Hyvonen
Junior Member
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Posts: 15
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I wrote a relatively long response to this, but unfortunately I lost it due to an unfortunate copy/paste incident. :-/ So, I'll here's a short version - please ask if you need details.
In any LC-based oscillator, the losses in the tank are caused by nonidealities such as parasitic resistance. These are modeled with tank Q - the higher the better. This tank Q also represents the "tank resistance" Rp sapphire mentioned; a quick approximation would be Rp=Qtank*Ztank, where Ztank=2*pi*fres*L=1/(2*pi*fres*C) and fres=1/(2*pi*L*C).
To achieve and sustain oscillation a method of cancelling or compensating for the tank losses is needed - that's what the negative resistance is for. Assuming it is sufficient to overcome the losses, the oscillation amplitude will increase from cycle to cycle. This would be similar to an RC discharge, where the voltage slowly discharges, and the voltage could be calculated at any time point if we knew the RC time constant and the initial value, only that this time the value of R would be negative.
And this is where it gets tricky; to know how the amplitude changes over time, we need the time constant, the oscillation frequency etc. But to know the exact value of the amplitude, we would also need to know the initial value. In practice, the "initial value" depends on the amount of noise in the circuit, and that is somewhat hard to analyze - because of this, there is no simple answer to your question. However, the trends are definitely there - the higher your tank Q is, the faster the amplitude will ramp up, and the "more effective" negative resistance (e.g., the mode current you burn in your active transistor), the faster the amplitude will ramp up.
Overall, the only way to make the amplitude predictable is if you could have a well-defined initial value for the circuit at start-up (e.g., having an uneven initial bias voltages for a differential negative-R stage in a differential oscillator) - you could use that as the starting point for the calculation instead of a much less reliable estimation of noise in the circuit.
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