Alessandro Lo Schiavo

Analytical Approach to the Study of Injection-Locked Frequency Dividers

An analytical method for the nonlinear analysis of injection-locked frequency dividers (ILFDs) is presented based on the method of the slowly-varying amplitude and phase. We introduce a general model of ILFDs and derive the associated averaging equations describing their first-order dynamical behavior. These equations are solved in closed form for two typical injection techniques of ILFDs, namely the injection via tail device and the direct injection. The oscillation amplitude and phase in the locked states and during the transient, as well as the locking range are obtained in explicit form. Numerical simulations validate the presented formulas, which provide useful design insights.

Nonlinear dynamics of divide-by-two injection-locked frequency dividers in locked operation mode

A method for analyzing the nonlinear dynamics of the injection-locked frequency dividers in synchronized operation mode is presented, including the stability analysis of locked states. We use a specific divide-by-two circuit, namely a differential LC CMOS divider with a complementary topology, as a guideline for presentation, showing that the sizing of the devices significantly affects the synchronization mechanism of the divider, which exhibits a very rich dynamical behavior. We provide closed-form expressions to determine the amplitude and the phase in the locked state, as well as the locking range, leading to accurate results, which are validated by numerical simulations. The presented analysis of the frequency divider dynamics enables us to establish that stable locked oscillations occur on the whole locking range predicted by the well-known Adlers equation and that these are possible also beyond that range. Copyright

A CMOS Injection-Locked Frequency Divider Optimized for Divide-by-Two and Divide-by-Three Operation

This paper proposes a simple and effective modification of the conventional divide-by-two injection locked frequency divider (ILFD) with direct-injection aimed at allowing both the divide-by-two and the divide-by-three modes of operation. The proposed circuit does not employ additional inductors as usual in divide-by-three ILFDs, but exploits the combined effect of two independent injection techniques. The resulting locking range for the divide-by-three mode is comparable in size to that for the divide-by-two. Thus, the proposed circuit can be an optimum alternative to existing dividers, due to the flexibility of two division ratios and due to the absence of additional inductors. An intuitive explanation of the locking mechanism underlying this ILFD and a quantitative analysis are provided, allowing one to predict the amplitude and phase of oscillation in the locked mode, as well as the locking range, with approximate closed-form expressions. Measurements on a circuit prototype and results from SPICE simulations demonstrate the effectiveness of the circuit and validate the theoretical model and the resulting formulas.

On the Synchronization Condition for Superharmonic Coupled QVCOs

A steady-state nonlinear analysis of quadrature voltage-controlled oscillators (QVCOs) comprising two VCOs mutually coupled at their second-harmonic frequency through a direct coupling circuit is presented. The analysis is based on an accurate prediction of the behavior of each VCO, which is analyzed separately as an injection locked oscillator taking into account both higher order harmonics of the differential tank voltage and the effect of the common-mode voltage at the drain terminals. We show that synchronization of the two VCOs is possible only at a frequency, derived in closed-form, which differs appreciably from the tanks resonant frequency.

A Deep Investigation of the Synchronization Mechanisms in LC-CMOS Frequency Dividers

The behavior of LC-CMOS frequency dividers is investigated in depth obtaining a new and clearer insight into the synchronization phenomenon. We show that it occurs according to two mechanisms, which are referred to as “synchronization by mixing” and “synchronization by harmonics.” This latter, still not known, may be the dominant mechanism in some practical circuits, and justifies the presence of locked states when the well known mechanism by mixing fails to predict them. A new model of CMOS frequency dividers that includes both mechanisms is presented, and its utility is demonstrated by analyzing the locked states of a divider by 3 and of a divider by 4, which are predicted in closed form. The calculation results show a close correspondence between the theory and the circuit simulations and measurements.

On the Theory of Quadrature Oscillations Obtained Through Parallel

This paper presents a theory of parallel quadrature voltage-controlled oscillators, which are widely used to generate in-phase and quadrature signals in communication systems. The theory is developed without making any hypothesis on the circuit topology and allows us to thoroughly analyze the synchronized oscillations, as well as the effect of unavoidable parameter mismatches, by means of closed-form expressions for the amplitude and phase of oscillations and their orbital stability. The stability of oscillations is assessed analyzing the relevant eigenvalues, which are obtained in a closed form. The validity of the presented results has been verified through circuit simulations and experiments.

LC

We present a nonlinear analysis of the basic circuit of a dual-band voltage controlled oscillator (VCO), composed by an active one-port and by a double-tuned LC circuit. Both the frequencies and the amplitudes of the two modes of sinusoidal oscillation are calculated by closed-form expressions by which a detailed analysis of the steady-state and dynamical behavior of circuit is performed, including the stability analysis of the modes. We show the appearance of the phenomenon of the frequency hysteresis allowing us to explain the mechanism of the switching between modes and predict the VCOs performance over the whole operation range. The theoretical results, validated by Spice simulations of a VCO realized in 0.13 μm MOS technology and by measurements on a circuit prototype, provide useful design insights.

VCOs

An analytical model and a methodology are presented for the analysis of CMOS injection-locked frequency dividers with direct injection, which are used in modern wireless communication systems. The amplitude and phase of the oscillation in the synchronous operation mode, as well as the locking range, are found in explicit form. The width of the locking range is determined by the condition of the existence and stability of synchronous oscillations. The accuracy of the model and of the presented formulas is validated through a comparison with experimental results, which are in good agreement with the analytical ones.

Analysis and Design of Dual-Mode CMOS LC-VCOs

SUMMARY 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 Adlers 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 systems 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

Modeling, analysis, and experimental validation of frequency dividers with direct injection

We present a study of dual-band injection locking frequency dividers (ILFDs), based on a nonlinear analysis. We develop a quasi-normal model of these dividers suitable for applying the method of averaging, which allowed us to derive in a simple and expressive manner the first-approximation equations for the amplitudes and phases of the locked modes, both in transient and in steady state. The phase equations have the same form of the Adler’s equation, and represent the generalization of that well-known equation to higher-order frequency dividers. These equations allowed us to derive the locking ranges in a simple explicit form, useful for design purposes. The theoretical results are validated by Spice simulations, and by measurements on a circuit prototype.