Since A and L are mechanical constants, in this example this formulia reduces to
which is analogous to
Note: This law applies only to normal solid thermal conductors. The specially designed
“heat pipes” that depend on a change of state (i.e., latent heat of vaporization of the internal
coolant) for their heat conduction effect will have a very nonlinear thermal resistance, and
will not follow this equation.
In heat pipes, the thermal resistance R
will go very low at the transition temperature.
This must be considered when using this type of thermal shunt.
For the more commonly used heat sink metals, the variation of thermal resistance with
temperature is negligible at normal semiconductor temperatures. It has been neglected in
16.4.4 Thermal Resistance R
(Analogous to Resistance R)
In the above example, the junction is dissipating 10 J/s (and hence Q 10 W). This heat
flow (analogous to a current flow of 10 A) will develop a temperature difference T
between each interface, depending on the thermal resistance R
between each interface
and the heat flow.
(The electrical analogue shows a potential difference V between each interface, depend-
ing on the resistance R between each interface and the current flow.)
When steady-state conditions have been established, the temperatures at the various
interfaces may be calculated by considering the heat flow and thermal resistances in the
heat transfer path.
In this example, it is assumed that the free air, by virtue of its nearly infinite bulk and
free flow, will remain at a constant ambient temperature of 20°C at the surface of the
finned heat exchanger. Since the temperature at this interface is constant, the temperature
of the other junctions with respect to this interface can be calculated from right to left in
(The electrical analogue of a free air temperature of 20°C is a ground potential of 20 V,
in this example.)
There are three thermal resistors to be considered in Fig. 3.16.2b. First (and usually
the most important, because it is the largest), there is the thermal resistance of the free
air interface itself, that is, from the finned heat exchanger surface to the surrounding
free air. (This is designated R
Second, there is the thermal resistance from the finned heat exchanger surface, through
the mica insulator, to the case of the diode. (This is designated R
Finally, there is the thermal resistance from the case of the diode to the internal junc-
tion. (This is designated R
(The electrical analogue shows resistors R3, R2, and R1 in the same positions.)
For convenience, the thermal resistance of each section will be considered separately,
starting with the heat exchanger interface.
From the manufacturer’s data, the finned heat exchanger has a thermal resistance R
of 4°C/W in free air.
The diode is mounted on an insulator to provide electrical isolation. This mica insulator
also has a defined thermal resistance between the diode case and the mounting surface of
the heat sink R
Finally, from the diode mounting surface to the internal junction (where the heat is being
generated), the resistance R
is given in the manufacturer’s data as 0.5°C/W for this diode.
(The electrical analogues are 4 7, 0.5 7, and 0.5 7.)
16. THERMAL MANAGEMENT
Hence, the total thermal resistance R
from junction to free air is the sum of these three,
or 4 0.5 0.5 5°C/W (or 5 7), and this total resistance (R
) is used to calculate the total
temperature difference T between the junction and free air.
From the previous equation,
10 5 50 C
temperature rise (above ambient), °C
Q dissipation in junction, W
total thermal resistance, junction to free air, °C/W
10 5 50 V
is the temperature rise above ambient, the junction temperature will be 70°C, and
the analogous voltage would be 70 V.
Clearly the electrical analogue is hardly necessary in this simple example; however,
it serves to demonstrate the principle and will be found very useful in more complex
applications. The engineer will make few errors in thermal design if this simple model is
kept in focus.
16.5 HEAT CAPACITY C
TO CAPACITANCE C)
The concept of heat capacity tends to get little attention in thermal design, although it is
significant in magnitude. It is the confusion between thermal capacity (specific heat) and
true thermal resistance that leads to a common error. It is often assumed that a copper
heat exchanger will perform better than, say, an aluminum heat exchanger with the same
surface area. This error stems from the fact that the copper does not appear to get hot as
quickly as the aluminum. In fact, what is being observed here is the effect of the larger
thermal capacity of the copper. The copper heat exchanger will eventually end up at the
same temperature. (Although copper is a better heat conductor, it is the surface area which
predominates and defines the thermal resistance.)
This effect will become clear from the more complete model shown in Fig. 3.16.2d.
The various thermal capacitors, C
, and C
, have been included, together with the
previously neglected direct heat losses from the surfaces of the various bodies, R
The heat losses from the surface of the components, R
, are normally neglected,
as the direct heat loss is negligible because of the small exposed area of the diode and
insulator. However, this is not the case with the thermal capacitors C
, and C
. In the
example shown, the electrical analogue capacitance will effectively be hundreds of farads.
(Even at 10 W input it can take several minutes for the heat exchanger to reach final thermal
From Table 3.16.1, it will be noted that the heat capacity of common heat conductors
can be very large (for example, 57.5 J/°C for a 1-in copper cube). Hence, for the example
shown in Fig. 3.16.2, if 10 in3 of copper were used in the construction of the heat exchanger
(quite realistic), then with a heat input of 10 W (10 J/s), it would take 57 s for the tempera-
ture to increase by only one degree. Thus, it would take several minutes to reach the final
temperature. The heat sink’s thermal mass (thermal capacitance) will not affect the value of
the steady-state temperature, only the time taken to reach thermal stability.
TABLE 3.16.1 Heat Storage Capacity and Thermal Resistance of
Common Heat Exchanger Metals
(block I" r I"),
However, if the heat input is of a transient nature, with a small duty ratio (allowing
plenty of cooling time), then the larger thermal capacity (or greater specific heat) will be
effective in reducing the maximum variation in temperature during a thermal load transient.
Since thermal capacity will not affect the final steady-state temperature, it is not considered
further in this example.
16.6 CALCULATING JUNCTION TEMPERATURE
In the previous example, the diode junction temperature was easily established because the
dissipation was known. However, in practice the dissipation in switchmode applications
can often be very difficult to establish, as some factors, like diode reverse recovery losses,
can be difficult to establish with any real degree of confidence. Under these conditions, any
known thermal resistance in the heat conduction path can be used to establish the heat flow
(and hence the junction dissipation) by measuring the temperature differential across the
interface of the known thermal resistance.
Consider again the electrical analogue shown in Fig. 3.16.2c. In the same way that the
voltage difference between two parts of a circuit is given by IR, the temperature difference
is given by the product of heat flow in joules per second (watts) and thermal resistance.
For the example shown in Fig. 3.16.2b and c, the heat flow is known, and the temperature
difference for each element of the thermal shunt can be calculated as follows:
$T Q R
where $T temperature difference (across element)
heat flow (power dissipated at the junction)
thermal resistance (of element)
The temperatures at the various interfaces may be calculated as follows:
Temperature of heat sink surface T
j h a
heat exchanger to ambient thermal resistance
ambient air temperature, °C
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16. THERMAL MANAGEMENT
Temperature of diode surface T
4 0 5
thermal resistance from device case to heat exchanger surface
The junction temperature T
will be the total temperature difference T
across all the various
series heat shunt elements, plus the ambient temperature; hence
4 0 5 0 5
20 70 C
It has been shown that if the power dissipated in the junction and the thermal resistance to
the heat shunts or heat exchanger are known, then the junction and interface temperatures
can be calculated. Clearly, if the temperature of the heat exchanger is measured and the
thermal resistance is known, then the heat flow and junction dissipation can be calculated.
16.7 CALCULATING THE HEAT SINK SIZE
In many practical cases the power dissipated in the junction will be known, and the thermal
resistance of the heat exchanger will need to be calculated for a defined junction tempera-
ture rise. The design procedure would be as follows.
Assume that a finned heat exchanger as shown in Fig. 3.16.3 is to be used to free-air
cool a T0-3 transistor dissipating 20 W, and that the junction temperature is not to exceed
136°C when the ambient air temperature is 50°C.
FIG. 3.16.3 Thermal resistance example, showing a T03 transistor on a
finned heat exchanger.
From the manufacturer’s data, the thermal resistance between junction and case of the T0-3
, is 1.5°C/W. An insulating mica washer is also to be used, and this has a ther-
mal resistance of, say, 0.4°C/W. (The thermal resistance of the insulator may be established
from the basic material properties in Table 3.16.2 or from Table 3.16.3.)
TABLE 3.16.2 ThermalResistance, Maximum Operation Temperatures, and Dielectric Constant of
Common Insulating Materials
(block 1" r 1"),
*Warning: Beryllium oxide is highly toxic if fragmented into small particles.
TABLE 3.16.3 Typical Thermal Resistance of Case to Mounting Surface of T0–3 and T0–220
Transistors When Using Standard Insulator Kits and Materials with Thermal Mounting Compound
The maximum temperature permitted at the interface of the insulator and the heat sink
when the junction temperature is 136°C can be calculated as follows:
The thermal resistance R
from junction to insulator-heat exchanger interface is
15 0 4 1 9
The temperature difference $T between the junction and the heat exchanger interface is the
product of the thermal resistance and heat flow Q:
T R Q
19 20 38
16. THERMAL MANAGEMENT
The temperature at the heat exchanger interface T
will be T
, less the difference $T
from junction to heat exchanger:
136 38 98 C
The maximum permitted temperature difference $T
from the heat exchanger surface to
free air at 50°C is
98 50 48
The thermal resistance of the heat exchanger, R
, is the temperature difference divided by
the heat flow:
Hence a heat exchanger extrusion of 2.4°C/W will be chosen. The manufacturer’s data
provide information on the heat exchanger thermal resistance for various extrusions or heat
exchanger designs, and a suitable size can be calculated.
16.8 METHODS OF OPTIMIZING THERMAL
CONDUCTIVITY PATHS, AND WHERE TO USE
“THERMAL CONDUCTIVE JOINT COMPOUND”
In the example shown in Fig. 3.16.3, the largest thermal resistance is from the heat sink to
free air, R
. (This will often be the case with convected cooling.) Since the total thermal
resistance of the heat shunt from the junction to free air is the sum of the various elements,
this final large thermal resistance swamps the effects of all the others. For example, a
50% increase or decrease in the resistance of the mounting arrangements would affect the
temperature at the junction by only 2.5°C. Hence, in this example, there would be little
advantage to using thermal compound to reduce the thermal resistance of the mounting
arrangements—the effect would be negligible.
It is interesting to note from the above example that the messy (and expensive) practice
of using thermal mounting compound on small air-cooled heat exchangers is probably not
very effective in most cases.
The designer should locate the interface with maximum thermal resistance and reduce
this to a value compatible with the other elements in the path. In the above example, a large
improvement would come from an increase in the heat exchanger surface area or an increase
in cooling air flow, but not very much from a reduced mounting interface resistance.
In the second example, Fig. 3.16.4a, a highly dissipating transistor (for example, an active
load) is to be mounted on an efficient water-cooled heat exchanger. This heat exchanger
can be considered an infinite heat sink. (For practical purposes, it may be assumed that the
surface temperature of the heat exchanger will not exceed 20°C regardless of how much
heat is conducted to it.)
Assume that the transistor dissipation is 100 W. The equivalent thermal resistance dia-
gram is shown in Fig. 3.16.4b. In this example, the junction-to-case thermal resistance is
0.5°C/W, and the case-to-heat-sink thermal resistance (because an insulator is used) is
higher, 1°C/W. (Since the heat exchanger in this example is the infinite heat sink, its thermal
resistance is zero.)
FIG. 3.16.4 (a) Thermal resistance example, showing a T0-3 transis-
tor mounted on a water-cooled (near-infinite) heat sink. (b)Equivalent
thermal resistance circuit model for T0-3 transistor on “near infinite”
heat sink. (c) T0-111 stud transistor on “near infinite” heat sink, with
a copper header.
16. THERMAL MANAGEMENT
With 100 W dissipated, the temperature drop across the insulator will be 100°C, giving a
case temperature of 120°C. The temperature rise within the transistor from case to junction
will be 50°C, giving a junction temperature of 170°C.
In this example, there would indeed be a considerable reduction in temperature at the
junction if the thermal properties of the mounting arrangement were improved. (The insula-
tor now has the highest thermal resistance in the series chain.)
Figure 3.16.4c shows a suitable modification to reduce the mounting thermal resistance
while retaining galvanic isolation. The transistor (now in a T0-59 case) is screwed directly
into a copper block, and the copper block is then insulated from the heat sink. Thermal com-
pound would be used on all interfaces to exclude any air voids, and the mounting screws
should be tightened to the recommended torque (see Fig. 3.16.5).
FIG. 3.16.5 Effective thermal resistance of interface between T0-3 tran-
sistor and heat sink when using a standard mica insulator, as a function of
screw torque, with and without heat sink compound.
The area of the unavoidably high thermal resistance insulation interface is now 5 times
greater than it was in the previous example, and the effective thermal resistance of the
insulator interface is now only 0.2°C/W. Hence, for the same dissipation conditions and
insulating material thickness, the junction temperature will now be 90°C, a considerable
improvement. Alternatively, an insulating material of lower thermal resistance, such as
beryllium oxide, may be used (see Table 3.16.2).
This example demonstrates the importance of identifying the point of maximum thermal
resistance. This resistance is the one which should be reduced if effective improvement is to
be obtained. (In any series circuit, it is the highest resistance that predominates.)
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