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Crystal and MEMS Oscillators (XTAL)

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Crystal and MEMS Oscillators (XTAL) Berkeley Crystal and MEMS Oscillators (XTAL) Prof. Ali M. Niknejad U.C. Berkeley Copyright c 2014 by Ali M. Niknejad Niknejad Advanced IC's for Comm Crystal Resonator C0 L1 C1 R1 quartz L2 C2 R2 t L3 C3 R3 Quartz crystal is a piezoelectric material. An electric field causes a mechanical displacement and vice versa. Thus it is a electromechanical transducer. The equivalent circuit contains series LCR circuits that represent resonant modes of the XTAL. The capacitor C0 is a physical capacitor that results from the parallel plate capacitance due to the leads. Niknejad Advanced IC's for Comm Fundamental Resonant Mode Acoustic waves through the crystal have phase velocity v = 3 103 m=s. For a thickness t = 1 mm, the delay time through× the XTAL is given by τ = t=v = (10−3 m)=(3 103 m=s) = 1=3 µs. × This corresponds to a fundamental resonant frequency 1 f0 = 1/τ = v=t = 3 MHz = p . 2π L1C1 The quality factor is extremely high, with Q 3 106 (in vacuum) and about Q = 1 106 (air). This∼ is much× higher than can be acheived with electrical× circuit elements (inductors, capacitors, transmission lines, etc). This high Q factor leads to good frequency stability (low phase noise). Niknejad Advanced IC's for Comm MEMS Resonators The highest frequency, though, is limited by the thickness of the material. For t 15 µm, the frequency is about 200 MHz. ≈ MEMS resonators have been demonstrated up to GHz frequencies. MEMS resonators are an active research∼ area. Integrated MEMS resonators are fabricated from polysilicon beams (forks), disks, and other mechanical structures. These resonators are electrostatically induced structures. We'll come back to MEMS resonators in the second part of the lecture Niknejad Advanced IC's for Comm Example XTAL Some typical numbers for a fundamental mode resonator are C0 = 3 pF, L1 = 0:25 H, C1 = 40 fF, L1 R1 = 50 Ω, and f0 = 1:6 MHz. Note C that the values of L and C are 0 C 1 1 1 modeling parameters and not physical R1 inductance/capacitance. The value of L is large in order to reflect the high quality factor. The quality factor is given by !L 1 Q = 1 = 50 103 = R1 × !R1C1 Niknejad Advanced IC's for Comm XTAL Resonance Recall that a series resonator has a phase shift from 90◦ to +90◦ as the impedance changes from capacitive to inductive− form. The phase shift occurs rapidly for high Q structures. It's easy to show that the rate of change of phase is directly related to the Q of the resonator ! dφ Q = s 2 d! !0 For high Q structures, the phase shift is thus almost a \step" function unless we really zoom in to see the details. Niknejad Advanced IC's for Comm XTAL Phase Shift +90◦ +45◦ L1 +0◦ ω0 2Q C1 R 45◦ ∆ω 1 − 90◦ − ω0 In fact, it's easy to show that the 45◦ points are only a ± distance of !s =(2Q) apart. ∆! 1 = !0 Q For Q = 50 103, this phase change requires an only 20 ppm change in frequency.× Niknejad Advanced IC's for Comm Series and Parallel Mode C0 C0 L1 C1 R1 L1 R1 low resistance high resistance Due to the external physical capacitor, there are two resonant modes between a series branch and the capacitor. In the series mode !s , the LCR is a low impedance (\short"). But beyond this frequency, the LCR is an equivalent inductor that resonates with the external capacitance to produce a parallel resonant circuit at frequency !p > !s . Niknejad Advanced IC's for Comm Crystal Oscillator Leff Leff In practice, any oscillator topology can employ a crystal as an effective inductor (between !s and !p). The crystal can take on any appropriate value of Leff to resonate with the external capacitance. Topoligies that minmize the tank loading are desirable in order to minize the XTAL de-Qing. The Pierce resonator is very popular for this reason. Niknejad Advanced IC's for Comm Clock Application CLK CLK Note that if the XTAL is removed from this circuit, the amplifier acts like a clock driver. This allows the flexbility of employing an external clock or providing an oscillator at the pins of the chip. Niknejad Advanced IC's for Comm XTAL Tempco The thickness has tempco t 14 ppm=◦C leading to a variation in frequency with temperature.∼ If we cut the XTAL in certain orientations (\AT-cut") so that the tempco of velocity cancels tempco of t, the overall tempco is minimized ◦ and a frequency stability as good as f0 0:6 ppm= C is possible. ∼ Note that 1 sec=mo = 0:4 ppm! Or this corresponds to only 0:4 Hz in 1 MHz. This change in thickness for 0:4 ppm is only −6 −6 −3 −10 δt = 0:4 10 t0 = 0:4 10 10 m = 4 10 . That's about× 2 atoms!× × × × The smallest form factors available today's AT-cut crystals are 2 1.6 mm2 in the frequency range of 24-54 MHz are available.× Niknejad Advanced IC's for Comm OCXO ∆f f 15ppm The typical temperature variation of the XTAL is shown. The variation is minimized at room 55◦C 25◦C 125◦C − temperature by design but can be as large as 15 ppm at the extreme ranges. 15ppm To minimize the temperature variation, the XTAL can be placed in an oven to form an Oven Compensated XTAL Oscillator, or OCXO. This requires about a cubic inch of volume but can result in extremely stable oscillator. OCXO 0:01 ppm=◦C. ∼ Niknejad Advanced IC's for Comm TCXO analog or digital In many applications an oven is not practical. A Temperature Compenstated XTAL Oscillator, or TCXO, uses external capacitors to \pull" or \push" the resonant frequency. The external capacitors can be made with a varactor. This means that a control circuit must estimate the operating temperature, and then use a pre-programmed table to generate the right voltage to compensate for the XTAL shift. This scheme can acheive as low as TCXO 0:05 ppm=◦C. Many inexpensive parts use a DCXO, or a ∼ digitally-compensated crystal oscillator, to eliminate the TCXO. Often a simple calibration procedure is used to set the XTAL frequency to within the desired range and a simple look-up table is used to adjust it. A Programmable CrystalNiknejad OscillatorAdvanced (PCXO) IC's for Comm is a combined PLL and XTAL together. XTAL Below !s 1 1 1 jXc = = + j!Lx j!C !C jj j!C eff 0 x C + C 0 x 1 1 2 = 1 ! Lx Cx ωs j!C0 jjj!Cx − Below series resonance, the equivalent circuit for the XTAL is a capacitor is easily derived. The effective capacitance is given by Cx Ceff = C0 + 2 1 ! − !s Niknejad Advanced IC's for Comm XTAL Inductive Region Leff ωs ωp Past series resonance, the XTAL reactance is inductive 1 !s 2 jXc = j!Leff = j!Lx 1 j!C0 jj − ! The XTAL displays Leff from 0 H in the range from ! 1 !s !p. ! Thus for any C, the XTAL will resonate somewhere in this range. Niknejad Advanced IC's for Comm Inductive Region Frequency Range We can solve for the frequency range of (!s ;!p) using the following equation 2 1 !s j!pLeff = j!pLx 1 j! C jj − ! p 0 p ! . ! C p = 1 + x ! C s r 0 Example: Cx = 0:04 pF and C0 = 4 pF Since C0 Cx , the frequency range is very tight !p = 1:005 !s Niknejad Advanced IC's for Comm XTAL Losses Leff vi ro C2 RL′ RB C1 Bias XTAL Loss !"#$ Consider now the series losses in the XTAL. Let X1 = 1=(!C1) and X2 = 1=(!C2), and jXc = j!Leff . − 0 − Then the impedance ZL is given by 0 jX1(Rx + jXc + jX2) ZL = Rx + (jXc + jX1 + jX2) ≡0 at resonance | {z } Niknejad Advanced IC's for Comm Reflected Losses It follows therefore that jXc + jX2 = jX1 at resonance and so − 0 jX1(Rx jX1) X1 ZL = − = (X1 + jRx ) Rx Rx Since the Q is extremely high, it's reasonable to assume that Xc Rx and thus X1 + X2 Rx , and if these reactances are on the same order of magnitude, then X1 Rx . Then 2 0 X1 ZL ≈ Rx This is the XTAL loss reflected to the output of the oscillator. Niknejad Advanced IC's for Comm Losses at Overtones Since X1 gets smaller for higher !, the shunt loss reflected to the output from the overtones gets smaller (more loading). The loop gain is therefore lower at the overtones compared to the fundamental in a Pierce oscillator. For a good design, we ensure that A` < 1 for all overtones so that only the dominate mode oscillates. Niknejad Advanced IC's for Comm XTAL Oscillator Design The design of a XTAL oscillator is very similar to a normal oscillator. Use the XTAL instead of an inductor and reflect all losses to the output. C1 A` = gmRL C2 0 RL = R RB ro Ljj jj jj · · · For the steady-state, simply use the fact that G R C1 = 1, or m L C2 Gm=gm = 1=A`. Niknejad Advanced IC's for Comm XTAL Oscillator Simulation For a second order system, the poles are placed on the circle of radius !0. Since the envelope of a small perturbation grows like σ1t v0(t) = Ke cos !0t Q 2 where σ1 = 1/τ and τ = . !0 A`−1 For example if A = 3, τ = Q .
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