Tingdong Zhou

Lossy transmission line simulation based on closed-form triangle impulse responses

Analytical frequency-domain expressions for single and coupled transmission lines with triangular input waveforms are first developed. The inverse Fourier transform is then used to obtain an expression for the time-domain triangle impulse responses for frequency-independent transmission line parameters. The integral associated with the inverse Fourier transform is solved analytically using a differential-equation-based technique. Closed-form expressions for the triangle impulse responses are given in the form of incomplete Lipschitz-Hankel integrals (ILHI) of the first kind. The ILHI can be efficiently calculated using existing algorithms. Combining these closed-form expressions for the triangle impulse responses with a time-domain convolution method using a triangle impulse as a basis function, provides an accurate and efficient simulation method for very lossy transmission lines embedded within linear and nonlinear circuits.

Triangle impulse response (TIR) calculation for lossy transmission line simulation

Triangle impulse responses (TIRs) for lossy transmission lines are accurately calculated using both an inverse fast Fourier transform algorithm and an accelerated inverse Laplace transform algorithm. Frequency dependent transmission line parameters, i.e., R, L, G, and C, are employed to model the skin effect and the frequency dependent electrical properties of the substrate material. The calculated TIR can be further used to carry out time domain simulations for a large number of lossy transmission lines. Frequency dependent line parameters, R, L, G, and C should be used in specific cases to assure the causality of signal waveform, the accuracy of the time delay, and the amplitude of the waveform evaluations in the time domain.

Application of the Pade approximation via Lanczos (PVL) algorithm to electromagnetic systems with expansion at infinity

ROMES (reduced-order modeling of electromagnetic systems) was developed based on the frequency domain finite difference (FDFD) method in our lab. Previously, the Pade via Lanczos (PVL) algorithm was used for the reduced-order modeling of the linear system with finite frequency values used for the expansion point. In this paper, the PVL method, with expansion at infinity, has been used to enhance the performance of ROMES. The advantage of this method is avoidance of the LU decomposition step that is costly both in speed and memory. It also provides better wide frequency band results than PVL with expansion at a finite frequency, which only gives correct results near the expansion point. The disadvantage is that the dimension of the reduced-order model must be higher relative to PVL with finite value expansion for accurate approximation of the original electromagnetic system. Although it suffers from this disadvantage, PVL with expansion at infinity makes it possible to solve some complicated electromagnetic problems efficiently.

Closed-form representations for triangle impulse responses associated with single and coupled lossy transmission lines

A fast simulation method was proposed for single and coupled transmission lines that are connected to linear and non-linear circuit elements by Z. Chen et al. (1999) [1]. In that method, the time-domain voltages and currents at the ends of the lines are approximated by a series of triangular expansion functions. A time-stepping procedure can then be employed for the circuit simulation provided that the triangle impulse responses for the lines are known. In [1], a triangle impulse response database for the lossy transmission lines is employed. Other simulation tools are used to calculate the required lossy transmission line triangle impulse responses numerically. The numerical results for the triangle impulse responses are then used with a convolution algorithm to carry out the circuit simulation. In our work, analytic frequency-domain expressions for single and coupled transmission lines with triangular input waveforms are first developed. The inverse Laplace transform is then used to obtain an expression for the time-domain triangle impulse responses. The integral associated with inverse Laplace transform is solved analytically using a differential-equation-based technique. Closed-form expressions for the triangle impulse responses are given in the form of incomplete Lipschitz-Hankel integrals (ILHIs) of the first kind. The ILHIs can be efficiently calculated using algorithms developed by S.L. Dvorak and E.F. Kuester (1990). Combining these closed-form expressions for the triangle impulse responses with the method proposed in [1], provides an accurate and efficient simulation method for transmission lines embedded within linear and non-linear circuits.

Application of subspace projection approaches for reduced-order modeling of electromagnetic systems

The requirements for fast switching speeds, low voltage, low power, tightly integrated, and mixed analog-digital circuits, have brought the need for electromagnetic computer-aided design capability to integrated circuit design, especially for packaging and interconnect design. For clock frequencies in the GHz regime and rise times of a few tens of ps, full wave electromagnetic analysis may be required for packaging design and interconnection simulation. Subspace projection approaches, including the Pade via Lanczos (PVL), Krylov, and rational Krylov algorithms, have been used for reduced-order modeling of wide-band electromagnetic systems. A frequency segmentation technique has also been used with the Lanczos algorithm to obtain benchmark data and for scattering parameter extraction from the electromagnetic system. From the various techniques, the combined PVL/frequency segmentation technique is the most promising for efficient and accurate modeling of electromagnetic systems.

Characterization of electromagnetic systems using Pade/spl acute/ approximation with expansion at infinity

The Pade/spl acute/ via Lanczos (PVL) method, with expansion at infinity, has been used to enhance the performance of the simulation program reduced-order modeling of electromagnetic systems (ROMES), developed based on the frequency domain finite difference (FDFD) method and the PVL algorithm with a finite frequency value used as the expansion point. The advantage of this new method is avoidance of the LU decomposition step that is costly both in speed and memory. It also provides better wide frequency band results than PVL with expansion at a finite frequency, which only gives correct results near the expansion point. The disadvantage is that the dimension of the reduced-order model must be higher relative to PVL with finite value expansion for accurate approximation of the original electromagnetic system. Although it suffers from this disadvantage, PVL with expansion at infinity makes it possible to solve some complicated electromagnetic problems efficiently.

High-end server system partitioning for cost reduction

In this paper, we demonstrate the use of finite-dimension linear programming to maximize the number of partial good multicore processor chips in a symmetric multiprocessing (SMP) node of a given logical size and physical footprint. It is asserted that to the first order the cost of a productized processor chip will be proportional to the scrap of a processor chip containing good cores but being unusable for the implementation of an SMP node. Therefore, the tradeoff between the number of processing units (PUs) on a chip and the total number of PUs on an SMP node is examined. This paper shows that an optimized SMP offering can be found so that the total chip cost of a high-end system can be minimized. However, such cost reduction will limit the SMP node size for a given processor chip yield. It will also be shown that as the chip yield improves the SMP node size that can be profitably implemented will increase.

Crosstalk Analysis for High-Speed Pulse Propagation on Frequency-Dependent Lossy Electrical Interconnections

Frequency-domain expressions for coupled transmission lines with triangular input waveforms are first developed. The inverse Fourier transformation (IFFT) and an accelerated inverse Laplace transformation (AILT) are then used to obtain the time-domain triangle impulse responses (TIR). Compared with the IFFT, AILT needs much less CPU running time. The triangle impulse responses can then be used in a time-domain convolution approach to accurately and efficiently simulate the crosstalk of frequency-dependent lossy electrical interconnections.