I seek an intuitive understanding of a coupled inductor in a system. My professors, a generation ago, bemoaned that they'd had four semesters of magnetic circuits but had time to give us just one course including motors. This, I suspect, is what I missed. Please start with a general description before diving into the differential equations!
Take a core (such as a ferrite toroid) with two windings. Assume we will not saturate this core. Designate one winding primary and one secondary. The secondary has a non-reactive load. It may be linear (resistive) or non-linear (eg. resistor plus PN junctions).
Apply a voltage to the primary from a current-limited DC source. A voltage will appear across the secondary, and current will flow as the primary current increases.
What is happening with the fields from each winding? What energy does each hold?
Now the power source reaches its current limit. With no increase, how do the voltage and current in the secondary act? How does this affect conditions in the primary? As time goes to infinity, I expect the secondary current to go to zero and the primary's field to store energy equivalent to that stored in a simple inductor. Is this correct? How do the fields, currents, and voltages progress through time to reach this state?
If it will simplify the description, feel free to replace the voltage source with a variable current source which is ramped up at a slow, constant rate from zero to the final value.
Sorry it took so many words to describe. I hope the problem is clearly described. Thank you for aiding my education.