# Research and Design of the RF Cavity for an 11 MeV Superconducting Cyclotron
## 2 Miniaturization of RF Cavity for an 11 MeV Superconducting Cyclotron  
### Structure of the Cyclotron's RF Cavity Using the Stepped Impedance Resonator (SIR) Approach  
Figure 1 illustrates the design of a quarter-wavelength coaxial cavity of a classical cyclotron. The short-circuit (S.C.) end is located at the Stem's end, while the open-circuit (O.C.) end is at the Dee's head, where the Dee is the accelerating electrode, and the Stem is the component that supports and fixes the Dee, and senses and balances the electric capacity  
of the Dee during resonance. This entire cavity can be conceptualized as an SIR structure, composed of two transmission line sections with distinct characteristic impedances. The characteristic impedances at the short-circuit and open-circuit ends of the transmission line are \(Z_{1}\) and \(Z_{2}\), respectively. Their electrical lengths are represented by \(\theta_{1}\) and \(\theta_{2}\), where \(\theta=\beta l\). In this equation, \(\beta\) stands for the propagation constant, and \(l\) signifies the length of the transmission line. For simplicity in analysis, the characteristic impedances \(Z_{1}\) and \(Z_{2}\) of the structure are assumed to be uniformly distributed [10; 14; 15].  
Figure 1 can be regarded as the basic structure of the SIR, which includes an open-circuit end, a short-circuit end, and a step-junction surface. Here, O.C. denotes the open-circuit end, S.C. represents the short-circuit end, \(Z_{\text{left}}\) is the input impedance on the left side of the reference surface, and \(Z_{\text{right}}\) is the input impedance on the right side of the reference surface. According to the input impedance formula [16]:  
\[Z_{\text{in}}=Z_{0}\frac{Z_{\text{L}}+jZ_{0}\tan\theta}{Z_{0}+jZ_{\text{L}} \tan\theta} \tag{1}\]  
where \(Z_{0}\) is the characteristic impedance of the transmission line and \(Z_{\text{L}}\) is the characteristic impedance of the end load. The left and right input impedances can be calculated by selecting the step junction surface as the input reference surface. When the step junction discontinuity and the edge capacitance effect at the open-circuit end are neglected, the total susceptance at the reference surface is zero, and the impedance ratio \(R_{Z0}\) is calculated as:  
\[R_{Z0}=\frac{Z_{2}}{Z_{1}}=\tan\theta_{1}\tan\theta_{2} \tag{2}\]  
The resonance conditions of the structure depend on \(\theta_{1}\), \(\theta_{2}\), and the impedance ratio \(R_{Z0}\). Before the introduction of the SIR structure, the resonance condition of a typical quarter-wavelength coaxial cavity depended only on the length of the transmission line, which had one less degree of design freedom compared to the SIR structure.  
After normalizing the total length \(L\) of the cavity, \(L\) obtains a minimal value of \(L_{\text{n}}\) at \(0<R_{Z0}<1\):  
\[L_{\text{n}}=\frac{4\text{arctan}\sqrt{R_{Z0}}}{\pi} \tag{3}\]  
According to this formula, the impedance ratio \(R_{Z0}\) critically affects the length of the normalized resonator. Thus, theoretically, a smaller impedance ratio can be utilized to reduce the cavity's length. The design strategy involves increasing the transmission line's transverse dimension to compensate for a reduction in its longitudinal dimension. This adjustment aligns with the design needs of a superconducting cyclotron that requires a smaller radial dimension. Consequently, cavity miniaturization can be achieved by optimizing the shape and structural parameters of the RF cavity's Dee and Stem.  
Figure 1: The electrical parameters of the cyclotron Dee’s SIR structure.