Abstract
The adverse impact of temperature on device/material reliability has been emphasized often in this book. The degradation rate for most devices/materials tends to accelerate exponentially with increasing temperature. Therefore, for reliability reasons, lower temperature device operation is usually preferred. However, many devices (both electrical and mechanical) can generate significant amounts of heat as they are being operated. Once device operation begins, the rate of increase in temperature of the device/material will depend upon on the heat generation within the device, the heat capacity of the materials, and the heat dissipation from the device to the heat sink (which is often the ambient).
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Notes
- 1.
In Chap. 11 (Sect. 11.7) a failure mechanism (Hot Carrier Injection) was discussed that can be contrary to this general statement.
- 2.
The term external energy is usually reserved for any relative motion of the macroscopic system and/or any system energy associated with external fields.
- 3.
Often the specific heat is subscripted as c v (specific heat at constant volume) or c p (specific heat at constant pressure). This textbook will use c = c v .
- 4.
In general, the internal energy of a system can be changed by the flow of Heat (into or out of system) and/or by Work (done on or by system). The conservation of energy statement \( ({\text{d}}U = \delta{Heat}+\delta{Work})\) is often referred to as the First Law of Thermodynamics. δHeat and δWork imply that these are not exact differentials. Thus, before we can integrate, we must know details of the processes by which the heat is changed and/or how the work is performed.
- 5.
Using Eq. (3b) and solving for ΔT, you can see that water (due to its relatively large specific heat) is capable of absorbing significant amounts of heat with only relatively small changes in its temperature. Now you can perhaps better understand why water is an excellent cooling agent and widely used—from nuclear reactors and combustion engines to fire fighting.
- 6.
Recall that the divergence theorem states:
\( \int\limits_{V} {{\vec{\nabla}}} \cdot \vec{J}\,{\text{d}}V = \int\limits_{A} {\mathop J\limits^{ \to } } \cdot {\text{d}}\vec{A}, \) where V is the volume of interest which is bounded by a surface of area A.
- 7.
In heat flow analysis, the heat sink fixes the temperature at specified locations and plays a role similar to the use of ground in electrical circuits.
- 8.
Actually, not all of the heat flow (from metal plate bottom surface area A 0) is vertically downward. Part of the heat flow will be from the metal plate spreading laterally. Thus, the true heat flow across the surface area A, which bounds the volume V of interest, will be through an effective area A eff such that A eff > A 0. However, if S 0 is much less than the metal plate dimensions (length and width), then A eff ≈ A 0.
- 9.
Remember, the chip power P is given simply by: P = (Current) × (Voltage).
- 10.
Recall that the volume V of interest contains all of the materials in the generation region plus all the materials between the generation region and the heat sink/ambient. A is the area of the surface that bounds the volume V of interest.
- 11.
Identity used: \( A = \int\limits_{0}^{A} {{\text{d}}A} \).
- 12.
Since T sink is assumed to be a fixed temperature, then we have used: \( \frac{{\partial \left[ {T(x,t) - T_{\text{Sink}} } \right]}}{\partial t} = \frac{{\partial \left[ {T(x,t)} \right]}}{\partial t}. \)
- 13.
Thermal relaxation after a time t = 5τ is given by: \( \frac{{\Updelta T_{0} - \Updelta T(t = 5\tau )}}{{\Updelta T_{0} }} = 1 - \exp \left( {\frac{5\tau }{\tau }} \right) = 0.993 = 99.3\;\% \). After t = 10τ, the thermal relaxation is 99.995 % complete.
- 14.
The actual power pulse could be more complicated than a simple rectangular shape; however, in Chap. 13 we learned how to convert complicated waveforms into rectangular equivalents.
- 15.
Systems with large thermal time-constants are often referred to as systems with large thermal inertia. Such systems, with large thermal inertia, have great difficulty responding quickly to short-duration power pulses. This can be good or bad depending on the details of device application.
- 16.
Newton’s law for cooling states that the rate of heat loss from a body is proportional to the temperature difference between the body and its surroundings:
\( \frac{{{\text{d}}({\text{Heat}}_{{{\text{Loss}}\;{\text{from}}\;{\text{Body}}}} )}}{{{\text{d}}t}} \propto [T_{\text{Body}} - T_{\text{Ambient}} ] \). However, for convection cooling in general, deviations from this simple linear dependence can be observed.
- 17.
Note that the solar specific power density delivered to earth is equivalent to fourteen one-hundred watt light bulbs per square meter of the earth’s surface and represents a lot of power!.
Bibliography
Arpacz, V., Conduction Heat Transfer, Addison-Wesley Publishing, 1966.
Carslaw, H. and J. Jaeger, Conduction of Heat in Solids, 2nd Ed., Oxford Press, 1959.
Kittel, C. and H. Kroemer, Thermal Physics, 2nd Ed., W.H. Freeman and Co., 1980.
Sears, F and G. Salinger, Thermodynamics, Kinetic Theory, and Statistical Thermodynamics, 3rd Ed., 1975.
Thomas, L., Fundamentals of Heat Transfer, Prentice-Hall Inc., 1980.
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McPherson, J.W. (2013). Heat Generation and Dissipation. In: Reliability Physics and Engineering. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00122-7_16
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DOI: https://doi.org/10.1007/978-3-319-00122-7_16
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