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A thermal management example (part one)

The purpose of this page is to illustrate the process of making basic thermal analysis calculations common in electronics applications, and finding info in product literature that's relevant to that process.

For sake of example, we'll assume that we've got an IRL3713 transistor in a TO-220 package attached to an HS278-ND heat sink via a BER183-ND thermal interface pad, looking similar to the model at right.

In the context of electronics design, one usually has an electronic device, some sort of environment in which that device will live, and some sort of design constraint.  One or more of those is often unknown, and the point of the thermal analysis is to either fill in the blanks or inform a related decision.

In any case, a typical practice is to model the physical assembly as an electrical circuit, and assume the system is in a steady-state thermal condition–in other words, that all temperatures in the system have stabilized and aren't changing appreciably.  While that assumption is not generally accurate, it simplifies things and tends to produce results that err on the side of caution.

The process can be summarized briefly as follows, with an exploded cross-section view of the system at right shown alongside the corresponding thermal model circuit.

1. Draw nodes & resistances: Each component’s active region, each involved mechanical surface, and the environment into which waste heat will ultimately be dumped are designated as nodes in an electrical circuit. Resistances are drawn between the nodes, corresponding to their physical connections.
2. Pick a ground reference: Decide what point in the circuit/system you want the results to be referenced to.  Absolute zero is used as a reference in the diagram at right (note the added node in the circuit) but it is also common to use the ambient environment as a reference point–more on this near the end of the page.
3. Add voltage sources: Draw voltage sources between nodes with defined temperature differences. The difference between ambient temperature and absolute zero in this example is considered "defined" as whatever the room's thermostat is set at.
4. Add current sources: Thermal power dissipation is modeled as a current source, connecting between ground and the node in the circuit that represents the structure that's doing the dissipating–usually a semiconductor junction.
5. Set component values: Values are assigned to each component using measured data, design constraints, or manufacturer's characterizations, adapted as needed to represent actual application conditions.
6. Solve for unknowns: The circuit thus developed is analyzed, and the “voltage” at each node corresponds to the predicted temperature at that node.

Drawing nodes & resistances: This objective here is to identify physical points in the system where the temperature is a matter of interest, either directly or indirectly.  In a typical case involving a semiconductor device, the temperature of the actual, functional portion of the device (commonly referred to as the "junction") buried inside the packaging is the key item of interest–it's this temperature that influences the operational behaviors of the device.  This temperature is usually different than that of the outside of the package, which different from the surface of a heat sink that it's attached to, and different still from the temperature of the surrounding atmosphere.

The three-resistance model shown is used quite generally for many through-hole and chassis mounted components that are used with added heat sinks, power transistors and solid state relays being two examples.  The evaluation process for surface-mounted components often uses a simpler model.

Setting component values: This is the part of the process that can seem like a dark art and, truth be told, often does involve some educated guesstimation.  There are three resistances in the model representing three different items: the electronic device in question, the heat sink it's getting attached to, and whatever sort of thermal interface material one chooses.  Info on each thermal resistance is generally found in the corresponding datasheet, in many/most cases, represented with the symbol Rϴ  to indicate a thermal resistance, with additional characters in the subscript indicating what specific thermal resistance is being referenced. Components usually used with a separate heat sink will commonly list a figure for RϴJC, representing the junction-to-case thermal resistance; the thermal resistance between the device's internal active region and the point on the outside of its package to which a heat sink is designed to be attached.

Another common expression is RϴJA, which refers to a junction-to-ambient thermal resistance, usually applied to components not used with an added heat sink that are simply hanging out in free air, and describing an all-in one thermal resistance between a device's innards and the outside world.  These figures should be regarded with strong skepticism, for a few reasons:

• In the context of surface-mounted components, the specification is frequently based on a measurement made with a device mounted on a board of a specific size, specific component footprint, and specific orientation during testing, which actual application conditions are unlikely to replicate.
• The natural-convection heat transfer process that is wrapped up in the number given tends to be quite sensitive to application variables.

A less-common but helpful piece of information is a junction-to-solder-point resistance, often represented as RϴJS. This is analogous to an RϴJC figure, describing thermal resistance between a device's functional innards and the point where it gets soldered to a circuit board, which functions as a heat sink.  Because both RϴJS and RϴJC figures are based on factors that are relatively predictable and consistent (the materials from which a device is made and how it's put together) these figures are the most useful for comparing different devices, and are most likely to give calculated results that align with experimental observation.

Above: relevant excerpts from the IRL3713 datasheet.

This datasheet offers info on four different thermal resistances:

1. the junction-to-case resistance (applicable to all package styles)
2. an estimated interface resistance resistance for the TO-220 package type,
3. junction-to-ambient thermal resistances for the TO-220 package in free air
4. junction-to-ambient thermal resistances for the D2Pak under specified mounting conditions.

The importance of understanding what each figure means (and using the correct one) should be apparent from the fact that the largest is nearly 140 times that of the smallest, and that there is roughly a 50% difference between two figures with the same symbol.

Interface resistances:

At right is a snippet from the datasheet for the thermal interface pad chosen for this example, which consists of a chunk of Bergquist's Sil-Pad 900S material cut to fit a TO-220 package. It's thermal properties are characterized in three different ways:

Thermal Conductivity (W/m*K): This describes the thermal characteristics of the bulk material, based on the ASTM D5470 test method.  Both the thickness of the material sample in question and the area involved in the interface need to be known to use this figure.

Thermal Impedance (°C*in2/W): This series of numbers characterizes the thermal resistance ('resistance' and 'impedance' are sometimes used interchangeably in context of thermal models) of the material as a function of contact area involved, and is tabulated for varying values of applied clamping pressure. (Note thdistinction between a clamping force and a clamping pressure.) It's similar to a thermal conductivity figure expressed in inverted form, but only the interface area and clamping pressure need be known in order to use the figure–material thickness and it's variability under different clamping pressures are rolled in.  These figures are the ones to look to if using this specific interface material for an arbitrary contact area: look up the thermal impedance based on the applied clamping pressure, divide by the area of the interface, and a thermal resistance figure results.

TO-220 Thermal Performance (°C/W): This set of figures characterizes observed thermal resistance measured using a TO-220 test package. It's worth noting that even though the contact area involved doesn't change appreciably with clamping pressure, the ratio between the TO220 thermal performance and Thermal Impedance values does.  This discrepancy can be attributed to differences in the test methods employed, and illustrates the need for empirical testing to verify design calculations in matters of this type.

Since our example part is in a TO-220 package, we'll use the TO-220 Thermal performance data.  Understanding the amount of clamping pressure that is applied by the chosen attachment mechanism is necessary.  In the case of spring-type clamping mechanisms this may or may not be obtainable from manufacturer-provided data.  If using a threaded fastener, it's user-determined based on the amount of torque prescribed for the mounting bolt during assembly.  For purposes of this example, the data for a 50 PSI mounting pressure will be used.  Based on typical dimensions of a TO-220 package, this correlates to approximately 12 pounds (53 Newtons) of clamping force, which is not unreasonable.  Under these circumstances, the thermal resistance of the mechanical interface between transistor and heatsink in this example would be estimated at about 2.9°C/W.

Above: excerpt from the BER183-ND datasheet.

Heat sink thermal resistances:

It's not uncommon for heat sink manufacturers to provide two different sets of characterization curves for their products, to describe their behaviors with natural and forced convection–in fan-less and fan-equipped applications respectively, loosely speaking.  It's also not uncommon for multiple products to be characterized in the same chart, which can make identifying the information of interest a bit difficult.

In the catalog snippet at right, 5 products are characterized using 4 different curves.  The 530614 product was chosen for the example, making the dashed lines pertinent, and the arrows attached to the solid curves indicate which set of axis labels apply to similarly shaped curves.  Taking that into account, it can be seen that two different sets of information are given; thermal resistance as a function of air velocity, and a temperature rise as a result of power dissipated.

Though not explicitly stated, the implication is that the former is used in situations where forced airflow is present, and the second applies to natural convection, in which the buoyancy of the air heated by the heat sink is the only force that's causing air movement across the heat sink's surface.

In terms of coming up with an Rϴsink-ambient value to plug into the model, in the forced-air case, one would read that value off the right-side scale in the graph, from the appropriate curve as determined by product selection, at the point corresponding to the rate at which air will be flowing across the heat sink.  Assuming an air velocity of 500 feet/minute, the estimated  thermal resistance of this heat sink would be a bit under 7°C/W.

Should the forced air fail for some reason or not be provided in the first place, one needs to know how much thermal power is being moved through the heat sink–which is the value of PThermal in the model here. In such a case, the bottom and left scales are used with the appropriate curves.  Note that rather than a thermal resistance (which has units of °C/W) this chart provides a temperature rise at the mounting surface, which simply has units of °C. Since the value of PThermal in the model must be known to look up a value, the manufacturer here is providing the result of  PThermal*Rϴsink-ambient in the thermal model.  If desired, one simply has to divide the temp rise by the heat dissipated value in order to come up with a thermal resistance value.  For example, if one needs to get rid of 4 watts in a natural convection situation, doing so would cause the chosen heat sink's mounting surface to increase in temperature by about 80°C relative to the ambient.  80°C/4W=20°C/W.  This is nearly three times greater than the 7°C/W achieved with 500feet/minute of forced airflow.

Above: Excerpt from the HS278-ND (Aavid P/N 530614B00000G) product catalog

Calculating thermal power inputs:

Calculating the amount of thermal power that will be generated by a device in operation can range from being a simple task (such as when applying a DC voltage across a fixed resistor) to being a rather complex affair, as in the case of transistors used in high-frequency switching circuits. It's also a topic better discussed in a separate article.  For purposes of this example, it will be taken as a given that 4 watts of power are being dissipated in the transistor.

Solving for unknowns:

With values for the various components in the thermal model established, one can proceed to use basic circuit analysis concepts such as Ohm's law to predict temperatures at the various nodes in the system.  In this example, the transistor's internal temperature is the one that ultimately matters.  That's easy enough to solve for by adding up all the thermal resistances, multiplying it by the the thermal power dissipation number, and adding the result to the ambient temperature.

Under the circumstances used for this example, the innards of the transistor might be expected to linger at around 66°C with the decided-on 500 FPM air flow across the heat sink–a rather comfortable number that's well-within the rated operating range for the device.  If the device creating that airflow failed and the system reverted to a natural convection mode of operation, a junction temperature closer to 120°C would be expected.  While much hotter, it's still within the rated operating range of the part with a bit of room left before the rated maximum is encountered.  From a thermal standpoint, the proposed operating conditions in this example would appear quite reasonable.

Note that this model basically has two parts: one representing the system itself (the sum of Rϴ values multiplied by PThermal)  and the other (TAmbient) representing the environment in which it's operating.  Since environments vary, it's quite common for people to assign the ambient environment's temperature as the ground reference in the model, and find the temperature difference between a device's inner (junction) temperature and the ambient environment, represented as ΔTJ-A.  This is a somewhat more portable and generalized way of expressing the results of thermal calculations, as only one number needs to be communicated to express something of broad meaning; to say "the transistor will be 41°C warmer than the surrounding environment" is simpler than saying "the transistor's temperature will be 66°C when the ambient temperature is 25°C."

In conclusion, the importance of verifying calculated results with physical testing bears repeated mention. There are abundant opportunities for error and inaccuracy, especially with regard to agreement of the model being developed with the system of interest. Factors such as the speed of the airflow across a heatsink or the amount of clamping pressure applied can be difficult to measure or predict with accuracy, and errors in these inputs can have substantive effects on the results of the calculations.

Above: Thermal circuit model with example values and resulting junction temperature calculations shown.

Below: The same model simplified, with calculations for a temperature difference relative to the ambient environment.

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