Studying the Influence of the Thermoelectric Materials Parameters on the Dynamics of Singlecascade Cooling Devices

The effect of the variants of parameters of the original thermoelectric materials of the same efficiency on the operational dynamics of a single-cascade thermoelectric cooler has been examined. The variants differ by the coefficients of thermoEMF, electrical conductivity, and thermal conductivity. The study has been carried out in the range of changes in the working temperature, the rated heat load at the predefined geometry of thermoelement branches.<br><br>The analysis was performed for the characteristic current modes of operation: maximum refrigerating capacity Q0max, maximum refrigerating capacity at the predefined current (Q0/I)max, maximum refrigerating factor (Q0/I2)max, minimum failure rate λmin.<br><br>We have established the relationship between the cooler dynamics and the basic parameters and reliability indicators for different current modes of operation. A possibility has been shown to reduce the time to enter a stationary mode of operation for a variant with the increased electrical conductivity of a material, by 9‒10 %, compared to the basic variant calculated for the averaged electrophysical parameters. The minimum time to enter a stationary regime is achieved under a mode of maximum refrigerating capacity.<br><br>The economic feasibility of using the starting source materials with enhanced electric conductivity relates not only to the improved dynamic and reliable characteristics. Designing thermoelectric coolers is also associated with a decrease in the cost of a cooler by using materials that were considered substandard.<br><br>The rational design of thermoelectric coolers for the systems that enable the thermal modes of electronic equipment accounts for a set of restrictive requirements. These include energy consumption, weight, and size, performance speed, reliability indicators, etc., which are inherently contradictory. The proposed selection of compromise variants of the current modes of operation for different operating conditions allows the optimized design of thermally-loaded equipment.


Introduction
The operational dynamics of thermoelectric cooling devices (TEDs) are inextricably linked to the quality of the starting thermoelectric materials and, first of all, their efficiency. As the efficiency of the source materials improves, the time to enter a stationary mode of operation is reduced. As evidenced by the world practice of thermoelectric engineering, it is not possible to significantly improve the efficiency of thermoelectric materials at present. The issue relates to the fact that the creation of new materials with higher efficiency does not automatically improve the quality of thermoelectric coolers as the mechanical strength and thermal conductivity of the thermoelements' material can neutralize the results obtained. The relevance of improving the dynamic characteristics in the application of industrially produced materials, for which the technology of cooler fabrication has been tested and the climatic and mechanical tests have been performed, is beyond any doubt.

Literature review and problem statement
Paper [1] shows that the systems that enable thermal modes are an integral part of thermally-loaded electronic equipment without which its operation is impossible. However, there are unresolved issues related to the consideration of differences in the heat output of components of the equipment. For concentrated heat sources, such as semiconductor lasers or intense radiation receivers, the requirements to the reliability of tools that enable their thermal modes to become similar to the thermally-loaded elements [2]. The reason is that from the point of view of the theory of reliability, they are enabled consistently and a minimum of failure rate is achieved with equal reliability indicators. It was this approach that made it possible to distinguish thermoelectric coolers as the most promising ones in comparison with compression ones, including their mass-size, dynamic, and operational characteristics [3]. The optimization of the energy interaction between a cooled object and thermal modes is considered in work [4]. The study was conducted only for the static modes of operation and the requirements for stricter operating conditions for on-board systems required further research to improve reliability indicators [5]. One of the most important structural parameters that affect the reliability indicators of thermoelectric coolers is the variation of geometric parameters and the structural integrity of thermoelectric modules, which was addressed in [6]. The dynamic indicators were not considered because thermoelectric coolers have a significant advantage compared to air and liquid cooling systems. The modern approach to the systems that enable the thermal modes implies the inclusion of a thermoelectric cooler in the feedback chain of a temperature control system, so its dynamic characteristics become significant. At the same time, it is known that tests of thermoelectric coolers for operational reliability are carried out in a cyclical mode of heating and cooling [7]. A given mode, which makes it possible to reduce the mean time between failures by an order of magnitude, should become a working mode of thermoelectric modules, albeit in a softer manner. The link between dynamic characteristics and reliability indicators is a fundamental problem, and the need to improve the dynamics of a thermoelectric cooler included in the control chain is obvious. Studying the dynamic characteristics and the connection with reliability indicators were addressed in work [8]. It considers the issues related to decreasing the time to enter a stationary mode by a cooler depending on temperature changes and the current modes of operation. However, the impact on the dynamics and reliability indicators exerted by the coolers' structural parameters remained unsolved. Further research was aimed at analyzing the effect of the structural parameters of thermoelements in the range of working temperature changes and the current modes of operation on the time constant of the cooler and reliability indicators [9]. However, the issues related to the connection between the thermal-physical parameters of thermoelements and the inertia of a thermoelectric cooler remained unsolved. A previous study tackled the effect of the physical parameters of a thermoelectric material of thermoelements on reliability indicators [10]; the possibility of the effective impact of these parameters on the performance indicators of the cooler was shown. Since reliability indicators and dynamic performance are in direct contradiction, the challenge is to find a tradeoff between improving the dynamic characteristics and the acceptable reliability indicators that satisfy a specific task.

The aim and objectives of the study
The aim of this work is to study the effect of the combination of parameters of a thermoelectric material of the same efficiency on the operational dynamics of single-cascade cooling devices.
To accomplish the aim, the following tasks have been set: -to investigate a dynamic model of a thermoelectric cooling device for different variants of combinations of the parameters of the original materials of thermoelements; -to analyze the dynamic characteristics and reliability indicators for the main current modes of thermoelectric coolers.

A dynamic model of a thermoelectric cooler in terms of the efficiency of thermoelectric materials
The rated spread of parameters for the branches of thermoelements, used in the manufacture of unified thermoelectric modules, lies within the following limits of average values: the thermoEMF ratio -220-180 mcV/K, e = electrical conductivity -800 -1200 S/cm = σ at the original materials' efficiency ż=2.4·10 -3 1/K at T=300 K.
At the same time, as experience has shown, a possible range of change in the parameters in the manufacture of thermoelectric materials is much wider and can be within 250-165 V/K, e = µ 550 -1500 S/cm = σ at T=300 K. Materials with such boundary parameters were considered substandard and were not used in the manufacture of modules.
Let us consider the possible (experimentally obtained for the conditions of mass production) variants of the combination of parameters of the source materials in a module at T=300 K, ż=2.4·10 -3 1/K, l/S=10 cm -1 , ∆T=0, given in Table 1, for the purpose of their potential use. Using e and σ as the basic important parameters of thermoelectric materials gives quite complete information about the cooling capabilities of modules assembled on their basis.
Consider the model of the relationship between the main characteristics, the indicators of reliability and operational dynamics of a single-cascade TED with the original material parameters e and .
σ Paper [8] reported the ratios for determining the time to enter a stationary mode of operation τ; the authors comprehensively described the impact of the structural and technological elements (STE) on the basic TED parameters for the geometry of thermoelements branches l/S=10 cm -1 . We shall use the ratio to determine the time to enter a stationary mode of operation τ depending on the relative working current B K : R are, respectively, the maximum working current, A, and the electrical resistance of a thermoelement branch, Ohm, at the beginning of the cooling process at τ=0; I maxK , R K are, respectively, the maximum working current, A, and the electrical resistance of a thermoelement branch, Ohm, at the end of the cooling process τ; e H , e K are, respectively, the thermoEMF factor of a thermoelement branch at the beginning and end of the cooling process, V/K; B H =I/I maxH is, respectively, the relative working current at τ=0; B K =I/I maxK is, respectively, the relative working current at τ; if the currents are equal at the beginning and end of the cooling process: (2) T 0 is the temperature of heat-absorbing welding joint at the end of the cooling process, K; T is the temperature of heat-absorbing welding joint at the beginning of the cooling process, K; Θ=∆T/T max is the relative temperature difference; ∆T=T-T 0 is the relative difference in a TED temperature, K; is the maximum temperature difference, K; ż is the averaged value of the thermoelectric material efficiency in a module, 1/K; I is the magnitude of working current, A; = ae / ( / ) K K K l S is the heat transfer factor, W/K; ae K is the averaged thermal conductivity ratio, W/(cm·K); is the total magnitude of the product of heat intensity by the mass of STE components at the predefined geometry of thermoelement branches l/S=10 cm -1 . The number of thermoelements n can be determined from ratio where Q 0 is the magnitude of heat load, W; is the maximum power of thermoelectric cooling.
The power of consumption W K by a TED can be determined from expression: Voltage drop is Refrigerating factor E can be determined from formula The relative failure rate λ/λ 0 can be determined from formula [10]: is the relative heat load; 1 T K is the significant lower temperature factor; λ 0 =3·10 -8 1/h is the rated failure rate. The probability of a failure-free operation P by a TED can be determined from expression where t=10 4 h is the designated resource.
We calculate the dynamics of the main parameters and reliability indicators of a single-cascade TED for different variants of the combinations of the source material parameters (1) to (5) and the current modes of operation from Q 0max to λ min .

1. Mode Q 0max
The results of calculating the basic parameters taking into consideration the temperature dependence, reliability indicators, operational dynamics of a single-cascade TED for variants of the combination of the source material parameters (1) to (5) are given in Table 2. Calculations were performed at T=300 K, temperature variations from 10 to 60 K, Analysis of the results of calculating the main parameters, reliability indicators and the operational dynamics of a single-cascade TED for different variants of the combinations of the source material parameters (1) to (5), given in Table 2, has shown that an increase in the temperature difference ∆T leads to the following: -the magnitude of refrigerating capacity per thermoelement Q 0 /n is reduced for the assigned variant of the combination of the original material parameters (1) to (5). The largest increase in Q 0 /n is observed at ∆T=10 K ( Fig. 1) and is 19.2 % for variant 5 compared to variant 1 and 69 % for variant 5 compared to traditional variant 3; -the magnitude of the maximum temperature difference ∆T max decreases and does not depend on the combination of the original material parameters (1) to (5); -the number of thermoelements n increases for all combination variants (1) to (5). At a predetermined temperature difference ∆T, for example, ∆T=40 K, n is reduced from variant 1 to variant 5 by 13.6 %, and from variant 3 to variant 5 -by 7 %; -the magnitude of the relative temperature difference Θ is increasing and does not depend on the combination of the original material parameters (1) to (5); -the magnitude of the maximum working current I maxK is reduced for different combination variants (1) to (5); -the magnitude of the refrigerating factor E decreases and does not depend on the combination of the source material parameters (1) to (5); -the magnitude of the working current I decreases for all variants of the combination of the original material parameters (1) to (5) (Fig. 2); at the assigned temperature change, for example, ∆T=40 K, the magnitude of the working current I increases from variant 1 to variant 5 by 73 %; -the magnitude of the relative working current B K =1.0 remains almost unchanged under a Q 0max mode and does not depend on the variant of the combination of the source material parameters (1) to (5), while B Н decreases; -the failure rate λ/λ 0 increases for all variants of the combination of the source material parameters (1) to (5); at the assigned temperature change, for example, ∆T=40 K, the failure rate λ/λ 0 decreases for the variant of combination 5, compared to variant 1, by 13.9 %, and, compared to 3, by 10.6 %; -the probability of failure-free operation P is reduced for all variants of the combination of the original material parameters (1) to (5); at the assigned temperature difference, for example, ∆T=40 K, the probability of failure-free operation P increases from the variant of combination 1 to variant 5; -the time to enter a stationary mode τ is increased (Fig. 3) for all variants of the combination of the original material parameters (1) to (5); at the assigned temperature change, for example, ∆T=40 K, the time to enter a stationary mode τ is reduced for the variant of combination 5, compared to variant 1, by 13 %, and, compared to 3, by 7.7 %; -the relative magnitude of the time to enter a stationary mode Δτ/τ=(τ 1 -τ 5 )/τ 1 % -pos. 1 for the variant of combination 5, compared to variant 1, Δτ/τ=(τ 3 -τ 5 )/τ 3 %pos. 2 for the variant of combination 5, compared to variant 3, decreases (Fig. 4); -the relative magnitude of the time to enter a stationary mode Δτ/τ=(τ 1 -τ 4 )/τ 1 % -pos. 1 for the variant of combination 4, compared to variant 1, and Δτ/τ=(τ 3 -τ 4 )/τ 3 %pos. 2, compared to variant 3, decreases (Fig. 5); at the assigned temperature change, for example, ∆T=40 K, the magnitude Δτ/τ is reduced for the variant of combination 4, compared to variant 1, by 9 %, and, compared to 3, by 3 %. Table 2 Mode Q 0max ; В K = 1.0; T=300 K; Q 0 =2.0 W; l/S=10 cm -1 ;  4. Dependence of the relative magnitude of the time to enter a stationary mode ∆τ/τ in a single-cascade TED on temperature difference ∆T for different variants of the combination of the original material parameters at T=300 K, 1 - The minimum number of thermoelements n is provided by the variant of the combination of parameters 5 of the source material and is n=14.6 pcs. This is achieved at ∆T=40 K, refrigeration factor E=0.216, the magnitude of working current I=6. 6, the maximum working current I max =6.6 А, and voltage drop U=1.4 V.
Thus, the choice of the variant of the combination of parameters 5 with the increased electrical conductivity of the source material is the most appropriate when constructing the thermoelectric systems to enable the electronic equipment thermal mode.

2. Mode (Q 0 /I) max
The results of calculating the basic parameters taking into consideration the temperature dependence, reliability indicators, operational dynamics of a single-cascade TED for different variants of the combination of the original material parameters (1) to (5) are given in Table 3. Calculations were performed at T=300 K, temperature difference ∆T from 10 to 60 K, heat load Q 0 =2.0 W, l/S=10 cm -1 , Thus, we have established the relationship between the operational dynamics of a single-cascade TED and the basic parameters and reliability indicators under a Q 0max : at T=300 K; Q 0 =2.0 W; l/S=10 cm -1 .
The minimum time to enter a stationary mode of operation τ min is provided by the variant of the combination of parameters 5 of the source material and is, at ∆T=40 K, τ min =6 s.
The lowest failure rate λ/λ 0 and, accordingly, the largest probability of failure-free operation P, are provided by the on the variant of the combination of the original material parameters (1) to (5), while B Н decreases; -the magnitude of the working current I increases (Fig. 7) for all variants of the combination of the original material parameters (1) to (5). At the assigned temperature change, for example, ∆T=40 K, the magnitude of the working current I increases from variant 1 to variant 5 by 70 %, and, by 28 %, from variant 3 to variant 5; -the magnitude of the voltage drop U increases; for all variants of the combinations of the original material parameters (1) to (5). At the predefined temperature difference, for example, ∆T=40 K, the magnitude of the voltage drop U decreases, for the variant of combination 5, compared to 1, by 42 %, and, compared to 3, by 23 %; -the failure rate λ/λ 0 increases for all variants of the combination of the source material parameters (1) to (5); at the assigned temperature change, for example, ∆T=40 K, the failure rate λ/λ 0 decreases for the variant of combination 5, compared to variant 1, by 13.2 %, and, to 3, by 6.1 %; Our analysis of the results of calculating the basic parameters, reliability indicators, and the operational dynamics of a single-cascade TED for different variants of the combination of the original material parameters (1) to (5), given in Table 3, has revealed that the increase in temperature difference ∆T leads to the following: -the functional dependence Q 0 /n=f(∆T) has a maximum at ∆T=20 K (Fig. 6). The highest refrigerating capacity per thermoelement Q 0 /n is observed for variant 5, under which Q 0 /n increases for variant 5, compared to variant 1, by 17.6 %, and, for variant 5 compared to 3, by 8.1 %; -the functional dependence of the number of thermoelements n=f(∆T ) has a flat minimum for ∆T=20 K; at the predetermined temperature change, for example, ∆T=40 K, the number of thermoelements n decreases from variant 1 to variant 5 by 13.2 %, and, from variant 3 to variant 5, by 6.8 %; -the relative working current B К and B Н increases at the beginning and end of the cooling mode; B К does not depend Table 3 Mode (Q 0 /I) max ; T=300 K; Q 0 =2.0 W; l/S=10 cm -1 ;  The minimum time to enter a stationary mode of operation τ min is provided by the variant of the combination -the probability of failure-free operation P is reduced; at the assigned temperature change, for example, ∆T=40 K, the probability of failure-free operation P increases from the variant of combination 1 to variant 5; -the time to enter a stationary mode τ is increased (Fig. 8) for all variants of the combination of the original material parameters (1) to (5). At the assigned temperature change, for example, ∆T=40 K, the time to enter a stationary mode τ is reduced for variant 5, compared to 1, by 14.5 %, and, compared to 3, by 9 %; -the relative magnitude of the time to enter a stationary mode Δτ/τ=(τ 1 -τ 5 )/τ 1 % -pos. 1 for variant 5, compared to 1, Δτ/τ=(τ 3 -τ 5 )/τ 3 % -pos. 2 for variant 5, compared to 3, decreases (Fig. 9); -the relative magnitude of the time to enter a stationary mode Δτ/τ=(τ 1 -τ 4 )/τ 1 % -pos. 1 for variant 4, compared to 1, and 1, and Δτ/τ=(τ 3 -τ 4 )/τ 3 % -pos. 2, compared to 3, decreases (Fig. 10). At the assigned temperature difference, for example, ∆T=40 K, the magnitude Δτ/τ is reduced for the variant of combination 4, compared to variant 1, by 9 %, and, compared to 3, by 3 %. Therefore, the choice of the variant of the combination of parameters 4, 5 with increased electrical conductivity of the source material is the most appropriate.

3. Mode (Q 0 /I 2 ) max
The results of calculating the basic parameters taking into consideration the temperature dependence, the time to enter a stationary mode, reliability indicators for the (Q 0 /I 2 ) max mode and for different temperature difference ∆T are given in Table 4. of parameters 5 of the source material and is, at ∆T=40 K, τ min =7 s.
The lowest failure rate λ/λ 0 is provided by the variant of the combination of parameters 5 of the source material and is λ/λ 0 =13.8.
The number of thermoelements n, provided by the variant of combination 5 of the parameters of the source material is, ∆T=40 K, n=17.7 pcs. at refrigerating factor E=0.34, the magnitude of working current I=4.6 А, the maximum working current I max =6.57 А, and a drop in voltage U=1.27 V. Table 4 Mode (Q 0 /I 2 ) max ; T=300 K; Q 0 =2.0 W; l/S=10 cm -1 ;  The lowest time to enter a stationary mode of operation τ min is provided by the variant of the combination of parameters 5 of the source material and is, at ∆T=40 K, τ min =10 s.
The lowest failure rate λ/λ 0 and, therefore, the highest probability of failure-free operation P is provided by the Our analysis of the results of calculating the basic parameters, reliability indicators, and the operational dynamics of a single-cascade TED for different variants of the combination of the original material parameters (1) to (5), given in Table 4, has revealed that the increase in temperature difference ∆T leads to the following: -the functional dependence Q 0 /n=f(∆T) has a maximum for ∆T=40 K (Fig. 11). The highest refrigerating capacity per thermoelement Q 0 /n is observed for variant 5, under which Q 0 /n increases for variant 5, compared to variant 1, by 15.4 %, and, for variant 5, compared to 3, by 7.6 %; -the relative working current B К and B Н increases at the beginning and end of the cooling mode; B К does not depend on the variant of the combination of the original material parameters (1) to (5), while B Н decreases; -the magnitude of the working current I (Fig. 12) increases for all variants of the combination of the original material parameters (1) to (5). At the assigned temperature change, for example, ∆T=40 K, the magnitude of working current I increases from variant 1 to variant 5 by 73 %, and by 30 % from variant 3 to 5; -the functional dependence of the number of thermoelements n=f(∆T) has a minimum for ∆T=40 K. At the assigned temperature difference, for example, ∆T=40 K, the number of thermoelements n decreases from variant 1 to variant 5 by 13.3 %, and, from variant 3 to variant 5, by 7 %; -the magnitude of refrigerating factor E decreases and does not depend on the variant of the combination of the source material parameters (1) to (5); -the magnitude of voltage drop U increases; at the assigned temperature change, for example, ∆T=40 K, the magnitude of voltage drop U decreases for the variant of combination 5, compared to variant 1, by 42 %, and, to 3, by 23.6 %; -the failure rate λ/λ 0 increases for all variants of the combination of the source material parameters (1) to (5). At the assigned temperature change, for example, ∆T=40 K, the failure rate λ/λ 0 decreases for the variant of combination 5, compared to 1, by 13.5 %, and, to 3, by 6.7 %; -the probability of failure-free operation P is reduced; at the assigned temperature difference, for example, ∆T=40 K, the probability of failure-free operation P increases from the variant of combination 1 to variant 5; -the time to enter a stationary mode τ is increased (Fig. 13) for all variants of the combination of the original material parameters (1) to (5). At the assigned temperature difference, for example, ∆T=40 K, the time to enter a stationary mode τ is reduced for variant 5, compared to 1, by 15.3 %, and, to 3, by 9.1 %; -the relative magnitude of the time to enter a stationary mode of operation Δτ/τ decreases for different variants of combinations 1 -Δτ/τ=(τ 1 -τ 5 )/τ 1 % and 2 -Δτ/τ= =(τ 3 -τ 5 )/τ 3 % (Fig. 14); -the relative magnitude of the time to enter a stationary mode Δτ/τ=(τ 1 -τ 4 )/τ 1 % -pos. 1 for variant 4, compared to 1, and Δτ/τ=(τ 3 -τ 4 )/τ 3 % -pos. 2, compared to 3, decreases (Fig. 15). At the assigned temperature difference, for example, ∆T=40 K, the magnitude Δτ/τ is reduced for the variant of combination 4, compared to variant 1, by 10 %, and, to 3, by 3.5 %.
Thus, we have established the relationship between the operational dynamics of a single-cascade TED and the main parameters and reliability indicators under a (Q 0 /I 2 ) max mode at T=300 K; Q 0 =2.0 W; l/S=10 cm -1 .
namics of a single-cascade TED for different variants of the combination of the original material parameters (1) to (5), given in Table 5, has revealed that the increase in temperature difference ∆T leads to the following: -the functional dependence Q 0 /n=f(∆T ) has a maximum for ∆T=40 K (Fig. 16). The highest refrigerating capacity per thermoelement Q 0 /n is observed for variant 5, and is, at ∆T=40 K, Q 0 /n=0,0463, that is, it is larger by 15.8 % compared to variant 1, and, compared to 3, by 7.7 %.
-the relative working current B К and B Н increases at the beginning and end of the cooling mode; B К does not depend on the variant of the combination of the original material parameters (1) to (5), while B Н decreases; -the magnitude of the working current I (Fig. 17) increases for all variants of the combination of the original material parameters (1) to (5). At the assigned temperature difference, for example, ∆T=40 K, the magnitude of working current I increases from variant 1 to variant 5 by 72 %, and by 32 % from variant 3 to 5; -the functional dependence of the number of thermoelements n=f(∆T) has a minimum for ∆T=40 K. At the assigned temperature difference, for example, ∆T=40 K, the number of thermoelements n decreases from variant 1 to variant 5 by 13.63 %, and, from variant 3 to variant 5, by 7.1 %; -the magnitude of refrigerating factor E decreases and does not depend on the variant of the combination of the source material parameters (1) to (5); -the magnitude of voltage drop U increases; at the assigned temperature change, for example, ∆T=40 K, the magnitude of voltage drop U decreases for the variant of combination 5, compared to variant 1, by 42 %, and, compared to 3, by 23.5 %; -the failure rate λ/λ 0 increases for all variants of the combination of the source material parameters (1) to (5). At the assigned temperature change, for example, ∆T=40 K, the failure rate λ/λ 0 increases, compared to the variant of combination 5, compared to 1, by 14.1 %, and, to 3, by 7.6 %; -the probability of failure-free operation P failure is reduced; at the assigned temperature change, for example, ∆T=40 K, the probability of failure-free operation P increases from the variant of combination 1 to variant 5; -the time to enter a stationary mode τ is increased (Fig. 18) for all variants of the combination of the original material parameters (1) to (5). At the assigned temperature difference, for example, ∆T=40 K, the time to enter a stationary mode τ is reduced, for variant 5 compared to 1, by 15 %, compared to 3, by 9.4 %. For the variant of combination 4, compared to variant 1, by 9.5 %, and, compared to 3, by 3.6 %; -the relative magnitude of the time to enter a stationary mode of operation Δτ/τ decreases for different variants of combinations 1 -Δτ/τ=(τ 1 -τ 5 )/τ 1 % and 2 -Δτ/τ= =(τ 3 -τ 5 )/τ 3 % (Fig. 19); -the relative magnitude of the time to enter a stationary mode Δτ/τ=(τ 1 -τ 4 )/τ 1 % -pos. 1 for the variant of combination of 4, compared to 1, and Δτ/τ=(τ 3 -τ 4 )/τ 3 % -pos. 2, compared to 3, decreases (Fig. 19). At the assigned temperature difference, for example, ∆T=40 K, the magnitude Δτ/τ is reduced for the variant of combination of 4, compared to variant 1, by 9.5 %, and, to 3, by 3.5 %.
variant of the combination of parameters 5 of the source material and is λ/λ 0 =1.8; Р=0.99946 at ∆T=40 K. The number of thermoelements n, provided by the combination of parameters 5 of the source material, is n=29.3 pcs. The calculations were performed for ∆T=40 K at refrigerating factor E=0.38, the magnitude of working current I=3.3 A, the maximum working current I max =6,67 А, and the drop in voltage U=1.6 V.
Thus, the choice of the variant of the combination of parameters 4, 5 with increased electrical conductivity of the source material is the most appropriate.

4. Mode λ min
The results of calculating the basic parameters taking into consideration the temperature dependence, the time to enter a stationary mode, reliability indicators for the λ min mode and for different temperature change ∆T are given in Table 5.
Our analysis of the results of calculating the basic parameters, reliability indicators, and the operational dy-   Thus, we have established the relationship between the operational dynamics of a single-cascade TED and the main parameters and reliability indicators under a λ min mode: at T=300 K; Q 0 =2.0 W; l/S=10 cm -1 .
The shortest time to enter a stationary mode of operation τ min is provided by the variant of the combination of parameters 5 of the source material and is, at ∆T=40 K, τ min =12.5 s.
The smallest failure rate λ/λ 0 and the highest probability of failure-free operation P are provided by the variant of the combination of parameters 5 of the source material and are λ/λ 0 =1.3; P=0.99960 at ∆T=40 K.
The number of thermoelements n, provided by the variant of the combination of parameters 5 of the source material, is n=43.2 pcs. The calculations were performed for ∆T=40 K at refrigerating factor E=0.34, the magnitude of working current I=2.8 А, the maximum working current I max =6.57 А and the drop in voltage U=2.1 V.
Thus, the choice of the variant of the combination of parameters 4, 5 with increased electrical conductivity of the source material is the most appropriate.