Thermally Coupled NTC Chip Thermistors: Their Properties and Applications

Abstract

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 B_10/30 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 T_a 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.

1. 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 NiMn₂O₄, (NiMn)₃O₄, (NiMnCo)₃O₄, (NiMnFeCo)₃O₄, and (Fe,Ti)₂O₃. They are usually doped by a low percentage of oxides such as CuO, ZnO, CoO, LiO, RuO_2, ZrO₂, Y₂O₃, 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 Ni²⁺ and octahedral ion Mn³⁺ can exchange their sites in the lattice. Octahedral ions can change their valence such as 2Mn³⁺→(Mn²⁺, Mn⁴⁺). In that way, the Mn²⁺ ion fits the vacancy in the lattice where ion Ni²⁺ is missing. The main carriers in NiMn₂O₄ 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 NiMn₂O₄ powder in molds and sintered at 1100 °C to 1200 °C/2 h in air [19,20]. The electrodes for disk and chip thermistors 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 BaTiO₃, Fe₂TiO_5, Co₃O₄, CaTiO₃, Mg(Al_1−xCr_x)₂O₄, Bi₂Zr₃O₇, ZrO₂/CaO, and complex oxides such as MgAl₂O₄–LaCr_0.5Mn_0.5O₃, Ni_1.0Mn_2−xZr_xO₄, Ce_1.2(1−x)Mn_1−xSi_x, Sr₇Mn₄O₁₅, 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 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.

2. Experiment and Results

2.1. 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 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]:

$$R(T)=R_0\exp \left[-B·(\frac{1}{T_0}-\frac{1}{T})\right]$$

(1)

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):

$$B=\left[\frac{T_{10}·T_{30}}{T_{30}-T_{10}}\right]·\ln \left[\frac{R_{10}}{R_{30}}\right]$$

(2)

Using Equation (2) and the results in Figure 2, the chip thermistor coefficient was determined as B_10/30 = 3908.3 K. 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):

$$T_1=\left[\left(B_1·T_0/(B_1+T_0\ln (\frac{R_1}{R_{01}}))\right)\right]$$

(3)

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):

$$T_2=\left[\left(B_2·T_0/(B_2+T_0\ln (\frac{R_2}{R_{02}}))\right)\right]$$

(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.

2.2. 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 I₁ of thermistor Th₁, and output resistance R₂ of thermistor Th₂.

The self-heating current I₁ of thermistor Th₁ 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 P₁ = U∙I₁ of the self-heating thermistor Th₁ 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.

2.3. 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 T_a was used a parameter. The obtained results are given in Figure 6.

2.4. 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 I₁ of the chip thermistor Th₁ 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 I₁ 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 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 T_a = 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.

2.5. 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 Th₁ and Th₂, and the input voltage U on thermistor Th₁ governs with 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 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 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 V_out = F(T₂) and power to voltage conversion V_out = 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:

$$\mathrm{A}:V_{out}=\frac{R_0}{R_0+R_2}·V$$

(5)

$$\mathrm{B}:V_{out}=\frac{R_2}{R_0+R_2}·V$$

(6)

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 V_out = F(T₂) and power to voltage conversion V_out = 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, each 1 °C of T₂ 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).

$$P_1=a_0+a_1·T_2+a_2·T_2^2; {\mathrm{T}}_{\mathrm{a}}=25 °\mathrm{C}$$

(7)

$$P_1=b_0+b_1·T_2+b_2·T_2^2; {\mathrm{T}}_{\mathrm{a}}=20 °\mathrm{C}$$

(8)

The dashed curves are introduced gradually with P_n polynomials n = 1,2, … 4, as given by Equation (9):

$$P_{1n}=(b_0+n·{\Delta }_0)+(b_1+n·{\Delta }_1)·T_2+(b_2+n·{\Delta }_2)·T_2^2$$

(9)

$${\Delta }_0=(a_0-b_0)/5; {\Delta }_1=(a_1-b_1)/5; {\Delta }_2=(a_2-b_2)/5$$

∆—increments of polynomial coefficients: n = 1,2 … 4.

The function P₁ = F(T₂) 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 T₂ 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 P₁ = F(T₂) 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 T₂ in Figure 9.

3. Discussion

3.1. 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 Th₁ (Figure 3) enables the self-heating current I₁ and generates power P₁, which are of an exponential/logarithmic type, as given in Figure 4. The chip thermistor Th₂ is only a heat receiver (passive thermistor). The resistances R₁ and R₂ of the chip thermistors Th₁ and Th₂ 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 T₁ and T₂ of the chip thermistors Th₁ and Th₂, respectively, are also dependent on the supply voltage U and ambient temperature T_a (as presented in Figure 6). The curves of T₁ are more exponential compared to the curves of T₂ as the heat generated in Th₁ or power P₁ with input temperature T₁ is only partly transferred to the thermistor Th₂ through the epoxy resinto cause an appearance of output temperature T₂ on the thermistor Th₂. As a consequence, the chip thermistor Th₂ (heat receiver) has a much lower temperature T₂ compared to T₁. 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 ΔT = T₁ − T₂ 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 ΔT is a function of the input supply voltage U as ΔT(U) = T₁(U) − T₂(U), and using Equations (3) and (4), it can be defined as in Equation (10):

$$\Delta T=B_1·T_0/(B_1+T_0\ln (\frac{U/I_1}{R_{01}}))-B_2·T_0/(B_2+T_0\ln (\frac{R_2}{R_{02}}))$$

(10)

where the input resistance R₁ = U/I₁ and output resistance R₂ are measured by a multimeter.

3.2. Heat Transfer in TCCT Device

The self-heating current I₁(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 I₁(t). The delay time t_d ≈ 30 s from the switch on of the supply voltage U was estimated from the behavior I₁(t) in the region below the knee of curves. As the horizontal curves I₁(t) at T_a = 20 °C are not ideally horizontal, especially for higher values of U, the criterion of the stability of I₁(t) in the horizontal region was introduced as k = I₁(t = 40)/I₁(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 I₁. The aim was to limit the input power P₁ in accordance with the criterion of k > 0.97. The behavior of the resistances R₁ and R₂ and temperatures T₁ and T₂ 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):

$$q_1=q_2+q_3+q_4$$

(11)

The heat generation in the self-heating thermistor Th₁ is a product of electrical power U·I₁ and time Δt, as given by Equation (12):

$$q_1=U·I_1·\Delta t$$

(12)

The heat transfer from the Th₁ to Th₂ 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):

$$q_2=-K_e·A·[(T_2-T_1)/d_i]·\Delta t$$

(13)

Hence, the sum of the heat loss q₃ on the boundary of the thermistor Th₁/epoxy/air corresponding to T₁ − T_a and q₄ on the boundary of the thermistor Th₂/epoxy/air corresponding to T₂ − T_a is given by Equation (14).

$$q_3+q_4=U·I_1·\Delta t+K_e·A·[(T_2-T_1)/d]·\Delta t$$

(14)

The dissipation surface for q₃ and q₄ is A₃ = A₄ = A₀ − A, where A₀ is the total outlook surface of the chip thermistor, T₁ and T₂ are the thermistor temperatures, respectively, the epoxy coating thickness of the chip is d₃ = d₄ = 0.5 mm, and the heat dissipation length d_ain the surrounding air is unknown. Further, using an approximation of the temperatures on the epoxy coatings T_e as T_e1 ≈ T₁ and Te₂ ≈ T₂ for the coated chip thermistors Th₁ and Th₂, respectively, q₃ and q₄ are given as (15) and (16):

$$q_3=-K_a·A_3·[(T_a-T_1)/d_a]·\Delta t$$

(15)

$$q_4=-K_a·A_4·[(T_a-T_2)/d_a]·\Delta t$$

(16)

where K_a is the thermal conductivity of the air. Now, by replacing q₃ and q₄ in Equation (14), the heat balance equation is formed at room temperature T_a = 20 °C (17):

$$-K_a·A_3·[(T_a-T_1)/d_a]·\Delta t+-K_a·A_4·[(T_a-T_2)/d_a]·\Delta t=U·I_1·\Delta t+K_e·A·[(T_2-T_1)/d]·\Delta t$$

(17)

The unknown heat dissipation length d_a in the surrounding air is given by (18).

$$d_a=\{U·I_1+K_e·A·[(T_2-T_1)/d]\}/\{-K_a·A_3[2·T_a-T_1-T_2]\}$$

(18)

Finally, adding the values for U, I₁, K_a, K_e, T₁, T₂, T_a, d_i, A, and A₃, 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 Th₁ and Th₂ can be defined using (19):

$$k_{1,2}=[(T_2-T_a)/(T_1-T_a]$$

(19)

The thermal coupling k_1,2 between the chip thermistors depends on the U, I₁, and Ta; for example, at a room temperature of 20 °C, U = 8 V, I₁ = 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.

3.3. Comparison of TCCT with TF TCT and TCDT Devices

The properties of the TCCT device were compared in

Table 1. Comparison of TF TCT, DISK TCT, and TCCT device properties.

Properties	TF TCT	DISK TCT	TCCT
R0 [Ω] nominal resistance at 20 °C	562	586	2600
B [K] temperature exponential factor	3353	3186	3908
type of device geometry	planar	cubic	cubic
device dimensions [mm]	25.4 × 12.7 × 1	D = 4.6; l = 6.5	2.2 × 1.4 × 1
V [mm3] device volume	323	108	3.08
S [mm2] occupied surface on board	323	30	3.6
Ta [°C] ambient temperature	1 to 40	1 to 40	1 to 40
Umax [V] supply voltage	16 to 8	14 to 4.5	14 to 5
I1max [mA] self-heating current	15 to 48	16 to 24	2.5 to 9.5
P1max [mW] nominal input power	500	450	100
R1min [Ω] input resistance	905 to 155	956 to 105	5378 to 461
R1max [Ω] input resistance	1375 to 173	1412 to 384	6399 to 1091
R2min [Ω] output resistance	9855 to 1550	11,450 to 3060	5843 to 743
R2max [Ω] output resistance	13,950 to 2992	11,950 to 3750	6499 to 1034
ΔT[K] temperature difference	6 to 10	4.5 to 18	2 to 16
td [s] self-heating time *	10 to 30	10 to 30	** 10 to 30
k1,2 relative thermal coupling	0.55	0.35	0.25
DC Insulation resistance [Ω] *	>1012	>109	>109
insulation resistance [Ω] * at 50 Hz	>200 M	>250 M	>200 M
parasitic capacitance [pF] *	<12.5	<3	<1

* measured at room temperature (20 °C), ** measured at 20 °C, U and I₁ given in bar diagram in Figure 7.

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 Th₁ and Th₂ thermistors. The TCCT device sensitivity can be defined as Δ₁ = ΔT₂/ΔU (from Figure 6) using limitations for U (in Figure 7) and Δ₂ = ΔP₁/ΔT₂ (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 P₁ by the output temperature T₂. 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. Comparison of TF TCT, DISK TCDT, and TCCT device sensitivity.

Properties	TF TCT	DISK TCT	TCCT
Ta [°C] ambient temperature	1 to 40	1 to 40	1 to 40
Δ1min [K/V] min sensitivity	0.116 to 0.321	0.112 to 0.308	0.076 to 0.362
Δ1max [K/V] max sensitivity	0.927 to 2.573	0.625 to 1.562	0.201 to 1.45
Δ2min [mW/°C] power sensitivity	15.9 to 12.25	37.15 to 18.51	1.011 to 1.31
Δ2max [mW/°C] power sensitivity	39.2 to 29.35	60.9 to 41.66	4.95 to 7.85

The inaccuracy in measurements of chip TCCT device response such as output temperature T₂ depends on partial inaccuracies such as the following: Δ_T of temperature T, Δ_MS of supply voltage U, self-heating current I₁, and resistance R₂, Δ_B of exponential factor B and Δ_SH of using the Steinhart–Hart equation in the first approximation.

.

The inaccuracy ΔB 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 ΔB is now determined by Equation (20):

Δ_B = 2·ΔR/R + 2·ΔT/T

(20)

The inaccuracy ΔB measured in the climatic test chamber does not exceed 0.3%.

The temperature inaccuracy Δ_T according to Equation (3) depends on resistance R and ΔB from the previous equation. It is defined as in (4):

Δ_T = 2·ΔR/R + ΔB/B

(21)

The temperature inaccuracy Δ_T does not exceed 0.5%.

The inaccuracy in the measurement of the self-heating electrical current I₁ is dependent on the measuring of the input voltage U, ambient temperature T_a, and time t as I₁(U, T_a, t). The self-heating current has an inaccuracy of Δ_sc given by the next Equation (22):

Δ_sc = ΔI/I + ΔU/U + Δt/t

(22)

It does not exceed 0.3%. The inaccuracy of the measuring system Δ_MS (only digital instruments in Figure 2) depends on the measuring of U, I, R, and the power source Ups, as given in (23):

Δ_MS = ΔU/U + ΔI/I + ΔR/R + ΔU/U_PS

(23)

The inaccuracy of the full measuring system Δ_MS does not exceed 0.4%. After a year of testing, there was no change in the thermistor nominal resistances R₀₁ and R₀₂ and the exponential factor B.

4. 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 Table 1 and Table 2). The main advantages are summarized and grouped as follows.

• 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 R₀ ≈ 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 >10⁹ 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 I₁ was less than 30 s.
• 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).
• 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.