## 4 Implementation Details
### Time-Based Sensor  
For the sake of design validation, a set of five RC variants with different resistor and capacitor values have been implemented. The five variants, denoted as \(R\)-\(h\)C, \(h\)R-C, \(R\)-C, \(2R\)-C, and \(R\)-\(2C\), represent combinations of resistors and capacitors at 0.5\(\times\) (notated as \(h\)), 1\(\times\), and 2\(\times\) their nominal values. In these configurations, the nominal values for \(R_{metal}\) and \(C_{metal}\) are 6 k\(\Omega\) and 1.5 pF, respectively. As discussed in Section 2.4, the auxiliary  
Figure 10: (**a**) State machine of the timing controller. (**b**) Partial timing controller circuits for simplicity. Figure 9: Counter. components are chosen to be 10 times larger than \(R_{metal}\) and \(C_{metal}\) to significantly reduce the quantization error, leading to \(R_{poly}\) at 60 k\(\Omega\) and \(C_{MIM}\) at 16 pF for the \(R\)-\(C\) variant. Detailed parameters for \(R\)-\(C\) variant are listed in Table 2.  
To estimate the distribution of the discharge cycle after fabrication, the parasitic extracted time-based sensors are simulated with 200-point Monte Carlo simulations. The simulation results for the three configurations and the derived routing time constant are summarized in Table 3. For the purpose of brevity, only the \(R\)-\(C\) variant is reported. The simulated mean value (\(\mu\)) and standard deviation (\(\sigma\)) for each configuration are tabulated, providing a statistical characterization of the discharge cycle distribution. The \(RC_{design}\) column represents the RC time-constants used in simulations, while the \(RC_{cal}\) column provides the calculated time-constants derived from the mean value of the discharge cycle. Discrepancies between these values are presented as a percentage difference in the \(Error\) column. As discussed in Section 3.1, the charge redistribution effect is influenced by several factors, including the storing capacitor \(C_{store}\), the parasitic capacitors \(C_{p1}\) and \(C_{p2}\), the resistors \(R_{poly}\) and \(R_{metal}\), and the number of the discharge cycle. Consequently, while the calculated time-constant in each configuration is smaller than the designed value, the level of the charge redistribution effect varies among them. For instance, the \(R_{metal}(C_{metal}+C_{MIM})\) configuration, with an error of 11.5%, exhibits less discrepancy than the \((R_{metal}+R_{poly})C_{metal}\) configuration, which has an error of 26.6%, due to its larger storing capacitor. However, in the \((R_{metal}+R_{poly})(C_{metal}+C_{MIM})\) configuration, despite having a large storing capacitor, it accumulates the highest number of discharge cycles. This frequent occurrence of charge redistribution results in more significant charge loss from \(C_{store}\) and, consequently, leads to the largest error among the three configurations. The discharge cycle of the routing time-constant, \(R_{metal}C_{metal}\), is derived in Table 3 based on Equation (11). As expected from Equation (17), the error is significantly reduced through the calculation of the quotient \((R_{metal}+R_{poly})C_{metal}\times R_{metal}(C_{metal}+C_{MIM})/(R_{metal}+R _{poly})(C_{metal}+C_{MIM})\), thereby mitigating the influence of the charge redistribution effect. Consequently, the derived routing time-constant yields a more accurate result with a minimal error of 0.89%. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline \(V_{DD}\) & 1.5 V & \(C_{MIM}\) & 16 pF \\ \(f_{CLK}\) & 500 MHz & \(C_{metal}\) & 1.5 pF \\ \(R_{poly}\) & 60 k\(\Omega\) & \(V_{1}\) & 0.4 V \\ \(R_{metal}\) & 6 k\(\Omega\) & \(V_{2}\) & 0.8 V \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Design parameters of \(R\)-\(C\) variant.