I understand that the equivalent circuits describe the behavior of amplifier for signals of low amplitude that allow us to assume that the circuit behaves linearly. 
My questions are:

1. Why are all the DC voltage and current sources that aren't varying with time zeroed out? I don't understand the statement- "As far as the signal is concerned, all DC sources have no effect on operation".

![enter image description here](https://i.stack.imgur.com/EZK7K.png)

As in the above figure, the small signal equivalent model shows that the r**esistance tied between drain and VDD(which is shorted) to be in parallel with the the current source gmVgs. However the voltage across resistance and the voltage VDS(Vo) are not quite the same. How is this accounted in the above model?** 

Is there any other way to derive expressions for gain and I/O impedance? I tried to draw an equivalent circuit with all the dc sources present.Note that the voltage across RD is not VDS(as it should not be). Is it correct?

![enter image description here](https://i.stack.imgur.com/clWda.png)

Applying KVL to output loop:

VDD - gmVgsRD - Vout = 0

VDD/Vgs - gmRD = Vout/Vgs

Voltage gain = Av = Vout/Vgs = VDD - gmRD.

**But the expression for gain derived by shorting VDD is -gmRD.** 

Where am I going wrong?