Qinwei Xu

Accurate modeling of lossy nonuniform transmission lines by using differential quadrature methods

This paper discusses an efficient numerical approximation technique, called the differential quadrature method (DQM), which has been adapted to model lossy uniform and nonuniform transmission lines. The DQM can quickly compute the derivative of a function at any point within its bounded domain by estimating a weighted linear sum of values of the function at a small set of points belonging to the domain. Using the DQM, the frequency-domain Telegraphers partial differential equations for transmission lines can be discretized into a set of easily solvable algebraic equations. DQM reduces interconnects into multiport models whose port voltages and currents are related by rational formulas in the frequency domain. Although the rationalization process in DQM is comparable with the Pade approximation of asymptotic waveform evaluation (AWE) applied to transmission lines, the derivation mechanisms in these two disparate methods are significantly different. Unlike AWE, which employs a complex moment-matching process to obtain rational approximation, the DQM requires no approximation of transcendental functions, thereby avoiding the process of moment generation and moment matching. Due to global sampling of points in the DQM approximation, it requires far fewer grid points in order to build accurate discrete models than other numerical methods do. The DQM-based time-domain model can be readily integrated in a circuit simulator like SPICE.

Transient analysis of lossy interconnects by modified method of characteristics

In this paper, the modified method of characteristics (MMC) is presented to analyze lossy interconnects. The method applies lower order Taylor approximation to model the characteristic admittance, and further applies lower order Pade approximation to model propagation functions of a transmission line. However, different from the single-point or multipoint moment-matching approaches, this method does not generate a reduced-order model of a transmission line prior to constructing the system matrix. Instead, it takes a special expansion and directly incorporates the coefficients of the expansion into the modified nodal admittance (MNA) matrix. On the basis of the specific expansion, a set of time-domain recursive formulae is derived, which concerns only the quantities at the ends of transmission lines. The recursive formulae are similar to those of method of characteristics (MC), with some added modifications. Based on the formulae of MMC, an equivalent time-domain macromodel of a uniform lossy transmission line is obtained. It can be easily implemented within the framework of an existing circuit simulator such as SPICE. The examples indicate that the method gives accurate time-domain simulation of interconnects in high-speed IC systems.

Modeling of transmission lines by the differential quadrature method

The differential quadrature (DQ) method is employed to model transmission lines. The DQ method is a direct numerical technique. It is based on the ideas that the derivative of a function with respect to a coordinate direction can be expressed by a weighted linear sum of all function values at every mesh points along that direction. To achieve the same accuracy, the DQ method requires fewer grid points than other numerical methods such as finite-different method. This method leads to passive macromodels. As a direct numerical method to model transmission lines, it provides general formulation, which can be applied in extensive cases. Numerical results show that high accurate solutions can be calculated rapidly.

Equivalent-circuit interconnect modeling based on the fifth-order differential quadrature methods

This paper introduces an efficient and passive discrete modeling technique for estimating signal propagation delays through on-chip long interconnects that are represented as distributed RLC transmission lines. The proposed delay model is based on a less frequently used numerical approximation technique, called the differential quadrature method (DQM). The DQM can compute the partial derivative of a function at any arbitrary point located within a prespecified closed domain of the function by quickly estimating the weighted linear sum of values of the function at a relatively small set of well-chosen grid points within the domain. By using the fifth-order DQM, a new approximation framework is constructed in this paper for discretizing the distributed RLC interconnect and thereafter modeling its delay. Due to high efficiency of DQM approximation, the proposed framework requires only few grid points to achieve good accuracy. The presented equivalent-circuit model appears like the ones derived by the finite difference (FD) method. However, it has higher accuracy and less internal nodes than generated by the FD-based modeling. The fifth-order DQM modeling technique is shown to preserve passivity. It has linear forms that are compatible with the passive order-reduction algorithm for linear network. Numerical experiments show that the proposed modeling approach leads to high accuracy as well as high efficiency.

Time-domain modeling of high-speed interconnects by modified method of characteristics

In this paper, a new model of lossy transmission lines is presented for the time-domain simulation of high-speed interconnects. This model is based on the modified method of characteristics (MMC). The characteristic functions are first approximated by applying lower order Taylor series in the frequency domain, and then a set of simple recursive formulas are obtained in the time domain. The formulas, which involve tracking performances between two ends of a transmission line, are similar to those derived by the method of characteristics for lossless and undistorted lossy transmission lines. The algorithm, based on the proposed MMC model, can efficiently evaluate transient responses of high-speed interconnects. It only uses the quantities at two ends of the lines, requiring less computation time and less memory space than required by other methods. Examples indicate that the new method has high accuracy and is very efficient for the time-domain simulation of interconnects in high-speed integrated circuits.

Efficient Macromodeling for On-Chip Interconnects

The improved T and improved /spl Pi/ models are proposed for on-chip interconnect macromodeling. Using global approximations, simple approximation frames are derived and applied to modeling of on-chip distributed RC interconnects. The applications lead to equivalent circuit models for on-chip interconnects, which are represented by the improved T and improved /spl Pi/ models. By matching the first three moments of an open-ended interconnect, the improved /spl Pi/ model with AWE is consequently obtained, which retains the symmetric structure. The new models for distributed RC interconnects are independent of CMOS gates, and therefore can be directly incorporated into SPICE frames. Numerical experiments show that for current feature sizes, the improved T and improved /spl Pi/ modeling methods can be used to accurately evaluate on-chip interconnect effects, while the computational costs are comparable to the original T and original /spl Pi/ modeling. The presented macromodeling approaches are useful for quick simulation and layout optimization.

Novel macromodeling for on-chip RC/RLC interconnects

The improved T and improved /spl Pi/ models are proposed for on-chip interconnect macromodeling. Using global approximations, simple approximation frames are derived and applied to modeling of on-chip distributed RC/RLC interconnects. The applications lead to equivalent circuits, which are represented by the improved T and improved /spl Pi/ models. The new models for distributed RC/RLC interconnects are independent of CMOS gates, and therefore can be directly incorporated into SPICE frames. Numerical experiments show that for current feature sizes, the improved T and improved /spl Pi/ modeling methods can be used to accurately evaluate onchip interconnect effects, while the computational costs are comparable to the original T and /spl Pi/ modeling. The presented macromodeling approaches are useful for quick simulation and layout optimization.

Transmission line modeling by modified method of characteristics

In this paper a new model of lossy transmission lines is presented for the time-domain simulation of high-speed interconnects. This model is based on modified method of characteristics (MMC). The characteristic functions are first approximated by applying lower-order Taylor series in the frequency domain, and then a set of simple recursive formulas are obtained in the time domain. The formulas, which involve tracking performance between two ends of the transmission line, are similar to those derived by the method of characteristics (MC) for lossless or undistorted lossy transmission lines. The algorithm based on the model can efficiently evaluate transient responses of high-speed interconnects. It only uses the quantities at two ends of the lines, requiring less computation time and less memory than other methods. Examples indicate that the new method has high accuracy and is very efficient for the time-domain simulation of interconnects in high-speed integrated circuits.

Efficient Modeling of Transmission Lines With Electromagnetic Wave Coupling by Using the Finite Difference Quadrature Method

This paper proposes an efficient numerical technique, called the finite difference quadrature (FDQ) method, to model the transmission line with radiated electromagnetic (EM) wave noise coupling. A discrete modeling approach, the FDQ method adapts coarse grid points along the transmission line to compute the finite difference between adjacent grid points. A global approximation scheme is formulated in the form of a weighted sum of quantities beyond the local grid points. Unlike the Gaussian quadrature method that computes numerical integrals by using global approximation framework, the FDQ method uses a global quadrature method to construct the approximation schemes for the computation of, however, numerical finite differences. As a global approximation technique, the FDQ method has superior numerical dispersion to the finite difference (FD) method, and, therefore, needs much sparser grid points than the FD method to achieve comparable accuracy. Equivalent voltage and current sources are derived, exciting the transmission line at the grid points. Equivalent circuit models are consequently derived to represent the transmission line subject to radiated electromagnetic wave noise. The FDQ-based equivalent models can be integrated into a simulator like SPICE.

Rational ABCD Modeling of High-Speed Interconnects

Introduces a new numerical approximation technique, called the differential quadrature method (DQM), in order to derive the rational ABCD matrix representing the high-speed interconnect. DQM is an efficient differential equation solver that can quickly compute the derivative of a smooth function by estimating a weighted linear sum of the function values at few mesh points in the domain of the function. Using DQM, the s-domain Telegraphers equations of interconnect are discretized as a set of easily solvable algebraic equations, which lead to the rational ABCD matrix. The entries of ABCD matrix take the form of rational approximations with respect to s, rather than the conventional ABCD matrix whose entries are complex transcendental functions in s. Although the rationalization result is comparable with Pade approximation of AWE, DQM does not require moment-generating or moment-matching. For both uniform and nonuniform interconnects, DQM-based rational ABCD matrices lead to high accuracy as well as high efficiency for transient analysis of high-speed interconnects.