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A Time-Dependent Analytical Analysis of Heat Transfer in a PCB During a Thermal Excursion
October 24, 2012 |Estimated reading time: 6 minutes
This paper was originally presented at IPC APEX EXPO in February 2012 in San Diego and appeared in the September issue of The PCB Magazine.
A great deal of work has already been done to determine the equilibrium temperature of a PCB when exposed to a heat source such as the thermal environment of reflow soldering. This study will go beyond an equilibrium condition and explore the temperature-time distribution of the board when a variable temperature heat source is applied to both outer surfaces. For simplicity, the model will be a two-sided board. Obviously, the model board has two material interfaces. An interesting observation is that anywhere within the board, including the material interface, thermal energy must be conserved. There is not a similar requirement for the temperature.
Consequently, at the material interfaces, we can expect the thermal properties of the board to change in a profound manner. A similar situation occurs when a fluid passes through a shock wave. This will be reflected in such board properties as the thermal stresses in the various layers and the resulting warp. This phenomenon also explains and quantifies why a thermal shock can be devastating, while a slow temperature rise to the same endpoint may well be tolerated.
The analysis will use a one-dimensional, time-dependent model (i.e., there are two independent variables). This necessitates a partial differential equation to describe the temperature variation within the board. The boundary conditions are the outer temperature of the board, which is the temperature of the heat source on both outer surfaces. The third boundary condition is at the copper epoxy interface where conservation of thermal energy is required.
Introduction
Techniques for measuring the surface temperature of a PCB have been available for some time. Measurements for assessing the temperature at various positions in the laminate are at least problematic. Inserting thermal couple wires well into a thin laminate will more than often distort the temperature as they now become a part of the thermal mass of the system—the so-called Helmholtz effect (to some extent the tool used to measure any physical quantity will disturb the measurement).
In this case, the thermocouple wire will conduct heat locally, away from the laminate, cooling that area of the package. The larger the thermocouples, the greater the issue. In situations of this nature, it often is best to use analytical models to estimate the value of the desired measurement (i.e., the internal temperature history of a PCB as it goes through a heat excursion). If the temperature history distribution in the PCB package can then be estimated, the internal shear stress can be quantified. Once this is accomplished, the likelihood of forming an internal delamination can be established. Developing such a procedure and using it to determine the internal laminate history as the PCB is exposed to a temperature ramp, such as in reflow, is the objective of this analysis.
Analysis
To simplify the analysis, the investigative package will be a symmetrical, double-sided PCB. The copper thickness is defined to be 1.7 mils. The thickness of the internal laminate is 60 mils and composed of FR-4, (Figure 1). The PCB will be exposed to several temperature ramps, all ending at 600°F.
Figure 1. The symmetrical double-sided PCB for investigation features a copper thickness of 1.7 mils and an internal laminate thickness is 60 mils, and is composed of FR-4 laminate.
It will be assumed that the board has been preheated to a uniformed temperature of 3,000°F before the temperature ramp.
According to first principals, heat conduction through a solid is governed by Fourier’s law of heat conduction. The assumption is made that the heat flow is one-dimensional. Consequently, there are two independent variables: time and position. For one-dimensional, time-dependent heat transfer, Fourier's law becomes
Equation 1
A unique solution requires three boundary conditions:
- At t=0, T=3,000°F for the entire board
- At x=0, T is specified by the ramp temperature
- Along the axis of symmetry of the board,
Unfortunately, since the second geometric derivative of temperature appears in (1), the first derivate must be continuous. Conservation of energy requires a consistent heat-flow rate across the interface of copper and FR-4. Since the thermal properties of the material change at the interface, conservation of energy requires a discontinuous adjustment in the temperature gradient. To avoid the issue we will introduce the following transformation1:
Equation 2
X=k β
thus making β the new geometric variable.
After making the transformation, Equation 1 becomes:
Equation 3
A closed-form solution exists for Equation 1 modified by the above transformation, but it involves several infinite series, which defy extraction. A better course of action is to use a numerical solution. For the finite difference approximation to Equation 3, the time derivative will be replaced by a forward difference and the geometric derivative by a centered difference. The result is:
Equation 4
It is shown in Reference 1 that a numerical solution for Equation 4 will converge provided:
Equation 5
Results
A numerical integration of Equations 4 and 5 will now be examined for temperature ramps of 2, 4, and 6 degrees Fahrenheit/second. Initially, the board is at a uniform temperature of 3,000°F. The temperature ramp is completed when the surface temperature of the copper reaches 600°F. The temperature ramps are shown in Figure 2.
Figure 2. Copper surface temperature.
The integration of Figures 3 and 4 also defines the temperature within the board as it passes along the temperature ramp.
Figure 3 shows the board’s temperature profile when the surface of the board first reaches 600°F.
Figure 3. Temperature profile at surface equilibrium.
As seen, the temperature gradient is large in the outer portion of the board. Then, beyond the copper/FR-4 interface, the gradient rapidly approaches zero. This is caused by the favorable heat transfer properties of copper and the adverse heat transfer properties of FR-4. As seen later, this abrupt behavior causes large shear stresses to form at the interface, which can result in copper delamination.
In order to estimate the shear stress, it is first necessary to calculate the difference in the temperatures of the copper and the substrate. For this purpose, the characteristic temperature of the PCB component is defined as the average temperature of each component. The result is presented in Figure 4, which shows the temperature difference as a function of temperature ramp. The temperature differential is obviously nonlinear. The rate of change in the temperature differential rapidly changes as the temperature ramp increases. It then becomes nearly constant at the high end.
In Reference 2, it is shown that the shear stress at the interface is:
Equation 6
Where is the coefficient of thermal expansion, L is the characteristic length of the copper feature and is the temperature differential. The shear stress is presented as a function of the copper feature size and ramp rate in Figure 5. It will be noticed that the shear stress increases rapidly as the feature size diminishes. In fact, one can expect to experience shear stresses in excess of 30K psi for present day PCBs.
Figure 4. Average temperature difference between copper and laminate.
Figure 5. Shear stress at Cu/laminate interface.
Summary
The analysis above has developed a technique for numerically integrating the one-dimensional, time-dependent Fourier heat conduction equation for a PCB which is stressed by a heating ramp. This in turn will quantify the relationship between the process and design parameters and the stresses incurred during a thermal excursion. The analysis shows there are substantial temperature gradients at the interface of the copper and FR-4 components. It is furthermore shown that these temperature gradients can produce very substantial shear stress at the copper/FR-4 interface. This stress is inversely proportional to the size of the copper feature and directly proportional to the gradient of the heating ramp.
References
1. Max Jakob, Heat Transfer, John Wiley and Sons, New York, 1962
2. J. Lee Parker, Proceedings of IPC APEX, April 2011
J. Lee Parker retired from Lucent Bell Laboratories as a Distinguished Member of the technical staff after a 27-year career. He has more than 30 years in electronics and is now an industry consultant. He was a joint recipient of the Best Technical Paper Award at the 1996 IPC Expo and is a past chairman of the IPC Flexible PC Design Guide Subcommittee.