# Absolute Capacitance Measurement by Direct Digital Fitting of Proportional Coefficient
## 4 Analysis of Measurement Results
### Stability Testing  
The reference resistor \(R\) is selected to be 100 k\(\Omega\), the sampling time interval \(\Delta\) is 1.78571 us, and the square wave excitation signal period is 16 ms. A total of 100 repetitive measurements are made under five consecutive square wave signal inputs for a variety of capacitors of different magnitudes selected from the range of 60 pF-60,000 pF, and the average measurement results and stability are shown in Table 5.  
The measurement results were evaluated for precision, and the curve in Figure 11 shows the relative standard deviation (RSD) variation curve of the capacitance measurement. As shown in Figure 11, the reference resistor \(R\) is 100 k\(\Omega\), and under five consecutive square wave signal inputs, the RSD of the capacitance is better than 0.2% when the measured capacitance is 800 pF, while the RSD of the capacitance is better than 0.03% when the measured capacitance is 45 nF, and the RSD is even smaller in the case of measuring capacitance of high capacitance values, and the method of direct fitting of the proportional factor has a high stability. Since the system proposed in this paper is designed for measurement scenarios involving large capacitance variations, such as capacitance boxes, the measured data spans from 60 pF to 60,000 pF. Theoretically, the system is capable of measuring even larger capacitance values. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Nominal Value of** & **Commercial Instrument** & **Scaling Factor \(k\)** & **Actual Measurement** \\
**Capacitance/pF** & **Measurements/pF** & **Scaling Factor \(k\)** & **Result/pF** \\ \hline
800 & 818.881 & 0.0218 & 820.361 \\
1200 & 1181.61 & 0.0151 & 1179.79 \\
2000 & 1904.22 & 0.00942 & 1896.03 \\
3500 & 3371.01 & 0.00529 & 3378.13 \\
15,000 & 15,249.3 & 0.00117 & 15,261.2 \\
25,000 & 25,463.9 & 0.000701 & 25,459.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Calculation of data for different capacitance values in a single measurement. Figure 10: Fitted image of \(k\) for different capacitance values. After reviewing the references, we can compare with other existing capacitance measurement methods and show the main information of each method in Table 6. The literature [28] proposes an unsourced dynamically balanced AC bridge based on a De-Sauty configuration, achieving a deviation of less than 10% over a measurement range of 900 pF to 1100 pF. The literature [29] describes a system where the control unit is implemented using a microcontroller and a phase-delay-to-time converter. This system offers a wide measurement range from 100 pF to 2000 pF with a full-scale deviation of less than 6%. The literature [30] introduces a technique using a relaxation oscillator to integrate the difference between the two arms of an active De-Sauty bridge. This method can measure capacitances from 1 pF to 619 pF, with a measurement deviation of less than 2.33% in the 500 pF to 619 pF range. The literature [31] presents a dual-slope phase method with reference and sensor phases, capable of measuring capacitances from 66 pF to 506 pF with a relative deviation of less than 1% within 1 s. The circuit can also measure capacitances from 1 pF to 506 pF with similar accuracy. The literature [32] presents a capacitance relaxation time circuit that determines capacitance values based on the response time of a voltage-stepped RC circuit. This method achieves a measurement deviation of less than 1% full-scale range (FSR) in the 2 pF to 10 pF range and 8% FSR in the 100 pF to 1 nF range. Compared to these emerging capacitance measurement methods, the method proposed in this paper, based on direct fitting of proportionality coefficients, demonstrates superior performance  
Figure 11: Relative standard deviation of capacitance measurement.