Modelling and analysis of differential VCOs

We present a systematic non‐linear analysis of differential voltage controlled oscillators (VCOs), both bipolar and MOS. Using the standard device models, we derive the second‐order non‐linear equation describing the behaviour of these oscillators, which is formulated in a perturbation form. The solution of this equation is obtained as a particular case of the solution of the most general equation of second‐order oscillators, which is solved through a suitable perturbation method. Unlike a pure numerical analysis, simple analytical relationships are derived for predicting the steady‐state oscillation, its transient behaviour and for ascertaining the existence of a stable oscillation in differential VCOs. These relationships, leading to results which well agree with the SPICE simulations, are useful in both analysis and design. Copyright © 2004 John Wiley & Sons, Ltd.     

On the perturbation analysis of the limit cycle in oscillators with shifting bias

We propose a method of perturbation analysis of nearly sinusoidal oscillators with shifting bias, obtained by generalizing a method recently discussed in the literature [1] (Buonomo A, Di Bello C. IEEE Transactions on Circuits and Systems, 1996; CAS‐43: 953–963). The problem of periodic oscillations is formulated as a regular perturbation problem, P_ϵ(x) =0, whose peculiarity is that the limiting linear problem, P₀(x)=0, obtained when the perturbation parameter ϵ tends to zero, has a non‐purely harmonic solution x₀=B₀+A₀ cos ϑ. We give a simple condition for the existence of a periodic oscillation and an analytical method for constructing it in the form of a power series in ϵ. Unlike the existing perturbation methods, the method here proposed, which remains in the spirit of the bifurcation process of Poincaré, allows us to obtain the coefficients of the series solution, to an order in ϵ as great as we want, using recurrence formulae. The results of the analysis of a typical LC oscillator are given to show that these formulae are very useful as a practical method for determining all of the characteristics of the periodic oscillation, such as the harmonic content and the frequency correction due to the non‐linear effect. Copyright © 2000 John Wiley & Sons, Ltd.     

Floquet theory and non‐linear perturbation analysis for oscillators with differential‐algebraic equations

Oscillators are key components of electronic systems. In RF communication systems, they are used for frequency translation of information signals and for channel selection, and in digital electronic systems, they are used as a time reference, i.e. a clock signal, in order to synchronize operations. Undesired perturbations in practical electronic systems adversely affect the spectral and timing properties of oscillators, which is a key performance limiting factor, being a major contributor to bit‐error‐rate (BER) of RF communication systems, and creating synchronization problems in clocked and sampled‐data systems. Characterizing how perturbations affect oscillators is therefore crucial for practical applications. The traditional approach to analysing perturbed nonlinear systems (i.e. linearization) is not valid for oscillators. In this paper, we present a theory and efficient numerical methods, for non‐linear perturbation and noise analysis of oscillators described by a system of differential‐algebraic equations (DAEs). Our techniques can be used in characterizing phase noise and timing jitter due to intrinsic noise in IC devices, and evaluating the effect of substrate and supply noise on the timing properties of practical oscillators. In this paper, we also establish novel results for periodically time‐varying systems of linear DAEs, which we rely on in developing the above theory and the numerical methods. Copyright © 2000 John Wiley & Sons, Ltd.     

Design equations for fractional‐order sinusoidal oscillators: Four practical circuit examples

Four practical sinusoidal oscillators are studied in the general form where fractional‐order energy storage elements are considered. A fractional‐order element is one whose complex impedance is given by Z = a(jω)^±α, where a is a constant and α is not necessarily an integer. As a result, these oscillators are described by sets of fractional‐order differential equations. The integer‐order oscillation condition and oscillation frequency formulae are verified as special cases. Numerical and PSpice simulation results are given. Experimental results are also reported for a selected Wien‐bridge oscillator. Copyright © 2007 John Wiley & Sons, Ltd.     

A procedure for the computation of accurate PWL approximations of non‐linear dynamical systems

In this paper we propose a variational method to find out piecewise‐linear (PWL) approximations of non‐linear dynamical systems in view of their circuit implementations. The method is based on some significant trajectories of the dynamical system and provides reasonably accurate PWL approximations with a relatively low number of parameters. The effectiveness of the method is validated by applying it to the approximation of limit cycles (both stable and unstable) in the Bautin system. Copyright © 2006 John Wiley & Sons, Ltd.     

On simplex‐based piecewise‐linear approximations of non‐linear mappings

Simplex‐based piecewise‐linear (PWL) approximations of non‐linear mappings are needed when the robust PWL analysis is used to directly solve non‐linear equations. This paper proposes a straightforward technique for transforming the well‐known approximations into another form. This new form is computationally more efficient, since it preserves the sparse structure of the original Jacobian matrix. Furthermore, this new form of PWL approximation explicitly relates the simplex‐based PWL analysis to the conventional formulation of the Katzenelson algorithm. The proposed transform technique is also extended to treat groupwise‐separable mappings and, finally, non‐separable but sparse mappings that arise in real‐life simulation of large electronic circuits. In this paper, all these (transformed) simplex‐based PWL approximations are compared in terms of their generality and efficiency. The computational efficiency of the PWL approximation that utilizes sparsity is validated with realistic simulations. Copyright © 2005 John Wiley & Sons, Ltd.     

Accelerating the transient simulation of semiconductor devices using filter‐bank transforms

Simulation of high frequency semiconductor devices, where non‐local and hot carrier transport cannot be ignored, requires solution of Poisson's equation and at least the first three moments of the Boltzmann transport equation (hydrodynamic transport model). These equations form non‐linear, coupled and time‐dependent partial differential equations. One of the most efficient solvers of such system of equations is decoupled solver. In conventional decoupled methods, the fully implicit, semi‐implicit and explicit methods are used to solve the equations. In fully or semi‐implicit schemes, the method is unconditionally stable for any Δt or for very large Δt compared to explicit scheme. Thus, these schemes are very suitable and efficient for transient simulations. But, using these techniques leads to a large system of linear equations. Here for the first time, a filter bank‐based preconditioning method is used to facilitate the iterative solution of this system. This method provides efficient preconditioners for matrices arising from discretizing of the PDEs, using finite difference techniques. Numerical results show that the condition number and iteration number are significantly reduced. The most important advantage of this preconditioner is its low computational complexity which can be reduced to O(N). Copyright © 2006 John Wiley & Sons, Ltd.     

Non‐linear performance comparison for FD and PD SOI MOSFETs based on the integral function method and Volterra modelling

The harmonic and intermodulation distortions of both fully‐depleted (FD) and partially‐depleted (PD) silicon‐on‐insulator (SOI) MOSFETs are studied. The analysis is based on the recently developed integral function method and the results are compared to a third‐order Volterra model of the MOSFET. This modelling helps us to understand the non‐linear mechanisms of the considered devices and to predict their frequency behaviour. The models are validated through large‐signal network analyser measurements. The devices performances are discussed. Copyright © 2005 John Wiley & Sons, Ltd.     

Fundamentals of fractional‐order LTI circuits and systems: number of poles,stability, time and frequency responses

This paper investigates some basic concepts of fractional‐order linear time invariant systems related to their physical and non‐physical transfer functions, poles, stability, time domain, frequency domain, and their relationships for different fractional‐order differential equations. The analytical formula that calculates the number of poles in physical and non‐physical s‐plane for different orders is achieved and verified using many practical examples. The stability contour versus the number of poles in the physical s‐plane for different fractional‐order systems is discussed in addition to the effect of the non‐physical poles on the steady state responses. Moreover, time domain responses based on Mittag‐Leffler functions for both physical and non‐physical transfer functions are discussed for different cases, which confirm the stability analysis. Many fractional‐order linear time invariant systems based on fractional‐order differential equations have been discussed numerically in both time and frequency domains to validate the previous fundamentals. Copyright © 2016 John Wiley & Sons, Ltd.     

A method for the approximate synthesis of cellular non‐linear networks—Part 2: Circuit reduction

In this paper, we face the problem of model reduction in piecewise‐linear (PWL) approximations of non‐linear functions. The reduction procedure presented here is based on the PWL approximation method proposed in a companion paper and resorts to a strategy that exploits the orthonormality of basis functions in terms of a proper inner product. Such a procedure can be favourably applied to the synthesis of the resistive parts of cellular non‐linear networks (CNNs) to reduce the complexity of the resulting circuits. As an example, the method is applied to a case study concerning a CNN for image processing. Copyright © 2003 John Wiley & Sons, Ltd.     

Analytical synthesis and comparison of voltage‐mode Nth‐order OTA‐C universal filter structures

Complementary single‐ended‐input operational transconductance amplifier (OTA)‐based filter structures are introduced in this paper. Through two analytical synthesis methods and two transformations, one of which is to convert a differential‐input OTA to two complementary single‐ended‐input OTAs, and the other to convert a single‐ended‐input OTA and grounded capacitor‐based one to a fully differential OTA‐based one, four distinct kinds of voltage‐mode nth‐order OTA‐C universal filter structures are proposed. TSMC H‐Spice simulations with 0.35µm process validate that the new complementary single‐ended‐input OTA‐based one holds the superiority in output precision, dynamic and linear ranges than other kinds of filter structures. Moreover, the new voltage‐mode band‐pass, band‐reject and all‐pass (except the fully differential one) biquad structures, all enjoy very low sensitivities. Both direct sixth‐order universal filter structures and their equivalent three biquad stage ones are also simulated and validated that the former is not absolutely larger in sensitivity than the latter. Finally, a very sharp increment of the transconductance of an OTA is discovered as the operating frequency is very high and leads to a modified frequency‐dependent transconductance. Copyright © 2010 John Wiley & Sons, Ltd.     

Synchronous and asynchronous operation in dual‐band VCOs

We present a comprehensive analysis of the asynchronous and synchronous operations of fourth‐order oscillators underlying dual‐band voltage‐controlled oscillators. We analyze the occurrence of the self‐synchronization phenomenon (internal resonance) if the ratio of normal frequencies is nearly a ratio of integers, which is 1:3 in the cubic approximation of the nonlinear oscillator characteristic. In this case, we have the simultaneous presence of 2 oscillations with a frequency ratio 1:3, which was demonstrated to be very effective in generating high‐frequency signals in mm‐wave range. The analysis is carried out by developing an analytical approach relying on the averaging principle, as it follows the van der Pol method. The averaging equations, derived simply by a quadrature, allow us to analyze easily the stationary and transient oscillations, and their stability, both in asynchronous and synchronous operations. Expressions for the amplitudes and phases were derived for a cubic nonlinearity and verified by Spice simulations.     

Two‐dimensional non‐stationary numerical model for MSM structures

In this paper the solution of the non‐stationary model for MSM structures is obtained numerically. The two‐dimensional model consists of three singularly perturbed non‐linear partial differential equations. The alternating‐direction method for discretization in time and the non‐oscillatory streamline upwind method on a piecewise uniform grid for discretization in space are used to eliminate the interior and boundary layer oscillations. The described model is used for the analysis of the time response of a GaAs n‐type MSM structure to a Heaviside function form of the applied voltage. For the stationary case the I–V characteristic of the structure is determined. The numerical results confirm that the applied method is convenient for solving convection–diffusion problems. Copyright © 1999 John Wiley & Sons, Ltd.     

High input impedance voltage‐mode first‐order all‐pass sections

This paper proposes six new first‐order voltage‐mode all‐pass sections (VM‐APSs) based on three general topologies. Each circuit uses two differential voltage current conveyors and three grounded passive components. All the circuits possess high input impedance and easy control of pole frequency either by a simple matching of resistors (two equal‐valued resistors) for the three canonical circuits or by a single resistor for three non‐canonical circuits. PSPICE simulation results using real device 0.5µ CMOS parameters are given to validate the proposed circuits. Copyright © 2007 John Wiley & Sons, Ltd.     

A systematic method for the development of a three‐phase transformer non‐linear model

In this work, a novel three‐phase transformer non‐linear model is developed. The proposed model takes into account the magnetic core topology and the windings connections. The non‐linear characteristic curve of the core material is introduced by its magnetization curve or by its hysteresis loop using the mathematical hysteresis model proposed by Tellinen or the macroscopic hysteresis model proposed by Jiles–Atherton. The eddy currents effects are included through non‐linear resistors using Bertotti's work. The proposed model presents several advantages. An incremental linear circuit, having the same topology with the magnetic circuit of the core, is used in order to directly write the differential equations of the magnetic part of the transformer. The matrix L _d that describes the coupling between the windings of the transformer is systematically derived. The electrical equations of the transformer can be easily written for any possible connection of the primary and secondary windings using the unconnected windings equations and transformation matrices. The proposed methods for the calculation of the coupling between the windings, the representation of the eddy currents and the inclusion of the core material characteristic curve can be used to develop a transformer model appropriate for the EMTP/ATP‐type programs. Copyright © 2009 John Wiley & Sons, Ltd.     

Adaptive output‐feedback stabilisation of an uncertain second‐order linear systems

The adaptive stabilisation of uncertain second‐order linear systems is addressed, under a lack of information from both states and parameters. The only standard assumptions are no zeros and the sign of the high frequency gain known. To the best of the authors' knowledge, there has been only an explicit solution proposed in the literature so far with proven stability. Despite the simplicity of the system, it does not fit in any of the standard nonlinear control methodologies available. Thus, this work is a complementary contribution providing a mixed control design strategy based on a reduced‐order observer, adaptive Immersion & Invariance and Backstepping approaches. Hence, this solution depicts a transversal outlook of those nonlinear control strategies and provides a breakthrough for the generalisation of this non‐trivial control problem. Numerical simulations are reported to assess the effectiveness of the adaptive strategy. Copyright © 2016 John Wiley & Sons, Ltd.     

A numerical model for the oscillation frequency,the amplitude and the phase‐noise of MOS‐current‐mode‐logic ring oscillators

This paper presents a new model for the frequency of oscillation, the oscillation amplitude and the phase‐noise of ring oscillators consisting of MOS‐current‐mode‐logic delay cells. The numerical model has been validated through circuit simulations of oscillators designed with a typical 130 nm CMOS technology. A design flow based on the proposed model and on circuit simulations is presented and applied to cells with active loads. The choice of the cell parameters that minimize phase‐noise and power consumption is addressed. Copyright © 2009 John Wiley & Sons, Ltd.     

Adaptive output tracking for a class of non‐linear time‐delay systems

In this paper the output tracking control problem for a class of non‐linear time delay systems with some unknown constant parameters is addressed. Such a problem is solved in the case that the non‐linear time‐delay system has full delay relative degree and stable internal dynamics. It is supposed moreover that the output and its time derivatives until n?1, where n is the length of the state vector (euclidean part), do not depend explicitly on the unknown parameters. This work is the first step towards the application of the methodologies of adaptive control for non‐linear delayless systems, based on tools of differential geometry, to non‐linear time‐delay systems too. Copyright © 2004 John Wiley & Sons, Ltd.     

On the uniqueness of the numerical solution of non‐linearly loaded lossy transmission lines

The purpose of this paper is to investigate the uniqueness of the solution of lossy lines with frequency‐dependent parameters terminated with non‐linear resistors. Several solutions that satisfy the same initial conditions may exist if the terminal resistors are locally active. In these cases the uniqueness of solution is assured adding parasitic capacitances in parallel to the voltage controlled resistors and parasitic inductances in series to the current controlled resistors. In this way, among all the possible solutions, the only one that assures the time continuity of the current and voltage waveforms at the ends of the line is captured. In the light of these results, the properties of numerical models of these distributed circuits based on convolution techniques have been studied, and conditions assuring the uniqueness of the numerical solution have been found. Numerical simulations, when based on qualitative information of this type, enable us to obtain the quantitative properties in an efficient manner. In particular, a simple numerical method that enforces artificially the time continuity of the solution is proposed to circumvent the need of adding parasitics. Copyright © 1999 John Wiley & Sons, Ltd.     

Fast and accurate analysis of one‐dimensional arrays of dynamic PWL cells

This paper describes a new method for the time‐domain analysis of one‐dimensional arrays of dynamic piecewise linear cells. The method exploits the local connectivity, typical of cellular arrays, and the piecewise linear behaviour of the v–i characteristic of the non‐linear elements to obtain a piecewise analytical expression of the solution. Examples demonstrate the accuracy and the efficiency, in terms of CPU‐time, of the proposed method with respect to standard simulation tools as SPICE and the numerical integration of a system of ordinary differential equations. Copyright © 2001 John Wiley & Sons, Ltd.