# An NFC on Two-Coil WPT Link for Implantable Biomedical Sensors under Ultra-Weak Coupling
## 4 Minimizing Interactions between NFC and WPT
### Minimizing Interactions between NFC Uplink and Power Link
#### 4.1.3 Selecting the Optimal \(R_{\rm d}\) and \(C_{\rm d}\)  
\(R_{\rm d}\) and \(C_{\rm d}\) are optimized to achieve the highest PTE. At the same time, the selection of \(R_{\rm d}\) and \(C_{\rm d}\) should prevent the uplink receiving filter from being saturated and supply sufficient received uplink signal power for the demodulation. Equation (9) provides the constraint to prevent the uplink filter from being saturated.  
The PTE \(\eta\) is the ratio between \(P_{\rm load}\) (the power delivered to \(R_{\rm load}\)) and \(P_{\rm dl}\) (the transmitted power of the source \(V_{\rm dl}\)). It is calculated in Equation (13), where \(V_{\rm load}\left(j\omega_{\rm dl}\right)/V_{\rm dl}\left(j\omega_{\rm dl}\right)\) and \(I_{\rm i1}\left(j\omega_{\rm dl}\right)/I_{\rm o1}\left(j\omega_{\rm dl}\right)\) are given in Equations (1) and (2).  
\[\eta=\frac{P_{\rm load}}{P_{\rm dl}}=\frac{V_{\rm load}\left(j\omega_{\rm dl} \right)I_{\rm i1}\left(j\omega_{\rm dl}\right)}{V_{\rm dl}\left(j\omega_{\rm dl }\right)I_{\rm o1}\left(j\omega_{\rm dl}\right)}. \tag{13}\]  
Figure 5: The link budget analysis of the uplink.  
The received uplink signal power equals the power of the uplink carrier on \(C_{\rm d}\), because the input impedance of the out-body receiver is much higher than that of \(C_{\rm d}\). The received uplink signal power could be calculated in Equation (14), where \(V_{\rm d}({\rm j}\omega_{\rm ul})\) is the uplink carrier amplitude on \(C_{\rm d}\). The \(V_{\rm d}({\rm j}\omega_{\rm ul})\) can be calculated by using the output voltage (\(V_{\rm ul}=3.3\) V) of the in-body uplink PA and the uplink transfer function of \(\frac{V_{\rm d}(s)}{V_{\rm ul}(s)}\) in Equation (4).  
\[P_{\rm d}\left({\rm j}\omega_{\rm ul}\right)={\rm ABS}\left(\frac{1}{2}{V_{ \rm d}}^{2}\left({\rm j}\omega_{\rm ul}\right)\omega_{\rm ul}C_{\rm d}\right), \tag{14}\]  
Figure 6 shows the PTE and the received uplink signal power versus \(R_{\rm d}\) and \(C_{\rm d}\), when \(k=0.01\), which is the lowest value in the operating range. When the value of \(k\) is lowest, the received uplink signal power is smallest, and the power carrier interference is strongest due to the adaptive power adjustment mentioned in Section 2.1. That is, the uplink SIR is the lowest in this situation, which is the worst case for the uplink communication. If the uplink could communicate when \(k\) is the lowest value in the operating range, the uplink can be reliable in the whole operating range. Figure 6 denotes that the change of \(R_{\rm d}\) affects both the PTE and the uplink received power, while \(C_{\rm d}\) only affects the uplink. The value of \(C_{\rm d}\) is selected to get higher uplink received power, as reducing \(C_{\rm d}\) could improve the uplink received power. However, too small \(C_{\rm d}\) will violate the constraint of avoiding the uplink receiving filter from being saturated (in Equation (9)). The groups of \(C_{\rm d}\) and \(R_{\rm d}\) that violate the constraint are enclosed by the pink polygon in Figure 6. In order to obtain a high uplink received power without saturating the uplink filter, \(C_{\rm d}\) is selected as 20 pF. When \(R_{\rm d}\) becomes larger, the PTE increases while the uplink received power decreases remarkably, because the RC divider also scales down the uplink carrier. \(R_{\rm d}\) is selected to be 768 k\(\Omega\) because it benefits the higher PTE, and at the same time, it makes the received uplink signal power always higher than the sensitivity limit of the out-body receiver in the operating range.