The physics
A cyclotron has two hollow D-shaped halves, the dees, in a strong field. Inside a dee the proton moves on a half circle.
r = m v / (q B) T = 2π m / (q B)
A voltage across the gap gives the proton a kick each time it crosses. A faster proton goes round a bigger circle. One turn still takes the same time. So one fixed frequency stays in step.
f = q B / (2π m)
The field does no work. Only the gap gives energy: qV at each crossing, two crossings a turn.
The proton leaves when its circle reaches the edge, R.
K = q² B² R² / (2m) = 2 N q V