The 11th Symposium on Accelerator Science and Technology, Harima Science Garden City, 1997

A Method of Beam-based Calibration for Beam Position Monitor
Mitsubiro MASAKI and Storage Ring Commissioning Group
Japan Synchrotron Radiation Research InstituteSPring-S, Kamigori, Ako-gun, Hyogo, 67S-12, Japan
Abstract

A method of beam-based calibration for beam position
monitor is proposed instead of mapping by use of anantenna or a wire. The proposed method was applied to thebeam position monitors of SPring-S storage ring.
§ 1 Introduction

A typical beam position monitor (BPM) consists of fourbutton electrodes. The map function of such BPM isusually measured by use of an antenna or a wire withoutbeam. The map obtained without beam is generally different
from the map by the beam passing, because a coaxial wave
guide is formed by an antenna or a wire. The TEM mode is
the most dominant wave guide mode excited by beampassing. The electric field distribution of the TEM mode
excited by an antenna with finite diameter is different fromthat by beam which is like line charge. Therefore the mapmeasured by an antenna bas to be corrected. The correction
will be more complicated if higher order modes are excited.
The electric field distributions of the. higher order modes in
the coaxial wave guide with an antenna or a wire are
essentially different from that in the ordinary wave guide.
For example, in case of SPring-S storage ring, the BPMs
are mounted on the vacuum chamber whose cross-section iscomplex with ante-chamber, and the beam signal is detected
at the rf frequency of 50SMHz which is higher than thecutoff frequency (about 350MHz) of the lowest higher ordermode TE 10. It will be difficult to perform the correction forthe maps. To evade the difficulty, a method of beam-based
calibration for the BPM is proposed. In section 2, a methodof the calibration with beam is described. In section 3, themethod is applied to the BPMs of SPring-S storage ring.

Vl(x,y)- V2(x,y)
a(x,y) = Vl(x,y) + V2(x,y)(eq. 1)V 4(x,y)- V3(x,y)(3(x,y) = V 4(x,y) + V3(x,y)(eq. 2)(eq. 3)V2(x,y)- V3(x,y)
o(x,y) = V2(x,y) + V3(x,y) .

(eq. 4)

Parameters a and ~ are sensitive to the horizontal deviationof beam. Parameters y and o are sensitive to the verticaldeviation of beam.

yV2

Vl

V3

V4

Figure 1 Schematic view of a BPM with four electrodes
and the coordinate system.
In case that a BPM is ideally completed, we can get thefollowing four pairs of two-dimensional map functions by
calculation,Xl(~,B), Y1(~,0)(eq. 5)Y2(~, 'Y)(eq. 6)x3(a, y), y3(a, y)X4(a, o), Y4(a, o)(eq. 7)

X2(~, y),§2 A method of beam-based calibration

As shown in Fig. 1, a beam position monitor generally
consists of four button electrodes. The following valuesa.(x,y), ~(x,y), y(x,y), o(x,y) are defined by the output
voltages Vl(x,y), V2(x,y), V3(x,y), V4(x,y) from eachelectrode,Vl(x,y)- V4(x,y)'Y(x,y) = Vl(x,y) + V4(x,y)(eq. 8)This means that there are four ways to obtain beam
positions, by use of the output voltages from threeelectrodes chosen out of the four electrodes. These four pairsof beam positions should coincide with each other if theBPM is ideal. When the four pairs of positions arecalculated by the voltages Vl'(x,y), V2'(x.y), V3'(x,y),V4'(x,y) measured by a real BPM with beam. thesepositions are generally different from each other. Because areal BPM bas such errors as manufacturing error of thevacuum chamber, imbalance of the gain of each electrodeand non-linearity of the signal processing circuit. The-83-The 11th Symposium on Accelerator Science and '(echnology, Harima Science Garden City, 1997

corrections of the voltages for these errors are generally as
follows,Vl(x, y) = Cl(x, y; lb, G) Vl'(x, y)(eq. 9)V2(x, y) = C2(x, y ; lb, G) V2'(x, y)(eq. IO)

N

X(f2,f3,f4) 2= C3(x, y ; lb. G) V3'(x, y)(eq. 11)V4(x, y) = C4(x, y ; lb, G) V4'(x, y),(eq. I2)

V3(x, y)obtained by the method of least squares. The chi-square isdefined by the pairs of(~. ~(a, y)) and (0, o(a, y)) derived
from voltage data sets (VI', V2', V3', V4') at several
different beam positions as follows,

where Ib is the beam current and G is the gain of signal
processing circuit. If x, y, lb and G dependence of thecorrection functions CI, C2, C3, C4 is negligible, thecorrections are performed by the following constantparameters FI, F2, F3, F4,
Fl = Cl(x, y ; lb, G)(eq. I3)

F2 = C2(x, y ; lb, G)(eq. I4)

F3 = C3(x, y ; lb, G)(eq. I5)

F4 = C4(x, y ; lb, G).

(eq. I6)

Since a, ~. y, o are defined by the ratios of the voltages
VI, V2, V3, V4, the constant parameters FI, F2, F3, F4
can be normalized by FI, as follows,n=l+[On- O(<Xn, 'Yn)] 2 },(eq. 24)

where N is the number of the beam positions. N must be
more than four. The parameters f2, f3, f4 can be optimized
by minirilizing the chi-square.

§3 Application to the BPMs of SPring-S
storage ring
The method developed in section 2 was applied to all ofthe 288 BPMs of SPring-S storage ring. The electron beam
was perturbed by a single dipole kick of the steering
magnets to the horizontal and vertical directions. The
steering magnets were excited in order as the phase of thecoherent oscillation rounds at all BPMs. The amplitude of
the oscillation was about 1.5mm. The typical distributionof beam positions at a BPM is shown in Fig. 2.

(eq. I7)

2~~~~~~~~~~~~

a, ~. y, oare reproduced by use of the measured voltages
VI', V2', V3', V4' corrected by the normalized parametersf2. f3, f4,a.(x, ) = Vl '(x,y)- f2 V2'(x,y)
yVl'(x,y) +f2V2'(x,y)

f3(f 4V4'(x,y) + f3V3'(x,y)

x,yVl'(x,y)- f4V4'(x,y)'Y(x,y) = Vl'(x,y) + f4V4'(x,y)

5::(u x,y)=,f2 V2 (x,y) + f3 V3 (x,y)

o.

. (eq. 2I)oTwo parameters are independent in a, ~. y, ~ and arederived from a and yby elirilination of parameters x, y,~=~(a, y)o =o(a., y) .

:~ ~-~~-~r :-1-f:l~~ : : ~~~· L~:~!=·!~:~~(eq. I9)-~2 -1.5(eq. 20)

f2 V2'(x,y)- f3 V3'(x,y),I(eq. 18)) = f4V4'(x,y)- f3V3'(x,y)= L {[f3n-f3(<X.n, 'Yn)] 2-1-0.5

0 0.5
x (mm)

1

1.5

2Figure 2 Typical distribution of beam positions
at a BPM. There are 36 points.
The functions ~(a, y) and o(a, y) were obtained by fitting
the numerical solution of· the static Laplace equation byboundary element method to seventh-orders polynomials,

7(eq. 22)

7-84-(eq. 25)

k=O n=O(eq. 23)The relationships are obtained from mapping data by acalculation. As equations 22 and 23 are satisfied regardlessof beam position, the correction parameters f2, f3, f4 arekJ3(a, 'Y) = L L bn,k-n<XD-y k-nO(<X., 'Y) =kL L dn,k-n<XD-y k-n.
k=O n=O(eq. 26)

The Ilth Symposium on Accelerator Science and Technology, Harima Science Garden City, 1997

The chi-square (equation 24) was calculated from equations
18, 19, 20, 21, 25 and 26. The correction parameters f2, f3,and f4 for each BPM were obtained by minimization of thechi-square.
The differences Ax, Ay of the four pairs of beam positions
which are derived from equations 5,6,7 and 8 are defined asfollows,~X(32241J.m, 46671J.ffi). All of the pairs of (Ax, Ay) with thecorrection converge less than (441J.m, 63Jl.m). The
correction parameters f2, f3, f4 disperse from 0.77 to 1.30.
The origin of large correction factors is mainly thereflection of beam signal in connectors of cables whichconnect electrodes to the signal processing circuits. The
reflection is not calibrated on the off-line bench test

= max(Xl, X2, X3, X4) - min(Xt. X2, X3, X4)- - - y with conection- - x with correction- -•- - x without correction - -a- - y without c:onection(eq. 27)~y4 ~~~~~~~~~~~~~~~~ 10= max(yt. Y2· Y3· Y4)- min(yr. Y2· Y3· y4).

(eq. 28)

The map functions, equations 5, 6, 7 and 8, were also
obtained by fitting the ni.Jmerical solution of the static
Laplace equation to seven-orders polynomials, for example,z806i._, -z~7Xt(J3, 8) =L L x~.~-nJ3n8k-n (eq. 29)-4

k=O n=O-67Yt(J3, 8)

kk= L L y<n1,~-nJ3n8k-n0-80(eq. 30)

50150
zoo
100
BPM serial number

Z50

k=O n=OFigure 4 The comparison between CODs with and without
the corrections by parameters f2, f3, f4.
Ax, Ay with and without the correction by f2, f3, f4 arecompared in Fig. 3.
0.7i!
r:l3o.sz =-=...=0 :2..= o.z0.3;!

i0.1~0050

150
100
zoo
BPM serial number

Z50-1a-zaI=I=~-3 !
300

0.7~

0.6

40.5
0.4=,z s.
...=1 :i!

0.2

0a-1i""'0.1<I

01~.=0.3i1
....

,s.

tl= 0.4ta.

~

0.6

050

150100zoo
BPM serial number

Z50

The comparison of the closed orbit distortions (CODs) withand without the correction is shown in Fig. 4. The CODs
with the correction become remarkably smooth. The root
mean squares (RMSs) of the horizontal COD with andwithout the correction are 0.32mm and 0.50mm, respectively. The RMSs of the vertical COD with and without
the correction are 0.26mm and 0.69mm, respectively .
Especially, the vertical COD was drastically improved. The
vertical beam positions are more sensitive to the various
errors than the horizontal positions. This is due to theallocation of the four button electrodes on the beamchamber whose cross-section is 70x40mm ellipse like.

~-z ......

300

Figure 3 The comparisons between Ax, Ay with andwithout the corrections by parameters f2, f3, f4.
Many pairs of (Ax, Ay) without the correction have values
larger than a few hundreds micron, and the maximum is§4 Conclusions
A method of beam-based calibration for beam position
monitor was developed in the section 2. The method wasapplied to all of the 288 BPMs of SPring-S storage ring.
The RMS of the measured COD was improved. The
improvement was mainly owing to the correction for thereflections of beam signals in connectors of cables. Thismethod is effective in the linear corrections for BPM output
voltages, however, is ineffective in the non-linear corrections, for example the non-linearity of the beam signal
processing circuits. To correct the non-linearity, we have totake account of the x, y, lb and G dependence of thecorrection functions C1, C2, C3, C4.
The mapping for all BPMs by an antenna or a wire spends
much time and is laborious. The correction of the map will
be difficult due· to the different electric boundary condition
from.beam passing. We propose this method as a usefulprocedure instead of mapping by use of an antenna or a wireon the off-line.

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