Three-dimensional magnetic field of a superconducting quadrupole with rectangular coil blocks and constant-perimeter end windings

Transcription

1 Thee-dimensional magnetic field of a supeconducting quadupole with ectangula coil blocks and constant-peimete end windings G. Dugan Conell Univesity In this note, I descibe the use of the field calculation techniques outlined in CB 96-5 to calculate the magnetic field of a quadupole coil wound with a "constant peimete" constaint in the ends. The coil is composed of thee coil blocks, which have a ectangula two-dimensional coss section, as opposed to the cylindical-secto shape associated with Ruthefod cable. This note descibes the details of the geomety needed to apply the CB 96-5 model to this case, and gives some esults fo a constant-peimete coil vey simila to that cuently poposed fo the CESR Q1/Q quadupole. 1. Rectangula Coil-block geomety Figue 1 defines some quantities to be used to descibe the ectangula coil coss section. The coil block is taken to be made fom flat cables, each of thickness d and width T. A given flat cable is located by specifying its aimuthal angle α, inne adius R, and inclination angle β(α) with espect to the x-axis. The cylindical coodinates of an element of the cable at distance t along the width (<t<t) ae ρ(α,t) R + t + RtCos[α β(α)] Tan[φ(α,t)] RSin[α] + tsin[β(α)] RCos[α] + tcos[β(α)] y t β(α) R ρ α φ x Fig. 1: Flat cable geomety A complete cable block, made fom flat cables of width d each, has a width wd. Fig. shows a complete ectangula cable block, extending fom α to α 1, with fixed inne adius R 1. Geneally the whole cable block has the same inclination angle β and inne adius R 1 ; fo geneality, howeve, I allow fo β to vay with α. This allows the "ectangula" block to be wedge-shaped. If β vaies with α, then of couse the cable block width w must also vay with t. In what follows below, we shall concentate on the ectangula block case, whee w and β ae fixed fo a given coil block. y 4 w β1 Block α1 α 1 R1 T β 3 x Fig. Cable block geomety 1 May 3,1996 8:16 PM

2 The stat of the block is defined by R 1, α ; The aimuthal angle of the end of the block is α 1 β 1 + Sin 1 Sin[α β 1 ] + wcos[β 1 β ]. The block extends at α fom point 1 to point 3, along the line shown; this is the distance T. At α 1, the block extends fom point to point 4, also a distance T. The points (1,,3,4) in fig. which define the cable block have cylindical coodinates: {ρ 1,φ 1 } {R 1,α }; {ρ,φ } {R 1,α 1 }; { ρ 3,φ 3 } {R 1 + T + R 1 TCos[α β(α)],tan 1 [ R 1Sin[α ] + TSin[β(α )] R 1 Cos[α ] + TCos[β(α )] ]} { ρ 4,φ 4 } {R 1 + T + R 1 TCos[α 1 β(α 1 )],Tan 1 [ R 1Sin[α 1 ] + TSin[β(α 1 )] R 1 Cos[α 1 ] + TCos[β(α 1 )] ]} As the vaiable t anges fom to T, and α anges fom α to α 1, we span the whole coil block, and can calculate the cylindical coodinates ρ,φ at each point fom the above equations. ote that ρ 3 and ρ 4 ae slightly diffeent (typically by less than 1%) R 1. Constant peimete end constaints Each flat cable must make a tansition aound the coil end in such a way as to keep the peimete of the edges of the cable the same fo all t fom t to tt. y s l s 1 1 l 1 x Fig. 3 Constant peimete flat cable: l 1 l. Figue 3 shows a flat cable as it tuns aound the end of the magnet. The constant peimete equiement is that the lengths l 1 and l be equal. 1 f 1 s 1 + (α) f s l l 1 ζ 1 (α,t ) ζ s 1 s(α, t ) s s(α,t) s Fig 4. Coil layout at fixed α May 3,1996 8:16 PM

3 In fig 4., the edges of the flat cable ae shown in sρθ vs. space. (θ/4 φ). The flat coil is located at aimuth α, with inclination β(α). The edge with peimete l 1 stats at t; the edge with peimete l stats at T. The ac-length distances s 1 and s ae s 1 s(α,t) ρ(α,t) 4 φ(α,t) R 1 + t + R 1 tcos[α β(α)] 4 R 1 Sin[α] + tsin[β(α)] Tan 1 R 1 Cos[α] + tcos[β(α)] s s(α,t) ρ(α,t) 4 φ(α,t) R 1 + T + R 1 TCos[α β(α)] 4 R 1 Sin[α] + TSin[β(α)] Tan 1 R 1 Cos[α] + TCos[β(α)] On the suface of a cylinde, each end of the flat coil follows an ellipse as shown in fig. 4. The end at tt stats at s and intesects the -axis at f s. The end at t stats at s 1, follows a staight line (on the cylindical suface) a distance (α,t), and then follows an ellipse, intesecting the - axis at 1 f 1 s 1 + (α,t). The distance (α,t) and the eccenticities of the ellipses f 1,f ae adjusted so as to make the peimetes (path lengths in fig.4) l 1 and l equal. In geneal, the cuves can be paameteied using the angle ζ shown in the figue as The diffeential path length is dl ds + d and s 1 (ζ 1 ) s 1 Cos[ζ 1 ] 1 (ζ 1 ) f 1 s 1 Sin[ζ 1 ] + (α,t) s (ζ ) s Cos[ζ ] (ζ ) f s Sin[ζ ] which makes the diffeential along the elliptical path ds ssin[ζ]dζ d fscos[ζ]dζ dl s Sin [ζ] + f Cos [ζ] The total path lengths shown in fig.4 ae then In tems of the complete elliptic integal we obtain Fo a constant peimete, we must have l 1 l : l 1 dl 1 (α,t) + s 1 Sin [ζ] + f 1 Cos [ζ] dζ l dl s Sin [ζ] + f Cos [ζ] dζ E(m) 1 msin [φ] dφ g(m) me(m) l 1 (α,t) + s 1 g(f 1 ) l s g(f ) (α,t) + s 1 g(f 1 ) s g(f ) which specifies a elation between f 1, f, and (α,t). We conside only cases in which f 1 f f is independent of t; then (α,t) (s s 1 )g(f) (s(α,t) s(α,t))g(f) With this fom fo (α,t), we will have a constant peimete fo the flat coil. In geneal, the ellipticity paamete f will be a function of α. The fom of f(α) is fixed if we assume a "constant width" coil. 3 May 3,1996 8:16 PM

4 y s 3 s x Fig 5. Coil block edge at tt Fig. 5 shows the layout of a piece of the tt edge of the coil block as it goes aound the magnet end; the piece vaies in α fom α 1 to α. 4 f(α)s 4 3 f(α 1 )s 3 s 3 s(α 1,T) s 4 s(α,t) s Fig 6. shows the same layout, in the s- suface. We have Fig. 6: Coil end layout at fixed tt The -distances ae s 3 s(α 1,T) s 4 s(α,t) 3 f(α 1 )s 3 4 f(α)s 4 The coil block width is s 4 -s 3. If this width is held constant as the coil block edge goes aound the end of the magnet, then we must have 4 3 f(α)s 4 f(α 1 )s 3 s 4 s 3 f(α) 1 + s 3(f(α 1 ) 1) 1 + s(α 1,T)(f 1) s 4 s(α, T) in which we have set f(α 1 ) f. The quantity f is the atio of the half-axis of the "uppe ellipse" at α 1 to the ac length s 3 : f The constant-peimete equiement then becomes Half axis of uppe ellipse of cable block Ac length fom 45 o to cable block (α,t) (s(α,t) s(α,t))g(1 + s(α 1,T)(f 1) ) s(α, T) 3 s 3 A coss-sectional cut of the coil block in a plane fomed by the -axis and a line at 45 o in the x-y plane is shown in figue 7: 4 May 3,1996 8:16 PM

5 w ρ ρ 3 ρ 4 ρ 1 θ θ 1 ρ θ Fig. 7: Inclination angles θ 1,θ In geneal, the -coodinate of an element of the block at (α,t) is given by (α,t) (α,t) + f(α)s(α,t) In fig. 7, the edges of the coil block ae given by 1 (α (α + f(α )s(α (α 1 (α 1 + f(α 1 )s(α 1 3 (α,t) (α,t) + f(α )s(α,t) f(α )s(α,t) 4 (α 1,T) (α 1,T) + f(α 1 )s(α 1,T) f(α 1 )s(α 1,T) The inclination angle of the coil block in this plane is given by Since we have Tan[θ 1 ] ρ 4 ρ 4 ρ 4 ρ f(α 1 )s(α 1,T) (α 1 f(α 1 )s(α 1 ρ 4 ρ f s(α 1,T) s(α 1 (α 1 (s(α 1,T) s(α 1 )g(f) ρ Tan[θ 1 ] 4 ρ ( f g(f) ) s(α 1,T) s(α 1 ( ) ( ) (α 1 and Tan[θ ] ρ 3 ρ ρ 3 ρ 1 f(α )s(α,t) (α f(α )s(α ρ 3 ρ 1 f(α ) s(α,t) s(α ( ) (α 5 May 3,1996 8:16 PM

6 3. Implementation in the model descibed in CB 96-5 y α 1 Block t i φ i β α φ 1i R1 ρ 1i ρ i x Fig 8: Division of coil block into segments The model descibed in CB 96-5 can calculate the field fom coil blocks which ae "wedge-shaped" in aimuthal coss-section, and extend in aimuth fom φ 1 to φ,and in adius fom R 1 to R. Each such coil block has a staight section of length L, with ends composed of staight extensions of length L, and a "constant-width" coil end geomety with ellipse paamete f (fo the inne edge of the ellipse). In ode to model the ectangula, constant peimete coil block descibed above, we divide the ectangula block into M segments in t, of width δtt/m, whee t i (i-.5)*t/m is the cente of the segment, and i uns fom 1 to M. The adial extent of each segment is fom ρ 1i to ρ i, whee ρ 1i ρ i R 1 + (t i δt ) + R 1 (t i δt )Cos[α β[α ]] R 1 + (t i + δt ) + R 1 (t i + δt )Cos[α β[α ]] The aimuthal extent of each segment is taken as unning fom φ 1i to φ 1, whee R φ 1i Tan 1 1 Sin[α ] + t i Sin[β[α ]] R 1 Cos[α ] + t i Cos[β[α ]] R φ i Tan 1 1 Sin[α 1 ] + t i Sin[β[α 1 ]] R 1 Cos[α 1 ] + t i Cos[β[α 1 ]] We will use the model of CB 96-5 fo each of these segments, and then supeimpose the esults. To descibe the constant-peimete ends, we use the ellipse paamete f; the constant-peimete equiement is imposed by giving each segment a length extension L i (α,t ) + (α,t ) 1 i i + in which is the oveall shift of the whole coil block, if any. Because (α,t) is only weakly dependent on α, the aveage indicated above intoduces little eo. Magnetic field and hamonics: Let n be the index of the coil block (n uns fom 1 to ), and let i be the index of the segment of the nth block (i uns fom 1 to M). The aimuthal extent of this segment is fom φ 1i to φ i ; the adial extent is fom ρ 1i to ρ i. The magnetic field due to this segment is The field in hamonic fom is given by B i µ I ( ) i I[ρ,φ, + L i ] φ1i,φ i,f,l 4 ρ i ρ 1i φ s,i 6 May 3,1996 8:16 PM

7 B i (;k) 4µ ( I) i φ s,i ρ ρ V k (φ) ρ k 1+ δ(,k,χ + L i ) φ 1i,φ i,f,s in which ( I) i I M is the atio of the numbe of amp-tuns in block n to the numbe of segments, M, and Sin[φ s,i ]Cos[φ 1i + φ s,i ] We ae assuming that the cuent is constant in each coil block. The total field fo coil blocks is then and in hamonic expansion fom, B tot 4µ ρ In tems of the gadient G, we have B tot µ 4 I M 1 I B tot Gρ Q M M i1 M i1 I M φ s,i φ s,i M i1 k, k, I[ρ,φ, + L i ] φ1i,φ i,f,l ( ρ i ρ 1i )φ s,i ρ V k (φ) ρ ρ V k (φ) ρ k 1+ k 1+ δ(,k,χ + L i ) φ 1i,φ i,f,s δ(,k,χ + L i ) φ 1i,φ i,f,s whee the gadient is given by The integated field in hamonic fom is G 4µ Q 4µ I M M i1 φ s,i 1 B tot d Gρ Q I M M i1 φ s,i k, ρ V k (φ) ρ k 1+ δ(,k) φ1i,φ i,f,s The integated hamonics ae in which The effective length is b k 1 QL eff L eff b k I M L eff 1 Q M i1 1 I Q M M i1 ρ φ s,i I i (n;k) I M φ s,i M i1 [ ] φ1i δ tot,end ρ (,k) + δ tot,body ρ (,k) φ s,i k,φ i,f,s s i Sin[kφ ] Sin[kφ ] k 1 i 1i + ( 1) I i (n;k) k dδ(fθ 1i + δ) dαsin[kh i (n;α,δ)]sin[α] s i Sin[φ ] Sin[φ ] i 1i + I i (n;1) φ s,i 4. Results fo a Q1/Q candidate design A thee-block coil design is being consideed fo the CESR Q1/Q supeconducting IR quadupole being poduced by TESLA Engineeing fo Conell. The basic geometical paametes of the design ae given in table 1. The common body length is 44 mm; the coil develops a field gadient of T/m, with kamp tuns. The nominal TESLA design contains 364 tuns, and so equies 15 A to get this field gadient. The effective length is mm. 7 May 3,1996 8:16 PM

8 Table gives the integated hamonic content. The hamonics ae all well within the specified toleances. Coil block # 1 3 R 1 (mm) T(mm) w(mm) I(kAmp) α (ad) β(ad) α 1 (ad) ρ 1 (mm) ρ (mm) ρ 3 (mm) ρ 4 (mm) φ 1 (ad) φ (ad) φ 3 (ad) φ 4 (ad) f (mm) Oveall length(mm) Table 1: Geometical paametes Hamonic numbe (k) b k ˆb k Table : Hamonic content of the field (in units; 1 unit1-4 ) Finally, we list in table 3 the peak fields in the body and ends fo each coil block. Compaing these numbes with the -coil design descibed in CB 96-5, we see that the peak fields in the ends have been educed by.3 T, and the peak body fields ae down by close to.5 T. This is due in pat to the use of thee coils, athe than two as in the CB 96-5 design, and also to the fact that a smalle inne coil adius has been used. Coil block # 1 3 End peak field (T) ρ (mm) φ (ad) (mm) Body peak field (T) ρ (mm) φ (ad) (mm) Table 3: Peak fields in the coils 8 May 3,1996 8:16 PM

9 Fo a moe complete compaison of this design with othe magnets, we pesent in tables 4 though 9 vaious magnet paametes fo this design and those of simila magnets, calculated as outlined in CB The peak field fo this design (called "CESR Q1/Q") includes the end field enhancement, the effect of skew quadupole winding (as in CB 95-13), and the effect of the CLEO solenoid field (in the body only). Even including all these effects, the design has magin in excess of 4%, because of the geneous use of high-quality supeconducto. Descipto ISR LEP FAL RHIC LEP CESR Q1/Q Tuns/pole Wie aea (mm ) a1 (mm) a(mm) Re(mm) I (amp) g(t/m) J(amp/mm) B-peak(T) B-ion (T) length(m) L(H) W-stoed (kj) Table 4 Model design paametes Quadupole T B Jc Tb Bop Tc(Bop) Bc(Tb) Jc(Bop,Tb) o K T A/mm o K Bpeak o K T A/mm T ISR LEP CESR Q1/Q FAL RHIC LEP Quads Table 5 Citical Tempeatues, Fields and Cuent Densities 9 May 3,1996 8:16 PM

10 Quadupole Tbath o K Bpeak T Iop A Ic(Bop,Tb) A/mm Iquench A Table 6 Cuent and tempeatue magins magi n % ISR LEP CESR Q1/Q FAL RHIC LEP Τ ο K Quadupole J op P (mm) G c (mwat (A/mm ) t/mm 3 ) α l min (mm) ISR LEP CESR Q1/Q FAL RHIC LEP Table 7 Cyostability paametes Quadupole v (m/sec) t Q (sec) Τ max ( o K) V max (V) ISR LEP CESR Q1/Q FAL RHIC LEP Table 8 Quench paametes Quadupole Aimuthal foce/length (/mm) a 1, φφ 1. Radial foce/length (/mm) a 1, φ Aimuthal pessue(/m m ) a 1, φφ 1. Table 9 Electomagnetic Foces Radial pessue (/mm ) a 1, φ ISR LEP CESR Q1/Q FAL LEP May 3,1996 8:16 PM

11 5. Diect compaison with ROXEI pogam Finally, fo the Q1/Q design descibed above, we pesent in table 1 a diect compaison between the fields calculated by the techniques descibed hee and in CB 96-5, and the fields calculated by the pogam ROXEI, which is being used by TESLA to design the quadupoles. The field points chosen ae all inside the coil. ROXEI does a diect Biot-Savat integation ove the exact geomety of the wie aay, but omits the field contibution fom the stand which passes though the field point, so we might expect some small diffeence due to this. Othe diffeences may be due to appoximations used in the model geomety. In geneal, the esults ae seen to agee to typically bette than.1 T. ROXEI This Pape Diff:(This Pape-ROXEI) x y B x B y B B x B y B Bx B y B Table1: Compaison of ROXEI fields and fields calculated by the technique descibed in this pape 11 May 3,1996 8:16 PM