# Category Archives: Thermal resistance

## Optimizing Heat Sink Base Spreading Resistance to Enhance Thermal Performance

(This article was featured in an issue of Qpedia Thermal e-Magazine, an online publication dedicated to the thermal management of electronics. To get the current issue or to look through the archives, visit http://www.qats.com/Qpedia-Thermal-eMagazine.) Heat sinks are routinely used in electronics cooling applications to keep critical components below a recommended maximum junction temperature. The total resistance to heat transfer from junction to air, Rja, can be expressed as a sum of the following resistance values as shown in Equation 1 and displayed in Figure 1. Where, Rjc is the internal thermal resistance from junction to the case of the component. RTIM is the thermal resistance of the thermal interface material. Rf is the total thermal resistance through the fins. The final term in Equation 1 represents the resistance of the fluid, e.g. air, going through the heat sink where m is the mass flow rate and Cp is the heat capacity of the fluid. As Equation 2 shows, Rcond and Rconv are the conduction and convection resistance respectively through the heat sink fins respectively. Fig. 1 – Resistance network of a typical heat sink in electronics cooling. 

Rs stands for the spreading resistance that is non-zero when the heat sink base is larger than the component. The next few sections show the full analytical solution for calculating spreading resistance, followed by an approximate simplified solution and the amount of error from the full solution and finally the use of these solutions to model and optimize a heat sink.

Lee et al.  derived an analytical solution for the spreading resistance. Figure 2 shows a cross-section of a circular heat source with radius a on the base with radius b and thickness t. The heat, q, originates from the source, spreads out over the base and dissipates into the fluid on the other side with heat transfer coefficient, h. For heat transfer through finned heat sinks, the effective heat transfer coefficient is related to thermal resistance of the fins, Rf as shown in Equation 3. For square heat source and plates, the values of a and b can be approximated by finding an effective radius as shown in equations 4 and 5. Where,
h = heat transfer coefficient [W/m2K]
a = effective radius of the heater [m]
Aheater = area of the heater [m2]
b = effective radius of the heat sink base [m]
Abase = total area of the heat sink base [m2]

The derivation of the analytical solution starts with the Laplace equation for conduction heat transfer and applying the boundary conditions. Equation 6 shows the final analytical solution for spreading resistance. The values for the eigenvalue can be computed by using the Bessel function of the first kind at the outer edge of the plate, r=b as shown in Equation 7. Where,
k = Thermal Conductivity of the plate or heat sink [W/mK]
J1 = Bessel function of the first kind
λn = Eigenvalue that can be computed using Equation (3) at r = b
t = thickness of the heat sink base [m]

Lee et al.  also offered an approximation as shown in Equation 8 along with the approximation for the eigenvalues as shown in Equation 9. This approximation eliminates the need for calculating complex formulas that involve the Bessel functions and can be computed by a simple calculator. Approximation vs. Full Solution

Simons  compared the full solution (Equations 6 and 7) with the approximations shown in (Equations 8 and 9). The problem contained a 10 mm square heat source on a 2.5 mm thick plate with a conductivity of 25 W/mK, 20 mm width and varying length, L as shown in Figure 3. Figure 4 shows that the percentage error increases with length but stays relatively low. Less than 10% error is expected for lengths up to 50 mm; five times the length of the heater. This is acceptable for most engineering problems since analytical solutions are first-cut approximations that should later be verified through empirical testing and/or CFD simulations. However, the full analytical solution should be used if the heater-to-heat sink base area difference gets much larger or if a more accurate solution is desired. Fig. 3 – Example problem for comparing analytical and approximate solutions for spreading resistance. Fig. 4 – Percent error between the analytical and the approximate solution of spreading resistance for the example shown in Figure 3. 

Optimizing Heat Sink Performance

The goal of any electronic cooling solution is to lower the component junction temperature, Tj. For a given Rjc and RTIM, the objective is to maximize heat sink performance by reducing the spreading resistance, Rs, and the fin resistance Rf.

The spreading resistance can be reduced by increasing base thickness. However, most electronics applications are limited by total heat sink height and thus any increase in base thickness leads to shorter fins which reduce the total area of the fins Afins. For a fixed heat transfer coefficient (the heat transfer coefficient is a function of fin design and air velocity and we can assume it is fixed for this exercise) a reduction in the fin area increases Rf as shown in Equation 2. Equation 10 shows this combined heat sink resistance, Rhs, as a function of the spreading and fin resistance. Thus, for a given fin design, the thermal engineer must choose the appropriate heat sink base thickness to optimize heat sink performance. To illustrate this point, let’s take an example of an application with the parameters as shown in Table 1.

Figure 5 shows a graph of the total thermal resistance of the heat sink, Rhs and spreading resistance, Rs as a function of base thickness for copper and aluminum material. (Note that the final term from Equations 1 and 10 is ignored because it is constant and does not contribute to the understanding of spreading resistance). The graph shows that spreading resistance improves monotonically with increased base thickness. However, the total thermal resistance has an optimal point between 2-4 mm base thicknesses. For base thicknesses less than 2 mm, there is a sharp increase in spreading resistance which leads to a higher overall thermal resistance. Fig. 5 – Total and spreading resistance of the example shown in Table 1 for a 50 mm heat sink.

On the other hand, increasing the base thickness above 4 or 5 mm gives diminishing marginal returns; the improvement in spreading resistance is minimal compared to the increase in thermal resistance due to the reduced fin area. Additionally, the graph also shows that higher conductivity materials such as copper, improves thermal performance across the entire domain.

Conclusion

The heat spreading resistance is an important factor when designing a heat sink for cooling electronics components. The full analytical solution for calculating the spreading resistance, shown in Equations 6 and 7, can be substituted with the approximations shown in Equations 8 and 9 with minimal error. The error increases with increased difference between the heat sink base and heater size and the complete analytical model should be used if needed. The analytical model can be used to choose the right heat sink base thickness that optimizes heat sink performance as shown in Figure 5.

Techniques such as higher conductivity materials, embedded heat pipes, vapor chambers etc. are available if the spreading resistance is major obstacle in the cooling. Thermal engineers must balance the increased weight and cost of such techniques against the benefits for each application.

References
1. “Spreading Resistance of Single and Multiple Heat Sources,” Qpedia. September 2010
2. Seri Lee et al. “Constriction/Spreading Resistance Model for Electronics Packaging,” 1995.

For more information about Advanced Thermal Solutions, Inc. (ATS) thermal management consulting and design services, visit https://www.qats.com/consulting or contact ATS at 781.769.2800 or ats-hq@qats.com.

## Thermal Resistance and Component Temperature

To maintain operation, the heat must flow out of a semiconductor as such a rate as to ensure acceptable junction temperatures. This heat flow encounters resistance as it moves from the junction throughout the device package, much like electrons face resistance when flowing through a wire. In thermodynamic terms, this resistance is known as conduction resistance and consists of several parts. From the junction, heat can flow toward the case of the component, where a heat sink may be located. This is referred to as ÎJC, or junction to case thermal resistance. Heat can also flow away from the top surface of the component and into the board. This is known as junction to board resistance, or Î˜JB. Source: JESD51-2, Integrated Circuits Thermal Test Method – Natural Convection, JEDEC, March 1999.

Î˜JB is defined as the temperature difference between the junction and the board divided by the power when the heat path is from junction to board only. To measure Î˜JB, the top of the device is insulated and a cold plate is attached to the board edge (Figure 1). This is the true thermal resistance, which is the characteristic of the device. The only problem is that, in a real application one does not know how much power is being transmitted from different paths.

Due to the multiple heat transfer paths within a component, a single resistance cannot be used to accurately calculate the junction temperature. The thermal resistance from junction to ambient must be broken down further into a network of resistances to improve the accuracy of junction temperature prediction. A simplified resistor network is shown in Figure 2. As board layouts become denser, there is a need to design optimized thermal solutions that use the least amount of space possible. Simply put, there is no margin to allow for over-designed heat sinks with tight component spacing. Accounting for the effect of board coupling is an important part of this optimization. The possibility for using an oversized heat sink exists only if the junction to case heat transfer path is considered.

To ensure a 105°C junction temperature at 55°C ambient a typical component (see Table 1) needs a heat sink resistance of 2.05°C/W (if we ignore board conduction). When board conduction is taken into account, the actual junction temperature could be as low as 74°C, assuming the board temperature is the same as the air temperature. This indicates a heat sink that is larger than necessary. From this example, it is clear that all heat transfer paths from the component junction must be considered. Using just the Î˜JC and Î˜CA values can lead to a larger than optimal heat sink and may not accurately predict operating junction temperatures. Using the proposed correlation can also predict junction temperature when the board temperature is known from experimentation, as shown in Figure 3. 