Evaluation and comparison of GHz‐range LC oscillators using time‐varying root‐locus

This paper presents a comprehensive comparison between complementary metal‐oxide‐semiconductor (CMOS) LC‐oscillator topologies often used in GHz‐range transceivers. The comparison utilizes the time‐varying root‐locus (TVRL) method to add new insights into the operation of different oscillators. The paper focuses on the treatment of the TVRL trajectories obtained for different oscillators and establishes links between the trajectories and physical phenomena in oscillators. The evaluation of the root trajectories shows the advantages of the TVRL method for comparing oscillator topologies, which is also extended towards the analysis of voltage‐controlled oscillators. The necessary circuit simplifications required in closed‐form root‐locus analysis are avoided by the TVRL, which allows precise oscillator comparison and reveals details on the topology specifics. The derived conclusions have been verified by the Cadence Spectre‐RF simulator on 130‐nm CMOS process. Copyright © 2011 John Wiley & Sons, Ltd.     

Generalized technique for designing broadband varactor-tuned negative resistance oscillators

A powerful method for designing broadband varactor-tuned negative resistance oscillators is presented. The CAD procedure stems from a novel way of employing Hilbert transform techniques. The method is capable of realistically taking into account the decisive factors governing oscillator performance and directly processes measured large-signal data of the active device without relying on an approximate circuit model for the device. Device—circuit interactions at higher harmonic frequencies are not considered in the context. A transferred-electron oscillator design example is used to illustrate the method.     

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.     

A new hardware-efficient digital sinusoidal oscillator with low- and uniform-frequency spacing

In this paper, a new low-frequency digital sinusoidal oscillator with uniform-frequency spacing is proposed, and its performance is evaluated. The proposed oscillator structure, P, requires a single, short word-length multiplier for its hardware implementation. For the same frequency range, the single multiplier utilized can be implemented using an 8-bit word length, assuming that the multipliers in the previously reported oscillators, such as the multiple-output direct form digital oscillator and its modified versions, are implemented using 16 bits. The saving in multiplier word length results in a reduction in the silicon area and power dissipation. The proposed oscillator structure utilizes a multiple-output direct form digital oscillator with a single, short word-length multiplier in addition to a very simple switching and sign changing operation. The proposed oscillator structure is capable of generating sinusoidal signals with a large number of samples per cycle. The phase of the generated sinusoids is continuous at the switching points, and the difference between any two adjacent generated frequencies is approximately constant. Simulation results are presented to verify the analysis and demonstrate the performance as measured in terms of total harmonic distortion. It is evident from the simulation results that the performance of the proposed oscillator is essentially the same as those of the previously reported oscillators with less hardware complexity.     

Improved switching mode oscillators employing generalized switching lines

The present work proposes an unusual method for controlling the activation of the switches of switching mode oscillators. The proposed method is in fact a tool for developing switching mode oscillators that are controlled by full variable feedback. The methods developed in the present article rely on characterizing the switching procedure by intentionally moulding the related switching line (the switching line is the line in the state space, where the switches change polarity). The switching line in most of the switching mode oscillators is relatively simple. For example, in many cases the switch changes state in accordance with the output current sign. Hence, the line ‘output current’ is zero, is the switching line in this case. Relatively many advantages are attained by the present moulding of the switching line. For example, oscillators that oscillate even with very low‐quality factor (large load) are obtainable. The proposed method enables a simple way for controlling the output voltage and the frequency of the switching mode oscillator. Furthermore, current limiter property of the oscillator output can be obtained by the method, even in cases where the original oscillator is inherently a voltage source. The output impedance is also made controllable by the method. The successfully improved characteristics are demonstrated by investigating a particular structure, namely, a switching mode series resonant oscillator. Copyright © 2000 John Wiley & Sons, Ltd.     

Sinusoidal shaping of the ISF in LC oscillators

A new method to decrease the phase noise of the sinusoidal oscillators is proposed. The proposed method is based on using a dynamic transistor biasing in a typical oscillator topology. This method uses the oscillator impulse sensitivity function (ISF) shaping to reduce the sensitivity of the oscillator to the transistor noise and as a result reducing the oscillator phase noise. A 1.8 GHz, 1.8 V designed oscillator based on the proposed method shows a phase noise of ?130.3dBc/Hz at 1 MHz offset frequency, thereby showing about 6 dB phase noise decreasing in comparison with the typical constant bias topology. This result is obtained from the simulation based on 0.18u CMOS technology and on‐chip spiral inductor with a quality factor equal to 8. Copyright © 2007 John Wiley & Sons, Ltd.     

Realization of tunable RC-active oscillators using grounded capacitors and voltage amplifiers

A systematic procedure is given to generate a set of tunable RC oscillators, each using two finite gain voltage controlled voltage sources, three resistors and two grounded capacitors. These realizations are shown to be minimal with respect to passive and active elements under the constraint that (one end of) all the capacitors remain grounded. The set consists of four oscillators and is shown to be complete, that is, no additional oscillator circuit can be generated under the given assumptions. A simple approach is then introduced for testing the structures for latch up behaviour. This approach also suggests a way of redesigning the oscillator circuits to render them free of latch up problems, if they exist. Experimental results are found to be in good agreement with the theory.     

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.     

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.     

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.     

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.     

Fully nonlinear oscillator noise analysis: an oscillator with no asymptotic phase

Oscillators exist in many systems. Detailed and correct characterization and comprehension of noise in autonomous systems such as oscillators is of utmost importance. Previous approaches to oscillator noise analysis are based on some kind of perturbation analysis, some linear and some nonlinear. However, the derivations of the equations for perturbation analysis are all based on information that is produced by a linearization of the oscillator equations around the periodic steady‐state solution, where it is assumed that the oscillator is orbitally stable and it has the so‐called asymptotic phase property. In this paper, we first discuss these notions from a qualitative perspective, and demonstrate that the asymptotic phase property is crucial in validating all of the previous approaches. We then present the case of a simple oscillator that is orbitally stable but without asymptotic phase, for which previous approaches fail. We then present a fully nonlinear noise analysis of this oscillator. We derive and compute nonlinear, non‐stationary and non‐Gaussian stochastic characterizations for both amplitude and phase noise. We arrive at results that are distinctly different when compared with the ones obtained previously for oscillators with asymptotic phase. We compare and verify our analytical results against extensive Monte Carlo simulations. Copyright © 2006 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.     

Analysis and design of LC oscillators associated with frequency multipliers: a phase noise perspective

When a local oscillator signal generation system is based on an LC oscillator and a frequency multiplier, the question of determining the optimal multiplication factor is a key issue. In this paper, the problem is addressed in order to minimize the 1/f ² phase noise within a tuning range constraint. The analysis, with a practical graphical representation, reveals the oscillator phase noise dependence on the oscillating frequency in the transition from two different regimes, named the inductor‐limited quality factor and the capacitor‐limited quality factor. The results obtained enable the evaluation of the phase noise performance of systems based on a sub‐harmonic and super‐harmonic oscillators and how they compare with an oscillator in the fundamental mode. Crucial questions like the phase noise improvement that these systems can achieve are analytically answered. A design methodology is thus proposed and verified through measurements on a frequency source at 31 GHz, composed by a sub‐harmonic voltage‐controlled oscillator followed by an injection‐locked frequency tripler, dedicated to backhauling applications, designed on a BiCMOS process technology. The tuning range is 10%, and the phase noise at a 1‐MHz offset is −112 dBc/Hz. Copyright © 2017 John Wiley & Sons, Ltd.     

New structures of four‐phase oscillators obtained by strongly interweaving mono‐phase limit‐cycle oscillators

The present work is a part of our effort of developing multiphase oscillators. The particular system dealt with here is that of strongly nonlinearly coupled four oscillators that form a multiphase source. Such sources possess potential applications in power electronics, in phased‐array antennas, and in modern methods of modulation and especially in demodulating multi‐phased modulated signals. The present system can be interpreted as embracing four two‐phase oscillators. Nevertheless, as a result of the strong coupling, the second state equation of each oscillator merges with the first equation of the following oscillator. The resulted four‐phase source is, therefore, represented by merely four state equations. The applications related to communications (especially those related to receivers) may be susceptible to the noise performance of the source. We believe that the presently suggested system, which relies on strong coupling of oscillators, is advantageous in its noise performance in comparison to more straightforward recently described multiphase sources, which incorporate loosely coupled oscillators. Copyright © 2007 John Wiley & Sons, Ltd.     

First harmonic analysis: a circuit theory point of view

The first harmonic method is usually applied to nonlinear system analysis, being particularly adequate for the study of oscillations. As shown in this paper, oscillators analysis can be performed by using only basic concepts of circuit theory. The nonlinearities present in the oscillator electric circuits are approximated, via the first harmonic method, by resistance or source-controlled equivalent circuit elements, and the resulting linear circuits are entirely analyzed via conventional circuit theory. As a result, the analysis of electric oscillators and the first harmonic method are introduced in a comprehensible electric circuit context     

The averaging perturbation method for the dominant behaviour analysis of adiabatic single-controlled oscillators

Electrical oscillators are called single-controlled when their non-linear elements depend on a unique variable, while they are termed adiabatic when their transient operation signal can be represented as a narrowband-modulated Fourier series with only a finite number of dominant components, including the DC one. the novel method presented here allows one to analyse the dominant dynamics, steady state and dynamical stability of such oscillators. the approach devised is perturbative and is termed averaging, since it moves from the differential equation in the oscillator signal to provide a set of simultaneous differential equations in the complex envelopes of the dominant components of that variable. Owing to the treatment generality, investigable circuit topologies comprise a linear lumped time-invariant network of any order with bias sources, and any number of non-linear elements, with or without memory, with a common control variable. to illustrate the use of the method, formulae concerning a seventh-order oscillator are derived.     

On the development of symmetrically stabilized five-phase oscillators and some implications

The paper deals with the development of a symmetrically stabilized five-phase oscillator. the development results from two origins. It employs on one hand a five-phase conservative oscillator. On the other hand the stabilization process of the oscillator is due to similar non-linear damping terms in earlier developments of stabilized three-phase oscillators. A new feature of the present system is that it possesses an interesting steady state limit-cycle behaviour. the two possible steady state oscillations are similar, since both of them consist of a five-phase balanced set of phasors which are arranged symmetrically. However, each mode of possible steady state oscillation possesses its own unique frequency, although the amplitudes are equal. Another interesting feature of the oscillator is related to the way in which the various possible solutions (oscillatory, steady state, stable and unstable dynamic behaviours) evolve from initial conditions. There seem to exist regions where the system solutions appear sensitive to minute changes in initial conditions. It appears that completely different types of dynamic behaviours can develop from initial conditions in close proximity, which may lead eventually to the development of slightly modified systems with chaotic dynamics. the paper concludes by suggesting an application of multiphase oscillators for feeding phased array antennas.     

基于Duffing振子的弱信号频率检测研究

现有Duffing振子弱信号检测方法检测频率信号时存在检测频率范围窄的缺点,在检测宽频带信号时需要用到多振子阵列,增加了实现的复杂度。文章充分利用周期信号频率特性和Duffing振子的弱信号检测技术,提出了一种新的Duffing振子检测频率信号的方法,该方法摒弃了传统外加周期策动力的方式,将待测信号直接送入Duffing振子,然后对振子输出量作频谱分析,降低了调谐的难度并增加了振子运行的稳定性。仿真结果表明,该方法可实现对信号频率的精确检测,具有精确度高（10-4数量级）、复杂度低的优点。     

A generalized family of memristor‐based voltage controlled relaxation oscillator

Recently, memristive oscillators are a significant topic in the nonlinear circuit theory where there is a possibility to build relaxation oscillators without existence of reactive elements. In this paper, a family of voltage‐controlled memristor‐based relaxation oscillator including two memristors is presented. The operation of two memristors‐based voltage relaxation oscillator circuits is demonstrated theoretically with the mathematical analysis and with numerical simulations. The generalized expressions for the oscillation frequency and conditions are derived for different cases, where a closed form is introduced for each case. The effect of changing the circuit parameters on the oscillation frequency and conditions is investigated numerically. In addition, the derived equations are verified using several transient PSPICE simulations. The power consumption of each oscillator is obtained numerically and compared with its PSPICE counterpart. Furthermore, controlling the memristive oscillator with a voltage grants the design an extra degree of freedom which increases the design flexibility. The nonlinear exponential model of memristor is employed to prove the oscillation concept. As an application, two examples of voltage‐controlled memristor‐based relaxation oscillator are provided to elaborate the effect of the reference voltage on the output voltage. This voltage‐controlled memristor‐based relaxation oscillator has nano size with storage property that makes it more efficient compared with the conventional one. It would be helpful in many communication applications.