## 5 Measurement Results
### Direct Measurement Array  
The DMA-resistor array comprises three types of resistor cells labeled as _hR_, \(R\), and _2R_, with values of 3 k\(\Omega\), 6 k\(\Omega\), and 12 k\(\Omega\), respectively. These values correspond to 0.5\(\times\), 1\(\times\), and 2\(\times\) of the nominal resistance values. Each cell type includes 27 identical elements, bringing the total to 81 elements within the resistor array.  
The external current source is supplied by a Keithley-2400 unit (Keithley Instruments, Inc., Solon, OH, USA), configured to provide a current of 10 uA, while the voltage measurements are carried out using a Keysight-34420A device (Keysight Technologies, Inc., Santa Rosa, CA, USA). Given that the leakage current observed from simulations is at the pA level, the impact of the leakage current is negligible.  
Measured results from the first wafer, encompassing four chips, are presented in Figure 13a,c,e. The displayed bimodal shape of the histogram is attributed to die-to-die variation and the small sample size. As the sample size increases, it is expected that the distribution would converge toward a Gaussian distribution. Similarly, results from the second wafer, involving seven chips, are plotted in Figure 13b,d,f. Notably, the second wafer's three resistance cell mean values consistently exhibit a 10% elevation compared to those of the first wafer, likely due to process variation. When compared to circuit simulations in the typical-typical (TT) corner, the mean values of the measured results  
Figure 12: Die photo highlighting the time-based sensor layout.  
across two wafers are approximately 22% higher for all three resistor cells, as summarized in Table 4. This discrepancy may be attributed to the difference between models and real manufacturing process as well as process variations. This discrepancy highlights the critical need for the DMA measurements, as simulation results alone are insufficient.  
### Direct Measurement Array  
_Capacitor Array_  
The DMA-capacitor array comprises three types of capacitor cells denoted as _hC_, \(C\), and _2C_, with values of 0.73 pF, 1.45 pF, and 2.88 pF, respectively, corresponding to 0.5\(\times\), 1\(\times\), and 2\(\times\) capacitance values. Each cell type contains 36 identical elements, yielding a total 108 elements within the capacitor array.  
The clock signal is generated using Agilent E8267D equipment (Keysight Technologies, Inc., Santa Rosa, CA, USA), employing on-chip buffers to produce a square wave with a 50% duty cycle at 20 MHz.  
The voltage sources \(V_{P}\) and \(V_{N}\) are provided by Keithley-2400 unit set at 1.5 V and 0 V, respectively. Since half of the clock period is 25 ns, significantly larger than the delay of the inverter chain, the measured error due to the shortened charging period is negligible. Figure 14 presents the measured results separately for each wafer. Due to die-to-die variation and a small sample size, the histograms deviate from a Gaussian distribution. Notably, unlike the resistor array, the mean values across the three capacitor cells from  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Cell** & **Simulation Results** &
\begin{tabular}{c} **Measurement** \\ **Results** \\ \end{tabular} & **Process Shift** \\ \hline \(hR\) & 3 k\(\Omega\) & 3.67 k\(\Omega\) & 22\% \\ \(R\) & 6 k\(\Omega\) & 7.28 k\(\Omega\) & 21\% \\ \(2R\) & 12 k\(\Omega\) & 14.8 k\(\Omega\) & 23\% \\ \hline \hline \end{tabular}
\end{table}
Table 4: Summary of DMA-Resistance array.  
Figure 13: Measurement results of DMA-resistor array. (**a,c,e**) results are from the first wafer, and (**b,d,f**) from the second wafer.  
both wafers are notably similar. When compared to circuit simulations in the TT corner, the mean values of the measured results closely match, with a maximum error of 2.5%. These results are summarized in Table 5.