# Instrumentation & Measurement Magazine 24-9 - 26

```Δ AB
Σ



AB R

4sin



2

with x being the calibrated horizontal beam position with
reference to the beam pipe center (typically expressed in
mm, and α being the coverage angle of the electrodes, intercepting
some fraction of the wall (image) current Jw
. In
general, the position characteristic is non-linear, however,
for narrow electrodes sin / 2 / 2
placements x2
+ y2
approximatively:
Δ2
Σ

R
with 2 / R = kBPM
1 / kBPM
x
(12)
sometimes called the monitor constant, and
expresses the sensitivity of the BPM pickup in the
linear region near the beam pipe center and serves as linear calibration
factor. From (5) and (7) we can also find a closed form
expression for the normalized position characteristic, now in
Cartesian coordinates:
Δ
Σ
with:
s x yR, , ,


 
  
  





/2
/2
 arctan
2
J R d
R x y
w
,Φ Φ
2
 

2

x yR

 arctan
 

R x y2

x yR

being the sensitivity function.
An example of the horizontal position characteristic based
on (13) and (14) is illustrated in Fig. 4b as parametric plot with
contours of constant Δ/Σ, and in Fig. 4c as function of the horizontal
beam position. The small covering angle (α = 30°) of
the electrodes lead to high non-linear " cushion " effects in the
position characteristic, with substantial cross-coupling to the
vertical plane, however, most of these non-linear position effects
can be corrected in the BPM data post-processing by
look-up tables or polynomial-fit functions, assuming also a set
of vertical BPM electrodes is available. Historically, the splitplane
type BPM pickup, sometimes called " shoe-box " BPM,
was used to deliver a perfect linear position characteristic, but
has several other drawbacks and is a bit outdated.
The analytical expressions given in this section for the position
characteristic of a broadband BPM pickup in a beam-pipe
of circular cross-section can be modified for a different arrangement
of the electrodes, e.g., electrodes rotated by 45°, as
well as for a different normalization procedure, e.g., applying
the ratio of the two opposite BPM electrodes as difference of
the logarithmic BPM electrode signal levels:
26
22 2
22 2
tan / 4 2Ry
tan / 4 2Ry
(14)
i
bunch
  
t q t it
  w
(16)
which is cancelled by the wall current iw(t), originated by the
image charges qw
In most other cases the longitudinal distribution of the
particles in the RF bucket is approximated by an analytic
function ibunch



(t) to describe the envelope function of the
longitudinal particle current distribution vs. time, e.g., in
most cases by a Gaussian, sometimes by a cos2
or similar
particle density distribution function (see (17) and Fig. 5,
with Fig. 5c indicating the spectral lines, e.g., in this example
spaced by 100 MHz, in a logarithmic scaled magnitude
display similar to that of a spectrum analyzer when observing
a single bunch circulating in a ring accelerator of small
circumference).
time-domain
t
i
Gauss
t 


zeN
e
2t
it     bb 
c s
 

22
cos bb
 , /2t tt /2
tt
Nt
1o
0,
2
cos
elsewhere

2t
2
2
frequency-domain
2
I f eNe
Gauss
 
 t
f
2
= −q distributed around the azimuth of the
beam pipe wall. In some cases, e.g., very short bunches, this
δ-signal approach for the bunch current is an acceptable approximation,
with the Dirac (δ) function defined as  1t
 .

AB s x y R s xyR
AB s xyR s xy R



  
  
,, ,  ,, ,
, ,,  , ,,


(13)
<< R2
and small beam disthe
beam position response follows
 x
 higher order terms
(11)
Δ
Σ
 vv (15)
vB
20 log  20 log AB

10


vA
10
 20 log10
  
For cross-section geometries different from the discussed
circular BPM pickup, e.g., elliptical, rectangular or arbitrary
shape, also to include the details of the shape of the BPM electrodes
itself, a numerical analysis is required. Still, in most
cases a solution of the electrostatic problem, i.e., solving the
Laplace equation for one of the BPM electrodes in two or three
dimensions is sufficient, followed by applying Green's reciprocation
theorem to compute the scalar potential of a pair or
quadruple of symmetric arranged electrodes, thus the position
characteristic for the normalized horizontal / vertical BPM
signals [14].
Bunched Beam Signals
For the analysis of the position characteristic of a broadband
BPM pickup, based on the sensitivity function s(x, y), the discussed
line charge approach is usually sufficient. However,
to estimate the waveform and signal power delivered by the
BPM pickup electrode in (3), the bunch current signal ibunch
(t)
(ω) needs to be known-as discussed in this section-
along with the transfer impedance Z(ω) of the pickup electrode
- discussed in the next section.
A point-like charge q = zeN, with z being the charge state
(for ions), e ≈ 1.6 · 10−19
or Ibunch
C the elementary charge and N the number
of particles, traveling with relativistic velocity β = v / c = 1
in a perfectly conducting vacuum chamber (see Fig. 3d) has a
bunch current:
I f  ft ft
2
eN ft
 bb
sin  b
1


(17)


IEEE Instrumentation & Measurement Magazine
December 2021
```

# Instrumentation & Measurement Magazine 24-9

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