Erratum: “Understanding Thermal Lagging Behaviors in Thermoelectric Elements With the Dual-Phase-Lag Model” [ASME J. Electron. Packag., 2022, 144(3), p. 031011; DOI: 10.1115/1.4052948]

This erratum pertains to elements that were in error in the original published article. The authors gratefully acknowledge the reader who informed us on the typographical error with the current density (⁠$je$⁠) unit in the Nomenclature section. The typo leads to the issue of this erratum. The corrected forms are documented herein.

1. The last sentence at the end of Sec. 3.1 (page 031011-3) should be changed to read:

   The system operation is subjected to a constant current density (⁠$je=I/Ac$⁠).

2. The last sentence at the end of Sec. 5.1 (page 031011-5) should be changed to read:

   Note that the current density (⁠$je$⁠) is defined as the ratio of the current and the element cross-sectional area (I/A_c) in which $Ac=πd2/4$⁠.

3. The title of Fig. 3 (page 031011-7) should be updated using $τq instead of τg$⁠, which can be rewritten as follows:

   Figure 3 Comparison of C–V (⁠$τcv$= 10⁻⁹) and DPL (⁠$τq=10−9, τT=10−10)$ models: (a) spatial temperature, (b) temporal temperature, (c) spatial heat flux, and (d) temporal heat flux.

4. The title of Fig. 4 (page 031011-8) should be updated using $τq instead of τg$ as shown below:

   Figure 4 Effect of temperature gradient on temperature and heat flux profiles using DPL model (⁠$τq=10−9, τT=10−8 versus τT=10−10):$(a) spatial temperature, (b) temporal temperature, (c) spatial heat flux, and (d) temporal heat flux.

5. In the Nomenclature section (page 031011-9), the unit for the current density (⁠$je$⁠) needs to be corrected to read:

   $je=current density, A/m2$

   The current density (j_e) is defined as the ratio of the electrical current (I) and the cross-sectional area (⁠$Ac=πd2/4$⁠) of the thermoelectric element as presented in Secs. 3.1 and 5.1 using the engineering data in Table 5.1; therefore, the unit of j_e has to be in ampere per area. The current density was never entered in any of the computations. All computational results were generated based on the current (I) and cross-sectional area $Ac$ (⁠$=πd2/4, see$ Table 1) rather than the current density (j_e) alone. Therefore, the typo of the current density unit in the Nomenclature has no bearing on the results of this paper.
6. For the purpose of consistency, the dimensionless heat flux (⁠$Q$⁠) as it appears in the Nomenclature section (page 031011-10) may be modified to read:

   $Q = dimensionless heat flux,qτqαt/kTc$

Other than those mentioned above, all mathematical equations had also been verified for typos. Using the same symbol for the current density (j_e) throughout the paper and the proper engineering unit along with the engineering data from Table 1 for numerical simulation, it is expected that this technical paper will be free of mystification. The aforementioned changes will not impact the key contributions of the paper.