Solenoidal core:

A pair of cylindrical coils, wound around a rod shaped piece of powdered iron or ferrite, was once a common configuration for the transformers at the radio frequencies. At times this kind of transformer is still seen, although it is used most often as a loopstick antenna in the portable radio receivers and in radio direction finding equipment.

The coil windings are placed one atop the other, or they can be separated to reduce capacitance between primary and secondary.

Figure--   Solenoidal-core transformer.

In loopstick antenna, the primary serves to pick up the radio signals. The secondary winding gives the best impedance match to 1st amplifier stage, or the front end, of radio.