TECHNICAL NOTE 30 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 V current and low voltage battery operated portable V1 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. Figure 1 - Basic Pierce Oscillator Circuit

In this paper, the design equations used in a Crystal Characteristics typical crystal controlled pierce oscillator circuit In order to analyze the quartz crystal oscillator, we design are derived from a closed loop and phase must first understand the crystal itself. Figure 2 analysis. The frequency, gain and crystal drive shows the electrical equivalent circuit of a quartz current equations are derived from this method. crystal. The L1, C1 and R1 are generally referred to as the electrical equivalent of the mechanical parameters; inertia, restoring force and friction, Basic Crystal Oscillator respectively. These parameters can be measured The basic quartz crystal CMOS Pierce oscillator using a crystal impedance meter or a network circuit configuration is shown on Figure 1. The analyzer. C0 is the shunt capacitance between crystal oscillator circuit consists of an amplifying terminals and the sum of the electrode section and a feedback network. For oscillation to capacitance of the crystal and package occur, the Barkhausen criteria must be met 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 R1 Motional Resistance L1 Motional Inductance output voltage of the amplifier and is used to C1 Motional Capacitance C0 Shunt Capacitance reduce the crystal drive level. Figure 2 – Crystal Electrical Equivalent Circuit

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This equivalent circuit can effectively be simplified From equation (1) and (2), an example of the as a resistance (Re) in series with a reactance magnitude of Re and Xe as a function of frequency (Xe) at a frequency f as shown in Figure 3. 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 (Δf/f). These two graphs are very Figure 3 - Effective Electrical Circuit of a Quartz useful in the analysis of the crystal oscillator. Crystal

1.E+06

Re (f) and Xe (f) as a function of frequency are as follows: C0=3.5pF C0=1.4pF 1.E+05 (ohms) R1 e R (f) = R e (1) R1 22Xm + -1 ()XO ()XO 1.E+04 1 10 100 1000 10000

Δf/f above fs (ppm)

Figure 4 - Re (ohms) vs. Δf/f (ppm)

(2) 1.E+08

Co=3.5pF where (3) 1.E+07

1.E+06 The series resonant frequency of the crystal is Co=1.4pF (ohms)

defined as e X 1.E+05 (4)

1.E+04

The quality factor Q is defined as 1.E+03 ω 1 10 100 1000 10000 SL1 1 Q = = (5) Δf/f above fs (ppm) R1 ωSRC11

Figure 5 - Xe (ohms) vs. Δf/f

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Crystal Oscillator Design IX The AC equivalent circuit of the amplifier and 2 2 Ree + X ’ feedback network of a pierce oscillator is shown in Re Figure 6. For the following analysis, RA is omitted θ and will be reintroduced later. ’ Xe X e’ XG

’ Xe ’ = X eG - X

Re X’e sinθ = cosθ = 2 ’ 2 , 2 ’ 2 Ree + X ’ Ree + X ’ Figure 6 - Pierce Oscillator AC Equivalent Circuit Figure 8 - Impedance Phase Diagram

From Figure 6; , , Frequency Equation From the imaginary part of the current phase and , diagram (y-axis)

, FRV θ , ; , 2 VLQ θ (6)

The phase and amplitude relationship of the and from the equations derived from the oscillator voltage, current and impedance are equivalent circuit, the voltage and impedance shown in Figures 7 and 8. Assume that the phasor diagram equation (6) becomes oscillator is oscillating at a frequency f and the ’ Xe ’ Re amplifier output current ID is 180° out of phase IXXX = I + I XD RO with the oscillator input voltage V1. ’ VOLTAGE from Xe ’ = X eG - X I 2 ’ 2 X V = IX Ree + X ’ . VRe θ V V’ Xe Xe V1 Then . (7)

Assuming R 2 X 2 R1 Xm 9;H 9;H9Ã 1 << m - 1 and << - 1 ()’ ()’ XXmO’ X ’ XO XO O 9 9;H9 95H

Figure 7 - Current and Voltage Phase Diagram

Equation (2) becomes s

Xm s Xe (f) = X (7a) 1 - m ’ XO 1 Then (10) where XOʼ = and COʼ = CO + Cs . ωCO’

where CL = CS + CLʼ . Cs is the circuit stray capacitance across the crystal. Equation 10 is the oscillating frequency of the crystal oscillator. CL is called the load capacitance Let (7b) of the oscillator. With a specified CL , the crystal manufacturer can then match the crystal to the customerʼs circuit to obtain the desired oscillation and . 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 From eq. 7a and 7b one can obtain the oscillation frequency of the crystal oscillator.

XXmO’’ = X C ’ (XOm - X ) L In a typical CMOS oscillator RO generally decreases as the supply voltage increases. This XC L ’XO’ Xm = causes a decrease in load capacitance and an XO’ + X CL’ increase in the oscillation frequency. Figure 9 shows the effective load capacitance Then (8) (CL) changes as the output resistance (RO) changes.

From eq. (3) and (4) 14 13

12 CD = 20pF

CG = 30pF (pF) 11 L V C C&SM = 1.0pF 10 CO = 1.4pF

9 R1 = 28kΩ 8 From equation (8) 0.01 0.1 1 10

RO (MΩ)

Figure 9 - Effective Load Capacitance (CL) vs. Output Resistance (RO)

. (9)

Gain Equation It is important to note here that in most analyses, From the real part of the current phase diagram only the first term of equation (12) is used. The (x-axis); second term must be taken into account especially for low frequency application were the ID = IO cos θ + I1 sin θ (11) 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) 6 becomes 5 gm2 4

3 (u mhos) m

g 2 gm1 1

0 01234

RO (MΩ)

Figure 10 - Comparison of minimum gm requirements vs. Amplifierʼs and from Xʼe = Xe – XG and output resistance (RO). , Where eq. (7) 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.

(12) Using equation (12), Figures 11 and 12 show the change in the minimum gm requirements due to C’O 2 where Re1 ~ R 1 + change in either CD or CG, while maintaining the ~ ()CL’ other capacitor constant. For a 32.768kHz oscillator, as shown in Figure 11, trimming the output capacitor (CG) will produce more change in Equation (12) gives the minimum gm required for gm than the input capacitor (CD). As shown in the oscillator to maintain oscillation. In practice, 5 Figure 12, a decrease in the amplifiersʼ output to 10 times the calculated value is required to resistance (RO) increases the minimum gm insure fast start of oscillation. This equation also requirement. aids the designer in selecting the component values for CD and CG to match the CMOS amplifier and the crystal.

2.5 Where; and, 2 CG= 30pF )

1.5 (u mhos 1 CD=30pF m g 0.5

0 0 10 20 30 40

CD or CG (pF) Figure 13 - Oscillator AC equivalent circuit with the crystal electrical Figure 11 - For RO= 2.5MΩ gm comparison equivalent circuit. between CD and CG, where CS=1.1pF, CO=1.4pF, R1=28KΩ, The crystal voltage, current and impedance phase fO=32.768kHz relationships are shown in Figure 14 and 15; VOLTAGE CURRENT

6.7

5.7 CG= 30pF

4.7

3.7 (u mhos) m

g 2.7 CD= 30pF

1.7

0.7 0 10 20 30 40 Figure 14 - Voltage and current phase CD or CG (pF) relationship with the circuit equivalent Ω Figure 12 - For RO= 500k gm comparison CRYSTAL IMPEDANCE between CD and CG, where CS=1.1pF, CO=1.4pF, R1=28KΩ, fO=32.768kHz

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 crystalʼs electrical equivalent circuit as shown in Figure 13. The crystal drive current is ib , and ia is the current through the Figure 15 – Crystal impedance phase shunt capacitance COʼ. diagram

V e Typical Effects Of RA In The Oscillator From; |ia| = and, XO’ Circuit In many cases, a resistor RA is introduced 22 ia =IXee () R + X (13) between the amplifier output terminal and the XO’ crystal input terminal as shown in Figure 1. The use of RA will increase the frequency stability, 1 1 since it provides a stabilizing effect by reducing where X O’ = ω = ω . ( CO ’) ( COs ’ + Cm ) the total percentage change in the amplifier output resistance RO and also increases the effective From the current phase diagram of Figure 14 and output impedance by RA as shown on Figure 9. RA also stabilizes the output voltage of the the relationship oscillator and is used to reduce the drive level of the crystal. . The complete AC equivalent circuit of Figure 1 is and from the crystal impedance phase diagram shown in Figure 16, where Xd is the total output Figure 15 capacitance of the amplifier. R Xe φ e φ sin = 2 2 ; cos = 2 2 . Using the same analytical approach, the Ree + X Ree + X frequency, gain and crystal drive current equations with RA are derived. Substituting sin φ and cos φ and ia from equation (13)

Figure 16 - Pierce oscillator AC equivalent circuit with RA included or ib =

From the frequency equation (10); Substituting; (10)

(14) where; CL = CS + CLʼ and, where Xʼe = Xe – XG

From eq. (14) the crystal drive can be calculated from; The gain equation is; (in Watts) 2 2 Re gmG > 4π fC (CD + C de ) R + C d + C d R A [] () R O where R1 = crystalʼs motional resistance.

References: [1] S.S. Chuang and E. Burnett, “Analysis of CMOS Quartz Oscillator”, Proc. 9th Int. Congress Chronometry (Stuttgart, W. Germany), Sept. 1974 where paper C2.2 [2] E. Vittoz, “High-Performance Crystal Oscillator The crystal drive current; circuits: Theory and Application” IEEE Journal of Solid state circuits, vol. 23, No. June 1988 pp.774- 2 Xe R 2 783. |V| 1 + + e ( X O ) ( X ’) O th ib = A version of this paper was presented at the 18 2 R 2 X’e e Piezoelectric Devices Conf. in Aug. 1996 by Jim ReA + R 1 - + XeA ’ + R X [ ( X O ’ )] [ D ] Varsovia

C’O 2 where and, R e1 ~ R 1 + . ~ ()CL’

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.