﻿ 变面积型圆柱面电容式倾角传感器的输出特性
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 大地测量与地球动力学  2023, Vol. 43 Issue (4): 429-434  DOI: 10.14075/j.jgg.2023.04.018

### 引用本文

WANG Fuqian. Output Characteristics of Variable Area Cylindrical Capacitive Inclination Sensor[J]. Journal of Geodesy and Geodynamics, 2023, 43(4): 429-434.

### 项目来源

2020年四川省教育科研课题(SCJG20A275)；四川西南航空职业学院2022年度校级科研课题(XNHY-2022-01)。

### Foundation support

2020 Education and Scientific Research Project of Sichuan Province, No.SCJG20A275; 2022 School Level Scientific Research Project of Sichuan Southwest Vocational College of Civil Aviation, No.XNHY-2022-01.

### 第一作者简介

WANG Fuqian, professor, majors in sensing technology and computer simulation, E-mail: 13935511796@163.com.

### 文章历史

1. 四川西南航空职业学院机务学院，成都市学府大道 996号，610401

 图 1 变面积型圆柱面电容式倾角传感器结构示意图 Fig. 1 Structural diagram of variable area cylindrical capactive tilt sensor
1 圆柱面电容倾角传感器的电场

 图 2 变面积型圆柱面电容式倾角传感器的横截面 Fig. 2 Cross section of variable area cylindrical capactive tilt sensor

 $w_1=i \ln (z)$ (1)

 $\left\{\begin{array}{l} z_A=-R_1 \sin \left(\theta_1 / 2+\theta\right)+i R_1 \cos \left(\theta_1 / 2+\theta\right) \\ z_B=R_1 \sin \left(\theta_1 / 2-\theta\right)+i R_1 \cos \left(\theta_1 / 2-\theta\right) \\ z_C=R_2 \sin \left(\theta_2 / 2\right)+i R_2 \cos \left(\theta_2 / 2\right) \\ z_D=-R_2 \sin \left(\theta_2 / 2\right)+i R_2 \cos \left(\theta_2 / 2\right) \end{array}\right.$ (2)
 $\left\{\begin{array}{l} w_{1 A}=i \ln \left(z_A\right), w_B=i \ln \left(z_B\right) \\ w_C=i \ln \left(z_C\right), w_D=i \ln \left(z_D\right) \end{array}\right.$ (3)
 图 3 变换后的带形区域 Fig. 3 Transformed ribbon region

 $w_2=w_1-i v_{1 A}$ (4)

 图 4 再变换后的带形区域 Fig. 4 Retransformed ribbon region

 $\zeta=\mathrm{e}^{\frac{{\rm{ \mathsf{ π}}}}{h} w_2}$ (5)

 图 5 再变换后的上半平面 Fig. 5 Retransformed upper half plane

 $\left\{\begin{array}{l} \nabla^2 u=0 \\ u=-U_0, \eta=0, \xi_A \leqslant \xi \leqslant \xi_B \\ u=U_0, \eta=0, \xi_C \leqslant \xi \leqslant \xi_D \end{array}\right.$ (6)

 $\begin{gathered} u(\xi, \eta)=\frac{\eta}{{\rm{ \mathsf{ π}}}} \int_{-\infty}^{+\infty} \frac{f\left(\xi^{\prime}\right) \mathrm{d} \xi^{\prime}}{\left(\xi^{\prime}-\xi\right)^2+\eta^2}= \\ \frac{\eta}{{\rm{ \mathsf{ π}}}} \int_{\xi_C}^{\xi_D} \frac{U_0 \mathrm{~d} \xi^{\prime}}{\left(\xi^{\prime}-\xi\right)^2+\eta^2}+\frac{\eta}{{\rm{ \mathsf{ π}}}} \int_{\xi_A}^{\xi_B} \frac{\left(-U_0\right) \mathrm{d} \xi^{\prime}}{\left(\xi^{\prime}-\xi\right)^2+\eta^2}= \\ \frac{U_0}{{\rm{ \mathsf{ π}}}}\left(\arctan \frac{\xi_D-\xi}{\eta}-\arctan \frac{\xi_C-\xi}{\eta}\right)- \\ \frac{U_0}{{\rm{ \mathsf{ π}}}}\left(\arctan \frac{\xi_B-\xi}{\eta}-\arctan \frac{\xi_A-\xi}{\eta}\right) \end{gathered}$ (7)
 $\begin{gathered} w_2=i \ln (z)-i \ln \left(R_1\right)= \\ i \ln \left(\sqrt{x^2+y^2} \cdot \mathrm{e}^{i \arctan \frac{y}{x}}\right)-i \ln \left(R_1\right)= \\ i\left[\frac{1}{2} \ln \left(x^2+y^2\right)+i \arctan \frac{y}{x}\right]-i \ln \left(R_1\right)= \\ \frac{i}{2} \ln \left(x^2+y^2\right)-\arctan \frac{y}{x}-i \ln \left(R_1\right) \\ \zeta=\xi+i \eta=\mathrm{e}^{\frac{{\rm{ \mathsf{ π}}}}{h} w_2}=\mathrm{e}^{\frac{{\rm{ \mathsf{ π}}}}{h}\left[\frac{i}{2} \ln \left(x^2+y^2\right)-\arctan \frac{y}{x}-i \ln \left(R_1\right)\right]}= \\ \mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \arctan \frac{y}{x}} \cdot\left\{\cos \left[\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{1}{2} \ln \left(x^2+y^2\right)-\ln \left(R_1\right)\right)\right]+\right. \\ \left.i \sin \left[\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{1}{2} \ln \left(x^2+y^2\right)-\ln \left(R_1\right)\right)\right]\right\} \end{gathered}$ (8)

 $\left\{ \begin{array}{l} \xi_A=\mathrm{e}^{\frac{{\rm{ \mathsf{ π}}}}{h} \arctan \left[\cot \left(\frac{\theta_1}{2}+\theta\right)\right]} , \\ \ \ \ \ \xi_B=\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \arctan \left[\cot \left(\frac{\theta_1}{2}-\theta\right)\right]}\\ \xi_C=-\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \arctan \left[\cot \left(\frac{\theta_2}{2}\right)\right]}, \\ \xi_D=-\mathrm{e}^{\frac{{\rm{ \mathsf{ π}}}}{h} \arctan \left[\cot \left(\frac{\theta_2}{2}\right)\right]} \end{array} \right.$ (9)

 $\begin{array}{c} u(x, y)=\frac{U_0}{{\rm{ \mathsf{ π}}}}\left\{\arctan \frac{\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_2}{2}\right)}+\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \arctan \frac{y}{x}} \cdot \cos \left[\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{1}{2} \ln \left(x^2+y^2\right)-\ln \left(R_1\right)\right)\right]}{\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \arctan \frac{y}{x}} \cdot \sin \left[\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{1}{2} \ln \left(x^2+y^2\right)-\ln \left(R_1\right)\right)\right]}-\right. \\ \left.\arctan \frac{\mathrm{e}^{\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_2}{2}\right)}+\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \arctan \frac{y}{x}} \cdot \cos \left[\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{1}{2} \ln \left(x^2+y^2\right)-\ln \left(R_1\right)\right)\right]}{\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \arctan \frac{y}{x}} \cdot \sin \left[\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{1}{2} \ln \left(x^2+y^2\right)-\ln \left(R_1\right)\right)\right]}\right\}- \\ \frac{U_0}{{\rm{ \mathsf{ π}}}}\left\{\arctan \frac{\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta}{2}+\theta\right)}-\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \arctan \frac{y}{x}} \cdot \cos \left[\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{1}{2} \ln \left(x^2+y^2\right)-\ln \left(R_1\right)\right)\right]}{\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \arctan \frac{y}{x}} \cdot \sin \left[\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{1}{2} \ln \left(x^2+y^2\right)-\ln \left(R_1\right)\right)\right]}-\right. \\ \left.\arctan \frac{\mathrm{e}^{\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_1}{2}-\theta\right)}-\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \arctan \frac{y}{x}} \cdot \cos \left[\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{1}{2} \ln \left(x^2+y^2\right)-\ln \left(R_1\right)\right)\right]}{\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \arctan \frac{y}{x}} \cdot \sin \left[\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{1}{2} \ln \left(x^2+y^2\right)-\ln \left(R_1\right)\right)\right]}\right\} \end{array}$ (10)

 $\begin{gathered} u(\rho, \varphi)= \\ \frac{U_0}{{\rm{ \mathsf{ π}}}}\left\{\arctan \frac{\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_2}{2}\right)}+\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \varphi} \cdot \cos \left[\frac{{\rm{ \mathsf{ π}}}}{h}\left(\ln \rho-\ln R_1\right)\right]}{\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \varphi} \cdot \sin \left[\frac{{\rm{ \mathsf{ π}}}}{h}\left(\ln \rho-\ln R_1\right)\right]}-\arctan \frac{\mathrm{e}^{\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_2}{2}\right)}+\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \varphi} \cdot \cos \left[\frac{{\rm{ \mathsf{ π}}}}{h}\left(\ln \rho-\ln R_1\right)\right]}{\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \varphi} \cdot \sin \left[\frac{{\rm{ \mathsf{ π}}}}{h}\left(\ln \rho-\ln R_1\right)\right]}\right\}+ \\ \frac{U_0}{{\rm{ \mathsf{ π}}}}\left\{\arctan \frac{\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_1}{2}+\theta\right)}-\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \varphi} \cdot \cos \left[\frac{{\rm{ \mathsf{ π}}}}{h}\left(\ln \rho-\ln R_1\right)\right]}{\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \varphi} \cdot \sin \left[\frac{{\rm{ \mathsf{ π}}}}{h}\left(\ln \rho-\ln R_1\right)\right]}-\arctan \frac{\mathrm{e}^{\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_1}{2}-\theta\right)}-\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \varphi} \cdot \cos \left[\frac{{\rm{ \mathsf{ π}}}}{h}\left(\ln \rho-\ln R_1\right)\right]}{\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \varphi} \cdot \sin \left[\frac{{\rm{ \mathsf{ π}}}}{h}\left(\ln \rho-\ln R_1\right)\right]}\right\} \end{gathered}$ (11)

 $\begin{array}{c} \boldsymbol{E}=-\left(\frac{\partial u}{\partial \rho} \boldsymbol{e}_\rho+\frac{1}{\rho} \frac{\partial u}{\partial \varphi} \boldsymbol{e}_{\varphi}\right)=\frac{U_0}{h}\left\{\frac{\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \varphi}}{\rho} \frac{\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \varphi} \cos \left[2\left(\frac{{\rm{ \mathsf{ π}}}}{h}\left(\ln \rho-\ln R_1\right)\right)\right]+\mathrm{e}^{\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\rho_2}{2}\right)} \cos \left(\frac{{\rm{ \mathsf{ π}}}}{h}\left(\ln \rho-\ln R_1\right)\right)}{\mathrm{e}^{-2 \frac{{\rm{ \mathsf{ π}}}}{h} \varphi}+\mathrm{e}^{2 \frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\rho_2}{2}\right)}+2 \mathrm{e}^{\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\rho_2}{2}\right)} \mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \varphi} \cos \left(\frac{{\rm{ \mathsf{ π}}}}{h}\left(\ln \rho-\ln R_1\right)\right)}-\right. \\ \frac{\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \varphi}}{\rho} \frac{\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \varphi} \cos \left[2\left(\frac{{\rm{ \mathsf{ π}}}}{h}\left(\ln \rho-\ln R_1\right)\right)\right]+\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_2}{2}\right)} \cos \left(\frac{{\rm{ \mathsf{ π}}}}{h}\left(\ln \rho-\ln R_1\right)\right)}{\mathrm{e}^{-2 \frac{{\rm{ \mathsf{ π}}}}{h} \varphi}+\mathrm{e}^{-2 \frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_2}{2}\right)}+2 \mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_2}{2}\right)} \mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \varphi} \cos \left(\frac{{\rm{ \mathsf{ π}}}}{h}\left(\ln \rho-\ln R_1\right)\right)}+ \\ \frac{\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \varphi}}{\rho} \frac{\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \varphi} \cos \left[2\left(\frac{{\rm{ \mathsf{ π}}}}{h}\left(\ln \rho-\ln R_1\right)\right)\right]-\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_1}{2}+\theta\right)} \cos \left(\frac{{\rm{ \mathsf{ π}}}}{h}\left(\ln \rho-\ln R_1\right)\right)}{\mathrm{e}^{-2 \frac{{\rm{ \mathsf{ π}}}}{h} \varphi}+\mathrm{e}^{-2 \frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_1}{2}+\theta\right)}-2 \mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_1}{2}+\theta\right)} \mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \varphi} \cos \left(\frac{{\rm{ \mathsf{ π}}}}{h}\left(\ln \rho-\ln R_1\right)\right)}- \\ \left.\frac{\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \varphi}}{\rho} \frac{\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \varphi} \cos \left[2\left(\frac{{\rm{ \mathsf{ π}}}}{h}\left(\ln \rho-\ln R_1\right)\right)\right]-\mathrm{e}^{\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_1}{2}-\theta\right)} \cos \left(\frac{{\rm{ \mathsf{ π}}}}{h}\left(\ln \rho-\ln R_1\right)\right)}{\mathrm{e}^{-2 \frac{{\rm{ \mathsf{ π}}}}{h} \varphi}+\mathrm{e}^{2 \frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_1}{2}-\theta\right)}-2 \mathrm{e}^{\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_1}{2}-\theta\right)} \mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \varphi} \cos \left(\frac{{\rm{ \mathsf{ π}}}}{h}\left(\ln \rho-\ln R_1\right)\right)}\right\}\boldsymbol{e}_{\rho}- \\ \frac{U_0}{{\rm{ \mathsf{ π}}}}\left\{-\frac{\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \varphi}}{\rho} \frac{\mathrm{e}^{\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_2}{2}\right)} \sin \left[\frac{{\rm{ \mathsf{ π}}}}{h}\left(\ln \rho-\ln R_1\right)\right]}{\mathrm{e}^{\frac{2 {\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_2}{2}\right)}+\mathrm{e}^{-2 \frac{{\rm{ \mathsf{ π}}}}{h} \varphi}+2 \mathrm{e}^{\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_2}{2}\right)} \mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \varphi} \cos \left[\frac{{\rm{ \mathsf{ π}}}}{h}\left(\ln \rho-\ln R_1\right)\right]}+\right. \\ \frac{\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \varphi}}{\rho} \frac{\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_2}{2}\right)} \sin \left[\frac{{\rm{ \mathsf{ π}}}}{h}\left(\ln \rho-\ln R_1\right)\right]}{\mathrm{e}^{-2 \frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_2}{2}\right)}+\mathrm{e}^{-2 \frac{{\rm{ \mathsf{ π}}}}{h} \varphi}+2 \mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_2}{2}\right)} \mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \varphi} \cos \left[\frac{{\rm{ \mathsf{ π}}}}{h}\left(\ln \rho-\ln R_1\right)\right]}- \\ \frac{\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \varphi}}{\rho} \frac{\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_1}{2}+\theta\right)} \sin \left[\frac{{\rm{ \mathsf{ π}}}}{h}\left(\ln \rho-\ln R_1\right)\right]}{\mathrm{e}^{-2 \frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_1}{2}+\theta\right)}+\mathrm{e}^{-2 \frac{{\rm{ \mathsf{ π}}}}{h} \varphi}-2 \mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_1}{2}+\theta\right)} \mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \varphi} \cos \left[\frac{{\rm{ \mathsf{ π}}}}{h}\left(\ln \rho-\ln R_1\right)\right]}+ \\ \left.\frac{\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \varphi}}{\rho} \frac{\mathrm{e}^{\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_1}{2}-\theta\right)} \sin \left[\frac{{\rm{ \mathsf{ π}}}}{h}\left(\ln \rho-\ln R_1\right)\right]}{{{\rm{e}}^{2\frac{{\rm{ \mathsf{ π}}} }{h}\left( {\frac{{\rm{ \mathsf{ π}}} }{2} - \frac{{{\theta _1}}}{2} - \theta } \right)}} + {{\rm{e}}^{ - 2\frac{{\rm{ \mathsf{ π}}} }{h}\varphi }} - 2{{\rm{e}}^{\frac{{\rm{ \mathsf{ π}}} }{h}\left( {\frac{{\rm{ \mathsf{ π}}} }{2} - \frac{{{\theta _1}}}{2} - \theta } \right)}}{{\rm{e}}^{ - \frac{{\rm{ \mathsf{ π}}} }{h}\varphi }}\cos \left[ {\frac{{\rm{ \mathsf{ π}}} }{h}\left( {\ln \rho - \ln {R_1}} \right)} \right]}\right\} \boldsymbol{e}_{\varphi} \end{array}$ (12)

 图 6 圆柱面电容倾角传感器的电场 Fig. 6 Electric field of cylindrical capactive tilt sensor
2 圆柱面角位移电容传感器的输出特性 2.1 圆柱面角位移电容传感器的电容

 $\begin{gathered} \sigma=\left|-\varepsilon_0 \varepsilon_r E_\rho\right|_{\rho=R_1} \mid=\varepsilon_0 \varepsilon_r \frac{U_0}{h R_1} \cdot \\ {\left[\frac{\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \varphi}}{\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \varphi}+\mathrm{e}^{\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\partial_2}{2}\right)}}-\frac{\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \varphi}}{\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h} \varphi}+\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\partial_2}{2}\right)}}\right]} \end{gathered}$ (13)

 $\begin{gathered} Q=\int_{\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_1}{2}+\theta\right)}^{\left(\frac{{\rm{ \mathsf{ π}}}}{2}+\frac{\theta_1}{2}+\theta\right)} \sigma \cdot R_1 \cdot L \mathrm{~d} \varphi= \\ \varepsilon_0 \varepsilon_r \frac{U_0 L \ln \left(R_1\right)}{2 {\rm{ \mathsf{ π}}} h}\left\{\ln \left[\frac{\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}+\frac{\theta_1}{2}+\theta\right)}+\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_2}{2}\right)}}{\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_1}{2}+\theta\right)}+\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_2}{2}\right)}}\right]-\right.\\ \left.\ln \left[\frac{\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}+\frac{\theta_1}{2}+\theta\right)}+\mathrm{e}^{\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_2}{2}\right)}}{\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_1}{2}+\theta\right)}+\mathrm{e}^{\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_2}{2}\right)}}\right]\right\} \end{gathered}$ (14)

 $\begin{array}{c} C=\frac{Q}{\Delta U}=\frac{Q}{U_0}= \\ \varepsilon_0 \varepsilon_r \frac{L \ln \left(R_1\right)}{2 {\rm{ \mathsf{ π}}} \ln \left(R_2 / R_1\right)}\left\{\ln \left[\frac{\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}+\frac{\theta_1}{2}+\theta\right)}+\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_2}{2}\right)}}{\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_1}{2}+\theta\right)}+\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_2}{2}\right)}}\right]-\right. \\ \left.\ln \left[\frac{\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}+\frac{\theta_1}{2}+\theta\right)}+\mathrm{e}^{\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_2}{2}\right)}}{\mathrm{e}^{-\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_1}{2}+\theta\right)}+\mathrm{e}^{\frac{{\rm{ \mathsf{ π}}}}{h}\left(\frac{{\rm{ \mathsf{ π}}}}{2}-\frac{\theta_2}{2}\right)}}\right]\right\} \\ \end{array}$ (15)

2.2 圆柱面电容式倾角传感器的输出特性曲线

 图 7 变面积型圆柱面电容式倾角传感器的输出特性曲线 Fig. 7 Output characteristic curve of variable area cylindrical capactive tilt sensor
2.3 圆柱面电容式倾角传感器输出特性分析 2.3.1 输出特性与θ1θ2的关系

MATLAB软件的数值模拟结果表明：1)当θ1θ2θ1θ2时，该传感器的线性度和灵敏度均较差(图 8(a))；2)θ1=θ2时，该传感器的平均灵敏度及线性度均有较大幅度的提高(图 8(b))；3)当θ1θ2θ1θ2时，该传感器具有较好的线性度和灵敏度，输出性特性比较理想(图(c)~(d))，且在一定的范围内随着θ2θ1差值的增大，该传感器灵敏度和线性度均得到提高，输出特性得到改善。

 图 8 圆柱面电容倾角传感器的输出特性曲线 Fig. 8 Output characteristic curve of cylindrical capactive tilt sensor
2.3.2 输出特性与R1R2的关系

1) θ1θ2。MATLAB软件的数值模拟结果表明，当θ1θ2保持不变且θ1θ2时，随着传感器内、外半径差的减小，传感器的输出特性变差(图 9)。

 图 9 圆柱面电容倾角传感器的输出特性曲线 Fig. 9 Output characteristic curve of cylindrical capactive tilt sensor

2) θ1=θ2。数值模拟结果表明，当θ1θ2保持不变且θ1=θ2时，随着传感器内、外半径差的减小，传感器的输出特性变差(图 10)。

 图 10 圆柱面电容倾角传感器的输出特性曲线 Fig. 10 Output characteristic curve of cylindrical capactive tilt sensor

3) θ1>θ2。数值模拟结果表明，当θ1θ2保持不变且θ1>θ2时，在θ1θ2取值合适的情况下，随着传感器内、外半径差的减小，传感器的灵敏度和线性度均提高，输出特性得以改善(图 11)。

 图 11 圆柱面电容倾角传感器的输出特性曲线 Fig. 11 Output characteristic curve of cylindrical capactive tilt sensor
2.3.3 改善传感器输出特性的途径

3 结语

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Output Characteristics of Variable Area Cylindrical Capacitive Inclination Sensor
WANG Fuqian1
1. Maintenance College, Sichuan Southwest Vocational College of Civil Aviation, 996 Xuefu Road, Chengdu 610401, China
Abstract: To study the output characteristics of the variable area cylindrical capacitive inclination sensor considering the edge effect, this paper combines the conformal transformation method, Green's function method and computer numerical simulation to discuss the electric field of the variable area cylindrical capacitive inclination sensor, give the expression of its capacitance, draw the variation curve of the sensor capacitance with the inclination, and analyze the factors affecting the linearity and sensitivity of the sensor. It points out how to improve the output characteristics of the sensor. The numerical simulation results show that when the center angle of the outer plate of the sensor is smaller than that of the inner plate, reducing the distance between the two plates can increase sensor linearity and sensitivity at the same time. Because the lateral edge effect of the sensor plate is considered in this paper, the results obtained have high accuracy.
Key words: variable area cylindrical capacitive inclination sensor; edge effect; conformal transformation method; Green function method; output characteristics