LFNM2005, 1547 September 2005, Yalta, Climea, Ukraine

PROBABILISTIC MODEL OF A CRYSTAL OSCILLATORAT LOW DRIVE LEVELS

(Invitedpaper)

Yuriy S. Shmaliyl, Rdmi Brendel2

'Guanajuato University,FIMEE, 36730, Salamanca, Gto, Mexico,

Phone: +52-464-6470195, Fax: +52-464-6472400, e-mail. shnndaliyCsya1anncya.u,gio.ttin2Franche-Comt( Electronique, Mecanique, Thermique et Optique - Sciences et Technologies(FEMTO-ST) LPMO,

32 avenue de l'Observatoire, F-25044 Besancon CedexPhone: +33 (0)3 81 85 39 54, Fax: +33 (0)3 81 85 39 98, e-mail . remi.lbrendel(cvlpm?o.cedul

Abstract - The probability density of the envelope of crystal oscillator oscillations is studied at low drives whenself-excitation becomes not available due to the large and nonlinear friction of a piezoelectric resonator that occursafter a long storage. We show that the effect know as "sleeping sickness" is accompanied with the noise-inducedoscillations at a new steady state far from the equilibrium. The stochastic differential equation is given for the enve-lope of the oscillations induced by the interplay of the nonlinear resonator friction and multiplicative noise. TheFokker-Plank equation is discussed in a sense of Stratonovich. The stationary probability density of the envelope isderived and investigated.Keywords: crystal oscillator, low drive levels, envelope, probability density

INTRODUCTION

Crystal oscillators are now used widely in different applications whenever a low cost pre-cision and accurate frequency source is needed. When stored for a few days-decades withoutconnection to a power supply, however, the crystals often develop a condition known as "sleep-ing thickness" [1]-[7]. Their facility for self-oscillation then becomes strongly attenuated andmay even disappear. The origin of this phenomenon is not well understood, although it is knownto be associated with increased resonator losses caused, in particular, by a surface contamination.An overview of possible courses and observable effects associated with this phenomenon wasrecently given in [8]. To circumvent, provision often has to be made for an increased drivingamplitude at switch-on. In this paper we present a probabilistic model of a crystal oscillator atlow drive levels. Based upon, we give a stochastic explanation of sleeping sickness, viewing it asa nonlinear phenomenon self-induced by the Johnson noise in the circuit with an insufficientfeedback. As we show, the picture we propose seems to fit principle features of sleeping sicknessreported in the literature. We consider, in particular, what happens at very low drives where thenatural frequency of the crystal will be least "pulled" by the external circuit, focusing our atten-tion on physics close to the noise floor where the amplitude of the oscillatory drive is compara-ble with the internal noise intensity.

PROBABILISTIC MODEL

In [6], it was experimentally shown that the following approximation fits reasonably wellthe toward drive level dependence (DLD) of the crystal resonator losses R, ifwe start measuringfrom the noise floor and then gradually increase the piezoelectric current up to the normal drives,with which the losses are RIO. In the fractional term R(Ir)=RI(r) R10, where AR, is theDLD increment of the losses, such a "toward" DLD is

0-7803-91 47-O/051$20.00 © 2005 IEEE

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=R= YR (1)

where yR is the DLD coefficient, Ir is the pick amplitude of the piezoelectric current, and adigital n ranges from I to 4, depending on the crystal resonator quality factor Q. It is importantto notice that when coming backward, from the normal drives to the noise floor, the DLD coeffi-cient measures almost zero and only after a long time the effect recurs. Such a hysteresis phe-nomenon, however, is not studied yet.

An electric equivalent of the piezo-electro-mechanical system of a resonator with a feed-back and low drives is drawn in Fig. 1. The motional inductance (mass) L,, capacity (spring) Cl,and losses (dashpot) R, are responsible for the resonator series resonance natural frequencyW,a (Z =1/ /LTC and bandwidth 27 = RI L,. The static capacity C0 induces a parallel reso-

nance of the natural frequency op I/ L1C1C0/(C1 +C0), which tends toward infinity if CO be-comes zero. The amplitude equilibrium in the oscillator (Fig. 1) is obtained by the nonlinearnegative losses -,6(i4) of the feedback. Finally, the phase equilibrium is provided with anequivalent negative inductance - La to make possible for the oscillator to operate at the desiredfrequency co within an inter resonance gap that corresponds to some positive phase angle q' be-tween the feedback voltage v(t) and piezoelectric current i(t).

R,K(i,) -El(t,ir 'eg(t) l

ci ~~~~~~Ra

Piezoelectric resonator Feedback

Fig. 1 - Electric equivalent of a crystal oscillator with a nonlinear resonator at low drives

It may be shown that the motion equation of an oscillator (Fig. 1) with the nonlinear lossesRI(Ir) = R1j[1 + JR(Ir)], can be written as follows

V+.i 27{ (f R)(I+ ) - 27o (2)

where v = i/A, v)= dvldt, v(i) = v(i)/Rl0A, p = C,/Co <<1, 77 is a damping coefficient, andA is a normalizing constant. The DLD noise s(t,v) is combined by the additive and multiplica-tive terms, which are of Johnson's thermal origin (additive and colored). The colored noise, weassume, does not contribute very much to excitation dynamics, so that the function may be per-formed by the narrowband random model q(t, v) = (eas +e,)cos oot + (eac +eTC)sin cot, where

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eTs (3R)a eTc (5R) and eas a eac are the quadrature components of the normalized multiplicativeand additive noises of a resonator and feedback, respectively.

In our studies near the noise floor, it is convenient to substitute the normalizing constantA by the noise intensity induced by the crystal resonator with normal drives. within its band-width Af =qliz, namely by A=l2kT1/,R0 . Thereafter, the Markov approach, applied byStratonovich to the oscillator fluctuation differential equation, transforms (2) to the stochasticdifferential equations for the oscillations envelope and phase. The phase process is not associatedwith sleeping sickness and, therefore, we study only the random envelope, which SDE is derivedfor the linear feedback with account ofthe phase noise to be

a(-fIx)x+ ( n ) +n

g S, (3)dx f(x)dt + q(x)dw, (4)

where x = Ir / A > 0 is the normalized envelope of the piezoelectric current; x= dx dr; -r =r;a=GL -1 is the linear friction coefficient; a, = (1+ g2) /2; r= rR(&/ 2kTi7)n12 r, y2a= ;

g = //1?R; ; and f(x) and q(x) are readily defined by comparing (3) and (4). The noise ~(r)in (4) now is mean-zero, (S(T)) = 0, white Gaussian, having a uniform power spectral density

N0 /2 =1. Finally, w(t) is the mean-zero, (w(t)) = 0, Wiener process with a variance

w2 (t)) = t . It may also be shown that the linear gain of a closed system (Fig. 1) is calculated byGL = g2(cosp-sin.o), (5)

where K= a RCO is the resonator K -factor. An ordinarily limiting case of (4) follows straight-forward: having a linear resonator, y =0, degenerates (4) to the Langevin equation describingthe Rayleigh process:

ox+ L+ +2Sa, x > 0. (6)x

The Fokker-Plank equation corresponding to (4) is written below in a sense of Stratono-vich, which calculus is appropriate for the delta-correlated processes, namely in a form of

a a IF( ___ n(yI+r y 11a r P(X) ax LK ccn )x xn) 2( x) x"' p(X)J

2 {[2 +4xn +g]p(x)}. (7)

A solution of (7) entails difficulties for arbitrary n. On the other hand, various values of n do notcontribute with essential peculiarities to the dynamics picture. We, therefore, investigate belowthe case of n = 2 which yields the following stationary solution of (7), satisfying the conditionfor the probability flux to be equal zero,

a2 2I(=x x) X2 +barctan+gx(8

ps,()=Cx (a1X+ 2~X2 + r2)2 e2a x (8)

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where the constant C is determined to tend (8) to unit area by the integral relation

Jp21(x)dx = 1 (9)0

and the coefficients are

a=i1 + ,(10)4 Y2a:)(3

b = ra(1 -a1) + a, 112ga,2

It now may easily be observed that the linear case, y =0, degenerates (8) to the Rayleighprocess (6) associated with the probability density

a x2p,(x)

a xel , x>O a<O, (12)

namely: by changing a variable y = x- a/a, , we go to PS,L(Y) = yeY /2. The nth order initialmoment, mean value, and variance corresponding to (12) are, respectively,

(xn= -2aLJr n21 (13)a)" 2r(

a,(X)L -2Rs (14)2 a

CX9 =2~1.) (15)a ( 4)

An important point is that the linear friction coefficient a may be either negative or posi-tive in (8) depending on the excitation conditions. Therewith, the probability of the noise-induced oscillations is not fully absorbed at infinity meaning that even a negligible chance mayexists for the oscillations to be self-excited without bounds for the oscillator with a linear feed-back. Certainly, it could be supposed if to recall that the excitation value xo is random and, thus,it may causally overcome the threshold produced by the resonator nonlinear friction. To demon-strate noise-induced excitations in a crystal oscillator, we show in Fig. 2 two typical surfaces ofthe stationary probability density in a wide range of the DLD coefficient y. Even a quick look atFig. 2 leads to the conclusion that the Rayleigh distribution, corresponding to zero oscillationamplitude with y = 0 (pure noise) evolves, by y > 0, to the near Rice distribution with a nonzerooscillation amplitude and does not seem to be Gaussian even by large values of y>> I. Let usnow examine Fig. 2 in more detail. Assuming zero linear gain (5), this is a = -1 (Fig. 2a), wevirtually have an open system. Here, with y =0, the oscillation amplitude inherently becomeszero. Contrary, the case of y > 0 makes it to be nonzero at a new state that is stable. It is impor-tant to note that this new state does not exceed the noise intensity bound xo calculated by the fol-lowing equation, if to suppose that the initial excitation value is equal to the noise intensity,

X0= 1I4 +9. (16)

We depict this bound in Fig. 2 by circles.

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a) as= -I 5) a= o

6 P",00 60 ~~~~~~~~~~~~~~~~~~~~~~~~~~~p,,(x)

0.2 2~~~~~~~~~~~~~~~~.08

15~~~~~~~~~010 x 0 1

at) b)

Fig. 2 - Evolution of the stationary probability density of the amplitude by increasing y for 0, IC= 10-3,and g calculated by (5): (a) a =-1 and (b) a = 0 .

The situation is changed with -1 < a by involving the feedback energy owing to whichthe noise-induced oscillations possess larger arnplitude as y growths and exceed the noise inten-sity bound (see Fig. 2b). The critical value of the linear friction coefficient a = 0 demonstratesthat having y = 0 results in self-excitation owing to the nonlinear friction. Herewith, increasingy gradually turns the probability density to the aforementioned new steady state of the sleepingsickness. A generalization of an analysis is illustrated in Fig. 3, where the effect of noise-inducedoscillations is shown for the particular case of a = -0.03.

110 I10 30

60 t-<50 b6)a) 15 70

0.2 4 0.2-- Psi2~~~~~~.20 10

0 1 15 0 10

a.) b)

Fig. 3 - Probability density of the envelopc of the noise-induced oscillations in a crystal oscillator for a =-0.03,0=O, K = 103, and g calculated by (5). A linear case of =O is subject to the Rayleigh distribution: (a)

small values of 7 and (b) large values of 7.

It is neatly seen that low values of y (Fig. 3a) squeeze the probability density functionand turn it toward zero until some critical point, whereas high values of y produce an inverseeffect: the probability density shifts toward greater amplitudes (Fig. 3b). In either of the above-mentioned cases the envelope distribution concentrates around a new state that is stable. It mayalso be observed that approaching zero the linear friction coefficient a gradually elevates thisnew state point up to higher amplitudes owing to the feedback energy. This virtually means thatthe nonlinear friction works as a filter for the noise, making the oscillation envelope to be notfully random as in a case of y =0 and a < 0. In other words, a crystal oscillator with a nonzerocoefficient y generates oscillations induced by noise. The noise in our numerical studies was

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simulated as multiplicative and additive (by setting y = 0 in the noisy term). Based upon theachieved results, as an addendum, we confirm a known conclusion regarding the role of the addi-tive noise in phase transitions, namnely that the interplay of the nonlinearity in question and addi-tive noise produces the transitions.

CONCLUSIONS

In this paper we show that noise-induced oscillations in crystal oscillators with largenonlinear resonator friction and low drives accompany the effect known as "sleeping sickness".The latter seems now not a total "sleep" but rather a new steady state induced by noise with thenoise-level amplitude.

ACKNOWLEDGEMENT

The first author thanks Prof. Peter V. E. McClintock of the School of Physics and Chem-istry of Lancaster University, UK for valuable comments and assistance in reading this paper.

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