Theoretical Investigation of Multiple Bunch Signals

The multiple bunch signal was simulated from simply adding decaying complex oscillators at the separation of the bunch length offset by the bunch separation. The simulation parameters were:

Parameter Value
Number of bunches 150
Charge per bunch/nC 0.05
Sensitivity V/nC/mm 14.8
Decay time/ns 5

The simulation was performed with two different resonant frequencies, 15.01 GHz and 15.50 GHz and in both cases, demodulated using a 15.00 GHz digital local oscillator in order to determine the phase.

plus10MHz.png plus10MHz_zoomamp.png
plus10MHz_zoom.png

plus500MHz.png

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.

Mathematical Description

The multiple bunch signal in time t, is given by

 \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*}
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
 \begin{equation*} \omega=m \omega_b+\delta \omega, \end{equation*}
where m is an integer and ωb is the bunching frequency, sampling the signal once per bunch gives
  \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*}
Therefore, if the expression
 \begin{equation*} \sum^{N}_{n=0}\exp \left[-\frac{nt_b}{\tau}\right] \sin(n \delta \omega t_b), \end{equation*}
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.

Convergence Tests


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Topic revision: r2 - 31 Aug 2012 - FrancisCullinan

 
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