Brian J. Vaughn | Searcharxiv

arXiv2024

Approximating Accelerator Impedances with Resonator Networks

It is common in the accelerator community to use the impedance of accelerator components to describe wake interactions in the frequency domain. However, it is often desirable to understand such wake interactions in the time domain in a general manner for excitations that are not necessarily Gaussian in nature. The conventional method for doing this involves taking the inverse Fourier Transform of the component impedance, obtaining the Green's Function, and then convolving it with the desired excitation distribution. This method can prove numerically cumbersome, for a convolution integral must be evaluated for each individual point in time when the wake function is desired. An alternative to this method would be to compute the wake function analytically, which would sidestep the need for repetitive integration. Only a handful of cases, however, are simple enough for this method to be tenable. One of these cases is the case where the component in question is an RLC resonator, which has a closed-form analytical wake function solution. This means that a component which can be represented in terms of resonators can leverage this solution. As it happens, common network synthesis techniques may be used to map arbitrary impedance profiles to RLC resonator networks in a manner the accelerator community has yet to take advantage of. In this work, we will use Foster Canonical Resonator Networks and partial derivative descent optimization to develop a technique for synthesizing resonator networks that well approximate the impedances of real-world accelerator components. We will link this synthesis to the closed-form resonator wake function solution, giving rise to a powerful workflow that may be used to streamline beam dynamics simulations.

physics.acc-ph↗

arXiv2023

A Simple, Flexible Technique for RF Cavity Wake-Field Calculations

It is typical in the accelerator field to model machine components, especially RF cavities, as parallel RLC resonators. To properly model wake-fields, knowledge of the time-domain voltage resulting from beam excitation is often necessary. While analytical and quasi-analytical expressions are available to accomplish this for common bunch distributions such as the Gaussian, analogous results for less standard distributions can be difficult or computationally-taxing to obtain using direct methods, which opens the door for the development of a more generalized technique. In this paper, a formulation is created that allows for the simple computation of the time-domain voltage waveform of an RLC resonator. The formulation uses the Cauchy Residue Theorem to extract the convolution result from the Fourier Domain, and if current distribution Fourier Transform has no poles, knowledge of its value is only required at one specific evaluation point. This greatly simplifies the computation of the time domain voltage for a large amount of bunch distributions both common and uncommon. Accuracy considerations for this technique and the approximation of accelerator components as RLC resonators are also discussed, resulting the development of a figure of merit for quantifying the robustness of this type of approximation.

physics.acc-ph↗