Hi guys,
I am trying to mathematically find out the frequency of oscillation of this Miller crystal oscillator circuit.

But I can't seem to get the right answer. I believe it has to do with the top part of the circuit. The frequency should be around 9MHz.
The configuration is from: TFM Verónica do Amaral Rosal.pdf (uvigo.es) at page 51.
Please help!
Thanks for reading my post.

 

I think you need to start with the motional parameters of the Crystal. They are appropriately named Lm and Cm. These two values tell you the series resonance frequency of the crystal.
\[ f_s\;=\;\frac{1}{2\pi\sqrt{L_mC_m}} \]
When I substitute the values which I can I can barely read, I get 9.7772 MHz. as the series resonant frequency.
EDIT: the value of Rm can be used in conjunction with the impedance of Lm at resonance to determine the Q.
Using a simulator to characterize a crystal is something you can do as well.
The subcircuit description of this particular crystal shows how the motional parameters for the model can be calculated.
I drew the schematic from the subcircuit to illustrate what is happening in modeling a crystal. The series resonant peak is on the left and the parallel resonance is on the left. Additional load capacitance is required to make the crystal operate at the parallel resonance.

 

I think you need to start with the motional parameters of the Crystal. They are appropriately named Lm and Cm. These two values tell you the series resonance frequency of the crystal.
\[ f_s\;=\;\frac{1}{2\pi\sqrt{L_mC_m}} \]
When I substitute the values which I can I can barely read, I get 9.7772 MHz. as the series resonant frequency.
EDIT: the value of Rm can be used in conjunction with the impedance of Lm at resonance to determine the Q.
Using a simulator to characterize a crystal is something you can do as well.

Thank you for your answer. I can't edit my post so I uploaded a better resolution version of the circuit here.

- I am aware that the circuit is using a 9MHz crystal that is why the value of Lm and Cm when substituting into your equation will give 9MHz.
- However, when I replaced Lm, Cm, Rm, Cp with new values which represent a 5MHz crystal, and ran this simulation in my PC, the result of the oscillation frequency of the circuit at Vout (pin 6 of the IC) was still around 9MHz.
- I tried to change the value of the L1 and C1, and this time, the oscillation of the circuit did change significantly.

 

Thanks for the better resolution schematic. Drawing the OPA660 as a square box is less than helpful in figuring out what is going on. If you change the parameters of the alleged crystal and the frequency does not change then I would conclude that crystal model has no effect on the operation of the circuit. What parameters did you use for the lower frequency crystal?

 

I used Lm=30mH, Cm=33fF, Rm=400Ω, and Cp=20pF to represent a 5MHz crystal.
I found a datasheet for the operational transducer amplifier OPA660 at this link: Burr-Brown_SPICE_Based_Macromodels (ti.com) on page 35.
I will post the schematic here as well. I believe you can click on the picture to view it better.

 

In fig 31 below, one leg of a fundimental cut crystal connects to a parallel resonant tank.
There are parallel and series type oscillators. The Colpitts oscillator tends to favor series resonance and Pierce tends to favor parallel.
The parallel cut crystal in a parallel variety crystal oscillator circuit will approach frequency tolerance error in ppm.
A precise frequency can be pulled by adjusting capacitance but the crystal's piezo effect reduces the symmetrical shape of the wave.
The resonance of the parallel tank in the miller oscillator along with the feed back can improve the wave shape.


https://www.industrial-electronics.com/crystal_osc_5.html
It also suggest that Miller design is rich in harmonic content. By filtering out the harmonics the fundamental remains as a symmetrical sine.