ANALYSIS OF 1/f2 AND 1/f FREQUENCY NOISES IN QUARTZ RESONATORS

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ANALYSIS OF 1/f2 AND 1/f FREQUENCY NOISES IN QUARTZ RESONATORS

J. Gagnepain, G. Théobald, J. Uebersfeld

To cite this version:

J. Gagnepain, G. Théobald, J. Uebersfeld. ANALYSIS OF 1/f2 AND 1/f FREQUENCY NOISES IN QUARTZ RESONATORS. Journal de Physique Colloques, 1981, 42 (C8), pp.C8-201-C8-209.

�10.1051/jphyscol:1981824�. �jpa-00221719�

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JOURNAL DE PHYSIQUE

Colloque C8, suppldment au nO1 2, Tome 42, dgcernbre 1981 page C8-201

A N A L Y S I S OF l/f AND i/f FREQUENCY NO1 SES I N QUARTZ RESONATORS J.J. Gagnepain, G. ThBobald and J. Uebersfeld

Laboratoire de Physique e t MgtroZogie des Oscillateurs du C. N. R. S.

*,

32, avenue de 2 'Observatoire, 25000 Besangon, France.

Abstract.- The two main noise contributions to quartz frequency standard instabilities are l/fz and l/f frequency noises. l/f2 frequency random walk is correlated with temperature fluctuations through temperature gradients. l/f noise level depends on the resonator Q-factor as it follows from experimental data at room and low temperatures. A first theoretical interpretation of this dependance is proposed.

Introduction.- In quartz frequency standards three fundamental noise mechanisms are responsible of frequency and phase fluctuations. They correspond to white noise, l/f2 noise and l/f noise. White noise sources are found only in the electronic components ; In quartz crystal their levels are too small and hided by the other noises. Therefore they are not attainable by measurements. l/f2 noise is essentially found in quartz crystals as frequency random walk fluctuations correlated with temperature fluctuations. l/f noise, also called flicker noise, appears in both electronic components and quartz resonator. Indeed l/f noise is an ubiquitous phenomenon, which as a rule appears in almost all physical sys- tems, as well in the natural ones as in electronic devices. It is generally agreed upon, that its origin and its mechanisms still remain not well understood.

Frequency noise measurements.- With an oscillating loop it is difficult to dis- tinguish in the output signal the respective contributions of the electronics

(amplifiers, buffers,

...

⁾ and of the quartz crystai resonator to the phase fluctuations. A new technique was presented by F. Walls

,

whlch allowed to measure the inherent frequency fluctuations of a quartz crystal in a passive n transmission network with a large rejection of the noise usually associated with electronics. This measurement system is shown on fig. 1.

A frequency source with a good spectral purity drives in transmission two crystals, as identical as possible. A phase bridge composed of a double balanced mixer and a 90° phase shifter delivers a voltage proportional to the relative phase fluctuations of the two transmitted signals. By careful adjustments of the resonator frequencies and Q-factors the phase bridge is balanced in order to reject the phase fluctuations of the source by 50 dB to 60 dB.

The typical frequency noise spectrum of a 10 MHz quartz resonator is presented on fig. 2 . On this figure the filtering effect of the resonator was cor- rected out by taking into account the resonator phase-frequency response. For lower Fourier frequencies (below 1 Hz) the l/f2 frequency random walk noise is predominant and for intermediate Fourier frequencies l/f noise appears. The white noise which is observed at higher frequencies is entirely due to the mea-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981824

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C8-202 JOURNAL DE PHYSIQUE

surement system itself. Both l/f2 and l/f noises are characteristic of the resonator and define its frequency stability.

l/fz frequency noise.- There is a direct correlation between l/f2 frequency nois- se and temperature fluctuations. Simultaneous measurements of both frequency and temperature fluctuations were performed on a 10 MHz resonator and a high resolu- tion quartz temperature sensor. The 10 MHz resonator and the temperature probe were thermally coupled and placed in the same oven. Autocorrelation and crosscor- relation functions of the two random signals proportional to their respective frequency fluctuations were calculated. The coherence function was evaluated for three time intervalls corresponding to the Fourier frequencies of 0.01, 0.1 and 1 Hz. It is given in Table I.

Fourier Frequency

1

0.01 Hz

I

0.1 Hz

I

1 Hz

I I I

cohgrence

function 30 X

I

6 %

I

Table I

Coherence function between the frequency fluctuations of a 10 MHz resonator and the temperature fluctuations.

This shows that the resonator frequency noise is correlated to the temperature fluctuations for the lower Fourier frequencies, which indeed correspond in the spectrum of fig. 1 to the frequency random walk noise. This correlation depends on the sensor position with respect to the resonator, i.e. on their thermal coupling, and the best way would be to combine resonator and sensor in the same enclosure. This can be done by using at the same time the C and B modes of an SC-cut quartz resonator.

It is important to notice that the l/f2 frequency noise does not depend on the mean temperature of the crystal and for instance is not minimized by ope- rating at the turn-over temperature point of the static frequency-temperature characteristic.

In fact any external temperature variation is followed by heat diffusion, between the surrounding medium and the crystal itself, along the connecting wi- res, the supports and the electrodes. Consequently temperature within the crystal has a spatial distribution and the thermal behavior includes at the same time spatial and temporel gradients. It follows that thermal stresses and strains take place and induce frequency shifts by non linear coupling with the high frequency wave.

This dynamic thermal behavior leasto frequency fluctuations much la ger than would be expected from the static one as it has been shown by Ballato' who proposed a phenomenological model by introducing an additional term, time dependant, in the first order temperature coefficient of the static f-T effect.

A theoretical model was built up by G. ~h6obald~. It consists in a crystal plate of thickness 2h, of infinite lateral sizes along xl and xg and uniformly plated on its two faces. Let T(t) be the external temperature and O(x2,t) the temperature distribution within the crystal as shown on fig. 3. The frequency variation Aw/w is evaluated as a function of the mean tempera'ture of the crystal Q ( O = (1/2h)

7

h ^Q(x2,t)dx2) and of the temperature gradient f (x2,t).

-

-h ^+h ^2wx

Aw/w = a O

+

k66

1

f(x2,t) cos

-

^dx₂ (1)

-h "0

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a is the static temperature coefficients, i.e. when the crystal is in thermal e6uilibriurn and the temperature uniformly distributed. k is a coefficient characteristic of the temperature gradient sensititivy. 1t68epends on the crystal orientation through the second and third order elastic constants, but it is in- dependant of the shape of the gradient. This is an interesting point, because thus it is possible to minimize the effect by choosing appropriate crystal orien- tations, whatever the temperature distribution is. This is the case of the SC- cut.

After complete calculation the frequency versus temperature relation takes the simple form

Aw/w =

2

+ a [ ~ ( t ) - ~ d

+

higher order terms dt

where

2

is called the "dynamic temperature cbefficient", a being the static one.

-5 -7

For AT cut resonators =

-

¹⁰ s/OC and for SCcut resonators = s/OC. This means that to achieve frequency stability of the order of 1 x 10 with AT-cut, it is necessary to maintain the temperature fluc uations of the

5

crystal at a level which does not exceed OC/s and/or 10- OC/day. With SC-cut the fluctuations can be hundred times larger. This phenomenon is one of the main causes of instability of quartz oscillators at short and medium term.

This is illustrated by fig. 4. Therefore the improvement of high stability quartz oscillators requires very accurate temperature controlled ovens. This can be done by using quartz resonators as temperature sensors (for instance LC-cut with bulk waves and LST-cut with surface waves) and a digital control system4. The performances of such a thermostat are shown on fig. 5. The residual temperature fluctuations can be reduced to a level of the order of .1 v°C.

l/f frequency noise.- Experimental results

...

As shown on fig. ^1,quartz crystal resonators do not escape from l/f noise, and the ultimate stability of oscillators will be given by the level of the l/f fluctuations of the resonator's resonance frequency. This level was measured on pairs of identical resonators in a frequency range from 1 MHz to 25 MHz, including different technologies of fabrication, various sizes and shapes of electrodes and mountings.

The noise level measured at 1 Hz from the carrier is plotted on fig. 6 as a function of the resonator Qfactor.

By linear regression with logarithmic scale curves were fitted to those experimental points, following the relation between the fractional frequency power density spectrum, the Q-factor and the Fourier frequency

The values obtained for the level coefficient A and for the Q-exponent n depend on the selected experimental points among the available values. After mathematical treatment it was found that 3.8 < n < 4.3. ThereTore it is proposed to consider that the most probable value will be n=4. However a small variation of n induces a drastic change of the level coefficient A. This is not sur- prising because of the number of decades involved in the Q-factor and frequency scales. For the previous range of n the corresponding values of A are

0.1 <

x

< 63. A reasonable fitting would correspond to X close to unity when n is an integer equal to 4. Therefore the averaged phenomenological law which is proposed is

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s

(f)

=l.

Y Q* f

This is the first and only one law which was found between l/f noise levels and resonator parameters. Attempts were made to correlate l/f noise and the resonance frequency, but without decisive success.

All these data were collected from different quartz crystal units, and therefore it is difficult to assume that even at constant frequency and constant temperature only one parameter -the Q-factor- was changed. However the acoustic attenuation of elastic waves being temperature dependant, it is possible to no- ticeably modify the Q-factor by cooling the resonator down to liquid helium teT- perature. Fig. 7 shows this variation from 300K to 4K. The internal losses (Q- ) present a peak near 20K. This peak corresponds to a maximum of phonon interactions. For lower temperatures the losses decrease and a large improve ent of

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Q-factor can be observed. For instance Q-factors as large as 1.5 x 10 were obtained with 5 MHz, 5th overtone, resonators made with high quality synthetic quartz.

The l/f noise of a pair of 5 MHz - 5th overtone

-

AT-cut quartz resonators was measured at 300K, 4.2K and 1K. The difference between their resonance frequencies at 4.2K was equal to 1 KHz. The noise levels reduced to one resonator are given on table 11.

Table I1

l/f frequency noise levels as a function of temperature These results confirm the dependa ce

a 05

l/f noise versus Q-factor, even if the data do not strictly follow the 1/Q law

.

Theoretical interpretation of the 1/Q law 4

...

The lattice anharmonicities due to the nonlinear nature of the interato- mic bonds, and the corresponding three and four phonon interaction processes are at the origin of fundamental properties of the crystal such as thermal ex- pansion, finite thermak conductivity and acoustic attenuation, among its general nonlinear behavior

.

Calculation of the attenuation can be made by using two different theoretical methods. In the Landau-Rumer theory the sound wave is considered as a beam of phonons which are scattered by collisions with thermal phonons. This theory does not take into account interactions between thermal phonons and is valid only in the low temperature range, when the phonon life time is large enough and therefore the energy and momentum well defined. At higher temperatures the life time becomes shorter. The uncertainties of energy and momentum increase and the selection rules arising from energy and momentum conservation laws break down. In this case, following the theory of Akhieser the sound wave is represented macroscopically in the classical form. The thermal phonon system is disturbed from equilibrium by the sound wave strain, but tends to return to equilibrium because of the collisions. The phonons are localized and their mean free path has to be small when compared to the sound wave length.

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where R is the angular frequency of the sound wave and T the absolute temperature. ?I and k respectively are the Planck and Boltzmann constants. A detailed description

09

the method can be found in ref. 7.

A lengthly calculation using a perturbation method enables to determine the complex velocity of the wave, i.e. the acoustic attenuation a and the velocity change (s-so), s being the mean velocity of the wave in the absence of thermal phonons

.

~ h e ~ O t a k e the general form

where C is the specific heat, p the specific mass, R the unperturbed sound wave frequency, and yeff is an effective Gruneisen congtant. <Y:

d

is an average over all the normal modes of the crystal. And r is the mean refaxation time of the thermal phonons.

Using the low frequency Young modulus E =os2 the quantity AE=CT<Yeff>, and

0 0'

the relation between attenuation and Q-factor a = - Ro

2Qs0 ^{( 8 )}

and between velocity and frequency S-s R-R

- - - s - a

0

one obtains

R-R R2.c2

o AE o - - - -

62 -

2Eo l+R2r2

A fluctuation 6r of the relaxation time T will produce a frequency fluctuation 6Q of the resonator resonance frequency Ci

Introducing the power spectral density S (f) of the fractional frequency fluctuation ~ = G R / R ~ , and using eq. (10) in eq.'(12) yields

which exhibits the 1/Q law. 4

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It is now to be assumed, first, that r is fluctuating, and, second, its fluctuations exhibit a l/f spectrum. The fluctuation of r can be understood if one considers that any given mode of the crystal interacts with a large number of other modes through each elementary three phonon processes. One relaxation time is attached to each elementary process, and therefore the sumation of all these processes will involve a dispersion of the values of r, equivalent to a fluctuation in time, which could be evaluated. Of course the l/f nature of this fluctuation is still to be demonstrated.

The preceding derivation of the Q -4 experimental law is based on the particular form (eqs. 10 and 11) of the relation between the attenuation coefficient a and the velocity change s-s

.

This relation expresses the fact that the mechanism (described by the phonon gelaxation time r) which produces the losses, also produces a change in the velocity of sound in the quartz crystal. Basical- ly, this connection between dissipation and velocity is expressed by the Kramers-Kronig relations between the real and imaginary parts of the complex mechanical susceptibility

contained in a complex Young modulus E = E (1+K ) . 0 M

By applying the Kramers-Kronig relations to K (w4, one can again show that the assumption of fluctuations in r leads to the M ~ - law.

The use of a mechanical complex susceptibility is a particular case of the following complex susceptibility method. Let us assume that a resonator which stores the oscillation energy (mechanical or electromagnetical) is filled with a material of complex susceptibility K(w). If X is a field, Y the causal and linear response of the material, and W=XY the stored energy density in the material, any fluctuation of the loss rate in the material will produce a fluctuation in the resonant frequency. The quantitative connection between these two fluctuations is given by the Kramers-Kronig relations applied to the complex susceptibility K, which expresses the causal and linear relation between X and Y.

The assumptions of the Debye-form, as in equation (14) for the mechanical susceptibility, are not essential for the Q- result. Any complex susceptibility governed by a siagle time constant T , and obeying the Kramers-Kronig relations, gives the Q lax in the limit of w r << 1.

References

.-

(1) F.L. Walls, A.E. Wainwright, "Measurement of the short term stability of quartz crystal resonators and the implications for crystal oscillator design and applications", IEEE Trans. Inst. Meas., IM-24, no 1, 15, (1975).

( 2 ) A. Ballato, J. Vig, "Static and dynamic frequency-temperature behavior of singly and doubly rotated, oven-controlled quartz resonators". Proc. 32nd Ann. Freq. Cont. Symp., Fort Monmouth (1978).

(3) G. Thsobald et al., "Dynamic thermal behavior of quartz resonators". Proc.

33rd Ann. Freq. Cont. Symp., Fort Monmouth (1979).

(4) G. Marianneau, J.J. Gagnepain, "Digital temperature control for ultrastable quartz oscillators". Proc. 34th Ann. Freq. Cont. Symp., Fort Monmouth (1980).

(5) G. Goujon, "Mesure des fluctuations de frgquence des rksonateurs 5 quartz", Doct. Ing. Thesis, no 107, LPMO-Besan~on (1980).

(6) J.J. Gagnepain, "Nonlinear properties of quartz crystal and quartz resonators : a review". Proc. 35th Ann. Freq. Cont. Symp., Fort Monmouth (1981).

(7) H.J. Maris, "Interaction of sound waves with thermal phonons in dielectric crystals", Physical Acoustics, vol. VIII, Academic Press, (1971).

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DBM

IS0

Fig. 1 noise manrement system for pars of identical resonators at room temwalue

0.1 10. 10C

Fo.xler frequency (Hz)

Fig. 2 : Frequency nolse spectrum of a 10 MHz quartz resonator

Fig. 3 : One-dimensionnal thermal diffusion model of a thickness shear vibrating quartz resonator

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JOURNAL DE PHYSIQUE

Fig. 4 : Frequency and Temperature fluctuations of a 5 MHz AT cut o s c i l l a t o r

Temperature probe : EVA SC c u t

1 a r l t h r e e f o l d oven

- -oi 7

?

&

- z

'x; \

lo-@

.

1

,

1

U a

10 100 1000 ^d

T ( s e c )

F i g . 5 : D l g i t v l temperature c o n t r o l l e d oven temperature s t a b i l i t y

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2 3 4 5 8 7 9 0 6 2 3 4 5 6 7 6 0

10

id

Fig.6

1/F frequency noise level a t 1 Hz from t h e c a r r i e r versus Q.

a t ' room temperature

F

0

loe6

16'

1 0 - ~

T E M P E R A T U R E ('K)

Fig.7 htemal losses of qrah aykd wsus temperatwe

-

100 200 300