5 Stored Energy, Magnetic Forces and Dynamic Effectsuspas.fnal.gov/materials/16Austin/lecture05.pdfMagnetic Forces US Particle Accelerator School –Austin, TX –Winter 2016 3 •

Mauricio Lopes – FNAL

5

Stored Energy, Magnetic Forces

and Dynamic Effects

Outlook

US Particle Accelerator School – Austin, TX – Winter 2016 2

• Magnetic Forces

• Stored Energy and Inductance

• Fringe Fields

• End Chamfering

• Eddy Currents

Magnetic Forces


• We use the figure to illustrate a simple example.

w a

x

B0

j = uniform current density

FF

Magnetic Force on a Conductor


We use the expression, dvBjF ×= ∫Integrated over the volume of the conductor.

hLdxdvw

xBB

w

B

wh

NIj=

hBNI

==

==

0

0

0

0

0

µµL is the length into the paper.

0

20

2

02

20

002

20

00

0

22

µµµ

µhLBw

w

hLBxdx

w

hLB

hLdxw

xB

w

BB dvjF

w

===

==

∫

∫∫

0

20

2

µB

hL

F

Area

Forcepressurerepulsive ===

Magnetic Force on a Pole


We use the expression, dvHF ⋅= ∫ ρ

h

Bρ

0=where the magnetic charge density ρ is given by,

The force is in the same direction as the H vector and is attractive.

dv=aLdyh

yBH

0

0

µ=

0

20

2

02

20

002

20

0

00

2=

2==

µµµµρ aLBh

h

aLBydy

h

aLBaLdy

h

yB

h

BdvHF

h

∫∫∫ ==

0

20

2

µB

aL

F

Area

Forcepressureattractive ===

Pressure


The magnitudes of the repulsive pressure for the current and the attractive

pressure at the pole are identical. In general, the pressures parallel to the

field lines are attractive and the forces normal to the field lines are

repulsive. The pressure is proportional to the flux density squared.

0

20

2µB

pressure =

Example


Let us perform a calculation for a flux density of 5 kG = 0.5T.

27

2

0

20 472,99

1042

5.0

2 m

NewtonsBpressure =

××== −πµ

atmosphere 143.14472,99 220.5T @ ≈==

in

lb

m

Newtonpressure f

Magnet Stored Energy


• The magnet stored energy is given by; ∫= HBdvU2

1

the volume integral of the product of H and B.

Consider a window frame dipole field (illustrated earlier) with uniform field in

the space between the coil. If we ignore the field in the coil and in the iron,

volumemagneticB

U 2 0

20 ×=

µ

Magnet Inductance and Ramping Voltage


Inductance is given by, 2

2

I

UL =

tI

URI

t

I

I

URI

dt

dILRIV

∆+=

∆×+=+= 22

2

In fast ramped magnets, the resistive term is small.

volumeB

tItI

UV ××

∆=

∆≈

0

20

2

22

µvolume=ahL

N

hBI

0

0

µ=

t

NaLBahL

B

tN

hBV

∆=××

∆≈ 0

0

20

0

0

1

µµ

Substituting,

Units and Design Options


The units are,

VoltVoltm

m

WebersTm

t

NaLBV =−=×==

∆≈

sec

sec

secsec

2

2

20

t

NaLBV

∆≈ 0

Given the field = B0, pole width = a, Magnet Length = L and ramp time, ∆ t, the

only design option available for changing the voltage is the number of turns, N.

Other Magnet Geometries


Normally, the stored energy in other magnets (ie. H dipoles, quadrupoles

and sextupoles) is not as easily computed. However, for more complex

geometries, two dimensional magnetostatic codes will compute the

stored energy per unit length of magnet.

Effective Length


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-300 -250 -200 -150 -100 -50 0 50 100 150 200 250 300

Bn

(a

u)

Z (mm)

Ideal

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-300 -250 -200 -150 -100 -50 0 50 100 150 200 250 300

Bn

(a

u)

Z (mm)

Ideal Real

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-300 -250 -200 -150 -100 -50 0 50 100 150 200 250 300

Bn

(a

u)

Z (mm)

Ideal

Real

Effective length

�� =� ��(�)��

��

��(� = 0)

z

Fringe Fields and Effective Lengths

• Often, canonical rules of thumb are adopted in order to estimate

the effective length of magnets.

– Dipole fringe field length = 1 half gap at each end

– Quadrupole fringe field length = 1/2 pole radius at each end.

– Sextupole fringe field length = 1/3 pole radius at each end.


Three Dimensional Fringe Fields

The shape of the three dimensional fringe field

contributes to the integrated multipole error of a

magnet.


• Dipole Fringe Field

– Typically, the fringe

field is longer at the

center of the magnet

and drops off near the

edges.

– This distribution is

approximately

quadratic and the

integrated multipole

field looks like a

sextupole field.

h

h = "canonical" fringe field length

edge of pole

Fringe field length is typically longer in the center for an "unchamfered" pole end.

z

B

z

x


Virtual Field Boundary


beam

Magnet

virtual field boundary

ρβtan1 −=

Fx

ββ

• The photograph shows a removable insert with a machined chamfer installed on the SPEAR3 prototype gradient magnet.

• The shape of the chamfer depth was determined empirically and was approximately parabolic. It was designed to reduce the integrated sextupole field.

• The chamfer shape was machined onto the end packs of subsequently manufactured production magnets.


Quadrupole 3-Dimensional Fringe Field

• The quadrupole “gap” is largest at its center. The gap decreases as the distance from its center (g1>g2>g3>g4>g5>g6). Since the fringe field is roughly proportional to the magnet gap, it is longest near the magnet pole center. g1 g2

g3 g4 g5 g6x

y


Quadrupole Chamfer

• The quadrupole pole chamfer is a straight angled cut, which shortens the pole at its center. The angle is cut such that the pole is longer near its edge.

• Again, this cut was determined empirically by trial an error, minimizing the n=6 integrated multipole error.


-10

-8

-6

-4

-2

0

2

4

0 3 4 5 6

B6B10B14

end chamfer length (mm)

R = 25 mmB

n/B

2. 104

Harmonics as function of the end

chamfer length


Ideal end chamfer length as function

of quadrupole length


R=20mm

3.0

3.5

4.0

4.5

5.0

5.5

6.0

100 150 200 250 300 350 400 450 500 550 600

Quadrupole length (mm)

Idea

l en

d c

ham

fer

len

gth

(m

m)

Effective length as function of the end

chamfer length


Q530

563.0563.5564.0564.5565.0565.5566.0566.5567.0

0 1 2 3 4 5 6

Chamfer length (mm)

Eff

ecti

ve l

eng

th (

mm

)

Multipoles as function of the coil

excitation


Dynamic effects


dt

BE

∂−=×∇

h

w

g th

w

g tk B

x

y

jB .̂. tieBo ω=

jBj ˆˆ &−=

∂∂−

∂∂

x

E

z

E zx

oz Bi

x

E ω=∂

∂

xBidxBiEz oo ωω == ∫EJ σ=

xBiEJ ozz ωσσ ==

Dynamic effects


o

e

Feo

gBdl

Bdl

BI

dlHI

µµµ=+=

=

∫∫

∫

..

.

Be

stands for the magnetic

field due to the eddy currents

w1

2

3g tk

31

21

321

2 III

II

IIII

+==

++=

∫∫= JdxdyI i

( )2

0

1 2

1wtBixdxBitI

w

kook ∫ == ωσωσ

( )23 2

1k

w

tw

oo twtgBixdxBigIk

−== ∫−

ωσωσ

−+=22

2 gtwgw

g

tBiB kkoo

e

σωµ

Be (x=0) (theoretical) 0.0129 (T)

Be (x=0) (FEMM) 0.0127 (T)

Bo(T) 0.145

frequency (Hz) 100

tk (mm) 3

g (mm) 25w (mm) 28σ (MS/m) 1.334

Real-case example


Real-case example


thickness = 1 mm

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

-30 -20 -10 0 10 20 30x (mm)

By

(T)

400 Hz

350 Hz

300 Hz

250 Hz

200 Hz

150 Hz

100 Hz

90 Hz

80 Hz

70 Hz

60 Hz

50 Hz

40 Hz

30 Hz

20 Hz

10 Hz

thickness = 2 mm

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

-30 -20 -10 0 10 20 30x (mm)

By

(T)

400 Hz

350 Hz

300 Hz

250 Hz

200 Hz

150 Hz

100 Hz

90 Hz

80 Hz

70 Hz

60 Hz

50 Hz

40 Hz

30 Hz

20 Hz

10 Hz

thickness = 3 mm

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

-30 -20 -10 0 10 20 30x (mm)

By

(T)

400 Hz

350 Hz

300 Hz

250 Hz

200 Hz

150 Hz

100 Hz

90 Hz

80 Hz

70 Hz

60 Hz

50 Hz

40 Hz

30 Hz

20 Hz

10 Hz

thickness = 4 mm

-0.070000

-0.060000

-0.050000

-0.040000

-0.030000

-0.020000

-0.010000

0.000000

-30 -20 -10 0 10 20 30x (mm)

By

(T)

400 Hz

350 Hz

300 Hz

250 Hz

200 Hz

150 Hz

100 Hz

90 Hz

80 Hz

70 Hz

60 Hz

50 Hz

40 Hz

30 Hz

20 Hz

10 Hz

x = 20 mm

0.00

0.20

0.40

0.60

0.80

1.00

0 100 200 300 400

f (Hz)

By(

f)/B

y(0

Hz) 4 mm

3 mm

2 mm

1 mm


x = 0 mm

0.00

0.20

0.40

0.60

0.80

1.00

0 100 200 300 400

f (Hz)

By(

f)/B

y(0

Hz) 4 mm

3 mm

2 mm

1 mm

x = -20 mm

0.00

0.20

0.40

0.60

0.80

1.00

0 100 200 300 400

f (Hz)

By(

f)/B

y(0

Hz) 4 mm

3 mm

2 mm

1 mm

Real-case example

Phase delay


σωµµδ

or

2=

[ ]radians δ

φ kt=

Attenuation due to lamination on the

material


)2/tan(2

kdkdo

=ψψ

ωµσik −=2

µ = 2200.µo

σ = 9.93 MS/m

Example

Bo

Bmin

Bmean

d2

minBBmean =

Real-case example summary


Vacuum

chamber

thickness

2 mm 3 mm

Core

lamination

thickness

0.60 0.54

0.5 mm 0.67 0.40 0.36

1.0 mm 0.46 0.27 0.25

@400 Hz

Induced sextupolar field due to eddy

currents


xBEJ zz&σσ ==

h

w

g th

w

g t B

x

y

2xBtI k&σ=o

BhI

µ=

2)( xt

B

h

txB k

o ∂∂= σµ

t

B

Bh

tFk k

o ∂∂=

ρσµ 1

22

( ) dxxw

gxF

2/11

0

22

2 12∫

−

+=

F=1 when g<<w, and F=2 for a circular chamber

)()()( tfBBBtB iei −+=

The sextupole strength (or moment) is defined by

( ))cos(12

1)( ttf ω−=

Example – Alba booster dipole


ρ 11.4592 (m)

µo

4π10-7 (H/m)

σ 1.344 (MS/m)

Freq. 3 (Hz)

Be

0.029 (T)

Bi

0.873 (T)

20 mm

30 mm

17 mm

50 mm

VC1

VC2

F

VC1 1.5739

VC2 1.2141

Summary


This chapter showed a collection of “loose-ends” calculations:

• Magnetic Forces

• Stored Energy and Inductance

• Fringe Fields

• End Chamfering

• Eddy Currents

Those effects have an impact in the magnet design but also need to be

taken into consideration into the magnet fabrication, power supply

design, vacuum chamber design and beam optics.

Next…


Magnetic measurements

Documents

5 Stored Energy, Magnetic Forces and Dynamic Effectsuspas.fnal.gov/materials/16Austin/lecture05.pdfMagnetic Forces US Particle Accelerator School –Austin, TX –Winter 2016 3 •