# Research and Design of the RF Cavity for an 11 MeV Superconducting Cyclotron
## 5 Calculation and Verification of the RF Cavity for 11 MeV Superconducting Cyclotrons  
### Calculation and Verification of the RF Cavity with Different Structural Parameters  
After determining all the circuit parameters, the transcendental equation is substituted and the resonance frequency of the circuit is calculated by the iterative method and also utilizing the numerical simulation software to calculate the resonance frequency of the RF cavity. The results obtained by the two calculation methods are compared and verified, and the errors between them are within the acceptable range. At the same time, the influence trend of the structural parameters of the RF cavity on the resonant frequency is determined.  
The effect of the structural parameters of the round coaxial resonator on the resonant frequency of the cavity is shown in Figure 10, where it can be seen that the longer the length, the larger the radius of the outer conductor, and the smaller the radius of the inner conductor, the smaller the resonant frequency of the RF cavity. At the same time, it can be seen that the numerical calculation results and the equivalent circuit calculation results are consistent and show good correlation; only the change in the radius of the inner conductor of the round coaxial will produce a relatively large error; the maximum error reaches about 2.7%. The main reason is that changing the radius of the inner conductor of a round coaxial resonator results in a change in the dimensions of the transition coaxial resonator part. When the dimensions of the rectangular part of the transition coaxial and the dimensions of the round part are similar, the error in the results of the characteristic impedance of the transition coaxial calculated by using the empirical formula is small. However, when there is a large difference between the dimensions of the rectangular part and the round part, the results obtained from the empirical formula tend to produce certain errors, which can eventually result in the generation of resonant frequency calculation errors. This has been taken into account in the design when modeling based on the target frequency, so that the error at the target frequency is small, about 0.1%.  
The effects of changing the dimensions of some structure parameters of the transition coaxial transmission line and the rectangular coaxial transmission line on the resonance frequency can be observed in Figures 11 and 12. It can be seen that the results of the numerical simulation calculations are in good agreement with the results of the equivalent circuit model. When the transition coaxial length is longer and the outer conductor radius is larger, the resonant frequency of the cavity is smaller. For the rectangular coaxial transmission line, the longer the length and the lower the height, the lower the resonant  
Figure 9: Field distribution of the RF cavity: (**a**) electric field and (**b**) magnetic field.  
frequency of the RF cavity. The overall maximum error is around 1% and the error at the target frequency is about 0.13%.  
Figure 11: Impact of transition coaxial line parameters on the resonant frequency of the cavity. (**a**) Line length and (**b**) radius of line outer conductor.  
Figure 12: Impact of rectangular coaxial line parameters on the resonant frequency of cavity. (**a**) Line length and (**b**) line height.  
Figure 10: Round coaxial line parameters versus resonant frequency of cavity. (**a**) Line length, (**b**) radius of line outer conductor, (**c**) radius of line inner conductor radius.