 Long-channel MOSFET current law

For an n-channel MOSFET, assume a gradual channel, constant charge carrier mobility $\mu$, gate capacitance per area $C_{\mathrm{ox}}$, width $W$ and length $L$. With overdrive $V_{\mathrm{ov}}=V_{GS}-V_T>0$, the inversion-sheet charge magnitude is $C_{\mathrm{ox}}(V_{\mathrm{ov}}-V(x))$. The drift-current relation $I_D=\mu W{C}_{\mathrm{ox}}(V_{\mathrm{ov}}-V)V^{\prime }$ is constant along the channel. Integrating from $V=0$ to $V_{DS}$ over length $L$ proves the displayed expression for $0\le V_{DS}\le V_{\mathrm{ov}}$. At the drain pinch-off condition $V_{DS}=V_{\mathrm{ov}}$, it yields $I_{D,\mathrm{sat}}=\mu C_{\mathrm{ox}}W{V}_{\mathrm{ov}}^2/(2L)$. This ideal saturation neglects channel-length modulation and short-channel velocity saturation.