Quartz tuning fork oscilators

Certainly the LC product (frequency), and the R * sqrt(C/L) product (Q) will depend heavily on the tuning fork properties. I suspect that the magnitude of the motational inductance and capacitance, however, will not only depend on the mechanical properties of the resonator, but also the way that you're coupling it to the electric circuit.

You've done the obligatory web search?

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Tim Wescott
Wescott Design Services
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Tim Wescott
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What significant non-linearities do you believe are operative in real crystals in normal oscillator applications? Did you perhaps mean that the lumped model (as opposed to a distributed model) is inaccurate?

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--Larry Brasfield
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Larry Brasfield

Quartz tuning forks, such as those used in electronic watches etc., can be modeled mechanically as a damped, driven harmonic oscillator and electrically as a series RLC driven by an external voltage. The differential equations corresponding to these two models are of the same form with correspondances L Mass, R frictional loss coefficient, and 1/C spring constant. If the dimensions of the tines of the tuning fork are changed, the mass, the spring constant and hence the resonant frequency can be calculated easily. My question is whether the correspondance is strong enough that the changes in mass and spring constant can be used to calculate the changes in the motional inductance, L and the motional capacitance, C. For example, if the lengths of the tines of a tuning fork are doubled, the mass is doubled and the spring constant is reduced by 8, and the resonance frequency is reduced by a factor of 4. Have L and C increased by the corresponding factors of 2 and 8?

Any help including suggested references appreciated. I am trying to understand how changing the size of a quartz tuning fork would affect the performance for sensing forces such as in uses for atomic force microscopy.

thanks, Bret Cannon

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Bret Cannon

In article , Tim Wescott wrote: [.. modeling crystal as a LC ..]

Also: None of the values for the real crystal are going to be nice and linear like the LC model would suggest.

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kensmith@rahul.net   forging knowledge
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Ken Smith

In article , Larry Brasfield wrote: [...]

No, I really mean non-linear.

At very low drive levels, the Q of the crystal is lower than at normal drive levels. As the drive level increases the resonant frequency of an AT crystal increases. The OP had a tuning fork in his question. I don't off hand know if the tuning forks also increase but I'd expect them to.

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kensmith@rahul.net   forging knowledge
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Ken Smith

I have done several web searches and some searches on Web of Science. I haven't found much on the web, not even data where I can match the sizes of tuning forks and their motional capacitance.

I have found papers in the journal Vacuum where finite element analysis has been used to do studies on the sensitivity of frequency, stray capacitance and the resistance of a tuning fork to fabrication tolerances. Those papers don't go into the scaling with geometry except for the frequency nor deal with motional inductance or capacitance. The frequency of a tuning fork is well modeled as the vibration of a cantilevered beam and I have papers that do more sophisticated treatments of the flexture at the base of the tuning fork, but that work is all dealing with the frequency. There are also some interesting papers discussing the increase of R with pressure that date fom the 1980's where tuning forks were explored as pressure gauges accurate to about 10% between 1E-5 to 1 atmosphere.

As for non-linearity, I'm interested in the case when the motion of the tips of the tuning fork is a few nanometers, the current flow into a transimpedance amp is a picoamp or so, and the excitation is at the resonance frequency of the tuning fork, which is the resonance frequency of the series RLC. This is well below where the drive is large enough to cause frequency shifts larger than a fraction of 1 Hz out of 32768 Hz.

Bret Cann>

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Bret Cannon

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