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Research Papers: Conduction

# A Fixed-Angle Dynamic Heat Spreading Model for (An)Isotropic Rear-Cooled Substrates

[+] Author and Article Information
Bjorn Vermeersch1

Department of Electronics and Information Systems, Ghent University, Sint Pietersnieuwstraat 41, 9000 Gent, Belgiumbjorn.vermeersch@elis.ugent.be

Gilbert De Mey

Department of Electronics and Information Systems, Ghent University, Sint Pietersnieuwstraat 41, 9000 Gent, Belgiumdemey@elis.ugent.be

1

Corresponding author.

J. Heat Transfer 130(12), 121301 (Sep 23, 2008) (9 pages) doi:10.1115/1.2976557 History: Received October 02, 2007; Revised May 07, 2008; Published September 23, 2008

## Abstract

During a period of almost 40 years already, various fixed-angle heat spreading models have been developed in the literature. These models are commonly used by thermal engineers as approximations for the thermal steady-state resistance of a heat source on a rear-cooled substrate. In this paper, an extension of these models to dynamic (time-dependent) phenomena is proposed. The heat dissipated by a square source (side $a$) is assumed to spread out into the substrate (thickness $b$) under an angle $ϕ$. An analytical solution for the complex thermal impedance $Zth(jω)$ in phasor notation is derived. The obtained expression, in which $ϕ$ is used as a fitting parameter, is compared with accurate analytical results. A very good agreement is observed (average relative error less than 6%) for a wide range of the normalized thickness $λ=b/a$. A compact expression for the optimal heat spreading angle as a function of $λ$ is given. Also the temperature response to a heat power step is investigated, and a simple formula for the thermal rise time is provided. Finally, the model can be easily extended to anisotropic media, which often appear in electronic packaging applications. Overall the proposed model allows a thermal designer to make quick yet accurate estimations about the dynamic behavior of the device.

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## Figures

Figure 1

Fixed-angle heat spreading model for a square heat source on a rear-cooled substrate: (a) geometry; (b) distributed network of the pyramidal body (phasor notation)

Figure 2

Heat source sandwiched between two identical rear-cooled substrates for calculation of temperature distribution (cross section view)

Figure 11

Thermal resistance (as fraction of isotropic value) as function of anisotropy factor for various substrate thicknesses

Figure 12

Heat flow in thin substrates

Figure 13

Heat flow in thick substrates

Figure 3

Interpretation of the error function e(ϕ)

Figure 4

Average relative fitting error as function of heat spreading angle ϕ for various substrate thicknesseses

Figure 5

Optimum heat spreading angle and corresponding average relative fitting error as function of normalized substrate thickness

Figure 6

Normalized thermal impedance curves for various substrate thicknesses

Figure 7

Normalized response to a power step for various substrate thicknesses

Figure 8

Normalized rise time as function of normalized substrate thickness

Figure 9

Comparison between proposed analytical model and lumped network approximation: (a) thermal resistance; (b) thermal rise time

Figure 10

Heat sink/spreader operation and typical occurrence in electronic package

Figure 14

Thermal rise time (as fraction of isotropic value) as function of anisotropy factor for various substrate thicknesses

## Errata

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