{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T21:52:33Z","timestamp":1777499553431,"version":"3.51.4"},"reference-count":31,"publisher":"MDPI AG","issue":"18","license":[{"start":{"date-parts":[[2021,9,20]],"date-time":"2021-09-20T00:00:00Z","timestamp":1632096000000},"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 inductive displacement sensor is widely used in active magnetic bearing (AMB) systems to detect rotor displacement in real time, and the performance of the sensor directly affects the performance of AMB. At present, most theoretical studies on the working principle of inductive displacement sensor are based on a traditional mathematical model, ignoring the influence of the core magnetic resistance and core eddy current, which will lead to a certain error between the theoretical analysis of the sensor output characteristics and the actual situation. In this regard, based on the theory of electromagnetic field and circuit, an improved theoretical model of the inductive sensor was established in this paper by introducing the complex permeability, by which the influence of core eddy current on magnetic field can be taken into account. In order to verify the improved model, an eight-pole radial self-inductive displacement sensor with an air gap of 1 mm was designed. Then the electromagnetic field of the designed sensor was simulated by a finite element software and the GW LCR-6100 measuring instrument was used to measure the changes of the inductance and resistance of the designed sensor core coils with the rotor displacement at 20\u2013100 kHz. The results demonstrated that there is a good linear relationship between the impedance change of the sensor coils and the rotor displacement within the measurement range of \u22120.4 ~ +0.4 mm. At the same time, compared with the traditional model, the sensitivity of the improved theoretical model is closer to the results from FEM and experiment, and the accuracy of the sensitivity of the improved theoretical model can be approximately doubled, despite there are certain differences with the experimental situation. Therefore, the improved theoretical model considering complex permeability is of great significance for studying the influence of core eddy current on the coil impedance of sensor.<\/jats:p>","DOI":"10.3390\/s21186292","type":"journal-article","created":{"date-parts":[[2021,9,21]],"date-time":"2021-09-21T22:35:20Z","timestamp":1632263720000},"page":"6292","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Research on Coil Impedance of Self-Inductive Displacement Sensor Considering Core Eddy Current"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8110-9654","authenticated-orcid":false,"given":"Zongqiang","family":"Ren","sequence":"first","affiliation":[{"name":"School of Electrical Engineering, Shandong University, Jinan 250061, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7897-3756","authenticated-orcid":false,"given":"Hongwei","family":"Li","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering, Shandong University, Jinan 250061, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wentao","family":"Yu","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering, Shandong University, Jinan 250061, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,9,20]]},"reference":[{"key":"ref_1","unstructured":"Schweitzer, G., and Maslen, E.H. (2009). Magnetic Bearing\u2014Theory, Design, and Application to Rotating Machinery, Springer."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1959","DOI":"10.1109\/TMAG.2011.2180731","article-title":"Integral Design and Analysis of Passive Magnetic Bearing and Active Radial Magnetic Bearing for Agile Satellite Application","volume":"48","author":"Han","year":"2012","journal-title":"IEEE Trans. Magn."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2196","DOI":"10.1109\/TIE.2016.2626238","article-title":"A Novel 4-DOF Hybrid Magnetic Bearing for DGMSCMG","volume":"64","author":"Sun","year":"2017","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2205","DOI":"10.1109\/TIE.2016.2627021","article-title":"Weight-Reduction Design Based on Integrated Radial-Axial Magnetic Bearing of a Large-Scale MSCMG for Space Station Application","volume":"64","author":"Han","year":"2017","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"3319","DOI":"10.1109\/TIE.2016.2542786","article-title":"Stable Control of High-Speed Rotor Suspended by Superconducting Magnetic Bearings and Active Magnetic Bearings","volume":"64","author":"Tang","year":"2017","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2617","DOI":"10.1109\/JSEN.2017.2677526","article-title":"Demodulation Techniques for Self-Oscillating Eddy-Current Displacement Sensor Interfaces: A Review","volume":"17","author":"Chaturvedi","year":"2017","journal-title":"IEEE Sens. J."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Dong, Z., Xu, F., Sun, X., and Liu, W. (2019). A Laser-Based On-Machine Measuring System for Profile Accuracy of Double-Headed Screw Rotor. Sensors, 19.","DOI":"10.3390\/s19235059"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1538","DOI":"10.1109\/JSEN.2007.908232","article-title":"New Noncontacting Inductive Analog Proximity and Inductive Linear Displacement Sensors for Industrial Automation","volume":"7","author":"Fericean","year":"2007","journal-title":"IEEE Sens. J."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Filatov, A.V., and Hawkins, L.A. (2010, January 14\u201318). An Axial Position Sensor for Active Magnetic Bearings. Proceedings of the ASME Turbo Expo: Power for Land, Sea and Air, Glasgow, UK.","DOI":"10.1115\/GT2010-22524"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"12640","DOI":"10.3390\/s140712640","article-title":"Inductive Displacement Sensors with a Notch Filter for an Active Magnetic Bearing System","volume":"14","author":"Chen","year":"2014","journal-title":"Sensors"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"4378","DOI":"10.1109\/JSEN.2017.2710135","article-title":"Optimized Differential Self-Inductance Displacement Sensor for Magnetic Bearings: Design, Analysis and Experiment","volume":"17","author":"Wang","year":"2017","journal-title":"IEEE Sens. J."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"5256","DOI":"10.1109\/JSEN.2018.2839730","article-title":"An Inductive Linear Displacement Sensor Based on Planar Coils","volume":"18","author":"Tang","year":"2018","journal-title":"IEEE Sens. J."},{"key":"ref_13","first-page":"1","article-title":"Magnetic Bearings and Synchronous Magnetic Axial Coupling for the Enhancement of the Driving Performance of Magnetic Wireless Pumps","volume":"50","author":"Kim","year":"2014","journal-title":"IEEE Trans. Magn."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2784","DOI":"10.1109\/TMECH.2016.2582644","article-title":"Modeling and Identification of a Solid-Core Active Magnetic Bearing Including Eddy Currents","volume":"21","author":"Zhou","year":"2016","journal-title":"IEEE\/ASME Trans. Mechatron."},{"key":"ref_15","first-page":"2027","article-title":"Loss Calculation and Thermal Analysis of Rotors Supported by Active Magnetic Bearings for High-Speed Permanent-Magnet Electrical Machines","volume":"63","author":"Huang","year":"2016","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1341","DOI":"10.1109\/20.996021","article-title":"Analytical method for eddy current loss in laminated rotors with magnetic bearings","volume":"38","author":"Sun","year":"2002","journal-title":"IEEE Trans. Magn."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"3021","DOI":"10.1109\/TMAG.2012.2208944","article-title":"Advanced Iron-Loss Estimation for Nonlinear Material Behavior","volume":"48","author":"Eggers","year":"2012","journal-title":"IEEE Trans. Magn."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1109\/TMAG.2015.2449905","article-title":"Effects of Eddy Current in Electrical Connection Surface of Laminated Cores on High-Speed PM Motor Supported by Active Magnetic Bearings","volume":"51","author":"Fang","year":"2015","journal-title":"IEEE Trans. Magn."},{"key":"ref_19","first-page":"1","article-title":"Prediction of 3-D High-Frequency Eddy Current Loss in Rotor Magnets of SPM Machines","volume":"52","author":"Nair","year":"2016","journal-title":"IEEE Trans. Magn."},{"key":"ref_20","first-page":"1992","article-title":"Three-dimensional Analytical Model of Eddy Current Loss of Permanent Magnet in High Frequency Axial Flux Permanent Magnet Machine","volume":"41","author":"Tong","year":"2021","journal-title":"Trans. Chin. Electr. Soci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1248","DOI":"10.1109\/TMAG.2005.844847","article-title":"Analytic model for a nonlaminated cylindrical magnetic actuator including eddy currents","volume":"41","author":"Lei","year":"2005","journal-title":"IEEE Trans. Magn."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"803","DOI":"10.1109\/TPWRD.2009.2036358","article-title":"The Influence of Complex Permeability on the Broadband Frequency Response of a Power Transformer","volume":"25","author":"Mitchell","year":"2010","journal-title":"IEEE Trans. Power Del."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1378","DOI":"10.1109\/TMAG.2010.2081352","article-title":"A New Formulation of Anisotropic Equivalent Conductivity in Laminations","volume":"47","author":"Wang","year":"2011","journal-title":"IEEE Trans. Magn."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"807","DOI":"10.1049\/iet-smt.2014.0276","article-title":"Application of an advanced eddy--current loss modelling to magnetic properties of electrical steel laminations in a wide range of measurements","volume":"9","author":"Hamzehbahmani","year":"2015","journal-title":"IET Sci. Meas. Technol."},{"key":"ref_25","first-page":"1","article-title":"Impedance Linearity of Contactless Magnetic-Type Position Sensor","volume":"53","author":"Yang","year":"2017","journal-title":"IEEE Trans. Magn."},{"key":"ref_26","first-page":"1","article-title":"Comparison of Subdomain, Complex Permeance, and Relative Permeance Models for a Wide Family of Permanent-Magnet Machines","volume":"57","author":"Tang","year":"2021","journal-title":"IEEE Trans. Magn."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1109\/74.997888","article-title":"Fractal antennas: A novel antenna miniaturization technique, and applications","volume":"44","author":"Gianvittorio","year":"2002","journal-title":"IEEE Antenn. Propag. M."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"731","DOI":"10.1109\/TAP.2007.891553","article-title":"A Hybrid Optimization Method to Analyze Metamaterial-Based Electrically Small Antennas","volume":"55","author":"Erentok","year":"2007","journal-title":"IEEE Trans. Antenn. Propag."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Syaifudin, A.R.M., Mukhopadhyay, S.C., and Yu, P.L. (2009, January 14\u201316). Electromagnetic field computation using COMSOL Multiphysics to evaluate the performance of novel interdigital sensors. Proceedings of the 2009 Applied Electromagnetics Conference (AEMC), Kolkata, India.","DOI":"10.1109\/AEMC.2009.5430661"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"156","DOI":"10.1109\/60.50827","article-title":"An analytical solution for the field of a hysteresis motor based on complex permeability","volume":"5","author":"Zaher","year":"1990","journal-title":"IEEE Trans. Energy Conver."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1016\/j.physb.2004.06.059","article-title":"Hysteresis and initial permeability behavior of vanadium-substituted lithium\u2013zinc\u2013titanium ferrite","volume":"352","author":"Maisnam","year":"2004","journal-title":"Phys. B"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/18\/6292\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:02:25Z","timestamp":1760166145000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/18\/6292"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,9,20]]},"references-count":31,"journal-issue":{"issue":"18","published-online":{"date-parts":[[2021,9]]}},"alternative-id":["s21186292"],"URL":"https:\/\/doi.org\/10.3390\/s21186292","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,9,20]]}}}