Conventional Magnets for Accelerators Lecture 1
Ben Shepherd Magnetics and Radiation Sources Group ASTeC Daresbury Laboratory
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 1 Course Philosophy
An overview of magnet technology in particle accelerators, for room temperature, static (dc) electromagnets, and basic concepts on the use of permanent magnets (PMs).
Not covered: superconducting magnet technology.
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 2 Contents – lectures 1 and 2
• DC Magnets: design and construction • Introduction • Nomenclature • Dipole, quadrupole and sextupole magnets • ‘Higher order’ magnets • Magnetostatics in free space (no ferromagnetic materials or currents) • Maxwell's 2 magnetostatic equations • Solutions in two dimensions with scalar potential (no currents) • Cylindrical harmonic in two dimensions (trigonometric formulation) • Field lines and potential for dipole, quadrupole, sextupole • Significance of vector potential in 2D
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 3 Contents – lectures 1 and 2
• Introducing ferromagnetic poles • Ideal pole shapes for dipole, quad and sextupole • Field harmonics-symmetry constraints and significance • 'Forbidden' harmonics resulting from assembly asymmetries • The introduction of currents • Ampere-turns in dipole, quad and sextupole • Coil economic optimisation-capital/running costs • Summary of the use of permanent magnets (PMs) • Remnant fields and coercivity • Behaviour and application of PMs
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 4 Contents – lectures 1 and 2
• The magnetic circuit • Steel requirements: permeability and coercivity • Backleg and coil geometry: 'C', 'H' and 'window frame' designs • Classical solution to end and side geometries – the Rogowsky rolloff • Magnet design using FEA software • FEA techniques and codes – Opera 2D, Opera 3D • Judgement of magnet suitability in design • Magnet ends – computation and design • Some examples of magnet engineering
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 5 DC Magnets INTRODUCTION
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 6 Units
• SI Units Variable Unit Force, F Newton (N) Charge, q Coulomb (C) Flux density, B Tesla (T) or Gauss (G) (commonly referred to as ‘field’) 1 T = 10,000 G Magnetic field, H Amp/metre (A/m) (magnetomotive force produced by electric currents) Current, I Ampere (A) Energy, E Joule (J) • Permeability of free space = × 10 T.m/A = 1.6−7× 10 C • Charge of 1 electron0 1 eV = 1.6x10-19𝜇𝜇J 4𝜋𝜋 −19 𝑒𝑒 − Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 7 Magnetic Fields Flux Density
Value Item 0.1 - 1.0 pT human brain magnetic field 24 µT strength of magnetic tape near tape head 31-58 µT strength of Earth's magnetic field at 0° latitude (on the equator) 0.5 mT the suggested exposure limit for cardiac pacemakers by American Conference of Governmental Industrial Hygienists (ACGIH) 5 mT the strength of a typical refrigerator magnet 0.15 T the magnetic field strength of a sunspot 1 T to 2.4 T coil gap of a typical loudspeaker magnet
1.25 T strength of a modern neodymium-iron-boron (Nd2Fe14B) rare earth magnet. 1.5 T to 3 T strength of medical magnetic resonance imaging systems in practice, experimentally up to 8 T 9.4 T modern high resolution research magnetic resonance imaging system 11.7 T field strength of a 500 MHz NMR spectrometer 16 T strength used to levitate a frog 36.2 T strongest continuous magnetic field produced by non-superconductive resistive magnet 45 T strongest continuous magnetic field yet produced in a laboratory (Florida State University's National High Magnetic Field Laboratory in Tallahassee, USA) 100.75 T strongest (pulsed) magnetic field yet obtained non-destructively in a laboratory (National High Magnetic Field Laboratory, Los Alamos National Laboratory, USA) 730 T strongest pulsed magnetic field yet obtained in a laboratory, destroying the used equipment, but not the laboratory itself (Institute for Solid State Physics, Tokyo) 2.8 kT strongest (pulsed) magnetic field ever obtained (with explosives) in a laboratory (VNIIEF in Sarov, Russia, 1998)
1 to 100 MT strength of a neutron star 0.1 to 100 GT strength of a magnetar
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 8 Why magnets?
Analogy with optics
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 9 Magnets are like lenses…
… sort of
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 10 Magnet types
Dipoles to bend the beam
Quadrupoles to focus it
Sextupoles to correct chromaticity
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 11 Magnets - dipoles
To bend the beam uniformly, dipoles need to produce a field that is constant across the aperture. But at the ends they can be either:
Sector dipole Parallel-ended dipole They have different focusing effect on the beam; (their curved nature is to save material and has no effect on beam focusing).
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 12 Dipole end focusing
Sector dipoles focus horizontally
The end field in a parallel ended dipole focuses vertically Off the vertical centre line, the B field component normal to the beam direction produces a B vertical focusing beam force.
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 13 Magnets - quadrupoles
Quadrupoles produce a linear field variation across the beam.
Field is zero at the ‘magnetic centre’ so that ‘on-axis’ beam is not bent.
Note: beam that is horizontally focused is vertically defocused.
These are ‘upright’ quadrupoles.
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 14 Skew Quadrupoles
Beam that has horizontal displacement (but not vertical) is deflected vertically.
Horizontally centred beam with vertical displacement is deflected horizontally.
So skew quadrupoles couple horizontal and vertical transverse oscillations.
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 15 Sextupoles
In a sextupole, the By field varies as the square of the displacement. 0 0 x • Off-momentum particles are incorrectly focused in quadrupoles (e.g., high momentum particles with greater rigidity are under- focused), so transverse oscillation frequencies are modified – chromaticity. • But off momentum particles circulate with a horizontal displacement (high momentum particles at larger x) • So positive sextupole field corrects this effect – can reduce chromaticity to 0.
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 16 Higher order magnets e.g. – Octupoles: Effect?
Octupole field3 induces Landau 𝐵𝐵𝑦𝑦 ∝ 𝑥𝑥 damping: • Introduces tune-spread as a function of oscillation By amplitude • De-coheres the oscillations 0 0 x • Reduces coupling
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 17 Describing the field MAGNETOSTATICS IN FREE SPACE
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 18 No currents, no steel - Maxwell’s static equations in free space
. = 0 × = In the absence of currents: 𝜵𝜵 𝑩𝑩 = 0 Then we can put: 𝜵𝜵 𝑯𝑯 = 𝒋𝒋 So that: 𝒋𝒋 = 0 (Laplace’s equation) 𝟐𝟐𝑩𝑩 − 𝜵𝜵𝜙𝜙 𝜵𝜵 𝜙𝜙 Taking the two dimensional case (i.e. constant in the z direction) and solving for polar coordinates (r, θ): = + + ln +
𝜙𝜙 ∞ 𝐸𝐸 cos𝐹𝐹 𝜃𝜃 𝐺𝐺+ 𝐻𝐻 𝑟𝑟sin + cos + sin 𝑛𝑛 𝑛𝑛 −𝑛𝑛 −𝑛𝑛 � 𝐽𝐽𝑛𝑛𝑟𝑟 𝑛𝑛𝑛𝑛 𝐾𝐾𝑛𝑛𝑟𝑟 𝑛𝑛𝑛𝑛 𝐿𝐿𝑛𝑛𝑟𝑟 𝑛𝑛𝑛𝑛 𝑀𝑀𝑛𝑛𝑟𝑟 𝑛𝑛𝑛𝑛 𝑛𝑛=1
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 19 In practical situations
The scalar potential simplifies to:
= ∞ cos + sin 𝑛𝑛 𝑛𝑛 Φ � 𝐽𝐽𝑛𝑛𝑟𝑟 𝑛𝑛𝑛𝑛 𝐾𝐾𝑛𝑛𝑟𝑟 𝑛𝑛𝑛𝑛 with n integral and𝑛𝑛=1 Jn, Kn a function of geometry.
Giving components of flux density:
= ∞ cos + sin 𝑛𝑛−1 𝑛𝑛−1 𝐵𝐵𝑟𝑟 − � 𝑛𝑛𝐽𝐽𝑛𝑛𝑟𝑟 𝑛𝑛𝑛𝑛 𝑛𝑛𝐾𝐾𝑛𝑛𝑟𝑟 𝑛𝑛𝑛𝑛 𝑛𝑛=1 = ∞ sin + cos 𝑛𝑛−1 𝑛𝑛−1 𝐵𝐵𝜃𝜃 − � −𝑛𝑛𝐽𝐽𝑛𝑛𝑟𝑟 𝑛𝑛𝑛𝑛 𝑛𝑛𝐾𝐾𝑛𝑛𝑟𝑟 𝑛𝑛𝑛𝑛 𝑛𝑛=1 Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 20 Physical significance
This is an infinite series of cylindrical harmonics; they define the allowed distributions of B in 2 dimensions in the absence of currents within the domain of (r,θ).
Distributions not given by above are not physically realisable.
Coefficients Jn, Kn are determined by geometry (remote iron boundaries and current sources).
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 21 In Cartesian Coordinates
To obtain these equations in Cartesian coordinates, expand the equations for φ and differentiate to obtain flux densities cos = cos sin cos = cos 3 cos sin sin = 2 sin2 cos 2 sin = 3 sin3 cos sin 2 2𝜃𝜃 𝜃𝜃 − 𝜃𝜃 3𝜃𝜃 𝜃𝜃 − 2 𝜃𝜃 3 𝜃𝜃 2𝜃𝜃 𝜃𝜃 𝜃𝜃 3𝜃𝜃 𝜃𝜃 𝜃𝜃 − 𝜃𝜃 cos = cos + sin 6 cos sin sin = 4 sin4 cos 4 4 sin 2cos 2 4𝜃𝜃 𝜃𝜃 3 𝜃𝜃 − 3 𝜃𝜃 𝜃𝜃 4𝜃𝜃 𝜃𝜃 𝜃𝜃 − 𝜃𝜃 𝜃𝜃 etc (messy!); = cos = sin
𝑥𝑥 = 𝑟𝑟 𝜃𝜃 𝑦𝑦 = 𝑟𝑟 𝜃𝜃 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝐵𝐵𝑥𝑥 − 𝐵𝐵𝑦𝑦 − 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 22 n = 1: Dipole field
Cylindrical: Cartesian: = cos + sin = = sin + cos = 𝑟𝑟 1 1 𝑥𝑥 1 𝐵𝐵 = 𝐽𝐽 cos𝜃𝜃 +𝐾𝐾 sin𝜃𝜃 𝐵𝐵 = 𝐽𝐽 + 𝐵𝐵𝜃𝜃 −𝐽𝐽1 𝜃𝜃 𝐾𝐾1 𝜃𝜃 𝐵𝐵𝑦𝑦 𝐾𝐾1 𝜙𝜙 𝐽𝐽1𝑟𝑟 𝜃𝜃 𝐾𝐾1𝑟𝑟 𝜃𝜃 𝜙𝜙 𝐽𝐽1𝑥𝑥 𝐾𝐾1𝑦𝑦
So, J1 = 0 gives vertical dipole field: φ = const. B
K1 =0 gives horizontal dipole field.
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 23 n = 2: Quadrupole field
Cylindrical: Cartesian: = 2 cos + 2 sin = 2( + ) = 2 sin + 2 cos = 2( ) 𝑟𝑟 2 2 𝑥𝑥 2 2 𝐵𝐵 = 𝐽𝐽 𝑟𝑟cos 2𝜃𝜃 + 𝐾𝐾 𝑟𝑟sin 2𝜃𝜃 𝐵𝐵 = (𝐽𝐽 𝑥𝑥 𝐾𝐾 )𝑦𝑦+ 2 𝜃𝜃 2 2 𝑦𝑦 2 2 𝐵𝐵 − 𝐽𝐽2 𝑟𝑟 2𝜃𝜃 𝐾𝐾2 𝑟𝑟 2𝜃𝜃 𝐵𝐵 𝐾𝐾 𝑥𝑥 − 𝐽𝐽 𝑦𝑦 2 2 2 2 𝜙𝜙 𝐽𝐽 𝑟𝑟 2𝜃𝜃 𝐾𝐾 𝑟𝑟 2𝜃𝜃 𝜙𝜙 𝐽𝐽2 𝑥𝑥 − 𝑦𝑦 𝐾𝐾2𝑥𝑥𝑥𝑥 Line of constant J2 = 0 gives 'normal' or ‘right’ quadrupole field. scalar potential
Lines of flux density K2 = 0 gives 'skew' quad fields (above rotated by π/4).
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 24 n = 3: Sextupole field Cylindrical: Cartesian: = 3 cos + 3 sin = 3( ( ) + 2 ) = 3 2 sin + 3 2 cos = 3( ( 2 2 ) 2 ) 𝑟𝑟 3 3 𝑥𝑥 3 3 𝐵𝐵 = 𝐽𝐽 𝑟𝑟cos2 3𝜃𝜃+ 𝐾𝐾 𝑟𝑟sin2 3𝜃𝜃 𝐵𝐵 = (𝐽𝐽 𝑥𝑥 2−3 𝑦𝑦 2) + 𝐾𝐾(𝑥𝑥𝑥𝑥 ) 𝜃𝜃 3 3 𝑦𝑦 2 3 𝐵𝐵 − 𝐽𝐽3 𝑟𝑟 3𝜃𝜃 3𝐾𝐾 𝑟𝑟 3𝜃𝜃 𝐵𝐵 𝐾𝐾 3 𝑥𝑥 − 2𝑦𝑦 − 𝐽𝐽 𝑥𝑥𝑥𝑥 2 3 𝜙𝜙 𝐽𝐽3𝑟𝑟 3𝜃𝜃 𝐾𝐾3𝑟𝑟 3𝜃𝜃 𝜙𝜙 𝐽𝐽3 𝑥𝑥 − 𝑦𝑦 𝑥𝑥 𝐾𝐾3 3𝑦𝑦𝑥𝑥 − 𝑦𝑦 -C J3 = 0 giving 'normal' or +C +C ‘right’ sextupole field.
Line of constant scalar potential -C -C Lines of flux density +C
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 25 Summary: variation of By on x axis
Dipole - constant field By
By x
Quad - linear variation x
By
Sextupole - quadratic variation
0 0 x
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 26 Alternative notation (most lattice codes)
= ∞ ! 𝑛𝑛 𝑘𝑘𝑛𝑛𝑥𝑥 𝐵𝐵 𝑥𝑥 𝐵𝐵𝐵𝐵 � 𝑛𝑛=0 𝑛𝑛 Magnet strengths are specified by the value of kn (normalised to the beam rigidity) order n of k is different to the 'standard' notation:
dipole is n = 0 quad is n = 1 etc. k has units: -1 k0 (dipole) m -2 k1 (quadrupole) m
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 27 Significance of vector potential in 2D
We have: = × (A is vector potential) and . = 0 Expanding: 𝑩𝑩 = 𝛁𝛁 × 𝑨𝑨 𝛁𝛁 𝑨𝑨 = + 𝑩𝑩 𝛁𝛁 𝑨𝑨 + 𝜕𝜕𝐴𝐴𝑧𝑧 𝜕𝜕𝐴𝐴𝑦𝑦 𝜕𝜕𝐴𝐴𝑥𝑥 𝜕𝜕𝐴𝐴𝑧𝑧 𝜕𝜕𝐴𝐴𝑦𝑦 𝜕𝜕𝐴𝐴𝑥𝑥 − 𝒊𝒊 − 𝒋𝒋 − 𝒌𝒌 where i,𝜕𝜕 j,𝜕𝜕 k are𝜕𝜕𝜕𝜕 unit vectors𝜕𝜕𝜕𝜕 in x,𝜕𝜕𝜕𝜕 y, z. 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 In 2 dimensions = 0 and ∂ / ∂ = 0 So = = 0 𝐵𝐵𝑧𝑧 𝑧𝑧 and 𝐴𝐴𝑥𝑥 = 𝐴𝐴𝑦𝑦 𝜕𝜕𝐴𝐴𝑧𝑧 𝜕𝜕𝐴𝐴𝑧𝑧 A is in the z𝑩𝑩direction,𝜕𝜕𝜕𝜕 𝒊𝒊 normal− 𝜕𝜕𝜕𝜕 𝒋𝒋 to the 2D problem. Note: . = = 0 2 2 𝜕𝜕 𝐴𝐴𝑧𝑧 𝜕𝜕 𝐴𝐴𝑧𝑧 𝜵𝜵 𝑩𝑩 𝜕𝜕𝑥𝑥𝑥𝑥𝑥𝑥 − 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 28 Total flux between two points ∝ ∆A
In a two-dimensional problem the magnetic flux between two points is proportional to the difference between the vector potentials at those points.
B
( )
2 1 A1 𝜙𝜙 ∝ 𝐴𝐴 − 𝐴𝐴 A2
Φ Proof on next slide.
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 29 Proof
Consider a rectangular closed path, length λ in z direction at (x1,y1) and (x2,y2); apply Stokes’ theorem: = . = × . = . B A dS 𝜙𝜙But A∬is 𝑩𝑩exclusively𝑑𝑑𝑺𝑺 ∬ in 𝜵𝜵the z𝑨𝑨direction,𝑑𝑑𝑺𝑺 ∮ 𝑨𝑨 𝑑𝑑𝒔𝒔 and is constant in this direction. So: λ
. = [ , , ] z ds y � 𝑨𝑨 𝑑𝑑𝑺𝑺 𝜆𝜆 𝐴𝐴 𝑥𝑥1 𝑦𝑦1 − 𝐴𝐴 𝑥𝑥2 𝑦𝑦2 x (x1, y1) (x2, y2)
= [ , , ]
𝜙𝜙 𝜆𝜆 𝐴𝐴 𝑥𝑥1 𝑦𝑦1 − 𝐴𝐴 𝑥𝑥2 𝑦𝑦2 Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 30 Going from fields to magnets STEEL POLES AND YOKES
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 31 What is the perfect pole shape?
What is the ideal pole shape? • Flux is normal to a ferromagnetic surface with infinite µ:
curl H = 0 µ = ∞ therefore ∫ H.ds = 0 in steel H = 0 µ = 1 therefore parallel H air = 0 therefore B is normal to surface. • Flux is normal to lines of scalar potential: = ∇φ
• So the lines of scalar potential are the perfect𝑩𝑩 pole− shapes! (but these are infinitely long!)
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 32 Equations for the ideal pole
Equations for Ideal (infinite) poles; ( = 0) for normal (ie not skew) fields: Dipole: 𝐽𝐽𝑛𝑛 = ± (g is interpole𝑔𝑔 gap) 𝑦𝑦 2 Quadrupole: R
= ± 2 𝑅𝑅 Sextupole:𝑥𝑥𝑥𝑥 2 2 3 = ± 3
Ben Shepherd,3𝑥𝑥 ASTeC𝑦𝑦 − 𝑦𝑦 Cockcroft 𝑅𝑅Institute: Conventional Magnets, Autumn 2016 33 Combined function (CF) magnets
'Combined Function Magnets' - often dipole and quadrupole field combined (but see next-but-one slide):
A quadrupole magnet with B physical centre shifted from magnetic centre.
= Characterised by 'field index' n, 𝜌𝜌 𝜕𝜕𝜕𝜕 𝑛𝑛 − positive or negative depending 𝐵𝐵0 𝜕𝜕𝜕𝜕 ρ is radius of curvature of the on direction of gradient - beam do not confuse with harmonic n! Bo is central dipole field
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 34 Combined Function Magnets
CERN Proton Synchrotron
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 35 Combined function geometry
Combined function (dipole & quadrupole) magnet: • beam is at physical centre
• flux density at beam = B0 • gradient at beam = 𝛿𝛿𝐵𝐵 • magnetic centre is at B and X = 0 𝛿𝛿𝑥𝑥 • separation magnetic to physical centre = X0
x X0
magnetic centre, physical centre X = 0 x = 0
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 36 Pole of a CF dipole & quad magnet
Ref geometry as on previous slide:
Flux density at beam centre: B0 [T] Gradient across beam: g [T/m]
Displacement of beam from quadrupole centre: X0 ‘Local’ displacement from beam centre: X;
So: 0 = 0
𝐵𝐵0 = 𝑔𝑔 𝑋𝑋 𝐵𝐵0 And 𝑋𝑋 = 𝑔𝑔 + 0 Quadrupole equation: = 𝑋𝑋 𝑥𝑥 2 𝑋𝑋 𝑅𝑅 So pole equation ref beam centre: 𝑋𝑋𝑋𝑋= 2 2 𝑅𝑅 1 Adjust R to satisfy beam dimensions. 𝑦𝑦 2 𝑥𝑥+𝑋𝑋0
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 37 Other combined function magnets
Other combinations:
• dipole, quadrupole and sextupole • dipole & sextupole (for chromaticity control) • dipole, skew quad, sextupole, octupole (at DL) Generated by • pole shapes given by sum of correct scalar potentials • amplitudes built into pole geometry – not variable • multiple coils mounted on the yoke • amplitudes independently varied by coil currents
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 38 The SRS multipole magnet
Could develop: • vertical dipole • horizontal dipole • upright quad • skew quad • sextupole • octupole • others
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 39 The Practical Pole
Practically, poles are finite, introducing errors; these appear as higher harmonics which degrade the field distribution. However, the iron geometries have certain symmetries that restrict the nature of these errors. Dipole: Quadrupole:
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 40 Possible symmetries
Lines of symmetry: Dipole Quad Pole orientation y = 0 x = 0 determines whether pole y = 0 is normal or skew.
Additional symmetry x = 0 y = ± x imposed by pole edges.
The additional constraints imposed by the symmetrical pole edges limits the values of n that have non-zero coefficients Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 41 Dipole symmetries
Type Symmetry Constraint
Pole orientation φ θ = φ( θ) all Jn = 0
Pole edges φ(θ) = −φ(π − θ) Kn non-zero only for − n = 1, 3, 5, etc.
+φ +φ +φ
-φ
So, for a fully symmetric dipole, only 6, 10, 14 etc. pole errors can be present.
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 42 Quadrupole symmetries
Type Symmetry Constraint
Pole orientation φ(θ) = φ( θ) All Jn = 0 φ(θ) = φ(π θ) − − Kn = 0 for all odd n − − Pole edges φ(θ) = φ( θ) Kn non-zero 𝜋𝜋 for n = 2, 6, 10, etc. 2 −
So, for a fully symmetric quadrupole, only 12, 20, 28 etc. pole errors can be present.
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 43 Sextupole symmetries
Type Symmetry Constraint
Pole orientation φ(θ) = φ( θ) All Jn = 0 φ(θ) = φ( θ) Kn = 0 where n is φ(θ) = −φ(2𝜋𝜋− θ) 3 not a multiple of 3 − 4𝜋𝜋 − − 3 − Pole edges φ(θ) = φ( θ) Kn non-zero only 𝜋𝜋 for n = 3, 9, 15, etc. 3 − So, for a fully symmetric sextupole, only 18, 30, 42 etc. pole errors can be present.
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 44 Summary - ‘Allowed’ Harmonics
Summary of ‘allowed harmonics’ in fully symmetric magnets: Fundamental ‘Allowed’ geometry harmonics Dipole, n = 1 n = 3, 5, 7, ...... (6 pole, 10 pole, etc.) Quadrupole, n = 2 n = 6, 10, 14, .... (12 pole, 20 pole, etc.) Sextupole, n = 3 n = 9, 15, 21, ... (18 pole, 30 pole, etc.) Octupole, n = 4 n = 12, 20, 28, .... (24 pole, 40 pole, etc.)
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 45 Asymmetries generating harmonics (i)
Two sources of asymmetry generate ‘forbidden’ harmonics: i) magnetic asymmetries - significant at low permeability:
e.g. C core dipole not completely symmetrical about pole centre, but negligible effect with high permeability. Generates n = 2,4,6, etc.
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 46 Asymmetries generating harmonics (ii) ii) asymmetries due to small manufacturing errors in dipoles:
n = 2, 4, 6 etc. n = 3, 6, 9, etc.
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 47 Asymmetries generating harmonics (iii)
ii) asymmetries due to small manufacturing errors in quadrupoles:
n = 3 n = 2 (skew) n = 4 -ve n = 3 n = 4 +ve These errors are bigger than the finite µ type; can seriously affect machine behaviour and must be controlled.
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 48 Current Affairs FIELDS DUE TO COILS
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 49 Introduction of currents
Now for ≠ 0 ∇ × =
To expand,𝒋𝒋 use Stoke’s Theorem𝑯𝑯 𝒋𝒋 dS for any vector V and a closed curve s: . = × . ds V Apply this∫ 𝑽𝑽 to:𝑑𝑑𝒔𝒔 ∇ ×∬ 𝜵𝜵= 𝑽𝑽 𝑑𝑑𝑺𝑺 then in a magnetic circuit: (Ampere’s equation) 𝑯𝑯 𝒋𝒋 . =
N I (Ampere-turns) is total current∫ 𝑯𝑯 𝑑𝑑𝒔𝒔 cutting𝑁𝑁𝑁𝑁 S
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 50 Excitation current in a dipole B is approximately constant round the NI/2 loop made up of λ and g, (but see λ below); But in iron, 1 g = and NI/2 𝜇𝜇 ≫ 𝑎𝑎𝑎𝑎𝑎𝑎 𝐻𝐻 µ >> 1 So =𝐻𝐻𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝜇𝜇 𝜇𝜇0𝑁𝑁𝑁𝑁 𝜆𝜆 𝐵𝐵𝑎𝑎𝑎𝑎𝑎𝑎 g, and λ/µ are the 'reluctance'𝑔𝑔+𝜇𝜇 of the gap and iron. Approximation ignoring iron reluctance ( ): 𝜆𝜆 = 𝜇𝜇 ≪ 𝑔𝑔 𝐵𝐵𝐵𝐵 𝑁𝑁𝑁𝑁 − 𝜇𝜇0 Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 51 Excitation current in quad & sextupole
For quadrupoles and sextupoles, the required excitation can be calculated by considering fields and gap at large x. For example: Quadrupole: y
Pole equation: = 2 On x axes =𝑅𝑅 𝑥𝑥𝑥𝑥 2 x B where g is gradient in T/m 𝐵𝐵𝑦𝑦 𝑔𝑔𝑔𝑔 The same method for a At large x (to give vertical lines of B): Sextupole (coefficient g ) = S 2 𝑅𝑅 gives: = i.e.𝑁𝑁𝐼𝐼 𝑔𝑔𝑔𝑔 2𝑥𝑥𝜇𝜇0 2 (per pole) = (per pole) 𝑔𝑔𝑔𝑔 3 𝑠𝑠 𝑁𝑁𝐼𝐼 2 𝜇𝜇0 𝑔𝑔 𝑅𝑅 𝑁𝑁𝐼𝐼 3𝜇𝜇0 Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 52 General solution for magnets order n
In air (remote currents!) = µ0 = ∇φ y Integrating over a limited path 𝑩𝑩 𝑯𝑯 𝑩𝑩 −φ –φ (not circular) in air: = 1 2 φ , φ are the scalar potentials at two points in air. 1 2 𝑁𝑁𝑁𝑁 𝜇𝜇0 Define φ = 0 at magnet centre; φ = µ0 NI then potential at the pole is: µ
0 Apply the general equations for magnetic 𝑁𝑁𝐼𝐼 field harmonic order n for non-skew φ = 0 magnets (all Jn = 0) giving: = 1 1 𝐵𝐵𝑟𝑟 𝑛𝑛 Where: 𝑛𝑛−1 𝑁𝑁𝑁𝑁 𝑛𝑛 𝜇𝜇0 𝑅𝑅 𝑅𝑅 NI is excitation per pole R is the inscribed radius (or half gap in a dipole) is magnet strength in T/m(n-1) 𝐵𝐵𝑟𝑟 𝑛𝑛−1 𝑅𝑅
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 53 Further Reading
• CERN Accelerator School on Magnets Bruges, Belgium; June 2009 https://arxiv.org/html/1105.5069v1 • United States Particle Accelerator School Magnet and RF Cavity Design, January 2016 http://uspas.fnal.gov/materials/16Austin/austin- magnets.shtml • J.D. Jackson, Classical Electrodynamics • J.T. Tanabe, Iron Dominated Electromagnets
Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn 2016 54