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The Thermal Territory Concept |
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On Line Thermal territory for a component
Figure 1
A thermal territory is defined as the smallest rectangular
area of a PCB that a component needs for its cooling.
Introduction
The thermal territory concept is not new. An engineer
employed at Ericsson, Goran Flodman,introduced it in the
late 1980:s. It happened at a time when the heat density
suddenly took a big leap upwards and when multi-layer PCBs
started to appear. This was no coincidence. These events are
actually closely related and one could say that the birth
of the thermal territory concept also marks the breakpoint
between
component oriented and PCB oriented thermal design.
Although the method has been published a couple of times it
has only thrived within the Ericsson Company. The reasons for
this are an enigma to the author. The method is certainly not
the solution to all the problems in the world but at times
it is just the proper one to use.
The subject is too large to be covered in one single article.
The basics are given in this the first part. Some additional
problems and typical applications follow in the next
part.
Definition
A thermal territory is defined as the smallest rectangular
area of a PCB that a component needs for its cooling.
Figure 1 shows some examples. The method is to some extent
based on backward thinking. A thermal territory can actually
be looked at as small PCB but instead of calculating the
temperatures in that sub-PCB the method takes the reverse
way and calculates the size of the sub-PCB that is needed
to maintain a specified temperature.
The simple thermal resistance analogy in figure 1 may help
to clarify things. One part of the generated heat is
dissipated from the top of the component the other part is
dissipated from the PCB. In the normal case it is interesting
to determine the chip temperature, Tj, for which the thermal
resistances Rja, Rjb and Rba need to be known. In the
thermal territory case it is Rba that needs to be determined
and it is assumed that Tj, Rja and Rjb are known. Once Rba
has been determined it is a matter of calculating the
corresponding area on the PCB. A closer look at the problem
also discloses that Tb can replace Tj.
The size of a thermal territory can vary considerably. It can
some times be smaller than the component itself and other
times quite large. Each thermal territory also has the side
ratio. According to the definition this ratio must have the
value that minimises its surface. The side ratios for small
territories are consequently near the side ratio of the heat
source but for larger territories they successively approach
the value 1.0.
Figure 2
Thermal efficiency definition.
Thermal efficiency
2 shows a temperature profile example for a thermal
territory. The borders are by definition adiabatic. The
profile must therefore always have zero temperature gradients
at its extremes. The middle parts of the profile reflect the
character of the heat source, which in figure 2 is a
rectangle with uniform heat flux generation.
The thermal efficiency is defined as the average temperature
difference in the thermal territory over its maximum
temperature difference. As the name suggests it is a measure
of how well the territory is used for cooling.
Figure 3
The thermal territory size increases rapidly when the
heat dissipation is high. The thermal efficiency decreases
accordingly.
The temperatures in a thermal territory always decrease with
the distance from the heat source. For very extreme cases
they can in fact decrease so much that their difference
relative to the ambient air temperature becomes negligible.
It is obvious that adding some extra area in those cases not
will provide any additional cooling. In a plot of the thermal
territory size as a function of the heat dissipation, this
phenomenon typically appear as an asymptotic limit that the
heat dissipation not can exceed, figure 3. A plot of the
thermal efficiency also reflects the same phenomenon.
This effect is sometimes important in applications. It
simply states that there is an upper heat dissipation limit
that not can be exceeded without violating the temperature
criterion, no matter how large the thermal territory is made.
In an application it is of coarse highly undesirable to even
near that limit but it unfortunately happens.
Figure 4
Conventional methods can be used to calculate thermal
territories.
Conventional calculation methods
All thermal territory calculations are based on a
temperature criterion. It is important to note that this
criterion must be defined in such a way that the territory
can be integrated with a thermal resistance model of a
component. Various types of average definitions are
therefore inevitable. For the case in figure 4 this
criterion is the average temperature below the heat source.
A thermal territory can be looked at as a small PCB with a
heat source in its centre. It is therefore always possible
to use conventional temperature calculation methods to
determine its size. The fact that the size is the unknown
parameter and not the temperature does however make things
somewhat complicated. The basic calculation procedure must
as a consequence be embedded in two iteration loops,
figure 4. To use a numerical method of the finite element
type as central calculation engine in this process is
therefore always possible but it is both a complex and a
slow approach.
An alternative approach could be to use an
analytical temperature calculation method. These
methods are often both faster and simpler than the
numerical ones but they are also less flexible. They will
as a consequence function well for flat heat sources on
the surface but not for actual components.
Figure 5
The wing method.
Alternative calculation methods
There are some unconventional methods that can be used for
the sole purpose of calculating thermal territories. The
simplest one is the wing method, figure 5. It is an
approximate approach that only can be used for small and
medium sized territories but it has the advantage of being
simple. The basic idea is to treat the territory as if it
was composed of 5 rectangular structures, the footprint of
the heat source and 4 wings. The equations in figure 5 are
valid for a flat surface heat source but it is not difficult
to go one step further and integrate it with a component
model.
The surface below the component is treated as if it had been
isothermal. Its heat dissipation is consequently easy to
calculate. The wings handle the remaining part of the heat
flow and they can simply be treated as rectangular fins. The
method has a reasonable accuracy down to a thermal
efficiency of about 80%. Most components on a typical PCB
qualify for this criterion but there are quite a few
exceptions. The method can therefore never be a candidate
for large-scale use but it is attractive because of its
simplicity.
Figure 6
The circular approximation method.
Another approach is the circular approximation method,
figure 6. Its basic idea is to approximate a rectangular
heat source with a circular one. The approximation can be
made on the bases of equal area or equal circumference or
something in between. The calculation is performed step
by step on small annular areas. Outside the heat source it
is a simple matter of successively calculating the heat
dissipated by convection and the temperature decrease. The
calculation is stopped when all initial heat flow has been
dissipated and the corresponding radius defines the size
of the thermal territory. Inside the heat source the
procedure is basically the same except that the heat flux
gain from the source must be considered.
The procedure is basically numeric and is therefore flexible
enough to be easily integrated with a component model. There
are nevertheless some problems. The result does not reveal
anything about the side ratio and it is somewhat difficult
to insert a temperature criterion. The side ratio problem
can be reasonably well solved by the last equation in
figure 6. It is empirical but it has a surprisingly good
accuracy.
The temperature criterion is a slightly more difficult
issue. As explained above this criterion is always defined
as some type of average value below the heat source. The
calculation procedure for the circular approximation must
nevertheless always start with a temperature assumption for
its centre. The only way to ensure that the temperature
criterion is fulfilled is therefore to iterate through inner
heat source area several times and adjust the centre
temperature until the temperature criterion is met.
Figure 7
Comparisons of alternative calculation methods.
Figure 7 shows a comparison of the alternative calculation
methods. The wing method starts to deviate considerably
below 80% thermal efficiency. The circular approximation
methods do much better. Of the 3 alternatives it is the one
that uses a 50/50 average of equal area and equal
circumference that does best and it is surprisingly accurate
down to efficiencies of about 40%. This limit value is
definitely sufficiently low to cover the vast majority of
PCB applications.
Figure 8
The component integration can be handled with some quite
simple correlations.
Integrating a component
To integrate a component with a thermal territory does raise
some problems. The first is obviously that the component
must be sufficiently well characterised. This is a very
important concern but it is here assumed that sufficient
data is available to create a simple analogy such as shown
in figure 8. Given this, the temperature criterion and the
generated heat it is easy to determine the heat flow that
must be handled by the territory.
The second problem is that the PCB below a component only is
cooled from one side whereas it outside the component
normally is cooled from two sides. This is not problematic
if the basic calculation engine is numerical but it does
make it difficult to use analytical methods.
A third and a more aesthetic problem is how to define a
thermal territory that is smaller than the component itself.
It does not matter much from an application point view but
if not reasonably managed it can cause undesirable jumps and
break points in curves and diagrams. A simple solution is
to define the size by linear comparison of the heat that
could have been dissipated from a territory of the same size
as the component and the actual heat dissipation.
As a fourth problem it can be noted that a thermal territory
actually can be negative. All components with zero heat
dissipation belong to this category which also can include
components with large heat sinks. Most negative thermal
territories are so small that their impact can be neglected
but there are, as always, exceptions.
Summary
Although the definition of a thermal territory is quite
simple and there is no lack of possible calculation
methods, it is not easy to find the best one.
A thermal territory has a thermal efficiency, which is a
measure on how well the territory is used for cooling.
Integrating a component with a thermal territory raises
several problems that must be properly addressed.
Ake Malhammar
