## 2 Modeling and Simulation Methods  
### Design of the Verification Device  
Here, we present the design of a square coaxial low-pass filter (SCLPF) that can operate in the L-band. For low-frequency circuits, low-pass or high-pass filters are composed of lumped parameter reactive components (inductors and capacitors). For high-frequency devices, we use microwave structures to simulate lumped parameter elements. This microwave structure can be called a \"quasi-lumped element.\" The specific design process uses the insertion loss method. First, select the appropriate order (i.e., the number of lumped elements) according to the prototype of the equal-ripple low-pass filter to match the requirements of the filtering performance. Next, according to the normalized element values obtained from the Chebyshev polynomial solution, solve the actual values of the elements using the target cut-off frequency, and obtain the lumped circuit topology. Finally, use the appropriate microwave structure to approximately correspond to the lumped element values one by one to complete the design of the microwave SCLPF [34]. For the SCLPF, the method of approximating the lumped element values of the microwave structure is shown in Figure 1. It illustrates the T-shaped and the \(\prod\)-shaped equivalent circuits of the coaxial line, as well as the calculation method of the component values. For the equivalent circuit in (b), if high impedance coaxial lines are connected at both ends of it, due to the small value of the electrical reactance in (b), the equivalent circuit can be further simplified to a separate parallel capacitor. Similarly, the two electrical capacitances with small value in (c) can be ignored when two low impedance coaxial lines are connected at both ends of it. As a result, we can obtain the approximate equivalent structures shown in (d) and (e). With the known values of the lumped circuit elements of the filter, we can apply Formulas (1) and (2) to select appropriate impedance and coaxial line length, where \(v\) is the wave speed, \(l\) is the length of coaxial line, and \(w\) is the angular frequency [35]. \[B=Y_{0}\sin\frac{\omega l}{v} \tag{1}\]  
\[X=Z_{0}\sin\frac{\omega l}{v} \tag{2}\]  
In the actual design process, compromises have to be made sometimes, which depends heavily on the engineer's practical engineering experience. Here, we use the optimized function of CST software (version number, CST2020) to achieve the preferable size parameters for each part [36]. The aforementioned design method is applicable to various microwave transmission structures, such as circular coaxial, microstrip, rectangular waveguide, etc., as long as their characteristic impedance characteristics are known. For the square coaxial structure operating in the transverse-electromagnetic-wave mode [34], the inner and outer conductors form a parallel plate gap with an electric field perpendicular to the surface. This configuration is particularly suitable for analyzing multipactor phenomena. Therefore, we have chosen the square coaxial structure for this purpose. The SCLPF designed in this article is shown in Figure 2. It is a 13 th-order filter with an inner conductor being arranged alternately with rectangular blocks of different side lengths. In the central low-impedance area, there is the narrowest distance between the inner and outer conductors. Extra dielectric sheets are placed in this area to create a gap of 1 mm for checking the multipactor property of the device. In other low-impedance areas, the gaps are filled with PTFE to \"block\" them, ensuring that multipactor can only occur in the narrowest central section. Figure 1: (**a**) An approximate lumped element circuit of the coaxial line; (**b**,**c**) two equivalent circuits with corresponding formulas; (**d**,**e**) represent further simplified circuit models. Alumina and PTFE are chosen as the dielectric sheets for checking the multipactor threshold in this work.