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The bobbin is the means of providing magnetic coupling between the primary and secondary transformer windings. Secondary voltage and current will be affected if the bobbin (core) saturates magnetically. At saturation, no further coupling of energy from primary to secondary can occur. This is how magnetic amplifiers work - a control winding is used to saturate the core to change the coupled primary to secondary energy.
Both the size and the material of the bobbin (core) will determine when it reaches magnetic saturation.
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