Lecture 4 - Magnets ACCELERATOR PHYSICS Melbourne E. J. N. Wilson

Transcription

1 Lecture 4 - Magnets ACCELERATOR PHYSICS Melbourne E. J. N. Wilson Lecture 4 - E. Wilson 5-Feb-08 - Slide 1

2 Lecture 4 Magnets - Contents Magnet types Multipole field expansion Taylor series expansion Dipole bending magnet Diamond quadrupole Various coil and yoke designs Power consumption of a magnet Magnet cost v. field Coil design geometry Field quality Shims extend the good field Flux density in the yoke Magnet ends Superconducting magnets Magnetic rigidity Bending Magnet Fields and force in a quadrupole Lecture 4 - E. Wilson 5-Feb-08 - Slide

3 Components of a synchrotron RING.GIF, Fig. sans nom 1_PULSE, Annexe1C Lecture 4 - E. Wilson 5-Feb-08 - Slide 3

4 Magnet types By x Dipoles bend the beam B y x Quadrupoles focus it 0 0 Sextupoles correct chromaticity Lecture 4 - E. Wilson 5-Feb-08 - Slide 4

5 Multipole field expansion (polar) Scalar potential φ (r,θ) obeys Laplace φ x + φ y = 0 or 1 r φ θ + 1 r r r φ = 0 r whose solution is φ = φ n r n sin nθ n=1 Example of an octupole whose potential oscillates like sin 4θ around the circle Lecture 4 - E. Wilson 5-Feb-08 - Slide 5

6 Taylor series expansion φ = φ n n=1 r n sin nθ Field in polar coordinates: B r = φ r, B θ = 1 φ r θ B r = φ n nr n 1 sinnθ, B θ = φ n nr n 1 cosnθ To get vertical field B z = B r sin θ + B θ cosθ = φ n nr n 1 [ cosθ cosnθ + sinθ sin nθ] = φ n nr n 1 cos( n 1)θ = φ n nx n 1 (when y = 0) Taylor series of multipoles B z = φ + φ.x + φ.3x 0 1 Bz = B0 + x + 1! x Dip. Quad 3 B x Sext + φ.4x z 4 x Bz + x 3 3! x Octupole Lecture 4 - E. Wilson 5-Feb-08 - Slide 6 Fig. cas 1.c

7 Multipole field shapes Lecture 4 - E. Wilson 5-Feb-08 - Slide 7

8 Dipole bending magnet Lecture 4 - E. Wilson 5-Feb-08 - Slide 8

9 Diamond dipole Lecture 4 - E. Wilson 5-Feb-08 - Slide 9

10 Diamond quadrupole Lecture 4 - E. Wilson 5-Feb-08 - Slide 10

11 Various coil and yoke designs ''C' Core: Easy access Less rigid λ g NI/ μ >> 1 NI/ H core : Symmetric; More rigid; Access problems. ''Window Frame' High quality field; Major access problems Insulation thickness Lecture 4 - E. Wilson 5-Feb-08 - Slide 11

12 Power consumption of a magnet λ NI/ g μ >> 1 NI/ B air = μ 0 NI / (g + λ/μ); g, and λ/μ are the 'reluctance' of the gap and iron. A is the coil area, l is the length σ is conductivity Approximation ignoring iron reluctance (λ/μ << g ): NI = B g /μ 0 ln R = A σ Power = I R = g B l μ Aσ o Lecture 4 - E. Wilson 5-Feb-08 - Slide 1

13 Magnet cost v. field 9 Variation of cost with field B Coil costs Running costs Pole costs Yoke costs Total Field B (T) Pressure from need to save real estate Constraint from saturation or critical current, synchrotron radiation Lecture 4 - E. Wilson 5-Feb-08 - Slide 13

14 Coil design geometry Standard design is rectangular copper (or aluminium) conductor, with cooling water tube. Insulation is glass cloth and epoxy resin. Amp-turns (NI) are determined, but total copper area (Acopper) and number of turns (N) are two degrees of freedom and need to be decided. Lifetime cost Current density: j = NI/A copper Optimum j determined from economic criteria. 0.0 running Lecture 4 - E. Wilson 5-Feb-08 - Slide 14 Current density j

15 Field must be flat to 1 part per Lecture 4 - E. Wilson 5-Feb-08 - Slide 15

16 Shims extend the good field Shim A Quadrupole To compensate for the non-infinite pole, shims are added at the pole edges. The area and shape of the shims determine the amplitude of error harmonics which will be present. Lecture 4 - E. Wilson 5-Feb-08 - Slide 16

17 Flux density in the yoke Lecture 4 - E. Wilson 5-Feb-08 - Slide 17

18 Magnet ends The 'Rogowski' roll-off: y = g/ +(g/π) exp ((πx/g)-1); Three dimensional computer calculation Lecture 4 - E. Wilson 5-Feb-08 - Slide 18

19 Bending Magnet Effect of a uniform bending (dipole) field l lb sin( θ ) = = ρ ( Bρ) If θ << π / then lb θ Bρ Sagitta ρ ± Lecture 4 - E. Wilson 5-Feb-08 - Slide 19 ( 1 cos( θ ) ) ρθ 16 lθ = 16 ( )

20 Fields and force in a quadrupole No field on the axis Field strongest here B x (hence is linear) Force restores Gradient Normalised: k = 1 B. y x ( Bρ ) x B y Defocuses in vertical plane POWER OF LENS lk l B y 1 =. = B x f ( ρ ) Lecture 4 - E. Wilson 5-Feb-08 - Slide 0 Fig. cas 10.8

21 Summary Magnet types Multipole field expansion Taylor series expansion Dipole bending magnet Diamond quadrupole Various coil and yoke designs Power consumption of a magnet Magnet cost v. field Coil design geometry Field quality Shims extend the good field Flux density in the yoke Magnet ends Superconducting magnets Magnetic rigidity Bending Magnet Fields and force in a quadrupole Lecture 4 - E. Wilson 5-Feb-08 - Slide 1