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.     

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.     

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.     

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.     

Harmonic oscillators realized using current amplifiers and grounded capacitors

New configurations of harmonic oscillators, realized using current amplifier blocks and only grounded capacitors, are introduced in this article. The proposed configurations are based on a grounded inductor simulator scheme and on a loop constructed from first‐order sections, respectively. Comparison with the already published topologies shows that the new configurations have attractive characteristics concerning their implementation in integrated form. Copyright © 2006 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.     

CMOS digitally programmable quadrature oscillators

CMOS digitally programmable quadrature oscillators based on digitally controlled current followers and voltage followers are proposed. The proposed designs provide the advantage of programmability similar to the operational transconductance amplifier‐based oscillators while offering improved linearity. In mixed analog/digital systems, the digital tuning feature allows direct interfacing with the digital signal processing part. Novel realizations that provide both voltage‐mode and current‐mode quadrature sinusoidal signals are presented. Employing only grounded capacitors the designs achieve independent control of the frequency and condition of oscillation that can be tuned digitally. Experimental results obtained from a 0.35 µm CMOS chip fabricated using standard CMOS process are given. Copyright © 2008 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.     

Using modal series to analyze the transient response of oscillators

In this paper, a new form of Modal series is used to obtain the transient time‐domain response of oscillators. It is applicable to n‐dimensional systems and is not dependent on the existence of a small parameter in circuit's model. In addition, it provides an approximate analytical expression for the transient response instead of numerical solutions. It is valuable since the transient response of oscillators is not frequency stationary and therefore the FFT of numerical methods may not be so useful. The Colpitts oscillator is selected as a case study and a closed‐form expression for its transient response is derived which approximates the real response up to the steady state. Copyright © 2009 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.     

Analysis and synthesis of perturbed Duffing oscillators

Analysis and synthesis of perturbed Duffing oscillators have been presented. The oscillations in such systems are regarded as limit cycles in perturbed Hamiltonian systems under polynomial perturbations of sixth degree and are analysed by using the Melnikov function. It has been proved that there exists a polynomial perturbation depending on the zeros of the Melnikov function so that the system considered can have either two simple limit cycles, or one limit cycle of multiplicity 2, or one simple limit cycle. A synthesis of such oscillators based on the Melnikov's theory has been proposed. Copyright © 2006 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.     

Sinusoidal oscillators with explicit current output employing current‐feedback op‐amps

Employing a state‐variable synthesis, a number of new current‐mode oscillators with explicit current output have been derived, which can be practically implemented from commercially available current‐feedback op‐amps (CFOA). The workability of the proposed structures has been confirmed by experimental results using AD844‐type CFOAs and some sample results have been presented. Copyright © 2008 John Wiley & Sons, Ltd.     

A method for analysing the transient and the steady‐state oscillations in third‐order oscillators with shifting bias

We provide an asymptotic method for systematically analysing the transient and the steady‐state oscillations in third‐order oscillators with shifting bias. The method allows us to construct the general solution of the weakly non‐linear differential equation describing these oscillators through an iteration procedure of successive approximations typical of perturbation methods. The approximation to first order is obtained solving a system of two first‐order non‐linear differential equations in the leading terms of solution (dc component and fundamental harmonic), whereby the dominant dynamics, the stationary states and their stability can be easily analysed. Unlike existing approaches, our method also enables us to determine the higher harmonics as well as the frequency shift from the system's natural frequency in the exact solution through analytical formulae. In addition, formulae for higher‐order approximations of the above quantities are determined. The proposed method is applied to a practical circuit to show its usefulness in both analysis and design problems. Copyright © 2001 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.     

Evaluating the spectrum of periodic pulling in subharmonic resonant LC circuits

We analyze the features of the oscillations arising in forced inductor–capacitor (LC) oscillators when they operate in the periodic pulling mode, under the action of a weak injection signal. In radio frequency integrated circuits, both voltage‐controlled oscillators subject to undesired couplings and injection‐locked frequency dividers behave like forced LC oscillators. These are modeled as second‐order driven oscillators working in the subharmonic (secondary) resonance regime. The analysis is based on the generalized Adler's equation, which we introduce to describe the phase dynamics of dividers of any division ratio and to derive closed‐form expressions for the spectrum components of the system's oscillatory response. We show that the spectrum is double‐sided and asymmetric, unlike the single‐sided spectrum of systems with primary resonance. Numerical and experimental results are given to validate the presented results, which significantly generalize those available in the literature. Copyright © 2014 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.     

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.     

On hyperbolic laws of capacitor discharge through self‐timed digital loads

A new model to predict the dynamic behavior of a self‐timed autonomous digital system powered by a capacitor is derived. The model demonstrates the hyperbolic shape of the discharging process on the capacitor. It allows a symbolic analysis of the discharging process for complex digital loads comprised of series (stack) and parallel configurations of digital circuits. For example, for a stack configuration, important non‐trivial relationships between the hyperbolic discharging rates have been derived based on the knowledge of the velocity saturation index (alpha) of the semiconductor devices used in the digital part. For a realistic (modern complementary metal oxide semiconductor (CMOS) devices) value of alpha = 1.5, the discharging process for a stack of two identical circuits proceeds nearly three times slower than that of any of the stand‐alone circuits. This shows a potential way of extending the lifetime of the energy sources by means of stacking self‐timed circuits. Although the analysis is based on configurations consisting of ring oscillators in CMOS technology, the analysis method can be extended to other types of self‐timed systems and other semiconductor technologies in which the instantaneous switching activity of the digital load is determined by the instantaneous voltage levels provided by the capacitive power transfer mechanism. The analytical derivations have been validated by simulations and experiments carried out with real hardware. © 2014 The Authors. International Journal of Circuit Theory and Applications published by John Wiley & Sons, Ltd.