Why does the length of an RG-316 coaxial cable affect the crystal resonator frequency response?

Question

I was measuring a 32 [kHz] crystal resonator using a ZURICH HF2LI with 50 ohm input and found that the length of the coaxial cable will affect the frequency response or have a little difference at some frequency range (green dashed circle) especially for phase response, no matter how many times I measured it.

In a nutshell, I believe it is the cable length that caused it but I have no clue where to start it.

My questions are

1. What causes it?
2. Is there a way to fix it?
3. Is it possible to simulate this effect?

The measurement setup and result are shown below

EDIT:2023/10/23

After reference from Qrk's, Antonio's and Andy's advices, I extracted the equavilant circuit of crystal resonator and add a capacitor "c_medium" about 20 pF (0.02 cm * 95 pF/m) parallel with the input resistor of instrument.

However, the frequency response didnt change before and after i added "c_medium" in the circuit even I change 20 pF to 1000 pF the frequency response remain the same, the LENGTH effect of RG-316 seems can't be represented by just adding a capacitor.

The result is shown below, blue line and red line are measurement result using 10 cm and 20 cm RG-316 cable respectively and the green dotted line is the simulation result with the circuit used ("c_cable" = 20 pF)in below figure.

Answer 1

What cause it?

The cable for sure. Specifically the cable capacitance. At 32 kHz it will be the dominant impedance. At DC the series resistance will dominate and, above 100 kHz to 1 MHz it will be both inductance and capacitance that share dominance (producing a resistive characteristic impedance): -

Image from here. The picture is a graph of typical telephone cable based on it's lumped component specifications. Coaxial cable will be somewhat different but generally, it will follow the same curve. You can find all the values for the cable you used by reading its data sheet.

Is there a way to fix it?

There's nothing that needs fixing. I mean we don't ever use a 32.768 kHz crystal connected this way AND, the important area of the characteristic is here: -

Which appears to be unaffected by the cable. Well, maybe marginally affected. So, nothing to fix.

Is it possible to simulate this effect?

It sure is. If you know the equivalent circuit for the crystal you can do a simulation just like what I did here for a 10 MHz series resonant crystal: -

Image from my basic website (crystals).

Answer 2

First thing to do is draw a diagram which will help you understand what is happening.

^{simulate this circuit – Schematic created using CircuitLab}

At these low frequencies the coax will act like a capacitor (\$C_{cable}\$). Different lengths of cable will have different values of \$C_{cable}\$ which will affect your measurement results.

In all, this test doesn't show much, but with some software you should be able to convert this in to an impedance measurement. Normally software of this sort will have the ability to null the "fixture" parasitics where a resistor of a known value (often 50 ohms) is put in place of the device under test, namely the crystal. Did this instrument come with such software?

Answer 3

The cable is a part of the test fixture. The VNA should let you calibrate the effects of test fixture, so that they can be removed from the measurement (to the degree that is possible).