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UDC 53.072.8
Doi: 10.31772/2587-6066-2019-20-3-334-343
For citation: Derevyanko V. A., Makukha A. V. Measurement of temperature distribution using a three-wire system of sensors based on thermistors. Siberian Journal of Science and Technology. 2019, Vol. 20, No. 3, P. 334-343. Doi: 10.31772/2587-6066-2019-20-3-334-343
Для цитирования: Деревянко В. А., Макуха А. В. Измерение распределения температуры с помощью трех-проводной системы датчиков на основе термисторов // Сибирский журнал науки и технологий. 2019. Т. 20, № 3. С. 334-343. Doi: 10.31772/2587-6066-2019-20-3-334-343
MEASUREMENT OF TEMPERATURE DISTRIBUTION USING A THREE-WIRE SYSTEM
OF SENSORS BASED ON THERMISTORS
V. A. Derevyanko*, A. V. Makukha
Institute of computational modeling SB RAS 50/44, Akademgorodok, Krasnoyarsk, 660036, Russian Federation *E-mail: ks1@icm.krasn.ru
Improving the reliability and increasing the avionics resource is associated with possibility of continuous control of temperature fields ofprinted circuit boards. This problem can be solved only with the use of a large number of temperature sensors. It raises the problem of connecting the measuring elements and recording equipment. Several methods with their own advantages and disadvantages are proposed.
One of the implemented and patented methods is using a set of resistive diode sensors installed in series on a three-wire line. The temperature sensors are pairs of counter - connected diodes with a sequential survey when applying sawtooth voltage. The system is simple and easy to implement, but its main drawback is the method of determining the temperature by measuring the amplitude of the total reverse currents of diode pairs. It determines the large measurement errors, especially in the temperature range less than 20°C.
The article deals with a similar design of a three-wire circuit, but with a fundamentally different approach to temperature measurement. The temperature sensor here is not diode pairs, but thermistors with a well-known dependence of resistance on temperature and high accuracy, and diode pairs record only the moment of coincidence of the sawtooth voltage with the voltage on the thermistors.
This approach allows using mathematical methods of signal processing to accurately determine the voltage drop on the thermistor, and this ensures the accuracy of the resistance/temperature and the expansion of the temperature range.
Given the fact that thermistors are increasingly used to measure temperature, simplifying their inclusion in a large number will allow to register the temperature field of electronic units, which is extremely important for spacecraft.
The proposed version of a three-wire circuit for connecting temperature sensors at several points was tested experimentally, including at negative temperatures.
Keywords: thermistor, counter-connected diodes, three-wire temperature measurement circuit.
ИЗМЕРЕНИЕ РАСПРЕДЕЛЕНИЯ ТЕМПЕРАТУРЫ С ПОМОЩЬЮ ТРЕХПРОВОДНОЙ СИСТЕМЫ
ДАТЧИКОВ НА ОСНОВЕ ТЕРМИСТОРОВ
В. А. Деревянко*, А. В. Макуха
Институт вычислительного моделирования СО РАН Российская Федерация, 660036, г. Красноярск, Академгородок, 50/44
*E-mail: ks1@icm.krasn.ru
Повышение надежности и увеличение ресурса радиоэлектронной аппаратуры космических аппаратов связывают с возможностью непрерывного контроля температурных полей печатных плат. Такая задача может быть решена только с использованием большого количества датчиков температуры. При этом возникает проблема подключения измерительных элементов и регистрирующей аппаратуры. Предложено несколько способов, имеющих свои достоинства и недостатки.
Одним из реализованных и запатентованных способов является использование набора резистивно-диодных датчиков, установленных последовательно на трехпроводной линии. Датчиками температуры являются пары встречно включенных диодов с последовательным опросом при подаче пилообразного напряжения. Система простая и легко реализуемая, но её основной недостаток заключается в способе определения температуры по измерению амплитуды суммарных обратных токов диодных пар. Именно этим определяются большие погрешности измерения, особенно в области температур меньше 20 °С.
В статье рассматривается аналогичная конструкция трехпроводной цепи, но с принципиально иным подходом к измерению температуры. Датчиком температуры здесь являются не диодные пары, а термисто-ры с хорошо известной зависимостью сопротивления от температуры и высокой точностью, а диодные пары фиксируют только момент совпадения величины пилообразного напряжения с напряжением на терми-сторах.
Такой подход позволяет использовать математические методы обработки сигналов для точного определения падения напряжения на термисторе, а это обеспечивает и точность определения сопротивления / температуры и расширения температурного диапазона измерений.
Учитывая то, что термисторы все чаще используются для измерения температуры, упрощение схемы их включения в большом количестве позволит регистрировать температурное поле радиоэлектронных блоков, что крайне важно для космических аппаратов.
Предложенный вариант трехпроводной схемы подключения датчиков температуры в нескольких точках проверен экспериментально, в том числе и при отрицательных температурах.
Ключевые слова: термистор, встречно включенные диоды, трёхпроводная схема измерения температуры.
Introduction. Many years of experience in testing electronic equipment in the space industry showed that 80 % of electromechanical failures are associated with various thermal effects on the equipment and 20 % of failures are caused by various types of vibrations and impact, therefore the close attention is paid to monitoring the thermal operating conditions of avionics.
The method currently used provides for temperature control with help of temperature sensors installed in the immediate vicinity of radio electronic components.
To get real information about the temperature field distribution of electronic components and to monitor the thermal conditions of avionics in real time a large number of temperature sensors are required [1-5] (several dozen sensors on each board and several hundred ones in the device).
There are devices for measuring temperature fields with thermoelectric sensors in the form of paired heterogeneous conductors [6]. The disadvantages of these devices include the complexity of their implementation (the need to isolate electrical conductors from the object and stabilize the temperature of the sensors leads).
Widespread resistance thermometers [7] have a complex system of separate power supply and registration which complicates to use a large number of these sensors on an electronic circuit board. This is due to the complexity of the connector wiring as well as sensor addressing and interrogation.
There are devices for measuring the temperature fields using a thermoelectric sensor in the form of a grid with quadrangular cells of m and n heterogeneous conductors connected at the intersection and forming columns and rows, respectively, by measuring thermal electromotive force between the output ends of conductors [8].
In [9] it was proposed to use distributed resistive diode sensors which allow installing up to tens of sensors with a sequential survey in series on a three-wire line. This method is based on the measurement with temperature-sensitive elements Ti consisting of counter-connected identical diodes DDi and two equal resistors R (fig. 1). The number of lead wires in this measurement scheme is three. The constant voltage source 1 is connected to the thermal sensitive elements via wires U1 and U3. The harmonic voltage source 2 and the signal recorder 3 are connected via wires U1 and U2.
The principle of temperature measuring is as follows: since the constant voltage source 1 is connected to a voltage divider of 2N identical resistors with resistance R the total differential resistance of the on-board diodes becomes minimal when the harmonic signal from the source 2 increases at the moment when the voltage across the diode pair DDi is equal to zero. The change in current through the diode pair is measured by the recorder 3 and processed. The signal is differentiated and the temperature of the thermal sensitive element Ti which number is determined by the order of the following peaks is calculated by the peak amplitude of the derivative. The measuring tool in this case is a diode pair. This construction highly facilitates the sensors placement and simplifies the system of interrogation and registration.
However, semiconductor diodes act here as a thermal sensitive element and its reverse saturation current depends on temperature. The main disadvantage of this method is the need of preliminary calibration of diode pairs and nonlinear dependence of the temperature on the signal magnitude. Radiation-resistant silicone or galliumarsenide diodes are used in space industry. Their reverse currents are much smaller than germanium diodes ones and therefore they can be used in the amplitude measurement method on resistive-diodes circuits only in the temperature range above 20 °C. Measurements errors are larger at lower temperatures.
Problem formulation. An undoubted advantage of distributed resistive-diodes temperature sensors is the possibility of their sequential polling on a three-wire line with the temporary addressing. The disadvantage is the amplitude method for recording the total saturation current of diodes which magnitude depends on many factors: the type of diode, the recorded temperature, the coincidence of the diode pair reverse currents, the influence of parallel-connected diode pairs and the interference taking into account large values of the diodes inverse resistance.
Taking into account all the problems of the semiconductor diodes the thermistors were considered as temperature sensors [10-12] Rx1, ..., RxN (fig. 2), and diode pairs are used to fix the moment when the harmonic signal voltage Upi coincides with the voltage Uxi in the corresponding point of the voltage divider [13]. The Rx0 resistor (with minimal temperature resistance coefficient) acts as a calibration resistance.
Fig. 1. Measurement scheme in the patent Рис. 1. Схема измерений в патенте
Fig. 2. Measurement scheme in the experiment Рис. 2. Схема измерений в эксперименте
The constant voltage U0 is supplied through one wire to a divider composed of N thermistors Rxi. The sawtooth voltage Up is supplied through the second wire to diode pairs DDi which are connected to the voltage divider. The signal Ux0 from the resistor Rx0 on the third wire is differentiated and the voltage between the two thermistors is determined by the maximum peak amplitude of the derivative. The number of thermistor is determined by the order of the following peaks. Thus, the thermistor voltage is determined by the time of the diode pair peaks and it significantly increases the measurement accuracy.
Experiment results. To verify the method effectiveness an installation based on the measuring complex LTR was assembled. Series connected thermistors operating in the temperature range from +20 to +80 X were replaced by calibrated resistances measured to the fourth digit. For definiteness the values of calibrated resistances and the corresponding temperature calculated for MF521021k thermistors with resistance R0 = 1.0 kilo-ohm ± 1.0 % when ^ = 25 X or R0 = 510 ohm ± 1.0 % and the temperature coefficient В = 4000 1/^ are shown on tab. 1. The temperature value TxN for each RxN is calculated by the formula
i = +Ln TxN T0 B
f Rx ^
R
о J
The sawtooth voltage Up and the constant voltage to the divider U0 = 10 V came from the LTR34-8 digital-to-analog converter. The Up and Ux0 voltage at the calibration resistance was measured by a 14-bit LTR11 AD converter (fig. 3).
The figure also shows the time derivative dUx0/dt of the measured signal on an enlarged scale as well as Ux0. Each peak of the derivative dUx0/dt is marked with the corresponded diode pair DDi on which the signal increment Ux0 and the sawtooth voltage value Upt = Uxt at a given time passed.
Since the current through the diode pairs was three orders of magnitude less than the current through the divider they were neglected in the calculations. Therefore, from the constancy of the current through the thermistors and the calibration resistance, the thermistor resistance was calculated by the formula
Ux, -Ux, ,
Rxt = —*-i-1 Rx0..
Ux0
As diode pairs germanium and silicone diodes of different purposes were used (tab. 2). The reverse current JD was measured at a reverse voltage of 20 V and an ambient temperature of +20 °C.
In the tab. 3 the results of five measurements, the average measured value of resistance and the deviation from the exact value are shown.
The selection of diode pairs exerts a great influence on the measurement accuracy. A preliminary selection for the same reverse current is required. At different currents with differentiation Ux0 asymmetric peaks were obtained at Uxi = Up. Because of this there was a shift in one or another direction from the peak maximum and, accordingly, a distortion of measurement results. It is clearly demonstrated by the data given in tabl. 2 and 3. An imbalance of reverse currents in the diode pair DD1 affects the Rxi, Rx2 and DD4 as well as Rx4 measurement accuracy.
Temperature range check. Testing of the proposed temperature measurement scheme was carried out in a narrow temperature range of the board and diodes from 20 to 25 °C. Of special interest is the measurement of devices thermal regimes at negative temperatures. For this purpose in fig. 2 all diodes were replaced by diode pairs from D311A with the same reverse current according to tab. 4.
TP018-02 thermistor was glued to the board base. Its resistance was measured by the H-27R-100 submodule and converted to temperature by a computer. In range from 10 to 20 °C all measurements were carried out according to the well-functioning scheme. At lower temperatures the voltage gain AUx0 = Ux0(t) - Ux0min is reduced by repeatedly reducing the reverse current of diode pairs. Thus, the signal itself decreased many times and the peaks of the derivative dUx0/dt couldn't be distinguished against the background of interference due to the discreteness of digitalization (fig. 4).
In this case it is necessary to apply averaging over K measurement. Random noise during averaging decreased 4K times and the useful signal was clearly distinguished. The averaging results over 16 and 1024 measurements and the temperature TR = 9.2 °C and TR = -10.3 °C, respectively, are shown on fig. 5 and 6.
Experimental data processing was reduced to the time derivative dUx0/dt transition, to the dependence not on time but on the sawtooth voltage F(Up) = dUx0/dt with the received signal post-processing. Since, due to a small change in the signal Ux0 and the discreteness of its digi-talization large interference noise arose and there were problems with determining the exact voltage value Uxi = Up, a special technique was used [14; 15].
Table 1
Temperature matching for experimental resistors
Rx1 Rx2 Rx3 Rx4 Rx5 Rx6 Rx7
R0 911.7 121.4 702.8 391.8 452.4 217.4 453.5
1 kilo-ohm 27.1 80.6 33.0 47.4 43.7 63.3 43.7
510 ohm 12.6 60.7 18.0 31.0 27.7 45.2 27.6
Fig. 3. Results of measurement of Up and Ux0(^V) versus time
Рис. 3. Результаты измерения Up и Цг0(мкВ) в зависимости от времени
Table 2
Types and characteristic of diode pairs
DDo DD1 dd2 DD3 DD4 dd5 DD6 dd7
Type IN5819 D311A D311A D311A SR360 IN5819 2D213A D223
jd(hA ) 5.3 12.4 16 15.0 7.9 2.5 4.8 1.8
5.1 20.2 15.1 14.8 4.5 2.6 5.2 4.6
Table 3
Accurate and measured characteristics of experimental resistors
Exact value Experimental value Average value Error, %
Rxo 597.2 № 1 № 2 № 3 № 4 № 5 <Rx>
Rx1 911.7 926.2 924.5 928.1 934.3 929.2 928.46 1 84
Rx2 121.4 119.2 118.8 118.7 118.2 118.4 118.66 -2,26
Rx3 702.8 705.2 704.6 705.9 710.1 707 706.56 0.54
Rx4 391.8 400.8 399.7 400.6 402.3 402.1 401.1 2.37
Rx5 452.4 455.9 457.2 456.7 458.7 458 457.3 1,08
Rx6 217.4 214.4 212.9 214.2 216.5 213.2 214.24 -1.45
Rx7 453.5 453.9 454.4 456.1 455 457.4 455.36 0.41
Table 4
Diode pairs with the same reverse current
DD0 DDi dd2 DD3 DD4 dd5 DD6 dd7
тУРе D311A D311A D311A D311A D311A D311A D311A D311A
jdd(^A ) 18.5 18.5 24.0 16.0 17.2 19.2 20.0 21.5
Fig. 4. The results of Ux0 measurement and processing depending on the time at the board temperature TR = -5.7 °C.
The scale for Ux0 and dUx0/dt is increased 10 times
Рис. 4. Результаты измерения и обработки их0 в зависимости от времени при температуре платы ТЯ = -5,7 °С.
Масштаб по Пх0 и dUx0/dt увеличен в 10 раз
Fig. 5. Averaging results AUx0 = Ux0(t ) - Ux0min and dUx0/dt over 16 measurements Рис. 5. Результаты усреднения AUx0 = Ux0(t) - Ux0min и dUx0/dt по 16 измерениям
Fig. 6. Results of averaging Д Ux0 = Ux0(t) - Ux0min and dUx0/dt over 1024 measurements. The scale for Ux0 and dUx0/dt is increased 100 times
Рис. 6. Результаты усреднения Д^0 = Ux0(t) - Ux0min и dUx0/dt по 1024 измерениям. Масштаб по Ux0 и dUx0/dt увеличен в 100 раз
For this, cross-correlation function kFG(Up) between F(Up)and
G (upi - Up) = e-a\Up-Up, (2)
was estimated by the formula
X(F (Up) -( F (Up)) )x :{G(Upt - Up) - {G(Upt - Up)))
kFo {Up)=-
(Up) -( F (Up)) )2 x : X (G(Upt - Up) - (G(Up1 - Up)))2
(3)
Where the function G(Upi - Up) looked like the peak in the derivative of voltage Ux0, and
N
{G(Up, - Up) ^ - Upk)
k=1
<f m)=iF (Upk >
k=1
N
is average values of correlating functions over the summed range and Upt is the voltage at which dUx0/dt reaches maximum for each peak.
The variable a in the function (2) changed during the method development for more accurate determination of thermistor resistance. It has been experimentally established that the best value for 1/a in the formula (2) is a value in range 0,1 ^ 0,2 V.
At lower values the peaks broaden and, accordingly, the exact determination of the maximum worsens. In addition, at low values of thermistor resistance an overlap of two adjacent peaks is observed which leads to the de-
crease in the calculated value Rxi. At larger values false peak arise due to interference in the numerical determination of the signal Ux0.
For checking purposes the cross-correlation function kFG(Up) was constructed according the measurement results (fig. 4).
The maximum value kFG(Up) didn't exceed the value 0.15 (fig. 7), and it was later used as a criterion for the results reliability (fig. 8, 9).
Fig. 7. Cross-correlation function according to the measurement results in fig. 4 Рис. 7. Кросскорреляционная функция по результатам измерения рис. 4
Fig. 8. Cross-correlation function according to the measurement results in fig. 5 Рис. 8. Кросскорреляционная функция по результатам измерения рис. 5
Fig. 9. Cross-correlation function according to the measurement results in fig. 6 Рис. 9. Кросскорреляционная функция по результатам измерения рис. 6
Statistical processing results for 7 thermistors
Table 5
N 16 32 64 128 512 1024
Tr(°C) 9.2 7.6 3.8 -3.3 -3.9 -10.3
Rx1 914.6 917.6 910.9 911 889.7 892.4 885.6
Rx2 119.2 116.1 116.9 117.4 112.2 114.9 108.1
Rx3 703.9 707.9 707.8 703.2 698.7 703.5 658.1
Rx4 393.9 391.5 391 391.8 397.4 394 387.8
Rx5 224.5 216.6 220.4 217.3 218.5 215.1 217.3
Rx6 453.4 462.4 458.2 454.8 470.5 459.5 455.3
Rx7 454.4 444.3 443.8 456.7 427.5 451.2 448.8
ARx1/Rx1 0.328 % -0.405 % -0.394 % -2.723 % -2.427 % -3.171 %
ARx2/Rx2 -2.601 % -1.930 % -1.510 % -5.872 % -3.607 % -9.312 %
ARx3/Rx3 0.568 % 0.554 % -0.099 % -0.739 % -0.057 % -6.507 %
ARx4/Rx4 -0.609 % -0.736 % -0.533 % 0.889 % 0.025 % -1.549 %
ARx5/Rx5 -3.519 % -1.826 % -3.207 % -2.673 % -4.187 % -3.207 %
ARx6/Rx6 1.985 % 1.059 % 0.309 % 3.772 % 1.345 % 0.419 %
ARx7/Rx7 -2.223 % -2.333 % 0.506 % -5.920 % -0.704 % -1.232 %
Scp(N) 1.690 % 1.263 % 0.937 % 3.227 % 1.765 % 3.628 %
The lower the temperature the greater number of N averages must be carried out in order to get a correct result. The lowest temperature at which it was still possible to process the signal was -10 °C at K = 1024 (fig. 6, 9).
Tab. 5 shows the measurement of thermistor simulators resistance at different temperatures TR(°C) of the board depending on the number of N averages and the relative measurement error ARx1/Rx1. The last line shows the absolute average relative measurement error
§op (N) = x LflARx-l/Rx- )/7
depending on N measurements. It is clearly seen that at N = 128 and N = 512 and approximately equal temperature
~ 3,227 ^ /512 _ 2
128 _ .
5ср (128) SCp (512)
1,765
1
This example confirms the correctness of the chosen statistical averaging method.
Conclusion. Thus, the use of thermistors as temperature sensors and diode pairs determining the addressing of the measuring elements allows the three-wire temperature measurement scheme in the range from -10 °С and above with an error of less than 4 % to be implemented.
Acknowledgments. The study was carried out with the financial support of the Russian Foundation for Basic Research, the Government of the Krasnoyarsk Territory, the Krasnoyarsk Regional Fund for Support of Scientific and Technological Activities in the framework of the project No. 18-47-242005.
Благодарности. Исследование выполнено при финансовой поддержке РФФИ, правительства Красноярского края, Красноярского краевого фонда поддержки научной и научно-технической деятельности в рамках проекта №18-47-242005.
References
1. Derevyanko V. A., Latypov A. F. [Recovery of the temperature distribution of the medium in the two-dimensional domain by the tomographic method from the result of measurements of the electrical resistance of oriented conductors]. Vychislitel'nye tekhnologii. 2018, No. 23 (3), P. 31-38 (In Russ.).
2. Vasil'ev E. N., Derevyanko V. A. [Analysis of the effectiveness of the use of thermoelectric modules in cooling systems of radioelements]. VestnikSibSAU. 2013, No. 4 (50), P. 9-13 (In Russ.).
3. Korneeva A., Shaidurov V. Computational algorithms for analysis of data from thin-film thermoresistors on a radio-electronic printed circuit board. Conference Proceeding of American Institute of Phisics. 2016, No. 1759, Art. 020048.
4. Korneeva A. A., Shaidurov V. V. [Numerical analysis of temperature data from film resistors of electronic board]. Vychislitel'nye tekhnologii. 2017, No. 22 (3), P. 32-44 (In Russ.).
5. Shaidurov V. V., Korneeva A. A. [Numerical method for the analysis of non-local data from film thermistors of electronic boards]. Sibirskie elektronnye mate-maticheskie izvestiya. 2017, No. 14, P. 914-926 (In Russ.).
6. Lineveg F. Izmerenie temperatury v tekhnike [Temperature measurement in the technique]. Moscow, Metallurgiya Publ., 1980, P. 320-346 (In Russ.).
7. GOST 6651-2009. Gosudarstvennaya sistema obe-specheniya edinstva izmereniy (GSI). Termopreobra-zovateli soprotivleniya iz platiny, medi i nikelya. Obshchie tekhnicheskie trebovaniya i metody ispytaniy [State system for ensuring the uniformity of measurements. Platinum, copper and nickel resistive temperature transducers. General technical requirements and test methods]. Moscow, Standartinform Publ., 2011.
8. Temperature measuring conductor network - uses thermoelectric effect at crossing points of conducting strips of two different materials. Patent France FR2315689, G0№7/02, 1977.
9. Bashirova A. G., Evdokimov Yu. K., Kraev V. V., Natfulov F. Kh. Ustroystvo izmereniya prostranstvennogo raspredeleniya temperatury [A device for measuring the spatial distribution of temperature]. Patent RU2079822 C1. 1994.
10. Wood S. D., Mangum B. W., Fillben J. J., Tillett S. B. An investigation of the stability of thermistors. J. Research of the National Bureau of Standarts. 1978, No. 83 (3), P. 247-263.
11. Precise and reliable measurement, optimization and management. Germany: Heraeus Sensor Technology,
2017, 38 p.
12. Yadevich A. I. [Platinum thin film sensor for resistance thermometers]. Mir izmereniy. 2010, No. 3, P. 52-55 (In Russ.).
13. Neiman L. R., Demirchyan K. S. Teoreticheskie osnovy elektrotekhniki [Theoretical foundations of electrical engineering]. Vol. 1. Leningrad, Energoizdat, 1981, 536 p.
14. Bronshtein I. N., Semendyaev K. A. Spravochnik po matematike dlya inzhenerov i uchashchikhsya VTUZov [Handbook of mathematics for engineers and students of high schools]. Moscow, Nauka Publ., 1986, P. 464-465.
15. Korn G., Korn T. Spravochnik po matematike dlya nauchnykh rabotnikov i inzhenerov [Mathematical handbook for scientists and engineers]. Moscow, Nauka Publ., 1973, 684 p.
Библиографические ссылки
1. Деревянко В. А., Латыпов А. Ф. Восстановление распределения температуры среды в двумерной области томографическим методом по результатам измерений электрического сопротивления ориентированных проводников // Вычислительные технологии.
2018. № 23 (3). С. 31-38.
2. Васильев Е. Н., Деревянко В. А. Анализ эффективности применения термоэлектрических модулей в системах охлаждения радиоэлементов // Вестник СибГАУ. 2013. № 4 (50). С. 9-13.
3. Korneeva A., Shaidurov V. Computational algorithms for analysis of data from thin-film thermoresistors on a radio-electronic printed circuit board // Conference Proceeding of American Institute of Phisics. 2016. No. 1759. Art. 020048.
4. Корнеева А. А., Шайдуров В. В. Численный анализ температурных данных с пленочных резисторов электронных плат // Вычислительные технологии. 2017. № 22 (3). C. 32-44.
5. Шайдуров В. В., Корнеева А. А. Численный метод анализа нелокальных данных с пленочных терморезисторов электронных плат // Сибирские электронные математические известия. 2017. № 14. C. 914-926.
6. Линевег Ф. Измерение температуры в технике. М. : Металлургия, 1980. C. 320-346.
7. ГОСТ 6651-2009. Государственная система обеспечения единства измерений (ГСИ). Термопреобразователи сопротивления из платины, меди и никеля. Общие технические требования и методы испытаний. М. : Стандартинформ, 2011.
8. Temperature measuring conductor network - uses thermoelectric effect at crossing points of conducting strips of two different materials. Patent France FR2315689, G01K7/02, 1977.
9. Пат. RU2079822 C1. Устройство измерения пространственного распределения температуры / А. Г. Баширова, Ю. К. Евдокимов, В. В. Краев,
Ф. Х. Натфулов ; заявитель КНИТУ-КАИ. № 94038391/28 ; заявл. 11.10.1994 ; опубл. 27.03.1997.
10. An investigation of the stability of thermistors / S. D. Wood, B. W. Mangum, J. J. Fillben et al. // J. Research of the National Bureau of Standarts. 1978. No. 83 (3). P. 247-263.
11. Precise and reliable measurement, optimization and management. Germany : Heraeus Sensor Technology, 2017. 38 p.
12. Ядевич А. И. Тонкопленочные платиновые чувствительные элементы для термисторов сопротивления // Мир измерений. 2010. № 3. С. 52-55.
13. Нейман Л. Р., Демирчан К. С. Теоретические основы электротехники. Т. I. Л. : Энергоиздат, 1981. 536 с.
14. Бронштейн И. Н., Семендяев К. А. Справочник по математике для инженеров и учащихся втузов. М. : Наука, 1986. С. 464-465.
15. Корн Г., Корн Т. Справочник по математике для научных работников и инженеров. М. : Наука, 1973. 684 с.
i-:: Derevyanko V. A., Makukha A. V., 2019
Derevyanko Valerii Aleksandrovich - Cand. Sc., Senior researcher of the Department of information and computational modeling; Institute of computational modeling SB RAS. E-mail: dv@icm.krasn.ru.
Makukha Aleksandr Vasilievich - Leading engineer of the Department of information and computational modeling; Institute of computational modeling SB RAS. E-mail: ks1@icm.krasn.ru.
Деревянко Валерий Александрович - кандидат физико-математических наук, старший научный сотрудник отдела информационно-вычислительного моделирования; Институт вычислительного моделирования СО РАН. E-mail: dv@icm.krasn.ru.
Макуха Александр Васильевич - ведущий инженер отдела информационно-вычислительного моделирования; Институт вычислительного моделирования СО РАН. E-mail: ks1@icm.krasn.ru.
