Rational Function Interpolation of Electromagnetic Transfer Functions of High-Speed Interconnect Systems from Discrete Time-Domain and Frequency-Domain Data

We present new methodologies to generate rational function approximations of broadband electromagnetic responses of linear and passive networks of high-speed interconnects, and to construct SPICE-compatible, equivalent circuit representations of the generated rational functions. These new methodologies are driven by the desire to improve the computational efficiency of the rational function fitting process, and to ensure enhanced accuracy of the generated rational function interpolation and its equivalent circuit representation. Toward this goal, we propose two new methodologies for rational function approximation of high-speed interconnect network responses. The first one relies on the use of both time-domain and frequency-domain data, obtained either through measurement or numerical simulation, to generate a rational function representation that extrapolates the input, early-time transient response data to late-time response while at the same time providing a means to both interpolate and extrapolate the used frequency-domain data. The aforementioned hybrid methodology can be considered as a generalization of the frequency-domain rational function fitting utilizing frequency-domain response data only, and the time-domain rational function fitting utilizing transient response data only. In this context, a guideline is proposed for estimating the order of the rational function approximation from transient data. The availability of such an estimate expedites the time-domain rational function fitting process. The second approach relies on the extraction of the delay associated with causal electromagnetic responses of interconnect systems to provide for a more stable rational function process utilizing a lower-order rational function interpolation. A distinctive feature of the proposed methodology is its utilization of scattering parameters. For both methodologies, the approach of fitting the electromagnetic network matrix one element at a time is applied. It is shown that, with regard to the computational cost of the rational function fitting process, such an element-by-element rational function fitting is more advantageous than full matrix fitting for systems with a large number of ports. Despite the disadvantage that different sets of poles are used in the rational function of different elements in the network matrix, such an approach provides for improved accuracy in the fitting of network matrices of systems characterized by both strongly coupled and weakly coupled ports. Finally, in order to provide a means for enforcing passivity in the adopted element-by-element rational function fitting approach, the methodology for passivity enforcement via quadratic programming is modified appropriately for this purpose and demonstrated in the context of element-by-element rational function fitting of the admittance matrix of an electromagnetic multiport.

Ultrasonic wave reflection measurements on self-compacting pastes and concretes

The objective of this study was to extend the use of combined longitudinal (P-wave) and shear (S-wave) ultrasonic wave reflection (UWR) to monitor the setting and stiffening of self-compacting pastes and concretes. An additional objective was to interpret the UWR responses of various modified cement pastes. A polymeric buffer with acoustic impedance close to that of cement paste, high impact polystyrene, was chosen to obtain sensitive results from the early hydration period. Criteria for initial and final set developed by our group in a prior study were used to compute setting times by UWR. UWR results were compared with standard penetration measurements. Stiffening behavior and setting times for normal cement pastes, pastes modified with mineral and chemical admixtures, self-compacting pastes, and concretes were explored using penetration resistance, S-wave and P-wave reflection. All three methods showed that set times of pastes varied linearly with w/c, that superplasticizer and fly ash delayed the set times of pastes, and that differences in w/cm, sp/cm, and fa/cm could be detected. Final set times determined from UWR correlated well with those from penetration resistance. Initial set times from S-wave reflection did not correlate very well with those from penetration resistance. Final set times from P-wave and S-wave reflection were roughly the same. Pastes with different chemical admixtures were tested, and the effects of these admixtures on stiffening were determined using UWR. Self-compacting concretes were studied using UWR, and their response and setting times were largely similar to that of corresponding self-compacting pastes. The P-wave reflection response was explored in detail, and the phenomenon of partial debonding and the factors affecting it were explained. Partial debonding is probably caused by autogenous shrinkage at final set, and was controlled and limited by water. The extent of partial debonding was higher with the transducers placed on the side as opposed to the bottom, and the S-wave transducer seemed to promote debonding in the P-wave reflection, whereas the P-wave transducer seemed to reduce debonding in the S-wave reflection. Simultaneous formwork pressure testing and UWR were performed; however, no clear correlation was seen between the two properties.