Commit 7bffcef0 authored by snuverink_j's avatar snuverink_j
Browse files

replace sb and intertext

parent 1ac56c43
......@@ -54,15 +54,15 @@ t[i] = t[0] + (z[i] - z[0]) / (c * b[i])
\end{verbatim}
By doing so we assume that the kinetic energy, K, increases linearly and proportional to the maximal voltage. With this model for the progress of time we can calculate $\varphi$ according to Equation~\ref{rulelag}. Next a better model for the kinetic Energy can be calculated using
\begin{alltt}
K[i] = K[i-1] + q \(\Delta\)z[i](cos(\(\varphi\))(Ez[i-1](\(\Gamma\sb{21}\)[i] - \(\Gamma\sb{22}\)[i]) + Ez[i]\(\Gamma\sb{22}\)[i])
\(\,\,\)- sin(\(\varphi\))(Ez[i-1](\(\Gamma\sb{11}\)[i] - \(\Gamma\sb{12}\)[i]) + Ez[i]\(\Gamma\sb{12}\)[i])).
K[i] = K[i-1] + q \(\Delta\)z[i](cos(\(\varphi\))(Ez[i-1](\(\Gamma_{21}\)[i] - \(\Gamma_{22}\)[i]) + Ez[i]\(\Gamma_{22}\)[i])
\(\,\,\)- sin(\(\varphi\))(Ez[i-1](\(\Gamma_{11}\)[i] - \(\Gamma_{12}\)[i]) + Ez[i]\(\Gamma_{12}\)[i])).
\end{alltt}
With the updated kinetic energy the time model and finally a new $\varphi$, that comes closer to the actual maximal kinetic energy, can be obtained. One can iterate a few times through this cycle until the value of $\varphi$ has converged.
\section{Traveling Wave Structure}
\index{Autophase Algorithm!Traveling Wave Structure}
\begin{figure}
\includegraphics[angle=0,width=0.9\linewidth]{figures/field_crop}
\includegraphics[angle=0,width=0.9\linewidth]{figures/field_crop.png}
\caption{Field map 'FINLB02-RAC.T7' of type 1DDynamic}
\label{fig:tws}
\end{figure}
......@@ -88,7 +88,7 @@ For this example we find
V_\text{core} &= \frac{V_{0}}{\sin(2.0/3.0 \pi)} = \frac{2 V_{0}}{\sqrt{3.0}}\\
\varphi_\text{c1} &= \frac{\pi}{6}\\
\varphi_\text{c2} &= \frac{\pi}{2}\\
\varphi_\text{ef} &= - 2\pi \cdot(\text{\texttt{NUMCELLS}} - 1) \cdot \text{\texttt{MODE}} = 26\pi
\varphi_\text{ef} &= - 2\pi \cdot(\mathbf{NUMCELLS} - 1) \cdot \mathbf{MODE} = 26\pi
\end{align*}
\subsection{Alternative Approach for Traveling Wave Structures}
If $\beta$ doesn't change much along the traveling wave structure (ultra relativistic case) then $t(z,\varphi)$ can be approximated by $t(z,\varphi)=\frac{\omega}{\beta c}z + t_{0}$. For the example from above the energy gain is approximately
......@@ -142,7 +142,11 @@ where
E_z^{(1)}(z) \qquad & 0 \le z \le 3\cdot s\\
0 & \text{otherwise}
\end{cases}
\intertext{and}
\end{align*}
and
\begin{align*}
-\frac{\omega}{\beta c} \xi &= \varphi.
\end{align*}
Here we also used some trigonometric identities:
......
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