Longitudinal Coupling Impedance Research Articles

Longitudinal coupling impedance of a small hole in a coaxial liner near the cutoff frequencies

We recently developed a general analysis for an azimuthally asymmetric rectangular slot in the inner conductor of a coaxial liner, which allowed us to investigate the coupling impedance numerically. In the present paper we obtain analytic expressions for a small hole of arbitrary shape. Specifically, we go beyond the quasistatic (Bethe) approximation to explore and understand the structure of the impedance in the frequency region near the cutoffs of the inner beam pipe and outer coaxial structure. Finally, we extend our analytic analysis to a hole in a wall of finite thickness.

Analytic and numerical analysis of the longitudinal coupling impedance of a rectangular slot in a thin coaxial liner

Beam pipes of high-energy superconducting colliders require a shielding tube (liner) with pumping slots to screen cold chamber walls from synchrotron radiation. Pumping slots in the liner walls are required to keep high vacuum inside the beam pipe and provide for a long beam lifetime. As previously discussed [Fedotov and Gluckstern, Phys. Rev. E 54, 1930 (1996)], for a long narrow slot whose length may be comparable with the wavelength, the usual static approximation for the polarizability and susceptibility that enter into the impedance is a poor one. Therefore, finding semianalytic expressions for the impedance of a rectangular slot in a broad frequency range is highly desirable. We develop a general analysis based on a variational formulation, which includes both the realistic coaxial structure of the beam-pipe and the effect of finite wavelength, in order to calculate the coupling impedance of a rectangular slot in a liner wall of zero thickness. We then present a numerical study of the frequency dependence of the coupling impedance of a transverse rectangular slot. Numerical results for a small square hole are presented for frequencies above and below cutoff, and compared with the results of other calculations.

Longitudinal coupling impedance imposed by a beam feedback in a synchrotron

Commonly, a longitudinal beam feedback processes a slowly varying signal at zero intermediate frequency (a phase offset, an amplitude departure). Often, only a portion of the data confined in a picked-up band-pass beam signal is retained (like, say, in a purely phase feedback). Sometimes, a beam feedback employs different RF bands to pick up beam data and return a correction back to the beam. All the manipulations thus involved with signal spectra result in cross-talk between various beam-current and electric-field waves propagating along the orbit, which is shown to be described by an impedance matrix with, at most, three non-trivial elements per row. It is this matrix which gives the intuitive notion that a linear feedback is seen by a beam as an artificial coupling impedance controlled from the outside from a quantitative basis. This (impedance) approach has at least two plain advantages: (i) It allows one to mount the feedback's effect into the well-established theory of longitudinal coherent instabilities to use most of its inventory: beam transfer functions, threshold maps, handling of coupled-bunch motion, etc. (ii) The destabilizing effect of the beam environment, being available in standard terms of coupling impedances, is naturally taken into account since the early stages of feedback R&D.

A new method to compute the longitudinal coupling impedance of a drift tube

In this paper an effective analytical-numerical approach to study the electromagnetic interaction between a bunch of particles and the drift tube of the vacuum chamber of a particle accelerator is presented. Particular attention is dedicated to the longitudinal coupling impedance, which is expanded as a Neumann series. Numerical calculations can be easily performed in a wide range of frequency.

A real time bunch-length monitor using the beam spectrum and measurements of bunch lengthening

A bunch length monitor used for detecting two frequency components of a beam spectrum has been developed for the TRISTAN accumulation ring (AR). The principle, design and performance of the monitor are described. The monitor comprises a stripline electrode, a coaxial cable and two identical synchronous detectors. Calibration of the monitor was carried out using the natural bunch length at a low beam current. The bunch length was measured at an injection energy of 2.5 GeV by the monitor and a streak camera. The imaginary part of the longitudinal coupling impedance was estimated by the potential well bunch lengthening and synchrotron tune shift.

Coupling impedance of beam pipes of general cross section.

We have derived expressions for the longitudinal and transverse resistive wall coupling impedances for a beam pipe of arbitrary cross section in the ultrarelativistic limit. These expressions involve the integral of the square of the tangential magnetic fields along the wall, which can be obtained from the solution of the two-dimensional Poisson equations with a monopole and dipole singularity, respectively. Explicit results are given for beam pipes of elliptical and rectangular cross sections, including the limiting cases of a circle and a pair of parallel plates.

Coupling impedance of a single hole in a thick-wall beam pipe.

The coupling impedance of a hole in a thick-wall beam pipe has been calculated for hole dimensions that are small compared to the wavelength. In particular, the longitudinal and transverse coupling impedances are shown to be proportional to the difference between the transverse magnetic susceptibility and the electric polarizability of the inside surface of the hole, and these quantities are themselves proportional to the cube of the hole dimensions

Measurement of the Bunch Length on the UVSOR Storage Ring

We have measured the bunch length in single-bunch operation of the UVSOR (Ultraviolet Synchrotron Orbital Radiation) storage ring in a beam current region from 0.1 to 70 mA at energies ranging from 500 to 750 MeV. The momentum compaction factor of the present operation point is determined experimentally from the bunch length and the synchrotron oscillation frequency measured simultaneously at a low beam current. The bunch length increases gradually as the beam current increases, and bunch lengthening is larger when the beam energy is lower. This behavior of bunch lengthening agrees well with a prediction of the potential-well distortion theory. Based on the theory, the effective longitudinal coupling impedance is estimated as a function of bunch length.

Longitudinal coupling impedance of a circular iris

In this paper a method is proposed for solving the problem of the computation of the longitudinal coupling impedance for a particle which passes through the center of a round aperture in a perfectly conducting metallic plane (iris). It is shown that the solution of the problem can be expressed in quadrature by means of one auxiliary function that satisfies a one-dimensional Fredholm integral equation with a continuous kernel. For small values ofkR 0 (k is the wave number andR 0 is the radius of the aperture) the impedance can be expressed in the form of converging series of this parameter. An asymptotic expansion is also given for the real part of the impedance.

Radiation from a disk and loop of charge in a cylindrical pipe with multiple step changes in wall radius

A theory is developed to model diffraction and Bremsstrahlung radiation from a loop and a disk of charge in a cylindrical pipe with multiple step changes in wall radius supporting all azimuthally symmetric transverse magnetic modes. The beam kinetics are assumed a priori. A correction in phase due to the abrupt change in beam velocity is incorporated at each step change in wall radius. A longitudinal coupling impedance based on energy and power considerations is employed. Loop charge radiation from cavity and collimator geometries is examined. By suitably choosing the loop charge radius, the energy backscattered in the beam tunnel problem can be minimized. The loading effects of the cavity onto the beam is sensitive to its length.

Transmission-line impedance measurements for an advanced hadron facility

High-intensity synchrotrons such as the proposed Advanced Hadron Facility (AHF) [1] and the Los Alamos Proton Storage Ring (PSR) [2] must have a low beam-coupling impedance for stability. The conflicting requirement of low eddy-current effects in the rapid cycling AHF makes the AHF beam-pipe design a particularly difficult problem. A ceramic pipe deposited with conducting stripes has been proposed for AHF. It can have adequately small eddy currents, but it may have an unexpectedly large coupling impedance. Therefore, a facility for measuring transmission-line impedances has been established and is described in this paper. Measurements of transverse and longitudinal coupling impedances for ceramic/metal and other beam-pipe segments are reported; a ceramic/metal chamber can have adequately small impedance. Results of impedance measurements for other AHF and PSR devices, such as a PSR beam pickup and bumper magnet, are also reported.

Reactive impedance of a smooth toroidal chamber below the resonance region.

We evaluate the longitudinal coupling impedance of a smooth toroidal vacuum chamber in the domain of frequencies below the first synchronous resonant mode. The chamber has rectangular cross section. With infinite wall conductivity, as assumed here, the nonresonant impedance is purely reactive. It consists of a space-charge term proportional to $\frac{1}{{\ensuremath{\gamma}}^{2}}$ and a curvature term which survives even when $\ensuremath{\gamma}\ensuremath{\rightarrow}\ensuremath{\infty}$. In the entire subresonant domain, the curvature term is well represented as a quadratic function of frequency: namely, $\frac{Z}{n}=i{Z}_{0}{(\frac{h}{\ensuremath{\pi}R})}^{2}[A\ensuremath{-}3B{(\frac{\ensuremath{\nu}}{\ensuremath{\pi}})}^{2}]$, where $h$ is the height of the chamber and $R$ is the trajectory radius and $\ensuremath{\nu}=\frac{\ensuremath{\omega}h}{c}$. The constants $A$ and $B$ are of order 1, being nearly equal to 1 if the width of the chamber is somewhat greater than its height. Thus, $\frac{\mathrm{Im}Z}{n}$ from curvature is typically a very small fraction of an ohm below the resonance domain, which begins when $\ensuremath{\nu}>{(\frac{R}{h})}^{\frac{1}{2}}$. Consequences for beam stability, if any, arise from high-frequency resonances which can produce values of several ohms for $\frac{Z}{n}$.

High-frequency behavior of the longitudinal impedance for a cavity of general shape.

An ultrarelativistic beam bunch traveling along the axis of an azimuthally symmetric cavity of general shape connected to a beam pipe generates wake fields. An integral equation is derived for the longitudinal electric component of these wake fields at the beam-pipe radius. The kernel for this equation is approximated at high frequency for several different cavity shapes and the resulting integral equation is solved for the electric field at the beam-pipe radius. The longitudinal coupling impedance is obtained and shown to vary inversely as the square root of the frequency, in agreement with recent results obtained by others using different approximation techniques. The results are also shown to depend only on the axial extent of the cavity at the beam-pipe radius, but otherwise are independent of cavity shape for four different cavity geometries. In addition, the results are shown to be in agreement with the analytic properties of the coupling impedance as a function of complex frequency required by causality for an ultrarelativistic beam.

Bunch Lengthening in Single-Bunch Mode of UVSOR Storage Ring

The bunch lengthening and widening in the single-bunch mode of the UVSOR electron storage ring were measured. Certain thresholds were observed in the measurement and they were consistent with the thresholds determined by the balance between the bunch lengthening due to the distortion of the acceleration potential well and the microwave instability. The longitudinal coupling impedance of the UVSOR ring was obtained with the threshold current. The horizontal bunch widening of the beam was also observed above the threshold current.

Longitudinal Coupling Impedance for a Beam Pipe with a Cavity

A method is presented for the computation of the longitudinal coupling impedance of an azimuthally symmetric obstacle of general shape in a long beam pipe. The method involves a numerical calculation of the fields at mesh points in the cavity alone, from which the additive contribution of the cavity to the coupling impedance of the beam pipe can be obtained as a surface integral confined to the cavity wall. Numerical work for obstacles of various geometries is in progress.

A Low Coupling Impedance Double Helix Structure for Use in a Ferrite Kicker Magnet

In a machine such as the CBA, the ejection ferrite kicker magnet has a very large longitudinal and transverse coupling impedance which could destroy the beam. Using a double helix structure that surrounds the beam, the beam induced fields are confined within the helix and, therefore, decoupled from the kicker; but at the same time the helix is transparent to the external fields of the kicker. At first, this may seem paradoxial that the helix is opaque to the fields generated inside the structure by the beam and simultaneously transparent to the external fields generated by the kicker.

The longitudinal coupling impedance of the electron ring with its environment

The present paper is devoted to the problem of determination of the electron ring coupling impedance which characterizes the ring stability against azimuthal perturbations of its density (self-bunching). The calculations results for the coupling impedance of the ring with various structures as well as the experimental results obtained by using an analogous method are presented.

A Method of Measuring and Calculating the Longitudinal Microwave Coupling Impedance of ISABELLE

Modes above the cutoff frequency of the ISABELLE vacuum chamber have been investigated to ascertain the microwave longitudinal coupling impedance. (The investigation is limited to those modes that have fields in the beam pipes.) Perturbation measurements of the electric fields were made between 2.6 and 2.8 GHz. A phenomenological method of calculating these impedances was developed. There was excellent agreement between the calculated and measured impedance. The theoretical results were then extrapolated to 5.5 GHz.

On the Coupling Impedance of a Resonant Cavity

Longitudinal coupling impedance of a resonant cavity is studied by the normal mode analysis using the Coulomb gauge. The vector potential giving a solenoidal electric field leads to a resonant form described by a parallel LCR circuit. The coupling impedance at resonance is equal to the shunt impedance of the cavity. The scalar potential giving an irrotational electric field results in a non-resonant and purely imaginary coupling impedance. Since this latter impedance plays no role in power consumption, it has no relation to the shunt impedance of a cavity. Explicit expressions of the coupling impedance are given for a pill-box cavity considered by Keil et al. It is shown that their result on higher-order mode loss is derived from the real part of the coupling impedance considered here.

A Measurement of the Longitudinal Coupling Impedance in the Brookhaven Ags

The imaginary part of the longitudinal coupling impedance has been measured as a function of energy from 5 to ~ 28 GeV. This impedance is proportional to ?f = (fq - 2fd) where fd is the coherent dipole frequency and fq the coherent quadrupole frequency. These frequencies are obtained by stimulating coupled bunch oscillations. If the dominant impedance is due to inductive wall plus space charge effects, then one has (Z/n) = j[?oL - goZo/2s?2]1 where L is the inductance per turn and ?o = 2?fo the particle rotation frequency. The expression (Z/n) = 4j?f?2hVocos?sB3/ 3Iofd can be used to find the impedance if the synchrotrop phase space distribution is proportional to (1 - r2)?. Io is the current per bunch, B = fo × Tl the bunch length and Vo is the external voltage. For a distribution given by (1 - r2) the right hand side should be multiplied by 27/4?2. If the latter is assumed, an inductive impedance of 20.4 ? is obtained with a null at ? GeV (?tr = 8.5) for a transverse emittance of 22 ?? rad-m. At 5 GeV the reactance is negative but larger than the simple relation assumed for (Z/n) would predict. If the bunches are parabolic, then the inductive impedance would be 29.7 ? with a null again at 6.6 GeV but only for an emittance of 2.5 ?rad-m. Again the .5 GeV reactance is much too large. The significance of these results is discussed.