## 2 Modeling of the PDSH Circuit
### The Measurement Speed of the PDSH Circuit  
The measurement speed of the PDSH circuit is a critical factor determining whether the PDSH circuit can accurately measure the peak value of a pulse. Therefore, an analysis of the measurement speed of the PDSH circuit is conducted here. The input pulse signal \(V_{i}\) is assumed to consist of three stages: rising period, plateau period, and falling period, as shown in Figure 3. In the rising period, the input voltage \(V_{i}\) can be described by (5), where \(t_{r}\) represents the rise time. And in the falling period,  
Figure 2: The schematic diagram of the PDSH circuit based on OTA. can be described by (6), where \(t_{f}\) is the fall time. During the plateau period, \(V_{i}\) maintains a constant peak value \(V_{m}\), and the duration is \(t_{h}\). \[V_{i}(t)=\frac{V_{m}}{t_{r}}t \tag{5}\]  
\[V_{i}(t)=-\frac{V_{m}}{t_{f}}(t-t_{r}-t_{h}-t_{f}) \tag{6}\]  
When the assumed pulse signal \(V_{i}\) is applied as the input to the PDSH circuit, the output voltage of the PDSH circuit without considering delay is shown in Figure 3.  
Initially, the OTA operates in the linear region, and the output voltage \(V_{o}\) can be expressed by (7). \[C_{h}\frac{dV_{o}}{dt}=g(V_{i}-V_{o}) \tag{7}\]  
Using the difference between the input voltage \(V_{i}\) and the output voltage \(V_{o}\) as the variable, (7) can be rewritten as (8). \[\frac{d\Delta V}{dt}+\frac{g}{C_{h}}\Delta V=\frac{V_{m}}{t_{r}} \tag{8}\]  
According to the initial conditions \(\Delta V(0)=0\), the solution of the equation is expressed as  
\[\Delta V=\frac{C_{h}V_{m}}{gt_{r}}(1-e^{-\frac{g}{C_{h}}t}) \tag{9}\]  
The difference between the input voltage and the output voltage is equal to \(C_{h}V_{m}/gt_{r}\). The condition for the OTA to not enter saturation region can be expressed as (10). According to (2), the condition can also be expressed as (11)  
\[\frac{C_{h}V_{m}}{gt_{r}}<V_{th} \tag{10}\]  
\[\frac{I_{om}}{C_{h}}>\frac{V_{m}}{t_{r}} \tag{11}\]  
The difference between the input voltage and output voltage is no more than \(V_{th}\) at this time. Since \(V_{th}\) is very small, the measurement speed of the PDSH circuit meets the requirements for detecting the peak value of the pulse signal when (11) is satisfied. However, when the OTA enters the saturation region, the difference between the input voltage and the output voltage gradually increases, indicating that the measurement  
Figure 2:0 The voltage response process of the PDSH circuit with respect to changes in the input voltage. The red line is the input signal, and the blue line is the output signal of the PDSH circuit. speed of the PDSH circuit is insufficient, and the output voltage cannot keep up with the input changes. Assuming that the PDSH circuit enters the saturation region at time \(t_{0}\) and returns to the linear region at time \(t_{1}\), the output voltage of the PDSH circuit during the time interval from \(t_{0}\) to \(t_{1}\) can be expressed by (12). Figure 2:1  
The solution of (12) can be obtained by incorporating the output voltage at time \(t_{0}\), but it is not intuitive to observe the main factors that affect the PDSH output voltage. Therefore, a simplification of (12) is necessary. As the value of \(k\) is very small, Equation (12) can be simplified to  
Figure 2:2  
Furthermore, due to the short time that OTA works in the linear region and the relatively small output current, it can be assumed that the output voltage of the PDSH circuit is described by (12) during the entire charging phase of the sampling capacitor. At this point, the output voltage \(V_{o}\) can be expressed as  
Figure 2:3  
Therefore, the measurement speed of the PDSH circuit can be described as  
Figure 2:4  
The main factors affecting the measurement speed of the PDSH circuit are the maximum output current of the OTA \(I_{om}\) and the sampling capacitance \(C_{h}\), as indicated by (11) and (15). The error mainly stems from two simplification processes during the charging of the sampling capacitor.