# Absolute Capacitance Measurement by Direct Digital Fitting of Proportional Coefficient
## 1 Introduction  
Capacitance sensors are widely used in a variety of detection systems to measure physical quantities such as level [1], pressure [2], displacement [3], acceleration [4], etc. In some specialized applications, such as capacitance generators [5] providing simulated capacitance in circuits, the variation in capacitance often ranges from just a few picofarads to several tens of nanofarads. For the calibration and measurement of capacitance in such devices, it is essential to meet requirements for a wide measurement range, fast response, and high precision. This paper aims to improve capacitance measurement methods to address the needs of these specific applications. Commonly used capacitance measurement techniques [6, 7, 8] include charge amplification [9, 10], AC excitation [11, 12], and charge-discharge methods. The capacitive measurement circuit based on the charge amplification principle utilizes the high gain characteristics of operational amplifiers to amplify the minute charge variations during the charging and discharging process of a capacitor, thereby indirectly measuring the capacitance value. The charge amplification method offers significant advantages in measuring small capacitances and high-precision dynamic signals. However, it alsohas drawbacks such as a narrow measurement range, sensitivity to parasitic capacitance, and relatively high cost. The AC excitation method applies an alternating voltage to the capacitor under test using an AC power source and measures the capacitor's response to these voltages. Capacitive detection using AC excitation features low power consumption, is resistant to direct current interference, and has a high signal-to-noise ratio. However, it also has disadvantages such as high circuit complexity and stringent requirements for the frequency of the AC signal. The charge-discharge capacitance detection circuit [13; 14; 15] can effectively mitigate the influence of stray capacitance on measurement accuracy, making it suitable for detecting weak capacitances. The circuit structure is simple, requiring only basic resistors, a power supply, and voltage measurement equipment, which results in a low-cost solution. However, compared to other high-speed measurement methods, the response speed may not be as fast. It is also susceptible to environmental factors such as temperature changes and power supply fluctuations. There are generally two charging methods in the charge-discharge technique: using a current source [16; 17] and using a voltage source [18; 19]. The current source method involves applying a constant current to charge and discharge the capacitor, while the voltage source method applies a constant voltage for charging and discharging. As the capacitor voltage changes, the current also changes. The voltage source method is generally easier to implement than the current source method. Both methods achieve capacitance measurement through the measurement of charging times. The charge-discharge method requires precise control of both the charging and discharging times to ensure accurate measurement of the capacitance value. Measuring the charging time can be challenging in terms of improving time resolution, while in the voltage source method, the time measurement is influenced by the noise at the starting point. Additionally, the initial voltage of the capacitor during charging and discharging must be either zero or at full voltage, which can lead to longer measurement times and a waste of time. For research on the charge-discharge method for capacitance measurement, [PERSON] [20] introduced a charge-discharge circuit with charge injection compensation for capacitance value measurement. [PERSON] et al. [21] used a charge transfer method without a switch to obtain a similar functionality to the conventional charge-discharge circuit, which simplifies the design even more. [PERSON] et al. [22] introduced a square-wave-excited charge-discharge method for capacitance chromatography imaging for small capacitance measurements. [PERSON] et al.