# Eddy currents

```Eddy currents
(in accelerator magnets)
CAS Magnets, Bruges, June 16‐25 2009
Introduction
•
Definition
According to Faraday‘s law a voltage is induced in a conductor loop, if it is
subjected to a time-varying flux.As a result current flows in the conductor, if
there exist a closed path.
‚Eddy currents‘ appear, if extended conducting media are
subjected to time varying fields. They are now distributed in the conducting
media.
•
Effects
• Field delay (Lenz &acute;s Law), field distortion
• Power loss
• Lorentz-forces
– Beneficial in some applications (brakes, dampers, shielding, induction
heating, levitated train etc.)
– Mostly unwanted in accelerator magnets:
to avoid them / to minimize the unwanted effects.
G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009
appropriate design
Outline
• Introduction
– Definition, Effects (desired, undesired)
• Basics
– Maxwell-equations
– Diffusion approach
• Analytical solutions: Examples
• Numerical solutions: Introduction of numerical codes
– Direct application of Maxwell-equations (small perturbation)
•
Eddy currents in accelerator magnets
– Yoke, mechanical structure, resistive coil, beam pipe
• Design principles / Summary
• Appendix (references)
G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009
Basics
Maxwell-equations
Ampere‘s Law
 H  j
B
 E  
t
•Quasistationary approach
•No excess charge
E  0
B  0
Material properties
B  0  r H
G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009
 H  ds   j  dA
A

 E  ds   t A B  dA
 E  dA  0
A
 B  dA  0
A
j E
Lenz‘s law
&quot;the emf induced in an electric circuit always acts in such a direction that the current it drives around a closed circuit produces a magnetic field which opposes the change in magnetic flux.&quot; Lenz‘s Law: Reason for field delay slow diffusion process
G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009
Field diffusion equation
one way of eddy current calculation
vanish
 
 H  j


j  E


    H    j   (  )


H
2
 H  
t
1

 


B
 E  
t
Magnetic diffusivity
•Assumption: σ uniform in space
•Diffusion equation does also exist also for current density j, magnetic Induction B and
magnet Vector Potential A !
•Having solved the differential equation for H, the eddy current density j can be
calculated by Ampere‘s Law and consequently the power loss P.
Analytical solutions: Half-space conductor (1D-approach)(1)
(following closely H.E. Knoepfel ‚Magnetic Fields‘)
H z
2H z
 
2
t
x
Application of external field: Hz(t)
Solution Hz(x,t) = ? Boundary conditions:
Hz(0,t)=0 t&lt;0
Hz(0,t)= Hz(t) t≥0
Hz(x,0)= 0 0&lt;x&lt;∞
1. Step‐function field Hz(t) =H0=constant
2. Transient linear field Hz(t) = H0/t0*t
3. Transient sinusoidal field Hz(t) = H0*sin (ωt)
G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009
Analytical solutions: Half-space conductor (1D-approach) (2)
1. Step‐function field Hz(t) =H0=constant
Diffusion time constant
Hz(x,t)=H0*S(x,t)
With response function S(x,t)
and S(x,0)=0 and S(x,t→∞)=1

 d   1  S ( x, t ) dt
0
1.0
9
0.8
= 110 , 0
H0 (A/m)
0.6
0.4
0.5 cm
1 cm
2 cm
0.2
0.0

x
2 t
Similarity variable
-0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18
t (sec)
Hz ( x , t )  H 0 (1  erf )
Special response function S(x,t)
G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009
Analytical solutions: Half-space conductor (1D-approach) (3)
2. Transient linear field Hz(t) = H0/t0*t
0,007
0,006
Hz (A/m)
0,005
excitation
0.1 cm
0.2 cm
0.3 cm
H0 
2
2
 2 
H z  x, t  
t  1  2 erfc  
e 
t0 



0,004
0,003
0,002

 H ( x, t )  H ( x, t )
0,001
0,000
0,000 0,001 0,002 0,003 0,004 0,005 0,006 0,007
t (sec)
s
z
t
z
H0
H0
t


t
x

  d
 H z ( x, t )
t0
t0
0, if t &gt;&gt;τd
Field lag
H ed (x)
Analytical solutions: Half space conductor (1D-approach) (4)
3. Transient sinusoidal field Hz(t) = H0*sin (ωt)
H z ( x, t )  H ( x, t )  H ( x, t )
s
z
t
z
stationary
H ( x, t )  H 0  e
s
z
transient

x

x
 sin(t  )


2

Harmonic skin depth
G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009
Skin depth as function of frequency and conductivity
Refer: Knoepfel, fig. 4.2‐5
G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009
Analytical solutions: Slab conductor (lamination)
Boundary conditions:for step function field
Hz(&plusmn;d,t)=0 t&lt;0
Hz(&plusmn;d,t)= H0
t≥0 Hz(x,0)= 0 ‐d&lt;x&lt;+d
Refer to Knoepfel 4.2, Table 4.2‐I
Step‐function field (1D)
nx


cos

t

 
2
d
n
H z ( x, t )  H 0 1  4

n 1 e
n 1


n (1) 2


4
n 
n
2
2

d2

n odd
LC iron, 1mm , &micro;r=1000
 1  1m sec
Step‐function field (2D)
nx
my

cos
cos



2a
2b
H z ( x, y, t )  H 0 1  4
2
  f (n.m)
n 1 m 1





 n ,m
e 


t
 n,m
G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009
 2  n 2 m 2 
 4 /    2  2  n,m odd
b 
a

Analytical solutions: Field in the gap of an iron-dominated Cdipole
For this special case:
z
y
H
l
2H 2H



0
x 2
y 2
g   l  t
 r 
G. Brianti et al., CERN SI/Int. DL/71‐3 (1971)
g
b
l
a
x
Case1 : g=0 Standard diffusion equation
l  g
Case 2: Special diffusion equation

r
li  H
1 H
2H 2H

  0

2
2
g t
 1 t
y
x
1
g
1 

 0 l
With this diffusivity κ1 we can use the slab solutions!
G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009
Analytical vs. numerical methods
pros
Analytical Methods
cons
physical
understanding
•simple geometry
(mainly1D/ 2D)
•Homogeneous,
isotropic and linear
materials
•simple excitation
Numerical Methods •complex geometry long computing
(3D)
times
•inhomogeneous,
anisotropic and
nonlinear materials
•complex excitation
G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009
Vector Potential A - the most common way of numerical eddy
current calculation
B   A
Field calculation




 A 
B

 E  
    A      
t
t
 t 



A
J  E  
t



2
  B   A  J


A
2
 A  
Find vector potential
t
Diffusion equation for Vector potential A
Sometimes the current vector potential T is used:
j  T
Refer: MULTIMAG ‐ program for calculating and optimizing magnetic 2D and 3D fields in accelerator magnets (Alexander Kalimov [kalimov@sptu.spb.su])
Widely used numerical codes for the calculation of eddy
current in magnets
•
•
•
Opera (Vector Fields Software, Cobham Techn. Services, Oxford)
www.vectorfields.com
– FEM
– Opera 2d,AC and TR, Opera 3d, ELEKTRA , (TEMPO-thermal and
stress-analysis)
ROXIE (Routine for the Optimization of Magnet X-Sections, Inverse Field
Calculation and Coil End Design) (S. Russenschuck, CERN)
https://espace.cern.ch/roxie/default.aspx
– BEM/FEM
– Optimization of cosnθ-magnets, coil coupling currents only
ANSYS (ANSYS Inc.) www.ansys.com/
– Finite Element Method
– Direct and in-direct coupled analysis (Multiphysics)
• eddy current  heat  rising temperature  change resistivity 
change eddy current
– “This feature is important especially in the region of cryogenic
temperature. Because most of physical parameters depend highly on
temperature in that region” .
G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009
Direct application of maxwell equations
-another way of eddy current calculation
Eddy Current effect handled as small perturbation:
•geometrical dimension &lt;&lt; skin depth (high resistivity!)
•low field ramp rate
• assumptions
•Field Bz only, uniform
•d,h&lt;&lt;l
•d,h&lt;&lt;penetration depth s
(magnetically thin!)
1. Eddy currents in a rectangular thin plate
z
l
y
x
d
B=(0,0, Bz)
h
‐d/2
0 +d/2

j y ( x)  
Bz

x
neglecting the resistance
contribution of the ends,
since 2d&lt;&lt;l
 2
From Ampere‘s law:
H
eddy
z
Bz  2 d 2 
 x  
( x) 
2 
4 
G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009
Top/bottom of a
rectangular beam pipe!
And then for the loss dP
in an area A=hdx:
dP  
l
2
2
  j y ( x)  A    l  j y ( x)  h  dx
A
d 2
lh d 3  2
P  2  dP 
Bz
12 
0
After integration
Same formula as for a thin slab!
8000
SST,  = 5E-7 m
6
Joule heat (watt/m)
Joule heat (watt/m)
8
or
formula
FEM
4
2
CU,  = 5E-10 m
6000
formula
FEM
4000
1 d2  2
P / volume 
Bz
12 
For Copper: No small perturbation
anymore!
2000
0
0
0
1
2
3
4
0
5
1
2
3
4
dB/dt (T/sec)
dB/dt (T/sec)
2. Eddy loss in a long,thin cylinder (radius r, length l,
thickness d (r&gt;&gt;d))
j
5
r cos 


B
r3  2
P  B dl or

(round thin beam pipe)
2
r
P 
B 2
V 2
3. Eddy loss of round plate/disk (radius r, thickness d , r&gt;&gt;d)
G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009
2
r
P 
B 2
V 8
Resistivity ρ (Ohm*m) @300K/4K (typical)
300K
4K
LC steel (3% Silicon)
590  10‐9
440 x 10‐9
Stainless steel
720  10‐9
490  10‐9
Copper
17.4  10‐9
0.156  10‐9
G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009
Avoid copper!
Eddy currents in magnets
Different Magnet types
WF
Eddy currents in all conductive elements, especially in:
•iron yoke (low carbon iron)
•mechanical structure (low carbon iron
•
or stainless steel)
•Coil (Copper, superconductor)
• beam pipe (stainless steel
Iron‐dominated
magnets
Coil dominated cos nθ‐ magnet
G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009
Laminated yoke (no isotropy!!)
Recap: thin slab
P~d2*σ
Def.: Laminated magnets with insulated laminations and low conductivity!!
Packing factor fp = Wi / (Wi + Wa )
Wi ‐ thickness single lamination
Wa ‐ thickness of insulation Conductivity: σz=0 σxy≠0 Rel. permeability of laminations μr
Effective permeability: &micro;z, &micro;xy
FLUX DIRECTION
z
LAMINATION
For laminations tangential to the flux:
μeff = fp &middot; (μr - 1 ) + 1
FLUX DIRECTION
z
LAMINATION
For laminations normal to the flux:
μeff = μr / (μr ‐ fp ∙ (μr ‐ 1))
Laminated yoke : &micro;xy, &micro;z
•
z , xy different for laminated
magnets (highly anisotropic!!)
μr ~ 4080
xy
– xy = r(H)
– z
• fp = 1 : z = r(H) μ ~ 670
• fp &lt; 1: z = 1 / (1 - fp)
z = 15 – 50
μr → 1
r
Electrical Yoke steel 3414 μr ~ 20
z for fp = 0.95
μr ~ 15
μr ~ 10
μr → 1
(Courtesy of E. Fischer)
Laminated yoke : choice of iron
Low carbon silicon steel reduces
Eddy current losses due to higher resistivity
Hysteresis losses due to lower coercivity
but
P. Shcherbakov et al., Design Report SIS 300 6T dipole (2004) Practical limit
G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009
Iron losses (steel supplier)
Note: Steel suppliers give typically total losses at 50 Hz
Total losses = eddy losses (~ν2)
+ hysteresis losses (~ν)
+ anomalous losses (~ν1.5)
Eddy losses:
 
PW
 2 2
Bpd
3 
m
6
2
2
ν‐frequency (Hz)
d ‐lamination thickness (m)
ρ‐resistivity (Ohm*m)
Bp‐ Induction amplitude (T)
Hysteresis losses: from measurements wit a permeameter
Anomalous losses = rest
G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009
P. Fabbriccatore, et al. Technical design Report SIS 300 4.5T model dipole Yoke design of pulsed magnets
• 2D‐design (ideal)
No Bz, only Bx, By
– Appropriate lamination thickness d (practical limit 0.3 mm)
– Low steel conductivity
– Low coercivity (to reduce the hysteresis
losses)
• 3D‐design exist Bz
eddy currents in the lamination sheet surface
– Yoke end region
– Areas with low packing factor
z
G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009
z
Yoke end region
Large Bz components (up to 2.5T depending on the magnet)
Source of eddy currents near the magnet end
GSI: SIS18 dipole
Courtesy of F. Klos
Results
20
t = 0.3 s
2
Current density, A/cm
A.Kalimov, et al., IEEE Transactions on Applied Superconductivity., vol.12, No.1 pp. 98‐101, 2002 (MT17)
25
t = 1.1 s
15
t = 1.5 s
t = 2.0 s
10
5
0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
z, m
Eddy current lines on the yoke backside surface. Yoke end
. Current density distribution on the pole surface along the central line of the magnet
En d of ramp
0.05%
Field integral: difference between static and
Aperture field induced by the eddy currents dynamic value. field integral max: 48000 Gs.m
SIS100 sc dipole model – laminated yoke
&micro;r(X,Y) ↔ original B(H) curve
&micro;r(Z) = 15
eddy current power in laminated yoke (S. Koch, H. de Gersem, T. Weiland ( TU Darmstadt))
Eddy current power in the yoke along the yoke
Vectors of BZ in yoke vs time (&micro;z=25)
G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009
R. Kurnyshov et al., Report on FE‐R&amp;D, Contract No. 5, GSI, October 2008
Vectors of eddy current density in yoke vs time (&micro;z=25)
G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009
R. Kurnyshov et al., Report on FE‐R&amp;D, Contract No. 5, GSI, Field relaxation at different longitudinal positions
SIS 100 dipole
SIS100
Z=0 (Center of the magnet)
Yoke
-4
1.2x10
0.689 m
-4
1.0x10
-4
3.0x10
1.17 m
-5
-4
2.0x10
-5
6.0x10
-4
-5
4.0x10
1.0x10
By (T)
By (T)
8.0x10
&gt;0, max. 1G
-5
2.0x10
Around 0, 0.0
-4
-1.0x10
0.00
By (T)
SIS100 dipole
•Linear ramp, up with 4 T/s,
•1.9 T
•All curves start at t=0 at the
end of the ramp
0.05
0.10
0.15
0.20
0.25
-4
-2.0x10
time (sec)
0.00
0.05
0.10
1.4x10
0.20
0.25
-3
-1.0x10
1.01 m
-3
1.2x10
-3
-2.0x10
-3
-3
-3.0x10
1.0x10
-4
8.0x10
&gt;0, max. 14 G
-4
6.0x10
1.319 m
-3
-4.0x10
-3
-5.0x10
-3
-6.0x10
-4
4.0x10
&lt;0, max.60 G -3
-7.0x10
-4
2.0x10
0.00
0.0
S. Y. Shim, to be published
0.15
time (sec)
-3
By (T)
0.0
0.05
0.10
0.15
time (sec)
0.00
0.05
0.10
0.15
time (sec)
0.20
0.25
0.20
0.25
Transient field behaviour after a linear ramp
y
x
Z
CNAO Scanning Dipole Magnet
max. center field: 0.3 T
500 T/s
Lamination thickness: 0.3 mm
z= 0 magnet center,
z=22.0 cm: end of yoke
Fitting function

Bt   B  1  e  t /

Field delay
Diffusion time constant
Magnet center
By_max  0.7 % of maximum operation field
Time constant  some milliseconds
Remark: In a medical accelerator like CNAO the beam energy is varied in small time steps less then the diffusion time constant!
Courtesy of S. Y. Shim
SIS 300 Dipole- eddy currents(direction and current density
in the magnet ends
4.5 T, 1T/s
3.0 T, 1T/s
M. Sorbi, et al., “Electromagnetic Design of the Coil‐Ends for the FAIR SIS300 Model Dipole,” in Proc. ASC’08, Chicago, 2008
G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009
Variation of the packing factor along
the magnet
(Synchrotron dipole of the HIT-facility Heidelberg)
•Variation of the DC‐field
•Field reduction by eddy currents (induced by local Bz components) in the AC‐case
Courtesy of F. Klos
Eddy currents in mechanical structure
–
–
–
–
Brackets
Endplates
Collar pins, Collar keys, Rods
Shield, shell,
Try to avoid closed flux loops ! (for example by welding seams at the pole)
LHC dipole cold mass
Example: eddy curents in the copper shield of a sc dipole
R&amp;D magnet GSI001
Bmod at 4T nominal field
Power density (W/m3) in the copper shield (resistivity: 2.5 nm) at 4T nominal field and a ramp rate of 4T/s.
Maximum power density: 270 kW/m3
Courtesy of H. Leibrock
G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009
Eddy currents in Pins, Rods and Keys (SIS 300 dipole)
If possible: insulate them
to avoid closed flux looops!
Flux‐loop minimized!!
G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009
M. Sorbi et al. Technical Design Report SIS 300 4.5T model dipole Eddy currents in resistive coil
SIS 18 dipole
A. Asner et al., SI/Int.DL/69-2 9.6.1969
(Booster Bending Magnet)
Application of Biot‐Savart gives the eddy current contribution ito the field n the magnet gap
This case:
Eddy currents improve field quality!! G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009
Elliptical beam pipe (SIS 100 dipole)
simulation of the eddy current loss density in the beam pipe
Thickness 0.3 mm
Average loss: 4.9 W/m (0-2T, 4T/s)
For the use as cryopump: need to be cooled!)
R. Kurnyshov et al., Report on FE‐R&amp;D, Contract No. 5, GSI, October 2008
Elliptical beam pipe (SIS 100 dipole)
- complete 3D model with ribs and cooling pipe
Designed to avoid closed flux loops
Current density (A/m2), central part
Cycle:0 ‐2T, 4 T/s
Average
loss (W/m):
4.9 pipe only
8.1 with tubes
8.7 with tubes and ribs
Current density (A/m2), end part
Courtesy of S. Y. Shim
Summary: Pulsed magnet design principles
(to minimize eddy current effects)
• Insulated laminations
• Choice of iron
• Appropriate magnet design
–
–
–
–
–
Avoid saturation (&micro;r&gt;&gt;1)
Rogowski‐profile of the magnet pole ends
Slits in the end laminations
Non‐conductive material at the magnet ends
‚long‘ magnets (also from the eddy current aspect!)
• Appropriate design of the mechanical structure
– Choice of materials (non‐conductive wherever possible)
– Avoid ‚bulky‘ components
– Avoid magnetic ‚flux loops‘
• Field Control (‚B‐Train‘)
G.Moritz, 'Eddy Currents', CAS Bruges, June 16 ‐ 25 2009
References
•Books
•Heinz E. Knoepfel, ‘Magnetic Fields’, John Wiley and
Sons, INC. , New York…., 2000
•Jack T. Tanabe, Iron dominated electromagnets:
design, fabrication, assembly and measurements,
WORLD Scientific 2005
•Y. Iwasa, ‘Case studies in superconducting magnets’,
Plenum Press, New York and London, 1994
•Reviews
•K. Halbach, ‘Some eddy current effects in solid core
magnets’, Nuclear Instruments and Methods, 107
(1973), 529-540
•E.E. Kriezis et al., 'Eddy Currents: Theory and
Applications', Proceedings of the IEEE, Vol. 80, NO. 10.
October 1992, p.1599-1589
Acknowledgement:
I am greatly indebted to S.Y. Shim for his help
during the preparation of this talk.
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