Formulating hybrid equations and state equations for nonlinear circuits using SPICE

Hybrid equations are often used in the theoretical study of nonlinear resistive circuits because they have an easy‐to‐analyze structure. They are also advantageous in the numerical analysis of nonlinear resistive circuits because they are separable and consist of a relatively small number of variables. However, the hybrid equations are seldom used in practical applications because their formulation is complicated. In this letter, we propose a simple method for formulating the hybrid equations using SPICE. In the proposed method, we only perform the transient analysis of SPICE on a linear circuit that is obtained through a small modification to the original circuit. It is also shown that state equations for nonlinear dynamic circuits can also be formulated by using the proposed method. Copyright © 2011 John Wiley & Sons, Ltd.     

Non-iterative nodal solution for transient simulation of nonlinear and time-varying elements

A non-iterative algorithm based on nodal equation formulation is proposed for transient simulation of nonlinear and time-varying elements. Terminal equations of inductors and capacitors are transformed into algebraic equations using the trapezoidal rule of integration by treating all nonlinear elements as linear and nodal equations are formulated as a set of algebraic equations. LU factorization is used for the solution of nodal equations. Variation of system elements is represented by renewing the nodal conductance matrix at each time step of the solution accordingly. In the first illustrative example, a nonlinear oscillator circuit is considered. In the second, the transient response of a transmission line with a surge arrester is computed by including the corona effects and in the third, a time-dependent primary arc model of a faulty transmission line is examined. Obtained results are compared with those obtained using EMTP and state-space method. Change of simulation time with respect to the step length of the numerical integration is also investigated.     

Switching‐mode counterparts of the Rayleigh and Van‐der‐Pol oscillators

The present work deals with two fundamental oscillator models. One of them can be regarded as a switching‐mode counterpart of the Rayleigh oscillator, while the other can be regarded as a switching‐mode counterpart of the Van‐der‐Pol oscillator. The models are investigated by several means. Their structure is discussed by treating their circuit models. It is also shown that the related differential equations can be solved analytically and explicit forms of exact solutions are attained by employing recursive algebraic processes. The latter solutions are successfully compared to comprehensive direct simulations based on the original differential equations. Furthermore, the exhibited solutions of the switching‐mode counterparts for small and moderate values of ϵ are shown to be closely similar to the solutions of their parent oscillators. Possible applications of the present oscillatory models are discussed. It is argued that the models can favourably serve for simply representing biological and other systems that rely on oscillatory processes. Copyright © 2000 John Wiley & Sons, Ltd.     

Adaptive switched capacitor biquadratic filter with high-frequency applications

This paper considers the design of an adaptive biquadratic switched capacitor (SC) filter. It uses a low-sensitivity SC integrator, a gain stage and four-quadrant analogue multipliers (4-QAMs). The resulting circuit exhibits reduced sensitivity to op amp non-idealities (DC offset, finite op amp gain and bandwidth) associated with analogue adaptive filters. The techniques used are specific to high-precision and high-frequency operation. Test results obtained from a prototype circuit verify the viability of the proposed architecture.     

CMOS weak‐inversion log‐domain glycolytic oscillator: a cytomimetic circuit example

This paper presents a 3 V, 1.21μW subthreshold log‐domain circuit which mimics the oscillations observed during the biochemical process of glycolysis due to the phosphofructokinase enzyme. The proposed electronic circuit is able to simulate the dynamics of the glycolytic oscillator and represent the time‐dependent concentration changes of the reactants and the products of the chemical process based on nonlinear differential equations which describe the biological system. By modifying specific circuit parameters, which correspond to certain chemical parameters, good agreement between the biochemical and electrical model results has been reached. The paper details the similarities between the equations that describe the biochemical process and the equations derived from the circuit analysis of a transistor and a source‐connected linear capacitor, a topology also known as the Bernoulli Cell. With the use of the Bernoulli Cell formalism, the chemical equations which describe the biochemical system have been transformed into their electrical equivalents. The analog circuit, which implements the whole process, has been synthesised, and simulation results including Monte Carlo analysis are provided, in order to verify the robustness of the proposed circuit and to compare its dynamics with prototype biological behaviour. Copyright © 2012 John Wiley & Sons, Ltd.     

Advanced filter design technique based on equivalent circuits and coupling matrix segmentation

This paper presents a technique for the efficient design of bandpass waveguide microwave filters using a segmentation technique applied to an equivalent circuit. The technique is based on first developing an equivalent circuit that synthesizes the desired transfer function. Then, the different parts of the real physical structure are optimized by segmenting this equivalent circuit. The technique was originally developed for in‐line filters, and the main contribution of this paper is in the combination of this technique with the coupling matrix formalism. In this way, we adapt for the first time this design strategy to the design of complex coupling topologies, beyond the in‐line configuration. As an example, a complex sixth‐order dual‐mode filter, implemented in all‐inductive waveguide technology, is designed using the new coupling matrix segmentation technique, showing the effectiveness of the presented theory. A prototype of the filter has been manufactured, and the accuracy of the design technique is verified by measurements on the real prototype.     

Model order reduction of coupled circuit‐device systems

We consider model order reduction of integrated circuits with semiconductor devices. Such circuits are modeled using modified nodal analysis by differential‐algebraic equations coupled with the nonlinear drift‐diffusion equations. A spatial discretization of these equations with a mixed finite element method yields a high dimensional nonlinear system of differential‐algebraic equations. Balancing‐related model reduction is used to reduce the dimension of the decoupled linear network equations, whereas the semidiscretized semiconductor model is reduced using proper orthogonal decomposition. Because the computational complexity of the reduced‐order model through the nonlinearity of the drift‐diffusion equations still depends on the number of variables of the full model, we apply the discrete empirical interpolation method to further reduce the computational complexity. We provide numerical comparisons that demonstrate the performance of the presented model reduction approach. Copyright © 2012 John Wiley & Sons, Ltd.     

A general tool for circuit analysis based on wavelet transform

This paper deals with a generalized method for the time‐domain solution of electrical networks. The aim is to explore a new technique for the numerical simulation of circuits considering linear, time‐varying and non‐linear cases. By using the wavelet transform of the electrical quantities, differentiation and integration in the time domain are replaced by matrix multiplication. Then, the classical circuit differential equations obtained by mesh‐current, node‐voltage or state variables methods are transformed in algebraic equations. The numerical efficiency of wavelets makes the method effective for a fast and reliable numerical circuit simulation. Comparisons with analytical solutions and PSPICE simulations are presented in order to evaluate the numerical characteristics of the proposed method. Copyright © 2000 John Wiley & Sons, Ltd.     

Low‐power,low‐noise,active‐RC band‐pass filters using a ‘lossy’ LP–BP transformation

A new and straightforward design procedure for simple canonical topologies of allpole, active‐RC, low‐selectivity band‐pass (BP) filters, with low sensitivity to component tolerances is presented. The procedure is primarily intended for discrete‐component, low‐power filter applications using just one amplifier for relatively high‐order filters. The design procedure starts out with an ‘optimized’ low‐pass (LP) prototype filter, yielding an ‘optimized’ BP filter, whereby the wealth of ‘optimized’ single‐amplifier LP filter designs can be exploited. Using a so‐called ‘lossy’ LP–BP transformation, closed‐form design equations for the design of second‐ to eighth‐order, single‐amplifier BP filters are presented. The low sensitivity, low power consumption, and low noise features of the resulting circuits, as well as the influence of the finite gain‐bandwidth product and component spread, are demonstrated for the case of a fourth‐order filter example. The optimized single‐opamp fourth‐order filter is compared with other designs, such as the cascade of optimized Biquads. Using PSpice with a TL081 opamp model, the filter performance is simulated and the results compared and verified with measurements of a discrete‐component breadboard filter using 1% resistors, 1% capacitors, and a TL081 opamp. Copyright © 2011 John Wiley & Sons, Ltd.     

Global unique solvability for memristive circuit DAEs of Index 1

Known solvability results for nonlinear index‐1 differential‐algebraic equations (DAEs) are in general local and rely on the Implicit Function Theorem. In this paper, we derive a global result which guarantees unique solvability on a given time interval for a certain class of index‐1 DAEs with certain monotonicity conditions. Based on this result, we show that memristive circuit DAEs arising from the modified nodal analysis are uniquely solvable if they fulfill certain passivity and network topological conditions. Furthermore we present an error estimation for the solution with respect to perturbations on the right‐hand side and in the initial value. Copyright © 2013 John Wiley & Sons, Ltd.     

On the symmetry features of some electrical circuits

The use of symmetry of some nonlinear circuits, composed of similar resistive (more generally, algebraic) elements, is considered for the analysis of the input resistive function of such a circuit. The focus is on recursively obtained (‘fractal’‐type) 1‐ports, analysed using the concept of α‐circuit introduced by Gluskin. The methods under study should be of interest for the analysis and calculation of complicated nonlinear resistive (algebraic) 1‐port structures, e.g. grid cuts for different symmetry conditions. Copyright © 2006 John Wiley & Sons, Ltd.     

A frequency‐domain approach to the analysis of stability and bifurcations in nonlinear systems described by differential‐algebraic equations

A general numerical technique is proposed for the assessment of the stability of periodic solutions and the determination of bifurcations for limit cycles in autonomous nonlinear systems represented by ordinary differential equations in the differential‐algebraic form. The method is based on the harmonic balance (HB) technique, and exploits the same Jacobian matrix of the nonlinear system used in the Newton iterative numerical solution of the HB equations for the determination of the periodic steady state. To demonstrate the approach, it is applied to the determination of the bifurcation curves in the parameters' space of Chua's circuit with cubic nonlinearity, and to the study of the dynamics of the limit cycle of a Colpitts oscillator. Copyright © 2007 John Wiley & Sons, Ltd.     

Global and local parametric diagnosis of analog short‐channel CMOS circuits using homotopy‐simplicial algorithm

The paper offers an algorithm for local and global parametric diagnosis in nonlinear analog circuits, including both identification of the faulty parameters and determination their values. The algorithm exploits a nonlinear algebraic type test equations which may possess multiple solutions, corresponding to different sets of the parameters values which meet the test. To find the solutions, the homotopy concept is applied. Since the test equation is not given in explicit analytical form, the simplicial method is used to trace the homotopy path. The proposed approach can be applied to a broad class of analog circuits, including the complementary metal–oxide–semiconductor circuits fabricated in nanometer technology. The developed diagnostic procedure has been implemented in DELPHI, whereas the required by the algorithm repeated circuit analyses are carried out using IsSPICE 4 and both environments have been joined together. For illustration, two numerical examples are given. Copyright © 2013 John Wiley & Sons, Ltd.     

Index characterization of differential–algebraic equations in hybrid analysis for circuit simulation

Modern modeling approaches in circuit simulation naturally lead to differential–algebraic equations (DAEs). The index of a DAE is a measure of the degree of numerical difficulty. In general, the higher the index, the more difficult it is to solve the DAE. The modified nodal analysis (MNA) is known to yield a DAE with index at most two in a wide class of nonlinear time‐varying electric circuits. In this paper, we consider a broader class of analysis method called the hybrid analysis. For linear time‐invariant RLC circuits, we prove that the index of the DAE arising from the hybrid analysis is at most one, and give a structural characterization for the index of a DAE in the hybrid analysis. This yields an efficient algorithm for finding an optimal hybrid analysis in which the index of the DAE to be solved attains zero. Finally, for linear time‐invariant electric circuits that may contain dependent sources, we prove that the optimal hybrid analysis by no means results in a higher index DAE than MNA. Copyright © 2008 John Wiley & Sons, Ltd.     

Balanced Gm‐C filters with improved linearity and power efficiency

A novel G_m‐C filter design technique is presented. It is based on floating‐gate metal oxide semiconductor (FGMOS) transistors and consists in a topological rearrangement of conventional fully differential G_m‐C structures without modifying the employed transconductors at transistor level. The proposed method allows decreasing the number of active elements (transconductors) of the filter. Moreover, high linearity is obtained at low and medium frequencies of the pass band. Drawbacks inherent to the use of FGMOS transistors are analyzed, such as large occupied area, high sensitivity to mismatch, or parasitic zeros in transfer functions. The features of the proposed technique are fully exploited in all‐pole G_m‐C filter design, specially implementing unity gain Butterworth transfer functions. Thus, two low‐power second‐order Butterworth G_m‐C filters have been designed and fabricated to compare the proposed FGMOS technique with their equivalent topologies obtained by a conventional design method. Measurement results for a test chip prototype in a 0.5‐µm standard complementary MOS process are presented, confirming the advantages of the proposed FGMOS design technique. Copyright © 2014 John Wiley & Sons, Ltd.     

Circuit models for symmetric systems of linear equations

Electric circuit models are constructed for general symmetric systems of linear algebraic equations. The modelling procedure is based on the node‐voltage analysis of linear circuits under steady‐state conditions. These electric circuit models can, in principle, be physically realized and used for the solution of the systems of equations by simply measuring the electric voltage at the circuit nodes. Copyright © 2000 John Wiley & Sons, Ltd.     

A practical guide to multidimensional wave digital algorithms using an example of fluid dynamics

The wave digital concept for numerical integration of partial differential equations leads to algorithms with highly advantageous features as robustness, full localness and massive parallelism. However, the required synthesis of an internally multidimensionally passive reference circuit, from which the algorithm is derived, usually demands an in‐depth knowledge of circuit theory and a high level of intuition. In this practical guide, a step‐by‐step approach for the synthesis of such reference circuits is introduced to relax these requirements, using the nonlinear fluid dynamic equations as a nontrivial example. General implementation issues for the wave digital algorithm are discussed as well as applying arbitrary passive linear multistep methods in place of the commonly used trapezoidal rule. As an example, we take the well‐known numerically critical shock tube problem, the solution of which is problematic when the trapezoidal rule is used as unwanted oscillations occur. These oscillations are suppressed when using the second‐order accurate Gear method instead. Copyright © 2010 John Wiley & Sons, Ltd.     

A dual filtering scheme for nonlinear active noise control

Active noise control problems are often affected by nonlinear effects such as distortion and saturation of measurement and actuation devices, which call for suitable nonlinear models and algorithms. The active noise control problem can be interpreted as an indirect model identification problem, due to the secondary path dynamics that follow the control filter block. This complicates the weight update mechanism in the nonlinear case, in that the error gradient depends on the secondary path gradient through nonlinear recursions. A simpler and computationally less demanding approach is here proposed that employs the updating scheme of the standard filtered‐x least mean squares (LMS) or filtered‐u LMS algorithm. As in those schemes, the calculation of the error gradient requires a signal filtering through an auxiliary system, here obtained through a secondary adaptation loop. The resulting dual filtering LMS algorithm performs the adaptation of the controller parameters in a direct identification mode and can therefore be easily coupled with adaptive model structure selection schemes to provide online tuning of the model structure, for improved model robustness. Copyright © 2013 John Wiley & Sons, Ltd.