Nonlinear Terminations Research Articles

Successive asymptotic waveform evaluation of lossy distributed networks with nonlinear terminations

A new technique is presented for the transient simulation of high-speed VLSI interconnects, modelled by lossy transmission lines with nonlinear terminations. The asymptotic waveform evaluation (AWE) technique is extended from analysing linear circuits to nonlinear ones. Significant central processing unit (CPU) speedup over conventional simulators such as SPICE is achieved.

Analysis of nonuniform, frequency-dependent high-speed interconnects using numerical inversion of Laplace transform

Interconnects in high-speed VLSI circuits and systems exhibit transmission line effects. Due to the complex geometries of interconnections, coupling between various layers, and inhomogeneous insulating materials, these high-speed interconnects need to be modelled as nonuniform frequency dependent transmission lines. In this paper, we describe a method of simulating the transient response of nonuniform high-speed interconnects in its most general form, i.e. nonuniformly distributed, lossy, coupled, multiple lines with frequency-dependent parameters with linear and nonlinear terminations. Transmission line equations are formulated in the frequency domain as an initial value problem and solved using numerical integration. A new algorithm is proposed for overcoming the inherent initial value instability encountered in the solution of transmission line equations. The time domain response is obtained by Numerical Inversion of Laplace Transform (NILT). Nonlinear networks containing nonuniform high-speed interconnects are analyzed using the Piecewise Decomposition Technique. The accuracy and efficiency of the proposed method is illustrated by appropriate examples and comparisons with published results.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Scattering parameter-based simulation of transients in lossy nonlinearly terminated packaging interconnections

A new and efficient approach is presented for the transient analysis of coupled transmission line structures frequently encountered in microelectronic packaging applications. Frequency-dependent parameters for the transmission lines, as well as nonuniformities in the cross-section of the structures are taken into account. The method is based on the scattering parameter representation of the lossy and/or nonuniform interconnects, An important advantage of this approach compared to previous scattering parameter formulations is that the scattering parameter models have been fully integrated with the standard SPICE models of nonlinear semiconductor devices such as bipolar and CMOS transistors, thus allowing realistic modeling of the driver and receiver circuits terminating the transmission lines. Moreover, measured scattering parameters and non-TEM models can also be included in the simulation. To illustrate the capabilities of this new method, several simulation studies showing the effects of losses and nonuniform interconnect geometries on the propagating signals in transmission line circuits with nonlinear terminations are included in the paper.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Analysis of nonlinear termination networks for coupled lossy and dispersive transmission lines

Based on an algorithm developed by the authors, multiple transmission lines with skin effect losses and dispersive characteristics are analyzed by the volume equivalent principle, and the scattering matrix (S omega ) and characteristic impedance matrix (Z/sub 0 omega /) of the transmission lines are obtained. The (S omega ) and (Z/sub 0(w)/) are then transformed by the inverse fast Fourier transform (FFT) into the time domain. The scattering matrix representation is multiplicative in nature, which leads to a time-domain formulation as a set of convolution integrals. Instead of attempting to solve a set of coupled convolution integral equations by the multivariable Newton-Raphson method, which may occasionally be unstable, the authors generate a set of object functions and apply a multivariable optimization technique, referred to as the modified Levenberg-Marquardt algorithm, to attain the solutions. The new method, which is quite general, reduces to the special cases derived in many previous publications.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Delay and crosstalk simulation of high-speed VLSI interconnects with nonlinear terminations

A method for analysis of VLSI interconnects that contain both lossy coupled transmission lines and nonlinear components is presented. An equivalent time-domain macromodel is derived for the linear subnetworks that contain lossy coupled transmission lines. The macromodel takes the form of a set of ordinary differential equations. The method takes full advantages of the asymptotic waveform evaluation (AWE) technique, which offers two to three orders of magnitude speed-up relative to other methods with comparable accuracy. In addition, it does not require explicit evaluation of the dominant poles of the network. The proposed technique can be easily implemented within the framework of an existing conventional circuit simulator such as SPICE.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Lossy transmission lines: time domain formulation and simulation model

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. >

Transient analysis of coupled, tapered transmission lines with arbitrary nonlinear terminations

A fast and efficient method of simulating the time-domain transient response of coupled, tapered transmission lines is presented. A time-domain scattering parameter formulation is used to derive the simple closed-form expression for the voltage variables for uniform lossless lines; this expression is applied to tapered lines by dividing the lines into many uniform section. Computational efficiency and stability are achieved using recursive time-domain algorithms. When a quasi-TEM propagation mode is assumed, the method is applicable to nonlinear terminations and inhomogeneous dielectric media. Memory requirements are minimized and are independent of the number of time steps. Simulation results showed good agreement with experimental results. >

Analysis of the time response of nonuniform multiconductor transmission lines with a method of equivalent cascaded network chain

Nonuniform multiconductor transmission lines are considered to be equivalent to a cascaded chain of many multiport subnetworks which are made of short sections of uniform lines. The ABCD matrices of the subnetworks can be obtained by the matrix series expansions of their analytic expressions. As long as the number of the subnetworks is large enough to reflect the line nonuniformity fully, the expansions will converge so fast that a few of low-order series terms will be good approximations. After the overall ABCD matrix of the cascaded network chain is evaluated from that of each subnetwork, the time of response of transmission lines can be analyzed. The lines may have frequency-dependent parameters and arbitrary nonlinear terminals. Furthermore, transmission systems with branches uniform and nonuniform transmission lines can be studied with this method conveniently. The analysis accuracy and efficiency are discussed in detail.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Simulation of arbitrary transmission line networks with nonlinear terminations

The transient response of a high-speed digital circuit is determined by the interaction of a lossy distributed transmission line network and lumped nonlinear circuits. A robust and accurate method for the analysis of such a system is reported, and its implementation in an analog circuit simulator is discussed. The method uses a convolution technique and a time-domain impulse response to perform a transient simulation of a transmission line network. The time-domain impulse response is derived using a Fourier transform of modified frequency-domain scattering parameters. Implementation of the technique is verified by comparison with measured results.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Transient analysis of lossy multiconductor transmission lines in nonlinear circuits

Using the frequency-dependent transmission line parameters, two time-domain models are developed for lossy multiconductor transmission lines. It is shown that the end-points of a lossy multiconductor line can be represented at each time step by discretized Thevenin or Norton equivalent circuits. Because these models contain only lumped elements, they can be easily implemented in a general circuit analysis program for simulating the transient responses of nonlinear circuits. The analysis procedure developed makes exclusive use of infinite line impulse responses in the formulation of the time-domain models. Because infinite lines are matched, there are no reflections in the impulse responses. The result is that these impulse responses are relatively short. Results are shown for the most general case of lossy multiconductor transmission lines with discontinuities and nonlinear terminations in which the modal transformation matrices are necessarily frequency dependent.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Computation of transients in lossy VLSI packaging interconnections

An efficient method for analyzing the dynamic behavior of lossy electrical interconnects (with frequency-dependent parameters) in VLSI Systems is presented. The method allows for inclusion of the electrical interconnects which are terminated by networks of lumped passive (R, L, C) and active nonlinear devices (diodes, and bipolar and MOS transistors). The method consists of deriving the circuit model for a transmission line from impulse-response data and incorporating this model into the UANTL (University of Arizona simulator for nonlinearly terminated transmission-line networks) computer program, which performs time-domain analysis for coupled transmission lines with nonlinear terminations. Several numerical experiments with this method were performed. Comparisons were made between the results obtained using this method and other published results. >

Nonlinear transient analysis of coupled transmission lines

A novel approach for the analysis of transients in coupled transmission lines is presented. The method uses a modal scattering parameter formulation and is applicable to the case of lossy lines and nonlinear terminations. The advantage of this method over other techniques resides in the computational stability, in its flexibility in the choice of a reference system, and in the simplicity of the solutions in the case where losses are neglected. An analysis of different types of coupled-line problems is presented, and the scattering parameter approach is developed to lead to the solutions for various special cases. Experimental and computer simulations are analyzed in order to evaluate the efficiency, accuracy, and computational speed of the developed scheme.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Transient Analysis of Frequency-Dependent Transmission Line Systems Terminated with Nonlinear Loads

A new method for analyzing frequency-dependent transmission line systems with nonlinear terminations is presented. The generalized scattering matrix formulation is used as the foundation for the time domain iteration scheme. Compared to the admittance matrix approach proposed in a previous paper, it has the advantage of shorter impulse response which leads to smaller computer memory requirement and faster computation time. Examples of a microstrip line loaded with nonlinear elements are given to illustrate the efficiency of this method.

Analysis of chemical kinetic systems over the entire parameter space - II. Isothermal oscillators

Two simple but similar isothermal kinetic schemes are studied by using recently developed qualitative techniques with the aim of finding all of the qualitatively distinct behaviour that the system can exhibit. The first scheme, which can be regarded as one of the simplest isothermal models of interest, is based on the ‘quadratic’ autocatalytic step A + B → 2B rate = k q ab , where a and b are the concentrations of A and B respectively, combined with a nonlinear termination step of the form B → C rate = k t b /(1+ rb ), where r is some parameter. The second scheme is a simplified (by treating one parameter as a constant) version of a model that has been proposed for the oxidation of carbon monoxide (CO). This scheme has a termination step of the same form as above, and like the ‘quadratic’ model, the highest nonlinearity apart from this term is quadratic. For the quadratic model we find 10 qualitatively distinct phase portraits and, with our choice of bifurcation parameter, 22 distinct bifurcation diagrams. The most complex phase portraits contain 3 steady states and 2 limit cycles or 2 steady states and 3 limit cycles. For the CO model we find 8 distinct phase portraits but leading to only 5 separate bifurcation diagrams. In both models much of the behaviour occurs over only extremely small regions in parameter space.

Nonlinear Termination of Transmission Lines

The technique of using nonlinear resistances to terminate interconnecting transmission lines has been designed and tested, and is recommended for use in very high-speed digital circuits. This technique can save power, simplify wiring, and eliminate the problem of ringing in branched networks. Line matching by nonlinear resistance is investigated theoretically, and the properties, possibilities, and limitations of this method are discussed. Schottky diodes are recommended for this purpose.

Multiple Reflections on Pulse Signal Transmission Lines-Model for Computer Solution

A mathematical model for multiple reflections on lossless, distortionless signal transmission lines is developed for step changes in voltage or current at the source. The model is quite general and is valid for a source driving either one or two lines each loaded with an arbitrary number of discontinuities at arbitrary distances. It will give an exact solution for discontinuities and terminations which are purely resistive and allows for resistances of any size, none of which need be alike. Although an exact solution is valid only when the capacitances are negligible, a method is presented which will partially compensate for a capacitively distorted wave if the capacitive-characteristic resistance time constant is reasonably small compared to the transit time of the line. Considerations for digital computer simulation of the mathematical model are discussed and several examples including open backplane wiring comparing actual and computer results are given. The model and simulation are especially useful for lines which are not terminated in the characteristic impedance at the source or load end and also where modification of the model can be made to investigate reactive and nonlinear terminations.

Transient analysis of transmission lines with nonlinear terminations

Transient analysis of transmission lines with nonlinear terminations