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

Transient-Response Curves

The transient-response curve mentioned previously simply refers to what you get if you're tracking the temperature at some point in your system, in response to a step of constant power applied to some (perhaps other) point in your system. Fig. 3 illustrates a set of transient-response curves for a two-source system.

Two of the curves are labeled “self-heating curves” and one is labeled “interact.” In a completely general way, there should be four curves (since there are four matrix elements for a 2-by-2 matrix). Although not mentioned in part one of the article series, when dealing with truly linear networks, the Reciprocity Theorem states (for our purposes) that the theta matrix, in theory, will be symmetric. That is, in our 2-by-2 matrix the two source-to-source interaction curves will be identical — true even when the self-heating curves themselves are not at all identical.

I've tested lots of real-life systems, and this theoretical reciprocity/symmetry is usually a good approximation for semiconductor devices. Obviously, if you have any doubts, try to get both curves on their own. But for this example, we will use a single interaction curve interchangeably between the two sources.

Characterizing the System

Part one of this article discussed how you could use either experiments or some sort of simulation of your steady-state self-heating and interaction responses to fill in the theta matrix. The same applies here. Thus, your first step is either to actually go down to the lab with your prototype application and do some live thermal transient measurements (an art in itself), or create a representative Cauer network model of your system and use it as a lab to run a set of experiments, which means you'll need a circuit simulator.

A third option is to use high-powered finite-element or computational fluid dynamics code to do a transient simulation of your system. However you go about it, the goal is to generate a set of thermal transient-response curves at each point of interest to unit-steps of constant power at each heat source. Reciprocity can save a lot of work, but when in doubt, measure your interaction curves both ways. Remember to normalize the curves so they are on a per-watt basis.

Once you have a complete set of transient-response curves, tabulate the data on a logarithmic time scale. In my lab and simulations, I typically pick a ratio, say ¹⁵√10, and then space my data points as close as I can to that ratio from 100 µs all the way out to 1000 sec. For the data in Fig. 3, that would give me 15 equally spaced points per decade times seven decades, which would equal 105 points total. (In my lab, I actually log data at a uniform 8-µs interval and then throw most of it away. In simulations, I can more easily solve for and save just the required points.)

Here is when Excel can really be put to use. To fit Foster network models to the data of Fig. 3, you probably don't need more than one time constant for each order of magnitude of time data available to you. (I've rarely seen data for which this doesn't provide a decent fit, though using more hardly costs any additional effort; unless you go overboard, Excel can easily handle more.)

With the curves in Fig. 3, that would mean choosing time constants at 0.0001 sec, 0.001 sec and all the way up to 100 sec. Fig. 4 shows how: list the time constants (taus) in a column (I3:I10) and, next to them, allow three more columns for the Foster resistances (J3:L10). Columns E through G are the computed fits, each column coming from its own specific version of Eq. 3. Cells E1, F1 and G1 are root-mean-squared errors (RMSE) for each column; E1, for instance, contains the formula:

{=SQRT(SUMSQ(E4:E100-B4:B100)/COUNT(E4:E100))}.

Now select Tools:Solver. For each curve in turn, point Set Target Cell: to its RMSE value, click Min, select the corresponding column of resistances for By Changing Cells: and hit Solve. Almost any starting guesses will do; the resistances shown in cells J3:L10 of Fig. 4 are the converged, least-squares best-fit values. You saw how well this worked in Fig. 3 if you noticed the curves superimposed on top of the raw data points: Those are the best fit results.

Single-Source Time Superposition

This next technique should just be a review of something you learned a long time ago — namely, how to apply time-domain superposition to the analysis of a time-varying heat source (or current source using the electrical analogy). If you understand how this works for a single heat source, it won't be such a stretch to see how it adapts to the multiple heat-source situation. Fig. 5 illustrates the basic stages:

1. Begin by “squaring up” your actual time-varying power input. The resolution of the squaring basically depends on how accurate you need to be at critical points. Basically, replace large regions of complicated stuff with a single rectangular power pulse that contains the same energy, anywhere you don't care about the temperature details within that region.

2. Convert those individual square power pulses into steps of increase or decrease as you move from one constant region to the next (including zero-power regions).

3. Each step-change of power turns into its own power-scaled version of the transient-response curve for the heat source. It starts when the change is applied and ends when the simulation ends.

4. Sum up these individual response curves algebraically (meaning that increases add and decreases subtract). Clearly, only pulses up through any particular moment of interest affect the cumulative temperature change; the farther along in time you go, the more steps you have to track.