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TRAC: A Thermal Resistance Advanced Calculator for Electronic Packages^{ †}

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

- Delphi BCI CTMs can only consider fixed values of geometrical dimensions and thermal properties;
- often they are not as accurate as requested;
- their extraction can be very time-consuming, and most of the effort is spent for coping with boundary conditions (BCs) that are not relevant for the extraction of JEDEC thermal metrics;
- they cannot be used for electronic components with multiple heat sources (HSs) [1].

## 2. Parametric Detailed Thermal Model

- a sequence of parallelepipeds of chosen size and position, including the one representing the dissipation region, hereinafter referred to as HS;
- the thermal conductivity of all materials;
- the BCs.

_{JA}, Ψ

_{JB}, Ψ

_{JCtop}, ϑ

_{JB}, ϑ

_{JCtop}, ϑ

_{JCbottom}[2] in 4 ambients, which differ in terms of thermal path followed by the heat generated within the HS to emerge from the die; more specifically: the ambient to evaluate ϑ

_{JCbottom}requires a cold plate in intimate contact with the package backside; for the computation of ϑ

_{JCtop}the plate is located over the top surface; in the ambient for determining ϑ

_{JB}, a cold ring surrounds the package; no cooling systems are exploited in the ambient common to ϑ

_{JA}, Ψ

_{JB}, Ψ

_{JCtop}. In all ambients, the board over which the package is mounted is thermally modeled with a single finely-meshed parallelepiped with a thermal conductivity adjusted to account for the aggregate effect of metal traces and vias, the detailed representation of which would have unnecessarily made the thermal problem much more complex. As shown in Figure 1, the metrics are calculated from the temperatures probed in four positions, namely: (1) the point of the die where the peak (“junction”) temperature is reached; (2) the center of the top of the case; (3) the center of the bottom of the case; and (4) at the foot of the package lead half way along the side of the package (QFP) or within 1 mm of the package body (QFN). ϑ

_{JA}is computed from (1) and the ambient temperature; Ψ

_{JB}from (1) and (4); Ψ

_{JCtop}from (1) and (2); ϑ

_{JB}from (1) and (4); ϑ

_{JCtop}from (1) and (2); ϑ

_{JCbottom}from (1) and (3). As far as the metric ϑ

_{JCtop}is concerned, a calibrated layer was interposed between the epad and the high-conductivity cold plate to emulate the epad-plate contact resistance. For all other metrics, the heat emerging from the die flows through the low-conductivity board, and the contact resistance epad-board was not accounted for, since it plays a negligible role.

**ϑ**(

**p**) is the N × 1 vector with the DoFs of the temperature rise distribution,

**K**(

**p**) is the N-order stiffness matrix,

**G**(

**p**) is a N × n power density matrix,

**p**is the

**p**× 1 parameter vector varying in a set $\mathcal{P}$, and

**P**is the n × 1 vector containing the powers P

_{i}(with i = 1,…, n) dissipated by the n independent HSs in the structure. These equations define the pDTM.

## 3. Parametric Compact Thermal Model

**U**is determined, which allows approximating the N × 1 temperature rise vector in the form

**p**ϵ $\mathcal{P}$, in which $\widehat{\mathit{\vartheta}}(p)$ is a q × 1 vector with q « N. The pCTM is derived from Equation (1) using Equation (2) and the Galerkin’s projection. In this way it results in

**p**, the relative residual ξ does not exceed the assigned value

**Ξ**, so that the pCTM does not vary any longer.

Algorithm 1. Parametric model-order reduction (MOR) iteration. | |

Step | |

1 | Pick up a value of p ϵ $\mathcal{P}$ |

if a pCTM has already been extracted then | |

solve pCTM Equations (3) for $\widehat{\mathit{\vartheta}}(p)$ | |

determine ϑ(p) from Equation (2) using $\widehat{\mathit{\vartheta}}(p)$ | |

2 | Determine the relative residual ξ of Equations (1) using ϑ(p) |

ifξ > Ξ then | |

3 | Solve pDTM Equations (1) for ϑ(p) |

4 | Update U using ϑ(p) |

5 | Update the pCTM using U |

**p**ϵ $\mathcal{P}$ are chosen equal to the values of the parameters defining the specimen in the family of packages for which the JEDEC thermal metrics must be computed. This strategy minimizes the time needed to evaluate the metrics for this case.

**V**being the N-order diagonal matrix with the measures of the N volumes introduced in the FIT discretization.

**U**matrix is updated by appending a column orthonormal to the columns of the initial

**U**matrix in the ‖•‖

_{V}norm, such that the columns of the final

**U**matrix span

**ϑ**(

**p**).

**U**matrix is used for reconstructing the pCTM. Exploiting the fact that the last column of

**U**is changed, the update of the pCTM does not require recomputing the whole model.

## 4. Numerical Results

^{2}package, a 4.5 × 4.5 mm

^{2}epad, and a 2 × 2 mm

^{2}die; as can be seen, the mesh leading to ϑ

_{JA}= 28.87 K/W (1.487 × 10

^{6}DoFs for a quarter of the structure) was chosen to avoid the huge number of DoFs (>50 × 10

^{6}) required to obtain a negligibly more accurate ϑ

_{JA}value.

_{JA}, ϑ

_{JB}, and ϑ

_{JCtop}(the others were not represented not to overcrowd the graphs) for a 10 × 10 mm

^{2}LQFP and a 14 × 14 mm

^{2}TQFP (Figure 3a), as well as for a 6 × 6 mm

^{2}QFN (Figure 3b). The slight discrepancy between TRAC and the FV software must be attributed: (i) to the different mesh styles of the compared tools; and (ii) to the fineness degree adopted in both of them, which was not extremely high to prevent unnecessarily long CPU times. As can be seen, the thermal metrics decrease with increasing the die size (i.e., the HS size) due to the lower dissipated power density and the enhanced lateral heat spreading.

_{JA}and ϑ

_{JCtop}as a function of die size for 10 × 10 mm

^{2}LQFPs and TQFPs commonly sharing a 6 × 6 mm

^{2}epad. It is inferred that the impact of the package thickness is significant only for ϑ

_{JCtop}(for TQFPs, a 25–35% reduction of this metric is observed with respect to LQFPs), since in this case the heat generated in the die flows toward the cold plate placed on the top crossing the whole package; a marginal influence is instead found for all other metrics (including ϑ

_{JA}), where the heat propagates mostly downward.

_{JCtop}remains almost unaffected.

_{JA}as a function of HS size, the HS being centered in the die (which offers the possibility of meshing and simulating one quarter of the structure), for a 10 × 10 mm

^{2}LQFP with a 4.5 × 4.5 mm

^{2}epad and a 2 × 2 mm

^{2}die; the assigned dissipated power amounts to 1 W regardless of the HS size. As expected, ϑ

_{JA}markedly increases with reducing the HS size, which implies a growth in power density. This analysis allows concluding that a correct representation of the HS geometry (which depends on the specific application) is of utmost importance for an accurate evaluation of the thermal metrics of electronic packages.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

ϑ_{JA} (K/W) | junction to ambient thermal resistance |

Ψ_{JB} (K/W) | thermal characterization parameter to report the difference between junction temperature and the temperature of the board measured at the top surface of the board |

Ψ_{JCtop} (K/W) | thermal characterization parameter to report the difference between junction temperature and the temperature at the top center of the outside surface of the component package |

ϑ_{JB} (K/W) | junction to board thermal resistance |

ϑ_{JCtop} (K/W) | junction to case top thermal resistance |

ϑ_{JCbottom} (K/W) | junction to case bottom thermal resistance |

TRAC | thermal resistance advanced calculator |

BC | boundary condition |

BCI | BC independent |

CTM | compact thermal model |

pCTM | parametric CTM |

DTM | detailed thermal model |

pDTM | parametric DTM |

DoF | degree of freedom |

epad | exposed pad |

FIT | finite integration technique |

FV | finite volume |

HS | heat source |

JEDEC | joint electron device engineering council |

QFP | quad flat package |

LQFP | low-profile (thick) QFP |

TQFP | thin QFP |

QFN | quad flat no-leads package |

MOR | model-order reduction |

## References

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**Figure 1.**Specimens of the LQFP (

**a**) and QFN (

**b**) families sharing the same size of package (6 × 6 mm

^{2}), epad (4.5 × 4.5 mm

^{2}), and die (2 × 2 mm

^{2}). Black circles represent the temperature probes needed to determine the thermal metrics.

**Figure 2.**JEDEC thermal metric ϑ

_{JA}as a function of the number of DoFs, as evaluated through the pDTM for a quarter of the 10 × 10 mm

^{2}LQFP with a 4.5 × 4.5 mm

^{2}pad and a 2 × 2 mm

^{2}die. The selected discretization is indicated.

**Figure 3.**Some JEDEC thermal metrics against die size, as determined from the pDTM (blue) and a FV commercial software (red): (

**a**) 10 × 10 mm

^{2}LQFP (circles) and 14 × 14 mm

^{2}TQFP (triangles), both with a 6 × 6 mm

^{2}epad; (

**b**) 6 × 6 mm

^{2}QFN (squares) with a 4.7 × 4.7 mm

^{2}epad.

**Figure 4.**JEDEC thermal metrics ϑ

_{JA}and ϑ

_{JCtop}vs. die size evaluated with the pDTM; results obtained for 10 × 10 mm

^{2}LQFPs (blue circles) are compared with the TQFP counterparts (orange triangles); in both cases, a 6 × 6 mm

^{2}epad is considered.

**Figure 5.**JEDEC thermal metric ϑ

_{JA}vs. HS size for a square HS centered in the die, as evaluated through the pDTM for a 10 × 10 mm

^{2}LQFP with a 4.5 × 4.5 mm

^{2}epad and a 2 × 2 mm

^{2}die.

**Figure 6.**Comparison of the JEDEC thermal metric ϑ

_{JA}provided by both a pDTM and a pCTM for various QFPs.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Codecasa, L.; Race, S.; d’Alessandro, V.; Gualandris, D.; Morelli, A.; Villa, C.M. TRAC: A Thermal Resistance Advanced Calculator for Electronic Packages. *Energies* **2019**, *12*, 1050.
https://doi.org/10.3390/en12061050

**AMA Style**

Codecasa L, Race S, d’Alessandro V, Gualandris D, Morelli A, Villa CM. TRAC: A Thermal Resistance Advanced Calculator for Electronic Packages. *Energies*. 2019; 12(6):1050.
https://doi.org/10.3390/en12061050

**Chicago/Turabian Style**

Codecasa, Lorenzo, Salvatore Race, Vincenzo d’Alessandro, Donata Gualandris, Arianna Morelli, and Claudio Maria Villa. 2019. "TRAC: A Thermal Resistance Advanced Calculator for Electronic Packages" *Energies* 12, no. 6: 1050.
https://doi.org/10.3390/en12061050