Derivation of time constant expression

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My hunch is that you wonder what frequency shall be found stamped on the crystal's package? Falstad's simulator supports a crystal modeled very much like your diagrams. A sample circuit is found in the menu by navigating to Circuits > Mosfets > Cmos Crystal Oscillator.

Falstad.com/circuit

Default frequency is 10MHz. Right-Click Edit on the crystal to bring up the edit fields. You can alter values and see what frequency results.

 

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I presume that you misunderstood the purpose of this thread.

Did you check out equation (13) of the paper mentioned in my original question ?
 

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Exponential decay respectively increase time constant is the outcome of harmonic oscillator transient solution. The author lazily "skips" respective equations.

It's electric engineering basic studies stuff. Continue post #1 "natural RLC circuit response" equations by deriving time domain solution.
 
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    promach

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Use LaPlace, stim your circuit with a step response sine, and do the algebra.


Knight
 

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The author mentions the critical transconductance to oscillation where the dV/dt =0 of the envelope. Yet no mention of how loop gain accelerates the envelope to saturation or the bias delay time to make the feedback amplifier operate in linear mode.

The feedback R and load C controls the inverter DC bias RC=tau value.
The series R controls the L/R envelope startup.
The gm or RdsOn of the inverter gate affects the gain and time constant of the output which is faster than the input with gain to reach max envelope first.

Which envelope rise time is significant for settling time? L/R with the external series R is the most significant factor and also is recommended to reduce Xtal power dissipation often rated at 50 uW.


sim
 
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