Calculation of Frequency-Dependent Effective Roughness Dielectric Parameters for Copper Foil Using Equivalent Capacitance Models

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Applying the same image processing technique as for the roughness magnitude extraction, but performing summation for each column of pixels, one can get the volume concentration of metallic inclusions in the transition between pure dielectric to pure metal. Figure 5 shows the function for different types of foils. It is seen that 0% concentration corresponds to dielectric matrix, while 100% to smooth copper. The transitions are comparatively smooth – the left-hand front corresponds to the foil side, and the right-hand side to the "oxide" side. The smoother the conductor side, the more abrupt the metallic concentration slope is.

Figure 5: Volume concentration of metallic inclusions in black oxide STD (a), VLP (b), and HVLP (c) foil on PPO blend substrate.

The profiles on the foil and oxide sides can be fitted using exponential or polynomial functions as is shown in Figures 6-8. For simplicity of calculating integrals analytically in (5), (6), and (12), the exponential approximation  will be further used. Note that the parameter b herein is the same as K₁ in (1). The approximation data for a number of studied samples of black-oxide foils on PPO Blend substrates are presented in Table 1. The parameter δrms herein is the root-mean-square error at the approximation.

Figure 6: Approximation of volume concentration of metallic inclusions as a function of D distance from the smooth conductor: "foil" side (a) and "oxide" side (b) on STD foil.

Figure 7: Approximation of volume concentration of metallic inclusions as a function of distance from the smooth conductor: “foil” side (a) and “oxide” side (b) on VLP foil.

Figure 8: Approximation of volume concentration of metallic inclusions as a function of distance from the smooth conductor: “foil” side (a) and “oxide” side (b) on HVLP foil.

Table 1: Exponential approximation of profile functions on “foil” and “oxide” sides of copper foils.

Calculation of ERD Parameters Using the Proposed Analytical Model

The proposed equivalent capacitance model was applied to calculate the ERD parameters of the three types of foils as in Table 1. Figures 9-11 show the calculated frequency dependences for DK_r and DF_r of the corresponding ERD layers. The thicknesses of the layers are also determined from the metallic concentration profiles. Note that in the previous publications [4], [6], [13], [14], the ERD parameters were independent of frequency. However, the new analytical model shows that there is frequency dependence. The ERD parameters for the STD foil on its “foil” and “oxide” sides differ significantly because the “foil” side is much rougher than the “oxide” side. The corresponding differences for the sides on the VLP and HVLP foils do not differ that much, though they are not equal. Though the extracted ERD parameters for the VLP foil herein are close to those of the HVLP, the thicknesses of the layers to be modeled differ: the HVLP layers are thinner than VLP. Note that the calculated ERD results are not the same as reported in [10], because the test samples studied herein are different from those in [13].

In the present study, the roughness parameters of HVLP and VLP samples are not much different, while in [13] the VLP and HVLP foils are quite distinct.

Figure 9: Effective roughness dielectric parameters as functions of frequency for STD foil: DK_r (a) and DF_r (b).

Figure 10: Effective roughness dielectric parameters as functions of frequency for VLP foil: DK_r (a) and DF_r (b).

Figure 11: Effective roughness dielectric parameters as functions of frequency for HVLP foil: DK_r (a) and DF_r (b).

Numerical Simulations Based on Analytically Calculated ERD

The measured insertion loss |S₂₁|, dB and time delay t on a transmission line, i.e., a single-ended stripline, increase as conductor roughness magnitude increases, and hence, the values DK_r and DF_r of the corresponding ERD layers increase. This is illustrated by Figure 12.

Figure 12: Insertion loss (a) and time delay (b) on 16-inch stripline with PPO blend dielectric and different foil types.

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