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Tutorial on Frequency Control and Timing Application of Quartz Crystal Resonators and Oscillators (Part 4)

Oct 25,2024
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Causes of Short Term Instabilities

l Johnson noise (thermally induced charge fluctuations, i.e., "thermal emf” in resistive elements)

l Phonon scattering by defects & quantum fluctuations (related to Q)

l Noise due to oscillator circuitry (active and passive components)

l Temperature fluctuations - thermal transient effects activity dips at oven set point

l Random vibration

l Fluctuations in the number of adsorbed molecules

l Stress relief, fluctuations at interfaces (quartz, electrode, mount, bond)

l Shot noise in atomic frequency standards

l ? ? ?

Allan Deviation

Also called two - sample deviation ,or square-root of the " Allan variance ," it is the standard method of describing the short term stability of oscillators in the time domain. It is denoted by σy(τ ),

where

The fractional frequencies, are measured over a time

interval, τ; (yk+1 - yk) are the differences between pairs of

successive measurements of y, and, ideally, < > denotes a time average of an infinite number of (y k+1 - y k)2 . A good estimate can be obtained by a limited number, m, of measurements

(m≥100). σy(τ ) generally denotes · , i.e.,

Why σy(τ)?

Classical variance:

diverges for some commonly observed noise

processes, such as random walk, i.e., the variance increases with increasing number of data points.

l Allan variance:

• Converges for all noise processes observed in precision oscillators.

• Has straightforward relationship to power law spectral density types.

• Is easy to compute.

• Is faster and more accurate in estimating noise processes than the Fast Fourier Transform.

Frequency Noise and σy(τ )

Time Domain Stability

*For σy(τ ) to be a proper measure of random frequency fluctuations, aging must be properly subtracted from the data at long τ’s.

Power Law Dependence of σy(τ )

Below the flicker of frequency noise (i.e., the “flicker floor”) region, crystal oscillators typically show τ-1 (white phase noise) dependence. Atomic

standards show τ-1/2 (white frequency noise) dependence down to about the

servo-loop time constant, and τ-1 dependence at less than that time constant.

Typical τ’s at the start of flicker floors are: 1s for a crystal oscillator, 103s for a

Rb standard and 105s for a Cs standard. At large τ’s, random walk of frequency and aging dominate.

Pictures of Noise

Plots show fluctuations of a quantity z(t), which can be,e.g., the output of a counter (Δf vs. t) or of a phase detector (φ[t] vs. t). The plots show simulated time-domain behaviors

corresponding to the most common (power-law) spectral densities; h α is an amplitude

coefficient. Note: since S Δf = f 2S φ , e.g. white frequency noise and random walk of phase are equivalent.

Spectral Densities

In the frequency domain, due to the phase deviation, φ(t), some of

the power is at frequencies other than ν0 . The stabilities are

characterized by "spectral densities." The spectral density, SV(f), the

mean-square voltage <V2(t)> in a unit bandwidth centered at f, is not a good measure of frequency stability because both ε (t) and φ(t) contribute to it, and because it is not uniquely related to frequency fluctuations

(although ε (t) is often negligible in precision frequency sources.)

The spectral densities of phase and fractional-frequency fluctuations, Sφ(f) and Sy(f), respectively, are used to measure the stabilities in the

frequency domain. The spectral density Sg(f) of a quantity g(t) is the mean square value of g(t) in a unit bandwidth centered at f. Moreover,

the RMS value of g2 in bandwidth BW is given by

Mixer Functions

Phase Detector

Phase Noise Measurement

Frequency - Phase - Time Relationships

Types of Phase Noise

Noise in Crystal Oscillators

l Frequency multiplication by N increases the phase noise by N2 (i.e., by 20log N, in dB's).

l Vibration-induced "noise" dominates all other sources of noise in many applications (see acceleration effects section, later).

l Close to the carrier (within BW of resonator), Sy(f) varies as 1/f, Sφ(f) as 1/f3 , where f = offset from carrier frequency, ν . Sφ(f) also varies as 1/Q4 , where Q = unloaded Q. Since

l Qmaxν = const., Sφ(f) ∝ ν4 . (Qmaxν )BAW = 1.6 x 1013 Hz; (Qmaxν )SAW = 1.05 x 1013 Hz.

regions where σy(τ) varies as τ-1 and τ-1/2 (τ-1/2 occurs in atomic frequency standards),

σy(τ) ∝ (QSR)- 1 , where SR is the signal-to-noise ratio; i.e., the higher the Q and the signal- to-noise ratio, the better the short term stability (and the phase noise far from the carrier, in the frequency domain).

l It is the loaded Q of the resonator that affects the noise when the oscillator sustaining circuitry is a significant noise source.

Noise floor is limited by Johnson noise; noise power, kT = -174 dBm/Hz at 290。K.

l Higher signal level improves the noise floor but not the close-in noise. (In fact, high drive levels generally degrade the close-in noise, for reasons that are not fully understood.)

Low-Noise SAW and BAW Multiplied to 10 GHz

(in a nonvibrating environment)

Low-Noise SAW and BAW Multiplied to 10 GHz

(in a vibrating environment)

TCXO Noise

The short term stabilities of TCXOs are temperature (T) dependent, and are generally worse than those of OCXOs, for the following reasons:

• The slope of the TCXO crystal’s frequency (f) vs. T varies with T. For example, the f vs. T slope may be near zero at ~20oC, but it will be ~1ppm/oC at the T extremes. T

fluctuations will cause small f fluctuations at laboratory ambient T’s, so the stability can be good there, but millidegree fluctuations will cause ~10-9 f fluctuations at the T extremes. The TCXO’s f vs. T slopes also vary with T; the zeros and maxima can be at any T, and the

maximum slopes can be on the order of 1 ppm/oC.

• AT-cut crystals’ thermal transient sensitivity makes the effects ofT fluctuations depend not only on the T but also on the rate of change of T (whereas the SC-cut crystals

typically used in precision OCXOs are insensitive to thermal transients). Under changing T conditions, the T gradient between the T sensor (thermistor) and the crystal will aggravate the problems.

• TCXOs typically use fundamental mode AT-cut crystals which have lower Q and larger C1 than the crystals typically used in OCXOs. The lower Q makes the crystals inherently noisier, and the larger C1 makes the oscillators more susceptible to circuitry noise.

• AT-cut crystals’ f vs. T often exhibit activity dips (see “Activity Dips” later in this

chapter). At the T’s where the dips occur, the f vs. T slope can be very high, so the noise due to T fluctuations will also be very high, e.g., 100x degradation of σy(τ) and 30 dB degradation of phase noise are possible. Activity dips can occur at any T.

Quartz Wristwatch Accuracy vs. Temperature

Frequency vs. Temperature Characteristics

Resonator f vs. T Determining Factors

Primary: Angles of cut

l Secondary:

• Overtone

• Blank geometry (contour, dimensional ratios)

• Material impurities and strains

• Mounting & bonding stresses (magnitude and direction)

• Electrodes (size, shape, thickness, density, stress)

• Drive level

• Interfering modes

• Load reactance (value & temperature coefficient)

• Temperature rate of change

• Thermal history

• Ionizing radiation

Frequency-Temperature vs. Angle-of-Cut, AT-cut

Desired f vs. T of SC-cut Resonator for OCXO Applications

OCXO Oven’s Effect on Stability

A comparative table for AT and other non-thermal-transient compensated cuts of oscillators would not be meaningful because the dynamic f vs. T effects would generally dominate the static f vs. T effects.

Oven Stability Limits

• Thermal gains of 105 has been achieved with a feed-forward compensation technique (i.e., measure outside T of case & adjust setpoint of the thermistor to anticipate and

compensate), and with double ovens. For example, with a 105 gain, if outside ΔT = 100oC, inside ΔT = 1 mK.

• Stability of a good amplifier ~ 1μK/K

• Stability of thermistors ~1mK/year to 100mK/year

• Noise < 1μK (Johnson noise in thermistor + amplifier noise + shot noise in the bridge current)

• Quantum limit of temperature fluctuations ~ 1nK

• Optimum oven design can provide very high f vs. T stability

Warmup of AT- and SC-cut Resonators

TCXO Thermal Hysteresis

Apparent Hysteresis

OCXO Retrace

In (a), the oscillator was kept on continuously while the oven was cycled off and on. In (b), the oven was kept on continuously while the oscillator was cycled off and on.

TCXO Trim Effect

In TCXO ’s,temperature sensitive reactances are used to compensate for f vs. T

variations. A variable reactance is also used to compensate for TCXO aging. The effect of the adjustment for aging on f vs. T stability is the “trim effect” . Curves show f vs. T

stability of a “0.5 ppm TCXO,” at zero trim and at ±6 ppm trim. (Curves have been vertically displaced for clarity.)

Why the Trim Effect?

Effects of Load Capacitance on f vs. T

Effects of Harmonics on f vs. T

Amplitude - Frequency Effect

At high drive levels, resonance curves become asymmetric due to the nonlinearities of quartz.

Frequency vs. Drive Level

Drive Level vs. Resistance

Second Level of Drive Effect

Activity Dips

Activity dips in the f vs. T and R vs. T when operated with and without load

capacitors. Dip temperatures are a function of C L, which indicates that the dip is caused by a mode (probably flexure) with a large negative temperature coefficient.

Frequency Jumps