1 The structure of the vanes

The metal vanes have been found very effective in dispersion shaping for the circular helix and the PH-SEC ^[4]. By considering the structure of vanes, the radian of the vane is concentric with the circular helix in helix TWTs, and for the PH-SEC, the cross section of the vane is rectangle. As the angle log-periodic slow wave structure has the trend of divergence in radial direction, the cross section of the vanes used in this kind of SWS is designed to be isosceles trapezoid. The formation of vanes is shown in Fig.3, two half-lines with an angle of θ_vane launch out from original point O, which is the center of the angular log-periodic meander-line, and they are on the symmetry of the middle axis. There are four intersections a, b, c and d which are formed by two half-lines and the first and the last segments of the angular log-periodic meander-line. Connect four points to form the cross section of the vane in the r-θ plane. The single solid vane is formed by stretching the cross section a thickness of h_vane in the longitudinal direction, shown in Fig.4(a). With O as the center, rotating the single vane an angle of θ_rotate with four times repeat, forming the multi-vane structure, shown in Fig.4(b). The number of the vanes is n, which is 5 in Fig.4(b).

Fig. 3 The formation of the vane

图3 翼片形成示意图

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Fig. 4 The structure of (a) single vane, and (b) multi-vane

图4 （a）单翼片结构, （b）多翼片结构

2 The structure of the vane-loaded strip meander line SWS

The establish procedure of the angular log-periodic strip meander line is similar to the method mentioned in Ref. [6]. Figure 2 shows the multi-vane-loaded SWS. The material of the meander line and vanes is set as perfect electrically conducting (PEC), and the material of the dielectric support sheets is Boron nitride (BN), relative dielectric constant εr=4 ，dielectric loss tangent tanδ=0.0005 . The relative position of the vanes and the angular log-periodic strip meander line of two methods are shown in Fig.5 (vertical view and side view). The open angle of the vane is θ_vane, the thickness of the vane is h_vane, and the vane length is depended on the length of the strip meander line SWS. The height of vacuum chamber at one side of the SWS is h_air, and the gap between the vanes and SWS is the difference between h_air and h_vane.

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Fig. 5 The relative position of the vanes and the angular log-periodic strip meander line (a) vertical view of multi-vane loaded, and (b) side view of multi-vane loaded structure

图5 翼片与角度对数曲折线的相对位置（a）多翼片加载结构俯视图 ，（b）多翼片加载结构侧视图

The structure parameters we used are listed in table 1. The parameters a, θ, b, w and t are initial radius, the open angle, growth rate, strip width, and strip thickness of the angular log-periodic strip meander line, respectively. The dielectric support sheets’ thickness and width are t_sup and w_sup. The length of dielectric support sheets is determined by the length of the strip SWS.

3 Dispersion relation and interaction impedance of the slow wave structure

Angular log-periodic meander-line is not the standard periodic structure, but a logarithmic periodic structure. The method of calculating the dispersion relation and interaction impedance of this kind of structures is proposed in Ref. [9]. According to this method, the phase velocity of the structure mentioned in this paper can be obtained from vp=∆r/∆t ,where ∆r is the distance between the input and output ports, ∆t is the time of signal propagating from input port to output port. By changing the frequency of the signal, the vp varies with the frequency can be obtained, which means the dispersion relation curves can be obtained. As shown in Fig.6, multi-vane loaded structure have lower normalized phase velocity than the structure without vane, the normalized phase velocity of multi-vane structure is lower than 0.145, which can predict the lower operated voltage. It shows that metal vanes are effective in reducing normalized phase velocity. This result can be interpreted as follows: for the vane-loaded structure, vanes are equivalent to the electrical boundary, the distribution of the electric field is compressed towards slow wave line. The distribution capacitance between vanes and slow wave line increases, the phase velocity will decrease. In summary, vane loaded technology is effective in dispersion control, which can get lower phase velocity.

Fig. 6 Comparison of normalized phase velocity of multi-vane-loaded structure and no vane structure：In order to study on how the structure parameters of vanes effect on the dispersion relation, the variation of dispersion relation with the number of vanes, the thickness of vanes and the open angle of vanesare given in Fig.7 (a-c).

图6 翼片加载结构与无翼片加载结构的归一化相速对比

Figure 7 (a) shows the variation of dispersion relation with the number of vanes when h_vane=0.1 mm, θ_vane=0.5 °, the normalized phase velocity will decrease with the number of vanes increases. Figure 7 (b) shows the variation of dispersion relation with the thickness of vanes when n=5, θ_vane=0.5 °, the normalized phase velocity will decrease with the thickness of vane increases. Figure 7 (c) shows the variation of dispersion relation with open angle of the vane when n=5, h_vane=0.1 mm,it can be seen that the open angle of vane has little influence on dispersion relation. Based on these results, the structure parameters of vanes are set as n=5, h_vane=0.1 mm and θ_vane=0.5 °.

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Fig. 7 Dispersion relation varies with (a) the number of vanes, (b) the thickness of vanes, (c) the open angle of vanes

图7 色散曲线与（a）翼片个数,（b）翼片厚度,（c）翼片张角的关系

The interaction impedance of these structures can be calculated by the formula kc=Ez2/2β2P . In Ref. [9], Ez and P can be obtained by setting the integral surface and integral lines, β can be calculated from vp=ω/β . The comparison of interaction impedance of multi-vane-loaded structures and no vane structure in the frequency range from 10 GHz to 40 GHz is shown in Fig. 8. It can be found out that the vanes have no significant influence on interaction impedance, and in the Ka-band which is from 26 GHz to 40 GHz, the interaction impedance of two structures approaches from 6.3 Ω to 10.3 Ω.

Fig. 8 Comparison of interaction impedance of multi-vane-loaded structure and no vane structure

图8 多翼片加载结构与无翼片结构的耦合阻抗对比

4 The transmission characteristics of vane-loaded slow wave structure

The transmission characteristics of SWS are usually expressed in S parameters, S₁₁ is the reflection coefficient of port 1 when port 2 matches the load. S₂₁ is the transmission coefficient of port 1 to port 2 when port 2 matches the load. The S parameters of the multi-vane loaded azimuthal supported angular log-periodic strip meander line SWS are calculated by CST and shown in Fig. 9. In the frequency range from 26 GHz to 40 GHz, S₁₁ is less than -12dB, S₂₁ is greater than -0.07 dB. The vanes have no significant influence on S parameters of the SWS mentioned in this paper.

Fig. 9 S parameters of the vane-loaded SWS

图9 翼片加载结构的S参数

5 Beam-wave interaction simulation results

To investigate the amplification performance of the multi-vane loaded azimuthal supported angular log-periodic strip meander line SWS, the beam-wave interactions of the models are simulated by PIC solver of CST Particle Studio. Two ideal cathodes are settled at the input side of the strip meander line with the thickness of 0.2 mm and on the symmetry of the strip meander line, both side at distance 0.02 mm over of the strip meander line surface, the position of the cathodes is shown in Fig.5. Both of cathodes emit electron beam with dimensions 4°×0.2 mm propagated along the radial axis. The synchronous voltage of the electron beam is determined by the dispersion relation of the slow wave structure, as the phase velocity decreases, the required synchronous voltage decreases, which means the work voltage decreases. The work voltages of the structure without vane and multi-vane loaded structure are optimized to be 5450 V and 4650 V, respectively. Beam current is 0.3 A of each electron beam, total beam current is 0.6 A.

In order to choose appropriate input power, the saturation power curves of two structures are obtained, as shown in Fig. 12. The saturation output power of two structures are similar, close to 490 W, which makes a great agreement with the results of interaction impedance. As two structures have similar interaction impedance, the saturation power will be similar. The vane-loaded structure reaches saturation with input power of 12 W, and the structure without vanes reaches saturation with input power of 24 W. Considering the power available from solid-state devices, the input power of two structures are set as 5 W, and the beam-wave interaction results are obtained.

Fig. 12 Comparison of gain of multi-vane loaded structure and no vane structure

图12 多翼片加载结构与无翼片加载结构增益对比

Fig. 10 The saturation output power curve of multi-vane loaded structure and no vane structure

图10 多翼片加载结构与无翼片加载结构的饱和输出功率曲线

The beam-wave interaction results of multi-vane loaded structure at 38 GHz is shown in Fig.11(a-c). Fig.11 (a) shows the electron bunching in the direction of motion, which is the necessary condition for beam-wave interaction. Fig.11 (b) shows the electron beam kinetic energy variation along the radial distance, the kinetic energy of the electron beam is transferred to the electromagnetic field, so that the energy of electromagnetic field increases, the excitation signal is amplified. Fig.11 (c) shows the frequency spectrum of the output signal obtained by Fourier transformation. It is relatively pure at 38 GHz.

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Fig. 11 (a) Electron bunching in the direction of motion, (b) electron beam kinetic energy variation along the radial distance, (c) frequency spectrum of the output signal

图11 （a）运动方向电子群聚情况,（b）电子动能随径向距离的变化,（c）输出信号频谱图

With the input power of 5 W, the beam-wave interaction results of two kinds of structures are obtained. The comparison of gain of these structures are presented in Fig.12. The gain of the structure without vane and structure with multi-vane loaded is 14.5 dB and 17.9 dB, respectively. The 3-dB bandwidth of structure without vane is 2.5 GHz, which is from 38 GHz to 40.5 GHz. The 3-dB bandwidth of structure with multi-vane loaded is broadened to 3 GHz from 38 GHz to 41 GHz.

As simulation results shown, the multi-vane loaded SWS obtains better amplification performance, the voltage is 800 V lower than the structure without vane, which makes a great agreement with the dispersion relation. The maximum gain of multi-vane loaded SWS is 3.4 dB higher and the 3-dB bandwidth is 0.5 GHz broader than those of the structure without vane.

The beam-wave interaction results mentioned above are obtained based on lossless models. After that, the material of mender line and vanes is set as oxygen-free high-conductivity copper (OFHC), the conductivity σ=5.8×107 S/m, the beam-wave interaction results are shown in Fig.13. The output power of the lossy structure is slightly lower than the lossless structure.

Fig.13 Comparison of output power of lossless vane-loaded structure and lossy vane-loaded structure

图13 考虑损耗的翼片加载结构与无损耗翼片加载结构的输出功率对比

6 Conclusion

The multi-vane-loaded azimuthal supported angular log-periodic strip meander line slow-wave structure is studied in this paper. Multi-vane loaded structure has lower normalized phase velocity than the structure without vane, and the vane has no significant influence on interaction impedance of SWS. According to the beam-wave interaction results, the multi-vane loaded SWS reaches saturation with lower input power than the structure without vanes. The work voltage of multi-vane loaded SWS is 4650 V, which is 800 V lower than the voltage of SWS without vane. The 3-dB bandwidth of multi-vane loaded structure is 0.5 GHz broader than that of structure without vane. As the results shown, vane loading method can expand the bandwidth and reduce the work voltage of angular log-periodic strip meander line slow-wave structure.

References