JAERI-Conf 95-021

PB08
NUMERICAL CALCULATION OF THE ELECTROMAGNETIC COUPLING
STRENGTH BETWEEN THE ELECTRODES OFA BEAM-POSffiON MONITOR
Tsuyoshi SUWADA
KEK, National Laboratory for High Energy Physics
1-1 Oho, Tsukuba, lbaraki, 305 Japan
Abstract

2. Brief overview of the charge-simulation method

The PF 2.5-GeV linac is now being upgraded forthe. KEKB project. A stripline-type beam-position
monitor is under development in order to easily handle
the beam orbits of primary high-current electron beamsfor producing a sufficient number of positrons. The
spatial dimensions of the mechanical monitor
components were decided during its design. In themonitor design, it is particularly important to determine
the opening angle of the electrode, because larger
electromagnetic couplings between the electrodes
generate a larger nonlinearity concerning pick-up
signals. The opening angle of the electrode was decided
based on a numerical calculation which considered thecharge-simulation method. This report summarizes themethod and results of the numerical calculation in detail.

A numerical analysis ·based on the chargesimulation method is briefly discussed here using asimple example. Two conductor rods and a groundpotential plane are arranged by some geometricalconfiguration (Fig. 1). The electrostatic potentials andcharges on rods #1 and #2 are given as (V1, Q1) and (V2,QJ, respectively. Rod #1, #2 and the ground-potential
plane are mutually electrically coupled throughequivalent capacitors (C1i).
Conductor rod #1 (VI' Qt)

n .

yt"O
cloT

Conductor rod #2 (V2, QJ. I

1. Introduction
The linac is required to stably accelerate primary
high-current electron beams in order to produce asufficient number of positrons in the KEKB project[!].
The beam-position monitor (BPM) is important in orderto easily handle orbits of high-current electron beams so
as to suppress any beam break-up generated by a largetransverse wake field. A stripline-type BPM wasdesigned in order to perform this function. In its design,
it is required to determine the mechanical spatial
dimensions of the BPM components, that is, the bore
radius, the stripline length, the opening angle of theelectrode etc.. The opening angle of the electrode is
generally designed from the point of view of the signalto-noise ratio of the pick-up signals and theelectromagnetic coupling strength between theelectrodes. That is, although a large opening angle isdesirable from the former point of view, it generates alarger electromagnetic coupling between the electrodes
through equivalent capacitors, which are onlydetermined by the geometrical configuration of theelectrodes and the bore radius. The capacitive couplings
between the electrodes do not preserve a good linearityof the pick-up voltage induced by beams, and, thus, asmall opening angle of the· electrode is desirable fromthis point of view. A numerical calculation of theelectromagnetic coupling strength between the electrodes
was performed in order to determine the optimum
opening angle of the electrode on the basis of the charge
simulation method[2]. This report describes the methodand results of the numerical calculation in detail.

1Ct2:;= c20

Ground-potential plane
Fig. 1. Schematic drawing of a simple three-conductor
system comprising two conductor rods and a groundpotential plane.
These parameters can be related by the following
formula:Ql = ciO vi + c 12 rvl - v2 ),(2- 1)

Q2 = c20 v2 + C2JV2 -vi),(2- 2)C OJ.. = C JO.. (i :;t j :;t 0).

(2- 3)

Here, C10 and C ii ( j:;t:O) are called the self-capacitanceand the partial capacitance, respectively. The couplingstrength between the rods is given by the ratio of thepartial capacitance to the self-capacitance (C/C 10). Thisratio is determined only by the geometrical configurationof the conductor system. The charge-simulation method
rearranges the charge (Q J as a finite number ofimaginary charges, q1 (Q 1 = I. q1 ), on the conductor
Isurface. The configuration of the imaginary charges ischosen so as to give an electrostatic potential (V;) on the-269-

JAERI-Conf 95-021

surface of each conductor which can be uniquelydetermined by the electrostatic theorem.
3. Application to the beam-position monitor
The capacitive couplings of the stripline-type BPM
are shown in Fig. 2.

circles i~dicate the outline points on the pipe andelectrode surfaces, the cross points being imaginary
charges. The outline points on both the inner (radiusc=28.5mm) and outer (radius d=32.5mm) surface of thepipe are configured in equal (n) parts in the azimuthal
direction. The electrode surface (inner radius, a=20mm;
outer radius, h=21.5mm) is segmented in the azimuthal
direction so as to have an azimuthal angle which is equal
to the segmentation of the pipe, and is segmented inequal m parts in the radial direction. The coordinates (x;.
y~ of the outline points are given as:

clO

v3, Q3 <<1>=..;..;1t"-~)~-+-~+-~b;--+-+-v~~· Ql (<1>=0)x 1 = r cos(21Ci In),

(3 -1)

y 1 = r sin(21Ci In),

(1 SiS n). (3- 2)

The coordinates (XitYJ of the induced charges are given
as:Fig. 2. Cross-sectional view of the stripline-type BPM.
The equivalent capacitive couplings are also shown
inside the pipe.
It is a conventional stripline-type BPM with 1C/2
rotational symmetry. The interval between the electrodeand the inner surface of the pipe is chosen in order tomake a 500 transmission line. The opening angle {A<I>)of the electrode should be determined in terms of boththe signal-to-noise ratio of the pick-up signals and thecoupling strength between the electrodes.
The electric field generated by relativistic beams
inside the pipe can be treated as a two-dimensional
electrostatic field because the field is almost boosted in
the transverse direction to the beam axis. Thus, only theelectrostatic field is treated in the analysis.
-Outline point on the pipe surface\~
Imaginary charge

X, = (r ±B) cos(21Ci In),(3- 3)Y = (r ±B) sin(21Ci In),

(1 S i S n). (3- 4)

1Here, r indicates the pipe and electrode radius (r=a+(ba) j/m, (1 s j s m)) is used in the case of the electrodeside surfaces), and B is the interval between the outline
point and the imaginary charge to the radial direction,
which must be inside the outline point. The parameter B
is generally given by the following formula:

B =EX f,

(3- 5)

where e is twice the interval length between theadjoining outline points and/ is a free parameter to betuned (to be generally chosen within 0.2-1.5). The
electrostatic potential (~) on each outline point can becalculated by superposing the potentials generated by allof the imaginary charges, as follows:(3- 6)(3 - 7)(3- 8)I

Outline point on
the electrode surface

where v1 is the potential generated by one imaginary
charge (q~ and P li is the potential coefficientapproximated for a line charge with infinite length. The
coupling strength (CS) between electrodes #1 and #2 iscalculated using:

IOpening angle{A<I>)
Beam centerFig. 3. Segmentation of the beam pipe and electrodesurface.

(3- 9)= I.q./I- 11~.•iFirst of all, the surface of the pipe and eachelectrode are segmented as shown in Fig. 3. The black-270-li(3 -10)JAERI-Conf 95-021

Here, the summations of the imaginary charge ( q; andQ;) are on electrode #1 and on all of the electrodes,respectively.

are approximately estimated only by taking account of
the nearest-neighbor electrode using equation (2-1 ), as
follows:

4. Check of the numerical calculation(4- 3)The parameter f was tuned so as to produce goodsymmetrical and constant electrostatic potentials on theelectrode surfaces. Figure 4 shows the result of acalculation in terms of the electrostatic equipotential
field lines on the use of the parameter /=1, which gives
the best result. The segmentation numbers (n and m)were also tuned by checking the convergence of the total
induced charges on both the electrode (Q 1) and pipe
surfaces (I. ~ ). A segmentation number of n= 120 (to= cs

Table 1. Several parameters used in the checkcalculation.
0
Pipe potential V0 (Volt)
1Electrode potential V1-V4 (Volt)
60-140Azimuthal segmentation n3
Radial segmentation m
0.2-1.5
Free parameter f
60
The O-p£ning a~sie of the electrode {deg)(4- 4)'I

a2 _ A2
Voc
I,(4-5)22a +A - 2aA cos(cjl 1 - <p)where A and <p are the displacement and azimuthal
angle of the beam, and <jl; is the azimuthal angle of theelectrode. Figure 5 shows the variations (dot lines) of
the nonlinearity (AQdQ 1) as a function of the opening
angle of the electrode.
25~================~-----,10!-Coupling Strength(%)

I- ·A-· Nonlinearity (A,=3mm)•- ·•- • Nonlinearity (N=Smm) , '70·---.. __ ..,---""

ill'60"-, ..
,.
~,.,--.
__ ,,...

0~~~~~~~~~~~~~050

04080
60
4020Opening Angle (degree)

100Fig. 5. Variations of the coupling strength (solid line)
and nonlinearity (dot lines) as a function of the opening
angle of the electrode.

302010v.

where AQ 1 is the charge induced by electrode #2. The
pick-up voltage (V;) is determined by the well-known
wall-current formula:I

be fixed on the segmentation number m=3) was obtainedby a convergency calculation, the accuracy of which wasdeduced to be -2% from the convergency. The
parameters used in the calculation are summarized in thefollowing table.

(v;-~)... ......... ...

5. Conclusions

0~----------------~
Fig. 4. Calculated electrostatic potential field lines (solidlines) inside the BPM pipe. The black points indicate theoutline points of the pipe and the electrode surface.

The opening angle of the stripline-type BPM wasanalyzed on the basis of the charge simulation method.
The coupling strength between the electrodes was -11%
at the opening angle 60° which generated a 2.3%
nonlinearity of the pick-up signals induced by the beams.

4. Results of the numerical calculation

References

The coupling strength between the electrodes wasobtained by using equation (3-10 ), in which theparameters derived in the section 3 were used. Figure 5
shows the variation (solid line) of the coupling strength
as a function of the opening angle of the electrode. Thecoupling strength is about 11% for an opening angle of60°. The linearity of the pick-up signals is changed by
the induced charges generated through the capacitive
couplings. The induced charges through the couplings[1] A Enomoto et al., "REFORMATION OF THE PF
2.5-GEV LINAC TO 8GEV", Procs. of the 17th
International Linac Conference, Tsukuba, Japan, Aug.
21-26, 1994, pp.184-186; S. Kurokawa et al.,"Accelerator Design of the KEK B-Factory", KEK
Report 90-24, 1991.
[2] T. Kohno and T. Takoma, Numerical Calculation
Method of Electric Field (CORONA PUBLISHING CO.,LTD., 1980).

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