No disrespect intended, but I think you're blowing smoke.
Well, it cant be that because i quit smoking years ago
The swing may have been a bad example however.
I think what you are after is sustained oscillations, and i too am a little
puzzled about the results i am seeing. I would like to get the equation
into a form which allows me to look into this a little further but i
need to take an inverse Laplace transform of a very big equation and
i dont feel like doing it by hand.
I tried this with the original circuit values by using a CCVS (H) as the amplifier (with and without a diode limiter in the loop), but the DC feedback (see edit at the end of this post) always made the output run off to teravolts. A cap in series added a third frequency where the phase passed through zero, so it wanted to oscillate at that frequency. As a test, I removed L2, making the network a simple damped series LC circuit. It still has a DC path, but I was able to get oscillations at the zero phase shift frequency:
Fres=90.394MHz (1/(2pi√LC))
Fpeak=90.356MHz (shift from Fres due to 139Ω damping resistor)
FΦ=88.4873MHz (zero phase shift)
Fosc=88.4877MHz (period measurement)
Fosc=88.4722MHz (FFT)
Ok, when we say 'ideal' amp we mean one with a perfect gain constant
and infinite input impedance, which also brings up a problem with the
network because the coil L1 can not work into an infinite impedance,
so i tried various resistors of 1meg down to 10k and these provided
some impedance for L1 to work into. I got resonance even with L1
equal to 150nH, but i can not yet explain why this worked. BTW the
10k connects to a voltage source of 1v to kick start the oscillator.
Without a kick it wont start because everything is zero at first. In
the real world a little white noise would start up a normal osc like
this.
I think i can add a little more confusion by saying that we might
get unsustained oscillation with other values too, but i dont think
that was the point you were trying to make, and i think your
point is valid. I also think that w0 has significance that we shouldnt
overlook and that goes without saying even if a circuit oscillates or
not because there is a lot of literature which will resort to this
type of reasoning too. Maybe your circuit proves that there is
a very good physical significance to this w0, but what you would
need to do is to define your circuit a bit better...after all a perfect
feedback amp has infinite input impedance and so no network will work
into this input correctly. Using a CCVS
may not be appropriate because we never wanted to ground one
side of the network, which is the same as loading it with 0 ohms.
I thought we wanted to place this in parallel with a regular voltage
amplifier.
I had tried ringing the circuit's bell with a fast step also, but I could not get enough consistency in time domain period measurements or FFTs to be certain that the ring frequency was different from both the others.
It's not easy to measure, but perhaps you can try some other values where the |Z| max is
more different than the Im(Z)=0 frequency. What i do is run a simulation with a very short
pulse that goes positive, then at the end of the damped oscillations i inject a very short
pulse that goes negative, then right after that another pulse that goes positive again, then
repeat for several cycles. i then zoom in on one of the single cycles being very careful
to observe where the zero crossing is so as to measure the time between zero crossings.
EDIT: The "oscillator" went to teravolts on the original circuit because the phase shift and impedance of the network are both zero at DC.
Ok, mine broke into oscillation (150nH) but as i said i cant yet explain
exactly why this worked. If you want to try to explore this, try
a voltage gain of 1 and an input resistor of 10k.
Note that without any resistor a voltage on the right side of the
network produces exactly the same voltage on the left side of
the network with no phase shift or other alteration no matter what
the values of the components are because the cap is a short and
the two inductors dont have any difference in voltage across them.
I'd call on Barkie but this network seems to oscillate so
Conclusions so far:
1. The |Z| max frequencies seem very important because the amplitude is always important.
2. The Im(Z)=0 frequencies seem more historical than anything else, so far.
We do have this one circuit with the amp with gain=1 but we cant decide on
how exactly the circuit is constructed (yet). We need to nail this down before
we can proceed, but we also have to keep in mind that classical circuit theory
is going to have a lot of this in it. We also have to keep in mind that it is going
to cause confusion because there is a lot of theory that confuses this with |Z| max
and i believe this is because the resistance R is relatively small in many circuits so that
these two frequencies are very close to each other.
3. The ring frequency FreeOsc(Z) seems to be interesting and have physical significance
when the network is driven by a pulse such as in a converter circuit.
I guess there is always the possibility that we are getting too far off the track here,
and we should let the end user decide what way they want to analyze their circuits
But then again, do we have a definitive answer when someone asks: "What is the
resonant frequency of my network?" ?