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October 2017 • The PCB Design Magazine 29 centers of the spheres, then the total height of the Cannonball stack is equal to the height of two pyramids plus two radii. Through simple geometry and a little bit of algebra we can approximate the radius of a sin- gle sphere (r) as [2] : Equation 1 And base area (A flat ) as: Equation 2 Because the model assumes the ratio of A matte /A flat = 1, and there are only 14 spheres, the Cannonball-Huray model can therefore be simplified as: Equation 3 Where: K SR (f) = roughness correction factor, as a function of frequency; δ (f) = skin-depth, as a function of frequency in meters; r = the radius of spheres in meters; A flat = base area in sq. meters. Effective Dk Due to Roughness Everyone involved in the design and man- ufacture of PCBs knows that one of the most important properties of the dielectric material is D k . We often assume the value reported in manufacturers' data sheets is the intrinsic prop- erty of the material. But in fact, it is the effective dielectric constant (D keff ) generated by a specific test method. When you compare simulation against measurements, you will often see a dis- crepancy in D keff , due to increased phase delay caused by surface roughness. If D k and R z roughness parameters from the manufacturers' data sheets are known, then the effective D k due to roughness (D keff_rough ) of the fabricated core laminate can now be easily esti- mated by [1] : Equation 4 Where: H smooth is the thickness of dielectric from data sheet; R z is 10-point mean roughness from data sheet; Dk is dielectric constant from data sheet. As I mentioned earlier, there are many EDA tools that implement the Huray model. The Po- lar Instruments Si9000e 2017 field solver [5] now includes the Huray roughness model, and it is now one of my go-to tools for helping me get an "OK answer now." I especially like it because the user interface is so intuitive and easy to use. Practical Modeling of a High-speed Backplane Channel Case Study A traditional high-speed serial link back- plane channel model has three separate parts. They are two plug-in circuit cards and a back- plane. Neglecting vias, the high-speed chan- nel can be quickly modeled as three separate transmission line segments with connectors in- between. Each transmission line segment is modeled separately using Polar Si9000e field solver. The S-parameters are then saved as touchstone for- mat to be brought into a channel modeling software tool like Keysight ADS [6] . The best way to demonstrate this is through a practical case study example. I will use the Amphenol-FCI Examax demo platform. This is a platform I helped design in 2013 to showcase the Examax connector performance at 28GB/s NRZ. A picture of the platform is shown in Fig- ure 3. Among other test structures designed into the backplane, there are four channels with dif- ferent overall lengths. For simplicity only one channel topology shown will be used for com- parison in this case study. The PCBs were fabri- cated with Nelco N4000-13epsi material [7] clad with VSP foil from Oak-Mitsui [8] . The respective transmission line design and data sheet param- eters are summarized in Table 1. The first step is to determine the effective D k due to roughness for the cores and prepregs used on the daughter cards and backplane. By applying Equation 4 to the respective values in Table 1, the D keff results are summarized in Figure 4. The next step is to determine the radius of spheres and base tile area for Huray model, as summarized in Figure 5. Because electro-depos- ited (ED) foil has a matte side and drum side, PRACTICAL MODELING OF HIGH-SPEED BACKPLANE CHANNELS