Algebraic Deriation of the Oscillation Condition of High Q Quartz Crystal Oscillators NICOLAS RATIER Institut FEMTO ST, CNRS UMR 67 Département LPMO a. de l Obseratoire, 50 Besançon FRANCE nicolas.ratier@femto-st.fr REMI BRENDEL Institut FEMTO ST, CNRS UMR 67 Département LPMO a. de l Obseratoire, 50 Besançon FRANCE remi.brendel@femto-st.fr Abstract: A symbolic numeric method to simulate a high Q quartz crystal oscillator is deried in this paper. The proposed method can take into account the isochronism defect of the resonator when high precision simulations are needed. The oscillator circuit is expressed under a canonical 5 terminal network which can be easily handled. The oscillator behaior is then described under the form of a nonlinear characteristic polynomial whose coefficients are functions of the circuit components and of the oscillation amplitude. The steady state oscillation amplitude and frequency are determined by soling the polynomial in the frequency domain. Moreoer, if tolerance is assigned to each circuit component, efficient worst case analysis can be performed. Key Words: Quartz Crystal Oscillator, Nonlinear, Simulation, Characteristic Polynomial, n Terminal Network Introduction ISSN: 09-7 9 We hae deelopped [9][0][][][] a symbolic numeric method to simulate high Q quartz crystal oscillator. This method is based on rewritting the amplifier part of the oscillator under a reduced form. To obtain this reduced form, the transistor is replaced by an equialent admittance circuit, then each component of the circuit is progressiely integrated into this equialent circuit. Finally, the oscillation condition can be easily stated. Although this method gies accurate simulation results, it has two drawbacks. The first one is the number of equations to manage, indeed, at each step of the reduction, the relations between both equialent circuits hae to be saed. The second problem is that we are not sure that eery oscillator circuit can be analysed by this method. The work described here was initiated to oercome these problems while saing the efficiency of our initial method. The oscillator is now separated into parts (Fig. : the quartz crystal resonator, the actie component (transistor and a 5 terminal network which links the two preious elements. The paper is organized as follow: we first recall the standard equialent circuit of the resonator and define a large signal equialent circuit for the transistor. Next, we show that the network N can always be expressed under a canonical form. Thus, the transformation in the so called reduced form of an oscillator becomes straighforward. The oscillation condition and characteristic polynomial are then obtained, which lead eentually to the equation system which represents the oscillator behaior with respect to all circuit components. Æ ÉÙ ÖØÞ Æ ØÛÓÖ ÌÖ Ò ØÓÖ Figure : 5 terminal representation of a quartz crystal oscillator. Resonator equialent circuit The classical equialent circuit of a quartz crystal resonator is shown in Fig. [5][6]. It comprises a motional arm (R q, L q, C q in parallel with a capacitance (C p. The resonator behaior is characterized by the series resonant frequency f q, the inductance L q, the Issue, olume 7, February 008
series resistance R q, and the parallel capacitance C p. Y q R q L q C q C p Figure : Equialent circuit of the resonator. Some high precision simulation needs to take into account the isochronism defect of the quartz crystal resonator. In that case, the actual resonant frequency f q depends on the drie leel [7] according to the law: f q = f q ( + ap a ( P a = Re {Z q } i q ( where P a is the actie power within the crystal, a is the isochronism effect parameter of first order and i q is the mean square alue of the current through the crystal. The computation of P a is detailed in appendix. The capacitance C q has to be replaced by C q which is calculated from f q and L q (Eq.. of a table or computed at run time. y i ( u = g i ( u + sc i ( u ( y r ( = g r ( + sc r ( (5 y f ( u = g f ( u + sc f ( u (6 y o ( = g o ( + sc o ( (7 Processing of the 5 terminal network The theory of network circuits proes that any n terminal passie network can be represented by n (n / admittances connected between each n terminal nodes. An example is gien in Fig. for some n terminal networks. Y Representation of some n terminal net- Figure : works. Y Y Y Y Y Y Y Y Y C q = π L q f q ( It can be shown [] that the n (n / admittances Y kl (k l can be computed using the following formula: Transistor equialent circuit The transistor is modelled [6] by a large signal admittance parameter circuit (Fig.. The admittances y i, y r, y f and y o enable to modelize accurately the nonlinear behaior of the component. Y kl = (S k + S l S kl (8 The admittance S k is the driing point admittance when all the terminals except k are shorted and when a oltage is applied to terminal k. These conditions are represented in Fig. 5 on a 5 terminal network for k =. u y i y r. y f.u y o u Figure : y parameter representation of a transistor. S 5 N From their definition, it results that y i and y f depend on the input oltage u while y r and y o depend on the output oltage ( (7. The 8 functions g and c are calculated [] for a gien bias and temperature conditions by using the electrical simulator SPICE []. They can be represented under the form ISSN: 09-7 50 Figure 5: Computation or measurement of the admittance S k. The admittance S kl is the driing point admittance when all the terminals except k and l are shorted Issue, olume 7, February 008
and when the same oltage is applied to terminals k and l. These conditions are represented in Fig. 6 on a 5 terminal network for k = and l =. 5 N The canonical circuit in Fig. 7 can be transformed into the reduced form shown in Fig. 8. Under this form, the computation of the oscillation condition becomes triial. We shall express the new coefficients Y i, Y r, Y f and Y o as functions of the transistor admittances y i, y r, y f, y o and of the network admittances. I i I o S Figure 6: Computation or measurement of the admittance S kl. U Y i Y r. Y f.u Y o The computation of the driing point admittances S k, S l and S kl is straightforward by using topological methods. The common feature of these methods is that the network is represented by a graph that resembles the network, and the calculation of any network function is transformed into a subgraph enumeration problem. We recall some graph theory definitions: A tree T of a graph G is a subgraph of G if T is connected, if T contains all nodes of G and if T has no loops. A tree admittance product of N is the product of the admittances of a tree of N. The driing point admittance of an RLC network N is gien [8][] by (9 where N is deried from N by short circuiting the two input terminals, and then remoing any short circuited branches. sum of all tree admittance products of N Y = sum of all tree admittance products of N (9 It is clear that the calculus of the sum of all tree admittance products is a disguised method to compute the determinant of a node admittance matrix. As opposed to regular algebraic method to inerse the linear equation system of node admittances, the final answer of the determinant is here directly written down, i.e. without cancellation of terms. Reduced form of the amplifier We hae shown that the network N and the transistor part of an oscillator can always be represented as in Fig. 7. The networks is replaced by the 5 (5 / = 0 equialent admittances Y, Y,..., Y 0, and the transistor is substituted by grayed admittances y i, y r, y f and y o. ISSN: 09-7 5 Figure 8: Reduced form of the amplifier. The relation beetween the oltages (u, across the transistor and the oltages (U, across the resonator is calculated by writing nodal equations at nodes and. This leads to (0, ( u = ( C C C C with the following coefficients: ( U (0 = A A A A ( C = (A Y A Y 5 / ( C = (A Y A Y 6 / ( C = (A Y 5 A Y / ( C = (A Y 6 A Y / (5 The inerse of Eq. 0 is gien here because it will be used into the oscillation condition deriation section. From the same nodal equations at nodes and, or by inerting (0, one can obtain: ( U = ( B B B B with the following coefficients: ( u (6 δ = Y Y 6 Y 5 Y (7 B = (Y 6 A Y A /δ (8 B = (Y 6 A Y A /δ (9 B = (Y A Y 5 A /δ (0 B = (Y A Y 5 A /δ ( The deriation of C ij and B ij is detailed in appendix. The coefficients A ij are defined in both cases by Issue, olume 7, February 008
I i I o Y Y Y 5 U Y u Y Y 6 Y 9 Y 7 Y 8 y i y r. y f.u y o Y 0 Figure 7: Canonical form of the amplifier. the following equations (Eq. 5. A = Y 8 + y i + Y + Y + Y 9 ( A = y r Y 9 ( A = y f Y 9 ( A = Y 0 + y o + Y 6 + Y 5 + Y 9 (5 By writing nodal equation at nodes and, replacing the oltages (u, by their expressions as a fonction of (U,, and finally equating with node equations of Fig. 8, one can express the relation between the amplifier admittances (Y i, Y r, Y f, Y o and the transistor admittances (y i, y r, y f, y o. Y i = Y + Y + Y ( C + Y 5 ( C (6 Y r = Y Y C Y 5 C (7 Y f = Y Y 6 C Y C (8 Y o = Y + Y 7 + Y 6 ( C + Y ( C (9 5 Oscillation condition deriation We hae shown that the circuit of a quartz crystal oscillator can be represented as shown in Fig. 9. The resonator is connected across the amplifier part of the oscillator which is itself composed of the network N and the transistor. Under this reduced form, the deriation of the oscillation condition is straighforward. The application of Kirchhoff s law at both input and ouput parts of the circuit (Fig. 9 leads to the following relations (0 (. (Y i + Y q U + (Y r Y q = 0 (0 (Y f Y q U + (Y o + Y q = 0 ( ISSN: 09-7 5 U Y q Y i Y r. Y f.u Y o Figure 9: Reduced form of the oscillator. 5. Gain of the transistor The system (0 and ( gies two ways to compute the gain of the transistor (ratio /u. By replacing into these equations the oltage (U, as a fonction of (u, (Eq. 6, the oltage is expressed as a fonction of the oltage u. Obiously, these two equations must gie the same result. u = B (Y i + Y q + B (Y r Y q B (Y i + Y q + B (Y r Y q ( u = B (Y f Y q + B (Y o + Y q B (Y f Y q + B (Y o + Y q ( 5. Oscillator characteristic polynomial The system (0 and ( admits a nontriial solution only if its determinant is null, this gies the oscillation condition of the circuit (. The admittances Y i, Y r, Y f and Y o are functions of the circuit components (inside the network N and of the transistor representatie components (conductances g and capacitances c. The admittance Y q is expressed by its equialent Issue, olume 7, February 008
circuit (R q, L q, C q and C p. Y i Y o Y f Y r + (Y i + Y r + Y f + Y o Y q = 0 ( Replacing the admittances by their expressions, which are rational functions in the Laplace s ariable s, leads to the oscillator characteristic polynomial (5. Each coefficient a k of this polynomial is expressed as a function of the component alue of the circuit. The degree K of the characteristic polynomial depends on the component number and on the topology of the network N. K a k s k = 0 (5 k=0 To obtain the steady state frequency and amplitude of the oscillation, the Laplace s ariable s is replaced by the harmonic ariable jω, this splits (5 into real and imaginary part. One of the two equations or must be added to compute the oltage as a function of u. K α k (u, ω k = 0 (6 k=0 K β k (u, ω k = 0 (7 k=0 B (Y i + Y q + B (Y r Y q B (Y i + Y q + B (Y r Y q = u (8 The numerical calculation of the ariables u, and ω which satisfies the system (6 (8 determines the frequency and the amplitude of the oscillation. The numerical computation is independent of the time constants of the circuit. If the designer would eidence relatie frequency changes of about f/f 0 6, the isochronism defect of the resonator can be taken account by modifying its equialent circuit. 6 Conclusion The core of the analysis of an oscillator circuit is done by computing the 0 equialent admittances of the canonical form described here, and by simulating the equialent admittances of the transistor. All the other computations are common to all oscillator circuits with one transistor and can be thus coded once for all. The characteristic polynomial coefficients are expressed as functions of all circuit components. Therefore, the influence of a change of any component alue on the oscillation amplitude and frequency as well as ISSN: 09-7 5 on the resonator excitation leel can be directly calculated. In addition, if tolerance and temperature coefficient are assigned to each component, our method can be used to perform efficient worst case analysis and to simulate the signal sensitiity at a gien temperature with respect to eery circuit component. Appendix : Computation of the isochronism defect By definition, the actie power P a within the crystal is (9, where i q is the mean square alue of the current through the crystal. For usual alue of quartz Re {Z q } R q. P a = Re {Z q } i q R q i q (9 (0 The gains of the amplifier G and of the transistor g are defined by respectiely ( and (. G = Y i + Y q Y r Y q ( g = B (Y i + Y q + B (Y r Y q B (Y i + Y q + B (Y r Y q ( The current i q can be expressed from the reduced form of the oscillator (Fig. 9. The oltage U is then replaced by its expression in (6. i q = Y i U + Y r ( = (Y i + GY r U ( = (Y i + GY r (B u + B (5 = (Y i + GY r (B + gb u (6 Finaly, the actie power is expressed as (7. P a = R q Y i + GY r B + gb u (7 Appendix : Deriation of reduced form equations This appendix details how to obtain the equations of section in compact form. The nodal equations of the Issue, olume 7, February 008
canonical form of the amplifier (Fig. 7 are: 0 = Y 8 u + y i u + y r + Y (u U + Y (u + Y 9 (u (8 0 = Y 0 + y o + y f u + Y 6 ( + Y 5 ( U + Y 9 ( u (9 I i = Y (U + Y U + Y (U u + Y 5 (U (50 I o = Y ( U + Y 7 + Y 6 ( + Y ( u (5 The key point is to rewrite (8 and (9 under the following symetric form: ( ( ( ( A A u Y Y = U A Y 6 A Y 5 (5 Equation (5 determines the coefficients A ij. The resolution of (5 with respect to (u, gies the coefficients C ij. The resolution of (5 with respect to (U, gies the coefficients B ij. References: [] R. Brendel, F. Djian, and E. Robert. High precision nonlinear computer modelling technique for quartz crystal oscillators. In Proc. of the 5th Annual Symposium on Frequency Control (AFCS, pages 5, Los Angeles, California, May 9-99. [] R. Brendel, N. Ratier, L. Couteleau, G. Marianneau, and P. Guillemot. Slowly arying function method applied to quartz crystal oscillator transient calculation. IEEE Trans. on Ultrasonics, Ferroelectrics and Frequency Control, 5(:50 57, March 998. [] R. Brendel, N. Ratier, L. Couteleau, G. Marianneau, and P. Guillemot. Analysis of Noise in Quartz Crystal Oscillators by Using Slowly arying Functions Method. IEEE Trans. on Ultrasonics, Ferroelectrics and Frequency Control, 6(:56 65, March 999. [] L. Couteleau, R. Brendel, N. Ratier, and P. Guillemot. Analysis of Parametric Noise in Quartz Crystal Oscillators. In Joint Meeting of the th European Frequency and Time Forum and the 999 IEEE International Frequency Control Symposium, pages 58 5, Besançon, France, April -6 999. [5] an Dyke. The electric network equialent of a piezoelectric resonator. Phys. Re., 5:895, 95. ISSN: 09-7 5 [6] M.E. Frerking. Crystal Oscillator Design and Temperature Compensation. New York: an Nostrand Reinhold, 978. [7] J.-J. Gagnepain and R. Besson. Nonlinear effect in piezoelectric quartz crystals. Physical Acoustics, XI:66 78, 975. [8] P.M. Lin. Symbolic Network Analysis. Elseier, New York, 99. [9] N. Ratier, R. Brendel, and P. Guillemot. Quartz oscillators: deriing oscillation condition by symbolic calculus. In Proc. of the Tenth European Frequency and Time Forum, pages 6, Brighton, United Kingdom, March 5-7 996. [0] N. Ratier, L. Couteleau, R. Brendel, and P. Guillemot. Automatic formal deriation of the oscillation condition. In International Frequency Control Symposium (IEEE, pages 95 9, Orlando, Florida, May 8-0 997. [] N. Ratier, M. Markoa. Symbolic Computation of the Kirchhoff s Topological Formulas. WSEAS Transactions on Circuits and Systems, 5(8:50 55, August 006. [] Steen M. Sandler and Charles Hymowitz. SPICE Circuit Handbook. McGraw Hill Professional, 006. [] Louis Weinberg. Network Analysis and Synthesis. Krieger Publishing Company, second edition, 975. Issue, olume 7, February 008