What is the difference between these two transformers:
Imagine them with bobbin.
Why is air gap important in flyback transformer?
$$\color{darkblue}{\text{Short story}}$$
A gapped core has a lower peak flux density for the same inductance carrying the same current. This means it is less likely to suffer from core saturation problems. Put another way, it can tolerate a higher energy transfer to the secondary winding if primary current is increased.
$$\color{purple}{\text{Longer story}}$$
For a flyback transformer, the most fundamental thing to get right is the primary inductance. So, with an ungapped core, you might use 10 turns to get a required inductance of 100 μH.
$$\color{orange}{\text{I'm using convenient numbers to illustrate the point}}$$
When you introduce an air-gap, the core permeability drops and, to counter this, you need more turns to get the original inductance value. So, if the permeability reduces by a factor of four (due to the air-gap), 10 turns only gets you 25 μH.
To restore the inductance from 25 μH to 100 μH, you need to double the turns to 20.
$$\color{red}{\text{This is because inductance is proportional to turns squared}}$$
So, your gapped transformer has 20 primary turns and the ungapped transformer has 10.
"So what" you might say?
$$\color{blue}{\text{Here's where the benefit is seen (reduction of magnetic saturation)}}$$
The "things" that cause the magnetic core to saturate are current, turns and permeability. But, of course, the current in the ungapped core still has to match the current in the gapped core to ensure the correct level of energy is transferred each switching cycle.
We can't do anything about that; current and inductance make the energy to be transferred.
So, the gapped transformer (compared to the ungapped transformer) has: -
Hence, the peak flux density in the gapped core is half the peak flux density of the ungapped core. That is why we gap many, many magnetic components.
If we are going to wrap all the windings of transformer to the airgap what happens?
Because of fringing fluxes around the air-gaps, we tend to be careful about applying windings in those areas because the copper wire local to those spots can heat up excessively. So, just be careful about this.
It's better to apply several smaller gaps than one big gap because fringing is less with a smaller gap. And, of course, we can find ferrite materials that have low permeability (in effect, the gap is homogeneously distributed around the core).
How should we consider the equivalent circuit of a standard flyback transformer?
I treat a flyback transformer no differently to any other transformer; two (or more) highly coupled coils. How we think of the flyback transformer in a circuit is a little different because we need to recognize that a flyback design doesn't push significant energy through the secondary circuit when the primary inductance is being charged hence, we use dot notation on circuit diagrams and ensure we are diligent when we wind the transformer.
Why is air gap important in flyback transformer?
Because a flyback's transformer (actually a coupled inductor), unlike other converters', is used as an energy storage component: When the switch is on no current can flow in secondary, so the primary current builds up energy in primary winding's magnetic field according to \$E=0.5 \ LI^2\$, where L is the primary inductance and I is the peak of the primary current (ramp). And when the switch opens this energy is transferred (partially or fully) to the secondary. Without air gap no energy is stored. So an air gap is a must in a flyback transformer.
If we are going to wrap all the windings of transformer to the airgap what happens?
Quite possibly there's going to be a high-temperature spot on your windings because the losses will be high around the gaps. To reduce the effect, you can
How should we consider the equivalent circuit of a standard flyback transformer?
No different than others.
A flyback transformer must store energy during the primary 'charging' part of the cycle, in order to release energy into the secondary during the flyback phase.
If you are going to buy a lump of magnetic material for your transformer, you will want it to store as much energy as possible.
The amount of energy stored in a given volume is proportional to BH, the magnetic flux B, and the magnetising field H.
The maximum B field is limited by the material properties, up to 2 T for iron and the low 100s of mT for ferrite.
With the high permeability available from either iron or ferrite, very few ampere turns (H field) are required to reach the max B field, and the copper winding will be operated at only a small fraction of its current capability.
We can make the core require more H field for max B field by introducing an airgap, which reduces the permeability. The core now stores more energy, and the copper is used at its optimum current.
