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

60 4. CIRCUIT MODEL
Remembering the expression of T N with errors, in the calculations
of the expression (4.57), we obtain a formula similar to the formula
(4.43), but in this case the product is limited to the last N − i + 1
terms. And therefore, we have dealings with
N
= ( ˙ T + Pi + Qi)( ˙ T + Pi+1 + Qi+1) . . . ( ˙ T + PN + QN)
l=i
Tl
∼ = ˙
T N−i+1 +
N
T˙ q−i Pq ˙ T N−q +
q=i
<strong>and</strong> in a similar way, we obtain
N
= ˙ T N−i+1 + ˙ T N−i
N
ε + q +
l=i
Tl
q=i
cosh x
sinh x
N
T˙ q−i Qq ˙ T N−q
q=i
N
ε + q [♥ + ♦] +
q=i
N
q=i
(4.58)
ε − q ♣
(4.59)
where the cards symbols represents the following matrices
⎛
⎞
sinh (N − i)x jωM sinh x cosh (N − i)x
♥ = ⎝
⎠
cosh (N−i)x
sinh (N − i)x
jωM sinh x
⎛
⎞
− sinh (2q − N − i)x jωM sinh x cosh (2q − N − i)x
♦ = ⎝
⎠
−1 cosh (2q−N−i)x
sinh (2q − N − i)x
jωM sinh x
⎛
⎞
cosh (N − 2q + i)x jωM sinh x sinh (N − 2q + i)x
♣ = ⎝
⎠
− cosh (N − 2q + i)x
<strong>and</strong> therefore it is
Ii(ε L p , ε R p ) =
sinh (N−2q+i)x
− jωM sinh x
= cosh (N − i + 1)x +
−
N
q=i
ε + q
sinh (N − 2q + i)x
tanh x
cosh (N − i)x +
sinh (N − i)x N
ε
tanh x
q=i
+ q
+ ε −
q cosh (N − 2q + i)x
(4.60)
As stated in the previous section, the machining tolerances are represented
by a displacement of the variable x → jπ/2 + ∆x, where ∆x is
the one of the expression (4.52), namely N p=1 ε∗p/N. After a substitution
of the perturbed x in the previous expression <strong>and</strong> noting that we
are interested in the odd cavities, which are the accelerating ones, one