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Saturation Region of Operation :

When we increase the drain to source voltage further the assumption that the channel voltage is larger than the threshold all along the channel does not hold and the drain current does not follow the parabolic behaviour for V_DS > V_GS - V_TH as shown in
Figure below. In fact as shown in Figure I_DS becomes relatively constant and the device operates in the saturation region.

In order to understand the phenomenon of saturation consider the Equation (8.3.6) again which is given as :

Q_i (x) = - C_ox [V_GS - V (x) - V _TH]

i.e. Inversion layer charge density is proportional to (V_GS - V (x) - V_TH). Thus if V (x) approaches V_GS - V _TH then Q_i (x) drops to zero. In other words as shown in Figure below if V_DS is slightly greater than V_GS - V_TH then the inversion layer stops at x £ L and we say that the channel is 'pinched off'.

As V_DS increases further the point at which Q_i (x) equals zero gradually moves towards the source. Thus at some point along the channel the local potential difference between the gate and the oxide silicon interface is not sufficient to support an inversion layer.

Therefore for saturated device the integral limits on the left hand side of the
Equation (8.3.13) must be taken from x = 0 to x = L¢, where L¢ is the point at which Q_i (x) drops to zero and that on the right side from V (x) = 0 to V (x) = V_GS - V_TH.

As a result of Integration we get,

I_DS = m_n C_ox (V_GS - V _TH)² …(8.3.21)

The above equation of I_DS indicates that I_DS is independent of V_DS if L¢ remains close
to L.