The cyclotron can achieve higher energies (about a hundred MeV), in a smaller space than is possible with a linac by accelerating the particles in circular paths.
A voltage applied to the dees creates an electric field that accelerates the particles as they cross the gap between them. In the dees, the particles follow a semi-circular path (because of the magnetic field) bending through 180° to emerge once again at the gap between them. The voltage between the dees is alternated in synchronisation with the particles motion so that as the particles reach the gap the electric field has reversed and can accelerate the particle once more.
As the particles become more energetic it becomes harder to bend them so as they are accelerated the diameter of their circular path in the dees becomes larger.
The particles therefore follow a spiral path from the source to emerge from a hole in one of the dees at high energies.
The maximum energy attainable depends on how many times the particles can cross the gap before leaving the cyclotron. This in turn depends on how tightly the particles can be bent in the dees which is governed by the strength of the magnetic field.
For protons, with a ordinary cyclotron, this would be about 10 million electron volts (10MeV).