Can cardboard be used as core of the transformer?
Cardboard's magnetic properties are much the same as air, or vacuum. 

If you try to make a mains transformer (50/60 Hz) on a cardboard core, you'll find the magnetising current to be so high as to be unusable. The iron in a conventional transformer core allows a much lower H field (primary ampere turns) to develop a decent B field, the variation of which induces the voltage in all the windings.

Once you get up to 100s of kHz and above, air/cardboard cored transformers become more practical.