I don't think you understood what I said (meant to say) so I'll try again.
Imagine that you have a little black box that contains the function that creates the distortion. This box has an input and an output. If the contents of that box produces an output that depends only on the instantanious value of the input, that box must create balanced side bands.
The filtering of the sidebands will only happen in the high Q tuned circuit, assuming that we are trying to make a low drift oscillator. We can also assume on that basis that the design is done such that the effects of JFET parameters on tuning have been minimized. When this is the case, the sidebands will remain balanced.
When the upper and lower side band components are at 180 degrees to each other, you have phase modulation. When they are in phase, you have amplitude modulation. Any pair of side bands can be broken down into the amplitude modulation component and the phase modulation component. Then you can disregard the amplitude part.
Actually, the 1/F^3 rise continues into the part near the carrier where the tuned circuit curve flattens. For that matter, there is often a sudden increase in the slope at that point to well above the cubed factor. A frequency drift will appear as a 1/F in the graph. A 1/F noise modulation of a capacitance will appear as a sqrt(1/F)
[...]The OP is talking about a lowish frequency oscillator. Even at high frequencies, the desire is to make the ratio higher. To make the ratio higher you make the impedance even lower. The rule still applies it just gets harder to follow.
Once again I think you've misunderstood or perhaps I've misunderstood you. Remember that I suggested that the amplifier section (FET) be one that has a much higher band width than needed. This and the low terminal impedance to to prevent the modulation of FET parameters from being a problem.
Huh? Do you mean non-linearities in the phase or nonlinearities in the more normal sense of the word.
If you mean in the more normal sense of the word then I disagree as stated above. Non-linearities that operate on the instantanious value always make equal sidebands.
This is a different case. The amplifier in an oscillator has its terminals connected to the frequency determining circuit. The measurement circuit does not.
If we take a perfectly impractical set of cases I think you will see:
Oscillator #1:
We use a simple 2 capacitor divider Voltage on inductor = 20V RMS Voltage from gate to source = 10VRMS
Oscillator #2:
We use the 3 capacitor divider Voltage on the inductor = 10 billion VRms Voltage from the gate to source = 10VRMS
Both have the same signal to noise at the gate of the FET but the second one has about 10^12 less ability for the FET to control the frequency.
Lets say 10fA/sqrt(Hz) and few 100K impedance.
10fA/sqrt(Hz) * 100K = 1nV/sqrt(Hz) so we are in the same range as the noise voltage of a low noise JFET. I have run into this fact in practical circuits.
No, the harmonics are in the noise voltage of the gate of the FET they do not pass through the tuned circuit before they hit the non-linearity in the FET.
It is fairly straight forward if you take a very simplified case:
Imagine we have an extremely non-linear amplifier. The amplifier is assumed to be noiseless and the noise is in a generator, added to the signal just before the amplifier. The input to output function of the amplifier can be represented by a series. Lets just take the first few terms:
Y = X + AX^2
Where: X is the input A is a constant Y is the output
Now we consider X as the sum of (S)ignal and (N)oise.
Y = (S + N) + A(S + N)^2
Y = S + N + AS^2 + 2ANS + AN^2
It is the 2NS that does the dirty work. It will mix noise near the second harmonic down to near the operating frequency. The more nonlinear things are, the bigger A will be and the more 2nd harmoic noise gets shifted down. The higher terms bring the higher frequencies down. As a result, the more nonlinear the circuit is the noisier it is.
Reference left for later look up.
[...]Actually the gain must be exactly one if the amplitude is constant.
The gain around the loop must be exactly one at the operating frequency. The amplifier's gain does not effect the bandwidth of the system unless we are taking the case of a poorly designed oscillator where the transistor controls the frequency.