# An Extensive Unified Thermo-Electric Module Characterization Method

^{*}

## Abstract

**:**

## 1. Introduction

## 2. State of the Art and Proposed Method

#### 2.1. Thermal Characterization

_{2}O

_{3}) ceramic external plates in the module fabrication, makes the correction (6) negligible, since the joint combination of a high thermal conductivity (one order of magnitude higher than Bi

_{2}Te

_{3}) and a small thickness (one order of magnitude lower than Bi

_{2}Te

_{3}pellets) lead to a thermal resistance ${\mathsf{\Theta}}_{cer}$ value that is usually negligible as furtherly demonstrated in Equation (20).

#### 2.2. Electrical Characterization

#### 2.3. Proposed Method

## 3. Electrical Measurement

#### 3.1. Current Sweep (CS) Method

#### 3.2. Small Signal (SS) Method

#### 3.3. Analytical Comparison among CS and SS Method

## 4. Thermal Characterization

^{®}System Identification Toolbox [39].

- ${k}_{ref}$ has no associated uncertainty since it is a calibration factor
- the four temperatures have the same uncertainty ${u}_{T}$ because they are measured with the same kind of temperature sensor (type-J thermocouple), they are acquired using the same DAQ board and referred to the same cold-junction sensor
- all variables are uncorrelated, except ${I}_{st}$ and ${V}_{th}$ that are obviously correlated

## 5. Automatic Test

- ${V}_{L}$ is the voltage at the TEM terminals
- ${V}_{s}$ is the voltage across the shunt resistor ${R}_{S}$
- ${V}_{a}$ is the climate chamber ambient temperature measured by means of a LM35 temperature sensor (Texas Instruments, Dallas, TX, USA)
- ${V}_{1},{V}_{2},{V}_{3},{V}_{4}$ are the voltages generated by the four different J-type thermocouples each one inserted into one surface of the heat-flux sensors
- ${V}_{cj}$ is the cold junction compensation temperature acquired by means of a further LM35 sensor placed close to the DAQ board

^{®}Simulink Auto-tune tool. Although both methods produce good results, the latter has been chosen because of its ability to produce a desired response in a smaller time with less overshoot.

- (1)
- steady-state: the last driving current value is applied and the acquired values are used to compute statistics over ${I}_{st}$.
- (2)
- SS: a small current 10 Hz sinusoidal stimulus is added to ${I}_{st}$, with an amplitude equal to the ratio of standard deviation threshold to the static gain of Equation (19):$${I}_{SS}=\frac{\sqrt{2}{\sigma}_{th}}{{H}_{\mathsf{\Delta}T}\left(0\right)}$$
- (3)
- CS: the driving current ${I}_{st}$ is swept to its symmetric value $-{I}_{st}$ with a ramp-like signal that is finally switched back to the initial value.

- ${V}_{L}$ is left as it is.
- ${V}_{s}$ is divided by ${R}_{s}$ to obtain $I$.
- ${T}_{a}$ and ${T}_{cj}$ are obtained from respective voltages using the LM35 nominal sensitivity S = 10 mV/(°C).
- ${T}_{1},{T}_{2},{T}_{3},{T}_{4}$ are computed using the coefficients provided by National Institute of Standards and Technology (NIST) and applying a software cold-junction compensation.
- ${T}_{c}$ and ${T}_{h}$ are derived by ${T}_{2}$ and ${T}_{3}$ computing the temperature drop on the ceramic layers induced by the computed heat fluxes.

- ${V}_{th}$ and ${R}_{IN}$ are derived using Equation (10), applying a linear regression to acquired values $\left\{I,{V}_{L}\right\}$.
- ${\alpha}_{s}$ is then obtained as ratio of ${V}_{th}$ to $\mathsf{\Delta}T$.
- ${\mathsf{\Theta}}_{IN}$ is computed using Equation (7).

## 6. Experimental Results

#### 6.1. Module Identification

#### 6.2. SS Method Analysis

#### 6.3. CS Method Analysis

#### 6.4. Experimental Comparison among CS and SS Method

## 7. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Typical TEM voltage profile using transient method: applied current I (

**Bottom**) and resulting voltage V (

**Top**) vs. time.

**Figure 7.**Measured step response $\mathsf{\Delta}{T}_{32}$ (black) vs. simulated (red) of the identified model ${H}_{32}$.

**Figure 9.**(

**a**) Normal probability plot of the residuals of the fitted model; (

**b**) Plot of residual vs. lagged residuals for serial correlation analysis.

**Figure 10.**(

**a**) ${R}_{IN}$ standard deviation in all working conditions for SS method; (

**b**) ${V}_{th}$ standard deviation in all working conditions for SS method.

**Figure 11.**(

**a**) Normal probability plot of the residuals of the fitted model: linear (blue) and quadratic (red); (

**b**) Leverage plot of data and model

**Figure 12.**(

**a**) ${R}_{IN}$ standard deviation in all working conditions for CS method; (

**b**) ${V}_{th}$ standard deviation in all working conditions for the CS method.

**Figure 15.**(

**a**) Internal thermal resistance ${\mathsf{\Theta}}_{IN}$; (

**b**) Figure of merit $Z\overline{T}$.

Transfer Function | Thermal Time Constant | NRMSE ^{1} |
---|---|---|

${H}_{32}\left(s\right)=\frac{3.75}{s+0.0876}$ | 11.5 | 88.79% |

${H}_{41}\left(s\right)=\frac{0.095}{s+0.0136}$ | 73.4 | 66.51% |

^{1}Normalized Root Mean Square Error [42].

Symbol | Parameter | Mean Relative Error ${\overline{{\mathit{e}}_{\mathit{r}}}}_{\mathbf{\%}}$ | Standard Deviation ${{\mathit{\sigma}}_{\mathit{r}}}_{\mathbf{\%}}$ | ||
---|---|---|---|---|---|

Linear | Quadratic | Linear | Quadratic | ||

${\alpha}_{s}$ | Seebeck coefficient | 4.68 | 2.63 | 0.17 | 0.15 |

${R}_{IN}$ | Electrical resistance | 3.80 | 1.65 | 0.11 | 0.11 |

${\mathsf{\Theta}}_{IN}$ | Thermal resistance | 5.50 | 3.03 | 0.21 | 0.19 |

$Z\overline{T}$ | Figure of merit | 7.64 | 3.91 | 0.25 | 0.24 |

Symbol | Description | Value | |
---|---|---|---|

$n$ | Number of thermocouples | 127 | |

A | Single module area [$\mathrm{mm}$] × [mm] | 30 × 30 | |

${T}_{h}$ | Hot side temperature at environment [°C] | 27 | 50 |

$\mathsf{\Delta}{T}_{\mathrm{max}}$ | Temperature Difference when cooling capacity is zero at cold side [°C] | 68 | 76 |

${V}_{\mathrm{max}}$ | Voltage applied to the module at $\mathsf{\Delta}{T}_{\mathrm{max}}$ [V] | 15.5 | 17.4 |

${I}_{\mathrm{max}}$ | DC current through the modules at $\mathsf{\Delta}{T}_{\mathrm{max}}$ [A] | 3.5 | 3.5 |

${Q}_{{C}_{\mathrm{max}}}$ | Cooling capacity at cold side of the module under $\mathsf{\Delta}T=0$ °C [W] | 34.1 | 37.4 |

${R}_{in}$ | Module resistance under AC [$\mathsf{\Omega}$] | 3.5$~$3.9 | 3.8$~$4.3 |

${R}_{ds}$ | Internal resistance ^{1} [$\mathsf{\Omega}$] | 3.42 | 3.79 |

${\alpha}_{ds}$ | Seebeck coefficient ^{1} [$\mathrm{mV}/\mathrm{K}$] | 51.7 | 53.7 |

${\mathsf{\Theta}}_{ds}$ | Thermal resistance ^{1} [$\mathrm{K}/\mathrm{W}$] | 3.24 | 3.27 |

^{1}Data derived using method in [30].

Units | Function | ${\mathit{R}}^{2}$ |
---|---|---|

$\left[\frac{mV}{K}\right]$ | ${\alpha}_{S}=\frac{\mathsf{\Delta}T}{38.55}+\frac{\overline{{T}_{c}}}{15.52}+48.63$ | 0.9967 |

$\left[\mathsf{\Omega}\right]$ | ${R}_{IN}=\frac{\mathsf{\Delta}T}{102.3}+\frac{\overline{{T}_{c}}}{52.48}+3.256$ | 1.0000 |

$\left[\frac{K}{W}\right]$ | ${\mathsf{\Theta}}_{IN}=\frac{\mathsf{\Delta}T}{57.43}+\frac{\overline{{T}_{c}}}{{10}^{8}}+2.929$ | 0.9992 |

$[-]$ | $Z\overline{T}=\frac{\mathsf{\Delta}T}{255.4}+\frac{\overline{{T}_{c}}}{673.4}+\frac{\mathsf{\Delta}{T}^{2}\text{}\overline{{T}_{c}}}{3.4e6}-\frac{\mathsf{\Delta}T\text{}{\overline{{T}_{c}}}^{2}}{1.24e6}+0.556$ | 0.9996 |

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**MDPI and ACS Style**

Attivissimo, F.; Guarnieri Calò Carducci, C.; Lanzolla, A.M.L.; Spadavecchia, M.
An Extensive Unified Thermo-Electric Module Characterization Method. *Sensors* **2016**, *16*, 2114.
https://doi.org/10.3390/s16122114

**AMA Style**

Attivissimo F, Guarnieri Calò Carducci C, Lanzolla AML, Spadavecchia M.
An Extensive Unified Thermo-Electric Module Characterization Method. *Sensors*. 2016; 16(12):2114.
https://doi.org/10.3390/s16122114

**Chicago/Turabian Style**

Attivissimo, Filippo, Carlo Guarnieri Calò Carducci, Anna Maria Lucia Lanzolla, and Maurizio Spadavecchia.
2016. "An Extensive Unified Thermo-Electric Module Characterization Method" *Sensors* 16, no. 12: 2114.
https://doi.org/10.3390/s16122114