# Research and Design of the RF Cavity for an 11 MeV Superconducting Cyclotron
## 3 Equivalent Circuit of the RF Cavity  
### Equivalent Circuit Model of RF Cavity  
According to the principle of equivalent transmission line, the waveguide-based RF cavity of the cyclotron can be decomposed and equated into multiple transmission lines. In this way, the boundary-value problem of the three-dimensional electromagnetic field can besimplified into the boundary-value problem of the two-dimensional Helmholtz equation, which is satisfied by the transverse-mode function and the problem of the one-dimensional coupled transmission line equation, which is satisfied by the mode voltages and the mode currents. Since only the longitudinal component of the propagation constant of the TEM mode exists in the coaxial cavity of the cyclotron, the longitudinal propagation constant is equal to the propagation constant \(k_{0}\) of a plane wave in free space, and the transverse component is zero [12; 13]. The 11 MeV superconducting cyclotron RF cavity can be viewed as consisting of the inner wall of the cavity, the Stem, the Dee, and the Dummy Dee, excluding the frequency tuning structure and so on. One side of the Stem is connected to the end of the chamber's inner wall, and the other side is connected to the Dee. The Dee and the Dummy Dee act as the accelerating electrodes, which are electrically connected to each other. According to the transmission line theory, the cavity wall and Stem can be equated to a section of the transmission line, and the main mode of the transmission line is the TEM mode. Similarly, the cavity wall and Dee can be equated to a section of the transmission line. Due to the characteristics of the TEM mode, when propagating inside the coaxial cable, the phase constants of the two different segments are equal to the phase constants of the free space [17]. According to the above theory, the equivalent of the Stem and the cavity wall can be divided into three segments: a rectangular coaxial segment, a transitional coaxial segment, and a round coaxial segment. As a result, the RF cavity can be equated to four coaxial transmission line segments. Because the difference in characteristic impedance across transmission lines produces discontinuities and consequently higher modes, it can be replaced by the equivalent capacitance \(C_{\text{d}}\). In addition, the distributed capacitance \(C_{\text{a}}\) between Dee and Dummy Dee needs to be considered. The equivalent circuit of the cavity is shown in Figure 2.  
The microwave equivalent circuit of the RF cavity can be expressed using the coaxial transmission line with distributed parameters and the lumped parameter capacitance based on the preceding theory. According to the transverse resonance method, it is known that the sum of the total susceptance at any reference plane of the equivalent circuit at resonance is zero. As a result, a transcendental equation containing the resonant frequency can be obtained, and the resonant frequency of the cavity can be obtained by solving this transcendental equation using the iterative method after the relevant circuit parameters of the microwave equivalent circuit have been determined [18]. For the convenience of calculation, the contact surface between the rectangular coaxial transmission line and the transitional coaxial transmission line is selected as the reference surface. The input admittance is then calculated according to [16],  
\[Y_{in}=Y_{0}\frac{Y_{L}+jY_{0}\tan\beta l}{Y_{0}+jY_{L}\tan\beta l} \tag{7}\]  
Figure 2: The equivalent circuit of an 11 MeV superconducting cyclotron’s RF cavity. where \(Y_{\text{in}}\) is the input admittance, and \(Y_{0}\) and \(Y_{\text{L}}\) are the characteristic admittance and load admittance of the transmission line, respectively.