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A previous formula for longitudinal beam impedance was developed by H. Tsutsui and consists of 13 terms. The formula by H. Tsutsui is developed up to terms of second order in the transverse offsets. In this report we develop a new formula, which use symmetries from the Lorentz reciprocity principle and the multipolar decomposition of transverse fields to reduce the number of terms from 13 to 8. This new formula is also developed up to second order, but could of course be developed up to any higher order. The transverse beam impedances (for horizontal and vertical planes) can be by obtained by differentiation of this new formula., CERN Yellow Reports: Conference Proceedings, Vol 1 (2018): Proceedings of the ICFA Mini-Workshop "Impedances and Beam Instabilities in Particle Accelerators"

The NA61 experiment, aims to study the properties of the onset of deconfinement at low SPS energies and to find signatures of the critical point of strongly interacting matter. A broad range in T-μB phase diagram will be covered by performing an energy (13A-158AGeV/c) and system size (p+p, Be+Be, Ar+Ca, Xe+La) scan. In a first phase, fragmented ion beams of 7Be or 11C produced as secondaries with the same momentum per nucleon when the incident primary Pb-ion beam hits a thin Be target will be used. The H2 beam line that transports the beam to the experiment acts as a double spectrometer which combined with a new thin target (degrader) where fragments loose energy proportional to the square of their charge allows the separation of the wanted A/Z fragments. Thin scintillators and TOF measurement for the low energy points are used as particle identification devices. In this paper results from the first test of the fragmented ion beam done in 2010 will be presented showing that a pure Be beam can be obtained satisfying the needs of the experiment.

The implementation of the new Multi-Turn Extraction (MTE) at the CERN Proton Synchrotron required major hardware changes for the nearly 50-year old accelerator. The installation of new Pulse Forming Networks (PFN) and refurbished kicker magnets for the extraction, new sextupole and octupole magnets, new power converters, together with an in-depth review of the machine aperture leading to the design of new vacuum chambers was required. As a result, a heavy programme of interventions had to be scheduled during the winter shut-down 2007-8. The newly installed hardware and its commissioning is presented and discussed in details.

We propose to install a beam stopper in the TT10 transfer line from the PS to the SPS. This beam stopper will replace one of the three sets of magnets in the present access safety chain. This will make the access safety chain more robust against a possible common mode failure in the powering system of the magnets. We have investigated three scenarios for the installation of a beam stopper and found that a PS type beam stopper, in front of the radiation shielding at the start of the TT10 line, is the best solution.

Since 2001 considerable effort has been devoted to the study of a possible replacement of the continuous-transfer extraction mode from the PS to the SPS. Such an approach, called Multi-Turn Extraction (MTE), is based on capture of the beam inside stable islands of transverse phase space, generated by sextupoles and octupoles, thanks to a properly chosen tune variation. Both numerical simulations and measurements with beam were performed to understand the properties of this new extraction mode. The experimental study was completed at the end of 2004 and by the end of 2005 a scheme to implement this novel approach in the PS machine was defined and its performance assessed. This design report presents the outcome of the studies undertaken both in terms of technical issues as well as of resources necessary to implement the proposed scheme.

The proposed under study in the PS machine requires magnets generating nonlinear magnetic fields, e.g. sextupoles and octupoles. These devices are used for generating stable islands in transverse phase space where particles are trapped by means of an appropriate tune variation inducing a crossing of a low-order resonance. Observations made during the experimental sessions performed throughout the 2003 PS run seems to indicate that the PS octupoles are the main culprit for the measured beam losses. In this note two options to overcome the difficulties encountered so far and for pursuing the tests during the 2004 run are presented and discussed.

Since the year 2001 considerable efforts were devoted to the study of a possible replacement of the Continuous Transfer (CT) extraction mode from the PS to the SPS. Such an approach is based on the use of stable islands of phase space, generated by sextupoles and octupoles, to capture the beam inside, thanks to a properly chosen tune variation. Both numerical simulations and measurements with beam were performed to understand the properties of this new extraction mode. Recently, a Study Group was set-up with the mandate of studying this novel extraction in view of using it as a replacement for the CT, as well as technical issues related with this approach. The results of the analysis carried out so far, and the conclusions of the Study Group are presented and discussed in this report.

The so-called "AC-Dipole" principle allows the excitation of transverse oscillations to large (several sigma) excursions without emittance blow-up. The idea was originally proposed and tested at BNL for resonance crossing with polarised beams, using an orbit corrector dipole with an excitation frequency close to the betatron tune, hence "AC-Dipole". This method of beam excitation has several potential applications in the LHC, such as phase advance and beta measurements, dynamic aperture studies and the investigation of resonant driving terms. The technique was recently tested in the CERN SPS using the transverse damper as an "AC-Dipole" providing the fixed frequency excitation. Results from this experiment are presented, along with a brief explanation of the underlying principle.

The so-called "AC-Dipole" principle allows the excitation of transverse oscillations to large (several sigma) excursions without emittance blow-up. The idea was originally proposed and tested at BNL for resonance crossing with polarized beams, using an orbit corrector dipole with an excitation frequency close to the betatron tune, hence "AC-Dipole". This method of beam excitation has several potential applications in teh LHC, such as phase advance and beta-measurements, dynamic aperture studies and the investigation of resonance strengths. The technique was recently tested in the CERN SPS using the transverse damper as an "AC-Dipole" providing the fixed frequency excitation. Results from this experiment are presented, along with an explanation of the underlying principle.

The precise measurement of the local value of the beta-function at the place of a beam size monitor is necessary for the precise determination of the beam emittance. We developed a new method for the measurement of the beta-function by using of continuous square-wave modulation of the force of the quadrupole and by continuous tune tracking. Measurements were performed at LEP in order to evaluate the precision that can be achieved with this method in the LHC. The paper describes the method and discusses in details the results obtained at LEP for colliding and non-colliding beams.

The proof of all the standard Fourier transformations: Discrete Fourier Transform, Fourier Series, Discrete-Time Fourier Transform and Fourier Transform is presented here in very simple mathematics. The emphasis is on giving a clear understanding of Fourier transforms, the relationship between the different flavours of Fourier transforms and the transformations between time and frequency domain. An overview of all the relations are shown graphically on one page.

This is a list of different filters - low pass, high pass, 1'st and 2'nd order - which are used in the LEP Q-meter. The PLL algorithm is also explained. All filters are described in detail and their formula, used for digital signal processing, are derived from the Laplace transfer function. The PLL algorithm is also derived from its Laplace transfer function, which in turn comes from a functional diagram.

The PLL (Phase Locked Loop) has two modes: search mode and locking mode. The mode is determined by how far apart the internal frequency is from the external frequency. When the internal - and exter nal frequencies are close, the PLL is in locking mode. While locking on the external frequency, the PLL can largely be expressed as a linear term superimposed with a small "noise" term. We can t herefore calculate a transfer function that will completely describe the performance of the PLL apart from the precision, which is determined by the noise term. Two ideas are presented on how to re duce the noise term. The first idea is to compensate the noise term with an equal value of opposite sign. This completely removes the noise when the PLL locks on a steady frequency. Unfortunately it can not be used for the LEP Q-meter which is plagued by the presence of satellite frequencies. For other equipment with cleaner spectra, the method can improve their precision dramatically. T he second idea reduces the noise level with a factor 2 but limits the locking range. This will be tested out on the LEP Q-meter.

In normal operation LEP is ramped from injection energy, typically 22 GeV, to energies of over 90 GeV where physics taking takes place. Effective control of the betatron during the ramp is essential t o ensure good transmission of stored current. The LEP Q-loop is a feedback system used to control the tunes during the energy ramp. By following a pre-programmed tune function it provides a means of avoiding dangerous resonances and thus beam loss. The basic components the Q-loop will be described, and operational results presented. Emphasis will be given to the problems encountered and the solut ions found.

By slightly exciting one of two colliding bunches in LEP, it is possible to enhance the eigenfrequencies of the resonant system of the two bunches coupled by the space charge force. The LEP Qmeter has been adapted to detect, among these excited frequencies, the so called s- and p- modes, whose distance is proportional to the luminosity. A real time display of these quantities provides the Operators with an effective way of finely optimizing the luminosity.

The LEP tune meters have been upgraded by replacing the original Motorola 68020 processor cards by Texas Instruments TSM320C30 Digital Signal Processor cards with floating point arithmetic and by creating an optional connection to a more sensitive beam position monitor. This upgrade has lead to a considerable increase in speed and accuracy. The new instrument can generate a continuous real time display of the beam motion in the frequency domain which is well suited to monitor dynamic phenomena occurring during injection and acceleration of the LEP collider. The dynamic phenomena can also be stored for off line analysis. The paper describes the functionality of the instrument in terms of user interface and covers some aspects of code debugging and process synchronization for DSP's connected to the standard control system of an accelerator.

After the 2011 run, actions were put in place during the 2011/2012 winter stop to limit beam induced radio frequency (RF) heating of LHC components. However, some components could not be changed during this short stop and continued to represent a limitation throughout 2012. In addition, the stored beam intensity increased in 2012 and the temperature of certain components became critical. In this contribution, the beam induced heating limitations for 2012 and the expected beam induced heating limitations for the restart after the Long Shutdown 1 (LS1) will be compiled. The expected consequences of running with 25 ns or 50 ns bunch spacing will be detailed, as well as the consequences of running with shorter bunch length. Finally, actions on hardware or beam parameters to monitor and mitigate the impact of beam induced heating to LHC operation after LS1 will be discussed.

In the framework of the High-Luminosity LHC project, the CERN Proton Synchrotron (PS) would require a major upgrade to match the future beam parameters requested as pre-injector of the collider. The different beam dynamics issues, from space-charge limitations to longitudinal instabilities are discussed, as well as the proposed technical solutions to overcome them, covering the increase of the injection energy to RF related improvements.

In particle accelerator components, parasitic resonances must be reduced because they heat up the equipment and cause beam instabilities. In this paper, a method for designing and characterizing Higher Order Mode (HOM) couplers for reducing such resonances in a broad bandwidth is proposed. A case study is considered for a specific component, called QuattroTank, showing geometrical discontinuities and thus causing significant electro-magnetic resonances. Results of numerical simulation and experimental emulation prove the capability of the proposed method to reduce the peaks and the Q-factor of the resonances. In particle accelerator components, parasitic resonances must be reduced because they heat up the equipment and cause beam instabilities. In this paper, a method for designing and characterizing Higher Order Mode (HOM) couplers for reducing such resonances in a broad bandwidth is proposed. A case study is considered for a specific component, called QuattroTank, showing geometrical discontinuities and thus causing significant electro-magnetic resonances. Results of numerical simulation and experimental emulation prove the capability of the proposed method to reduce the peaks and the $Q-factor of the resonances.