Спосіб лінеаризації температурних характеристик NTC-термісторів за допомогою поліномінального методу
The objective of this research is to develop a circuit-based method for linearizing the temperature characteristics of thermistors using a polynomial digital technique for NTC-thermistor temperature characteristics, together with the characteristics of the designed measurement channel, by means of a...
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The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute"
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| author | Matvienko, Sergey Tymchyk, Grygoriy |
| author_facet | Matvienko, Sergey Tymchyk, Grygoriy |
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| description | The objective of this research is to develop a circuit-based method for linearizing the temperature characteristics of thermistors using a polynomial digital technique for NTC-thermistor temperature characteristics, together with the characteristics of the designed measurement channel, by means of an original MATLAB Simulink model. A model of a temperature-measurement device employing a thermistor-based sensor is proposed. The polynomial digital method for linearizing the temperature characteristics of NTC thermistors in the developed device has been simulated. To improve measurement accuracy, the components of the measurement channel were selected such that they enable simulation with minimal error. Methods for introducing correction-coefficient values into the model have been determined to compensate for error-inducing factors, including the thermistor’s self-heating effect. The proposed data-processing algorithm offers advantages for implementation in a low-cost, low-power microcontroller. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2026.1.04 |
| first_indexed | 2026-04-20T01:00:21Z |
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S.N. Matvienko, G.S. Tymchyk, 2026
58 ISSN 1681–6048 System Research & Information Technologies, 2026, № 1
TIÄC
ПРОБЛЕМНО І ФУНКЦОНАЛЬНО ОРІЄНТОВАНІ
КОМП’ЮТЕРНІ СИСТЕМИ ТА МЕРЕЖІ
UDC 536.531
DOI: 10.20535/SRIT.2308-8893.2026.1.04
POLYNOMIAL-BASED METHOD FOR LINEARIZING THE
TEMPERATURE RESPONSE OF NTC THERMISTORS
S.N. MATVIENKO, G.S. TYMCHYK
Abstract. The objective of this research is to develop a circuit-based method for lin-
earizing the temperature characteristics of thermistors using a polynomial digital
technique for NTC-thermistor temperature characteristics, together with the charac-
teristics of the designed measurement channel, by means of an original MATLAB
Simulink model. A model of a temperature-measurement device employing a ther-
mistor-based sensor is proposed. The polynomial digital method for linearizing the
temperature characteristics of NTC thermistors in the developed device has been
simulated. To improve measurement accuracy, the components of the measurement
channel were selected such that they enable simulation with minimal error. Methods
for introducing correction-coefficient values into the model have been determined to
compensate for error-inducing factors, including the thermistor’s self-heating effect.
The proposed data-processing algorithm offers advantages for implementation in a
low-cost, low-power microcontroller.
Keywords: temperature measurement, NTC thermistor, MATLAB Simulink, linearization.
INTRODUCTION
Temperature measurement devices play a critical role across various industrial
and consumer domains, particularly within Internet of Things (IoT) networks that
leverage Smart technologies, including smart cities, smart homes/digital houses,
smart grids, and smart sensors. A fundamental requirement for temperature sen-
sors in such applications is high measurement accuracy over a wide temperature
range, which is primarily determined by the type and characteristics of the sensor.
Negative Temperature Coefficient (NTC) thermistors are widely employed
as temperature sensors due to their broad operational temperature range, capabil-
ity for remote monitoring, resistance to strong magnetic fields, and compact phys-
ical dimensions. However, one of the primary limitations of thermistors, shared
by many sensor types — is the nonlinear nature of their resistance–temperature
characteristic )(TR , which adversely affects measurement accuracy.
Additionally, the transfer characteristics of the thermistor interface circuitry
and the voltage signal amplifier are also nonlinear. Since nonlinearities introduce
measurement errors, improving the linearity of the transfer function of the entire
measurement channel is essential for achieving high accuracy. The task of linear-
izing the temperature dependence of a measurement system can be addressed
Polynomial-based method for linearizing the temperature response of NTC thermistors
Системні дослідження та інформаційні технології, 2026, № 1 59
through analog, digital (software-based), or mixed hardware-software digital
techniques.
Analog methods can reduce measurement error to some extent but typically
only within a narrow temperature range. In contrast, digital linearization tech-
niques not only improve measurement accuracy but also significantly extend the
effective temperature measurement range. These techniques are implemented via
software or through the use of analog-to-digital interface circuits that generate a
linear signal response.
Among analog approaches, passive compensation networks are the most
common; they enable the creation of quasi-linear segments on the TR curve
within specific temperature intervals. Analog methods are generally more cost-
effective compared to digital techniques, which require microcontrollers, field-
programmable gate arrays (FPGAs), digital signal processors (DSPs), or personal
computers. Nevertheless, digital methods provide a substantial improvement in
accuracy over a wider temperature range.
With advancements in digital signal processing and the availability of in-
creasingly capable digital integrated circuits, the implementation of such methods
has become significantly more practical and accessible.
RELEVANCE OF THE WORK
Increasing the accuracy of temperature measurement is achieved by optimizing the
design of the measuring probe and selecting the appropriate characteristics of the
thermistor. Accuracy can also be increased by setting the optimal parameters for the
measuring channel and implementing effective digital data processing algorithms.
During the design phase of a temperature measurement device, it is essential to con-
duct mathematical modeling to determine the optimal linearization method, select
component parameters, and define the appropriate data processing algorithm.
The development of mathematical models enables the identification of suita-
ble characteristics for the components of the measurement channel and the data
processing algorithms, tailored to the specific requirements of the application.
In this research, a custom-designed model implemented in MATLAB Simulink is
used to investigate a digital method for linearizing the device characteristics by
applying polynomial functions to the measured data. The model facilitates the
estimation of measurement errors and the development of calibration recommen-
dations aimed at improving overall measurement accuracy.
ANALYSIS OF RECENT RESEARCH AND PUBLICATIONS
Linearized sensor characteristics significantly simplify the design and calibration
processes and improve measurement accuracy. To compensate for the inherent
nonlinearity of thermistor characteristics and the measurement channel, both ana-
log and digital linearization methods have been widely reported in the literature.
Researches such as [1–3] provide an in-depth analysis of these techniques and
offer a comprehensive overview of various approaches used for sensor character-
istic linearization.
It is noted that digital methods, particularly when combined with software-based
processing, yield superior results in terms of flexibility and performance. Software
S.N. Matvienko, G.S. Tymchyk
ISSN 1681–6048 System Research & Information Technologies, 2026, № 1 60
algorithms implemented in digital systems have proven to be more effective, practi-
cal, and adaptable than traditional analog methods. Common software-based lineari-
zation techniques include spline fitting, polynomial curve approximation, and ad-
vanced intelligent methods such as artificial neural networks (ANNs) [4].
Among these, the use of polynomial functions remains the most widespread
technique for correcting measured data [1]. A single high-order polynomial can
be used to model the full sensor response. However, to reduce computational
complexity, the entire temperature range is often divided into smaller sub-ranges,
each approximated with a lower-order polynomial. This segmentation improves
processing efficiency without sacrificing accuracy.
Additionally, alternative enhanced techniques based on lookup tables are al-
so employed, including piecewise linear interpolation (PWLI), piecewise linear
equations (PWLE), and programmable gain amplifiers (PGA) [1]. In PWLE, the
processor selects the appropriate linear equation based on the input value and re-
trieves stored coefficients from memory. In some implementations, the processor
dynamically determines the applicable linear equation for each measurement
based on pre-calibrated data points.
Depending on the complexity of the data processing algorithm, various
hardware platforms can be used, such as microcontrollers, FPGAs, DSPs, or PCs.
More complex algorithms typically require higher-cost solutions and result in in-
creased power consumption. A comparison of selected algorithms can be found
in [5]. Therefore, it is critical to choose the most suitable linearization approach
and its hardware implementation based on the specific performance requirements
of the temperature measurement device.
Mathematical modeling of the device and its linearization method can signif-
icantly streamline the design process, enabling efficient selection of component
parameters and processing algorithms to meet the required accuracy specifica-
tions. Manufacturer-provided models [6–8] are also beneficial in selecting appro-
priate thermistor types for specific applications and offer detailed characteristic
data that can be used to address )(TR nonlinearity.
Researchers and engineers have proposed a wide range of algorithms to ad-
dress the nonlinearity problem of the )(TR characteristic [1; 3–5; 9–11]. Given
the diversity of available digital linearization techniques, selecting the optimal
algorithm and its hardware implementation can be challenging and depends on the
specific application requirements.
During the modeling stage, it is essential to evaluate the achievable measure-
ment accuracy, the impact of hardware element characteristics, and the feasibility of
implementing calibration procedures. In this research, a classical linearization ap-
proach is explored, using polynomial correction of measured data with segmentation
of the full measurement range into smaller sub-ranges. This approach reduces the re-
quired polynomial order, thereby lowering the computational load on the processing
hardware. Consequently, cost-effective microcontrollers can be used, and device cali-
bration functionality can be feasibly integrated into the system.
Therefore, the purpose of our work is to develop a mathematical model of
the measurement channel of a temperature sensing device based on an NTC ther-
mistor within the MATLAB/Simulink environment. Using the developed model
of digital linearization methods for NTC thermistor temperature characteristics —
based on polynomial approximations — it is intended to determine the required
Polynomial-based method for linearizing the temperature response of NTC thermistors
Системні дослідження та інформаційні технології, 2026, № 1 61
specifications of the components within the measurement channel. Additionally,
an analysis of the sources of measurement error will be conducted to improve
temperature measurement accuracy across a wide temperature range.
The aim of this work is to investigate digital methods for linearizing the
temperature characteristics of NTC thermistors and the associated measurement
channel components using the developed simulation models.
RESEARCH METHODOLOGY AND RESULTS
Measuring temperature using an NTC thermistor involves using the dependence
of the thermistor’s resistance on its temperature, that is, on the temperature of the
environment surrounding the thermistor.
The dependence of the electrical resistance of an NTC thermistor on temper-
ature has the form [12]:
N
N TT
RTR 11exp)( , (1)
where )(TR — is the resistance of the NTC thermistor at temperature T in K; NR —
NTC thermistor resistance at nominal temperature NT in K; T — the current
temperature (in K) value at which the thermistor resistance NR is calculated;
NT — nominal temperature (in K), i.e. the reference or standard temperature at
which the nominal resistance of the thermistor is known; B — is a constant coeffi-
cient that depends on the thermistor material in К.
This dependence is nonlinear, which creates certain difficulties in creating
temperature measurement devices and introduces additional error. The value of
the error will depend on the linearization method used.
Fig. 1 shows the developed structural and mathematical model of a device
for measuring temperature using an NTC thermistor in the MATLAB Simulink
environment.
Fig. 1. Functional diagram of a temperature measurement device in the MATLAB Sim-
ulink environment
S.N. Matvienko, G.S. Tymchyk
ISSN 1681–6048 System Research & Information Technologies, 2026, № 1 62
The model consists of 2 groups of blocks for modeling the electrical circuit
of the measuring channel, these are the sensor interface group “Sensor interfacing
circuit” and the “Amplifier” group, 3 software modules “Subsystem” — “Subsys-
tem Microcontroller”, “Subsystem U-T” and “Subsystem self-heating”.
The group “Sensor interfacing circuit” is a diagram of connecting the ther-
mistor model — Rth to the differential amplifier model “Fully Differential Op-
Amp” of the group “Amplifier” using a correspondingly configured diagram of
connecting the resistor models “Resistor” R1,.. R6.
thR can be connected to the Wheatstone bridge arm R1, R2, R3, thR to the
“Fully Differential Op-Amp”. The Wheatstone bridge arms R1-R3 and R2- thR are
connected to the “Voltage Source” model.
In the MATLAB Simulink environment there is a model “Thermistor” in the
library that simulates the operation of a thermistor according to the set parame-
ters, but in this model the resistance of the thermistor is calculated by formula (1)
depending on the temperature. In a real thermistor, the coefficient B is not a con-
stant value, but varies depending on the temperature. Fig. 2 shows the difference
between the TR characteristic of the thermistor of the “Thermistor” MATLAB
Simulink model and the characteristic built from tabular data for a thermistor type
RH18 6Y103 Mitsubishi Materials.
If the thermistor is used within a narrow temperature range and moderate
measurement accuracy is acceptable, the built-in “Thermistor” block in
MATLAB Simulink may be employed for modeling purposes. However, when
the temperature range is wide and high measurement accuracy is required, the
simulation should rely on tabulated data provided by the manufacturer for the
specific thermistor type.
Fig. 2. Graph of the difference in )(TR characteristics for the thermistor model “Ther-
mistor” and tabulated data for the thermistor type RH18 6Y103 Mitsubishi Materials in
the range from 233 K to 383 K
Polynomial-based method for linearizing the temperature response of NTC thermistors
Системні дослідження та інформаційні технології, 2026, № 1 63
In the presented model, the thermistor is implemented using the “Variable
Resistor”, whose resistance is defined based on the manufacturer’s tabular data via
three “Repeating Sequence” blocks. These sequences correspond to the nominal,
minimum, and maximum resistance values of the thermistor. The resistance value
used in the simulation “Rmax”, “Rnom” and “Rmin” is selected using switch S1.
To convert the temperature value measured by the thermistor into a
corresponding voltage output, a widely adopted namely “Sensor interfacing
configuration”, the Wheatstone bridge (Fig. 1) — is utilized. To reduce the
nonlinearity of the thermistor’s resistance-temperature characteristic )(TR , a
circuit-level linearization technique is applied by connecting a resistor R4 in
parallel with the thermistor (Fig. 1). This approach minimizes the deviation
between the linearized and actual nonlinear characteristics of the “Sensor
interfacing circuit” output, thereby enhancing the accuracy of polynomial
correction methods.
The resistance of the parallel resistor used to linearize the TR characteris-
tic is determined according to the following formula [13]:
N
N
thN TB
TBRR
2
2
4
,
where 4R — resistance of a parallel-connected resistor, Ohm; thNR — NTC
thermistor resistance at nominal temperature NT in K; T, NT — temperature in K;
B — is a constant coefficient that depends on the thermistor material in K.
The total resistance of a thermistor with a resistor connected in parallel is de-
termined by the formula:
thN
thN
p RR
RRR
4
4 .
For thermistor RH18 6Y103F Mitsubishi Materials with 85/25B = 3435 К
and resistance 10 kOhm±1% at temperature NT = 298 K (25 oC) R4 = 7043 Ohm a
pR =4142 Ohm.
The “Amplifier” group consists of the “Fully Differential Op-Amp” and the
ideal operational amplifier model “Op-Amp”, which together provide the required
voltage gain G for the Wheatstone bridge imbalance signal.
The software module “Subsystem U-T” is used to obtain the uncorrected ana-
log signal, whose voltage is proportional to the measured temperature. This output
is necessary for evaluating the effectiveness of the digital linearization method.
The block diagram of the “Subsystem U-T” module is shown in Fig. 3, a.
The instantaneous voltage at the output of the “Amplifier” block, which is
proportional to the set temperature, is converted by the “PS-Simulink Converter”
into a signal compatible with the Simulink environment. This signal is then scaled
using the “Divide” block by an appropriate gain coefficient and converted into the
corresponding temperature value in K. The computed thermistor self-heating tem-
perature, determined by the “Subsystem self-heating”, is then added to this value.
The “Subsystem self-heating” module is employed to calculate the self-
heating temperature of the thermistor as a function of its operating temperature.
The internal structure of this module is depicted in Fig. 3, b.
S.N. Matvienko, G.S. Tymchyk
ISSN 1681–6048 System Research & Information Technologies, 2026, № 1 64
The thermistor’s self-heating temperature is influenced by the magnitude of
the current flowing through it, the materials and structural design of the sensor, as
well as the thermal conductivity of the surrounding medium [12; 14]. This
phenomenon is utilized in devices for measuring thermophysical properties of
materials [15–17], as well as in systems for determining fluid flow velocity [18].
Self-heating in NTC thermistors causes an additional decrease in their
resistance, which leads to distortion in the measurement result. Therefore, in
temperature measurement systems, a correction must be applied to the measured
value. This correction is computed using the following formula:
thth
A
TRIT
R
UTT
)(
(T)
22
, (2)
where NT — actual value of the controlled temperature; T — Measured tempera-
ture value; U — instantaneous value of the voltage on the thermistor, I — instan-
taneous value of the current flowing through the thermistor; )(TR — the value of
the resistance of the thermistor corresponding to the temperature Т; th — heat
dissipation coefficient in the measuring medium.
Using the voltage sensor model “Voltage Sensor” the voltage values on the
“Variable Resistor” at the current time are recorded, and using the current sensor
a
b
Fig. 3. Functional diagram “Subsystem UT” of the device module — a; functional diagram
of the “Subsystem self-heating” device module — b
Polynomial-based method for linearizing the temperature response of NTC thermistors
Системні дослідження та інформаційні технології, 2026, № 1 65
models “Current Sensor” the current value passing through the “Variable Resis-
tor” is recorded. From these data, the self-heating temperature of the thermistor
for air is determined by formula (2). This data is used to correct the measurement
data in the “Subsystem Microcontroller” module.
Fig. 4. Functional diagram of the “Subsystem Microcontroller” device module
For correction of measured data in the “Subsystem Microcontroller” module
“The model implements” a digital linearization algorithm using polynomials with
the division of the full measurement range into small subranges. Fig. 4 shows
a model of the software module “Subsystem Microcontroller”.
Fig. 5. Graphs of the linear characteristic NTlin compared to the nonlinear characteristic
)(NT
N by means of a 16-bit ADC “Idealized ADC quantizer”. Due to the nonlin-
earity ( )R T of the thermistor characteristic, the nonlinear transfer characteristic of
the Wheatstone bridge circuit, the differential amplifier and the ADC, the charac-
S.N. Matvienko, G.S. Tymchyk
ISSN 1681–6048 System Research & Information Technologies, 2026, № 1 66
teristic of the source code at the ADC output is TN will also be nonlinear. The
task of the “Subsystem Microcontroller” module is to form a linear characteristic
( )linT N .
Fig. 5 shows the generated linear characteristic )(NTlin compared to the non-
linear characteristic )(NT .
Formation of linear characteristic )(NTlin so that it connects the starting
point of the nonlinear characteristic )(NT , when the value at the ADC output
minN corresponds to the temperature minT = 233 К , and the end point of the
range, when the value at the ADC output maxN corresponds to the temperature
maxT = 383 К, i.e., it corresponds to the expression:
min( )linT N a N T ,
where a — characteristic slope coefficient, which is determined by the formula:
minmax
minmax
NN
TTa
.
Then, for the correcting function should be equal to:
T (N ) T(N ) Tlin (N ) .
That is, to obtain the measured temperature value at the output of the “Mi-
crocontroller Subsystem” module, it is necessary to add to the calculated tempera-
ture value by the linear function )(NTlin the correction value at point N, which
corresponds to T at point N. This value can be calculated using the polynomial
function:
P(N) cn N n cn1 N n1 cn2 N n2 c1N1 c0
where 0121 ,,...,, ccccc nnn — polynomial coefficients, which are determined by
the polynomial trend line of the function )(NT . In this model, this is done using
Excel using saved time series data or an array in the basic MATLAB workspace
by the “Simout ADC” module group.
Therefore, to calculate the measured temperature value at the output of the
“Microcontroller Subsystem” module, the value of the polynomial )(NP at point
N is added to the calculated temperature value according to the linear function
min)( TNaNTlin , which corresponds to the value of T at point N, and
therefore the measured temperature value is equal to:
)()( min NPTNaNT .
To do this, the value N from the ADC output is fed to 6 polynomial eval-
uation modules, which calculate the value of the polynomial )(NP with a giv-
en polynomial array of coefficients c depending on the value of N. The poly-
nomial array of coefficients for each of the subranges is determined
separately. Depending on the current value of N, the corresponding module
)(uP is connected, where the value of the polynomial )(uP is cleared with the
corresponding subrange given polynomial array of coefficients. These values
Polynomial-based method for linearizing the temperature response of NTC thermistors
Системні дослідження та інформаційні технології, 2026, № 1 67
are added to the formed linear characteristic )(NTlin . Also, the “Microcontrol-
ler Subsystem” module implements the possibility of correcting the output
characteristic according to the calibration results, and introducing correction
coefficients for the self-heating correction of the thermistor depending on the
thermal conductivity of the environment.
The “Solver Configuration” module defines general simulation parame-
ters. The obtained data of each of the obtained values is recorded by the oscil-
loscope “Scope” (Fig. 6) and entered using the “Simout” module group into
the specified time series or array in the basic MATLAB workspace for further
data processing.
“Calibration Data” block consists of “Constant” modules, in which the val-
ues of the coefficients 298K , 298K and /298T and heatingselfK , which allows
the calibration data to be used to adjust the original measurement data accordingly
to compensate for the influence of possible causes of error.
To generate the current temperature value over a time series, the “Repeating
Sequence” model is used, where a sequence of numbers time-temperature value is
entered. Next, the temperature values are converted from Simulink format to
physical signal data-temperature in K using the “Simulink-PS Converter”. This
data is used to compare the measured temperature value with the set value corre-
sponding to the current thermistor resistance value.
Fig. 6. Graphs of changes in parameter values during the simulation of temperature meas-
urement in MATLAB Simulink
S.N. Matvienko, G.S. Tymchyk
ISSN 1681–6048 System Research & Information Technologies, 2026, № 1 68
RESULTS OF THE RESEARCH
The research was conducted over the full temperature range of the thermistor,
from 233 K to 383 K (i.e., from –40°C to +110°C). The operational temperature
range and parameters of the thermistor corresponded to those of an RH18-type
thermistor manufactured by Mitsubishi, with a nominal resistance of k10)(R
at 25°C and a 85/25B constant of 3435 K. This thermistor is encapsulated in
epoxy resin and features compact dimensions (1.8 mm in diameter and 7 mm in
length), making it particularly sensitive to self-heating effects (thermal dissipation
constant CmW/1) th in air). This characteristic enables a comprehensive
analysis of the impact of linearization methods on measurement error across the
entire temperature range, as well as the determination of correction values ac-
counting for the self-heating effect of the thermistor.
During simulation, temperature measurement error was evaluated at the ana-
log output of the amplifier, both without correction and with the application of
digital signal processing using a high-order polynomial function for data correc-
tion over the full temperature range. Additionally, the measurement range was
divided into three and six smaller subranges, where lower-order polynomial func-
tions were applied, respectively, to evaluate the influence of segmentation on
measurement accuracy.
Fig. 7 illustrates the dependence of measurement error on the current tempera-
ture value. The corresponding measurement data are presented in tabular form.
As shown in Fig. 7 and the data table, increasing the number of subranges into
which the full measurement range is divided results in a reduction of measurement
error and allows for the use of lower-degree polynomial correction functions.
Table 1 presents the simulation results for the digital signal processing algo-
rithm used to correct the measured data, based on polynomial functions of varying
orders applied over the full temperature range and over its division into three and
six smaller subranges.
Fig. 7. Dependence of measurement error on the current temperature value
Polynomial-based method for linearizing the temperature response of NTC thermistors
Системні дослідження та інформаційні технології, 2026, № 1 69
T a b le 1 . Research results data
Number
of subbands Adjustment
Error
Average value ( T ), К Root mean square value , К
1 (Full) No adjustment 9.589 9.518
1(Full) 6th order polynomial 0.145 0.859
3 subbands 4th order polynomial 0.111 0.055
6 subbands 3rd order polynomial -0.002 0.036
Fig. 8 presents the actual temperature measurement error values at resistance
points minR , nomR and maxR , obtained using third-order polynomial correction
functions within each of the six subranges. The values of minR nomR , and maxR ,
over the temperature range from 233 K to 383 K (– 40°C to +110°C) were taken
from the datasheet provided by the manufacturer of the RH18 6Y103 thermistor
(Mitsubishi Materials).
As observed from the plots, the minimum measurement error occurs at the
nominal temperature of 298 K (+25°C), while the error increases as the tempera-
ture approaches the lower minT and maxT bounds of the range. Calibration of the
device allows for the determination of additional correction coefficients, denoted as
298K , 298K and /298T aimed at reducing the measurement error.
The coefficient 298K — compensates for the slope deviation of the linear
characteristic within the range from 233 K to 298 K (–40°C to +25°C), while 298K
serves the same purpose in the range from 298 K to 383 K (+25°C to +110°C). The
coefficient /298T represents a temperature offset at the nominal point NT =298 K
(+25 oC), as determined during calibration. The developed model provides the capa-
bility to input these coefficients to enhance measurement accuracy.
Fig. 8 demonstrates the potential for reducing measurement error through the
inclusion of the correction coefficients 298K , 298K and /298T .
Fig. 8. Actual value of temperature measurement error at minR , nomR and maxR when
correcting measured data using 3rd order polynomial functions in each of the 6 subbands
S.N. Matvienko, G.S. Tymchyk
ISSN 1681–6048 System Research & Information Technologies, 2026, № 1 70
Fig. 9 shows the actual temperature measurement error under conditions of a
reference voltage shift in the Wheatstone bridge circuit, refU on refU = ±0.05V
(±0.5 %), using third-order polynomial correction functions in each of the six
subranges. The results are shown for both uncalibrated conditions and after apply-
ing the additional calibration coefficients.
Fig. 9. Actual value of temperature measurement error when shifting the reference voltage
of the Wheatstone bridge refU at refU = ±0.05 V
Fig. 10 presents the actual temperature measurement error resulting from a
deviation of the amplifier gain coefficient from its ideal value by G = ±0.01,
using third-order polynomial correction functions within each of the six subrang-
es. The results are shown both without and with the application of additional cali-
bration coefficients.
Fig. 10. The actual value of the temperature measurement error when the actual value of
the amplifier gain deviates from its ideal value by G = ±0.01
Polynomial-based method for linearizing the temperature response of NTC thermistors
Системні дослідження та інформаційні технології, 2026, № 1 71
Fig. 11 illustrates the actual temperature measurement error due to thermistor
self-heating, corrected using third-order polynomial functions within each of the
six subranges, with the inclusion of various values of the heatingselfK coefficient:
0, 0.25, 0.5, 0.75, and 1.0.
Fig. 11. Actual value of temperature measurement error taking into account thermistor
self-heating when correcting measured data using 3rd order polynomial functions in each
of 6 subranges with the input of heatingselfK = 0; 0.25; 0.5; 0.75; 1.0
The summarized data of the research results are presented in Table 2.
To investigate the measurement error caused by the self-heating effect of the
thermistor and the potential for compensating this error, the model includes a ded-
icated module, “Subsystem self-heating”. This module determines the self-heating
temperature of the thermistor at a given ambient temperature under still air condi-
tions, which is used as the baseline reference. This approach is justified by the
fact that thermistor manufacturers typically specify the thermal dissipation con-
stant th only for still air.
When the thermistor operates under different environmental conditions (e.g.,
in liquids or moving air), the actual th value must be determined experimentally.
Subsequently, the self-heating correction coefficient heatingselfK should be ad-
justed proportionally relative to the value of heatingselfK determined under still
air conditions.
In the case of stationary water, which has a significantly higher thermal con-
ductivity compared to air, the thermal dissipation constant th increases by a fac-
tor of 2 to 5, depending on the thermistor’s design, the material of its protective
coating, and other structural factors. Consequently, the self-heating temperature
of the thermistor is reduced, and a correction must be introduced by applying a
corresponding self-heating compensation coefficient heatingselfK typically rang-
ing from 0.2 to 0.5.
S.N. Matvienko, G.S. Tymchyk
ISSN 1681–6048 System Research & Information Technologies, 2026, № 1 72
T a b le 2 . Summary of research results
Parameter
Calibration factors Error
298K 298K /298T
heatingselfK T ,
К
, К
nomR - -0.002 0.036
minR
- 0.678 0.305
2.5*10-5 -2.0*10-13 0.25 K - -0.018 0.057
maxR - -0.689 0.309
-2.5*10-5 2.0*10-13 -0.25 K - 0.016 0.057
refU = 10.00 V - -0.002 0.036
refU =+0.05
V (+0.5%)
- 0.896 0.525
-1.2*10-5 -2.0*10-13 0.1 K - 0.011 0.119
refU = –0.05
V (–0.5%)
- -0.198 0.529
1.5*10-5 2.5*10-13 -0.2 K - -0.018 0.108
G = 1.6 - -0.002 0.036
G = +0.01
(+0.63%)
- 0.258 0.695
-1.5*10-5 -2.6*10-13 -0.1 K - 0.026 0.150
G = –0.01
(–0.63%)
- -0.246 0.659
1.7*10-5 2.5*10-13 0.1 K - 0.026 0.140
heatingselfT -
0 1.141 0.551
0.25 0.855 0.413
0.5 0.569 0.276
0.75 0.282 0.141
1.0 -0.002 0.036
CONCLUSIONS
The issue of sensor characteristic linearization is critically important for real-time
applications across various fields, as most sensors exhibit nonlinear behavior.
Digital linearization methods offer greater flexibility and improved measurement
accuracy over a wide range of measured quantities. These methods can be imple-
mented using software on a personal computer or via dedicated hardware plat-
forms such as microcontrollers, FPGAs, or DSP processors.
This work proposes a method for linearizing the temperature characteristics
of NTC thermistors using digital correction based on polynomial functions. The
proposed approach can be implemented on low-power, cost-effective microcon-
trollers through sequential execution of programmed arithmetic operations.
The method was validated using a newly developed simulation model of a
temperature measurement device based on an NTC thermistor, created in the
MATLAB Simulink environment. This model includes a data processing algo-
rithm capable of generating all necessary polynomial functions, incorporating cor-
rection coefficients obtained through calibration.
Polynomial-based method for linearizing the temperature response of NTC thermistors
Системні дослідження та інформаційні технології, 2026, № 1 73
Simulation results demonstrated that the highest measurement accuracy over
a wide temperature range can be achieved using a polynomial correction algo-
rithm that divides the full measurement range into multiple subranges. Increasing
the number of subranges enables the use of lower-order polynomial functions
within each subrange, which in turn enhances overall measurement accuracy.
Specifically, when the full temperature range from 233 K to 383 K was divided
into six subranges, third-order polynomial functions were used for data pro-
cessing. Under these conditions, the measurement error did not exceed 0.1 K
(RMS).
The research identified several factors that influence measurement accuracy:
deviations of the actual thermistor resistance from its nominal value due to the
temperature dependence of the B-parameter; instability of the Wheatstone bridge
reference voltage refU ; deviation of the amplifier gain from its ideal value ΔG;
and errors introduced by self-heating of the thermistor. To achieve a measurement
error within 0.1 K (RMS), the accuracy of the B-parameter, refU , and G must
be within 0.5%, and the thermistor current thI must be limited to no more than
100 µA. At this current level, self-heating of the thermistor does not exceed 0.1 K
and therefore does not significantly affect measurement results.
To further enhance the effectiveness of temperature characteristic lineariza-
tion of NTC thermistors based on the proposed method, future research will focus
on improving the mathematical model. This will support the design and analysis
of various temperature measurement circuits aimed at achieving the desired
measurement accuracy using optimally selected sensors and components of the
measurement channel.
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Received 11.06.2025
INFORMATION ON THE ARTICLE
Sergey N. Matvienko, ORCID: 0000-0002-7547-4601, National Technical University of
Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Ukraine, e-mail: s.matvienko@kpi.ua
Grygoriy S. Tymchyk, ORCID: 0000-0003-1079-998X, National Technical University
of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Ukraine, e-mail: deanpb@kpi.ua
Polynomial-based method for linearizing the temperature response of NTC thermistors
Системні дослідження та інформаційні технології, 2026, № 1 75
СПОСІБ ЛІНЕАРИЗАЦІЇ ТЕМПЕРАТУРНИХ ХАРАКТЕРИСТИК NTC-
ТЕРМІСТОРІВ ЗА ДОПОМОГОЮ ПОЛІНОМІНАЛЬНОГО МЕТОДУ /
С.М. Матвієнко, Г.С. Тимчик
Анотація. Мета роботи – розроблення схемотехнічного способу лінеаризації
температурних характеристик термісторів із використанням поліномінального
цифрового методу температурних характеристик NTC-термісторів та характе-
ристик розробленого вимірювального каналу за допомогою розробленої оригі-
нальної моделі в MATLAB Simulink. Запропоновано модель пристрою для ви-
мірювання температури із сенсором на базі термістора. Виконано
моделювання поліномінального цифрового методу лінеаризації температурних
характеристик NTC-термісторів розробленого пристрою. Для підвищення точ-
ності вимірювань вибрано елементи вимірювального каналу з параметрами,
що дають змогу здійснювати моделювання вимірювань із мінімальною похиб-
кою. Визначено методи внесення значення корекційних коефіцієнтів у модель
для компенсації факторів впливу на похибку включно з ефектом саморозігріву
термістора. Запропонований алгоритм оброблення даних має переваги за реа-
лізації в мікроконтролері невисокої вартості та низької потужності.
Ключові слова: вимірювання температури, NTC-термістор, MATLAB
Simulink, лінеаризація.
|
| id | journaliasakpiua-article-358065 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-04-20T01:00:21Z |
| publishDate | 2026 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/e8/4d6d096b111fdc32edf47779b80440e8.pdf |
| spelling | journaliasakpiua-article-3580652026-04-19T21:53:19Z Polynomial-based method for linearizing the temperature response of NTC thermistors Спосіб лінеаризації температурних характеристик NTC-термісторів за допомогою поліномінального методу Matvienko, Sergey Tymchyk, Grygoriy вимірювання температури NTC-термістор MATLAB Simulink лінеаризація temperature measurement NTC thermistor MATLAB Simulink linearization The objective of this research is to develop a circuit-based method for linearizing the temperature characteristics of thermistors using a polynomial digital technique for NTC-thermistor temperature characteristics, together with the characteristics of the designed measurement channel, by means of an original MATLAB Simulink model. A model of a temperature-measurement device employing a thermistor-based sensor is proposed. The polynomial digital method for linearizing the temperature characteristics of NTC thermistors in the developed device has been simulated. To improve measurement accuracy, the components of the measurement channel were selected such that they enable simulation with minimal error. Methods for introducing correction-coefficient values into the model have been determined to compensate for error-inducing factors, including the thermistor’s self-heating effect. The proposed data-processing algorithm offers advantages for implementation in a low-cost, low-power microcontroller. Мета роботи – розроблення схемотехнічного способу лінеаризації температурних характеристик термісторів із використанням поліномінального цифрового методу температурних характеристик NTC-термісторів та характеристик розробленого вимірювального каналу за допомогою розробленої оригінальної моделі в MATLAB Simulink. Запропоновано модель пристрою для вимірювання температури із сенсором на базі термістора. Виконано моделювання поліномінального цифрового методу лінеаризації температурних характеристик NTC-термісторів розробленого пристрою. Для підвищення точності вимірювань вибрано елементи вимірювального каналу з параметрами, що дають змогу здійснювати моделювання вимірювань із мінімальною похибкою. Визначено методи внесення значення корекційних коефіцієнтів у модель для компенсації факторів впливу на похибку включно з ефектом саморозігріву термістора. Запропонований алгоритм оброблення даних має переваги за реалізації в мікроконтролері невисокої вартості та низької потужності. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2026-03-31 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/358065 10.20535/SRIT.2308-8893.2026.1.04 System research and information technologies; No. 1 (2026); 58-75 Системные исследования и информационные технологии; № 1 (2026); 58-75 Системні дослідження та інформаційні технології; № 1 (2026); 58-75 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/358065/343995 |
| spellingShingle | вимірювання температури NTC-термістор MATLAB Simulink лінеаризація Matvienko, Sergey Tymchyk, Grygoriy Спосіб лінеаризації температурних характеристик NTC-термісторів за допомогою поліномінального методу |
| title | Спосіб лінеаризації температурних характеристик NTC-термісторів за допомогою поліномінального методу |
| title_alt | Polynomial-based method for linearizing the temperature response of NTC thermistors |
| title_full | Спосіб лінеаризації температурних характеристик NTC-термісторів за допомогою поліномінального методу |
| title_fullStr | Спосіб лінеаризації температурних характеристик NTC-термісторів за допомогою поліномінального методу |
| title_full_unstemmed | Спосіб лінеаризації температурних характеристик NTC-термісторів за допомогою поліномінального методу |
| title_short | Спосіб лінеаризації температурних характеристик NTC-термісторів за допомогою поліномінального методу |
| title_sort | спосіб лінеаризації температурних характеристик ntc-термісторів за допомогою поліномінального методу |
| topic | вимірювання температури NTC-термістор MATLAB Simulink лінеаризація |
| topic_facet | вимірювання температури NTC-термістор MATLAB Simulink лінеаризація temperature measurement NTC thermistor MATLAB Simulink linearization |
| url | https://journal.iasa.kpi.ua/article/view/358065 |
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