# Research and Design of the RF Cavity for an 11 MeV Superconducting Cyclotron
## 2 Miniaturization of RF Cavity for an 11 MeV Superconducting Cyclotron
### Effects of Discontinuity  
The above results are achieved by neglecting discontinuities in the structure shown in Figure 1. In order to obtain a more accurate resonant frequency formulation, transmission line discontinuities, including open-end capacitance and step junction effects, must be considered. The open-end capacitance can be directly replaced by an additional length of transmission line during the actual design process. The step junction effects are mainly the discontinuity capacitance generated by the step junction in the transmission line [10]. The reason for this is that the connection of two transmission lines with different characteristic impedances excites higher TM modes due to impedance discontinuities, but these higher TM modes are cut off, and the resulting effect can be replaced by an equivalent capacitance. In the calculation, this equivalent capacitance can be obtained either by numerical analysis of the electromagnetic field distribution at the step junction surface or by using the theoretical formulas in the conclusions of existing studies [12; 13]. After considering the above discontinuities, a more accurate expression for the resonant frequency can be further derived.  
The step junction surface is selected as the reference surface, and the input admittance at the open-circuit end (\(Y_{\text{O,C}}\)), at the short-circuit end (\(Y_{\text{S,C}}\)), and the admittance due to the impedance discontinuity (\(Y_{\text{S}}\)) are, respectively:  
\[\begin{array}{c}Y_{\text{O,C}}=j\frac{1}{Z_{3}}\tan\theta_{2}\\ Y_{\text{S,C}}=-j\frac{1}{Z_{1}}\cot\theta_{1}\\ Y_{\text{S}}=j\omega C_{\text{d}}\end{array} \tag{4}\]  
where \(\omega\) is the angular resonant frequency and \(C_{\text{d}}\) is the step capacitance. The resonance condition can be obtained according to the susceptance method as:  
\[B=-\frac{1}{Z_{1}}\cot\theta_{1}+\frac{1}{Z_{2}}\tan\theta_{2}+\omega C_{\text {d}}=0 \tag{5}\]  
Therefore, the SIR resonance condition, which accounts for the discontinuity capacitance of the transmission line step junction, can be obtained from the derivation as follows:  
\[R_{Z}=\frac{Z_{2}}{Z_{1}}=\frac{\tan\theta_{1}\tan\theta_{2}}{1-\omega C_{ \text{d}}Z_{1}\tan\theta_{1}} \tag{6}\]  
where \(R_{Z}\) is the impedance ratio after considering the effects of discontinuity.  
According to the above equation, the resonant frequency of the cavity can be calculated after determining the characteristic impedance of the different transmission lines and the step junction capacitance.  
The above calculations are mainly for the division of the cavity into two transmission lines. When the cavity structure is more complex or to achieve greater accuracy, it becomes necessary to perform a more detailed equivalence of the cavity. This analysis includes determining the location of the step junction surface and dividing the cavity into multiple segments of transmission lines, each with different characteristic impedances. Subsequently, all these equivalent transmission lines are integrated, ultimately yielding the equivalent circuits of the complex cavity. As a result, the transcendental equations, including the cavity's resonant frequency derived from an equivalent circuit model with a more intricate structure, will be more elaborate. However, the computational methods and principles remain consistent with the previously described approach.