The Linearization of the Equations of Motion
81
and
T (x|δ) di + (a|δ) di = − [T (l|a) di + (l|x) di ]
=
1
R 0
tan
φ
2
1 + η 0
2 + η 0
R 0 (1 − cos φ) +
1 + η 0
2 + η 0
sin φ
=
1 + η 0
2 + η 0
tan
φ
2
(1 − cos φ) + sin φ
=
1 + η 0
2 + η 0
· 2 tan
φ
2
.
The matrix of the y-b block is
ˆ
M y =
1 0
−T 1
1 R 0 φ
0 1
1 0
−T 1
=
1 − T R 0 φ
R 0 φ
−T (2 − T R 0 φ) 1 − T R 0 φ
,
where
1 − T R 0 φ = 1 − φ tan
φ
2
,
−T (2 − T R 0 φ) = −
1
R 0
tan
φ
2
2 − φ tan
φ
2
.
In summary, we obtain
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
x f
a f
y f
b f
l f
δ f
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
1 R 0 sin φ
0
0
0 ( x|δ)
0
1
0
0
0 (a|δ)
0
0
1− φ tan (φ/2)
R 0 φ
0
0
0
0
(b|y)
1− φ tan (φ/2) 0
0
(l|x) (l|a)
0
0
1 ( l|δ)
0
0
0
0
0
1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
x i
a i
y i
b i
l i
δ i
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
,
where the abbreviated matrix elements are
(b|y) = −
1
R 0
tan
φ
2
2 − φ tan
φ
2
,
(x|δ) = −(l|a) =
1 + η 0
2 + η 0
R 0 (1 − cos φ) ,
(a|δ) = −(l|x) =
1 + η 0
2 + η 0
· 2 tan
φ
2
,
(l|δ) = −R 0
η 0
2 + η 0
φ −
1 + η 0
2 + η 0
2
sin φ
.
Note that the determinant of the matrix is unity, which also can be deduced
from that the determinant of all the contributing matrices is unity.
To conclude, let us compare the characteristic effects of the rectangular
dipole and the sector dipole in the limit of small deflection angle φ. The x-a
matrix and the y-b matrix of both dipoles can then be approximated as
Sector dipole:
ˆ
M xa →
1
R 0 φ
−φ/R 0 1
,
ˆ
M yb →
1 R 0 φ
0 1
,
Rectangular dipole: ˆ
M xa →
1 R 0 φ
0 1
,
ˆ
M yb →
1
R 0 φ
−φ/R 0 1
.
Thus we have the interesting effect that the characteristic behavior in the
horizontal plane and the vertical plane is exchanged.