Design for Thermal Issues

Unit 2: Sources of heat

In this Unit we are expanding on Unit 1’s consideration of the sources of heat, in particular about the way in which heat is generated within the circuit. You will recognise applications of the Second Law of Thermodynamics, and we will be introducing some ideas about modelling, although detailed discussion on that topic is in later Units. As before, we are looking at sources of heat from a holistic standpoint, as affecting almost all aspects of component, board and equipment.


Unit contents


Mechanisms of heat generation

Not all heating by electricity is bad, a by-product of the conversion of electrical energy into rotation, air movement or light generation. In fact, electrical heating is used in a number of general industrial situations such as the heating of food, as well as in electronics assembly for “melting and joining”. As an example at the materials level, induction heating is a key process in making the silicon ingots used for semiconductors. However, for the purposes of our module, we are only considering the ways in which unwanted heat is generated in conductors, semiconductors and dielectric materials. As you will discover, in some cases the heat is generated because materials are less than perfect, whilst in other cases the heat generated is a consequence of fundamental electronic and electromagnetic principles.

Conductors

When a current I flows through a material, its resistance will produce a voltage drop V, and energy will be dissipated in the form of heat. This is the phenomenon of ‘Joule heating’. Applying Ohm’s Law, the dissipation W is given by:

W = V \times I = (I \times R) \times I = I^2 R

Joule heating is very obvious in, for example, a resistor. However, it occurs to some extent in all materials which carry current, since all materials offer some resistance to current flow. Heat will be generated by the current flowing in:

We will be looking more at the semiconductor package in Units 10 and 12, and the heating associated with cabling is generally outside the scope of this module. However, bear in mind that heat dissipation within a cable conductor can be significant, and cables need to be ‘de-rated’, with a maximum loading that depends on the application conditions and the material used for insulation.

But the local temperature rise that occurs with a wire or cable applies in the case of the PCB at lower currents and with more significant effects.

Activity

Read this information on track resistance, and explore the commercial calculators referenced there.

Then read this paper on current-carrying capacity (PDF file, 333KB). How does the work on IPC-2152 liberate our calculations?

 

Regrettably, the work that Mike Jouppi started has stalled in recent years, and the IPC machine has yet to disgorge the final version of the new standard. Hopefully, when it does, it will be possible to use more accurate models of the thermal effects of ohmic heating to generate better estimates of temperature rise, allowing us to dispense with the earlier ‘rules of thumb’ that have proven to be too conservative for today’s applications.

The illustrations in Jouppi’s paper also indicate how heat spreads from a hot area – a smooth but not necessarily linear function – and show that thermal gradients exist across both conductors and board. It is these thermal gradients that produce stresses which may impact on the reliability of the assembly.

Not only do “imperfect” conductors generate heat when current is passed through them, but so may the interconnections on an assembly. A good solder joint may have a resistance in the 1–10mΩ region; the resistance exhibited by connectors and terminals may be even lower. However, not all joints are good, and terminals and connectors in particular may exhibit high resistance, caused by mechanical factors or changes in the surface.

Although this aspect is generally not modelled, because imperfections of this type are not intentionally present, the possibility of overheating from these sources should not be ignored by designer and assembler. So, for terminals and connectors, correct decisions have to be made about terminal rating, especially for “demountable” connections if they are intended for disconnection during operation, and attention to correct soldering practice is required in order to avoid high resistance joints.

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Resistors

In contrast to a conductor, a resistor intentionally impedes current flow, so some Joule heating is inevitable. In terms of the numbers used, the majority of resistors are surface-mount chip devices, using thick film (cermet) technology, and dissipate very small amounts of power. Their construction is shown at this link. Typically all but the smallest sizes will be coded with their resistance value, and normal practice is to mount them with the coding upwards to aid inspection. As the dissipative surface is directly under the glaze and coding, this means that the source of the heat generated will be on the top surface. From here it will be cooled both by convection and by conduction through the ceramic body and joints to the PCB.

Cermet chips are available for dissipations up to 1W; above this, surface-mounted parts use adaptations of other technologies. A leaded format may be preferable, because this gives a larger surface area, so more heat can be lost by convection, but a limited amount of conduction also takes place through the leads. Resistors are available in a number of different technologies, but most axial components are still tubes of ceramic with conductive end-caps and with a resistive material coating the outer surface (Figure 1).

Figure 1: Film resistor construction (carbon- or metal-film)

Film resistor construction (carbon- or metal-film)

 

Whether carbon film, metal film or metal oxide, such a resistor will be adjusted to value by trimming the film, often with a spiral cut. As with the thick film chip component, the current flow in the resistor will be distorted by the trim cut, with the result that the resistor will have a higher current density in some areas than in others. Higher current density equates to higher local power dissipation, and thus to ‘hot spots’. Whilst these areas are most vulnerable to change in value and to catastrophic breakdown, in practice there is sufficient conduction of heat through the substrate for this not to be a reliability issue except where components are subjected to high transient1 stresses.

1 Transient failures can happen; see Joachim von der Ohe’s paper Transient induced failure of miniature resistors (PDF file, 114KB).

 

For high values of dissipation, many resistor types are based on winding resistive wire on a ceramic former, usually with a heat-resistant coating of silicone or a ceramic ‘cement’. Intended for operation at high surface temperatures, these sometimes need to be spaced away from the PCB. In other styles, the resistive element is bonded to a metal plate to disperse the heat that is generated.

Figure 2: Different formats of power resistor

Different formats of power resistorDifferent formats of power resistorDifferent formats of power resistor

Source: Vishay

Some packages are similar in appearance to heat sinks (Unit 13), and have their resistive element located in a cylindrical cavity within a metal cooling fin structure.

A key item on the specification of a resistor will be its maximum dissipation, coupled with an operating ambient temperature. As the ambient temperature increases, so the allowable dissipation in the component reduces, usually in line with some model of the rate at which heat will leave the component, and is thus linked to a nominal maximum temperature for the resistive element.

Activity

Look at the power rating information on these data sheets for three different types of resistor (PDF files, total size 548KB):

Trimmable resistor chip

Heatsink-encased wirewound power resistor

Thick film power resistor

What conclusions can you draw about the limiting conditions of operation of the three types? And about the different specifications that are applied?

Compare your answer with our comments.

 

As your study of this module progresses, you will see that this type of specification is “full of holes”! Fortunately, most resistors operate well inside their safe limits, and high-wattage types are usually spaced sufficiently far from other components to avoid any problem with over-temperature.

We can understand intuitively that there will be a temperature differential between the heat-generating element itself and the surface of the resistor, and that the magnitude of this difference will depend on the materials involved, on the thermal effectiveness of their contact with each other, and on whether there are any voids in the assembly.

The temperature difference is also a function of time. If you haven’t already done so, we recommend you read this short paper, which makes it clear that the ability of a component to absorb energy depends on its thermal mass, and the ability of a component to absorb pulses of heat depends on their duration – the longer the pulse, the more chance heat has to travel into and along adjoining materials.

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Semiconductors

The dissipation in a semiconductor will be the sum of the dissipation in the different junctions – for integrated circuits, the sum of the dissipation in many junctions. But these junctions are not ohmic; that is the relationship between voltage and current is non-linear, and typically of the form shown in Figure 3. The dissipation at any point is given by the V×I product.

Figure 3: Forward and reverse conduction of a semiconductor junction

Forward and reverse conduction of a semiconductor junction

 

A complication with semiconductor junctions is that their characteristics change with temperature; both the voltage at which the device turns on and the slope of the V/I curve during the conduction phase, reduce with temperature. So a hotter device is easier to “turn on”, and exhibits a lower resistance. The latter mirrors the bulk resistance of silicon, which has a substantial negative temperature coefficient, in contrast to the high positive values exhibited by metals.

Diodes occur in many different forms, in terms both of circuit function and package format. At this link there is a good summary of the different types. Note the range in characteristics, and the explanations given for this, and in particular the very different characteristics of a Schottky diode, with its low forward voltage drop and very fast switching speed.

But where in the semiconductor are the sources of heat positioned? For this, we have to look at the way in which a semiconductor is made, and the mechanism by which it operates. Our description is for a bipolar transistor, but similar considerations apply to other device types.

Figure 4: Diagrammatic representation of an npn transistor, showing typical dimensions and doping densities

Diagrammatic representation of an npn transistor, showing typical dimensions and doping densities

 

As shown in Figure 4, the material of which a transistor is made is not homogenous; the silicon matrix may still be a single crystal, but the deliberate diffusion of impurity atoms has generated areas with different resistivities as well as some areas rich in electrons (n-type) and those rich in acceptor positions (p-type, or “holes”). The electrical resistivity of the material is a direct function of the concentration of impurity atoms1, so that the materials vary widely in resistivity. This is why, in this npn transistor, the silicon has been highly doped just below the collector junction.

1 For an explanation of this, go to this link.

The thinning of the slice that takes place immediately prior to dicing and assembly also plays a part in minimising the electrical and thermal resistance between collector and back surface. So, apart from ohmic heating from the current passing through the bulk silicon, most of the heat is actually dissipated at the junctions.

Figure 5 shows a schematic of an npn transistor under equilibrium conditions, with no external potential applied. As we have seen from Figure 4, the base is thin and lightly-doped; depletion regions are formed at the base-emitter and base-collector boundaries by diffusion of electrons from n-type material to p-type and of holes from p-type to n-type. This creates a potential barrier, and carrier flow stops, in the same way as with a junction diode.

Figure 5: Schematic of an npn transistor under equilibrium conditions

Schematic of an npn transistor under equilibrium conditions

 

However, when we provide bias as in Figure 6, so that the emitter-base junction is forward-biased, and the collector-base junction reverse-biased (the “common emitter” mode), this reduces the width of the emitter-base junction, and increases the width of the base-collector junction, compared with equilibrium values. Electrons are injected from emitter to base under this forward bias, and at the same time holes are injected from base to emitter, giving a net flow of current from base to emitter. Because the emitter region is more heavily-doped than the base, injected electrons diffuse to the base-collector junction and are swept across the high potential barrier to create a collector current.

Figure 6: Schematic of an npn transistor under bias

Schematic of an npn transistor under bias

 

Keeping the doping level low creates a thin depletion layer (of the order of 1µm), which enhances diffusion across it and reduces the chance of electrons recombining with holes, so that most of the electrons entering the base are swept across to the collector. This means that the collector current is large, while the base current is comparatively much smaller. From the characteristics of the two junctions (Figure 6), we see that the collector-base junction is also associated with a higher voltage differential, and therefore is the one that dissipates the bulk of the heat in a transistor.

As well as bipolar devices, which are most common in linear integrated circuits, an important type of device is the “field effect” transistor (FET), a unipolar device that relies on only one carrier type and is voltage-controlled, unlike the bipolar transistor, which is current-controlled. But for this module you don’t have to understand the detail of operation of any device – if you want to do so any electronic textbook will give you information. What is important is that you should appreciate the way in which, for all types of devices, the energy is dissipated in a relatively small region in the slice. The only reason that the whole of the die is heated is that heat is conducted through the silicon.

As an example, Figure 7 shows a view of an integrated circuit, which differentiates between the heat flux in different areas of the chip and the resulting temperature; the areas of high heat flux represent concentrations of power-dissipating elements. As you will see from the temperature contours, some lateral conduction of heat has taken place, so that the contours are merged. However, it must be borne in mind that certain areas of the die are becoming highly-stressed, and there may well be a thermal gradient within the die that is not apparent from the surface.

Figure 7: Heat flux and temperature distribution in an integrated circuit

Heat flux and temperature distribution in an integrated circuit

 

Whilst it would be good to spread out the high-dissipation elements over the whole chip, this is only practicable in special cases such as power devices; for digital circuits, in particular microprocessors, the trend is to reduce the distance between high-speed components, on the die as well as on the printed circuit board, and this compounds the power management problem.

We will be looking at the physically modelling of a semiconductor package in more detail in Unit 12. At this stage, you should be aware that there are a number of different active device constructions, especially power devices for control and RF applications. Examples are Schottky diodes, which have a metal-semiconductor junction and a lower forward voltage, and the gallium arsenide devices used for RF amplifiers. However, whichever types are being considered, they will all have specific areas within the die where the heat is dissipated, and from which heat is transferred by conduction to adjacent surfaces. A visual example is given at this link, which shows the uneven light distribution from an LED supplied with different currents, the distribution of current and heat mirroring the visual appearance.

The problem of equalising the current density as far as possible in order to minimise temperature rise, is one that has received most attention for power semiconductors. For example, typical practice with power devices is to distribute the current between a number of similar elements in parallel.

Activity

Skim read these two articles:

MOSFETs Basics

Application Training Guide: Device cross-section

There are a number of reasons for asking you to skim read these data sheets on power MOSFETs:

 

So far we have concentrated on heat that is dissipated during study state conditions, but much of the heat in an integrated circuit is dissipated during switching because, as you will have seen from the MOS material, there is no such thing as a perfect switch, and switches take time to change between states. This time equates to an opportunity to dissipate energy. It should be no surprise therefore that the dissipation in many types of devices depends on the switching speed. Which is why digital circuits are often designed to have a ‘sleep’ state, and why modern microprocessors have fast-increasing dissipation, as we saw in Unit 1.

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The conductor as fuse

Although not directly related to heating, a designer should be aware that, under extreme current density conditions, conductors may go open circuit. But not in every case is the mechanism the kind of thermal rupture that occurs in a cartridge fuse; exposed to a high current density, material can be physically moved by “electromigration”. This causes local thinning of the conductor and “hillocks” in other areas, eventually leading to open circuit conditions.

A function of current density and temperature, electromigration may be modeled by Black’s Equation:

t_{50}  = C \cdot J^{ - n} e^{\left( {E_a /kT} \right)}

where

t50 = the median lifetime of the population of lines subjected to electromigration
C = a constant based on metal line properties
J = the current density
n = an integer constant from 1 to 7 (many experts believe that n = 2)
T = temperature in K
k = the Boltzmann constant
Ea = 0.5–0.7 eV for pure aluminium

See this link for some pictures of electromigration, and follow this link for a fuller explanation of the driving forces. Note how the lifetime reduces with both current density and temperature, and that there is thus a “vicious circle”, whereby a thinning of a track creates a higher current density, which increases the temperature, and hence accelerates the growth of the defect.

Supplementary information

For an insight into electromigration, and the way in which surfaces can change under the influence of electrons, visit this link.

 

As Cadence say, electromigration “has been a persistent problem since the early days of IC manufacturing”, and electromigration is certainly a major issue at the chip level. Visit this link for an insight into the control of this problem as an inbuilt part of the design process.

For 30 years, electromigration has been known to be a way in which chip metallisation could fail, but the situation is becoming worse as devices get smaller. It can also be exacerbated by current crowding; an example of this with objects as thermally massive as flip-chip bumps is reported in Tu’s paper (PDF file, 128KB). The bump investigated is connected to the chip by metallisation with a small cross-sectional area; electromigration has led to void formation and eventual failure of the connection to the bump near the point of entry of the trace.

Most fuses are designed to rupture within a given envelope of over-current and time – the higher the over-current, the shorter is the time to rupture. However, as this is a thermal effect, you should not be surprised that fuses designed to fail at a relatively small percentage of over-current normally run quite warm. Depending on the fuse style, the dimensions of the fusible element, and the average current carried under operating conditions, the internal temperature of the fuse may be quite high. You should always assume that a fuse will be just as much a source of heat as a resistor.

If you have ever compared the fusing characteristic of a wire suspended “in free air” with that of PCB conductor of equivalent cross-sectional area (or if you have read the earlier material relating to IPC-2152) you will know that a copper trace can carry a current that is higher than one would predict from the free-air situation. This is explained by its higher surface area and the heat capacity of the board. However, though it might be tempting to save money by doing so, do not use your copper trace as a fuse! There are four very good reasons at this link why this is not good practice.

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High-frequency losses

Although in domestic use only for the past 25 years, microwave heating for cooking purposes is now widespread. But why does this heating take place? And why do dielectric losses occur in components such as capacitors? There are two helpful papers by Gavin Whittaker, an introduction at this link, and much more scientific explanation at this link, but a simplistic view is that losses occur when there is a phase difference between the applied field and the effect of that field on the material. Whilst the mechanisms of loss will depend on the particular situation, the amount of energy transformed into heat will be frequency-dependent, increasing with frequency.

Another effect that impacts on loss as frequency increases is the ‘skin effect’ in conductors. This is explained at a theoretical level in this article by Howard Johnston (PDF file, 79KB), which shows how magnetic fields cause the current to be squeezed against the surface of the conductor, decreasing the useful current-carrying cross-section and raising the effective resistance. But how much of a problem is this?

Activity

Take a look at this correspondence, and draw your conclusion as to how important skin effect will be in an application operating around 1GHz.

 

Now look at our comments.

Losses occur in all kinds of dielectric, both in the board and in capacitors. These loses are important in the context of modelling impedance, and in signal integrity, but the thermal magnitude is relatively small, given that the values of tan δ are typically in the low per cents; as with skin effect, this is something to be aware of, but will only need applying in the case of power components. If you have forgotten what tan δ means, there are details at this link.

Of course some components have deliberate high-frequency losses as part of the circuit function. Among these are the ferrite beads used for applications such as isolating supplies. As with dielectric losses, these are rarely a thermal problem.

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Electrolytic capacitors

All capacitors, but especially electrolytic types, will pass some leakage current. This is at a very low level (µA), so is not generally a source of heat. However, all capacitors have some losses, normally modelled as the ESR (Effective Series Resistance), as their impedance is not pure reactance, but has a resistive (that is ‘lossy’) component. The ESR is the sum of all resistive components within a capacitor. Expressed mathematically:

where f is the frequency in Hz and C the value of the capacitor in Hz. If we use typical values for dissipation factor, the values of ESR for different dielectrics can be seen in the following table. Note that the aluminium electrolytic capacitor is the lossiest of the group, but it is far and away the cheapest per µF.

Dielectric Dissipation factor ESR (1µF capacitor)
aluminium electrolytic (50V)
10% at 120Hz
132Ω
tantalum electrolytic (50V)
4% at 120Hz
53Ω
polyester
1% at 1kHz
1.6Ω
polypropylene
0.1% at 1kHz
0.16Ω

ESR is certainly something that designers of power supplies need to take into account when specifying their main smoothing capacitors. Placed immediately after the rectifier, these are subjected to high “ripple currents” either at the supply frequency or at double that frequency in the case of a full-bridge rectifier.

As with semiconductors, the characteristic is not linear, but I2R heating takes place and can be very significant. For this reason, electrolytic capacitors frequently have a corrugated exterior to maximise the surface area, and some styles have integral heat sinks. Manufacturers’ web sites are a good source of information on the likely heat dissipation of electrolytic capacitors.

Note that the total capacitor ripple current in a typical power supply is a combination of ripple currents at different frequencies, whose total rms value determines the heating of the capacitor. However, one cannot just add the squares of the ripple currents at the different frequencies, because the ESR drops as ripple frequency increases. The correct procedure is to scale any higher-frequency ripple current to 100Hz, and use the square of the scaled currents to determine the actual current load. Reputable manufacturers provide a well-defined relationship between ripple current loads, ambient temperature and life expectancy.

Supplementary information

Particularly with high rated power supplies, using appropriate circuitry, managing the heat dissipation, and choosing the right capacitor become crucial. You will find a great deal of published information on this, but we suggest reading these short papers.

The paper by Rifa (PDF file, 196KB) on their long-life electrolytic capacitors is particularly interesting in defining when end-of-life occurs from the three components of ESR, the resistance in the aluminium tabs and foil, the resistance of the oxide layer, and the resistance of the dielectric.

The CapSite article at this link contains some useful comparisons of the reliability of capacitor types, together with a formula for estimating electrolytic capacitor life.

Note the comment that “The most common failure in an aluminium electrolytic at least for through-hole aluminiums, is not loss of capacitance or leakage, but increase in ESR due to loss of water from the electrolyte. This is temperature dependent”.

The writer also comments that “The higher the operating temperature, the shorter their life, and running aluminium too hot seems to be a common design mistake. This can be caused by things like excessive ripple current, poor ventilation, too high a system ambient, and/or locating them too close to a hot power supply component”. You have been warned!

Finally, this paper by Lief Eliasson suggests ways in which a thermal model is useful in optimising capacitor selection.

 

Whilst the discussion above has centred on the problems on an aluminium electrolytic capacitor, the same issues are true of any capacitor with significant loss. Data for tantalum capacitors at this link (PDF file, 440KB) shows how the permissible ripple current reduces for a typical component at 125°C to only 40% of its value at room temperature.

Supplementary information

If you need a reminder about the construction of electrolytic capacitors, visit this link.

 

When designing, bear in mind that the characteristics of capacitors are temperature-related, and that the series resistance of a capacitor can increase significantly if the electrolyte “dries out” with extended life, especially at high temperature. Although the large electrolytics to which these comments primarily apply are generally part of the system assembly rather than board-mounted, the heat that they dissipate needs to be considered in thermal modelling, and if possible they should be placed away from major sources of heat.

ESR is not just limited to electrolytic types and is an important factor at high frequencies. However, typical values are of the order of 0.1Ω at 1GHz, so are not a thermal issue.

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Heat from the surroundings

A component generates heat, but it also receives heat from its immediate environment. This environment is influenced by other components on the module, the boards in the overall system, and the way that these are arranged.

Local sources

When components are packed closely together, conduction between components may lead to additional thermal gain. As a simple example of thermal interaction, take a single surface-mounted IC which reaches a final operating temperature of 40°C above ambient through self-heating in normal operation. Figure 8 shows the steady–state temperature contour profile obtained by computer simulation, caused by a single component on a circuit board. This is sometimes termed the component’s ‘thermal footprint’ or ‘thermal territory’.

Figure 8: Thermal footprint of a single IC

Figure 1: Thermal footprint caused by a single IC

The temperature contours shown are in °C rise above ambient

Now, if we bring another identical IC, and mount it close to the first, the heat conducted between the two components will cause the steady-state temperature of each to increase. Figure 9 is the simulated thermal footprint resulting from two components placed close together.

Figure 9: Thermal footprint caused by two identical ICs placed closely together

Figure 2: Thermal footprint caused by two identical ICs placed closely together

 

Not only does this change the shape of the contours, but the simulation now predicts that both components will suffer temperature rises of 50oC rather than 40oC – a 25% increase in the temperature rise of each. Bearing in mind that the temperatures shown are temperature rises above ambient, bringing the ICs close together may jeopardise reliable operation, or at least cause additional drift.

It is clearly important to know a component’s thermal territory in order to be able to assess how closely it can be placed to another.

Thermal interaction can also occur in double-sided assemblies. Boards are relatively thin, the two layers of components being typically 0.8–1.6mm apart, so components placed on opposite sides directly above one another are likely to interact thermally. The heat flow between the sides will be greatly increased if thermal vias are used, and this technique will be one of the cooling strategies considered later in the module.

Self Assessment Question

Compile a list of factors that would affect the amount of steady state thermal interaction by heat conduction?

Find the answer here

 

To minimise thermal interaction by conduction, we can alter the relative component positions and change the PCB layer structure. For example, putting extra copper ground planes in a PCB enables the heat to spread out more rapidly before it reaches the board surface, where it can escape to the ambient air. This results in lower component and board temperatures. More about this in later Units.

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The equipment environment

Although the previous section has featured integrated circuits, and these are indeed major contributors to heat flux on a board, much of the heat that affects electronic assemblies comes from the environment within the equipment. For example, it has already been remarked that electrolytic capacitors frequently fail to have a long life simply because they are placed in areas of the enclosure where they become overheated. So it is this aspect that will be the focus for most of the rest of this module. However, our enclosure is part of a wider world, and another important factor is the heat that comes from this external environment.

Suggested reading

It is very easy to sit in an air-conditioned office, and think of the world ranging from comfortable to slightly warm. The reality is different, as we explain at this link.

Whilst the real environment is enormously varied, in practice it is possible to define a restricted range of reproducible conditions that are representative of the broader spectrum, in terms both of temperature and other aspects of the environment. How this is done is explained at this link.

 

The second of those papers finished by looking at some of the tests that are applied to equipment in order to assure continued functionality for the intended life. This is an aspect of enclosure design that we will be bearing in mind throughout this module, and especially in Unit 16.

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Steady-state and transient conditions

At various points throughout this Unit we have given examples of where the thermal performance of a component or assembly depends on the length of its ‘dissipation experience’. Typically components, like people, can withstand a high stress for a short period, but only a lower level of continuing stress!

These differences are due to the time element that is associated with thermal diffusion. There will be more about this in Units 8 and 9 and in the practical simulations, but the time-related aspect of heat flow is very important when we come to considering the effects of temperature cycling or power cycling – as it takes time for a steady state to be achieved, we need to allow at least this dwell time if our temperature cycling regime is to exercise the assembly to its fullest extent.

The concept of ‘thermal capacitance’ is important here; an analogue of the electrical situation, the thermal capacitance represents the amount of energy that is needed to raise the temperature of a body by a given amount. Mathematically equivalent to the specific heat multiplied by the mass, the rate of increase of temperature depends on this ‘thermal mass’ and the rate of heat input.

Thermal capacitance is responsible for ‘thermal lag’, the phase shift in temperature that occurs between the source of heat and a point at some distance from the source. Only when all heat flows have been allowed sufficient time to stabilise and be an equilibrium, can a ‘steady-state’ set of measurements or simulations be taken.

In many cases, systems do not reach equilibrium, especially when cooling, because the temperature differences are reducing, and the rate of heat transfer becomes vanishingly small. However, when heating, there can be some ‘build up’ of heat, so that continued power cycling, for example, does not achieve what was intended, because the equipment under test never regains its initial cold state.

Most of the simulations we will be carrying out involve the calculation of steady-state conditions, as these are less computationally intensive. However, you should remain aware of the potential for the situation in the short term to be considerably different from the eventual equilibrium state.

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Resources for this Unit

Each of these lists is in the order in which the material is referenced in the Unit text. However, note that links to SAQ answers are not included!

Needed for activities

Recommended supplementary material

Optional links and supplementary information

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