Periodic Steady State analysis for class C amplifier

I am working on a simple Class C amplifier which is non-linear, therefore conventional AC analysis could not be used. Instead, Periodic Steady State analysis has to be used.

The context of this question is understanding PSS analysis throughoutly before applying it to analyse my class C amplifier.

For page 2 in Computer-Aided Circuit Analysis Tools for RFIC Simulation: Algorithms, Features, and Limitations , could anyone further explain the following statement ?

consider the simulation of high-Q oscillators. These also require very long transient simulations; a Q of 10 000 suggests that the turn-on transient time starting from a zero initial state will be of the order of 10 000 cycles of the oscillation period

Edit: For clarification purpose.

The second screenshot below shows that the turn-on transient time is about two microseconds.

Using the equation (8.56) given in the third screenshot below gives Qtot = 628.318530718

However from the paper discussed above, the author claimed that the turn-on transient time will be of the order of 'Qtot' cycles of the oscillation period (in this case, period is 1/15.9154943092Meg)

What I have from calculation is 31.8309886184 cycles (2us divided by 1/15.9154943092Meg).

Could anyone suggest why I am getting two different 'Qtot' values using different calculation methodology ?

Please correct me if I have mistake in any of the calculation steps.

In high Q oscillators, it is very difficult to change the amount of energy which is involved in the oscillation.

Take a simple low-Q RLC oscillator like this one:

In this type of circuit the Q of the LC tank (mainly L1 and C1) is not very high. Something like a Q of 100 should be achievable at RF frequencies. On a chip or for very high frequencies Q might be much lower like Q = 10.

Having a low-Q tank means that if there was no active circuit to keep the oscillation going, the oscillation would die out after only a few cycles (of the oscillation frequency). The task of the active circuit is to pump energy into the tank to keep the oscillation going.

You can easily simulate such circuits, also in a PSS simulation, because the oscillator is quick to respond (within a few cycles) to any changes.

Not let's look at a high-Q oscillator like this Crystal oscillator:

As the tank it uses a Crystal which can be modeled as a high-Q RLC tank like this:

A crystal behaves as a very low-loss (and therefore high-Q) RLC tank. This is modeled by having a model with a very small C1 (a few femto Farad) and a very large L1 (a few milli Henry). It is not uncommon for Crystals to have a Q factor in the order of 100000 (100k). Compare that to a factor Q = 100 which is already quite good for a circuit only using capacitors and inductors.

This makes for a very high Q tank and the result of this is that it takes many cycles for changes to take place regarding the signals in the tank. It is like the energy is "trapped" inside the tank (the energy alternates between C1 and L1) and only small portions of energy can be added or removed from the tank in each cycle.

The result is that it is very impractical to simulate very high-Q oscillators using a PSS simulation or even a transient simulation. Many cycles need to be simulated to simulate for example the startup behavior. For a 25 MHz crystal oscillator I worked on, the startup time was measured to be around 10 ms so that means that you would need to simulate more than 250000 (250k) cycles to simulate this.

What I do as a workaround is make a lower-Q version model of the Crystal with for a example a Q of 1000. Then I do my time simulations (transient or PSS) using this model. I can still use the high-Q model in an AC analysis to determine loopgain and frequency response though.