Antonio Buonomo

Modelling and analysis of differential VCOs

We present a systematic non-linear analysis of differential voltage controlled oscillators (VCOs), both bipolar and MOS. Using the standard device models, we derive the second-order non-linear equation describing the behaviour of these oscillators, which is formulated in a perturbation form. The solution of this equation is obtained as a particular case of the solution of the most general equation of second-order oscillators, which is solved through a suitable perturbation method. Unlike a pure numerical analysis, simple analytical relationships are derived for predicting the steady-state oscillation, its transient behaviour and for ascertaining the existence of a stable oscillation in differential VCOs. These relationships, leading to results which well agree with the SPICE simulations, are useful in both analysis and design. Copyright

Perturbation analysis of nonlinear distortion in analog integrated circuits

A method for predicting the distortion in weakly nonlinear analog circuits is presented, which relies on the classical theory of regular perturbation. Accordingly, a nonlinear circuit is described and analyzed as a perturbation of its linearized model, and the response to a periodic signal is analytically calculated through frequency-domain recurrent formulas. The method is simple and quite straightforward to apply, as it involves the calculation of frequency-domain transfer functions and of Fourier coefficients only, making it easily adaptable to any circuit topology. The method can be a valid alternative to the Volterra series method. A relationship between the proposed method and the Volterra series method is established, showing that they lead to very similar approximants to the solution. The method has been numerically tested in practical circuits wherein the devices are modeled by polynomial and exponential nonlinearities.

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.

Finding the Tuning Curve of a CMOS— LC VCO

A nonlinear perturbation model of a complementary LC-tuned voltage-controlled oscillator is derived, which consists of two mutually-coupled second-order differential equations. The first-order approximate periodic solution of the describing equations is found, obtaining closed-form expressions for both the amplitude and the harmonics of oscillation, as well as for the correction of the oscillation frequency due to the nonlinear effect of varactors. This allows us to find the tuning curve in explicit form. The accuracy of presented formulas was validated by circuit simulations.

Analyzing the dynamic behavior of RF oscillators

Using an asymptotic perturbation method, we perform a full nonlinear analysis of second- and third-order oscillators, widely used in radio frequency (RF) applications. The steady-state oscillation and its orbital stability, as well as the initial transient response are determined through relatively simple and expressive relationships making it possible to establish design criteria. Some practical circuits of RF voltage-controlled oscillators (VCOs), which are representative of wide classes of oscillators, are analyzed. Moreover, a nonlinear perturbation model of differential VCOs is developed to make their nonlinear analysis possible.

Nonlinear Analysis of Voltage-Controlled Oscillators: A Systematic Approach

A method for the nonlinear analysis of voltage-controlled oscillators (VCOs) is presented, accounting for the nonlinear effect of both the varactors and the active devices. A nonlinear perturbation model of VCOs is presented, as well as a method for finding their steady-state oscillation, overcoming the limitation of previous analyses which rely on the solution of the tank balance equation only. Simple formulas for predicting the oscillation and its frequency correction due to the effect of nonlinearities are presented, allowing us to accurately predict the tuning curve of VCOs as well as the amplitude to phase-noise conversion due to the varactors. Their accuracy was validated by circuit simulations.

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.