The workings of a cyclotron

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I am having trouble understanding the workings of a cyclotron. Supposing the typical design with two D shaped objects, changing potential differences between the D's, and a source of charges in the center, I can see that the particles would move in arcs (circles) at a constant energy. I just don't quite understand why they end up gaining energy, moving in a spiral of increasing radius, rather than a circle. Can someone help me to understand this?

The hyperphysics link, bottom of page, seems to describe the simple logic. The particle would move in a circle in the magnetic field at a given velocity. Each time it crosses the gap between the two D's, an electric field accelerates the particle across the gap. The circle gets wider each time the particle accelerates.

Some links including basic principles:

(Note I scrapped my initial comments)
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The main feature that makes the cyclotron so versatile is that the particle revolution frequency, qB/(2 pi m), is independent of particle energy until it becomes relativistic. So the orbit radius and the particle energy grow by the voltage gained from the Dees every turn. One requirement, not often mentioned, is that for vertical focusing, the magnetic field has to decrease slightly at larger radii, or else the particles will hit the top or bottom of the Dees.
Bob S
[added] This FREE online book by Stan Humphries is a good source of information on "The Principles of Charged Particle Accelerators"
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