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

The Theta Matrix

So, where do we get the values that go into the theta matrix? We're going to assume that there's no way you're going to find datasheet values for most of the numbers in the matrix. (Even if you had curves showing θJA versus copper spreader area and airflow for the FETs, and you believed they were a good fit to your board, all those interactions between devices are totally beyond anything you'll find from a component manufacturer.) If you're handy in the lab, this is the most direct approach to take:

Step 1: Power up FET#1 without putting power anywhere else, and measure all five temperatures of interest. Be sure that you also establish any other significant boundary conditions. It will not do to make a several measurements in still air with the box wide open, when you are ultimately trying to predict the performance of your system with a big fan pushing air through your box.

Step 2: Compute the five coefficients of the first column of the theta matrix from those measurements, namely:

Note that all five of these values are computed using the actual power you supplied to FET#1 during this characterization experiment, and they're also all based on whatever ambient happened to be in the lab at the time. Fig. 4 illustrates how easily all five of these calculations can be done using a single Excel array formula.

Step 3: Turn off FET#1 and turn on FET#2. Measure all five temperatures, and compute the second column of the theta matrix in a similar way.

Step 4: Turn off FET#2 and put some power in the coil. Calculate the third and final column of the theta matrix.

Once you've got the theta matrix, you can use this model to predict temperatures at all your locations of interest, at any other power distribution you choose: Just plug different values into column H, of the Excel model shown in Fig. 2, and new predictions appear in column A.

Contingencies

Of course, the real world isn't always so cooperative, and the experimental methodology I've outlined may cause you some trouble. So let's talk about some alternatives.

First, what if you can't put enough dc power into the coil to heat up anything? After all, these are dc measurements I'm talking about, and the coil's resistance is pretty low. Maybe you can't put significant power into the coil without running the power supply at frequency, under load. But obviously if you do that, you're also dissipating power in the FETs, and the premise that you're only powering up one component at a time falls apart.

Here's one choice. Temporarily replace the coil with something that can dissipate a lot of dc heat — maybe just a 1-W axial-leaded resistor. The board doesn't really care what the heat source at the coil's location looks like, especially if you can send the heat into the same footprint. Since we don't care about the coil temperature itself, it really doesn't have to be a coil that puts the heat into the board at the coil's location.

Second, this does not have to be done experimentally. It can be done just as easily, sometimes more easily, through a simulation. A thermal simulation solves some problems we've alluded to: It's really easy to turn on the heat sources independently, and you can put as much heat into a coil as you want. Just follow the method outlined previously to generate your theta matrix, using the simulator as your lab.

Third, you don't have to power up everything completely independently. All you need are linearly independent power vectors to stimulate your system. So, if you can at least partially shift the power distribution from one device to another, then you can modify the data-reduction technique to solve for the theta matrix coefficients as if they were the 15 unknowns in a set of 15 equations.

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© 2007 Penton Media, Inc.