Frequency-domain analysis of the periodically-forced Josephson-junction circuit

In this paper, a new frequency domain technique to analyze the Josephson-junction circuit dynamics is presented. This technique overcomes some of the limitations inherent to the analytical and time-integration techniques used in previous works. The technique can be extended to the analysis of this device when combined with distributed elements in microwave systems. It allows an efficient analysis of the different types of steady-state solutions and the bifurcation loci in the presence of a periodic driving current source. No restriction is imposed to the driving source amplitude, enabling an accurate analysis of the influence of this parameter on the device superconducting properties. The technique has also been applied to analyze the quasi-periodic states present in this device together with the synchronized solutions to the driving current source.This work was supported by the Spanish Ministry of Economy and Competitiveness under Contract TEC2011-29264-C03-01

Noise analysis of super-regenerative oscillators in linear and nonlinear modes

A rigorous analysis of noise effects in super-regenerative oscillators (SROs), operating in both linear and nonlinear modes, is presented. For operation in the linear mode, two different analysis methods are presented. One is based on the calculation of linear-time variant (LTV) transfer function with respect to the input signal and the noise sources. The second method is based on a compact semianalytical formulation of the pulsed oscillator under the effect of the quench signal. The compact formulation also enables the analysis of the SRO in the nonlinear mode. It constitutes a fully new mathematical description of SROs, with general applicability, as it is not restricted to a particular oscillator topology. It relies on a numerical nonlinear black-box model of the stand-alone free-running oscillator, extracted from harmonic-balance simulations. This model is introduced into an envelope-domain formulation of the SRO at the fundamental frequency. Both the method based on LTV transfer functions and the semianalytical formulation take into account the cyclostationary nature of the SRO response to the noise sources. In the nonlinear mode, the variances of the amplitude and phase are calculated linearizing the formulation of the pulsed steady-state solution. The particular time variation of the phase variance is explained in detail and related to the onset and extinction of the oscillation in the presence of an RF input signal. The new analysis methods have been validated with both independent circuit-level simulations and measurements.This work was supported by the Spanish Ministry of Economy and Competitiveness and the European Regional Development Fund (ERDF/FEDER) under Project TEC2017-88242-C3-1-R

Cyclostationary noise analysis of superregenerative oscillators

A rigorous analysis of noise effects in super regenerative oscillators (SRO) operating in linear mode is presented. The analysis takes into account the cyclostationary nature of the SRO response to the noise sources, due to the effect of the quench signal. It is based on the determination of envelope-domain linear-time-variant (LTV) transfer function with respect to each noise source, plus the application of a detailed stochastic analysis of the SRO output. Initially, the autocorrelation of the output signal is calculated, which varies at two different time scales, and is periodic with respect to the quench signal, so it can be expressed in terms of the frequency-dependent harmonic components of the LTV transfer functions. This enables the calculation of the output spectral density, depending on these harmonic components. Once the spectral density is known, the signal-to-noise ratio can be obtained in a straightforward manner. The analysis method has been validated with both independent circuit-level simulations and measurementsThis work was supported by the Spanish Ministry of Science, Innovation and Universities and the European Regional Development Fund (ERDF/FEDER) under the research project TEC2017-88242-C3-1-R

Piecewise semi-analytical formulation for the analysis of coupled-oscillator systems

A new simulation technique to obtain the synchronized steady-state solutions existing in coupled oscillator systems is presented. The technique departs from a semi-analytical formulation presented in previous works. It extends the model of the admittance function describing each individual oscillator to a piecewise linear one. This provides a global formulation of the coupled system, considering the whole characteristic of each voltage-controlled oscillator (VCO) in the array. In comparison with the previous local formulation, the new formulation significantly improves the accuracy in the prediction of the system synchronization ranges. The technique has been tested by comparison with computationally demanding circuit-level harmonic balance simulations in an array of Van der Pol-type oscillators and then applied to a coupled system of FET-based oscillators at 5 GHz, with a very good agreement with measurements.This work was supported by the Spanish Ministry of Economy and Competitiveness and the European Regional Development Fund (ERDF/FEDER) under Project TEC2014-60283-C3-(1/2)-R and Project TEC2017-88242-C3-(1/2)-

Nonlinear technique for the analysis of the free-running oscillator phase noise in presence of an interference signal

A new methodology for the prediction of oscillator phase noise under the effect of an interference signal is presented. It is based on a semi-analytical formulation in the presence of the noisy interferer, using a realistic oscillator model extracted from harmonic-balance simulations. The theoretical analysis of the phase process enables the derivation of key mathematical properties, used for an efficient calculation of the interfered-oscillator phase noise spectrum. The resulting quasi-periodic spectrum is predicted, as well as the impact of the interferer phase noise over each spectral component, in particular over the pulled oscillation frequency. It is demonstrated that under some conditions, the phase noise at this component is pulled to that of the interference signal. Resonance effects at multiples of the beat frequency are also predicted. The analyses have been validated with experimental measurements, obtaining excellent agreement.This work was supported by the Spanish Ministry of Economy and Competitiveness under the research project TEC2014-60283-C3-1-R, the European Regional Development Fund (ERDF/FEDER) and Juan de la Cierva Research Program IJCI-2014-19141, and by the Parliament of Cantabria under the project Cantabria Explora 12.JP02.64069

Generalized stability criteria for power amplifiers under mismatch effects

Potential instability of power amplifiers (PAs) under mismatch effects is analyzed, with emphasis on the ease and generality of application of the stability criteria. The methodology is based on the evaluation of a large-signal version of the μ factor, considering mismatch effects in the fundamental frequency and three relevant sidebands: the baseband, the lower sideband and the upper sideband. This requires an outer-tier scattering-type conversion matrix of order 3 × 3 to be obtained, with the rest of sideband equations acting as an inner tier. It is taken into account that the circuit behaves nonlinearly with respect to the termination at the fundamental frequency. The consideration of three sidebands will enable the prediction of the two major forms of large-signal instability: incommensurable oscillations and frequency divisions by two. The analysis is preceded by an evaluation of the circuit own stability properties (proviso) under open and short circuit terminations at the sidebands, for all possible values of the termination at the fundamental frequency. Three different μ factors can be defined between any two ports of the scattering matrix. The analysis of the relationships between these factors and their continuity properties will allow the derivation of a single number able to characterize the PA potential instability for each fundamental-frequency termination. Results have been exhaustively validated with independent circuit-level simulations based on pole-zero identification and with measurements, using a variable output load and loading the PA with an antenna.This work has been supported by the Spanish Government under contract TEC2014-60283-C3-1-R and the Parliament of Cantabria (12.JP02.64069

Nonlinear analysis of cycle slips in injection-locked oscillators

A method is presented for the analysis of cycle slips in injection-locked oscillators. This nonlinear phenomenon gives rise to a temporal desynchronization between the injected oscillator and the input source due to noise perturbations. It involves very different time scales, so even envelope-transient based Monte Carlo analyses may suffer from high computational cost. The method presented is based on the extraction of a semi-analytical nonlinear model of the injected oscillator. This reduced order model enables an efficient stochastic analysis of this oscillator, based on the use of the associated Fokker-Planck equation in the phase probability density. The analysis allows predicting the parameter-space regions that are best protected against cycle slips. The method has been applied to an injection-locked oscillator at 5 GHz, with good agreement with commercial software simulations and measurements.This work was supported by the Spanish Ministry of Economy and Competitiveness under Contract TEC2011-29264-C03-0

Effects of noisy and modulated interferers on the free-running oscillator spectrum

A new methodology for the prediction of oscillator phase dynamics under the effect of an interference signal is presented. It is based on a semianalytical formulation in the presence of a noisy or modulated interferer, using a realistic oscillator model extracted from harmonic-balance simulations. The theoretical analysis of the phase process enables the derivation of key mathematical properties, used for an efficient calculation of the interfered-oscillator spectrum. The resulting quasi-periodic spectrum is predicted, as well as the impact of the interferer phase noise and modulation over each spectral component, in particular over the one at the fundamental frequency. It is demonstrated that under some conditions, the phase noise at this component is pulled to that of the interference signal. Resonance effects at multiples of the beat frequency are also predicted. In addition, the effects of interferer phase and amplitude modulation on the oscillator phase dynamics have been studied and compared. For that analysis, efficient simulation techniques have been developed. The analyses have been validated with experimental measurements in an FET-based oscillator at 2.5 GHz, obtaining excellent agreement.This work was supported in part by the Spanish Ministry of Economy and Competitiveness and the European Regional Development Fund (ERDF/FEDER) under research projects TEC2014-60283-C3-1-R and TEC2017-88242-C3-1-R, in part by the Juan de la Cierva Research Program under Grant IJCI-2014-19141, and in part by the Parliament of Cantabria through the project Cantabria Explora under Grant 12.JP02.6406

Analysis of the transient dynamics of coupled-oscillator systems

A realistic reduced-order formulation of systems containing several transistor-based oscillators, such as coupledoscillator networks, is presented. The formulation is able to predict the oscillation build-up and other transient effects for the first time to our knowledge. The individual oscillator models are constructed from a nonlinear admittance function extracted from circuit-level harmonic-balance simulations. These models are used to derive a nonlinear differential-equation system able to describe the transient behavior of the entire structure. For illustration, the method has been applied to a coupled-oscillator system at 5 GHz, obtaining very good agreement with circuit-level envelope transient (when applicable) and with measurements.This work was supported by the Spanish Ministry of Science, Innovation and Universities and the European Regional Development Fund (ERDF/FEDER) under the research project TEC2017-88242-C3-1-R

Coupled-oscillator systems: efficient simulation with harmonic-balance based oscillator models

A two-level simulation methodology for coupled-oscillator systems is presented. The inner level corresponds to the harmonic-balance analysis of the elementary oscillator in free-running regime, so as to obtain a one-port reduced-order model. The outer level corresponds to the analysis of the coupled system, deriving a formulation that explicitly relates the oscillation-frequency deviation and the amplitude and phase distributions to magnitudes characterizing the coupling network and oscillator elements. A stability analysis based on this formulation is also carried out obtaining a mathematical condition that determines the stable phase-shift interval. The formulation provides valuable insight into the system behavior and allows the derivation of realistic criteria for an optimum coupled-system performance. The two-level analysis has been extended to injection-locked systems, which enables an investigation of the impact of the number of oscillator elements and relative position of the injection signal on the locking bandwidth.This work was supported by the Spanish Ministry of Economy and Competitiveness under Contract TEC2011-29264-C03-01