Issue link: https://iconnect007.uberflip.com/i/509724
May 2015 • The PCB Design Magazine 35 CANNONBALL STACk FOR CONDuCTOR ROuGHNESS MODELING continues (TD), compared to the fixed spatial length of the trace. The H&J roughness correction factor (K HJ ), at a particular frequency, is solely based on a mathematical fit to S. P. Morgan's power loss data and is determined by [2] : equation 2 Where: K HJ = H&J roughness correction factor; ∆ = RMS tooth height in meters; δ = skin depth in meters. Alternating current (AC) causes conductor loss to increase in proportion to the square root of frequency. This is due to the redistribution of current towards the outer edges caused by skin- effect. The resulting skin-depth (δ) is the effec- tive thickness where the current flows around the perimeter and is a function of frequency. Skin-depth at a particular frequency is deter- mined by: equation 3 Where: δ = skin-depth in meters; f = sine-wave frequency in Hz; μ 0 = permeability of free space =1.256E-6 Wb/A-m; σ = conductivity in S/m. For annealed copper σ = 5.80E7 S/m. The model has correlated well for microstrip geometries up to about 15 GHz, for surface roughness of less than 2 RMS. However, it proved less accurate for frequencies above about 5 GHz for very rough copper [3] . In recent years, the Huray model [4] has gained popularity due to the continually in- creasing data rate's need for better modeling accuracy. It takes a real-world physics approach to explain losses due to surface roughness. The model is based on a non-uniform distribution of spherical shapes resembling snowballs and stacked together forming a pyramidal geom- etry, as shown by the SEM photo in Figure 2. By applying electromagnetic wave analysis, the superposition of the sphere losses can be used to calculate the total loss of the structure. Since the losses are proportional to the surface area of the roughness profile, an accurate esti- mation of a roughness correction factor (K SRH ) can be analytically solved by [1] : equation 4 Where: K SRH (f) = roughness correction factor, as a function of frequency, due to surface roughness based on the Huray model; = relative area of the matte base compared to a flat surface; a i = radius of the copper sphere (snowball) of the i th size, in meters; = number of copper spheres of the i th size per unit flat area in sq. meters; δ (f) = skin-depth, as a function of frequency, in meters. Figure 2: seM photograph of electrodeposited copper nodules on a matte surface resembling snowballs on top of heat treated base foil. Photo credit Oak-Mitsui. article