# Absolute Capacitance Measurement by Direct Digital Fitting of Proportional Coefficient
## 4 Analysis of Measurement Results  
### Comparison of Capacitance Measurements with Different Excitation Cycles  
In the method introduced in this paper, under square wave excitation, the input and output signals are simultaneously sampled using a dual-channel ADC. Based on the sampled data, the proportional coefficient k is calculated, which can then be used to determine the capacitance under test. In this method, there is no need to care about the initial state of the capacitor as well as no need to wait for the capacitor charging and discharging to be completely finished compared to the conventional time constant.  
The capacitance measurement is performed with a capacitor with a nominal value of 18,000 pF, a reference resistance of 100 k\(\Omega\), and square wave excitation signals with periods of 6 ms, 20 ms, and 40 ms, respectively. The three cycles correspond to charging and discharging not reaching zero full voltage state, just reaching zero full voltage state, and fully reaching zero full voltage state, respectively. In order to ensure that the same number of excitations are sampled after the period is longer, it is necessary to adjust the appropriate sampling frequency at the same time. The response signals under three different cycles are shown in Figure 9.  
Figure 9: Response signals under different excitation signal cycles. (**a**) Not reaching zero full voltage state (cycle time 6 ms). (**b**) Just reaching zero full voltage state (cycle time 20 ms). (**c**) Fully reaching zero full voltage state (cycle time 40 ms).  
The average results of 100 measurements of an 18,000 pF capacitor under different excitation signal cycles by adjusting the sampling frequency are shown in Table 3. This instrument was used in all the comparisons of commercial instruments in the text.  
Taking the measurement results when the excitation period is 20 ms as the standard, the deviation of the measurement results is 0.13% at 6 ms and 0.27% at 40 ms. Our analysis shows that the direct fitting proportional coefficient method for capacitance measurement offers significant advantages over the traditional time constant method. The period of the excitation signal has minimal impact on the measurement results. Moreover, this method allows for capacitance measurement without requiring the charging and discharging process to reach a zero-full state, making it independent of the initial and final states of the capacitance.  
The experimental data also demonstrate another key advantage of the direct fitting method based on the ratio coefficient. Since this method eliminates the need to wait for the completion of charging and discharging, it substantially reduces measurement time compared to traditional approaches. For instance, in the case of measuring an 18,000 pF capacitance, our method can utilize an excitation signal with a period of 6 ms. In contrast, the traditional charge-discharge method requires a more appropriate period of 40 ms. Using a 6 ms period with the traditional method would lead to significant measurement errors and even prevent capacitance measurement. Furthermore, as capacitance increases, the measurement time for the traditional charge-discharge method grows considerably, while our method remains largely unaffected. Within a cycle of the excitation signal, the direct fitting method based on the proportional coefficient saves approximately 34 ms compared to traditional methods. Additionally, at the same sampling frequency, our method requires less sampling data and simplifies capacitance calculation. In practical measurement and calculation scenarios, to obtain more accurate data, we typically use multiple cycles of square wave signal excitation and repeat measurements to take average values. The time saved by this method will increase multi-fold. In conclusion, compared to the traditional charge-discharge method, the capacitance measurement method based on the proportional coefficient shows substantial improvements in measurement time.