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.     

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.     

Mode dynamics in fourth‐order LC oscillators

We present a complete analysis of single and concurrent modes in fourth‐order LC‐voltage‐controlled oscillators ( VCOs), which are increasingly applied in dual‐band communication systems. We give a procedure based on the averaging method that simplifies the derivation of the abridged equations, which are derived without resorting to a change of co‐ordinates. The amplitudes of the oscillatory modes in steady state and in transient are found in explicit form. Conditions for the stability of the single and concurrent modes are derived, which apply to any active one‐port dual‐band LC‐VCO and allow one to predict the nonlinearities ensuring the occurrence of a stable concurrent mode. Numerical and experimental results show a good accuracy of the presented formulas. Copyright © 2016 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.     

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.     

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.     

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.     

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.     

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 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.     

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.     

Current Control Method of Achieving Wide‐Range Power Driving for Linear Synchronous Motor with Half‐Wave‐Rectified Self‐Excitation

In a previous paper, we proposed a novel linear synchronous motor with half‐wave‐rectified self‐excitation. A long‐stator‐type linear synchronous motor was built and its performance was verified by experiments. This paper presents a new current control method for the linear synchronous motor in order to achieve a wide range of speeds and high‐power operations. First, we propose a current control method for high‐thrust operation in the constant‐thrust region. This operation is realized by using the reluctance thrust resulting from the saliency of the linear synchronous motor. We also propose a control method that maximizes the ratio of the thrust to the voltage; this method can be used to expand the operating range. Wide‐range‐speed operation can be achieved by applying this new control method along with field‐weakening control. The thrust and operating characteristics of the proposed control methods are estimated by performing experiments and coupled electric and magnetic analysis.     

The frequency tuning characteristics of transistor voltage‐controlled oscillators with distributed parameters

The frequency varactor tuning characteristics of voltage‐controlled oscillators (VCO) on elements with distributed parameters are analysed and presented in a generalized form. The recommendations on the choice of transmission line type, circuit and varactor parameters for the purposes of widening and linearization of the voltage frequency tuning characteristics are given. Large amounts of numerical and experimental results are performed to illustrate the theoretical assumptions. Copyright © 1999 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.     

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.     

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.     

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.     

Asynchronous sum‐of‐products logic minimization and orthogonalization

We propose a method of the asynchronous sum‐of‐products (SOP) logic simplification that comprises of minimization and orthogonalization. The method is based on a transformation of the conventional single‐rail SOP synchronous logic into the dual‐rail asynchronous one operating under so‐called modified weak constraints. We formulate and prove the product terms constraint that ensures a correct logic behavior. We have processed the MCNC benchmarks and generated the asynchronous SOP logic. The complexity of the logic obtained is compared with the state‐of‐the‐art representation. Using our approach, we achieve a significant improvement. Copyright © 2012 John Wiley & Sons, Ltd.     

同步电动机转矩角矢量控制原理

通过分析异步电动机转差频率矢量控制的特点,阐述了异步电动机转差频率和同步电动机转矩角之间的本质联系.类比于异步电动机转差频率矢量控制原理,提出了同步电动机转矩角矢量控制原理.该原理适用于控制精度要求不太高的同步电动机的控制系统,具有较高的性能价格比.     

云南异步电网中频率振荡的分段线性模型分析方法

云南异步联网试验中部分频率振荡现象发生在调速器死区附近,为分析此问题,文中建立了考虑调速器死区的分段线性系统模型。通过稳态和动态行为分析阐明了调速器死区对频率稳定性的影响。研究结果表明,增强型死区可能导致系统不存在平衡点,引起频率在死区附近振荡;同时分段线性化系统的稳定性由各频率区间对应的线性系统特征值决定,系统的负阻尼效应也会导致频率失稳,采用简化系统模型和云南电网实际模型均能仿真再现试验中的不同频率振荡现象。