Inexpensive Rf Modeling and Analysis Techniques as Applied to Cyclotrons

A review and expansion of the circuit analogy method of modeling and analysing multiconductor TEM mode rf resonators is described. This method was used to predict the performance of the NSCL K500 and K1200 cyclotron resonators and the results compared well to the measured performance. The method is currently being applied as the initial stage of the design process to optimise the performance of the rf resonators for a proposed K250 cyclotron for medical applications. Although this technique requires an experienced rf modeller, the input files tend to be simple and small, the software is very inexpensive or free, and the computer runtimes are nearly instantaneous. 1 THE MODELING TECHNIQUE The modeling technique applied is well known and fundamentally consists of slicing the TEM-mode resonator up into pieces that can be accurately described by equivalent linear circuit elements. The response of the resulting circuit is then equivalent to the response of the actual resonator within some predictable margin of error. To be successful, the modeler must meet two fundamental requirements; 1) the modeler must be able to qualitatively understand the standing-wave pattern of the structure and describe it terms of an equivalent circuit, and 2) the modeler must be able to determine accurate circuit models for the resonator elements. The circuit elements used to model the resonator are linear and include resistors, capacitors, inductors, and transmission lines. The lumped elements are normally used to describe discontinuities and coupling elements while the transmission lines are used to describe distributed effects. In many cases, the transmission line segments of the circuit are both non-uniform (the transverse cross-section varies along the direction of wave propagation) and non-standard (the transverse crosssection does not have a handbook solution). Since the transmission line is normally the dominant element used in the model, the central issue to this method involves efficiently determining accurate representations of nonuniform and non-standard transmission line segments. This paper shall therefore describe in detail an accurate model for non-uniform, non-standard transmission lines. 2 THE TRANSMISSION LINE MODEL The detailed derivations from first principles for the transmission line model described here as well as formulas for many other components of the equivalent circuit such as bends, discontinuities, couplers, etc. may be found in reference[1]. Formulas and techniques for standard components and systems can be found elsewhere[2]. The technique requires that numerical analysis be performed on the cross-section at each end of the transmission line segment to determine the transmission line parameters. Normally this analysis is performed with electrostatic software since the fundamental TEM mode supports this field solution, however, with the proper software this technique is extensible to higher order transmission line modes. The parameters thus obtained are fed to formula to parameterise a standard uniform transmission line of the same length with characteristic impedance and filler/conduction loss parameters in a manner that attempts to accurately represent the electric stored energy, magnetic stored energy and losses. The technique is extended in this paper to better maintain the stored energy through the addition of lumped elements expressly added for this purpose. The final circuit describing the segment thus accurately models the structure for stored energy (both electric and magnetic), time delay, and losses Figure 1 illustrates the process of converting a mechanical segment of transmission line to its equivalent circuit illustration. It is assumed that the segment has been taken short enough such that the characteristic impedance can be assumed to vary linearly from “Face A” to “Face B”. The conduction losses are maintained by using a formulation that maps the losses to an “Equivalent Width” of conductor (wi, wo) with constant current density. The constant characteristic impedance (Zo) of each line in the model is calculated based on a formulation that attempts to best account for the magnetic and electric stored energy. The lumped elements are used to account for the magnetic and electric energy left unaccounted for by the transmission line segments. The first step in modeling a transmission line segment requires determining the parameters of each face. This is typically accomplished by invoking a numerical analysis program. A program that does a particularly CP600, Cyclotrons and Their Applications 2001, Sixteenth International Conference, edited by F. Marti © 2001 American Institute of Physics 0-7354-0044-X/01/$18.00 293