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\title{\Huge 4 Gs/s Excitation System for SPS Machine Physics}


%\author{O. Turgut, C.H. Rivetta, Sho Uemura, J.D. Fox (SLAC), W. Hofle (CERN)}
\author[1,2]{Sho Uemura}
\author[2,3]{Ozhan Turgut}
\author[2]{Claudio H. Rivetta}
\author[2]{Ivo Rivetta}
\author[2]{Jeff Olsen}
\author[2,4]{John D. Fox}

\affil[1]{Department of Physics, Stanford University}
\affil[2]{SLAC}
\affil[3]{Department of Aeronautics and Astronautics, Stanford University}
\affil[4]{Department of Applied Physics, Stanford University}
%\institute{Stanford University}
%\conference{2011 INTERNATIONAL PARTICLE ACCELERATOR CONFERENCE, SEPTEMBER 4 - SEPTEMBER 9 , 2011, San Sebastian, Spain}
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\begin{document}
\maketitle

%\large 
\begin{multicols}{3}
\section*{Abstract}
{We present an electronic excitation system used to study single-bunch vertical dynamics in the CERN SPS. The system uses a 4 GS/sec/ fast digital excitation system, in conjunction with 400 W of 20 - 1000 MHz amplifiers and a vertical kicker to excite vertical motion. A related beam receiver system digitizes the beam response on 4 strip-line electrodes and we record the beam response. Our long-term goal is to design a wide-band feedback system to control/stabilize the bunch in the presence of e-cloud and transverse mode coupling instabilities (TMCI). This initial work validates the back-end functions and is part of a dynamics effort to characterize how the presence of electron clouds impacts the proton bunch dynamics.}
	%\columnseprule=0mm
%\section*{Motivation}
%
%\begin{itemize}
%
%\item The SPS at high intensities exhibits transverse single-bunch instabilities with signatures consistent with transverse mode coupling and e-cloud driven instabilities (Fig. ~\ref{fig:disp1}~\ref{fig:disp2} and~\ref{fig:disp3}).
%
%%\begin{figure}[h]
%%\centering
%%\includegraphics[width=0.9\columnwidth,height=65mm]{WEP199f1.eps}
%%\caption{Measured tune shift for turns 100--200 due to e-cloud .}
%%\label{fig:disp1}
%%\end{figure}
%%
%%\begin{figure}[h]
%%\centering
%%\includegraphics[width=0.9\columnwidth,height=65mm]{verticalDisp.eps}
%%\caption{Measured vertical displacement for a stable bunch ( data from a single turn).}
%%\label{fig:disp2}
%%\end{figure}
%%
%%\begin{figure}[h]
%%\centering
%%\includegraphics[width=0.9\columnwidth,height=65mm]{verticalDisp2.eps}
%%\caption{Measured instability in vertical motion illustrating head -- tail displacement.}
%%\label{fig:disp3}
%%\end{figure}
%
%\item A high frequency feedback system is a promising approach to control transverse bunch instabilities induced  by e-clouds and transverse mode coupling instability (TMCI) anticipated for high current SPS and LHC operations ~\cite{ecloud10}.
%\item The application of feedback control to stabilize the bunch is challenging because it requires a bandwidth sufficient to sense the transverse position and apply correcting fields to multiple sections of a nanosecond-scale bunch.
%\item The \textit{model-based design technique} allows addressing problems associated with designing complex control systems and provides an efficient approach to quantify stability margins of the system, impact of external perturbations, uncertainties and parameter variations of the system, etc.
%
%\end{itemize}


%\section*{Basic Concepts : System Identification}
%
%\begin{figure}
%\centering
%\includegraphics[width=0.9\columnwidth,height=65mm]{WEP199f6.eps}
%\caption{Block diagram for system identification process}
%\label{fig:identblock}
%\end{figure}
%
%\begin{itemize}
%%\item System identification uses statistical methods to build mathematical models of dynamical systems from observed input and output data.
%
%\item A discrete time-domain signal $u(k)$, with enough frequency content (\textit{persistent excitation}) to drive the dominant modes to be identified in the system is applied and the output signal $y(k)$ in response to that excitation is measured.
%
%\item A discrete dynamical system can be represented by following representation,
%
%\begin{equation}
%\begin{array}{1}
%\displaystyle x_{k+1} = f_{k} (x_{k}, u_{k}, \theta)\\ \\
%\displaystyle y_{k} = g_{k} (x_{k}, u_{k}, \theta)
%\end{array} 
%\label{eq:nonlinear}
%\end{equation}
%\\
%
%\item If system is linear then system can be represented by following state space representation,
%
%\begin{equation}
%\begin{array}{2}
%\displaystyle x_{k+1} = Ax_{k} + Bu_{k}\\ \\
%\displaystyle y_{k} = Cu_{k} + Du_{k}
%\end{array} 
%\label{eq:linear}
%\end{equation}
%
%\item The main goal in identification process is to define the system parameters embedded in ${A,B,C,D}$  by using the measured input and output data.
%
%\end{itemize}
\section*{Identification of Bunch Dynamics for CERN SPS Ring}

\begin{itemize}

\item The identification of the individual bunch dynamics in the SPS ring involves the development of wide-bandwidth hardware to be able to drive the individual sections of the bunch.

\item Starting with August 2011 MD, we've started to use a deterministic excitation system that can create an arbitrary synchronized excitation signal.

\item This approach allows new beam diagnostic techniques which give us an opportunity to validate nonlinear macroparticle simulation codes with real measurements taken from machines.


%\end{itemize}
%
%\begin{figure}[h]
%\centering
%\includegraphics[width=0.9\columnwidth,height=65mm]{WEP199f5-eps-converted-to.pdf}
%\caption{Block Diagram.}
%\label{fig:blockdiagram}
%\end{figure}

%\subsection* { Simulation Results for Time Varying Parameters Identification}
%\begin{itemize}
%
%\item The nonlinear effects and synchrotron motion of the particles within the bunch requires a reduced model that it is not linear to represent the dynamics of the bunch. The simplest model to include this effect is a linear-time-variant model (LTV).
%
%\item An LTV approach has time-varying  matrices {A,B,C} in eqn. \ref{eq:linear}, with the representation in state space, 
%
%\begin{equation}
%\begin{array}{2}
%\displaystyle x_{k+1} = A_{k}x_{k} + B_{k}u_{k}\\ \\
%\displaystyle y_{k} = C_{k}u_{k}
%\end{array} 
%\label{eq:LTV}
%\end{equation}
%where the matrices ${A(k),B(k),C(k)}$ are slowly varying with $k$.
%
%\item Based on the set of multi-input/multi-output (MIMO) signals obtained from  simulation codes or machine measurements, we develop an identification algorithm based on matrix fraction description (MFD) ~\cite{identification}.
%\item The class of observable canonical forms ~\cite{identification} are used to represent ${A,B,C,D}$. 
%\item Results from the identification algorithm are depicted in Fig. ~\ref{fig:results}, where the original data from C-MAD code and estimated by the identification algorithm are compared. The vertical motion of 3 particular slices of the bunch calculated by C-MAD code is depicted by the solid line, while the output data generated by the reduced model is shown in dotted lines for 300 turns in the SPS machine.  
%\item In this example, the window technique is used to estimate different sets of parameters corresponding to each of the 5 time windows over the interval.
%
%
%\usepackage{subfig}
% 
%\begin{figure}
%\centering
%\subfloat{\label{fig:gull}\includegraphics[width=0.9\textwidth,height=65mm]{results2.eps}}                
%\subfloat{\label{fig:tiger}\includegraphics[width=0.9\textwidth,height=65mm]{results1.eps}}
%\caption{Results of modified system identification technique to capture time varying system parameters, comparing the vertical motion of different bunch slices. \textit{Solid lines: Vertical displacement from C-MAD}, \textit{dotted lines}: Vertical displacement estimated by the reduced model. Reduced model reproduces almost same output signals for randomly picked 6 samples ( each sample is a slice in a bunch )}
%\label{fig:results}
%\end{figure}
%
%\begin{figure}[h]
%\centering
%\includegraphics[width=0.9\columnwidth,height=110mm]{results2.eps}
%\caption{Results of modified system identification technique to capture time varying system parameters, comparing the vertical motion of different bunch slices. \textit{Solid lines: Vertical displacement from C-MAD}, \textit{dotted lines}: Vertical displacement estimated by the reduced model.}
%\label{fig:identblock}
%\end{figure}
%
%\begin{figure}[h]
%\centering
%\includegraphics[width=0.9\columnwidth,height=110mm]{results1.eps}
%\caption{Results of modified system identification technique to capture time varying system parameters. Reduced model reproduces almost same output signals for randomly picked 3 samples ( each sample is a slice in a bunch )}
%\label{fig:results}
%\end{figure}
%
\end{itemize}

\subsection*{Hardware Development for Excitation System}
%

To allow arbitrary wide-band synchronized excitation we have developed a system (figure~\ref {fig:channel}) capable of driving user-defined 4 GS/sec. waveforms on selected bunches during the SPS machine cycle. This was installed at the SPS in July/August 2011.

The system uses a synchronized master oscillator to drive data streams at 4 GS/sec. from a local memory.

\begin{figure}[h]
\centering
\includegraphics[width=0.9\columnwidth,height=100mm]{Blockdiagram}
\caption{Hardware configuration for excitation and measurements .}
\label{fig:channel}
\end{figure}

\begin{figure}[h]
\centering
\includegraphics[width=0.9\columnwidth,height=110mm]{DSC_0033}
\caption{Excitation system with synchronised master oscillator developed at SLAC for use at CERN}
\label{fig:excitationbox}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.9\columnwidth,height=110mm]{MG_0057}
\caption{4 GS/sec. doublet signal showing 250 ps transition }
\label{fig:signal}
\end{figure}

%\vspace{-2.5cm}

A trigger and synchronization system allows the selection of a single or multiple bunches in the SPS with respect to a revolution fiducial, and for each bunch a 16 or 32 sample/turn excitation (15,000 turn length) is generated after synchronization to an injection trigger.

\begin{figure}[h]
\centering
\includegraphics[width=0.9\columnwidth,height=100mm]{GUI2}
\includegraphics[width=0.9\columnwidth,height=100mm]{GUI1}
\caption{GUI for excitation system. Random signal (top) and phase modulated signal at a tune of 0.185 (bottom).}
\label{fig:GUI}
\end{figure}

The baseband D/A signal is amplified by four 100W 20--1000 MHz RF amplifiers, which drive an existing wide-band pickup electrode as a vertical kicker.

\begin{figure}[h]
\centering
\includegraphics[width=0.9\columnwidth,height=145mm]{cernamps006}
\caption{Power amplifiers in test at CERN.}
\label{fig:amps}
\end{figure}

\begin{figure}[h]
\centering
\includegraphics[width=0.9\columnwidth,height=140mm]{tunnel1}
\caption{Four 100W amplifiers are placed in the tunnel under a bend magnet.}
\label{fig:tunnel}
\end{figure}

\begin{figure}[h]
\centering
\includegraphics[width=0.9\columnwidth,height=130mm]{kicker}
\caption{An exponentially-tapered pickup array (BPW 319.01) is used as a kicker in the tunnel for our MD measurements. A similar array (BPW 321) is used as a pickup.}
\label{fig:kicker}
\end{figure}


The basic goal of the first experiment was to excite different internal modes to test our excitation system in the CERN SPS tunnel. We were successful in exciting mode 0 (barycentric motion, in Figure~\ref{fig:mode0}) and mode 1 (head-tail motion, in Figure~\ref{fig:mode1}).

\begin{figure}[h]
\centering
\includegraphics[width=0.9\columnwidth,height=100mm]{Mode0}
\caption{Excitation of mode 0 motion from a excitation file with all bunch slices in phase. Plot shows vertical dipole moment for 25 turns.}
\label{fig:mode0}
\end{figure}

\begin{figure}[h]
\centering
\includegraphics[width=0.9\columnwidth,height=100mm]{Mode1}
\caption{Excitation of head-tail (mode 1) motion from an excitation file with 180 degrees phase difference between head and tail of bunch. Plot shows vertical dipole moment for 25 turns.}
\label{fig:mode1}
\end{figure}

This is a very promising step that shows we can excite the beam even with the existing kicker structure in the tunnel. 

\begin{figure}[h]
\centering
\includegraphics[width=\columnwidth]{110803_172151_evo_spectrogram_80}
\caption{Spectrogram showing the interaction of the beam modes (wavy bands) with the excitation (red line at 0.185). The different beam modes are excited at different times as tune shifts bring each of the modes into resonance with the excitation.}
\label{fig:spectrogram}
\end{figure}



\section*{Future Work and Conclusion}

The immediate next step is to excite the beam again with band-limited noise and use the excitation/response measurements in the identification process. These experiments and data analysis are necessary adjuncts to the design and specification of a useful wide-band feedback system, and help specify critical issues of necessary bandwidth, required amplifier power, specifications of kicker structure parameters, and the overall strategy for a control filter.

Our group is fabricating a proof-of-principle 4 GS/sec. feedback control system, and developing a  wide-band vacuum kicker structure for tests at the SPS in 2013.
\section*{Acknowledgement}
		The authors would like to thank U. Wehrle, B. Salvant (CERN), M. Pivi (SLAC) for their help, and the SLAC ARD, CERN AB RF departments and the US LHC Accelerator Research Program (LARP) for the support.
\\
				
		\textbf{Work is supported by the U.S. Department of Energy under contract  DE-AC02-76SF00515 and the US LHC Accelerator Research Program LARP}

%\begin{thebibliography}{14}   % Use for  1-9  references
%%\begin{thebibliography}{99} % Use for 10-99 references
%
%\bibitem{epac08} G. Rumolo et al., Experimental Study of Electron Cloud Instability in the CERN-SPS, EPAC 08, Genoa Italy, pp TUPP065 June 2008.
%%\bibitem{ecloud} G. Arduini et al. 31st Advanced ICFA Beam Dynamics Workshop, Napa, CA, USA, CERN-2005-001
%\bibitem{ecloud} B. Salvant et al., Transverse Mode-coupling Instability in the CERN SPS: Comparing MOSES Analytical Calculations and HEADTAIL Simulations with Experiments in the SPS, EPAC08-TUPP067, CERN-AB-2008-018, Jun 24, 2008. 3pp.
%\bibitem{pac10} J.D. Fox et al.,  SPS Ecloud Instabilities - Analysis of Machine Studies and Implications for Ecloud Feedback, IPAC 10, Kyoto, Japan / May 23-28 2010.
%
%\bibitem{pac11} O.Turgut et al., Estimation of Ecloud and TMCI Driven Vertical Instabilitiy Dynamics From SPS MD Measurements - Implications For Feedback Control, PAC 11, NY, USA.
%
%\bibitem{ecloud10} C. Rivetta et al.,  Feedback Control of SPS E-cloud / Transverse Mode Coupling Instabilities, Ecloud 10, Cornell University, Ithaca, NY, USA.
%
%\bibitem{cmad} M.T.F. Pivi, C-MAD: A new self-consistent parallel code to simulate the electron cloud build-up and instabilities, PAC07, Albuquerque, New Mexico, USA
%
%\bibitem{warp} Jean-Luc Vay et al., Update on Electron-Cloud Simulations Using the Package WARP-POSINST, PAC 09 FR5RFP078.
%
%\bibitem{ident} L. Ljung, Perspectives on System Identification, July 2008, \url{http://www.control.isy.liu.se/~ljung/seoul2dvinew/plenary2.pdf}
%
%\bibitem{identification} N.F. Al-Nuthairi, S. Bingulac and M. Zribi, Identification of discrete - time MIMO systems using a class of observable canonical-form.
%
%\end{thebibliography}


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