Accelerator Magnet System

Transcription

1 Accelerator Magnet System Friday, 9 July, 010 in Beijing, 16:15 18:15 Ching-Shiang Hwang ( 黃清鄉 ) (cshwang@srrc.gov.tw) gov National Synchrotron Radiation Research Center (NSRRC), Hsinchu

2 Outlines Introduction to lattice magnets The magnet features of various accelerator magnets Magnet code for field calculation Field error source of magnet Magnetic field analysis of lattice magnet Introduction of pulse magnet Introduction of magnet field measurement

3 Introduction 1. Basic theorem for accelerator magnet type of electromagnets in the synchrotron radiation source.. The main properties and roles of accelerator magnets. 3. Magnet design should consider both in terms of physics of the components and the engineering constraints in practical circumstances. 4. The use of code for predicting flux density distributions and the iterative techniques used for pole face and coil design. 5. What parameters for accelerator magnet are required to design the magnet? 6. An example of NSRRC magnet system will be described and discussed.

4 Introduction of magnet arrangement

5 The magnet features of various accelerator magnets-1 Lorentz fource r F = r v v q( E + B) Although, the electric field appears in the Lorentz forces, but a magnetic field of one Tesla gives the same bending force as an electric field of 300 million Volts per meter for relativistic particles with velocity v c. Magnets in storage ring Dipole - bending and radiation Quadrupole - focusing or defocusing Sextupole - chromaticity correction Corrector - small angle correction

6 The magnet features of various accelerator magnet- Magnetic field features of lattice magnets m+ 1 ( B m ( z ) + i A m ( z ))( x+ iy ) Φ ( x, yz, ) = m=0 ( m + 1 ) Two sets of orthogonal multipoles with amplitude constants Bm and Am which represent the Normal and Skew multipoles components. The fields strength can be derived as B r ( x, y, z) = r Φ( x, y, z) Therefore, the general magnetic field equation including only the most commonly used upright multipole components is given by B B x y A 3 3 = A x + ( ) + ( 3 + ) A x x 6 x 3 3 = + + ( ) + B B B x B ( 3 ) y y x x x o 6 y where B 0,B 1 (A 1 ), B (A ),and B 3 (A 3 ) denote the harmonic field strength components and call it the normal (skew) dipole,quadrupole,sextupole and so on. A y

7 The magnet features of various accelerator magnet- Magnetic spherical harmonics derived from Maxwell s equations. Maxwell s equations for magneto-statics: v B v = 0, v H v = v j No current excitation: j = 0 Then we can use : v v B = Φ M Therefore the Laplace s equation can be obtained as: Φ = 0 where Φ M is the magnetic scalar potential. Thinking the two dimensional case (constant in the z direction) and solving for cylindrical coordinates (r,θ): Φ M = n n n n ( E + Fθ)( G + H ln r) ( a n r cos nθ + b n r sin nθ + a n r cos nθ + b n r sin nθ) n = 1 In practical magnetic applications, scalar potential becomes: Φ M = n n ( a n r cos n θ + b n r sin n θ ) n with n integral and a n, b n a function of geometry. This gives components of magnetic flux density: n 1 n 1 n 1 n 1 B r = ( na nr cos nθ + nb nr sin nθ) Bθ = ( na nr sin nθ + nb nr cos nθ) n n

8 Field distribution features of varied multipole magnet

9 Dipole Magnet-1 Dipole field on dipole magnet expressed by n=1 case and and field polarity will be changed when the pole π polarity changed. ( ) Cylindrical Br 1 1 = a cosθ + b sin θ Bθ = a1 sin θ + b1 Φ M cosθ = a1r cos θ + b1r sin θ 1 Cartesian B x = a 1 B x = a1 + ( a + b ) y B = B y = b + a b ) x y b 1 1 ( + Φ = x + b y Φ = a x + b y + ( a + b )xy M a1 + 1 M So, a 1 = 0, b 1 0 gives vertical dipole field: y y B Φ M = constant x x Pure dipole magnet Combine function magnet

10 Dipole Magnet- π b 1 = 0, a 1 0 gives horizontal dipole field (which is about rotated ) 1 B y B x ( ) (B x ) B y x ( y ) y (x) Normal term in separate function dipole Skew term in combine function dipole

11 Quadrupole Magnet-1 Quadrupole field given by n = case and field polarity will be changed when the pole polarity changed ΔΘ= =π /( ). Cylindrical Cartesian = a r cosθ + b r sin θ Bx = ( a x + b y) r B B θ = a r sin θ + b r cos θ Φ = a r cos θ + b r sin θ M Φ ( a y b x) BY = + ( x y ) b xy M = a + These are quadrupole distributions, with a = 0 giving normal quadrupole field. y y Φ M = - C Φ M = - C Φ M = + C θ x Φ M = + C Φ M = + C x Φ M = + C Φ M = - C normal skew Θ= π /4 Θ= 0 Φ M = - C

12 Quadrupole Magnet- π Then b = 0 gives skew quadrupole fields (which is the above rotated by ) By (Bx) Bx (By) X(Y) X(Y) Normal term Skew term

13 Sextupole Magnet-1 Sextupole field given by n = 3 case. and field polarity will be changed when the pole polarity changed ΔΘ= π /(3 ). Cylindrical Cartesian B r = 3a 3r cos3θ + 3b 3r sin 3θ B x = 3a 3 ( x y ) + 6b 3xy Bθ = 3a r sin 3θ + 3b r cos3θ Φ 3 3 M = a3r + 3r 3 cos 3θ b sin 3θ 3 Φ M B y = a ( ) y 6a 3xy + 3b 3 ( x y ) 3 3 ( x 3y x) + b ( 3yx y ) = 3 3 y y Φ M = - C Φ M = + C Φ M = - C Φ M = + C Φ M = + C Θ B x Φ M = - C Φ M = + C x Φ M = - C Φ M = + C Φ M = - C Φ M = + C Normal Skew Θ= π /6 Θ= 0

14 Sextupole Magnet- ( ) For a 3 = 0, b 3 0, B y x, B give normal sextupole y = 3b 3 x y π For a 3 0, b 3 = 0, give skew sextupole (which is about rotated ) 3 By (Bx) By (Bx) X (y) Y (x) Normal term Skew term

15 Error of dipole magnet Practical dipole magnet. The shape of the pole surfaces does not exactly correspond to a true straight line and the poles have been truncated laterally to provide space for coils. Skew quadrupole will come from the bad coil. n= 1(m+1) m=0,1,, , n is the allow term for the dipole magnet (1) Quadrupole (n=) : pole tilt or one bad coil -----Forbidden term () Sextupole (n=3) : lateral truncation -----Allow term (3) Octupole (n=4) : pole tilt or one bad coil -----Forbidden term (4) Decapole (n=5) : lateral truncation -----Allow term

16 Error of quadrupole magnet Practical quadrupole. The shape of the pole surfaces does not exactly correspond to a true hyperbola and the poles have been truncated laterally to provide space for coils. n= (m+1) m=0,1,, , n is the allow term for the quadrupole magnet (1) Sextupole (n=3) : a=b=c d, or one bad coil -----Forbidden term () Octupole (n=4) : A=B, a=d, b=c, but a b -----Forbidden term (3) Decapole (n=5) : Tilt one pole piece -----Forbidden term (4) Dodecapole (n=6) : a. lateral truncation b. a=b=c=d, A=B R (5) 0-pole (n=10) : a. lateral truncation b. a=b=c=d, A=B R -----Allow term -----Allow term

17 Error of sextupole magnet n= 3(m+1) m=0,1,, , n is the allow term for the sextupole magnet If dipole field (n=1) changed, the field for the allow term of dipole (n=m+1) would also be changed. The dipole perturbation in a sextupole field generated by geometrical asymmetry (a=b<c). The dipole and skew octupole perturbation in a sextupole field generated by the bad coil (1 and 4 is bad). (1) Octupole (n=4) : a=b c, or one pair coil bad -----Forbidden term () Decapole (n=5) : a=b=c R -----Forbidden term (3) 18-pole (n=9) : a. lateral truncation ba=b=c R b. b (4) 30-pole (n=15) : a. lateral truncation b. a=b=c R -----Allow term -----Allow term

18 Sextupole magnet with corrector mechanism 3 1 (1) Horizontal corrector (vertical field): N turns of Coil & 5 combine together th with N turns of Coil 3 & 4 & 1& 6. () Vertical corrector (horizontal field): N turns of Coil 1 & 6 combine together with N turns 6 of Coil 3 & 4. (3) Skew quadrupole corrector: N turns of Coil & 5.

19 Symmetry constraints in normal dipole, quadurpole and sextupole geometries Magnet Symmetry Constraint t Dipole Quadrupole Sextupole φ(θ) = -φ(π-θ) All a n = 0; φ(θ) ) = φ(π-θ) ) b n non-zero only for: n=(j-1)=1,3,5,etc; (n) φ(θ) = -φ(π-θ) b n = 0 for all odd n; φ(θ) = -φ(π-θ) All a n = 0; φ(θ) = φ(π/-θ) b n non-zero only for: n=(j-1)=,6,10,etc; (n) φ(θ) = -φ(π/3-θ) b n = 0 for all n not multiples of 3; φ(θ) = -φ(4π/3-θ) φ(θ) = -φ(π-θ) All a n = 0; φ(θ) = φ(π/3-θ) b n non-zero only for: n=3(j-1)=3,9,15,etc. (n)

20 Ideal pole shapes are equal magnetic line At the steel boundary, with no currents in the steel: H = 0 Apply Stoke s theorem to a closed loop enclosing the boundary: v v ( H ) ds = H Hence around the loop: v v v H d l = 0 v d v l B dl Steel, μ = dl Air But for infinite permeability in the steel: H = 0; Therefore outside the steel H = 0 parallel to the boundary. v v Therefore B in the air adjacent to the steel is normal to the steel surface at all points on the surface should be equal. Therefore v v from =, the steel surface isan iso-scalar-potential ti l line. B Φ M

21 Contour equations of ideal pole shape For normal (ie not skew) fields pole profile: Dipole: y = ± g/; (g is interpole gap). Quadrupole: xy = ± R /; (R is inscribed radius). This is the equation of hyperbola which is a natural shape for the pole in a quadrupole. Sextupole: 3x y y 3 = ± R 3 (R is inscribed radius).

22 Ampere-turns in a dipole B is approx constant round loop l & g, and H iron = H air / μ; B air = μ 0 NI / (g + l/μ); g, and l/μ are the reluctance of the gap and the iron. μ >> 1 l g NI/ NI/ Ignoring iron reluctance: NI = B g /μ 0

23 Ampere-turns in quadrupole magnet Quadruple has pole equation: xy = R / On x axes B y = G x, G is gradient (T/m). At large x (to give vertical field B): ( )( G x R / x) 0 NI = B l / μ = μ 0 / x B ie NI = G R / 0 μ (per pole)

24 Ampere-turns in sextupole magnet Sextupole has pole equation: 3 3 R y y x 3 ± = On x axes B y = G s x, G s is sextupole strength (T/m ). At large x (to give vertical field ): 3 3 R R Gs 3 Gs / μ μ μ R x R x B NI = = = l

25 Judgment method of field quality For central pole region: ( g x ) By ( x ) By ( 0 ) B ( x ) By ( 0 ) B ( x) B ( x) Dipole: calculate and plot in pure dipole or in y y + combine function dipole magnet, g is gradient field in dipole magnet db Quadrupole: calculate and plot y x By x [ bo + b1x] = g( x) and and dx By ( x) d B ( ) ( ) y g ( x) g( 0) g( 0) ( x ) By = S( x) ( x) [ bo + b1 x + bx ] S( x) S( 0) By ( x) S( 0) y Sextupole: calculate and plot and and For integral good field region: Dipole: B Quadurpole: ( ) ( ) B ( x) ds dx y x ds By 0 ds By ( x) ds ( ) or By 0 B ( x) y ( x) ds ( b + B( x) ds B [ o b1x) ds] [ ds gxds] + y ds and Sextuple: B ( x ) ds [ ( bo + b1x + bx ) ds ] and S( x) ds S( 0) B( x) ds S( 0) ds ( x ) ds g ( 0 ) g( 0) ds g 0 ds ds

26 Magnet geometries and pole shim of dipole magnet C Type: Advantages: Easy access for installation; Classic design for field measurement; Disadvantaged: Pole shims needed; Asymmetric (small); Less rigid; Shim Detail H Type: Advantages: Symmetric of field features; More rigid; Disadvantages: Also needs shims; Access problems for installation. Not easy for field measurement.

27 Magnet geometries and pole shim of multipole magnet Quadrupole and Sextupole Type: Shim However shim design should consider the assembly accuracy and However, shim design should consider the assembly accuracy and the engineering factor

28 Effect of shim and pole width on distribution

29 Magnet chamfer for avoiding field saturation at magnet edge Chamfering at both sides of dipole magnet to compensate for the sextupole components and the allow harmonic terms. Chamfering at both sides of the magnet edge to compensate for the 1-pole (18-pole) on quadrupole (sextupole) magnets and their allow harmonic terms. Chamfering means to cut the end pole along 45 on the longitudinal axis (z-axis) to avoid the field saturation ti at magnet edge.

30 Square ends * display non linear effects (saturation); * give no control of radial distribution in the fringe region. y Saturation z

31 Chamfered ends or shim ends * define magnetic length more precisely; * prevent saturation; * control transverse distribution; * prevent flux entering iron normal to lamination (vital for ac magnets). y z

32 Comparison of four commonly used magnet computation codes MAGNET: Advantages * Quick to learn, simple to use; * Only D predictions; Disadvantages * Small(ish) cpu use; * Batch processing only-slows down problem turn-round time; * Fast execution time; * Inflexible rectangular lattice; POISSON: * Inflexible data input; * Geometry errors possible from interaction of input data with lattice; * No pre or post processing; * Similar computation as MAGNET; * Harder to learn; * Poor performance in high saturation; * Interactive input/output possible; * Only D predictions. * More input options with greater flexibility; * Flexible lattice eliminates geometery errors; * Better handing of local saturation; * Some post processing available.

33 Comparison of four commonly used magnet computation codes TOSCA: Advantages Disadvantages * Full three dimensional package; * Training course needed for familiarization; * Accurate prediction of distribution and * Expensive to purchase; strength in 3D; * Extensive pre/post-processing; * Large computer needed. * Multipole function calculation lation * Large use of memory. * For static & DC & AC field calculation * Run in PC or workstation RADIA: * Cpu time is hours for non-linear 3D problem. * Full three dimensional package * Larger computer needed; * Accurate prediction of distribution and * Large use of memory; strength in 3D * Careful to make the segmentation; * With quick-time to view and rotat 3D structure * DC field calculation; * Easy to build model with mathematic * Fast calculation & data analysis * Run in PC or MAC * Can down load from ESRF ID group website.

34 Field calculation by Tosca code Without trim coil charging With trim coil charging Magnet pole and return yoke should be large enough to avoid field saturation The maximum current density is within 1 A/mm of the corrector on the sextupole magnet

35 AC &DC Corrector design I(t)=I 0 Sin(ωt) ] ) ( ) ( [ ) ( ) ( R t i dt t di L t i V t i P + = = I(t)=I 0 R t i t i V t i P ) ( ) ( ) ( = T average dt t P T P 0 ) ( 1 Average power = AC corrector R t i t i V t i P = = ) ( ) ( ) ( DC corrector

36 Dipole shim or chamfer Effect End shim Chamfer End shim Iy Normaliz zation Cf0 Cf6 Cf8 Cf10 Cf1?B 0 ds??bds / Cf0 Cf6 Cf8 Cf10 Cf Harmonic xaxis (mm)

37 Field analysis of dipole magnet Lamination mapping Radial mapping Lx = 100, Chamfer = 6 mm, Ja =.4 A/mm^ 30 Lx = 100, Chamfer = 6 mm, Ja =.4 A/mm^ (T/m) -0. m^) uadrupole strength Q Radial Lamination Sextu upole strength (T/ 0 10 Radial Lamination z-axis (mm) z-axis (mm) The weak edge focus strength is around 0.06 T (0.006 m37-1 )

38 Quadrupole &sextupole magnet chamfer effect Norm malize of Bds (b) Lx = 61 mm, Ja = 3.0 A/mm^, Chamfer = 5.6 Angle = 35 Angle = 45 Angle = 55 Angle = 60 Norm malize of Bds (a) Lx = 61 mm, Ja = 3.0 A/mm^, Chamfer Angle=60 Chamfer = 8 mm Chamfer = 9 mm Chamfer = 11 mm Chamfer = 1 mm Harmonic Harmonic Quadrupole magnet Sextupole magnet

39 Quadrupole chamfered ends or shim ends effect (a) ΔB/B Δ Δ Bds/ Bds 未加鐵心修正量增加鐵心修正量 無 chamfer 有 chamfer X-axis (mm) ( ) Lx = 61 mm, Ja = 3.0 A/mm^, Chamfer Angle=60 (a) X-axis (mm) (b) Lx = 61 mm, Ja = 3.0 A/mm^, Chamfer = f Bds Normalize of Chamfer = 8 mm Chamfer = 9 mm Chamfer = 11 mm Chamfer = 1 mm Normalize of B Bds Angle = 35 Angle = 45 Angle = 55 Angle = Harmonic Harmonic

40 Quadrupole field analysis by FFT method Br Sin(θ) Fast Fourier Transform

41 Sextupole field analysis by FFT method Br Sin(3θ) Fast Fourier Transform

42 B-H function of quadrupole magnet with iron yoke cutting 5 (a) D and 3D Gradient field compare with yoke cutting 6 (b) Short and long magnet-integral of Gradient field G (T/m) D 3D_Lx61 3D_Lx561 Gds (T) short long Gds (T) Jc (A/mm ) Jc (A/mm ) There isa big difference of the quadrupole field strength th between D and 3D at high field region Saturation had happened with yoke cutting at the nominal field Field saturation in the short quadrupole magnet is larger than the long quadrupole magnet Operating at 3.5GeV is impossible on the 3D calculation with yoke cutting and without any compensation

43 Magnet effective length definition 145D008 Longitudinal Field Distribution B (T) Iron Length =145meters 1.45 Effective Length = meters z (m) Effective length Dipole magnet L eff = Bds/B o Quadrupole magnet L eff = Gds/G o Sextupole magnet L eff = Sds/S o

44 Magnet end chamber or shim on lattice magnet Sextupole strength ( T/m ) Without Shim at 1.3 GeV With Shims at 1.3 GeV Dipole end shim s-axis (m) Dipole end shim 1 With 5-5 Shims Without at 1.5 Shim GeVat 1.5 GeV no chamfer chamfer trength ( T/m ) Sextupole st trength (T/m**5 5) Dodecapole st s-axis (m) z-axis(m) Dipole end shim Quadrupole end chamber

45 Magnet end chamber & shim at the pole end Quadrupole integral field strength distribution w & w/o end chamber Dipole integral field strength distribution w & w/o end shim

46 Dipole magnet analysis of the field measurement Dipole magnet shown in Fig1-8 Fig1 The field deviation as a function of x of the two dimensional theoretical calculation, and Fig NSRRC magnet measure dipole data the measurement results of the combined for magnet field effective length BL/B0 function bending magnet. Where the field deviation definition is [ B( x) ( a a x) ] B( x) Δ B B = / / + 1

47 Quadrupole magnet analysis of the field measurement Fig3 The gradient field deviation of the -D Fig4 After and before - shims calculation and measurement results at the measurement results of the integrated t magnetic center. field deviation

48 The results of field measurement NSRRC dipole magnet Fig5 The dipole field distribution of the radial and lamination mapping along the beam direction Fig6 The gradient field distribution of radial and lamination mapping along the beam direction

49 The results of field measurement SRRC dipole magnet Fig7 The sextupole distribution of the radial and lamination mapping along the beam direction Fig8 The sextupole field distribution along the beam direction of the after and before - shims measurement results

50 Dipole end shim 1 16 Without Shim at 1.3 GeV With Shims at 1.3 GeV 1 16 With 5-5 Shims Without at 1.5 Shim GeV at 1.5 GeV trength ( T/m ) Sextupole st Sextupole stren ngth ( T/m ) s-axis (m) s-axis (m)

51 The results of field measurement NSRRC quadrupole magnet Quadrupole magnet shown in Fig9-10 Fig9 The field deviation as a function of transverse direction of the -D and measurement results on the midplane Fig10 The deviation of integrated field with different thickness chamfering