2.2. Mathematical Model of the Temperature Measurement Sensor
First, we developed a mathematical model of the temperature measurement sensor to find the analytical and graphical representation of the transformation function F(T).
Mathematical modeling of the measuring sensor of human body temperature was performed for the circuit based on the metal–pyroelectric–semiconductor structure (
Figure 2).
The circuit of the frequency-measuring converter consists of bipolar and field-effect transistors, which are powered by a constant voltage source
Up.s. (
Figure 2). In this BJT-MOS transistor structure, there is negative resistance on the collector-drain electrodes of transistors
VT1 and
VT2. Connecting an external inductance
L to the collector-drain electrodes allows one to create a generator of electrical oscillations, the generation frequency of which clearly depends on the temperature.
Since the SPT is a dynamic system, the analysis of the studied structure will be carried out within the duration of the active pulse of heat radiation [
12]. When the signal hits the sensitive element, the pyroelectric heats up, which results in the arising signal charging both the pyroelectric capacitor itself and the transistor input capacitance. The heat radiation power required to trigger the sensor is in the range of microwatts [
13]. Therefore, a bipolar transistor with a pyroelectric film has high sensitivity. A potential of a certain polarity, which is applied to the base of the bipolar transistor, changes the voltage of the transistor structure with a pyroelectric film:
where
U0 = 2.4 V;
θ is the correction factor;
Ube is the base-emitter voltage.
If the potential on one surface of the pyroelectric is constant and the potential on the other varies, then
where
p is the pyroelectric coefficient;
δ is the thickness of the pyroelectric; Δ
T is the temperature change;
ε is the dielectric constant of the pyroelectric material; and
ε0 is the dielectric constant. By substituting the expression for the temperature change in the pyroelectric in (2), we obtain
where
A is the area of the absorbing layer on the sensing element surface;
η is the sensor emission coefficient;
α is the coefficient characterizing the heat transfer by thermal conductivity and radiation;
τ is a constant that does not depend on temperature and time;
t is time; and
T is the temperature of the incident radiation.
Combining (1) and (3), we obtain the dependence of the base-emitter voltage of a bipolar transistor with a pyroelectric film on temperature
Assuming that the sensor is switched on at the moment of radiation appearance, (4) can be rewritten as
Changes in the human body temperature affect various parameters and characteristics of the transistor structure [
14], namely, the width of the band gap of the semiconductor, the contact potential differences in transistor junctions, and the capacitances of transistor junctions [
4]. This is reflected in the change in elements of the equivalent circuit, which is shown in
Figure 3.
The equivalent circuit in
Figure 3 uses the following notation:
U1 is the power supply;
R1 is the internal resistance of the power supply;
L is the passive inductance;
CPE is the nonlinear capacitance of the pyroelectric structure;
C0 is the capacitance of the pyroelectric structure;
RPE is the resistance of the pyroelectric structure;
E(
T) is the source of electric power of the pyroelectric structure; and
C1 and
C2 are circuit capacitances. Regarding elements of the equivalent circuit of the bipolar transistor,
RB,
RE, and
RC are the volumetric resistances of the base, emitter, and collector;
Cbe and
Cbc are capacitances of the emitter and collector;
If is the nonlinear source of the direct current of a bipolar transistor; and
Idf and
Idr are currents of the transistor’s internal transitions. Regarding elements of the equivalent circuit of a field-effect transistor,
RD,
RSD, and
RS are resistances of the drain, source-drain, and source;
CD,
CSD, and
CG are capacitances of the drain, source-drain, and gate; and
Ipt is the nonlinear current source of the field-effect transistor.
We then consider the temperature dependences of the elements of the equivalent circuit of the transistor structure with negative resistance, which are necessary to build a mathematical model of the measuring transducer.
The dependence of the band gap width of the semiconductor on temperature is described by the following expression [
4]:
where
T is the temperature of the human body, s is the area of the p-n junction,
EG0 is the width of the band gap at a normal temperature of 23 °C, and
a and
b are the temperature coefficients of the p-n junction. The contact potential difference of the emitter junction is determined by the following formula [
5]:
where
T0 is a normal temperature of 23 °C;
Vt =
kT/q is the junction temperature potential;
k is the Boltzmann constant;
q is the electron charge; and
Uje is the contact difference of emitter junction potentials.
The contact difference of the collector junction potentials is represented by
where
Ujc is the contact difference of the collector junction potentials.
Substituting Equation (6) into Equations (7) and (8), we obtain the equation of dependence of the junction voltage of the transistor structure of the measuring transducer on the human body temperature
The calculation of Equation (9) makes it possible to obtain a graphical representation of the dependence of the transistor structure of the sensor junction voltages on temperature changes.
As can be seen from the conversion scheme of
Figure 3, the change in the voltage of the transistor structure transitions causes a change in the capacitance of the corresponding transistor structure transitions of the measuring converter. The change in the transistor junction capacitance, in turn, causes changes in the barrier capacitance of the sensor and the base-collector junction capacitance. Next, we determine these dependencies in analytical form.
The dependence of the collector junction capacitance at zero offset on the human body temperature is described by the following expression [
6]:
where
Mjc is a coefficient that takes into account the smoothness of the collector junction and
Cjc is the collector junction capacitance at zero bias.
The dependence of the emitter junction capacitance on temperature at zero bias is described by the following expression [
4]:
where
Mje is a coefficient that takes into account the smoothness of the emitter junction and
Cje is the capacitance of the emitter junction at zero bias.
The barrier capacity is described by the following expression [
4]:
where
Ube is the voltage between the base and the emitter of the bipolar transistor and
Fc is the nonlinearity coefficient of the barrier capacitances of forward-biased junctions.
The capacitance of the base-collector junction depending on the temperature is as follows:
where
Ubc is the voltage between the base and the bipolar transistor collector.
Substituting Equations (10) and (11) into Equations (12) and (13), we obtain the dependence of the capacity of the transistor structure of the measuring transducer on the change in the contact potential difference of the transistor structure sensor
Changing the capacitance of the transistor structure causes a change in the output frequency of the converter according to the Thompson formula
where
Leqv is the equivalent inductance of the measuring transducer.
For the measuring transducer circuit shown in
Figure 3, expression (15) will have the following form:
Formula (16) is the conversion function of the lambda-type temperature transducer.
The analytical dependence of the output frequency on the temperature of the human body F = f(T) can be obtained by successive substitutions of dependencies (9) and (14) into equation (16).
The sensitivity of the measuring transducer for a mobile robotic system is determined from Equation (16) as the first derivative of the temperature transformation function
To model the output signal of the measuring transducer, we use the method of alternating states [
7].
The equivalent circuit of the frequency-measuring transducers is shown in
Figure 3. Next, we design a circuit where capacitances and inductances are replaced by ideal sources (
Figure 4).
The direction of the injected sources
Uck(
t),
Ucds(
t),
Ucp(
t), and
iL(
t) is consistent with the positive direction for the state variables. In this case, the directions of the emf
UC(
t) are opposite to the direction of the current
iC(
t), and the directions of the current sources
iL(
t) coincide with
uL(
t). Since in the transformed equivalent circuit (
Figure 4), there is no loop from voltage sources or cut from current sources, we proceed to the second stage.
Now, it is necessary to determine the currents through the input voltage sources (
ic(
t)) and voltages at the input current sources (
UL(
t)). To do this, we solve the system of Equation (18) for the circuit, which is formulated according to the laws of Ohm and Kirchhoff
After replacing the left side of the system of Equation (18) with the derivatives of the state variables, we obtain the system of states Equation (19) in an orderly form
The system of Equation (19) is nonlinear since it contains nonlinear dependent sources of currents
Ipt,
Ibit,
Idf, and
Idr and capacitances
Ce and
Cc as constituent elements. The calculation of nonlinear parameters was carried out using Equations (20)–(26).
where
Ir is the nonlinear reverse current source of the bipolar transistor;
IS is the saturation current;
Vt is the junction temperature potential;
NF and
NR are the imperfection factors in normal and inverted modes; and
BF and
BR are the maximum current transfer coefficients in the circuit with the
CE in normal and inverted modes.
VBE =
VB −
VE and
VBC =
VB −
VC—voltages are the internal points of the base emitter and base collector (for a p-n-p transistor, it is taken with the reverse sign as
VBE = −(
VB −
VE) and
VBC = −(
VB −
VC).
where
Ibit is the current of the combined source and
QB is the transistor transition imperfection coefficient.
where
Idf and
Idr are the currents of internal transitions of the base emitter and base collector.
where
Ipt is the piecewise nonlinear current source of the field-effect transistor;
β is the steepness of the transient response;
VTO is the threshold voltage;
KP is the specific steepness;
Wd is the channel width;
Ld is the channel length; and
VGS =
VG −
VS and
VDS =
VD −
VS are the voltages at the internal gate-drain and drain-source points (for a transistor with a
p-channel,
VGS = −(
VG −
VS),
VDS = −(
VD −
VS) is taken with the opposite sign).
The bipolar transistor has diffusion and barrier capacitances. For emitter junction
and for the collector
where
CJE and
CJC are the capacitances of the emitter and collector junctions at zero offset, respectively;
MJE and
MJC are the smoothness coefficients of the junctions; and
VJE and
VJC are the contact potential differences in the junctions.
To verify the developed model, we used a program to calculate the output frequency of generation from the temperature value, acting on the sensitive element in the language of the Maple 13 Release 13.02 software package.
Theoretical and experimental studies have shown that the conversion function of the measuring transducer for the non-contact measurement of the human body temperature is almost linear. The relative deviation between the theoretical and experimental characteristics does not exceed 0.15%. Thus, the developed mathematical model of the measuring transducer is adequate.
2.3. Assessment of Measurement Errors
Errors exist in any measurement. Based on practical needs, a decision is made regarding the level of accuracy required. From this, we can conclude that measurement is characterized not only by the result, which is the numerical value of the measured quantity, but also by the error that is obtained.
Next, we determined the total error in the process of determining the value of the human body temperature using a non-contact frequency-measuring transducer. The causes of error are numerous and have a diverse nature associated with the following factors: the use of a non-contact temperature measurement method, the influence of the electrical circuit of the device, the inertia of the measurement process, and errors in converting the measured frequency into a digital code.
Next, we analyzed the errors that arise when determining the phase transitions of substances from the use of temperature-measuring instruments based on their radiation. The expressions of the corresponding error component can be determined from the temperature measurement equations
by expanding them into a Taylor series in the vicinity of the measurement results of the arguments
Ii,
where
is the component of the methodological error caused by the discrepancy in the functional relationship between the quantities.
The instrumental component of the error depends on the quality of the measuring instrument itself and arises due to the imperfection of measuring instruments and the dependence of their properties on the influence of external conditions [
10]. When measuring temperature by radiation, the instrumental error is caused by the inaccuracy of measuring the radiation flux from the object under study due to the influence of the parameters of the optical system, the electrical circuit, features of the radiation receiver, and changes in their characteristics over time.
The expression of the components of the radiation temperature measurement error is defined as the sum of partial derivatives of the expression of the conditional temperature by radiation flux intensities [
8], which are perceived by a pyrometric transducer based on transistor structures with negative resistance. The general expression that describes the error component due to changes in the intensity of the radiation flux perceived by a non-contact frequency-measuring transducer is
where
is the transfer coefficient of the instrumental error component of temperature measurement, the expression and value of which depend on the corresponding pyrometry method;
depending on the method used; and
i is the number of working spectral channels.
For the developed frequency-measuring transducer, based on the reactive properties of transistor structures with negative resistance, the error value (28) is 0.25%.
The accuracy of temperature measurement using radiation is also significantly affected by the methodological error of measurement. According to the theory of errors [
3], methodological error is a component of measurement error caused by the inadequacy of the measurement object and its model used in the measurement. The main factors that cause methodological error in measuring temperature using radiation are the theoretical simplifications used (in particular, the use of the Wien formula and the failure to take into account the non-monochromaticity of spectral channels), the lack of reliable information about the radiation properties of the object under study, and neglecting the influence of background radiation and the intermediate medium.
The methodological error of frequency devices used to determine the phase transitions of substances is caused by the influence of the radiation properties of the object, the use of a limited spectral region when using frequency devices based on TSNR, and the influence of the intermediate medium through which radiation passes from the object to the measuring instrument.
The expression of the methodological error component of temperature measurement using radiation in determining the phase transformations of substances is defined as the sum of partial derivatives of the expression for determining the thermodynamic temperature using this method. The general expression of the methodological error component will be
where
is the transfer coefficient of the methodological error component of temperature measurement, the values of which depend on the pyrometry method, and
m and
p are the coefficients of influence of the error components.
The value of the methodological error of the developed frequency-measuring transducer according to Equation (29) will be 0.33%. The methodological errors include an error due to the introduction of an incorrect value of the emissivity of the object; an error arising from a change in the intensity of the radiation flux; an error due to the dependence of the measurement results on the distance to the measurement object; an error due to incomplete filling of the field of view of the sensitive element by the measurement object; and an error due to the temperature of the sensitive element housing. The total methodological error of temperature measurement will be 0.42%.
Errors arising from the influence of the electrical circuit of the frequency-measuring converter are of the following natures:
δ1—measurement error arising from the instability of the generator frequency.
δ2—measurement error resulting from changes in ambient temperature.
δ3—error due to instability of the power supply of the transistor structure.
δ4—error due to own noise and external interference to the input circuit of the electronic frequency meter.
The value estimate is determined on the basis of the equation
where
is the characteristic resistance of the circuit;
L is the external inductance;
C is the equivalent capacitance of the transistor structure; and
A0 is the relative value of the oscillation amplitude in the zero approximation
where
is the differential negative resistance;
,
is the resistance of the inductive element;
Q is the quality factor of the circuit, (Q = 150); and
,
,
,
, and
are the approximation coefficients determined from the system of equations
where
;
;
;
and
are the maximum and minimum current values on the descending section of the static frequency response of the frequency converter;
and
are the voltages corresponding to
and
;
is the offset voltage, which is counted from the origin in the figure of the device’s frequency response; and
is the cyclic frequency.
According to the calculations made by Equations (30)–(32), the measurement error is 0.3%.
The measurement error resulting from the changes in ambient temperature is determined by
where
is the frequency of generation when the ambient temperature changes by 1 °C and
is the carrier frequency of the PFM.
If thermal stabilization of the frequency converter circuit is used, one can achieve a change in ambient temperature within 0.1 °C. Then, the error due to frequency deviation is
To determine the error
associated with the change in the generation frequency due to fluctuations in the supply voltage, it is necessary to first find the change in the output frequency of the VFD due to a 1% change in the supply voltage. For this VFD scheme, the supply voltage is = 3 V, and then its change will be 0.03 V, which corresponds to a change in the output frequency of generation by 3000 Hz. The power supply instability error will be determined by
The LM7805 voltage regulator was used to stabilize the supply voltage. The linear stabilization of this chip is 5 mV, which corresponds to a change in the generation frequency of 300 Hz. Then, the error will be equal to
The error
arising from intrinsic noise and extraneous interference is estimated experimentally. We denote the performance of the original instrument due to noise and interference as
and the performance of the original instrument during the calibration process as
. Then, the distribution of a random variable
allows us to determine the mathematical expectation
and standard deviation
, hence
The total error of the frequency-measuring temperature transducer of non-crystalline semiconductors is defined as the result of the found errors
The total measurement error is equal to the sum of instrumental and methodological errors and is 0.7%.
Since the measured value is a function of time, due to the inertia of the measuring instruments and other reasons, a dynamic error occurs in the device for determining the phase transformations of materials, which is a component of the error [
5]. For the developed frequency-measuring transducers based on the reactive properties of semiconductor materials, the switch-on time is about 20 ns, so the dynamic error resulting from determining the phase transformation temperature is five orders of magnitude less than the static errors of the device itself. In this case, the dynamic errors do not significantly affect the resulting error of the device so they are not considered in this paper.
2.4. Mobile Robotic System with the Proposed Pyroelectric Temperature Measurement Sensors
The proposed high-precision temperature sensor was installed on a mobile portable robot for the non-contact measurement of the human body temperature (MPRMT). The approach to the use of temperature sensors, combining them with other robotic and medical structures, is not new, but in contrast, has become more common in recent years [
13,
14,
15]. The robot has a manipulator designed and printed on a 3D printer, as well as a Bluetooth module (this allows the control of the robot from a smartphone). The LCD display allows for the display of information about the current temperature and recommendations for further action. The block diagram of the mobile robotic system with a proposed pyroelectric temperature measurement sensor is shown in
Figure 5. It consists of a microcontroller board (MC), which controls the MPRMT movement with the help of the motor unit (MU), controls the manipulator (M) movement, processes data from the sensor, and displays the measured temperature with certain recommendations on the LCD display (D). The MC used the Arduino Uno board, which has also become a popular technical solution in recent years [
16,
17,
18,
19,
20]. The main reason for using the Arduino Uno board as the MC of the MPRMT is the advantages of the ATMEGA microcontroller, which are listed in [
20]. The motor control unit (MtCU) and the manipulator control unit (MpCU) serve as interfaces between the MC and the MD and between the MC and the M, respectively. The Sensor Unit (SU) contains sensors that allow the MPRMT to move correctly, avoiding obstacles.
Figure 6 shows the hardware implementation of the mobile robotic system with a proposed pyroelectric temperature measurement sensor. It consists of three main parts: the mobile platform, the temperature sensor proposed by the authors, and the control unit.
The mobile platform serves as the foundation of the robotic system, providing mobility and stability. It is typically an Automated Guided Vehicle (AGV) or a similar mobile unit equipped with wheels and motors for navigation. The platform is designed to move smoothly across different surfaces and avoid obstacles using integrated sensors.
The chassis is constructed from lightweight yet durable materials to ensure stability and ease of movement. Motors are used to drive the wheels, enabling forward and backward movements.
Ultrasonic sensors and LiDAR are employed for obstacle detection and navigation. These sensors help the robot to map its environment and plan its path effectively.
The core component for temperature measurement is the proposed non-contact pyroelectric measurement sensor. This sensor measures the temperature of objects by detecting heat radiation and operates over the I2C protocol, allowing easy integration with microcontrollers like the Arduino Uno.
The control unit, consisting of the Arduino Uno Rev3 microcontroller equipped with the ATmega328p (Arduino S.r.l., Santa Liberata, Italy), is responsible for processing sensor data, managing navigation, and controlling the overall operation of the robot. It handles communication between sensors and actuators, ensuring real-time response. Integrated with the ESP-01 Wi-Fi module, the system gains access to a Wi-Fi network, enabling remote control and data transmission. This module allows the robot to send temperature data to a central server or a mobile device for further analysis and monitoring.
The software architecture includes algorithms for processing raw data from the temperature sensor and other navigation sensors, involving noise filtering, environmental compensation, and the conversion of sensor readings into meaningful temperature values. The software calculates body temperature based on the heat radiation detected by the sensor, compensating for ambient temperature and other factors that might affect the readings.
Operators can monitor and control the robot remotely through a user interface, which provides real-time feedback on the robot’s status, temperature readings, and navigation path. Developed on the Python Kivy framework, the mobile application connected via Wi-Fi enables users to control the robot and view temperature data, offering an intuitive interface for easy operation and monitoring. The robot is also equipped with a display screen to show temperature readings and other relevant information, while audio signals provide feedback on the robot’s status and ensure optimal positioning for temperature measurement.
External system analysis involved gathering and sending information for further evaluation and integration.