Difference: ClicAverageTheory (1 vs. 11)
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Theoretical Investigation of Multiple Bunch Signals | |||||||||||||||||||||||||||||||
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Revision 910 Sep 2012 - FrancisCullinan
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Theoretical Investigation of Multiple Bunch Signals | ||||||||
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Down-Mixed Signals | ||||||||
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| < < | The same thing can still be seen with the down-mixed signals. The dominant frequency is now at the nearest multiple of the bunching frequency (15 GHz) minus the local oscillator frequency (14.8 GHz). | |||||||
| > > | The same thing can still be seen with the down-mixed signals. The signals used to generate the plots above were run through the CTF3 prototype test electronics simulation that contains a mixer using an LO frequency of 14.8 GHz. The dominant frequency is now at the nearest multiple of the bunching frequency (15 GHz) minus the local oscillator frequency (14.8 GHz). | |||||||
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Mathematical Description | ||||||||
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Revision 810 Sep 2012 - FrancisCullinan
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Theoretical Investigation of Multiple Bunch Signals | ||||||||
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| It can be seen that in both cases, the dominant frequency in the spectrum is at 15 GHz. The signal tends to being periodic at the bunching frequency and oscillates at the cavity resonant frequency between bunches. The extra harmonics of the bunching frequency arise from the sawtooth-like variation in the signal amplitude and there is extra substructure to each peak coming from the start and end of the signal. In the second case, the resonant frequency is strong enough and far enough from an exact multiple of the bunching frequency to be resolved in the spectrum. | ||||||||
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| > > | Down-Mixed SignalsThe same thing can still be seen with the down-mixed signals. The dominant frequency is now at the nearest multiple of the bunching frequency (15 GHz) minus the local oscillator frequency (14.8 GHz). | |||||||
Mathematical DescriptionAssuming a uniform train of identical bunches with a single offset and no tilt, the signal coming from the first resonant dipole mode excited in a cavity beam position monitor (BPM) over time t, is given by | ||||||||
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Theoretical Investigation of Multiple Bunch Signals | ||||||||
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Theoretical Investigation of Multiple Bunch Signals | ||||||||
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| The factors given by Eq. (?? REFLATEX{4} not defined in eqn list ??) for different values of τ and ω after 1000 bunches are shown below. For a given frequency, it can be seen that %BEGINLATEX{label="5"}%\begin{equation*} | ||||||||
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| < < | \lim_{\tau \to \infty}\sum^{N}_{n=0}\exp \left[-\frac{nt_b}{\tau}\right] \sin (n \delta \omega t_b)=(\sinh (\delta\omega t_b))^{-1}. | |||||||
| > > | \lim_{\tau \to \infty}\sum^{N}_{n=0}\exp \left[-\frac{nt_b}{\tau}\right] \sin (n \delta \omega t_b)=(\sinh (\delta\omega t_b))^{-1} | |||||||
| \end{equation*}%ENDLATEX% | ||||||||
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| > > | which are the values indicated on each graph by the dashed line. | |||||||
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Revision 504 Sep 2012 - FrancisCullinan
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Theoretical Investigation of Multiple Bunch Signals | ||||||||
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| The factors given by Eq. (?? REFLATEX{4} not defined in eqn list ??) for different values of τ and ω after 1000 bunches are shown below. For a given frequency, it can be seen that %BEGINLATEX{label="5"}%\begin{equation*} | ||||||||
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| < < | \lim_{\tau \to \infty}\sum^{N}_{n=0}\exp \left[-\frac{nt_b}{\tau}\right] \sin(n \delta \omega t_b)=\frac{1}{\sin(\delta\omega t_b)}. | |||||||
| > > | \lim_{\tau \to \infty}\sum^{N}_{n=0}\exp \left[-\frac{nt_b}{\tau}\right] \sin (n \delta \omega t_b)=(\sinh (\delta\omega t_b))^{-1}. | |||||||
| \end{equation*}%ENDLATEX%
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Revision 403 Sep 2012 - FrancisCullinan
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Theoretical Investigation of Multiple Bunch Signals | ||||||||
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Convergence Tests | ||||||||
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| < < | The plot below shows how the 'summation factor' given by Eq. (?? REFLATEX{4} not defined in eqn list ??) converges as the number of bunches is increased. | |||||||
| > > | The plot below shows how the 'summation factor' given by Eq. (?? REFLATEX{4} not defined in eqn list ??) converges as the number of bunches is increased for different frequency offsets and different decay times. | |||||||
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| The factors given by Eq. (?? REFLATEX{4} not defined in eqn list ??) for different values of τ and ω after 1000 bunches are shown below. For a given frequency, it can be seen that %BEGINLATEX{label="5"}%\begin{equation*} | ||||||||
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Revision 303 Sep 2012 - FrancisCullinan
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| < < | It can be seen that in both cases, the dominant frequency in the spectrum is at 15 GHz. The signal tends to being periodic at the bunching frequency and oscillates at the cavity resonant frequency between bunches. The extra harmonics of the bunching frequency arise from the sawtooth-like variation in the signal amplitude and there is extra substructure to each peak coming from the start and end of the signal. In the second case, the resonant frequency is far enough from an exact multiple of the bunching frequency to be resolved in the spectrum. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| > > | It can be seen that in both cases, the dominant frequency in the spectrum is at 15 GHz. The signal tends to being periodic at the bunching frequency and oscillates at the cavity resonant frequency between bunches. The extra harmonics of the bunching frequency arise from the sawtooth-like variation in the signal amplitude and there is extra substructure to each peak coming from the start and end of the signal. In the second case, the resonant frequency is strong enough and far enough from an exact multiple of the bunching frequency to be resolved in the spectrum. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Mathematical Description | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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| < < | The multiple bunch signal in time t, is given by %BEGINLATEX% | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| > > | Assuming a uniform train of identical bunches with a single offset and no tilt, the signal coming from the first resonant dipole mode excited in a cavity beam position monitor (BPM) over time t, is given by %BEGINLATEX{label="1"}% | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| \begin{equation*} S=A\sum^{N}_{n=0} \exp\left[-\frac{t-nt_b}{\tau}}\right] \sin(\omega t - n \omega t_b). \end{equation*} %ENDLATEX% | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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| < < | where A is the magnitude of the multibunch signal, N is the number of bunches, tb is the bunch separation in time, ω is the frequency of the cavity BPM and τ is the signal decay time. By expressing the cavity frequency as %BEGINLATEX% | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| > > | where A is the amplitude of each single bunch excitation, N is the number of bunches, tb is the bunch separation in time, ω is the mode frequency and τ is the signal decay time. By expressing the mode frequency as %BEGINLATEX{label="2"}% | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| \begin{equation*} \omega=m \omega_b+\delta \omega, \end{equation*} %ENDLATEX% where m is an integer and ωb is the bunching frequency, sampling the signal once per bunch gives | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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| < < | %BEGINLATEX% | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| > > | %BEGINLATEX{label="3"}% | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| \begin{equation*} S(N) = A\sum^{N}_{n=0}\exp \left[-\frac{(N-n)t_b}{\tau}\right] \sin((N - n) \delta \omega t_b) \end{equation*} %ENDLATEX% | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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| < < | Therefore, if the expression %BEGINLATEX% | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| > > | The expression %BEGINLATEX{label="4"}% | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| \begin{equation*} \sum^{N}_{n=0}\exp \left[-\frac{nt_b}{\tau}\right] \sin(n \delta \omega t_b), \end{equation*} %ENDLATEX% | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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| < < | where the order of the summation has been reversed, converges in the limit of a large number of bunches N, the phase advance for each bunch will be zero and the signal will oscillate at an exact multiple of the bunching frequency. This is necessarily true since the sine function is oscillating and the exponential term forms a monotonically decreasing sequence with limit 0. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| > > | where the order of summation has been reversed from in Eq. (?? REFLATEX{3} not defined in eqn list ??), converges in the limit of an infinite number of bunches N, since the sine function is oscillating and the exponential factor is a monotonically decreasing sequence with limit 0. The phase advance per bunch, therefore, tends to zero and the dominant signal frequency tends to an exact multiple of the bunching frequency. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Convergence Tests | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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| > > | The plot below shows how the 'summation factor' given by Eq. (?? REFLATEX{4} not defined in eqn list ??) converges as the number of bunches is increased.
Latex rendering error!! dvi file was not created. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Revision 231 Aug 2012 - FrancisCullinan
Revision 131 Aug 2012 - FrancisCullinan
View topic | History: r11 < r10 < r9 < r8 | More topic actions...   Copyright © 2008-2023 by the contributing authors. All material on this collaboration platform is the property of the contributing authors. Ideas, requests, problems regarding RHUL Physics Department TWiki? Send feedback | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
![\begin{equation*} \lim_{\tau \to \infty}\sum^{N}_{n=0}\exp \left[-\frac{nt_b}{\tau}\right] \sin(n \delta \omega t_b)=\frac{1}{\sin(\delta\omega t_b)}. \end{equation*}](/twiki/pub/PP/JAI/ClicAverageTheory/latex2b21dd9df06a835ca9ade636d694788e.png)
![\begin{equation*} S=A\sum^{N}_{n=0} \exp\left[-\frac{t-nt_b}{\tau}}\right] \sin(\omega t - n \omega t_b). \end{equation*}](/twiki/pub/PP/JAI/ClicAverageTheory/latexa6921d0eae2789bfa1b25e085c4583e3.png)
![\begin{equation*} S(N) = A\sum^{N}_{n=0}\exp \left[-\frac{(N-n)t_b}{\tau}\right] \sin((N - n) \delta \omega t_b) \end{equation*}](/twiki/pub/PP/JAI/ClicAverageTheory/latexf6f976a570802a3b53969313924c3e22.png)
![\begin{equation*} \sum^{N}_{n=0}\exp \left[-\frac{nt_b}{\tau}\right] \sin(n \delta \omega t_b), \end{equation*}](/twiki/pub/PP/JAI/ClicAverageTheory/latex5669184c2da80c2a402861fa4db1d158.png)