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A Bessel function Solution |
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Forced Convection Cooled PCB on Line
Back ground article....:
A Fourier Series Solution
Figure 1
The 3 steps needed to model a rectangular heat source on a rectangular
plate.
Introduction
There are many ways to calculate the temperature distribution on
convection cooled plates. This article presents yet another one. It
is fundamentally based on a modified Bessel function solution to the
temperature field that surrounds a circular heat source on an
infinite surface. This fundamental solution is then step by step
adapted to describe the temperature filed on a rectangular plate
with rectangular heat sources.
The method is a bit particular in the sense that it does not appear
to be attractive at a first glance. Bessel functions are known to be
heavy to calculate and the circular approach may also seem awkward.
These disadvantages can however be effectively bypassed and make the
method fast, very fast.
The solution path
Figure 1 shows the basic path. It starts with a solution for a
circular heat source on an infinitely large plate. The next step is
to model a rectangular source as an array of circular sources and
the last step is to shape the rectangular plate with a mirror process.
Finding the temperature in a specific point on the plate is therefore
a matter of adding the contributions from all these sources.
Figure 2
Temperature field solution for a circular heat source on an
infinitely large plate cooled by convection. Double sided cooling
is assumed.
The temperature field solution for the first step is shown in
figure 2. It holds several Bessel functions and could therefore
appear to be both complicated and heavy to calculate. It should
nevertheless be noted that the functions C1 and C0 only depend on the
radius of the heat source, r0 . If the purpose is to calculate the
entire temperature field on the plate, they therefore only need to be
evaluated once. The functions that do need to be re-evaluated
repeatedly are consequently only K0(kr) for external points and
I0(kr) for internal points.
Figure 3
The circular heat source approximation error increases with the side
ratio but decreases rapidly with the distance from the source.
Subdivision
To analyse the best way to approximate a rectangular heat source
with an array of circular heat source is quite complicated. The
problem is strongly non-linear, which makes a complete analysis
almost impossible. In an applied approach one can therefore only
focus on the most important matters. The general tendency is that the
approximation error decreases with the size of the rectangle and
increases with its side ratio. For side ratios up to about 5 the
maximum error is found just outside the longest side of the rectangle
and it decreases rapidly for more distant points, figure 3.
One issue that always is brought up in this context is whether the
approximating circle should have the same area or the same
circumference as the rectangle. This analysis is based on the same
area. For typical PCB applications there are however indications that
a weighted average between these two alternatives is better. The '
proposed solution can therefore be improved but the potential gain is
not radical.
Figure 4
The circular heat source approximation error can be decreased by
subdivision.
Given these general tendencies it is evident that the approximation
error always can be decreased by subdivisions, figure 4. It is also
evident that the subdivision process should strive to produce sub
rectangles with side ratios as near 1.0 as possible. An empirical
correlation for the relative error, which accounts both for the side
ratio and the size effect, is included in figure 4. For overview
applications it is usually possible to accept a 10% error. The
corresponding maximum heat source size, for a typical multi-layer PCB
cooled by forced convection, can then be estimated to be of the order
20x30 mm. If the error level instead is set to 5% the maximum size
decreases to 7x10 mm. The number of subdivision needed to simulate a
typical component on a PCB is therefore in many cases zero and in any
case quite limited!
The mirroring process
The purpose of the mirroring process is to model a rectangular plate
with adiabatic sides, figure 1. It is a well know method but it is
only effective if the image count is reasonably low. The fact that
the temperature difference in a point decreases very rapidly with the
distance from its heat source is very helpful in this particular case.
The product of this distance, r, and the k-value is the critical
parameter. The value of this parameter, at which the temperature
contribution can be neglected, depends on the size of the heat source.
A reasonably good criteria for typical PCB cases kr>6.
A typical forced convection cooled multi-layer PCB has a k-value on
the 30 1/m level. Using the rule above the corresponding critical
r-value can be determined to 200 mm. A 200x200 mm PCB can therefore
be modelled with 1 - 2 mirror images at each side, which indeed is a
limited number. Other applications can be less favourable and if the
mirror count becomes too large an alternative calculation method
should be considered.
Another issue of some importance is whether the heat sources on the
mirror images need to be subdivided or not. This matter is not easy
to address. A superficial analysis indicates that subdivisions only
are needed for components that are near the plate edges and for
extreme cases of size or side ratio. A simple strategy would be to
always subdivide on the nearest layer of mirror images and only for
extreme cases on the other ones.
Calculation speed
The basic equation shown in figure 2 essentially holds two terms.
One that depends on the approximation radius and one that depends on
the distance to the centre of the heat source. The former only has to
be calculated once but the latter must be recalculated for every
calculation point. If the purpose is to map the temperature
distribution on a PCB this needs to be done on a grid of the order
20x20. If there further are 50 components that each need to be
mirrored 25 times, it would all in all imply evaluating a Bessel
function some 500 thousand times. That would be quite a heavy task
even for a modern computer.
A solution out of this dilemma is to do what engineers did before
the computer age, to use tables. To keep a table with some 1000
values is not a difficult task for a modern computer and such a
table can in addition easily be stored in the cash memory, which
further speeds up the process.
The proposed calculation procedure therefore consists of some
initial procedures that are both complicated and heavy but only has
to be done once. Once that basic structure is in place it is easy
to calculate a temperature contribution from a heat source. All that
is needed is to determine the distance to the its centre, interpolate
in a Bessel function in a table and multiply the result with a value
that is characteristic for the heat source. These operations are
all very simple and can consequently be done fast.
Figure 5
Comparison of a finite element calculation and a Bessel function
approximation.
Some application aspects
The procedure is ideal for creating temperature overviews on PCBs. A
particularly attractive possibility is to get a fast result when the
position of a component is changed. How the super position principle
can be used for this purpose is explained in the reference article.
The procedure is actually so fast that the major part of the waiting
time usually is consumed by the screen graphics.
There are nevertheless also limitations. The heat that is dissipated
directly from the components to the air is always a problematic issue.
Compensations, with various degrees of sophistication, can be used to
handle this impact but they will never be perfect. The method
therefore works best for cases in which the component-to-air
dissipation is much smaller than the part dissipated by the PCB.
Designs that have heat sinks or PCBs with a very low thermal '
conductivity are therefore better addressed with other methods.
Another problem is that the uniform heat source assumption does not
always comply well with the actual conditions. Devices that contact
the PCB around their circumference are best modelled as a combination
of positive and negative heat sources. Modelling that will however
always slow down the calculation speed.
Despite these difficulties it is often possible to attain results
that are surprisingly good. Figure 5 shows a comparison with a
finite element method. It is evident that the method not can replace
more accurate methods but it can indeed give a designer a good
overview of the impact of different layout alternatives.
It could be of some interest to compare the calculation speed of the
proposed method with that of the Fourier series method in the
reference article. For a 10x10 mm heat source on a 200x200 mm PCB
the Bessel function method uses the sum of some 25 terms whereas
the Fourier series method uses the sum of some 900 much more
complicated terms. The speed difference is coarsely a factor 40!
This advantage will however decline for larger heat sources and at
sizes of the order 60x60 mm it is reversed. The two methods are
therefore complementary. The Bessel function solution is favourable
for small heat sources, the Fourier series solution for large heat
sources. Both methods can of coarse also be combined.
Ake Malhammar