Almudena Suarez

Nonlinear analysis tools for the optimized design of harmonic-injection dividers

New nonlinear analysis tools for harmonic-injection dividers are presented based on bifurcation concepts. The advantage of these tools is their application simplicity and efficiency, which has enabled their use for actual circuit design and optimization. The tools allow control over the divided frequency and output power and predict the variation of the synchronization bands versus the circuit element values, which facilitates design correction. They have been extended to the analysis and optimization of phase-locked harmonic-injection dividers, which contain a low-frequency feedback loop. The use of this loop, together with the accuracy of the analysis, has enabled the implementation of novel frequency-division functions, such as the division of variable order, versus a circuit parameter, or the division by fractional order. The output noise of the frequency dividers is analyzed through the conversion-matrix approach, studying the noise variation along the division bands. The new techniques have been applied to the design of a frequency divider by order 4 and 5, with 18-GHz input frequency, and excellent agreement with experimental results has been obtained.

Synchronization analysis of autonomous microwave circuits using new global-stability analysis tools

A new spectral-balance technique for the global-stability analysis of autonomous circuits is presented in this paper. This technique relies on the introduction of measuring probes into the circuit and it allows a simple determination of both bifurcation diagrams and bifurcation loci as a function of any suitable parameter. Through the proposed algorithms, this kind of analysis can be easily added to any existing software, since it is performed externally to the harmonic-balance (HB) calculation. Due to its local nature, it also allows an easy selection of the bifurcation parameters, which spreads the simulation possibilities. Both periodic and quasi-periodic regime simulations are possible, and bifurcations are detected in both operating modes. The synchronization phenomenon in injected oscillators and frequency dividers is also analyzed in detail for an accurate prediction of the operating bands. The simulation techniques are illustrated by means of their application to a cubic nonlinearity oscillator. They are then used for the stability analysis of a monolithic microwave integrated circuit (MMIC) divider by two operating in the millimetric range. A very good agreement has been obtained with the experimental results.

Phase-Noise Analysis of Injection-Locked Oscillators and Analog Frequency Dividers

In-depth investigation of the phase-noise behavior of injection-locked oscillators and analog frequency dividers is presented. An analytical formulation has been obtained, which allows a better understanding of the shape of the output phase-noise spectrum of these circuits. The simplicity of this formulation is also helpful for circuit design. Approximate expressions for the corner frequencies of the spectrum are determined, identifying the most influential magnitudes and deriving design criteria. In particular, a technique has been developed to shift the frequency of the first corner of the phase-noise spectrum, up to which the output phase noise follows the input one. The expressions for the corner frequencies can be introduced in either in-house or commercial harmonic-balance software, thus allowing an agile design, as no separate phase-noise analysis is required. The validity of the analytical techniques is verified with the conversion-matrix approach and with measurements using two field-effect-transistor-based circuits: a 4.9-GHz injection-locked oscillator and a frequency divider by 2 with 9.8-GHz input frequency.

New Techniques for the Analysis and Design of Coupled-Oscillator Systems

An in-depth analysis of the nonlinear dynamics of coupled-oscillator arrays is presented for a better understanding of their complex autonomous behavior. In one-dimensional arrays, the constant inter-stage phase shift is varied by simultaneously detuning the two outermost oscillators in opposite directions. Thus, the array can be considered as a two-parameter system. Here, a two-parameter stability analysis of the oscillator array is carried out, investigating the phenomena that limit the achievable values of constant inter-stage phase shift under both weak and strong coupling conditions. The gradual evolution of the system behavior with increasing coupling strength is studied. A semianalytical approach is presented for an efficient design of the oscillator array, avoiding the computational expensiveness of harmonic balance (HB) in systems with a high number of oscillator elements. The proposed method, valid for weak coupling, uses a perturbation model of the elementary oscillator obtained with HB so it is of general application to any oscillator topology with accurate descriptions of its linear and nonlinear components. Approaches for the stability and phase-noise analyses based on this formulation are also presented. The new techniques have been applied to the design of a coupled system of three oscillators at 6 GHz. The results have been successfully compared with full HB simulations and with measurements

Analytical comparison between time- and frequency-domain techniques for phase-noise analysis

In the literature, different techniques have been presented for the phase-noise analysis of free-running oscillator circuits. In order to give some insight into the relationships existing between them, an analytical comparison is carried out in this paper between three different approaches. Two of them are time-domain approaches, based on Floquets theory and the impulse sensitivity function, respectively, and the third one is the carrier modulation approach, in frequency domain. The application of Floquets theory enables the calculation of periodic sensitivity functions to the noise perturbations. Here, the possibility to determine these functions through harmonic balance is demonstrated. This allows applying the whole stochastic characterization of phase noise, obtained from time-domain analysis, to circuits simulated through harmonic balance. For illustration, calculations in a cubic-nonlinearity oscillator are presented.

A 60-GHz HEMT-MMIC analog frequency divider by two

This paper describes an analog frequency divider by two working in the millimeter wave frequency range around 60 GHz. This circuit is analyzed with a new method that allows one to determine the steady-state regime of any synchronized circuits with standard CAD commercial software. The method proposed relies upon the concept of open loop systems and is applicable to any feedback transistor circuits. The designed circuit was processed using a standard 0.25-/spl mu/m HEMT technology. Four transistors were used for realizing the frequency division function as well as the input and output amplification. More than 10% frequency lock-in bandwidth was achieved, and conversion gain was obtained using input and output buffers. Measured results were found to be in good agreement with simulated ones. >

Analysis of noise effects on the nonlinear dynamics of synchronized oscillators

The higher sensitivity to noise of nonlinear systems near the onset of instability is analyzed here. The analysis is particularized to synchronized oscillators, studying the influence of the proximity to Hopf and Saddle-Node bifurcations. The calculations are compared with former scaling relationships and with results from time-domain integration. The average shift of bifurcation points due to noise perturbations is also analyzed. Two examples are shown: a cubic-nonlinearity oscillator and a 5 GHz hybrid oscillator, for experimental verifications.

Nonlinear Design Technique for High-Power Switching-Mode Oscillators

A simple nonlinear technique for the design of high-efficiency and high-power switching-mode oscillators is presented. It combines existing quasi-nonlinear methods and the use of an auxiliary generator (AG) in harmonic balance. The AG enables the oscillator optimization to achieve high output power and dc-to-RF conversion efficiency without affecting the oscillation frequency. It also imposes a sufficient drive on the transistor to enable the switching-mode operation with high efficiency. Using this AG, constant-power and constant-efficiency contour plots are traced in order to determine the optimum element values. The oscillation startup condition and the steady-state stability are analyzed with the pole-zero identification technique. The influence of the gate bias on the output power, efficiency, and stability is also investigated. A class-E oscillator is demonstrated using the proposed technique. The oscillator exhibits 75 W with 67% efficiency at 410 MHz

General envelope-transient formulation of phase-locked loops using three time scales

In this paper, a novel time-frequency method for the analysis of phase-locked loops (PLLs) has been developed. It enables an efficient and realistic simulation, taking into account the spurious frequency components generated at the phase detectors and intrinsic to their nonlinear performance. The variables of the loop equations are expanded in a Fourier series, at these spurious frequencies, with time-varying phasors. This generalizes the envelope-transient analysis to loops containing three different time scales, provided by the frequency of the voltage-controlled oscillator, external signals, and noise or modulations, respectively. The new technique substantially reduces the computational cost in the simulation of transients and acquisition times in the common case of a narrow-band loop. It also allows considering low-frequency noise perturbations, while taking into account the spurious frequency components. In the absence of modulation, the harmonic-balance (HB) formulation of the loop equations enables an efficient analysis of the variation of the PLL solution versus any parameter of interest. In addition, the conversion matrix approach, based on this HB formulation, can be used to determine the output noise spectrum from frequency-domain descriptions of the input noise sources, while taking the spurious content into account. To show the generality of application of the techniques, two different PLLs have been considered here: a frequency synthesizer with a tri-state comparator and charge pump, and an automatic frequency-control loop. In each case, the results have been successfully compared with time-domain simulations and measurements.

Phase and Amplitude Noise Analysis in Microwave Oscillators Using Nodal Harmonic Balance

In this paper, a nodal harmonic balance (HB) formulation is presented for the phase and amplitude noise analysis of free-running oscillators. The implications of using different constraints in the resolution of the perturbed-oscillator equations are studied. The obtained formulation allows the prediction of the possible spectrum resonances without ill conditioning at low frequency offset from the carrier. The noise spectrum is meaningfully expressed in terms of the eigenvalues of a newly defined matrix, obtained from the linearization of the nodal HB system about the steady-state solution. The cases of real or complex-conjugate dominant eigenvalues are distinguished. The developed phase-noise formulation is extended to a system of two coupled oscillators. The phase and amplitude noise analyses have been applied to a push-push oscillator at 18 GHz, a bipolar oscillator at 1 GHz, and a coupled system of two field-effect transistor oscillators at 6 GHz.