Physics
Cyclotron
One fixed radio frequency, a spiral that grows for free, and a ceiling put there by relativity rather than by engineering.
What happens
A slow particle takes the same time to go round whatever its speed: a bigger circle at a higher speed, and the two cancel. So one fixed frequency keeps kicking every turn and the spiral grows for free - that sentence is the whole patent. Then the mass rises with the energy, the turn takes longer, the kick arrives early, shrinks, and reverses. The ceiling that follows does NOT depend on the magnetic field: the field sets how big the machine is and what frequency it runs at, and nothing about how fast the particle ends up going. Doubling the dee voltage buys the square root of two. Two knobs a control room has: trim the radio frequency a little below f₀ and the phase drifts ahead first, relativity brings it back, and the kick stays positive for many more turns - until the trim is too much and the phase reaches −90° before it turns, at which point the beam dies early. The injection phase does the same with the same cliff.
Equations used in the module
Equations f = qB / (2π γ m) r = p / (qB), p c = √(T² + 2Tmc²) Δφ per turn = 2π(γ − 1) T_max ∝ √V (and not on B at all)
