AC analysis of a oscillator

This question relates to my previous question.

I am interested in deriving an analytical relationship for the oscillation frequency of this oscillator. As the first step, I have initiated the frequency domain simulations and results are as follows. In order to see the effect of the oscillation frequency, I sweep the value of C2 from 0.5pF to 6pF (oscillation sustains within this range).

The AC model is: (Mosfet capacitances are extracted from the datasheet) The phase and magnitudes of voltages at secondary inductor and Gate voltage are as follows. I have marked resonances close to the oscillation frequency. simulations is done for different secondary capacitor values 0.5pF,and 5PF. In both the cases three resonances can be seen (say f1, f2, and f3)

When C2=5pF (oscillation frequency in time domain simulation is 8.3MHz) When C2=0.5pF (oscillation frequency in time domain simulation is 25.1MHz) From these results, it can be seen that the resonances in AC simulation and the oscillation frequencies are significantly different, particularly for high frequencies.

My questions are:

1. Am I moving in the right directions with the simulated AC model?
2. What could be the reasons for the disparities between AC model and time domain simulations?
3. I understand that mosfet capacitances can be varying with frequency and voltages. In addition, certain capacitances may not effective throughout the oscillation period (when the switch is fully on). How can I model the mosfet capacitances in this circuit?
4. As I am biasing the transistor just enough to start the oscilation (Vt0=4.486V and V(bias)=4.26V), can we still assume mosfet provides a 180 degrees phase shift? How can I incorporate this in the AC simulation.

An Update

I found a similar analysis for a slightly different oscillator in "class-e mosfet tuned power oscillator design procedure". In the analysis, they have measured mosfet capasitances at the frequency of interest and phase shift introduced by the mosfet was known (phase shit was 196 degrees in their study). If I can approximate these parameters ((i.e., Cgs, Cgd, Cds, and phase shift introduced by the MOSFET) at each oscillation frequency, I should be able derive an analytical (rather a semi-analytical) form of solution.

Is there a simulation based method that I can approximate MOSFET capacitances and phase shift introduced by the amplifier stage?

Update 2

Trying to express the loop gain expression

First, I tried to simplify the coupled network for the fundamental frequency component as follows. Where reflected impedance are calculated as (Mij represents the mutual inductance between ith and jth inductor)

Now my goal is to write loop gain equations to obtain the oscillation condions But I am not sure how should I differentiate the amplifier and feedback network

• It might be worth the effort to replace the MOSFET in the transient analysis with a voltage controlled current source surrounded by the appropriate values of CGS and CGD and CDS. Then see what it oscillates at. Apr 24 '18 at 13:46
• AC simulation needs an ac source, right? Where is the source?
– LvW
Apr 24 '18 at 14:26
• Moreover, I think, for identification of the oscillating frequency in the frequency domain you must do an analysis of the LOOP GAIN response.
– LvW
Apr 24 '18 at 14:31
• You have so much data that this is hard to understand. I would strongly advise you to simplify and just show graphs of one value of capacitance the demonstrates the problem then explain (by drawing a "marker" on that same graph) of what the real oscillation frequency is. Apr 24 '18 at 14:37
• Pojj, be aware that a resonant point does not necessarily means that at this frequency an oscillation takes place. Such a point of resonance must fulfill the oscillation criterion. For this purpose, you need the loop gan analysis.
– LvW
Apr 24 '18 at 20:56

I'm not on my computer, but here's a quick setup of what LvW suggests: You can see more of this topology in LTspice's Examples directory (in My Documents/LTspiceXVII/examples/Educational, see LoopGain.asc and LoopGain2.asc, and the links in their descriptions).

• Thanks, it gives exact oscillation frequency where total phase become zero. (just that the polarity of L1 is other way then phase become zero at oscillation frequency). However, How can we deduce transistor characteristics (i.e., Cgs, Cgd, Cds, and phase shift introduced by the MOSFET) at different operating conditions from this? Here, I am interested in deriving an analytical solution. (or a semi-analytical form where numerical values of mosfet parameters are defined at each parametric sweep)
– Pojj
Apr 25 '18 at 6:58
• @Pojj You could explode the inner workings of the MOSFET model, but I don't think it's the way, rather, you could use an ako model of your MOSFET and .step some of its parameters, something like this: .model BSP89x ako:BSP89 Cgs={x}. Honestly, I think you're trying to over-simplify this, but, if it works for you, ... Apr 25 '18 at 7:37
• Thanks, I used your method to see the effect of MOSFET model parameters (Cgdmax, Cgdmin, Cgso, and Cjo) to the oscillation frequency. And I noticed that only Cgd affect the oscillation frequency significantly.
– Pojj
Apr 25 '18 at 12:57
• Rather, I would start with a very simple transistor model (ideal voltage controlled current source - perhaps with a finite output resistance. This is the first step with the aim to check if the rest (feedback factor, loop gain) is correct. Such an approach makes sense because the feedback path is not a simple one.
– LvW
Apr 25 '18 at 18:43
• @LvW I tried to simplify the feedback path, but not sure how the feedback factor and loop gain is defined with this representation.
– Pojj
Apr 26 '18 at 15:52