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EXAMPLE 4-5 A metric that is used in the study of colliding gravitational waves is the KahnPenrose metric (see Fig. 4-1). The coordinates used are (u, v, x, y). With u 0 and v < 0, this metric assumes the form ds 2 = 2 du dv (1 u)2 dx 2 (1 + u)2 dy 2 Find the Christoffel symbols of the rst kind (using (4.15) for this metric.

Balancing material properties with PWB manufacturability is critical. Materials that exhibit excellent properties have failed because of difficulties experienced when fabricating the PWB, such as fracturing in drilling, routing or scoring, difficulty in texturing drilled holes for copper plating, resin recession, or hole wall pull-away during thermal stress.

10.6.1 Example of Material Types and Properties versus Assembly Reliability To highlight a couple of these conclusions, consider the following test. First, multilayer PWBs made from the materials in Table 10.3 were processed through Infra-Red (IR) reflow cycles at different peak temperatures. The PWB was a 10-layer, 0.093-in. (2.6 mm) thick board designed to fail, meaning the copper weights and patterns, construction, and resin contents were chosen so that the board would be more sensitive to thermal cycles. In addition, the dwell time at the peak temperature was 1.5 minutes. This allowed differences in material performance to be detected more clearly. Figure 10.21 graphs the percentage of boards surviving six reflow cycles without any evidence of blisters, measles, or delamination.

260C

Notably, the first material to exhibit defects is the conventional dicy-cured, high-Tg FR-4 material.This material began to exhibit defects when the peak temperature reached 240 C.At a peak temperature of 260 C, the conventional dicy-cured FR-4 materials, both the 175 C Tg and the 140 C Tg products, all exhibited evidence of defects. On the other hand, the materials with higher decomposition temperatures, both the 150 C Tg and the 175 C Tg products, all survived six cycles to 260 C. 10.6.2 Example of Material Types and Properties versus Long-Term Reliability In another test, three high-Tg materials were evaluated through Interconnect Stress Test (IST) testing.11 This particular test provided insight into the effect of thermal expansion and decomposition temperature on long-term reliability, as assessed by the IST method. The IST test method uses an electric current to heat test coupons that contain a network of plated through holes. The test samples are generally preconditioned several times to simulate exposure to

TABLE 10.4 Materials Evaluated through IST Testing Glass Transition Temp. ( C) 175 175 175 Decomposition Temp. ( C) 310 335 335 % Expansion, 50 260 C (40% Resin Content) 3.5 3.4 2.7

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assembly processes, and then cycled back and forth between an elevated temperature, most commonly 150 C, and room temperature The samples are cycled in this manner until failure occurs, with failure generally being defined as a change in measured resistance of 10 percent The materials evaluated in this example are described in Table 104 Note that the Tg values of these materials are the same, but differences exist in decomposition temperatures and thermal expansion values Product D* is similar to product D except that it has a higher level of thermal expansion Product D* exhibits approximately the same thermal expansion as Product C, but Product D* has a higher decomposition temperature Product D has both a high decomposition temperature and a very low level of thermal expansionThe PWB tested was a 14-layer, 0120-in (31 mm) thick multilayer with 0012-in (0.

30 mm) diameter plated through holes The average copper plating in the via was 08 mil (203 micron), although 10 mil (254 micron) had been requested Figure 1022 charts the average number of cycles to failure (10 percent resistance change in the plated via net) for each material type at each preconditioning level: as is (no preconditioning), three cycles to 230 C, six cycles to 230 C, three cycles to 255 C, and six cycles to 255 C Clearly, the two materials with improved decomposition temperatures exhibit much better performance than the conventional high-Tg product Also, in comparing product D to product D*, it appears that the lower thermal expansion of product D does offer improvement in the number of cycles to failure, but this improvement is smaller in comparison to the improvement due to the higher decomposition temperature, at least for this PWB design.

The benefit of reduced thermal expansion becomes more important as the thickness of the PWB increases In addition, the technique used to reduce the CTE values in this case also provides benefits in PWB manufacturability..