Derivation of I_d of a self-biased JFET
Edit:  For a derivation of the JFET equation from physical principles, there are plenty of online resources.  Google:  "JFET device physics" and, for example, find this.  The derivation is not trivial.



I'm not sure where you're getting your information but  \$I_{DSS}\$ isn't given by the equation you've written.  The following is from Marshall Leach's JFET notes:

In the saturation region of operation, the JFET total drain current is given by:

\$i_D = \beta (v_{GS} - V_{TO})^2\$ for \$v_{GS}&gt; V_{TO} \$

Here, \$V_{TO}\$, the threshold or pinch-off voltage, is negative.

This can be written as:

\$i_D = I_{DSS}(1 - \dfrac{v_{GS}}{V_{TO}})^2\$

From the above, it is clear that \$i_D = I_{DSS} \$ when \$v_{GS} = 0 \$; it is the drain current when the gate and source have the same voltage.



Solving for \$I_{DSS} \$ gives:

\$I_{DSS} =  \beta V_{TO}^2\$