + |
| 35 | + \pause |
| 36 | + \vspace{2em} |
| 37 | + This looks ok, but how are these values defined? |
| 38 | + \begin{align*} |
| 39 | + z_1, z_2, z_3, z_4, z_{L_+}, z_{L_-}, z_{R_+}, \text{ and } z_{R_0} |
| 40 | + \end{align*} |
| 41 | +\end{frame} |
| 42 | + |
| 43 | +\begin{frame}{Model Definition (2/2)} |
| 44 | + \vspace{-1em} |
| 45 | + The smallest non-negative solutions to the following implicit equations |
| 46 | + \begin{subequations} |
| 47 | + \begin{align} |
| 48 | + (q \cdot \cos(\theta) + \mu \cdot \chi) \cdot e^{\lambda \cdot z_{L_+}} |
| 49 | + & = q \cdot \cos(\theta + z_{L_+}) + \chi \label{equ:setup.og.def.impl.1.A} \\ |
| 50 | + (q \cdot \cos(\theta) + \mu \cdot \chi) \cdot e^{\lambda \cdot z_{L_0}} |
| 51 | + & = q \cdot \cos(\theta + z_{L_0}) - \chi \\ |
| 52 | + (q \cdot \cos(\theta) - \mu \cdot \chi) \cdot e^{\lambda \cdot z_{R_+}} |
| 53 | + & = q \cdot \cos(\theta + z_{R_+}) - \chi \\ |
| 54 | + (q \cdot \cos(\theta) - \mu \cdot \chi) \cdot e^{\lambda \cdot z_{R_0}} |
| 55 | + & = q \cdot \cos(\theta + z_{R_0}) + \chi \label{equ:setup.og.def.impl.1.D} |
| 56 | + \\ |
| 57 | + (q \cdot \cos(\theta + z_{L_+}) + \chi + 1) \cdot e^{\lambda \cdot z_1} - 1 |
| 58 | + & = q \cdot \cos(\theta + z_{L_+} + z_1) + \mu \cdot \chi \label{equ:setup.og.def.impl.2.A} \\ |
| 59 | + (q \cdot \cos(\theta + z_{L_0} + z_2) - \chi - 1) \cdot e^{\lambda \cdot z_2} + 1 |
| 60 | + & = q \cdot \cos(\theta + z_{L_0} + z_2) - \mu \cdot \chi \\ |
| 61 | + (q \cdot \cos(\theta + z_{R_+}) + \chi + 1) \cdot e^{\lambda \cdot z_3} - 1 |
| 62 | + & = q \cdot \cos(\theta + z_{L_+} + z_1) + \mu \cdot \chi \\ |
| 63 | + (q \cdot \cos(\theta + z_{R_0} + z_4) - \chi - 1) \cdot e^{\lambda \cdot z_4} + 1 |
| 64 | + & = q \cdot \cos(\theta + z_{R_0} + z_2) - \mu \cdot \chi \label{equ:setup.og.def.impl.2.D} |
| 65 | + \end{align} |
| 66 | + \end{subequations} |
| 67 | + \begin{flushright} |
| 68 | + Definition from \cite{akyuz2022} |
| 69 | + \end{flushright} |
| 70 | +\end{frame} |
| 71 | + |
| 72 | +\begin{frame}{Unusual Bifurcation Structure} |
| 73 | + \begin{figure} |
| 74 | + \only<1>{ |
| 75 | + \includegraphics[width=0.45 \textwidth]{../Figures/2/2.3/result.png} |
| 76 | + } |
| 77 | + \only<2>{ |
| 78 | + \stackunder[5pt]{ |
| 79 | + \includegraphics[width=0.3 \textwidth]{../Figures/2/2.4a/result.png} |
| 80 | + }{$A:\:\A^3\B^3\C^3\D^3$} |
| 81 | + \stackunder[5pt]{ |
| 82 | + \includegraphics[width=0.3 \textwidth]{../Figures/2/2.4b/result.png} |
| 83 | + }{$B:\:\A^3\B^3\C^2\D^4,\:\A^2\B^4\C^3\D^3$} |
| 84 | + \stackunder[5pt]{ |
| 85 | + \includegraphics[width=0.3 \textwidth]{../Figures/2/2.4c/result.png} |
| 86 | + }{$C:\:\A^2\B^4\C^2\D^4$} |
| 87 | + } |
| 88 | + \end{figure} |
| 89 | + \pause |
| 90 | + \vspace{1em} |
| 91 | + Symmetry $F(\theta + \pi) = F(\theta) + \pi \mod 2\pi$ \hfill \cite{akyuz2022} |
| 92 | +\end{frame} |
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