{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,10]],"date-time":"2026-01-10T01:31:26Z","timestamp":1768008686129,"version":"3.49.0"},"reference-count":43,"publisher":"MDPI AG","issue":"14","license":[{"start":{"date-parts":[[2021,7,16]],"date-time":"2021-07-16T00:00:00Z","timestamp":1626393600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100008530","name":"European Regional Development Fund","doi-asserted-by":"publisher","award":["POCI-01-0247-FEDER-037902"],"award-info":[{"award-number":["POCI-01-0247-FEDER-037902"]}],"id":[{"id":"10.13039\/501100008530","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["PD\/BD\/128142\/2016"],"award-info":[{"award-number":["PD\/BD\/128142\/2016"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["PD\/BDE\/135102\/2017"],"award-info":[{"award-number":["PD\/BDE\/135102\/2017"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>An open-source tool that allows for a fast and precise analytical calculation of multi-layer planar coils self-inductance, without any geometry limitation is proposed here. The process of designing and simulating planar coils to achieve reliable results is commonly limited on accuracy and or geometry, or are too time-consuming and expensive, thus a tool to speed up this design process is desired. The model is based on Grover equations, valid for any geometry. The validation of the tool was performed through the comparison with experimental measurements, Finite Element Model (FEM) simulations, and the main analytical methods usually used in literature, with errors registered to be below 2.5%, when compared to standard FEM simulations, and when compared to experimental measurements they are below 10% in the case of the 1-layer coils, and below 5% in the 2-layer coils (without taking into consideration the coil connectors). The proposed model offers a new approach to the calculation of the self-inductance of planar coils of several layers that combines precision, speed, independence of geometry, easy interaction, and no need for extra resources.<\/jats:p>","DOI":"10.3390\/s21144864","type":"journal-article","created":{"date-parts":[[2021,7,18]],"date-time":"2021-07-18T21:18:52Z","timestamp":1626643132000},"page":"4864","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["A Fast and Precise Tool for Multi-Layer Planar Coil Self-Inductance Calculation"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7117-1478","authenticated-orcid":false,"given":"Andreia","family":"Faria","sequence":"first","affiliation":[{"name":"ALGORITMI Center, University of Minho, 4800-058 Guimar\u00e3es, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7477-6934","authenticated-orcid":false,"given":"Lu\u00eds","family":"Marques","sequence":"additional","affiliation":[{"name":"Centre of Physics of Minho and Porto Universities, 4710-057 Braga, Portugal"}]},{"given":"Carlos","family":"Ferreira","sequence":"additional","affiliation":[{"name":"PhD AESI Bosch Car Multimedia, 4705-820 Braga, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3439-8235","authenticated-orcid":false,"given":"Filipe","family":"Alves","sequence":"additional","affiliation":[{"name":"Integrated Micro and Nanotechnologies, International Iberian Nanotechnology Laboratory, 4715-330 Braga, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9954-9746","authenticated-orcid":false,"given":"Jorge","family":"Cabral","sequence":"additional","affiliation":[{"name":"ALGORITMI Center, University of Minho, 4800-058 Guimar\u00e3es, Portugal"},{"name":"CEiiA\u2014Centro de Engenharia e Desenvolvimento, 4450-017 Matosinhos, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,16]]},"reference":[{"key":"ref_1","unstructured":"Pospisilik, M., Kouril, L., Motyl, I.V.O., and Adamek, M. (2011, January 23\u201325). Single and Double Layer Spiral Planar Inductors Optimisation with the Aid of Self-Organising Migrating Algorithm. Proceedings of the Recent Advances in Signal Processing, Computational Geometry and Systems Theory, Florence, Italy."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"7195","DOI":"10.1109\/JSEN.2020.3045461","article-title":"Technologies and Applications of Angle Sensors: A Review","volume":"21","author":"Kumar","year":"2020","journal-title":"IEEE Sens. J."},{"key":"ref_3","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_4","unstructured":"Howard, M. (2021, July 06). Choosing the Right Position Sensor, Celera Motion. Available online: https:\/\/www.zettlex.com\/articles\/choosing-right-position-sensor."},{"key":"ref_5","unstructured":"Cambridge Integrated Circuits Ltd. (2021, July 06). Resonant Inductive Position Sensors. Available online: https:\/\/www.sensorland.com\/."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Karimov, T., Druzhina, O., Karimov, A., and Butusov, D. (2020, January 23\u201324). Axial Movement Sensor Based on Chaotic Oscillator and Planar Coil. Proceedings of the Conference of Open Innovation Association, FRUCT, Yaroslav, Russia.","DOI":"10.23919\/FRUCT48808.2020.9087471"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"560","DOI":"10.1109\/TPEL.2015.2411618","article-title":"A Systematic Approach to Modeling Impedances and Current Distribution in Planar Magnetics","volume":"31","author":"Chen","year":"2016","journal-title":"IEEE Trans. Power Electron."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Lazarus, N., Meyer, C.D., Bedair, S.S., Nochetto, H., and Kierzewski, I.M. (2014). Multilayer liquid metal stretchable inductors. Smart Mater. Struct., 23.","DOI":"10.1088\/0964-1726\/23\/8\/085036"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Khan, S.R., Pavuluri, S.K., Cummins, G., and Desmulliez, M.P. (2020). Wireless power transfer techniques for implantable medical devices: A review. Sensors, 20.","DOI":"10.3390\/s20123487"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"5102","DOI":"10.1109\/TPEL.2019.2944194","article-title":"Modeling of a Magnetic Coupler Based on Single- and Double-Layered Rectangular Planar Coils with In-Plane Misalignment for Wireless Power Transfer","volume":"35","author":"Real","year":"2020","journal-title":"IEEE Trans. Power Electron."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"674","DOI":"10.1109\/TPEL.2020.2998186","article-title":"A Self-Compensated Planar Coil for Resonant Wireless Power Transfer Systems","volume":"36","author":"Wang","year":"2021","journal-title":"IEEE Trans. Power Electron."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"4069","DOI":"10.1109\/TIA.2018.2836098","article-title":"Inductive Power Transfer for Automotive Applications: State-of-the-Art and Future Trends","volume":"54","author":"Cirimele","year":"2018","journal-title":"IEEE Trans. Ind. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1109\/TBME.2017.2778729","article-title":"Near-field inductive-coupling link to power a three-dimensional millimeter-size antenna for brain implantable medical devices","volume":"65","author":"Manoufali","year":"2018","journal-title":"IEEE Trans. Biomed. Eng."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Reddy, B.P., Murali, A., and Shaga, G. (2017, January 14\u201316). Low cost planar coil structure for inductive sensors to measure absolute angular position. Proceedings of the 2017 2nd International Conference on Frontiers of Sensors Technologies (ICFST), Shenzhen, China.","DOI":"10.1109\/ICFST.2017.8210463"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Anil Kumar, A.S., George, B., and Mukhopadhyay, S.C. (2020, January 25\u201328). An eddy current based non-contact displacement sensor. Proceedings of the I2MTC 2020\u2014International Instrumentation and Measurement Technology Conference, Dubrovnik, Croatia.","DOI":"10.1109\/I2MTC43012.2020.9128506"},{"key":"ref_16","unstructured":"Renesas (2021, July 06). IPS2200 Inductive Position Sensors a New Era in Motor Commutation. Available online: https:\/\/www2.renesas.cn\/cn\/en\/document\/ovr\/ips2200-family-overview?language=en."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"98","DOI":"10.1016\/j.mejo.2019.01.012","article-title":"Mutual inductance calculation between two coaxial planar spiral coils with an arbitrary number of sides","volume":"85","author":"Tavakkoli","year":"2019","journal-title":"Microelectron. J."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"711","DOI":"10.1109\/TVLSI.2018.2881075","article-title":"Design and optimization of inductive-coupling links for 3-D-ICs","volume":"27","author":"Fletcher","year":"2019","journal-title":"IEEE Trans. Very Large Scale Integr. Syst."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Fletcher, B.J., Das, S., and Mak, T. (2018, January 19\u201323). A high-speed design methodology for inductive coupling links in 3D-ICs. Proceedings of the 2018 Design, Automation and Test in Europe Conference and Exhibition, Dresden, Germany.","DOI":"10.23919\/DATE.2018.8342059"},{"key":"ref_20","first-page":"37","article-title":"A new calculation for designing multilayer planar spiral inductors","volume":"55","author":"Zhao","year":"2010","journal-title":"Edn"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Nejadpak, A., Barzegaran, M.R., Sarikhani, A., and Mohammed, O.A. (2011, January 6\u201311). Design of planar inductor based Z-source inverter for residential alternate energy sources. Proceedings of the IEEE Applied Power Electronics Conference and Exposition, Fort Worth, TX, USA.","DOI":"10.1109\/APEC.2011.5744824"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Pacurar, C., Topa, V., Munteanu, C., Racasan, A., and Hebedean, C. (2012, January 25\u201327). Spiral inductors inductance computation and layout optimization. Proceedings of the 2012 International Conference and Exposition on Electrical and Power Engineering, Iasi, Romania.","DOI":"10.1109\/ICEPE.2012.6463850"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1109\/TMTT.2016.2645571","article-title":"Optimized toroidal inductors versus planar spiral inductors in multilayered technologies","volume":"65","author":"Vidal","year":"2017","journal-title":"IEEE Trans. Microw. Theory Tech."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Ulvr, M. (2018). Design of PCB search coils for AC magnetic flux density measurement. AIP Adv., 8.","DOI":"10.1063\/1.4991643"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Pacurar, C., Topa, V., Giurgiuman, A., Munteanu, C., Constantinescu, C., Gliga, M., and Andreica, S. (2019, January 21\u201323). The Construction of a Wireless Power Supply System using Planar Spiral Inductors. Proceedings of the 2019 8th International Conference on Modern Power Systems, MPS 2019, Cluj-Napoca, Cluj, Romania.","DOI":"10.1109\/MPS.2019.8759779"},{"key":"ref_26","unstructured":"Musunuri, S., and Chapman, P. (2003, January 12\u201316). Multi-layer spiral inductor design for monolithic DC-DC converters. Proceedings of the 38th IAS Annual Meeting on Conference Record of the Industry Applications Conference, Salt Lake City, UT, USA."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1016\/j.sna.2017.12.060","article-title":"Robust and high-performance soft inductive tactile sensors based on the Eddy-current effect","volume":"271","author":"Wang","year":"2018","journal-title":"Sens. Actuators A Phys."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1016\/j.sna.2016.01.035","article-title":"A novel sensing chip with dual-coil inductance for determining raw milk quality","volume":"241","author":"Li","year":"2016","journal-title":"Sens. Actuators A Phys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"410","DOI":"10.1016\/j.sna.2004.05.020","article-title":"Planar coils with magnetic layers for optimized energy transfer in telemetric systems","volume":"116","author":"Leuerer","year":"2004","journal-title":"Sens. Actuators A Phys."},{"key":"ref_30","unstructured":"Xiao, Q., Luo, T., Shi, Y., Chen, D., Ye, H., Hu, S., and Ren, Z. (2008, January 2\u20135). Simple and accurate radio frequency inductance expression for on-chip planar spiral inductors. Proceedings of the ISAPE 2008\u2014The 8th International Symposium on Antennas, Propagation and EM Theory Proceedings, Kunming, China."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1016\/j.sna.2016.04.065","article-title":"Analytical study of mutual inductance of hexagonal and octagonal spiral planer coils","volume":"247","author":"Tavakkoli","year":"2016","journal-title":"Sens. Actuators A Phys."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"246","DOI":"10.1016\/j.sna.2017.05.043","article-title":"An inductive salt solution concentration sensor using a planar coil based on a PQCR-L circuit","volume":"263","author":"Ding","year":"2017","journal-title":"Sens. Actuators A Phys."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1109\/TPHP.1974.1134841","article-title":"Design of Planar Rectangular Microelectronic Inductors","volume":"10","author":"Greenhouse","year":"1974","journal-title":"IEEE Trans. Parts Hybrids Packag."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Durmus, F., and Karagol, S. (2018, January 19\u201321). Mutual Inductance Calculation Formula for Planar Square Coils. Proceedings of the ISMSIT 2018\u20132nd International Symposium on Multidisciplinary Studies and Innovative Technologies, Ankara, Turkey.","DOI":"10.1109\/ISMSIT.2018.8567040"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1419","DOI":"10.1109\/4.792620","article-title":"Simple accurate expressions for planar spiral inductances","volume":"34","author":"Mohan","year":"1999","journal-title":"IEEE J. Solid State Circ."},{"key":"ref_36","first-page":"1398","article-title":"Simple inductance formulas for radio coils","volume":"16","author":"Wheeler","year":"1928","journal-title":"Proc. Inst. Radio Eng."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Cheng, Y., and Shu, Y. (2014). A New Analytical Calculation of the Mutual Inductance of the Coaxial Spiral Rectangular Coils. IEEE Trans. Magn., 50.","DOI":"10.1109\/TMAG.2013.2290972"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Biswas, D.K., Sinclair, M., Le, T., Pullano, S.A., Fiorillo, A.S., and Mahbub, I. (2020). Modeling and characterization of scaling factor of flexible spiral coils for wirelessly powered wearable sensors. Sensors, 20.","DOI":"10.3390\/s20082282"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Hsu, L.C., Kadomoto, J., Hasegawa, S., Kosuge, A., Take, Y., and Kuroda, T. (2016, January 25\u201328). Analytical thruchip inductive coupling channel design optimization. Proceedings of the Asia and South Pacific Design Automation Conference, ASP-DAC, Macao, China.","DOI":"10.1109\/ASPDAC.2016.7428098"},{"key":"ref_40","first-page":"2393","article-title":"Dynamic Modeling and Calculation of Self and Mutual Inductance between a Pair of Coils for Wireless Power Transfer Applications using ANSYS Maxwell","volume":"2","year":"2015","journal-title":"Int. Adv. Res. J. Sci. Eng. Technol."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"215","DOI":"10.4028\/www.scientific.net\/MSF.792.215","article-title":"Simplified calculation method of planar coil impedance considering the eddy current distribution by using finite element method","volume":"792","author":"Wakiwaka","year":"2014","journal-title":"Mater. Sci. Forum"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Snyder, J.N., and Grover, F.C. (1964). Inductance Calculations Working Formulas and Tables, Dover Publications, Inc.","DOI":"10.2307\/2003443"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"560","DOI":"10.1109\/16.824729","article-title":"Physical modeling of spiral inductors on silicon","volume":"47","year":"2000","journal-title":"IEEE Trans. Electron Devices"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/14\/4864\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:30:55Z","timestamp":1760164255000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/14\/4864"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,16]]},"references-count":43,"journal-issue":{"issue":"14","published-online":{"date-parts":[[2021,7]]}},"alternative-id":["s21144864"],"URL":"https:\/\/doi.org\/10.3390\/s21144864","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,7,16]]}}}