## 2 Research Methodology
### Theoretical Magnetron Analysis
#### 2.2.2 The Analysis of Resonant Frequency by PIC Simulation Method  
Generally, CST Particle Studio (CST Studio Suite, License No. 5065f3510 dac, Suranaree University of Technology) is used to calculate and perform particle dynamics in the 3D electric field with high accuracy. This commercial code consists of several solvers, such as the eigenmode solver, particle tracking and particle-in-cell (PIC). Each solver has different features for simulation. The tracking particle provides the charged particle trajectory in electrostatic, magnetostatic or eigenmode fields. The wake field is used to compute the fields generated from the interaction between the charged particles traveling with relativistic velocity and structure around them. The PIC solver deals with simulation of the charged particle in the time domain and performs consistent simulation of particles and electromagnetic fields. Using these features of CST, many devices, particularly high-power magnetrons, have been designed and simulated based on this code. In addition, it is possible to import all geometric parameters of a structural magnetron via Solid Works (Solid Works 2019, License No. 9020009109154195 PTMG9 JBC, Synchrotron Light Research Institute, Thailand) into the code and define the device materials needed for the simulation. Thus, the twelve-hole-slot-type magnetron is modeled and simulated using the particle-in-cell (PIC) method in CST. The normal procedures of the particle-in-cell (PIC) method are used for the interaction space between the electron motion in the electromagnetic field of the magnetron. The basic procedure behind the electromagnetic PIC solver is very similar to that demonstrated in the tracking simulation workflow. On the other hand, the tracking solver simulates particles in a self-consistent field by using an integration scheme for particles and electromagnetic fields, in line with the fundamentals of the PIC method algorithm, as shown in Figure 7[22]. The magnetron operation is simulated by the PIC simulation method. The magnetron consists of four components, namely the anode block with a hole-slot type and tuner slot distance component, cathode emission, double-strapped ring and vacuum components. These components were illustrated in 3D coordinates using Solid Works(r) and examples are shown in Figure 8.  
Figure 8: The twelve-hole-slot-type magnetron modeling. (**a**) 3D visual model; (**b**) tuning system of magnetron. Figure 7: A block diagram of particle-in-cell (PIC) simulation technique. In Table 1, the parameters of the structural dimensions measured by the CMM are used to simulate magnetron operation in dominant mode. The resonant frequency is tuned by adjusting the tuner slot distance via the actuator. The tuner slot can mechanically move in and out with a range of 5 mm, which is done by a stepper motor. In the simulation, the particle distribution is uniform, based on an explosion emission model with an initial kinetic energy of 3 eV and particle density from 1830 emission points. In the simulation, conditions based on PIC are 45 kV DC voltage and a constant magnetic field density of 0.152 tesla. An anode voltage is applied for 300-ns duration with a rise time of 1 ns at 3 eV energy [23; 24]. In addition to the analysis of the anode configuration, it can be defined by the properties of the materials, namely the electrical conductivity and electrical resistivity. Both electrical properties of the material under the ambient temperature of 20 Celsius can be analyzed by using the linear equation if the temperature is slightly changed. Thus, the equation of the electrical conductivity and resistivity can be expressed as follows [25].