Thermally Coupled NTC Chip Thermistors: Their Properties and Applications

Negative temperature coefficient (NTC) chip thermistors were thermally coupled to form a novel device (TCCT) aimed for application in microelectronics. It consists of two NTC chip thermistors Th1 and Th2, which are small in size (0603) and power (1/10 W). They are in thermal junction, but concurrently they are electrically isolated. The first thermistor Th1 generates heat as a self-heating component at a constant supply voltage U (input thermistor), while the second thermistor Th2 receives heat as a passive component (output thermistor). The temperature dependence R(T) of NTC chip thermistors was measured in the climatic test chamber, and the exponential factor B10/30 of thermistor resistance was determined. After that, a self–heating current I1 of the input thermistor was measured vs. supply voltage U and ambient temperature Ta as a parameter. Input resistance R1 was determined as a ratio of U and I1 while output thermistor resistance R2 was measured by a multimeter concurrently with the current I1. Temperatures T1 and T2 of both thermistors were determined using the Steinhart–Hart equation. Heat transfer, thermal response, stability, and inaccuracy were analyzed. The application of thermally coupled NTC chip thermistors is expected in microelectronics for the input to output electrical decoupling/thermal coupling of slow changeable signals.


Introduction
A thermistor is a type of resistor, the resistance of which can have a positive temperature coefficient (PTC) or negative temperature coefficient (NTC), i.e., it increases or decreases its resistance with the temperature increase.Typical thermistor geometries are the disk, chip, and thick film.The resistance values, temperature range, and temperature coefficient B depend on the semiconductor oxides used for their preparation [1,2].They are mass-produced by pressing thermistor powders in molds and after that, sintering compacts the thermistor at different temperatures in air [3,4].NTC thermistors are divided into two groups: low temperature application range thermistors (−50 • C to +250 • C) [5,6] and high temperature application range thermistors for +250 • C to +1000 • C [7,8].The semiconductive oxides used in the production of low temperature application range NTC thermistors are based on nickel manganese spinel and its modifications such as NiMn 2 O 4 , (NiMn) 3 O 4 , (NiMnCo) 3 O 4 , (NiMnFeCo) 3 O 4 , and (Fe,Ti) 2 O 3 .They are usually doped by a low percentage of oxides such as CuO, ZnO, CoO, LiO, RuO 2, ZrO 2 , Y 2 O 3 , and other oxides to form complex thermistor materials [9][10][11][12][13][14].The nickel manganese spinel is partly inverse as the tetrahedral ion Ni 2+ and octahedral ion Mn 3+ can exchange their sites in the lattice.Octahedral ions can change their valence such as 2Mn 3+ →(Mn 2+ , Mn 4+ ).In that way, the Mn 2+ ion fits the vacancy in the lattice where ion Ni 2+ is missing.The main carriers in NiMn 2 O 4 are small polarons (partly polarized) electrons and the energy gap has a relatively low value (around 0.5 eV) [15][16][17][18].The disk and chip NTC thermistors for the low temperature application range are pressed out of NiMn 2 O 4 powder in molds and sintered at 1100 • C to 1200 • C/2 h in air [19,20].The electrodes for disk and chip thermistors Sensors 2024, 24, 3547 2 of 13 are printed out of thick film conductive PdAg paste on both sides.After that, they are dried at 150 • C/10 min and sintered at 850 • C/10 min in air [21,22].The semiconductive oxides used in the production of NTC thermistors fora higher temperature application range are based on BaTiO 3 , Fe 2 TiO 5, Co 3 O 4 , CaTiO 3 , Mg(Al 1−x Cr x ) 2 O 4 , Bi 2 Zr 3 O 7 , ZrO 2 /CaO, and complex oxides such as MgAl 2 O 4 -LaCr 0.5 Mn 0.5 O 3 , Ni 1.0 Mn 2−x Zr x O 4 , Ce 1.2(1−x) Mn 1−x Si x , Sr 7 Mn 4 O 15 , etc. [23][24][25][26][27][28][29].
A chip thermistor is a leadless component designed for surface mounting technology (SMT).It is small in size and small in power and serves for the measurement of the temperature in the measuring point in air, liquids, the ground, and solid bodies.Their outlook and thermal coupling are given in Figure 1.
A chip thermistor is a leadless component designed for surface mounting technology (SMT).It is small in size and small in power and serves for the measurement of the temperature in the measuring point in air, liquids, the ground, and solid bodies.Their outlook and thermal coupling are given in Figure 1.A novel chip TCCT device is based on NTC chip thermistors.The thermistors were joined laterally using epoxy resin, i.e., they are thermally coupled through epoxy but they are electrically decoupled (galvanically insulated).The input thermistor Th1 serves as a self-heating thermistor powered by DC or AC voltage U, while the output thermistor Th2 acts as a heat receiver and changes its higher electrical resistance R2.The main property of the chip TCCT device is that the temperature of the output chip thermistor follows the temperature of the input self-heating chip thermistor.Moreover, the input and output thermistors are galvanically isolated and the electrical noise caused by switching power electronics is not transferred from the input to output thermistor.The comparatively low (electrical) coupling capacity between the two NTCs and (2) their thermal inertia allows (1) to reduce the disturbing influence of the voltage or current peaks on the temperature measurement and (2) to smoothen the ripple of the temperature signal.The output thermistor can be connected to an operational amplifier and used in different ways: feedback to input power, output temperature to input power convertor, electro-thermal potentiometers, etc.In comparison to the thick film (TF) TCT device with segmented thermistors and disk TCDT device which were realized recently in our previous work [30,31], the TCCT device has the following advantages: it occupies the smallest surface and has a simpler construction, lower cost, smaller power, and a high availability of chip thermistors with different nominal resistances, which is suitable for the realization of different resistance ratios in a thermal coupling device.A novel chip TCCT device is based on NTC chip thermistors.The thermistors were joined laterally using epoxy resin, i.e., they are thermally coupled through epoxy but they are electrically decoupled (galvanically insulated).The input thermistor Th 1 serves as a selfheating thermistor powered by DC or AC voltage U, while the output thermistor Th 2 acts as a heat receiver and changes its higher electrical resistance R 2 .The main property of the chip TCCT device is that the temperature of the output chip thermistor follows the temperature of the input self-heating chip thermistor.Moreover, the input and output thermistors are galvanically isolated and the electrical noise caused by switching power electronics is not transferred from the input to output thermistor.The comparatively low (electrical) coupling capacity between the two NTCs and (2) their thermal inertia allows (1) to reduce the disturbing influence of the voltage or current peaks on the temperature measurement and (2) to smoothen the ripple of the temperature signal.The output thermistor can be connected to an operational amplifier and used in different ways: feedback to input power, output temperature to input power convertor, electro-thermal potentiometers, etc.In comparison to the thick film (TF) TCT device with segmented thermistors and disk TCDT device which were realized recently in our previous work [30,31], the TCCT device has the following advantages: it occupies the smallest surface and has a simpler construction, lower cost, smaller power, and a high availability of chip thermistors with different nominal resistances, which is suitable for the realization of different resistance ratios in a thermal coupling device.

Main Properties of Chip Thermistors
The chip thermistors given in Figure 1a (EPCOS NTC 0603, TDK Europe, Munchen, Germany) have the following main properties: a dissipation power of 1/10 W, nomi-nal resistance of around 2600 Ω at a room temperature of 25 • C, and exponential factor B 25/75 ≈ 3900 K (given by the producer).For our experiments, nominal resistance was additionally measured at 20 • C and B 10/30 was determined in the range of 10 • C to 30 • C. Electrical resistance R was measured in the climatic test chamber with the temperature increase T as the temperature behavior R(T) of the NTC chip thermistor and given in Figure 2. It can be approximated bythe Steinhart-Hart equation in the first article (1) [32]: Sensors 2024, 24, x FOR PEER REVIEW 3 of 14

Main Properties of Chip Thermistors
The chip thermistors given in Figure 1a (EPCOS NTC 0603, TDK Europe, Munchen, Germany) have the following main properties: a dissipation power of 1/10 W, nominal resistance of around 2600 Ω at a room temperature of 25 °C, and exponential factor B25/75 ≈ 3900 K (given by the producer).For our experiments, nominal resistance was additionally measured at 20 °C and B10/30 was determined in the range of 10 °C to 30 °C.Electrical resistance R was measured in the climatic test chamber with the temperature increase T as the temperature behavior R(T) of the NTC chip thermistor and given in Figure 2. It can be approximated bythe Steinhart-Hart equation in the first article (1) [32]: Resistance R0 is the nominal value of the chip resistor at 20 °C, T0 is the room temperature at 20 °C (293,16 K), B is the exponential temperature coefficient of the chip thermistor, and T is the current temperature of the thermistor.The thermistor exponential factor B was obtained in the vicinity of room temperature using the values of chip resistance R10 and R30 measured at temperatures T10 = 10 °C and T30 = 30 °C, respectively.The thermistor exponential factor B is derived from Equation (1) and given in Equation (2): Using Equation (2) and the results in Figure 2, the chip thermistor coefficient was determined as B10/30 = 3908.3K. Nominal resistance at 20 °C was measured with the digital multimeter Fluke 179 as R = 2680 Ω.Also, using Equations ( 1) and ( 2), the unknown current temperature T of the thermistor is a function of current thermistor resistance R. In that way, the temperature T1 of the self-heating thermistor Th1 (input thermistor) is a function of resistance R1, coefficient B1, and can be calculated by Equation (3): Resistance R 0 is the nominal value of the chip resistor at 20 • C, T 0 is the room temperature at 20 • C (293,16 K), B is the exponential temperature coefficient of the chip thermistor, and T is the current temperature of the thermistor.The thermistor exponential factor B was obtained in the vicinity of room temperature using the values of chip resistance R 10 and R 30 measured at temperatures T 10 = 10 • C and T 30 = 30 • C, respectively.The thermistor exponential factor B is derived from Equation (1) and given in Equation (2): Using Equation ( 2) and the results in Figure 2, the chip thermistor coefficient was determined as B 10/30 = 3908.3K. Nominal resistance at 20 • C was measured with the digital multimeter Fluke 179 as R = 2680 Ω.Also, using Equations ( 1) and ( 2), the unknown current temperature T of the thermistor is a function of current thermistor resistance R. In that way, the temperature T 1 of the self-heating thermistor Th 1 (input thermistor) is a function of resistance R 1 , coefficient B 1 , and can be calculated by Equation (3): whereas resistance R 1 = U/I 1 is the resistance of the self-heating thermistor Th 1 (input resistance) and R 01 is the nominal value of thermistor Th 1 measured at 20 • C. Hence, thermistor Th 2 in the thermally coupled device has temperature T 2 calculated by thermistor resistance R 2 , coefficient B 2 , and is given in Equation (4): The electrical resistance R 2 of thermistor Th 2 is measured using a multimeter in the climatic test chamber, and R 02 is the thermistor nominal value measured at 20 • C. In our partial case, Th 1 and Th 2 were made out of the same material and B 1 = B 2 = B and R 01 ≈ R 02 .Before the measurement of thermistor resistance, the climatic test chamber was twice recalibrated by a PT-1000 platinum thermometer to lower the inaccuracy of measuring temperature T to around ±0.020 • C.

Thermal Coupling of Chip Thermistors
At first, thin wire leads were soldered onto the chip thermistors and then the chip thermistors were insulated using epoxy resin.Finally, a pair of insulated chip thermistors were placed in lateral contact and joined together using a thin layer of epoxy between them to form the TCCT device as given in Figure 1b,c.The main properties of the TCCT device were measured in the climatic test chamber.The measuring setup is given in Figure 3; it consists of thermally coupled chip thermistors TCCT and three multimeters used for the measuring of the input supply voltage U, self-heating current I 1 of thermistor Th 1 , and output resistance R 2 of thermistor Th 2 .
whereas resistance R1 = U/I1 is the resistance of the self-heating thermistor Th1 (input resistance) and R01 is the nominal value of thermistor Th1 measured at 20 °C.Hence, thermistor Th2 in the thermally coupled device has temperature T2 calculated by thermistor resistance R2, coefficient B2, and is given in Equation ( 4): The electrical resistance R2 of thermistor Th2 is measured using a multimeter in the climatic test chamber, and R02 is the thermistor nominal value measured at 20 °C.In our partial case, Th1 and Th2 were made out of the same material and B1 = B2 = B and R01 ≈ R02.Before the measurement of thermistor resistance, the climatic test chamber was twice recalibrated by a PT-1000 platinum thermometer to lower the inaccuracy of measuring temperature T to around ±0.020 °C.

Thermal Coupling of Chip Thermistors
At first, thin wire leads were soldered onto the chip thermistors and then the chip thermistors were insulated using epoxy resin.Finally, a pair of insulated chip thermistors were placed in lateral contact and joined together using a thin layer of epoxy between them to form the TCCT device as given in Figure 1b,c The self-heating current I1 of thermistor Th1 in the TCCT device vs. supply voltage U was measured at different ambient temperatures Ta as a parameter in the range from 0 °C to 40 °C (in steps of 5 °C).The supply voltage U was increased in steps of 1 V each 20 s to observe the behavior of the self-heating process.The input power P1 = U•I1 of the self-heating thermistor Th1 was defined for different ambient temperatures Ta in the climatic test chamber, which was also changed in steps of 5 °C.The results of these measurements are given in Figure 4.The self-heating current I 1 of thermistor Th 1 in the TCCT device vs. supply voltage U was measured at different ambient temperatures T a as a parameter in the range from 0 • C to 40 • C (in steps of 5 • C).The supply voltage U was increased in steps of 1 V each 20 s to observe the behavior of the self-heating process.The input power P 1 = U•I 1 of the selfheating thermistor Th 1 was defined for different ambient temperatures T a in the climatic test chamber, which was also changed in steps of 5 • C. The results of these measurements are given in Figure 4.

Resistances and Temperatures in TCCT Device
The resistance R1 of the chip thermistor Th1 is defined from the self-heating current I1 and supply voltage U as R1 = U/I1, while the resistance R2 of the chip thermistor Th2 is measured by a multimeter at the same time as U and I1.The results of these measurements are given in Figure 5.

Resistances and Temperatures in TCCT Device
The resistance R 1 of the chip thermistor Th 1 is defined from the self-heating current I 1 and supply voltage U as R 1 = U/I 1 , while the resistance R 2 of the chip thermistor Th 2 is Sensors 2024, 24, 3547 5 of 13 measured by a multimeter at the same time as U and I 1 .The results of these measurements are given in Figure 5.

Figure 4.
The self-heating current I1 and self-heating power P1 of the chip thermistor Th1 in TCCT device vs. supply voltage U changed in steps of 1 V. Ta-ambient temperature as a parameter.

Resistances and Temperatures in TCCT Device
The resistance R1 of the chip thermistor Th1 is defined from the self-heating current I1 and supply voltage U as R1 = U/I1, while the resistance R2 of the chip thermistor Th2 is measured by a multimeter at the same time as U and I1.The results of these measurements are given in Figure 5.The temperatures T1 and T2 of the chip thermistors Th1 and Th2, respectively, are defined using Equations ( 3) and ( 4), and the values of the resistances R1 and R2 are given above in Figure 5.The ambient temperature Ta was used a parameter.The obtained results are given in Figure 6.The temperatures T 1 and T 2 of the chip thermistors Th 1 and Th 2 , respectively, are defined using Equations ( 3) and ( 4), and the values of the resistances R 1 and R 2 are given above in Figure 5.The ambient temperature T a was used a parameter.The obtained results are given in Figure 6.

Resistances and Temperatures in TCCT Device
The resistance R1 of the chip thermistor Th1 is defined from the self-heating current I1 and supply voltage U as R1 = U/I1, while the resistance R2 of the chip thermistor Th2 is measured by a multimeter at the same time as U and I1.The results of these measurements are given in Figure 5.The temperatures T1 and T2 of the chip thermistors Th1 and Th2, respectively, are defined using Equations ( 3) and ( 4), and the values of the resistances R1 and R2 are given above in Figure 5.The ambient temperature Ta was used a parameter.The obtained results are given in Figure 6.

Self-Heating Current Stability vs. Time of TCCT Device
Using the same measuring setup as given in Figure 3, the behavior of the self-heating current I 1 of the chip thermistor Th 1 vs. time t from the switch on moment (t = 0) of the fixed supply voltage U was measured.The supply voltage U was changed as a parameter in steps of 1 V in the climatic test chamber.The I 1 curves from the switch on to saturation level are given in Figure 7.The delay time from the switch on supply voltage U marked as t d ≈ 30 s was estimated from the behavior of the self-heating current I 1 (t) in Figure 7 in the region below the knee for a bundle of curves.The criterion of the stability of I 1 in the horizontal region was introduced as k = I(t = 40)/I(t = 80) > 0.97.Using this criterion of stability, the bar diagram in Figure 7 was formed for the values of the supply voltage U and self-heating current I 1 to follow the criterion k > 0.97 in the ambient temperature range from 0 • C to 40 • C.
Sensors 2024, 24, 3547 6 of 13 saturation level are given in Figure 7.The delay time from the switch on supply voltage U marked as td ≈ 30 s was estimated from the behavior of the self-heating current I1(t) in Figure 7 in the region below the knee for a bundle of curves.The criterion of the stability of I1 in the horizontal region was introduced as k = I(t = 40)/I(t = 80) > 0.97.Using this criterion of stability, the bar diagram in Figure 7 was formed for the values of the supply voltage U and self-heating current I1 to follow the criterion k > 0.97 in the ambient temperature range from 0 °C to 40 °C.The self-heating cycle from t = 0 s to t = 180 s and the cooling cycle from t = 180 s to t = 360 s for the TCCT device was measured using the same measuring setup as given in Figure 3 and the fixed input supply voltage U = 8 V at ambient temperature Ta = 20 °C.The resistances R1 and R2 and temperatures T1 and T2 of the chip thermistors Th1 and Th2, respectively, are given in Figure 8.The self-heating cycle from t = 0 s to t = 180 s and the cooling cycle from t = 180 s to t = 360 s for the TCCT device was measured using the same measuring setup as given in Figure 3 and the fixed input supply voltage U = 8 V at ambient temperature T a = 20 • C. The resistances R 1 and R 2 and temperatures T 1 and T 2 of the chip thermistors Th 1 and Th 2 , respectively, are given in Figure 8.
saturation level are given in Figure 7.The delay time from the switch on supply voltage U marked as td ≈ 30 s was estimated from the behavior of the self-heating current I1(t) in Figure 7 in the region below the knee for a bundle of curves.The criterion of the stability of I1 in the horizontal region was introduced as k = I(t = 40)/I(t = 80) > 0.97.Using this criterion of stability, the bar diagram in Figure 7 was formed for the values of the supply voltage U and self-heating current I1 to follow the criterion k > 0.97 in the ambient temperature range from 0 °C to 40 °C.The self-heating cycle from t = 0 s to t = 180 s and the cooling cycle from t = 180 s to t = 360 s for the TCCT device was measured using the same measuring setup as given in Figure 3 and the fixed input supply voltage U = 8 V at ambient temperature Ta = 20 °C.The resistances R1 and R2 and temperatures T1 and T2 of the chip thermistors Th1 and Th2, respectively, are given in Figure 8.

Application of TCCT Device
The applications of the TCCT device with thermally coupled/electrically insulated chip thermistors are related to microelectronics.For example, two applications with TCCT are proposed and given in Figure 9. Block schemes A and B can be classified as voltage dividers/slider-less potentiometers or an electro-thermal potentiometer limited with a factor of stability of k > 0.97 (without a moving part known as the slider) based on chip thermistors Th 1 and Th 2 , and the input voltage U on thermistor Th 1 governs with electrical resistance R 2 in thermistor Th 2 via generated and transferred heat (thermal route).In the other block scheme in Figure 9, the resistance R 2 of thermistor Th 2 of the TCCT device is placed in a modified bridge with an operation amplifierIC 1 (TLV 9002) for the linearization of the output V out .The fixed resistor in the bridge, such as the 1.5 kΩ resistor placed in parallel with a 1 kΩ resistor +5 kΩ potentiometer, is chosen based on the nominal value of the NTC chip thermistor Th 2 and the temperature range.A parallel connected capacitor such as the 100 nF with a fixed resistor of 1.5 kΩ (input to output feedback) limits bandwidth, improves stability, and helps to reduce noise.The range of temperature T 2 linearization is room temperature 20 • C ± 25 • C. The bridge enables a temperature to voltage conversion V out = F(T 2 ) and power to voltage conversion V out = F(P 1 ).Other applications are related to the control of small power AC/DC converters, thermostat function in air conditioning equipment, level detection, LED switching, etc. electrical resistance R2 in thermistor Th2 via generated and transferred heat (thermal route).In the other block scheme in Figure 9, the resistance R2 of thermistor Th2 of the TCCT device is placed in a modified bridge with an operation amplifierIC1 (TLV 9002) for the linearization of the output Vout.The fixed resistor in the bridge, such as the 1.5 kΩ resistor placed in parallel with a 1 kΩ resistor +5 kΩ potentiometer,ischosen based on the nominal value of the NTC chip thermistor Th2 and the temperature range.A parallel connected capacitor such as the 100 nF with a fixed resistor of 1.5 kΩ (input to output feedback) limits bandwidth, improves stability, and helps to reduce noise.The range of temperature T2 linearization is room temperature 20 °C ± 25 °C.The bridge enables a temperature to voltage conversion Vout = F(T2) and power to voltage conversion Vout = F(P1).Other applications are related to the control of small power AC/DC converters, thermostat function in air conditioning equipment, level detection, LED switching, etc.The electro-thermal potentiometer functions A and B are given in Equations ( 5) and ( 6), respectively: Output resistance R2 = f(U), R0-chosen fixed resistor, R0 = R2 only at room temperature.
The input supply voltage U governs with the self-heating current I1 in the thermistor Th1 and generates heat P1 transferred to the thermistor Th2, which governs with electrical resistance R2, and output temperature T2.Consequently, T2 is a function of P1 and P1(T2) given in Figure 10, and is used to measure input power as a function of output temperature.Hence, the TCCT in the bridge configuration enables a temperature to voltage conversion Vout = F(T2) and power to voltage conversion Vout = F(P1).
The function P1 = F(T2) given in Figure 10 consists of measured curves (solid curves), for which each step of ambient temperature Ta was around 5 °C, and interpolated curves, The electro-thermal potentiometer functions A and B are given in Equations ( 5) and ( 6), respectively: B : Output resistance R 2 = f(U), R 0 -chosen fixed resistor, R 0 = R 2 only at room temperature.The input supply voltage U governs with the self-heating current I 1 in the thermistor Th 1 and generates heat P 1 transferred to the thermistor Th 2 , which governs with electrical resistance R 2 , and output temperature T 2 .Consequently, T 2 is a function of P 1 and P 1 (T 2 ) given in Figure 10, and is used to measure input power as a function of output temperature.Hence, the TCCT in the bridge configuration enables a temperature to voltage conversion V out = F(T 2 ) and power to voltage conversion V out = F(P 1 ).
The function P 1 = F(T 2 ) given in Figure 10 consists of measured curves (solid curves), for which each step of ambient temperature Ta was around 5 • C, and interpolated curves, each 1 • C of T 2 as auxiliary curves (dashed curves).They were interpolated between two measured (solid) curves by using a polynomial of the third order as given in Equations ( 7) and (8).
Sensors 2024, 24, 3547 8 of 13 Figure 10.Input power P1 (DC regime) of self-heating thermistor Th1 as a function of output te perature T2 of the thermistor Th2.Ta-ambient temperature as a parameter (solid curves).dashed curves are interpolated using a polynomial of the third order.The limitations of self-heat current I1 are based on the criterion of stability k > 0.97 in saturation regime.
The function P1 = F(T2) given in Figure 10 consists of measured curves (solid curv each around 5 °C of ambient temperature Ta, and interpolated curves, each 1 °C of T auxiliary curves (dashed curves).They were interpolated between two measured (so curves by using a polynomial of the third order as given in Equations ( 7) and (8).
The described application is related to the control of small power AC/DC convert using the function P1 = F(T2) given in Figure 10.Other applications have to be develop using the applications' electro-thermal potentiometers and bridges for the linearization the output temperature T2 in Figure 9. T a -ambient temperature as a parameter (solid curves).The dashed curves are interpolated using a polynomial of the third order.The limitations of self-heating current I 1 are based on the criterion of stability k > 0.97 in saturation regime.
The function P 1 = F(T 2 ) given in Figure 10 consists of measured curves (solid curves), each around 5 • C of ambient temperature Ta, and interpolated curves, each 1 • C of T 2 as auxiliary curves (dashed curves).They were interpolated between two measured (solid) curves by using a polynomial of the third order as given in Equations ( 7) and (8).
The described application is related to the control of small power AC/DC converters using the function P 1 = F(T 2 ) given in Figure 10.Other applications have to be developed using the applications' electro-thermal potentiometers and bridges for the linearization of the output temperature T 2 in Figure 9.

Operating Point of TCCT Device
Small NTC chip thermistors 0603 (EPCOS), as given in Figure 1, with a dissipation power of 1/10 W, nominal resistance ofaround 2680 Ω at 20 • C, and exponential factor B = 3908 K, were used in the forming of thermally coupled chip thermistors-the TCCT device-for the first time.The slope of the chip thermistor NTC curve is moderate (Figure 2).The input supply voltage U of the self-heating thermistor Th 1 (Figure 3) enables the selfheating current I 1 and generates power P 1 , which are of an exponential/logarithmic type, as given in Figure 4.The chip thermistor Th 2 is only a heat receiver (passive thermistor).The resistances R 1 and R 2 of the chip thermistors Th 1 and Th 2 are also of an exponential/logarithmic type dependent on the supply voltage U and ambient temperature Ta as a parameter, as given in Figure 5.The temperatures T 1 and T 2 of the chip thermistors Th 1 and Th 2 , respectively, are also dependent on the supply voltage U and ambient temperature T a (as presented in Figure 6).The curves of T 1 are more exponential compared to the curves of T 2 as the heat generated in Th 1 or power P 1 with input temperature T 1 is only partly transferred to the thermistor Th 2 through the epoxy resinto cause an appearance of output temperature T 2 on the thermistor Th 2 .As a consequence, the chip thermistor Th 2 (heat receiver) has a much lower temperature T 2 compared to T 1 .The insulation thickness of around 1 mm consists of two epoxy layers which enable the transfer of a part of the heat from the self-heating to the heat receiving thermistor.The temperature difference ∆T = T 1 − T 2 changes from around 2 • C to around 16 • C in the range of the ambient temperature Ta from 0 • C to 40 • C. The temperature difference ∆T is a function of the input supply voltage U as ∆T(U) = T 1 (U) − T 2 (U), and using Equations ( 3) and ( 4), it can be defined as in Equation (10): )) (10) where the input resistance R 1 = U/I 1 and output resistance R 2 are measured by a multimeter.

Heat Transfer in TCCT Device
The self-heating current I 1 (t) in Figure 7 is a current in a forced regime depending on the fixed supply voltage U.The heat generation and heat dissipation are in balance on the horizontal part of the bundle of curves I 1 (t).The delay time t d ≈ 30 s from the switch on of the supply voltage U was estimated from the behavior I 1 (t) in the region below the knee of curves.As the horizontal curves I 1 (t) at T a = 20 • C are not ideally horizontal, especially for higher values of U, the criterion of the stability of I 1 (t) in the horizontal region was introduced as k = I 1 (t = 40)/I 1 (t = 80) > 0.97 to limit the voltage U and heat generation.Using the same criterion k > 0.97 of stability for other ambient temperatures T a in the ambient temperature range from 0 • C to 40 • C, the bar diagram in Figure 7 was formed with limited values of the supply voltage U and self-heating current I 1 .The aim was to limit the input power P 1 in accordance with the criterion of k > 0.97.The behavior of the resistances R 1 and R 2 and temperatures T 1 and T 2 in the TCCT device vs. time given in Figure 8 show that the self-heating cycle (U = 8 V) and cooling cycle (U = 0 V) differ in shape; the self-heating cycle is a forced regime caused by the supply voltage U, and the cooling cycle is a natural process of heat dissipation.The initial heating and cooling time to the heating/cooling balance is a delay time t d ≈ 30 s from the switch on of the supply voltage U.
The heat balance equation in the local equilibrium is given by ( 11): The heat generation in the self-heating thermistor Th 1 is a product of electrical power U•I 1 and time ∆t, as given by Equation ( 12): The heat transfer from the Th 1 to Th 2 chip thermistor through a thin epoxy layer interface d i ≈ 1 mm, surface value A, and thermal conductivity K e is given by Equation ( 13): Hence, the sum of the heat loss q 3 on the boundary of the thermistor Th 1 /epoxy/air corresponding to T 1 − T a and q 4 on the boundary of the thermistor Th 2 /epoxy/air corresponding to T 2 − T a is given by Equation (14).
The dissipation surface for q 3 and q 4 is A 3 = A 4 = A 0 − A, where A 0 is the total outlook surface of the chip thermistor, T 1 and T 2 are the thermistor temperatures, respectively, the epoxy coating thickness of the chip is d 3 = d 4 = 0.5 mm, and the heat dissipation length d a in the surrounding air is unknown.Further, using an approximation of the temperatures on the epoxy coatings T e as T e1 ≈ T 1 and Te 2 ≈ T 2 for the coated chip thermistors Th 1 and Th 2 , respectively, q 3 and q 4 are given as ( 15) and ( 16): Sensors 2024, 24, 3547 10 of 13 where K a is the thermal conductivity of the air.Now, by replacing q 3 and q 4 in Equation ( 14), the heat balance equation is formed at room temperature T a = 20 • C (17): The unknown heat dissipation length d a in the surrounding air is given by (18).
Finally, adding the values for U, I 1 , K a , K e , T 1 , T 2 , T a , d i , A, and A 3 , the heat dissipation length d a is derived as d a ≈ 38.5 mm in the air at room temperature.The purpose could be that any other heat source within the vicinity of the TCCT (the "size" of this vicinity is defined by d a ) could affect the characteristic behavior of the TCCT and, therefore, this d a value must at least be roughly estimated.The TCCT device has negligible thermal contact with the board; practically chip thermistors hang on the wires in the air.
The relative thermal coupling k 1,2 between chip thermistors Th 1 and Th 2 can be defined using ( 19): The thermal coupling k 1,2 between the chip thermistors depends on the U, I 1 , and Ta; for example, at a room temperature of 20 • C, U = 8 V, I 1 = 5.7 mA, its value is k 1,2 ≈ 0.25.The obtained k 1,2 value is smaller compared to the thermal coupling in thick film TF TCT and disk TCDT devices due to the fact that chip thermistors have a five times smaller nominal dissipation power.

Comparison of TCCT with TF TCT and TCDT Devices
The properties of the TCCT device were compared in Table 1 with the thick film (TF) TCT device and disk TCDT properties which were published recently in [30,31].The TCCT is a cubic device and has around 30 times lower volume compared to the TCDT device and around 100 times compared to the thick film TF TCT.The power of TCCT device is 5 times lower and relative thermal coupling k 1,2 is 1.5 to 2 times lower compared to TF TCT and TCDT devices.This result is a consequence of the heat transfer from the self-heating to passive chip thermistor through the small interface surface and the device body.The surface on the board of the TCCT is around 30 to 100 times smaller and the heat transfer path is only 1 mm.This result of k 1,2 is a consequence of high alumina thermal conductivity, which is much higher than the thermal conductivity of the epoxy insulation layer between the Th 1 and Th 2 thermistors.The TCCT device sensitivity can be defined as ∆ 1 = ∆T 2 /∆U (from Figure 6) using limitations for U (in Figure 7) and ∆ 2 = ∆P 1 /∆T 2 (from Figure 10).In brief, the reason for this is the output to input feedback and practical use of the TCCT for the control of the input power P 1 by the output temperature T 2 .The comparison of the TF TCT, disk TCT, and TCCT devices' sensitivity for the ambient temperature T a in the range of around 1 • C to 40 • C is given in Table 2.The inaccuracy ∆B of the exponential factor B according to Equation (2) given above in Section 2.1 is dependent on the inaccuracies in the measuring of resistances R and temperature T in the climatic test chamber.The inaccuracy ∆B is now determined by Equation (20): The inaccuracy ∆B measured in the climatic test chamber does not exceed 0.3%.The temperature inaccuracy ∆ T according to Equation (3) depends on resistance R and ∆B from the previous equation.It is defined as in (4): The temperature inaccuracy ∆ T does not exceed 0.5%.The inaccuracy in the measurement of the self-heating electrical current I 1 is dependent on the measuring of the input voltage U, ambient temperature T a , and time t as I 1 (U, T a , t).The self-heating current has an inaccuracy of ∆ sc given by the next Equation (22): It does not exceed 0.3%.The inaccuracy of the measuring system ∆ MS (only digital instruments in Figure 2) depends on the measuring of U, I, R, and the power source Ups, as given in (23): ∆ MS = ∆U/U + ∆I/I + ∆R/R + ∆U/U PS (23) The inaccuracy of the full measuring system ∆ MS does not exceed 0.4%.After a year of testing, there was no change in the thermistor nominal resistances R 01 and R 02 and the exponential factor B.

Conclusions
In this work, the electrical and thermal properties of the chip thermistor TCCT device (thermally coupled chip thermistors) were measured and analyzed and after that, compared with the previous thick film (TF) TCT device and disk TCDT device (the main properties are given in Tables 1 and 2).The main advantages are summarized and grouped as follows.

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The main properties: The chip TCCT is a resistive device based on the thermal junction of two small NTC chip thermistors: a chip size of 0603 and dissipation power of 100 mW, nominal resistance R 0 ≈ 2600 Ω, and exponential factor B = 3908 K.The input voltage U was limited using a practical criterion k > 0.97 (Section 2.2, bar diagrams in Figure 7) to keep the stability of the self-heating current according to the ambient temperature T a in the range from around 1 • C to 40 • C. Practically, the self-heating current is limited to a maximum of 10 mA and power to a maximum of 25 mW.The various chip thermistors' nominal resistances can be chosen to form the TCCT with a resistances' ratio R input /R output from 1:1 to 1:5 or more.The input supply voltage can be DC, AC, or slow changeable impulses, and the output chip thermistor has a thermal response with a thermally coupling factor k 1,2 ≈ 0.25.The insulation resistance of the epoxy between the chip thermistors was >10 9 Ω, and the delay time of the chip thermistor response from the switch on of the supply voltage U to a stable self-heating current I 1 was less than 30 s.

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The main advantages: The chip TCCT is a small cubic type of device made out of modified nickel manganese chip thermistors (NTC).The thermal coupling (heat transfer) from the self-heating thermistor to heat receiving thermistor is weaker (k 1,2 ≈ 0.25) compared to the TF TCT and disk TCDT devices (realized recently in our previous works), but the volume and occupied area on a PCB are much smaller.The production of TCCT is cheaper.The TCCT device operates like an electro-thermal potentiometer (slider-less).

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The core conclusions are the following: The chip TCCT device has a small volume and simple construction, is cheap to use, and enables different applications, such as the measurement of the input power through the output temperature.It can be coupled with operational amplifiers for output linearization and used in microelectronics and power microelectronics as feedback in small power AC/DC convertors, car automation electronics, home appliances, thermo switches, thermostats, air conditioning, etc.

Figure 2 .
Figure 2. The resistance R of NTC chip thermistor (0603) vs. temperature T measured in the climatic test chamber: Rm-measured values, R*-fitted curve (solid line).

Figure 2 .
Figure 2. The resistance R of NTC chip thermistor (0603) vs. temperature T measured in the climatic test chamber: R m -measured values, R*-fitted curve (solid line).

Sensors 2024 , 14 Figure 4 .
Figure 4.The self-heating current I1 and self-heating power P1 of the chip thermistor Th1 in TCCT device vs. supply voltage U changed in steps of 1 V. Ta-ambient temperature as a parameter.

Figure 4 .
Figure 4.The self-heating current I 1 and self-heating power P 1 of the chip thermistor Th 1 in TCCT device vs. supply voltage U changed in steps of 1 V. T a -ambient temperature as a parameter.

Figure 5 .
Figure 5. Resistance R1 of the self-heating chip thermistor Th1 and resistance R2 of the chip thermistor Th2 in TCCT device vs. supply voltage U changed in steps of 1 V. Ta-ambient temperature as a parameter.

Figure 6 .Figure 5 .
Figure 6.The temperatures T1 and T2 of the chip thermistors Th1 and Th2 in TCCT device, respectively, vs. supply voltage U changed in steps of 1 V. Ta-ambient temperature as a parameter.

Figure 4 .
Figure 4.The self-heating current I1 and self-heating power P1 of the chip thermistor Th1 in TCCT device vs. supply voltage U changed in steps of 1 V. Ta-ambient temperature as a parameter.

Figure 5 .
Figure 5. Resistance R1 of the self-heating chip thermistor Th1 and resistance R2 of the chip thermistor Th2 in TCCT device vs. supply voltage U changed in steps of 1 V. Ta-ambient temperature as a parameter.

Figure 6 .
Figure 6.The temperatures T1 and T2 of the chip thermistors Th1 and Th2 in TCCT device, respectively, vs. supply voltage U changed in steps of 1 V. Ta-ambient temperature as a parameter.

Figure 6 .
Figure6.The temperatures T 1 and T 2 of the chip thermistors Th 1 and Th 2 in TCCT device, respectively, vs. supply voltage U changed in steps of 1 V. T a -ambient temperature as a parameter.

Figure 7 .
Figure 7. Self-heating current I1 of the chip thermistor Th1 in TCCT device vs. time t from the switch on of the input supply voltage U. Ta-fixed ambient temperature.Input voltage U changed in steps of 1 V as a parameter.The bar table for the limitations of U and I1 vs. Ta based on the criterion of self-heating current stability k > 0.97 in saturation regime.

Figure 8 .
Figure 8. Resistances R1 and R2 and temperatures T1 and T2 in the TCCT device for the self-heating cycle from t = 0 s to t = 180 s and cooling cycle (*) from t = 180 s to t = 360 s.Fixed input supply voltage U = 8 V, Ta-ambient temperature Ta = 20 °C.In the self-heating cycle, supply voltage was fixed to U = 8 V, while in the cooling cycle, the supply voltage U is switched off (U = 0).

Figure 7 .
Figure 7. Self-heating current I 1 of the chip thermistor Th 1 in TCCT device vs. time t from the switch on of the input supply voltage U. T a -fixed ambient temperature.Input voltage U changed in steps of 1 V as a parameter.The bar table for the limitations of U and I 1 vs. Ta based on the criterion of self-heating current stability k > 0.97 in saturation regime.

Figure 7 .
Figure 7. Self-heating current I1 of the chip thermistor Th1 in TCCT device vs. time t from the switch on of the input supply voltage U. Ta-fixed ambient temperature.Input voltage U changed in steps of 1 V as a parameter.The bar table for the limitations of U and I1 vs. Ta based on the criterion of self-heating current stability k > 0.97 in saturation regime.

Figure 8 .
Figure 8. Resistances R1 and R2 and temperatures T1 and T2 in the TCCT device for the self-heating cycle from t = 0 s to t = 180 s and cooling cycle (*) from t = 180 s to t = 360 s.Fixed input supply voltage U = 8 V, Ta-ambient temperature Ta = 20 °C.In the self-heating cycle, supply voltage was fixed to U = 8 V, while in the cooling cycle, the supply voltage U is switched off (U = 0).

Figure 8 .
Figure 8. Resistances R 1 and R 2 and temperatures T 1 and T 2 in the TCCT device for the self-heating cycle from t = 0 s to t = 180 s and cooling cycle (*) from t = 180 s to t = 360 s.Fixed input supply voltage U = 8 V, T a -ambient temperature T a = 20 • C. In the self-heating cycle, supply voltage was fixed to U = 8 V, while in the cooling cycle, the supply voltage U is switched off (U = 0).

Figure 9 .
Figure 9. Application of TCCT device: thermo-electrical potentiometers (A and B) with chip thermistors.Th 1 -self-heating thermistor, Th 2 -heat receiver (thermally coupled), U-input voltage, I 1 -input (self-heating) current, V-DC voltage, V out -output voltage, R 0 -fixed resistor.Thermally coupled thermistors Th 1 and Th 2 in the modified bridge with IC 1 OP amplifier for the linearization of output temperature T 2 (temperature to voltage convertor).

Figure 10 .
Figure10.Input power P 1 (DC regime) of self-heating thermistor Th 1 as a function of output temperature T 2 of the thermistor Th 2 .T a -ambient temperature as a parameter (solid curves).The dashed curves are interpolated using a polynomial of the third order.The limitations of self-heating current I 1 are based on the criterion of stability k > 0.97 in saturation regime.

Table 1 .
Comparison of TF TCT, DISK TCT, and TCCT device properties.
* measured at room temperature (20 • C), ** measured at 20 • C, U and I 1 given in bar diagram in Figure7.

Table 2 .
Comparison of TF TCT, DISK TCDT, and TCCT device sensitivity.The inaccuracy in measurements of chip TCCT device response such as output temperature T 2 depends on partial inaccuracies such as the following: ∆ T of temperature T, ∆ MS of supply voltage U, self-heating current I 1 , and resistance R 2 , ∆ B of exponential factor B and ∆ SH of using the Steinhart-Hart equation in the first approximation.