Gentec 203605, Руководство пользователя

Страница: 55 / 58

Beamage-M² User Manual Revision 6 48
𝜃
𝑦
=
𝑑
𝜎
𝑦
𝑓
𝜃
𝜎
=
𝑑
𝜎
𝑓
In the laboratory or usual system of coordinates (X’, Y’, Z’), the coordinates of the latest calculated position
of the centroid for both X’ and Y’ axes are given by the following equations:
𝑥
′
̅ (𝑧) =
∫ ∫
𝐸(𝑥
′
, 𝑦
′
, 𝑧
′
)𝑥
′
𝑑𝑥
′
𝑑𝑦
′
∞
−∞
∞
−∞
∫ ∫
𝐸(𝑥
′
, 𝑦
′
, 𝑧
′
)𝑑𝑥
′
𝑑𝑦
′
∞
−∞
∞
−∞
𝑦′
̅ (𝑧) =
∫ ∫
𝐸(𝑥′, 𝑦′, 𝑧′)𝑦′𝑑𝑥′𝑑𝑦′
∞
−∞
∞
−∞
∫ ∫
𝐸(𝑥′, 𝑦′, 𝑧′)𝑑𝑥′𝑑𝑦′
∞
−∞
∞
−∞
The coordinat
es of the mean position of all computed centroids for both X’ and Y’ axes are thus defined by
the following equations, which are simple arithmetic means:
𝑥̅
𝑀
′
=
∑ 𝑥̅
𝑖
′
𝑖
𝑛
𝑦̅
𝑀
′
=
∑ 𝑦̅
𝑖
′
𝑖
𝑛
where
𝑥′
̅
𝑖
(𝑧)
and
𝑦̅′
𝑖
(𝑧)
are the centroid coordinates for the
X’ and Y’ axes already saved in the buffer, and
n is the number of computed centroid positions saved in the buffer.
The azimuth angle, which is the angle between the usual X’ axis and all computed centroids, is given by
the following equation:
𝜓 =
1
2
𝑎𝑟𝑐𝑡𝑎𝑛 (
2𝑠
𝑥𝑦 ̃
2
𝑠
𝑥̃
2
− 𝑠
𝑦̃
2
)
where we have the following definitions:
𝑠
𝑥
= √
∑ 𝑥̅
𝑖
2
𝑖
𝑛 − 1
𝑠
𝑦̃
2
=
∑ (𝑦̅
𝑖
′
− 𝑦̅
𝑀
′
)
2
𝑖
𝑛 − 1
𝑠
𝑥𝑦 ̃
2
=
∑ (𝑥̅
𝑖
′
− 𝑥̅
𝑀
′
)(𝑦̅
𝑖
′
− 𝑦̅
𝑀
′
)
𝑖
𝑛 − 1
In the beam axis coordinate system (X, Y, Z), the beam positional stability values in the azimuthal direction
(X) and perpendicularly to the azimuthal direction (Y), which are 4 times the standard deviations of all
computed centroid values, are given by the following equations:
Δ
x
(z) = 4s
x
Δ
y
(z) = 4s
x
The overall positional stability is given by:
Δ(𝑧) = 2√2𝑠