A. Buonomo

Asymptotic formulas in nearly sinusoidal nonlinear oscillators

We give an iterative procedure for calculating the steady-state output waveform of nearly sinusoidal nonlinear feedback-oscillators whose order as well as the shape and the degree of the nonlinearity is assumed to be of the greatest generality. The procedure inherits its foundation from the classical perturbation theory and makes it possible to determine both the frequency and the complex amplitude of the harmonics of the oscillation by very simple recurrence formulas, by which a complete analytical representation of the nonlinear oscillation is obtained. These asymptotic formulas are valuable for both the analysis and synthesis purposes. The application of the above formulas to some typical oscillator circuits is given to show their usefulness in practical problems.

Nonlinear dynamics of divide-by-two injection-locked frequency dividers in locked operation mode

A method for analyzing the nonlinear dynamics of the injection-locked frequency dividers in synchronized operation mode is presented, including the stability analysis of locked states. We use a specific divide-by-two circuit, namely a differential LC CMOS divider with a complementary topology, as a guideline for presentation, showing that the sizing of the devices significantly affects the synchronization mechanism of the divider, which exhibits a very rich dynamical behavior. We provide closed-form expressions to determine the amplitude and the phase in the locked state, as well as the locking range, leading to accurate results, which are validated by numerical simulations. The presented analysis of the frequency divider dynamics enables us to establish that stable locked oscillations occur on the whole locking range predicted by the well-known Adlers equation and that these are possible also beyond that range. Copyright

On the periodic solution of the van der Pol equation for small values of the damping parameter

We give a purely analytical perturbation method of solution of the van der Pol equation, whereby the periodic solution can be actually developed in the form of a power series in the damping parameter e up to any desired order and thus analysed in detail. The coefficients of the series solution, which are given in explicit form by recurrent analytical formulae, are calculated up to the order e500 to accurately determine the radius of convergence of the series solution, for which the value of 1·89 (correct to two decimal places) was found. We have then investigated the well-known Shohat expansion in order to elucidate an unsolved question concerning its validity, which we show to be restricted to the narrower interval e<2·70 of e⩾0.

A criterion of existence and uniqueness of the stable limit cycle in second-order oscillators

Conditions that ensure the existence of a single stable limit cycle in second-order oscillators are found. These conditions enable to characterize both the required nonlinear element and the range of values of the linear circuit parameters in a simple way. It will be shown how these conditions can be applied to the design of the commonly used oscillators.

A Collocation Algorithm for Calculating the Periodic Solutions of Non-linear Oscillators

An approximation method is given for calculating the periodic solutions of non-linear oscillators based upon the well-known method of collocation, which belongs to the class of projection methods. Unlike available techniques, the method leads to a simple and efficient algorithmic procedure which consists of solving a system of non-linear equations given in a simple explicit form. To demonstrate the effectiveness of the proposed algorithm, the results of some typical oscillator circuits are provided.

On the evaluation of higher harmonics in nearly sinusoidal oscillators

An approximation method for calculating the periodic oscillations of nearly sinusoidal oscillators is presented which is based upon the classical pertubation method. Both the frequency and the amplitude of the harmonics of the oscillation are calculated to a first approximation by very simple relations which are useful for deriving a design criterion. To demonstrate the possibilities of the proposed method, some illustrative examples are provided.

Nonlinear two-dimensional impurity diffusion in semiconductors: A quasi-linear numerical analysis

A two-dimensional numerical simulation program is proposed which enables the concentration profiles of diffused dopants in semiconductors to be calculated. This program takes the nonlinear phenomena typical of high concentrations into account; however, since the corresponding nonlinear diffusion model is made quasi-linear by means of a suitable transformation of the variable, it becomes almost as easy and efficient as a linear numerical simulation program. The numerical algorithm developed is based on the finite difference Alternating Direction Implicit (ADI) method proposed by Peaceman and Rachford. As a practical application, the two-dimensional doping profiles of arsenic into silicon are calculated for a predeposition process and for a drive-in process following an ion implantation.

Some remarks about the use of the charge-neutrality approximation in the simulation of diffusion processes

Doping profiles in bipolar devices are numerically calculated taking into account the actual space charge produced during the diffusion process, which is described by standard models based upon effective diffusivities. The influence of this charge on the diffusion of dopants was examined for arsenic/boron double diffusion in order to evaluate the errors on profiles obtained using the well-known approximation of space-charge neutrality. Numerical results show that this approximation satisfactorily predicts practical doping profiles except for short-time low-temperature diffusion. >

Symbolic analysis of nearly sinusoidal oscillators

We show that the nonlinear analysis of nearly sinusoidal oscillators can be performed by a very simple symbolic program, whereby the periodic oscillation as well as its frequency is determined in symbolic form in terms of both linear and nonlinear circuit parameters. The program is an useful design tool, as it enables the harmonic content and the frequency sensitivity to circuit parameters to be obtained in analytic form.

A lower bound for the half-power frequency in RC amplifiers

It is shown that in RC amplifiers there exists a simple relationship between the half-power frequency \omega_{h} and the so-called firstorder time moment; a more precise estimate of \omega_{h} than that usually adopted is, therefore, given.