Instrumentation & Measurement Magazine 24-9 - 25

Fig. 3. Electromagnetic field of a point charge in a beam-pipe with circular cross-section: (a)-(c) E-field and wall current density for an off-center beam travelling
with β = 0; 0.5; 0.8; (d) E- and H-field, and image charges qw
illustrated for a centered point charge, travelling at a relativistic velocity β = 1.
transverse electro-magnetic (TEM) field ( " pancake-like " field)
and the related image (or wall) charges qw
, which are distributed
symmetrically on the inside surface of the metallic beam
pipe of circular cross-section if the beam of charge q travels in
the center of the vacuum chamber.
In the following we assume this case, a so-called " pencil "
beam (with infinite small transverse dimensions), Ibeam
, traveling
at or near a relativistic velocity (β → 1), which therefore
has only transverse electromagnetic field components (TEM
field), in a conductive beam pipe with circular cross-section
of radius R. The EM-field problem is now reduced to a twodimensional
electrostatic problem of a line charge located at
(x = r cos φ, y = r sin φ) in a conductive cylinder (see Fig. 4a.)
The solution of the related Laplace equation based on the image
charge method yields the wall current distribution as closedform
expression [13]:
JR r    ,Φ, ,
w
I
beam
Rr

2R R r Rr 
22 2 cos Φ
 
which also can be expressed in form of a series expansion [12]:

JR r  
2 RR
w
,Φ, ,


 beam
 cos Φ
I 
n



1
n 1



r
(6)
(5)
22
B
/ , ,
4 we find:
I
I
elec  
2 s rR/ ,,
beam
elec
(8)
with the sensitivity functions for the A and B electrodes based
on (6):
s rR    cos sin 
2
A
 4nRn 1 
/ , ,


1
  cos sin n
s rR  4  
n 1 nR


1 
n
r

n 



2
rn 

n
n
 
(9)
A wall current related fraction Ielec
trode covering an arc α:
I R JR r d


elec


/2
 ,Φ, , Φ
/2
w
(7)
For the two horizontal arranged electrodes A and B of Fig.
is induced on a BPM elec(10)
Following
the intensity normalization concept of (2) for our
two symmetric, horizontal electrodes A and B, using (6), (8),
(9), and (10), we can approximate the horizontal position characteristic
for a circular beam pipe of radius R as:
Fig. 4. Position characteristic of a " pencil " beam in a beam pipe with circular cross-section. (a) Image charge distribution on the beam pipe wall (R,Φ) for a
line charge located at (r,φ); (b) Position characteristic f (x, y) = const Δ/Σ for horizontal electrodes covering α = 30°; and c) Δ/Σ = f (x) for different vertical beam
displacements in the beam pipe of R = 12.5 mm.
December 2021
IEEE Instrumentation & Measurement Magazine
25

Instrumentation & Measurement Magazine 24-9

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