## 4 Conclusions  
This paper presents the analysis results of a twelve-hole-slot-type magnetron designed to fulfil the suitable parameters based on dominant-mode (pi-mode) operation. In the experiments, in both relativistic and theoretical magnetrons, all parameters and boundaries of the suitable resonant frequency could be determined, which are usually used for the design and manufacturing of a magnetron prototype to reach a desired tolerance in the radius of the cavity. The following conclusions about the estimated resonant frequency can be drawn from this work:  
1. The estimation of the operating point of the magnetron in dominant mode was conducted, generating a resonant frequency of 2.9982 GHz, after measuring the geometry of the inner structure of the magnetron using a CMM with a resolution of 0.5 \(\upmu\)m. The results of the normal cavity radius yielded a tolerance value of 3.195 \(\pm\) 0.00025 mm. The results of the boundary condition to operate the magnetron were an anode voltage range between 41.0 and 74.2 kV and a permanent magnetic field of around 0.152 tesla.
2. The estimation of the resonant frequency was done by using the equivalent parallel circuit of the magnetron in operation modes 1 to 11. This resulted in a resonant frequency in pi-mode at the desired anode voltage of 45 kV with T\({}_{\text{SLOT}}\) position of 3.97 mm. The measured cavity radius of 3.195 \(\pm\) 0.00025 mm has a relative error of 0.02% with reference to the target frequency load of 2.9982 GHz.
3. The resonant frequency can be estimated by using PIC simulation in the CST Particle Studio program. The results of the resonant frequency in pi-mode with T\({}_{\text{SLOT}}\) position  
Figure 21: The fuzzy system response: (**a**) X-ray response; (**b**) AFC signal response; (**c**) Motor position response; (**d**) Temperature response.  
as 3.97 mm and a cavity radius of 3.195 \(\pm\) 0.00025 mm yield a relative error of 0.03% with reference to the target frequency of the accelerator tube.
* The design and the manufacturing of the prototype of the magnetron with tolerance are required for PIC simulation. Based on the standard measured cavity radius of 3.195 \(\pm\) 0.00025 mm, a cavity radius in the range of -8 um to +20 um and T\({}_{\text{SLOT}}\) tuning of 3.22 to 8.22 mm are sufficient to tune the resonant frequency of the magnetron equivalent to a LINAC frequency of 2.9982 GHz. Thus, the manufacturing of the prototype of the magnetron at a resonant frequency of 2.9982 GHz is required by a CNC machine which has a tolerance of less than \(\pm\)14 um, approximately. The advantages of a reverse engineering technique are studied to reach the structural dimensions and solve the operating point of the magnetron in dominant mode.
* The control system for magnetron is based on the fuzzy logic of the Takagi-Sugeno method agrees well with the LINAC's frequency with an anode voltage supply in pi-mode of 45-kV and magnetic field of permanent magnet of 0.15 T.  
Conceptualization, N.Y., S.C. and J.S.; methodology and validation, N.Y., J.S., S.C. and N.R.; software and formal analysis, N.Y., S.C., N.R. and J.S.; investigation, N.Y. and J.S.; writing--original draft preparation, N.Y.; writing--review and editing, N.Y., J.S., S.C. and N.R. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Ministry of Higher Education Science Research and Innovation, grant number 5401/1324.  
Not applicable.  
The authors would like to thank the Ministry of Higher Education Science Research and Innovation, Synchrotron Light Research Institute, and Suranaree University of Technology for granting budgets, equipment, and locations.  
The authors declare no conflict of interest.