## 2 Research Methodology
### Magnetron Structure Analysis  
One of the most efficient sources of high-power microwave is a relativistic magnetron. A relativistic magnetron has the structure of a diode which mainly consists of the cathode and anode polarities confining electrons in between. The anode of the magnetron is usually manufactured by a cylindrical copper block configuration. The cathode and filament are installed in the central area of the anode block and supported by the filament leads. The filament leads are rigid in order to maintain the position of the cathode and filament. The anode is indirectly supplied with DC voltage to heat up the filament. This leads to electrons being emitted from the cathode surface, which is constructed from a high-emission material around the area of the cavity. In general, the cavity magnetron has been designed to possess eight or twenty cavities distributed around the cathode. It is called the resonant cavity, where each cavity forms a parallel resonant circuit, as shown in Figure 1.  
Figure 1 shows the magnetron configuration, which consists of several resonant cavities, a space between the anode block and cathode called the interaction space and a slot gap which runs from each cavity into the central part of the cathode. The overall resonant cavity is enclosed by metal walls equivalent to inductance, while the slot gap represents the capacitive portion of the equivalent parallel resonant circuit. Thus, to simplify the study, the lumped equivalent circuit of a resonator is used to represent the cavity with the parallel combination of an inductor and a capacitor, while the dimension and shape of the cavity represent the resonant frequency. Electrons emitted from indirect heat in the cathode travel radially towards the anode if the magnetic field produced by a permanent magnet is zero.  
Figure 1: The structures and resonant cavity of the magnetron.  
If the magnetic field strength is slightly increased, the magnetic force will bend the electron path as a circular path to the outer ring, which is the anode. However, the magnetic field's strength, at some point, prevents the electrons from reaching the anode and returns them to the cathode, causing overheating in the cathode. Thus, this results in zero current in the anode. The circular path of the electron will return to the cathode without reaching the anode's surface when the magnetic field is less than the critical magnetic field (B\({}_{\text{C}}\)). Because the electrons' trajectory changes during travel to the anode, they will lose their energy by radiating electromagnetic energy. This radiated electromagnetic wave is the natural resonant frequency of the cavity. In addition, these electrons create a charge on the anode's surface, leading them to radiate an electromagnetic wave at the resonant frequency. As a result, the cavity is excited and then it induces electromagnetic energy oscillation in the adjacent cavity with a phase of 180 degrees. The resonant frequency operate like a slow-wave structure or multi-cavity traveling-wave magnetron. In the cathode part, the transfer of the electron energy that is emitted by filaments moves to the interaction space, where the electric and magnetic fields interact perpendicularly together and cause a force to hit the electron, resulting in a loss of the energy of the electron partly to the area of the cavities. The output of the resonant frequency is coupled by coaxial coupling to the target load.