I am wondering what equation governs the operation of an n-channel JFET when Vg is positive relative to the drain and/or source. I understand that if Vgs is positive but less than Vt, where Vt is the turn-on voltage of the PN junction, the depletion region will shrink, enhancing the channel as compared to when Vgs = 0. However, once Vgs = Vt, the depletion region is gone, and the channel can be enhanced no further. In this case, the gate will draw current. I understand that this is not what JFETs are normally used for, but I appreciate any help I can get. Thanks!

Since a JFET is symmetrical, Vgs and Vds can be interchangeable. What equations govern each current when Vd, Vs < Vg-Vt?

How about when Vs < Vg-Vt, but Vd > Vg-Vt?

 

At direct gate currents, the transistor starts to operate in a similar manner to a bipolar transistor. About 30 years ago I found formulas for this mode in one Russian book. I don't think you'll find it now, but I think it's on the Internet. Once it was considered as a promising logical element with very little switching work. If you don't need it for idle curiosity, you will find it. There is a special type of field-effect transistors with zero threshold voltage (with a normally closed channel) that work similarly to bipolar transistors. In Russian they are called BSIT.
https://dmkpress.com/files/PDF/978-5-94120-042-9.pdf

 

At direct gate currents, the transistor starts to operate in a similar manner to a bipolar transistor. About 30 years ago I found formulas for this mode in one Russian book. I don't think you'll find it now, but I think it's on the Internet. Once it was considered as a promising logical element with very little switching work. If you don't need it for idle curiosity, you will find it. There is a special type of field-effect transistors with zero threshold voltage (with a normally closed channel) that work similarly to bipolar transistors. In Russian they are called BSIT.
https://dmkpress.com/files/PDF/978-5-94120-042-9.pdf

Does this mean that the JFET will conduct approximately Ids = ~β*Igs, assuming Vgs, Vds > 0 and β is some positive constant?

 

Actually, there are two mechanisms working. Normal field and bipolar. The current gain depends on the current. At low gate currents, the gain is higher.