## 2 Modeling of the PDSH Circuit
### The Calibration of Voltage Step Error  
Measurement accuracy is another important requirement of the PDSH circuit. The OTA-based PDSH circuit exhibits a low pedestal voltage and a small overshoot voltage due to the adjustable output current of the OTA with respect to the input voltage. However, due to the presence of diodes, the OTA-based PDSH circuit has relatively large measurement errors caused by the voltage step phenomenon.  
When the PDSH circuit completes the peak value measurement of the pulse, the output voltage of the OTA is equal to the negative saturation voltage of the OTA [10; 24]. At this moment, the diode switches from forward bias to reverse bias. When a diode is forward-biased, current flows through the diode, causing the diffusion of electrons and holes and giving rise to the existence of the diffusion capacitance. In the case of reverse bias, the diode is in a cutoff state, but due to the presence of a certain amount of charge carriers in the depletion region, the depletion capacitance exists at this time. During the transition of a diode from forward bias to reverse bias, both the diffusion capacitance and depletion capacitance must absorb charge from the sampling capacitor. As a result, the sampling capacitor will discharge when the diode changes from forward bias to reverse bias, resulting in the step voltage error as shown in Figure 3. Due to the nonlinearity in the variation of depletion capacitance with reverse bias voltage, the analysis of voltage step processes becomes highly intricate. To simplify the analysis, an equivalent capacitor was used to realize the equivalent process of the diode from forward bias to reverse bias. Therefore, during the transition of the diode from forward bias to reverse bias, an ideal diode and an equivalent capacitor \(C_{\text{eq}}\) were used in parallel for equivalent substitution, as shown in Figure 4.  
In order to correct the voltage step error, it is necessary to establish the relationship between the final output voltage \(V_{f}\) and the peak voltage \(V_{m}\).  
When the PDSH circuit completes the peak value measurement of a pulse signal, the voltage on the sampling capacitor is equal to the peak voltage \(V_{m}\), and the output voltage of the OTA is higher than the peak voltage \(V_{m}\) by the forward voltage of a conduction diode \(V_{m}\). Subsequently, the output voltage of OTA rapidly changes to the negative saturation voltage of OTA \(V_{m}\), and the sampling capacitor charges the equivalent capacitor until reaching the final voltage \(V_{f}\).  
According to the law of conservation of charge, the following equation holds.  
\[C_{eq}V_{on}+C_{eq}(V_{f}-V_{n})=C_{h}(V_{m}-V_{f}) \tag{16}\]  
The final voltage \(V_{f}\) can be expressed as  
\[V_{f}=V_{m}-\frac{C_{eq}}{C_{eq}+C_{h}}(V_{m}+V_{on}-V_{n}) \tag{17}\]  
The value of the voltage step can be described as  
\[V_{step}=\frac{C_{eq}}{C_{eq}+C_{h}}(V_{m}+V_{on}-V_{n}) \tag{18}\]  
Therefore, the important factors affecting the voltage step error include the equivalent capacitance of the diode, the sampling capacitance, the peak value of the pulse voltage, the forward voltage of the diode, and the negative saturation voltage of the OTA. The paper measures the final voltage \(V_{f}\) at different sampling capacitance \(C_{h}\) and fits the data by using (18) to determine the equivalent capacitance of the diode.  
Furthermore, the magnitude of the voltage step can be calculated using (18), which can be used to correct the measurement results of the PDSH circuit and reduce the voltage step error.