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

Putting It All Together

In the final step in our analysis, we convert matrix Eq. 2 back into two separate equations that describe the temperatures at the two junctions of our 2-by-2 example:

where θ_{J1_SELF}(t) and θ_{J2_SELF}(t) represent the two curves in Fig. 3 labeled “q₁ self avg” and “q₂ self avg,” respectively, and ψ_1-2(t) represents the “interact” curve.

Recall that these equations predict the temperature rise at the two locations of interest to us here, relative to some reference (in this case, it's going to be ambient). As I've already suggested, the idea is that any time either of the power sources change value, either by increasing or decreasing, you have to start adding in new step solutions on top of however many previous steps you've made — and this is crucial: Don't stop adding in the previous steps just because you've started a new step.

You can also see why you have to start a new time-step solution for every location, whenever any power source changes, because every location is affected by every power source. Eq. 6 is a mathematical statement of these ideas, as applied to Eq. 4, utilizing the concept of the unit-step function, and where the summation is taken over the entire set of times at which power changes at either source:

where u(x) = 0 for x < 0 and u(x) = 1 for x ≥ = 0.

The way I usually implement this in Excel is illustrated in Fig. 6, which includes liberal use of variable names to make the formulas as mnemonic as possible. The following tables, shown in Fig. 6, highlight major points:

Table 1 is simply the Foster model we derived earlier.

Table 2 is the input table of all the times at which either of the power sources change value, along with the power values. I've plotted those two power versus time functions on the chart in Fig. 6 on the secondary Y axis. Cells O7:Z8 immediately below Table 2 are the changes in power at each of those times, needed in subsequent formulas. (Note that there's no correlation between the waveforms in Fig. 6 and those in Fig. 5.)

Table 3 implements the unit-step function using an array-formula “if” statement (where time is the name of cell O11 and change_at is the name of the entire block O4:Z4):

{= if ( time > change_at, time - change_at, 0)}.

Table 4 uses the following two array formulas to compute the contribution of each individual time-step of Table 3 toward the total (compare these with Eqs. 4 and 5):

{=SUM((P7*q1_R_s+P8*interact_R_s)*(1-EXP(-P11/taus)))}

{=SUM((P7*interact_R_s+P8*q2_R_s)*(1-EXP(-P11/taus)))}.

The first of those two formulas actually appears in cell P13, and though it is an array formula, it returns just the single value seen in cell P13. The formula is then copied into every other cell of row 13 in Table 4, and similarly with the second formula and row 14. Conveniently, (1-EXP(-P11/taus)) contributes exactly zero for a zero argument, thus simplifying the use of the unit-step function, which stuffs a zero into every relative time value of Table 3 for times later than the time of interest).

Finally, Table 5 is the only true Excel “table,” via the formula: {=TABLE(,O11)}.

Table 5 calculates the final values of T_J1 and T_J2 at the selected times. To construct this table, you list in cells J14:J35 all the times for which you'd like results, and in cells K13 and L13, input the (nonarray) formulas:

=SUM(T1_terms)+ambient, and =SUM(T2_terms)+ambient.

Then select cells J13:L35, pick Data:Table from the menu, and for the Column Input Cell, pick O11. Note in the example that the yellow highlighted time entries happen to be some, but not all, of the exact times at which power levels changed in Table 2. I added some extra time values between some of the change_at points where I was interested in more detail. If you want results at more time points, simply lengthen Table 5.

Extending the Technique

Though this example was just a two-source problem, the technique can be extended to an arbitrary number of power sources. The main points to emphasize in adding more sources are these: You must have interaction curves for all pairs of source and location (so that the number of columns of Table 1 increases geometrically with the number of sources — though the principle of reciprocity may be able to reduce the number somewhat), and you need an entry in Table 2 each time any power source changes value. Finally, if you take advantage of Excel's naming capability, you can keep the formulas fairly mnemonic, and thus easy to maintain.

When reading part one of this article series, you may have scratched your head a bit and wondered why you'd even bother breaking down a sophisticated steady-state simulation into individual linear responses, when you could probably just as easily change your power inputs in the simulation and rerun it (especially if you've got a nonlinear simulation). Well, that was really just to set the stage for the thermal transient problem, because when you throw time into the mix, the possibilities are more than endless.

Not only can you change the power inputs at different locations, but you can turn them on and off in different sequences; they might be synchronized and periodic in one application, and totally random or asynchronous in another. A full-blown transient simulation, even in a completely linear system, can take hours to run. Having to rerun it every time a single power source changes amplitude or timing quickly becomes prohibitive.

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