Field singularities and coil fields

Transcription

Field Computation and Magnetic Measurements for Accelerator Magnets Superconducting Accelerator Magnets
(Field singularities and coil fields)
Stephan Russenschuck, 2015
Stephan Russenschuck, CERN TE-MSC-MM, 1211 Geneva 23
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Where we are
➔ We have studied the mathematical foundations of magnetic fields –
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Vectorfields Lumped circuit techniques for normal conducting magnets The Laplace equation Harmonic fields ➔ So far we assumed that the fields on the domain boundary where known (from measurements or calculations) ➔ Now we need to address how to calculate these fields – The field of line currents and the Biot Savart law – Development of these field solutions into the Eigenfunctions
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Rutherford (Roebel) Kabel, Strand, Nb-‐Ti Filament
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Cabling Machine for Rutherford Cables
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Turk’s Head
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The Field of Line Currents
Why bother?
Reciprocity; except for
sign it does not matter if
we exchange the source
and field points
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Greens Functions of Free Space
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Green’s Functions of Free Space
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Green’s Functions of Free Space
But what if boundaries are present?
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Biot-‐Savart’s Law
This works only in Cartesian Coordinates
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Biot Savart’s Law
But wait a minute: Are we finished? Are we sure that the
divergence of the vector potential is zero as it was required for
the Laplace equation?
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Biot Savart’s Law
But wait a minute: Are we finished? Are we sure that the
divergence of the vector potential is zero as it was required for
the Laplace equation?
Current loops must always be closed and must not leave the problem domain
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Biot-‐Savart’s Law for Line Currents
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Vector Potential of a Line Current
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Field of a Line Current Segment
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Field of a Line Current (Infinitely Long)
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Field of a Line Current
Appearance of elliptic integrals:
To be solved numerically.
On axis:
In the center:
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Magnetic Dipole Moment
Far field approximation
Definition
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Solid Angle and Magnetic Scalar Potential
Solid angle (easy to compute) gives the magnetic scalar potential of a current
loop
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The Imaging Current Method
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Notes on the Imaging Method
➔ Domain 1: Domain with current sources ➔ Domain 2: Highly permeable material – All imaging currents must be in domain 2 – The sources and the images must create a field that satisfies the continuity conditions at the interface between domains 1 and 2 – The image of the image must be the original source – The field generated in domain 1 is identical to the source field plus the field from the (iron) magnetization. – The field generated in domain 2 has no physical significance
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The Field of a Line Current (2D)
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Imaging Method
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Ideal Current Distributions
Exercise: Do this for the quadrupole and compare the results in terms of bore radius
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Ideal Current Distributions
Exercise: Do this for the quadrupole and compare the results in terms of bore radius
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Nested Helices
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Nested Helices
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Coil-‐Block Approximations
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Generation of Multipole Field Errors
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The LHC Magnet Zoo
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Finite-‐Element / Boundary-‐Element Coupling
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Corrector Magnets
Octupole
Sextupole-spool pieces
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Magnet Extremities
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The CERN Field Computation Program ROXIE Stephan Russenschuck, CERN TE-MSC-MM, 1211 Geneva 23
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Numerical Field Computation
➔ Principles of numerical field computation – Formulation of the Problem – Weighted residual – Weak form – Discretization – Higher order elements – Numerical example ➔ Weak forms in 3-‐D ➔ Element shape functions – Global shape functions – Barycentric coordinates ➔ Mesh generation
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Field singularities and coil fields

Transcription

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