The voltage induced by a point-charge, , according to the fundamental theorem of beam loading given by

(1) |

where is the angular frequency of mode, , is the velocity of the particle as a fraction of the speed of light and is given by

(2) |

where is the stored energy in the mode and the electric field along the beam axis. Due to losses, the amplitude of the voltage will decay according to

(3) |

where is the voltage decay time constant with as the loaded quality factor of the complete system and is the inverse sum of all the subsystems

(4) |

For a superconducting cavity, the natural is of the order of meaning that that is dominated by and so the approximation that is made.

In pulsed operation the voltage after one pulse with bunches and no HOM voltage present before the arrival of the first bunch, is given by

(5) |

where is the time between bunches. After pulses with pulse period , the voltage present in the cavity is given by:

(6) |

(7) |

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Topic revision: r4 - 06 Jun 2012 - TomCraneAdmin

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