# Research and Design of the RF Cavity for an 11 MeV Superconducting Cyclotron
## 4 Design and Simulation of the RF Cavity
### Structural Analysis of the RF Cavities  
Varied geometries for RF cavities result in cavities with varied characteristic parameters. In general, it is necessary to consider the parametric margin of the cavity at the beginning of the design to ensure the flexible design of related structures (including tuning capacitors, shorting plates, couplers, and others). In addition, the compact design requires that the overall dimensions and losses be minimized while maintaining the main design requirements. Therefore, the results of the different shapes of resonant cavities are compared under this design premise. According to the previous analysis, it can be known that after changing the shape of the cavity, the equivalent characteristic impedance of the structural part in which it is located will be changed, and thus different characteristic parameters of the resonant cavity will be obtained. Referring to the design of the RF cavity of the KIRAMS-13 cyclotron [24; 25], the rectangular inner conductor of the Stem was replaced by a cylindrical inner conductor. As shown in Figure 3, the radius of the conductor inside the cylindrical Stem should not be too large in order to prevent the connection between the Dee and the Stem from \begin{table}
\begin{tabular}{c c} \hline \hline
**Cyclotron Parameters** & **Value** \\ \hline Energy & 11 MeV \\ Resonant Frequency & 45.026 MHz \\ Harmonic No. & 1 \\ Extraction Radius & 161 mm \\ Dee Voltage & 30 kV \\ Coupling Type & Inductive \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Conceptual design parameters for an 11 MeV superconducting cyclotron’s RF cavity. creating additional impedance discontinuities. The calculation ensures that the length of the Stem's inner conductor remains constant while only the diameter changes. From Figure 4, the maximum resonant frequency of the RF cavity with the circular Stem structure does not exceed 46 MHz, while the resonant frequency of the RF cavity with the rectangular Stem structure can easily reach 46 MHz for the same length. In addition, when the resonant frequency of the RF cavity with the rectangular Stem structure is 45.321 MHz, the resonant frequency of the RF cavity with the circular Stem structure under the equivalent radius is 44.952 MHz. Because the characteristic impedance of a circular Stem is greater than that of a rectangular Stem, the resonance frequency of the cavity is reduced while the length remains constant. According to the calculation formula for the characteristic impedance of the circular coaxial transmission line whose filling medium is air, \(Z_{\rm c}=138\lg(D/d)\ \Omega\), where \(d\) is the diameter of the inner conductor and \(D\) is the diameter of the outer conductor. When the diameter of the circular Stem increases, the radius ratio of the circular coaxial line decreases, the characteristic impedance becomes smaller, and the resonant frequency of the cavity increases. Therefore, the designed target resonant frequency value can be achieved when the diameter of the circular Stem inner conductor is adequate. However, increasing the diameter leads to an increase in the impedance discontinuity at the connection between the Dee and the Stem inner conductor, resulting in an increase in the step capacitance, which leads to a decrease in the resonant frequency of the RF cavity. It is calculated that the RF cavity resonant frequency decreases from 45.224 MHz to 42.001 MHz when the step capacitance is increased from 1 pF to 10 pF. However, when the step capacitance is increased from 10 pF to 100 pF, the RF cavity resonant frequency decreases rapidly to 26.145 MHz. In addition, the increase in the diameter of the conductor inside the circular Stem increases the step junction surface loss and the RF cavity surface loss. After comprehensive consideration, a rectangular Stem is used to connect to the Dee, making the step capacitance at the contact surface as small as possible.