Appendice C Espressioni analitiche dei flussi magnetici
APPENDICE C: ESPRESSIONI ANALITICHE DEI FLUSSI MAGNETICI
La risoluzione del sistema (1.16):
⎪⎪
⎭
⎪⎪
⎬
⎫
⎪⎪
⎩
⎪⎪
⎨
⎧
⎥⎥
⎥⎥
⎥⎥
⎦
⎤
⎢⎢
⎢⎢
⎢⎢
⎣
⎡
ℜ + ℜ ℜ
−
ℜ
− ℜ + ℜ ℜ
−
ℜ
− ℜ ℜ ℜ
−
ℜ
− ℜ ℜ
=
⎪⎪
⎭
⎪⎪
⎬
⎫
⎪⎪
⎩
⎪⎪
⎨
⎧
+ +
4 3 2 1
0 0
2 0
0 2
0 0
4 3 2 1
d d d d
i s i
i i
s i s i
i i
s
i i
Ni Ni Ni Ni
ϕ ϕ ϕ ϕ
(C.1)
consente di risalire alle espressioni dei flussi dispersi ϕd1, ϕd2, ϕd3 e ϕd4, che sono del tipo:
(C.2)
∑=
= 4
1 j
j i ij
d N P i
ϕ
con i = 1, , 4. I coefficienti Pij assumono le espressioni:
( 3 2 2 )
11 44
5 6
P P
3
s s i s i i
ℜ + ℜ ℜ + ℜ ℜ + ℜ
= =
∆ (C.3)
( 2 2)
12 21 34 43
P P ℜ ℜ + ℜ ℜ + ℜi s 3 s i i P P
= = = =
∆ (C.4)
( )
2
13 31 24 42
P P ℜ ℜ + ℜi s i P P
= = =
∆ = (C.5)
C-1
Appendice C Espressioni analitiche dei flussi magnetici
3
14 41
P =P =ℜi
∆ (C.6)
( )( 2 2)
22 33
P P ℜ + ℜ ℜ + ℜ ℜ + ℜs i s 3 s i i
= =
∆ (C.7)
( )2
23 32
P P ℜ ℜ + ℜi s i
= =
∆ (C.8)
con
( 3 6 2 10 2 4 3)
s s s i s i
∆ = ℜ ℜ + ℜ ℜ + ℜ ℜ + ℜi
j
(C.9)
La risoluzione del sistema:
(C.10)
⎪⎪
⎪
⎭
⎪⎪
⎪
⎬
⎫
⎪⎪
⎪
⎩
⎪⎪
⎪
⎨
⎧
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎦
⎤
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎣
⎡
ℜ ℜ ℜ
ℜ
ℜ ℜ ℜ
ℜ ℜ ℜ
ℜ ℜ
ℜ ℜ ℜ
ℜ
− ℜ
ℜ ℜ ℜ
=
⎪⎪
⎪⎪
⎭
⎪⎪
⎪⎪
⎬
⎫
⎪⎪
⎪⎪
⎩
⎪⎪
⎪⎪
⎨
⎧
∑ − +
+ +
+ + +
+
=
u ls rs l r
l r l
r
m m
m m
l m l
v m
r m
r v m
j j
z z
i
N ϕ
ϕ ϕ ϕ ϕ
0 0
0 0
0
0 0
0
0 0
0 0
4
1 0 0 0 0
F F F F
consente il calcolo dei flussi ϕu ,ϕr ,ϕl ,ϕdr e ϕdl, che assumono espressioni del tipo:
( ) ( ) 4
0 0 1
1
F
u s s
j
g x g x N i ϕ
=
= + ∑ (C.11)
C-2
Appendice C Espressioni analitiche dei flussi magnetici
( ) ( ) 4
0 0 1
1
F
r s s
j
h x h x N i ϕ
=
= + ∑ j (C.12)
( ) ( ) 4
0 0 1
1
l s F s
j
l x l x N i ϕ
=
= + ∑ j (C.13)
( ) ( ) 4
0 0 1
1
r F
d s s
j
m x m x N i ϕ
=
= + ∑ j (C.14)
( ) ( ) 4
0 0 1
1
l F
d s s
j
n x n x N i ϕ
=
= + ∑ j (C.15)
Le funzioni g0,1, h0,1, l0,1, m0,1, n0,1 hanno le seguenti espressioni:
( ) 2( ) ( )
0
z m v m v z r l
g xs =⎡⎣ℜ ℜ + ℜ + ℜ ℜ ℜ ⎤ ℜ − ℜ
∆ ⎦ (C.16)
( ) ( )( ) ( )
( )
( )( )
2 2
1
2
2 2
2
2
m r l v z r l v z
s
z m v r l r l v
z m v r l r l v m v
g x
⎡ ⎤
ℜ ⎣ ℜ + ℜ ℜ + ℜ + ℜ ℜ + ℜ + ℜ ⎦
= +
∆
⎡ ⎤
ℜ ℜ ℜ ℜ + ℜ + ℜ ℜ + ℜ⎣ ⎦
+ +
∆
⎡ ⎤
ℜ ⎣ ℜ + ℜ ℜ + ℜ + ℜ ℜ + ℜ + ℜ ℜ ⎦
+ ∆
(C.17)
( ) ( ) ( ) ( )
0 0
z m v z v m r l
s s
h x =l x =ℜ ℜ ℜ + ℜ ℜ + ℜ⎡⎣ ⎤⎦ ℜ + ℜ
∆ (C.18)
( ) ( ) ( )( )2
1
r m z m z v l m z
h xs
⎡ ⎤
ℜ ℜ ℜ ℜ + ℜ + ℜ + ℜ ℜ + ℜ⎣ ⎦
= ∆ (C.19)
C-3
Appendice C Espressioni analitiche dei flussi magnetici
( ) ( ) ( )( )2
1
l m z m z v r m z
l xs
⎡ ⎤
−ℜ ℜ ℜ ℜ + ℜ + ℜ + ℜ ℜ + ℜ⎣ ⎦
= ∆ (C.20)
( ) ( ) ( ) ( )
( )( )
0 0
2
2 2
m r l v z z v r l
s s
v z m r l z r l v
m x =n x =ℜ ⎡ ℜ ℜ ℜ + ℜ + ℜ ℜ ℜ + ℜ ⎤⎣ ⎦
∆
ℜ ℜ + ℜ ℜ + ℜ + ℜ ℜ ℜ ℜ
+ ∆
+
(C.21)
( ) ( )( )
1
r m m z v l z m
m xs = −ℜ ℜ ⎡ ℜ + ℜ⎣ ℜ + ℜ + ℜ ℜ ⎤⎦
∆ (C.22)
( ) ( )( )
1
l m m z v r z m
n xs =ℜ ℜ ⎡ ℜ + ℜ⎣ ℜ + ℜ + ℜ ℜ ⎤⎦
∆ (C.23)
con
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
4 3 3 2 2
2 2 2
2
2 2 2 4
3 2
2
2 2
2 2 2
4 2
2
2 2
m r m r m v m r l r m m v
r m l m v m r z m v m r z l
r m l z m v r m z m v
r l z m v r z m v m l
m l m v m l z m v
m
∆ = ℜ ℜ − ℜ ℜ ℜ + ℜ − ℜ ℜ ℜ + ℜ ℜ ℜ + ℜ + + ℜ ℜ ℜ ℜ + ℜ − ℜ ℜ ℜ ℜ + ℜ − ℜ ℜ ℜ ℜ + + ℜ ℜ ℜ ℜ ℜ + ℜ + ℜ ℜ ℜ ℜ + ℜ
+ ℜ ℜ ℜ ℜ + ℜ + ℜ ℜ ℜ + ℜ + ℜ ℜ +
− ℜ ℜ ℜ + ℜ − ℜ ℜ ℜ ℜ + ℜ +
+ ℜ ℜl(ℜ + ℜm v)2+ ℜ ℜ ℜ ℜ + ℜ2 m l z( m v)2+ ℜ ℜ ℜ + ℜl z2( m v)2
(C.24)
C-4
