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

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     (13)

the corresponding time delay is calculated as,

     (14)

The corresponding insertion loss on the line is approximately

     (15)

where the total loss constant α for the TEM mode on the line is comprised of the conductor and dielectric loss parts [25],

     (16)

and

     (17)

Note that herein ε_r is the dielectric constant (real part of permittivity) of the effective dielectric inside the transmission line, which includes both the substrate dielectric matrix and ERD.

Model Setup for Numerical Simulations

Similar to the previous studies [13], the stripline simulation numerical electromagnetic model has been created. It includes the dielectric matrix material characterization of the substrate dielectric obtained through the extrapolation DERM technique as in [20], [22], and thin layer-like objects representing conductor surface roughness as the ERD: above the trace (“foil” side) and below the trace ("oxide" side). The corresponding ERD layers are also placed on the reference (ground/return) planes. Figure 13 shows a cross-sectional view of the numerical model setup. The line length of the stripline structure was 391.414 mm (15.4 inches); stripline traces were 17.5 mm thick (0.5-oz copper) and 340 mm (13.5 mil) wide. The impedance of the single-ended line was 50 Ω. Cross-sectional dimensions for all the three test lines were identical, except for the foil roughness. Since the length of the modeled line is comparatively long, to make the models more computationally efficient, each model was subdivided into two equal segments, and then cascading of the corresponding ABCD matrices was done.

Figure 13: Numerical model setup.

Numerical Simulations Results with Homogenized Frequency-dependent ERD

The models were simulated using the Finite Integral Technique (FIT), a time domain solver [9]. Time domain solvers are suited to capturing phase results across wide frequency bands. A mesh size in the models was about 2 million cells. The waveguide ports were used for excitation. The measured and modeled data are shown in Figures 14-16. The dielectric parameters of the homogeneous ERD layers in these models are as shown in Figures 9-11. The agreement of the modeled and measured results for all the three test scenarios with STD, VLP, and HVLP foils validate the proposed analytical approach. The high-frequency behavior is captured better as compared to the non-dispersive models as in [13], [14] due to the frequency-dependent DKr and DFr parameters extracted using the equivalent capacitance ERD model.

Figure 14: Measured and modeled S₂₁ results for a stripline structure with STD foil.

Figure 15: Measured and modeled S₂₁ results for a stripline structure with VLP foil.

Figure 16: Measured and modeled S₂₁ results for a stripline structure with HVLP foil.

Numerical Model of Layered ERD Structure

Another way of roughness dielectric numerical modeling was also tested. The “layered” model was set up in the same way as the other models with the only one difference. The foil layer is specified differently from the previous models with homogeneous ERD parameters. In the "layered" model for the STD foil, the roughness dielectric properties are split into three parts: the top 1/3^rd part (close to metal) has independent of frequency DK_r = 16, the middle 1/3^rd part has DK_r= 12, and bottom 1/3^rd part (next to matrix) has DK_r =8. DFr is set as 0.17 in all the three sublayers. In this case, each sublayer is very thin, adding significant mesh count and therefore increasing simulation time.

However, herein, when space mapping for the “layered” model is applied, the object is not split into three separate layers/objects with homogeneous dielectric constants, but the object material properties change depending on the position within the object according to the specified "space map". Note that a "space map" based model does not introduce a new kind of material, but is used to define, for a normal (or anisotropic) material, a generic spatial distribution. This allows for modeling complicated and arbitrary materials. In this work, the ERD itself is specified this way within the matrix material.

In Figure 17, the measured phase is compared to the modeled using “space map” of the ERD layer. The tested cases are the dielectric constants of all three ERD sublayers having first DK_r =12; then all of them having DK_r =16; and finally, the “layered” roughness dielectric “space map” object with three different DK_r values defined consequently. Loss tangent DF_r=0.17 in all the layers. Phase results are chosen for comparison because they are the most sensitive to the model parameters choice. As Figure 17 shows, there is an excellent agreement between the measured and the layered model results. From Table 2, the difference between these two results is indeed small (within a few degrees) when compared to the overall phase.

Figure 17: The phase of measured and modeled structures over a narrow frequency band of ~24–26GHz; ERD with DKr=12, with DKr=16, and with space map layered structure.

Table 2: Phase of analytical and layered ERD model at a number of frequencies.

Conclusions

In this work, an analytical model to calculate effective roughness dielectric (ERD) parameters for conductor surface roughness of a PCB foil is presented. Based on the microscopic analysis of the roughness profile, a concentration dependence of metallic inclusions in the transition between the ambient dielectric matrix and copper is obtained. Using such a concentration dependence, the equivalent capacitance associated with the roughness layer is calculated analytically. Then the parameters of the effective roughness dielectric are extracted from this equivalent capacitance. The ERD parameters obtained from the analytical model are frequency dependent unlike in the previous works; therefore, they describe the high-frequency behavior (at data rates of a few dozen Gbps) of PCB interconnects more accurately than the frequency-independent models. The proposed model is applied to three stripline test scenarios with three different types of foils—STD, VLP, and HVLP, and is validated by an excellent agreement between the full-wave FIT numerical modeling and measurements. Two types of numerical models are obtained: using homogeneous effective roughness dielectric and using space mapping when modeling a "layered" ERD. The "layered" ERD provides the closest to the measured result when S₂₁ phases are compared.

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This paper, with original title “Equivalent Capacitance Approach to Obtain Effective Roughness Dielectric Parameters for Copper Foils,” was first presented at the 2018 IPC APEX EXPO Technical Conference and published in the 2018 Technical Conference Proceedings.

* Marina Koledintseva was with Oracle during the writing and presentation of this paper.

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