Hi,
In basic transformer theoretical explanations, there is a primary wound on one side of a soft iron core and secondary coil wound on other side of the core. The explanation will be like the flux generated in the primary wound that is linked to the secondary through the square-shaped soft iron core. (Note: In theoretical explanation, they will not have the explanation showing the primary wound over the secondary.) But in practical construction, the primary is wound over the secondary. In this case, why do we need to have a soft iron core when the primary magnetic field is directly going to get coupled with secondary?
Also, adding over Mathiew LAmbert's explanation, in order to minimize losses (although I would have used a more empirical one: magnetic fields like to reside in high permeability unless it spreads away and hence losses apear), the magnetic material has to form a close circuit that fits very much the magnetic field distribution in air. Thus, for very little losses they even made toroid - like transformers (these are said to be the most efficient although winding is difficult) or transformers with metallic bands bent in two directions, very much as the magnetic field looks like in air medium.
... And it must withstand a lot before going into saturation, which is a construction limit.
Behaviour of magnetic flux in air is similar to the behaviour of electric current in saline water. You need the magnetic flux to completely encircle the coils for better results.
The reason for having a soft iron core is that it is capable of producing internal magnetic flux densities some 10,000 (relative permeability) times stronger than in air. This is the basic principle of induction where, as engineers, we exploit the magnetic properties of the soft iron core material. The mmf (ampere-turns) of the primary winding is used to create (induce) significantly higher (>10,000 times) magnetic flux densities within the iron than is available or possible in air. Thus if we were to couple the two coils in air without any iron core the flux density of the surrounding magnetic fields would differ by the relative permeability of the iron core and would be some ~10,000 times smaller.
This is why magnetic materials are used in transformers and motors as it has been found they can produce extremely high magnetic flux densities, which we may exploit. In a sense the soft iron core acts like a "magnetic field amplifier" and where the term magnetic amplifier arose. The core allows a great saving in copper to be achieved.
However, Nicola Tesla used air-cored transformers to great effect generating voltages up to some 20 million volts for his wireless transmission of power experiments in the 1890's.
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