HFSS15: Introduction to Causality Issues for Simulations

Signal integrity (SI) engineers are challenged to efficiently design and verify high-speed PCBs and interconnects by simulation. When working in the multi-GHz frequency range, simulation typically requires full-wave electromagnetic analysis. The Finite Element method (FEM) is the most common way to rigorously and accurately simulate 3D structures in a broad frequency range and extract the S parameters, which will then be used in a transient SPICE simulation as black boxes to predict the signals in the overall system. A critical consideration for these is the causality behavior of the resulting S-parameter models. Causality means that an output signal cannot start to change before the input signal changes. A necessary and sufficient condition for the causality is that the frequency domain transfer functions (S parameters) have to satisfy the Kramers - Kronig relations [1]. This technical note presents the best practices for getting accurate and causal solutions and not how to enforce causality of existing broad band S parameters. The Kramers - Kronig relationship will just be applied as a causality checker.

Physical Parameters Compared to Simulation

There might be two sources of the discrepancies between the physical parameters and the results of the simulation: one is the improper mathematical model used for the FEM calculations and the other is the numerical solution itself. Both discrepancies can result in causality violation. The paper focuses on the proper mathematical models. It is supposed that the adaptive mesh refinement has converged for the finite element solution, so the mesh is fine enough to approximate the exact solution well.

Modeling Issue: Solve Inside and Surface Impedance Boundaries

One of the most important modeling issues is how the solution should be treated in metals when the frequency range is broad. It is well known that the skin depth varies between infinity and very small value. A large skin depth typically necessitates solving the field inside the metal objects. This solve inside option is impractical for very small skin depths (high frequency), because a large mesh is needed to model the very strong decay of the field. A widely used method is to apply the infinite half space skin impedance as a Surface Impedance Boundary Condition (SIBC), replacing the solution inside the metal objects. It is accurate for high frequencies, MHz to GHz for typical interconnect and PCB geometries, but very inaccurate/unphysical for low frequencies when the finite thickness of the object approaches the skin depth d, because the real part of the impedance goes to zero with decreasing the frequency instead of going to an appropriate finite DC value. An obvious option would be to combine two methods; apply solve inside at low frequencies and use SIBC at high frequencies. However problems would arise in the transition region, strongly violating causality by introducing jumps or non-physical smoothing. The paper presents a method, which uses the analytical impedances of the conductors with finite thickness in the whole frequency region. The method doesn't violate the causality, since the analytical formulas satisfy the Maxwell's equations and the DC point can also be approximated with arbitrary accuracy.

Modeling Material Properties

A common dielectric material model is to assume constant properties across the frequency range. However this is not a physical model of the loss mechanism in dielectrics and violates the causality of the results. Fitting a Debye or Djordjevic - Sarkar material model [3] to the input data will provide a causal model definition.

Modeling Surface Roughness

Another issue is how to take the loss mechanism of surface roughness into account. Two methods for obtaining causal surface roughness models will be presented and discussed in the paper.

When extracting S parameters of the system, ports are inserted into the simulation model. Ports won't introduce any error when they model perfect transmission line terminations, which is possible in FEM codes, unlike Method of Moment (MoM) codes, where so called port calibration is always necessary. In geometries where a transmission line port is not possible, the FEM method allows for non-transmission line terminations (e.g. lumped gap ports). Such a situation most often occurs when there is not even room for a complete transmission port in the design. The user is still interested in extracting S parameters as if the model was terminated by perfect transmission line wave ports. A method will be presented to deembed the parasitic effects introduced by using lumped gap ports.

If all the above mentioned techniques are used in a FEM extraction and the adaptive meshing procedures have converged within a reasonably small tolerance, the S parameters will be accurate and causal. Real-life problems will be presented to demonstrate this using ANSYS HFSS, where all the techniques are implemented.

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Surface Impedance Boundary Condition (SIBC) for Metal Traces of Finite Thickness