A CIRCUIT FOR ALL SEasONS Behzad Razavi The Crystal Oscillator Most electronic systems rely on a pre- oscillators have found new impor- cise reference frequency or time base 15 tance for their low phase noise 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 crystals also proves critical 1 17 18 mentation, and the electronic watch. in most applications. The crystal oscillator 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 resistor covered “piezoelectricity” [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” voltage if subjected to mechanical In addition to a precise resonance resistance, inductance, and capaci- force. 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 oscillation 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, capacitance presented to the crystal to Cady’s 1922 paper [2]. Cady pro- the phase noise of the crystal oscil- by the printed circuit board 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 amplifier through two coupled piezoelectric resonators. 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 structure 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 SOLID-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 frequencies. 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 negative resistance, (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 resonator 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 feed back 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 leads 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 sensors—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- quartz crystal operating in parallel reso- Actual crystals also exhibit resonances 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 transistor related to the first. Thus, the topology in this case? circuitry,” U.S.