开云体育

I use this formula for the motional capacitance:

Cm = (f1 - f2)/(2*PI*f1*f2*Rt)

Where f1 and f2 are the upper and lower 3 dB points, so you could just say
BW for f1 - f2.

In the denominator, f1*f2 is not far from fc^2 where fc is the resonant
frequency of the crystal so you can use fc^2 if you like.

Rt is the total resistance of the circuit. So assuming you have a 50 ohm
generator plus 50 ohm detector and you measured the loss resistance of the
crystal at resonance at 55 ohms, your value is 155 ohms.

Having found Cm, Lm is the value of inductance that has the same reactance
as Cm at the resonant frequency. But a formula for that is:

Lm = 1/(4 * PI^2 *f^2 * Cm)

Where f is just the resonant frequency of the crystal.

You'll probably wind up with Cm in femtofarads and Lm in millihenries.

All of this depends of the resolution of the VNA. Being able to get down to
1 Hz resolution can help. I tried one crystal and didn't think I got good
results. But in the nanoVNA notes by Wes Hayward linked here recently, he
seemed to get some good crystal measurements. And Hayward is the guru of
crystal measurements although he wasn't deriving motional parameters in his
paper.

In doing crystal measurements, people often use transformers or resistive
pads to put the crystal in a 12.5 ohm environment, meaning the crystal sees
12.5 ohms looking in either direction. I'm not sure how advantageous this
is, but it does lower the BW and so would seem to make fine resolution even
more important.

For parallel capacitance, generally using a low frequency capacitance meter
like the AADE or eBay clones works.


73-

Nick Kennedy, WA5BDU