Characteristic analysis of coupled HTS interconnects withtwo-dimensional FDTD

In this work, the frequency-dependent RLGC parameters of high-speed coupled high T_c superconductor (HTS) interconnects are extracted with a two-dimensional (2-D) FDTD algorithm. The response signals of an HTS interconnect circuit and a normal Al interconnect circuit are simulated and compared, showing that not only the signal dispersion, delay, and magnitude decay of HTS interconnects are smaller than that of Al interconnects, the crosstalk of HTS interconnects is much smaller, too     

有耗互连线频变电感提取及等效电路模型

采用二维电感模型，计算了带接地导体的有耗互连线的频变阻抗。阻抗函数用分式多项式近似，并表达为串接的并联电阻、电感的福斯特电路形式。考虑互连线分布电容参数，并根据多个频点阻抗值，用有限数量极点，综合得到互连线单位长度的等效电路模型。采用该模型进行互连线时域响应分析，其结果与改进特征法结果吻合较好，且便于与其他电路模型结合，进行大规模电路的时域分析。     

Crosstalk and transient analyses of high-speed interconnects andpackages

A multiconductor interconnect is modeled using resistors and linear-dependent current and voltage sources. The analysis of a high-speed circuit including lossy interconnection buses is then reduced to simulation of the circuit together with the equivalent circuits of the interconnects. The authors present a new method for the crosstalk and transient analysis of lossy interconnects with arbitrary termination circuits. In order to analyze an interconnect containing N signal conductors, they derive closed-form formulas to determine its transfer functions, and they apply the inverse Fourier transform to obtain its time-domain pulse response functions. Two types of equivalent circuit models can be formulated once the pulse response functions of the interconnect are found. The circuit schematics of the models depend on the number of the signal conductors, irrespective of the physical parameters of the interconnect. These models are compatible with standard circuit simulation tools since they consist of linear resistive networks and linear-dependent sources only. Two example circuits are studied to examine the accuracy and efficiency of the method     

Optimization of Driver Preemphasis for On-Chip Interconnects

In modern digital systems, on-chip interconnects have become the system bottleneck, limiting the performance of high-speed clock distributions and data communications in terms of speed and power dissipation. An inverse signaling analysis is developed to optimize the driving signal waveforms for lossy interconnects. By specifying the performance parameters, i.e., the signal swing and edge rate of the interconnect output signal, the corresponding input signals can be derived analytically. The result can be used to guide and optimize the design of interconnect preemphasis drivers. Numerical examples are shown for both lossy RC and RLC distributed lines. Analysis shows that optimized driving voltage and current can increase the interconnect bandwidth without voltage overshoot at the output. The significance of an interconnect inductance is also evaluated with this technique.     

Novel closed-form resistance formulae for rectangular interconnects

Two closed-form formulae for the frequency-dependent resistance of rectangular cross-sectional interconnects are presented.The frequency-dependent resistance R(f) of a rectangular interconnect line or a interconnect line with a ground plane structure is first obtained by a numerical method.Based on the strict numerical results, a novel closed-form formula R(f) for a rectangular interconnect alone is fitted out using the Levenberg-Marquardt method.This R(f) can be widely used for analyzing on-chip power grid IR-drop when the frequency is changing. Compared to the previously published R(f) formula for an interconnect,the formula provided here is more accurate during the frequency transition range.Also for a bigger width to thickness ratio,this formula shows greater accuracy and robustness.In addition,this paper fits out the closed-form R(f) formula for a micro-strip-like interconnect (an interconnect with a ground plane),which is a typical structure in the on-chip or package power delivery system.     

Passive model reduction of multiport distributed interconnects

Signal integrity analysis has become imperative for high-speed designs. In this paper, we present a new technique to advance Krylov-space-based passive model-reduction algorithms to include distributed interconnects described by telegrapher's equations. Interconnects can be lossy, coupled, and can include frequency-dependent parameters. In the proposed scheme, transmission-line subnetworks are treated with closed-form stamps obtained using matrix-exponential Pade, where the coefficients describing the model are computed a priori and analytically. In addition, a technique is given to guarantee that the contribution of these stamps to the modified nodal analysis formulation leads to a passive macromodel     

Simulation of lossy package transmission lines using extracted data from one-port TDR measurements and nonphysical RLGC models

In this paper, the frequency-dependent characteristic impedance and propagation constant of lossy transmission lines have been extracted from one-port time-domain reflectometry (TDR) measurements. Nonphysical resistance (R), inductance (L), conductance (G) and capacitance (C) (RLGC) models have been developed for simulating lossy transmission lines using the extracted data. The extraction method has been demonstrated for transmission lines on an organic substrate such as coplanar lines. Using the extracted data and nonphysical RLGC models, the simulation results show good correlation with TDR measurements for coplanar lines.     

Some measurement results for frequency-dependent inductance of ICinterconnects on a lossy silicon substrate

Frequency dependent measurements of scattering (S) parameters using a vector network analyzer (VNA) have been performed on IC interconnects on a lossy silicon substrate. The multiline calibration method has been used to perform the de-embedding of the line parameters, from which the line inductance is extracted. A highly accurate closed-form approximation for frequency-dependent impedance per unit length of a lossy silicon substrate for IC interconnects has been used to compare with the measurements performed     

Transient analysis of lossy transmission lines: an efficient approach based on the method of Characteristics

This paper is devoted to transient analysis of lossy transmission lines characterized by frequency-dependent parameters. A public dataset of parameters for three line examples (a module, a board, and a cable) is used, and a new example of on-chip interconnect is introduced. This dataset provides a well established and realistic benchmark for accuracy and timing analysis of interconnect analysis tools. Particular attention is devoted to the intrinsic consistency and causality of these parameters. Several implementations based on generalizations of the well-known method-of-characteristics are presented. The key feature of such techniques is the extraction of the line modal delays. Therefore, the method is highly optimized for long interconnects characterized by significant propagation delay. Nonetheless, the method is also successfully applied here to a short high/loss on-chip line, for which other approaches based on lumped matrix rational approximations can also be used with high efficiency. This paper shows that the efficiency of delay extraction techniques is strongly dependent on the particular circuit implementation and several practical issues including generation of rational approximations and time step control are discussed in detail.     

A novel closed-form resistance model for trapezoidal interconnects

A closed-form model for the frequency-dependent per-unit-length resistance of trapezoidal cross-sectional interconnects is presented.The frequency-dependent per-unit-length resistance R(f) of a trapezoidal interconnect line is first obtained by a numerical method.Using the method we quantify the trapezoid edge effect on the resistance of the interconnect and the current density distribution in the cross section.Based on this strict numerical result,a novel closed-form model R(f) for a single trapezoidal ...     

An accurate and efficient analysis for transient plane wavesobliquely incident on a conductive half space (TE case)

A method which allows for the analytical evaluation of the inverse Laplace transform representations for a transient TE plane wave, obliquely incident on a conductive half-space, is discussed. We assume that the permittivity and conductivity of the dispersive half-space are independent of frequency. Starting with the equations for the transmitted wave in the Laplace domain, the corresponding time-domain expressions are first represented as inverse Laplace transforms. The transient fields are shown to consist of two canonical integrals f(β) and c(β). The canonical integrals, in turn, are solved analytically, thereby yielding solutions involving incomplete Lipschitz-Hankel integrals (ILHTs). The ILHIs are computed numerically using efficient convergent and asymptotic series expansions, thus enabling the efficient computation of the transient fields. The solutions are verified by comparing with previously published results and with results obtained using standard numerical integration and fast Fourier transform (FFT) algorithms     

Modeling and Analysis of On-Chip Single and H-tree Distributed RLC Interconnects

This paper presents novel methods for modeling and analysis of on-chip Single and H-tree distributed resistance inductance capacitance interconnects. The matrix pade-type approximation and scaling and squaring methods are employed for the numerical estimation of delay in single interconnect, and H-tree interconnects. The proposed models, which are based on these methods, provide rational function approximation for obtaining a passive interconnect model. Multiple single input single output model approximated transfer functions are developed for H-tree interconnects structure. With the equivalent reduced order lossy interconnect transfer functions, finite ramp responses are obtained, and line delay is estimated for various line lengths, input ramp rise times, source resistances, parasitic capacitances and load capacitances. In order to demonstrate the accuracy of proposed models, the estimated 50 % delay values are compared with the standard HSPICE W-element model and are found to be in good agreement. The proposed models worst case 50 % delay errors of single interconnect are 0.27 and 0.24 % respectively, while the worst case 50 % delay errors of H-tree structure are 5.73 and 3.94 % respectively.     

Investigation of tapered multiple microstrip lines for VLSIcircuits

The analysis of tapered, coupled microstrip transmission lines is presented. These lines, used as interconnects between integrated circuit devices, are modeled using an iteration-perturbation approach applied in the spatial domain. From this model, a frequency-dependent scattering parameter characterization is determined. A time-domain simulation of pulse propagation through the tapered, coupled microstrip lines is performed. The frequency-domain scattering parameters are inverse Fourier transformed to obtain the time-domain Green's function. The input pulse is convolved with the Green's function, and a Newton-Raphson algorithm is applied to account for nonlinear loads. Some experimental results are shown, and a simulation approximation is proposed     

Modeling and electrical analysis of seamless high off-chipconnectivity (SHOCC) interconnects

Scaling down on-chip interconnect cross-sectional dimensions results not only in higher circuit wiring density, but also in the long lossy line problem, wherein the long lines become highly resistive and have unacceptable delays. One possible solution to the problem of long lossy lines is to transfer these lines off-chip using seamless high off-chip connectivity (SHOCC) technology. In this work, me modeled and studied the electrical performance of SHOCC signal lines. The performance of SHOCC interconnects was compared with that of typical on-chip interconnerts. Modeling and simulation results, along with recommendations with regards to driver sizes and the type of interconnect that should be used, are presented     

Frequency-dependent mutual resistance and inductance formulas for coupled IC interconnects on an Si-SiO2 substrate

A highly accurate closed-form approximation of frequency-dependent mutual impedance per unit length of a lossy silicon substrate coplanar-strip IC interconnects is developed. The derivation is based on a quasi-stationary full-wave analysis and Fourier integral transformation. The derivation shows the mathematical approximations which are needed in obtaining the desired expressions. As a result, for the first time, we present a new simple, yet surprisingly accurate closed-form expression which yield accurate estimates of frequency-dependent mutual resistance and inductance per unit length of coupled interconnects for a wide range of geometrical and technological parameters. The developed formulas describe the mutual line impedance behaviour over the whole frequency range ( i.e. also in the transition region between the skin effect, slow wave, and dielectric quasi-TEM modes). The results have been compared with the reported data obtained by the modified quasi-static spectral domain approach and new CAD-oriented equivalent-circuit model procedure.     

Efficient sensitivity analysis of lossy multiconductor transmissionlines with nonlinear terminations

An efficient approach for sensitivity analysis of lossy multiconductor transmission lines in the presence of nonlinear terminations is described. Sensitivity information is extracted using the recently developed closed-form matrix-rational approximation of the distributed transmission-line model. The method enables sensitivity analysis of interconnect structures with respect to both electrical and physical parameters. An important advantage of the proposed approach is that the derivatives of the modified nodal admittance matrices with respect to per-unit-length parameters are obtained analytically     

A comprehensive 2-D inductance modeling approach for VLSI interconnects: frequency-dependent extraction and compact circuit model synthesis

Although three-dimensional (3-D) partial inductance modeling costs have decreased with stable, sparse approximations of the inductance matrix and its inverse, 3-D models are still intractable when applied to full chip timing or crosstalk analysis. The 3-D partial inductance matrix (or its inverse) is too large to be extracted or simulated when power-grid cross-sections are made wide to capture proximity effect and wires are discretized finely to capture skin effect. Fortunately, 3-D inductance models are unnecessary in VLSI interconnect analysis. Because return currents follow interconnect wires, long interconnect wires can be accurately modeled as two-dimensional (2-D) transmission lines and frequency-dependent loop impedances extracted using 2-D methods . Furthermore, this frequency dependence can be approximated with compact circuit models for both uncoupled and coupled lines. Three-dimensional inductance models are only necessary to handle worst case effects such as simultaneous switching in the end regions. This paper begins by explaining and defending the 2-D modeling approach. It then extends the extraction algorithm to efficiently include distant return paths. Finally, a novel synthesis technique is described that approximates the frequency-dependent series impedance of VLSI interconnects with compact circuit models suitable for timing and noise analysis.     

Accurate closed-form expressions for the frequency-dependent line parameters of on-chip interconnects on lossy silicon substrate

Accurate closed-form expressions for the complete frequency-dependent R, L, G, C line parameters of microstrip lines on lossy silicon substrate are presented. The closed-form expressions for the frequency-dependent series impedance parameters are obtained using a complex image method. The frequency-dependent shunt admittance parameters are expressed in closed form in terms of the shunt capacitances obtained in the low and high frequency limits. The proposed closed-form solutions are shown to be in good agreement with the electromagnetic solutions.     

Application of recursive convolution to transient simulation of interconnects using a hybrid phase-pole macromodel

A hybrid phase-pole macromodel (HPPM), which explicitly includes phase shifts (time delays) in addition to the system poles, was recently developed for modeling interconnects. In this paper, the HPPM is applied to the transient simulation of interconnects. First, the time-domain source waveform is expanded in terms of triangular expansion functions. Knowledge of the triangle impulse response (TIR) for an interconnect, which is represented in the form of an HPPM, then allows for the time domain simulation of the interconnect. A recursive convolution algorithm is adopted to carry out the required convolutions efficiently in the transient simulator. Combining this transient simulator with an HPPM extractor yields a transient interconnect signal analysis tool.