Transient Analysis Example with Heating Curves and Structure Functions
For this demonstration example, test vehicles were constructed by mounting a MOSFET on three different PWBs: 30x40mm PWB with 2 inner layers, 15x20mm PWB with 2 inner layers, and 15x20mm PWB with no inner layers. The other side of the test coupons were then bonded to a heat sink. All impedances measured are referenced to the heat sink surface temperature. Previous tests of the MOSFET directly mounted to the heat sink revealed a junction-to-sink thermal resistance of 0.66°C/W. The following heating curve plot compares the transient response of the three test vehicles from initial heating to steady state.
The next figure shows the differential structure function plot associated with this heating curve data. Starting from the left, all three cases show virtually identical diff. structure function plots from 0 to 3.5°C/W. The common peak at about 0.2°C/W which is associated with thermal interface between the die and the copper heat spreader inside the MOSFET. Moving to the right, we next see a valley at about 0.6-0.7°C/W which is indicative of heat flow transitioning out of the component's copper heat spreader. This is consistent with the previously measured thermal resistance of 0.66°C/W for the direct-mounting to the heat sink.
Moving further rightward, the peak at 1.0°C/W indicates the transient heat flow across the thermal interface between the transistor body and the PWB. The peak is associated with both the transit of material properties as well as the significant change in the heat flow geometry. Continuing further to the right, a broad anti-peak occurs between 2-3.5°C/W indicating the transient two-dimensional spreading within the PWB. Note that all three cases show virtually identical structure function plots from 0 to 3.5°C/W despite the fact that from 1 to 3.5°C/W heat is substantially flowing within the three different PWBs.
From the differential structure function plot, data for A and B diverge at about 3.5°C/W and the data for B and C diverge at about 4.3°C/W. Looking at the heating curve, we see a similar divergence at 8 seconds between cases A and B at an impedance of 4.0°C/W and a divergence between B and C at about 10.5 seconds at about 5.5°C/W. So there is a slight disagreement of divergence points between the structure function and the heating curve. This result is expected because the differential structure function detects divergences in the heat flux patterns which occur before the heat reaches the interface of interest when the interface is substituted with an alternate material. So, despite the sensitivity of the differential structure function, the reported cumulative resistances of the divergences between multiple configurations are not tightly bound to the geometrical interfaces due a broader distortion of the heat flux networks.
At impedances above 4.5°C/W, the three cases are diverging with each showing a similar peak but at different impedances. This peak indicates saturation of the transient lateral spreading within PWB and that heat is beginning to move out of the PWB in a transverse direction. Looking further to the right, the vertical asymptotes of 7.7°C/W, 9.5°C/W, and 10.5°C/W equal the final equilibrium values shown on the heating curve and would also equal the thermal resistances measured with a simple steady-state test method.
In summary, the combination of heating curve and structure function analysis when applied to a few variations of component design, allows accurate interpretation of the thermal performance of various thermal design implementations. Without multiple variations in component design, the transient response interpretations are much more speculative. Importantly, the structure function does not independently differentiate between changes in transient heat-flow geometry within a consistent material and consistent transient flow-geometry within a body composed of material with different thermal properties (conductivity and heat capacity). The peaks and anti-peaks of the structure function are indicative of both transits in material properties and variation in heat-flow-geometry. Features of the differential structure function are not generally tightly bound to the physical/geometrical interfaces due to the broad distortion in the heat-flow-network that occurs when aspects of the thermal design are changed. Direct interpretation of the heating curve or cumulative structure function offer the benefit of less distortion in interface-resistances although will less inherent sensitivity. And lastly, the final vertical asymptote of the structure function is simply the steady state thermal resistance of the component in the particular cooling configuration.