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So after the channel is pinched off in a MOSFET the equation that relates current and voltage through the MOSFET changes. What I am confused about though, is why in the derivation of this new formula the voltage is integrated from V = 0v to V = Vgs - Vt where Vgs is the potential difference between the source and the gate and Vt is the threshold voltage (at least that's what I think that's what those represent).

Anyway here is an visual representation of the circuit mentioned: enter image description here

Here is also the integral I mentioned (equation 6.16): enter image description here

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Actually the integrals for the linear region and saturation region are the same. The result for saturation is just obtained as a special case with \$V_{ds} = V_{gs} - V_T\$.

The reason why \$V_{gs} - V_T\$ is used is simply because that is the maximum voltage that can drop along the channel. Put differently, this is the maximum possible \$V_{ds}\$ for the linear region which marks the transition to the saturation region. As a first approximation the current in the saturation region is constant, so the current at this boundary is exactly the current for saturation.

In a nutshell, the current in saturation is the maximum possible current in the linear region.

The same result is obtained, if the expression for the linear region is used with \$V_{ds}\$ being equal to \$V_{gs} - V_T\$.

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