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## Analytical model of electrostatic force generated by edge effect of a Kelvin capacitor based on conformal transformation

Li Shi-Song, Zhang Zhong-Hua, Zhao Wei, Huang Song-Ling, Fu Zhuang
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• #### 摘要

提出了一种用于弱力测量的带电Kelvin电容器产生静电力的新型工作方式. 该工作方式的优点是在机械上可明显简化静电力的实现, 避免了带电Kelvin电容器在传统工作方式下需要同步移动中心电极和保护电极的困难; 缺点是因Kelvin 电容器的边缘效应, 会不可避免地随之产生一个小的静电力分量, 需要准确建立它的数学模型, 方可清晰地评估它对测量结果的影响. 为解决此问题, 本文采用部分电容模拟和保角变换, 将二维电场的计算转换为一维电场计算问题, 并基于此, 得到了一种合理表征Kelvin电容器边缘效应所产生静电力分量的解析模型. 通过与一个具体计算实例的有限元计算求解结果相比较, 分析了所建立的解析模型的准确性.

#### Abstract

We present a new realization of weak electrostatic force based on Kelvin capacitor, which, compared with the conventional realization, simplifies the mechanical design without synchronously moving the central and guard-ring electrodes. However, a residual force due to the edge effect of the capacitor should be accurately known. In this paper, an analytic model of the electrostatic force generated from the edge effect is presented based on conformal transformations. The accuracy of the model is verified by the finite element calculation based on a micro-Newton weak force generation.

#### 作者及机构信息

###### 1. 清华大学电机系, 北京 100084; 2. 中国计量科学研究院, 北京 100029
• 基金项目: 国家科技支撑计划(批准号: 2011AA06Z000)、国家自然科学基金(批准号: 51477160)和国家质检总局公益性行业科研专项项目(批准号: 201010010)资助的课题.

#### Authors and contacts

###### 1. Department of Electrical Engineering, Tsinghua University, Beijing 100084, China; 2. National Institute of Metrology, Beijing 100029, China
• Funds: Project supported by the National Key Technology R&D Program (Grant No. 2011AA06Z000), the National Natural Science Foundation of China (Grant No. 51477160), and the National Department Public Benefit Research Foundation, China (Grant No. 201010010).

#### 参考文献

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#### 施引文献

•  [1] Chan H B, Aksyuk V A, Kleiman R N, Bishop D J, Federico C 2001 Science 291 1941 [2] Craighead H G 2000 Science 290 1532 [3] Gross L, Mohn F, Moll N, Schuler B, Criado A, Guitián E, Pena D, Gourdon A, Meyer G 2012 Science 337 1326 [4] Solin S A, Hines D R, Rowe A C H, Tsai J S, Pashkin Y A, Chung S J, Goel N, Santos M B 2002 Appl. Phys. Lett. 80 4012 [5] Peng Y Z, Li Y, Ru B, Huo D X, Qian Z H 2014 Chin. Phys. B 23 097503 [6] Filippov A E, Klafter J, Urbakh M 2004 Phys. Rev. Lett. 92 135503 [7] Bustamante C, Bryant Z, Smith S B 2003 Nature 421 423 [8] Juan M L, Righini M, Quidant R 2011 Nature Photon. 5 349 [9] Huang X F, Li S J, Zhou D H, Zhao G J, Wang G Q, Xu J R 2014 Acta Phys. Sin. 63 178802 (in Chinese) [黄雪峰, 李盛姬, 周东辉, 赵冠军, 王关晴, 徐江荣 2014 物理学报 63 178802] [10] Rugar D, Budakian R, Mamin H J, Chui B W 2004 Nature 430 329 [11] Beer W, Fasel W, Moll E, Richard P, Schneiter U, Thalmann R, Egger J 2002 Metrologia 39 263 [12] Fang M H, Wei Z Y, Zhang Z X, Zhu L, Fu Y, Shi M, Li G W, Guo G 2013 Chin. Phys. B 22 116105 [13] Huan Q, Hu H, Pan L D, Xiao J, Du S X, Gao H J 2010 Chin. Phys. B 19 080517 [14] Liang P B, Lei J J, Liu Z H, Zhang Y, Yuan L B 2014 Chin. Phys. B 23 088702 [15] Ren H L, Ding P F, Li X Y 2012 Acta Phys. Sin. 61 210701 (in Chinese) [任洪亮, 丁攀峰, 李小燕 2012 物理学报 61 210701] [16] Berman G P, Doolen G D, Hammel P C, Tsifrinovich V I 2000 Phys. Rev. B 61 14694 [17] Girard G 1994 Metrologia 31 317 [18] Li S S, Zhang Z H, Zhao W, Li Z K, Huang S L 2014 Chin. Phys. B (accept for publication) [19] Li S S, Han B, Li Z, Lan J 2012 Measurement 45 1 [20] Zimmerman N M 2010 Phys. Today 63 68 [21] Pratt J R, Kramar J A, Newell D B, Smith D T 2005 Meas. Sci. Technol. 16 2129 [22] Hamilton C A 2000 Rev. Sci. Instrum. 71 3611 [23] Trapon G, Thévenot O, Lacueille J C, Poirier W 2003 Metrologia 40 159 [24] Heerens W C, Vermeulen F C 1975 J. Appl. Phys. 46 2486 [25] Li W 2011 Chin. Phys. B 20 116201 [26] Bai Z W 2004 Acta Phys. Sin. 53 2472 (in Chinese) [白占武 2004 物理学报 53 2472] [27] He D W, Cheng X H, Wang Z J, Xu D W, Song Z R, Yu Y H 2011 Chin. Phys. B 20 010210
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##### 出版历程
• 收稿日期:  2014-09-24
• 修回日期:  2014-10-13
• 刊出日期:  2015-03-05

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