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.

In situ characterization of sediment oxygen demand across the spectrum of non-resuspension to full resuspension with varying bed shear stress

Sediment oxygen demand (SOD) can be a significant oxygen sink in various types of water bodies, particularly slow-moving waters with substantial organic sediment accumulation. In most settings where SOD is a concern, the prevailing hydraulic conditions are such that the impact of sediment resuspension on SOD is not considered. However, in the case of Bubbly Creek in Chicago, Illinois, the prevailing slack water conditions are interrupted by infrequent intervals of very high flow rates associated with pumped combined sewer overflow (CSO) during intense hydrologic events. These events can cause resuspension of the highly organic, nutrient-rich bottom sediments, resulting in precipitous drawdown of dissolved oxygen (DO) in the water column. While many past studies have addressed the dependence of SOD on near-bed velocity and bed shear stress prior to the point of sediment resuspension, there has been limited research that has attempted to characterize the complex and dynamic phenomenon of resuspended-sediment oxygen demand. To address this issue, a new in situ experimental apparatus referred to as the U of I Hydrodynamic SOD Sampler was designed to achieve a broad range of velocities and associated bed shear stresses. This allowed SOD to be analyzed across the spectrum of no sediment resuspension associated with low velocity/ bed shear stress through full sediment resuspension associated with high velocity / bed shear stress. The current study split SOD into two separate components: (1) SODNR is the sediment oxygen demand associated with non-resuspension conditions and is a surface sink calculated using traditional methods to yield a value with units (g/m2/day); and (2) SODR is the oxygen demand associated with resuspension conditions, which is a volumetric sink most accurately characterized using non-traditional methods and units that reflect suspension in the water column (mg/L/day). In the case of resuspension, the suspended sediment concentration was analyzed as a function of bed shear stress, and a formulation was developed to characterize SODR as a function of suspended sediment concentration in a form similar to first-order biochemical oxygen demand (BOD) kinetics with Monod DO term. The results obtained are intended to be implemented into a numerical model containing hydrodynamic, sediment transport, and water quality components to yield oxygen demand varying in both space and time for specific flow events. Such implementation will allow evaluation of proposed Bubbly Creek water quality improvement alternatives which take into account the impact of SOD under various flow conditions. Although the findings were based on experiments specific to the conditions in Bubbly Creek, the techniques and formulations developed in this study should be applicable to similar sites.