Dear Sir/Madam
I am designing a toroid core inductor for a current control circuit shown in the attached file (1.jpg). The voltage of power supply is 5V and the frequency of mosfet switching is selected as 10 kHz. The resistance is measured as 0.05Ohm. I am going to provide 40A current through resistors. Thus the current passing though inductor is 80A. The minimum inductance of 40uH yields good results in simulation as shown in the attached file (2.jpg). My question is about the flux density passing through toroid core. The flux density can be calculated based on Faraday's law or Ampère’s law, as shown in attached file (3.j. The first one yield a reasonable value lower than the saturation flux (0.4T) of ferrite material, while the second one yields very high value flux which is very greater than the saturation flux. Which equation should be used for design of toroid core of inductor? Why?

 

It's the second one. It's a DC choke. If you go by ΔV, the positive going ΔV is greater than the negative [EDIT] until the output voltage stabilises, so when it reaches its steady state there is standing DC flux.
Ferrite won't work for a DC choke unless you gap it (Dremel and diamond disc cutter!). You need an iron powder core.

 

Download Micrometals' inductor design software. It also takes into account the change in inductance with DC current, which a simple μr-based calculation doesn't.
Use the old DOS software
http://www.iec-international.com/micrometals/micrometals/software.html
the fancy new version on Micrometals website gives you 10,000 spurious answers for the one useful one.
If you want to use ferrite, I can send you my spreadsheet which calculates them.