Amplifier Limitation Mechanisms in Quartz Crystal Oscillators

Transcription

1 SETIT th International Conference: Sciences of Electronic, Technologies of Information and Telecommunications March -9, TUNISI mplifier Limitation Mechanisms in Quartz Crystal Oscillators Farid Chirouf *, Mahmoud ddouche ** and Réda Yahiaoui ** * SERF, Université de Haute-lsace, rue des Frères Lumières 9 Mulhouse Cedex France. ** Département LPMO, Institut FEMTO ST, UMR CNRS., venue de l Observatoire Besançon Cedex France. Farid.Chirouf@eturs.u-strasbg.fr bstract: This work presents some results in connection with the amplifier limitation mechanisms in uartz crystal oscillators. These mechanisms were studied using the so called non linear dipolar method. This techniue makes it possible to determine the close-loop operating conditions of the oscillator using open loop transient analyses. The well understanding of these limitation mechanisms should allow us to custom made uartz crystal oscillators, specifically for metrology and space applications. Key words: Dipolar method, dipolar impedance, uartz crystal oscillator, high-q factor, time-domain analysis, SPICE. INTRODUCTION In past, the ultra-stable oscillator development for space and military applications was essentially based on practical and experimental knowledge of specialists. However, several simulation tools appeared since the advent of mini and micro computer sciences witch radically changed the computer science landscape and more particularly the computer-aided design (CD). This work focused on the study of some amplifier limitation mechanisms of uartz crystal oscillators. We tent to achieve a non exhaustive classification of these different limitation mechanisms. In fact, the amplitude and to a lesser extent the freuency are determined through the non linear behavior of the amplifier. From this point of view, we can classify the amplifiers from their limitation mechanisms such as saturation or cut off, soft or hard limitations, symmetrical or non- symmetrical limitation and freuency band width limitation. We can also consider the combination of these mechanisms. The whole of these mechanisms were studied using the so called non linear dipolar method [M,M-FC]. This techniue makes it possible to determine the closeloop operating conditions of the oscillator using open loop analyses obtained with SPICE-like simulator [LN-TQ9]. The well understanding of these limitation mechanisms should allows us to custom made uartz crystal oscillators, specifically for metrology and space applications. This paper is split on five sections. fter an introduction on section I, operating principle, characteristics and performances of the uartz crystal oscillator are presented section II. Next section looks over the use of non linear dipolar method to analyze uartz crystal oscillator. The different limitation mechanisms such as amplitude and freuency limitations are described in the forth section. Finally, last section concludes this work and outlines the results of this study.. Quartz crystal oscillators circuits.. Characteristics and performances of uartz oscillator The growing reuirements of the scientific and technical applications, need increasingly powerful oscillators in terms of stability, precision and accuracy. These three criteria remain most determining in an oscillator whatever the application for which it is intended. The characteristics of uartz crystal oscillator such as the oscillation amplitude and freuency are strongly sensitive to the environment. Some of these influent factors are: the drive level of the resonator (isochronisme defect) [JJG], the electrode stress [EPE], the temperature effect [MFE], the acceleration sensitivity [JMP], the electroelastic effects and impurity relaxation [RB], the ionizing radiations effects [RC,MB9] the influence of a magnetic field [RB9]. - -

2 SETIT.. Basic configurations s shown in Figure. the oscillator may be split into two main elements: - Element R represents the resonant circuit (uartz crystal). Its main task is to fix the greatest part of the oscillation freuency. - Element is the amplifier. It is composed of one or several non-linear components (transistors, diodes...). Its main task is to guarantee energy transfer and losses compensation on the one hand and to fix a limit to the amplitude oscillations on the other hand. φ R R = φ φr = kπ, V V Figure. General representation of oscillator.. Oscillation condition The oscillator operation is determined from the euality condition of V and V voltages. This condition is known as the Barkhausen criterion of oscillation [HB]. What results in the euation system () implying the gain ( and R) and also the phase (φ and φ R ): k Z R () The Barkhausen criterion imposes an unitary gain and a null phase into the loop. Thus, we can differentiate three cases:. when R <, the oscillations diminish and the oscillator stops.. when R >, the oscillations are amplified and grow indefinitely.. when R =, steady-state is reached. In a real oscillator, the amplifier non-linearities involve the gain reduction when the amplitude increases. This phenomenon brings back the loop gain towards the unit... Electric model of the resonator The euivalent electrical schematic of a uartz crystal resonator in the vicinity of its freuency of resonance is shown in figure. This schematic suggested by VN DYKE in 9 [VD] is drawn up from the fundamental euations that describe the resonator response subjected to a periodic excitation. This excitation which is applied on the resonator electrodes brings the vibrations of the uartz V V φ crystalline structure by inverse piezoelectric. The direct piezoelectric effect generates an electrical field between the resonator electrodes. Conseuently, the resonator impedance change according to the freuency excitation C p Figure.. Electrical model of uartz resonator. W.G. CDY [WGC] demonstrates that the uartz resonator spectrum is similar to the second order resonant circuit (Figure ). The euivalent electrical schematic is composed of two parallel impedances. The first one consists on a motional branch which made up serial RLC circuit (R, C and L ). The second impedance corresponds to the capacitance (C p ) of the electrodes. The values of the motional branch elements depend on the crystal cut and the vibration mode of the resonator. While the oscillation angular freuency ω is very close to the resonant angular freuency ω S of the resonator, we can write: Δω = ω ω S with Δω << ω S () The total admittance Y of the uartz is given by: Y = = jc pω S R () jx Δ R jq ω ω S The total reactance X of the resonator worth zero for two freuencies, the resonant freuency ω r and the antiresonant freuency ω an : and Motional branch Quartz resonator R i ll L C ω = ω r ω an S = ω S R C L. Dipolar nalysis background.. Dipolar representation p p C R ( C p C ) C L () () The non-linear dipolar method consists in representing uartz crystal oscillator by the motional branch of the resonator connected across a non-linear dipole amplifier (Figure ). The oscillator behavior modeling is performed by analyzing separately the motional branch of the resonator and the amplifier part. Let s note that the parallel capacitance of the resonator is included on the amplifier part (Figure. ). This disjoined analysis overcomes the expensive computing time imposed by the high-q factor of the uartz resonator. - -

3 SETIT Quartz mplifier Figure.. Dipolar representation of the uartz crystal oscillator The nonlinear behavior of the amplifier can be determined in open loop by using transient analysis carried out by using an electric Spice-like simulator. The nonlinear euivalent impedance of the amplifier is obtained by computing the complex impedance of the non-linear dipole (R d X d ). The simulator results allow us to determine accurately the nonlinear euivalent impedance of the amplifier, i.e. its impedance vs the freuency and amplitude. The analysis consists in seeking the closed loop solution by using a series of open loop dipolar analyses with an iterative algorithm. This method leads to an accurate calculation of the operating conditions of the oscillator [M, M, FC]: freuency, amplitude, drive level in transient and steady-state operation, and also amplitude and phase noise spectra... nalysis principle Z =R jx R L C C p mp Z d =R d jx d The resonator is considered as an RLC serial circuit connected in parallel with the capacitance Cp which is included in the amplifier circuit. The impedance of the non-linear dipole amplifier strongly depends on the current amplitude and weakly depends on the current freuency. It can be represented by a non-linear resistance Rd in series with a non linear reactance X d (Figure ). Because of the resonator high uality factor, the loop current is almost perfectly sinusoidal. Thus, it is possible to replace the RLC motional branch of the resonator by a harmonic current source i(t) with amplitude a [M, M, FC], and perform a set of transient analysis with increasing amplitudes by using an electric SPICE like simulator (Figure ). When the amplifier dipole is connected to the resonator, in steady-state operation, the non-linear dipolar impedance Z d is exactly eual and opposite to the resonator impedance Z [KK9]. The non-linear differential euation of the oscillation loop current i is given by () assuming that the condition L >>L d is always satisfied [M, M, FC]. d i di L ( a) ( ( )) = d R Rd a ω i () dt L dt L Where ω is the series resonance angular freuency, and a the amplitude of the harmonic current source. The solution of euation () takes the form () where a(t) and ϕ(t) are slowly varying functions of time: ( ω t ϕ( )) i( t) = a( t) cos t () R d i(t) v dip - X d Z d =R d jx d By performing a Fourier analysis of the voltage across the dipole, it is possible to obtain the complex impedance Z d as a function of the current amplitude a [M, M, FC]. Euation () admits an increasing amplitude solution when the damping term is negative at low motional current amplitude. Then the start-up oscillation condition is given by: R R < () ds where R ds is the value of the dipolar resistance at very low current amplitude. The dipolar resistance increases as the oscillation amplitude increases and the steady-state is obtained when: R R = (9) with R = R ( ) () d a where a is the steady-state oscillation amplitude. In this case, the oscillation angular freuency is given by () [M, M, FC]: L d ( a ) ω ω () L The second order differential euation (9) can be transformed into a first order non-linear system, the solution of which gives the motional current amplitude and freuency transient as well as their steady-state operation values [M]. Dipolar resistance Rd (Ohms) Dipolar reactance Xd (Ohms) Curent amplitude loop () Start-up : Curent amplitude loop () Steady-state : Figure.. Non-linear dipolar analysis of the amplifier part : Dipolar resistance, Dipolar reactance - -

4 SETIT. mplifier Limitation mechanisms in uartz crystal oscillators In oscillator circuits, it is well know that the oscillation amplitude is determined by the non-linear behavior of the amplifier part. From this point of view, the amplifier circuits can be split in to several categories such as: bandwidth limitation, saturation or cut-off, soft or hard limitations, symmetrical or nonsymmetrical limitation, with possible combination of these various limitation mechanisms... mplitude limitation We can classify the amplifiers according to several criteria but it is essential to indicate some key properties of the amplifiers. s figure.a shows it, Z and Z are respectively the input and output impedance of the amplifier. Z represents the feedback impedance which include the parallel capacitance C p. lthough it is possible to include the Z effect into the Z and Z impedances, it is more judicious to keep it out of these input and output impedances because of the particular part that C p play in the dipolar impedance. i(t) Quartz u u mp Z Z Figure.. mplifier representation: General representation, Dipolar representation The amplifier can be represented either by a controlled voltage source or a controlled current source (Figure.b). Let us note that the voltage reference node is not necessary the circuit ground (Figures.a and.b). We can obtain the general expression of small-signal dipolar impedance by replacing the resonator motional branch with a current source having the same freuency (Figure.b). By expressing the dipolar voltage v dip according to the loop current i, we obtain the small-signal dipolar impedance as: = Z Z GZ () Z ds Z From the dipolar analysis point of view, the transconductance G may have a real or complex value and may be linear or non-linear. In addition, the whole variables of the right hand side of euation () might be function of the current amplitude a. So that the non-linear dipolar impedance takes the form: Z ( a) = R ( a) jx ( a) () d d d u - v dip - Gu Z Z Z d =R d jx d u - Output voltage u (V)... Case of soft saturation limiting The Van der Pol oscillator (Figure ) is a simple example of soft limitation mechanism. In this case, the limitation is due to the non-linear transfer function () of the amplifier as shown Figure.a. u = u ( ε ) () u u Quartz u u u R Figure.. Van der Pol oscillator : General schematic, Dipolar reduction. The small-signal gain and the non-linear coefficient ε are supposed real and positive. We can demonstrate in this case that the non-linear impedance (Figure..b) takes the form: εr a Z d = ( ) R () s the amplitude increases, the impedance Z d remains real and increases according to a parabolic law (Figure.b). This is true as long as the amplifier does not comprise a reactive element Input voltage u (V) u (V) (c) Output voltage Vdip (V) i(t) v dip - Figure.. Saturation limitation in the case of Van der Pol oscillator: DC transfer function, Dipolar resistance, (c) Output voltage waveform vs the gain, (d) Dipolar voltage waveform vs the gain Dipolar resistance Rd (Ohms) R d X d Z d =R d jx d - 9 Curent loop amplitude (m) (d) Dipolar voltage - -

5 SETIT Output voltage u (V) The sinusoidal signal that is injected into the input of the amplifier is deformed as it is shown in Figure.c and.d. Thus, the gain variation may also deforms the signal shape at the output u and also across the resonator electrodes (dipolar voltage v dip ).... Case of hard saturation limiting In this subsection, we consider the oscillator configuration of the Figure with an amplification gain which is real, linear and symmetrically saturated (Figure.a). We also assume that the voltage amplifier is ideal and have null output impedance u (V) U sat -U (c) U Figure.. Hard saturation limitation : DC transfer function, Dipolar resistance, (c) Output voltage waveform vs the gain, (d) Dipolar voltage waveform vs the gain The non-linear transfer function of the amplifier is defined with the following conditions: U < u < U u U u U U sat Input voltage u (V) Curent loop amplitude (m) Output voltage Vdip (V) = u = U = U sat sat (d) () Dipolar voltage For the weak values of the loop current amplitude, the dipolar impedance takes the form: Zds = Rds = R( ) () This expression shown that the start-up dipolar impedance is a constant, real and negative as long as the gain is higher than the unity. s the loop current amplitude a increases, the output voltage u reaches the saturation level and becomes suare wave (Figure.c) when the dipolar voltage is deformed (Figure.d). Dipolar resistance Rd (Ohms) a L Output voltage u (V) We compute the dipolar impedance according to the loop current amplitude a by considering the fundamental harmonic. t it is shown Figure.b, the dipolar impedance becomes non-linear as long as the amplitude exceeds the limit a L which is expressed as: Vsat al = () R s no reactive element is present in the circuit, the dipolar impedance remains purely real ( X d ). In addition, the expression of the start-up dipolar impedance () demonstrates that it is impossible to use an inverter amplifier ( < ) when the input and output impedances are real.... Case of cut-off limitation The cut-off limitation is often implied in the oscillators. The amplifier transfer function (Figure 9.a) is given in this case by the following euation system: u U u U = = ( u U ) (9) With this kind of limitation, the dipolar voltages and also the voltages at the output are not any more symmetrical (Figure 9.c and 9.d). s in the case of the saturation mechanism, the dipolar start-up impedance remains constant as long as the loop current amplitude stays below a threshold value. This value is indicative of reaching the cut-off voltage. Starting from this threshold voltage, dipolar impedance increases according to the growing of loop current amplitude (Figure 9.b). Thus, this value corresponds to the beginning of the amplifier non-linear behavior Input voltage u (V) u (V) 9 Output voltage vdip (V) Curent loop amplitude (m) (c) (d) Figure. 9. Cut-off limitation : Transfer function, Dipolar resistance, (c) Output voltage waveform vs the gain, (d) Dipolar voltage waveform vs the gain Dipolar resistance Rd (Ohms) Dipolar voltage - -

6 SETIT.. Freuency limitation and parallel capacitance effect Oscillators circuits presented in section IV. are nothing else than ideal behavioral models without any reactive component (X d ). ny real amplifier has a reactive part at least because they have limited bandwidth or because of the capacitance that is included in the amplifier part as illustrate in section III.b.... Bandwidth amplifier effect In order to illustrate the bandwidth effect, we reuse the Van der Pol oscillator (Figure ) with a different gain that depends on the angular freuency as following: = with ω ω c >> j c ω () ω We assume that the amplifier bandwidth is much higher than the oscillation freuency. Thus, we demonstrate that in this case the start-up dipolar impedance takes the following form: R X ds ds R( ω τ c ) = ω τ c Rωτ c = ω τ c () When the amplifier has a limited bandwidth, it can be shown that the real part of the dipolar impedance doesn t depends strongly on this limitation mechanism. Nevertheless, the imaginary part is considerably affected since it becomes different from zero and decreases with the current amplitude. Dipolar reactance (Ohms) Dipolar reactance (Ohms) Limited bandwidth Curent loop amplitude Limited bandwidth Curent loop amplitude Figure.. Dipolar impedance of limited bandwidth amplifier. (Oscillation freuency: MHz, cut-off freuency: MHz).... Parallel capacitance effect s it was mentioned before, the parallel resonator capacitance C p is included into the amplifier part. However, it is not trivial to give a sense to the dipolar impedance of this capacitance separately since it is crossed by a non-sinusoidal current. Dipolar resistance (Ohms) Dipolar resistance (Ohms) Figure.. Influence of the paralelle capacitance C p : Dipolar resistance, Dipolar reactance Figure gives us the parallel capacitance effect on the dipolar impedance of a limited bandwidth Van der Pol amplifier. We can observe that the parallel capacitance has a weak influence on the real part of the dipolar impedance but it completely changes the dipolar reactance... Case of a transconductance amplifier with cutoff limitation mechanism Several simulations performed on transconductance amplifiers with different limitation mechanisms have shown that the input and output impedances (or time constants) have similar effects on the non-linear dipolar impedance. Thus, the attention will be focused only on the cut-off limiting mechanism often involved in the oscillator circuits. In this case, the non-linear transconductance represented in Figure.a is defined by the conditions (). u U u U Cp=pF Cp=pF Curent loop amplitude () Cp=pF Cp=pF Curent loop amplitude () ig = () i = G( u U ) G The output and dipolar waveforms of the transconductance amplifier with cut-off limiting for a given pair of impedances Z and Z are shown in Figure, and its dipolar impedance is represented in Figure. s it shown on Figures., for small signal current amplitude the dipolar resistance and reactance of the amplifier keep the same values. s the amplitude - -

7 SETIT Dipolar resistance (Ohms) current loop increases, the tow curves may have a uite different location. The parallel capacitance strongly modifies both the dipolar resistance and reactance of the amplifier (Figure.). s shown in (), it is obvious that a negative reactance corresponds to a positive freuency shift that can become very large if the resonator motional inductance L is not much greater than the dipolar inductance Ld of the amplifier. i(t) u u R C R u - C vdip ig Z Z u - Zd=RdjXd. Output curent () (R =R, C =C ) (R =.R, C =C ) Cp = Curent loop amplitude 9 9 Dipolar reactance (Ohms) Quartz (R =R, C =C ) (R =.R, C =C ) Cp = Curent loop amplitude.. Figure.. Oscillator transconductance with amplifier cut-off limiting: Dipolar resistance, Dipolar reactance. U.. Conclusion Output voltage (V) The dipolar analysis allowed us to study the most important and influent amplifier parameters on the oscillator operation. This work demonstrates the efficiency of this analysis method that is well adapted to the uartz crystal oscillators. ll the simulations are achieved using dedicated software (DOQ) whose efficiency and accuracy have been assessed experimentally [M, M]. Using this software, the designer can choose the right circuit for a given purpose or the right components for a given design. (c) Figure.. Transconductance oscillator with amplifier cut-off limiting: Basic schematic, Dipolar reduction, (c) DC transfer function. Let us note in addition that the input and output impedances of the amplifier stage may significatively modify the dipolar behavior of the oscillator in study. Output voltage (V) CKNOWLEDGMENT The authors are grateful to Prof. R. BRENDEL from the Département LPMO, Institut FEMTO ST, Besançon for his help and discussions Time (µs).. REFERENCES [JJG] J. J. Gagnepain Mécanismes non linéaires dans les résonateurs à uartz : Théories, expériences et applications métrologiues Thèse de doctorat, Université de Franche Comté, Besançon, France, 9. Dpolar voltage (V) [EPE] E. P. Ernisse Quartz resonator freency shifts arising from electrode stress in Proc. 9th IEEE FCS, pp., Time (µs).. [MEF] M. E. Frerking. Crystal oscillator design and temperature compensation. New York: Van Nostrand, 9. Figure.. Oscillator transconductance with amplifier cut-off limiting: Steady-state out put voltage waveform, Steady-state dipolar voltage waveform --

8 SETIT [JMP] J. M. Przyjemski Improvement in system performance using a crystal oscillator compensated for acceleration sensitivity in Proc. nd IEEE FCS, pp., 9. [RB] R. Brendel, J. J. Gagnepain Electroelastic effects and impurity relaxation in uartz resonators in Proc. th IEEE FCS, pp. 9, 9. [RC] R. Chévralis Effets des rayonnements ionisants sur les résonateurs à uartz Rapport techniue no / PN; ONR.9. [MB9] M. Brunet. Etude expérimentale sur la dérive des OUS spatiaux soumis aux irradiations. Rapport CNES, 99. [RB9] R. Brendel Influence of a magnetic field on uartz crystal resonators IEEE Trans. Ultrason., Ferroelect., Fre. Contr., vol., pp., 99. [HB] H. Barkhausen, Lehrbuch der Elektronen Rohre,.Band, RÄuckkopplung, Verlag S. Hirzel, 9. [VD] Van Dyke The electric network euivalent of piezoeletric resonator Phys. Rev., vol., pp. 9, 9. [WGC] W. G. Cady The piezoelectric resonator Phys. Rev., vol., pp., 9. [M] M. ddouche, N. Ratier, D. Gillet, R. Brendel, and J. Delporte, Experimental validation of the nonlinear dipolar method Proc. of the th EFTF, St. Petersburg, Russia, March. [M] M. ddouche, R. Brendel, D. Gillet, N. Ratier, F. Lardet-Vieudrin, and J. Delporte, Modeling of Quartz crystal oscillators by using nonlinear dipolar method IEEE Trans. UFFC, vol., pp. - 9, May. [FC] F. Chirouf, R. Brendel, M. ddouche, D. Gillet, N. Ratier, F. Lardet Vieudrin and J. Delporte Using dipolar method for CMOS oscillator analysis in Proc. nd IEEE SETIT: Sciences of Electronic,Technologies of Information and Telecommunications, pp., Sousse, Tunis,. [KK9] K. Kurokawa, Some basic characteristics of broadband negative resistance oscillator circuits Bell System Technical Journal, pp. 9-9, July ugust 99. [LN] L. Nagel SPICE: computer program to simulate semiconductor circuits Electron. Res. Lab., Univ.California at Berkeley, UCB/ERL M, 9. [TQ9] T. Quarles The SPICE implementation guide Electron. Res. Lab., Univ. California at Berkeley, UCB/ERL M 9/,