Assume an N channel MESFET with uniform doping and sharp depletion region shown in figure 1.
The depletion region is given by the depletion width for a diode. Where the voltage is the voltage from the gate to the channel, where the channel voltage is given for a position x along the channel as .
The current density in the channel is given by:
Substituting from equation 1:
One defines constant Β as the channel conductance with no depletion. And the work function to deplete the channel W00 :
We now define Vto, the voltage such that the channel is pinched off. d is the ratio of channel depletion to maximum depletion for the drain. s the ratio of channel depletion to maximum depletion for the source.
Equation 2 is Shockley's expression  for drain current in the linear region. When the device enters saturation, one end is pinched off(normally the drain). Thus $d=1$ and one may derive the equation for the saturation region:
General power law:
It was found that a general power law provided a better fit for real devices .
Where Q is dependent on the doping profile and a good fit is usually obtained for Q between 1.5 and 3. A general power law is approximately equal to Shockley's equation for Q = 2.4. Β is also empirically chosen and is proportion to the previous Β
Modelling the various regions is done though model binning. This however infers that a sharp transition exists from one region to another, which may not be accurate.
 A. E. Parker. Design System for Locally Fabricated Gallium Arsenide Digital Integrated Circuits. PhD thesis, Sydney University, 1990.
 W. Shockley. A unipolar field-effect transistor. IEEE Trans/ Electron Devices, 20(11):1365–1376, November 1952.
 I. Richer and R.D. Middlebrook. Power-law nature of field-effect transistor experimental characteristics. Proc. IEEE, 51(8):1145–1146, August 1963.