{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:27:14Z","timestamp":1760243234282,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2014,4,23]],"date-time":"2014-04-23T00:00:00Z","timestamp":1398211200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Ideally, the design of high-performance micro-hotplates would require a large number of simulations because of the existence of many important design parameters as well as the possibly crucial effects of both spread and drift. However, the computational cost of FEM simulations, which are the only available tool for accurately predicting the temperature in micro-hotplates, is very high. As a result, micro-hotplate designers generally have no effective simulation-tools for the optimization. In order to circumvent these issues, here, we propose a model for practical circular-symmetric micro-hot-plates which takes advantage of modified Bessel functions, computationally efficient matrix-approach for considering the relevant boundary conditions, Taylor linearization for modeling the Joule heating and radiation losses, and external-region-segmentation strategy in order to accurately take into account radiation losses in the entire micro-hotplate. The proposed model is almost as accurate as FEM simulations and two to three orders of magnitude more computationally efficient (e.g., 45 s versus more than 8 h). The residual errors, which are mainly associated to the undesired heating in the electrical contacts, are small (e.g., few degrees Celsius for an 800 \u00b0C operating temperature) and, for important analyses, almost constant. Therefore, we also introduce a computationally-easy single-FEM-compensation strategy in order to reduce the residual errors to about 1 \u00b0C. As illustrative examples of the power of our approach, we report the systematic investigation of a spread in the membrane thermal conductivity and of combined variations of both ambient and bulk temperatures. Our model enables a much faster characterization of micro-hotplates and, thus, a much more effective optimization prior to fabrication.<\/jats:p>","DOI":"10.3390\/s140407374","type":"journal-article","created":{"date-parts":[[2014,4,23]],"date-time":"2014-04-23T12:29:40Z","timestamp":1398256180000},"page":"7374-7393","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["An Accurate and Computationally Efficient Model for  Membrane-Type Circular-Symmetric Micro-Hotplates"],"prefix":"10.3390","volume":"14","author":[{"given":"Usman","family":"Khan","sequence":"first","affiliation":[{"name":"Department of Electronic Engineering, University of Tor Vergata, Via del Politecnico 1,  00133 Rome, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Christian","family":"Falconi","sequence":"additional","affiliation":[{"name":"Department of Electronic Engineering, University of Tor Vergata, Via del Politecnico 1,  00133 Rome, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2014,4,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0925-4005(00)00639-0","article-title":"Micromachined metal oxide gas sensors : Opportunities to improve sensor performance","volume":"73","author":"Simon","year":"2001","journal-title":"Sens. Actuators B Chem."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"644","DOI":"10.1109\/JSEN.2002.807780","article-title":"A Monolithic CMOS Microhotplate-Based Gas Sensor System","volume":"2","author":"Afridi","year":"2002","journal-title":"IEEE Sens. J."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1016\/j.snb.2006.09.022","article-title":"Electronic interfaces","volume":"121","author":"Falconi","year":"2007","journal-title":"Sens. Actuators B Chem."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"242","DOI":"10.1126\/science.1124005","article-title":"Piezoelectric nanogenerators based on zinc oxide nanowire arrays","volume":"312","author":"Wang","year":"2006","journal-title":"Science"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"465401","DOI":"10.1088\/0957-4484\/22\/46\/465401","article-title":"Piezoelectric potential in vertically aligned nanowires for high output nanogenerators","volume":"22","author":"Romano","year":"2011","journal-title":"Nanotechnology"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"4719","DOI":"10.1002\/adma.201104588","article-title":"Piezo-Semiconductive Quasi-1D Nanodevices with or without Anti-Symmetry","volume":"24","author":"Araneo","year":"2012","journal-title":"Adv. Mater."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1023\/A:1023224123925","article-title":"Featured Article : Use of Microhotplates in the Controlled Growth and Characterization of Metal Oxides for Chemical Sensing","volume":"9","author":"Cavicchi","year":"2002","journal-title":"J. Electroceram."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1209","DOI":"10.1016\/S0956-5663(03)00086-1","article-title":"Lung cancer identification by the analysis of breath by means of an array of non-selective gas sensors","volume":"18","author":"Macagnano","year":"2003","journal-title":"Biosens. Bioelectron."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1016\/S0925-4005(03)00422-2","article-title":"Feature Extraction of chemical sensors in phase space","volume":"95","author":"Martinelli","year":"2003","journal-title":"Sens. Actuators B Chem."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1016\/j.snb.2012.07.095","article-title":"1\/F Noise and Its Unusual High-Frequency Deactivation at High Biasing Currents in Carbon Black Polymers With Residual 1\/F\u03b3 (\u0393 = 2.2) Noise and a Preliminary Estimation of the Average Trap Energy","volume":"174","author":"Falconi","year":"2012","journal-title":"Sens. Actuators B Chem."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1109\/JSEN.2009.2034624","article-title":"Micromachined mid-infrared emitter for fast transient temperature operation for optical gas sensing systems","volume":"10","author":"Hildenbrand","year":"2010","journal-title":"IEEE Sens. J."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1016\/j.sna.2004.12.010","article-title":"Impact of environmental parameters on the emission intensity of micromachined infrared sources","volume":"121","author":"Schulz","year":"2005","journal-title":"Sens. Actuators A Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"051101","DOI":"10.1063\/1.3304835","article-title":"Localized heating induced chemical vapor deposition for one-dimensional nanostructure synthesis","volume":"107","author":"Sosnowchik","year":"2010","journal-title":"J. Appl. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"534","DOI":"10.1016\/j.snb.2011.10.072","article-title":"Self-adapted temperature modulation in metal-oxide semiconductor gas sensors","volume":"161","author":"Martinelli","year":"2012","journal-title":"Sens. Actuators B Chem."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"971","DOI":"10.1088\/0960-1317\/12\/6\/330","article-title":"Thermal optimization of micro-hotplates that have a silicon island","volume":"12","author":"Briand","year":"2002","journal-title":"J. Micromech. Microeng."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1016\/S0924-4247(02)00209-1","article-title":"Analytical modelling of steady-state temperature distribution in thermal microsensors using Fourier method: Part 1. Theory","volume":"101","author":"Kozlov","year":"2002","journal-title":"Sens. Actuators A Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1016\/S0924-4247(02)00210-8","article-title":"Analytical modelling of steady-state temperature distribution in thermal microsensors using Fourier method Part 2. Practical application","volume":"101","author":"Kozlov","year":"2002","journal-title":"Sens. Actuators A Phys."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/0924-4247(93)80142-4","article-title":"Optimization tool for the performance parameters of thermoelectric microsensors","volume":"36","author":"Baltes","year":"1993","journal-title":"Sens. Actuators A Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1016\/j.snb.2010.11.007","article-title":"A study of heat distribution and dissipation in a micromachined chemoresistive gas sensor","volume":"153","author":"Giberti","year":"2011","journal-title":"Sens. Actuators B Chem."},{"key":"ref_20","unstructured":"Li, T., Wu, L., Liu, Y., Wang, L., Wang, Y., and Wang, Y. (2006, January 20\u201325). Micro-Heater on Membrane with Large Uniform-Temperature Area. Daegu, Korea."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"102402","DOI":"10.1115\/1.2945904","article-title":"Measurement of the Thermal Conductivity and Heat Capacity of Freestanding Shape Memory Thin Films Using the 3\u03c9 Method","volume":"130","author":"Jain","year":"2008","journal-title":"J. Heat Transf."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"190","DOI":"10.1088\/0960-1317\/15\/1\/027","article-title":"3D nonlinear modeling of microhotplates in CMOS technology for use as metal-oxide-based gas sensors","volume":"15","author":"Graf","year":"2005","journal-title":"J. Micromech. Microeng."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1016\/j.snb.2012.11.007","article-title":"Temperature distribution in membrane-type micro-hot-plates with circular geometry","volume":"177","author":"Khan","year":"2012","journal-title":"Sens. Actuators B Chem."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"274","DOI":"10.1016\/j.snb.2013.04.098","article-title":"Micro-hot-plates without simply-connected hot-spots and with almost-circular temperature distribution","volume":"185","author":"Khan","year":"2013","journal-title":"Sens. Actuators B Chem."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"015014","DOI":"10.1088\/0960-1317\/21\/1\/015014","article-title":"Design and fabrication of micro-hotplates made on a polyimide foil: Electrothermal simulation and characterization to achieve power consumption in the low mW range","volume":"21","author":"Courbat","year":"2011","journal-title":"J. Micromech. Microeng."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1408","DOI":"10.1109\/JMEMS.2008.2007228","article-title":"Tungsten-Based SOI Microhotplates for Smart Gas Sensors","volume":"17","author":"Ali","year":"2008","journal-title":"J. Microelectromech. Syst."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"820","DOI":"10.1016\/j.ijthermalsci.2007.07.011","article-title":"Investigation of the natural convection boundary condition in microfabricated structures","volume":"47","author":"Hu","year":"2008","journal-title":"Int. J. Therm. Sci."},{"key":"ref_28","unstructured":"Pitts, D., and Sissom, L.E. (1998). Schuam's Outline of Heat Transfer, McGraw-Hill. [2nd ed.]."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1016\/0924-4247(92)80006-O","article-title":"A heated membrane for a capacitive gas sensor","volume":"32","author":"Hille","year":"1992","journal-title":"Sens. Actuators A Phys."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Ali, S.Z., Santra, S., Haneef, I., Schwandt, C., Kumar, R.V., Milne, W.I., Udrea, F., Guha, P.K., Covington, J.A., and Gardner, J.W. (2009, January 25\u2013\u201328). Nanowire hydrogen gas sensor employing CMOS micro-hotplate. Christchurch, New Zealand.","DOI":"10.1109\/ICSENS.2009.5398224"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1109\/84.557531","article-title":"Thermal characterization of surface-micromachined silicon nitride membranes for thermal infrared detectors","volume":"6","author":"Eriksson","year":"1997","journal-title":"J. Microelectromech. Syst."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"171912","DOI":"10.1063\/1.1921350","article-title":"Thermal and electrical conductivity of a suspended platinum nanofilm","volume":"86","author":"Zhang","year":"2005","journal-title":"Appl. Phys. Lett."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1088\/1742-6596\/15\/1\/005","article-title":"Design and simulation of resistive SOI CMOS micro-heaters for high temperature gas sensors","volume":"15","author":"Iwaki","year":"2005","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1393","DOI":"10.1016\/j.mejo.2008.05.004","article-title":"Micro-hotplates for thermal characterisation of structural materials of MEMS","volume":"40","year":"2009","journal-title":"Microelectron. J."},{"key":"ref_35","unstructured":"Futurlec Futurlec Gas Sensors. Available online: http:\/\/www.futurlec.com\/Gas_Sensors.shtml."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/14\/4\/7374\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:10:38Z","timestamp":1760217038000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/14\/4\/7374"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,4,23]]},"references-count":35,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2014,4]]}},"alternative-id":["s140407374"],"URL":"https:\/\/doi.org\/10.3390\/s140407374","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2014,4,23]]}}}