How is it that the impact of vibration on phase noise is defined in ppb/g?

In the world of oscillators specifications get passed around as a form of convenient shorthand and we get so accustomed to the terms used to describe the specifications that we often forget how they got that way.

In an earlier post we talked about contributions to jitter, and one of the topics we touched upon was vibration sensitivity.

In response to this post we were recently asked why an oscillator’s response to vibration, which is often characterized as an increase in the oscillators Phase Noise, should be measured in units of ppb/g?

It was shown a long time ago (since at least the 1960’s) that a quartz oscillator undergoes acceleration the resonant frequency of the oscillator shifts. The amount of frequency change is proportional to the acceleration and also depends on the direction of the acceleration:

$latex f(\vec{a})=f_{0}(1+\vec{\Gamma}\cdot\vec{a})&s=1$

This is where the ‘gamma’ that showed up in our earlier post starts out. It defines a sensitivity to acceleration and allows us to see how much an oscillators frequency changes under acceleration. Rearranging the equation a little it would be possible to see that gamma would be in units along the lines of ‘change in frequency’/’acceleration’. In the oscillator world ‘change in frequency’ is measured in fractional frequency offset (a ratio usually expressed in parts per million, ppm, or parts per billion, ppb)  and it is often convenient to express acceleration in units of ‘g’s (g = acceleration due to gravity, or 9.8m/s^2).

So we get a measure of sensitivity, called gamma, that is expressed in ppb/g. That’s it, right?

Not quite. So far we have talked about acceleration – what does this have to do with vibration and phase noise?

Well, vibration is acceleration that changes over time. So vibration makes the frequency change as well – just in a way that varies over time. And when we talk about frequencies being shifted by varying amounts over time, then we get into the territory of discussing frequency modulation. And once we do that we can look in the frequency domain and talk about phase noise due to vibration.

That, in very rough outline, is the conceptual link between acceleration sensitivity (in ppb/g), vibration (which can be thought of as acceleration that changes over time) and phase noise (which is a tool to talk about frequency changes over time). We can look at the phase noise over vibration equation and get a sense for what business gamma (typically in units of ppb/g) has doing in an equation for single side band PM noise, L(F), (in units of dBc for Sine vibration and dBc/Hz for random vibration).

$latex L(f_{\nu})=20log(\frac{\vec{\Gamma}\cdot\vec{a}}{2f_\nu}f_0)&s=1$

There is some fairly interesting physics underpinning this, but the mathematics is much more involved than we want to get into here and is better explained by some of the original publications on this subject:

http://www.ieee-uffc.org/frequency-control/learning/filler_paper.html

Supplement this with:

https://www.vectron.com/products/g_sensitivity/Vig-tutorial%20on%20g-sensitivity.pdf

http://tf.nist.gov/general/pdf/2328.pdf

http://tf.boulder.nist.gov/general/pdf/906.pdf