Abstract

INTRODUCTION
Most commonly used method for BPM is based on the evaluation of the signals induced on the pick-up electrodes (PUEs) by a circulating beam. Beam position is calculated from the signal amplitudes using delta over sum [1]. For the vertical plane BPM with two PUEs the equation is
Udown are the amplitudes, k is a scaling where Uup and determined by geometry. For a factor,   which 1S symmetrical system and beam in the center both signals have equal amplitudes and the corresponding position readback is zero. With two (or more) beams circulating inside the vacuum chamber we need to separate the signals and process them individually. For the colliders with beams moving in the opposite directions this task is solved by utilizing the striplines, which have directional properties. The signals appear on the different ports and conventional processing units can be utilized. This technique is not suitable for energy recovery linacs (ERL) where two or more beams co-propagate through a vacuum system and each beam has its own trajectory.
PROPOSED METHOD
For ERL with two co-propagating beams the time delay between accelerated and decelerated bunches is fixed by design and it becomes possible to employ the phase of the PUE signal to extract information on the position of each bunch. If bunches, separated by a flyby time At 12 , have different positions then each PUE sees different longitudinal "center of gravity" (see Fig. 1) and there is a phase shift between two signals. For a processing unit, utilizing signal processing at frequency ω, and small displacements of the first and the second bunches δ 1 and δ2 (Sδ1, Sδ2 << 1, where S=1/k is a sensitivity coefficient) we can write the linearized equations: *Work supported by Brookhaven Science Associates under Contract No. DE-AC02-98CH10886 with the U.S. DoE #pinayev@bnl.gov Figure 1: Signals induced on the pick-up electrodes by two bunches with different coordinates. Because of bunches' displacements the amplitudes of the induced voltages differ.
When both bunches have equal charges assumption for ERL) then we re-write Eq. 2 as amplitude PUE Substitution of the found amplitudes into the gives That means that using information on the amplitude w measuring the average position of the beam. Now we wil consider the phases of the signals. Using the same O assumptions we will find
and difference of the two phases gives us the difference between two positions
The processing frequency is usually equal to the RF frequency. Such a choice allows processing of any fill pattern because all bunches are separated by a multiple of the RF period. In the ERL distance between bunches can be different. It may be as small as half of the RF wavelength because the decelerated bunch should be in the opposite phase vs. accelerated one. Moreover, this distance can differ from the multiple of the half periods due to the shifts in the merge lines. Also, there is no revolution period in the ERL but only a round-trip time. Hence, choice of the processing frequency is s somewhat arbitrary and can be used for optimization of the system. There are frequencies that should be avoided: if cos(ωΔt 12) = 0 then there is no signal for the average position calculation and with sin(ωΔt 12) = 0 phases will be constant and difference in orbits can not be found. Therefore, the processing frequency should be in between + N ), where N is an integer. close to ω
ACCURACY
Modern digital beam position electronics have the required capability to process both amplitude and phase of the PUE signals [2]. To estimate the accuracy of the measurement let us consider vector diagram shown in Fig. 2. When the measurement error is not correlated with the signal the probable error will draw a circle around the end of the vector and r.m.s. phase error will be equal to the signal to noise ratio of the amplitude measurement Creative Commons Attribution 3.0 (C cc (
That means that achievable error in the difference of position is the same as for the average position. For the system described in [2] r.m.s. position error is 3 microns for k=10 mm, which means that noise is 3×10 –4 of the signal level and the ratio is close to the r.m.s. fluctuations of the phase measurements 0.01° (1.75×10 -4 radians). Figure 2: Vector diagram for the PUE signals. Two longer vectors indicate phases and amplitudes of the signal when two smaller ones show the noise amplitude. So far we have neglected phase shifts in the system associated with propagation of the signals the cables and in the processing electronics. The system transfer function will not be identical for all channels and a phase shift corresponding to the identical beam positions should be established. Otherwise these phase shifts will include a systematic error in the measurement of the difference of the two positions. Similarly, the unequal gain and/or losses in the channels generate systematic offset in the average position readback. Both phase and amplitudes can be calibrated using a RF generator with a splitter. In this case the whole chain from the cable to electronics can be characterized. For the phase calibration exclusively it is also possible to use exclusively the accelerated bunches (dumping the beam at high energy). In this case the BPM processing unit sees signal from a single bunch, and there are no ambiguities associated with the second bunch.
FOUR PUES SYSTEM
In the system with four pick-up electrodes, as shown in Fig. 3, the beam position is calculated with the modified equations
As before, for the small beam displacements we can write © 2012 by the respective authors — r U A = U 0 (1 − S x δx1 + S y δy1 ) sin ω(t + Δt12/2) + U 0 (1 — S x δx2 + S x δy2) sin ω(t — Δt 12/2) U B = U 0 (1 + S x δx1 + S y δy1) sin ω(t + Δt 12 /2) + - Δt 12 / 2) + Δt 22 /2) + At 12/2) + Δt 2/2) + U 0 (1 — S x δx2 — S x δx2) sin ω(t — Δt 12/2) (10) With the similar to the above calculations we can find that Eq. 9 is applicable for the calculation of the average beam position when the difference can be found from

SUPPRESSION OF THE CABLE DRIFTS
The temperature changes and other slow processes can change physical length of the cable or phase velocity of the signal in the cable. To suppress the drifts one can lower the processing frequency to the range where the sensitivity to the variations of delays is less. Unfortunately, it also lowers the sensitivity to the position difference due to the tan(ωΔt 12 /2) term. To have both high sensitivity to the differential beam position (and low noise) and to suppress influence of drifts we can convert down the PUE signals. In this case the sensitivity is defined by sum of the local oscillator frequency and processing frequency, while drifts are defined by processing frequency only. Local oscillator signal should be brought to the PUE with a single cable and split to required number of mixers with shortest connection.