Iron Dominated Electromagnets

Iron Dominated Electromagnets
Jack Tanabe

Introduction Because of the proliferation of light source synchrotrons, which store and accelerate medium charged energy particles with stiffness Br approximately equal to 10 T-m, there is new interest in iron dominated magnets whose maximum field amplitude is limited by iron saturation characteristics. The most important properties of magnets designed and fabricated for this application is the need for high quality and for magnet to magnet reproducibility sharing the same family. These requirements reflect the requirements for these accelerators to store high current beams for several hours. The long beam storage life requires magnets whose fields are not contaminated by fields other than the desired fields. For dipoles, the field must be constant throughout its volume. For quadrupoles, the field must be linearly distributed throughout its volum. For sextupoles, the field must be quadratically distributed throughout its volume. This class is given for those involved in the design, fabrication and measurement of magnets whose properties satisfy the demanding requirements of this and other applications demanding high quality reproducible magnets.

Scope Accelerator magnets include: superconducting magnets,
magnets using permanent magnet material, fast pulsed magnets specialized magnets such as septa used for injection and extraction. The scope of these lectures will be limited to conventional room temperature, iron dominated accelerator magnet design.

Schedule The schedule for this one week half week course calls for two lecture sessions per day, one in the morning and the second in the afternoon. Contractual obligations require >20 hours of contact during this period. Therefore, each day will include five hours of lecture and lab (whew!). 10 lectures have been organized for this course. Also, it is anticipated that at least one afternoon computer laboratory session will be scheduled in order to familiarize the student with POISSON, the two dimensional magnetostatic code used by many magnet scientists engineers and designers. Also, for those taking the course for credit, the last morning will be scheduled for a final examination. Unfortunately, the economies of flying require me to leave Cornell early on Friday morning to catch a flight. Therefore, someone from the USPAS staff will be assigned to monitor the test and mail it to me for scoring. Therefore, this tight schedule makes it necessary to schedule some of the sessions for more than one lecture.

Purpose The purpose of this course in conventional magnet design is to transfer the skills and knowledge required to understand the design, fabrication and measurement of high quality, conventional iron dominated, modest field accelerator magnets to the next generation of magnet design scientists and engineers. It is an opportunity to collect and accumulate the tools, practices and mathematical expressions used in magnet design.

History and Structure This course has evolved over the years.
The first courses, taught at Stanford and Duke University, covered a far broader scope than the limited scope of this course. The first courses were taught by Drs. Klaus Halbach and Ross Schlueter and Jack Tanabe and involved complex theory, including mathematical transformations and the design and fabrication of permanent magnet structures. A later course was taught at the Chinese Academy of Science by Dr. Ross Schlueter and myself. Subsequent courses, taught by myself at Rice, University of Arizona and UC Santa Barbara, isolated a portion of the broader course to cover iron dominated electromagnets. This course was based on the notes of an internal training course at Lawrence Livermore National Laboratory. The lecture notes for this course have evolved over the repeated courses. These lecture notes have been reorganized and published as a book, “Iron Dominated Electromagnets”, intended to supplement the lectures and the notes for this course. The lectures roughly parallel the different chapters in the book and, in order to maximize the benefits of this course, the student is strongly urged to read the chapters prior to the scheduled lecture.

What This Course Will Cover Part One
This course will describe the design of electromagnets whose fields are shaped by the iron yoke. The physics of particle beam optics are briefly discussed, if only to illustrate the requirements for the magnets. Magnet types and their functions are discussed. Forces on particle beams are described and conventions for determining polarities and means of electrically connecting the magnets to achieve the desired polarities are discussed.

General principles for the visualization of two dimensional magnetic fields are described.
These visualization tools are useful since they provide means for describing the field shapes when the shapes of the yoke and coils are generally known without the necessity of going into the mathematics of magnetic fields. Despite these visualization tools, it is always useful to understand the mathematics of magnetic fields. Mathematical functions are developed from the two dimensional second order differential equations, derived from Maxwell’s electromagnetic equations, which describe the magnetic fields and their errors. Although the two dimensional expressions are developed, it is claimed (without proof) that the expressions holds also for the two dimensional fields integrated in the third dimension over the domain where the fields are non-zero. The properties of the mathematical expressions are exploited in order to characterize properties of the magnetic fields which result from the symmetries of the magnetic design.

The properties of the mathematical expressions are further exploited and result in developing a tool, conformal mapping, which can be used to extend knowledge about a simple magnet geometry, the dipole, to more complex magnet types, the quadrupole and sextupole magnets. The mathematical expressions are also useful for understanding the fundamentals of one of the most useful means of magnet analysis, rotating coil measurements of various magnet types.

Stoke’s Theorem is applied to Maxwell’s non-homogeneous differential equations and result in expressions to determine the currents required to produce desired magnetic fields. These expressions are used to select specific coil configurations and to lay out the general shape of the two dimensional magnet cross section. The engineering parameters of selected coils are computed to ensure that the final magnet design satisfies power supply and facilities constraints (ie. the hydraulic cooling capabilities of the facility).

Perturbation Theory, a method of relating mechanical and electrical tolerances and errors to the parameters provided by the physics tolerances of the magnets (multipole errors) of the synchrotron lattice is discussed. Perturbation Theory is further exploited to design trim coil configurations required to introduce trim fields in cores designed for other fields.

POISSON, a two dimensional magnetostatic code, originally written by Dr. Klaus Halbach and Ron Holsinger and maintained at Los Alamos by James Billen, is described. Example POISSON problems are presented in order to expose the student to this program and to teach the student means used to exploit some of its features. At least two computer lab sessions will be held so that the student can practice using this tool.

Some time will be spent discussing rotating coil magnetic measurements
Some time will be spent discussing rotating coil magnetic measurements. These discussions will be brief and general since the subject is complex and the student can obtain most of what is needed to understand the technique by careful reading of the subject in the chapter written on this subject. The concept of magnetic stored energy is used to introduce magnetic forces, the requirements power supplies to control slowly changing fields in a magnet. (Slow meaning field variations in msecs.) Extremely fast cycled magnets (<< msec.) which require understanding transmission line theory is not covered. Eddy currents, although covered in the book, will not be discussed.

Part Two Part Two of this course is less mathematical and is meant for the engineer and designer translating the mathematical principles into manufactured hardware. The principles described in this part of the course is equally important to the mathematics since, if they are not properly observed, can result in magnets whose performance described by the mathematics to fall short of the physics specifications required for the synchrotron lattice.

Techniques for the manufacture of the yoke are described.
These techniques include shaping the ends of the poles to trim the fringe fields so that the field integral maintains the field quality of the two dimensional field. Techniques for the manufacture and testing of the coils are described. Means of assembling the coils to the cores and making the electrical connections among the coils and connecting the power supplies are described. Techniques of fiducialization, installation and alignment of magnets are described.

Skills The student will need to understand the mathematics of complex variables. The student should understand some of the physics concepts which determine the parameters which determine good synchrotron lattice performance. It is useful if the student understands some of the principles of electrical circuit design. Understanding the capabilities and limitations of shop manufacturing techniques is useful in order to complete the spectrum of understanding magnet design from concepts to final product.

Lecture One The first lecture describes the basic magnet types, describes forces on a particle beam, defines polarities and determines means of electrically connecting electro-magnets to achieve the desired polarities. An introduction to the orthogonal analog model as a means of visualizing the magnetic fields is presented. This tool is used to determine some properties of different magnet geometries. Much of the discussion will be devoted to understanding the properties of a dipole magnet. This discussion will be generalized in later lectures to generalize the knowledge gained about the design of dipole magnets into other magnet types. The Orthogonal Analog model is used as a tool to visualize the magnetic field and highlights differences among the two types of dipole magnets used in accelerators, window frame and H style dipole magnets. Much of the material for this lecture is covered in Chapters one and three (section 3.1.3) of the book.

Magnet Types Dipoles Quadrupoles Sextupoles
Specialized Magnets (to be covered later) Correctors Horizontal/Vertical Steering Magnets Skew Quadrupoles Injection Magnets (to be discussed later) Septa Bumps and Kickers

Polarities Electrical and Beam Currents
Polarities are defined for positive charges. A positive direction of positive current is indicated by the arrow in the following figures. The positive pole of a magnet is determined by applying the right hand rule to the direction of positive current. Magnetic flux flows from the positive pole of a magnet to the negative pole. The force on a positive beam direction is determined by a vector expression observing the right hand rule. The polarities illustrated in the figures for the three types of magnets create forces for positive beam currents (positrons, protons or heavy ion nuclei) into the screen. In order to achieve the illustrated polarities for the various magnets, the magnet power supplies must be connected so that the positive terminal of the power supply leads connects to the current in coil terminal and the negative power supply leads connects to the current out coil terminal. Polarities become confusing if the particle beam is negatively charged (ie. electrons). The force rule becomes a left hand rule, although the flux direction from the positive magnetic pole follows the right hand rule.

For this basic introduction, although the mathematical relationships have not yet been developed, the well know relations between current and magnetic fields are used to illustrate certain principles.

Integral form of the Magnetic Field Equation
The solution to Maxwell’s equations can be written in integral form. Applying the line integral vector equation to the illustrated case; The right hand rule applies for the field direction.

Magnetic Forces on a Particle Beam
We use the mks system of units on the vector equation. The “right hand” rule is used for vector directions.

The Dipole Magnet The dipole magnet has two poles, a constant field and steers a particle beam. Using the right hand rule, the positive dipole steer the rotating beam toward the left.

Dipoles PEPII Low Energy Positron Ring Dipole Magnet
SPEAR3 Gradient Magnet

The Quadrupole Magnet The Quadrupole Magnet has four poles. The field varies linearly with the the distance from the magnet center. 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.

A series of F-D or D-F magnets will focus the beam in both planes.

Quadrupole Magnets PEPII Low Energy Ring Quadrupole SPEAR3 Quadrupole

The Sextupole Magnet The Sextupole Magnet has six poles. The field varies quadratically with the the distance from the magnet center. It’s purpose is to affect the beam at the edges, much like an optical lens which corrects chromatic aberration. An F sextupole will steer the particle beam toward the center of the ring. Note that the sextupole also steers along the 60 and 120 degree lines.

The function of the sextupole magnet is to correct for momentum spread (the charged beam analogue to the optical chromaticity). In the following figure, a charged particle beam with differences in energy is bent by a dipole. Lower energy beam is bent more than the higher energy beam. Thus, when bent by the following quadrupole, the lower energy beam is focussed at a shorter focal length than the higher energy beam. The function of the sextupole, located behind the quadrupole, is to defocus the lower energy beam and focus the higher energy beam so that the focal length of the optical system is restored for all energy beams.

Sextupole Magnet SPEAR3 Sextupole Magnet
This magnet, in addition to the six coils exciting the six poles, two coils are added to the top and bottom poles. These coils are added as trims to create a trim field for the magnet. In this case, the trim fields are skew quadrupole fields. The skew quadrupole field is a quadratically distributed field which is horizontal, rather than vertical on the horizontal axis. Trim coils are often added to magnet whose yokes are designed to provide the main field. The trim coils can provide horizontal and vertical steering as well as skew quadrupole. Trim coils are added, especially when the lattice perimeter is limited and there is limited room for separate corrector magnets.

Correctors SPEAR3 Corrector
SPEAR3 is an upgrade of an existing accelerator and is a storage ring synchrotron used to exploit high energy photons which are produced when relativistic particles are bent in a magnetic field. Because of this application, the corrector has a yoke with a horizontal gap to provide clearance for the photon beamlines. Although the horizontal bending can be conventionally produced with a coil, the vertical bending (produced by horizontal fields) must be provided by pole face windings whose current distribution is tailored in order to enhance the field uniformity

Alternate Definition of Polarities
A number of things have to be remembered in order to install magnets and their power supply connections with appropriate polarities. Primarily, one needs to satisfy the physics needs of the magnets and bend the beam in the desired direction for dipoles and sextupoles and to either focus or defocus the beam (along the horizontal centerline) for the quadrupole. Particle beams can be positively or negatively charged. One must remember that power supply conventions define positive current flow from the positive to negative terminals.

All these factors are confusing and it is quite easy to make design errors and misconnect busses among coils in a magnet or make installation errors and misconnect power supply cables to magnets. Both types of errors are quite common and happen often during the installation and connection of accelerators magnets. The following uses an alternate method of determining the proper magnet polarity and requires remembering only one principal and its corollaries.

Magnetostriction The forces on parallel currents is illustrated in the following figure. The force on a charge moving with a given velocity through a magnetic field is expressed with the vector equation,

Alternate Polarity Definition
Currents with the same charge travelling in the same direction attract. Corollaries: Currents with opposite charge travelling in the same direction repel. Currents with the same charge travelling in the opposite direction repel. Currents with the opposite charge travelling in the opposite direction attract.

Orthogonal Analog Model
The name of the method for picturing the field in a magnet is called the Orthogonal Analog Model by Klaus Halbach. This concept is presented early in the lecture in order to facilitate visualization of the magnetic field and to aid in the visualization of the vector and scalar potentials.

A window frame dipole magnet is illustrated in order to demonstrate the principles.
Flow Lines go from the + to - Coils. Flux Lines are ortho-normal to the Flow Lines. Iron Surfaces are impervious to Flow Lines. Applying the model to the Window Frame Dipole, it can be seen that the field distribution in this geometry is very uniform.

Cosine Q Dipole Using the Orthogonal Analog Model and requiring the flux lines to be vertical and uniformly spaced in the circular yoke, the flow lines, connecting the current filaments, must be horizontal and also be uniformly spaced. It can be shown that the current filaments satisfy a Cosine Q distribution. This is an example of a magnet whose field is shaped by a tailored current distribution.

Three Block Cosine Q Distribution
When dividing the conductors among three blocks, approximating the cosine Q distribution, the POISSON plot looks like the illustration.

Application of the Orthogonal Analog model to the H magnet geometry reveals several field properties; The field falls off near the edges of the pole. The field at the pole corner is high and likely to saturate.

Choice of Dipole Type Why build H dipoles?
Window Frame Magnets require Saddle Coils wound on two axes. H Magnets can use Pancake Coils wound on a single axis.

H Magnet Field Uniformity
In general, the two dimensional magnet field quality can be improved by the amount of excess pole beyond the boundary of the good field region. The amount of excess pole can be reduced, for the same required field quality, if one optimizes the pole by adding features (bumps) to the edge of the pole.

The relation between the field quality and "pole overhang" are summarized by simple equations for a window frame dipole magnet with fields below saturation.

These expressions are very important since they give general rules for the design of window frame dipole designs. It will be seen later that these expressions can also be applied to quadrupole and gradient magnets.

Graphically:

Lecture 2 Lecture 2 will cover the mathematical characterization of the two dimensional magnetic field and discuss functions which describe the three basic magnet types, the dipole, quadrupole and sextupole. Chapter Two of the book should be read.

Lecture 2 Jack Tanabe Old Dominion University Hampton, VA January 2011
Mathematical Formulation of the Two Dimensional Magnetic Field Lecture 2 Jack Tanabe Old Dominion University Hampton, VA January 2011

Introduction An understanding of magnets is not possible without understanding some of the mathematics underpinning the theory of magnetic fields. The development starts from Maxwell’s equation for the three-dimensional magnetic fields in the presence of steady currents both in vacuum and in permeable material. For vacuum and in the absence of current sources, the magnetic fields satisfy Laplace’s equation. In the presence of current sources (in vacuum and with permeable material) the magnetic fields satisfy Poisson’s equation. Although three dimensional fields are introduced, most of the discussion is limited to two dimensional fields. This restriction is not as limiting as one might imagine since it can be shown that the line integral of the three dimensional magnetic fields, when the domain of integration includes all regions where the fields are non-zero, satisfy the two dimensional differential equations.

Maxwell’s Steady State Magnet Equations

Function of a Complex Variable
The derivation of the expressions to show that a function of the complex variable, F where F is a function of the two dimensional complex space coordinate z=x+iy is developed from Maxwell’s equations. This function satisfies Laplace’s and Poisson’s equations. The development of these expressions are developed in sections 2.1 to 2.5 (pages 19 to 25 of the text). This function is used to describe different two dimensional magnetic fields and their error terms. Poisson’s Equation Laplace’s Equation

Vector and Scalar Potentials
The function can be expressed as F=A+iV where A, the vector potential is the real component V, the scalar potential V is the imaginary component An ideal pole contour can be computed using the scalar equipotential. The field shape can be computed using the vector equipotential.

The two dimensional vector components of the magnetic field can be computed from the function.
Certain characteristics of the magnetic field can be determined by symmetry conditions using the function. The two dimensional characterization of the magnetic field is a subset of the formulation for the three dimensional magnetic field. The integrated three dimensional field distribution can be completely characterized by the two dimensional complex function. The concepts covered in this lecture will be useful later when discussing; Conformal mapping. Field perturbations. Magnetic Measurements.

Fundamental Relationships
The complex coordinate is, where z can be expressed as where and z can also be expressed in polar coordinates.

LaPlace’s Equation The function F describing the two dimensional magnetic field in vacuum satisfies LaPlace’s equation. The equation is satisfied in the absence of current sources and permeable material. One useful function (among many) which satisfies this equation is a function of the complex variable. This function is useful since it describes multipole magnets and their error terms.

Homework #1 Prove that F satisfies LaPlace’s Equation. Hint

Vector and Scalar Potentials
In this section, we describe a subset of functions which describe a class of magnets. These magnets are important in the synchrotron business since they supply the particle optics to steer, focus and correct the particle beam orbit. This subset of functions are the functions which are components of the Taylor’s series of the complex space coordinate z=x+iy. Although specific functions are used to describe ideal fields, the full Taylor series expansion is used to characterize the desired field as well as the unavoidable error fields. In general, F=Czn describes a class of two dimensional magnetic fields in air and in the absence of permeable material where n is any integer and C can be a real or complex constant. A=Vector Potential V=Scalar Potential.

Much can be learned about the magnet characteristics from the Function of the complex variable.
The pole shape can be determined. Flux Lines can be mapped.

Quadrupole Example C is a real constant n=Field index
=2 for quadrupoles A=Vector Potential =Real (F) V=Scalar Potential =Imaginary (F)

Vector Equipotential Hyperbolic curve with asymptotes at +- 45 deg.
Scalar Equipotential Hyperbolic curve with asymptotes along the x and y axes.

Quadrupole Equipotentials

Homework #2 Find the expressions for the poles and the flux lines for a dipole. (Hint n=1) Find the expressions for the poles and the flux lines for a skew quadrupole. (Hint C=iC, imaginary number)

Sextupole Example For the sextupole case, the function of a complex variable is written in polar form. This case is presented to illustrate that both polar and Cartesian coordinates can be used in the computation.

Vector Potentials Scalar Potentials

Sextupole Equipotentials

Multipole Magnet Nomenclature
The dipole has two poles and field index n=1. The quadrupole has four poles and field index n=2. The sextupole has six poles and field index n=3. In general, the N-pole magnet has N poles and field index n=N/2.

Even Number of Poles Rotational periodicity does not allow odd number of poles. Suppose we consider a magnet with an odd number of poles. One example is a magnet with three poles spaced at 120 degrees. The first pole is positive, the second is negative, the third is positive and we return to the first pole which would need to be negative to maintain the periodicity but is positive.

Characterization of Error Fields
Since Satisfies LaPlace’s equation, must also satisfy LaPlace’s equation. Fields of specific magnet types are characterized by the function where the first term is the “fundamental” and the remainder of the terms represent the “error” fields.

Allowed Multipole Errors
The error multipoles can be divided among allowed or systematic and random errors. The systematic errors are those inherent in the design and subject to symmetry and polarity constraints. Symmetry constraints require the errors to repeat and change polarities at angles spaced at p/N, where N is the index of the fundamental field.

In the figure, the poles are not symmetrical about their respective centerlines. This is to illustrate rotational symmetry of the N poles.

Using the “polar” form of the function of the complex variable:
Requiring the function to repeat and change signs according to the symmetry requirements: Using the “polar” form of the function of the complex variable:

In order to have alternating signs for the poles, the following two conditions must be satisfied.
Rewriting;

Therefore; The more restrictive condition is; Rewriting;

Thus, the error multipoles allowed by rotational symmetry are;
For the dipole, N=1, the allowed error multipoles are n=3, 5, 7, 9, 11, 13, 15, … For the quadrupole, N=2, the allowed error multipoles are n=6, 10, 14, 18, 22, … For the sextupole, N=3, the allowed error multipoles are n=9, 15, 21, 27, 33, 39, …

Magnetic Field from the Function of the Complex Variable
The field is a vector with both magnitude and direction. The vector can be described in complex notation since the x and y components can be described as the real and imaginary components of the complex function. Therefore, the field can be described as a function of F(z).

Dipole Example The complex conjugate of the field is given by;
Suppose; Suppose;

Homework #3 From the function; Find Bx and By using

Sextupole Example a quadratically varying field.

The Curl Equation We postulate that the B field can be completely determined by the vector potential A.

This is consistent with the vector equation;
Where in two dimensions; and

In general, the three dimensional field vector can be written as the vector equation;
The Vector Potential A is a vector quantity.

The Divergence Equation
We postulate that the B field can be completely determined by the Scalar potential V.

In general, the 3D field vector can be written as the vector equation,
This is consistent with; In general, the 3D field vector can be written as the vector equation, The Scalar Potential V is a scalar quantity.

Cauchy-Riemann Using one or the other potentials; which requires
which are the Cauchy-Riemann conditions and can only be satisfied for and not for

Complex Extrapolation
Using the concept of the magnetic potentials, the ideal pole contour can be determined for a desired field. Gradient Magnet Example The desired gradient magnet field requires a field at a point and a linear gradient. Given: A central field and gradient. The magnet half gap, h, at the magnet axis. What is the ideal pole contour?

The desired field is; The scalar potential satisfies the relation; Therefore; For on the pole surface, Therefore, the equation for the pole is, or solving for y,

Hyperbola with asymptote at,

Other Functions In this section, we discussed two dimensional “multipole” magnets, those with rotational symmetric fields. This is a small subset of all of the possible magnetic field distributions. Wigglers and undulators are magnets which are finding increasing use in light source synchrotrons. The two dimensional characterization of the magnetic fields from these magnets is longitudinally (rather than rotationally) periodic and represents another subset of possible magnetic fields. The characterization of the fields from these magnetic structures is well documented in the published literature and can be characterized by an analytic function. Although the full characterization of these fields has not been included in the text and should be covered in a separate course on magnetic structures, it should be emphasized that the analytic expression describing these fields can also be characterized by a father simple analytic function. The constants for this function are evaluated by computing the Fourier constants which satisfy the boundary conditions.

Lecture 3 Lecture 3 will cover conformal mapping and application of the tools to extend knowledge about the simple dipole magnet to the more complex quadrupole magnet. Section 2.1 should be reviewed and Chapter 3 should be read. It would also be helpful to read a part of Chapter 6 (section 6.6) on applications of conformal mapping to POISSON calculations.

Conformal Mapping Lecture 3 Jack Tanabe Old Dominion University Hampton, VA January 2011

Introduction This section introduces conformal mapping.
The means of ensuring dipole field quality is reviewed. Conformal mapping is used to extend the techniques of ensuring dipole field quality to quadrupole field quality. Conformal mapping can be used to analyze and/or optimize the quadrupole or sextupole pole contours in by using methods applied to dipole magnets. Conformal mapping maps one magnet geometry into another. This tool can be used to extend knowledge regarding one magnet geometry into another magnet geometry.

Mapping a Quadrupole into a Dipole
The quadrupole pole can be described by a hyperbola; Where V is the scalar potential and C is the coefficient of the function, F, of a complex variable. The expression for the hyperbola can be rewritten; We introduce the complex function;

Rewriting; since Therefore; the equation of a dipole since the imaginary (vertical) component is a constant, h.

Mapping a Dipole into a Quadrupole
In order to map the dipole into the quadrupole, we use the polar forms of the functions; and Since was used to convert the quadrupole into the dipole, \ therefore; and Finally;

Quadrupole Field Quality
The figure shows the pole contour of a quadrupole and its required good field region. The pole cutoff, the point at which the unoptimized or optimized quadrupole hyperbolic pole is truncated, also determines the potential field quality for the two dimensional unsaturated quadrupole magnet.

The location of this pole cutoff has design implications
The location of this pole cutoff has design implications. It affects the saturation characteristics of the magnet since the iron at the edge of the quadrupole pole is the first part of the pole area to exhibit saturation effects as magnet excitation is increased. Also, it determines the width of the gap between adjacent poles and thus the width of the coil that can be installed (for a two piece quadrupole). The field quality advantages of a two piece quadrupole over a four piece quadrupole will be discussed in a later section.

H Magnet Field Quality Review
The relation between the field quality and "pole overhang" are summarized by simple equations for a window frame dipole magnet with fields below saturation.

The required pole overhang beyond the good field region
are given by the following equations.

The relations can be presented graphically.

Given; (uc, vc) satisfying dipole uniformity requirements.
Find; (xc, yc) satisfying the same requirements for quadrupoles.

For the Dipole; Therefore;

Substituting a unitless (normalized) good field region,
and using the conformal mapping expressions, and the half angle formulae,

and substituting, we get,

Substituting the appropriate factors for the unoptimized and optimized dipole cases, we get finally for the quadrupoles;

The equations are graphed in a variety of formats to summarize the information available in the expressions. The expressions are graphed for both the optimized and unoptimized pole to illustrate the advantages of pole edge shaping in order to enhance the field. The quality at various good field radii are computed since the beam typically occupies only a fraction of the aperture due to restrictions of the beam pipe.

Since the field for the quadrupole
varies with the radius;

The Septum Quadrupole PEPII is a positron electron collider. In order to maximize the the number of collisions and interactions, the two beams must be tightly focused as close to the interaction region as possible. At these close locations where the final focus quadrupoles are located, the two crossing beams are very close to each other. Therefore, for the septum quadrupoles, it is not possible to take advantage of the potential field quality improvements provided by a generous pole overhang. It is necessary to design a quadrupole by using knowledge acquired about the performance of a good field quality dipole. This dipole is the window frame magnet.

The conformal map of the window frame dipole aperture and the centers of the separate conductors is illustrated. The conductor shape does not have to be mapped since the current acts as a point source at the conductor center.

Other Uses for Conformal Maps
Programs such as POISSON compute the two dimensional distribution of the vector potential. The vector potential function is computed using a relaxation method (for POISSON) or a modified matrix inversion (for PANDIRA) among neighboring mesh points defined by the magnet geometry.

The magnetic field distribution is then computed from the derivative of the vector potential.
For an ideal dipole field (Hy=constant and/or Hx=constant) the vector potential function is a linear function of z. For a quadrupole field, the vector potential function is a quadratic function of z. The vector potential for a sextupole is a cubic function of z.

When computing the field distribution, it is necessary to compute the derivative by interpolating the distribution of the vector potential function among several mesh points. The precision of the field calculations depends on the mesh density and the continuity of the interpolated values of the vector potential. Since the dipole function is simple (a linear distribution), the potential precision of field calculations is much higher than for quadrupole (quadratic) or sextupole (cubic) fields. (An accurate estimate of the derivatives for a linear distribution of a potential function can be obtained from fewer values from “neighboring” mesh points than for a quadratic or cubic distribution.)

Therefore, when high precision computations for magnetic field distribution have been required, a conformal transformation is often employed to convert the quadrupole and/or sextupole geometry to a dipole configuration.

However, there is a problem in the mapping of the
quadrupole and sextupole to the dipole space. Typically, Therefore, and in the mapped space.

When mapping from the quadrupole or sextupole geometries to the dipole space, the POISSON computation is initially made in the original geometry and a vector potential map is obtained at some reference radius which includes the pole contour.

The vector potential values are then mapped into the dipole (w) space and used as boundary values for the problem.

Quadrupole/Sextupole Pole Optimization
It is far easier to visualize the required shape of pole edge bumps on a dipole rather than the bumps on a quadrupole or sextupole pole. It is also easier to evaluate the uniformity of a constant field for a dipole rather than the uniformity of the linear or quadratic field distribution for a quadrupole or sextupole. Therefore, the pole contour is optimized in the dipole space and mapped back into the quadrupole or sextupole space.

The process of pole optimization is similar to that of analysis in the dipole space.
Choose a quadrupole pole width which will provide the required field uniformity at the required pole radius. The pole cutoff for the quadrupole can be obtained from the graphs developed earlier using the dipole pole arguments. The sextupole cutoff can be computed by conformal mapping the pole overhang from the dipole space using

Select the theoretical ideal pole contour.
for the quadrupole. for the sextupole. Select a practical coil geometry. Expressions for the required excitation and practical current densities will be developed in a later lecture. Select a yoke geometry that will not saturate. Run POISSON (or other 2D code) in the quadrupole or sextupole space. From the solution, edit the vector potential values at a fixed reference radius.

Map the vector potentials, the good field region and the pole contour.
Design the pole bump such that the field in the mapped good field region satisfies the required uniformity.

Map the optimized dipole pole contour back into the quadrupole (or sextupole) space.
Reanalyze using POISSON (or other 2D code).

Closure The function, zn, is important since it represents different field shapes. Moreover, by simple mathematics, this function can be manipulated by taking a root or by taking it to a higher power. The mathematics of manipulation allows for the mapping of one magnet type to another --- extending the knowledge of one magnet type to another magnet type. One can make a significant design effort optimizing one simple magnet type (the dipole) to the optimization of a much more difficult magnet type (the quadrupole and sextupole). The tools available in POISSON can be exploited to verify that the performance of the simple dipole can be reproduced in a higher order field.

Lecture 4 Lecture 4 will cover the POISSON computer code. This session will be followed by a computer laboratory session where the lessons learned in the lecture can be applied. Chapter 6 should be read prior to the lecture.

POISSON A Two-Dimensional Magnetostatic Solver Lecture 4 Jack Tanabe Old Dominion University Hampton, VA January 2011

POISSON POISSON/SUPERFISH are a family of electric and electromagnetic codes, written originally by Klaus Halbach and Ron Holsinger. Information regarding the code family and means of downloading the codes can be obtained by contacting James H. Billen It is a public access code (it’s free), maintained under contract with DOE by Los Alamos National Accelerator Laboratory (LANL) personnel. The LANL interest is mainly in the RF field, so the bulk of the development and maintenance of this family of codes is in the SUPERFISH area. Thus, maintenance and development of the magnetostatic capabilities is limited by funding and time.

Magnetostatic Elements
Four main code components are of interest to those involved in magnet design. Automesh.EXE Automesh is a automatic mesh generator. It takes input information written in a text file and generates a mesh of points written in several matrices used by POISSON or PANDIRA to solve the distribution of magnetic fields in a two dimensional magnet cross section. poisson.EXE POISSON uses the Automesh output and solves for the vector potential distribution in the two dimensional geometry by successive approximation using the relaxation method. It computes the field distribution from the vector potential distribution. pandira.EXE PANDIRA solves for the vector potential by diagonalizing the matrix. Wfsplot.EXE WFSPLOT is a graphics routine which can present either the lattice geometry from Automesh or the Vector equi-potentials resulting from POISSON or PANDIRA solutions.

Program Flow Chart The structure of the information flow for the POISSON/Pandira program group is shown in the diagram. A text file is written which includes all the information required to define the problem. If WFSPlot is run alt this point, the Tape35 file is used to graphically present the geometry information. AutoMesh is run using this file. Automesh overwrites the current Tape35 file. Poisson/Pandira is run using the current Tape35 file and overwrites the Tape35 file writeen by AutoMesh. WFSPlot is run using the current Tape35 file and creates a graphical output of the calculated results. Outpoi is a text file which is overwritten each time POISSON or Pandira is run. This file contains the information used in the program, including the BH curve called in the .txt file. Parts of this file (the results of the edit) can be copied and used to present the results of the calculations.

Text File A portion of a text file written for a dipole geometry is shown. The first two lines of the file are title lines which are carried throughout the calculations. All the lines shown are preceded by a ! sign, which indicates that these are comments. Sample Problem Simple Dipole !First two lines in the text file are the title and descriptors for the problem. !These titles will be carried in subsequent output files. !mat=1 means air region, =2 means iron region using the internal iron BH properties. !kprob=0 means POISSON problem !mode=0 means finite permeability using one of the tables !xmin, xmax, ymin, ymax are the physical horizontal and vertical limits of the problem. ! xreg1 and kreg1 are the x value for the mesh with kreg1 nodes. kmax is the no. of k !nodes. More regions defined by xreg2, kreg2, etc. are possible. ! yreg1 and lreg1 are the y value for the mesh with lreg1 nodes. lmax is the no. of l nodes. ! icylin=0 is Cartesian Coordinates, =1 is cylindrical coordinates. ! xminf, xmaxf, yminf, ymaxf are the physical limits for the data output limits. ktop and !ltop are the number of data output points over the limits. Example, xminf=-30, !xmaxf=30, ktop=13 means that the output edit data will have 13 entries from -30 to +30 !mm at 5 mm. increments. ienergy=1 means that the stored energy will be computed. ! nbsup, nrslo, nbsrt, nbslf are boundary condition for UPper, LOwer, RTside, LFside. ! =0 and 1 mean Dirichlet and Neumann Boundaries, respectively. ! conv=1 is the default for cm. =.1 is for mm, =2.54 is for inches. ! The following are the CONS for the harmonic analysis. ! ktype=124. First integer =1 dipole, 2 quadrupole. !Second integer =2 means multipole inteval. Third integer =1 means midplane !symmetry, =0 skew terms. ! nptc=30 number of arc points, nterm=number of terms, rint=35 mm interpolation radius, !rnorm=32 mm normalization radius. !angle=180 maximum angle, anglz=0 starting point for interpolation.

icylin=1 is cylindrical symmetry problem
This text file defines a dipole geometry. The first section defines the problem constraints. Note that $reg is used before and $ after this input section. Poisson/Pandira Problem Air Region Finite Permeability Using Table 0.1 is for mm. 1 is for cm. 2.54 is for inch $reg mat=1, kprob=0, mode=0,conv=.1, xmin=0,xmax=150., kmax=50, ymin=0, ymax=100., lmax=50, icylin=0, xminf=0., xmaxf=25,ktop=11 yminf=0.0, ymaxf=10,ltop=5 ktype=121,nptc=31 nterm=14,rint=20, rnorm=25, angle=90, anglz=0 ienergy=1, nbsup=0, nbslo=1, nbsrt=0, nbslf=0$ x and y limits k and l limits No cylindrical symmetry icylin=1 is cylindrical symmetry problem

The last part of the first section describes the edit constraints.
This section describes the limits of the output edit. The x limits are 0 to 25 mm, the y limits are 0 to 10 mm. 11 points are used for x (dx=2.5 mm) and 5 points are used for y (dx=2.0mm). $reg mat=1, kprob=0, mode=0,conv=.1, xmin=0,xmax=150., kmax=50, ymin=0, ymax=100., lmax=50, icylin=0, xminf=0., xmaxf=25,ktop=11 yminf=0.0, ymaxf=10,ltop=5 ktype=121,nptc=31 nterm=14,rint=20, rnorm=25, angle=90, anglz=0 ienergy=1, nbsup=0, nbslo=1, nbsrt=0, nbslf=0$ This line describes the edit for the Fourier analysis of the beam. Ktype=121 means (1=dipole, 2=multipole interval, 1= midplane Neumann boundary, third digit 0= midplane Dirichlet boundary. nptc=31 means number of points on the circle, rint=20 means interpolation on 20 mm radius arc, rnorm=25 means multipole normalization at 25 mm, angle=90 and anglez=0 means the edit points are from 0 to 90 degrees. nterm=14 means the maximum number of multipole terms. ienergy=1 means calculate the stored energy in joules/meter. ienergy=0 means don’t calculate the stored energy.

The last line in the first section describes the boundary conditions
The last line in the first section describes the boundary conditions. nbsup, nbslo, nbsrt and nbslf means the boundary condition at the upper, lower, right hand, and left hand boundaries. = 0 means Dirichlet (flux parallel) and =1 means Neumann (flux perpendicular) boundaries. $reg mat=1, kprob=0, mode=0,conv=.1, xmin=0,xmax=150., kmax=50, ymin=0, ymax=100., lmax=50, icylin=0, xminf=0., xmaxf=25,ktop=11 yminf=0.0, ymaxf=10,ltop=5 ktype=121,nptc=31 nterm=14,rint=20, rnorm=25, angle=90, anglz=0 ienergy=1, nbsup=0, nbslo=1, nbsrt=0, nbslf=0$

Geometry Definition. Note that all regions must close, that is the first and last coordinates are equal. Each line must begin and end with $ or & sign. !first region is air (mat=1), defines the problem limits. $po x=0, y=0$ $po x=150, y=0$ $po x=150, y=100$ $po x=0, y=100$ !Coil $reg mat=1 cur=-20000$ $po x=55, y=25 $ $po x=75, y=25 $ $po x=75, y=45 $ $po x=55, y=45 $ $po x=55, y=25$ !Iron Yoke $reg mat=2$ $po x=0, y=25 $ $po x=40, y=25$ $po x=50, y=50$ $po x=80, y=50$ $po x=80, y=0 $ $po x=120, y=0 $ $po x=120, y=80$ $po x=0, y=80$ $po x=0, y=25$ Problem rectangular boundary. It uses the material definition given in the first section, mat=1 means air or vacuum. Coil region is defined by its boundary, material specification mat=1 and the current value in Amps. Negative currents in the right hand coil gives positive flux on the horizontal centerline. Coil regions can be single points and/or lines. The iron yoke area uses mat=2, which uses the BH curve for a “generic” iron whose magnetic properties approximate the behavior of 1010 steel.

Lattice Text Writing Techniques
I find it easier to develop the geometry lattice using the Excel. =CONCATENATE(C$1,A5,C$2,B5,C$3) Copy and Paste

Symmetric Quadrupole $reg mat=1, kprob=0, mode=0,conv=.1
xmin=0, xreg1=50.0, xreg2=235, xmax=301.65, kreg1=60, kreg2=200, kmax=220, ymin=0.0, yreg1=30.0, yreg2=108, ymax=301.65, lreg1=40, lreg2=120, lmax=220, icylin=0, xminf=0, xmaxf=230, yminf=0, ymaxf=0, ktype=4, nbsup=0, nbslo=1, nbsrt=0, nbslf=1, ktype=241,nptc=46 nterm=14,rint=30, rnorm=32.5, angle=45, anglz=0$ !first region is air (mat=1) !problem type is poisson $po x=0.0, y=0.0$ $po x=301.65, y=0.0$ $po x=301.65, y=301.65$ $reg mat=1 cur=10700$ $po x= , y=55.152$ $po x=205.74, y=99.264$ $po x= , y=56.655$ $po x= , y=12.543$ $po x=140.4, y=33.848$

$reg mat=2$ $po x=24.75, y=24.75$ $po x=25.124,y=24.382$ $po x=26.482,y=23.129$ $po x=29.063,y=21.075$ $po x=31.731,y=19.296$ $po x=34.429,y=17.798$ $po x=37.159,y=16.679$ $po x=38.959,y=16.097$ $po x=39.768,y=15.647$ $po x=40.494,y=15.006$ $po x=41.171,y=14.227$ $po x=41.905,y=13.615$ $po x=42.7,y=13.215$ $po x=43.544,y=13.022$ $po x=45.902,y=13.0$ $po x=46.436,y=13.093$ $po x=46.683, y=13.206$ $po x=119.38, y=53.379$ $po x= , y=60.152$ $po x= , y= $ $po x=232.19, y=61.655$ $po x=232.19, y=0.0$ $po x=301.65, y=0.0$ $po x=301.65, y=301.65$ $reg ibound=0$ $po x=0., y=0.0$ $reg mat=1 cur=135$ $po x= , y=48.379$ $po x= , y=55.152$ $po x=140.4, y=33.848$ $po x= , y=27.075$ A line boundary region defines the Neumann boundary condition along a diagnoal line.

Using POISSON in Conformally Mapped Geometries
In addition to analyzing the performance of a high performance geometry, POISSON can be used analyze the performance of mapped geometry. However, the mapping algorithm is highly nonlinear. Therefore, when a dipole geometry (which is well understood) is mapped into a quadrupole geometry, the entire quadrupole area is highly distorted and the boundaries of the extremities of the geometry can be very large compared to the area where the fields need to be resolved. Because of this limitation, means of modeling a smaller boundary in the quadrupole space needs to be understood.

Conformal Mapping The figure shows the simple dipole (the baseline case) described in the earlier section of this lecture. The output edit instruction requested a vector potential edit on a circular boundary which enclosed the required good region and a portion of the pole. Mapping the dipole (w-space) into quadrupole (z-space) maps half the angle and generates a mesh where the corner of the dipole maps into a very large multiple of the magnet gap, h.

Review from Lecture #3 Mapping a Dipole into a Quadrupole
In order to map the dipole into the quadrupole, we use the polar forms of the functions; and Since was used to convert the quadrupole into the dipole, therefore; and Finally;

The upper figure is the POISSON solution for the full quadrupole
The upper figure is the POISSON solution for the full quadrupole. The text file is from the POISSON edit where the values of the vector potential is evaluated on a fixed radius enclosing the pole tip and the required good field region. The lower figure is the POISSON calculation using a mesh in which the vector potential values are used for the boundary condition.

A A is the quadrupole POISSON flux plot from the dipole mesh using the vector potential boundary. B is an “optimized” dipole contour. C is the POISSON flux plot from the mapped “optimized pole contour. B C

The final Poisson flux plot describes the field distribution using the actual quadrupole yoke and coil geometry.

Although there are many commercial software packages which perform two and three dimensional magnet calculations, POISSON uses the “language” of magnetics and allows one to compute the vector potential around a boundary and performs the harmonic analysis which are used to track the beam in existing lattice codes. Most of the accelerator community uses this tool, thus it is the software package of choice for me. The individual student should investigate other packages which may be more user friendly. In particular, the meshing package for POISSON is rather weak and often does not have the flexibility nor is robust enough to generate difficult detailed meshes easily.

Homework This lecture will be followed by a computer laboratory session. If you haven’t already done so, please read thoroughly and carefully chapter 6 of the text. Pay special attention to the section which presents the writing of the text file. An example writing the coordinates using a spreadsheet is illustrated by Fig. 4 on page 142 of the text. Familiarize yourself with Problem 6.1 on page 169 of the text. The examples of the conformally mapped geometry, although important, may take too much time to understand and to execute properly during this week. The student is encouraged to study the examples carefully and to run these, or similar cases, when he/she returns home. If you have the opportunity to do so, begin making the calculations of the parameters and the quadrupole poletip geometry and compute the current required to produce the required excitation in problem 6.1. An expression needed to calculate this excitation is eq in chapter 5, the chapter covered in lecture 6. We will work together writing the text file and running POISSON during the next session.

Lecture 5 Chapter 4 of the text covers the material which will be presented in lecture 5, Perturbations. Although much effort can be invested to define pole geometry which will result in a uniform magnet, errors in fabrication, assembly or design assymetry can introduce error fields which can compromise the magnet performance. This chapter and lecture will cover the multipole field errors introduced in an otherwise perfectly designed magnet. In many ways, these errors are much more important and damaging than those limited by the design. This is because these errors are lower order multipoles which vary as a lower power of the radius and thus damp less quickly than the “allowed” multipoles. Understanding the next lecture will be enhanced if chapter 4 is read before the next lecture.

Perturbations Lecture 5 Jack Tanabe Old Dominion University Hampton, VA January 2011 [1] Halbach, K., FIRST ORDER PERTURBATION EFFECTS IN IRON-DOMINATED TWO-DIMENSIONAL SYMMETRICAL MULTIPOLES, “Nuclear Instruments and Methods”, Volume 74 (1969) No. 1, pp [2] Halbach, K., and R. Yourd, TABLES AND GRAPHS OF FIRST ORDER PERTURBATION EFFECTS IN IRON-DOMINATED TWO-DIMENSIONAL SYMMETRICAL MULTIPOLES, LBNL Internal Report, UCRL-18916, UC-34 Physics, TID 4500 (54thEd.), May 1969.

The subject of Perturbations is covered in chapter 4 of the text and is one of the more important subjects covered in this course. This is because the performance of an accelerator lattice is dominated by the quality and reproducibility of the magnets fabricated/installed in the lattice. Perturbations are characterized by the multipole content of a magnet. A good magnet is characterized by the harmonic content of its integrated field. A perfect dipole is characterized by F1= C1 z A perfect quadrupole is characterized by F2= C2 z2 A perfect sextupole is characterized by F3= C3

Tables of coefficients and expressions are presented which make it possible to compute the magnitude of error multipoles resulting from fabrication, assembly and pole excitation errors associated with magnet manufacture. Pole excitation errors can result from shorted coil turns, errors in winding the coils or other sources. The other sources of pole excitation errors are poles which differ in length from other poles. Pole excitation errors can also be introduced intentionally in order to produce trim fields, those fields which do not normally exist in a particular yoke geometry. Physics requirements normally specify the maximum amplitude of the various multipole errors. As a magnet designer, these multipole errors must be translated into fabrication/assembly tolerances. Examples of tolerance calculations are given in Section of the text.

Effect of Mechanical Fabrication Errors on Error Multipole Content
In the previous lecture, we showed that the field distribution in a magnet can be characterized by a function of the complex variable, z. In particular;

Random Multipole Errors Introduced by Pole Excitation and Pole Placement Errors
Random multipole errors are introduced if the poles are improperly excited or assembly errors which displace poles are introduced. If one can identify these errors, one can predict the multipole content of the magnet. The means for calculating these errors are summarized in two papers published by Klaus Halbach. The first paper describes the derivation of the relationships, the second computes and tabulates the coefficients used to calculate the multipole errors from the perturbations derived in the first paper.

A portion of a table from UCRL by Halbach and Yourd is reproduced below. This table is for quadrupoles, N=2. n

This table of coefficients is used to estimate the error multipole due to excitation, radial, azimuthal and rotation errors in the location of a pole on the horizontal axis. The imaginary term in the denominator for the excitation (j) and radial (rd) terms indicate that errors in these quantities introduce skew terms. The tabulated coefficients are computed for poles centered on the positive horizontal axis. For the normal multipole magnet (one whose axis is rotated away from the horizontal axis), the expression for calculating the error multipole normalized to the desired fundamental, evaluated at the pole radius, is given by; where b is the angle of the pole(s) which have been perturbed.

Example Calculation Suppose we construct a 35 mm radius quadrupole whose first pole is radially offset by 1 mm. What is the effect on the n=3 (sextupole) multipole error. where n=3.

Meaning of the Result The calculation means that the n=3 (sextupole) error multipole normalized to the fundamental field at the pole radius (35 mm) is approximately 0.8% due to the radial displacement of the first pole at p/4 by 1 mm. Carrying the calculation further to determine the phases;

Further simplification;
This means that the real component of the sextupole error is out of phase with the fundamental field and that a positive skew component of the sextupole field also exists.

Evaluation at the Required Good Field Radius
We recall that the field for an n multipole varies as zn-1. Therefore if the good field radius is r0=30 mm, the n=3 normalized multipole error can be evaluated at this radius.

Other Errors The coefficient table can be used for other errors.

The Error Multipole Spectrum
In general, the table of coefficients includes entries for all the multipole indices. Therefore, although the sample calculation was performed only for the sextupole error field component, all the multipole errors due to the radial misalignment of the first pole exist. These errors include the error in the fundamental (n=2) as well as the n=1 dipole field.

Error Amplitudes as a Function of Radius

Lesson The lesson from this sample calculation is not the detailed calculation of the multipole error, but the estimate of the mechanical assembly tolerances which must be met in order to achieve a required field quality. In general, the coefficient is <0.5. Therefore in order to achieve a field error at the pole radius of 5 parts in (a typical multipole error tolerance), the following tolerance illustrated in the calculation must be maintained. A very small error.

Another Lesson The Magnet Center
We note that all the multipole errors are introduced by mechanical assembly errors. In particular, we look in detail at the dipole error term introduced by assembly errors. For the pure quadrupole field, the expression for the complex function is; If the magnet center is shifted by an amount Dz, the expression becomes; The first term in this expression is the quadrupole field (a linear function of z). The second term is a constant and, therefore, is the dipole field.

Rewriting the expression as the sum of two fields;
Evaluating the quadrupole field at the pole radius, r0; Equating the real and imaginary parts of the expression, the magnetic center shift can be evaluated from the dipole field.

These relationships may be more easily visualized with a figure

Effect of a Pole Excitation Error on the Magnetic Center
One of the many issues faced by the NLC project is the stability of the magnetic center of adjustable hybrid permanent magnet quadrupoles. A sample calculation is made to compute the required pole excitation precision. For n = 1 (dipole error) where

Suppose we have a 1% error on the excitation of a single pole in the first quadrant.
Equating the real and imaginary terms,

The Four Piece Magnet Yoke
The ideal assembly satisfies the rotational symmetry requirements so that the only error multipoles are allowed multipoles, n=6, 10, However, each segment can be assembled with errors with three kinematic motions, x, y and e (rotation). Thus, combining the possible errors of the three segments with respect to the datum segment, the core assembly can be assembled with errors with 3x3x3=27 degrees of freedom.

The Two Piece Magnet Yoke
This assembly has the advantage that the two core halves can be assembled kinematically with only three degrees of freedom for assembly errors. Thus, assembly errors are more easily measured and controlled.

Coefficients for a Two-Piece Quadrupole

Two Piece Quadrupole Error Computations
The computations of the multipole error fields due to assembly errors of the two piece quadrupole are similar but simpler than the computations for the four piece quadrupole. In the expressions below, h is the pole radius. The error terms are evaluated at the pole radius. er=Rotational Perturbation Why i?

Referring to the table;
The shear motion of the top half of the magnet with respect to the bottom introduces skew even multipole errors. The vertical motion introduces real even multipole errors. The rotational motion introduces real odd multipole errors.

Experiments Computations using the coefficients for the two piece magnet have been compared to experiments where the upper half of a magnet was intentionally displaced with respect to its nominal position.

Vertical Perturbation
Normal Assembly Vertical Perturbation

Vertical and Rotational Motion

Shear and Vertical Motion

Skew term due to vertical and shear perturbations.

Other Applications In crowded lattices, there is often insufficient room to place all the desired magnetic elements. An occasional solution to this problem, employed both at ALS and at APS, is to provide trim windings on a sextupole yoke in order to obtain horizontal and vertical steering fields and skew quadrupole fields without introducing a sextupole field. (The controls sextupole fields want to be independent of the horizontal and vertical steering and the skew quadrupole controls.) The design of such trim windings and the evaluation of the field quality which results when employing these techniques exploit Klaus Halbach’s perturbation coefficients.

n The table of coefficients for N=3 (sextupole) for pole excitation error, e, is reproduced from Klaus Halbach’s perturbation paper. where and

Vertical Steering Trim (Horizontal Flux Lines)
Vertical steering trim (horizontal flux lines) is achieved in a sextupole yoke by exciting the four horizontal poles.

The formulation of the expressions follows:

Equating real and imaginary terms for the horizontal steering trim;

Tabulating the results;

Required Vertical Steering Trim Excitation
From the table: and

Substituting; From the table, we note that the n=5 multipole error is >70% of the fundamental (horizontal dipole) field at the pole radius.

Horizontal Steering Trim (Vertical Flux Lines)
Horizontal steering trim can be achieved by exciting all six of the sextupole poles. The excitation of the vertical poles is twice the excitation of the horizontal poles.

Again, we can formulate the field in terms of the excitations of the various poles.

Tabulating the results,

Required Horizontal Steering Trim Excitation
From the table: and Substituting;

Skew Quadrupole Trim (Horizontal Flux Lines)
Skew quadrupole trim field can be achieved by exciting two of the sextupole poles. Again, we can formulate the field in terms of the excitations of the various poles.

Tabulating the results,

Required Skew Quadrupole Trim Excitation
From the table: and Substituting;

Predicted n=4, 8 and 10 Multipole Errors
Applying the perturbation theory, the multipole errors normalized to the skew quadrupole field, evaluated at the pole radius, h, can be computed. Magnetic measurements were performed on the SPEAR3 production sextupole magnets with skew quadrupole windings. These measurements were evaluated at the required good field radius, 32 mm. The predicted normalized multipole errors at 32 mm can be computed.

These prediction are compared with measurements.

Sextupole – Skew Quadrupole Field Error Measurements

Closure In many ways, this is one of the most important lectures. It is important that the student understands the chapter on Perturbations since successfully translating the performance of the mathematical design to the magnets manufactured and installed in a synchrotron requires that mechanical manufacturing and assembly errors translates into field errors which can threaten the performance of the synchrotron. Understanding the impact of mechanical fabrication and assembly errors on the magnet performance and thus, the physics impacts of these errors, can provide the understanding so that mechanical tolerances can be properly assigned.

Lecture 6 The material covered in lecture 6 is covered in chapter 5 of the text. Please read this chapter prior to the next lecture. Homework covered in this chapter will be assigned.

Integral Solution to Poisson’s Equation Coil Design System Design Water Flow Calculations Lecture 6 Jack Tanabe Old Dominion University Hampton, VA January 2011

Introduction This section develops the expressions for magnet excitation. The relationship between current density and magnet power is developed. An example of the optimization of a magnet system is presented in order to develop a logic for adopting canonical current density values. Engineering relationships for computing water flows for cooling magnet coils are developed.

Poisson’s Equation The Poisson equation is the nonhomogeneous version of the LaPlace equation and includes the term for the current. Both Poisson’s and LaPlace’s equations are two dimensional versions of Maxwell’s equations for magnetics.

Application of Stoke’s Theorem results in the more familiar integral form of Poisson’s equation.
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.

Dipole Excitation Along Path 1 and Therefore;

Along path 2, For iron; Therefore; Along path 3, and Therefore; Finally;

Current Dominated Magnets
Occasionally, a need arises for a magnet whose field quality relies on the distribution of current. One example of this type of magnet is the superconducting magnet, whose field quality relies on the proper placement of current blocks. This is the cosine distribution of current.

Two flux plots from a cosine block coil distribution are shown, one with a cylindrical iron yoke and the other without. The one without the shield appears to have poorer field uniformity. This is because the field along path2 is large. This is an artifact of the computation since this computed field would have been smaller had the boundaries been extended farther from the problem.

Cosine Block Distribution
The distribution of the block areas approximate the cosine distribution of the current. The example shown illustrates a solution with three blocks. Three blocks provide three parameters so that the first three dipole multipole errors can be minimized. What are these multipole indices?

Quadrupole Excitation
Using arguments similar to those used for the dipole; Along Path 1, and Therefore; Finally;

Sextupole Excitation Using arguments similar
to those used for the dipole; Along Path 1, and Finally;

Magnet Efficiency We introduce efficiency as a means of describing the losses in the iron. Use the expression for the dipole excitation as an example. For magnets with well designed yokes.

Units For magnet excitation, we use the MKS system of units.

Current Density One of the design choices made in the design of magnet coils is the choice of the coil cross section which determines the current density. Given the required Physics parameters of the magnet, the choice of the current density will determine the required magnet power. Power is important because they affect both the cost of power supplies, power distribution (cables) and operating costs. Power is also important because it affects the installation and operating costs of cooling systems.

Substituting; Calculating the coil power; we get the expression for the power per coil, Substituting, where,

But the required excitation for the three magnet types is,
Substituting and multiplying the expression for the power per coil by 2 coils/magnet for the dipole, 4 coils/magnet for the quadrupole and 6 coils/magnet for the sextupole, the expressions for the power per magnet for each magnet type are,

Note that the expressions for the magnet power include
only the resistivity r, gap h, the field values B, B’, B”, current density j, the average turn length, the magnet efficiency and m0. Thus, the power can be computed for the magnet without choosing the number of turns or the conductor size. The power can be divided among the voltage and current thus leaving the choice of the final power supply design until later. Reasonable magnet design can be obtained by using canonical values of some of the variables.

More Units Using a consistent set of units, the power is expressed in
Watts.

Magnet System Design Magnets and their infrastructure represent a major cost of accelerator systems since they are so numerous. Magnet support infrastructure include; Power Supplies Power Distribution Cooling Systems Control Systems Safety Systems

Power Supplies Generally, for the same power, a high current - low voltage power supply is more expensive than a low current - high voltage supply. Power distribution (cables) for high current magnets is more expensive. Power distribution cables are generally air-cooled and are generally limited to a current density of <1.5 to 2 Amps/mm2. Air cooled cables generally are large cross section and costly.

Dipole Power Supplies In most accelerator lattices, the dipole magnets are generally at the same excitation and thus in series. Dipole coils are generally designed for high current, low voltage operation. The total voltage of a dipole string is the sum of the voltages for the magnet string. If the power cable maximum voltage is > 600 Volts, a separate conduit is required for the power cables. In general, the power supply and power distribution people will not object to a high current requirement for magnets in series since fewer supplies are required.

Quadrupole Power Supplies
Quadrupole magnets are usually individually powered or connected in short series strings. Since there are so many quadrupole circuits, quadrupole coils are generally designed to operate at low current and high voltage.

Sextupole Power Supplies
Sextupole are generally operated in a limited number of series strings. Their effect is distributed around the lattice. In many lattices, there are a maximum of two series strings. Since the excitation requirements for sextupole magnets is generally modest, sextupole coils can be designed to operate at either high or low currents.

Power Consumption The raw cost of power varies widely depending on location and constraints under which power is purchased. In the Northwest US, power is cheap. Power is often purchased at low prices by negotiating conditions where power can be interrupted. The integrated cost of power requires consideration of the lifetime of the facility. The cost of cooling must also be factored into the cost of power.

Optimization As shown in a previous section, once the Physics requirements (field indices, B, B’, B”, gaps and magnet lengths) have been determined, there is only one parameter which can be chosen in order to optimize the lifetime costs of the magnet system for a facility. That parameter is the current density, j. Therefore, based on the various cost parameters, a design current density can be selected for different magnet systems.

Optimization Example The following example is purely fictional and serves to illustrate the considerations which are included in the selection of a magnet system current density. Magnet coil cost generally vary with the weight of the coil (its size) and thus varies inversely with the current density. For low current densities, the coil sizes and costs can increase exponentially.

Power costs generally vary linearly with the power.
Power supply costs vary at quantized levels and increase only when certain power thresh-holds are exceeded. Facility costs vary at quantized levels (substation costs) and take large increase increments at fairly high power levels.

For the illustrated example, the optimum is flat and appears to be j=4 Amps/mm2. However, a higher design value (the canonical j=10 Amps/mm2 value) is generally chosen. This is because the integrated power cost is generally regarded as an operating expense. Construction estimates are normally kept as low as possible in order to secure funding.

Coil Cooling In this section, we shall temporarily abandon the MKS system of units and use the mixed engineering and English system of units. Assumptions The water flow requirements are based on the heat capacity of the water and assumes no temperature difference between the bulk water and conductor cooling passage surface. The temperature of the cooling passage and the bulk conductor temperature are the same. This is a good assumption since we usually specify good thermal conduction for the electrical conductor.

Pressure Drop

Friction Factor, f We are dealing with smooth tubes, where the surface roughness of the cooling channel is given by; Under this condition, the friction factor is a function of the dimensionless Reynold’s Number.

For turbulent flow (Re>4000), the friction factor is gotten by
solving a transcendental equation. Normally, this type of equation can be solved only by successive iterations. However some Algebra can be used to simplify the solution.

Direct Solution of the Transcendental Equation
For turbulent flow, Re>4000; Substituting into the expression for the the friction factor;

which is an equation that can be solved directly for f.

Water Flow The equation for the pressure drop is, Solving for the
water velocity, Substituting the expression derived for, we get, finally,

Water Flow - Units The velocity is expressed in ft/sec. when g=32.2ft/sec2 and DP in (psi). In the expression, the units in the factors must be consistent. e, the surface roughness for a smooth tube, is 5x10-6 ft. Therefore, in the term, , d is expressed in ft.

More Units Similarly, since the water kinematic viscosity,
for the term , d is also expressed in ft. Finally for the term , d and L must be in the same units.

Coil Temperature Rise Based on the heat capacity of water, the water temperature rise for a flow through a thermal load is given by, Assuming good heat transfer between the water stream and the coil conductor, the maximum conductor temperature(at the water outlet end of the coil) is the same value. One more set of units has to be sorted out in order to compute the temperature rise.

Water Flow Spreadsheet
A CD is enclosed with the text. Among the files in the CD is a spreadsheet which can be used to reduce the drudgery involved in calculating the coil cooling. Two different spreadsheets are included in this file. One is written in metric units and the other is written in English units. Information in yellow are input data.

Results – Water Velocity
For water velocities > 15 fps, flow vibration will be present resulting in long term erosion of water cooling passage.

Results – Reynolds Number
Results valid only for Re > 4000 (turbulent flow).

Results – Water Flow Say, we designed quadrupole coils to operate at Dp=100 psi, four 0.30 gpm, total magnet water requirement = 1.2 gpm.

Results – Water Temperature Rise
Desirable temperature rise for Light Source Synchrotrons < 10 deg. C. Maximum allowable temperature rise (assuming 20 deg. C. input water) < 30 deg. C for long potted coil life.

Sensitivities Coil design is an iterative process.
If you find that you selected coil geometries parameters which result in calculated values which exceed the design limits, then you have to start the design again. Dp is too large for the maximum available pressure drop in the facility. Temperature rise exceeds desirable value. The sensitivities to particular selection of parameters must be evaluated.

Sensitivities – Number of Water Circuits
The required pressure drop is given by, where L is the water circuit length. K = 2, 4 or 6 for dipoles, quadrupoles or sextupoles, respectively. N = Number of turns per pole. NW = Number of water circuits. Substituting into the pressure drop expression, Pressure drop can be decreased by a factor of eight if the number of water circuits are doubled.

Sensitivities – Water Channel Diameter
The required pressure drop is given by, where d is the water circuit diameter. where q is the volume flow per circuit. Substituting, If the design hole diameter is increased, the required pressure drop is decreased dramatically. If the fabricated hole diameter is too small (too generous tolerances) then the required pressure drop can increase substantially.

Homework Do problems 5.1 and 5.2 on page 128 of the text.
Study problem The answer is given at the end of the text. Practice inputting the parameters in the problem and observe the results. Change some of the parameters and observe the changes in the results.

Lecture 7 Magnetic measurements is a specialized area.
Few (if any) of you will be involved intimately in this area and will need to understand the concepts in detail. However, the field is important since the quality of the magnets manufactured using the design principles covered in this course cannot be evaluated without a good magnetic measurements infrastructure. Few institutions maintains this infrastructure and often has to resurrect this capability whenever the needs arise. Some time will be invested in covering the material so that the student can gain some appreciation of the electronics which must be gathered and connected to take measurements and the mathematical rigor which underpins this field. The electronics required in the area of a small area of magnetic measurements, rotating coil measurements, and the mathematics used for the data reduction is covered in chapter 8.

Magnetic Measurements Lecture 7 Jack Tanabe Old Dominion University Hampton, VA January 2011

Introduction Magnetic measurements, like magnet design, is a broad subject. It is the intention of this lecture to cover only a small part of the field, regarding the characterization of the line integral field quality of multipole magnets (dipoles, quadrupoles and sextupoles) using compensated rotating coils. Other areas which are not covered are magnet mapping, AC measurements and sweeping wire measurements.

Voltage in a Coil

Therefore, substituting;
where A, the vector potential is a function of the rotation angle, q.

Measurement System Schematic

Digital Integrator The Digital Integrator consists of two elements.
Voltage to Frequency Converter. Up-Down (Pulse) Counter.

Using an Integrator on a Rotating Coil
Using an integrator simplifies the requirements on the mechanical system. The use of an integrator measures the angular distribution of the integrated field independent of the angular rotation rate of the coil.

Theory where L is the coil length and A is the vector potential, a function of the rotation angle q. The magnetic field can be expressed as a function of a complex variable which can be expressed, in general as ; Rewriting; The Vector Potential is, therefore;

Therefore, when we are measuring the integrated Voltage, we are actually measuring the real part of the function of a complex variable. We are measuring the rotational distribution of the integrated Vector Potential, AL. We really want to measure the distribution of the Field Integral.

Field Integral Let us take just one term of the infinite series.
Equating the real and imaginary parts of the expression;

In order to fully characterize the line integral of the magnetic field distribution, we need to obtain only |Cn| and yn from the measurement data. The graph illustrates the output from a quadrupole measurement. The integrator is zeroed before the start of measurement and the graph displays the result of a linear drift due to DC voltage generated in the coil.

Fourier Analysis In principal, it is possible to mathematically characterize the measured data by performing a Fourier analysis of the data. The Fourier Analysis is performed after the linear portion of the curve is subtracted from the data. Equating common terms,

or, finally, Separately, for the fundamental and error terms;

Fundamental and Error Fields
In general, the Fourier analysis of measurement data will include as many terms as desired. The number of terms is only limited by the number of measurement points. Earlier, we introduced the concept of the fundamental and error fields. The Vector potential can be expressed in these terms.

Compensated (Bucked) Coil
The multipole errors are usually very small compared to the amplitude of the fundamental field. Typically they are < 10-3 of the fundamental field at the measurement radius. The accuracy of the measurement of the multipole errors is often limited by the resolution of the voltmeter or the voltage integrator. Therefore, a coil system has been devised to null the fundamental field, that is, to measure the error fields in the absence of the large fundamental signal.

Consider the illustrated coil.
Two sets of nested coils with Mouter and Minner number of turns to increase the output voltage for the outer and inner coils, respectively, are illustrated.

Compensated Connection
The two coils are connected in series opposition. Define the following parameters: and We define the coil sensitivities; then,

Compensation (Bucking)
The sensitivities for the fundamental (n=N) and the multipole one under the fundamental (n=N-1) are considered. Why one under the fundamental? Consider the quadrupole, N=2

The classical geometry which satisfies the conditions for nulling the N=2 and N=1 field components in the compensated mode have the following geometry. Homework, show that s1 and s2 are zero for these values, compute the balance of the sensitivities and compare with the graph.

Compensated Measurements
Quadrupole measurements using the coil in the compensated configuration are typically as illustrated in the figure.

Bucking Ratio In the illustrated example of the compensated measurements, two properties can be readily seen. The drift is present. Usually, it is a larger portion of the signal than in the uncompensated measurements. This is because the DC voltage, usually due to thermocouple effects, is a larger fraction of the small compensated coil measurements. The signal is dominated by a quadrupole term. This is because of coil fabrication errors so that the quadrupole sensitivity is only approximately zero. The quality of the compensation is measured as a bucking ratio. Achieving a Bucking Ratio > 100 indicates a well fabricated coil.

Uncompensated Measurements
The magnet is also measured with the rotating coil wired in the uncompensated condition to measure the fundamental field integral and the multipole one below the fundamental. Where the sensitivities in the uncompensated condition are designated by capital S. For the Quadrupole;

Recalling the expression for the magnetic field components,
the amplitude of the fundamental field is, Solving,

Substituting into the expression for the fundamental amplitude;

Normalized Field Errors
The separate multipole field errors, normalized to the fundamental field amplitude can be computed from the measurement data. an and bn are from the compensated measurements and aN and bN are from the uncompensated measurements.

Reference Radius The expression for the normalized error multipole is evaluated at the outside radius of the inner coil, r1. This radius is limited by measurement coil fabrication constraints and, in general, is substantially smaller than the pole radius and generally smaller than the desired radius of the good field region, which might be > 80% of the pole radius. Therefore, the expression for the normalized error multipole is re-evaluated at a reference radius, r0.

The figure illustrates a 35 mm
The figure illustrates a 35 mm. pole radius quadrupole with a compensated rotating coil installed in the gap. The coil housing is < 35 mm. so that it will fit between the four poles. A half cylinder sleeve is placed around the housing to center the coil. As a result of these mechanical constraints, the maximum coil radius is < 27 mm.

The desired good field radius is 32 mm. , the maximum 10 s beam radius
The desired good field radius is 32 mm., the maximum 10 s beam radius. Therefore, in order to compute the field quality at this radius, the normalized field errors are recomputed at the required r0. and Therefore, and

Dipole Measurements The quadrupole coil configuration can also be used to measure a dipole magnet. Since the coil has no quadrupole sensitivity in the bucked configuration, a quadrupole error must be evaluated using the unbucked configuration. Since a quadrupole multipole is not an allowed multipole for a symmetric dipole magnet, this does not usually present a serious problem. However, if the dipole design constraints requires that the symmetry conditions be violated (ie. a “C” shaped dipole), the evaluation of the small quadrupole error present in this geometry may be marginal.

Sextupole Measurements
For sextupole measurements, it is desirable to make s3 and s2=0 for the compensated coil. This set of equations is difficult to solve algebraically. Therefore, the equations are solved transcendentally.

One of many solutions to these equations are,
The compensated sensitivities for these parameters are illustrated.

Relative Phase The calculation of the phase angles is based on an arbitrary mechanical angular shaft encoder zero datum, adjusted by aligning the measurement coil. Therefore, a phase of the fundamental field, yN, is always present. This angular offset can introduce large errors since small angular offsets between this datum and the zero phase of the fundamental field can result in large errors in the relative phase of the multipole error with respect to the quadrupole zero datum. Therefore, one normally computes a relative phase with respect to a zero phase for the fundamental field.

Sample Quadrupole Measurements
A one page summary of the multipoles for 15Q-001 measured at approximately 81 Amps is reproduced in the table. These measurements were made at IHEP in the PRC.

Two measurements are made at each current, one with the coil connected in the uncompensated mode and one in the compensated mode. The integrated voltage for each magnet is Fourier analyzed and the amplitudes of each coefficient are listed. The u1 and u2 amplitudes (PHI[n] in 10E-8 V-sec.) are the amplitudes of the coefficients for the cos q and cos 2 q terms from the uncompensated measurements. The balance of the amplitudes are the coefficients of the cos n q terms from the compensated coil measurements.

The next four columns include measured and computed values.
Angle The absolute phase angle of the nth Fourier term with respect to the shaft encoder zero datum. The same datum is used for both the uncompensated and compensated measurements. PHI[n]/PHI[2] The ratio of the compensated nth Fourier coefficient to the uncompensated 2nd Fourier coefficient. Coil Coef.[n] The coil sensitivities computed from the design radii of the various measurement coil wire bundles. B[n]/B[2] The computed (using the coil sensitivities) absolute value of the ratio of the multipole amplitude to the quadrupole field amplitude, evaluated at 32 mm.

Multipole Spectrum

Multipole Errors as Vectors

Distribution of n=6 Multipole Errors

Distribution of n=10 Multipole Errors

Distribution of n=3 First Random Multipole Errors

Iso-Errors The normalized multipole errors and their phases provide information regarding the Fourier components of the error fields. Often, however, one wants to obtain a map of the field error distribution within the required beam aperture. This analog picture of the field distribution can be obtained by constructing an iso-error map of the field error distribution. This map can be reconstructed from the normalized error Fourier coefficients and phases.

where is the phase angle of the multipole error with respect to the zero phase for the fundamental (quadrupole) field. Therefore,

and Where

The computations and contour map are programmed using MatLab. 15Q01 at 81 Amps.

The iso-error plot is replotted for only the allowed multipoles (n=6, 10, 14 and 18) and the first three unallowed multipoles (n=3,4 and 5). It can be seen that it is virtually identical with the previous plot, indicating that the unallowed multipole errors > 6 are not important. 15Q001 at 81 Amps

When the iso-error curve is replotted with the unallowed multipole errors reduced to zero and the allowed multipole phases adjusted to eliminate the skew terms, the DB/B <1x10-4 region is dramatically increased. This illustrates the importance of the first three unallowed multipole errors which are primarily the result of magnet fabrication and assembly errors. 15Q01 at 81 Amps. Unallowed multipole errors = 0. No skew phases for allowed multipoles.

Lecture 8 Lecture 8, describes techniques and principles for core fabrication. These descriptions are extremely important since the performance and quality of the magnetic field are dominated by the iron core of the manufactured magnets. The requirements are for the full population of magnets required for the synchrotron, not only for the individual magnets. This important subject is covered in chapter 9 of the text. Lecture 8 also describes magnet assembly and electrical bussing. Finally, fiducialization, installation and alignment are briefly described. These subjects are covered in chapter 12 of the text.

Coil Fabrication and Testing Coil Design/Fabrication Practices Electrical Safety Coil Quality Assurance (Specifications) Lecture 9 Jack Tanabe Old Dominion University Hampton, VA January 2011

Introduction Magnet design must incorporate practices which protect both the hardware and personnel operating and maintaining the magnet system. This section of the lecture will cover practices adopted at LBNL and used at many of the US accelerator laboratories to ensure safe, reliable operation and long life of accelerator magnets.

Whether coils are made in house or by vendors, certain quality standards must be met to ensure that magnets will operate as required. Specifications have been developed over the years to ensure magnet coil quality. These include material and fabrication requirements as well as performance tests. Rules governing electrical safety differ with different regions. The practices to ensure that magnet systems satisfy state and federal electrical safety requirements in California are reviewed.

Hydraulic Practices The accompanying figure illustrates several practices: Water hoses should be at least one meter long and use non-conducting material to prevent current leakage from the magnet. The water “in” fittings should be smaller than the water “out” fittings. Why? If a flow interlock (orifice plate) is used, it should be attached to the return manifold. Why?

Electrical Practices The figure illustrates some of the electrical design practices used in a typical magnet coil system.

Klixon Thermal Interlock
The Klixon thermal interlock system is a switch which opens when it senses a temperature higher than its set-point. The thermal interlock system is connected to the power supply and is designed to shut down power if a coil temperature exceeds the interlock set-point. The normal set-point of Klixons is about 89o C. It will generally reset at about 70o C, although a buyer can request special temperature values for these interlocks. One thermal interlock is installed on each water circuit. If a coils has more than one water circuit, multiple interlocks are installed to protect that coil. All the interlocks on one magnet are connected in series.

Interlocks for magnets in power supply series strings may or may not be connected in series.
If too many elements are connected in a series string, it is difficult to identify the particular magnet or circuit in that magnet. This becomes especially troublesome if the Klixon suffers intermittent failure. The Klixon is preferably mounted on the water return lead of the coil. Why? The Klixon is always mounted on the current carrying portion of the conductor (inside the electrical connection bus. Why? The Klixon is mounted to a block hard-soldered to the conductor. The interface between the Klixon and the mounting block is coated with a thermal conducting paste. Why hard-solder?

Internal Electrical Bussing
Practices for the internal electrical connection separate magnet coils are used to ensure safe and reliable magnet operation. Blocks to the connection of busses to the separate coils are hard soldered to the conductor. The busses are generally air cooled and therefore are sized so that the maximum current density is < 1.5 Amps/mm2. Since the busses are significantly heavier than the coil conductor, with current density < 10 Amps/mm2, insulating support blocks (usually epoxy fiberglass) are attached to the core to support the busses and protect the coil conductor from damage. No mechanical load should be carried by coil leads. The interfaces between the busses and the coil blocks are either silver plated or coated with an electrically conducting compound. Either split or Belleville (conical) washers are used under flat washers and either the bolt head or the nut. Why?

Power Supply Connections
Power Supply cables are very heavy. The design of the bus connection to the cables should be rigid and rigidly attached to the magnet core. The cable-bus interface should be coated with electrically conductive material. The fastener system should include a split washer or other energy storage device.

Electrical Safety Normally, an electrical safety inspection and audit is performed at a facility before it is allowed to turn on. It is extremely important that all electrical safety rules and standards be observed during the design stage of the project, since retrofit after the components are fabricated installed can be extremely expensive and time consuming.

Lead Insulation and Coil Covers
Coil covers or careful coil taping is required to prevent direct personnel contact with any magnet conductor or bus connection under the following conditions: IV > 150 V-Amperes or I > 30 Amps or V > 130 Volts or when the magnet stored energy is > 5 joules. Coil covers are recommended for all magnets regardless of their operating condition since it is difficult to audit safety practices when many different magnets are installed in the typical beamline. The hot end of the coil cooling tubes (the fittings) must also be enclosed in the cover. Safety rules require that any removable section of a coil cover must be attached by at least four screws.

Electrical Grounding Safety rules require all non-electrically powered accelerator components to be electrically grounded. Magnet cores are often assembled from laminations which may or may not be electrically connected to each other or to the core structure. As part of the core assembly procedure, a single weld bead is often specified, electrically connecting all magnet laminations together. A single connection point is provided to attach a metal strap grounding the magnet core to the support girder or other ground point.

Coil Fabrication Specifications
Specifications are written to assure coil quality and long life. If magnets are built “in house” or by vendors to designs completed by the customer, detailed specifications must be written. Certain design constraints and tests should be included in these specifications. The following list includes recommendations that may or may not be included in the specifications at the discretion of the magnet engineer/designer.

The specification should require that coils are wound in a clean environment and not in a shop environment where metal chips are present. If metal chips are embedded in the potted coils, they can cause intermittent shorts that are difficult to detect and can be extremely troublesome. A single water circuit in a coil assembly should be wound from a single continuous length of conductor. Splices “buried” within the potted insulation should not be allowed.

Hollow aluminum conductors should be tested for leaks prior to winding
Hollow aluminum conductors should be tested for leaks prior to winding. Aluminum conductors are usually assembled from four separate pieces which results in a conductor with four seams which run the length of the conductor. Leaks can occur in this type of fabrication. Copper conductor are extruded from a single piece of copper in a identical manner that small diameter tubes are fabricated. Therefore, one seldom encounters leaks in hollow copper conductor. Usually, the leak test is a pneumatic test where the cooling channel is pressurized and the pressure monitored for some time (usually several hours).

A ball test is specified for a conductor before it is wrapped into a magnet coil. This test consists of using high pressure air to blow a ball, no more than 80% of the diameter of the conductor hole, through the conductor water flow passage in order to ensure no blockage or collapsed passages. For long conductor lengths, or small cooling hole diameter, this test is omitted. (Pneumatic pressure drops can be very high during this test.) If the conductor vendor supplies long hollow conductor which is spliced, the location(s) of a splice should be clearly marked so that the continuity of the water cooling passage can be ensured.

Coil Tests An impulse test should be specified and photos of the scope trace should be required. The test involves applying a short voltage pulse across a combination of a capacitor and the coil being tested and observing the current waveform detected by a shielded pickup loop in the coil. The waveform will be a damped sinusoidal oscillation. Its frequency and damping rate will depend on the parameters of the coil and the capacitor in the tester. The procedure consists of starting with a low voltage pulse (~ 10 Volts/turn) and increasing the voltage until one reaches 200 Volts/turn or 2 kV maximum and observing the waveform. For the ordinary water cooled coil, the required turn to turn voltage is a small fraction of a volt per turn. Thus, impulse tests at the capacity of the tester (a few kV) is more than sufficient.

An impulse test is useful to determine whether intermittent shorts will occur since the turn to turn voltage during an impulse will exceed the voltage during DC operation. An impulse test is required in order to detect intermittent turn to turn shorts. This test can only be performed on a coil isolated from metallic surfaces and will not work once the coil is installed on the core. Core eddy currents and the iron permeability will mask the expected behavior of the electrical circuit.

A healthy coil will exhibit the same waveform as the voltage is increased from test to test with only a variation in the amplitude. A sick coil will exhibit waveforms whose frequency and/or damping rate changes as the voltage increases or will exhibit hash at the peak of the damped sinusoid. A dead turn to turn short cannot be detected with the impulse tests unless the waveform can be compared to a good coil of the identical design. Even then, the wave form of a good coil with many turns will vary only slightly from the wave form of a coil with a dead turn to turn short of the same design. Wave form for a coil with an intermittent short.

The impulse test must be performed on all coils after potting and prior to installation of a magnet. Optionally, an impulse test can be performed before it is potted. If the impulse test is performed on a coil both before and after potting, the voltage traces must be identical for the two tests. Specifications should require photographs made of all the scope traces (low and high voltage, optionally before and required after potting) and delivered to the customer as part of the Quality Assurance (QA) documentation.

Null Measurements For coils with a large number of turns, a precise resistance measurement prior to potting often cannot reveal the presence of a dead turn to turn short. This test is inaccurate since the conductor cross section may vary. For coils with a large number of turns of small conductor, a null measurement with two coils in a magnetic circuit can reveal differences in the number of turns of two otherwise identical coils.

The coils are connected in opposite polarities
The error in the number of turns can be measured using a Hall probe.

Hipot Test After potting, the coil should undergo a hipot test. The techniques used to hipot vary according to design. Vacuum potted coils with impermeable surfaces, can be immersed in salted water. A less risky technique is to wrap the insulated part of the coil in aluminum foil. Specifications normally call for raising the voltage in the coil to twice the operating voltage plus one kV. The maximum drainage current is usually set at 2 mAmps/kV.

Measurements Water flow calculations made for the preliminary design may be unreliable for a coil designed with many tight turns. This is due to the added flow impedance of tight radius turns. Thus, these measurements should be made carefully at the anticipated pressure differential available in the magnet water cooling system and the results compared for a number of coils with the same design and with design predictions. The ambient temperature should be recorded at the time of the measurement since the water flow is affected by the water viscosity which change substantially with temperature. Coil resistance should be carefully measured using a double bridge (so that current is not carried in the coil while measurements are being made) and compared with computed values. The ambient temperature when the resistance measurements were made should be recorded.

QA Travellers As part of the Quality Assurance procedure for coil fabrication and testing, a form (Traveller) is developed which is filled out, signed and dated by technicians performing coil fabrication and testing and signed and dated by the supervisor responsible for the fabrication and testing of coils.

At a minimum, the Traveller contains the following information:
Coil type and serial number. Name(s) of individual(s) inspecting and testing conductor and date of task completion. Inspection includes leak tests, ball tests (if required) and locating conductor splices (if any). Name(s) of individual(s) winding, insulating and ground wrapping the coil and date of task completion Name(s) of individual(s) potting the coil and date of task completion Name(s) of individual(s) finishing the coil and date of task completion Finishing includes lead bending, soldering Klixon interlock blocks and lead blocks and soldering water fittings.

Coil QA Traveller Data In addition to the previously described information, the traveller should include the following measured data. Impulse test scope photographs at low and high voltages. Hipot test results. Maximum Voltage Measured leakage current (max < 5 mAmps) Water Flow Test Results Pressure Drop Flow Temperature Resistance Value Ambient temperature during measurement

Lecture 10 Lecture 10 collects “loose ends” which have not been covered thus far. The subject of stored energy, and magnetic forces are covered in chapter 7. Stored energy is used to compute magnet inductance, which is used to compute power supply requirements for dynamic magnets, magnets with time varying magnetic fields. Although other dynamic effects (eddy currents) are covered in this chapter, this is a broad complex subject and will not be covered in the lectures. General principles for reducing fringe fields are covered. Means of minimizing errors in the magnet integrated field by chamfering to control the fringe fields are discussed.

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