Transmission Lines and Cables
Transmission Lines and Cables

Macromodeling
The “TransmissionLine” element is a fundamental device available in any standard circuit simulation environment. However, with the continuous increase of data rates, more and more refined modeling is required by the electronic designers. As an example, frequencydependent metal and dielectric losses are no longer negligible, since they cause high signal degradation at high frequencies. This is the reason why the models available in the most advanced circuit solvers are being and will be continuously improved over the years. The most critical aspects are accuracy, causality, passivity, robustness, applicability to a large number of line conductors and, of course, reduced computational cost. Our group contributed to this research by proposing several macromodeling approaches. All these approaches are aimed at the derivation of SPICEready subcircuits. Among the various contribution we highlight three examples. The first led to the inclusion of field coupling terms in the line macromodels, thus allowing to perform systemlevel EMI analyses. The second example is a SPICE macromodel for lowloss interconnects based on transient scattering representations. The third and most recent development is an advanced macromodel for frequencydependent lines based on the Generalized Method of Characteristics. This macromodel is named TOPLine and is currently available in IBM internal circuit simulation environment (PowerSPICE).

Nonuniformity (deterministic)
Several efforts have been devoted to the characterization and modeling of nonuniform transmission lines. These structures are characterized by a dominant direction of propagation, but their crosssection is not translationinvariant. As an example, in the picture you see a portion of a package structure with six signal conductors. If the variations of the cross section are small, a good approximation of the physical behavior is provided by the Telegraphers Equations with spacedependent perunitlength parameters. A few validations of this approach have been proposed. However, the main contribution by our group is focused on modeling and simulations. Several advanced algorithms have been proposed, including highorder finitedifference timedomain methods, timedomain spaceexpansionmethods, and timedomain waveletbased schemes allowing for time/space adaptivity of the discretization mesh.

Crosstalk and Field Coupling
One of the main reasons for the ubiquity of transmissionline structures in EMC studies is their susceptibility to unwanted interference. Crosstalk can be described as “internal interference” due to the intrinsic propagation properties of a multiconductor transmission line. Field coupling (EMI) leads instead to a behavior of the transmission line as a receiveing antenna, which captures any impinging electromagnetic field. Accurate predictions of these two effects are of paramount importance for EMC engeineers. Much of our research activity has been devoted to the development of modeling algorithms and tools for accurate and efficient simulations. These have been addressed both in standalone codes for algorithm prototyping and in macromodels to be employed in standard circuit simulation environments like SPICE.

Efficient Simulation Algorithms
Transmissionline modeling and simulation in frequency and in time domain is a classical problem. Nonetheless, a quick look at the recent literature on the subject suggests that more and more advanced and optimized methods are being continuously proposed. Our group has developed several of these techniques, addressing from time to time different aspects of the modeling procedure. We can mention frequencydomain simulation of transmission line networks including junction elements and field coupling, several transmissionline macromodels for standard circuit simulation environments (see above), and a number of timedomain techniques for transient analysis of transmission lines with different characteristics. Among the effects that were analyzed we include field coupling, arbitrary losses and dispersion, and line nonuniformities. In addition, advanced adaptive waveletbased techniques have been proposed. As an example, in the picture we report the typical outcome of the latter techniques applied to propagation of a single pulse along a matched uniform line. The location of the wavelet coefficients in the spacetime plane is depicted with a dot, clearly indicating the high degree of time/space adaptivity of the computational mesh to the solution being computed.

Random Bundles and Statistics
Wire bundles are widely used in several application areas. As an example, avionic or automotive electronic devices and apparatus are often linked by bundles with tens to hundreds of wires. These bundles are characterized by random twists and displacements of single wires within the crosssection, thus forming random nonuniform transmission lines. A deterministic approach for the modeling and simulation of propagation, crosstalk, and field coupling of such structures is therefore not adequate. A statistical approach is certainly more suitable. Our research activities focused on several aspects in the framework of statistical characterization of wire bundles. Fractal theory has been applied for the geometrical modeling of the random wire cross section. This approach combined with statistical prediction methods allowed to derive several results on crosstalk and field coupling.

PerUnitLength Parameters
Any transmission line structure is fully characterized by its perunitlength parameters of resistance, inductance, capacitance and conductance. Therefore, the precise calculation of these parameters for a given line geometry is an essential step for the subsequent analyses. The line parameters are computed by solving a 2D field problem. This is a classical problem with many solutions available in the literature. We have developed a tool based on Finite Elements (FEM) for the computation of these parameters. This tool (PULP) is more aimed at educational purposes rather than research. In fact, PULP has been extensively used in the labs of our EMC courses at Politecnico di Torino.