What is the effect of an amplifier on an LC tank?
An amplifier might be considered a voltage amplifier, whose input voltage provides an output voltage scaled by a unit-less &quot;gain&quot;. In that case, &quot;C&quot; at amplifier output does nothing, and can be eliminated.
Such an oscillator topology is not useful - at the resonant frequency of the remaining L &amp; C, amplifier feedback has the wrong phase.
However, a transconductance amplifier does provide a useful tool to examine oscillation. Now the &quot;C1&quot; capacitor is relevant and must be included in the equivalent circuit...current from amplifier output can produce a voltage where it meets &quot;L1&quot;. The resonator now consists of two &quot;C&quot; and one &quot;L&quot;. This kind of useful oscillator is ubiquitous - often called &quot;Pierce&quot;.

 I have added a resistor to illustrate resonator losses, which are compensated by gain of the transconductance amplifier. If you change amplifier gain from -0.001 to some other value, you can achieve either damped oscillation or continually-growing oscillation.