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PRINCIPLES OF
POWER ELECTRONICS
Second Edition

(Preliminary Draft KPVS22-12-6)

John G. Kassakian
David J. Perreault
George C. Verghese
Martin F. Schlecht

Copyright 2022 - Authorized Use Only

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25 Thermal Modeling and Heat Sinking

An unfortunate consequence of our preoccupation with things electrical is that the

problems of heat sinking and thermal management are frequently ignored until forced
on us by sound, sight, or smell. The insatiable need to make things smaller — and the
possibility of doing so by using higher frequencies and new components and materials —
aggravates the problem of heat transfer, because such improvements in power densities
are seldom accompanied by corresponding improvements in efficiency. Thus we are stuck
with the task of getting the same heat out of a smaller volume while disallowing any
increase in temperature.
The diversity of heat sources within power electronic apparatus produces a challeng-
ing cooling problem. Unlike signal processing circuits — where most heat-generating
components come in a common, small, and low-profile rectangular package — energy
processing circuits contain components of odd shapes and orientations. Even those of
the same type come in many different forms and packages. Inductors, for instance,
can be small or large, round or rectangular, and with loss dominated by core or cop-
per. Each possesses special requirements and presents a unique thermal problem. The
task of integrating these parts into a reliable piece of equipment becomes as much a
thermo-mechanical challenge as the circuit design was an electrical challenge.
Heat transfer occurs through three mechanisms: conduction, convection, and radia-
tion. In conduction the heat transfer medium is stationary, and heat is transferred by
the vibratory motion of atoms or molecules. Convective heat transfer occurs through
mass movement — the flow of a fluid (gas or liquid) past the heat generating apparatus.
In natural-convection, the buoyancy created by temperature gradients causes the fluid
to move; in a forced-convection system, the mass flow is created by pumps or fans.
Heat transfer by radiation turns the heat energy into electromagnetic radiation, which
is absorbed by other elements in the environment. Radiation as a mechanism of heat
transfer is important for space applications but less so for terrestrial power electronic
systems. Heat transferred through radiation is a function of the temperatures, TS and
TR respectively, of the radiating element’s surface and the receiving surface (which may
be a surface in the surrounding environment at a remove from the hot component).
Specifically,

Qrad ∝ TS4 − TR4

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834 Chapter 25: Thermal Modeling and Heat Sinking

Radiation may be important in equipment where this difference is large. However, the
strong nonlinearity of this relationship and the relatively low incidence of its importance
do not justify the complexity of considering radiation in detail here. Therefore in this
chapter we focus on conduction and convection.
If only these two mechanisms are considered, the design will be conservative, as
any heat transferred through radiation will reduce the temperature of the apparatus
below the design temperature. The exception is in enclosures, where radiation from
hot components may be absorbed by those at lower temperatures, causing these latter
components to operate at higher temperatures than anticipated. In such cases, radiation
shields — shiny metal partitions — can be employed to isolate the offending or affected
components.
The material in this chapter will not give you novel ideas for designing heat transfer
systems. The problem is too application-specific to permit such a discussion to be of
value. Instead, we describe first the parameters governing the performance of any such
system. Then we consider the modeling of both steady-state and transient thermal
behavior, as applicable to power electronic systems. Some straightforward examples of
specific designs will be presented to illustrate the discussions.

25.1 Static Thermal Models

Circuit theory is the lingua franca of engineering for good reason. The elegance and
simplicity of its canonical formulations (for example, KCL and KVL) permit complex
problems to be approached in an organized way, and the insights gained through such
formulations are extremely valuable in predicting system behavior. Therefore many
engineering problems in contexts other than electrical engineering — particularly in
the setting of ’flows’ driven by ’gradients’ — are cast in terms of circuit models before
being analyzed. One of these contexts is heat transfer.

25.1.1 Analog Relations for the DC Case

The rate at which heat energy is transferred by conduction from a body at temperature
T1 to another at temperature T2 is denoted by Q12 . It is well modeled as linearly pro-
portional to the temperature difference between the two bodies, T1 − T2 , and inversely
proportional to a physical parameter called the thermal resistance between them, Rθ :
T1 − T2 ∆T
Q12 = = (25.1)
Rθ Rθ
The analogy with Ohm’s law is evident, and we can make the following assignment of
variables:

T1,2 ⇐⇒ v1,2 Q12 ⇐⇒ i and Rθ ⇐⇒ R (25.2)

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25.1 Static Thermal Models 835

Note that the analog of thermal power is i, not vi. If heat leaves body 1 only through
the interface characterized by Rθ , then i is not only analogous to Q12 , but, because we
are considering only steady-state conditions, it also represents the rate at which energy
is being converted to heat in body 1. In the context of our interests, body 1 would
be a packaged electrical network, and Q12 would represent the rate at which electrical
energy is being converted to heat (dissipated) in the package, that is,

pdiss ⇐⇒ i

The thermal management problem is to design a heat transfer system (that is, Rθ ) that
constrains ∆T to the value dictated by component ratings and ambient conditions.
Figure 25.1 illustrates the electrical analog for the simple two-body system just dis-
cussed. The bodies are at temperatures T1 and T2 and are connected thermally through
the crosshatched interface, which can be characterized by a thermal resistance of value
Rθ . If the units of T are ◦ C and the units of Q are watts (W), then thermal resistance
has the units ◦ C/W. As with electric circuits, where parallel resistances can be com-
bined into a single equivalent resistance, parallel thermal paths can be characterized
by thermal resistances and combined into an equivalent single thermal resistance.

Figure 25.1 Electrical analog of simple two-body thermal system. The crosshatched region is the
thermal interface characterized by the longitudinal thermal resistance Rθ .

Convection is the mechanical transport of heat by a moving fluid. The fluid (air,
for instance) can move because of gravitational forces caused by density gradients, in
which case the process is called natural convection. Or the fluid can be driven (perhaps
by a fan), resulting in what is called forced convection. Convection is a somewhat more
complex process than conduction and can be described by the relation

Q12 = h(∆T, ν)A(T1 − T2 ) (25.3)

where ν is the fluid velocity. The parameter h(∆T, ν) is termed the film coefficient
of heat transfer; it depends on fluid velocity and the difference between the inlet and
outlet temperatures. The cross-sectional area of the interface is A. Over the range
of temperature differentials of interest in our application, h is fairly constant. With

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836 Chapter 25: Thermal Modeling and Heat Sinking

respect to fluid velocity, significant changes in h occur when the flow regime changes
from laminar to turbulent. Within each regime, however, h improves only slowly with
increased velocity. For forced convection, h is independent of ∆T . Within these limits,
the product hA may be modeled as constant, giving to (25.3) the same form as (25.1),
with Rθ = 1/hA. Thus the electrical analog shown in Fig. 25.1 is appropriate for
representing convective as well as conductive heat transfer.

25.1.2 Thermal Resistance

As we have just shown, a thermal resistance can be used to model both conductive and
convective heat transfer. The physics governing thermal conduction is much like that
for electrical conduction, and the thermal resistance or conductance can be described
in terms of parameters abstracted from the physics of the process (for example, con-
ductivity) and geometry. In fact, thermal and electrical conductivity of a material are
intimately related by the Wiedemann–Franz law, which states that the ratio of these
conductivities varies linearly with temperature T (so is fixed at a given T ). — materials
of high electrical conductivity are also good thermal conductors.
Convection, however, depends on parameters that are not so easily abstracted. For
instance, while conductivity can be adequately described in terms of material type and
temperature, h is a function not only of these parameters but also of velocity, surface
characteristics, and geometry. The latter parameter often also determines the Reynolds
number, a dimensionless number that indicates whether laminar or turbulent flow will
occur in a particular situation. Furthermore, the geometry of the convective part of the
system is invariably complex, consisting quite often of a finned aluminum extrusion.
Thus the equivalent thermal resistance model for convective transfer from a specific
heat sink type in a variety of environments (for example, forced or natural convection)
is tabulated by the manufacturer. For this reason, the following discussion is directed
at determining the equivalent thermal resistance for those parts of the system where
heat transfer is by conduction.
The analog of electrical resistivity (Ω-m, or more commonly, Ω-cm) is thermal resis-
tivity, ρθ , in units of ◦ C-cm/W (or K-cm/W as Kelvin is often used instead of Celsius).
In terms of ρθ and physical dimensions, the longitudinal thermal resistance of a piece
of material of cross-sectional area A and length l is
ρθ l ◦
Rθ = C/W (25.4)
A
An alternate definition of thermal resistance, used for sheet material, incorporates the
sheet thickness. The conductance through a sheet of unit area and specified thickness
is given by the parameter hc having units of W/K-cm2 . Therefore the resistance of an
area A of the sheet is
1 ◦
Rθ = C/W (25.5)
hc A

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25.1 Static Thermal Models 837

The thermal resistivities of various materials used in heat transfer paths in electronic
equipment are shown in Table 25.1. Mylar, and less commonly mica, is used to provide
electrical isolation between electrically hot components (for example, the semiconduc-
tor device package and the heat sink). Mica has a much higher dielectric strength, is
more impervious to mechanical puncture, and can be cleaved to produce thinner sheets
than Mylar—but is more expensive. Beryllia (BeO) and alumina (Al2 O3 ), and recently
aluminum nitride (AlN), are also used to provide electrical isolation, most frequently
within device packages. Silicone grease impregnated with metallic oxides, such as

Table 25.1 Thermal Resistivities of Materials Used in Electronic Equipment

MATERIAL THERMAL RESISTIVITY (◦ C-m/W)
Still air 30.50
Mylar 6.35
Silicone grease 5.20
Mica 1.50
Filled silicone grease 1.30
Filled silicone rubber 1.0
Alumina (Al2 O3 ) 0.06
Silicon 0.012
Beryllia (BeO) 0.01
Aluminum Nitride (AlN) 0.0064
Aluminum 0.0042
Copper 0.0025

ZnO2 , is used to fill imperfections such as scratches on mating surfaces in a heat trans-
fer path — between the bottom of a device package and the top of a heat sink, for
instance. The need to fill these voids with something other than air is apparent from
the table. Filled silicone grease is also referred to as thermal grease or thermal compound
(or “goop”, for reasons that become clear when you use it). Anodizing is frequently used
to create an attractive or black surface on aluminum components. Since the resulting
oxide is very thin, it contributes little to the thermal resistance of a path. Although
the oxide is a good insulator, it is unwise to rely on it in lieu of a dielectric material
for providing electrical isolation between surfaces.

Example 25.1 A Calculation Using An Electrical Analog

Figure 25.2(a) shows a resistor embedded in the center of a 10 cm long block of aluminum whose
ends are at temperature TA . What is the temperature of the resistor if it is dissipating 50 W and
the ambient temperature is TA = 75◦ C?

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838 Chapter 25: Thermal Modeling and Heat Sinking

(a)

(b)

Figure 25.2 (a) A thermal system consisting of a resistor embedded in the center of an aluminum
block. (b) The electric circuit analog for the thermal system of (a).

As the length of the block (10 cm) is much longer than the radius of the resistor (3 mm), we
can assume that the detailed pattern of heat flow in the vicinity of the resistor is unimportant.
The resulting analog circuit model is shown in Fig. 25.2(b), where RθL and RθR are the thermal
resistances of the aluminum bar to the left and right of center. The value of these resistances are

(0.42)(5)
RθL = RθR = = 2.1◦ C/W (25.6)
1

At a dissipated power of 50 W, the temperature of the resistor, TR , is

 
2.1
TR = 75 + 50 = 127.5◦ C
2

25.2 Thermal Interfaces

A critical part of any heat transfer system is the interface between mechanical compo-
nents in the thermal path. Some issues related to these interfaces in the context of our
application were raised in the previous section. The geometry of most interfaces can be
modeled as two parallel planes with material of a specific thermal resistivity between.
If the material is of thickness δ and of area A, the thermal resistance of the interface

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25.2 Thermal Interfaces 839

between the planes is

ρθ δ
Rθi = (25.7)
A
Consider for example a device in a TO-220 package mounted on a heatsink. Between
the device and heatsink is a 1.6 mm alumina pad because electrical isolation between
the device and heatsink is required. The mating surface area of a TO-220 package is
approximately 0.95 cm2 , giving a thermal resistance between the case and the sink of:

(6)(0.16)
RθCS = = 1.01◦ C/W (25.8)
0.95
Thus the difference between the case and sink temperatures increases by 1.01◦ C for each
watt of thermal power being transported across the interface. A dissipation of 15 W
is not unusual for a device in a TO-220 package, giving a temperature rise of 15.2◦ C
across just the alumina interface. This amount, which does not include the thermal
resistance of the interfaces between the alumina pad and the TO-220 case or heatsink,
is significant, and illustrates the price paid for requiring electrical isolation.

Example 25.2 A Thermal System

The physical structure depicted in Fig. 25.3 is typical of the thermal system that results from
mounting a semiconductor die. The device itself is bonded to the header using solder or epoxy;
the header is made part of a package that is mounted to a heat sink (perhaps with some interven-
ing insulating material); and the heat sink is thermally connected to the ambient environment,
generally through free or forced (fan) convection. A highly detailed model for this system is
shown in Fig. 25.4(a), where each part of the system is explicitly represented by its equivalent
thermal resistance. The model also shows the relationship between certain node voltages and the
temperatures they represent. The current source represents the rate at which electrical energy is
dissipated in the device, Pdiss . The physical location within the device of this dissipation is the
node to which the source is connected, the junction in this case.

Figure 25.3 Typical mechanical structure used for mounting semiconductor die.

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840 Chapter 25: Thermal Modeling and Heat Sinking

(a) (b)

Figure 25.4 (a) A static thermal model for Fig. 25.3. (b) A simplified model of the circuit in (a).

Some of the thermal resistances shown in Fig. 25.4(a) are so small relative to others that
they can be neglected. Because the header is made of copper or aluminum, its vertical thermal
resistance is negligible, as is that of the thermal grease (assuming that it is applied properly,
which it often is not!). Others of the identified resistances are frequently lumped together, such
as those for the silicon and bonding material, which are generally inaccessible to the circuit
designer. Implicit in the element RθSA are the thermal resistance of the sink extrusion between
the region on which the package is mounted and the surfaces from which heat is being removed by
convection, and the thermal resistance representing the convection process. Figure 25.4(b) shows
the simplified model.
The variable of interest in the models of Fig. 25.4 is Tj , the “junction” temperature of the
device. The term “junction” is used rather loosely to represent in lumped form the source of heat
in the device. In reality, this source seldom exists as a simple plane. In the MOSFET it is not
a junction at all. Nevertheless the term persists, and manufacturers determine RθjC empirically,
which takes into account the actual geometry of the heat-producing region. To determine Tj ,
we need to know Pdiss as well as the thermal resistances between the junction and the ambient
environment.
The dissipation in the device is a function of its electrical environment (for example, its current,
voltage, and switching loci). For the purposes of this example, we assume that these calculations
have been made for our device, and that the result is Pdiss = 25 W. The physical configuration
is a TO-247 package mounted on an extruded, finned, free convection-cooled sink without any
insulating interface but with thermal grease. Typical values of the thermal resistances are RθjC =
1.1, RθCS = 0.12, and RθSA = 1.8, all units being ◦ C/ W. The last parameter needed is the
ambient temperature, which is not “room temperature,” but that of the air in the vicinity of the
sink. We take it to be 40◦ C. We determine the temperature drop between nodes in the model by

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25.2 Thermal Interfaces 841

using Ohm’s law, obtaining

TS = 40 + (25)(1.8) = 85◦ C
TC = 85 + (25)(0.12) = 88◦ C
Tj = 88 + (25)(1.1) = 115.5◦ C
Is this an adequate design? The answer depends on the type of device being cooled. If it were a
Si MOSFET, the design gives a good margin between predicted junction temperature and typical
maximum limits of 150◦ C. For a thyristor the design is marginal.

25.2.1 Practical Interfaces

Mechanical interfaces are not parallel in practice. They contain surface imperfections,
such as scratches, and a characteristic called run-out. Run-out is the maximum deviation
from flatness that a surface exhibits over a specified lateral distance. It is measured
in (linear dimension)/(linear dimension), for example, cm/cm. A standard aluminum
extrusion exhibits run-out that is typically 0.001 cm/cm. Both scratches and run-out
degrade the thermal performance of an interface. Run-out is generally not under the
design engineer’s control. Therefore we must measure or estimate it and make proper
allowance for it. However, run-out is seldom an issue with modern commercial heat
sinks.
The use of thermal grease has already been mentioned. It is designed to reduce the
degrading effects of surface scratches and other small imperfections but is not designed
to remedy the effects of run-out. Although our primary focus is on basic principles,
thermal grease is so frequently misused that a brief departure from “principle” to “prac-
tice” is justified. The problem arises from a belief that if a little is good, a lot is better.
However, the art of applying thermal grease is much like that of watering plants—too
much and it’s dead. Silicone grease is highly viscous and refuses to “squish out” when
squeezed between header and sink by mounting hardware. In such cases, a thin layer of
grease can remain in the interface, giving rise to a significant thermal resistance that
was not anticipated in the thermal design. The grease should be applied sparingly, and
then wiped off, removing almost all traces. Thermal grease oozing from under device
packages is a sign of poor construction and potential thermal problems.
When electrical isolation between device and heat sink is required, a pad made of
silicone (or other conformable material) can be used as an interface material. These
“squishy” materials fill scratches and other surface imperfections when mounting pres-
sure is applied and are available as sheets, or in shapes conforming to most device
package geometries. A unique consideration when using these pads is that the resulting

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842 Chapter 25: Thermal Modeling and Heat Sinking

thermal resistance between the two surfaces is a function of mounting pressure. The rela-
tionship among 1/hc , thickness, and pressure is usually provided by the manufacturer† .

The printed circuit board (PCB) presents unique thermal design problems. One
can mount on the board heat sinks to which are attached the semiconductor devices,
however this occupies valuable board real-estate. If the thermal requirements are not
too severe, a common approach is to mount the device directly on the board over
vias† that thermally connect the upper and lower layers of copper foil. The device is
thermally connected to the upper foil layer, which spreads and dissipates the heat as
well as transferring heat through the vias, where it is spread on the lower foil layer.
This heat-sinking method for PC boards is illustrated in Fig. 25.5.

(a)

(b)

Figure 25.5 An illustration of the use of vias as heatsinks for components mounted directly to a
PC board: (a) top view of device showing location of 4 vias; (b) cross-section A–A
illustrating the copper plating in the vias.

25.2.2 The Convective Interface

Even though we showed that both conduction and convection processes could be mod-
eled by similar electrical analogs, our discussion so far has focused on conduction
interfaces. However, all conduction actually leads to a convective interface. Heat is

†
Manufacturers of sheet material often use the term thermal impedance instead of resistance to denote
1/hc .
†
A via is a hole through the board, the interior wall of which has been plated with copper. It may or may
not connect different layers of copper foil. Vias may optionally be filled with copper or another material to
reduce thermal resistance.

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25.3 Transient Thermal Models 843

removed from a conventional finned sink by air flowing over the fins. A more sophisti-
cated system incorporating a heat exchanger probably uses a liquid to move the heat
from one place to another. These are convective processes. As mentioned earlier, the
physics governing these processes is beyond our scope here, However, a short discussion
of the application of finned sinks is helpful.
The critical issue in the application of finned sinks is to ensure that air flow through
the fins is turbulent rather than laminar. Laminar flow, as the name implies, is the flow
of a fluid in such a way that strata can be defined; that is, all flow is in one direction,
with no mixing of strata. Turbulent flow, on the other hand, causes considerable mixing.
Without such mixing, the particular stratum of fluid in contact with the fin would
remain in contact with it for its entire length, resulting in a very low value for the
film coefficient of heat transfer h, discussed in Section 25.1.1 Stated another way, the
boundary layer next to the fin surface remains intact in laminar flow, preventing efficient
heat transfer from the fin to the moving stream of air (or other fluid).
The transition between laminar and turbulent fluid flow is a function of many vari-
ables, however fin geometry and flow rate are the critical ones for our application. The
relationship among fin spacing, flow rate, and the onset of turbulence is given by the
Reynolds Number. A high Reynolds Number is characteristic of turbulent flow; a low
number is characteristic of laminar flow. The Reynolds Number Re for a fluid flowing
at velocity ν through a channel of width w is
ρνw
Re =
η
where ρ is the fluid density, and η is its coefficient of viscosity. This expression shows
that fluid flowing in a wider channel will enter the turbulent flow regime at a lower
velocity than that through a narrower channel. The point here is that, like thermal
compound, more is not necessarily better. Because of reduced turbulence and flow
rate, many closely spaced fins and a large surface area could result in poorer thermal
performance than fewer, but more widely spaced fins and a smaller surface area.
Although we have been using the context of semiconductor heat sinks for this discus-
sion, it is equally appropriate to the cooling geometry associated with other components.
Closely spaced parts impede proper convective flow for the same reasons that too closely
spaced fins do.

25.3 Transient Thermal Models

So far our discussion and models have been limited to systems in which both the energy
being dissipated and the temperatures within the system are constant. Our models do
not represent the thermal processes associated with start-up, where dissipation may be
constant, but temperatures are climbing — or pulsed operation, where temperatures
may be constant, but dissipation is not. The latter situation is the more important,
for under such conditions the permissible instantaneous dissipation can be much higher

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844 Chapter 25: Thermal Modeling and Heat Sinking

than predicted by static thermal models. Essentially, the heat capacity of components
or their constituent parts creates a low-pass filter, which in the limit of small bandwidth
only responds to the dc in pdiss (t).
Heat capacity is a measure of the energy required to raise the temperature of a mass
by a specific amount. In SI units, it is specifically the energy in joules required to raise
one kilogram of the material one centigrade degree, and has the units /◦ C-kg. Water has
one of the largest thermal capacities of any fluid at room temperature: 4.2 × 103 J/◦ C-
kg. Masses in a thermal system, then, constitute thermal energy storage devices, and
thermal systems containing mass will exhibit dynamic behavior.

25.3.1 Lumped Models and Transient Thermal Impedance

Since thermal power is Q, the analog of thermal capacitance is electrical capacitance in
the circuit model for heat transfer. In its simplest form, then, the dynamic model for
a mass being supplied with heat energy is an RC circuit, as in Fig. 25.6. If the heat
source is constant, the final temperature is analogous to the final capacitor voltage,
that is, Rθ Q. Previously we have dealt with the steady-state solutions to such thermal
systems. Now we are concerned with the transients leading to these steady states.
The temperature curve of Fig. 25.6(d) predicts the temperature T1 as a function of
time for a step Po in thermal power. If this curve is normalized by the step amplitude
(Po in this case), the resulting vertical scale has the units of thermal impedance, that is,
◦
C/ W. An experimentally or theoretically determined normalized curve of this kind is
useful for predicting temperatures during thermal transients. The normalized quantity
is a function of time and is called the transient thermal impedance, denoted by Zθ (t):
T (t)
Zθ (t) = (25.9)
P0

It is important to note that in order to properly represent the physics of the situation,
thermal capacitances in a system model should always be connected to “ground”, that
is, a reference temperature.

Distributed Models Energy in a mass is stored in a continuum. However, as is done

for a transmission line, this continuum system may be modeled by an interconnection
of lumped electrical elements. Consider, for instance, the mass of Fig. 25.6 with a
source of heat energy applied at one end. The mass can be broken into an arbitrary
number of sections, each assumed to be at a uniform temperature. Each such section is
characterized by a heat capacity, and the sections are interconnected through a thermal
resistance. This thermal resistance is that of the mass section between the interfaces
with adjoining sections. This multi-lump model of the mass of Fig. 25.6 is shown in
Fig. 25.7. The number of ‘lumps’, 5, was chosen arbitrarily.
When a continuous system is modeled by lumps, each lump displays the aggregate
behavior of the physical piece of the system it represents. The model of Fig. 25.7 has

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25.3 Transient Thermal Models 845

(a) (b) (c) (d)

Figure 25.6 (a) A simple thermal system consisting of a mass at temperature T1 being supplied
heat Q and in contact with a sink at temperature TS . (b) A single -ump dynamic
model for the system shown in (a). (c) A step in thermal power exciting the thermal
system of (a). (d) The temperature response of node T1 to the excitation of (c).

(a) (b)

Figure 25.7 (a) The thermal system of Fig. 25.6(a) divided into five “lumps.” (b) The lumped
electrical analog model for the thermal system of (a).

been constructed so that the node voltages represent the section temperature aggregated
at the interface. The number of lumps that should be chosen to represent a system
depends not only on the spatial resolution of interest but on the bandwidth of the
behavior being modeled. For instance, if Q is constant, no dynamics are excited, the
bandwidth of the behavior is small, and a one-lump static model is adequate. However,
if Q varies with time at a rate much greater than (Rθ C)−1 for the segments of Fig. 25.7,
more lumps would be needed to accurately model the behavior of the system.
The device and package structure of Fig. 25.3 contains several thermal masses that
contribute dynamics to its thermal behavior. These dynamics are important when the
device is forced to dissipate high levels of power for short periods of time. “Short” is rel-
ative to the Rθ C time constant of the structure’s electrical model. For very short pulses,
the mass of the silicon is most important in determining the excursion of the junction
temperature Tj . As the pulse gets longer, the mass of the header and then the heat sink

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846 Chapter 25: Thermal Modeling and Heat Sinking

become important. Manufacturers usually provide transient thermal impedance curves

as functions of duty ratio and pulse width in the specification sheets for their devices.
An illustrative set of curves for an SiC MOSFET in a TO-247 package is shown in
Fig. 25.8. Only the bottom curve in this family is Zθ (t) as defined by (25.9). The other
curves are parametric in duty ratio for a series of pulses having a pulse width given on
the x-axis.

1
Junction To Case Impedance, ZthJC (oC/W)

0.5

0.3

0.1
100E-3
0.05

0.02

D = 0.01
10E-3 Single Pulse

1E-3
1E-6 10E-6 100E-6 1E-3 10E-3 100E-3 1 10
Time, tp (s)

Figure 25.8 Transient thermal impedance, Zθ (t), parametric in duty ratio and functions of pulse
width tp , for a 1200 V, 32 A Wolfspeed C3M0075120D SiC MOSFET in a TO-247
package. (Used with permission of Wolfspeed, Inc.)

Example 25.3 Transient Thermal Design for a MOSFET

The Wolfspeed C3M0075120D SiC MOSFET rated at VD = 1200 V and Tj = 175◦ C, whose
transient thermal impedance characteristics are shown in Fig. 25.8, is used in an 800 V clamped
inductive switching application. We consider an example in which it is subjected to repetitive 35 A
current pulses with a duty ratio of 0.1 at a frequency of 10 kHz. The gate is driven between -4 V
and +15 V. It has already been determined that the device will remain within its safe operating
area (SOA). We want to determine the maximum allowable heat sink thermal resistance, RθSA ,
to maintain the junction at a conservative temperature of 125◦ C for an ambient temperature of
40◦ C.
The device dissipation has two parts: on-state and switching losses. Since they occur at different
times during the pulse and are each short compared to a thermal time constant, they can be
treated independently and their results added.
2
Energy is lost during the on-state at a power of RDS(on) IDS . But RDS(on) is a function of both
junction temperature and drain current, so we must consult Fig. 25.9(a) which is taken from
the device data sheet. The figure shows RDS(on) at IDS = 35 A at Tj = 175◦ C and 25◦ C. We

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25.3 Transient Thermal Models 847

interpolate RDS(on) to a value of 121 mΩ at Tj = 125◦ C. The on-state power during a pulse is
then

Pon = (352 )(0.121) = 148 W (25.10)

The switching loss is determined from the loss vs IDS curves using the data sheet graphs
shown in Fig. 25.9(b). Switching loss is not a strong function of temperature, so the measurement
condition of Tj = 25◦ C instead of 125◦ C is relatively immaterial. The Etotal curve at IDS = 35 A
gives Etotal ≈ 2 mJ/cycle. Since the switching times are on the order of 10’s of ns for this device,
the instantaneous switching power is very high, though the average power loss associated with
switching is not.

180 3.0
Conditions: Conditions:
VGS = 15 V TJ = 25 °C
160 ETotal
tp < 200 µs VDD = 800 V
2.5
RG(ext) = 0 Ω
140 TJ = 175 °C VGS = -4V/+15 V
On Resistance, RDS On (mOhms)

FWD = C3M0075120D
120 2.0 L = 157 μH
EOn

Switching Loss (mJ)

100
1.5
TJ = -40 °C
80
TJ = 25 °C
60 1.0

40 EOff
0.5
20

0 0.0
0 10 20 30 40 50 60 0 5 10 15 20 25 30 35 40 45
Drain-Source Current, IDS (A) Drain to Source Current, IDS (A)

(a) (b)

Figure 25.9 Specifications for the C3M0075120D SiC MOSFET: (a) RDS(on) vs. temperature and
IDS ; (b) switching loss vs. IDS . (Used with permission of Wolfspeed, Inc.)

The transient thermal impedance Zθ presented to the 10 µs, 0.1 duty ratio current pulses is
given by Fig. 25.8 as approximately 0.12◦ C/W. But Zθ for the very short (10’s of ns) pulses of
power during switching is not available from Fig. 25.8. The time scale of these switching power
pulses is extremely short compared to both the available time constants of the system and of
the on-state power pulses. We can estimate temperature rise by including the switching energy
with the longer time scale of the on-state power pulses (as the on-state pulses are still very short
compared to the known system time constants). Distributed over the ton = 10 µs duration of the
conduction period, the switching energy provides an additional equivalent on-state power of

Psw,equiv = Etotal /ton = (2 × 10−3 )/(10 × 10−6 ) = 200 W (25.11)

Therefore we use Zθjc and Pon + Psw,equiv to determine the maximum TC allowed to maintain
Tj < 125◦ C:

∆Tjc = Zθ (Pon + Psw,equiv ) = 0.12(148 + 200) = 41.8◦ C

TC ≤ 125 − 41.8 = 83.2◦ C

The thermal power transferred through the case to ambient includes both the conduction and
switching loss. Since the case is thermally massive, it is considered an isotherm; and therefore
we use the average total power to be dissipated to determine RθCA . The average on-state loss is
⟨Pon + Psw,equiv ⟩ = 0.1 × 348 = 34.8 W. Using ∆TCA = 83.2 − 40 = 43.2◦ C, we can now calculate

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848 Chapter 25: Thermal Modeling and Heat Sinking

the maximum allowable heat sink thermal resistance RθCA .

∆TCA 43.2
RθCA ≤ = = 1.24◦ C/W (25.12)
⟨P ⟩ 34.8
Thermal resistance in this range is achievable with an appropriately specified extruded aluminum
heat sink using natural convection.

Example 25.4 Derating the Safe Operating Area

A device’s maximum allowed average power dissipation, PD(max) , is given in its data sheet but is
usually specified at a case temperature TC = 25◦ C, accompanied by a graph derating PD(max) for
higher values of TC , as shown in Fig. 25.10 (a). The Safe Operating Area (SOA) graph provided in
data sheets is derived from single pulse measurements (duty ratio D = 0) and also at TC = 25◦ C
with curves parametric in pulse width, as illustrated by Fig. 25.10(b). A case temperature of 25◦ C
seldom conforms to the application, where the case temperature is generally much higher. So we
need to modify the data sheet SOA to reflect our application, derating the device and producing
a smaller SOA. The process requires the use of the PD derating and transient thermal impedance
curves from the data sheet, shown in Figs. 25.10 (a) and (b), respectively.

140 100.00 a
Conditions: d
TJ ≤ 175 °C
120 Limited by RDS On
Maximum Dissipated Power, Ptot (W)

1 µs

10.00
Drain-Source Current, IDS (A)

100 c 10 µs

80 100 µs

1.00
60 1 ms

40 100 ms
0.10

20
Conditions: b
TC = 25 °C
D=0
0 0.01
-50 -25 0 25 50 75 100 125 150 175 0.1 1 10 18.6 100 1000
Case Temperature, TC (°C) Drain-Source Voltage, VDS (V)

(a) (b)

Figure 25.10 SiC MOSFET C3M0075120D specifications: (a) maximum power dissipation derating
curve; (b) the safe operating area, where the dashed line defines the boundary for a 100
µs pulse at TC = 100o C. (Used with permission of Wolfspeed, Inc.)

The maximum voltage and current constraints of the SOA are unchanged, as is the line con-
strained by RDS(on) . We need to determine new coordinates for the constant power constraints
for the various pulse widths, at our application TC = TCa , using the transient thermal impedance
curves. The coordinates are scaled by δp , the ratio of P (Tca ), the maximum permissible average
dissipation at TCa , to P (25◦ ), the allowable dissipation at TC = 25◦ C, numbers obtained from
Fig. 25.10(a).
We calculate the maximum allowable dissipation for our pulse if Tj = 175◦ C and TC = 25◦ C,
and scale it by the ratio δp to obtain Pp (TCa ), the maximum pulsed power. This allowable

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Notes and Bibliography 849

dissipation at TCa then allows us to calculate new VDS − Id coordinates on the SOA graph for
our application case temperature and pulse width.
We illustrate the process by derating the iso-power line in Fig 25.10(b) for a 100 µs pulse at
Tca = 100◦ C. Figure 25.10(a) gives us P (25◦ ) = 136 W, P (100◦ ) = 68 W, and δp = 0.5. From
Fig. 25.8 for a single 100 µs pulse, we estimate Zθ = 0.06◦ C/W which we use to calculate Pp (25◦ ),
the 100 µs pulse power if TC = 25◦ C and Tj = 175◦ C, which we scale by δp to give us Pp (100◦ ).
175 − 25
Pp (25◦ ) = = 2917 W (25.13)
0.06
◦ ◦
Pp (100 ) = Pp (25 ) × δp = 1488 W (25.14)
We can now calculate a pair of coordinates on the iso-power limit line for a 100 µs pulse with
TC = 100◦ C. Choosing ID = 80 A (the maximum specified pulse current),
Pp (100◦ ) 1488
VDS = = = 18.6 V (25.15)
ID 80
We now have one point on the line, but the line is iso-power with a slope of -1 so we can draw
the new constraint on the SOA graph, as indicated by the dashed line in Fig. 25.10(b).

Notes and Bibliography

The volume of work published in the general area of heat transfer is massive. The references
selected here are representative of those that are accessible to the nonspecialist. An undergraduate
text covering most topics of interest to the designer of electronic equipment, although not in
this context, is [1]. The book is liberally illustrated and contains numerous examples. Lienhard
and Lienhard [2] is a very comprehensive text with numerous examples and problems addressed to
juniors through graduate students. Among its unique inclusions are photographs of Ludwig Prandtl,
Osborne Reynolds, and Ernst Kraft Wilhelm Nusselt, whose namesakes are the Prandtl, Reynolds,
and Nusselt numbers, important parameters in heat transfer. It is inexpensive and available as an
e-book.
Lee [3] is focused on specific heat exchange technologies. The book includes extensive analyses of
the different devices used for heat transfer. Steinberg, [4], is short on theory but long on practical
applications. The numerous examples reflect the author’s own experience in the military/avionics
area. A lot of practical data is presented, and there is a good discussion of fluid-based heat transfer
systems, including heat pipes.
A concise discussion and mathematical statement of the Wiedman-Franz law can be found on
p. 150 of Kittel, [5].
The TI application note [6] provides an extensive discussion of using thermal vias for heat sinking
on printed circuit boards.

1. F. M. White, Heat Transfer, Addison-Wesley, 1984.

2. J. H. Lienhard IV and J. H. Lienhard V, A Heat Transfer Textbook, 5th ed. Dover Publications,
2019.
3. H. Lee, Thermal Design: Heat Sinks, Thermoelectrics, Heat Pipes, Compact Heat Exchangers,
and Solar Cells, Wiley, 2010).
4. D. S. Steinberg, Cooling Techniques for Electronic Equipment, 2nd ed. Wiley Interscience, 1991.
5. C. Kittel, Introduction to Solid State Physics, 6th ed. New York: Wiley , 1986.

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850 Chapter 25: Thermal Modeling and Heat Sinking

6. Texas Instruments, ANM-2020 Thermal Design By Insight, Not Hindsight, Application Report
SNVA419C-April 2010- Revised April 2013.

PROBLEMS

25.1 A double-insulated window is made of panes of glass 4 mm thick spaced 1 cm apart. Window
glass has approximately the same thermal resistivity as SiO2 , 100◦ C-cm/ W. If the interior
temperature of the building is 25◦ C and the outside temperature is 0◦ C, what is the rate of
heat lost by conduction in kW/m2 ?
25.2 The CRC Handbook of Chemistry and Physics (35th ed.) defines thermal conductivity of mate-
rials as “the quantity of heat in calories which is transmitted per second through a plate 1 cm
thick across an area of 1 cm2 when the temperature difference is 1◦ C.” The value for dry compact
snow is 0.00051. What is the thermal resistivity of dry compact snow in units of ◦ C-cm / W?
25.3 An isolating interface of alumina having a thickness of 1 mm is placed between the device
package and the heat sink in Fig. 25.3 (Example 25.2). What is the junction temperature Tj ,
if other parameters of the example remain unchanged?
25.4 Figure 25.11 shows two identical devices, Q1 and Q2 , mounted on a common heat sink.
The devices are in TO-220 packages and have a thermal resistance from junction to case of
RθjC = 1.2◦ C/ W. The interface between the case and sink has a thermal resistance of RθCS =
0.20◦ C/ W, and the thermal resistance between the sink and ambient is RθSA = 0.8◦ C/ W.
(a) Draw the static thermal model for the thermal system of Fig. 25.11.
(b) If the devices are dissipating the same power, and TA = 40◦ C, what is the maximum total
power that can be dissipated if Tj(max) = 150◦ C?
(c) What is the maximum possible power dissipated if only one of the devices is operating?

Figure 25.11 Two devices mounted on a common heat sink analyzed in Problem 25.4

25.5 A transistor in a TO-3 case has a junction-to-case thermal resistance of 1◦ C/ W and is to be

used in an environment having an ambient temperature of 60◦ C. The transistor is to be isolated
from its heat sink by a Mylar spacer having a thickness of 0.1 mm, and the available heat sink
has a specified value of sink-to-ambient thermal resistance of RθSA = 2◦ C/ W.
(a) Determine and draw the static thermal model for this system.
(b) What is the maximum power that can be dissipated by the device if its junction
temperature must be less than 150◦ C?
25.6 Figure 25.12 shows the internal structure and dimensions of a power diode mounted in an
axial lead package. The diode is cooled by conduction through its leads, which are soldered to

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Problems 851

terminals that are assumed to be at temperature TA . Heat is generated at the junction of the
diode, which is planar and centered between the two surfaces.
(a) Draw the analog circuit model for the thermal system of Fig. 25.12.
(b) If the maximum permissible junction temperature of the diode is Tj = 225◦ C, what is the
maximum permissible dissipation for TA = 75◦ C?

Figure 25.12 The axial lead packaged diode analyzed in Problem 25.6.

25.7 A superjunction MOSFET in a TO-247 package is mounted to a heatsink with a 0.5 mm thick
silicone pad as the interface. The thermal contact area of a TO-247 package is 2.5 cm2 .
(a) At the mounting pressure of 10 psi the pad has a thermal impedance, Zth , of 0.6◦ C-cm2 /W.
What is RθCS , the case to sink thermal resistance?
(b) The junction to case thermal resistance of the MOSFET is RθjC = 0.3◦ C/W and at a
junction temperature Tj of 150◦ C its on-state resistance, RDS(on) = 40 mΩ. If the heatsink
temperature can be maintained at 50◦ C, what is the maximum continuous current that
the device can conduct?
25.8 What is Tf in Fig. 25.6(d)?
25.9 The SiC MOSFET characterized by the transient thermal impedance curves of Fig. 25.8 is
subjected to an overload condition that is cleared by a protection circuit in 3 µs. The MOSFET
had been operating at a junction temperature of Tj = 150◦ C. How much energy can the device
be allowed to dissipate during the fault to maintain Tj ≤ 200◦ C?
25.10 Consider the “single pulse” thermal response of a system ( e.g., as illustrated in Fig. 25.8). This
response Z θ (t) is in fact the thermal step response of the system. That is, Z θ (t) represents the
temperature rise response over time to a unit step in input power at t = 0.
(a) Show that if one can treat the dynamic thermal system as a linear, time-invariant (LTI)
system (e.g., the circuit elements in the model of Fig. 25.6(b) are LTI), then we can write
the temperature response to a short pulse in power of amplitude P starting at t = 0 and
having duration t1 as:
∆TjC = P [Zθ (t) − Zθ (t − t1 )]

(b) For the same LTI system assumption, what would be the temperature rise response to a
sequence of two pulses of amplitude P, each of duration t1, one starting at t = 0 and the
second starting at t = t2 (> t1 )?

25.11 Using the CREE SiC SOA of Fig. 25.10(b), determine the derated limiting boundary for a 1 ms
pulse if the case temperature is 125◦ C. What is the maximum allowable ID ?

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