How to prove this problem?

[firsttimedesigning]

The picture shows an one-port oscillator. I am having trouble proving that when Rp is bigger than 2/gm, the magnitude of the loop gain will be bigger than 1. Basically, I am trying to see if the one-port oscillator fit into barkhausen criterion. I can prove the phase of this circuit is 0 but the magnitude is what kills me...

 

[saro_k_82]

You can try figuring out the differential impedance by looking from the resistor (just the active circuit). This will give you -2/gm. The proof of this is very simple and popular. You can find it in any RF book. It is easy to simulate and verify.
When the tank's equivalent impedance at resonance is greater than 2/gm, you circuit can sustain oscillations.

 

[LvW]

The picture shows an one-port oscillator. I am having trouble proving that when Rp is bigger than 2/gm, the magnitude of the loop gain will be bigger than 1. Basically, I am trying to see if the one-port oscillator fit into barkhausen criterion. I can prove the phase of this circuit is 0 but the magnitude is what kills me...

To analyze a one-port oscillator you cannot apply the Barkhausen criterion since it only can be used for two-port circuits. This criterion needs an amplifier and a frequency determining circuitry - both connected in a closed loop.
Then, the gain of the open lop is analyzed.
However - if you want - there are methods formally to transfer the one-port topology into a two-port topology.

 

[butterfish]

You can try figuring out the differential impedance by looking from the resistor (just the active circuit). This will give you -2/gm. The proof of this is very simple and popular. You can find it in any RF book. It is easy to simulate and verify.
When the tank's equivalent impedance at resonance is greater than 2/gm, you circuit can sustain oscillations.

agree with saro_k_82.
assume NR is the ac resistance between M1's gate and ac ground, and
if NR//Rp is negtive, the oscillation can be hold on.