Part One: Linear Superposition Speeds Thermal Modeling

Jan 1, 2007 12:00 PM
By Roger Stout, Senior Research Scientist, ON Semiconductor, Technology Development, Advanced Packag

There are a couple of important points to note. First, Eq. 2 is linear as long as the three coefficients multiplying the power dissipations are constant. This implies that if every heat source doubles, the temperature rise will double. Second, Eq. 2 is a direct expression of the idea of linear superposition. It says the temperature rise at junction 2 is a linear combination of three individual terms, each of which is the temperature rise that junction 2 would experience if that particular heat source were the only heat source.

From Eq. 2, you can also see that the effective θJA for device 2 isn't defined by its own properties alone. Note that we've used the term θJ2A as the coefficient for the power dissipated by device 2, because if device 2 was the only heat source present, we'd have the basic definition of its θJA expressed in the simpler equation:

If you solve Eq. 2 for the effective θJA as experienced by device 2 in the complete interacting system, you find:

Thus, what device 2 sees in isolation, θJ2A, is not necessarily anything like what it sees when other components are present and contributing power. Obviously, the magnitudes of the interaction terms (Ψ, Ψ) figure prominently. More significantly, observe that as device 2 power approaches zero, there is no upper limit for the effective value of its thermal resistance. This is why some component manufacturers are smart enough to refuse to put θJA in their absolute maximum ratings tables.

In Eq. 1, all the temperature vector elements are indicated as ΔTs, because in the simplest thermal systems there is a common thermal ambient (TA) to which all temperatures can be referenced. (All heat is flowing to the same ultimate thermal ground.) However, if you want to explicitly refer to ambient, you could have written Eq. 1 like this:

Spreadsheet Math

Let's jump right in to show how easy it is to code up a set of linear equations in Microsoft Excel. We'll use the Eq. 5 version of our model as the example, because this provides a basis for thermal modeling for cases where there's a whole set of reference temperatures rather than just a single value. Fig. 2 shows how you might set up your cell definitions in an Excel worksheet.

Given specific values for the theta matrix (cells D1:F5), power vector (cells H1:H3) and ambient (cells K1:K5), the values in column A arise from a simple formula typed into Excel, which appears in all five of the temperature cells (A1:A5), as shown in Fig. 3.

The squiggly braces around the formula up in the formula bar are not typed in; Excel inserts them automatically when you use Ctrl-Shift-Enter to input the formula after you type it in (with all five cells selected at once). They indicate that the formula is, in Excel terms, an array formula; correspondingly, the formula refers to entire ranges (arrays) of cells as its arguments. The MMULT, for matrix multiply, function is built into Excel and performs the magic of doing exactly what we've expressed in Eq. 5, all at once. So you don't have to code five separate equations as you might have thought from Eq. 2, and it should be clear that as far as Excel goes, you could just as easily have 12 heat sources and 17 temperature locations; you'd still have a single array formula, and you'd create it in one easy step.

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© 2007 Penton Media, Inc.