In a particle accelerator, charged particles are accelerated by applying an electric field across a gap. The force on a particle crossing this gap is given by the Lorentz force law:

�=�[�+(��)]

where q is the charge on the particle, E is the electric field, v is the particle velocity, and B is the magnetic flux density. It is not possible to accelerate particles using only a static magnetic field, as the magnetic force always acts perpendicularly to the direction of motion, and therefore can only change the direction of the particle, not the speed.^[30]

In practice, the magnitude of an unchanging electric field which can be applied across a gap is limited by the need to avoid electrostatic breakdown.^[31]: 21  As such, modern particle accelerators use alternating (radio frequency) electric fields for acceleration. Since an alternating field across a gap only provides an acceleration in the forward direction for a portion of its cycle, particles in RF accelerators travel in bunches, rather than a continuous stream. In a linear particle accelerator, in order for a bunch to "see" a forward voltage every time it crosses a gap, the gaps must be placed further and further apart, in order to compensate for the increasing speed of the particle.^[32]

A cyclotron, by contrast, uses a magnetic field to bend the particle trajectories into a spiral, thus allowing the same gap to be used many times to accelerate a single bunch. As the bunch spirals outward, the increasing distance between transits of the gap is exactly balanced by the increase in speed, so a bunch will reach the gap at the same point in the RF cycle every time.^[32]

The frequency at which a particle will orbit in a perpendicular magnetic field is known as the cyclotron frequency, and depends, in the non-relativistic case, solely on the charge and mass of the particle, and the strength of the magnetic field:

�=��2��

where f is the (linear) frequency, q is the charge of the particle, B is the magnitude of the magnetic field that is perpendicular to the plane in which the particle is travelling, and m is the particle mass. The property that the frequency is independent of particle velocity is what allows a single, fixed gap to be used to accelerate a particle travelling in a spiral.^[32]

Each time a particle crosses the accelerating gap in a cyclotron, it is given an accelerating force by the electric field across the gap, and the total particle energy gain can be calculated by multiplying the increase per crossing by the number of times the particle crosses the gap.^[33]

However, given the typically high number of revolutions, it is usually simpler to estimate the energy by combining the equation for frequency in circular motion:

�=�2��

with the cyclotron frequency equation to yield:

�=����

The kinetic energy for particles with speed v is therefore given by:

�=12��2=�2�2�22�

where r is the radius at which the energy is to be determined. The limit on the beam energy which can be produced by a given cyclotron thus depends on the maximum radius which can be reached by the magnetic field and the accelerating structures, and on the maximum strength of the magnetic field which can be achieved.^[8]

In the nonrelativistic approximation, the maximum kinetic energy per atomic mass for a given cyclotron is given by:

��=(���max)22��(��)2=�(��)2

where � is the elementary charge, � is the strength of the magnet, �max is the maximum radius of the beam, �� is an atomic mass unit, � is the charge of the beam particles, and � is the atomic mass of the beam particles. The value of K

�=(���max)22��

is known as the "K-factor", and is used to characterize the maximum kinetic beam energy of protons (quoted in MeV). It represents the theoretical maximum energy of protons (with Q and A equal to 1) accelerated in a given machine.^[34]

While the trajectory followed by a particle in the cyclotron is conventionally referred to as a "spiral", it is more accurately described as a series of arcs of constant radius. The particle speed, and therefore orbital radius, only increases at the accelerating gaps. Away from those regions, the particle will orbit (to a first approximation) at a fixed radius.^[35]

Assuming a uniform energy gain per orbit (which is only valid in the non-relativistic case), the average orbit may be approximated by a simple spiral. If the energy gain per turn is given by ΔE, the particle energy after n turns will be:�(�)=�Δ�Combining this with the non-relativistic equation for the kinetic energy of a particle in a cyclotron gives:�(�)=2�Δ����This is the equation of a Fermat spiral.

As a particle bunch travels around a cyclotron, two effects tend to make its particles spread out. The first is simply the particles injected from the ion source having some initial spread of positions and velocities. This spread tends to get amplified over time, making the particles move away from the bunch center. The second is the mutual repulsion of the beam particles due to their electrostatic charges.^[36] Keeping the particles focused for acceleration requires confining the particles to the plane of acceleration (in-plane or "vertical"^[a] focusing), preventing them from moving inward or outward from their correct orbit ("horizontal"^[a] focusing), and keeping them synchronized with the accelerating RF field cycle (longitudinal focusing).^[35]

The in-plane or "vertical"^[a] focusing is typically achieved by varying the magnetic field around the orbit, i.e. with azimuth. A cyclotron using this focusing method is thus called an azimuthally-varying field (AVF) cyclotron.^[37] The variation in field strength is provided by shaping the steel poles of the magnet into sectors^[35] which can have a shape reminiscent of a spiral and also have a larger area towards the outer edge of the cyclotron to improve the vertical focus of the particle beam.^[38] This solution for focusing the particle beam was proposed by L. H. Thomas in 1938^[37] and almost all modern cyclotrons use azimuthally-varying fields.^[39]

The "horizontal"^[a] focusing happens as a natural result of cyclotron motion. Since for identical particles travelling perpendicularly to a constant magnetic field the trajectory curvature radius is only a function of their speed, all particles with the same speed will travel in circular orbits of the same radius, and a particle with a slightly incorrect trajectory will simply travel in a circle with a slightly offset center. Relative to a particle with a centered orbit, such a particle will appear to undergo a horizontal oscillation relative to the centered particle. This oscillation is stable for particles with a small deviation from the reference energy.^[35]

The instantaneous level of synchronization between a particle and the RF field is expressed by phase difference between the RF field and the particle. In the first harmonic mode (i.e. particles make one revolution per RF cycle) it is the difference between the instantaneous phase of the RF field and the instantaneous azimuth of the particle. Fastest acceleration is achieved when the phase difference equals 90° (modulo360°).^{[35]: ch.2.1.3}  Poor synchronization, i.e. phase difference far from this value, leads to the particle being accelerated slowly or even decelerated (outside of the 0–180° range).

As the time taken by a particle to complete an orbit depends only on particle's type, magnetic field (which may vary with the radius), and Lorentz factor (see § Relativistic considerations), cyclotrons have no longitudinal focusing mechanism which would keep the particles synchronized to the RF field. The phase difference, that the particle had at the moment of its injection into the cyclotron, is preserved throughout the acceleration process, but errors from imperfect match between the RF field frequency and the cyclotron frequency at a given radius accumulate on top of it.^{[35]: ch.2.1.3}  Failure of the particle to be injected with phase difference within about ±20° from the optimum may make its acceleration too slow and its stay in the cyclotron too long. As a consequence, half-way through the process the phase difference escapes the 0–180° range, the acceleration turns into deceleration, and the particle fails to reach the target energy. Grouping of the particles into correctly synchronized bunches before their injection into the cyclotron thus greatly increases the injection efficiency.^[35]: ch.7

In the non-relativistic approximation, the cyclotron frequency does not depend upon the particle's speed or the radius of the particle's orbit. As the beam spirals outward, the rotation frequency stays constant, and the beam continues to accelerate as it travels a greater distance in the same time period. In contrast to this approximation, as particles approach the speed of light, the cyclotron frequency decreases due to the change in relativistic mass. This change is proportional to the particle's Lorentz factor.^{[30]: 6–9}

The relativistic mass can be written as:

�=�01−(��)2=�01−�2=��0,

where:

• �0 is the particle rest mass,
• �=�� is the relative velocity, and
• �=11−�2=11−(��)2 is the Lorentz factor.^{[30]: 6–9}

Substituting this into the equations for cyclotron frequency and angular frequency gives:

�=��2���0�=����0

The gyroradius for a particle moving in a static magnetic field is then given by:^{[30]: 6–9} �=���0���=��0���=�0���−2−�−2

Expressing the speed in this equation in terms of frequency and radius�=2���yields the connection between the magnetic field strength, frequency, and radius:(12��)2=(�0��)2+(��)2