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


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.

Down-Mixed Signals

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).

Mathematical Description

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

  \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*} (1)

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

  \begin{equation*} \omega=m \omega_b+\delta \omega, \end{equation*} (2)

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*} (3)

The expression

  \begin{equation*} \sum^{N}_{n=0}\exp \left[-\frac{nt_b}{\tau}\right] \sin(n \delta \omega t_b), \end{equation*} (4)

where the order of summation has been reversed from in Eq. (3), 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

The plot below shows how the 'summation factor' given by Eq. (4) converges as the number of bunches is increased for different frequency offsets and different decay times.

convproof.png freqproof.png

The factors given by Eq. (4) for different values of τ and ω after 1000 bunches are shown below. For a given frequency, it can be seen that

 \begin{equation*} \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*}(5)

which are the values indicated on each graph by the dashed line.

taufactor.png freqfactor.png

Latex rendering error!! dvi file was not created.

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Topic revision: r8 - 10 Sep 2012 - FrancisCullinan

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