Basic question about RLC circuits

[Baluncore]

Do you have an idea (or can you point me towards some readings) about how well a properly implemented feedback loop can keep the frequency of the setup stable (i.e. what changes in the resonant frequency should I expect in a given amount of time)?

The operating frequency of the resonator, would be as stable as the signal generator crystal, say 1:10⁵, (which could be GPS locked if needed to 1:10¹²). The frequency would always be correct, only the phase could change, because the resonator is driven by the signal generator through the PA.

If the peak of the resonator moved away from the signal generator frequency, the resonator continues at the signal generator frequency. The thing that changes with the tuning of the resonator, is the amplitude of the resonance. For a low-Q resonator that is not a problem, but for a high-Q resonator it could significantly reduce the amplitude of the voltage. By measuring the resonator phase shift deviation, the PLL would pull the resonator back onto the reference frequency, restoring the amplitude, with zero phase error.

When the PA is loosely coupled to the resonator, there will be a phase shift across the resistive coupling network. That occurs because the reactance of the resonator is no longer zero at the operating frequency. The task of the PLL is to recognise a non-zero phase shift, and to pull the resonator back to zero reactance.

 

[Malamala]

The operating frequency of the resonator, would be as stable as the signal generator crystal, say 1:10⁵, (which could be GPS locked if needed to 1:10¹²). The frequency would always be correct, only the phase could change, because the resonator is driven by the signal generator through the PA.

If the peak of the resonator moved away from the signal generator frequency, the resonator continues at the signal generator frequency. The thing that changes with the tuning of the resonator, is the amplitude of the resonance. For a low-Q resonator that is not a problem, but for a high-Q resonator it could significantly reduce the amplitude of the voltage. By measuring the resonator phase shift deviation, the PLL would pull the resonator back onto the reference frequency, restoring the amplitude, with zero phase error.

When the PA is loosely coupled to the resonator, there will be a phase shift across the resistive coupling network. That occurs because the reactance of the resonator is no longer zero at the operating frequency. The task of the PLL is to recognise a non-zero phase shift, and to pull the resonator back to zero reactance.

I am sorry for the confusion. What I meant to ask is how much does the resonant frequency of the resonator circuit is expected to change (for a well done feedback loop)? Of course the actual frequency will be the driving one, but I would like to know how much should I expect the Q-factor at that driving frequency to change. For reference, I need the amplitude of the electric field in between the parallel plates to be as stable as possible, thus I need the change in the difference between the resonant frequency of the circuit and the driving frequency to be minimized (and this difference should ideally be zero).

For example, for a PDH lock of a laser to a cavity, you can lock the laser frequency to 1/1000 of the cavity linewidth. I am looking for a similar estimate here, when locking the circuit itself to the fixed driving frequency. Basically, for a driving frequency of 35 MHz, should I expect a variation of the circuit resonant frequency (under the feedback loop) on the order of tens of kHz? Can I go lower than that?

 

[Baluncore]

What I meant to ask is how much does the resonant frequency of the resonator circuit is expected to change (for a well done feedback loop)?

The physical stability of the resonator's self-resonant-frequency, is decided by materials, construction, and environment.

The aim is to operate the resonator on the flat top of the Q curve. A PLL is able to bring the self-resonant-frequency, to the driven operating frequency, and to lock it there. By monitoring the PLL output voltage, you can confirm that the resonator has been phase locked to the drive signal.

Of course the actual frequency will be the driving one, but I would like to know how much should I expect the Q-factor at that driving frequency to change.

To minimise the variation in Q, over the range of PLL regulated operation, identify if it is the inductor or the capacitor that is most significant in causing the deviation. Then use the PLL output to correct that component.