The acceleration sensitivity of an oscillator that will be used in a high-vibration environment would typically be characterized by applying a known vibration level (time-dependent acceleration) to the oscillator and then observing the effect of the vibration on its output spectrum. When the applied vibration is a single-frequency sine wave, a discrete spurious product will be produced offset from the carrier by the vibration frequency. By measuring the level of this spurious signal relative to the power of the carrier, the amount of frequency deviation may be determined from FM theory. Normalizing by the magnitude of the vibration will then give the amount of frequency shift per g of acceleration in the axis of vibration. This is the definition of acceleration sensitivity or Γ. For a crystal oscillator where the fractional frequency shift is very small, Γ is usually expressed in parts per billion per g (1 ppb/g = 1 × 10-9/g). The level of the sideband that will be produced when the excitation is a single-frequency sine wave is closely approximated by Eq. 2:

SB level (dBc) = 20log[(a × Γ × fnom)/2fv]   (2)

or, solving for Γ gives:

Γ = (2fv/a × fnom × 10dBc/20   (3)

where a is the peak magnitude of the vibration force in the direction of Γ being measured, fv is the frequency of the sine wave vibration, and fnom is the nominal carrier frequency at rest. Figure 2 shows the spectrum of a 20-MHz crystal being vibrated with a peak acceleration of 10 g’s at 90 Hz.  As can be seen from the difference between the two markers, the vibration-induced sideband is 55.2 dB below the carrier. According to Eq. 3, the magnitude of Γ of this crystal in the axis being vibrated is therefore 1.56 ppb/g.

To determine the exact magnitude and the directional orientation of the vector, it is necessary to measure the response along three mutually perpendicular axes, usually selected to be normal to the faces of the oscillator package for use of alignment. But the Γ vector is typically not aligned exactly with any of the faces of the package; consequently, the data must then be analyzed to determine the orientation of the vector relative to the established reference plane. (Further evaluation is required to determine the polarity of the vector.) Referring to Fig. 3, the parameters of the vector relative to the established axes can be calculated as:

|Γ| = (Γx2 + Γy2 + Γz2)0.5   (4)

|Γ|= (Γx2 + Γy2)0.5   (5)

Φ = cos-1xxy   (6)

θ = sin-1z/|Γ|)   (7)

Equation 4 gives the maximum sensitivity that will be exhibited when the force applied is exactly parallel to the direction of the vector. When specifying a crystal, this is sometimes referred to as Γmax or the total vector. This value will always be greater than the magnitude measured along one of the orthogonal axes of the oscillator unless the vector is exactly aligned with an axis. Once the magnitude and the angular orientation of Γ have been determined, it is then possible to predict the oscillator performance under other vibratory conditions.

If the oscillator in use will experience vibration at a single frequency, then the previous formulas will show the level of the sidebands that will be produced if the applied g level is known. However, in most real-world applications, the vibration energy will be spread across a frequency band, producing a random vibration profile. The shape of this profile will be highly application specific with most of the energy typically ranging from as low as 5 or 10 Hz through as high as 10 to 20 kHz. There may be peaks in the response where the energy is concentrated depending upon the source of the vibration and the mechanical mounting of the components.

This noise-like random energy is expressed as a power spectral density with units of g2/Hz. Rather than producing a discrete spurious signal product in the spectrum as with sine vibration, random vibration will cause the phase-noise floor of the oscillator to rise across the frequency band of vibration. The same basic calculation is then performed to predict the effect on the oscillator spectrum but the vibration energy is now expressed as g2/Hz, giving the following formula for the single-sideband phase noise of the oscillator under vibration:

L(fv) = 20log[|Γ| · fnom(2PSD)0.5/2fv]   (8)

where PSD = the power spectral density of the vibration at frequency fv.

Under random vibration, the entire noise floor of the oscillator will increase across the range of vibration frequencies. The acceleration sensitivity could then be determined by measuring the phase noise of the oscillator while vibrating.