Why the Ferrite in an air gapped core? 

Thank you Professor. I watch a lot of your video tutorials. As you explain, the reason for a permeable core is to achieve a high energy density. Prof Cuk is right of course in implying that the airgap defines the inductance, but he doesn't seem to realise the resulting disadvantages of a totally air-cored inductor - the best form of which for power electronics would appear to be a toroidal winding. In 1982 I needed an inductor to handle some 20+ KVA at 3kHz. As I'd never embarked on such a design I was advised to approach an "expert" from outside the company (I was working for BAe Dynamics at that time). He must have been of the same opinion as Prof Cuk, because he said that an air-gapped magnetic core would need such a large gap as to spray leakage flux all about, and that I should use a stack of about 6 or seven air-cored toroids about 70cm dia. Big enough to fill a filing cabinet. Well, he'd had that! This had to fit inside a small underwater body. So I had to undertake the design for myself. I worked my way up from basic principles. One way to overcome having a large single gap was obviously to have a number of smaller ones, evenly distributed along the length of the core, and as I reflected, I had the idea of iron dust. That would give the ultimate distribution of air gaps! I found out that some quite large (carbonyl) iron dust cores were made, and calculated that those with a µr of 11 and an OD of 150mm could be used. I calculated that a stacked pair of these cores could be wound to provide 1/4 of my total inductance, and so a total of 4 such inductors worked fine - with no measurable loss of Q. (The Q of the power device to be tuned was only around 5 anyway). Straight from my notebook to the finished devices! Carbonyl iron saturates around 1.0 Weber, cf ferrite at 200 mWeber which reduces to 100 mWeber when hot. But I think ferrite would have lower losses at HF. A totally irrelevant fact: according to Microsoft Translator, Cuk is the Turkish for dick.

Another advantage of gapped ferrite core (besides size reduction) over air-core could be lesser EMI as most of the magnetic flux is contained within the ferrite core because of its high permeability. Yes, there is some noise due to the gap but not as much as the air core.

It would appear both gentlemen neglect the fact that the primary purpose of the core is to act as a field concentrator while the primary purpose of the gap is to maximize the core ability to act as such either in terms of maintaining appropriate relative permeability, or defining the geometry of the area in which the magnetic field is to be maintained. In general, thinking of inductance as a space property manifesting itself as the ratio between electric current and magnetic energy resulting from the currents presence is by far more beneficial than diving into arithmetic formulas.

An example of the two possible coils, how much bigger and how many more turns needed, without ferrite, that maybe could make it even more clear? But clear it is.

From personal experimental experience and watching DIY videos I use the very general rule that a ferite rod has about 10 times less inductance than a closed core and 10 times more than air core.

I wonder, what's about the total losses. If you reduce the size, do you also reduce the copper power losses due to shorter winding lengths? One could argue, that size is not critical, and he just wants to reduce powerlosses for a given ammount of money.

Dear Sam, you made this too complicated or am I missing something, The core provides space to wind the wires, without it the effective air gap length would be much higher and hence the resultant Inductance would be much lower.