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

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Linear superposition, as its name implies, is associated with linear systems. In your core undergraduate electrical circuits class, you learned that networks comprising only resistors, capacitors and inductors fall into this category. But an analogy can be made with thermal systems, which typically consist of only resistances and capacitances, so they, too, can look like linear systems. There are certainly situations where thermal resistances — and sometimes even capacitances — don't act very constant. However, the basic equations describing the behavior of elements are written as if they were constant, and this assumption provides a convenient starting point for thermal analysis.

Thermal linear superposition says that if you turn on each component in the system by itself, and measure its effect on all the other components, then when you turn them all on at the same time, you can simply add up the individual effects each had alone, and you'll get the correct overall effect.

Consider a simplified power-supply system as in Fig. 1, where we'll suppose that there are only three significant heat sources: two FETs (q₁ and q₂) and a coil (q₃). We're interested in five temperatures: the two FETs (T_J1 and T_J2), the case of the axial-leaded device (T_X), the ground pin on the small-outline IC (T_L1) and the board location midway between the FETs (T_B). This system may be expressed mathematically through the following matrix equation:

Eq. 1 is actually just a shorthand notation for five separate equations, relating the five temperatures of interest (a temperature vector consisting of ΔT_J1… ΔT_B) to the three identified heat sources (a power vector consisting of q₁, q₂ and q₃), through the array of 15 coefficient values in the middle (a theta matrix). If we were to write out these equations separately, the second equation, describing the junction temperature of FET#2, for instance, would be:

where ΔT_J2 is the rise in T_J2 for the given power vector and theta coefficients, Ψࡋ is the thermal interaction coefficient giving the temperature rise at FET#2 due to heat at FET#1 (q₁), θ_J2A is the thermal resistance from junction to ambient of FET#2, and Ψࡋ is the thermal interaction coefficient giving the temperature rise at FET#2 due to heat at the coil (q₃).

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