The Role of Air Gap in High Frequency Transformers- BZTrafo transformer

Simply speaking, air core is to prevent magnetic saturation, but it also increases leakage inductance and reduces efficiency. This video will help you to clear more about air core in transformers.

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Inductors are what store electrical charge in the form of the induced magnetic field, not the air gap within a ferrite core.

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Not much info to really master the engineering problem...