**AS PROMISED IN** **MY** editorial of the 2011 Fall issue, I will devote this issue’s column on the topic of heat spreading. The reason is that I, while writing a white paper on basic thermal management for LED applications, found some unexpected facts pointing at problems of interpretation of heat spreading data for dual layers when using the often-used heat spreading equations.

In this magazine heat spreading has been a frequently discussed subject, the references [1-5] testify to this assertion. Despite this, I think it worthwhile to add some more words. Facts and fairy tales on heat spreading are all over the place and many contradictory conclusions can be found in literature. This does not mean that some conclusions are wrong, it means that some conclusions have only limited validity and cannot be extended to other boundary conditions, or dimensions, or physical properties. The real problem is that these limits are rarely explicitly mentioned and in some cases the data are presented in the form of correlations, introducing the danger that the original data are gone forever and only the dimensionless form survives. As an example, take my own article [4] titled “Heat Spreading, Not a Trivial Problem.” Therein, an example was given extending the Song, Lee and Au (SLA) equations to a two-layer problem of an LED on a submount fixed to a heat spreader/sink. While constraints were given (such as h/k>10 m^{-1}) it was also stated that this rarely occurs in practice. This conclusion was right for the category discussed, but not for other categories such as a common one I encountered doing some work for a company that sells metal-core PCBs. I was asked to write a white paper for the APEX/IPC conference in Las Vegas, on the subject of basic thermal management of LEDs on MC-PCBs.

Most of you probably know what an MC-PCB basically consists of: a thin dielectric (say 0.1 mm) on top of a metal core (say 1.6 mm). The manufacturer was wondering which requirement from a thermal point of view his MC-PCB should have, and he was especially wondering if the leading manufacturers were right in stating that a higher thermal conductivity of the board would improve its thermal performance. Well, I doubted this statement from the very beginning, maybe fuelled by my bias towards the claims of many vendors. One of the excesses of capitalism is that plain lies are taken for granted. Applying a simple 1D series resistance network, it becomes immediately obvious that of all elements in the chain (LED, MC-PCB, TIM, heat sink, convection), the MC-PCB is the last one to attack, unless you are dealing with top-of-the-bill (and hence high-power) LEDs, and liquid cooling. Such a 1D approach is commonly used to address first-order LED thermal management but the question is if heat spreading effects are going to change the conclusions based on such an approach.

We are dealing here with a two-layer problem, hence, I initially thought I could simply fall back on the approach sketched in [4] to address the heat spreading issues. However, I soon realized this was a dead end. First of all, the conditions were not met to warrant their use. Second, the basic idea behind using the extended SLA equations [6] is that the second layer acts as an effective heat transfer coefficient boundary condition for the first layer, the implicit assumption being that the first layer acts as a heat source for the second layer. For the case submount/spreader as discussed in [4] we may indeed assume that the submount area acts as a heat source. The SLA equations calculate a total R_{th} based on two parts: a spreading resistance and a convective resistance, where the convective area is the whole area. However, this is not the right approach for the case at hand. Consider a very thin layer with low thermal conductivity, a small source on top of it and convection at the bottom. Obviously, there is almost no heat spreading, and the area that dissipates heat into the air is not the layer area but rather the source area. In order to apply the SLA double-layer equation this latter area is needed but this one is not calculated. By the way, the same is true for the very handy tool available at the website of the University of Waterloo [7]. When dealing with only one layer, as was the intention of the SLA equations, it should be realized that these calculate a thermal spreading resistance using the total area. While this is apparently wrong in the case of the thin dielectric layer, it does not matter for the calculation because the calculated total R_{th} is OK. However, it does matter for a two-layer case because then you need the area explicitly.

Hence, the problem boils down to the following. While the SLA equations and the UoW tool can be used for two-layer heat spreading, we cannot get a feeling for the magnitude of the individual contributions. From a designer’s point of view we need this split, because she wants to get a feeling for all elements in the chain in order to make the right design decisions. The approach chosen is the following: for the dielectric thermal resistance, a spreading resistance is used based on some spreading angle rule, from which also the source area follows that is subsequently used to calculate the heat spreading in the metal using the SLA equations. In practice, to get a first-order estimate, it is sufficient to take for the dielectric the LED area and neglect the spreading in the metal layer. Optimal spreading angle and other accuracy issues will be covered in a paper to be published soon [8].

Finally, please keep in mind that, looking at the wide range of boundary conditions, physical properties, and dimensions that occur in practice, no general rule of thumb can be given, simply because physics does not allow us to separate the convection and conduction parts.

Here is my final advice: for early design phases, use a spread sheet approach, to get a first (and usually pretty accurate) feeling. For the final design phase, use dedicated tools such as CFD codes in conduction-only mode.

**References**

[1] Lee S., “Calculating Spreading Resistance in Heat Sinks,” *Electronics Cooling,* January 1998.

[2] Guenin B., “The 45 Heat Spreading Angle — An Urban Legend?,” *Electronics Cooling*, November 2003.

[3] Simons R., “Simple Formulas for Estimating Thermal Spreading Resistance,” *Electronics Cooling*, May 2004.

[4] Lasance C., “Heat Spreading: Not a Trivial Problem,” *Electronics Cooling*, May 2008.

[5] Guenin B., “Heat Spreading Calculations Using Thermal Circuit Elements”, *Electronics Cooling*, August 2008.

[6] Song, S., Lee, S. and Au, V., “Closed-Form Equations for Thermal Constriction/Spreading Resistances with Variable Resistance Boundary Condition,” IEPS Conference, pp. 111-121, 1994.

[7] http://mhtlab.uwaterloo.ca/RScalculators.html

[8] Lasance C., “Two-Layer Heat Spreading Approximations Revisited,” to be presented at SEMI-THERM, San Jose, March 2012.