PCB designers always paid tremendous attention to dielectric and metal losses, the factors that define most of the physical limits for digital signal transmission. In recent years, modeling of losses has been constantly perfected. One goal was to improve model accuracy, to better match physical experiments and measurements. Another one was to give simulation tools wideband, mathematically consistent equations, convenient and suitable for time and frequency domain analysis.

I remember times when a dielectric model was defined by two numbers, relative permittivity and loss tangent, not related to any particular frequency, and simulators delivered a constant capacitance accompanied by a conductance linearity growing with frequency. It was enough to take an inverse Fourier transform from these equations to find out that the model is non-causal: a considerable portion of the impulse response occurred at negative times. Now we are fortunate to have a wideband causal Djordjevic-Sarkar dielectric model, and a number of its modifications. Not only is it mathematically sound, but it agrees well with physical observations.

As to metal models, the problem did not seem so bad at the beginning. As frequencies did not get beyond gigahertz, skin effect formula was good enough to describe metal losses. And it is causal, too, being a square root of the complex frequency. An inverse Laplace transform of this formula gives us a time response that is identically zero at negative times.

However, as speeds further increased, it turned out that rough surface of the metal combined with decreasing skin depth makes losses higher. Yes, the major focus was losses, and for this reason the first available roughness models, such as Hammerstad, Hemispherical, Huray aimed to predict loss increase due to metal roughness. All these models were originally considered frequency dependent multipliers to the skin resistance. However, a widespread practice was to apply these (purely real) multipliers to the complex metal impedance, thus increasing real and imaginary parts in the same proportion. Many of us did not see the hitch: if skin resistance is causal, why increasing it at high frequency could be a problem?   However, there was a problem: a purely real but frequency dependent multiplier is not a causal function, as so becomes its product with causal skin impedance.

The first publication that provided a causal computational model was 2012 DesignCon paper of Eric Bracken, where the author presented a causal Huray roughness model. This time, it was a complex causal frequency-dependent multiplier, providing the same resistive loss as its non-causal counterpart, but making a different contribution into complex inductance. The paper however did not show the derivation of the causal model.

While at DesignCon 2017,  I attended Bert Simonovich’s presentation, “A Practical Method to Model Effective Permittivity and Phase Delay Due to Conductor Surface Roughness”, where he demonstrated the use of his Cannonball roughness model (a special case of Huray’s roughness model, where  14 equal sized spheres are stacked in a cubic close packing arrangement).

The loss in Bert’s model perfectly matched the one in measured stripline S-parameters, but a minor discrepancy remained in phase delay. After the presentation, we had a conversation regarding the cause of this discrepancy: could the reason be a non-causal roughness model? That was a start of our work (together with Igor Kochikov, physicist and mathematician) on the paper “A Causal Conductor Roughness Model and its Effect on Transmission Line Characteristics”, presented at DesignCon 2018.

In this paper, we described the derivation of causal Cannonball-Huray and Hammerstad roughness models. We also proved that metal roughness increases the inductive portion of the complex impedance by a larger value than its resistive portion. Therefore, with a non-causal roughness model, we underestimate PUL inductance, propagation delay, and characteristic impedance of the conductor.

What remains unsolved? First of all, the Kramers-Kronig integral transformation in closed form can be found only for a narrow class of the roughness equations. Even if exists, it takes considerable efforts to find the integral. A wide class of analytical models (including Groiss, Hemispherical, Bushminsky) and the loss correction dependences found directly from measurements, do not belong to this class. Overall, most of the practically important cases cannot be handled by the analytical approach, and we need a more universal solution.

In the paper presented here, we restore the unknown real part of the complex inductance by fitting. Unfortunately, the rational functions do not make a suitable basis because metal impedance is described by fractional power of complex frequency, as we have even for the smooth metal due to skin impedance. For that reason, we propose another basis comprised from causal functions that contains fractional power of complex frequency. This basis works remarkably well: it was possible to find causal versions of roughness factors for the Groiss, Hemispherical, and Bushminsky models. The accuracy of the models can be verified by closeness of the fit to the known part of the complex inductance, and then to the given “loss correction factor.” We compare complex dependences describing causal Hammerstad, Cannonball-Huray, and the three models above side-by-side and analyze their important properties, some of which are not even visible from the loss correction dependencies. Most importantly, we can now work with arbitrary data, for example defined by a tabulated loss correction found from measurements.

This paper was presented at DesignCon 2020.