STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA

where the full 4×4 transfer matrix has been used to take into account coupled trans-
verse motion and represents another feature of this derivation which hitherto had
been neglected in analytic treatments of BBU. The HOM angular kick is expressed
as V⊥/(c/e)pb where pb is the momentum of the beam bunch at the cavity and with
86
V⊥ = − c Va
sin φ (4.13)
ω a
The average power deposited by the beam in the HOM is expressed as
˙Ubeam = 〈∆U1 + ∆U2〉 · fb
(4.14)
where fb is the bunch repetition frequency. While the beam deposits energy in the
HOM, power is also dissipated on the cavity walls and leaks through the couplers.
The total power dissipated is given by
Ptot =
a 2
ω
c
V 2
a
2 Rd
Qo
QL
(4.15)
where (Rd/Qo) is in Ohms and QL is the loaded quality factor of the mode. The
complete energy balance equation is given by
˙U = ˙ Ubeam − Ptot = 〈∆U1 + ∆U2〉 · fb − Ptot
Averaging with respect to the phase, φ, ultimately yields
where
˙U = Io
= −Io
qV 2
a
ωa2pb qV 2
a
ωa 2 pb
V 2
a
M ∗ 〈sin φ cos(φ + ωTr)〉 −
a2 (ω/c) 2 (Rd/Qo)QL
∗ sin(ωTr)
V
M −
2
2
a
a2 (ω/c) 2 (Rd/Qo)QL
(4.16)
(4.17)
(4.18)