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58 PCB007 MAGAZINE I JUNE 2019 coefficients for each equation were plotted. A trendline was again added to each of the graphs (Figure 5). Again, second-degree polynomial trendlines strongly correlated to the plotted coefficient data (R2>0.974). The trendline equations (Table 2) for these coefficients would help determine the coefficients for unknown aspect ratios. The variable X in these equations represents the aspect ratio. Therefore, any aspect ratio that is desired can be substituted into each of these equations to determine the first- and second-order coefficients and intercept for the quadratic equation that will ultimately deter- mine the resistance of a plated through-hole based on copper thickness. Table 3 shows the equations that were derived from this informa- tion. Graphs of resistance versus copper thickness at aspect ratios ranging from 8:1 to 40:1 gener- ated from these equations are depicted in Fig- ure 6. These graphs and equations were used to create a simple calculator in Excel that would output a maximum resistance value based on a given aspect ratio and minimum copper thick- ness. This value is then used as an upper spec- ification limit when processing boards through four-wire Kelvin testing. Plated through-holes that failed this specification limit (resistance measurements higher than this limit) would have copper thicknesses that are lower than the user-defined minimum. Conversely, the calculator can also output the copper thickness given a known resistance and aspect ratio. Figure 5: Quadratic equation coefficients with trendlines. Table 3: Resistance equations by aspect ratio. Table 2: Quadratic coefficient trendlines by coefficient.