Thermal Component models, part 3

Back ground article....: Thermal component models, part 2
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Introduction

Thermal component modelling is a difficult issue. It took me several years to come to reasonable good terms with some of its intricacy. In a way it is like mathematics, you can not understand it but you can get used to it. Having that experience I decided to go slow when writing these articles. The first article was about basic definitions and the second about a basic theory. A rough guess is that neither of those was very exiting for hands-on application engineers who have problems to solve. That background is however needed to fully understand the content in this article, which hopefully will more interesting.

The first subject is a model comparison. The ambition is not to be completely conclusive but rather to unveil some interesting tendencies. Complex logical models is the second subject. This item has been discussed for years and has also been worked on in applied research projects. A few lines of text can therefore not cover all that mass but it may point at some essentials. The third subject is model conversions.



Figure 1
The totally imaginary component used for the comparisons.

Comparison methodology used

It is quite difficult to make general and conclusive comparisons of thermal component models. Such projects need to span over a large variety of component types and boundary conditions. The method used must apart from that also be able to trap the consequences of the component-to-air heat flow error. This article takes a much simpler approach and uses one single component type. The result is therefore not conclusive but it does show some interesting tendencies.

Figure 1 shows a detailed computer model of a totally imaginary kind. It is essentially a chip surrounded by a body of a uniform material. The leads are simulated as thin strips of this material around the circumference and the PCB is simulated as an isothermal plate. A uniform heat transfer coefficient is applied to the top of the component. It should be noted that this arrangement overlooks all PCB interface problems.

The comparisons were done in 3 steps. The first step was to apply the boundary conditions needed to extract the data for the logical model examined. The second step was to perform finite element calculations covering a range of heat transfer coefficients. The third step was to apply these conditions to the logical model and to compare the results.

The heat transfer coefficient was varied from 5 – 30 W/m2K, which approximately covers the range from natural convection to forced convection on the 2 – 3 m/s level. The PCB temperature was fixed to 25 K above the ambient temperature. The thermal conductivity of the body material was selected so that the chip-to-pad resistance had typical values. The generated heat was adjusted to always result in a 25 K temperature difference between the chip and the PCB.

It should also be remarked that the quality measure used for the temperature prediction was a ratio based on the chip-to-pad temperature difference, (Tj-Tp). Form an applied point of view it could be argued that chip-to-ambient, (Tj-Ta), is better, in which case the temperature prediction errors found would be reduced with a factor 2.

Two component sizes were used, 20x20 mm and 60x60 mm. The idea behind that choice was to let the former represent a typical case and the latter an extreme case.




Figure 2
The 3-parameter model.

The 3-parameter model

Figure 2 shows this model. It should be reminded that R1 is an external resistance and that the model therefore only has 2 resistances in its basic version. The problem with this model is that R1 only operates on a part of the top surface. That part is difficult to determine experimentally and is therefore assumed to be 100%. This practice unfortunately introduces a prediction error.


Figure 3
Test results for various models.

Figure 3 shows the test results. The predictions are quite good for the small component but for the large one there is a 15% Tj error when the heat transfer coefficient is high. An error on that level would typically be regarded as being on the pain limit. It is therefore apparent that caution must be observed when applying the 3-parameter model to very large components. The thermal data used for the large component was R2=4.47 and R0=4.87 K/W.



Figure 4
JEDEC test data can be used in two different ways.

The JEDEC model

JEDEC has specified two cold wall measurement procedures, junction-to-case, Rjc, and junction-to-board, Rjb. As far as the author knows they have however not specified a thermal component model. The model that often is referred to as the JEDEC model is a star model composed of these two values. The junction-to-board thermal resistance, Rjb, is here replaced by the junction-to-pad resistance, Rjp, which in this context is regarded as the same thing.

It is well known that models based on these two values do not perform particularly well. This comparison confirms that observation, figure 3. It is less well known that the accuracy actually increases when the data is used in a 3-parameter model structure. The thermal data used for the large component was Rjc=0.528 and Rjp=9.33 K/W. An interesting observation when comparing with the values for the 3-parameter model above is that R0+R2=4.47+4.87=9.34 K/W. This agreement is natural since both data represents a junction-to-pad value even if the extraction methods are different.



Figure 5
The 4-parameter model is a complete model, which has two thermal resistances that interface with the ambient air.

The 4-parameter model

The 4-parameter model is essentially an extension of the 3-parameter model. It is here assumed that the area on which R1 operates is known and that the additional resistance, R3, handles the heat dissipation from the rest of the top. This model is difficult to extract experimentally but can easily be created from a detailed computer model. The process is very straightforward, figure 6. R0 and R2 are extracted as for the 3-parameter model. R0 and the sensitivity factor give R1. The heat transfer coefficient and R1 then determine the R1-area.


Figure 6
The area that R1 operates on can be extracted if the heat transfer coefficient is known.

It should be observed that R1 is a heat transfer coefficient dependent parameter. The R1-area is therefore strictly only given for each heat transfer coefficient. The variations in this respect are however often so small that a fixed value can be used.

The results for this model are very good indeed, figure 3, which strongly indicates that the background theory is correct. It should nevertheless be noted that the method used to determine R3 is approximate but that it can be improved by adding yet another parameter. Since the component-to-air heat prediction is quite good this model can also provide an accurate measure on the ratio between the component-to-air heat and the generated heat. For the small component it varied from 3% to 17% and for the large component from 20% to 84%. The thermal data for the large component was R2=4.47, R0=4.87 K/W and R1-area=62%.



Figure 7
The Y and Delta-models.
Y and Delta-models

It could be of interest to compare the 4-parameter model with other models that have the same parameter count. Figure 7 shows two alternatives. Tc is in both cases defined as the average temperature on the component top. Both models have 2 interface temperatures. The 3 thermal resistances can therefore be used to satisfy the characteristic equation and to adjust the component-to-air heat for 1 boundary condition.


Figure 8
Computed results for the Y and Delta-models on the 60x60 mm component. The 4-parameter model is shown as comparison.

Figure 8 shows the computed results for the 60x60 mm component. The 4-parameter model is included for comparison. Both models have a good performance. The prediction of the chip temperature is almost identical but the Delta-model does slightly better for the component-to-air heat. It should also be observed that a positive component-to-air heat error at PCB calculations contributes to a negative chip temperature error. Looking at the result in that perspective there is a small advantage for the Delta model. The comparison also shows that subdividing the convection surface, such as in the 4-parameter model, is an effective mean to reduce variations. More about this matter follows further down.

The model data used for the Y-model was R0=2.74, R2=6.15 and R3=-1.08 K/W. For the Delta model it was R0=-6.69, R2=2.64 and R3=1.19 K/W. Some of these values are negative which clearly shows that this is fully allowed in logical models.


Some additional comments

Even if this study not is conclusive it shows that simple thermal component models can be reasonably accurate over a large range of convection conditions. The idea that 2-resistor models perform poorly for large components is partly true but the critical size limit is probably as high as 60x60 mm.

The component-to-air heat for components of the size 20x20 mm is so low that simple 1-parameter calculations of the junction-to-pad type can be used for estimations.

It is always difficult to find good thermal component data. The backbone for both the 3 and the 4-parameter model is however Rjp which often is available.

Neither the 3-parameter nor the 4-parameter model has an accuracy window that includes both the convection and the heat sink case. An second set of parameters must therefore be used for the latter. The simplest way to do this is to replace R2 with a cold wall Rjc value.

The PCB interface problem will be brought up in a coming article.



Figure 9
The computer world opens up the possibility to make “non-measurable” temperature definitions.

Complex logical models

Leaving the experimental world and going into the computing world opens up new possibilities. More interface temperatures can be used and the air and the PCB interfaces can be excluded from the control volume, figure 9. The component can subsequently be subdivided into several boundary surface elements. The more element there are the better will be the final result. The characteristic equation has the same linear structure as for simple models but with the difference that there are more sensitivity factor terms. It can also be added that any interface temperature could be used as reference temperature.

The extraction procedure follows the same basic scheme as for simple models. Step one is to determine the characteristic equation for the detailed model. Step two is to create a logical network which replicates that equation. There are however a couple of additional difficulties. The first one has to do with the parameters in the characteristic equation. A logical model is composed of thermal resistances with fixed values. The parameters in its characteristic equation must therefore be constants. For a detailed model it is different. The terms in its characteristic equation are boundary condition dependent. A logical model is therefore exact only for one, or possibly a few, specific boundary condition set. The deviations found in figure 8 reflect this problem. There can in particular be large differences between the convection case and the heat sink case.

The primary measure in the development of a complex logical model is therefore to try to reduce the parameter variations in the characteristic equation. The mean for doing this is to adjust the boundary element sizes. This is easier said than done. The work needed to complete this process is substantial. It includes a cut and try method for the boundary element sizes and numerous determinations of the characteristic equation. The latter operation is also delicate. It is done by applying several boundary conditions to the model, which results in parameter sets that in themselves are boundary condition dependent! The procedure is so slow that it in practice must be assisted by various shortcuts and rules of thumb. The quality achieved in this phase is nevertheless crucial for the quality of the model.

If the surface adjustment phase is successful it results in a characteristic equation with approximately constant parameters. It is sufficient to use a simple star model to replicate that equation. There is however a complication, the component-to-air heat. As explained in the reference articles it is important to predict this heat reasonably well and that star models are problematic in this respect. The number of additional thermal resistances needed to adjust for it is a matter of ambition. The minimum number is nevertheless one.



Figure 10
An example of a complex logical model.

There are some general guidelines for complex model structures. Pure star models are excluded because of the resistance count but some kind of star model structure is needed to satisfy the characteristic equation. Based on the experience from simpler models there should also be a structure that resembles a 3-parameter model.

The model in figure 10 has those ingredients. Some of its thermal resistances may at extraction become negative. That is not a problem. The model can replicate the characteristic equation and in addition adjust the component-to-air heat for one boundary condition. If this is good enough can be questioned. The answer can only be found on the bases of a large number of application examples.

The are however many facets of the problem and those who have studied it in detail have found another solution.


Figure 11
The DELPHI compact model structure.

The DELPHI compact model, figure 11, is the result of many years of applied research and calibration against real components. It has an impressive performance and not the least software to support it. The associated extraction procedure seems to be different from the one outlined above even if there are similarities.

The main advantage with complex logical models is obvious: they have large accuracy windows and they can therefore be used for a large variety of boundary conditions. There are nevertheless also disadvantages. Accessibility, complexity and cost are evident points. The difficulty to use them for simple estimates is another. They are also often targeted for IC-components and may therefore be difficult to apply to other devices such as magnetic components or various modules.


Model conversions

Each thermal PCB program seems to use its own particular thermal component model. How to convert from one model to another is therefore an important issue. A general rule is of coarse that conversions should be made towards a lower thermal resistance count. It is nevertheless possible to go the other way but not without some kind of fill in. From this point of view it is therefore an advantage to have computer codes that can use simple models.

The best method for all conversions is to create an environment around the origin model and then determine the characteristic equation for the target model. It is sometimes possible to use a simple network reduction procedure but this method always involves an accuracy loss risk.

Ake Malhammar