2 Multipoles, Conformal Mapping, Pole tip design– Conformal mapping is used to extend the techniques of ensuring dipole field quality to quadrupole field quality. – Conformal mapping

Mauricio Lopes – FNAL

2

Multipoles, Conformal Mapping,

Pole tip design

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 fieldindex n=N/2.

US Particle Accelerator School – Austin, TX – Winter 2016 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 120degrees. 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 (N) is the “fundamental” and the remainder of the terms (n) represent the “error” fields.

∑=n

n zCF

∑≠

+=Nn

nn

NN zCzCF

nCzF =


∑≠

+=1

1n

nndipole zCzCF

∑≠

+=2

22

n

nnquadrupole zCzCF

∑≠

+=3

33

n

nnsextupole zCzCF

∑≠

+=Nn

nn

NNNpole zCzCF

22

∑≠

=Nn

nn zCsErrorField


Allowed Multipole Errors


• The error multipoles can be divided among allowed orsystematic and random errors.

• The systematic errors are those inherent in the design andsubject to symmetry and polarity constraints.

• Symmetry constraints require the errors to repeat andchange polarities at angles spaced at π/N, where N is theindex of the fundamental field.

• In the figure, the poles are not symmetricalabout their respective centerlines. This is to

illustrate rotational symmetry of the N poles.


• Requiring the function to repeat and change signs according to the symmetry requirements:

• USing the “polar” form of the function of the

complex variable:

( ) θπθ NpoleNpole FNF ∆−=

+∆

∑

∑

++

+=

=

+∆

+

Nni

NnzC

ezCN

F

n

n

Ninn

n

πθπθ

πθπθ

sincos

( ) ( )∑∑∑ +===∆ θθθ θ ninzCezCzCF nninnnnn sincos


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

• Rewriting;

( )θπθ nN

n coscos −≡

+ ( )θπθ nN

n sinsin −≡

+

( ) ( ) ( )

( ) ( ) ( )θπθπθπθ

θπθπθπθ

nN

nn

N

nn

Nn

nN

nn

N

nn

Nn

sinsincoscossinsin

cossinsincoscoscos

−≡+=

+

−≡−=

+


• Therefore;

integers. odd all , 7, 5, 3, ,1 1cos

integers. all , 4, 3, 2, ,1 0sin

L

L

=⇒−=

=⇒=

N

n

N

nN

n

N

n

π

π

• The more restrictive condition is;

integers. all , 4, 3, 2, ,1 ere wh

12integers odd all

L=

+==

m

mN

n

• Rewriting;

( ) integers. all , ,4 ,3 ,2 ,1 where12 L=+= mmNnallowedUS Particle Accelerator School – Austin, TX – Winter 2016 10

Examples


– For the dipole, N=1, the allowed errormultipoles are n=3, 5, 7, 9, 11, 13, 15, …

– For the quadrupole, N=2, the allowed errormultipoles are n=6, 10, 14, 18, 22, …

– For the sextupole, N=3, the allowed errormultipoles are n=9, 15, 21, 27, 33, 39, …

Magnet Field Uniformity

• In general, the two dimensional magnetfield quality can be improved by the amountof excess pole beyond the boundary of thegood field region.

• The amount of excess pole can be reduced,for the same required field quality, if oneoptimizes 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.


( )[ ]75.077.2exp100

1 −−=

∆x

B

B

dunoptimize

( )[ ]39.017.7exp100

1 −−=

∆x

B

B

optimized

gap

overhang" pole"

halfh

ax ==

90.0ln 36.0 −∆−=B

Bx dunoptimize

25.0ln 14.0 −∆−=B

Bxoptimized

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:

2.52.01.51.00.510 -6

10 -5

10 -4

10 -3

10 -2

10 -1

Optimized

Unoptimized

Dipole Field Quality as a Function of Pole Overhang

x = ah

∆ BB


Introduction to conformal mapping

• This section introduces conformal mapping.

– 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 onemagnet geometry into another magnet geometry.


Mapping a Quadrupole into a Dipole

• The quadrupole pole can be described by ahyperbola;


constant a2

==C

Vxy

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;

2

2hxy =

We introduce the complex

function; ( )

h

ivx

h

zivuw

22 +==+=

h

yxwu

22

Re−==

hh

xywv === 2Im

h

xyi

h

yxivuw

222 +−=+=Rewriting;

since

2

2hxy =

ihh

yxw +−=

22

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;


φieww = θiezz =and

Since was used to convert the quadrupole into the dipole,

φiewhhwz ==2h

zw

2

=

θφ

ii ezewhz == 2 whz =2

φθ =therefore; and

Finally; 2coscos

φθ whzx ==

2sinsin

φθ whzy ==

Mapping a Dipole into a Sextupole

• In order to map the Dipole into the Sextupole, we use the polar forms of the functions;


φieww = θiezz =and

Since was used to convert the quadrupole into the dipole,

φiewhwhz 223 ==

2

3

h

zw =

θφ

ii

ezewhz == 33 2 32 whz =

3

φθ =therefore; and

Finally; 3coscos 3 2

φθ whzx ==

3sinsin 3 2

φθ whzy ==

Mapping a Dipole into 2N-pole

• General formula to map a Dipole into a 2N-pole


Nwhx N N

φcos1−=

Nwhy N N

φsin1−=

=

+=

+=

u

v

vuw

ivuw

arctan

22

φ

• u and v are the dipole coordinates

• x and y are the coordinates for the 2N-pole

Example: Ideal Dipole


Ideal Dipole mapped in a Quadrupole


Ideal Dipole mapped in a Sextupole


Optimized Dipole


Optimized Dipole mapped in a Quadrupole


Optimized Dipole mapped in a Sextupole


The gradient magnet


xBB

hBxy

o

o

')(

+=

'B

BxX o+=

XB

hB

B

BXBB

hBXy o

oo

o

''

')( =

−+=

'2

2

B

hBHXY o==

'

22B

hBH o=

( )H

XYi

H

YX

H

iYX

H

Zivuw

2222 ++=+==+=

H

YXu

+=2

H

XYv

2=

Gradient into a Dipole Magnet


'2

'2

2

B

hB

yB

Bx

uo

o +

+=

'

22

B

hBH

H

XYv o===

Dipole into a Gradient Magnet

( )uwHB

BxX o +=+=

2'( )uwHyY −==

2

Gradient into a Dipole Magnet



(not optimized)



(optimized)


The Septum Quadrupole


• In order to maximize the number of collisions andinteractions in a collider, 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.


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 theunoptimized or optimized quadrupolehyperbolic pole is truncated, alsodetermines the potential field quality forthe two dimensional unsaturatedquadrupole magnet.


• 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.


• Given; (uc, v

c) satisfying dipole uniformity

requirements.

• Find; (xc, y

c) satisfying the same requirements for

quadrupoles.

h

rr

h

zw regionfieldgood

20

2

=⇒=


[ ]factor 90.0ln 36.0 dunoptimizehB

Bha dunoptimize −=

+∆−=

[ ]factor 25.0ln 14.0 optimizedhB

Bhaoptimized −=

+∆−=

[ ] h= and 2

02

0cc vfactorhh

ra

h

ru −=+=

For the Dipole;

Therefore;


ihh

yxivuw +−=+=

22

h

rr

h

zw regionfieldgood

20

2

=⇒=

h

r00 =ρSubstituting a unitless (normalized) good field region,

2sinsin

2coscos

φθ

φθ

whzy

whzx

==

==and using the conformal

mapping expressions,

2

cos1

2sin

2

cos1

2cos

φφ

φφ

−=

+=and the half angle formula,


[ ]( ) [ ]( )

[ ]( ) [ ]( )factorfactorh

y

factorfactorh

x

c

c

−−+−=

−++−=

20

220

20

220

2

11

2

1

2

11

2

1

ρρ

ρρwe get,

[ ] [ ] 12

1

24

1

2222

2

2

20

22

2

20

2222

+

−=

+

−=

+

=+

=

factorh

r

h

hfactor

h

r

h

v

h

u

h

vu

h

wccccc

and substituting,


+∆−−+

+∆−=

+∆−++

+∆−=

90.0ln 36.02

1190.0ln 36.0

2

1

90.0ln 36.02

1190.0ln 36.0

2

1

20

2

20

dunoptimize

20

2

20

dunoptimize

B

B

B

B

h

y

B

B

B

B

h

x

c

c

ρρ

ρρ

+∆−−+

+∆−=

+∆−++

+∆−=

25.0ln 14.02

1125.0ln 14.0

2

1

25.0ln 14.02

1125.0ln 14.0

2

1

20

2

20

optimized

20

2

20

optimized

B

B

B

B

h

y

B

B

B

B

h

x

c

c

ρρ

ρρ

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 formatsto summarize the information available in theexpressions. The expressions are graphed for boththe optimized and unoptimized pole to illustratethe advantages of pole edge shaping in order toenhance the field. The quality at various goodfield radii are computed since the beamtypicallyoccupies only a fraction of the aperture due torestrictions of the beampipe.


2.01.51.01.010 -6

10 -5

10 -4

10 -3

10 -2

as a Function of Pole Cutoff

∆ BB

xch

Optimized pole

Unoptimized pole

ρ0 = 0.9

ρ0 = 0.9

ρ0 = 0.8

ρ0 = 0.8

ρ0 = 0.7

ρ0 = 0.7

ρ0 = 0.6ρ0 = 0.6


h

r00 =ρ

10 -210 -310 -410 -510 -60.2

0.3

0.4

0.5

∆ BB

ych

ρ0 = 0.9

ρ0 = 0.9

ρ0 = 0.8

ρ0 = 0.8

ρ0 = 0.7

ρ0 = 0.7

ρ0 = 0.6

ρ0 = 0.6

Uno

ptim

ize

dO

ptim

ize

d


h

r00 =ρ

10 -210 -310 -410 -510 -61.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

Ratio of Peak Field to Poletip Field

∆ BB

BcutoffBpole

ρ0 = 0.9

ρ0 = 0.9

ρ0 = 0.8

ρ0 = 0.8

ρ0 = 0.7

ρ0 = 0.7

ρ0 = 0.6

ρ0 = 0.6 Unoptimized

Optimized


h

r00 =ρ

Typically, 1max >h

z

12

quadrupole

>>

=

h

z

h

w

dipole

1 3

sextupole

>>

=

h

z

h

w

dipole

and

Therefore,

in the mapped space.

However, there is a problem in the mapping of the

quadrupole and sextupole to the dipole space.


• When mapping fromthe quadrupole or sextupole geometriesto the dipole space, the FEMcomputation is initially made inthe original geometry and a vector potential map is obtainedat 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.

( ) ( )θφ ,, refref rawA =

quadrupole

refref h

rw

=

2

sextupole

refref

h

rw

=

2

3

quadrupoleθφ 2=

sextupoleθφ 3=


General guidelines for

Quadrupole/Sextupole Pole Optimization


• It is far easier to visualize the required shape of pole edge bumpson a dipole rather than the bumps on a quadrupole or sextupolepole.

• It is also easier to evaluate the uniformity of a constant fieldfor a dipole rather than the uniformity of the linear orquadratic field distribution for a quadrupole or sextupole.

• Therefore, the pole contour is optimized in the dipole spaceand mapped back into the quadrupole or sextupole space.

• The process of pole optimization is similar to that of analysisin the dipole space.

– Choose a quadrupole pole width which will provide therequired field uniformity at the required pole radius.

• The pole cutoff for the quadrupole can beobtained from the graphs developed earlier using the

dipole pole arguments.

• The sextupole cutoff can be computed by conformalmapping the pole overhang from the dipole space

using

( )cc yx ,

3 2whz =


• Select the theoretical ideal pole contour.

2

2hxy =

3323 hyyx =−

for the quadrupole.

for the sextupole.

• Select a practical coil geometry.

– Expressions for the required excitation and practicalcurrent densities will be developed in a later lecture.

• Select a yoke geometry that will not saturate.

• Run a FEM code in the quadrupole or sextupole space.

• From the solution, edit the vector potential values at a fixedreference radius.


• Map the vector potentials, the good field region and thepole contour.

• Design the pole bump such that the field in the mappedgood field region satisfies the required uniformity.


• Map the optimized dipole pole contour back into thequadrupole (or sextupole) space.

• Reanalyze using the FEM code.


Closure


• The function, zn, is important since it represents differentfield shapes. Moreover, by simple mathematics, thisfunction can be manipulated by taking a root or by takingit to a higher power. The mathematics of manipulationallows for the mapping of one magnet type to another,extending the knowledge of one magnet type to anothermagnet type.

• One can make a significant design effort optimizing onesimple magnet type (the dipole) to the optimization of amuch more difficult magnet type (the quadrupole andsextupole).

• The tools available in FEM codes can be exploited to verifythat the performance of the simple dipole can bereproduced in a higher order field.

Next…


• Perturbations

• Magnet excitation

Documents

2 Multipoles, Conformal Mapping, Pole tip design– Conformal mapping is used to extend the techniques of ensuring dipole field quality to quadrupole field quality. – Conformal mapping