SIMPLIFIED ANALYTICAL APPROACH TO THE BEAM RIPPLE GENERATIONMECHANISM OF THE J-PARC MR SLOW EXTRACTION BEAMS
Y. Shirakabe*, Institute of Particle and Nuclear Studies, KEK, Tsukuba, Japan
Abstract
The 30 GeV proton beams are being stably extracted at
the J-PARC Main Ring slow extraction system anddelivered to the downstream Hadron Experimental
Facilities, however presently, the large beam ripples
deteriorate the beam spill qualities, while the ripple
generation mechanisms are still not well understood yet.
In order to obtain better understandings on how suchserious ripples are being generated in MR, a simplified
analytical approach is taken, and some of the so far
obtained results are presented.
The origin of the beam ripples can mainly be attributed
to the magnetic field ripples of the main bending andquadrupole magnets of the MR. The simplified model ofthe magnet circuit allows the analytical solution with theramp pattern magnet current. The dependence of themagnetic field ripples on the circuit parameters is discussed.

INTRODUCTION
Beam ripples have been critically important in proton
synchrotron operations for more than half a century since
the invention of synchrotrons [1], and are still one of thehardest problems for today's J-PARC Main Ring beams.
In the present MR slow extraction operations, for example[2], the beam spills are mainly composed of sharp spikes,
as is shown in Figure 1, and they are found to be difficult
to improve the spill shape even after the various ripple
correction systems are turned on. The bottom of the spillin Fig. 1 is completely blackened by the spikes withvarious frequency component ripples ranging from tens tothousands Hz. The MR spill shapes are clearly worse

compared to, for example, those of CERN PS in Figure 2
where the spills have clearer bottoms even without corrections.
The origins of beam ripples have been searched foryears as is, for example, described in [4]. In the present
MR case, the biggest source of the beam ripples is
supposed to be the magnetic field ripples of the main
dipole and quadrupole magnets. Since the beam ripples
contain lots of frequency components that cannot
necessarily be explained to be originated from the magnet
power supplies [2], the mechanism on how the ripples
with various frequency components are generated in themagnet circuit systems should be investigated.
In order to approach the better understanding on how
the various ripples are generated in the MR magnet
circuits, a simplified circuit model with a ramp patterncurrent is analytically solved, and the dependence of theripples as a function of circuit parameters as inductance,
capacitance, a ramp rate etc., is discussed.

RAMP PATTERN OF MAGNET CURRENT
The dipole and quadrupole magnet currents in the
proton synchrotrons ramp up and down in accordancewith the proton acceleration cycles T0, as is shown in
Figure 3. The beam is injected at the flat-bottom current
IB, accelerated to the flat-top current IT, and then extracted.

Figure 3: A typical magnet current ramp pattern with acycle T0, a flat-bottom current IB, and a flat-top current IT.
A magnet current ramps up from IB to IT as is shown in
Figure 4. At first period of time [0], the current is IB. The
next period [1] is the smoothing time TS where the current
starts to swing up at tB for TS in parabolic curve. After
tB+TS the current increases linearly during the period [2].

2.0

2.5

3.0

3.5

4.0
time [sec]4.5

Figure 1: An example of the J-PARC MR slow extraction
spills. The spill feedback systems of EQ and RQ, and thetransverse RF are applied on the debunched MR beam [2].

0

0.5 0

time [sec] 0.5

Figure 2: CERN PS slow extraction spills before (left) andafter (right) the ripple correction system is turned on [3].
____________________________________________*yoshihisa.shirakabe@kek.jp

Figure 4: A magnet current ramp from IB to IT. The
current swings up and back at [1] and [3], respectively, inparabolic curves in order to avoid sudden current changes.

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The current starts to swing back at t T − TS for TS in the
period [3], again in parabola. At tT the current reaches IT,
the period [4]. Due to the parabolic curve, the derivative
DR, i.e. the ramp rate of the magnet current, becomes
I − IB, where TR = t T − t B(1)
DR = TTR − TS
Accordingly, the magnet current as a function of time,
I0(t), becomes as follows: At the period [0], I (t) ≡ IB ,and at the period [4], I (t) ≡ IT . At the ramp periods [1],[2], and [3],D
D (t − t B )[1] I (t) = IB ＋ R, I (t) = R (t − t B ) (2)TS2TS[2] I (t) = IB + DR t − t B +[3] I (t) = IT −TS
2

, I (t) = DR

(3)D
DR (t T − t)
, I (t) = R (t T − t) . (4)TS
2TS

SIMPLIFIED CIRCUIT MODEL
Each dipole or quadrupole magnet system, called afamily, has several to tens of magnet units connected inseries by bus bars or cables. In Table 1 shows the families
for the J-PARC MR. Each magnet unit, as well as eachconnection part, has its own inductance, resistance, andcapacitance. The schematic circuit model of one magnet
family, consequently, can be expressed as Figure 5, where
the circuit contains N components which may be either
the magnets or the connections. The current source I0(t)outputs the ramp current which is mathematically ideal
with no ripples as is defined in (2) to (4).

Figure 5: The schematic circuit model of one magnet family.
The N components include both the magnets and theconnections. The current source I0(t) supplies the ramp current.
The current I0(t) mainly flow through L1, R1, ···, LN, RN,
with fractional currents branching out at C1, ···, CN. At nthcomponent (n: 1 to N), the branch current to Cn represents
the oscillation with the circuit inductance, i.e. the LC
oscillation, with the amplitude that positively correlateswith Cn. This LC oscillation is the magnet field ripple itself.
The large capacitance generates the large ripple current.

Figure 6: The simplified circuit model with a single load.

Solving the Fig. 5 circuit directly is too complicatedand does not necessarily gives good insights on the circuit
behaviour with respect to such parameters as inductance,
capacitance, and the ramp rate. Instead, starting with thesingle load with L1, R1, and C1, as is shown in Figure 6,
the circuit behaviour becomes much simpler to understand.

CIRCUIT EQUATIONSIn the Fig 6, the electric charge Q1 stored on C1 is
Q (t) = C ∙ V (t) .
(5)
Since the C1 current IC(t) is the time derivative of Q1,
I (t) = Q (t) = C ∙ V (t) .
(6)
Using I0(t) and IC(t), the potential V1(t) is
V (t) = L I − IC + R (I − IC ) .
(7)
Substituting V (t) in (6) by (7), the equation for IC(t) is
obtained as follows:
IC + 2λ IC + ω IC = I + 2λ I ,(8)where λ and ω are the basic circuit constants defined asR(9)
λ = 2L [1/sec]: damping constant,
1 [rad/sec](10)
L C∶ angular frequency of LC oscillation.
The implications of the constants λ and ω will bementioned in the next section. The equation (8) is aconstant coefficient second-order ordinary differential
equation for the unknown function IC(t). The right side of(8) is obtained with the known function I0(t) defined by (2)to (4). The general solution of (8) can be obtained as a sumof following (a) and (b).
(a) The general solution of (8)'s homogeneous equation:
IC + 2λ IC + ω IC = 0 .
(11)(b) The particular solution of the nonhomogeneous
equation (8)(12)
The solution of (8) for [1] to [4] is obtained as follows.
ω =Table 1: J-PARC MR dipole & quadrupole magnet families.
The magnet units marked in red are connected byshielded cables, resulting in large cable capacitance [5].
N Length LR
C Cable Total
units [m] [mH] [mΩ] [nF]BM1 17 5.85 103.5 42.52 50.0 2920 3770
BM2 16 5.85 103.5 42.52 50.0 3020 3820
BM3 16 5.85 103.5 42.52 50.0 3200 4000
BM4 16 5.85 103.5 42.52 50.0 3170 3970
BM5 16 5.85 103.5 42.52 50.0 2880 3680
BM6 16 5.85 103.5 42.52 50.0 2860 3660
QDN 48 1.86
70.5 39.90 22.3 2650 3720
QFN 48 1.56
59.7 34.60 18.7 2710 3608
QFX 48 1.26
48.9 30.10 15.1 2360 3085
QDX 27 1.66
63.3 36.00 19.9 2440 2977
QFR 9 1.76
63.3 38.30 21.1 3120 3310
QDR 6 1.86
774
70.3 39.90 22.3 640
QDT 6 1.86
624
67.7 39.60 22.3 490
QDS 6 1.66
609
59.9 36.30 19.9 490
QFT 6 1.46
605
53.0 33.10 17.5 500
QFS 6 1.26
601
48.8 30.10 15.1 510
QFP 6 0.86
782
32.5 23.25 10.3 720- 542 -

The initial conditions for (25) is that the expression(25) in the period [1] smoothly connects to that in [0], i.e.,
The general solution of the homogeneous equation (11) is
I (t B ) = 0, IC (t B ) = 0 .
(26)
IC (t) = J cos √1 − ε ∙ ω t − φ ∙ e(13)Applying the condition (26) on (25), J and φ are
where J and φ are the arbitrary constants of integration obtained, and the expression of I (t) for the period [1] is;Cthat should be obtained by the initial conditions of theDR(1 − 4ε) + 2λ (t − t B )I (t) =
time periods [0] to [4], and ε is
ω TS
λ
ε= ω
.
(14)
− 1 cos √1 − ε ∙ ω (t − t B ) − φ e ( B )√1 − εThe solution I (t) behaves as the damped oscillation(27)
with the damping factor λ and the angular frequency ω .
ε
3 − 4ε.
(28)
φ = TanIf the load is assumed, from Table 1, to have1 − 4ε 1 − ε
L = 10 mH, R = 10 mΩ , C = 10 nF,(15)
It is easily confirmed that IC (t) in (27) satisfies (26).
then λ and ω become
The right side of (27) is composed of 3 terms; theR
λ = 2L = 0.5 [1/sec], t ≡ 1 = 2 [sec] (16) constant term (1 − 4ε) , the term proportional to theλelapsed time 2λ (t − t B ), and the damped oscillation term.
ω1rad
IC (t) obeys the proportional expression as= 10 sec , f ≡= 16[kHz]. (17)
ω =
2π
L C
DR
D
=L C R(29)
I (t) ∝
If the load is assumed to be 10 times bigger than (15) as
TSω TS
L = 100 mH, R = 100 mΩ , C = 100 nF, (18)This relation is essentially important. The magnitude ofthen λ is the same as (16), and ω becomes
the damped oscillation is proportional to the circuitω = 10 [rad/sec], f = 1.6[kHz] .
(19) inductance and capacitance L , C , and the ramp rate DR .
As the examples (15) and (18) show, the oscillations Solution in the period [2] ( + ≦ < − ) :damp slowly, with the damping time t = 2sec. If the
In the period [2], I (t)and I (t) are, from (3),condition L ∝ R roughly holds in different size systems,
I (t) = DR , I (t) = 0 .
(30)then the slow damping characteristic also always holds.
The initial conditions in the period [2] is
The oscillation frequency f changes in inverse
IC (t B + TS ) = IC (t B + TS ) in [1](27)(31)
proportion to the load size as are shown in (17) and (19).
IC (t B + TS ) = IC (t B + TS ) in [1](27) .
If the condition L ∝ C roughly holds, the oscillation
From (30) and (31), IC (t) is solved as
frequencies become lower in the bigger magnet systems.
DRS
2λ TS +In the realistic magnet systems in any size, ε defined in IC (t) =
1−ε
ω TS
√
(14) is generally a small number;( B TS )ε≪1 .
(20) − ∙ cos √1 − ε ∙ ω t − (t B + TS ) − φ e

DAMPED OSCILLATION SOLUTION

SOLUTIONS IN PERIODS [1] TO [4](32)The general solution of (8) is obtained as a sum of theparticular solution of (8) and the general solution (13).
The former can be obtained with I (t) at the ramp periods[1] to [4]. Unknown constants of integration, J and φ , canbe obtained by applying the initial conditions.
Solution in the period [1] ( ≦ < + ) :In the period [1], I (t)and I (t) are, from (2),DD
I (t) = R .
(21)
I (t) = R (t − t B ) ,TS
TS
With (21), the right side of (8) becomesDD(22)
I + 2λ I = R + 2λ R (t − t B ) .
TS
TS
From (22), the particular solution I (t) in the left side of(8) can be assumed as a first-degree function of t, such as
I (t) = a + b(t − t B ) .
(23)
Substituting (23) for the left side of (8), I (t) becomes
− D
2λ DR
(t − t B ) .
(24)
I (t) = 1 4ε R +TS ω TSω
From (24) and (13), the general solution of (8) can beobtained, while substituting t to t − t B in (13), as
− D
2λ DR
(t − t B )I (t) = 1 4ε R +TS ω TSω+J cos √1 − ε ∙ ω (t − t B ) − φ e ( B ) . (25)S =1 − 2 cos √1 − ε ∙ ω TS e

φ = Tan

TS + eTS(33)

sin φ + sin √1 − ε ∙ ω TS − φeTScos φ − cos √1 − ε ∙ ω TS − φ

e

TS

.

(34)
The solution (32) satisfies the initial conditions (31) as
is easily checked, and keeps the relation (29).
Solution in the period [3] ( − ≦ < ) :In the period [3], I (t)and I (t) are, from (4),DD
I (t) = − R .
(35)
I (t) = R (t T − t) ,TS
TS
The initial conditions in the period [3] is
IC (t T − TS ) = IC (t T − TS ) in [2](32)(36)
IC (t T − TS ) = IC (t T − TS ) in [2](32) .
From (35) and (36), IC (t) is solved asDS
IC (t) = R −(1 − 4ε) + 2λ (t T − t) +
ω TS
√1 − ε−∙ cos √1 − ε ∙ ω t − (t T − TS ) − φ
S =- 543 -e

( T TS )(37)

1 − 2 S cos √1 − ε ∙ ω (TR − 2TS ) + φ − φ(38)(TR TS )∙ e (TR TS ) + S ∙ esin φ−S ∙ sin √1 − ε ∙ ω (TR − 2TS ) − φ

DAMPED OSCILLATION AMPLITUDES

(TR

TS )

1 − 2 S cos √1 − ε ∙ ω TS + φ − φ e+S ∙ eTS∙eφ = Tan.
cos φ+S ∙ cos √1 − ε ∙ ω (TR − 2TS ) − φ∙ e (TR TS )(39)
The solution (37) satisfies (36), and keeps the relation (29).
Solution in the period [4] ( ≦ ) :In the period [4], I (t)and I (t) are
I (t) = I (t) = 0 .
(40)
The initial conditions in the period [4] is
IC (t T ) = IC (t T ) in [3](37)(41)
IC (t T ) = IC (t T ) in [3](37) .
From (40) and (41), IC (t) is solved asDS
cos √1 − ε ∙ ω (t − t T ) − φIC (t) = R
ω TS √1 − ε∙∙ e ( T ) (42)S =φ = TanTS(43)

sin φ +S sin √1 − ε ∙ ω TS − φ eTScos φ −S cos √1 − ε ∙ ω TS − φ e

TS

.

(44)
The solution (42) satisfies (41), and keeps the relation (31).

COMPARISON TO THE SIMULATIONS
The first part of the solution IC (t) is plotted in Figure 7.
The parabola smoothing time TS = 100msec and thecircuit parameters of (15) are assumed. In the lower two
figures, the analytical solutions (black) and the LTSpicesimulation results (red) are shown. They agree well,suggesting the validity of the analytical solutions.
As is shown in (16), the ripple IC (t) damps slowly. In
the time range of Fig. 7, the amplitude is almost constant.

0
tB0(TS=100msec)

Ic(t) Analytic
Ic(t) byLTSpice+0.1msec

tB
0

tB+TS

+0.2msec

Ic(t) Analytic
Ic(t) byLTSpice+0.1msec-0.1msectB+TS
Figure 7: Top: The obtained ripple current IC (t) plottedfor 150msec from the beginning of the current ramp t B .
Middle and bottom: The expanded plots for the 0.2msec
period at t B and t B + TS . The circuit parameters of (15)are assumed, where one cycle becomes 0.063msec.

In the parabola periods [1] and [3], the solutions of IC (t)in (27) and (37) contain the constant term (1 − 4ε) and theterm proportional to the elapsed time. In the linear ramp
period [2], the constant term 2λ TS is included in (32).
Apart from those terms, all the solutions from [1] to [4],marked in blue, contain the damped oscillation terms. If εand φ (n: 1 to 4) are ignored due to their relatively smallvalues, the damped oscillation term can be characterized by
S cos ω ⊿t ∙ e ⊿(45)
where S is the specific amplitude and ⊿t is the elapsed
time of each period.
In the expressions of S , (33), (38), and (43), the cosine
values are between -1 and 1. Hence the ranges of S
become|1 − e TS | ≦ S ≦ 1 + e TS
|1 − S e (TR TS ) | ≦ S ≦ 1 + S e (TR TS ) (46)
|1 − S e TS | ≦ S ≦ 1 + S e TS
If, for example, λ = 0.5sec and TS = 0.1sec, e TS
is close to unity because of the slow damping, and S is
0.0488 ≦ S ≦ 1.9512(47)
The value S largely changes by a factor of up to 40 if
the cosine value in (33), cos √1 − ε ∙ ω TS , changes.
The same property is applied to S , S , and to all theamplitudes in the following ramp cycles in Fig. 3.
In the J-PARC MR slow beam extractions, it isgenerally observed that the spill shapes, as is shown in
Fig. 1, change every cycle to cycle of the MR acceleration.
Sensitive fluctuations of S in connection to the cosine
values will be the origin of this irreproducibility.

CIRCUIT PARAMETER DEPENDANCEAs are shown in the solutions (27), (32), (37), and (42),the ripple current IC (t) always has the form of
⋅ ⋅
D
(48)
I (t) = R ⋅ ⋅⋅⋅ ⋅ω TS
where the terms inside the parentheses {} are thedimensionless numbers in the order of unity and define thedependence of ripple as a function of time.
Outside of the parentheses {} is the factor that signifies
the magnitude of the ripple. Here this factor is defined asDDIC ≡ R = L C T R .
(49)
S
ω TS
As is seen from (49), the ripple magnitude IC is
proportional to the circuit inductance L , the capacitanceC , the ramp rate DR , and inverse of the parabolasmoothing time TS .
In the originally designed requirements of MR, theexpected current ripple is ~10-6 at the extraction [7]. The
maximum designed bending magnet current is ~3000 A at
50 GeV. The corresponding ripple current is
3000 ⋅ 10 = 3 mA
(50)
Hence the ripple current IC should be in the order of mA.
Assuming simple numbers for example, from Table 1,
as follows,
L ∼ 1000 mH, C ∼ 1000 nF,(51)
DR ∼ 1000 A/ sec, TS = 0.1sec ,then the ripple current IC becomes
IC = L C DR /TS ∼ 10 mA .
(52)

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The estimation of (52) shows that, although theexpression of IC is based on the simplified circuit, Fig. 6,
the derived LC oscillation current formula (49) properly
represents the magnitude of the ripple current.
In the realistic MR case at present, however, the total
capacitance of each magnet family is several times bigger
than the sum of magnet capacitance. The cables used forthe electrical connections of the magnets bring large
capacitance into the circuit, as is shown in Table 1. The
resulted ripple in (49) will be IC ~30mA with C ~3000nF.
The beam ripple influence with the large capacitance isparticularly severe in the injection periods when the beam
size is large and the majority of the beam power loss isoccurring. If the ramp down rate just before the injection
period is assumed to be bigger than the ramp up rate as
DR = 2000 A/sec, and C ~3000nF, then I (t) ∼ 60 mA.
The bending magnet current at the injection period is~200A, hence the ripple rate is ~60mA/200A = 3 × 10 .
With an uncontrolled TS , it may even reach 10 .

BUS BAR AND CABLE CAPACITANCE
Since the inductance is mainly defined by the machine
dimensions such as magnet apertures or power line
lengths, it’s impossible to reduce L in the IC .formula(49). The only possible way to lower the ripple is to
reduce the capacitance of the entire circuit.
In order to fulfil this critical requirement, bus bar
connections are commonly employed in the major proton
synchrotrons, as are described in, for example, [3] and [6].
In bus bars, the capacitance per unit length is in generaltens of times smaller than that of cables [8]. In fact, if thebus bar capacitance is estimated as co-axial conductors
with an inner conductor radius r of 2cm and an outerconductor radius r of 0.5m, the capacitance per unit
length C becomesC ≡ 2πε /ln(r /r ) ~17 nF/km .
(53)
The assumed outer radius r may differ from place toplace in the synchrotron tunnel, but C in (53) is generally
insensitive to r when its size is assumed to be ≳ 0.1m.
In contrast to bus bars, high current cables have r only
~20% or so larger than r , resulting in a large factor of
1/ln(r /r ) > 5 . And there exists another factor
multiplied, that is the relative permittivity of ε ~2.26 forthe crosslinked polyethylene insulation layer between the
inner and outer conductors. As a result, the cablecapacitance per unit length CC, for example for the
500 mm2 shielded cables [9] used for the MR bending andquadrupole magnets, becomesCC≡ 2πε′ε /ln(r /r ) ~740nF/km .
(54)
The original design for the high current connection of
the MR magnets is naturally by bus bars which have thesame dimensions as the magnet’s hollow conductors [8].
The design was kept for a while in the MR construction
period [10] but later it was changed to the cables [11].
In the realistic case, the large cable capacitance isdivided and scattered in the MR tunnel along with themagnets as in Figure 5, forming lots of minor current
loops associated with the ripples of various LC oscillation
frequencies. In the simplified treatment at present with asingle load, the distributed feature of multiple loads isignored. The expansion to the multiple element circuit is
the next step subject and will be reported elsewhere.

RAMP PARAMETER DEPENDANCEAs is seen from (49), the ripple current IC (t) is related
to the current ramp parameters as
IC (t) ∝ DR /TS .
(55)
This factor appears in the right side of the circuit
equation (8), I + 2λ I , as is seen in (22). This featureholds in the circuits with multiple loads, although its proof
is skipped in this article. It is only pointed out here that the
proportional expression (55) always holds in any circuit.

SUMMARY
The beam ripple generation mechanism in proton
synchrotron magnet ramp currents is analyticallyinvestigated in a simplified circuit model.
It is revealed that the magnet current ripple is generated
by the magnet current ramp itself, even if the ramp
current as a function of time is mathematically ideal withno ripples. The magnitude of the resulting ripple current is
proportional to the circuit capacitance and the ramp rate.
The large capacitance and the faster ramp increase the ripples.

ACKNOWLEDGMENT
The author would like to thank Associate Professor M.
Tomizawa, Dr. T. Kimura, and Dr. T. Shimogawa for the
fruitful discussions and encouragements. The author also
would like to thank all the people involved in the MR slow
extraction beam development for their timeless efforts.

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Academy of Science USSR,” Atomic Energy, Vol. 1, Num.
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[3] S. Gilardoni et al., “Fifty years of the CERN PS
Synchrotron Volume I,” CERN-2011-004 (2011) 103;
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[6] P. Zhang et al., “COMMON MODE NOISE ON THE
MAIN TEVATRON BUS AND ASSOCIATED BEAMEMITTANCE GROWTH,” Proc. of PAC’91, San
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2000, KEK Proceedings 2002-15 (2003) p.131 (in Japanese).
[8] M. Muto, private communications.
[9] Hitachi Cable, Ltd. Cable Handbook: http://www.hitachicable.co.jp/ICSFiles/cable/ecocable/em_6000v_ce_f.pdf[10] T. Oogoe, “J-PARC 50 GeV Proton Synchrotron Main
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