Teoretičeskaâ i prikladnaâ nauka Theoretical & Applied Science 02 (94) 2021 International Scientific Journal Theoretical & Applied Science Founder: International Academy of Theoretical & Applied Sciences Published since 2013 year. Issued Monthly. International scientific journal «Theoretical & Applied Science», registered in France, and indexed more than 45 international scientific bases. Editorial office: http://T-Science.org Phone: +777727-606-81 E-mail: [email protected] Editor-in Chief: Hirsch index: Alexandr Shevtsov h Index RISC = 1 (78) Editorial Board: 1 Prof. Vladimir Kestelman USA h Index Scopus = 3 (38) 2 Prof. Arne Jönsson Sweden h Index Scopus = 10 (33) 3 Prof. Sagat Zhunisbekov KZ - 4 Assistant of Prof. Boselin Prabhu India - 5 Lecturer Denis Chemezov Russia h Index RISC = 2 (61) 6 Associate Prof. Elnur Hasanov Azerbaijan h Index Scopus = 8 (11) 7 Associate Prof. Christo Ananth India h Index Scopus = - (1) 8 Prof. Shafa Aliyev Azerbaijan h Index Scopus = - (1) 9 Associate Prof. Ramesh Kumar India h Index Scopus = - (2) 10 Associate Prof. S. Sathish India h Index Scopus = 2 (13) 11 Researcher Rohit Kumar Verma India - 12 Prof. Kerem Shixaliyev Azerbaijan - 13 Associate Prof. Ananeva Elena Pavlovna Russia h Index RISC = 1 (19) 14 Associate Prof. Muhammad Hussein Noure Elahi Iran - 15 Assistant of Prof. Tamar Shiukashvili Georgia - 16 Prof. Said Abdullaevich Salekhov Russia - 17 Prof. Vladimir Timofeevich Prokhorov Russia - 18 Researcher Bobir Ortikmirzayevich Uzbekistan - Tursunov 19 Associate Prof. Victor Aleksandrovich Melent'ev Russia - 20 Prof. Manuchar Shishinashvili Georgia - ISSN 2308-4944 0 2 © Сollective of Authors © «Theoretical & Applied Science» 9 7 7 2 3 0 8 4 9 4 2 1 8 International Scientific Journal Theoretical & Applied Science Editorial Board: Hirsch index: 21 Prof. Konstantin Kurpayanidi Uzbekistan h Index RISC = 8 (67) 22 Prof. Shoumarov G'ayrat Bahramovich Uzbekistan - 23 Associate Prof. Saidvali Yusupov Uzbekistan - 24 PhD Tengiz Magradze Georgia - 25 Dilnoza Azlarova Uzbekistan - 26 Associate Prof. Sanjar Goyipnazarov Uzbekistan - 27 Prof. Shakhlo Ergasheva Uzbekistan - 28 Prof. Nigora Safarova Uzbekistan - 29 Associate Prof. Kurbonov Tohir Hamdamovich Uzbekistan - 30 Prof. Pakhrutdinov Shukritdin Uzbekistan - Il'yasovich 31 PhD Mamazhonov Akramzhon Uzbekistan - Turgunovich 32 PhD Ravindra Bhardwaj USA h Index Scopus = 2 (5) 33 Assistant lecturer Mehrinigor Akhmedova Uzbekistan - 34 Associate Prof. Fayziyeva Makhbuba Uzbekistan - Rakhimjanovna 35 PhD Jamshid Jalilov Uzbekistan - 36 Guzalbegim Rakhimova Uzbekistan - 37 Prof. Gulchehra Gaffarova Uzbekistan - International Scientific Journal Theoretical & Applied Science ISJ Theoretical & Applied Science, 02 (94), 362. Philadelphia, USA Impact Factor ICV = 6.630 Impact Factor ISI = 0.829 based on International Citation Report (ICR) The percentage of rejected articles: ISSN 2308-4944 0 2 9 7 7 2 3 0 8 4 9 4 2 1 8 ISRA (India) = 4.971 SIS (USA) = 0.912 ICV (Poland) = 6.630 ISI (Dubai, UAE) = 0.829 РИНЦ (Russia) = 0.126 PIF (India) = 1.940 Impact Factor: GIF (Australia) = 0.564 ESJI (KZ) = 8.997 IBI (India) = 4.260 JIF = 1.500 SJIF (Morocco) = 5.667 OAJI (USA) = 0.350 QR – Issue QR – Article SOI: 1.1/TAS DOI: 10.15863/TAS International Scientific Journal Theoretical & Applied Science p-ISSN: 2308-4944 (print) e-ISSN: 2409-0085 (online) Year: 2021 Issue: 02 Volume: 94 Published: 02.02.2021 http://T-Science.org Gennady Evgenievich Markelov Bauman Moscow State Technical University Candidate of Engineering Sciences, associate professor, corresponding member of International Academy of Theoretical and Applied Sciences, Moscow, Russia [email protected] A WORKING MATHEMATICAL MODEL OF AN PTC THERMISTOR Abstract: A mathematical model of a positive temperature coefficient thermistor was obtained using a unified approach to building a working mathematical model. This mathematical model has sufficient properties of fullness, accuracy, adequacy, productivity and economy for the purposes of this study. Applying such a model reduces the costs and time spent on research and makes efficient use of the mathematical modelling capabilities. Key words: PTC thermistor, working mathematical model, properties of mathematical models, principles of mathematical modeling. Language: English Citation: Markelov, G. E. (2021). A working mathematical model of an PTC thermistor. ISJ Theoretical & Applied Science, 02 (94), 1-4. Soi: http://s-o-i.org/1.1/TAS-02-94-1 Doi: https://dx.doi.org/10.15863/TAS.2021.02.94.1 Scopus ASCC: 2604. Introduction in [5; 6]. An example of building a mathematical Vast educational and scientific literature is model with the necessary properties for a study is devoted to the technical characteristics of positive presented in [7]; some of the results of this study were temperature coefficient thermistors, basic principles published in [8–10]. The particular features of using a of their operation and methods of circuit design using unified approach to building mathematical models are these thermistors. There are numerous examples of described, for example, in [11; 12]. successful practical use of such devices in various fields. Statement of the problem The aim of this study is to build a working The thermistor is considered to be a body with mathematical model of a positive temperature high thermal conductivity, i.e. the dependence of the coefficient thermistor using a unified approach. temperature of the body on the spatial coordinates at The dependence of the resistance R of such a any time point is disregarded. Its temperature T at the thermistor on its temperature T is not linear over a initial time point t0 equals T0 , while TT 1. broad temperature range (for an example, see [1; 2]). Convective heat exchange with the environment In a relatively narrow temperature range, however, it occurs on the thermistor surface with area S. The can be assumed that ambient temperature is equal to T0 , and the heat R T= r1 +β, T − T ( ) ( 0 ) transfer coefficient is known and equal to α. For a relatively narrow temperature range from T to T , let where r is the thermistor resistance at TT= 0; β is a 0 1 positive constant. us assume that A unified approach to building a working R( T) = r1 +β,( T − T0 ) mathematical model that has necessary properties for a specific study is described in [3; 4]. Some properties of mathematical models are formulated, for instance, Philadelphia, USA 1 ISRA (India) = 4.971 SIS (USA) = 0.912 ICV (Poland) = 6.630 ISI (Dubai, UAE) = 0.829 РИНЦ (Russia) = 0.126 PIF (India) = 1.940 Impact Factor: GIF (Australia) = 0.564 ESJI (KZ) = 8.997 IBI (India) = 4.260 JIF = 1.500 SJIF (Morocco) = 5.667 OAJI (USA) = 0.350 C T= c1 +γ, T − T I0 ( ) ( 0 ) −1 δ I 0 where RT( ) and CT( ) are the resistance and total is met, formula (2) may be used to find the sought heat capacity of the thermistor; r and c are the value with a relative error not exceeding δ. resistance and total heat capacity of the thermistor at 0 Therefore, when the inequality TT= ; β and γ are positive constants. An electric 0 II+1 δ (5) current flows through the thermistor; its strength 00( ) equals is satisfied, mathematical model (2) sufficiently U possesses the properties of fullness, accuracy, I = , (1) adequacy, productivity and economy. r1+−β T T ( 0 ) Then let us define the conditions under which where U is the constant electrical potential difference mathematical model (3) is applicable. To do this, we between the poles of the thermistor. need to consider unsteady-state heat transfer. In this Let I be the value of interest in the study. Let case, the change in thermistor temperature over time us design a working mathematical model of the object t is described by a first-order ordinary differential of study that has sufficient properties of fullness, equation adequacy, productivity and economy. dT U 2 CTTTS( ) = −α,( − ) dt R( T ) 0 Solution and the initial condition is as follows: To solve the problem, we need to build a T t= T . hierarchy of mathematical models for this object of ( 00) study and determine the conditions under which we Given that can calculate the sought value with a relative error I0 I = , not exceeding δ.0 1+−β(TT0 ) If the difference TT− 0 is sufficiently small, let us formulate a Cauchy problem then, according to (1), the sought value can be dIβ I 2 αβSIIUI( −−) 2 = 0 , (6) calculated using the formula dt cIγβ( I−+ I) I U 00 I0 = . (2) r I( t00) = I . Let us define the conditions under which the Then let us calculate the time point resulting formula is applicable. To do this, let us cIγ * I0 consider steady-state heat transfer. In this case, the tt*= 0 + −1 +δ 0 + αβSII heat output of the thermistor's material is equal to the 0* heat flow from the thermistor, i.e. IIIIγ − +0 + 0 * 0 −1 U 2 2IIIII0−− *β2 0 * * =−α,(TTS 0 ) RT( ) I* I0 ln 2 − −δ0 − + where T is the steady-state thermistor temperature. III0 2 0− * The resulting equality allows us to easily calculate IIII− 2 γ 0 * 0 0 14βU + ln δ,0 TT =0 + −1 + 1 + , β2IIIII0−− * * 0 * 2βαSr for which and then find the sought steady-state value I It( ) = * . 2U * 1− δ I* = , (3) 0 r1++ 1 4βα U2− 1 S − 1 r − 1 Evidently, at tt * and for this temperature range III− δ(I ) =** = 1 − δ , U 2 *0II 1 +β(TT − ) . (4) αSr T− T 10 ( 10) and the value I* can be considered equal to It( ) with The relative error of I0 is a relative error not exceeding δ.0 Therefore, it is IIII− possible to use formula (3) to find the sought value δI =0 = 0 − 1 0 − 1. ( 0 ) with a relative error not exceeding δ. III 0 If the condition If condition (5) is not met, mathematical model (3) at tt sufficiently possesses the properties of fullness, adequacy, productivity and economy.