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1330
IEEE
TRANSACTIONS ON MAGNETICS,
VOL
33,
NO.
2,
MARCH
1997
Eddy
Current Effects
in erconducting Magnets
Eugene A. Badeaf, Ovidiu Craius+
+BSG Inc., 6503 Taimer Ct., Sugarland, Tx, 77479,
USA
++Polytechnic University
of
Bucharest,
313
Splaiul Independentei,
77206,
Bucharest, Romania
Abstract-Magnetic resonance imaging
(MRI),
as
well
as
some volume selected spectroscopy meth-
ods, use pulsed magnetic field gradients which induce
multi-exponentially decaying eddy currents in all non-
laminated conductive parts
of
the superconducting
magnets. This paper presents the analysis of the
z
gradient field distorsion due to the induced eddy cur-
rents and the corresponding correction in a
4
T
/
30
cm
bore superferric self-shielded magnet.
I.
INTRODUCTION
The gradient coils’ pulsed field induces eddy currents
in the cryostat walls and other metallic parts of the MRI
superconducting magnets, where they can persist for long
periods. These eddy currents degrade the ideal switching
performance of the gradient field and produce additional
losses
in the cryostat.
Distortions due to eddy currents are
a
major cause of
concern in MRI because of the resolution degradation,
misregistration and intensity and phase variations. In or-
der to improve the imaging resolution and speed, as well
as to raise the efficiency in MRI systems, it is essential to
minimize the eddy current effects. Three principle strate-
gies are frequently used. One is by increasing the distance
between gradient coils and surrounding conducting struc-
tures. The limitations are dictated by the cost estimates,
according to a larger volume
of
active materials involved.
The second choice is the use
of
active shielded gradients.
Its main disadvantage is that one cannot correct eddy
currents that are induced in structures such as Faraday
screens or radio frequency (RF) coils, located inside the
gradient coils. Finally the last and probably the most con-
venient one is preemphasis. Correspondingly, the gradient
and shim systems are excited with pulse profiles having
a
multi-exponential characteristic.
1x1
the
present paper the contribution
of
z
gradient coils
was considered. Both cases
of
the field distortion without
and with preemphasis were correspondingly reported.
Manuscript received February
14,
1996. Eugene
A.
Badea, e-mail
73654.2166@compuserve.com,
phone 713-565-2154, fax 713-565-
2231; Ovidiu
Craiu,
e-mail
ocraiu@alpha.amotion.pub.ro,
phone
40-16-53-18-00.
11.
MAGNET DESIGN
The study was done on
a
4
T
/
30
em
bore super-
conducting superferric self-shielded magnet designed and
built by Texas Accelerator Center
[l].
The term superfer-
ric refers to
a
superconducting magnet containing iron in
which the magnetic induction is driven beyond the satura-
tion value of 2.1
T.
Integrating the superconducting coils
with saturated iron allows the iron to restrict the fringe
field while improving field strength and homogeneity. The
prototype, compared to an unshielded magnet, required
20
%
less superconductor and the
5
Gauss
fringe field was
included within
1
m
of
the magnet bore along the
x
axis.
Homogeneity
of
the raw magnetic field was 10
ppm
over
30
%
of the magnet’s diameter after passive shimming.
The design for the
30
em
diameter magnet consisted of
a
solenoid with an extra Helmholtz coil and ring
of
iron
at each end to approximate the geometry of an infinite
solenoid. The iron rings were connected on the outside of
the coils by twelve long iron flux returns
(
1.33
m
long by
15.24
em
by
15.24
em
)
in order to reduce the external
fringe field.
Fig.1. Perspective of the MRI Superconducting Magnet
0018-9464/97$10.00
0
1997
IEEE
1331
The superconductor was made of filaments of NbTi al-
loy in
a
copper matrix
(
ratio of copper to superconductor
of about
14
:
1
)
which formed
a
cylindrical wire of
0.081
cm
diameter. The superconducting coils consisted of two
separate circuits:
-
the central solenoid of
3721
turns with
the inner radius
ri
=
20.32
cm
and overall dimensions
of
36.75x4.11
em2;
-
the two end Helmholtz coils series
connected, each with
1960
turns, having the inner radius
ri
=
20.32
cm
and forming blocks of current
16.18~5
cm2.
The longitudinal gaps between coils were
0.62
cm.
A
gradient system made by Bruker Instruments, Inc.
was used, where the
z
gradient coils were
at
z
=
18.2
cm
and
z
=
-18.2
cm,
r
=
14.5
cm,
having
26
turns
each.
Fig2
presents a longitudinal sectional view of the
magnet.
If
i
designates
a
mesh node and
Ni(r,
z)
are the given
shape functions then
A
=
Ai Ni(r,
z)
i
(3.1)
defines
A
in terms of the nodal values
Ai.
The reason
that the energy functional
F
was expressed in terms of
B2
is because
A
is the main unknown function and
B
=
curl(
A)
or:
(3.3)
A dA A, dNi
r
dr r,
Bz=-+-=-
+CAiX
i
To
express
B,
in terms of the
A,
values while avoiding
problems near the symmetry axis
r
=
0,
the accurate cen-
troid formulation described in
[2]
was used. Correspond-
ingly,
F
is fully expressed in terms of the
A,
values, and
can be minimized by solving the system
of
linear equa-
tions
=
0
after the proper boundary conditions were
included.
By
using the Newton-Raphson procedure for treating
the magnetic nonlinearity and Crank-Nicholson technique
for time increment, the following matrix equation is ob-
tained
[3]:
IRON
RING
VACUUM VESSEL
AGO
SUPERCONDUCTING C
2a
Z
GRADIENT COIL
([GKI:+Ly
+
[GKdIf++
+
3pI)
(AfZk
-
A:++)
=
LIQUID
HE
VESSEL
2a
=
TLt++i
+
~[c]
(At
-
AZ,q)
-
[GKlf++tA;++t
[GK]:+?,
[GKd];++
and
[C]
are standard matrices de-
pending on the finite element type, geometrical parame-
ters of the mesh, reluctivity
v
and its derivative
&
at
iteration k and time
t
+
9,
TLt+y
represents the source
current density contribution at time
t
+
9.
(4)
INFINITE ELEMENTS
Fig.2. Longitudinal Cross Section of the Magnet
111.
ANALYSIS
Maxwell’s equations are expressed only in terms of the
azimuthal component
A
of the magnetic vector potential
due to the axisymmetric geometry of the problem.
For
nonlinear quasi-steady state conditions the field equation
is given by:
(1)
-
d
[--I
vd(Ar)
+
-
d
[--I
vd(Ar)
=
-J+u-
dA
ar
r
dr
dz
r
dz
at
with its energy functional:
where
J,
B,
a
and
v
are the source current density, mag-
netic induction, electric conductivity and magnetic reluc-
tivity respectively.
IV.
NUMERICAL RESULTS
A.
Without Preemphasis
The problem was solved using
a
uniform mesh with
NN
=
2425
nodes and
NE
=
4608
first order trian-
gular ring elements for the computational domain defined
in Fig.2.
A
very accurate cubic spline interpolation al-
gorithm was used to approximate the iron magnetization
characteristic
[4].
The natural Dirichlet condition
A
=
0
was specified along the symmetry axis. Infinite elements
were attached to the opened boundaries
to
improve accu-
racy at
a
very low computational cost
[5].
The
source current density was defined as it follows:
-
Helmholtz coil
JH~~~
=
5147.8
A/cm2;
-
Main coil
JM~~~
=
4018.6
A/cm2.
The corresponding magnetic
induction in the center
of
the magnet was
2.193
T.
1332
21930.4
219302
21930
The gradient pulse without preemphasis was trape-
zoidal, having a magnitude of
JG~~~
=
217.3
A/cm2,
and
producing
a
gradient of 0.95
Gs/cm
at
the center. The
signal has
a
rise-time of
750
ps,
stays constant for 14.25
ms
and then falls off within 750
,us.
The only non-laminated and metallic parts in the mag-
net were the liquid helium vessel, the vacuum vessel and
the iron rings at the end of the magnet. The liquid he-
lium vessel
(
4.2
K
)
was 1.01
cm
thick and had an elec-
tric conductivity of
6
=
1.811~10~
Slm
which gives an
electrodynamic time constant
r1
of 232
p.
The vacuum
vessel
(
at room temperature
)
was 0.95
cm
thick, had
U
=
9.804~10~
S/m
and
72
=
111
ps.
(
The electrody-
namic time constant was calculated as
upg2,
where
g
is
the thickness.
)
The solid iron ring
(
at room tempera-
ture
)
at each end of the magnet had a cross-section
of
11.43
cm
(
along the symmetry axis
)
by 5.08
cm
(
along
the radius
),
an average radius
of
17.78
cm
and its eIectric
conductivity was
CT
=
9.091~10~
S/m.
When the mag-
net was operated at 2.193
T,
the ring was in
a
weak field
and had an average
y
of 700; the corresponding electro-
dynamic time constant is about
20
s.
Fig.3
shows the numerical results of magnetic induction
variation versus time at the center of the magnet, obtained
by using
a
time step of
At
=
125
p.
Five cases with
respect to different metallic non-laminated parts of the
magnet were analyzed:
1.
no conductive medium; 2. with
iron rings;
3.
with vacuum vessel; 4. with liquid helium
-
1
x
ma
-
"
xnaxxa
Ib
0001
I
m
I
s
I ~
currents:
n
21930.4
219302
21930
k=l
where
ai(t)
is the gradient generated by source current
i(t)
in the gradient coil,
akik(t)
is the gradient generated
by eddy current
ik(t)
and
n
is the number
of
coupled eddy
current loops. Current
ik
in loop
k
depends on current
z
according to:
-
-
Ib
where
Rk
and
Lk
are the resistance and self inductance
of loop
k;
ibfk
is
the mutual inductance with the gradi-
ent coil. Using the Laplace transform and substituting
all currents in the transform of the gradient
(5),
one can
obtain:
1
x
ma
"
xnaxxa
0001
I
m
I
s
I ~
where
Ck
Ukibfk/U.Lk
and
Wk
=
Rk/Lk.
In the Case
when preemphasis
is
used, the problem is to find the ad-
equate current pulse
i(t)
in order to get
a
step function
response
of
the total gradient,i.e.,
G(s)
=
ais.
Further
we will consider only the case of two decaying exponen-
tial terms. The operational solution
I(s)
results from (7)
and its original is given by:
Dl(42
1
(1
-
c1
-
c2)
D'(q1)
=
(Q2)
vessel and
5.
with all conductive media.
Eddy currents were mainly induced in the liquid helium
vessel. The contributions corresponding to the vacuum
+
-_.yZt]
(8)
i(t)
=
1
+
vessel and iron rings were much smaller.
where:
I
21030.8
1
m
1
m
219298
I
1
Fig.3. Finite Element Solution for a Trapezoidal Pulse
0
5
10
15
a
25
34
lime
(msec)
B.
With
Preemphasis
Dl(S)
=
(s
+
Wl)(S
+
WZ)
,
D(s)
=
s(s
-
Cll)(S
-
42)
(9)
and
(41,
qz)
result from solving the system:
(ClW2
+
CZWl
-
w1
-
wz)
(1
-
c1
-
c2)
41
+
42
=
The second exponential term from
(8)
can be approxi-
mated with its first order Taylor expansion for a
suffi-
ciently small value of
(421.
The modified gradient waveform used to simulate the
preemphasis
for
the
given
problem was analytically
de-
fined
(
in terms of current density expressed in
A/cm2
)
as
it follows:
t
JGTad
=
292.758~ 750x10-6
'
t
<
750ps
The multi-exponential decay of eddy currents can be
JG~~~
=
292.758~e(~~'~~'-~-~)
-
2.11~10~~
modeled by
L-R
series circuits, inductively coupled to the
gradient coil
[GI.
The total generated magnetic field is the
x
(t
-
750x
lou6)
,
750
psst
<
4.275
ms
superposition
of
the gradients generated by the various
JGTad
217.35
,
4.275mSSt
<
14.25ms
1333
21931.2
21931
21930.8
P21930.6
a
-
121930.4
21930.2
21930
21929.8
T
Shtmthgreemp~ls’
0
3x0
JGrad
=
lo3
~(5.695
-
384.431xt)
,
14.25 msst
<
15
ms
~Frgli
In-mhwl~preemphasls’
t
.
-
-
-
-
-
-
I1
-
25w
t
JGTad
=
-70.976~e(O.O~~-~)
-/-
1.426x104x(t
-
0.015)
,
15ms<t
<
19.95ms
JGTad
=
0
,
19.95msLt
(11)
Fig.4 shows the corresponding numerical solution
B,
=
B,(t) in the center of the magnet for both cases without
and with the selected preemphasis.
For completeness some experimental results are also re-
ported. These were useful in finding the proper preem-
phasis for compensating the eddy current effects.
The nuclear magnetic resonance
(NMR)
test was con-
sidered for measuring the field distortion, The method
used
a
small probe
for
collecting a series of free induction
decays
(
FID
)
after gradient switching. Seventy-six FIDs
were registered, each signal being recorded
for
a
specified
delay time in the range of 100
ps
to
1
s,
after switching
off
a
15 ms gradient of 0.8 Gslcm. The operating field
of the magnet was set to 2.193
T
and the transmitter fre-
quency was set for proton resonance,
fo
=
93.44
MHz.
To measure the shift in the signal frequency, the phase
@
was calculated for each FID; frequency shift versus time
was then computed from the phase derivative
Fig.5 shows the frequency shift decay due to eddy cur-
rents induced by the
z
gradient pulse with the phantom
3
cm
off
center along the symmetry axis. The experiment
was in agreement with the numerical solution presented
in Fig.4:
-
the frequency shift without preemphasis was
Af
=
3000
Hz,
which corresponds
a
field decay of
0.7
Gs;
-
with preemphasis,
Af
=
1000
Hz,
which corre-
sponds
a
field decay of 0.233 Gs.
It
must be noticed that
preemphasis offered
a
good compensation of the first ex-
ponential decay.
[7].
21931.4
1
0
0
0
0
21929.6
c
-I
0
5
10 15
m
I
30
I”
(mec)
Fig.4. Finite Element Solution with and
without Preemphasis
”**
0
1
5
10 15
20
Ime
(msec)
Fig.5. Frequency Shift Decay
-
NMR
Experiment
V.
CONCLUSIONS
The paper presents the numerical and experimental
analysis of eddy current effects due to the
z
gradient field
in
a
4
T
/
30
cm bore superferric self-shielded magnet.
The field degradation due to eddy currents induced in the
non-laminated and metallic parts
of
the magnet was es-
timated for a trapezoidal current waveform. The most
dominant were the ones induced in the liquid helium ves-
sel. The case of
a
corrected gradient pulse was also con-
sidered. The preemphasis offered
a
good compensation of
the first, most dominant exponential decay.
REFERENCES
[l]
F.
R. Huson, R. N. Bryan, et al., “A High-Field Superfer-
ric NMR Magnet,”, Texas Accelerator Center, HARC, Inter-
nal Report, The Woodlands, Tx, USA,
1991.
[2]
E. A. Badea,
M.
R. Teodorescu, C.
V.
Bala, “Comparative
Analysis
of
FEM Approximation Models
for
Axisymmetric
Electromagnetic Field Problems Under Specified Source Con-
ditions,” Proceedings. IGTE Svmposium, Graz, Austria, Oct.
1990.
[3]
C.
V.
Bala, E. A. Badea,
R.
M. Teodorescu
,
“The Transient
Parameters Estimation
of
Coils with Ferromagnetic Core,”
Proceedings, European TEAM Workshop, Oxford, UK, April
[4]
E. A. Badea,
S.
Pissanetzky, “Accurate Cubic Spline Inter-
polation
of
Magnetization Tables,” COMPEL,
Vol.
12, No.1,
London, UK, March
1993,
pp.
49-58.
[5]
E. A. Badea,
S.
Pissanetzky, “Infinite Elements
for
Cylindri-
cal Geometries,” COMPEL,
Vol. 13,
No.2, London, UK, June
[SI
P.
Jehenson, M. Westphal, N. Schuff, “Analytical Method
for the Compensation of Eddy Current Effects Induced
by Pulsed Magnetic Field Gradients in NMR Systems,”
Journal
of
Magnetic Resonance,
V01.90, 1990,
pp.264-278.
[7]
W. R. Riddle,
M.
R. Willcott,
S.
J. Gibbs, R.
R.
Price, “Using
the Phase
of
the Quadrature Signal in Magnetic Resonance
Spectroscopy to Evaluate Magnetic Field Homogeneity
and
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pp.
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pp.
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pp.
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