{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:16:07Z","timestamp":1760148967803,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"13","license":[{"start":{"date-parts":[[2023,6,23]],"date-time":"2023-06-23T00:00:00Z","timestamp":1687478400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deputyship for Research &amp; Innovation, Ministry of Education in Saudi Arabia","award":["ISP22-39"],"award-info":[{"award-number":["ISP22-39"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Capacitive sensors that utilize the Coplanar Interdigitated (CID) electrode structure are widely employed in various technical and analytical domains, such as healthcare, infectious disease management, pharmaceuticals, metrology, and environmental monitoring. The present exigency for lab-on-a-chip contrivances and the requisite for the miniaturization of sensors have led to the widespread adoption of CID sensors featuring multiple dielectric layers (DLs), either in the form of substrates or superstrates. Previously, we derived an analytical model for the capacitance of CID capacitive sensors with four distinct 1-N-1 patterns (namely, 1-1-1, 1-3-1, 1-5-1, and 1-11-1) using partial capacitance (PC) and conformal mapping (CM) techniques. The aforementioned model has been employed in various applications wherein the permittivity of successive layers exhibits a monotonic decrease as one moves away from the electrode plane, resulting in highly satisfactory outcomes. Nevertheless, the PC technique is inadequate for structures with multiple layers where the permittivity exhibits a monotonic increase as the distance from the electrodes increases. Given these circumstances, it is necessary to adapt the initial PC method to incorporate these novel configurations. In this work, we have discussed a new approach, splitting the concept of PC into partial parallel capacitance (PPC) and partial serial capacitance (PSC), where new CM transformations are proposed for the latter case. Thus, the present study proposes a novel methodology to expand upon our prior analytical framework, which aims to incorporate scenarios where the permittivity experiences a reduction across successive layers. The outcomes are juxtaposed with the finite element simulation and analytical findings.<\/jats:p>","DOI":"10.3390\/s23135838","type":"journal-article","created":{"date-parts":[[2023,6,26]],"date-time":"2023-06-26T05:11:54Z","timestamp":1687756314000},"page":"5838","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Expansion of the Analytical Modeling of Capacitance for 1-N-1 Multilayered CID Structures with Monotonically Increasing\/Decreasing Permittivity"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6259-9296","authenticated-orcid":false,"given":"Anwar Ulla","family":"Khan","sequence":"first","affiliation":[{"name":"Department of Electrical Engineering Technology, College of Applied Industrial Technology, Jazan University, Jazan 45142, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2023,6,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"434","DOI":"10.1109\/JSEN.2004.830300","article-title":"Design and fabrication of a microimpedance biosensor for bacterial detection","volume":"4","author":"Radke","year":"2004","journal-title":"IEEE Sens. J."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1662","DOI":"10.1016\/j.bios.2004.07.021","article-title":"A high density microelectrode array biosensor for detection of E. coli O157:H7","volume":"20","author":"Radke","year":"2005","journal-title":"Biosens. Bioelectron."},{"doi-asserted-by":"crossref","unstructured":"Booth, J.C., Mateu, J., Janezic, M., Baker-Jarvis, J., and Beall, J.A. (2006, January 11\u201316). Broadband permittivity measurements of liquid and biological samples using microfluidic channels. Proceedings of the IEEE MTT-S International Microwave Symposium Digest, San Francisco, CA, USA.","key":"ref_3","DOI":"10.1109\/MWSYM.2006.249720"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1109\/JERM.2018.2807984","article-title":"A planar one-port microwave microfluidic sensor for microliter liquids characterization","volume":"2","author":"Bao","year":"2018","journal-title":"IEEE J. Electromagn. RF Microw. Med. Biol."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"4171","DOI":"10.1109\/TMTT.2012.2222660","article-title":"Accurate nanoliter liquid characterization up to 40 GHz for biomedical applications: Toward noninvasive living cells monitoring","volume":"60","author":"Chen","year":"2012","journal-title":"IEEE Trans. Microw. Theory Tech."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"698","DOI":"10.1109\/TUFFC.2010.1467","article-title":"Study of the acoustoelectric effect for SAW sensors","volume":"57","author":"Fisher","year":"2010","journal-title":"IEEE Trans. Ultrason. Ferroelectr. Freq. Control."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"674","DOI":"10.1109\/TMTT.2010.2040347","article-title":"Spread spectrum orthogonal frequency coded SAW tags and sensors using harmonic operation","volume":"58","author":"Gallagher","year":"2010","journal-title":"IEEE Trans. Microw. Theory Tech."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/j.snb.2008.12.053","article-title":"A low cost novel sensing system for detection of dangerous marine biotoxins in seafood","volume":"137","author":"Syaifudin","year":"2009","journal-title":"Sens. Actuators B Chem."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2951","DOI":"10.1016\/j.bios.2008.10.001","article-title":"Interdigitated array microelectrodes based impedance biosensors for detection of bacterial cells","volume":"24","author":"Varshney","year":"2009","journal-title":"Biosens. Bioelectron."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"744","DOI":"10.1109\/JSEN.2005.848138","article-title":"A microfabricated biosensor for detecting foodborne bioterrorism agents","volume":"5","author":"Radke","year":"2005","journal-title":"IEEE Sens. J."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"930","DOI":"10.1109\/JPROC.2003.813570","article-title":"Microfluidics meets MEMS","volume":"91","author":"Verpoorte","year":"2003","journal-title":"Proc. IEEE"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"24812","DOI":"10.1109\/JSEN.2021.3112848","article-title":"Design and Fabrication of Fringing Field Capacitive Sensor for Non-Contact Liquid Level Measurement","volume":"21","author":"Islam","year":"2021","journal-title":"IEEE Sens. J."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2957","DOI":"10.1109\/JSEN.2011.2154327","article-title":"Measurements and Performance Evaluation of Novel Interdigital Sensors for Different Chemicals Related to Food Poisoning","volume":"11","author":"Syaifudin","year":"2011","journal-title":"IEEE Sens. J."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1028","DOI":"10.1109\/TMTT.1970.1127407","article-title":"Interdigital Capacitors and Their Application to Lumped-Element Microwave Integrated Circuits","volume":"18","author":"Alley","year":"1970","journal-title":"IEEE Trans. Microw. Theory Tech."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1109\/TMTT.1983.1131429","article-title":"Design of Interdigitated Capacitors and Their Application to Gallium Arsenide Monolithic Filters","volume":"31","author":"Esfandiari","year":"1983","journal-title":"IEEE Trans. Microw. Theory Tech."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"152","DOI":"10.1109\/JQE.1977.1069319","article-title":"Distributed capacitance of planar electrodes in optic and acoustic surface wave devices","volume":"13","author":"Wei","year":"1977","journal-title":"IEEE J. Quantum Electron."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"156","DOI":"10.1109\/77.317831","article-title":"Voltage tunable capacitors using high temperature superconductors and ferroelectrics","volume":"4","author":"Wu","year":"1994","journal-title":"IEEE Trans. Appl. Supercond."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"896","DOI":"10.1109\/22.506449","article-title":"CAD models for multilayered substrate interdigital capacitors","volume":"44","author":"Gevorgian","year":"1996","journal-title":"IEEE Trans. Microw. Theory Tech."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1016\/j.sna.2004.01.040","article-title":"Analytical evaluation of the interdigital electrodes capacitance for a multilayered structure","volume":"112","author":"Igreja","year":"2004","journal-title":"Sens. Actuators A Phys."},{"unstructured":"Hoffman, R.K. (1987). Handbook of Microwave Integrated Circuits, Artech.","key":"ref_20"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"392","DOI":"10.1016\/j.sna.2011.09.033","article-title":"Extension to the analytical model of the interdigital electrodes capacitance for a multilayered structure","volume":"172","author":"Igreja","year":"2011","journal-title":"Sens. Actuators A Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"9502410","DOI":"10.1109\/TIM.2023.3235458","article-title":"Analytical Evaluation of a Coplanar Interdigitated Sensor Capacitance for 1-n-1 Multilayered Structure","volume":"72","author":"Khan","year":"2023","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"186","DOI":"10.1016\/j.snb.2007.07.021","article-title":"Single chip interdigitated electrode capacitive chemical sensor arrays","volume":"127","author":"Kitsara","year":"2007","journal-title":"Sens. Actuators B Chem."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1318","DOI":"10.1039\/c0lc00543f","article-title":"3D numerical simulation of a lab-on-a-chip- increasing measurement sensitivity of interdigitated capacitors by passivation optimization","volume":"11","author":"Jungreuthmayer","year":"2011","journal-title":"Lab A Chip"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1002\/mop.4650090419","article-title":"Analytical formulas for calculating the effective dielectric-constants of coplanar lines for Oic applications","volume":"9","author":"Zhu","year":"1995","journal-title":"Microw. Opt. Technol. Lett."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2287","DOI":"10.1109\/22.808972","article-title":"Microwave modeling and characterization of thick copla-nar waveguides on oxide-coated lithium niobate substrates for electrooptical applications","volume":"47","author":"Ghione","year":"1999","journal-title":"IEEE Trans. Microw. Theory Tech."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2007","DOI":"10.1109\/TMTT.2003.815873","article-title":"Revisiting the partial-capacitance approach to the analysis of coplanar transmission lines on multilayered substrates","volume":"51","author":"Ghione","year":"2003","journal-title":"IEEE Trans. Microw. Theory Tech."},{"key":"ref_28","first-page":"988","article-title":"Development and experimental verification of analytical models for printable interdigital capacitor sensors on paperboard","volume":"1\u20133","author":"Feng","year":"2009","journal-title":"IEEE Sens."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"631","DOI":"10.1109\/TMTT.1977.1129179","article-title":"Transmission-line properties of a strip on a dielectric sheet on a plane","volume":"25","author":"Wheeler","year":"1977","journal-title":"IEEE Trans. Microw. Theory Tech."},{"unstructured":"Abramowitz, M., and Stegun, I.A. (1970). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover.","key":"ref_30"},{"unstructured":"Brown, J.W., and Churchill, R.V. (1996). Complex Variables and Applications, McGraw-Hill. [6th ed.].","key":"ref_31"},{"unstructured":"Blume, S.O.P. (2014). A Multi-Electrode Array for Impedance Spectroscopy in a Digital Microfluidic Device. [Master\u2019s Thesis, University of Toronto].","key":"ref_32"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/13\/5838\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:59:09Z","timestamp":1760126349000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/13\/5838"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,23]]},"references-count":32,"journal-issue":{"issue":"13","published-online":{"date-parts":[[2023,7]]}},"alternative-id":["s23135838"],"URL":"https:\/\/doi.org\/10.3390\/s23135838","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2023,6,23]]}}}