# Research and Design of the RF Cavity for an 11 MeV Superconducting Cyclotron
## 3 Equivalent Circuit of the RF Cavity
### Determination of Equivalent Circuit Parameters  
After selecting the reference surface, the transmission line model can be divided into four segments: the Dee segment, the rectangular coaxial transmission line, the transitional coaxial transmission line, and the round coaxial transmission line. Using the input admittance formula and taking into account the step junction capacitance and the Dee-Dummy Dee local capacitance, a transcendental equation including the resonance frequency can be derived. For calculating the resonant frequency, the characteristic impedance, total step junction capacitance, and local capacitance of each section of the equivalent transmission line must be determined. The characteristic impedance of each part of the coaxial transmission line can be calculated using the following formula, which includes the round coaxial transmission line (\(Z_{\text{c0}}\)), the rectangular coaxial transmission line (\(Z_{\text{r0}}\)) [20], and the transition coaxial transmission line (\(Z_{\text{cr}}\)):  
Where an approximate method is used to calculate the characteristic impedance of the transition coaxial transmission line. The characteristic impedances of the coaxial transmission line with rectangular and circular inner conductors are obtained numerically and analytically, respectively. Then, the ratio of the average value of the two to the minimum value between them is taken as the approximation coefficient of \(Z_{\text{cr}}\).  
\[\begin{split} Z_{\text{c0}}=60\ln\left(\frac{D}{d}\right)\Omega\\ Z_{\text{r0}}=\frac{\eta_{0}}{4\sqrt{\epsilon_{\text{r}}}}\left[ \frac{\overline{\mu/D}}{1-\overline{\mu/D}}+\frac{2}{\overline{\mu}}\frac{1}{ \overline{\mu}}\left(\frac{1}{1-\overline{\mu/D}}+\coth\frac{\overline{\mu}} {2D}\right)\right]\Omega\\ Z_{\text{cr}}=n_{\text{ac}}\cdot 60\ln\left(\frac{D}{d}\right) \Omega\end{split} \tag{10}\]  
where \(D\) and \(d\) are the diameters of the outer and inner conductors of the round coaxial transmission line, respectively. In addition, \(\eta_{0}=(\mu_{0}/\epsilon_{0})^{1/2}=376.73\)\(\Omega\), \(\epsilon_{\text{r}}\) is the relative dielectric constant, \(w\) and \(t\) are the length and width of the inner conductor of the rectangular coaxial transmission line, \(a\) and \(b\) are the length and width of the outer conductor of the rectangular coaxial transmission line, and \(n_{\text{ac}}\) is the approximation coefficient of \(Z_{\text{cr}}\).  
The step capacitance distribution per unit length can be obtained by utilizing the step capacitance formula as [21]:  
\[C(\alpha)=\frac{\epsilon}{\pi}\left(\frac{\alpha^{2}+1}{\alpha}\ln\frac{1+ \alpha}{1-\alpha}-2\ln\frac{4\alpha^{2}}{1-\alpha^{2}}\right) \tag{11}\]  
where \(\alpha=A/B\), where \(A\) is the minimum clearance distance between the inner and outer conductors and \(B\) is the maximum clearance distance between the inner and outer conductors. Therefore, the distributed capacitance due to discontinuities in the vertical (\(C_{\rm v}\)) and horizontal (\(C_{\rm h}\)) directions can be calculated separately:  
\[\begin{array}{l}C_{\rm v}=2 mC(\alpha)\\ C_{\rm h}=2 nC(\alpha)\end{array} \tag{12}\]  
where \(m\) is the length of the inner conductor in the vertical direction and \(n\) is the width of the inner conductor in the horizontal direction.  
For the Dee-Dummy Dee local capacitance part of the calculation, which can be approximated as a plate capacitor, an approximation can be obtained by using the plate capacitance calculation equation.