well i tell you a fact, with a mains transformer, in the ideal case, the mag field from the primary is equal and opposite in the core to the mag field from the secondary current.
Therefore, an ideal transformer will not saturate.
However, a real transformer could saturate because there is a magnetic field in the core due to the magnetising current, the magnetising curent is kind of what activates the transformer to do its thing.
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The attached has virtually all the magnetics equations that you need in order to design basic SMPS.
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Be careful how you read, -many books are written more for the vanity of the author than anything else.
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Magnetism is a model invented by humans to describe physical phenomena....magnetism may not be real at all......but as long as the model allows us to design electrical goods, then many dont care whether the model is truly correct or not. Some say that magnetism is wrong....and that string theory would be a better, more accurate way to describe what we term as magnetic phenomina.
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In general engineering, you dont necessarly care whether the model is correct or not, but just want to use it to help you to get a product designed and sold. If possible, you will not use magnetic theory at all, or as little as possible, and will just seek to use whatever means to get products finished and sold out of the door.
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It was Einstein who discovered that there is actually a big inconsistency in all the modern theories such as magnetism....but often, its "who cares", you can use the model to design stuff that sells.