Theory, Design and Tests on a Prototype Module of a Compact ...

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58 4. CIRCUIT MODEL For the derivative with respect to ε L,R p , we have to consider all the terms, <strong>and</strong> by considering the following trigonometric identities sinh (N − 2p + 1)x = j(−1) p cos N π cosh (N − 2p + 1)∆x 2 cosh x = j sinh ∆x sinh (N − 1)x = j sin (N − 1) π cosh (N − 1)∆x 2 cosh Nx = cos N π cosh N∆x 2 cosh (N − 1)x = j sin (N − 1) π (4.50) sinh (N − 1)∆x 2 the expression (4.48) becomes N Ti = −ωM cos N π N cosh ∆x sinh N∆x + j sin (N − 1)π 2 2 i=1 12 · [− cosh ∆x cosh (N − 1)∆x − sinh ∆x sinh (N − 1)∆x] N − ε p=1 + p sinh ∆x cosh (2p − N − 1)(j π + ∆x) 2 N −j p=1 ε − p cosh ∆x(−1) p cos N π cosh (N − 2p + 1)∆x 2 = f(∆x, ε L p , ε R p ) p=1 ε + p (4.51) In the derivative with respect to ε L,R p , the only term different from zero comes from i = p. For example, the term in ε L p is ∂f ∂εL = −j p ωM 2 · sin (N − 1) π [cosh ∆x cosh (N − 1)∆x + sinh ∆x sinh (N − 1)∆x] 2 − 2j sinh ∆x cosh (2p − N − 1)(j π + ∆x) 2 cosh ∆x(−1) p cos N π cosh (N − 2p + 1)∆x 2 And finally, the resonance equation is jωM cos N π N ε 2 + p + jN∆x + (−1) p N ε − p = 0 p=1 <strong>and</strong> therefore the unknown is ∆x = j 1 N L εp + ε N R p 2 p=1 + (−1) p εR p − ε L p 2 p=1 = j N p=1 ε∗ p N (4.52)
4. THE PERTURBED AXIAL FIELD 59 where we introduced the new error parameter N L εp + εR p ε ∗ p = p=1 2 + (−1) p εR p − ε L p 2 (4.53) In order to evaluate the frequency error <strong>and</strong> remembering the equation (4.11), one obtains 1 ∆ω = ω0 √ − 1 ∼ K = −jω0 sinh ∆x (4.54) 1 − jK sinh ∆x 2 Since ∆x is of the same order of magnitude of all the ε L,R p , it is accept- able the following approximation K ∆ω = −jω0 2 sinh j 1 N N p=1 ε ∗ p ∼ = ω0K 2N N p=1 ε ∗ p (4.55) Note that the quantity ε ∗ p is equal to ε R p for even p, <strong>and</strong> for odd p it is equal to ε L p . Then, ∆ω is proportional to the following sum ε L 1 + ε R 2 + ε L 3 + ε R 4 + ε L 5 + . . . (4.56) which is a sum over the odd cavities. This is a general result since it states that the frequency error of the whole chain depends only on the errors in the accelerating cavities, which are the charged ones for the mode π/2 mode. It is worth noting that this result is obtained under the hypothesis of negligible nonadjacent cavities coupling, since in the other case the stop-b<strong>and</strong> concept is involved <strong>and</strong> the errors in the coupling cavities become important. Furthermore, the expression (4.55) does not contain product of different cavity errors, as we arrested the expansion to the first order, namely we neglect the correlation among the errors. Finally, under the previous hypothesis, it is also clear that the frequency error is independent of the relative positions of the cavities <strong>and</strong> that the variance of the frequency error is N times the one of the variables ε L,R p . 4. The perturbed axial field In the circuit model, the voltage on the capacitance of a generic cell is equal to the accelerating voltage. Then, except for the term 1/(jω2Ci) (1/(jωCi) for the lateral half cells), we look for the current Ii which is N Ii = , for i = 1, 2, . . . , N (4.57) l=i Tl 22 Note that such an expression comes from the hypothesis IN+1 = 1 which means that the last half cell is short-circuited <strong>and</strong> we arbitrarily fixed the current to one. In the following, we always use this hypothesis, <strong>and</strong> i = 1, 2, . . . , N.
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UNIVERSITÀ DEGLI STUDI DI NAPOLI F
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ii CONTENTS List of Figures 105 Ack
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iv INTRODUCTION In the state of art
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2 1. INTRODUCTION TO THE HADRONTHER
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Page 31 and 32: 1. GLOBAL PARAMETERS OF THE ACCELER
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Page 107: CONCLUSIONS AND OUTLOOK 101 Finally
Page 110 and 111: 104 BIBLIOGRAPHY [23] U. Amaldi, B.
Page 112 and 113: 106 LIST OF FIGURES 2.12 Drawing of
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108 LIST OF FIGURES methods, in add
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110 ACKNOWLEDGEMENTS Naples, Italy
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