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We know that:

$$V_{in} < V_{out} + V_{t0}$$ and $$V_{in} >= V_{t0}$$

Where \$V_{in}\$ is the input voltage (say voltage gate source)

\$V_{out}\$ is the output voltage say \$V_{ds}\$ (voltage drain source)

\$V_{t0}\$ is the threshold voltage

Say we are neglecting all the higher order effects and also assume that \$V_{sb}\$ (voltage between substrate and source) is zero.

Then is my logic correct of interpreting the above equation?

That \$V_{gs}\$ is proportional to \$I_d\$ (drain current) i.e. it sets a limit that it can be the maximum allowable current before the drain current saturates and starts to pinch off.

And \$V_{ds}\$ is also proportional to \$I_d\$. Hence when we increase the \$V_{ds}\$ the \$I_d\$ will increase and after some point it will start to saturate hence at that point \$V_{ds} + const\$ (\$V_{t0}\$) must be greater than \$V_{gs}\$ as \$V_{gs}\$ did set the upper limit on the amount of allowable \$I_d\$ before pinchoff.

Please correct me if I am wrong.

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\$V_{gs}\$ defines the thickness of the channel under the Gate (in a 3D mosfet model). Think of this channel as a hallway whose width you can increase as you increase \$V_{gs}\$. Think of people walking through the hallway as electrons ( \$I_{d}\$). Think of yourself as a god who can force people to have to run through this hallway (force = applied \$V_{ds}\$, which creates \$I_{d}\$)

If the hallway is wider (\$V_{gs}\$ is higher), you can fit more running people (electrons) before people start getting stuck in a tight trafficy environment (current saturation).

If you keep the hallway constant (\$V_{gs}\$ is constant), the there is only a finite amount of people (finite \$I_{d}\$) you can force (increase \$V_{ds}\$) before people start getting stuck and jammed up (\$I_{d}\$ saturates). If you try to force more people into the jammed hallway (try to increase \$I_{d}\$ by increasing \$V_{ds}\$), you will barely be able to. It will be a lot harder for you because the doorway is already jam-packed (\$I_{d}\$ = saturated) with people (current).

I like to think in analogies because all of these phenomena are already experienced by us in everyday life. This analogy gives a rough idea of whats happening under the hood. :) Hope it helps!

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