FIELD QUALITY IN ACCELERATOR MAGNETS

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Jemima Jefferson
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1 FIELD QUALITY IN ACCELERATOR MAGNETS S. Russenschuck CERN, 1211 Geneva 23, Switzerland Abstract The field quality in the supercnducting magnets is expressed in terms f the cefficients f the Furier series expansin f the field in the aperture, at a reference radius. Scaling laws fr different cefficients and reference radii are presented and the field generated by line currents in 2 and 3 dimensins is derived frm basic principles. 1 Furier series expansin f The magnetic field errrs in the aperture f the supercnducting acceleratr magnets are expressed as the cefficients f the Furier series expansin f the radial field cmpnent at a given reference radius (in the 2-dimensinal case). In the 3-dimensinal case, the transverse field cmpnents are given at a lngitudinal psitin r integrated ver the entire length f the magnet. Fr beam tracking it is sufficient t cnsider the transverse field cmpnents, since the effect f the z-cmpnent f the field, which is present in the magnet ends, n the particle mtins can be neglected. Assuming that the radial cmpnent f the magnetic flux density at a given reference radius inside the aperture f a magnet is measured r calculated as a functin f the angular psitin (if nthing else is stated, the lcal crdinate system f aperture 2 [right ne seen frm the cnncetin side] is used and the index is mitted), we get fr the Furier series expansin f the field "!#%'& )(+*,!#.- (1) If the field cmpnents are related t the main field cmpnent / we get fr 0 =1 diple, 0 =2 quadruple etc.: /.1 2!#%'3 )(+*4!#.- (2) The are called the nrmal and the & the skew cmpnents f the field given in 5 1, are the nrmal relative, and 3 the skew relative field cmpnents. They are dimensinless and are usually given in units f 79: at a 17 mm reference radius. 2 The field cmpnents Let us nw cnsider a single cil centered in an irn yke with circular inner aperture and a unifrm high permeability. The cil can be accurately described by a set f line currents at the psitin f the supercnducting strands. It will be shwn belw, that fr a set f!#; f these line currents at the psitin =>2+ carrying current? the cefficients are given by Ä@ B C? D CIHJ@ C H (+*4!#> (3) 210

2 & P B C? C C H 2!#> (4) where is the inner radius f the irn yke with the relative permeability C H. It is reasnable t fcus n the fields generated by line currents since the field f any current distributin ver an arbitrary crss-sectin can be apprximated by summing the fields f a number f line currents distributed within the crss-sectin. As supercnducting cables are made f strands with a diameter f abut 1 mm, a gd cmputatinal accuracy can be btained by representing each cable by tw layers f equally spaced line currents (same number as strands). Thus the grading f the current density in the cable due t the different cmpactin n its narrw and wide side is autmatically cnsidered. With equatin (3) and (4), a semi-analytical methd fr calculating the fields in supercnducting magnets is given. The irn yke is represented by image currents (secnd term in the parentheses). At lw field level, the saturatin f the irn yke is lw and this methd is sufficient fr ptimizing the cil crss-sectin. Under that assumptin sme imprtant cnclusins can be drawn: Q Fr a cil withut irn yke the field errrs scale with R where n is the rder f the multiple and is the mid radius f the cil. It is clear, hwever, that the cnsequence f an increase f cil aperture is a linear drp in diple field. Other limitatins f the cil size are the beam distance, the electrmagnetic frces, the yke size, and the stred energy which results in an increase f the ht-spt temperature during a quench. Q The relative cntributin f the irn yke t the ttal field (cil field plus irn magnetizatin) is fr a nn saturated yke apprximately SUTWVYXYZ. Fr the main diples with [ ]\4^-`_ cmpnent abut 19 % f cntributin frm the yke, mm and bac d mm we get fr the whereas fr the cmpnent the influence f the yke is nly abut 0.07 %. Q Fr certain symmetry cnditins in the magnet, sme f the multiple cmpnents vanish i.e. fr an up-dwn symmetry in a diple magnet (psitive current?= at > and ) n & terms ccur. If there is in additin a left-right symmetry, nly the dd f d hg -`cmpnents appear. It is therefre apprpriate t ptimize fr higher harmnics first, using the analytical apprach, and nly at a later state calculate the effect f irn saturatin n the lwer-rder multiples. When the LHC magnets are ramped t their nminal field f.4 T in the aperture, the yke is highly saturated, and numerical methds have t be used t replace the imaging methd. Then it is advantageus t use numerical methds that allw a distinctin between the cil-field and the irn magnetizatin effects, t cnfine bth mdelling prblems n the cils and FEM-related numerical errrs t the 20 % f field cntributin frm the irn magnetizatin. Cllabrative effrts with the University f Graz, Austria, and the University f Stuttgart, Germany, have been undertaken fr this task. Using the methds f reduced vectr-ptentials r the BEM-FEM cupling methd yields the reduced field in the aperture caused by the magnetizatin f the irn yke and avids the representatin f the cil in the FE-meshes, see cmpanin papers in this yellw reprt. In rder t avid field apprximatins by differential qutients, it is useful t use the vectrptentials &ji instead f the cmpnents in the Furier series expansin. In rder t d s, transfrmatin laws are derived. 211

3 ~ p r x x 3 The slutin f the Laplace equatin With Maxwell s equatins (+kmlni q (5) s t 7 () fr magnetstatic prblems and the cnstitutive equatin p p p x p x p C r C üc Ivw üc (7) and tgether with the vectr-ptential frmulatin ÿ(+kmlne we get (+kmlnz &{ C () (+kmlni &}@ (9) (+kmlǹ (+kmlnz &{ '(+kmln2 (10) In the tw-dimensinal case with ~ 7, & has nly a -cmpnent. In the absence f irn magnetizatin, we get the scalar Pissn differential equatin & ihä@ C i r i (11) and fr current free regins Eq. (11) reduces t the Laplace equatin which reads in cylindrical crdinates & i [ & i &ji 7 - (12) The general slutin f this hmgeneus differential equatin (which is valid nly inside the aperture f the magnet cntaining neither irn nr currents) is derived using the methd f separatin and reads &ji j. E.ƒ "!# ƒ E (+*4!#.- (13) Cnsidering that &ji is finite at =0 the E have t be zer fr the vectr ptential inside the aperture f the magnet. Fr the slutin in the area utside the cil all are zer. Rearranging Eq. (13) yields: At a reference radius we get: With ~ ˆ Š ~ Œ e &ji # &ji j 2!#%[ (+*4!#.- (14). 2!#% ƒ (+*4!#.- (15) the radial field cmpnent can be expressed as h 4Ž. "!# (+*4!#. (1) 2!#%'& (+*,!# /.1 2!#%'3 (+*m!#.- (17) 212

4 Œ ˆ / Œ 1 The small 3 are / the multiples related t the main field which is fr the diple, fr the quadruple etc. are given in 5, 5 RG, 5hRG etc., are given in 5 1, and are dimensinless and usually given in units f 79: at a reference radius f 17 mm. Fr the cmpnent we get jä@ ~ ˆ Š Ž.W 2!#%@ (+*m!# and therefre ~ ü Ž. - (1) Œ 4 Sme scaling laws E Frm equatin (1) it can be seen that the magnitude f a! -ple field cmpnent des nt depend n and scales with f. A cmpnent prduces n the -axis a field that rises with between the cefficients in Eq. (15) and (17) is as fllws: & #!. The relatin ƒ.- (19) These scaling laws can be used t calculate the field cmpnents frm the Furier series expansin f the vectr-ptential which is mre accurate when numerical field cmputatin methds are applied. Fr the scaling f different reference radii we get: & P 3 & (20) 1.- (21) Of curse the prblem still remains hw t calculate the field harmnics frm a given current distributin. 5 The field f a line current As previusly explained, it is reasnable t fcus n the fields generated by line currents as the field f any current distributin ver an arbitrary crss-sectin can be apprximated by summing the fields f a number f line currents distributed within the crss-sectin. In the 3-dimensinal case, Eq. (11) can be separated and we get fr the & š& where and the &ji cmpnent the slutin applying Greens therem, & C \ F r (22) with the surce pint and the field pint. Assembling the cmpnents we &žä& š Ÿ š '& Ÿ '& i Ÿ i C \ F r (23) and therefre b(+kmln & C \ F (+k4ln r b@ C r \ F l s C \ F r f (24) which is called Bit-Savart s law. The integral can be apprximated by the integratin ver segments f line currents f finite length which are used t apprximate the current distributin in the magnet. In 2-d the required particular slutin fr & i is &jih C r i n h & Ä@ 213 C? n (25)

5 Ÿ Ÿ with the surce pint > and the field pint and law The E (+* ª@'>e (2) can be rewritten as «Ž Œ «Ÿ Ž Œ (27) and therefre n Än2e E «Ž Œ I E With the Taylr series expansin f which gives fr the magnet), P b@! «Ÿ Ž Œ.- (2) (r inside the aperture f (29) Eq. (25) can be transfrmed t & ihä@ C? ne C?! (+*! %@'>e.m- (30) The r cmpnent f the magnetic field is then Ä@ C?! ª@ >e. (31) b@ C? 2!#>.- (32) Cmparisn f the cefficients with equatin (17) yields P b@ & P C? C? (+*,!#>e (33) 2!#> - (34) The effect f an irn yke with cnstant permeability and perfect circular inner shape with radius is taken int accunt by means f the imaging methd. The image current f the strength ±³²? is lcated ± ²Y at the same angular psitin and the radius µz R. Thus PÄ@ C? C C H (+*,!#>e (35) & P C 9? C CIHJ "!#>e- (3) 214

6 Fr the calculatin f the integrated multiple cntent in the cil-end regin, n analytical equatin exists. The cefficients, & can be estimated by means f the Furier series expansin f the field which is calculated with the Bit-Savart integrals fr the cil cntributin and with numerical methds (BEM-FEM cupling methd) fr the field generated by the irn magnetizatin. The integratin f the transverse field cmpnents is sufficient as the effect f the The magnetic length f the cil-end is given by ¹.º» ¼ ½ OW¾ i B i / (37) where» ¼ ½ OW¾ is the main field cmpnent in the magnet crss-sectin. ; is the starting pint and Ÿ the end pint f the integratin path. The field harmnics prduced by the cil-end can then be calculated by integrating the and & cmpnents alng the -axis and dividing by À ¹.º vj» ¼ ½ OW¾. Fr the calculatin f the field generated by a line current segment in 3 dimensins we nw assume that the line current starts in the rigin and ends at a pint 9+Á9 9. The field pint is +Á. With w we get: ;Â Ÿ Á ŸÁÃ Ÿ Ÿ C? \ F v Ÿ Á9 Ÿ Ÿ (3) Á9 Ÿ (39) f 3jŸÈš@ 1 Ÿ ÉzŸ i³ š9å ÇÆ š Å (40) Å where 3Ê [Á É ŸÂ 2 Á Á9 Ëª ÌÁ Á9 (41) - (42) The integral in eq. (40) yields: š9å ÍÆ š Å Å f b@ Ÿ E Ÿ Ë (43) 215

(1.1) V which contains charges. If a charge density ρ, is defined as the limit of the ratio of the charge contained. 0, and if a force density f

(1.1) V which contains charges. If a charge density ρ, is defined as the limit of the ratio of the charge contained. 0, and if a force density f 1.0 Review f Electrmagnetic Field Thery Selected aspects f electrmagnetic thery are reviewed in this sectin, with emphasis n cncepts which are useful in understanding magnet design. Detailed, rigrus treatments