## 2 Proposed Time-Based Sensor
### Three-Configuration Measurement  
In light of the measurement error issue highlighted in the proceeding section, this section presents a \"three-configuration\" measurement technique aimed at improving the accuracy of the measured RC time-constant. This approach's primary objective is to reduce the measurement error by utilizing three larger RC time-constants to derive the routing time-constant, \(R_{metal}C_{metal}\). The realization of the \"three-configuration\" measurement technique involves incorporating two additional components, namely \(R_{poly}\) and \(C_{MIM}\), as depicted in Figure 4. These elements, realized through a poly-resistor and a metal-insulator-metal (MIM) capacitor, facilitate the achievement of high resistance and capacitance values within a compact area, thereby minimizing area overhead. While it is possible to reduce measurement error by using larger values of \(R_{metal}\) and \(C_{metal}\), pursuing this approach would result in a significantly greater area consumption compared to the \"three-configuration\" method. The \"three-configuration\" approach follows the same operation concept as described in Section 2.2, albeit employing different components. Three larger time-constants are measured by closing and opening the resistor and capacitor switches, \(S_{R}\) and \(S_{C}\), in Figure 4, and can be expressed as follows:  
\[(R_{metal}+R_{poly})C_{metal}\approx\frac{(N_{DC,1})T_{CLK}}{\ln 2} \tag{7}\]  
\[R_{metal}(C_{metal}+C_{MIM})\approx\frac{(N_{DC,2})T_{CLK}}{\ln 2} \tag{8}\]  
\[(R_{metal}+R_{poly})(C_{metal}+C_{MIM})\approx\frac{(N_{DC,3})T_{CLK}}{\ln 2} \tag{9}\]  
Figure 4: In situ time-based sensor block diagram for three-configuration measurement. where \(N_{DC,n}\) denotes the discharge cycle, with \(n\) being an index corresponding to one of the three larger RC time-constants. After measuring the three larger time-constants, the desired routing time-constant, \(R_{metal}C_{metal}\), can be derived by multiplying the first two larger RC time-constants and dividing by the third:  
\[R_{metal}C_{metal}\] \[=\frac{[(R_{metal}+R_{poly})C_{metal}][R_{metal}(C_{metal}+C_{MIM })]}{[(R_{metal}+R_{poly})(C_{metal}+C_{MIM})]} \tag{10}\]  
\[R_{metal}C_{metal}\approx\left(\frac{N_{DC,1}N_{DC,2}}{N_{DC,3}}\right)\left( \frac{T_{CLK}}{\ln 2}\right) \tag{11}\]  
where \(\frac{N_{DC,1}N_{DC,2}}{N_{DC,3}}\) represents the effective discharge cycle of \(R_{metal}C_{metal}\). However, it is important to note that each of the three measurements is subject to its own quantization error. To account for this, Equations (7)-(9) can be adjusted as follows:  
\[(R_{metal}+R_{poly})C_{metal}=\frac{(N_{DC,1}-q_{1})T_{CLK}}{\ln 2} \tag{12}\]  
\[R_{metal}(C_{metal}+C_{MIM})=\frac{(N_{DC,2}-q_{2})T_{CLK}}{\ln 2} \tag{13}\]  
\[(R_{metal}+R_{poly})C_{metal}\approx\frac{(N_{DC,1})T_{CLK}}{\ln 2} \tag{7}\]0  
Here, \(q_{n}\) (where \(0<q_{n}\leq 1\)) represents the quantization error associated with each of the three large time-constants. By substituting Equations (12)-(14) into Equation (10), the routing time-constant \(R_{metal}C_{metal}\), alongside its corresponding quantization error, can be determined. The complete derivation can be found in Appendix A.  
\[(R_{metal}+R_{poly})C_{metal}\approx\frac{(N_{DC,1})T_{CLK}}{\ln 2} \tag{7}\]1 \[(R_{metal}+R_{poly})C_{metal}\approx\frac{(N_{DC,1})T_{CLK}}{\ln 2} \tag{7}\]2 \[(R_{metal}+R_{poly})C_{metal}\approx\frac{(N_{DC,1})T_{CLK}}{\ln 2} \tag{7}\]3  
It is worth noting that the second term, \(\frac{q_{1}N_{DC,2}+q_{2}N_{DC,1}}{N_{DC,3}}\), in the above equation accounts for the quantization error arising from the \"three-configuration\" measurement technique. To significantly mitigate this quantization error, the values of the auxiliary components in this prototype are selected to be \(10\times\) those of the routing resistance and capacitance.