dynamic effects in superconducting magnets · dynamic effects in superconducting magnets animesh...
TRANSCRIPT
Dynamic Effects in Superconducting Magnets
Animesh JainBrookhaven National Laboratory
Upton, New York 11973-5000, USA
US Particle Accelerator School on Superconducting Accelerator Magnets
Phoenix, Arizona, January 16-20, 2006
USPAS, Phoenix, January 16-20, 2006 Animesh Jain, BNL1
Introduction• Any change in the field near a superconductor
induces eddy currents. These eddy currents are of a persistent nature due to zero resistance in the superconductor.
• Such eddy currents produce field distortions (harmonics) depending on several factors, such as superconductor properties, ramp direction, ramp rate, etc.
• An understanding of these effects is important in the measurements of superconducting magnets.
USPAS, Phoenix, January 16-20, 2006 Animesh Jain, BNL2
Hysteresis in Harmonics• The dependence of eddy currents on the ramp
direction produces a hysteresis in the harmonics.
• Generally, the eddy currents have symmetries similar to the main field harmonic. Thus, only the allowed harmonics are commonly affected.
• For multilayer magnets with several multipoles, (e.g. corrector packages) the external field may have a symmetry different from the main field. In this case, hysteresis may be seen in someunallowed harmonics also.
USPAS, Phoenix, January 16-20, 2006 Animesh Jain, BNL3
Hysteresis in an Allowed TermIn a dipole magnet, a large negativesextupole is produced at low fields on the UP ramp. This changes to a large positivesextupole on the DOWN ramp.-20
-15
-10
-5
0
5
10
15
20
250
1000
2000
3000
4000
5000
6000
7000
Current (A)
Nor
mal
Sex
tupo
le (
unit
s at
25
mm
)
DRG60123.101 UP RAMP
DRG60124.101 DN RAMP
USPAS, Phoenix, January 16-20, 2006 Animesh Jain, BNL4
Hysteresis in an Unallowed TermCross section of a quadrupole magnet built at BNL for HERA, DESY. The magnet has concentric layers of normal and skew dipole, normal and skewquadrupole and normalsextupole,
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
X (cm)
Y (
cm)
Field seen by conductors in a given layer does not necessarily have the symmetry of that layer
USPAS, Phoenix, January 16-20, 2006 Animesh Jain, BNL5
Hysteresis in an Unallowed Term
Hysteresis in theoctupole term in thequadrupole layer of a multi-layer magnet consisting of several concentric layers of normal and skewmultipole magnets.
-1.2
-0.8
-600 60
0
-400
-200 20
00
400
-0.4
0.0
0.4
0.8
1.2
1.6
Current (A)
Nor
mal
Oct
upol
e (u
nit
at 3
1 m
m)
USPAS, Phoenix, January 16-20, 2006 Animesh Jain, BNL6
Critical State Model
• The hysteresis in certain harmonics resulting from the ramp direction dependence of eddy currents can be understood qualitatively by a simple model known as the Critical State Model.
• In this model, it is assumed that eddy currents, with a density equal to the critical current density, Jc, are induced in the superconductor to counteract any change in the external field.
USPAS, Phoenix, January 16-20, 2006 Animesh Jain, BNL7
Superconductor in an External Field
For low external fields (less than the Penetration Field), persistent currents are localized within a thickness, t, sufficient to shield the interior from the external field.
Bext +Jc
–Jc
X
Bext< Bp
By
t
x = +ax = –a
Bext
=
c
ext
JB
t0
µ
Superc
onducto
r
in a
n e
xte
rnal
magnetic fie
ld
Bp = µ0Jc a
USPAS, Phoenix, January 16-20, 2006 Animesh Jain, BNL8
Superconductor in an External Field
At a certain field (the Penetration Field), persistent currents span the entire volume of the superconductor.
The field at the center is still zero.
Bext
+Jc–Jc
Bext= Bp
By
x = +ax = –a
Xa
Superc
onducto
r
in a
n e
xte
rnal
magnetic fie
ld
Bp = µ0Jc a
Bext = Bp
USPAS, Phoenix, January 16-20, 2006 Animesh Jain, BNL9
Superconductor in an External Field
At external fields higher than the Penetration Field, persistent currents span the entire volume of the superconductor, but the interior is not completely shielded.
Bext
+Jc–Jc
Bext >Bp
By
x = +ax = –a
Xa
Superc
onducto
r
in an exte
rnal
magnetic fie
ld
Bp = µ0Jc a
Bext > Bp
USPAS, Phoenix, January 16-20, 2006 Animesh Jain, BNL10
Filament Magnetization: 1st Up RampWhen the external field is increased from zero to a small value, Ba, shielding currents are set up such that the field inside is zero. The boundary of shielding currents may be approximated by an ellipse.
/20 ; ;2
/)(sincos
1
0 π≤α≤≤
πµ=
αα
α−=
pac
p
p
a
BBaJ
B
B
B
dI = Jc rdr.dφ
+dI–dI
–dI +dI
φ
rr0
a
+ Jc– Jcb X
Y a
αφφ
cos/;sincos 222
0 ==+
= abee
br
USPAS, Phoenix, January 16-20, 2006 Animesh Jain, BNL11
Magnetic Moment: 1st Up Ramp
The persistent currents produce a magnetic moment. The Magnetization, or the magnetic moment per unit volume, can be calculated by integrating over elemental loops.
−µ
π−=⌡
⌠⌡⌠
πµ−=
−
2
2
020 1
342
2
1
a
baJdxxdy
a
JM c
x
x
a
a
c aJM c0peak 34
µ
π=
Magnetization is proportional to the critical current density, and the filament diameter.
dI= Jc dx.dy
+dI–dIa
+Jc– Jc b X
Y a
y
x–x x1 x2
)/(1 22 ay–1 bx =
222 yax –=
USPAS, Phoenix, January 16-20, 2006 Animesh Jain, BNL12
Full Magnetization: Up Ramp
aJMaJ
B cc
p 0peak0
34
; 2 µ
π=
πµ=
At fields above the penetration field, the filament is fully magnetized. In this simple model, the magnetization does not change as the field is increased further.
1peak
appliedinside
applied
−=
−=
≥
M
M
BBB
BB
p
pExample:
Jc = 2 x 104 A/mm2
dia. = 2a = 6 µm
Bp = 0.048 Tesla– Jc + Jc
USPAS, Phoenix, January 16-20, 2006 Animesh Jain, BNL13
Magnetization on the Down Ramp
When the field is reduced from the maximum field by an amount B0, new shielding currents are induced, which are opposite in sign to the earlier currents.
The geometry of the new shielding currents is such that a field of +B0 is produced inside the filament.
– Jc– Jc + Jc+ Jc
USPAS, Phoenix, January 16-20, 2006 Animesh Jain, BNL14
Magnetization on the Down Ramp
New distribution at Ba
= +
Distribution at Bmax
New shielding currents(Density = 2xJc)
ab
aJM
aJB
c
c
/cos
sin38
/)(sincos
14
20
00
=α
αµ
π=
αα
α−
πµ=
aJM
MM
c0peak
peak
34
1/
µ
π=
−=Ba = Bmax – B0
= +
– Jc– Jc + Jc+ Jc – Jc + Jc –2Jc+2Jc
USPAS, Phoenix, January 16-20, 2006 Animesh Jain, BNL15
Full Magnetization: Down Ramp
As the field is reduced from a certain maximum value, Bmax, the superconducting filaments are fully magnetized again in the opposite direction. This happens at a field of Bmax – 2Bp. This continues until the minimum field, Bmin, as long as the direction of field change is not reversed.
1
2
peak
max
+=
−≤
M
M
BBB pa
Starting from anunmagnetized state, the filament ends up with a magnetization of +Mpeak after a cycle to Bmax and back.
+dI
– Jc+ Jc
USPAS, Phoenix, January 16-20, 2006 Animesh Jain, BNL16
Magnetization on the 2nd Up RampIf the field is increased again after reaching Bmin, new shielding currents are induced, which are opposite in sign to the earlier currents.
If the applied field isBmin+B0, the geometry of the new shielding currents is such that a field of –B0is produced inside the filament.
– Jc– Jc + Jc+ Jc
USPAS, Phoenix, January 16-20, 2006 Animesh Jain, BNL17
Magnetization on the 2nd Up Ramp
New distribution at Ba > Bmin
= +
Distribution at Bmin
New shielding currents(Density = 2xJc)
ab
aJM
aJB
c
c
/cos
sin38
/)(sincos
14
20
00
=α
αµ
π−=
αα
α−
πµ=
aJM
MM
c0peak
peak
34
1/
µ
π=
+=Ba = Bmin + B0
= +
– Jc– Jc + Jc+ Jc
+dI
– Jc+ Jc –2Jc +2Jc
USPAS, Phoenix, January 16-20, 2006 Animesh Jain, BNL18
Magnetization Vs. Applied Field
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0 0.5 1 1.5 2 2.5 3 3.5Normalized Applied Field (B a /B p )
Nor
mal
ized
Mag
neti
zati
on (
M/ |
Mpe
ak|)
Initial Up RampDn Ramp2nd Up Ramp
aJMaJB ccp
πµ
=
πµ
=3
4 ;
2 0peak
0
USPAS, Phoenix, January 16-20, 2006 Animesh Jain, BNL19
Spatial Variation of Magnetization
Regions of magnet coil with field less than 1 Tesla in a RHIC arc dipole at its maximum operating field of 3.45 T. Some regions inthe midplane turns can be below the penetration field.
Even at the maximum field, some regions of coil may still be well below the penetration field.Reversing current ramp direction can create a considerably complex shielding current pattern in such regions.
USPAS, Phoenix, January 16-20, 2006 Animesh Jain, BNL20
Calculating Harmonics from Persistent Currents
• Divide the entire magnet coil cross section into a suitable number of small segments. For example, each turn can be represented by N strands, uniformly distributed, where N is approximately equal to the actual number of strands in the cable.
• Calculate the local field (magnitude and direction) at each coil segment due to the transport current in the magnet.
• Calculate the critical current density, Jc, at each coil segment, based on experimental data and/or empirical parameterization.
USPAS, Phoenix, January 16-20, 2006 Animesh Jain, BNL21
Harmonics from Persistent Currents (Contd.)
• Calculate the magnetic moment of each segment, based on excitation history, copper to superconductor ratio, and filament diameter. The magnetization of each segment is also scaled by a factor , where Isegment is the transport current carried by the segment and is the critical current for that segment. A simple model, such as one based on elliptical boundaries, can be used, although more complex algorithms have also been used.
• Calculate the harmonics produced at the center of the magnet by the magnetic moment of each segment.
( )segmentcsegment II−1
segmentcI
USPAS, Phoenix, January 16-20, 2006 Animesh Jain, BNL22
Field due to a Magnetic MomentComplex magnetic moment, m, is:
θ–I
+I
m = µ0Id*
my= m cosθ
mx= – m sinθ d
a
zP
X
Y
*d
m
IiId
imm xy
00 )sin(cos µ=θ−θµ=
+=
))((2
112
)(
0
0
0
0
dazaz
d
dazazzB
d
d
−−−−
πµ=
−−−
−
πµ=
→
→
ILt
ILt
Complex field at point P is:
2)(2)(
az
mzB
−π−= *
USPAS, Phoenix, January 16-20, 2006 Animesh Jain, BNL23
Harmonics due to a Magnetic Moment
1
12
2
2
2
12
)(
−∞
=
−
π
−=
−
π
−=
∑n
n
na
z
a
m
a
z
a
mzB
*
*
[ ]φ+−
π
−=
+−
)1(exp2
1
2 nia
R
a
n
iABn
ref
nn
*m
Field at the point P is given by:
Harmonics are thus given by:
φ
mm=my+imx
a
zP
X
Y
θ
USPAS, Phoenix, January 16-20, 2006 Animesh Jain, BNL24
Magnetic Moment in Iron Yoke
–I
+Im = µ0 Id*
da
X
Yd'
+I'
–I'
Ryoke
a'
*ma
m
2
1
1
+
−−=′ yokeR
µµ
m
m'
*
2
aa' yokeR
=
–I
+Im = µ0 Id*
da
X
Yd'
+I'
–I'
Ryoke
a'
*ma
m
2
1
1
+
−−=′ yokeR
µµ
m
m'
*
2
aa' yokeR
=
{ } { }
φ−−
+µ−µ−φ+−
×
π
−=+−
)1(exp11
)1(exp
22
1
2
niR
ani
a
R
a
niAB
n
yoke
n
refnn
mm *
Main and Image moments give different φdependences.
Main and Image moments give different φdependences.
USPAS, Phoenix, January 16-20, 2006 Animesh Jain, BNL25
-21.0
-20.5
-20.0
-19.5
-19.0
-18.5
-18.0
-17.5
0
200
400
600
800
1000
1200
1400
1600
1800
2000
2200
2400
2600
2800
3000
Measured Values
Two Exponentials Fit
Nor
mal
Sex
tupo
le (
unit
s at
25
mm
)
Time since reaching 300 A (s)
DMP402 (Left)
Time Decay of Harmonics
While sitting at a constant field, e.g. at injection, the persistent current induced harmonics decay with time.
This effect has to be considered while measuring superconducting magnets.
USPAS, Phoenix, January 16-20, 2006 Animesh Jain, BNL26
• Flux Creep: is caused by thermal activation and Lorentz forces due to transport current, effectively reducing Jc with time. This is temperature dependent, and produces a logarithmic decay.
• Boundary Induced Coupling Currents: Different strands in a cable may carry different currents (e.g. due to spatial variations in time derivative of the field). These coupling currents produce periodic axial variation of the field. As the variations decay, regions which were at a higher field jump to the down ramp branch. Can vary magnet to magnet.
Time Decay of Harmonics
USPAS, Phoenix, January 16-20, 2006 Animesh Jain, BNL27
Snapback of Harmonics
-23.5
-23.0
-22.5
-22.0
-21.5
-21.0
-20.5
-20.0
-19.5
-19.0
-18.5
280
290
300
310
320
330
340
350
360
370
380
Current (A)
Stop at 304 A & Ramp Again
Continuous Ramp at 1 A/s
Time decayat 304 A
Snapback onramping again
at 1 A/s
Nor
mal
Sex
tupo
le (
unit
s at
25
mm
)
When the field is ramped up again, the decay is recovered very quickly, and the harmonics follow the usual current dependence.
USPAS, Phoenix, January 16-20, 2006 Animesh Jain, BNL28
Ramp Rate Effects• The field quality is typically measured with
rotating coils under DC excitation.
• If the magnet is being ramped at a high ramp rate, eddy currents can cause significant distortion of the field, and thus generate harmonics.
• The extent of distortion depends on the ramp rate and inter-strand resistance in multi-strand cables.
USPAS, Phoenix, January 16-20, 2006 Animesh Jain, BNL29
Ramp rate dependence of the allowed 12-pole term measured in a 80 mm aperture quadrupole for RHIC.
-10.0
-9.5
-9.0
-8.5
-8.0
-7.5
-7.0
-6.5
600
800
1000
1200
1400
1600
1800
2000
2200
2400
2600
2800
3000
Current (A)
Nor
mal
Dod
ecap
ole
(un
it a
t 25
mm
)
0 A/s
20 A/s
60 A/s
70 A/s
USPAS, Phoenix, January 16-20, 2006 Animesh Jain, BNL30
2.2
2.4
2.6
2.8
3.0
3.2
3.4
3.6
3.8
0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2Dipole Field (T)
[B3
(Dn)
- B
3(U
p)] (
Gau
ss a
t 25
mm
)
DC
10 A/s20 A/s
40 A/s (0.028 T/s)
5 A/s
D1L103: Ramp Rate & Magnetization Effects
Ramp rate effects are best seen in the width of the hysteresis curve.
USPAS, Phoenix, January 16-20, 2006 Animesh Jain, BNL31
Measurements of Dynamic Effects• The harmonics decay rather rapidly in the
beginning.
• The snapback occurs within a few seconds.
• The primary field is also changing with time, particularly when the current is continuously being ramped at a high rate.
• All these factors present unique challenges in the measurement of dynamic effects. The key measurement issue is time resolution.
USPAS, Phoenix, January 16-20, 2006 Animesh Jain, BNL32
Techniques for “Fast” Measurements• Rotate a harmonic coil as fast as practical to
improve time resolution. This allows measurements with ~ 1 s typical resolution.(OK for time decay and snapback studies)
• The same technique can be used to measure harmonics during very slow current ramps.
• The technique can be extended to somewhat higher ramp rates by refining the analysis.
• This method is impractical for very high ramp rates, or very short time scales.
USPAS, Phoenix, January 16-20, 2006 Animesh Jain, BNL33
Techniques for “Fast” Measurements• One could use non-rotating probes to overcome
the time resolution problem.
• Without a rotating probe, one needs a multiple probe system to get harmonic information.
• A system of 3 Hall probes, for example, can measure the sextupole component. Similarly, NMR arrays have been built with many probes.
• Intercalibration of individual probes and non-linear behavior are some of the problems that must be addressed in such techniques.
USPAS, Phoenix, January 16-20, 2006 Animesh Jain, BNL34
A Harmonic Coil Array(Developed at BNL)
16 Printed Circuit coils, 10 layers6 turns/layer300 mm long0.1 mm lines with0.1 mm gapsMatching coils selected from a production batchRadius =
26.8 mm (GSI)35.7 mm (BioMed)
USPAS, Phoenix, January 16-20, 2006 Animesh Jain, BNL35
Sextupole Hysteresis Vs. Ramp Rate
0.0E+00
2.0E-04
4.0E-04
6.0E-04
8.0E-04
1.0E-03
1.2E-03
0 1 2 3 4 5 6 7Current (kA)
Sext
upol
e H
yste
resi
s (T
) DC 1.5 T/s2 T/s 3 T/s4 T/s
Prototype Dipole for GSI
USPAS, Phoenix, January 16-20, 2006 Animesh Jain, BNL36
Summary• Persistent currents in superconductors produce
history dependent harmonics.
• These harmonics decay with time, but also snapback as soon as the ramp is resumed.
• These dynamic effects demand particular care in the measurements of superconducting magnets. Good time resolution is also required.
• Various techniques have been employed for dynamic measurements, but no single technique seems to be the “best”. (Promising R&D area.)