Half Bridge SMPS, gapped transformer ?

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no, you want as high as possible magnetising inductance . a transformer shall only transfer energy, not store it as the flyback coupled inductors.

thus no air gap. typical primary inductance ie magnetising inductance is in the range of mH for switching transformers operating off-line but it depends of course on A lot of factors.

General rule the magnetising current should not exced 10% of max load current (primary side) but again , it depends.

I will dig up a few links where this is explained more in detail, if you are serious about going to transformer driven converter it is important to understand them to prevent the magic smoke from escaping or worse...
 
Air gap is only needed when a continuous DC magnetizing field is inside the core: Fly Back, Buck, Boost, polarity inverter, and forward converters using 1 or 2 transistors. And it is sometimes used a very small gap in push pull topology if there exists risk of DC core imbalance because of different on times, off time, DC wire resistance or RDSon in FET´s or VCEsat in BJT´s or IGBT´s.

But in HB mode converters, either hard or resonant, the blocking cap ussually connected between the center main bulk caps and the transformer, prevents core imbalance, so no gap is needed, in fact make it a worse transformer.
 
Ok, then no gap at all.

The "cold" wire of the transformer, is returned via a non-polarized cap to the middle point of two bulk caps in series, or one to each rail?

Quasi resonant usually uses the leakage inductance, normal or deliberately included in the transformer.
 
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This design has a DC blocking capacitor.
Even with a capacitor, the complete absence of air-gap can lead to some form of periodic flux walking.
The simplest cure is to include a very marginal air gap, even if it looks counter-intuitive.
See this document p 4-6:
http://www.ti.com/lit/ml/slup126/slup126.pdf
Damping can also be a solution but a small airgap is free from a technical and BOM perspective.
IEEE Xplore - Transformer saturation and unusual system oscillation in capacitively coupled half-bridge or full-br...
 
Yes, fex when doing some math on Epcos cores with the ue and ui data given one finds that the cores without gap actually have a gap of about 10-20um, which is no surprise as the force clamping the cores is not that high, and the cores are flexing a bit under load.

I guess it would be possible to lap the mating surfaces even more, but not cost effective.

One could benefit from a tiny air gap in forwards converters aswell, to reduce remanence, but at the price of higher magnetising current and probably worse voltage peaks.

Here is a few links
Runo's Power Design

And some more material:
Revision to Fundamentals of Power Electronics
 
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