compass needle

berkeman

At really high values of H, the curve will still bend over (lowering the effectiveness of the transformer). So what are the advantages and disadvantages of using an air gap in the transformer?

copresh1

advantages:the core can be magnetized further
disadvantages:lowers the efectiveness by inc the reluctance

berkeman

That's pretty much it. To withstand the DC current and still act like a transformer, you have two choices -- make the core a lot thicker so that you are still on the linear part of the B-H curve (usually expensive and a problem for the product's size), or introduce an air gap in the transformer design, and live with the fact that the inductance of the transformer is a lot lower (and hence the passband of the transformer is narrowed some, depending on the source and load impedances).