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38 The PCB Design Magazine • May 2015 square tile base. Nine spheres are on the first row, four spheres in the middle row, and one sphere on top. Because the Cannonball model assumes the ratio of A matte /A flat = 1, and there are 14 spheres, Equation 4 can be simplified to: equation 5 Where: K SR (f) = roughness correction factor, as a function of frequency, due to surface roughness based on the Cannonball model; r = sphere radius in meters; δ (f) = skin-depth, as a function of frequency in meters; A flat = area of square tile base surrounding the 9 base spheres in sq. meters. As shown in Figure 5, there are 5 square- based pyramids connecting the centers of all 14 spheres forming a stacked lattice structure. A single pyramid, labeled ABCDE, is shown for reference. Given that each side of the pyramid ABCDE = 2r, it can be shown that: Since: Then the radius of a single sphere is: And the area of the square flat base is: CANNONBALL STACk FOR CONDuCTOR ROuGHNESS MODELING continues Figure 4: Cannonball model show- ing a stack of 14 uniform size spheres (left). Top and front views (right) shows the area (a flat ) of base, height (h rMs ) and radius of sphere (r). Figure 5: Cannonball model with pyramid lattice structure. Five pyramids form a stacked lattice structure connecting the centers of all 14 spheres. Total height (h rMs ) equals the stacked height of 2 pyramids plus the diameter (2r) of a single sphere. article