With no core you can increase the duty cycle too, up to some point where the magnetic field strength no longer intensifies. Once you add core material, this will change quite a bit I would expect.
To strengthen a magnetic field, you can increase the current and/or increase the number of turns, that's your only options. Adding a core will focus the flux lines into a particular space. The total magnetic field you generate won't actually get any stronger, it will just stay where you put it.
How this will effect your back EMF...? I'm not certain. Also not certain what you are trying to get the back EMF to actually do.
adding ferrite to increase inductance.
more inductance would translate to a stronger magnetic field befor the coil saturates.
without ferrite the inductance is low, and the coil quickly saturates. this limits the production of back emf.
the back emf must be powerfull(high voltage) enough. to overcome the loss of the diode, and to fill the cap up with enough (back emf) energy.
I suspect we need to reach a treshold before we really see an effect (like a diode needs a treshold voltage before we see it emitting light).
As can be seen in Nelsons vids, he used kV pulses. (another clue).
so Matt, Im just guessing. but my intuition says we need a big voltage.
More important Matt, do you know at what phase angle we should inject/discharge the back emf filled capacitor into the already existent magnetic resonant sine, to see a increase in magnitude/output?