Abstract
Abstract
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.