Transient 2-D Solutions for a Convection Cooled Plate

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Introduction

Any thermal designer is sooner or later faced with a transient cooling problem. Estimating the heat up time for a PCB is a typical example. Problems involving power-switching components are also common. There is no lack of accurate calculation tools for these purposes. Their common drawback however, is that they are both complicated and slow. Faced with a transient problem it is therefore often advantageous to start with a simple but approximate method.

This article is about a group of analytical transient procedures for convection cooled plates. All the members are exact solutions to well defined problems. It is therefore the complexity of real world problems not defaults in the calculation procedures that make them approximate. The solutions may at a first glance seem complicated but they are actually both simple to program and relatively fast.


Figure 1
Solution for a sudden step in heat dissipation at time zero.

Step change

Figure 1 shows the solution for a sudden step in heat dissipation at time zero. As all other members of the group it is based on a double Fourier series. There are both problems and possibilities with this type of solution. The article: A Fourier Series Solution for the Temperature Distribution on Convection Cooled Plates with Discrete Heat Sources, covers several of them more in depth.


Figure 2
Typical result for the step solution.

Figure 2 shows a typical result. The curve resembles those for simple RC-circuits but it is a far more complex function. It is nevertheless sufficiently similar to motivate the same definition of the time constant: the time elapsed until the temperature has reach 63.2 %, (1-exp(-1) ), of its final value.

The discussion that follows is best understood by opening the referenced calculator. The time constant, "Tconst=", is shown in the lower part of the image and is easy to check. Clicking on a heat source activates it. Changing the input parameters is a simple way to discover various impacts. The effects of the thermal properties are rather obvious. Other tendencies are more difficult to explain, such as why the time constant varies with the position of the heat source. The phenomenon seems to be related with the temperature gradients. One explanation could be that dull gradients make the heat travel longer paths and thereby causes longer time delays. This observation is anyway well in line with the approach taken in the article Simplified transient model for IC packages, where one important parameter is the distance from the heat source at which the temperature difference has decreased to half of its maximum value.

It can also be noted that the time constant increases with the size of the source. If the source is made as large as the plate it should equal the time constant for the entire plate. This value can alternatively be found by using the equation m*cp/(h*2*B*H).

If the plate is a PCB there is of coarse a complication, the components. One way to approximately account for them is to increase the density and the specific heat of the plate but not the thermal conductivity. Components also tend to flatten the tops of the temperature profile. To base the time constant on the average, not the maximum, temperature below a source could therefore possibly be an improvement.

The transient response on the PCB level is sometimes of importance but in most cases it is the responses inside the components that are of interest. The time constants on that level are always longer than on the PCB. The referenced article above presents a method for dealing with this problem.


Figure 3
Solution changes needed for simulating a rectangular wave function.

Rectangular wave

Once the solution for a step change is known, the superposition principle can be used to generate solutions for almost any pattern of abrupt changes. A single rectangular pulse can for example be generated by adding a negative step after time zero. A rectangular wave function can be generated as a sum of a large number of positive and negative steps. Figure 3 shows the solution changes needed to simulate that case.


Figure 4
Typical rectangular wave response below a heat source and at some distance.

Figure 4 displays a typical result. A characteristic for rectangular wave responses is that the cooling cycle mirrors the heating cycle. Any periodic wave function also has the tendency to converge towards a sin-function at large distances from the source. This is because the higher frequencies in the spectrum always diffuse faster than lower frequencies.

A frequent problem is to estimate the impact of components that switch on and off. If the frequency is very high or very low it is evident that it can be treated as a static case. The only difference is that the average heat dissipation should be used in the former case and the maximum heat dissipation in the latter case. It is therefore essential to be able to determine the critical frequencies beyond which a static approach can be used.

In this context it could also be convenient to discuss the difference between maximum feed power and heat dissipation. The former needs to be provided even if it only lasts a millisecond. The latter has a time constant on the minute level, which completely eliminates all millisecond variations. The critical heat dissipation for a PCB is therefore not always the maximum feed power, (there might in addition also be a significant power in the output signals). Confusions on this point are common. It is good thermal design practice to always give it a thought.

The referenced calculator is not intended as a tool for solving all transient problems. It is merely a demonstrator of the fact that analytical solutions can be useful for surfacing general tendencies and identifying critical parameters. The calculator displays a parameter named "(max dT)/(static dT=)" in the lower part of the image. This parameter is defined as the ratio of the maximum temperature difference and the static temperature difference at maximum heat dissipation. Thus, 100% signifies changes so slow that there is no dynamic impact and 50% signifies changes so rapid that all dynamic parts are diffused. It is interesting to study the performance of this ratio. The main parameters are naturally the cycle time and the thermal properties but the distance from the heat source is also important. This fact is essential because it is sometimes crucial to respect the critical distance between oscillating power components and more temperature sensitive logical components.


Figure 5
Solution changes needed for simulating a sin-wave function.

Sine wave

There is also a sin-wave solution, figure 5. This solution makes it possible to simulate any periodic function by superimposing sin-waves of different frequencies. It should be observed that the source in this case oscillates between positive and negative values. The solution must therefore always be combined with a static solution to make any applied sense. A Mathcad-file and a corresponding PDF file for the solutions presented can be downloaded here.


Figure 6
Typical sin-wave response below a heat source and at some distance.

Figure 6 shows an example. This case is convenient for studying the time delay effect. The heat dissipation in the example peeks at 300 seconds. The time delay below the heat source is therefore around 50 seconds and at some distance more than 100 seconds.

Conclusions

There are analytical solutions to the transient 2-D problem. They are simple to program and they can expose all major tendencies. They are however not intricate enough to deal with the entire complexity on a PCB but they can be very helpful in finding solutions to those problems.

The superposition principle makes it possible to simulate almost any periodic or non-periodic heat dissipation variation.