Lossy transmission lines: time domain formulation and simulationmodel

The time domain quasi-TEM equations for lossy transmission lines with R, L, C, and G parameters is reformulated and solved to relate directly the currents and voltages at the line terminations, at present and past times. This allows a computer model to be set up for simulating circuits with nonlinear terminations in the time domain using general circuit simulators. This formulation describes propagation of two dynamic forward and backward waves and is the extension of the method of characteristics to the lossy case. Distortion and impedance changes are generated by finite convolutions with past history information at the line terminations. For constant R, L, C, and G, and for a skin effect approximation, the kernels of Green's functions for these convolutions are derived as analytic expressions     

Sensitivity analysis of lossy coupled transmission lines

An analysis method, based on the numerical inversion of the Laplace transform, is described for the evaluation of the time domain sensitivity of networks that include lossy coupled transmission lines. The sensitivity can be calculated with respect to network components and parameters of the transmission lines. Sensitivity analysis is useful for waveform shaping and optimization. Examples and comparisons with sensitivity determined by perturbation are presented     

FDTD法分析高速集成电路芯片内互连线

本文首次利用时域有限差分(FDTD)法分析了高速集成电路芯片内半导体基片上的有耗互连传输线的电特性.文中提出了有耗吸收边界条件,推导了不同媒质交界面上的边界条件通用格式.在FDTD分析的基础上,得到传输线各种参数的频变特性,为芯片内电路模拟提供了可靠的参数.     

Time-domain analysis of lossy coupled transmission lines

A novel method based on numerical inversion of the Laplace transform is presented for the analysis of lossy coupled transmission lines with arbitrary linear terminal and interconnecting networks. The formulation of the network equations is based on a Laplace-domain admittance stamp for the transmission line. The transmission line stamp can be used to formulate equations representing arbitrarily complex networks of transmission lines and interconnects. These equations can be solved to get the frequency-domain response of the network. Numerical inversion of the Laplace transform allows the time-domain response to be calculated directly from Laplace-domain equations. This method is an alternative to calculating the frequency-domain response and using the fast Fourier transform to obtain the time-domain response. The inversion technique is equivalent to high-order, numerically stable integration methods. Numerical examples showing the general application of the method are presented. It is shown that the inverse Laplace technique is able to calculate the step response of a network. The time-domain independence of the solution is exploited by an efficient calculation of the propagation delay of the network     

Transient Simulation of Lossy Interconnects Based on a Dispersive Hybrid Phase-Pole Macromodel

A transient simulator for interconnect structures that are modeled by lossy transmission lines is outlined in this paper. Since frequency-dependent RLGC parameters must be employed to correctly model skin effects and dielectric losses for high-performance interconnects, we first study the behaviors of various lossy interconnects that are characterized by frequency-dependent line parameters (FDLPs). We then developed a frequency-domain dispersive hybrid phase-pole macromodel (DHPPM) for such lines, which consists of a constant RLGC propagation function multiplied by a residue series. The basic idea is to first extract the dominant physical phenomenology by using a propagation function in the frequency domain that is modeled by frequency-independent line parameters (FILPs). A rational function approximation is then used to account for the remaining effects of FDLP lines. By using a partial fraction expansion and analytically evaluating the required inverse Fourier transform integrals, the time-domain DHPPM can be decomposed as a sum of canonical transient responses for lines with FILP for various excitations (e.g., trapezoidal and unit step). These canonical transient responses are then expressed analytically as closed-form expressions involving incomplete Lipshitz-Hankel integrals of the first kind and Bessel functions. The closed-form expressions for these canonical responses are validated by comparing with simulation results from commercial tools like HSPICE. The DHPPM simulator can simulate transient results for various input waveforms on both single and coupled interconnect structures. Comparisons between the DHPPM results and the results produced by commercial simulation tools like HSPICE and a numerical inverse fast Fourier transform show that the DHPPM results are very accurate.     

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.     

不等长非均匀有损耗传输线FDTD瞬态分析

以分析等长均匀无损耗多导体传输线的时域有限差分(FDTD)法为基础,在考虑传输线损耗的情况下,对不等长非均匀多导体传输线进行分析。首先,在考虑传输线损耗的情况下给出传输线上各点电压和电流的迭代计算公式;其次,利用该公式对不等长非均匀有损耗传输线模型进行数值计算和理论分析;最后,通过仿真实验,其结果表明所提计算方法是正确和有效的。该方法对不等长非均匀有损耗传输线的研究提供理论计算参考。     

The generalized method of characteristics for waveform relaxationanalysis of lossy coupled transmission lines

The transient response of lossy coupled transmission lines is simulated by iterative waveform relaxation analyses of equivalent disjoint networks constructed with congruence transformers, fast Fourier transform (FFT) waveform generators, and characteristic impedances synthesized by the Pade approximation. A two order reduction of CPU time and one order savings in computer memory are achieved. A lossy directional coupler is simulated for illustration     

有损土壤上的多导体传输线的时域分析

将多导体传输线（ＭＴＬ）的土壤复数阻抗拓展为土壤运算阻抗,采用Ｐａｄｅ展开法,提出了计及土壤影响的多导体传输线的时域模型,建立了该模型的时域有限差分（ＦＤＴＤ）算法。通过对计及土壤影响的架空单导体和双导体传输线的波过程计算,表明本文方法的正确性,并可以应用于超高压变电站高压母线和超高压输电线路的瞬态电磁干扰计算。     

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 asymptotic waveform evaluation for analysis of skineffect in lossy interconnects

Asymptotic waveform evaluation (AWE) is a technique for time-domain analysis of electrical interconnects. AWE is a computationally efficient method that asymptotically approximates the response of a large system with a lower-order transfer function. Asymptotic waveform evaluation is used to analyze lossy interconnects including the skin effect. The internal impedance of the interconnect conductors varies as a function of the square root of the frequency. First, an overview of AWE is presented. The AWE formulation for modeling frequency dependent loss in the conductors is derived using two different series expansions of the system response at both s=0 and s≠0 in the Laplace domain. The expansions for s≠0 are determined using a transfer function formulated for inclusion of the frequency-dependent internal impedance. The network response is computed by extracting the dominant poles and residues using the Pade approximation. The proposed method is evaluated using time-domain examples of lossy multiconductor transmission lines     

Time-domain modeling of high-speed interconnects by modified methodof 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     

近地电缆屏蔽层对高空核电磁脉冲响应的研究

为了研究近地面电缆屏蔽层对高空电磁脉冲的耦合规律,依据传输线理论,计算了在高空核电磁脉冲作用下损耗大地上电缆屏蔽层上的感应电流。利用快速傅立叶变换技术,首先得到近地面电缆屏蔽层感应电流的频域解,然后采用快速傅立叶逆变换技术得到时域解。得到了感应电流随电缆长度、电缆架设高度、电磁脉冲极化角、方位角、俯仰角、端接阻抗和大地电导率等的变化规律。     

A resetting algorithm for transient analysis of coupledtransmission line circuits

The numerical inversion of the Laplace transform has been used as an important tool for time domain analysis of high speed VLSI interconnects modeled by transmission line networks. In this paper, a resetting algorithm based on the numerical inversion of Laplace transform with Pade approximation is described. The initial conditions of coupled transmission lines required by the resetting algorithm are also derived. The new method results in substantial improvement of the accuracy of the numerical inversion of Laplace transform for solving transmission line networks with long transients. The new method also bridges the gap between two types of circuit simulation techniques, i.e., the numerical inversion of Laplace transform and the numerical integration     

用沃尔什变换法进行高速数字传输线时域分析

本文首次把波形松驰迭代法和快速Walsh变换相结合,提出了时域分析具有任意终端的高速数字集成传输线的新方法。这种方法的模拟过程十分符合实际电路状态,且有分析精度高、运算稳定、可靠等优点,具有推广应用价值。     

非均匀传输线特性重构中的噪声影响分析

利用时域反射技术(Time Domain Reflectometry,TDR),由测量得到的时域反射信号,可以重构出非均匀传输线的一些特征参数。当反射信号中混有噪声时,对非均匀传输线特性参数的重构会产生影响。采用Zakharov-Shabat类型逆散射问题的数值反演算法,以指数型非均匀传输线为例,对时域反射信号中混有高斯白噪声情况下的非均匀传输线特性参数重构问题进行了数值实验。数值计算结果表明,在噪声干扰下,算法本身是稳定的,在较宽的信噪比范围内,能有效地重构出非均匀传输线的特征参数。     

Time-domain response of multiconductor transmission lines

Evaluation of the time-domain response of multiconductor transmission lines is of great importance in the analysis of the crosstalk in fast digital circuit interconnections, as well as in the analysis of power lines. Several techniques for the computation of the line response, starting from the known circuit-theory parameters, are presented and evaluated. These methods are: time-stepping solution of the telegrapher equations, modal analysis in the time domain, model analysis in the frequency domain, and a convolution technique which uses line Green's functions. The last method can treat the most general case of lossy transmission lines with nonlinear terminal networks. Numerical and experimental results are presented to illustrate these techniques and to give insight into the crosstalk problems in fast digital circuits.     

对垂直交叉线中串扰的分析

用修正后的传输线方程对垂直的导体间的串扰进行了频域模拟，通过反傅立叶变换得到时域信号。模拟表明，虽然垂直放置可使串扰幅度大幅度降低，但交叉线中仍有较强的干扰，串扰幅度与各端接入的负载有很大关系，对干扰的原因进行分析。为了证明结果的正确性，对平行导体间的串扰用时域有限差分法和本方法进行模拟，得到一致的结果。该文对实际应用有参考价值。     

Circuit-Based Analysis of Electromagnetic Field Coupling With Nonuniform Transmission Lines

This paper presents a new algorithm for simulating electromagnetic (EM) field coupling with nonuniform multiconductor transmission lines in a circuit simulation environment. The proposed algorithm is based on the concept of passive model-order reduction, whereby an algorithmically developed passive reduced-order model, coupled with a set of equivalent sources representing the incident filed, are shown to accurately capture the behavior of the transmission line under EM excitation. The reduced-order model is developed independently from the particular shape of the incident field pulse, in the sense that, in constructing the model, one does not need prior knowledge about the waveform of the incident pulse of the EM field. In addition, it is also shown that the model developed can be used to simulate the transmission line in the absence of the EM field. The derived equivalent sources, representing the field coupling, are given directly in the time domain, thereby making simulation under nonlinear circuit terminations an easy task. Although the proposed work is aimed mainly at simulating nonuniform transmission lines, it can be applied to uniform lines as a special case. The proposed algorithm has been validated numerically with several examples.