## 4 Analog Front-End
### Impedance Matching  
Impedance matching networks are essential to transform the amplifier's input impedance \(R_{\text{a,in}}\) to the optimal load impedance of the resonators \(R_{\text{L,par}}\) given in Table 1. In the given application,  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \(i\) & \(r_{\text{b,i}}\) & \(\sigma_{\text{b,i}}\) & \(\sigma_{\text{b,i}}\) & \(p_{i}\) & \(w_{i}\) & \(N_{i}\) & \(s_{\text{b,i}}\) & \(L_{i}\) & \(R_{\text{cept,i}}\) & \(R_{\text{L,par,i}}\) & \(C_{\text{p,i}}\) \\ \hline
3 & 15 mm & 2 mm & 10\({}^{\circ}\) & 1 mm & 0.3 mm & 3 & 0.16 mm & 651 nH & 17.7 \(\Omega\) & 192 \(\Omega\) & 192 pF \\ \hline
4 & 0.55 mm & 0.6 mm & 10\({}^{\circ}\) & 0.3 mm & 0.15 mm & 3 & 0.14 mm & 263 nH & 7.1 \(\Omega\) & 77 \(\Omega\) & 475 pF \\ \hline \hline \end{tabular}
\end{table}
Table 1: Dimensions and electrical parameters of the designed resonators.  
Figure 7: Analog front-end of the custom NFC transceiver.  
a wideband matching must be realized due to the required high bandwidth of the communication link. Moreover, the network shall effectively suppress the energy carrier at 40.68 MHz. Both of these requirements imply a multi-stage network with low-pass characteristics, which is realized by three cascaded L networks as shown in Figure 7. According to [32], wideband matching can be established by introducing virtual intermediate resistances \(R_{\text{int,}i}\) and by dimensioning the individual L networks in order to maintain a constant ratio \(r\) of input and output impedance to be matched:  
\[r=\frac{R_{\text{L,par},i}}{R_{\text{int,1}}}=\frac{R_{\text{int,1}}}{R_{\text {int,2}}}=\frac{R_{\text{int,2}}}{R_{\text{a,in}}}. \tag{23}\]  
This results in the virtual intermediate resistances of  
\[R_{\text{int,1,}i}=(R_{\text{L,par},i})^{2/3}\cdot(R_{\text{a,in}})^{1/3}, \tag{24}\]  
\[R_{\text{int,2,}i}=(R_{\text{L,par},i})^{1/3}\cdot(R_{\text{a,in}})^{2/3}. \tag{25}\]  
As the impedance is matched over a relatively large frequency range and as the cutoff frequency of the resulting filter should be as low as possible (suppressing the power carrier at 40.68 MHz while still passing \(f_{0}+f_{\text{symb}}\)), the nominal frequency to compute inductors and capacitors was set to 9 MHz instead of 13.56 MHz; the component values obtained from this method are given in Table 2, transfer function, and input impedance provided to the resonators are summarized in Figure 8 for the assumption of ideal components.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline  & & \multicolumn{4}{c}{**Impedance Matching**} & \multicolumn{4}{c}{**Receiving Amplifier**} \\ \hline \(i\) & \(R_{\text{L,par},i}\) & \(L_{\text{MI}}\) & \(L_{\text{M2}}\) & \(L_{\text{M3}}\) & \(C_{\text{MI}}\) & \(C_{\text{M2}}\) & \(C_{\text{M3}}\) & \(R_{\text{a,in}}\) & \(R_{\text{a,1}}\) & \(R_{\text{a,2}}\) & \(R_{\text{a,3}}\) & \(R_{\text{a,4}}\) \\ \hline
3 & 192 \(\Omega\) & 2.5 \(\mu\)H & 5.0 \(\mu\)H & 4.8 \(\mu\)H & 44 pF & 30 pF & 12 pF & 700 \(\Omega\) & 20 k\(\Omega\) & 5 k\(\Omega\) & 100 k\(\Omega\) & 10 k\(\Omega\) \\ \hline
4 & 77 \(\Omega\) & 1.2 \(\mu\)H & 2.9 \(\mu\)H & 5.0 \(\mu\)H & 94 pF & 44 pF & 17 pF & 700 \(\Omega\) & 20 k\(\Omega\) & 5 k\(\Omega\) & 100 k\(\Omega\) & 10 k\(\Omega\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Component values of the analog front-end.  
Figure 8: Frequency behavior of voltage transfer function and input impedance of the matching and filtering network.