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44 The PCB Design Magazine • May 2015 1. Determine H RMS_drum and H RMS_matte from Equation 6 and Equation 7. 2. Determine the radius of spheres for drum and matte sides: 3. Determine the area of the square flat base for drum and matte sides: 4. Determine K SR_drum (f) and K SR_matte (f) : 5. Determine the average K SR_drum (f) and K SR_matte (f): 6. Apply Equation 8 to determine total in- sertion loss of the PCB trace. Summary and Results The results are plotted in Figure 10. The left plot compares the simulated vs measured inser- tion loss for data sheet values and design pa- rameters. Also plotted is the total smooth in- sertion loss (crosses) which is the sum of con- ductor loss (circles) and dielectric loss (squares). Remarkably there is excellent agreement up to about 30 GHz by just using algebraic equations and published data sheet values for Dk, Df and roughness. The plot shown on the right is the simu- lated (blue) vs measured (red) effective dielec- tric constant (Dkeff), and is determined by the equations shown. As can be seen, the measured curve has a slightly higher Dkeff (3.76 vs 3.63 @ 10 GHz) than published. According to [6], the small increase in the Dk is due to the anisotropy of the material. When the measured Dkeff (3.76) was used in the model, for core and prepreg, the IL results shown in Figure 11 (left) are even more remark- able up to 50 GHz! Figure 12 is a Comparison of the Cannon- ball model against the H&J model. The results show that the H&J is only accurate up to ap- proximately 15 GHz compared to the Cannon- ball model's accuracy to 50 GHz. Conclusions Using the concept of cubic close-packing of equal spheres to model copper roughness, a practical method to accurately calculate sphere size and tile area was formulated for use in the Huray model. By using published roughness pa- rameters and dielectric properties from manu- facturers' data sheets alone, it has been demon- strated that the need for further SEM analysis or experimental curve fitting, may no longer be required for preliminary design and analysis. When measurements from CMP-28 model- ing platform, fabricated with FR408HR and RT foil, was compared to this method, there was excellent correlation up to 50 GHz compared to the H&J model accuracy to 15 GHz. The Cannonball model looks promising for a practical alternative to building a test board and extracting fitting parameters from mea- sured results to predict insertion loss due to sur- face roughness. Supplemental Information Figure 13 are examples of close-packing of equal sphere models, and their respective equations. The model on the left is the hex- CANNONBALL STACk FOR CONDuCTOR ROuGHNESS MODELING continues article