## 2 Materials and Methods  
### Analysis of WPDs  
The S-parameters of a WPD can be calculated by using even-odd mode analysis. An equivalent circuit model for a conventional 2-way WPD is shown in Figure 1a. As can be seen from the figure, quarter-wavelength transmission lines having characteristic impedances of \(Z_{0}\sqrt{2}\) are used in both output paths. Furthermore, an isolation resistor of \(2Z_{0}\) is located between the two output ports. Based on this model, slow-wave transmission lines can be employed instead of the conventional quarter-wavelength transmission lines. Therefore, these models are the starting point of the entire design procedure.  
The S parameters of the WPD given in Figure 1a can be obtained by different approaches. In order to obtain S\({}_{11}\), the circuit model should be considered as illustrated in  
Figure 1: (**a**) Equivalent circuit model of a 2-way WPD. (**b**) Equivalent circuit model for the calculation of S\({}_{11}\).  
Figure 1b, since no current will flow through the isolation resistor. As can be seen from Figure 1b, S11 can be calculated from the following set of equations:  
\[\text{S}_{11}=\frac{\text{Z}_{\text{in}}-\text{Z}_{0}}{\text{Z}_{\text{in}}+ \text{Z}_{0}} \tag{1}\]  
\[\text{Z}_{\text{in}}=\frac{\text{Z}_{\text{Line}}}{2} \tag{2}\]  
\[\text{Z}_{\text{Line}}=\text{Z}_{0}\sqrt{2}\frac{\text{Z}_{0}+\text{j}\text{Z }_{0}\sqrt{2}\text{tan}\,\theta}{\text{Z}_{0}\sqrt{2}+\text{j}\text{Z}_{0}\text {tan}\,\theta} \tag{3}\]  
\[\theta=\frac{2\pi}{\lambda}.\frac{\lambda_{0}}{4} \tag{4}\]  
For equal power splitting, S21 and S31 can be obtained from S11. Furthermore, the isolation between the output ports can be calculated by using even-odd mode analysis. For this purpose, the even- and odd-mode half-circuits can be obtained by locating the magnetic and electric walls, respectively, on the symmetry axis shown in Figure 2a. Figure 2b,c illustrate the even- and odd-mode half-circuit models, respectively. It should be noted that the isolation resistor can be neglected for the even-mode circuit since the symmetry axis is open-circuited. In addition, the isolation resistor is \(\text{Z}_{0}\) for the odd-mode circuit due to the short-circuited symmetry axis. The isolation between the output ports, S32, can be found by deriving the even-mode and odd-mode input impedances. Here, the circuit model shown in Figure 1a should be considered a 2-port network, with the input port as port 2 and the output port as port 3. Thus, under the even- and odd-mode excitations, the input impedance seen from port 2 or 3 will be the even- and odd-mode input impedances, so that the isolation can be expressed as  
\[\text{S}_{32}=\frac{(\text{Z}_{\text{even}}-\text{Z}_{\text{odd}}\ )\text{Z}_{0}}{(\text{Z}_{\text{even}}-\text{Z}_{0}\ )(\text{Z}_{0}+\text{Z}_{\text{odd}}\ )} \tag{5}\]  
The even-mode input impedance shown in Figure 2b can be expressed as  
\[\text{Z}_{\text{even}}=\text{Z}_{0}\sqrt{2}\frac{\text{Z}_{0}+\text{j}\text{ Z}_{0}\sqrt{2}\text{tan}\,\theta}{\text{Z}_{0}\sqrt{2}+\text{j}\text{Z}_{0} \text{tan}\,\theta} \tag{6}\]  
The odd-mode input impedance can be calculated as  
\[\text{Z}_{\text{odd}}=\frac{\text{Z}_{\text{x}}\text{Z}_{0}}{\text{Z}_{ \text{x}}+\text{Z}_{0}} \tag{7}\]  
\[\text{Z}_{\text{x}}=\text{j}\text{Z}_{0}\sqrt{2}\text{tan}\,\theta \tag{8}\]  
Similar analyses can also be applied to 4- and 8-way WPDs. Since 2 and 3 quarter-wavelength transmission lines should be used in each output path for 4- and 8-way WPDs, respectively, they will excite 2 and 3 transmission poles within the power-dividing frequency band.