Nonlinear analysis of microwave FET oscillators using Volterraseries

A novel approach for determining the amplitude and frequency of nonlinear FET oscillators is presented. The nonlinear elements of the active device are modeled by the Volterra series method. The frequency and amplitude of oscillation are then calculated by solving two algebraic equations. Experimental results obtained from a constructed oscillator confirm the validity of the theory, the discrepancy between measured and calculated frequency and amplitude values being less than 10%     

Nonlinear antenna technology

Nonlinear antennas combine advances in nonlinear dynamics, active antenna design, and analog microelectronics to generate beam steering and beam forming across an array of nonlinear oscillators. Nonlinear antennas exploit two phenomena typically shunned in traditional designs: nonlinear unit cells and interelement coupling. The design stems from nonlinear coupled differential equation analysis that by virtue of the dynamic control is far less complex than the linear counterparts by eliminating the need for phase shifters and beam forming computers. These advantages arise from incorporating nonlinear dynamics rather than limiting the system to linear quasisteady state operation. A theoretical framework describing beam shaping and beam forming by exploiting the phase, amplitude, and coupling dynamics of nonlinear oscillator arrays is presented. Experimental demonstration of nonlinear beam steering is realized using analog microelectronics     

Buildup behaviour of IMPATT oscillators in linear and nonlinear ranges

The growth rate, frequency shift and starting amplitude of oscillations in IMPATT-diode oscillators are calculated in the linear and nonlinear range. It turns out that, with increasing r.f. amplitude, the frequency in such oscillators always decreases, and that this frequency shift cannot be compensated for by proper tuning of the load. The r.f. amplitude remains below the measurable level for a certain time depending on the oscillator tuning. This accounts for the observed delay of the onset of the r.f. pulse.     

General nonlinear analysis of second-order oscillators

A general nonlinear analysis of second-order oscillators is presented. The oscillator equation is given in perturbation form, involving a small perturbation parameter, and its periodic solution derived, if any exists, which is calculated using an asymptotic method. Analytical relations between design constraints and circuit parameters are deduced and applied     

Nonlinear Dynamics and Spectral Stability of Optoelectronic Microwave Oscillators

We use a nonlinear dynamics approach to study the deterministic behavior of ultrapure microwave generators referred to as optoelectronic oscillators. In conventional studies, the standard nonlinear effects are very strongly rejected because they generate harmonics of the microwave frequency that are definitely out of the selective oscillator bandwidth. However, we show that the nonlinearity still affects the slowly varying dynamics of microwave envelope, thereby inducing dynamical instabilities within the oscillator bandwidth. Starting from a full integro-differential model, we use the multiple timescales method to build a delay-differential equation for the slowly varying complex envelope of the microwave. Then, the corresponding stationary solutions are derived, and the stability of their amplitude and phase is investigated in detail as a function of the feedback gain. We evidence essential bifurcation phenomena, and, in particular, we demonstrate that the generated microwave may turn unstable if the gain is increased beyond a precise critical value. This nonlinear dynamics approach, therefore, demonstrates that the amplitude of the ultrapure microwave's amplitude does not monotonously increase with the gain. The theoretical study is confirmed by numerical simulations and experimental measurements.      

Large-Signal Technique for Designing Single-Frequency and Voltage-Controlled GaAs FET Oscillators

A systematic procedure is described for designing fixed-frequency and voltage-tuned GaAs FET oscillators for optimum large-signal performance. The approach is based on the use of a large-signal FET model for de-embedding dominant device nonlinearities, leading to a method which is both accurate and simple to apply. The viability of the technique is demonstrated with a 17-GHz fixed-frequency oscillator and a 7.4 to 13.1-GHz varactor-tuned oscillator. Design considerations as well measured performance characteristics are discussed in detail.     

Analytical Equations for Nonlinear Phase Errors and Jitter in Ring Oscillators

In this paper, we present a simple analytical equation for capturing phase errors in 3-stage ring oscillators. The model, based on a simple but useful idealization of the ring oscillator, is provably exact for small noise perturbations. Despite its simplicity and purely analytical form, our model correctly captures the time- dependent sensitivity of oscillator phase to external perturbations. It is thus well suited for estimating both qualitative and quantitative features of ring oscillator phase response to internal noises, as well as to power, ground and substrate interference. The nonlinear nature of the model makes it suitable for predicting injection locking as well. Comparisons of the new model with existing phase models are provided, and its application for correct prediction of thermal jitter demonstrated. Requiring knowledge only of the amplitude and frequency of the oscillator, the model is ideally suited for early design exploration at the system and circuit levels.     

Current amplitude control circuits suitable for current-modeoscillators

A novel current amplitude control circuit suitable for current-mode oscillators is proposed. The circuit is a modified version of the well-known Gilbert gain cell. The technique obtains independent control of oscillation amplitude and small-signal current gain. As an example, the amplitude control circuit is applied to a current-mode oscillator. Simulations were carried out using HSPICE with 0.8 μm Nortel BiCMOS technology and Motorola RF transistors. Simulated results demonstrate that the nonlinear current gain control circuit behaves in a well defined manner. Low distortion and high frequency oscillations are easily obtained when the circuit is applied to a current-mode oscillator     

Nonlinearities of multicavity klystron amplifiers

The nonlinearities of a five-cavity klystron for wide-band operation are investigated by making use of the large signal analysis and experiments. The variations of frequency-response characteristics with respect to the input signal level, and the amplitude and phase nonlinearities are presented in detail for two typical tunings. It is shown that the nonlinearities are affected by the tuning of the intermediate cavity. The amplitude nonlinearity at saturation is about -3 to -6 dB within a frequency range of 80 MHz at the 14-GHz band. There exists some discrepancy between the calculated and measured phase nonlinearities. The nonlinear mechanisms are discussed to explain the results obtained by categorizing them into four parts. In the discussions, an emphasis is placed upon the velocity modulation of the beam by the intermediate cavity which introduces the frequency dependence into the nonlinearities.     

RC oscillators with oscillation frequency independent of the op.-amp. gain-bandwidth product

Applying the determining equation proposed by Chua and Tang (1982), a frequency sensitivity problem in RC op.-amp. based oscillators is considered. A new expression for the sensitivity of oscillation frequency, ω_o, to changes in any oscillator parameter is developed. Then, the condition for this frequency to be insensitive to changes in the gain-bandwidth products (GB) of the op.-amp. used in the oscillator is formulated. Examples of two circuits exhibiting a zero sensitivity of ω_o to the GB changes (a very desirable feature) are presented. The first example represents oscillators with a single op.-amp. whose slew-rate effect is the only oscillator non-linearity. The second example concerns oscillators with a so-called composite amplifier where non-linear elements included in the oscillator feedback network are responsible for amplitude stabilization.     

Oscillator phase noise: a tutorial

Linear time-invariant (LTI) phase noise theories provide important qualitative design insights but are limited in their quantitative predictive power. Part of the difficulty is that device noise undergoes multiple frequency translations to become oscillator phase noise. A quantitative understanding of this process requires abandoning the principle of time invariance assumed in most older theories of phase noise. Fortunately, the noise-to-phase transfer function of oscillators is still linear, despite the existence of the nonlinearities necessary for amplitude stabilization. In addition to providing a quantitative reconciliation between theory and measurement, the time-varying phase noise model presented in this tutorial identifies the importance of symmetry in suppressing the upconversion of 1/f noise into close-in phase noise, and provides an explicit appreciation of cyclostationary effects and AM-PM conversion. These insights allow a reinterpretation of why the Colpitts oscillator exhibits good performance, and suggest new oscillator topologies. Tuned LC and ring oscillator circuit examples are presented to reinforce the theoretical considerations developed. Simulation issues and the accommodation of amplitude noise are considered in appendixes     

The effects of noise in oscillators

An explicit expression for the output signal from an oscillator with several noise sources in the circuit is derived. This formula describes qualitatively and quantitatively the manner in which thermal and shot noise act to corrupt the performance of an ideal oscillator. The statistical properties of the signal are then evaluated, as it emerges from the oscillator stage, after passage through an output filter and after being operated on by an ideal n-times multiplier. Expressions are derived for the short term frequency stability, the power spectral density, and the power spectrum of the signal, as well as for the spectral density of the signal phase. The key to the results reported is an apparently novel perturbation technique which does not require smoothing of the instantaneous nonlinearity in the basic differential equation. Discussion of the solutions shows that the instantneous nonlinearities cause the device to act simultaneously like a linear AGC oscillator and like a high Q passive tuned circuit, with each aspect accorded one half the total noise excitation. Possible implications of this effect for other types of transient conditions in oscillators are indicated briefly.     

Noise calculations and experimental results of varactor tunableoscillators with significantly reduced phase noise

The single-sideband phase noise of varactor tunable GaAs MESFET oscillators is investigated. Two oscillator circuits with different microstrip resonator circuits were designed and fabricated. Using a resonator consisting of coupled microstrip lines instead of a single microstrip line, which is a planar monolithically integrable structure, phase noise is reduced significantly because the quality factor is higher for the coupled resonator. The phase noise is calculated using a nonlinear time domain method, which solves the Langevin equations, describing the deterministic and stochastic behavior of an oscillator by perturbation methods. Calculated and measured phase noise agree within the accuracy of measurements. The very low phase noise of 95 dBc/Hz at 100 kHz offset frequency is achieved     

High-performance crystal oscillator circuits: theory andapplication

A general theory that allows the accurate linear and nonlinear analysis of any crystal oscillator circuit is presented. It is based on the high Q of the resonator and on a very few nonlimiting assumptions. The special case of the three-point oscillator, that includes Peirce and one-pin circuits, is analyzed in more detail. A clear insight into the linear behavior, including the effect of losses, is obtained by means of the circular locus of the circuit impedance. A basic condition for oscillation and simple analytic expressions are derived in the lossless case for frequency pulling, critical transconductance, and start-up time constant. The effects of nonlinearities on amplitude and on frequency stability are analyzed. As an application, a 2-MHz CMOS oscillator which uses amplitude stabilization to minimize power consumption and to eliminate the effects of nonlinearities on frequency is described. The chip, implemented in a 3-μm p-well low-voltage process, includes a three-stage frequency divider and consumes 0.9 μA at 1.5 V. The measured frequency stability is 0.05 p.p.m./V in the range 1.1-5 V of supply voltage. Temperature effect on the circuit itself is less than 0.1 p.p.m. from -10 to +60°C     

Synchronization of oscillators coupled through narrow-band networks

The ability of two coupled oscillators to synchronize depends critically on the coupling network. Previous analyses have accurately predicted the performance of quasi-optical microwave oscillator arrays for both weak and strong coupling, but have been limited to coupling networks with bandwidths considerably larger than the locking bandwidths of the oscillators. In this paper, the authors develop a method for deriving a suitable system of nonlinear differential equations describing the oscillator amplitude and phase dynamics using a generalization of Kurokawa's method. The method is applied to the case of two Van der Pol oscillators coupled through a resonant network for a wide range of coupling strengths and bandwidths. Simple approximate formulas are developed for the size of the frequency locking region as functions of the basic circuit parameters     

A continuum model of the dynamics of coupled oscillator arrays forphase-shifterless beam scanning

The behavior of arrays of coupled oscillators has been previously studied by computational solution of a set of nonlinear differential equations describing the time dependence of each oscillator in the presence of signals coupled from neighboring oscillators. The equations are sufficiently complicated in that intuitive understanding of the phenomena which arise is exceedingly difficult. We propose a simplified theory of such arrays in which the relative phases of the oscillator signals are represented by a continuous function defined over the array. This function satisfies a linear partial differential equation of diffusion type, which may be solved via the Laplace transform. This theory is used to study the dynamic behavior of a linear array of oscillators, which results when the end oscillators are detuned to achieve the phase distribution required for steering a beam radiated by such an array     

Noise in Single-Frequlency Oscillators and Amplifiers

A generalization of previous oscillator noise analyses has been developed to permit reliable noise characterization of active nonlinear devices. Effects due to sideband correlation in the equivalent noise source are included. A rotating wave approximation (RWA) developed by Lax is used in obtaining the amplitude and phase noise spectra. Conditions are given for phase stabilization of free-running oscillators and for minimum phase noise in phase-Iocked oscillators and amplifiers. Stability criteria, discussion of spurious sidetones, and effects of a noisy synchronizing signal are given. The noise measure is used to obtain alternative expressions for the noise spectra and the carrier-to-noise ratios of locked oscillators and amplifiers. It is shown that the noise power gain of AM fluctuations is usually much lower than the corresponding gain for FM noise. The theory should be useful in optimizing the noise performance of nonlinear RF generators, such as IMPATT, BARITT, and Gunn diode oscillators.     

Phase noise in LC oscillators

Analytical methods for the phase-noise analysis of LC-tuned oscillators are presented. The fundamental assumption used in the theoretical model is that an oscillator acts as a large-signal LC-tuned amplifier for purposes of noise analysis. This approach allows us to derive closed-form expressions for the close-to-carrier spectral density of the output noise, and to estimate the phase-noise performance of an oscillator from circuit parameters using hand analysis. The emphasis is on an engineering approach intended to facilitate rapid estimation of oscillator phase noise. Theoretical predictions are compared with results of circuit simulations using a nonlinear phase-noise simulator. The analytical results are in good agreement with simulations for weakly nonlinear oscillators. Complete nonlinear simulations are necessary to accurately predict phase noise in oscillators operating in a strongly nonlinear regime. To confirm the validity of the nonlinear phase-noise models implemented in the simulator, simulation results are compared with measurements of phase noise in a practical Colpitts oscillator, where we find good agreement between simulations and measurements     

Injection locking of oscillators

An oscillator can be locked in frequency by an external signal which is injected into the oscillator. In the oscillator model developed by Adler [1], the mechanism of the locking process depends upon the following. 1) The initial frequency difference between the oscillator and external signals. 2) The relative amplitude between the injected and the oscillator signals. 3) The circuit parameters. There are cases when the time required for locking must be known, particularly when an oscillator is being locked to a pulsed signal. In this paper, the work of Adler is extended to develop an equation which is useful for higher levels of locking signal, a case often encountered when an oscillator is being injection locked by a pulsed signal. Because the solution of this equation is unwieldy and difficult to understand intuitively, except in very special cases, curves describing the locking mechanism were obtained using a digital computer. These curves enable a designer to construct oscillators which will provide a desired performance. The curves were checked experimentally and showed a close agreement between predicted and measured results. The experimental data indicates that the theory describes the locking time remarkably well even at high levels of locking signal.     

Influence of noise intensity on the spectrum of an oscillator

This paper investigates the influence of high-intensity noise on the correlation spectrum of a two-dimensional (2-D) nonlinear oscillator. An exact analytical solution for the correlation spectrum of this 2-D oscillator is provided. The analytical derivations are well suited for oscillators with white noise of any intensity, but computational constraints on the solution of the partial differential equation may make it impractical for cases where the number of state variables exceeds three. The spectral results predicted by our analytical method are verified by numerical simulations of the noisy oscillator in the time domain. We find that the peak of the oscillator spectrum shifts toward higher frequencies as the noise intensity is increased, as opposed to the fixed oscillation frequency predicted in the existing literature. This phenomenon does not appear to have been reported previously in the context of phase noise in oscillators.