Superconducting Lecture V

Magnet Division

Magnetic Design
Yoke Optimization
Ramesh Gupta
Superconducting Magnet Division
Brookhaven National Laboratory

US Particle Accelerator School

Arizona State University
Phoenix, Arizona
January 16-20, 2006

January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 1 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Use of Iron Yokes in
Superconducting Conductor Dominated Magnets (1)
Magnet Division

Why should we use yokes in the conductor dominated magnets?

Especially why in high field magnets, where most of the field is
provided by coil? Yoke significantly increases the size and weight
of the magnet & increases the volume of the coldmass to be cooled.

Reason No. 1:
• For a variety of reasons, the magnetic field outside the magnet (fringe
field) should become sufficiently small.
• In almost all cases, and in virtually all accelerator magnets built so
far, the iron yoke has been found to be the most cost effective method
of providing the required magnetic shielding.
• Therefore, the iron yoke is used over the coil despite increasing the
size of the magnet.
January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 2 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Use of Iron Yokes in
Superconducting Conductor Dominated Magnets (2)
Magnet Division

Why should we use yokes in the conductor dominated magnets?

Reason No. 2:
• The magnetized iron gives an additional contribution to the field
generated by coils.

However, the gain does not come without any pain, particularly as we get
more and more ambitious (higher contribution, higher field). The iron starts
saturating at high field. That makes the field contribution non-linear and
field errors in the magnet (harmonics) depend on the central field.
• The trick is to develop techniques to benefit from the gain while
minimizing the pain.

The purpose of this course is to make you familiar with those techniques by
presenting the state-of-art.
January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 3 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
Superconducting
Shielding
Magnet Division

The shielding against the fringe field can be provided either by

the iron yoke or by additional outer coils having polarity opposite
to that of the main coils.

Home Assignment on the Shielding by Additional Coils:

• A thin cosine theta dipole coil is placed at a radius of 10 cm. This coil generates a central
field of 0.5 T. Compute the relative strength of an additional thin coil at a radius of 20 cm
that is placed to cancel the fringe field far away from coil regions. Also compute the change
in central field caused by this additional coils? Compute the change in central field, if instead
of an additional coil, a thick iron yoke is placed at a radius of 20 cm to provide the required
shielding. What happens if the yoke shielding is brought right up to 10 cm. In these
calculations, ignore saturation of the iron yoke and assume that it has infinite permeability.
• Do similar computations for quadrupole coils generating a field gradient of 5 T/mm.
• What would you use in your design for providing shielding and why?
•In which case would you prefer the additional coils and in which the yoke over coils?

Yet another possibility: Superconducting Meissner shield -- again, extra conductor.

January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 4 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 BH Table Used in Calculations
Superconducting
Magnet Division

January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 5 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Iron Yoke in RHIC Dipole
Superconducting
Magnet Division
Yoke can contain field lines at low fields Yoke cannot contain field lines at high
(~0.7 T, ~1 kA). No Fringe field outside. fields (~4.5 T, ~7 kA). Significant fringe
field outside. The design field is ~3.5 T.

January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 6 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Saturation in RHIC Arc Dipoles
Superconducting
Magnet Division

In RHIC dipoles, iron is closer to First Design

coil and contributes ~ 50% of the

coil field:
3.45 T (Total) ~ 2.3 T (Coil)
+ 1.15 (Iron)
That’s good.
Current Design
But the initial designs had bad saturation,
as conventionally expected when iron yoke
is so close to the coils and contributes such
a large fraction of coil field.

First Design
This course will teach you several
techniques to reduce the current-
dependence of field harmonics.
January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 7 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Consequences of the
Superconducting Saturation of the Iron Yoke
Magnet Division

An iron yoke provides good shielding against the fringe field. Moreover, the iron gets
magnetized such that it adds to the central field generated by the coil.
In cosine theta magnets (a=coil radius, Rf=yoke inner radius and Ra=outer radius):

At low fields, µ is large and (µ−1)/(µ+1) is nearly one. In principle, the yoke can
double the field. However, at high fields the iron magnetization becomes non-linear
and µ approaches one. This makes the relative contribution of the field from the iron
become smaller as compared to that of the coil. Moreover, the field distribution inside
the aperture changes, which in turn makes the field harmonics depend on the field.

January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 8 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 COS(mθ) Coil in Iron Shell
Superconducting
Magnet Division
Rf : Iron inner radius
Ra : Iron outer radius
a : Coil Radius

Note: µ is assumed to be constant in the entire iron yoke.

January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 9 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Field Enhancement for
Superconducting COS(mθ) Coil in Iron Shell
Magnet Division

January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 10 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Understanding Iron Saturation (1)
Superconducting
Magnet Division

January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 11 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Field and Saturation Parameters in
Superconducting RHIC Dipoles with Circular Iron Yokes
Magnet Division

Note the differences in

field between the pole and
midplane. The yoke at
pole has much larger Bmod (T)
magnitude of field.

(µ-1)/(µ+1) is a better variable

(µ-1)/(µ+1) than magnitude of the field as it
appears in the basic equations.
It has a range between 0 and 1.
0 means complete saturation
(µ=1) and 1 means little
saturation (µ is very large).
January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 12 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Understanding Iron Saturation (2)
Superconducting
Magnet Division

January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 13 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Understanding Iron Saturation (3)
Superconducting
Magnet Division

yoke saturation between midplane and pole

First order optimization: Force a similar
January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 14 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Method of Image Current to Understand and
Superconducting Minimize the Saturation-induced Harmonics
Magnet Division

The contribution of a circular iron yoke with constant permeability (µ) can be described with the help of
image currents. Note that in this case there will be only a radial component of the field at yoke inner
surface. The field of a line current (I) at a radius “a” inside a circular iron cavity of radius Rf is given by:

The image current will be at the same angular location, however, the
magnitude and the radial location are given by:

Note the appearance

of (µ-1)/(µ+1) in
various equations.

January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 15 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Conceptual Development of an Approach to
Superconducting Minimize Saturation-induced Harmonics
Magnet Division

The image current “I” of a line

current at a radius “a” will be at
the same angular location,
however, the magnitude and the
radial location are given by:

> The block may be described by a series of line currents and the image block by
a series of image currents. The image block will produce a field (and harmonics)
that are similar in shape to the main field, if the µ of the iron is constant.

> Real magnets have non-linear saturating iron. It seems intuitive that the change
in the field shape (and harmonics) as a function of excitation can be minimized by
minimizing the variation in µ. The quantitative deviation may be minimized by
minimizing (µ-1)/(µ+1), as this is the quantity that appears in most expressions.
January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 16 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 A Conceptual Model for Understanding and
Superconducting Minimizing the Saturation-induced Harmonics
Magnet Division
The contribution of a circular iron yoke with infinite permeability can be described with the
help of image currents. A series of image currents (second term in the following expression)
will retain the original angular distribution and the magnitude will be proportional to the
original current, if mu (µ) is constant in the iron (uniform magnetization across the iron).
In that case only the primary component depends on the magnetization and no other harmonics
will change. Moreover, the change in the primary component is related to (µ−1)/(µ+1).

The above theory does not work if the magnetization is not uniform. However, even in that
case one can still develop a conceptual understanding and minimize the saturation-induced
harmonics by using the following hypothesis. Describe the coil with a series of line currents
and assume that the image current is still at the same angular location but the magnitude is
related to the average mu in the vicinity of the angular location where the line currents are.
The variation in saturation induced harmonics may be minimized, if the
variation in iron magnetization, as measured by (µ−1)/(µ+1), is minimized.
January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 17 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Saturation in RHIC Arc Dipoles
Superconducting
Magnet Division
First Design
In RHIC dipole, iron is closer to
coil and contributes ~ 50% of
the coil field:
3.45 T (Total) ~ 2.3 T (Coil)
+ 1.15 (Iron)
Current Design

Initial design had bad saturation

as expected from conventional wisdom,
but a number of developments made the
saturation induced harmonics nearly zero!
First Design
Only full length magnets are shown.
Design current is ~ 5 kA (~3.5 T).

January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 18 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Saturation Control in RHIC Dipoles
Superconducting Variation in |B| in Iron Yoke
Magnet Division

Without holes
With holes

• Compare azimuthal variation in |B| with and without saturation control holes.
Holes, etc. increase saturation in relatively lower field regions; a more
uniform iron magnetization reduces the saturation induced harmonics.
• Old approach: reduce saturating iron with elliptical aperture, etc.
• New approach: increase saturating iron with holes, etc. at appropriate places.

January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 19 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Saturation Control in RHIC Dipoles
Superconducting Variation in (µ-1)/(µ+1) in Iron Yoke
Magnet Division

With out holes

With holes

• It is better to examine (µ-1)/(µ+1) instead of |B|. As it appears in various formula, e.g.

It also provides a better scale to compare the magnetization (see pictures).

• Compare the azimuthal variation in (µ-1)/(µ+1) with and without saturation control
holes, particularly near the yoke inner surface. A more uniform iron magnetization
reduces the saturation induced harmonics.
January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 20 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Current Dependence Beyond Design Field
Superconducting
Magnet Division
In all known major accelerator magnets (superconducting and iron dominated), the
harmonics decrease rapidly beyond the maximum design field. They could be made
relatively flat (small change) using this design approach . Please note the difference in
scale (50 units in an earlier b2 slide). It (a) shows a major impact of this design approach
on field quality and (b) may have relevance to a future RHIC energy upgrade as most
magnets in RHIC have ~30% quench margin over the maximum design field.
Current Dependence in RHIC Dipole DRG107 (DC Loop, Up Ramp)

2
b6
b2, b4, b6 (@25mm)

0
-2 b4

-4
b2
-6
b2
-8 Injection Field Max. Design Field
(~0.4 T, ~0.6 kA) (~3.5 T, ~5kA)
-10
-12
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Bo (Tesla)

January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 21 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Significant Difference With
Superconducting The Conventional Method
Magnet Division

In order to achieve a more uniform magnetization (iron saturation), one can force the field lines
to the region where the iron is less magnetized. This will increase the overall magnetization in
the iron, but the attempt should be to force a more uniform magnetization, particularly in the
iron region that is closer to aperture.

The conventional method called for not allowing the iron to saturate (too much magnetized).
Minimizing non-linear iron means minimizing the saturation induced harmonics. This meant
keeping the iron away from the coil as that is a high field region. However, that also meant
reducing the contribution of the iron to the total field as the iron near the aperture (coil)
contributes more. In brief, the old method relied on reducing the region of iron that saturates.

The major difference between the method used in RHIC magnets, as compared to the earlier
designs with which major accelerator magnets have been built, was that here the attempt was to
increase (force) the saturation (to make it uniform) and before the attempt was to decrease it.

The close-in iron for obtaining higher field need not compromise the field quality as long as the
iron saturation can be kept uniform, particularly in the iron region that is closer to the aperture.
January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 22 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Influence of Lorentz Forces
Superconducting on Field Harmonics
Magnet Division
The measured current dependence of field harmonics is a combination of
saturation-induced harmonics, and the Lorentz force-induced harmonics.
A typical Sextupole current dependence
due to Lorentz forces (schematic)
Low force/friction Current
b2
(practically no effect)

Radial motion
Azimuthal motion

Coil makes contact to collar

(maximum radial motion)
Assignment: Make a similar sketch for b4 (decapole).
A small radial gap in magnets (50-100 micron), could be
present due to tolerances in collar o.d. and yoke i.d. In
SSC that means ~-1 unit of sextupole. Such errors can be
accommodated in a flexible design - key material,etc.
January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 23 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Measured Current Dependence in Sextupole
Superconducting Harmonic in Various Full-length SSC Magnets
Magnet Division
Measurement of b2 current dependence in group of SSC magnets
Cross section of SSC 50 mm Dipole
Various SSC 40 and 50 mm dipoles
1.6
dss020 Yoke optimized for low saturation
1.2 KEK (Fe Key) SSC 50 mm
(BNL-built) dss010
SSC Specification
b2 (10 mm), US convention

0.8 dsa207
0.4 dca207
0
ds0202
dsa311
-0.4
dc0201
-0.8 KEK501
-1.2
SSC Specification
Lorentz forces
-1.6
2 3 4 5 6 7 8
Current (kA)

Near zero current dependence in b2 variation in the very

first design of BNL built SSC 50 mm long magnets. Non-magnetic key to force uniform saturation
Specifications was 0.8 unit. Could also have been used to adjust current
dependence after design, as in RHIC magnets.
A much larger value in earlier SSC 40 mm design.
b2 change from yoke magnetization & Lorentz forces. Major progress in reducing the
saturation-induced harmonics.
January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 24 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
Superconducting
Yoke Cross-section Optimization
Magnet Division

To do detailed magnetic design, you must use one of

several available computer codes.

Some popular codes that are currently being used for

designing accelerator magnets:
• POISSON, etc. (Developed in labs, public domain)
• OPERA, ANSYS, etc. (Commercial)
• ROXIE (Developed in labs, commercial & requires licensing)

In addition, various labs have written in-house computer codes to

meet their special requirements. For example, all RHIC coils, for a
variety of magnets, are designed with PAR2dOPT at BNL. And new
codes are being developed for racetrack coils.
January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 25 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Setting-up A Magnetic Model
Superconducting With A Minimum Geometry
Magnet Division

To make efficient use of the computer resources and to get more

accurate results in minimum time, set up the basic model with
proper boundary conditions.

For example, for a dipole magnet, usually you need to model only
a quadrant of the geometry, with the following boundary
conditions:
• field perpendicular boundary on the x-axis
• field parallel boundary on the y-axis
• infinite boundary condition on the other side(s), or else
extend the other boundary far away so that the field near the
end of boundary becomes very small.
Question: What will you do in the case of a quadrupole magnet?
January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 26 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Magnet Yoke Optimization
Superconducting
Magnet Division

Generally speaking, first determine the yoke envelope

• Yoke inner radius
Mechanical (Lorentz forces) & magnetic issues (iron saturation)

• Yoke outer radius

Mechanical (size and space consideration) & magnetic issues (iron
saturation, fringe fields)

… and then optimize the internal geometry

• Accommodate holes, etc for cooling, assembling and other
mechanical purpose
•Try to place above holes at strategic places and put extra
holes, etc., if necessary.
January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 27 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Yoke Inner Radius
Superconducting
Magnet Division

To first order, the yoke inner radius depends on the

mechanical design chosen to contain the Lorentz forces

• It is typically over 15 mm plus the coil outer radius, if stainless steel

or aluminum collars are used
Example: SSC or LHC dipoles
• It is typically 5-15 mm plus the coil outer radius, if the yoke is also
used as collar (material between the coil and yoke acts as an spacer)
Example: RHIC Dipole and Quadrupole Magnets

Smaller inner radius brings iron closer to the coil and adds to the field
produced by the coil alone. However, it also increases the saturation-
induced harmonic due to non-linear magnetization of iron at high fields.

January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 28 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Variation in Yoke Inner Radius
Superconducting in RHIC 80 mm Aperture Dipole
Magnet Division

January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 29 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Yoke Outer Radius
Superconducting
Magnet Division

The yoke outer radius should not be unnecessarily

large, as that:
• May increase the over all dimensions
• May increase the magnet weight
• May increase the over all cost
However, the yoke outer radius should not be too
small either, as that:
• May increase the fringe field
• May reduce the central field significantly
• May increase the saturation induced harmonics

January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 30 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Fringe Field for Various Outer Yoke Radii
Superconducting
Magnet Division

Fringe field in the SSC dipole at the design field of 6.6 T

outside the yoke for various values of yoke outer radius. These
models assume that there is no cryostat outside the coldmass.

January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 31 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Variation in Yoke Outer Radius
Superconducting in SSC 50 mm Aperture Dipole
Magnet Division

January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 32 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Elliptical Aperture to Reduce
Superconducting Saturation-induced Harmonics
Magnet Division

A model investigated for SSC 40 mm dipole magnet.

In order to reduce, the saturation-induced harmonics, the iron is selectively

removed from the region (pole), where it was saturating more due to higher field.
January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 33 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Saturation Control in RHIC IR Quads
Superconducting
Magnet Division

R = 92 mm

R = 87 mm

POISSON model of a quadrant of the Optimized design

130 mm aperture RHIC Insertion quadrupole.
Since the holes are less effective for controlling saturation in quadrupoles,
a 2-radius method was used.
January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 34 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Influence on T.F. and b9
Superconducting of 2-radius Design
Magnet Division

RHIC 13 cm aperture interaction region quadrupole

January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 35 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Influence of Notch/Tooth
Superconducting
Magnet Division

tooth

notch Influence of 5mm X 5 mm notch

or tooth as a function of angle.

January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 36 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Saturation Control Holes
Superconducting
Magnet Division

• The most powerful tool to control the saturation, or rather force a uniform
saturation, is to use saturation control holes.

• One can either use the holes that must be there for other purpose, or put some new
ones that are dedicated to the sole purpose of controlling saturation.

• Example of existing holes:

¾ Big helium holes for cooling (generally good flexibility in choosing the
location and some in choosing the size also).
¾ Pins for putting yoke laminations together (flexibility in choosing material,
magnetic steel or non-magnetic steel), and small flexibility in size and location.
¾ Yoke-yoke alignment keys (flexibility in choosing material, magnetic steel
or non-magnetic steel), and small flexibility in size and location.
¾ And some other in special cases, like tie rods, etc.

January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 37 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Influence of An Additional
Superconducting Saturation Control Hole in RHIC Dipole
Magnet Division

Saturation-induced harmonics in RHIC

Vary Radial Location
dipoles at the design current (5kA).

1 cm dia hole at 35 degree

Hole at r = 7.5 cm
Vary Angular Location and t= 35 degree

1 cm dia hole at 7.5 cm radius Vary Radial Size

January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 38 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 RHIC Arc Dipole
Superconducting (with saturation control features indicated)
Magnet Division

Magnetic Model of the

RHIC arc dipole with
saturation control
holes, etc.
January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 39 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Influence of Saturation Control Hole
Superconducting
Magnet Division

A RHIC 80 mm dipole
was rebuilt after
punching saturation
control holes in the
lamination.

A significant reduction
in the saturation-
induced (current
dependence of) field
harmonics can be seen.

This feature was

adopted in the RHIC
production magnets.

January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 40 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Current Dependence in Non-allowed
Superconducting (Un-allowed) Harmonics
Magnet Division
Non-allowed harmonics are those that are not allowed by the basic
magnet symmetry.
Current dependence in non-allowed harmonics implies that the iron
may not have the basic magnet symmetry.
Presence of non-allowed harmonics as a function of field may also be due
to loss of coil symmetry due to an asymmetry in Lorentz forces.
In addition it may also be due to the differences in the superconducting
properties of superconductors used in different coils.

Allowed harmonics in dipoles: Dipole (Bo), b2, b4, b6, …, b2n

Non-allowed harmonics in dipoles:
quadrupole, octupole, … (b2n+1) : left-right asymmetry
All skew harmonics an : top-bottom differences

Allowed Harmonics in quadrupole: Quadrupole gradient (b1), b5, b9, …

All others are not allowed
January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 41 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Current Dependence in
Superconducting Skew Quad Harmonic (a1) in Dipole
Magnet Division

Skew quadrupole harmonic (a1) in dipole reflects a top-bottom

asymmetry !

Suspect: Somehow the total amount of iron is not same on top and bottom
(at low field, not much iron is needed to contain the flux, so it matters less as long
as the geometry is the same)
Another source: asymmetric Lorentz forces (unlikely)

Type of variations in skew quad

• Integral : Overall asymmetry (or difference between top and bottom)
• Location-to-location : Local asymmetry (or difference between top and bottom)

January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 42 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Non-symmetric Coldmass
Superconducting Placement in Cryostat
Magnet Division

Design of the 80 mm
aperture RHIC dipole
coldmass in cryostat

Coldmass (yoke) is
made of magnetic steel
and cryostat is made of
magnetic steel.

What will happen at very high fields

when the magnetic flux lines cannot
be contained inside the iron yoke?

January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 43 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Leakage of Magnetic Flux Lines
Superconducting at High Fields in SSC Dipoles
Magnet Division

Cryostat

Yoke
What harmonics will
it create?

Note that the yoke

iron is not placed
symmetrically inside
the cryostat.

January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 44 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Measured Current Dependence
Superconducting in Two RHIC Dipoles
Magnet Division

1
Current
dependence of the skew
0 quadrupole term in the
dipoles DRG113 and
[units]

-1
DRG125. The magnitude
-2
of change between low
a 1(I ) - a 1 (1450A)

currents and 5000A is the

-3 largest in DRG125 and is
relatively small in
-4
DRG113.
-5 Seq. 15 (DRG113)
Seq. 27 (DRG125)
-6

-7
0 1000 2000 3000 4000 5000 6000

Magnet Current, I (A)

January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 45 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Reduction in Saturation-induced Skew
Superconducting Quad Harmonic (a1) in RHIC Dipoles
Magnet Division
weight of Top part - weight of Bottom part
asymmetry =
Average weight of Top and Bottom parts
3
The calculated current
2
dependence of skew quadrupole term for
a 1(5000A) - a 1(1450A) [units]

1
4
various values of the asymmetry between
0
the top and the bottom halves of the yoke.

Calculated Skew Quadrupole (units)

2
-1
0
-2
-2
-3

-4 -4

-5 -6 +1.0%
+0.5%
-6 0.0%
-8
-0.5%
-7
-10 -1.0%
-150 -100 -50 0 50 100 150
4
Yoke Weight Asymmetry (parts in 10 ) -12
1000 2000 3000 4000 5000 6000
Correlation between the yoke
Magnet Current (A)
weight asymmetry and the saturation-
induced a1.2006, Superconducting Accelerator Magnets
January 16-20, Slide No. 46 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Field lines in the SSC 2-in-1 Dipole
Superconducting (both aperture are excited at 6.6 T)
Magnet Division

A similar
situation in
LHC 2-in-1
arc dipoles

What field harmonics are allowed in this geometry?

At low fields and at high fields?
January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 47 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Field lines in the SSC 2-in-1 Dipole
Superconducting (two aperture are excited in 2:1 ratio)
Magnet Division

What field harmonics are allowed in this geometry?

At low fields and at high fields?
January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 48 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 Three Magnets with Similar Apertures
Superconducting (Tevatron, HERA and RHIC)
Magnet Division
RHIC Dipole
Tevatron Dipole HERA Dipole (80 mm bore)
(76.2 mm bore) (75 mm bore)

Consideration on systematic errors Wedges ( small higher order

harmonics expected).
No Wedges (large higher order Wedges ( small higher order Thin RX630 spacers to reduce cost
harmonics expected). - Iron close to coil (large saturation
systematic harmonics expected).
Al Collars - Iron away from coil from conventional thinking. But
S.S. Collars - Iron away from (small saturation expected). reality opposite: made small with
coil (small saturation expected). design improvements).

Collars used in Tevatron and HERA dipoles have smaller part-to-part dimensional variation (RMS
variation ~10 µ) as compared to RX630 spacers (RMS variation ~50 µ) used in RHIC dipoles.
Conventional thinking : RHIC dipoles will have larger RMS errors. But in reality, it was opposite.
Why? The answer changes the way we look at the impact of mechanical errors on field quality !

January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 49 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
Superconducting
Average Field Errors on X-axis
Magnet Division

COIL ID : RHIC 80 mm, HERA 75 mm, Tevatron 76.2 mm

At Injection Energy At Top Energy

0.0005 0.0005
<RHIC> <RHIC>
0.0004 0.0004
<HERA> <HERA>
0.0003 0.0003
<Tevatron> <Tevatron>
0.0002 0.0002
axis axis

dBy/Bo
0.0001 0.0001
dBy/Bo

0.0000 0.0000
-0.0001 -0.0001
-0.0002 -0.0002
-0.0003 -0.0003
-0.0004 -0.0004
-0.0005 -0.0005
-80 -60 -40 -20 0 20 40 60 80 -80 -60 -40 -20 0 20 40 60 80
Percentage of Coil Radius Percentage of Coil Radius

• Warm-Cold correlation have been used in estimating cold harmonics in RHIC dipoles (~20% measured cold and rest warm).
• Harmonics b1-b10 have been used in computing above curves.
• In Tevatron higher order harmonics dominate, in HERA persistent currents at injection. RHIC dipoles have small errors over entire range.

January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 50 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL
 SUMMARY: Yoke Optimization
Superconducting
Magnet Division

• The yoke iron used in accelerator magnets to reduce the fringe field outside
the magnet to an acceptable limit. This is the most cost effective method.
• The iron yoke also gives an additional contribution to field. The contribution
can be increased by bringing iron closer to the coil.
• It is generally expected that the close-in iron will increase the iron saturation
However, a number of techniques have been developed which demonstrate
that the yoke can be forced to saturate uniformly. These techniques keep
the saturation-induced harmonics to a small and acceptable value.

Therefore, one can now take the benefit of a good

enhancement in field from the close-in iron, without
sacrificing the field quality due to bad saturation
induced harmonics due to non-linear yoke saturation.

January 16-20, 2006, Superconducting Accelerator Magnets Slide No. 51 of Lecture 5 (Yoke Optimization) Ramesh Gupta, BNL