Due to the nonlinearity of \(C_{GS1}\) and \(C_{GS2}\) arising from large LO amplitudes, \(V_P\) does change with time but only at twice the LO frequency.
I don't quite understand where does this "twice" come from.
Then I check Problem 6.3.
To solve this problem I treat \(V_1\) and \(V_2\) one by one and then add the effects up (superposition). But I didn't take the nonlinearity of capacitance into account, I still used the linear relation: $$ I(s) = \frac{V(s)}{\frac{1}{sC}}$$ So I messed up with the results. To be precise, the relation between charge \(q(t)\), voltage that crosses the capacitor \(v(t)\) and capacitance \(C(v)\) still holds:
I try to build this circuit in LTSpice. Maybe the most difficult part is to learn how to model this nonlinear capacitor in SPICE. It turns out to be quite straightforward. Just write Cnnn n1 n2 Q=<expression> in the value section of the capacitor.
I have to tune a bit the parameters of the circuit before I can visualize the simulation results. Finally I got this schematic:
According to the SPICE simulation, if only first order effect of \(V\) on capacitance is taken into account, it results in the oscillation of \(V_{out}\) with a frequency \(\omega_{LO}\). Moreover, the oscillation is hardly seen.
Things get interesting when second order effect of \(V\) comes onto the stage. Observable oscillation with a frequency of \(2\omega_{LO}\) appears. I think this is due to
- Second order effect grows more rapidly.
- Second order effect mix \(\cos(\omega_{LO}t)\) to \(\cos(2\omega_{LO}t)\).
There are ways to do this simulation in Simulink as well. The most annoying part is still how to model the nonlinear capacitor. It seems that in Simulink, it has to be done through a look up table.