## 2 Research Methodology
### Theoretical Magnetron Analysis
#### 2.2.1 The Analysis of Resonant Frequency by Equivalent Circuit Method  
The resonant cavity system takes into account the passive electrical circuit that is used in the oscillator. The normally resonant circuit is divided into 2 parts that consist of the series and parallel resonant circuit. The type of coupling resonant cavity consists of a coaxial line and aperture. The magnetron is presented in Figure 4 using the structure of coupling to a coaxial line with the resonant cavity. Figure 3: An electron’s motion in an electromagnetic field. In Figure 4, the coupling type of the coaxial line is illustrated by inducing a current and voltage to transmit the coaxial line within matching impedance for the resonant circuit system. The energy of electromagnetic field coupling in each cavity of the resonant circuit is represented by a parallel circuit of capacitance, inductance and conductance; the equivalent circuit of the single-hole-slot magnetron resonant system is shown in Figure 5, where C, L and G are equivalent to capacitance, inductance and conductance, respectively. Thus, the resonant frequency of the equivalent parallel circuit of the magnetron is given by Equation (6). In Figure 5b, the consideration of an equivalent one-section network for the interaction space is described in terms of the network. It represents a simple magnetron operation that emits an electron from the cathode to interact with an electric field within the area of the interaction space. The equivalent network takes into account the capacitance (C\({}_{\text{AK}}\)) between the anode segment and cathode. The equivalent resonant circuit of the magnetron is shown in Figure 5c [17; 18; 19; 20]. \[\omega_{0}=\frac{1}{\sqrt{\text{LC}}}, \tag{6}\]  
Figure 4: Internal structures of twelve-hole-slot-type magnetron and equivalent circuit of resonant cavity. Figure 5: The equivalent resonant circuit network of the magnetron. (**a**) The equivalent circuit for the side resonators; (**b**) the equivalent circuit for the interaction space; (**c**) the equivalent circuit of the magnetron. Analysis of the admittance values shown in Figure 5a,b when the resonance of the circuits occurs was conducted. The admittance values of the two-section configuration are equal. This means that both circuits can be rearranged and then the resonant frequency that occurs can be analyzed, as shown in Figure 5c. Moreover, a simple equation expressing the equivalent resonant circuit of the magnetron in each mode of operation is given as Equation (7). \[\omega(\phi)=\frac{1}{\sqrt{\left(\mathrm{LC}\right)\left(1+\frac{\mathrm{C_{ AK}}}{2\mathrm{C}\left(1-\cos\phi\right)}\right)}}, \tag{7}\]  
where \(\phi=\frac{2\pi\mathrm{m}}{\mathrm{N}}\), \(\mathrm{n}=\frac{\mathrm{N}}{2}\), is mode of operation, \(\mathrm{N}\) is number of cavities. Using Equation (7), the capacitance \(\mathrm{C}\), \(\mathrm{C_{AK}}\) and inductance \(\mathrm{L}\) of the resonant system together with the interaction space shown in Figure 6 can be analyzed. The capacitance value is considered the electric distribution required to calculate the maximum electrical voltage change occurring on the adjacent anode block to the point at the voltage as zero or the upper area of the resonant cavity. This is expressed in Equations (8) and (9). For the inductance equation, it analyzes the area of the resonant cavity that is equivalent to the magnetic field in the conductor coil where the current flows in a closed loop. It can be described based on [PERSON]'s law so that the density of the current flow in the slot area and the resonant cavity of the magnetron are obtained. As a result, the inductance is identified by using the laws of Faraday, which are expressed in Equation (10). Thus, the resonant frequency of each mode of the twelve-hole-slot-type magnetron is given in Equation (11) [12; 17; 21].