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 Telegrapher's 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.     

Switch-factor based loop RLC modeling for efficient timing analysis

Timing uncertainty caused by inductive and capacitive coupling is one of the major bottlenecks in timing analysis. In this paper, we propose an effective loop RLC modeling technique to efficiently decouple lines with both inductive and capacitive coupling. We generalize the RLC decoupling problem based on the theory of distributed RLC lines and a switch-factor, which is the voltage ratio between two nets. This switch-factor is also known as the Miller factor, and is widely used to model capacitive coupling. The proposed modeling technique can be directly applied to partial RLC netlists extracted using existing parasitic extraction tools without advance knowledge of the return path. The new model captures the impact of neighboring switching activity as it significantly affects the current return path. As demonstrated in our experiments, the new model accurately predicts both upper and lower delay bounds as a function of neighboring switching patterns. Therefore, this approach can be easily implemented into existing timing analysis flows such as max-timing and min-timing analysis. Finally, we apply the new modeling approach to a range of activities across the design process including timing optimization, static timing analysis, high frequency clock design, and data-bus wire planning.     

考虑温度分布效应的RLC互连延时分析

基于等效Elmore延时模型和分段分布参数思想提出了一种RLC互连延时解析模型,该模型同时考虑了互连线温度分布效应和电感效应对延时的影响,更加贴近实际情况,在实际应用中具有重要意义. 仿真结果表明,对于简单的RLC互连树形结构而言,所提模型的延时误差在10%以内,且仿真效率高.     

Closed-Form RC and RLC Delay Models Considering Input Rise Time

The Elmore delay model is the most popular and efficient delay model used for analytical delay estimation. Closed-form delay formulas are useful for circuit design, timing-driven physical design, synthesis, and optimization. As signal rise time becomes faster and the line resistance becomes smaller from copper technology, the significance of inductance increases. Both RC and RLC delays are a strong function of signal rise time. We propose a novel and efficient delay modeling method based on nondimensionalization to consider finite input rise time as an improvement over the Elmore's approach. To further improve the accuracy of the delay model, a new correction method, effective distance correction factor (EDCF), is proposed to consider resistive shielding of downstream capacitance. EDCF can be used to correct the delays for both RC and RLC tree structures. The proposed delay modeling method was applied to a number of nets selected from an integrated circuit (IC) design, and the delay estimation results were compared with HSPICE simulations. The new delay model retains the efficiency and simplicity of the Elmore delay model with significantly improved accuracy.     

一种用于模拟高速互连线瞬态响应的高效数值方法

本文提出用积分求导法（ＤＱＭ）用于分析高速集成电路系统互连线的瞬态响应。ＤＱＭ是一种直接的数值方法,它将某坐标方向上的微分算子用一离散点的函数值加权逼近,将偏微分方程化为常微分方程求解,可有效地用于高速集成电路互连线系统的瞬态分析,其计算效率高于其它数值方法。     

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.     

亚阈值电路单元延时波动统计建模方法

集成电路产业的不断发展以及行业对高能效的不断追求使得工艺尺寸不断缩小，越来越多的电路工作在亚阈值区，工艺参数波动导致电路延时呈现非高斯分布。统计静态时序分析作为先进工艺下用于分析时序的新手段，采用将工艺参数和延时用随机变量表示的方法，可以加速时序收敛，显示预期成品率。文章主要研究了亚阈值电路单元延时波动的统计建模方法。分别对单时序弧和多时序弧的蒙特卡洛金标准数据进行建模研究。提出了单时序弧单元延时的分布拟合统计建模方法，其误差小于6.30%。提出了多时序弧单元延时人工神经网络统计建模方法，其误差小于4.95%。     

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.     

Performance evaluation of packet data services over cellular voice networks

In this paper we develop a Markov chain modeling framework for throughput/delay analysis of data services over cellular voice networks, using the dynamic channel stealing method. Effective approximation techniques are also proposed and verified for simplification of modeling analysis. Our study identifies the average voice call holding time as the dominant factor to affect data delay performance. Especially in heavy load conditions, namely when the number of free voice channels becomes momentarily less, the data users will experience large network access delay in the range of several minutes or longer on average. The study also reveals that the data delay performance deteriorates as the number of voice channels increases at a fixed voice call blocking probability, due to increased voice trunking efficiency. We also examine the data performance improvement by using the priority data access scheme and speech silence detection technique.     

Time domain modelled ADC BIST with ramp noise projection

This paper proposes a time domain modelled built-in self-test (BIST) with ramp noise projection and their effects on analogue to digital converter (ADC) in testing. A self-biased linear ramp generator has been proposed for high precision testing. Threshold inversion quantization (TIQ) comparator based fast switching flash ADC has considered under test. A time domain model of output response analysis technique has been proposed to calibrate the linearity errors of the converter. An ADC has been validated with different input frequencies to characterize the harmonic distortion and average delay of the system. The proposed testing technique requires less time to measures the uncertainties of the ADC since the full computation is performed within one ramp cycle. The testing results of the proposed BIST technique are aimed to characterize, validate and compare to the best results of the existing ADC BIST techniques for test accuracy.     

Reduced Order Modeling for RLC Interconnect Tree Using Hurwitz Polynomial

We investigate on-chip RLC interconnect reduced order modeling problem. A provably realizable and stable model order reduction approach is proposed. To guarantee stability of reduced order circuits, we first employ a realizable reduction for load approximation to preserve the first three driving-point admittance coefficients. Then, we use Hurwitz polynomials to approximate the denominators of original rational transfer functions. We prove that stability can be guaranteed during a hierarchical analysis while circuit response moments can still be matched implicitly. We also give some experimental results to show the accuracy and efficiency of the proposed approach.     

Effects of inductance on the propagation delay and repeaterinsertion in VLSI circuits

A closed-form expression for the propagation delay of a CMOS gate driving a distributed RLC line is introduced that is within 5% of dynamic circuit simulations for a wide range of RLC loads. It is shown that the error in the propagation delay if inductance is neglected and the interconnect is treated as a distributed RC line can be over 35% for current on-chip interconnect. It is also shown that the traditional quadratic dependence of the propagation delay on the length of the interconnect for RC lines approaches a linear dependence as inductance effects increase. On-chip inductance is therefore expected to have a profound effect on traditional high-performance integrated circuit (IC) design methodologies. The closed-form delay model is applied to the problem of repeater insertion in RLC interconnect. Closed-form solutions are presented for inserting repeaters into RLC lines that are highly accurate with respect to numerical solutions. RC models can create errors of up to 30% in the total propagation delay of a repeater system as compared to the optimal delay if inductance is considered. The error between the RC and RLC models increases as the gate parasitic impedances decrease with technology scaling. Thus, the importance of inductance in high-performance very large scale integration (VLSI) design methodologies will increase as technologies scale     

宽带延时宏模型高效延时提取方法

在高速电路信号完整性分析中,电大尺寸互连的建模仿真越来越普遍。而宽带延时宏模型以其仿真的高效性越来越受到重视,但现有延时提取方法比较耗时,限制了延时宏模型整体建模效率的提高。文中提出了一种基于傅里叶反变换IFT的高效延时提取方法。该方法不仅能从频域离散数据中提取多重延时项,而且可以较好地识别其中主要延时项以优化建模过程。文中在Matlab环境中实现该方法,并与目前常用的Gabor变换方法作对比。实验结果表明,文中方法大幅提高了延时提取效率,并且在宽带频域数据情况下具有较高的精度。     

Analytical models and algorithms for the efficient signal integrity verification of inductance-effect-prominent multicoupled VLSI circuit interconnects

Novel signal integrity verification models and algorithms for inductance-effect- prominent RLC interconnect lines are developed by using a traveling-wave-based waveform approximation (TWA) technique. The multicoupled line responses are decoupled into the eigenmodes of the system in order to exploit the TWA technique. Then, the response signals are mathematically represented by the linear combination of each eigenmode response based on TWA, followed by reporting the signal integrity models and algorithms for the multicoupled lines. The signal integrity of VLSI circuit interconnects is complicatedly correlated with input signal switching-patterns, layout geometry, and termination conditions. It is shown that the technique can be efficiently employed for complicated multicoupled interconnect lines with various termination conditions and the signal transients based on the technique have excellent agreement with SPICE simulations. Thus, with the proposed technique, the switching-dependent signal delay, crosstalk, ringing, and glitches of the inductance-effect-prominent RLC interconnect lines can be accurately as well as efficiently determined.     

Network theory without circuit elements

Network methods are by no means limited to lumped systems. Once the ports of a generalized physical structure are defined by use of a modal decomposition of signals, the structure can be analyzed using network techniques which extend beyond the domain of RLC systems and rational network functions. If the physical system is observed as a black box at its ports and various physical time-domain postulates such as linearity, energy, and power conservation theorems are applied in network terms, a variety of realizability relations are obtained for linear, passive, time-invariant structures. For example, one is led to generalizations of bounded real and positive real functions for distributed systems. The network technique also results in a number of interesting theorems for lossless structures such as a generalization of Foster's reactance theorem, and restrictions on minimum phase realizability, and on signal transmission and group delay in distrributed, lossless networks. These results apply in structures containing gyrotropic, dispersive media as well as in the reciprocal, nondispersive case.     

Addressing transient errors in passive macromodels of distributed transmission-line networks

Recently, several time-domain passive macromodeling algorithms were proposed for distributed transmission-line networks. most or them employ some kind of approximation in the frequency domain to match the response up to a maximum frequency of interest and the behavior after the highest frequency is generally not considered. This can cause significant errors in transient responses (especially in the early-time period). In order to address this difficulty, we will present a new algorithm to reduce high-frequency errors in time-domain macromodels, while preserving passivity. The proposed algorithm is very useful in eliminating spurious ripples in the flat delay portion of transient responses of distributed transmission-line networks without needing to increase the order of approximation.     

Comparison between two sets of basis functions for the currentmodeling in the Galerkin spectral domain solution for microstrips

A frequently used technique for the numerical full wave analysis of planar microwave passive structures is the method of moments (MoM). A very important determinant for the accuracy and efficiency of this technique is the choice of the basis functions used in the approximation of the unknown current components. The application of several types of entire and subsectional domain basis functions has been proposed in the literature. However, little comparison has been made between different types of basis functions. In this paper, a comparison is presented between two sets of frequently used subsectional domain basis functions for the canonical case of the microstrip transmission line. The emphasis in this paper is on the relative accuracy, the convergence and the computational efficiency of both sets of basis functions. The intent of the paper is primarily to be an eye opener for the implications of the choice of the basis functions     

Creating accurate multivariate rational interpolation models ofmicrowave circuits by using efficient adaptive sampling to minimize thenumber of computational electromagnetic analyses

A fast and efficient adaptive sampling algorithm for multivariate rational interpolation models based on convergents of Thiele-type branched continued fractions (BCFs) is presented in this paper. We propose a variation of the standard BCF that uses approximation to establish a nonrectangular grid of support points. Starting with a low-order interpolant, the technique systematically increases the order by optimally choosing new support points in the areas of highest error until the required accuracy is achieved. In this way, accurate surrogate models are established by a small number of support points without any a priori knowledge of the data. The technique is evaluated on several passive microwave structures     

Log-domain filtering by simulating the topology of passive prototypes

A systematic method for designing log-domain filter structures that simulate the topology of the corresponding passive prototypes is introduced in this paper. This has been achieved by transposing the i-/spl upsi/ characteristic equation of each passive element in the linear domain, to the corresponding one in the log domain. The transposition has been done in such a way that, the current that flows through the terminals of the passive element in the linear domain sustains its value in the corresponding log-domain configuration. In this way the linear operation of the whole filter is preserved. With regard to the voltage at each terminal of the passive prototype, this is logarithmically compressed in order to achieve filtering in the log domain. Following the above considerations, the log-domain equivalents of all passive elements of the prototype filter were derived. Having established the equivalents, the procedure for designing high-order log-domain filters is quite facilitated. The validity of the proposed technique is demonstrated through simulation results for a fifth-order elliptic low-pass filter.