- Email: [email protected]
A minimum volume microtron magnet
NUCLEAR
INSTRUMENTS
A N D M E T H O D S 6I
(t968) 347-348; © N O R T H - H O L L A N D P U B L I S H I N G
CO.
A M I N I M U M V O L U M E M I C R O T R O N MAGNET* G. A. PETERSONt
Instituut voor Kernphysisch Onderzoek, Amsterdam, The Netherlands Received 29 January 1968 A description and equations are given for an analytically-based 180° racetrack microtron magnet of minimum volume with independent pole and yoke construction.
A racetrack microtron with external injection and a linear accelerator driver section, as proposed by Wiik and Wilson1), should provide well-resolved, highcurrent, and high-duty-factor electron beams of several hundred MeV. They estimate that it can be built for 20 to 30% the cost of a linear accelerator having comparable beam qualities. If designs for higher energies are considered, the cost of the microtron increases substantially because the volume and cost of the 180 ° uniform field bending magnets is proportional to the cube of the final electron energy. We consider here a minimum volume magnet design, shown schematically in fig. 1, that uses 56% less steel than a conventional rectangular parallelopiped construction, and 23% less than a half-cylinder construction of optimum design. In fig. 2 is shown a sector of angle 0 of the magnet yoke. R is the distance between the magnet center (at e) and the outer coil edge (at a), r is the radial coordinate, fl is the angle between r and R, and x = r cosfl. An efficiency designed uniform field magnet, when * This work is supported by the Foundation for Fundamental Research on Matter (F.O.M.) and the Netherlands Organization for Pure Scientific Research (Z.W.O.). Presently on leave from Yale University New Haven, Connecticut, U.S.A.
operated near saturation, should have a constant flux density throughout the steel. From reluctance considerations, this requires a constant magnetic path length, as discussed below, and secondly, it requires that the area of the curved trapezoidal sheet abcd of fig. 2 should be a constant (½R20) as fl varies from 0 to zr: ½ [ ( R - x)O + RO]r = ½R20. This equation describes the surfaces of the upper and lower yoke sections and may be put in the form r =
R[-I + ( 1 - c o s t ? I - ' .
(1)
The volume of the upper (or lower) yoke was found to be ~zR3[ln(1 +~v/2)-k.¢2] by integration in cylindrical coordinates with point e as the origin. The total volume of steel in a magnet is
V = 2.579R3+½~HR2+½7r(R - W)2(H-G).
(2)
The first term is the volume of the upper and lower yoke, the second the volume of the center yoke of height H, and the third the volume of the pole pieces, each of radius Rp = R - W, and height ½ ( H - G), where Wis the coil width, and G is the average total of all air gaps. The poles are smaller in area than the central yoke by the amount ½7r(2RW-W2), but if the poles are
C
"-...
Fig. 1. A pictorial view of a 180° microtron bending magnet of minimum volume.
347
R ..- / - " ~
Fig. 2. A sector of the magnet yoke.
e
348
G . A . PETERSON
UPPER YOKE
~" ~
"
~\\
IHOMO(3ENIZING GAP
" " /~ . " {~---~V, .... . CPbLE / CENIER'YOKE" }1~,2 '~_' / / ' / , % , , ' "
.~/ \
, \ /.. / \ / . ' .\ "\ \ "" \/\I~G.t,.psP~cER
"\
\\
-
\\ \\
, . . LOWER YOKE',., -, . \ \
\
\ \. \ A N D VACUU_M
\
\ \ ".
-. \
" ]l
"~
\ \ \ SEAL~ \.'J
> \
/
'"
/
~
Fig. 3. A vertical cross section through the center of the magnet.
constructed of higher permeability steel than the yoke, then this small difference in area presents no saturation problem. The correct ratio of W to H for a minimum steel volume and a fixed total coil area A can be found by substituting R = R p + W and H = A~ W into eq. (2), and then setting ~V/~W=O. For R y e 2 m, and A _~ 0.03 m z, W/H ~ ½. The length of the upper curve of a yoke cross section is 1.89R. It can be seen from fig. 2 that the average total steel path length used in calculating the necessary magnetomotive force is given closely by
(R-x)xOdx
S= 2xl.89 W
(R-x)Odx +2H-G. W
Although Wiik and Wilson set only modest requirements of the order of 1% on field homogeneity, a better homogeneity may be achieved by incorporating certain features in the design as shown in the vertical cross section through the center of the magnet in fig. 3. The differences in potential that result from the large
differences in path length in the steel can be reduced in r the central gap~by the use of two additional air gaps z' 3) each of variable height g. If each gap is tapered according to the equation g = (1.89/#)(R-x), where/~ is the relative permeability of the yoke steel, then for the state of the steel corresponding to #, all magnetic path lengths are equal. If g is designed for the smallest value of ll to be used, then additional steel easily may be inserted to reduce g and to keep the magnetic path lengths approximately equal for several ranges of p. The use of homogenizing gaps has the additional advantage of separating the pole from the yoke construction3). Relatively crude tolerance can be placed on the yoke machining. Only the inner pole faces and the gap spacer need accurate machining. The gap spacer can be made conveniently from a large water-cooled conductor and can serve as the interior vertical vacuum wall of a double-walled separately pumped doublevacuum system. The pole surfaces serve as the top and bottom of the vacuum chamber. Window-flame type coil construction, as shown in fig. 1, exposes the front of the pole faces to permit vacuum connections and the use of fringing field correction devices such as suggested by BaNd and Sedla(:ek4). However, window frame construction may produce small excitation-dependent field nonuniformities near the entrance to the main gap that may be compensated for by multipole elements in the microtron drift paths or by pole face correction currentss). References 1) B. H. Wiik and P. B. Wilson, Nucl. Instr. and Meth. 56 0967) 197. ~) L. S. Goodman, Rev. Sci. Instr. 31 (1960) 1351. :3) H. A. Enge, Nucl, Instr. and Meth. 28 (1964) 119. 4) H. Babid and M. Sedla6ek, Nucl. Instr. and Meth. 56 (1967) 170. s) j. Sandweiss, Bubble and spark chambers 2 (ed. R. P. Shutt; Academic Press, New York, 1967) p. 239.
INSTRUMENTS
A N D M E T H O D S 6I
(t968) 347-348; © N O R T H - H O L L A N D P U B L I S H I N G
CO.
A M I N I M U M V O L U M E M I C R O T R O N MAGNET* G. A. PETERSONt
Instituut voor Kernphysisch Onderzoek, Amsterdam, The Netherlands Received 29 January 1968 A description and equations are given for an analytically-based 180° racetrack microtron magnet of minimum volume with independent pole and yoke construction.
A racetrack microtron with external injection and a linear accelerator driver section, as proposed by Wiik and Wilson1), should provide well-resolved, highcurrent, and high-duty-factor electron beams of several hundred MeV. They estimate that it can be built for 20 to 30% the cost of a linear accelerator having comparable beam qualities. If designs for higher energies are considered, the cost of the microtron increases substantially because the volume and cost of the 180 ° uniform field bending magnets is proportional to the cube of the final electron energy. We consider here a minimum volume magnet design, shown schematically in fig. 1, that uses 56% less steel than a conventional rectangular parallelopiped construction, and 23% less than a half-cylinder construction of optimum design. In fig. 2 is shown a sector of angle 0 of the magnet yoke. R is the distance between the magnet center (at e) and the outer coil edge (at a), r is the radial coordinate, fl is the angle between r and R, and x = r cosfl. An efficiency designed uniform field magnet, when * This work is supported by the Foundation for Fundamental Research on Matter (F.O.M.) and the Netherlands Organization for Pure Scientific Research (Z.W.O.). Presently on leave from Yale University New Haven, Connecticut, U.S.A.
operated near saturation, should have a constant flux density throughout the steel. From reluctance considerations, this requires a constant magnetic path length, as discussed below, and secondly, it requires that the area of the curved trapezoidal sheet abcd of fig. 2 should be a constant (½R20) as fl varies from 0 to zr: ½ [ ( R - x)O + RO]r = ½R20. This equation describes the surfaces of the upper and lower yoke sections and may be put in the form r =
R[-I + ( 1 - c o s t ? I - ' .
(1)
The volume of the upper (or lower) yoke was found to be ~zR3[ln(1 +~v/2)-k.¢2] by integration in cylindrical coordinates with point e as the origin. The total volume of steel in a magnet is
V = 2.579R3+½~HR2+½7r(R - W)2(H-G).
(2)
The first term is the volume of the upper and lower yoke, the second the volume of the center yoke of height H, and the third the volume of the pole pieces, each of radius Rp = R - W, and height ½ ( H - G), where Wis the coil width, and G is the average total of all air gaps. The poles are smaller in area than the central yoke by the amount ½7r(2RW-W2), but if the poles are
C
"-...
Fig. 1. A pictorial view of a 180° microtron bending magnet of minimum volume.
347
R ..- / - " ~
Fig. 2. A sector of the magnet yoke.
e
348
G . A . PETERSON
UPPER YOKE
~" ~
"
~\\
IHOMO(3ENIZING GAP
" " /~ . " {~---~V, .... . CPbLE / CENIER'YOKE" }1~,2 '~_' / / ' / , % , , ' "
.~/ \
, \ /.. / \ / . ' .\ "\ \ "" \/\I~G.t,.psP~cER
"\
\\
-
\\ \\
, . . LOWER YOKE',., -, . \ \
\
\ \. \ A N D VACUU_M
\
\ \ ".
-. \
" ]l
"~
\ \ \ SEAL~ \.'J
> \
/
'"
/
~
Fig. 3. A vertical cross section through the center of the magnet.
constructed of higher permeability steel than the yoke, then this small difference in area presents no saturation problem. The correct ratio of W to H for a minimum steel volume and a fixed total coil area A can be found by substituting R = R p + W and H = A~ W into eq. (2), and then setting ~V/~W=O. For R y e 2 m, and A _~ 0.03 m z, W/H ~ ½. The length of the upper curve of a yoke cross section is 1.89R. It can be seen from fig. 2 that the average total steel path length used in calculating the necessary magnetomotive force is given closely by
(R-x)xOdx
S= 2xl.89 W
(R-x)Odx +2H-G. W
Although Wiik and Wilson set only modest requirements of the order of 1% on field homogeneity, a better homogeneity may be achieved by incorporating certain features in the design as shown in the vertical cross section through the center of the magnet in fig. 3. The differences in potential that result from the large
differences in path length in the steel can be reduced in r the central gap~by the use of two additional air gaps z' 3) each of variable height g. If each gap is tapered according to the equation g = (1.89/#)(R-x), where/~ is the relative permeability of the yoke steel, then for the state of the steel corresponding to #, all magnetic path lengths are equal. If g is designed for the smallest value of ll to be used, then additional steel easily may be inserted to reduce g and to keep the magnetic path lengths approximately equal for several ranges of p. The use of homogenizing gaps has the additional advantage of separating the pole from the yoke construction3). Relatively crude tolerance can be placed on the yoke machining. Only the inner pole faces and the gap spacer need accurate machining. The gap spacer can be made conveniently from a large water-cooled conductor and can serve as the interior vertical vacuum wall of a double-walled separately pumped doublevacuum system. The pole surfaces serve as the top and bottom of the vacuum chamber. Window-flame type coil construction, as shown in fig. 1, exposes the front of the pole faces to permit vacuum connections and the use of fringing field correction devices such as suggested by BaNd and Sedla(:ek4). However, window frame construction may produce small excitation-dependent field nonuniformities near the entrance to the main gap that may be compensated for by multipole elements in the microtron drift paths or by pole face correction currentss). References 1) B. H. Wiik and P. B. Wilson, Nucl. Instr. and Meth. 56 0967) 197. ~) L. S. Goodman, Rev. Sci. Instr. 31 (1960) 1351. :3) H. A. Enge, Nucl, Instr. and Meth. 28 (1964) 119. 4) H. Babid and M. Sedla6ek, Nucl. Instr. and Meth. 56 (1967) 170. s) j. Sandweiss, Bubble and spark chambers 2 (ed. R. P. Shutt; Academic Press, New York, 1967) p. 239.