TECHNICAL NOTE 30 Design Guidelines for Quartz Crystal Oscillators Introduction A CMOS Pierce oscillator circuit is well known and is widely used for its excellent frequency stability and the wide range of frequencies over which they can be used. They are ideal for small, low current and low voltage battery operated portable products especially for low frequency applications. [1,2] When designing with miniaturized quartz crystals, careful consideration must be given to the frequency, gain and crystal drive level. In this paper, the design equations used in a typical crystal controlled pierce oscillator circuit design are derived from a closed loop and phase analysis. The frequency, gain and crystal drive current equations are derived from this method. Basic Crystal Oscillator The basic quartz crystal CMOS Pierce oscillator circuit configuration is shown on Figure 1. The crystal oscillator circuit consists of an amplifying section and a feedback network. For oscillation to occur, the Barkhausen criteria must be met V V1 Figure 1 - Basic Pierce Oscillator Circuit Crystal Characteristics In order to analyze the quartz crystal oscillator, we must first understand the crystal itself. Figure 2 shows the electrical equivalent circuit of a quartz crystal. The L1, C1 and R1 are generally referred to as the electrical equivalent of the mechanical parameters; inertia, restoring force and friction, respectively. These parameters can be measured using a crystal impedance meter or a network analyzer. C0 is the shunt capacitance between terminals and the sum of the electrode capacitance of the crystal and package capacitance. a) The loop gain must be equal to or greater than one, and b) The phase shift around the loop must be equal to an integral multiple of 2π. The CMOS inverter provides the amplification and the two capacitors, CD and CG, and the crystal work as the feedback network. RA stabilizes the output voltage of the amplifier and is used to reduce the crystal drive level. R1 Motional Resistance L1 Motional Inductance C1 Motional Capacitance C0 Shunt Capacitance Figure 2 – Crystal Electrical Equivalent Circuit REV A STATEK Corporation, 512 N. Main St., Orange, CA 92868 714-639-7810 FAX: 714-997-1256 www.statek.com Page 2 of 8 TN-30 This equivalent circuit can effectively be simplified as a resistance (Re) in series with a reactance (Xe) at a frequency f as shown in Figure 3. Figure 3 - Effective Electrical Circuit of a Quartz Crystal From equation (1) and (2), an example of the magnitude of Re and Xe as a function of frequency are shown in Figures 4 and 5 respectively for fs=32.768kHz, C1=2.4fF, and R1=28kΩ. The frequency is expressed in terms of part per million (ppm) above the series resonant frequency (fs) of the crystal (∆f/f). These two graphs are very useful in the analysis of the crystal oscillator. 1.E+06 Re (ohms) Re (f) and Xe (f) as a function of frequency are as follows: R1 Re(f) = R1 2 + XO ( ) ( 2 Xm -1 XO ) C0=3.5pF C0=1.4pF 1.E+05 (1) 1.E+04 1 10 100 1000 10000 ∆ f/f above fs (ppm) Figure 4 - Re (ohms) vs. ∆f/f (ppm) (2) 1.E+08 Co=3.5pF where 1.E+07 (3) Xe (ohms) 1.E+06 The series resonant frequency of the crystal is defined as Co=1.4pF 1.E+05 (4) 1.E+04 . The quality factor Q is defined as Q= ωSL 1 R1 = 1 ωSR 1C 1 1.E+03 1 (5) 10 100 1000 10000 ∆ f/f above fs (ppm) Figure 5 - Xe (ohms) vs. ∆f/f REV A STATEK CORPORATION 512 N. MAIN ST., ORANGE, CA 92868 714-639-7810 FAX: 714-997-1256 e-mail:[email protected] Page 3 of 8 TN-30 IX Crystal Oscillator Design The AC equivalent circuit of the amplifier and feedback network of a pierce oscillator is shown in Figure 6. For the following analysis, RA is omitted and will be reintroduced later. 2 2 Re + Xe’ Re θ Xe X e’ ’ XG ’ X e’ = Xe - XG sinθ = Re 2 ’ 2 Xe’ cosθ = , 2 Re + X e’ ’ 2 Re + X e’ Figure 6 - Pierce Oscillator AC Equivalent Circuit Figure 8 - Impedance Phase Diagram From Figure 6; , , , Frequency Equation and From the imaginary part of the current phase diagram (y-axis) ,FRVθ ,;,2VLQθ The phase and amplitude relationship of the oscillator voltage, current and impedance are shown in Figures 7 and 8. Assume that the oscillator is oscillating at a frequency f and the amplifier output current ID is 180° out of phase with the oscillator input voltage V1. V = IX 2 ’ 2 and from the equations derived from the equivalent circuit, the voltage and impedance phasor diagram equation (6) becomes ’ X e’ XD from VOLTAGE IX = IX + IX Re RO ’ X e’ = Xe - XG IX Re + X e’ θ VXe VXe’ (6) . VRe Then V1 . (7) Assuming 9;H 9;H9 (XR ’) << (XX ’- 1) 1 O 9 2 m O 2 and R1 << X mX O ’ Xm -1 XO’ 9;H995H Figure 7 - Current and Voltage Phase Diagram REV A STATEK CORPORATION 512 N. MAIN ST., ORANGE, CA 92868 714-639-7810 FAX: 714-997-1256 e-mail:[email protected] Page 4 of 8 TN-30 s Equation (2) becomes Xe(f) = Xm X 1- m XO’ s (7a) Then 1 where XO’ = ωC ’ and CO’ = CO + Cs . O where CL = CS + CL’ . Cs is the circuit stray capacitance across the crystal. Let (7b) and . From eq. 7a and 7b one can obtain X mX O’ = X C ’ (X O’ - Xm) L Xm = XCL ’X O ’ XO’ + XC ’ L Then (10) (8) Equation 10 is the oscillating frequency of the crystal oscillator. CL is called the load capacitance of the oscillator. With a specified CL , the crystal manufacturer can then match the crystal to the customer’s circuit to obtain the desired oscillation frequency. From the CL equation, the relationship between the other circuit parameters can be established (i.e. CD, CG, RO and CS) as it relates to the oscillation frequency of the crystal oscillator. In a typical CMOS oscillator RO generally decreases as the supply voltage increases. This causes a decrease in load capacitance and an increase in the oscillation frequency. Figure 9 shows the effective load capacitance (CL) changes as the output resistance (RO) changes. From eq. (3) and (4) 14 CL (pF) 13 CD = 20pF CG = 30pF 12 11 C&SM V = 1.0pF 10 CO = 1.4pF R1 = 28k Ω 9 8 0.01 From equation (8) 0.1 1 10 RO (MΩ) Figure 9 - . Effective Load Capacitance (CL) vs. Output Resistance (RO) (9) REV A STATEK CORPORATION 512 N. MAIN ST., ORANGE, CA 92868 714-639-7810 FAX: 714-997-1256 e-mail:[email protected] Page 5 of 8 TN-30 Gain Equation From the real part of the current phase diagram (x-axis); ID = IO cos θ + I1 sin θ (11) It is important to note here that in most analyses, only the first term of equation (12) is used. The second term must be taken into account especially for low frequency application were the second term becomes larger than the first term as shown in Figure 10, when RO is less than 1.2 MΩ. and from the equation derived from the voltage, and impedance phase diagram equation (11) becomes 6 g m (u mhos) 5 gm2 4 3 2 gm1 1 0 0 1 2 3 4 RO (MΩ) , and from X’e = Xe – XG and Figure 10 - Comparison of minimum gm requirements vs. Amplifier’s output resistance (RO). Where nd gm1 = first term and gm2 = 2 term of equation(12). For CD=20pF, CG=30pF, CS=1.1pF, CO=1.4pF, R1=28KΩ, fO=32.768kHz and CL=13pF. eq. (7) (12) ( CO’ where Re ~ R1 1 + CL ’ ) 2 Equation (12) gives the minimum gm required for the oscillator to maintain oscillation. In practice, 5 to 10 times the calculated value is required to insure fast start of oscillation. This equation also aids the designer in selecting the component values for CD and CG to match the CMOS amplifier and the crystal. Using equation (12), Figures 11 and 12 show the change in the minimum gm requirements due to change in either CD or CG, while maintaining the other capacitor constant. For a 32.768kHz oscillator, as shown in Figure 11, trimming the output capacitor (CG) will produce more change in gm than the input capacitor (CD). As shown in Figure 12, a decrease in the amplifiers’ output resistance (RO) increases the minimum gm requirement. REV A STATEK CORPORATION 512 N. MAIN ST., ORANGE, CA 92868 714-639-7810 FAX: 714-997-1256 e-mail:[email protected] Page 6 of 8 TN-30 2.5 Where; g m (u mhos) and, CG = 30pF 2 1.5 1 CD =30pF 0.5 0 0 10 20 30 40 CD or CG (pF) Figure 11 - For RO= 2.5MΩ gm comparison between CD and CG, where CS=1.1pF, CO=1.4pF, R1=28KΩ, fO=32.768kHz Figure 13 - Oscillator AC equivalent circuit with the crystal electrical equivalent circuit. The crystal voltage, current and impedance phase relationships are shown in Figure 14 and 15; VOLTAGE CURRENT 6.7 CG = 30pF g m (u mhos) 5.7 4.7 3.7 CD = 30pF 2.7 1.7 0.7 0 10 20 30 40 CD or CG (pF) Figure 12 - For RO= 500kΩ gm comparison between CD and CG, where CS=1.1pF, CO=1.4pF, R1=28KΩ, fO=32.768kHz Figure 14 - Voltage and current phase relationship with the circuit equivalent CRYSTAL IMPEDANCE Crystal Drive Current In order to analyze the current flowing through the crystal, the AC equivalent circuit from Figure 6 is redrawn to show the crystal’s electrical equivalent circuit as shown in Figure 13. The crystal drive current is ib , and ia is the current through the shunt capacitance CO’. Figure 15 – Crystal impedance phase diagram REV A STATEK CORPORATION 512 N. MAIN ST., ORANGE, CA 92868 714-639-7810 FAX: 714-997-1256 e-mail:[email protected] Page 7 of 8 TN-30 V Typical Effects Of RA In The Oscillator Circuit From; |ia| = e and, XO’ ia = IX (Re2 + Xe2) (13) XO’ where 1 1 = . ω(CO ’) ω(CO ’ + C sm ) X O’ = From the current phase diagram of Figure 14 and the relationship . and from the crystal impedance phase diagram Figure 15 sinφ = Re 2 2 ; cosφ = Re + Xe Xe 2 2 . Re + Xe Substituting sin φ and cos φ and ia from equation (13) In many cases, a resistor RA is introduced between the amplifier output terminal and the crystal input terminal as shown in Figure 1. The use of RA will increase the frequency stability, since it provides a stabilizing effect by reducing the total percentage change in the amplifier output resistance RO and also increases the effective output impedance by RA as shown on Figure 9. RA also stabilizes the output voltage of the oscillator and is used to reduce the drive level of the crystal. The complete AC equivalent circuit of Figure 1 is shown in Figure 16, where Xd is the total output capacitance of the amplifier. Using the same analytical approach, the frequency, gain and crystal drive current equations with RA are derived. Figure 16 - Pierce oscillator AC equivalent circuit with RA included or ib = From the frequency equation (10); Substituting; (10) (14) where X’e = Xe – XG From eq. (14) the crystal drive can be calculated from; (in Watts) where R1 = crystal’s motional resistance. CL = CS + CL’ where; and, The gain equation is; 2 2 [ gm > 4π f CG (CD + Cd) Re + Re (C + R C ) R ] d d A O REV A STATEK CORPORATION 512 N. MAIN ST., ORANGE, CA 92868 714-639-7810 FAX: 714-997-1256 e-mail:[email protected] Page 8 of 8 TN-30 References: [1] S.S. Chuang and E. Burnett, “Analysis of th CMOS Quartz Oscillator”, Proc. 9 Int. Congress Chronometry (Stuttgart, W. Germany), Sept. 1974 paper C2.2 [2] E. Vittoz, “High-Performance Crystal Oscillator circuits: Theory and Application” IEEE Journal of Solid state circuits, vol. 23, No. June 1988 pp.774783. where The crystal drive current; (1 + XX )+ (XR ’) R [R + R (1 - XX ’’)] + [X ’ + R X ] 2 e |V| O ib = e e e 2 O 2 A e e A O where and, ( th A version of this paper was presented at the 18 Piezoelectric Devices Conf. in Aug. 1996 by Jim Varsovia 2 D CO’ Re ~ R1 1 + C ’ L ) . 2 Summary By using the closed loop and phase diagram method, we were able to derive the frequency, gain and crystal drive current equations for a simple quartz crystal pierce oscillator. From the equations derived herein, it can be shown that the stray capacitance, minimum gain requirements and the output resistance of the amplifier must be carefully considered to obtain optimum oscillator performance. The minimum gain requirements should include consideration for the full range of operational temperature and voltage. The stray capacitance (CS) is especially critical due to negative feedback effects and will increase the minimum gain requirements of the oscillator [1]. As crystal manufacturers continue to miniaturize the crystal resonator, the oscillator designer must take into account the trade off in the crystal, amplifier and the circuit layout strays in order to select the appropriate component values to achieve proper crystal drive, start up, and a stable oscillation. REV A STATEK CORPORATION 512 N. MAIN ST., ORANGE, CA 92868 714-639-7810 FAX: 714-997-1256 e-mail:[email protected]