Positive‐real property of passive fractional circuits in W‐domain

Fractional circuits have attracted extensive attention of scholars and researchers for their superior performance and potential applications. Fractional circuits constitute a new challenge for the analysis and synthesis methods of traditional circuits theory. Passivity is the fundamental property of traditional circuits (integer order electric circuits). As is known to all, passivity is equivalent to positive realness in traditional linear circuits. However, this equivalence is broken down by introducing fractional elements into electrical networks in s‐domain. To address this issue, on the basis of s‐W transformation, we study the passive criteria of fractional circuits with rational order elements in this paper. Definitions of positive‐real (matrix) function in W‐domain are given, and the equivalence conditions of positive realness are derived. In addition, a conclusion is proposed in which the immittance (matrix) function of passive fractional circuits with rational order elements is positive real in W‐domain. The applications of passive criteria in circuit synthesis are shown.     

A matrix pencil approach to the local stability analysis of non‐linear circuits

This paper addresses local stability issues in non‐linear circuits via matrix pencil theory. The limitations of the state–space approach in circuit modelling have led to semistate formulations, currently framed within the context of differential‐algebraic equations (DAEs). Stability results for these DAE models can be stated in terms of matrix pencils, avoiding the need for state–space reductions which are not advisable in actual circuit simulation problems. The stability results here presented are applied to electrical circuits containing non‐linear devices such as Josephson junctions or MOS transistors. Copyright © 2004 John Wiley & Sons, Ltd.     

A note on stability of linear time‐variant electrical circuits having constant eigenvalues

The main aspect of this paper is to show that the stability of linear time‐variant systems cannot be estimated from the location of the eigenvalues. For this purpose, two simple time‐variant electrical circuits are presented, which have constant eigenvalues. As will be shown, the time‐variant circuits can be asymptotically stable although there is a positive eigenvalue and this circuit can be unstable despite negative eigenvalues only. The idea behind is a suited time‐variant state transformation of a linear time‐invariant system. An electrical interpretation of both systems and of state transformations allows for an energetic evaluation from an electrical point of view even though the analytical solution is not necessarily known. Copyright © 2012 John Wiley & Sons, Ltd.     

Semistate models of electrical circuits including memristors

In this paper we present several semistate or differential‐algebraic models arising in nodal analysis of nonlinear circuits including memristors. The goal is to characterize the tractability index of these models under strict passivity assumptions, a key issue for the numerical simulation of circuit dynamics. We show that the main model, which combines memristors' fluxes and charges, is index two. From a technical point of view, this result is based on the use of a projector along the image of the leading matrix, in contrast to previous index analyses. For charge‐controlled memristors, the elimination of fluxes yields an index one system in topologically nondegenerate circuits, and an index two model otherwise. Analogous results are also proved to hold for flux‐controlled memristors. Our framework accommodates coupling effects among resistors, memristors, capacitors and inductors. Copyright © 2010 John Wiley & Sons, Ltd.     

Rectifier circuits for optimal frequency conversion

It is proved that a circuit consisting of non-linear passive resistances and of any linear invariant passive elements cannot convert power from frequencies ω₁ and ω₂ into power at frequency mω₁ + nω₂ with an efficiency better than 1/(|m| + |n|)². Circuits attaining that efficiency are constructed for all m, n, so that the condition is both necessary and sufficient. For m = μt, n = vt, |μ| + |v| = 2^s (all literals are integers), the optimal circuit consists of a finite number of rectifiers and tuned circuits. For values of m, n that are not of the above form an infinite number of tuned circuits is necessary, but an efficiency better than 93 per cent of the optimum is attainable by simple finite circuits in all cases.     

Uniqueness of the steady state for small source amplitudes in non-linear non-autonomous circuitsc

The purpose of this paper is to give a sufficient condition for the uniqueness of the steady state for circuits that are composed of strictly locally passive capacitors and inductors, of time-dependent bounded voltage and current sources and of a resistive N-port that is strictly locally passive and eventually strictly passive. It is an extension of previously obtained results by the same authors, where strong local passivity was required. the new condition allows one to give a bound on the source amplitudes below which passive elements. It also gives a more explicit content to a theorem by Chua and Green which states that for sufficiently small source amplitudes the steady state is unique. the new condition is obtained by giving explicit uniform eventual bounds on the solutions as ^t → ∞, which are of independent interest. Two examples illustrate that the various explicit bounds can easily be computed in concrete examples.     

Non‐linear circuit modelling via nodal methods

We discuss in this paper several interrelated nodal methods for setting up the equations of non‐linear, lumped electrical circuits. A rather exhaustive framework is presented, aimed at surveying different approaches and terminologies in a comprehensive manner. This framework includes charge‐oriented, conventional, and hybrid systems. Special attention is paid to so‐called augmented node analysis (ANA) models, which somehow articulate the tableau and modified node analysis (MNA) approaches to non‐linear circuit modelling. We use a differential–algebraic formalism and, extending previous results proved in the MNA context, we provide index‐1 conditions for augmented systems, which are shown to be transferred to tableau models. This approach gives, in particular, precise conditions for the feasibility of certain state‐space reductions. We work with very general assumptions on device characteristics; in particular, our approach comprises a wide range of resistive devices, going beyond voltage‐controlled ones. Copyright © 2005 John Wiley & Sons, Ltd.     

A minimum five‐component five‐term single‐nonlinearity chaotic jerk circuit based on a twin‐jerk single‐op‐amp technique

A minimum 5‐component 5‐term single‐nonlinearity chaotic jerk circuit is presented as the first simplest chaotic jerk circuit in a category that a single op‐amp is employed. Such a simplest circuit displays 5 simultaneous advantages of (1) 5 minimum basic electronic components, (2) 5 minimum algebraic terms in a set of 3 coupled first‐order ordinary differential equations (ODEs), (3) a single minimum term of nonlinearity in the ODEs, (4) a simple passive component for nonlinearity, and (5) a single op‐amp. The proposed 5‐term single‐nonlinearity chaotic jerk circuit and a slightly modified version of an existing 6‐term 2‐nonlinearity chaotic jerk circuit form mirrored images of each other. Although both mirrored circuits yield 2 different sets of the ODEs, both sets however can be recast into a pair of twin jerk equations. Both mirrored circuits are therefore algebraically twin 5‐component chaotic jerk circuits, leading to a twin‐jerk single‐op‐amp approach to the proposed minimum chaotic jerk circuit. Two cross verifications of trajectories of both circuits are illustrated through numerical and experimental results. Dynamical properties are also presented.     

The overdamped double-scroll family. Part I: Piecewise-linear geometry and normal form

This paper is the first part of a two part work that aims to introduce a new family of piecewise-linear differential equations that exhibit chaotic behaviour in numerical simulations and to provide a rigorous mathematical proof of its chaotic nature. The new family presented here has the same eigenvalue distribution as the well-known Lorenz equations in its chaotic regime but has a different symmetry. It is a derivative of the recently investigated double-scroll family of piecewise-linear differential equations.¹ It differs, however, from the double-scroll family in that its members have all real eigenvalues at the origin. In particular, since the associated eigenspace E^s(0) is similar to that of an overdamped linear circuit, we will henceforth refer to this family as the overdamped double-scroll family. We establish Chua's circuit² as a member of this family under appropriate conditions. Preliminary numerical simulations exhibit some interesting phenomena such as period-doubling, periodic windows between chaotic behaviour, and strange attractors of differing geometric structure. Our approach in the rigorous analysis will be that employed for the double-scroll family in Reference 1. We derive a linearly equivalent class of piecewise-linear differential equations which includes the family simulated numerically as a special case. the necessary and sufficient condition for two piecewise-linear vector fields belonging to this new overdamped double-scroll family to be linearly equivalent is that their respective eigenvalues at respective equilibrium points be scalar multiples of each other. If the scalars are all identity, then we have linear conjugacy of the vector fields. an explicit normal form equation, in the sense of global bifurcation, is presented that is parametrized by its own eigenvalues. We find that linearly equivalent vector fields associated with the overdamped double-scroll family exhibit the same global behaviour even though their equivalence is based on the local concept of normalized eigenvalues. In addition, the piecewise-linear differential equations associated with these equivalent vector fields can be quite different from each other.     

Non-linear op-amp circuits: Existence and uniqueness of solution by inspection

Even simple circuits containing only one op amp and linear resistors can have multiple d.c. operating points. Using a realistic non-linear d.c. op-amp model which includes the saturation characteristics, this paper gives the necessary and sufficient conditions for an arbitrary op-amp circuit (containing op amps, linear resistors, strictly-increasing non-linear resistors and independent sources) to have a unique solution for all values of circuit parameters. These conditions are remarkable because they are couched strictly in topological terms. For many op-amp circuits (e.g. those containing only one op amp), the necessary and sufficient conditions can be checked by inspection.     

Designing non-linear single OP-AMP circuits: A cook-book approach

The inherent saturation non-linearity of the op amp is used to design circuits having a wide variety of useful non-linear v-i characteristics. These circuits are made of one op amp and 3 or 4 linear resistors which are passive under a rather mild assumption derived from the 3-port paramountcy condition. Explicit design formulae are given for each prototype circuit and numerous examples are given and validated by actual measurements.     

A unified set‐theoretic approach to the analysis of PWL resistive circuits and composite N‐ports

We consider the search for all DC solutions of resistive piecewise‐linear (PWL) circuits and the analysis of the characteristics of resistive PWL composite N‐ports. These problems are unified from a theoretical and operating point of view by introducing the so‐called N‐augmented circuit, obtained from the N‐port by closing its ports with N norators. Set‐theoretic approach is used to describe the properties of the N‐augmented circuits leading to the formulation of a general DC analysis algorithm, based on linear programming techniques. The examples at the end of the paper show some practical and efficient application of the general DC analysis algorithm. Copyright © 2010 John Wiley & Sons, Ltd.     

Equivalent circuits for small signal performance of spin ½ particles

A quantum system composed of a large number of non‐interacting spin ½ particles, biased by a large constant magnetic field along the z‐axis and excited by a small circularly polarized electromagnetic wave in the xy‐plane is considered. Its small signal reflectance about the equilibrium state resulting from the biasing field and a pumping activity is calculated in the complex frequency plane. It is suggested that the equilibrium state can be locally passive or active depending on the pumping sign and level. Equivalent circuits containing imaginary resistors, i.e. frequency independent reactances, are presented for both passive and active cases. Copyright © 2006 John Wiley & Sons, Ltd.     

Digital architectures realizing piecewise‐linear multivariate functions: Two FPGA implementations

Digital architectures for the circuit realization of multivariate piecewise‐linear (PWL) functions are reviewed and compared. The output of the circuits is a digital word representing the value of the PWL function at the n‐dimensional input. In particular, we propose two architectures with different levels of parallelism/complexity. PWL functions with n = 3 inputs are implemented on an FPGA and experimental results are shown. The accuracy in the representation of PWL functions is tested through three benchmark examples, two concerning three‐variate static functions and one concerning a dynamical control system defined by a bi‐variate PWL function. Copyright © 2009 John Wiley & Sons, Ltd.     

Synthesis of multiport resistors with piecewise‐linear characteristics: a mixed‐signal architecture

Non‐linear multiport resistors are the main ingredients in the synthesis of non‐linear circuits. Recently, a particular PWL representation has been proposed as a generic design platform (IEEE Trans. Circuits Syst.‐I 2002; 49 :1138–1149). In this paper, we present a mixed‐signal circuit architecture, based on standard modules, that allows the electronic integration of non‐linear multiport resistors using the mentioned PWL structure. The proposed architecture is fully programmable so that the unit can implement any user‐defined non‐linearity. Moreover, it is modular: an increment in the number of input variables can be accommodated through the addition of an equal number of input modules. Copyright © 2005 John Wiley & Sons, Ltd.     

New results on periodic symbolic sequences of second order digital filters with two's complement arithmetic

In this article, the second order digital filter with two's complement arithmetic in Reference 1 is considered. Necessary conditions for the symbolic sequences to be periodic after a number of iterations are given when the filter parameters are at b=a+1 and b=‐a+1. Furthermore, for some particular values of a, even when one of the eigenvalues is outside the unit circle, the system may behave as a linear system after a number of iterations and the state vector may toggle between two states or converge to a fixed point at the steady state. The necessary and sufficient conditions for these phenomena are given in this article. Copyright © 2003 John Wiley & Sons, Ltd.     

The CCII+ and the ICCII as basic building blocks in low‐pass filter realizations

Three new grounded capacitor current mode low‐pass filters using two inverting second‐generation current conveyor (ICCII) or one double output ICCII are given. The circuits employ the minimum number of passive circuit components, namely two resistors and two capacitors. The circuits are generated from three new voltage mode low‐pass filters realized with the ICCII. A new grounded capacitor CCII+ current mode low‐pass filter is generated from one of the new voltage mode low‐pass filters employing two ICCII?. A new grounded passive component low‐pass filter with independent control on Q and using three ICCII+ is also introduced. Spice simulation results based on using the 0.5 µm CMOS model are included to support the theoretical analysis. Copyright © 2007 John Wiley & Sons, Ltd.     

Realization of a general all‐pole current transfer function by using CBTA

In this article, a general all‐pole current transfer function synthesis procedure using current backward transconductance amplifiers (CBTAs) is proposed. The proposed configuration uses n current backward transconductance amplifiers and n grounded capacitors as the only type of passive elements. The circuit is eligible to realize any all‐pole transfer characteristics with a given strictly Hurwitz (stable) denominator polynomial. Further, it is straightforward to find the values of the passive elements from the coefficients of this polynomial by using the Routh–Hurwitz algorithm as in the realization of a two‐element kind passive network synthesis. In this sense and as far as the author's knowledge, it is the only active structure that can be synthesized like a passive two‐element kind Cauer circuit. The simulations that are performed using PSPICE exhibit satisfactory results coherent with the theory. Copyright © 2011 John Wiley & Sons, Ltd.     

Design of stable two-dimensional analogue and digital filters with applications in image processing

A method is proposed for the frequency domain design of linear two-dimensional analogue and digital filters with guaranteed stability. The technique used is based on the result that the numerator and the denominator of the input immittance of a two-variable network (which is passive and lossy) are strictly Hurwitz polynomials. One of these strictly Hurwitz polynomials is assigned to the denominator of a two-variable analogue transfer function and the network elements are then used as the variables of optimization thereby guaranteeing the stability of the analogue transfer function. The transfer function of the corresponding two-dimensional discrete (digital) filter is obtained from the analogue transfer function by the bilinear transformation. Examples illustrating the versatility of the technique in designing 2D digital filters of arbitrary order approximating a given magnitude and group delay response are presented. These filters are used to process a simple binary image. The results obtained demonstrate the importance of linear phase in image processing applications. The method presented here can be extended to the design of stable m-dimensional analogue and digital filters.