Quartz tuning fork oscilators

[Bret Cannon]

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

 

[Ken Smith]

In article <10uofh8ki12qv21@corp.supernews.com>,
Tim Wescott <tim@wescottnospamdesign.com> wrote:
[.. modeling crystal as a LC ..]

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.

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

--
--
kensmith@rahul.net forging knowledge

 

[Larry Brasfield]

"Ken Smith" <kensmith@green.rahul.net> wrote in message news:cshl2k$ht0$7@blue.rahul.net...

In article <10uofh8ki12qv21@corp.supernews.com>,
Tim Wescott <tim@wescottnospamdesign.com> wrote:
[.. modeling crystal as a LC ..]
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.

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

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?

--
--Larry Brasfield
email: donotspam_larry_brasfield@hotmail.com
Above views may belong only to me.

 

[Ken Smith]

In article <P5_Gd.36$G33.2573@news.uswest.net>,
Larry Brasfield <donotspam_larry_brasfield@hotmail.com> wrote:
[...]

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?

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.



--
--
kensmith@rahul.net forging knowledge

 

[Bret Cannon]

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 Cannon

"Tim Wescott" <tim@wescottnospamdesign.com> wrote in message
news:10uofh8ki12qv21@corp.supernews.com...

Bret Cannon wrote:

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
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?

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com