INS-NUMA-23
v. Quadrupole Magnet for TARN
V I V.
\/\ A. Noda, M. Mutou, T. Hattori, \ X \ V. Hirao, T. Hori, T. Katayama ^ J \ // \ '% and • / V V H.Sasaki #
May 1980
STUDY GROUP OF NUMATRON AND HIGH-ENERGY HEAVY-ION PHYSICS INSTITUTE FOR NUCLEAR STUDY UNIVERSITY OF TOKYO Midori-Cho 3-2-1, Tanashi-Shi, Tokyo. 188, Japan INS-NUMA-23
Quadrupole Magnet for TARN
A. Noda, M. Mutou, T. Hattori, Y. Hirao, T. Hori*, T. Katayama and H. Sasaki**
Institute for Nuclear Study, University of Tokyo
Abstract
The lattice structure of TARN is designed to be a strong focusing
FODO type with separated function. Its focusing elements consist of
sixteen quadrupole magnets, half of which are radially focusing and the
other half are radially defocusing. The core length and the radius of
the inscribed circle of the magnet pole are determined to be 200 mm and
65 mm, respectively on the basis of tha calculation of aperture
requirement.
The v-value of the ring is around 2.25 both in horizontal and
vertical directions and the field gradients to attain the above v-value
are 0.240 kG/cm and 0.435 kG/cm for Q and Q, respectively for NS+ beam
* On leave from Sumitomo Heavy Industries, Co. Ltd. ** National Laboratory for High Energy Physics (KEK). with the kinetic energy of 8.55 MeV/u. The field gradient up to 0.67 kG/cm
is attained for the maximum excitation current of 400 A.
The pole shape is basically a hyperbola and the both sides of the
hyperbolic pole are smoothly extended to its tangential line. The structure
of field gradient is measured by twin coils which move in horizontal
direction and is flat (deviation is less than ± 1 %) in the region of
± 95 mm in horizontal direction. For the purpose of obtaining the wide
flat region of the effective length, the ends of the poles are cut off with the dimension which is determined on the basis of experimental results
for the various cut-off-shapes. The flat region of the effective length is almost the same as the field gradient. The effective length of the magnet is measured to be 263 mm for the excitation current of 250 A. The magnet has a four-fold symmetry, which reduced the higher mutlipole fields.
1. Introduction
A storage ring of heavy ions (TARN — Ttest Accumulation JUng for the
NUMATRON project) is designed for the purpose of studying the possibility of the combined use of the multiturn injection and RF stacking method.1»z'
Due to the multiturn injection and RF stacking, the beam emittance in horizontal direction and the momentum spread of the accumulated beam are relatively large and special care is needed to attain wide aperture in horizontal direction.
The SF cyclotron at INS with a K number of 67 is usad as an injector for TARN, and it can provide various ions such as p, d, a and heavy ions. For example, N5+ can be accelerated up to the kinetic energy of 8.55 MeV/u. The magnetic rigidity of the beam is given as,
B
where B and p are flux density of the dipole magnet and a radius of
curvature, respectively, q and A are charge state and mass number of the
ion, respectively (e = q/A). Among the values of p, Lg (the length of the
dipole magnet) and Ng (the number of the dipole magnets), the following
relation should exist;
2irp = NB-LB . (2)
In our design, the ring has an eight-fold symmetry and the number of
periods equals to that of superperiods. A unit cell is designed as FODO
lattice and one dipole is included in each cell. Thus Ng is equal to 8.
The length Lg is determined to be 1.047 m and p is calculated to be
1.333 m. The length of the straight section is determined from the con-
sideration of the equipments to be installed. The length of the quadrupole
magnet is determined to be 0.2 m from the point of view of focusing
strength and available space. A unit cell is shown in Fig.l. The beta and
dispersion functions and their derivatives are given in Fig.2 for the
central orbit. The beam etnittance from the SF cyclotron is measured to be
4.8 IT and 7.3 TT mm-mrad in horizontal and vertical directions, respectively.
After the multiturn injection the horizontal emittance is estimated to
increase up to 54 ir-mm-mrad. The momentum spread of the beam from the
cyclotron is measured to be about ..,.--., but in our case due to RF stacking
it is considered to become up to + 3.7 %. In Fig.3, the si.'.e of the injected and the stacked beams in the Qp and dipole magnets is illustrated.
In table 1, the aperture requirements in the quadrupole magnet is given.
- 3 - From the consideration of this aperture requirements, the radius of the
inscribed circle of the magnet is determined to be 65 nun.
In this paper, the detailed design of the quadrupole magnet is
described in chapter 2. The method of the measurement of the field
gradient and its effective length is written in chapter 3 and the measured
field characteristics of the quadrupole magnets are given in chapter 4.
2. Design of the Quadrupole Magnet
2-1. Basic Design Principle of the Quadrupole Magnet
Let us consider the case where four line currents with infinite length are flowing in ± s direction as is shown in Fig.4, where s direction is perpendicular to the paper plane. From the consideration of the 2-fold symmetry about s axis, the complex potential W(£) can be written as3' I n=0 where E, = x + iy. The term for n = 1 (^E,2) represents the quadrupole field. The magnetic scalar potential
and can be written as
CO 2n K& = I a2 r sin2n6 , (4)
n=0 where r and 6 are polar coordinates of E,. Among the terms in the right
2 hand side of (4), a2r sin29 gives the quadrupole component and is of our concern. The curve which satisfies the relation;
2 a2r sin29 = k (const) (5) represents the equi-potential line of the magnetic potential if higher terms
can be neglected. The equation (5) represents a hyperbola
xy = •£- (const) . (6)
If the iron is assumed to have an infinitely large permeability, the surface
of the iron should coincide with the equi-potential surface. In the present
case, the radius of the inscribed circle of the magnet pole rQ, is
determined to be 65 mm, and the equation which describes the pole surface is
r 2 xy = -£- . (7)
If the iron can be assumed to have an infinitely large permeability, the following relation can be derived from the Ampere's law;
where yo is the permeability in the air, G is the field gradient of the quadrupole magnet and NI is an Ampere-turn per pole. In the present case, the numerical value is
NI = Jx^-* (0.065)2 x7 = 11770 AT (9) when the field gradient of the strength of 7 T/m is assumed. This value is assumed to attain v-value around 4 for the study of various operation points by the tuning of Qp and QD.
In the case of TARN, the horizontal aperture should be much larger than the vertical one and it might be considered to use a deformed pole shape which has a wider aperture in the horizontal direction. It is found, however, that a pole shape with complete four-fold symmetry is much
"""* 0 *""" better than the distorted pole shape from the point of view of field
property."*' If the pole shape has a complete four-fold symmetry, the
following relation can hold
Substituting (A) into (10), we obtain the relation;
2n a2nr {sin2n(6±|) + sin2n9} = 0 . (11)
In order to assure the relation (11) for arbitrary value of 9,
a2n = 0 for n = 2!l (A = 0, 1, 2 . .. ) (12)
and
2(H+1) (13)
As the magnetic field is obtained by the relation
B = - grad<|>m , (14)
y component of the field is written as
2 6 By = - y- [a2r sin26 + a6r sin49 + . . . ]
(15) 5 3 a = - [2a2x + a6(6x - 60x y + 30xy") +...], which becomes in the median plane
5 By = - [2a2x + 6a6x +...]. (16)
The relation (16) means that the higher raultipole component next to the quadrupole is 12-pole, that is to say, the octupole component is absent if
— 6 - no machining error exists. On the other hand, the octupole component can
appear if the pole shape does not have a four-fold symmetry, even if no
machining error exists, because the condition (10) does not hold at that
case. From the above consideration, the quadrupole magnet for TARN is
designed to have complete four-fold syrrmetry.
2-2. Design of the Quadrupole Magnet for TARN
In the previous section, it is assumed that the line current and it in
core continues infinitely in the s direction. In a real magnet, however,
the effect of the fringing field at the ends of the magnet should be taken
into account. Further the permeability in the iron is assumed to be
infinitely large in the previous section, but it is finite in the case of
a real magnet. For the design of a magnet, the numerical calculation of
the magnetic field with the computer program TRIM5'6' was executed. This
calculation also does not include the end effect, but it takes the finite
permeability of iron into account. In Fig.5, the MESH plot used for the
calculation is shown and the plot of the flux line by the field calculation
is given in Fig.6. In Fig.7, the radial variation of the field gradient i
shown for various pole widths at the case of hyperbolic pole. The flat
region becomes wider as the pole width becomes larger, but the flat region
is not wide enough even if the pole width W is 150 mm. In order to enlarge
the flat region, the both sides of the hyperbola are extended to the
tangential line of the hyperbola as is illustrated in Fig.8 and the position where the continuation is made; is searched empirically. In Fig.P, calculated radial variation of the field gradient is shown for several continuation points (xg) in the case of pole width of 150 nan. The figure
— 7 - shows that the optimum shape is xQ = 100 mm or xQ = 105 mm and we chose
x0 = 105 mm from the point of view to use the shape closer to the ideal
hyperbola.
The focusing strength of the quadrupole magnet is given by the
product of the field gradient and its effective length. In the above
discussion, the pole shape is studied to attain the flat region of field
gradient. In order to realize the flat region of effective length, we
used the experimental results obtained in collaboration with KEK. From
the result that the optimum end cut shape is As = 12.5 mm and Ah = 20.0 mm
for the quadrupole of r0 - 57 mm, we decided to make an end cut of As
= 14.3 mm and Ah = 22.8 mm from scaling for r0. In the fabrication of
magnet, the end cut shape is approximated by four steps as is shown in
Fig.10, because the quadrupole magnet is made by stacking laminated cores.
From the relation (8), the required Ampere-turn is 11770 AT, which
agrees well with the value of 11907 AT deduced from the numerical cal-
culation with TRIM. In real magnet, however, some additional Ampere-turns
should be required because of the fringing field at both ends. In the
present case, the coils are wound 30 turns per pole and each coil permits
the excitation current up to 400 A, therefore total ArapSre-turn is 12000 AT per pole.
The main specifications of the quadrupcle magnet is listed up in
table 2. The drawing of the magnet is shown in Fig.11, The overall view of the magnet is given in Fig.12.
The magnet core was made by stacking steel strips with the thickness of 0.5 mm (S30) punched out into the desired shape and the precision of the punched pole shape of each steel strip is better than 50 ma. The
- 8 - magnet is constructed by assembling the blocks, each of which consists of
four hundreds stacked steel strips of a quarter shape (Fig.13). The
precision of assembling these blocks is measured b the following way.
Four rods of the diameter of 45 mm with the accuracy of 10 ym are closely
attached to the pole gaps as shown in Fig. 14. The distances, A 'v. H are
measured by an inside micro-meter at both ends of the rods. In order to
reduce the higher multipole fields caused by the error of the pole
assembly, the difference between the distances C and D should be less ~han
0.25 mm and the deviation among the distances E, F, G and H should be less
than 0.15 mm. With these ac uracies, the contribution of the higher
multipole fields caused by assembly error to the field gradient is es-
timated less than 0.5 %. The measured maximum differences of the above
distances are 0.15 mm and 0.1 mm, respectively for the fabricated magnet.
The required precision for the dimensions A and B is less than 0.1 mm and
the obtained accuracy is better than 0.06 mm. In Fig.15, the B-H curve
and p characteristic is given, which was taken into account in the
numerical calculation with TRIM. The reason why we used stacked core is
to enable the fast tuning of v-value.
The inductance of the magnet can be estimated by the following way.
The induced voltage (g) at the both ends of the coil of the magnet by the dd) change of the magnetic flux (T^) can be written as
e=-2H|f, • (17) because the turn number of coils which cross a single closed magnetic circuit is 2N. From Ampere's law, the following relation is obtained;
— 9 - where Ba, i.a and ya are the magnetic field, the length of the magnetic
circuit and the permeability in the air, respectively and B^ 5,^ and y^ are
the corresponding values in the iron. In usual case where the iron is used
without extreme saturation, p. is much larger compared with y (y- ^ 1000 y ),
therefore the second term of the left hand side of equation (18) is
negligible. Hence we obtain the relation
2ya Ba = -r-^ NI , (19) where both B_ and St, have the dependence on x (radial displacement) . It seems not so bad to replace B by B , because the flux line is almost perpendicular to horizontal line (Fig.6). Then, the length of the magnetic circuit in the air H is given by 3.
r 2 *a = 2y = "T" • (20) The magnetic flux
x« Bv • Aeff - dx ,
where xc is the position where By falls down to zero and £eff is the ef- fective length of the magnet. According to the numerical calculation with
TRIM, B behaves as is shown in Fig.16, and as the value of xc let's use
14 cm, because this value is appropriate to give the same area by a right- angled triangle with the calculated line as is shown in the figure. By the combination of equations (17), (19), (20) and (21), the following relation is obtained;
Z • *eff -^) -§ • (22)
- 10 - From the definition, the inductance of the magnet (L) can be written as
2 L = 8N • Ua • fceff • fcV (H) , (23) o where a factor of 4 is multiplied considering series connection of the
coils wound to four poles. The numerical values in our case are
N =30
£eff = °-263 m
xc - 0.14 m
r0 = 0.065 m and we obtain 11.04 mH as the inductance of the coil. In Fig.17, the inductance of the quadrupole magnet is shown together with that of the dipole magnet. These values are obtained with the use of a multi-frequency
LCR meter (YHP4274A). The calculated value seems to agree quite well with the measured value.
3. Field Measurement of the Quadrupole Magnet
3-1. Measurement of the Absolute Value of the Field Gradient with a Hall-probe
For the purpose of measuring the absolute value of field gradient a Hall-probe is used which is temperature controlled and precisely positioned by a driving mechanism. The output Hall voltage is precisely calibrated in a dipole magnet with the use of NMR. The measuring system is described in detail elsewhere.8^ The overall view of the measuring system together with the quadrupole magnet is shown in Fig.18. The field
— 11 — gradient (G) is obtained by differentiating the vertical component of the
magnetic field B with respect to x (horizontal displacement in the
direction perpendicular to the magnet axis). The field gradient is 0.43
kG/cm for the excitation current of 242.5 A, which is in good agreement
with the calculated value of 0.433 kG/cm by the use of equation (8).
3-2. Measurement of the Radial Distribution of the Field Gradient by Moving Coils
In the measurement of the field gradient, the method described in
the previous section might have a possibility of introducing relatively
larger errors, because subtractions of experimental data generally cause
enlargement of error. Therefore the above mentioned method does not seem
to be adequate for the purpose of precise study of the structure of the
field gradient. We adopted the method to measure the difference of the
field gradient directly.9) The outline of the method is as follows. Two
coils of the same dimension are translated in horizontal direction by a
driving mechanism, called as "Coil Shifter" hereafter, which can hold the
coils horizontally throughout the translation. A schematic diagram is
given in Fig.19 and the overall view of the coil shifter and the quadrupole magnet is shown in Fig.20. The difference of the induced voltage at each
coil is measured by a VFC circuit.l0) Let's assume the effective areas of the two coils to be A1 and A2 and the induced voltage at these coils
to be e2 and e2. The distances of the. centers of the coils from the magnet axis are assumed to be xx and x2 before translation and after the transla-
tion of distance as large as Ax, they become xx + Ax and x2 + Ax as is illustrated in Fig.21. The time integrated induced voltages at these coils are written as;
- 12 - ( r dBVl Jei • dt = - nAi{By(xi+Ax) - By(x±)} = - nk% -£\ -Ax (24)
(i = 1, 2) ,
where n is the turn number of the coil. The measured value is proportional
to the difference of these voltages and can be written as
dBVl dBVl AV „ = C • n{A .-T-H - A .-r^ } • Ax , (25) up I dx ' 2 dx '
where C is the calibration constant of the detection "ircuit. The subscript
"up" is used due to the following reason. In order to cancel out the dif-
ference of the effective areas of the coils, the measurement was executed
with the coils 180 degrees rotated and in such a case the induced voltage
is given by
dBv. dBVl , AV = dmdowmn - Cn{A,.! —d •*x• ' - A,-—2 dx^ ' }Ax , (26) X2 Xj
as is easily known from the illustration of Fig.22. Adding up (25) and
(26), the following relation is obtained;
dBv dBvl , AV = AV,, + AV, = Cn(A, + A,) {—f | - —* } . t • . (27) Vupn + AVdowomn xi
In the relation (27), the factor C-n(Aj + Az) • Ax is constant all over the
dBy dBy. region of x and AV has a dependence on x only through I . '| - *| }.
Xj x2 In the measurement, two kinds of twin coils are used. One of them is
called "Long Coil" and has a dimension of 8 x 774 ram2, which is used to
/"dBy measure the radial dependence of ~^ ds and the other is called Reference
Coil" and has a dimension of 6.4 * 30 mm2, which is used to measure the
dBy x-dependence of —r2- in the central part of the quadrupole magnet with G.X respect to s direction. The formula (27) can be applied to the "Reference
-13- Coil" directly. The calibration constant C can be cancelled out by
dividing AV with twice of the output Vfl from the reference coil 1 set at
the position x = 0, i.e. we studied the structure of
dBVl dBv, , dBv, dBv Ao) {r r } r 7 C-n(A,+Ao) • {-r^ --r^ } ^ ^ ' dX dX 2A 2C2Cn^ V| " %%\ 1 dx 0 dx 0 A,+A, The factor , represents the difference of the effective areas of two coils and is measured experimentally to be 1.048. Hence from the equation
(28), the quantity ^= - is known from the measured value of -rrr-. dBy 2V0 "dx 'o A typical example of the measured structure of the field gradient is shown in Fig.23 for the excitation current of 243.5 A. In Fig.24, the difference of the radial distribution of the field gradient due to y coordinate
(height from median plane) is shown. The radial distribution of the field gradient for excitation currents of 242.4, 290.9 and 403.1 A are given in
Fig.25. The excitation curve of the field gradient, which is shown in
Fig.26, is obtained using the output of the VFC circuit and the circuit is almost linear10^ (Fig.27). The scale of the field gradient is calibrated by the use of the result for the excitation current of 242.5 A described in the previous section. The result is important to connect the excitation
11 currents of Qp and Qp with the work line. ) The deviation of the field gradients of sixteen quadrupole magnets for the same excitation current of
243 A is shown in Fig.28. The measured values about the field gradient are summarized in Table 3 for various magnets.
- 14 - 3-3. Measurement of the Radial Dependence of the Effective Length of the Field Gradient
For the purpose of studying the structure of the effective length,
"Long Coil" is used. Let's assume the widths of the twin long coils to be
dj and d2 and represent the vertical component of magnetic field by
B (x, s) where s is the coordinate taken along the magnet axis. The time
d voltage at thess coils, j\Jtdt and K dt, are given by
= " «d±[J" By (xi + Ax, s)ds (Xj, s)ds] •• -oo ' -00 (29)
= " NdJ ^f | • ds • Ax (i = 1, 2) t —oo xi where N is the turn number of the "Long Coil". If the difference of these
voltages are fed into the VFC circuit, the output gives
,L - - (" SBV, r SB AVu_p = C -NfdJ -sf| -ds-dJ -sf| • ds]Ax . (30)
The output count in the case when these coils are rotated by 180 degrees is
given by
L f° 8B8BV,V , f°f° 3B AVdown = "C ' [ .if if' ' • dS " d2 '" X2 '-" X2 ' -° where C is the calibration constant of the VFC circuit. Adding up (30) and (31), the following relation is obtained
.N.(d1+d2)[ -^j -ds- -ds]Ax . (32) -"-oo Xj J-
If this is divided by twice of the output count of long coil 1, VQ, the calibration constant C1 is factored out and the following relation is obtained;
- 15 - C'N(d,+4,)[d1+42)[ | -^-\ • ds- -r^| -ds]
2VL l°° 9Bv 0 2C'-Nd, -r^l • ds (33)
ds] J l "ds 0 d,+d, where the factor —r-.—- is experimentally measured to be 1.013. From the zd, AVL relation (33) and the measured values of —— for various radial position x,
the radial dependence of j (-^-)ds is known. The typical example of the
measurement is shown in Fig.29 for the excitation current of 242.4 A. The
radial distribution of I -r-^ ds is given for the excitation currents of ' -°° ox 242.4, 290.9 and 403.1 A (Fig.30) and the radial distributiona of /^-jr^ ds
for y = -15, 0, 15 mm are given in Fig.31. The deviation of f°° -—^ ds for J —c° 9x
the sixteei quadrupole magnet for the same excitation current of 243 A is shown in Fig.32.
The effective length of the quadrupole magnet at the radial position x is defined by the relation; f 3B-, J -
From the measured value -£rr- and ^r and relations (28), (33) and (34), the o 2VQ deviation of radial dependence of the effective length — .-. is obtained. AT r \ Lefft0} ALeff (x) In Fig.33, the typical example of — ._. is given for the excitation Lff - 16- 3-4. Measurement of the Distribution of the ?ield Gradient along the Magnet Axis In order to study the distribution of the field gradient along the magnet axis, the small coil with the dimension of 9 x 9 mm2 is used. The coil can be slided along the magnet axis. In Fig.34, the overall view of the coil is shown together with the "Reference" and "Long" coils. The measured structure along the magnet axis at the excitation current of 241.7 A is shown for the radial position of x = 0 (Fig.35), and from the data, the effective length of the quadrupole magnet is known to be 263 mm at the axis (x = 0), while the mechanical length of the magnet core is 200 mm. 4. Characteristics of the Quadrupole Magnet • 4-1. Excitation Characteristics The excitation curve of the field gradient is shown in Fig.26. In order to study the saturation effect of the iron core, it is convenient G r 2 to use the correlation between I and -r —~=rn , where the latter is expected I P"1 to be 1 if the magnetic resistance in the iron is negligible as is easily known from the equation (8). The result is shown in Fig.26 by open circles, where the effect of the reduction of the permeability in the iron core is slightly observed above 300 A. The effect of saturation of the iron core to this extent is negligible from the point of view of field property as is easily understood from Fig.25. The radial dependence of the field gradient is quite similar for the excitation currents of 242.4, 290.9 and 403.1 A. - 17- 4-2. Radial Distribution of the Field Gradient The field gradient of the quadrupole magnet can be expanded as 2 3 t G(x) =^(x) = G0[l + a1(-^)+a2(-r-) +a3(-^) +aI)(-^)' + . • • ] , (35) ax ro ro ro ro where Go is the field gradient at the magnet axis. The coefficients a^'s are dimensionless and are written as (36) They are useful as the measure of the contribution of higher tnultipoles. The relation (28) is rewritten by using the above expansion (35) as; 3 # ^T ^12^3^(f-) + . • •] -W , (37) 0 10 '0 0 0 where W is the distance between the centers of the coil 1 and coil 2 and is equal to 10 mm in the present case. The coefficients are obtained by fitting the measured data with the right hand side of equation (37). The results are listed up in Table 3. Using these coefficients, the radial distribution of the field gradient can be written as (35). In Table 4, the multipole components deduced from coefficients a^'s are summarized. The size of the sextupole component is 1.6 x 10~2G/cm2 a. 8.6 x 10~2G/cm2 and octupole component of the size of 1.4 x KT'G/cm3 ^ 2.0 x 10~1G/cm3 is observed. The relatively large octupole component compared with other component is considered to be the effect of a little distortion of the pole shape due to welding. But as is easily known from the typical example in Fig.23, the field gradient is flat enough (better than ± 1 %) in the radial region of x = - 95 ^ 95 mm. It is much better (better than ± 0.5 %) in the region of x = -85 - 18- 4-3. Radial Distribution of Integrated Focusing Strength Focusing strength of the quadrupole magnet should be evaluated not only by its field gradient at the inner part in the axial direction, but also by the effective length of the field gradient. As the measure of the focusing strength, the integrated field gradient dBv dBv r-fds s( "df J —00 should be considered. Using the relation (33), the derivative of this value can be fitted to the measured data by the relation; 1 T ^T ) 7 fb (^)+3b33(3)+4b,(^) + - • •] • W , (39) r r r 2VL la\ ro ro o ro 0 if the above integrated value is expanded as -r^ds= W . (40) The coefficients b^'s are listed in Table 5 for all the sixteen magnets. The typical example of the result is illustrated in Fig.29 and the integrated field gradient is also flat enough (better than ± 1.5%) in the region of -95 mm j< x <_ 95 mm. It is known that the flat region of the effective length is very narrow if no end cut is made7' even if the field gradient is flat at the inner part of the magnet in the direction of the magnet axis. Owing to the end cut, our quadrupole magnets can afford such a wide useful aperture as -95 mm <_ x _< 95 mm, which is required from the combination of the multiturn injection and RF stacking. It is known from Table 5, the octupole component (b2) is small compared with 12-pole component (b ) for the integrated field gradient and this is expected in Chapter 2 from four fold symmetry of the magnet. - 19- The multipole components of the integrated field gradient are deduced from coefficients b^ s aad are given in Table (?. 5. Aknowle dgemen t The authors are willing to present their thanks to Dr. N. Tokuda and Messrs. T. Nakanishi and M. Yoshizawa for their collaboration in the field measurement. Thanks are also due to the other members of the Study Group for the NUMATRON Project. The authors would like to present their sincere thanks to Mr. M. Kumada of KEK for his collaboration about *:he end cut shaping of the quadrupole magnet. The numerical calculation with the computer program TRIM was executed by HITAC 8800 at KEK, One of the authors (A.N.) would like to express his heartful thanks to Prof. K. Endo, Mr. A. Ando and Miss Y. Miura of KEK for their hospitality about the usage of KEK library TRIM. Numerical calculation of the measured data was partly executed by FACOM M180 EAD at INS. The authors are willing to present their thanks to Profs. K. Sugimoto and M. Sakai for their continuous encouragement. - 20 References 1) S. Yamada and T. Katayama, "Injection and Accumulation Method in the TARN", INS-NUMA-12 (1979). 2) A. Noda et al., "Lattice Structure and Magnet Design for the TEST RING", Proc. of 2nd Symp. on Ace. Sci. & Tech. (1978) 83. 3) H. Ikegami, Butsuri 34. (1*79) 778 (in Japanese). 4) G. Parzen, "Magnetic Fields for Transporting Charged Beams", Brookhaven National Laboratory, preprint, ISA-76-13. 5) A. M. Winslow, "Numerical Calculation of Static Magnetic Fields in an Irregular Triangle Mesh", UCRL-7784 (1964). 6) K. Endo, "Operational Manual of Two-Dimensional Magnetostatic Program 'TRIM'", KEK-ACCELERATOR-2 (1974). 7) M. Kumada, H. Someya, I. Sakai and H. Sasaki, "Wide Aperture Q Magnet with End Cut Shaping", Proc. of 2nd Symp. on Ace. Sci. & Tech. (1978) 75. 8) T. Hori et al., "Field Measurements of the Dipole Magnet for TARN", to be published by INS-NUMA. 9) M. Kumada, I. Sakai, H. Someya and H. Sasaki, "Flux Meter for Field Gradient with Pendulum", Proc. of 2nd Symp. on Ace. Sci. & Tech. (1978) 73. 10) M. Mutou, "High Precision Integration Circuit with VFC for Field Measurement of Quadrupole Magnet", to be published by INS-NUMA. 11) A. Noda et al., "Measurement of v-values for TARN by the RF Knock-out Method", to be published by INS-NUMA. -21 - Figure Captions Fig. 1 Lattice structure of a unit cell. Fig. 2 (a) Beta and dispersion functions in a unit cell. (b) Derivatives of beta-functions in horizontal and vertical directions in a unit cell. Fig. 3 Useful aperture requirement for the quadrupole and dipole magnets for TARN. Fig. 4 Line currents which induce the quadrupole field. Fig. 5 An example of a MESH PLOT used for the field calculation with the computer code TRIM. Fig. 6 The structure of magnetic flux obtained by the computer code TRIM. Fig. 7 Calculated distribution of the field gradient in the radial direction, x, for various pole width, W, with hyperbolic pole shape. r 2 Fig. 8 The pole shape of modified hyperbola. A hyperbola, xy = ? , where rQ is the radius of the inscribed circle of the magnet, 2 extends to its tangential lines xoy + xyQ = rQ and x x + y y 2 = r at the points (xo, yn) and (y , x^), respectively. Fig. 9 Calculated distribution of the field gradient in the radial direction, x for various continuation points x for the case of pole width of 150 mm. Fig.10 End-cut shape of the quadrupole magnet. The solid line represents a real boundary of laminated cores and the dashed line shows the optimum end-cut shape obtained from the experimental results for various end-cut shapes. Fig.11 Front and side views of the quadrupole magnet for TARN. - 22 - Fig.12 An overall view of the quadrupole magnet for TARN. Fig.13 The shape of the punched out quadrant steel strip. Fig.14 Distances measured to assure the correct assembly of four blocks made of stacked steel strips. Four rods with the diameter 45 mm are attached to the pole gaps. Fig.15 B-H characteristics of the used steel (solid line). Magnetic permeability, y is also given (dashed line). Fig.16 Calculated field distribution for the pole shape of final design with the computer code TRIM. Fig.17 Measured inductance of the dipole and quadrupole magnet for TARN. Fig.18 The mapping system of the magnetic field used for the measurement of the absolute value of the field gradient of the quadrupole magnet. A temperature controlled Hall-probe is ;ised as a sensor. Fig.19 Block diagram of the electronic measuring system of the field gradient. The coil shifter is translated by a trigger signal and the induced voltage at the coil is fed to the VFC circuits. Fig.20 An overall view of the coil shifter attached to the quadrupole magnet. Fig.21 A schematic illustration of the translation of coils with the coil shifter. Fig.22 A schematic illustration of the polarity of the induced voltages at twin coils. Fig.23 Measured distribution of the field gradient of the quadrupole magnet. The open circles are measured data of .ffi from -^r-. The black circles are obtained by integrating the above data with respect to x. -23 - Fig.24 Radial distributions of the field gradient in the different horizontal planes, where the crosses represent the data measured in the median plane and the black and open circles give the data obtained in the planes 15 mm up and down from the median plane, respectively. The latter two data are limited in the region of -65 nun _< x < 65 inn because the twin coils can move only in the region. Fig.25 Radial distributions of the field gradient for the various excita- tion currents, where the black circles, crosses and the open circles represent the data for the excitation currents of 242.4, 290.9 and 403.1 A, respectively. Fig.26 Excitation curve of the field gradient. The black circles re- present the measured field gradients and the dashed line represents 2 G-rn the relation (8) . The open circles represent the values of -z—*pp, which is expected to be unity if the magnetic path length in the iron is negligible. Fig.27 Linearity curve of the VFC circuit. Fig.28 Deviation of the field gradient among the magnet for the same excitation current of 243.00 A. Fig.29 Measured distribution of the integrated field gradient. The dashed (GL) ' line represents the fitted line to the measured value :,T ,ns and the GLI\\J) solid line is obtained by integrating the above line and represents Fig.30 Radial distributions of the integrated field gradient for the various excitation currents, where the black circles, crosses and open circles represent the data for the excitation currents of 242.4, 290.9 and 403.1 A, respectively. -24 - Fig.31 Radial distributions of the integrated field gradient in the different horizontal plane, where the crosses represent the data measured in the median plane snd the black and open circles give the data obtained in the planes 15 mm up and down from the median plane, respectively. Fig.32 Deviation of the integrated field gradient among the magnets for the same excitation currents oi 243.00 A. Fig.33 Distribution of the effective length of the field gradient in the radial direction for the excitation current of 242.4 A. Fig.34 Overall view of the coils used for the measurement. Upper one is the moving coils used for the measurement of the distribution of the field gradient in the direction of magnet axis. Lower one ' contains the "Reference" and "Long" coils, which are used in the measurement of. the field gradient and the integrated field gradient, respectively. Fig.35 Distribution of the field gradient along the magnet axis for the excitation current of 241.7 A. Table Captions Table 1 At?rture requirement for the quadrupole magnet for TARN. Table 2 Specifications of the quadrupole magnet for TARN. Table 3 The dimensionless coefficients of the polynomial expansion of the field gradient obtained by fitting to the measured data. Table 4 The multipole fields of the quadrupole magnet. Table 5 The dimensionless coefficients of the polynomial expansion of 25 - the integrated field gradient obtained by fitting co the measured data. Table 6 The .nultipole components of the integrated field gradient of the quadrupole magnet. -26- Table 1. Aperture Requirement for Quadrupole Magnet Horizontal Direction Vertical Direction Closed Orbit Distortion 11.2 mm 3.9 mm Betatron Oscillation 16.3 mm 6.3 mm Momentum Spread 42.5 mm Sum (Half Size) 89.8* mm 10.2 mm Clearance 5.2 mm 9.8 mm Design (Half Size) 95.0 mm 20.0 mm * Horizontal aperture requirement is not a simple sum. The detailed situation to obtain this value is illustrated in Fig.3. 27 - Table 2. Specifications of the Quadrupole Magnet Quadrupole Magnet Maximum Field Gradient 0.7 kG/cm Pole Length 0.2 m Bore Radius 65 mm Maximum Ampereturns per Pole 12000 AT Number of Turns per Pole 30 Turns Maximum Current 400 A Maximum Current Density 8.33 A/mm2 Resistance of the Coil per Pole 0.0115 JJ ( at 60°C ) Maximum Power Dissipation per Pole 1.84 kW Space Factor 0.593 Pressure Drop of Cooling Water per Pole 1.06 kg/cm2 Flowing Rate of Cooling Water per Pole 0.85 £/min -28 - Q Magnet Go Coefficients (I = 243.5 A) Number [G/cm] a a, i a2 a3 a5 x t 1644 436.10 7.14z0x10-* 8.5246*10"3 10.2831x;I -5.2861x10"3 -3.5OO7 l(T' 1645 437.16 7.8361 9.6492 -2. 8692 -5.3862 0.53921 1646 438.42 5.2121 7.9610 -2.8689 -4.6659 1.2556 1647 438.14 12.0657 8.6792 -4.8405 -5.3007 1.1748 1643 438.29 8.1773 6.7710 1.1837 -4.6413 -0.25240 1649 441.22 12.6980 7.8396 -8.3786 -4.7013 2.2535 1650 439.72 4.9285 8.3247 2.8656 -5.1177 -0.17165 1651 438.49 9.9069 6.7091 3.7244 -4.6349 ~1.5752 1652 439.99 7.9525 7.7036 -2.7788 -4.8142 2.1010 1653 438.53 8.1359 7.2788 2.2148 -4.9069 -0.44748 1654 440.39 2.9885 8.3495 -1.9563 -5.1344 1.5808 1655 439.13 -6.7883 6.9576 -3.8109 -4.7518 1.2330 1656 439.27 -9.0198 8.8772 1 6352 -5.1074 -0.06093 1657 441.20 -8.2878 9.^97? 0 40140 -5.8968 2.6469 1658 436.98 2.3175 7.4395 3 7590 -4.9803 -1.4926 1659 438.04 9.8849 8.3783 -4 3372 -5.2469 2.7259 i1. Table 3. Q Magnet Go Higher Ordei" Components (I = 243.5 A) 2 3 5 6 d By 2 d By d"Bv d Bv d Bv Number [G/cm] —T [G/cm ] —"T [G/cm3] —T [G/cm"] • . [G/ci:i ] dx dx dx dx X 3 1644 436.10 4.7917*1O~S 1.7598*10"1 9.7976 1O~ -3.0994xl0~Z -1.5789X10"3 45 437.16 5.2702 1.9968 -2.7404 -3.1658 0.24379 46 438.42 3.5155 1.6522 -2.7480 -2.7503 0.56930 47 438.14 8.1330 1.8001 -4.6336 -3.1225 0.53232 48 438.29 5.5139 1.4048 1.1335 -2.7350 -0.11441 49 441.22 8.6194 1.6374 -8.0768 -2.7889 1,0283 50 439.72 3.3341 1.7328 2.7530 -3.0256 -0.07806 51 438.49 6.6832 1.3926 3.5680 -2.7325 -0.71436 52 439.99 5.3831 1.6045 -2.6712 -2.8479 0.95604 53 438.53 5.4890 1.5110 2.1220 -2.8931 -0.20295 54 440.39 2.0248 1.7406 -1.8823 -3.0401 0.71998 55 439.13 -4.5861 1.4463 -3.6562 -2.8055 0.55999 56 439.27 -6.0956 1.845i 1.5693 -3.0164 -0.02 768 57 441.20 -5.6255 2.0045 0.38692 -3.4979 1.2078 58 436.98 1.5580 1.5389 3.5888 -2.9260 -0.67457 59 438.04 6.6615 1.7373 -4.1508 -3.0901 1.2349 Table 4 Q Magnet GL Coefficients (I = 242.6 A) 0 Number fern-G/cm) b b b b i 3 5 6 b7 "a x 3 x x 3 x x 2 1644 11556.3 -2.1178 10~ -6.6746X10"" 4.6324 10~" -1.6278 10~ -6.0618 10~" -1.0993 10~ 1.2273*10""* 7.0465X10"3 1645 11564.8 -1.0d66 -9.6152 -7.4636 -3.4127 5.8564 -0.9730 -2.5979 6.7960 1646 11562.3 2.2295 -7.5584 -1.9540 -3.3122 5.5304 -0.9541 -4.1551 6.5576 1647 11550.7 -0.3311 -11.8803 -0.6324 -1.5240 -1.2044 -1.1036 0.7052 7.0725 1648 11551.9 -1.5511 0.2546 -15.9379 -0.9734 16.0544 -1.1615 -5.2257 7.2175 1649 11569.3 2.2216 40.5657 0.5677 -0.8867 41.9759 -1.1826 7.1668 7.3487 1650 11543.9 3.0128 -5.7509 8.4497 -2.4794 -2.2322 -1.0486 -2.2961 7.0103 1651 11533.6 -2.6118 7.1773 -2.02 70 -4.4860 0.3387 -0.9168 0.1415 6.6813 1652 11572.0 3.4754 0.6165 -11.7144 -1.9839 6.8438 -1.1862 -4.6887 7.2616 1653 11563.8 -1.8403 3.8405 -1.7633 -3.3620 -4.7667 -0.9527 3.0293 6.6093 1654 11539.7 0.4651 -9.0606 -1.C330 -2.5581 -5.1832 -1.0206 0.9546 6.8635 1655 11563.2 -0.7498 9.2913 44.1272 -5.5628 -4.3053 -0.8580 0.4822 6.3739 1656 11566.6 -0.1553 -2.2654 -1.4084 -4.3737 -1.5274 -0.9484 0.1721 6.6528 1657 11557.4 4.2075 -5.1801 -4.8292 -4.0282 9.2587 -0.9052 -5.2816 6.5097 1658 11554.7 -0.4986 -4.8348 -0.5197 -2.1682 6.1413 -1.0708 -3.3179 7.0256 1659 11540.1 3.3804 -7.3210 6.8949 -3.3374 7.0528 -0.8929 -6.4654 6.3853 * 0 dx* i! (GL) Table 5 Q Magnet Higher Order Components (I = 242.4 A] d d3B f d"B d5B d B d?B d B d9B f BjL. f yJ f V f v f v f " v f V 3 S —-^ds —-fd5 s 6 e —fds Number fds j dx J dx* J dx J dx 1 dx' J dx J dx' 2 (cm-G/cm) (cm-G/cmJ dx 2) (cm-G/cm3) (cm-G/cm'*) (cm-G/cm5) (cm-G/cm6) (cm-G/cm7) (cm-G/cm8) (cm-G/cm9) 1644 11556.3 -3.7653 1.1696*10"' -2.5291* ->"' -7.2449x10~2 -1.2128 1.4581X1°"2 1.0304 1645 11564.8 -1.9333 -5.2638 -1.8858 -5.3063 7.0046 -1.0742 -3.0888 0.9945 1646 11562.3 3.9659 -4.1369 -0.4936 -5.1490 6.6132 -1.0531 -4.9393 0.9594 1647 11550.7 -0.5884 -6.4959 -0.1596 -2.3668 -1.4388 -1.2169 0.8374 1.0337 1648 11551.9 -2.7566 0.1392 -4.0225 -1.5119 19.1806 -1.2809 -6.2063 1.0550 1649 11569.3 3.9542 -5.7864 0.1435 -1.3792 -14.3294 -1.3062 8.5245 1.0758 1650 11543.9 5.3507 -3.1426 2.1311 -3.8482 -2.6650 -1.1556 -2.7251 1.0240 1651 11538.6 -4.6364 3.9203 -0.5110 -6.9594 0.4042 -1.0099 0.1678 0.9755 1652 11572.0 6.1872 0.3377 -2.9617 -3.0866 8.1907 -1.3104 -5.578Z 1.0633 1653 11563.8 -3.2740 2.1023 -0.4455 -5.2270 -5.7007 -1.0517 3.6015 0.9671 1654 11539.7 0.8257 -4.9494 -0.4117 -3.9689 -6.1860 -1.1244 1.1325 1.0022 1655 11563.2 -1.3339 5.0858 -3.5690 -8.6482 -5.1487 -0.9471 0.5733 0.9326 1656 11566.6 -0.2763 -1.2404 -0.3559 -6.8012 -1.8271 -1.0472 0.2047 0.9737 1657 11557.4 7.4812 -2.8340 -1.2194 -6.2593 11.0669 -0.9988 -6.2757 0.9520 1658 11554.7 -0.8863 -2.6445 -0.1312 -3.3683 7.2638 -1.1799 -3.9279 1.0272 1659 11540.1 6.0016 -3.9993 1.7384 -5.1782 8.3314 -0.9826 -7.6444 0.9324 Table 6 Fig. 1 B [7 1.047 *02 IO2|»- - Fig. 2 (b) f 1.(0,a) K M.J - -- Mk—-*-->J< «,J — J*^-1-* (].« tn P) tj fl.21) 13.51) 1. (-a.O) (a.O) . S9.B jlflMy 0(0,-a) 511,0 4} nranivt STWIIBC (i) oifoit untcr Fig. 4 Fig. 3 - 33 - FIELD PLOT Fig- 6 -34- GOO X(cm) Fig. 7 Fig. 8 Real boundary of laminated cores Optimum boundary extroctcd from ^.the modtt test in collaboration with KEK : 100 mm 0S 110mm •Hyperbola 0.9- 0*- 1 Pole Width 150mm -7,0- Mem) 105 4S Fig. 9 Fig. 10 - 35 - - 36 - Fig. 12 Fig. 13 Fig. 14 - 37 - H (Oersted) O.S 1.0 5 10 50 100 500 - 1000 10 50 100 500 1000 5000 10000 50000 100000 MAGNETIZING FORCE H(A/m) Fig. 15 \ \ \ - 38 - L(hnH) * Q MAGNET 15 • BENDING MARNET 10 1Hz 10 Hz 100 Hz 1kHz 10 kHz 100kHz Pig. 17 Fig. 1! - 39 - •Ch »|Gate VFC 888-8 I 888—8 Coil Shifter Gate 0.01 -9.99s Driver Clear Gene. Contr. Latch Strobe Fig. 19 Fig. 20 Fig. 21 -40 - (a) "Up" State (b) "Down" State Fig. 22 in Median Plane I = 243.5 A X(cm) - 41 - G(X)-6(0) 6(0) 1 = 243.5 A Y=15mm 0.5 *—*\ Y=0mm /7 \ H 1 -10 ^8 -6 -/, -2 8 10 X(ctn) -0.5-- Fig. 24 Excitation Current vs Distribution of G Q 1653 (G-Go)/Go (*/.) • • I = 2A2.AA —— i = 290.9A 1 =403.1A 1.0 •• 0.5 H 1 1- -10 -6 -6 -U -2 10(cm) -as-- -1.0-- Fig. 25 - 42 - VFC Count X105 1 - Fig. 26 0.005 0.01 0015 Vin(V) Fig. 2? Deviation of the Field Gradient at the Center of 16 Qmagnets I = 243. 00 A Go = 426.52G/cm 1.0- 0.5- X I I,, 1 1 i k i | k -0.5; io - 43 - I=2A2.AA X(cm) Fig. 29 •: 1=242.4 A • : l=290.9A • : I=A03.1A 10 X(cm) GLOQ- 6L(0) , ,. . G.(0) l 1.5 • 1=243.5 A 1.0 \ V =0 mm 0.5 4. I {•'fl /V=-15m m^ 1 l_ 1 1 -10• he/. '*•- "* .» 6 8 1 ••16• i -6 # V = 15*mm" "'" -0.5 I j + -1.0 • 1 ^ -1.S • Fig. 31 Deviation of the GLd at,the Center'qf the 16 Qmagnets I = 243.00A )§ C) 1.0 0.5 * * T •f » T . * » * 1 i -0.5- ID a Fig. 32 - 45-7- I =242.4 A Xfcm) Fig. 33 Fig. 34 -46- Effective Length of Q-Mag. 1=241.7 A G/Go U= 263.07 mm • • • • * • • • • • • 0.5 • • • • • • • • • • • • • • # • • • I -300 -200 -100 0 100 200 300 S(mm) Fig. 35