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Unit 5: Field harmonics 5.2. QUESTIONS. How to express the magnetic field shape and uniformityThe magnetic field is a continuous, vectorial quantityCan we express it, and its shape, using a finite number of coefficients ?Yes: field harmonicsBut not everywhere We are talking about electroma
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1. Unit 5Field harmonics Soren Prestemon and Paolo Ferracin
Lawrence Berkeley National Laboratory (LBNL)
Ezio Todesco
European Organization for Nuclear Research (CERN)
2. Unit 5: Field harmonics 5.2 QUESTIONS How to express the magnetic field shape and uniformity
The magnetic field is a continuous, vectorial quantity
Can we express it, and its shape, using a finite number of coefficients ?
Yes: field harmonics
But not everywhere
We are talking about electromagnets:
What is the field we can get from a current line ?
The mythical Biot-Savart law !
What are the components of the magnetic field shape ?
What are the beam dynamics requirements ?
3. Unit 5: Field harmonics 5.3 CONTENTS 1. Definition of field harmonics
Multipoles
2. Field harmonics of a current line
The Biot-Savart law
3. Validity limits of field harmonics
4. Field model and components
Geometric, persistent currents, iron saturation
5. Beam dynamics requirements on field harmonics
Random and skew components
Specifications (targets) at injection and high field
4. Unit 5: Field harmonics 5.4 1. FIELD HARMONICS: MAXWELL EQUATIONS Maxwell equations for magnetic field
In absence of charge and magnetized material
If (constant longitudinal field), then
5. Unit 5: Field harmonics 5.5 1. FIELD HARMONICS: ANALYTIC FUNCTIONS A complex function of complex variables is analytic if it coincides with its power series
on a domain D !
Note: domains are usually a painful part, we talk about it later
A necessary and sufficient condition to be analytic is that
called the Cauchy-Riemann conditions
6. Unit 5: Field harmonics 5.6 1. DEFINITION OF FIELD HARMONICS If
and therefore the function By+iBx is analytic
where Cn are complex coefficients
Advantage: we reduce the description of a function from R2 to R2 to a (simple) series of complex coefficients
Attention !! We lose something (the function outside D)
7. Unit 5: Field harmonics 5.7 1. DEFINITION OF FIELD HARMONICS
Each coefficient corresponds to a pure multipolar field
Magnets usually aim at generating a single multipole
Dipole, quadrupole, sextupole, octupole, decapole, dodecapole
Combined magnets: provide more components at the same time (for instance dipole and quadrupole) not very common
8. Unit 5: Field harmonics 5.8 1. DEFINITION OF FIELD HARMONICS Vector potential
Since one can always define a vector potential A such that
The vector potential is not unique (gauge invariance): if we add the gradient of any scalar function, it still satisfies
Scalar potential
In the regions free of charge and magnetic material
Therefore in this case one can also define a scalar potential (such as for gravity)
One can prove that is an analytic function in a region free of charge and magnetic material
9. Unit 5: Field harmonics 5.9 1. DEFINITION OF FIELD HARMONICS
The field harmonics are rewritten as (EU notation)
We factorize the main component (B1 for dipoles, B2 for quadrupoles)
We introduce a reference radius Rref to have dimensionless coefficients
We factorize 10-4 since the deviations from ideal field are ?0.01%
The coefficients bn, an are called normalized multipoles
bn are the normal, an are the skew (adimensional)
US notation is different from EU notation
10. Unit 5: Field harmonics 5.10 1. DEFINITION OF FIELD HARMONICS Reference radius is usually chosen as 2/3 of the aperture radius
This is done to have numbers for the multipoles that are not too far from 1
Some wrong ideas about reference radius
Wrong statement 1: the expansion is valid up to the reference radius
The reference radius has no physical meaning, it is as choosing meters of mm
Wrong statement 2: the expansion is done around the reference radius
A power series is around a point, not around a circle. Usually the expansion is around the origin
11. Unit 5: Field harmonics 5.11 1. FIELD HARMONICS: LINEARITY Linearity of coefficients (very important)
Non-normalized coefficients are additive
Normalized coefficients are not additive
Normalization gives handy (and physical) quantities, but some drawbacks pay attention !!
12. Unit 5: Field harmonics 5.12 CONTENTS 1. Definition of field harmonics
Multipoles
2. Field harmonics of a current line
The Biot-Savart law
3. Validity limits of field harmonics
4. Field model and components
Geometric, persistent currents, iron saturation
5. Beam dynamics requirements on field harmonics
Random and skew components
Specifications (targets) at injection and high field
13. Unit 5: Field harmonics 5.13 2. FIELD HARMONICS OF A CURRENT LINE Field given by a current line (Biot-Savart law)
using
!!!
we get
14. Unit 5: Field harmonics 5.14 2. FIELD HARMONICS OF A CURRENT LINE Now we can compute the multipoles of a current line
15. Unit 5: Field harmonics 5.15 2. FIELD HARMONICS OF A CURRENT LINE Multipoles given by a current line decay with the order
The slope of the decay is the logarithm of (Rref/|z0|)
At each order, the multipole decreases by a factor Rref /|z0|
The decay of the multipoles tells you the ratio Rref /|z0|, i.e. where is the coil w.r.t. the reference radius
like a radar we will see an application of this feature to detect assembly errors through magnetic field shape in Unit 21
16. Unit 5: Field harmonics 5.16 2. FIELD HARMONICS OF A CURRENT LINE Multipoles given by a current line decay with the order
The semilog scale is the natural way to plot multipoles
This is the point of view of Biot-Savart
But usually specifications are on a linear scale
In general, multipoles must stay below one or a fraction of units see later
This explains why only low order multipoles, in general, are relevant
17. Unit 5: Field harmonics 5.17 2. FIELD HARMONICS OF A CURRENT LINE Field given by a current line (Biot-Savart law) vector potential
18. Unit 5: Field harmonics 5.18 CONTENTS 1. Definition of field harmonics
Multipoles
2. Field harmonics of a current line
The Biot-Savart law
3. Validity limits of field harmonics
4. Field model and components
Geometric, persistent currents, iron saturation
5. Beam dynamics requirements on field harmonics
Random and skew components
Specifications (targets) at injection and high field
19. Unit 5: Field harmonics 5.19 3. VALIDITY LIMITS OF FIELD HARMONICS When we expand a function in a power series we lose something
Example 1. In t=1/2, the function is
and using the series one has
not bad with 4 terms we compute the function within 7%
20. Unit 5: Field harmonics 5.20 3. VALIDITY LIMITS OF FIELD HARMONICS When we expand a function in a power series we lose something
Ex. 2. In t=1, the function is infinite
and using the series one has
which diverges this makes sense
21. Unit 5: Field harmonics 5.21 3. VALIDITY LIMITS OF FIELD HARMONICS When we expand a function in a power series we lose something
Ex. 3. In t=-1, the function is well defined
BUT using the series one has
even if the function is well defined, the series does not work: we are outside the convergence radius
22. Unit 5: Field harmonics 5.22 3. VALIDITY LIMITS OF FIELD HARMONICS When we expand a function in a power series we lose something
Ex. 3. In t=-1, the function is well defined
BUT using the series one has
even if the function is well defined, the series does not work: we are outside the convergence radius
23. Unit 5: Field harmonics 5.23 3. VALIDITY LIMITS OF FIELD HARMONICS If we have a circular aperture, the field harmonics expansion relative to the center is valid within the aperture
For other shapes, the expansion is valid over a circle that touches the closest current line
24. Unit 5: Field harmonics 5.24 3. VALIDITY LIMITS OF FIELD HARMONICS If all coefficients are known with infinite accuracy, one can reconstruct the function from the first expansion wherever no singularities are present (analytic continuation)
Analytic function: local info gives global values (no need to travel )
f has a different convergence domain !
In practice the technique is not used since the measurement accuracy limits this possibility
Magnets with non-circular apertures are measured over several circular domains covering the interesting area
25. Unit 5: Field harmonics 5.25 3. VALIDITY LIMITS OF FIELD HARMONICS Field harmonics in the heads
Harmonic measurements are done with rotating coils of a given length (see unit 21) they give integral values over that length
If the rotating coil extremes are in a region where the field does not vary with z, one can use the 2d harmonic expansion for the integral
If the rotating coil extremes are in a region where the field vary with z, one cannot use the 2d harmonic expansion for the integral
One has to use a more
complicated expansion
26. Unit 5: Field harmonics 5.26 CONTENTS 1. Definition of field harmonics
Multipoles
2. Field harmonics of a current line
The Biot-Savart law
3. Validity limits of field harmonics
4. Field model and components
Geometric, persistent currents, iron saturation
5. Beam dynamics requirements on field harmonics
Random and skew components
Specifications (targets) at injection and high field
27. Unit 5: Field harmonics 5.27 4. OUTLOOK OF THE STATIC FIELD MODEL The final shape of the magnetic field is given by the sum of different contributions
The presence of some contributions and their magnitude depends on the design and on the excitation level
We will focus on superconducting magnets for the machine in the range of 0.1 to 15 T (and more?)
Detector magnets have a different phenomenology
Energy swing (from injection to high field) is in the range of a factor 5 to 20
Phenomena at injection and high field are different
We will come back on this in Unit 20
We first focus on static effects (no dependence on time)
We will follow the assembly stages of the magnet
28. Unit 5: Field harmonics 5.28 4. OUTLOOK OF THE STATIC FIELD MODEL 1 - Assembly of the coil: the collared coil
Collared coil: the coil surrounded
by its retaining structure
Room temperature: the current flows
in the copper part of the cable
Field is given by a sum of Biot-Savart contributions
Linear in the current
Multipoles independent of the current
The assembly is under forces that deform the coil: position is not the nominal one ? offset on main field and multipoles
Relevant contribution, tentative estimates with FEM (mechanical)
In case of collars with permeability different from 1 (stainless steel), a perturbation is added (offset to multipoles and main field)
Even with low permeability (??1.005) the contribution is not negligible
Can be compute with FEM methods (magnetic)
29. Unit 5: Field harmonics 5.29 4. OUTLOOK OF THE STATIC FIELD MODEL 2 - Assembly of the coil: the cold mass (collared coil + iron yoke)
Iron yoke is used to enhance the field,
to avoid fringe fields, and sometimes
as a mechanical support
Room temperature: the current flows
in the copper part of the cable
Iron yoke increases the main field by a factor ? (at most a factor two)
Iron yoke reduces multipoles by a factor ?, and adds an offset on multipoles
Iron yoke is not saturated ? linear, can be computed analytically for simple geometries or with magnetic FEM
Iron yoke also changes stress and coil deformations ? additional offset, can be computed with mechanical FEM
30. Unit 5: Field harmonics 5.30 4. OUTLOOK OF THE STATIC FIELD MODEL 3 - Cool-down to 2-4 K: the geometric
Additional deformations in the coil ? offset in the multipoles
Contraction of the structure rescaling, little effect
Change of pre-stress larger effect estimated with mechanical FEM
This gives the geometric
Field is linear in the current, multipoles independent of the current
Cannot be directly measured, but can be estimated from measurements
4 At injection energy: persistent currents
Magnetization of the superconductor large effect at low fields can be computed knowing the cable magnetization (see Unit 13)
5 At collision: iron saturation and Lorentz forces
If the field in the iron is larger than 2 T, the iron saturates and its permeability changes, going from ?103 to 1 this affects the main field and multipoles can be computed with magnetic FEM
At high field the induced Lorentz forces can deform the coil can be estimated with mechanical FEM
31. Unit 5: Field harmonics 5.31 4. OUTLOOK OF THE STATIC FIELD MODEL 6 Beam screen and cold bore
The cold bore and the beam screen inside affect the multipoles, depending on their permeability
This gives an additional offset on multipoles
Summary of static field model
Relevance estimated for the case of LHC dipoles
32. Unit 5: Field harmonics 5.32 CONTENTS 1. Definition of field harmonics
Multipoles
2. Field harmonics of a current line
The Biot-Savart law
3. Validity limits of field harmonics
4. Field model and components
Geometric, persistent currents, iron saturation
5. Beam dynamics requirements on field harmonics
Random and systematic components
Specifications (targets) at injection and high field
33. Unit 5: Field harmonics 5.33 5 BEAM DYNAMICS REQUIREMENTS Integral values for a magnet
For each magnet the integral of the main component and multipoles are measured
Main component: average over the straight part
(head excluded)
Magnetic length: length of the magnet as
If there were no heads, and the integrated
strength is the same as the real magnet
34. Unit 5: Field harmonics 5.34 5 BEAM DYNAMICS REQUIREMENTS Integral values for a magnet
Average multipoles: weighted average with the main component
Systematic and random components over a set of magnets
Systematic: mean of the average multipoles
Random: standard deviation of the average multipoles
35. Unit 5: Field harmonics 5.35 5 BEAM DYNAMICS REQUIREMENTS Beam dynamics requirements
Rule of thumb (just to give a zero order idea): systematic and random field harmonics have to be of the order of 0.1 to 1 unit
(with Rref one third of magnet aperture radius)
Higher order are ignored in beam dynamics codes (in LHC up to order 11 only)
Note that spec is rather flat, but multipoles are decaying !! Therefore in principle higher orders cannot be a problem
36. Unit 5: Field harmonics 5.36 5 BEAM DYNAMICS REQUIREMENTS Beam dynamics requirements: one has target values for
range for the systematic values (usually around zero, but not always)
maximum spread of multipoles and main component
The setting of targets is a complicated problem, with many parameters, without a unique solution
Example: can we accept a larger random b3 if we have a smaller random b5 ?
Usually it is an iterative (and painful) process: first estimate of randoms by magnet builders, check by accelerator physicists, requirement of tighter control on some components or of adding corrector magnets
37. Unit 5: Field harmonics 5.37 5 BEAM DYNAMICS REQUIREMENTS Beam dynamics requirements (tentative)
Dynamic aperture requirement: that the beam circulates in a region of pure magnetic field, so that trajectories do not become unstable
Since the beam at injection is larger of a factor ?(Ec/Ei) , the requirement at injection is much more stringent?? no requirement at high field
Exception: the low beta magnets, where at collision the beta functions are large, i.e., the beam is large
Chromaticity, linear coupling, orbit correction
This conditions on the beam stability put requirements both at injection and at high field
Mechanical aperture requirement
Can be limited by an excessive spread of the main component or bad alignment
38. Unit 5: Field harmonics 5.38 SUMMARY We outlined the Maxwell equations for the magnetic field
We showed how to express the magnetic field in terms of field harmonics
Compact way of representing the field
Attention !! Validity limits and convergence domains
We outlined the main contributions to field harmonics given by the successive assembly stages of a magnet
We outlined some issues about the beam dynamics specifications on harmonics
39. Unit 5: Field harmonics 5.39 COMING SOON
Coming soon
It is useful to have magnets that provide pure field harmonics
How to build a pure field harmonic (dipole, quadrupole ) [pure enough for the beam ] with a superconducting cable ? Which field/gradient can be obtained ?
40. Unit 5: Field harmonics 5.40 REFERENCES On field harmonics
A. Jain, Basic theory of magnets, CERN 98-05 (1998) 1-26
Classes given by A. Jain at USPAS
P. Schmuser, Ch. 4
On convergence domains of analytic functions
Hardy, Divergent series, first chapter (dont go further)
On the field model
A.A.V.V. LHC Design Report, CERN 2004-003 (2004) pp. 164-168.
N. Sammut, et al. Mathematical formulation to predict the harmonics of the superconducting Large Hadron Collider magnetsPhys. Rev. ST Accel. Beams 9 (2006) 012402.
41. Unit 5: Field harmonics 5.41 ACKOWLEDGEMENTS A. Jain for discussions about reference radius, multipoles in heads, vector potential
S. Russenschuck for discussions about vector potential
G. Turchetti for teaching me analytic functions and divergent series, and other complicated subjects in a simple way
www.wikipedia.org for most of the pictures