Cold-test parameters of a slow-wave structure can be obtained using CST MICROWAVE STUDIO® (CST MWS) Eigenmode solver without considering the input and output couplers. As depicted in Fig. 1, a single period helical slow-wave structure is simulated in the Eigenmode solver with periodic boundary defined in the longitudinal direction (z-axis).Figure 1: Geometry of a single period circular helix
The single period helix is extended periodically and infinitely in the longitudinal direction. The phase shift of the periodic boundary, which is corresponding to the phase constant, β, can be defined in the ‘Phase Shift/Scan Angles’ property page. The number of modes to be calculated can be specified in the Eigenmode solver setting. For each eigenmode, the field distribution of the mode and the corresponding eigenfrequency can be obtained. Figure 2 shows the field distributions for the fundamental mode....
Figure 2: Field distribution of the circular helix: (a) E field, (b) H field, and (c) surface current for the fundamental mode
The phase shift of the periodic boundary is defined as a parameter and sweeping of this parameter was performed from 5- to 175-degree with a step size of ten degrees. Template based post-processing can be defined to obtain several cold-test parameters such as phase constant, normalized phase velocity, normalized group velocity and power-flow. These results are shown in Figure 3. An application note on obtaining cold-test parameters using Eigenmode solver is available on the CST support website .Figure 3a
Interaction impedance is an important parameter for slow-wave structure design as it directly affects the gain of a traveling wave tube. In this example, the interaction impedance is obtained directly by evaluating the following formula through the template based post-processing:
where Ez,n(0) is the on-axis longitudinal electric field magnitude of the nth space harmonic; βn is the axial phase constant of the nth space harmonic; P is the power flow through the structure. Ez,n(0) can be obtained by performing Fourier analysis on the total on-axis axial electric field:
where βn can be obtained from:
where β is the fundamental axial phase constant and L is the helix pitch. The on-axis phase constant, βn, electric field, Ez,n(0), and interaction impedance, kn, for the fundamental and the non-fundamental space harmonics (n= 1, 0, and -1) are shown in Figure 4. The space harmonics at different off-axis positions  can be obtained by simply modifying the evaluation location of the E-field in the post-processing.Figure 4a Figure 4b Figure 4c Figure 4: Fundamental and higher order space harmonics analysis for (a) phase constant, (b) axial electric field and (c) interaction impedance. (x-axis: Frequency in GHz)
 CST Application Note: “Slow Wave Structure Postprocessing (#3359),” CST Knowledge Base.
 Ajith Kumar M.M., S. Aditya, and C. Chua, “Interaction impedance for space harmonics of circular helix using simulations,” IEEE Trans. Electron Devices, vol. 64, no. 4, pp. 1868-1872, Apl. 2017.