Why do so many sources say something along the lines "since a flyback transformer stores energy, an air gap is needed"? I have seen this reasoning in textbooks and app notes.
I thought air gaps cannot store energy and I thought also a flyback transformer stores energy with its inductance, and an air gap reduces inductance so I would think it also reduces an inductor/flyback's ability to store energy.
Where am I confused?
Unlike a forward-topology transformer (where the primary and secondary windings are conducting at the same time), the flyback transformer must store energy during the primary switch on-time, delivering it to the load during the primary switch off-time.
A forward-topology transformer doesn't need any gap since the peak flux density is a function of the applied volt-seconds only; the power being delivered 'through' the transformer isn't a variable (other than its effect on duty cycle). It's only the magnetizing current that moves the core along its hysteresis loop, which doesn't pose any saturation risk if everything is well-designed, since the primary and secondary ampere-turns cancel each other out.
A flyback transformer doesn't have the ampere-turn cancellation benefit of a forward converter, so the entire \$ \frac{1}{2}LI^2\$ primary energy moves the core up its hysteresis curve. The air gap flattens the hysteresis curve and allows more energy handling by decreasing the permeability of the core. You will of course need to add more turns to get your desired inductance compared to no-gap, but you avoid core saturation.
The key point here is that without an air gap an inductor will saturate if you try to put any current through it so inductance will fall and you can't store any energy.
The term "Flyback Transformer" is a little misleading and its more useful to consider it as coupled inductors rather than a transformer because the action is quite different with a conventional transformer energy is going into the primary and out of the secondary at the same time it does not store energy. With a "Flyback" transformer energy is first stored then released.
Taking some things we know about inductors
$$v = L \frac{di}{dt} = N A \frac {dB}{dt}$$
Where v is voltage, i is current, N is turns, B is flux density and A is the effective magnetic area.
Also
$$H = \frac {N \ i}{l} \Rightarrow i = \frac {H \ l}{N} $$
where H is the magnetic field strength, N is turns and l is magnetic path length
Finally permiability
$$ \mu = \frac {B}{H} \Rightarrow H = \frac {B}{\mu} $$
Thus
$$i = \frac{B \ l}{\mu \ N}$$
Now we can calculate Energy
$$ $$\begin{align} Energy & = \int{i \ v} \ dt\\ & = \int{\left( \frac{B \ l}{\mu \ N} \right) \ \left( N A \frac {dB}{dt} \right)} \ dt\\ & = \frac {A \ l}{\mu}\int{B} \ dB\\ & = \frac {A \ l}{\mu}\frac{B^2}{2}\\ \end{align}$$ $$
The energy storage is therefore only possible in the air gap and is proportional to be air gap volume and the square of the flux density.
Contrary to what most people think, including yourself, most of the useful energy is stored in the gap of the core.
For the case of ferrite, the gap is distributed between the tiny metallic particles so it too has as an effective gap used for calculations. This gap linearizes the BH loop and increases the current handling before saturation.
Air gaps are usually used for safety considerations. For a flyback transformer, you do not want arcs between the primary and secondary winding, and use an air gap.
