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.     

Current mode CCII oscillators using grounded capacitors and resistors

New realizations of grounded C, grounded R current mode oscillators using the current conveyor (CCII) are given. The proposed oscillators are classified into two classes depending on the number of feedback loops. In class I, there is a single current feedback loop, whereas in class II, there are two current feedback loops. Class I includes two types and it employs two CCIIs, three capacitors and three or four resistors. Class II employs two CCIIs having two-outputs each, two capacitors and three resistors and has independent control on the condition of oscillation and on the frequency of oscillation by varying two alternative resistors. PSpice simulations are included. Exact analysis based on the parasitic elements of the CCII is carried out indicating that class I has a third order characteristic equation. Class II has the advantage that the effects of the parasitic elements of the CCIIs can be absorbed in the circuit components. © 1998 John Wiley & Sons, Ltd.     

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.     

The effect of parameter mismatches on the output waveform of an LC‐VCO

The effect of parameter mismatches on the output waveforms of a popular voltage‐controlled oscillator is investigated, schematizing the circuit as a system of two mutually coupled oscillators, whose describing equations are derived in a perturbation form. The circuit is studied using the method of two time‐scales showing the existence of synchronization phenomena leading in presence of mismatches to a locking frequency, which significantly differs from the natural frequencies of the tanks, and to an oscillation amplitude different from that of the symmetric case. We also show that in‐phase and quadrature oscillations at the drain nodes can be generated with a proper parameter setting. Circuit simulations confirm the presence of a synchronized oscillation, which is consistent with the prediction of the presented analysis. Copyright © 2009 John Wiley & Sons, Ltd.     

RC oscillator admits negative frequency

The humble resistance–capacitance (RC) oscillator has held our attention for nearly seven decades. During that time many improvements have been made to enhance the performance and the range of applicability. In this letter we consider further refinements that extend the frequency range to include zero, and even negative frequencies. This allows the frequency to be swept through zero with no discontinuity in phase or amplitude. We then suggest an addition to the well‐known automatic level control (ALC) circuits to linearize the control function, with attendant benefits in modeling, simulation, and implementation. Closed‐loop dynamics for the improved ALC circuit are predicted, then verified by experiment. Copyright © 2010 John Wiley & Sons, Ltd.     

Experimental comparison of phase‐noise in cross‐coupled RC‐ and LC‐oscillators

Relaxation RC‐oscillators are notorious for their poor phase‐noise performance. However, there are reasons to expect a phase‐noise reduction in quadrature oscillators obtained by cross‐coupling two relaxation oscillators. We present measurements on 5 GHz oscillators, which show that in RC‐oscillators the coupling reduces both the phase‐noise and quadrature error, whereas in LC‐oscillators the coupling reduces the quadrature error, but increases the phase‐noise. A comparison using standard figures of merit indicates that quadrature RC‐oscillators may be a viable alternative to LC‐oscillators when area and cost are to be minimized. Copyright © 2009 John Wiley & Sons, Ltd.     

Systematic realization of a class of hysteresis chaotic oscillators

A systematic method for realizing a class of hysteresis RC chaotic oscillators is described. The method is based on direct coupling of a general second‐order sinusoidal oscillator structure to a passive non‐monotone current‐controlled non‐linear resistor. Owing to this passive non‐linearity, the power consumption, supply voltage and bandwidth limitations imposed upon the chaotic oscillator are mainly those due to the active sinusoidal oscillator alone. Tunability of the chaotic oscillator can be achieved via a single control parameter and the evolution of the two‐dimensional sinusoidal oscillator dynamics into a three‐dimensional state‐space is clearly recognized. The flexibility of this method is demonstrated by two examples using PSpice simulations and experimental results. Numerical simulations of derived mathematical models are also shown. Copyright © 2000 John Wiley & Sons, Ltd.     

A new CMOS astable multivibrator and its nonlinear analysis

We present a nonlinear analysis of a new inductively tuned astable multivibrator obtained by connecting a timing inductor across a composite nonlinear resistor with a characteristic of N‐type, which is made up of the parallel connection of two complementary pairs of cross‐coupled MOS devices. Some possible practical applications of the circuit are also envisaged. Closed‐form expressions for the amplitude and the period of the periodic oscillation are derived in both cases when the circuit exhibits a relaxation oscillation and in the more difficult case when, due to the effect of parasitic capacitances of the devices, the circuit has an almost‐discontinuous relaxation oscillation with a nonzero switching time. The accuracy of the presented formulas, which are useful for both the analysis and design, is validated through circuit simulations and experimental results. Copyright © 2009 John Wiley & Sons, Ltd.     

Chaotic oscillator configuration using a frequency dependent negative resistor

A chaotic oscillator configuration employing a frequency‐dependent negative resistor (FDNR) as the only active element is proposed. The configuration relies on a simple two‐terminal passive device; namely a general purpose signal diode, to provide the necessary non‐linearity. The structure requires no floating elements and is independent of any circuit specific realization of the FDNR. Experimental results, PSpice simulations and numerical simulations of the derived mathematical models are included. Copyright © 2000 John Wiley & Sons, Ltd.     

Design of ultra low‐power RF oscillators based on the inversion coefficient methodology using BSIM6 model

This paper presents a fast and accurate way to design and optimize LC oscillators using the inversion coefficient (IC). This methodology consists of four steps: linear analysis, nonlinear analysis, phase noise analysis, and optimization using a figure of merit. For given amplitude of oscillation and frequency, we are able to determine all the design variables in order to get the best trade‐off between current consumption and phase noise. This methodology is demonstrated through the design of Pierce and cross‐coupled oscillators and has been verified with BSIM6 metal oxide semiconductor field effect transistor compact model using the parameters of a commercial advanced 40 nm complementary metal oxide semiconductor process. Copyright © 2015 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.     

Sinusoidal oscillators with lower gain requirements at higher frequencies based on an explicit tanh(x) nonlinearity

Two novel sinusoidal oscillator structures with an explicit tanh(x) nonlinearity are proposed. The oscillators have the attractive feature: the higher the operating frequency, the lower the necessary gain required to start oscillations. A nonlinear model for the two oscillators is derived and verified numerically. Spice simulations using AMS BiCMOS 0.35µ model parameters and experimental results are shown. Copyright © 2009 John Wiley & Sons, Ltd.     

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.     

Truly random number generators based on non‐autonomous continuous‐time chaos

A novel non‐autonomous continuous‐time chaotic oscillator suitable for high‐frequency integrated circuit realization is presented. Simulation and experimental results, verifying the circuit feasibility, are given. Two methods for using this oscillator as the core of a random number generator are also proposed. Numerical binary data obtained according to the proposed methods pass the four basic tests of FIPS‐140‐2, while experimental data pass the full NIST‐800‐22 random number test suite. Copyright © 2008 John Wiley & Sons, Ltd.     

A frequency‐domain‐based master stability function for synchronization in nonlinear periodic oscillators

An efficient methodology to study conditions for stable in‐phase synchronization in networks of periodic identical nonlinear oscillators is proposed. The problem of investigating synchronization properties on periodic trajectories is reduced to an eigenvalue problem by means of the joint application of master stability function and harmonic balance techniques. The proposed method permits to exploit the periodicity of trajectories, reducing computational time with respect to traditional time‐domain approaches (which were designed to deal with generic attractors) and good accuracy. In addition, such method can easily deal with networks of nonlinear periodic oscillators described by differential‐algebraic equations, and then both static and dynamic coupling could be studied. Copyright © 2011 John Wiley & Sons, Ltd.     

Mutiphase oscillator using dissipatively coupled transmission lines with regularly spaced tunnel diodes

A generation scheme is proposed for multiphase oscillatory signals based on mutual synchronization of the oscillating pulse edges developed in point‐coupled transmission lines periodically loaded with tunnel diodes (TDs). When supplied with an appropriate voltage at the end of a TD line, a pulse edge is shown to exhibit a spatially extended limit‐cycle oscillation on the line. When two TD lines are dissipatively coupled, such edge oscillations are mutually synchronized such that two edges simultaneously pass the connecting cells. Based on this observation, the scheme employs multiple point‐coupled TD lines, wherein the first and last lines are linked at the same position. Its design criterion is obtained by applying the phase reduction method to the transmission equation of a TD line. In addition, several time‐domain calculations demonstrate the validity of the scheme. Copyright © 2016 John Wiley & Sons, Ltd.     

Quadrature VCOs based on direct second harmonic locking: Theoretical analysis and experimental validation

We propose a theoretical analysis of the class of quadrature VCOs (QVCOs) based on two LC‐oscillators directly coupled by means of the second harmonic. The analysis provides the conditions for the existence and stability of steady‐state quadrature oscillations and a simplified model for the phase noise (PN) transfer function with respect to a noise source in parallel to the tank. We show that the figure of merit defined as the product between PN and current equals that of the single VCO, confirming that quadrature generation is achieved by this class of QVCO without degrading that figure of merit. An analytical model for the phase quadrature error due to tank mismatches is also proposed. The validity of all analytical models is discussed against numerical simulations. A practical implementation at 3.26 GHz with ±20% tuning range in a 0.13µm CMOS technology is also presented, confirming the main theoretical findings. Copyright © 2009 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.     

An inductorless CMOS quadrature oscillator continuously tuneable from 3.1 to 10.6 GHz

Modern RF front‐ends require wide tuning‐range oscillators with quadrature outputs. In this paper we present a two‐integrator quadrature oscillator, which covers the whole bandwidth of UWB applications. A circuit prototype in a 130 nm CMOS technology is continuously tuneable from 3.1 to 10.6 GHz. The circuit die area is less than 0.013mm², leading to a figure‐of‐merit FOMA of ?176.7dBc/Hz at the upper frequency. The supply voltage is 1.2 V, and the power consumption is 7 mW at the lower frequency and 13 mW at the upper frequency. Copyright © 2010 John Wiley & Sons, Ltd.     

Hyperchaos from two coupled Wien‐bridge oscillators

This paper proposes a simple inductorless approach to generating hyperchaos by coupling two Wien‐bridge oscillators with a resistor and a diode. The whole circuit is investigated both numerically and experimentally. Also, its hyperchaotic dynamics is studied theoretically by a topological horseshoe with two‐directional expansions which provides an immediate evidence of hyperchaos. Copyright © 2007 John Wiley & Sons, Ltd.