analog - Frequency response of common source amplifier and Millers theorem

I was asked to draw the frequency response of a CS amplifier. The MOSFET has gate-drain capacitance between the input and output. This means that at high frequency, the output is shorted to input(the impedance of a capacitor decreases at high frequency). Which in turn says that there is a ZERO in the transfer function of the circuit.

But when I try it by using Millers theorem, the common capacitance between the input and output are resolved (see the pic). So there is no ZERO in the transfer function now.

So what does it mean? Whether the circuit has a zero or not?

Thanks

Answer

You are confused because you are assuming that the circuit transformation which you made where you connected the \$C_{gd}\$ to an equivalent (bigger) capacitor to ground is valid at all frequencies. In fact, this transformation is valid only at low frequencies, below the bandwidth of the amplifier.
If you consider this equation at higher frequencies, then you also need to incorporate the fact that the gain itself is frequency dependent. Thus, the equivalent impedance comes to be \$\frac{1}{sC_{gd}(1+A(s))} \$. Clearly, the equivalent impedance is capacitive only at low frequencies and there is no equivalent lumped capacitor at higher frequencies.
Since the frequency of zero is around \$\frac{g_m}{C_{gd}}\$ which is much beyond the bandwidth of the amplifier, your equivalent circuit is not applicable around that frequency region and will give you wrong conclusions. It's better to work with the original circuit in this region.