## 2 System Description
### Simultaneous Conjugate Impedance Matching
\begin{array}{cc}\left(2\sqrt{\text{Re}\left\{Z_{mS}\right\}} \right)^{-1}&0\\ 0&\left(2\sqrt{\text{Re}\left\{Z_{mL}\right\}}\right)^{-1}\end{array} \right) \tag{16}\]  
The power transfer efficiency of the matched channel, \(\eta_{mch}\), can be calculated by  
\[\eta_{mch}=\left|S_{m21}\right|^{2}\times 100\% \tag{17}\]  
where \(S_{m21}\) is the element of \(S_{m}\). The matching networks will be designed at the frequency, denoted as \(f_{mch,\text{ max}}\), at which the maximum efficiency of the matched channel, \(\eta_{mch,\text{ max}}\), occurs. Once \(Z_{m1}\) and \(Z_{m2}\) have been calculated at \(f_{mch,\text{ max}}\), impedance matching networks can be designed to transform each port impedance to the conjugate of a desired source or load impedance. The design method for each matching network is the same, so the following analysis focuses only on one channel port, the matched impedance of which is denoted simply as \(Z_{P}\). Figure 4 shows a Smith chart, which is normalized by the reference impedance \(Z_{0}\), and segmented into eight load regions (labelled A-H). Figure 4: Normalized smith chart for matching network design: (**a**) Load regions; (**b**) Matching network topologies. The port impedance of interest is likely to be complex such that \(Z_{P}=R_{P}+jX_{P}\). The normalized port impedance, \(z_{P}=Z_{P}/Z_{0}\), and the normalized port admittance, \(y_{P}=z_{P}^{-1}\), can be plotted on the normalized Smith chart shown in Figure 4a. The normalized port impedance will fall into one of the eight regions on the Smith chart in Figure 4a. In each of these regions, a finite set of possible one-element or two-element matching networks exist that can be used to match the port and source impedances, as illustrated in Figure 4b. Table 1 lists the impedance range of and the network topologies available for each region. In the network topologies, 'C' and 'L' represent a capacitor and an inductor, respectively. Besides,'s' represents that the reactance component after it, is connected in series with the channel port, while 'p' represents that the reactance component after it, is connected in parallel with the channel port. In two-element topologies, the second reactance component is close to the channel port. Figure 5 presents the set of fundamental one-element and two-element matching networks using component reactances \(X_{1}\) and \(X_{2}\) to generalize the specific L and C topologies. For all one-element matching network topologies (sX and pX), the single reactance can be calculated by  
\[X_{1}=-X_{P} \tag{18}\]  
For the sXpX topologies, the reactances can be calculated by  
\[X_{2}=\frac{-Z_{0}X_{P}+a\sqrt{Z_{0}R_{P}\left[R_{P}^{2}-Z_{0}R_{P}+X_{P}^{2} \right]}}{Z_{0}-R_{P}} \tag{19}\]  
\[X_{1}=-X_{2}\frac{R_{P}^{2}+X_{2}X_{P}+X_{P}^{2}}{\left(X_{2}+X_{P}\right)^{2} +R_{P}^{2}} \tag{20}\]  
where \(a=1\) for the sLpL and sCpL configurations and \(a=-1\) for the sLpC and sCpC configurations. For the pXsX topologies, the reactances can be calculated by  
\[X_{1}=bZ_{0}\sqrt{\frac{R_{P}}{Z_{0}-R_{P}}} \tag{21}\] \begin{table}
\begin{tabular}{c c c} \hline
**Region** & **Range of Normalized Port Impedance \& Admittance** & **Available Network Topologies** \\ \hline A & Re \(\left\{z_{P}\right\}>1\) & sCpL and sLpC \\ B & Re \(\left\{y_{P}\right\}>1\) & pLsC and pCsL \\ C & Re \(\left\{z_{P}\right\}=1\) and Im \(\left\{z_{P}\right\}>0\) & sC \\ D & Re \(\left\{y_{P}\right\}=1\) and Im \(\left\{z_{P}\right\}>0\) & pC \\ E & Re \(\left\{z_{P}\right\}<1\), Re \(\left\{y_{P}\right\}<1\) and Im \(\left\{z_{P}\right\}>0\) & sCpC, sLpC, pCsC and pLsC \\ F & Re \(\left\{z_{P}\right\}=1\) and Im \(\left\{z_{P}\right\}<0\) & sL \\ G & Re \(\left\{y_{P}\right\}=1\) and Im \(\left\{z_{P}\right\}<0\) & pL \\ H & Re \(\left\{z_{P}\right\}<1\), Re \(\left\{y_{P}\right\}<1\) and Im \(\left\{z_{P}\right\}<0\) & sLsL, sCPL, pLsL and pCsL \\ \hline \end{tabular}
\end{table} 
Table 1: Impedance range of and network topologies available for each region on Smith chart.