Part Two: Linear Superposition Speeds Thermal Modeling

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

In a thermally linear system, superposition may be applied to predict the transient response of the system to step changes in input power. This article is the second in a two-part series.

Part one of this series, published in the January 2007 issue, discussed how to use linear superposition in the steady-state analysis of multiple heat-source systems. However, linear superposition also can be used in the time domain, which is very useful. This means if you have a thermally linear system, you don't have to be limited to steady-state temperature predictions. The rules remain pretty much the same: The individual temperature responses in your system have to be proportional to individual heat-source values and, except for those heat sources, all the other boundary conditions have to stay fixed over the time and space of interest.

The Concept

Consider the case of a multiple heat-source system that can be described in steady state by a matrix equation. In such a case, the thermal transient response of the system can be determined merely by turning each matrix element into its corresponding time-varying counterpart. For example, the matrix equation cited in part one described a power system model (refer to Fig. 1 in part one) with three heat sources (q₁, q ₂and q)₃ and five points of interest on the pc board for which temperature calculations were desired (T_J1, T_J2, T_X, T_L1 and T_B). These temperatures are calculated by the following matrix equation:

where the theta coefficients (θ_J1A and θ_J2A) represent “self-heating” terms, and the psi coefficients (ψ₁₂, ψ₁₃, etc.) represent “interaction” terms.

We can use this same power system model to illustrate how the matrix is modified to obtain the transient response of the system. But to make the calculations easier to follow, we'll simplify the power system model by assuming a two heat-source system and calculate just two temperatures on the pc board. Making those changes along with the switch to time-varying elements gives us this transient matrix:

where Δq_1-STEP and Δq_2-STEP represent step changes made to the two heat (or power) sources.

Now, each of the theta matrix elements is a complete thermal transient response curve. The main complication is that every time any heat source changes its value, then every other response must also make a simultaneous adjustment — a fair amount more work than if you're just tracking the temperature of a single device.

So the bookkeeping may get tedious, but the method is straightforward. We can calculate the values of the time-varying temperature vector of Eq. 2 using Excel. But to get to that point will require a whirlwind course reviewing several separate, but ultimately related, topics including Cauer networks, Foster ladder networks, Reciprocity Theorem and time-domain superposition.

Thermal Transient Networks

Fig. 1 illustrates a hypothetical two-input grounded-capacitor, sometimes known as Cauer, thermal network. Cauer networks are of interest because in the thermal world, they are the natural representation of the physics. Nodes store energy, represented by capacitors to ground, and heat flow between nodes is represented by resistors. Unfortunately, although these are easy to draw, and the individual resistor and capacitor values may be easy to calculate (at least if you thought about where the heat was flowing when you drew the network), they're a pain to analyze with inexpensive tools.

Consider instead the ladder network shown in Fig. 2. These networks, called Foster ladders, are easy to analyze and implement. In response to a sudden application of constant power (Δq_J), the response of the end node is merely the sum of a bunch of exponentially decaying terms, as in:

where the Ri terms represent the resistors in the Foster networks, and the τ _iterms represent the associated R-C constants. Using Excel array notation and the Insert:Name:Define menu feature, this can be expressed very elegantly. For example, defining:

power	$E$2
resistances	$J$4:$J$10
taus	$I$1:$I$10,

the temperature rise at the time specified in cell A4 is computed in Excel by the array formula:

{=power*SUM(resistances*(1-EXP(-A4/taus)))}.

The saving grace of linear thermal networks such as Fig. 1, is that there is generally a mathematically equivalent Foster solution for every node in the system. Further, you probably only care about results at the heat-input locations, and maybe another location here or there. I do need to emphasize that when you find the corresponding Foster ladder that goes with the heat input at TJ1, whether it's from a Cauer model simulation or perhaps directly from an experimental curve, it will have a bunch of nodes or “rungs” as in Fig. 2 that don't mean anything physically; but, it will have one node, the end node, that behaves exactly like T_J1 over the entire time domain.

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