Nicolas Ratier

Modeling of quartz crystal oscillators by using nonlinear dipolar method

A quartz crystal oscillator can be thought of as a resonator connected across an amplifier considered as a nonlinear dipole the impedance of which depends on the amplitude of the current that flows through it. The nonlinear amplifier resistance and reactance are obtained by using a time domain electrical simulator like SPICE (Simulation Program with Integrated Circuit Emphasis): the resonator is replaced with a sinusoidal current source of the same frequency and a set of transient analyses is performed by giving the current source a larger amplitude. A Fourier analysis of the steady-state voltage across the dipolar amplifier is performed to calculate both real and imaginary parts of the dipolar impedance as a function of the current amplitude. From these curves, it is then possible to accurately calculate the oscillation amplitude and frequency without having to perform unacceptably long transient analyses needed by a direct oscillator closed loop simulation. This method implemented in the Analyse Dipolaire des Oscillateurs a Quartz or Quartz Crystal Oscillators Dipolar Analysis (ADOQ) program calculates the oscillation start-up condition, the oscillation steady-state features (oscillation amplitude and frequency), and the oscillator sensitivity to various parameters. The oscillation nonlinear differential equation is solved by using the slowly varying function method so that the program quickly and accurately calculates the current amplitude and frequency transients. Measurements performed on an actual amplifier show a very good agreement with the results obtained by the simulation program.

Slowly varying function method applied to quartz crystal oscillator transient calculation

By using an approach based on the full nonlinear Barkhausen criterion, it is possible to describe oscillator behavior under the form of a nonlinear characteristic polynomial whose coefficients are functions of the circuit components and of the oscillation amplitude. Solving the polynomial in the frequency domain leads to the steady state oscillation amplitude and frequency. In the time domain, the characteristic polynomial represents a nonlinear differential equation whose solution gives the oscillator signal transient. It is shown how symbolic manipulation capabilities of commercially available softwares can be used to automatically generate the coding of the oscillator characteristic polynomial from the SPICE description netlist. The numerical processing of such an equation in the time domain leads to unacceptable computer time because of the high quality factor of the oscillator circuits involved. Nevertheless, by using the slowly varying amplitude and phase method, it is possible to transform the initial nonlinear differential equation into a nonlinear first order differential equation system in the amplitude and phase variables. The solution of this system directly gives the designer the most relevant features of the oscillation; that is, the amplitude, phase, or frequency transients which can be accurately obtained within a short computer time by using classical numerical algorithms.

ADOQ: a quartz crystal oscillator simulation software

This paper presents the actual state of a computer program especially designed to simulate the behavior of quartz crystal oscillators. The program is based on the fact that the current through the quartz crystal is almost perfectly sinusoidal. Consequently the oscillator can be modeled by a resonator across a nonlinear impedance that depends only on the current magnitude through it. The resonator being replaced by a current source, the nonlinear impedance of the amplifier is computed from a series of transient analyses performed at the resonator frequency. When the steady state is reached, the resonator impedance is exactly equal and of opposite sign to the amplifier impedance. This identity allows one to compute the oscillation amplitude and the frequency shift with respect to the resonator frequency. This computation does not require to perform unacceptable long transient analyses in case of high-Q oscillator. Our program is intended to help the designer in checking or improving oscillator circuit design. From the Spice netlist, it enables the user to compute the steady state features of the oscillator, namely frequency and amplitude. Then, the user can study the effect of temperature change on any components or the influence of quartz characteristic. It is also possible to perform accurate oscillator sensitivity calculation to various parameters (component value, supply voltage, ...) as well as worst case analysis.

Automatic formal derivation of the oscillation condition

The behavior of a quartz crystal oscillator can be described by a nonlinear characteristic polynomial whose coefficients are function of the circuit parameters. Solving the polynomial in the frequency domain leads to the steady state oscillation amplitude and frequency. In the time domain, it gives the oscillator signal transient. Deriving the characteristic polynomial from the circuit description involves lengthy and tedious algebraic calculations if they are performed by hand. They may be now performed by using the symbolic manipulation capabilities of commercially available softwares. However, symbolic analysis using brute force method inevitably leads to an explosion of terms in equations. The paper will present a fully automatic method for generating the coding of an oscillator characteristic polynomial directly from the SPICE description netlist. The code thus generated is eventually compiled and takes place in an oscillator library. Then it is linked with the numerical main program that solves the polynomials. Solutions to overcome problems related to automatic symbolic calculations are presented and discussed. It is shown that the method used leads to concise and efficient code.

Analysis of noise in quartz crystal oscillators by using slowly varying functions method

By using formal manipulation capability of commercially available symbolic calculation code, it is possible to automatically derive the characteristic polynomial describing the conditions for oscillation of a circuit. The analytical expression of the characteristic polynomial is obtained through an encapsulation process starting from the SPICE netlist description of the circuit: by using a limited number of simple transformations, the initial circuit is progressively transformed in a simplified standard form. In this method, the nonlinear component is described by its large signal admittance parameters obtained from a set of SPICE transient simulations of larger and larger amplitude. The encapsulation process involving linear and nonlinear components as well as noise sources leads to a perturbed characteristic polynomial. In the time domain, the perturbed characteristic polynomial becomes a nonlinear nonautonomous differential equation. By using an extension of the slowly varying functions method, this differential equation is transformed into a nonlinear differential system with perturbation terms as the right-hand side. Eventually, solving this system with classical algorithms allows one to obtain both amplitude and phase noise spectra of the oscillator.

Analysis of parametric noise in quartz crystal oscillators

From the designers point of view, noise sources may act on oscillator frequency through two main mechanisms: additive noise and parametric noise. The first part of this paper describes how to obtain the analytical expression of the characteristic polynomial through a successive encapsulation process starting from the SPICE netlist description of the circuit. In the time domain, the perturbed characteristic polynomial becomes a non-autonomous differential equation. In contrast to additive noise, in which the excitation appears in the differential equation as inhomogeneity, parametric noise leads to a differential equation with time-varying coefficients. By using an extension of the slowly varying functions method, this differential equation is transformed into a nonlinear differential system in amplitude and phase variables with perturbation terms at the right hand side. Solving this system allows to obtain both amplitude and phase noise spectra of the oscillator.

AM and PM noise analysis in quartz crystal oscillators: symbolic calculus approach

Increasing performance of quartz crystal oscillators as well as predictability requirements when developing the devices need accurate analysis of noise sources. Our work is devoted to understand how an oscillator reacts to additive noise of an element in the electronic circuit. Up to now, oscillator designers often refer to the well-known Leesons model to explain the shape of phase noise spectral density. This physical model only allows one to obtain the global phase noise spectrum. By considering each noise source individually, we can obtain the comparative contribution of the sources. Then AM and PM noise source spectra can be related to the circuit architecture. The influence of an individual noise source can be obtained from the differential equation describing the oscillator behaviour. Nevertheless, setup of the differential equation from the inspection of the circuit involves lengthy and tedious algebraic calculations almost impossible to achieve by hand. By using symbolic calculation capability of formal calculus programs, it is possible to automatically derive the differential equation of the oscillator including noise sources from a SPICE netlist description of the circuit. The resulting expressions can be edited under the form of high level language code (Fortran, C, ...) which is eventually compiled and linked with the numerical programs calculating the noise spectra. This paper presents the method to construct the differential equations in a fully automatic way regardless of the studied oscillator circuit.

Quartz crystal oscillator classification by dipolar analysis

The dipolar method associated with a nonlinear time domain simulation program make up a powerful tool to analyze high Q-factor circuits like quartz crystal oscillators. After a brief remainder of the dipolar method, the paper will attempt to identify the main amplifier characteristics such as limitation mechanism, input and output impedances, etc. and to point out their influence on the amplifier dipolar impedance. The effect of the amplifier nonlinearities on the oscillator characteristics, as well as the particular role of the crystal parallel capacitance is particularly emphasized

Synthetic modeling of quartz crystal oscillator

This paper presents a new modeling technique to describe the nonlinear behavior of complicated oscillator circuits. The simulation program being developed first removes the resonator from the oscillation loop and call on SPICE to calculate the large signal /spl gamma/-parameters of the amplifier circuit considered as a nonlinear two-port circuits. The oscillation condition, obtained by reinserting the resonator across the two-port circuit, is expressed under the form of a complex polynomial in the harmonic variable j/spl omega/, the coefficients of this polynomial being nonlinear functions of the signal amplitude. Solving the real and imaginary parts of this characteristic polynomial by using nonlinear analysis algorithm, it is possible to accurately calculate both amplitude and frequency of the oscillation.

Symbolic harmonic analysis of quartz crystal oscillators

The nonlinear dipolar method is dedicated to the simulation of quartz crystal oscillators with high quality factor. In this method, the oscillator is considered as a resonator connected across an amplifier that behaves like a nonlinear dipole whose impedance evaluated at the resonators frequency depends on the current amplitude. This dipole allows us to compute very quickly the behavior of the oscillator. The computation time of the dipolar impedance by SPICE is of the order of seconds. To gain one order of magnitude in the simulation time of the oscillator, this paper propose a modification of the nonlinear dipolar method by changing the dipolar impedance SPICE calculation. that is the most time consuming part of the program, by a system of equations obtained through a symbolic manipulation of the circuit equations.