# Instrumentation & Measurement Magazine 24-9 - 27

```Fig. 5. Bunch signals with Gaussian and cos2
shape in time and frequency domain. (a) Time-domain; (b) Frequency-domain; and (c) 20 log ( ) log (0)
10 If
10
Output Signal of a " Button-style " BPM Pickup
Among the different types of BPM pickups, the " buttonstyle "
BPM is the most popular one because of its simplicity,
robustness, and compact design, and has well defined, reproducible
characteristics at relatively moderate costs.
Some basics have already been discussed in the section of
the button electrode of diameter d = 2r couples capacitively
to the beam. Together with the load impedance Rl
, which
in most applications is chosen to be 50 Ω, it forms a highpass
filter based on its intrinsic capacitance to ground (the
beam-pipe) Cb
. Fig. 6a shows the equivalent circuit of a button
electrode, with:
itbeam
v t it it
C
beam
 
12
2


beam
rr
bb
t
 D c


being the beam related generator voltage at the button, no load
impedance assumed [3]. This induced voltage is proportional
to the interception area of the button, given by the coverage
factor ϕ ≈ r / (2D) (see also (4), with D = 2R being the diameter
of the beam-pipe) and the transit time of the particles passing
the button Δt = 2r/c, and inverse proportional to the buttonto-ground
capacitance Cb
i.e., v c and a beam current signal ibeam
bunch length longer than the button diameter σs
. Eq. (18) holds for a relativistic beam,
(t) that resembles a
>> 2r. For a
centered beam x = y = 0 the transfer impedance of a button
electrode follows as:
ZR
I
bl
beam
 
age factor:
Vb 




with: 1/time constant: 
r
1
The button capacitance Cb is defined by its dimensions, di2D
ameter
2r and thickness, but also by the gap between button
and beam pipe, and other geometrical details. While Cb
can
be estimated analytically, fringe field effects may dominate,
and a numerical computation or a RF measurement will give
a more precise value. Fig. 6b shows the modulus of Zb
some variations of the parameters in (19), a transfer impedance
December 2021
(ω) for
Fig. 6. (a) Equivalent circuit and (b) transfer impedance |Z (f) | of a button
electrode.
IEEE Instrumentation & Measurement Magazine
27
1
RC
lb

 
1/
11
21
j
 

, 1/transit time: 2
c
  , cover2r
/
(19)

 beam
 DcC
r

2
(18)
button

I
.
of Zb ≈ 1 Ω @ f ≈ 1 GHz and a cut-off frequency in the range
f3dB
≈ 500-2000 MHz is typical.
Based on (3) and (19), Fig. 7 shows the output signal in
frequency and time-domain of a button BPM electrode of
2r = 8 mm diameter to a single bunch of Gaussian longitudinal
particle distribution, as given in (17), for three different bunch
lengths. The time domain voltage signal response of a button
electrode to a Gaussian bunch has the form:
v t  
 jfe

ff

f t
1  j
22
e df
j2ft
(20)
but unfortunately, there exist no closed form solution. The results
shown in Fig. 7 are found numerically by applying the
inverse discrete Fourier transformation (iDFT or iFFT). For
```

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