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

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54 4. CIRCUIT MODEL Under this hypothesis, we can write And considering that δω L 0i ω0 δω R 0i ω0 = − δCL i C L i = − δCR i C R i ω L 0i = ω R 0i − δω R 0i + δω L 0i ω R 0i = ω L 0i − δω L 0i + δω R 0i <strong>and</strong> after a substitution in the t11 expression (4.2), one obtains t L 11i = 1 1 − Ki ωL 0iωR 0i + ωL 0i(δωL 0i − δωR 0i) ω2 ∼ = ∼ 1 = 1 − Ki ω2 0 ω2 − 2δωL 0i = t11i ˙ + ε ω0 L i t R 11i = 1 1 − Ki ωR 0iωL 0i + ωR 0i(δωR 0i − δωL 0i) ω2 ∼ = ∼ 1 = 1 − Ki ω2 0 ω2 − 2δωR 0i = t11i ˙ + ε ω0 R i where we introduce the new error parameters ε L i = − 2 δω Ki L 0i ω0 ε R i = − 2 δω Ki R 0i ω0 . (4.35) (4.36) (4.37) (4.38) By neglecting the higher order terms, the matrix T becomes then Ti = ˙ Ti + ε L ⎛ ⎝ i 1 jωM 0 ⎞ t11 ˙ ⎠ + ε 0 R ⎛ ⎝ i 0 jωM 0 ⎞ t11 ˙ ⎠ 1 (4.39) where the variables with a dot on top indicate imperturbed quantities. The expression (4.39) can be rearranged as Ti = ˙ Ti + εLi + εR ⎛ i ⎝ 2 1 j2ωM ⎞ t11 ˙ ⎠ + 0 1 εLi − εR ⎛ ⎞ 1 0 i ⎝ ⎠ (4.40) 2 0 −1 <strong>and</strong> in a short way where we introduced ε + i = εL i + ε R i 2 Ti = ˙ Ti + ε + i Pi + ε − i Qi ε − i = εL i − ε R i 2 (4.41)
2. THE PERTURBED TRANSMISSION MATRIX 55 <strong>and</strong> Pi = I+B, I being the identity matrix <strong>and</strong> if one defines b = 2jωM then ⎛ 0 B = ⎝ 0 ⎞ b ⎠ , 0 ⎛ 1 Qi = ⎝ 0 ⎞ 0 ⎠ −1 (4.42) 2.2. The perturbed transmission matrix of the whole chain. The behavior of the whole chain is represented by the product of the transmission matrix for each cell; in the case of errors, this product is not given from T N because the T is different for each cell. It is given by the product of terms like T + Pp + Qp instead. Ttot = ( ˙ T + P1 + Q1)( ˙ T + P2 + Q2) . . . ( ˙ T + PN + QN) = ˙ T N + P1 T ˙ . . . ˙ T + (N−1)times ˙ T P2 T ˙ . . . ˙ T + . . . + (N−2)times ˙ T . . . ˙ T PN−1 (N−2)times ˙ T + ˙ T . . . ˙ T PN + Q1 T ˙ . . . (N−1)times ˙ T + (N−1)times ˙ T Q2 T ˙ . . . ˙ T (N−2)times + . . . + ˙ T . . . ˙ T QN−1 (N−2)times ˙ T + ˙ T . . . ˙ T QN + high order terms (N−1)times (4.43) where we neglected the product of two or more perturbed matrices. Then, in a short form we can write Ttot ∼ = ˙ T N + N T˙ p−1 Pp ˙ T N−p + p=1 The first term of the previous sum is N T˙ p−1 Qp ˙ T N−p p=1 T˙ p−1 Pp ˙ T N−p = UΛ p−1 U −1 Pp UΛ N−p U −1 = ε + p UΛ p−1 U −1 (I + B)UΛ N−p U −1 = T˙ N−1 + ε + p UΛ p−1 U −1 BUΛ N−p U −1 = ε + p (4.44) <strong>and</strong> since T can be expressed as the product of a diagonal matrix by the eigenfunctions matrices, one obtains Ttot = ˙ T N + ˙ T N−1 + N p=1 N ε + p + p=1 N p=1 ε − p UΛ p−1 U −1 QpUΛ N−p U −1 ε + p UΛ p−1 U −1 BUΛ N−p U −1 + (4.45) The expressions for the third <strong>and</strong> fourth terms of previous formula are obtained in a similar way. Then, the expression for the transmission
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UNIVERSITÀ DEGLI STUDI DI NAPOLI F
Page 4 and 5:
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
Page 10 and 11: 4 1. INTRODUCTION TO THE HADRONTHER
Page 12 and 13: 6 1. INTRODUCTION TO THE HADRONTHER
Page 14 and 15: 8 2. LINAC AND SCL ACCELERATORS par
Page 16 and 17: 10 2. LINAC AND SCL ACCELERATORS wh
Page 18 and 19: 12 2. LINAC AND SCL ACCELERATORS at
Page 20 and 21: 14 2. LINAC AND SCL ACCELERATORS 2.
Page 22 and 23: 16 2. LINAC AND SCL ACCELERATORS f
Page 24 and 25: 18 2. LINAC AND SCL ACCELERATORS Fi
Page 26 and 27: 20 2. LINAC AND SCL ACCELERATORS re
Page 29 and 30: CHAPTER 3 Design T
Page 31 and 32: 1. GLOBAL PARAMETERS OF THE ACCELER
Page 33 and 34: 2. ANALYSIS OF COUPLED CAVITIES 27
Page 35 and 36: 3. SINGLE CAVITY DESIGN 29 Spice ha
Page 37 and 38: 20mA 15mA 10mA 5mA 3. SINGLE CAVITY
Page 39 and 40: 3. SINGLE CAVITY DESIGN 33 • HFSS
Page 41 and 42: 4. BRIDGE COUPLERS DESIGN 35 4. Bri
Page 43 and 44: 5. COUPLING BETWEEN WAVEGUIDE AND B
Page 49: 5. COUPLING BETWEEN WAVEGUIDE AND B
Page 52 and 53: 46 4. CIRCUIT MODEL where the matri
Page 54 and 55: 48 4. CIRCUIT MODEL For the latest
Page 56 and 57: 50 4. CIRCUIT MODEL left side of th
Page 58 and 59: 52 4. CIRCUIT MODEL and</st
Page 62 and 63: 56 4. CIRCUIT MODEL matrix of the w
Page 64 and 65: 58 4. CIRCUIT MODEL For the derivat
Page 66 and 67: 60 4. CIRCUIT MODEL Remembering the
Page 68 and 69: 62 4. CIRCUIT MODEL the bars is pro
Page 70 and 71: 64 4. CIRCUIT MODEL is unitary, one
Page 72 and 73: 66 4. CIRCUIT MODEL 120 100 80 60 4
Page 74 and 75: 68 5. RADIOFREQUENCY MEASUREMENT Ri
Page 76 and 77: 70 5. RADIOFREQUENCY MEASUREMENT Fi
Page 78 and 79: 72 5. RADIOFREQUENCY MEASUREMENT re
Page 80 and 81: 74 5. RADIOFREQUENCY MEASUREMENT de
Page 82 and 83: 76 5. RADIOFREQUENCY MEASUREMENT In
Page 84 and 85: 78 5. RADIOFREQUENCY MEASUREMENT me
Page 86 and 87: 80 5. RADIOFREQUENCY MEASUREMENT pe
Page 88 and 89: 82 5. RADIOFREQUENCY MEASUREMENT Al
Page 92 and 93: 86 5. RADIOFREQUENCY MEASUREMENT ta
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Page 100 and 101: 94 5. RADIOFREQUENCY MEASUREMENT We
Page 102 and 103: 96 5. RADIOFREQUENCY MEASUREMENT ra
Page 105 and 106: Conclusions and Ou
Page 107: CONCLUSIONS AND OUTLOOK 101 Finally
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104 BIBLIOGRAPHY [23] U. Amaldi, B.
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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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