Calculation of Thermal Deformation of Electron Gun Components

The full model of the traveling wave tube electron gun includes hot wire, hot wire cover, hot wire support plate, hot wire ceramic, cathode, cathode tube, cathode tube support rod, heat shield tube, heat shield support frame, control electrode, control electrode support plate, anode, anode support plate, centering seat, centering seat support plate, metallized gun ceramic ring, gun end cover, end cover support tube, gun bottom cover and connecting sleeve. The cross-sectional view of the full model is shown in Figure 1.
Since the full model of the electron gun is relatively complex, modeling requires a lot of effort from designers and the calculation time is relatively long. Therefore, it is necessary to reasonably simplify and process the structure and geometric shape when establishing a simplified model. When the electron gun is in working state, the high temperature part is mainly concentrated in the three parts of the filament, cathode and cathode tube, and the main factor affecting the electron beam parameters is the thermal expansion of the cathode and cathode tube. Therefore, the simplified model only models the cathode and cathode tube. This can reduce the modeling workload and calculation time. The simplified model is shown in Figure 2.

Fig.1 Cross-section of the complete model of the traveling wave tube electron gunFig.2 Simplified model with only the cathode and cathode tube

1.1 Thermal calculation of complex models
In the electron gun, the filament is double-helix coiled. For the convenience of modeling, the cross section of the filament is changed to a rectangle. Considering that the components are in close contact, their contact thermal resistance can be ignored. When calculating the temperature field, the thermoelectric coupling unit solid69 is selected for the filament part, and solid70 is selected for the other parts of the model. When dividing the finite element mesh, hexahedral units are used for the regular-shaped parts, and tetrahedral units are used for the irregular-shaped parts. After meshing, 72 volume elements generate a total of 407,581 units and 523,169 nodes.
When applying boundary conditions, a fixed voltage is applied to both ends of the filament, and a convection boundary condition is applied to the electron gun shell. Since the interior of the electron gun is a vacuum, the internal heat transfer mainly depends on radiation and heat conduction. The cathode is heated by the thermal radiation of the filament, so the filament surface, cathode bottom surface and cathode tube inner surface are defined as one radiation group, and the cathode and cathode tube outer surfaces, heat shield tube and control electrode inner surface are another radiation group. The temperature of other components is relatively low, and heat transfer mainly relies on heat conduction, so their thermal radiation can be ignored.
When calculating thermal deformation, first convert the thermal analysis units solid70 and solid69 into structural analysis units solid45, add parameters such as Young's modulus and thermal expansion coefficient of each material, apply boundary conditions of zero displacement in all directions to the bottom surface of the electron gun, and then apply the results of thermal analysis as thermal loads to each node to solve and obtain the thermal deformation results.
Due to the large temperature variation range of the entire electron gun, its total temperature distribution cloud map cannot intuitively show the temperature distribution law of each component, so only the temperature distribution and thermal deformation results of the cathode assembly and its support rod are shown in Figure 3.
The factors affecting the temperature distribution of the electron gun are investigated by changing the boundary conditions. Table 1 shows the average temperature of the cathode and the minimum temperature of the electron gun shell when the filament voltage is different. Figure 4 shows the changes in the average temperature of the cathode and the minimum temperature of the shell when different convection heat transfer coefficients are added to the electron gun shell.
The temperature inside the electron gun is mainly transferred to the outside by the cathode tube, cathode tube support rod, heat shield tube, heat shield support frame, centering seat, centering seat support plate, metallized gun ceramic ring and gun bottom cover. In order to ensure that the cathode has a high enough temperature, the heat transfer between the inside and the shell of the electron gun should be reduced.

From the above calculation results, it can be seen that the filament voltage has the greatest impact on the cathode temperature and has a smaller impact on the outer shell temperature (it only varies within a few degrees); and the convection heat transfer coefficient added by the electron gun shell has little effect on the cathode temperature, indicating that the external environmental conditions when the traveling wave tube is working have little effect on the cathode temperature.

Figure 3 Temperature distribution and thermal deformation of cathode assembly and its support rod Figure 4 Effect of convection coefficient on cathode and shell temperature
Table 1 Effect of filament voltage on cathode and shell temperature

1.2 Thermal calculation of simplified model
The simplified model contains only two components: cathode and cathode tube. The cathode and cathode tube are selected from the full model. When calculating the temperature distribution, a constant temperature of 1 050℃ is directly applied to the bottom surface of the cathode for steady-state analysis. The same radiation boundary conditions as the full model are still applied to the surface of the cathode and cathode tube. After meshing, the total number of units is 6 994 and the total number of nodes is 2 120. The thermal analysis and thermal deformation calculation of the simplified model only take a few minutes.
1.3 Comparison of thermal calculation results of the two models
Table 2 lists the axial displacement of the cathode edge and cathode center point and the increase in the cathode radius due to thermal expansion. The thermal deformation of the cathode edge and cathode center in the full model in the axial direction is the value obtained by subtracting the thermal deformation of the heat shield tube base in the axial direction. Although the thermal analysis results show that the temperature distribution of the simplified model and the full model is somewhat different, the data in Table 2 show that their thermal deformation results are basically the same. At high temperatures, since the thermal expansion coefficient of the cathode tube is larger than that of the cathode material, the thermal expansion of the cathode assembly is mainly caused by the thermal expansion of the cathode tube, which shortens the cathode control distance. The distance the cathode moves in the axial direction is consistent with the adjustment value of the cathode control distance during the production and assembly of the traveling wave tube (about 0.05 mm).
In addition, the thermal expansion of the cathode edge is greater than the thermal expansion of the cathode center, resulting in a certain deformation of the cathode concave surface, which is equivalent to a reduction in the curvature radius of the cathode. At the same time, the radial thermal expansion of the cathode causes the cathode cross-sectional radius to increase, which increases the cathode semi-cone angle. These will lead to a reduction in the range, conductivity and injection waist radius of the electron beam, and an increase in the area compression ratio. However, the changes can be calculated by the electron optical system design software of the electron gun, such as TWTCAD, etc. [7]. Therefore, the length of the cathode tube and the curvature radius of the cathode can be appropriately reduced when designing the electron gun.
Table 2 Comparison of thermal deformation results of the two models