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

For instance, we've got five temperature locations to worry about, but each location depends on its own unique set of three theta-matrix coefficients. Consider the board temperature. You've made a series of measurements with three different power vectors. Those measurements, indicated by the hyphenated subscripts, represent three equations in the three unknown coefficients:

Or in matrix form:

Which is to say you can solve for the unknowns — the original coefficients — by inverting the matrix of power values:

Again, Excel makes this so easy that you almost don't even have to know from where the math comes. Fig. 5 shows how you might lay out this data-reduction problem. In this example, each row of data represents one power scenario, where we allow that the lab's ambient temperature drifts around between measurements. Here, you see two more of Excel's matrix-manipulating functions in action: MINVERSE, for matrix inverse, which does what you might expect, and TRANSPOSE, which turns a column vector into a row vector, or vice versa. If you compare the output results (10,10,5) of this data reduction example, with the fifth row of the original theta matrix in Fig. 2, you'll realize that I didn't choose my input values for the board and ambient temperatures by accident.

Speaking of matrix inversions, here's where the idea of linearly independent power vectors comes in. If the power vectors aren't linearly independent, then the power matrix will be singular and the inversion process will fail. If you're not sure you've created an obviously independent set of power vectors (and maybe even if you are), it's nice to make more measurements than you need — in effect, get more equations than you have unknowns for. You then use a different Excel approach to extract your model. Fig. 6 shows how to manage six power-vector scenarios, each with its own board and ambient temperature measurement, and use Excel's LINEST function to get the three coefficients we're after. LINEST performs a least-squares linear regression on a set of multiple, independent linear variables, as in y=m1x1 + m2x2 + m3x3 + b.

A couple of points regarding LINEST. First, observe that the third argument on the formula bar is zero; that forces LINEST to use a zero-intercept. After all, physically speaking, we know that if all power sources are turned off, all temperature rises must be zero. This intercept would have appeared in cell D10 in this layout, but as you can see, a zero was returned. LINEST also returns statistics on the coefficients (and the intercept, if used), which appear directly below each value (in row 11 of this example). Yet one additional statistic of interest (marked in red) is returned in cell A12, the infamous “r-squared” correlation coefficient; if the fit was perfect, this value would be exactly unity. You can see that my experimental data was pretty linear in this example. Finally, for reasons that remain obscure, LINEST returns the coefficients in the reverse order with respect to the input data. In other words, as you see in Fig. 6, q1, q2 and q3 appear from left to right in the input table (cells B2:D7), yet the three coefficients appear from right to left (as shown in cells A10:C10).

Note that the simultaneous power approaches build up the theta matrix row by row, rather than column by column as in the independent power method.

What if Nonlinearities Are Big?

If your system has significant nonlinearities (which could show up, for instance, as bad statistics in the LINEST analysis), most of the preceding techniques can still be applied, with a small modification. What we do is linearize the system behavior around a chosen operating point. Here's how:

First, choose a nominal operating point you think is close to where you're going to want to create predictions. Test, or simulate, your system at that point, and record all the power levels and temperature values you used to create it.

Now, systematically perturb your power vector from that operating point. For example, take each heat source in turn, and bump its value up (or down) a measurable amount from the nominal point, holding all the others constant. For each variation, record power levels and resulting temperatures. Your theta matrix will now be a set of coefficients that predict temperatures as small perturbations from the nominal values. In other words:

then the linearized matrix equation becomes:

Eq. 5 can be thought of as a special case of Eq. 16, where the nominal power dissipations are identically zero and the nominal temperatures are ambient. The practical difference between a linearized nonlinear system and the ideal, perfectly linear system is the addition of one more measurement. If the system is nonlinear, the farther you move the actual operating point from the nominal operating point, the less accurate the theta matrix becomes.

With modest effort, you can construct a linear superposition model of your system before you know your final power distribution, and without later rerunning experiments or detailed simulations, you can predict critical system temperatures over a range of actual operating points.

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© 2007 Penton Media, Inc.