## 3 Analytical Model
### Charge Redistribution Effect  
One critical factor that significantly impacts the discharge cycle of the time-based sensor is the presence of parasitic capacitances within the discharge path, specifically associated with the resistors (\(R_{metal}\) and \(R_{poly}\)) and the MOSFET switches (\(S_{discharge}\) and \(S_{R}\)). To establish a straightforward yet effective model, the parasitics of the \(R_{metal}\) and \(R_{poly}\), initially following a distributed RC model, are simplified to a \(\pi\)-model with two lumped capacitors, \(C_{sub}\), at both ends.  
Similarly, the junction capacitances, \(C_{j}\), of the MOSFET switches are modeled as capacitors at the source and drain terminals. To streamline the model, the overlapped capacitance, \(C_{ol}\), and channel capacitance, \(C_{channel}\), are simplified to two capacitors, \(C_{gs}\) and \(C_{gd}\), representing the gate-to-source and gate-to-drain capacitances, respectively.  
Considering that \(C_{j}\) and \(C_{sub}\) are both connected to the substrate, they can be merged. Consequently, the parasitic capacitances at internal nodes, \(V_{n1}\) and \(V_{n2}\), within the discharge path can be simplified into two lump parasitic capacitors, \(C_{p1}\) and \(C_{p2}\), as illustrated in Figure 5a. Specifically, \(C_{p1}\) includes the parasitic capacitance from \(S_{discharge}\), \(S_{R}\), and \(R_{poly}\), while C\({}_{p2}\) contains the parasitics of \(S_{R}\), \(R_{poly}\), and \(R_{metal}\).  
Additionally, to streamline the symbols, \(C_{metal}\), \(C_{MIM}\), and the parasitic capacitance of \(S_{C}\) and \(S_{rst}\) are combined into a single capacitor, denoted as \(C_{store}\), which is responsible for storing the charge on the \(V_{c}\) node, as shown in Figure 5a. The ensuing discussion will focus on the effects of \(C_{p1}\) and \(C_{p2}\), while the impacts of \(C_{gs}\) and \(C_{gd}\) of the discharge switch, which results in a clock feedthrough effect, will be covered subsequently.  
With a comprehensive understanding of the key parasitic capacitances in the TCG block, we now delve into the details of the discharge process. Upon closure of the switch \(S_{discharge}\), the charge stored on \(C_{store}\) redistributes, flowing through \(R_{poly}\) and \(R_{metal}\) to ground as well as to the parasitic capacitors, \(C_{p1}\) and \(C_{p2}\), as illustrated in Figure 5b. Voltages \(V_{n1}\) and \(V_{n2}\) are determined by the resistor-divider (\(R_{on,discharge}\), \(R_{poly}\), and \(R_{metal}\)) present in the discharge path. In contrast, when the discharge switch is opened, the charge flow from \(C_{store}\) ceases, and the charge stored on \(C_{p1}\) and \(C_{p2}\) discharges to ground, as depicted in Figure 5b. Given that the discharge switch OFF time being significantly longer than the time-constants of the parasitic capacitors, \(V_{n1}\) and \(V_{n2}\) nodes are reset to ground when the discharge switch is opened. Therefore, each cycle of closing and opening the discharge switch results in an additional loss of charge from \(C_{store}\), calculated as \(C_{p1}V_{n1}+C_{p2}V_{n2}\). As the cumulative number of discharge cycles increases, the charge redistribution effect becomes more pronounced, leading to an overall reduction in the number of discharge cycles.  
An analytical model of the TCG block can be developed by establishing Kirchhoff's Current Law (KCL) for nodes \(V_{c}\), \(V_{n1}\), and \(V_{n2}\):  
\[\begin{cases}-(C_{metal}+C_{MIM})\dfrac{dV_{c}(t)}{dt}=\dfrac{V_{c}(t)-V_{n1} (t)}{R_{on,discharge}}\\ \dfrac{V_{c}(t)-V_{n1}(t)}{R_{on,discharge}}=\dfrac{V_{n1}(t)-V_{n2}(t)}{R_{ poly}}+C_{p1}\dfrac{dV_{n1}(t)}{dt}\\ \dfrac{V_{n1}(t)-V_{n2}(t)}{R_{poly}}=\dfrac{V_{n2}(t)}{R_{metal}}+C_{p2} \dfrac{dV_{n2}(t)}{dt}\end{cases} \tag{18}\]  
where \(R_{on,discharge}\) is the on resistance of the discharge switch. To accurately capture the discharge behavior, the analytical model is solved for every discharge cycle.