﻿ 应用于二维倾斜仪中的圆弧柱面差动电容理论分析
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 大地测量与地球动力学  2022, Vol. 42 Issue (5): 545-550  DOI: 10.14075/j.jgg.2022.05.019

### 引用本文

HU Yuanwang, ZOU Tong, MA Wugang. Theoretical Analysis of Arc Cylindrical Plate Differential Capacitor Applied in Two-Dimensional Tiltmeter[J]. Journal of Geodesy and Geodynamics, 2022, 42(5): 545-550.

### Foundation support

Open Fund of Wuhan Gravitation and Solid Earth Tides, National Observation and Research Station, No.WHYWZ202101.

### Corresponding author

ZOU Tong, PhD, senior engineer, majors in geophysical instruments, E-mail: tong.zou@163.com.

### 第一作者简介

HU Yuanwang, assistant researcher, majors in geophysical instruments, E-mail: Hu19881003@163.com.

### 文章历史

1. 中国地震局地震研究所, 武汉市洪山侧路40号, 430071;
2. 武汉引力与固体潮国家野外观测研究站, 武汉市洪山侧路40号, 430071;
3. 地震预警湖北省重点实验室, 武汉市洪山侧路40号, 430071

1 基于圆弧柱面电容极板的二维倾斜仪模型

 图 1 基于圆弧柱面电容极板的二维倾斜仪模型 Fig. 1 Two-dimensional tiltmeter model based on arc cylindrical capacitor plate
2 圆弧柱面差动电容计算 2.1 差动电容工作原理

 图 2 平面差动电容示意图 Fig. 2 Schematic diagram of planar differential capacitor

 图 3 差动电容等效电路 Fig. 3 Equivalent circuit of differential capacitor

 $\frac{u-u_{0}}{\frac{1}{\omega C_{1}}}=\frac{u_{0}-(-u)}{\frac{1}{\omega C_{2}}}$ (1)

 $u_{0}=\frac{C_{1}-C_{2}}{C_{1}+C_{2}} u$ (2)
2.2 圆弧柱面极板的电容计算

 图 4 同轴圆弧柱面极板电容器模型 Fig. 4 Coaxial arc cylindrical plate capacitor model

 $\varPhi_{E}=\int E \cdot \mathrm{d} S=\frac{Q}{\varepsilon_{0}}$ (3)

 $E=\frac{Q}{\varepsilon_{r} \cdot \varepsilon_{0} \cdot S}=\frac{Q}{\varepsilon_{r} \cdot \varepsilon_{0}(r \cdot \theta \cdot L)}$ (4)

 $U=\int_{R_{1}}^{R_{2}} E \mathrm{d} r=\frac{Q}{\varepsilon_{r} \cdot \varepsilon_{0} \cdot \theta \cdot L} \ln \left(\frac{R_{2}}{R_{1}}\right)$ (5)

 $C=\frac{Q}{U}=\frac{\varepsilon_{r} \cdot \varepsilon_{0} \cdot \theta \cdot L}{\ln \left(\frac{R_{2}}{R_{1}}\right)}$ (6)

 $\mathrm{d} C=\frac{\varepsilon_{r} \cdot \varepsilon_{0} \cdot L}{\ln \left(\frac{R_{2}}{R_{1}}\right)} \mathrm{d} \theta$ (7)
 图 5 非同轴圆弧柱面极板电容器模型 Fig. 5 Non-coaxial arc cylindrical plate capacitor model

 $R_{2}^{\prime}=\sqrt{\left(x^{2}+R_{2}^{2}\right)-2 x R_{2} \cos \theta}$ (8)

 $\begin{gathered} C=\int_{\alpha}^{\beta} \frac{\varepsilon_{r} \cdot \varepsilon_{0} \cdot L}{\ln \left(\frac{R_{2}^{\prime}}{R_{1}}\right)} \mathrm{d} \theta= \\ \int_{\alpha}^{\beta} \frac{\varepsilon_{r} \cdot \varepsilon_{0} \cdot L}{\ln \sqrt{\left(R_{2}^{2}+x^{2}\right)+2 x R_{2} \cos \theta}-\ln R_{1}} \mathrm{~d} \theta \end{gathered}$ (9)

 $u_{0}=\frac{\int_{\alpha}^{\beta} \frac{\varepsilon_{r} \cdot \varepsilon_{0} \cdot L}{\ln \sqrt{\left(R_{2}^{2}+x^{2}\right)-2 x R_{2} \cos \theta}-\ln R_{1}} \mathrm{~d} \theta-\int_{\alpha}^{\beta} \frac{\varepsilon_{r} \cdot \varepsilon_{0} \cdot L}{\ln \sqrt{\left(R_{2}^{2}+x^{2}\right)+2 x R_{2} \cos \theta}-\ln R_{1}} \mathrm{~d} \theta}{\int_{\alpha}^{\beta} \frac{\varepsilon_{r} \cdot \varepsilon_{0} \cdot L}{\ln \sqrt{\left(R_{2}^{2}+x^{2}\right)-2 x R_{2} \cos \theta}-\ln R_{1}} \mathrm{~d} \theta+\int_{\alpha}^{\beta} \frac{\varepsilon_{r} \cdot\varepsilon_{0} \cdot L}{\ln \sqrt{\left(R_{2}^{2}+x^{2}\right)+2 x R_{2} \cos \theta}-\ln R_{1}} \mathrm{~d} \theta} \cdot u$ (10)

3 圆弧柱面电容参数对倾斜仪影响

 图 6 输出信号随倾角变化 Fig. 6 Output signal varies with tilt angle

 图 7 理论输出与拟合直线的差值 Fig. 7 Differences between theoretical and fitted lines

 $\begin{gathered} \delta=\frac{\Delta Y_{\max }}{k \times\left(x_{\max }-x_{\min }\right)} \times 100 \%= \\ \frac{1.427\ 3 \times 10^{-5}}{0.028\ 5 \times[20-(-20)]} \times 100 \% \approx 0.001\ 2 \% \end{gathered}$

3.1 动极板半径变化对倾斜仪输出影响

 图 8 灵敏度随动极板半径变化曲线 Fig. 8 Sensitivity varies with the radius of the moving plate

 图 9 非线性度随动极板半径变化曲线 Fig. 9 Nonlinearity varies with the radius of the moving plate
3.2 圆弧柱面定极板圆心角变化对倾斜仪输出影响

 图 10 灵敏度随定极板圆心角变化曲线 Fig. 10 Sensitivity varies with central angle of the fixed plate

 图 11 非线性度随定极板圆心角变化曲线 Fig. 11 Nonlinearity varies with central angle of the fixed plate
3.3 极板间距变化对倾斜仪输出影响

 图 12 灵敏度随极板间距变化曲线 Fig. 12 Sensitivity varies with the gap between capacitor plates

 图 13 非线性度随极板间距变化曲线 Fig. 13 Nonlinearity varies with the gap between capacitor plates
3.4 吊丝长度变化对倾斜仪输出影响

 图 14 灵敏度随吊丝长度变化曲线 Fig. 14 Sensitivity varies with the length of the hanger wire

 图 15 非线性度随吊丝长度变化曲线 Fig. 15 Nonlinearity varies with the length of the hanger wire
3.5 动极板高度变化对倾斜仪输出影响

 图 16 灵敏度随动极板高度变化曲线 Fig. 16 Sensitivity varies with the height of the moving plate

 图 17 非线性度随动极板高度变化曲线 Fig. 17 Nonlinearity varies with the height of the moving plate
4 结语

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Theoretical Analysis of Arc Cylindrical Plate Differential Capacitor Applied in Two-Dimensional Tiltmeter
HU Yuanwang1,2,3     ZOU Tong1,2,3     MA Wugang1,2,3
1. Institute of Seismology, CEA, 40 Hongshance Road, Wuhan 430071, China;
2. Wuhan Gravitation and Solid Earth Tides, National Observation and Research Station, 40 Hongshance Road, Wuhan 430071, China;
3. Hubei Key Laboratory of Earthquake Early Warning, 40 Hongshance Road, Wuhan 430071, China
Abstract: We introduce a two-dimensional tiltmeter based on a cylindrical pendulum body and an arc cylindrical plate; we theoretically analyze the arc cylindrical plate differential capacitor. The analysis shows that to improve the sensitivity of the tiltmeter, it is necessary to appropriately increase the radius, the height of the cylindrical pendulum body and the length of the hanger wire, and to reduce the central angle of the arc cylindrical plate and the distance between the moving plate and the fixed plate; to reduce the non-linearity of the tiltmeter, it is necessary to appropriately increase the radius of the cylindrical pendulum body and the distance between the moving plate and the fixed plate, and reduce the central angle of the arc cylindrical plate, the height of the cylindrical pendulum body, the length of the hanger wire. Based on the above principles, we select a set of typical parameters, and the theoretical calculation shows that the sensitivity of the tiltmeter is 0.028 5 V/(″), and the non-linearity is 0.001 2%.
Key words: two-dimensional; differential capacitor; arc cylindrical plate; tiltmeter