Automatic formal derivation of the oscillation condition

Proceedings of the 1998 IEEE International Frequency Control Symposium (Cat. No.98CH36165)

et al.

Self Cite

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 Leeson's 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 behavior. 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.

Section: Second Step: Oscillation Conditionmentioning

confidence: 92%

“…In practice, the modulus of each term in the sums (21, 22) is computed in the steady state of the noiseless oscillator [2][7] [8]. Then, the equation 20takes the form (23):…”

Section: Cqq Lqq Rqqmentioning

confidence: 99%

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

Ratier

Couteleau

Proceedings of the 1998 IEEE International Frequency Control Symposium (Cat. No.98CH36165)

et al.

Self Cite

Slowly varying function method applied to quartz crystal oscillator transient calculation

IEEE Trans. Ultrason., Ferroelect., Freq. Contr.

Ratier

Couteleau

et al. 1998

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.

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

IEEE Trans. Ultrason., Ferroelect., Freq. Contr.

Ratier²,

Couteleau³

et al. 1999

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.