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Home , Crystal oscillator, Piezoelectricity

A Circuit for All Seasons

Behzad Razavi

The Oscillator

Most electronic systems rely on a pre- oscillators have found new impor- cise reference or time base 15 tance for their low phase in 12 13 for their operation. Examples include 14 addition to their long-term frequency 4 19 16 20 Mwireless and wireline communication 5 stability. The low temperature coeffi- transceivers, computing devices, instru- 21 cient of also proves critical 1 17 18 mentation, and the electronic . in most applications. The has served this 6 2 3 purpose for nearly a century. In this 22 Crystal Model article, we study the design principles For circuit design purposes, we need of this circuit. Figure 1: Cady’s crystal oscillator. an electrical model of the electrome- chanical crystal. The mechanical res- Brief History onance is fundamentally represented In 1880, Pierre and Jacques Curie dis- high-precision time-base circuit moti- by a series RLC branch, with a covered “” [1], namely, vated extensive studies on crystal modeling the loss [Figure 3(a)]. These the ability of a device to generate a oscillators in that time frame [5], [7]. components are called the “motional” if subjected to mechanical In addition to a precise resistance, , and capaci- . In 1881, Lippman predicted frequency, piezoelectric devices ex­­ tance of the crystal, respectively. With that a converse effect must also exist, hibit extremely high quality factors this series branch, the crystal can act which was ­confirmed by the Curies (Qs), a property that has proved as a short circuit at resonance. In addi- shortly thereafter [1]. essential in communication trans- tion, since the crystal is formed by The use of a piezoelectric device— ceivers. While resonance frequency two parallel plates, a parallel capaci- a “crystal”—to define the drifts can be eventually compensated tance must also be included. The load frequency of a circuit can be traced as the received signal is processed, presented to the crystal to Cady’s 1922 paper [2]. Cady pro- the of the crystal oscil- by the and other

poses the oscillator shown in Fig- lator cannot. In other words, crystal devices can also be absorbed by CP . ure 1, which applies feedback around a three-stage through two coupled piezoelectric . 256 +V Crystal oscillators continued to 250 advance in the ensuing decades, natu- R2 252 rally migrating to bipolar and, even- 268 S P tually, MOS technologies. The interest D 260 in such oscillators was rekindled with a T1 1 F the conception of the electronic watch 254 D F – C in the 1960s and 1970s. In Figure 2, (a) P 1 N 258 shows a MOS realization reported by + S 259 Luscher as prior art in a patent filed Q 262 in 1969 [3], and (b) depicts a more C 264 familiar that dates back to R1 2 a patent filed by Walton in 1970 [4]. 266

The need for an extremely low-power, a2 (a) (b)

Digital Object Identifier 10.1109/MSSC.2017.2688679 Date of publication: 21 June 2017 Figure 2: The MOS crystal oscillators patented by (a) Luscher and (b) Walton.

IEEE -STATE CIRCUITS MAGAZINE Spring 2017 7 including M1, CX, and CY presents an impedance between X and Y |Z | cr given by Rs

gm C C 11 1 P ZXY =++ 2 , (2) CsXYCs CCXYs Zcr Zcr

L1 which, for sj= ~, reduces to a series

branch consisting of CX, CY , and a nega- s p 2 tive resistance equal to - gCmX/( CY ~ ) (a) (b) [Figure 4(b)]. For the circuit to oscil- late, this resistance must cancel the Figure 3: (a) A crystal model and (b) a crystal impedance plot showing series and parallel crystal’s loss. To arrive at a simple resonance . start-up condition, we compute the

real part of the impedance Z1 in Fig- ure 4(b) as [8] R C s VDD Re{}Z1 = L 1 M 1 2 CX CY -gCmXCY Rb , C 222 x ()gCmP ++()CCXY CCXP+CCPY ~ P Rs C –gm y 1 Z CP M (3) 1 2 1 CX M1 Cx L1 CXCY CY C y where ~ denotes the oscillation (a) (b) (c) frequency. Interestingly, this resis- tance is a nonmonotonic function of Figure 4: (a) A three-point oscillator consisting of a crystal and a , (b) an equivalent circuit of (a), and (c) a complete oscillator using an inverter. gm, reaching a maximum if [8]

CCXY gCmX=+CY + ~. (4) cmCP The series devices, L1 1//LC11CCPP()1 + C [Figure 3(b)]. and C1, in Figure 3(a) have peculiar These can also be obtained by neglect- Since Re{}Z1 appears in series with values, e.g., C1 . 5 fF, L1 . 50 mH ing RS and writing LC11,, and RS, we simply equate its for a series resonance frequency of magnitude to RS, obtaining the oscil- 2 10 MHz. This is because the quality LC11s + 1 lation condition as [8] Zcr . 2 . (1) factor, QL= ()1 ~ /,RS reaches several LCCs11PP++CC1 2 ~ ()CCXY++CCXP CCYP gm,crit = , thousand to several hundred thousand, QC1 CCXY (5) translating to large inductance values. Since CC1 % P, we have ~~ps.

The value of C1 is much less than CP, [/12+ CC1 ( P)]; that is, the two fre- where QR= 1/( S C1 ~). which is in the picofarad range. quencies differ by less than 1%. As The core amplifier of the oscillator The network shown in Figure 3(a) explained below, typical oscillators is typically configured as a self-biased exhibits a series resonance fre- operate at ~p . An important attribute of inverter [Figure 4(c)]. The feedback­­ quency, ~s = 1/,LC11 and a par- the crystal is that tolerances in CP only resistor, Rb, must be chosen large allel resonance frequency, ~p = negligibly affect ~p . For example, with enough not to degrade the crystal

C1 = 5 fF and CP = 2 pF, an error of 10% Q significantly.

in CP translates to a 0.01% change in It is interesting to explain why the RS ~p . On the other hand, this low sensi- topology of Figure 4(a) does not oscil- tivity also means that the crystal oscil- late at the crystal’s series resonance lator can be tuned only over a very frequency. Suppose it does. Then, the

narrow range by varying CP . circuit reduces to that shown in Fig- C X P ure 5. It can be proved that the phase Basic Crystal Oscillator shift around this loop is nonzero at If the crystal resonator in Figure 3(a) any frequency, thereby prohibiting Y M1 CX is attached to a negative resistan­ oscillation in this mode. ce, its loss can be compensated CY and oscillation can be sustained. A Start-Up Time common approach employs the The very high Q of crystals to Figure 5: An equivalent circuit of a three- “three-point” oscillator shown in a long start-up time. Of course, the ac- point oscillator in the case of series resonance. Figure 4(a). The one-port network tual oscillation growth rate is given by

8 Spring 2017 IEEE SOLID-STATE CIRCUITS MAGAZINE the net negative resistance in Figure 4(b), following an envelope given by C1 exp(/t x), where x = RLn /(2 1) and Rn is the absolute value of the net negative resistance. For example, a 10-MHz crys- C S1 A 2 B X tal oscillator with a Q of 5,000 can take Vin – roughly 0.5 ms to settle. This issue pos- A0 Vout + es several difficulties. In low-power ap- S4 S3 plications that operate with a low duty Cp1 Cp2 cycle—as in —the­ start-up time translates to a higher power consump- tion. Also, ­communication systems that come out of the sleep mode can- Figure 6: An integrator circuit including parasitics. not begin operation until the settling is completed. Finally, the simulation of the oscillator becomes a very lengthy C task, especially if the circuit must reach 1 steady state for its phase noise to be S computed accurately. S1 C2 2 X – Drive-Level Dependency Vin A0 Vout Crystals behave peculiarly if they + remain inactive: their equivalent series S4 S3 resistance rises considerably. The series resistance falls back to it origi- nal value after the crystal vibrates for some time. This effect is called drive- Figure 7: A noninverting integrator. level dependency. A crystal oscilla- tor that is turned on after a period of inactivity may fail unless the negative 2) How does the finite output imped- References [1] R. Bechmann, “Piezoelectricity – frequen- resistance is strong enough. As a rule ance of M1 and M2 in Figure 4(c) cy control,” in Proc. Annu. Symp. Frequen- of thumb, we select this resistance affect the oscillator’s performance? cy Control, May 1964, pp. 43–92. about four times RS in Figure 4(a). [2] W. Cady, “The piezoelectric resonator,” Answers to Last Issue’s ­Questions Proc. IRE, vol. 10, pp. 83–114, Apr. 1922. [3] J. Luscher, “Oscillator circuit including a Oscillation at Overtones 1) In the circuit of Figure 6, Cp2 ap- crystal operating in parallel reso- Actual crystals also exhibit pears in series with C2 when S3 nance,” U.S. Patent 3 585 527, June 15, 1971. at higher frequencies (overtones) turns off. Does the charge injected [4] R. Walton, “Electronically controlled time- that are approximately harmonically by S1 corrupt the sampled value piece using low power MOS related to the first. Thus, the topology in this case? circuitry,” U.S. Patent 3 664 118, Sept. 9, 1970. of Figure 4(c) can oscillate at an over- No, it does not. The charge [5] C. Fonjallaz and E. Vittoz, “Circuits tone, a property exploited in high- injected by S1 is later removed electroniques pour montres-bracelet a quartz,” in Proc. Int. Congress Chronom- frequency designs. On the other hand, by S4 . etry, 1969, pp. B244–1. low-frequency oscillators must avoid a 2) Given that the op amp in Figure 7 is [6] S. Eaton, “Micropower crystal-controlled solution at overtones. This is possible placed in an inverting configu- oscillator design using RCA COS/MOS in- verter,” RCA application note ICAN-6539, by inserting a resistor in series with ration, how do we intuitively ex- 1971. the output of the inverter in Figure 4(c) plain the noninverting operation [7] M. P. Forrer, “Survey of circuitry for wrist- so as to reduce the loop gain at higher of the integrator? ,” Proc. IEEE, vol. 60, pp. 1047– 1054, Sept. 1972. frequencies. This resistor can also limit The front-end passive sam- [8] E. Vittoz, M. Degrauwe, and S. Bitz, the crystal’s power dissipation, which, pling circuit in fact inverts the “High-performance crystal oscillator if excessive, could cause damage. signal. This can be seen by noting circuits: theory and application,” IEEE J. Solid-State Circuits, vol. 23, pp. 774–783, that, if S2 is absent, then the - June 1988. Questions for the Reader age generated on the right plate

1) Estimate the oscillation frequency of C2 in the hold mode is equal of Figure 2(a) if R1 and R2 are large. to -Vin .

IEEE SOLID-STATE CIRCUITS MAGAZINE Spring 2017 9