I. Introduction
Multipurpose cyclotrons [1]–[8] have proven themselves very useful as the constituents of advanced facilities dedicated to research and applications with accelerated ion beams. They are used either independently, for the production of ion beams in the low to medium energy range, or as elements in a chain of accelerating structures which deliver high energy ion beams. The design procedures of such machines grow more complex and time consuming with the increase in the number of ion beams chosen for acceleration. The design of the magnetic structure, electrodes in the central region, and injection and extraction systems of the machine involves geometrical shaping as well as setting the appropriate parameters for different modes of cyclotron operation. The proper design of these systems not only influences the efficiency, it enables the multipurpose operation of the machine as well and it is therefore very challenging. However, a beam spends the most time in the accelerating region; consequently, the well adjusted acceleration contributes mostly to the good overall efficiency of the machine. The maneuvering space for the improvement of the acceleration process is very narrow. There are only two parameters whose adjustment could lead to it: the frequency and the phase of the voltage applied to the accelerating radiofrequency (RF) electrodes, the RF frequency, , and RF phase, , respectively. These two parameters regulate the operation of the central region as well. The additional way to improve the quality of the acceleration is to center the accelerated test ion orbit well. The six-dimensional phase space coordinates of a test ion that correspond to the well centered orbit introduce limitations on the design of the central region geometry. The constraints put by optimal acceleration on the design of the central region could be accounted for by comprehending the full set of the ion coordinates defining the optimal acceleration. For a given test ion, the corresponding magnetic field and mode of the cyclotron operation we propose a method to adjust the RF electrode parameters, perform the orbit centering, and to define a complete continuous set of ion trajectories with the optimal acceleration.