INTRODUCTION to the DESIGN and FABRICATION of IRON- DOMINATED ACCELERATOR MAGNETS

Transcription

1 INTRODUCTION to the DESIGN and FABRICATION of IRON- DOMINATED ACCELERATOR MAGNETS Cherrill Spencer, Magnet Engineer SLAC National Accelerator Laboratory Menlo Park, California, USA Lecture # 1 of 2 Mexican Particle Accelerator School, October 2011

2 Overview of my Two Lectures, part 1 Lecture 1 Purpose of my lectures on electromagnets And steps of producing accelerator magnets How Maxwell s Equations help us design magnets for particle accelerators The steps of designing accelerator magnets Computer modelling to make a detailed design 2

3 Overview of my Two Lectures, part 2 Lecture 2 Choice of materials and fabrication techniques Fabricating steel yoke Fabricating coils Assembling the whole magnet, connecting it to power and cooling sources Testing & magnetically measuring a magnet Installing magnets in a beam line : alignment Resources where you can find out much more about making accelerator magnets 3

4 Purpose of my lectures; what I cannot do in the time available Why am I smiling? Give you an overview of topic, not enough time to go into much detail. Can answer your questions outside of the lecture time. Tell you about designing and fabricating accelerator magnets. What a magnet engineer does in order to produce magnets. 4

5 Steps in producing accelerator magnets Accelerator physicist creates a beam lattice, knows purpose of each magnet. Develops set of requirements for each magnet: integrated field strength, aperture size, approximate effective length, how it operates. Gives requirements to a magnet engineer (ME) ME begins discussions with power supply (PS) specialists and facilities personnel re sizes of PS, cooling water, size of tunnel ME performs the design of each style of magnet, on paper, with computer modeling [more details later] 5

6 More steps in producing accelerator magnets ME works with a mechanical designer to produce details drawings of all the parts of the magnet ME continues discussions with PS specialist and facilities personnel. Interfaces between magnet and rest of world must be agreed upon: power, water, supports. ME finds a magnet fabrication vendor; negotiates on choice of materials, cost, schedule ME monitors fabrication, monitors quality control data, answers questions from vendor ME supervises magnetic measurements -> field quality ME discusses magnetic data with AP->can be installed 6

7 Review some electromagnetism facts Electromagnetism is concerned with the long range interactions between electric charges Electric interactions occur between static OR moving charges Magnetic interactions occur only between moving charges By definition a FIELD is the FORCE created by a system of charges on one unit charge, q=1, having a velocity vector v=1, parallel to the velocity vector of a system of charges. Two fields derived from 2 kinds of force. electric force coming from the electric field, E magnetic force coming from the magnetic field, B, according to F = q(e + v x B) 7

8 Electric currents create magnetic fields Compass needle is affected by the bar magnet. It is forced into position: there is a magnetic field produced by the bar magnet. Can draw where lines of force are depending on orientation of the compass needle. Definition: lines of force, also called flux lines, or lines of magnetic induction, go from North to South. These lines have no beginning and no end; North poles and South poles always exist in pairs. MAXWELL S 2 nd EQUATION: B = 0 is math representation of above facts Electric current [positive ions moving] creates a magnetic field around the wire in a plane perpendicular to the wire: lines of force are B. MAXWELL S 4 th EQUATION x B = µ 0 J is math representation of this in free space with non-magnetic materials. 8

9 Units. Creating shapely magnetic fields. B is also called the Magnetic Flux Density and its units are Weber/meter 2 One Weber/meter 2 = one TESLA. We will use Tesla. Also 1 Tesla= 10kGauss J is a Current Density: amps/meter 2 B in Tesla Amps J in 2 m µ is the permeability of µ 0 is the permeability of air = 1 vacuum = 9 4π 10-7 T m Amp I can manipulate the current in space into strange shapes and the B flux will have its own shapes. What kind of shape do I want in magnets in a particle accelerator? As I wrote Maxwell s equations they are partial differential equations, do not yield the B field directly, we have to do some integration with some boundary conditions to be able to calculate values for B.

10 What shapes of magnetic fields do accelerator physicists want to guide & focus their beams? Explanation here of figure: Is a magnetic field pattern for a quadrupole that will focus the beam. Length of arrows indicate strength of field, at that point along an axis. Are not flux lines Can achieve this field shape with just electric currents but its easier and the currents needed for the same B are smaller if we use some ferromagnetic material to help shape the B 10

11 Representation of electromagnetic fields by potential functions When dealing with fields, is easier to represent them by POTENTIALS. Can do this with either a vector potential, A or a scalar potential Φ These potentials are caused by sources such as currents. Pay attention to where the field is relative to the source. e.g. Electric field intensity derived from electrostatic scalar potential V through E= - V Similarly, in a region where J=0, B can be derived from the magnetic scalar potential Φ [in cartesian coords]: B = - Φ. But also B = 0, so 2 Φ=0 (Laplace Eq) 11

12 Equipotential surfaces and ferromagnetism Metallic surfaces are equipotential surfaces for electric fields, what is the equivalent for magnetic fields? Consider ferromagnetic material such as iron or steel (=iron plus very small percentages of elements such as Carbon, Manganese, Nickel, Sulphur) Electrons in Fe give an Fe atom a magnetic moment, i.e. it has spin. In a collection of Fe atoms it is energetically favourable for the spins of adjacent atoms to be parallel -> say the domain is magnetized, it has a N and a S pole But the many domains in a piece of iron are randomly oriented, so overall it has no magnetization. M=0 If APPLY an external magnetic field then it will make the domains change orientation, piece of iron becomes magnetized. 12

13 Magnetizing a piece of steel The applied field, usually created by an external excitation current is called H: the magnetic field intensity. Has same units as M: ampere-turns/meter As H is increased so does the M, as domains get larger and rotate to align with H s direction. Eventually all the domains are parallel to the applied field and SATURATION has been reached. The overall flux density B is related to H and M: B = µ 0 (M + H) 13

14 Hysteresis loop : variation of M, B with H Saturation of iron occurs By definition the permeability is ratio between B and H µ = B/H To have a dimensionless parameter put µ = µ r µ 0 µ r is the relative permeability and µ 0 is the vacuum perm. It takes all the units! µ r =1 for air µ r for iron: is not constant, varies :1 to ~2000 USE THIS PROPERTY OF IRON IN MAKING MAGNETS: electromagnets use soft 14 materials: smaller H c

15 Consider the magnetic field in the aperture of an accelerator magnet 2 Φ=0 Solve Laplace s equation by separating variables and imposing boundary conditions that Φ be periodic in the angular coord and be finite at r =0. We know we want our focusing magnet to have 4 fold symmetry.- how to make this and a field that varies with radius? Series expansion of the scalar potential. Then take negative value of gradient of potential -> field For a given r the m-th term has m maxima and m minima as a function of azimuthal angle θ. These angular positions may be regarded as the locations of magnetic poles. 15

16 Continue analysis of 2D fields we could produce in a magnet Use De Moivre theorem to get expression into cartesian coords from polar coords Evaluate equations (1) and (2) for m from 1 to 5- see table on next slide 16 (1) (2) Study real and imaginary parts and if make Φ a constant then that equipotential defines an ideal pole shape for a magnet with 2m poles. m=2 gives a quadrupole

17 Potentials and field components for 2 D multipoles up to m=5 Evaluate the equations (1) and (2) on previous slide. Equation (1) gives the equipotential lines that define the ideal pole profile for a given type of magnet. 17

18 Consider m=2 equations in table Change of nomenclature : V= Φ C= -B 2 Scalar equipotential shape for an m=2 pole is a HYPERBOLA with its tip at some radius. Define an iron-free circular region with radius R and the equation for the quadrupole s pole is xy= R 2 /2 18

19 So a whole quadrupole looks like this The real quadrupole field is generated by the boundary condition where the magnet pole is located at the scalar potential hyperbola. Physical poles are made of steel. The field varies linearly with the distance from the magnet center (red lines). It focuses the beam along one plane while defocusing the beam along the orthogonal plane. An F or focusing quadrupole focuses the particle beam along the horizontal plane 19

20 We have dealt with iron pole shapes, now work out how much current we need x B = µµ 0 J Maxwell s 4 th equation in differential form x H = J Use H, will be more convenient Apply Stoke s Theorem and differential form can be written in integral form. Stokes Theorem - The line integral of a potential function around a closed boundary is equal to the area integral of the source distribution within that closed boundary. Suppose the source is a simple current Circumference of circle = 2π r I = H B = µ 0 dl 2π r 20

21 Important facts about fields at iron-air boundaries Examine Maxwell Eq 2 to find rule about B behavior at ironair interfaces: B = 0, can be re-written B x / x + B y / y + B z / z = 0 or B n / x =0 Can state from 2 nd expression that the B field is continuous along its vector direction, thus the perpendicular component of B field does not change across material interfaces. Examine Maxwell Eq 4 in integral form and apply it to an iron-air interface with no current sources. Can show that the parallel H vector H is continuous across such an interface. [look all this up in your favourite electromagnetism text book] 21

22 Use integral of H along flux path to calculate current needed to create B B, the gradient is constant Along Path 1, ( ) B r H r Therefore and B r = ( r) H = ' B' r µ 0 Consider path 2 B iron B h, L iron >10h and µ iron >1000, so integral along path 2 is ~0.01path1 and can be ignored for estimating current H Path1 Finally Path2 h dl = B' rdr µ 0 0 H dl + H H Path3 dl = NI = dl B' h 2µ 0 B' h 2µ

23 Use 2D computer modeling program to define pole and coil shapes, where to place them Define boundaries of steel core and coil in plane at right angles to beam direction. Beam passing here Program makes mesh of small triangles Would be too difficult and tedious to create a pole-tip shape and place some coils near it and calculate by hand the field distribution in the aperture where beam will pass In 1960s family of computer codes called POISSON developed to solve Poisson s equation and calculate the fields in a combination of steel and coil shapes input by the user. 23 ME inputs material properties and flux boundary conditions into Automesh

24 Figure showing what POISSON program produces, plus printout of field values TABLE FOR FIELD COEFFICIENTS FOR ATF2 QD0 with 2.5cm rad w/0.25"sideshim SQ end 7 Dec07 NORMALIZATION RADIUS = (BX - I BY) = I * SUM N*(AN + I BN)/R * (Z/R)**(N-1) N N(AN)/R N(BN)/R ABS(N(CN)/R) RATIO 2Npole/4 pole E E E E E E E E E E E E E E E E E E E E POISSON produces a flux plot, showing where the flux lines go. Closer together lines indicate higher magnetic fields. Note this happens at corners and sharply curved edges- steel will saturate there first E E E E-03

25 Here is a photo of the actual quadrupole being modeled in this POISSON run Steel core is painted red. Coils are painted light green 25

26 Typical set of magnet requirements from AP For A DIPOLE MAGNET Nominal bending angle Theta = mrad Nominal integral dipole field at 9 GeV B 0 L = kg-m Effective length (approx value) L = m Height of aperture 25.4 mm Operating field range from 15.5 to 16.6 kg Integrated strength operating range to kg-m Minimum full pole width for beam 210 mm Dipole field variation relative to average dipole field in four bends in e- or e+ chicane DeltaB 0 /Baver = +/-0.1% Multipole field tolerances at R = 10 cm (for coupled beams) B1/B0 = +/-0.21% (quadrupole) B2/B0 = +/-0.27% (sextupole) B3/B0 = +/-0.30% (octupole) B4/B0 = +/-0.33% (decapole) 26

27 Shape of top right hand part of new sector 10 dipole: my design in response to the requirements given to me by an AP Total width of poletip at gap is 26.0 cm Total width of core: 30 [76.2cm] Low carbon solid steel core Low carbon steel core Main coil: 20 turns of 0.44 sq hollow copper conductor with φ hole Trim coil: 88 turns of AWG#10 solid copper wire: <5% of main coil 1.27cm (0.5 ) half-gap Need model only one quarter of the whole magnet 2.54cm (1 ) wide, 30º chamfer improves the multipoles in the gap to well below the specs 27

28 POISSON 2D MODEL of new style dipole showing magnetic flux lines, with trim working In POISSON, because of symmetry of top and bottom halves and left and right halves, need to model only one quarter of the whole magnet Note the flux lines are perpendicular to the mid-plane of the gap where beam will pass [into the page] and enter the steel core at right angles to edge of steel 28

29 Photo of a dipole with dog-ear coils Top coil. Current in ends of coils does not contribute to the field in the magnet gap Dipole gap where beam passes through Current flowing in this end part of coil has no affect on the field in the gap. 29

30 Magnet Lecture #1 Homework Question 0.025m 0.30m 0.17m Develop an equation that connects the path integral of B through this section of a dipole and the total current in the section of the top coil enclosed by the path. If the half gap is 0.025m and B is to be one Tesla, what is the value of NI needed in this top coil? 30