Sourajeet Roy

Closed-Form Delay and Crosstalk Models for

In this paper, a closed-form matrix rational-approximation algorithm is proposed to efficiently model the delay and crosstalk noise of coupled RLC on-chip interconnects. A key feature of the proposed algorithm is that, for any rational order, the approximation is obtained analytically in terms of predetermined coefficients and the per-unit-length parameters. As a result, the proposed method is not limited to fixed number of poles and provides a mechanism to increase the accuracy for cases when inductive effects are significant, the length of the line increases, or when the rise time of the signal becomes sharper. An error criterion is provided to select the order of approximation. The algorithm is tested for various single- and coupled-interconnect scenarios. The 50% delay and overshoot results match that of SPICE with less than 2% average error. The crosstalk results also accurately match those of SPICE with less than 4% average error.

RLC

This paper presents a delay and crosstalk noise model for coupled resistance-inductance-capacitance (RLC) on-chip interconnects. The proposed algorithm, based on a modified Lie formula, is used to convert the solution of the transmission line network into delay algebraic equations to obtain the time domain response. The proposed algorithm is not limited to fixed number of coupled RLC lines or to specific topologies and can be used to model both identical and nonidentical multiconductor lines and loads. For the example presented in this paper, the 50% delay and crosstalk using the proposed method agrees with SPICE results to within 0.75% average error.

On-Chip Interconnects Using a Matrix Rational Approximation

In this paper, a waveform relaxation algorithm is presented for the efficient transient analysis of large power distribution networks. The network is modeled as an orthogonal grid of transmission lines where a delay-extraction-based macromodel is used to represent each line segment in the time domain. Novel partitioning and iterative techniques are proposed for fast convergence and improved scalability of the proposed relaxation algorithm. The overall algorithm is highly parallelizable and exhibits good scaling with both the size of the circuit matrices involved and the number of CPUs available. Numerical examples are presented to illustrate the validity and efficiency of the proposed algorithm.

Efficient Delay and Crosstalk Modeling of RLC Interconnects Using Delay Algebraic Equations

This paper presents an efficient approach for modeling irregular shaped power distribution networks (PDN) in high-speed packages. The proposed methodology is based on discretization of the plane into an orthogonal grid of transmission line segments. Using a delay-extraction-based model for each line segment, a compact circuit model is achieved where the size of the circuit matrices depend only on the nodes of the orthogonally discretized structure and all other internal nodes due to the macromodel are eliminated. This approach of eliminating the internal variable due to the transmission line macromodel is further extended to model skin effect losses without augmenting the circuit matrices. The proposed work has been successfully implemented for a variety of PDN structures and geometries and has been shown to yield significant savings in memory and run time costs compared to the existing simulation program with integrated circuits emphasis macromodels.

Delay-Extraction-Based Waveform Relaxation Algorithm for Fast Transient Analysis of Power Distribution Networks

With the use of low powered devices, susceptibility of high-speed interconnects to electromagnetic interference (EMI) is becoming a critical aspect of signal integrity analysis. For modeling the EMI in time domain, commercial circuit simulators like SPICE typically use longitudinal segmentation methodologies to discretize the interconnect network. For long lines as found in printed circuit board or cables, a large number of longitudinal segments are required to capture the response of the network leading to inefficient simulations. In this study, a waveform relaxation (WR) algorithm for the efficient EMI analysis of multiconductor transmission line networks is presented. Techniques to compress the size of the subcircuits, reduce communication overheads, and accelerate the convergence of the WR iterations are provided. The overall algorithm is demonstrated to be highly parallelizable and exhibits good scaling with both the size of the network involved and the number of central processing units available.

Efficient Modeling of Power/Ground Planes Using Delay-Extraction-Based Transmission Lines

In this paper, a waveform relaxation algorithm is presented for efficient transient analysis of large transmission-line networks. The proposed methodology represents lossy transmission lines as a cascade of lumped circuit elements alternating with lossless line segments, where the lossless line segments are modeled using the method of characteristics. Partitioning the transmission lines at the natural interfaces provided by the method of characteristics allows the resulting subcircuits to be weakly coupled by construction. The subcircuits are solved independently using a proposed hybrid iterative technique that combines the advantages of both traditional Gauss-Seidel and Gauss-Jacobi algorithms. The overall algorithm is highly parallelizable and exhibits good scaling with both the size of the network involved and the number of CPUs available. Numerical examples have been presented to illustrate the validity and efficiency of the proposed work.

Electromagnetic Interference Analysis of Multiconductor Transmission Line Networks Using Longitudinal Partitioning-Based Waveform Relaxation Algorithm

Typical modeling algorithms for multilayered irregular shaped power distribution networks are based on a finite difference solution of the Helmholtz equation. In this paper, the finite difference solution is demonstrated to be equivalent to a discretization of the Telegraphers partial differential equations for multiconductor transmission lines (MTL). With this concept, an efficient macromodeling algorithm for multilayered structures based on MTL theory is presented. The electromagnetic coupling between the plane layers due to wraparound currents is captured by the inductive and capacitive coupling between the multiconductor lines. A delay extraction-based macromodel is used to represent the MTL in SPICE that can better capture the distributed effects of the structure than existing lumped models. This approach is successfully implemented for multilayered structures with irregular geometries and is shown to be more accurate and efficient compared with existing SPICE lumped models.

Longitudinal-Partitioning-Based Waveform Relaxation Algorithm for Efficient Analysis of Distributed Transmission-Line Networks

This paper presents a numerical convolution based approach for transient simulation of distributed networks characterized by band limited frequency domain data when terminated with arbitrary nonlinear circuits. The proposed algorithm uses time-frequency decompositions to extract multiple propagation delays of a distributed network and the associated attenuation losses in a piecewise manner, and implements inverse fast Fourier transform to efficiently convert the frequency response into a sum of delayed time domain responses. Numerical examples illustrate that the proposed algorithm shows significantly more accurate results for networks with multiple long delays when compared to existing numerical convolution techniques.

Macromodeling of Multilayered Power Distribution Networks Based on Multiconductor Transmission Line Approach

In this paper, a waveform relaxation algorithm is presented to efficiently model power distribution networks. The proposed algorithm is based on physically partitioning the large circuit into smaller disjoint subcircuits. A key feature of the partitioning scheme is that it ensures that the noise injected by each transient source is localized within the subcircuit where the source resides thereby leading to efficient convergence of the algorithm. The iterative solution of the subcircuits is parallelizable and scales efficiently with the number of processors. A numerical example has been provided to illustrate the validity of the proposed algorithm over full SPICE simulations.

Transient Simulation of Distributed Networks Using Delay Extraction Based Numerical Convolution

This paper presents a model for fast transient analysis for single line resistance-inductance-capacitance (RLC) on-chip interconnects. The proposed algorithm, based on a modified Lie formula, is used to convert the solution of the transmission line network into delay algebraic equations, which can be solved in a closed form manner. The proposed algorithm is not limited to fixed number of coupled RLC lines or to specific topologies and can be used to model both identical and non-identical multi-conductor lines and loads.