{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T03:42:43Z","timestamp":1777347763405,"version":"3.51.4"},"reference-count":28,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T00:00:00Z","timestamp":1558915200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>The measurement results of a single-excitation petal-shaped capacitive encoder show strong periodic characteristics for nonlinear errors. This paper presents the analysis of periodic nonlinear errors in a single-excitation petal-shaped encoder in terms of three main aspects\u2014sensitive structure processing error, circuit demodulation error, and installation error. Analytical and simulation results confirm that the first-, second-, and fourth-periodic electrical errors are caused by the misalignment of circuit parameters, non-uniform segmentation of the processing error, and cross interference of the electric field, respectively. Further experimental investigation reveals that the mechanical periodic error is caused by installation misalignment. Based on these analytical, simulation, and experimental results, the design of the capacitive encoder was optimized and a method based on harmonic components was applied to compensate the periodic nonlinear error of the encoder. Measurement results shows that the prototype which has 180 petal-shaped numbers can achieve a reduction of periodic nonlinear errors to less than 0.02\u00b0 and its accuracy can be improved to 0.0006\u00b0 after compensation over the full measurement range.<\/jats:p>","DOI":"10.3390\/s19102412","type":"journal-article","created":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T11:19:27Z","timestamp":1558955967000},"page":"2412","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Periodic Nonlinear Error Analysis and Compensation of a Single-Excited Petal-Shaped Capacitive Encoder to Achieve High-Accuracy Measurement"],"prefix":"10.3390","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4030-0716","authenticated-orcid":false,"given":"Bo","family":"Hou","sequence":"first","affiliation":[{"name":"Engineering Research Center for Navigation Technology, Department of Precision Instrument, Tsinghua University, Beijing 100084, China"}]},{"given":"Bin","family":"Zhou","sequence":"additional","affiliation":[{"name":"Engineering Research Center for Navigation Technology, Department of Precision Instrument, Tsinghua University, Beijing 100084, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0018-3711","authenticated-orcid":false,"given":"Xiang","family":"Li","sequence":"additional","affiliation":[{"name":"Engineering Research Center for Navigation Technology, Department of Precision Instrument, Tsinghua University, Beijing 100084, China"}]},{"given":"Bowen","family":"Xing","sequence":"additional","affiliation":[{"name":"Engineering Research Center for Navigation Technology, Department of Precision Instrument, Tsinghua University, Beijing 100084, China"}]},{"given":"Luying","family":"Yi","sequence":"additional","affiliation":[{"name":"Engineering Research Center for Navigation Technology, Department of Precision Instrument, Tsinghua University, Beijing 100084, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3189-7562","authenticated-orcid":false,"given":"Qi","family":"Wei","sequence":"additional","affiliation":[{"name":"Engineering Research Center for Navigation Technology, Department of Precision Instrument, Tsinghua University, Beijing 100084, China"}]},{"given":"Rong","family":"Zhang","sequence":"additional","affiliation":[{"name":"Engineering Research Center for Navigation Technology, Department of Precision Instrument, Tsinghua University, Beijing 100084, China"}]}],"member":"1968","published-online":{"date-parts":[[2019,5,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"607","DOI":"10.1109\/JSEN.2003.817182","article-title":"A contactless capacitive angular-position sensor","volume":"3","author":"Gasulla","year":"2003","journal-title":"IEEE Sens. J."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1109\/TIM.2014.2328456","article-title":"A Capacitive Rotary Encoder Based on Quadrature Modulation and Demodulation","volume":"64","author":"Zheng","year":"2015","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"514","DOI":"10.1109\/TIM.2006.864243","article-title":"Capacitive Angular-Position Sensor with Electrically Floating Conductive Rotor and Measurement Redundancy","volume":"55","author":"Ferrari","year":"2006","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"282","DOI":"10.1016\/j.measurement.2014.12.026","article-title":"Development of a capacitive angular velocity sensor for the alarm and trip applications","volume":"63","author":"Azimloo","year":"2015","journal-title":"Measurement"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"382","DOI":"10.1108\/SR-11-2012-683","article-title":"A capacitive angular sensor with flexible digitated electrodes","volume":"34","year":"2014","journal-title":"Sens. Rev."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Kimura, F., Gondo, M., Yamamoto, A., and Higuchi, T. (2009, January 3\u20135). Resolver compatible capacitive rotary position sensor. Proceedings of the 2009 35th Annual Conference of IEEE Industrial Electronics, Porto, Portugal.","DOI":"10.1109\/IECON.2009.5414847"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1975","DOI":"10.1007\/s00202-018-0677-9","article-title":"A new capacitive rotary encoder based on analog synchronous demodulation","volume":"100","author":"Karali","year":"2018","journal-title":"Electr. Eng."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Jia, B., He, L., Yan, G., and Feng, Y. (2016). A Differential Reflective Intensity Optical Fiber Angular Displacement Sensor. Sensors, 16.","DOI":"10.3390\/s16091508"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"4408","DOI":"10.1109\/TIE.2014.2387794","article-title":"A Novel Absolute Magnetic Rotary Sensor","volume":"62","author":"Zhang","year":"2015","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1109\/JSEN.2014.2345587","article-title":"A Novel Approach of an Absolute Encoder Coding Pattern","volume":"15","author":"Dziwinski","year":"2015","journal-title":"IEEE Sens. J."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Paul, S., and Chang, J.H. (2016, January 23\u201326). Design of absolute encoder disk coding based on affine n digit N-ary gray code. Proceedings of the 2016 IEEE International Instrumentation and Measurement Technology Conference Proceedings, Taipei, Taiwan.","DOI":"10.1109\/I2MTC.2016.7520384"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Li, K., Li, Y., and Han, Y. (2017). An EM Induction Hi-Speed Rotation Angular Rate Sensor. Sensors, 17.","DOI":"10.3390\/s17030610"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Hou, B., Zhou, B., Song, M., Lin, Z., and Zhang, R. (2016). A Novel Single-Excitation Capacitive Angular Position Sensor Design. Sensors, 16.","DOI":"10.3390\/s16081196"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Hou, B., Tian, Z., Li, C., Wei, Q., Zhou, B., and Zhang, R. (November, January 29). A Capacitive Rotary Encoder with a Novel Sensitive Electrode. Proceedings of the 2017 IEEE SENSORS, Glasgow, UK.","DOI":"10.1109\/ICSENS.2017.8234143"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2614","DOI":"10.1109\/TMAG.2013.2245316","article-title":"A Demodulation Technique for Spindle Rotor Position Detection with Resolver","volume":"49","author":"Aung","year":"2013","journal-title":"IEEE Trans. Magn."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"326","DOI":"10.1109\/TIE.2011.2143370","article-title":"High-Accuracy All-Digital Resolver-to-Digital Conversion","volume":"59","author":"Gross","year":"2012","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_17","first-page":"2469","article-title":"An Annular Planar-capacitive Tilt Sensor with a 360\u00b0 Measurement Range","volume":"63","author":"Hu","year":"2016","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1596","DOI":"10.1109\/JSEN.2017.2654500","article-title":"Sensing Mechanism and Error Analysis of a Capacitive Long-Range Displacement Nanometer Sensor Based on Time Grating","volume":"17","author":"Peng","year":"2017","journal-title":"IEEE Sens. J."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"9595","DOI":"10.1109\/TIE.2017.2726982","article-title":"Advances in Capacitive, Eddy Current, and Magnetic Displacement Sensors and Corresponding Interfaces","volume":"64","author":"George","year":"2017","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1016\/j.sna.2007.07.007","article-title":"Voltage-induction type electrostatic film motor driven by two- to four-phase ac voltage and electrostatic induction","volume":"140","author":"Yamashita","year":"2007","journal-title":"Sens. Actuators A Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"7377","DOI":"10.1109\/TIE.2017.2698417","article-title":"Features of Capacitive Displacement Sensing That Provide High-Accuracy Measurements with Reduced Manufacturing Precision","volume":"64","author":"Peng","year":"2017","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1310","DOI":"10.5370\/JEET.2013.8.6.1310","article-title":"Software Resolver-to-Digital Converter for Compensation of Amplitude Imbalances using D-Q Transformation","volume":"8","author":"Kim","year":"2013","journal-title":"J. Electr. Eng. Technol."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"3382","DOI":"10.1364\/AO.20.003382","article-title":"Determination and correction of quadrature fringe measurement errors in interferometers","volume":"20","author":"Heydemann","year":"1981","journal-title":"Appl. Opt."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"794","DOI":"10.3788\/OPE.20152303.0794","article-title":"Dynamic extracting and compensation of system error for rotary inductosyn","volume":"23","author":"Yuan","year":"2015","journal-title":"Opt. Precis. Eng."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1190","DOI":"10.6113\/JPE.2016.16.3.1190","article-title":"Classification and Compensation of dc-offset error and scale error in Resolver Signals","volume":"16","author":"Lee","year":"2015","journal-title":"J. Power Electron."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Xiang, K., Wang, W., Qiu, R., Mei, D., and Chen, Z. (2017). A T-Type Capacitive Sensor Capable of Measuring5-DOF Error Motions of Precision Spindles. Sensors, 17.","DOI":"10.3390\/s17091975"},{"key":"ref_27","first-page":"31","article-title":"High-precision Time-grating Displacement Sensor Based on Harmonic Wave Correcting Method","volume":"27","author":"Peng","year":"2006","journal-title":"Chin. J. Sci. Instrum."},{"key":"ref_28","first-page":"611","article-title":"Error testing and compensation of an inductosyn based angular measurement system","volume":"56","author":"Li","year":"2016","journal-title":"J. Tsinghua Univ. (Sci. Technol.)"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/19\/10\/2412\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:55:20Z","timestamp":1760187320000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/19\/10\/2412"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5,27]]},"references-count":28,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2019,5]]}},"alternative-id":["s19102412"],"URL":"https:\/\/doi.org\/10.3390\/s19102412","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,5,27]]}}}