U.S. patent application number 09/774946 was filed with the patent office on 2002-09-26 for z gradient shielding coil for canceling eddy currents.
Invention is credited to Joseph, Peter M..
Application Number | 20020135369 09/774946 |
Document ID | / |
Family ID | 25102794 |
Filed Date | 2002-09-26 |
United States Patent
Application |
20020135369 |
Kind Code |
A1 |
Joseph, Peter M. |
September 26, 2002 |
Z GRADIENT SHIELDING COIL FOR CANCELING EDDY CURRENTS
Abstract
A cylindrical whole body magnetic resonance imaging system
gradient shielding coil having multiple windings which are
individually wound about a cylindrical support structure such that
the windings are evenly distributed in azimuthal angle and
interleaved with one another. The windings are preferably spaced in
azimuthal angle by 360.degree./N, where N is the number of
windings. In such a design, m azimuthal harmonic components of the
error field caused by the effects of eddy currents within the
imaging volume can be canceled. Canceling an increasing number of m
components requires increasing the N number of windings.
Inventors: |
Joseph, Peter M.; (Upper
Darby, PA) |
Correspondence
Address: |
Michael P. Dunnam, Esquire
WOODCOCK WASHBURN KURTZ
MACKIEWICZ & NORRIS LLP
One Liberty Place - 46th Floor
Philadelphia
PA
19103
US
|
Family ID: |
25102794 |
Appl. No.: |
09/774946 |
Filed: |
January 31, 2001 |
Current U.S.
Class: |
324/318 ;
324/309 |
Current CPC
Class: |
G01R 33/4215
20130101 |
Class at
Publication: |
324/318 ;
324/309 |
International
Class: |
G01V 003/00 |
Claims
I claim:
1. A cylindrical shielding coil for providing a canceling magnetic
field gradient to cancel an error magnetic field gradient created
within an imaging volume by eddy currents generated by a gradient
coil in a volume outside of said gradient coil comprising: a
non-magnetic electrically insulating cylindrical coil support
having an internal cavity which forms a volume for accepting said
gradient coil and said imaging volume; and N electrically
conductive cylindrical windings wound in a plurality of turns in a
substantially helical path about a surface of said coil support,
each turn of each winding being electrically spaced from each other
turn of each said winding such that spacing between respective
turns of each said winding decreases in approximate proportion to
the distance of said respective turns from a center of each of said
windings in a direction parallel to said axis of said coil, each of
said N windings being interleavingly wound in the same direction
about said surface of said coil support and separated from each
other winding of said N windings in an angular orientation of
approximately 360.degree./N about said coil support, whereby N is
determined so as to cancel harmonics of said error magnetic field
gradient up to and including the M.sup.th harmonic.
2. The shielding coil of claim 1 wherein said gradient coil is a
parallel loop gradient coil and N=M+1.
3. The shielding coil of claim 1 wherein said gradient coil does
not consist of parallel loops and N is no greater than 2M+1.
4. The shielding coil of claim 1 wherein a winding equation of
.theta.=2.pi.0 I.sub.k(z) of said windings is used to cancel
harmonics from -M to M of said error magnetic field.
5. The shielding coil of claim 4 wherein said gradient coil is a
parallel loop gradient coil and the winding equation of said
windings is given by: 4 I k ( z ) = 0 z z ' 0 ( z ' ) + ( k - 1 ) /
N for k=1 to k=N.
6. The shielding coil of claim 4 wherein said gradient coil does
not consist of parallel loops and the winding equation of said
windings is given by simultaneously solving the following
equations: 5 0 z z ' 0 ( z ' ) = ( 1 / N ) k = 1 N ( I k ( z ) - I
k ( 0 ) ) for m=0, and 6 0 z z ' m ( z ' ) = ( - 1 / 2 im N ) k = 1
N [ exp ( - 2 im I k ( z ) ) - exp ( - 2 imI k ( 0 ) ) ] for
non-zero m from -M to M.
7. The shielding coil of claim 1 wherein said windings are
displaced 360.degree./N with respect to each other.
8. The shielding coil of claim 1 wherein said shielding coil
includes at least one electrical connection that provides current
to said shielding coil.
9. The shielding coil of claim 1 further comprising: a second coil
support structure coaxial with said first coil support structure,
said first coil support structure located within said second coil
support structure, wherein at least one of the said N windings is
wound around said first coil support structure and at least one of
the said N windings is wound around said second coil support
structure.
10. A gradient coil set comprising: a cylindrical gradient coil;
and a cylindrical shielding coil comprising: a non-magnetic
electrically insulating cylindrical coil support having an internal
cavity which forms a volume for accepting said gradient coil; and N
electrically conductive cylindrical windings wound in a plurality
of turns in a substantially helical path about a surface of said
coil support, each turn of each winding being electrically spaced
from each other turn of each said winding such that spacing between
respective turns of each said winding decreases in approximate
proportion to the distance of said respective turns from a center
of each of said windings in a direction parallel to said axis of
said coil, each of said N windings being interleavingly wound in
the same direction about said surface of said coil support and
separated from each other winding of said N windings in an angular
orientation of approximately 360.degree./N about said coil
support.
11. The gradient coil set of claim 10 whereby N is determined so as
to cancel harmonics of an error magnetic field gradient created
within an imaging volume of said gradient coil by eddy currents
generated by said gradient coil in a volume outside of said
gradient coil up to and including the M.sup.th harmonic.
12. The gradient coil set of claim 10 wherein said gradient coil is
a parallel loop gradient coil and N=M+1.
13. The gradient coil set of claim 10 wherein said gradient coil
does not consist of parallel loops and N is no greater than
2M+1.
14. The gradient coil set of claim 10 wherein a winding equation of
.theta.=2.pi. I.sub.k(z) of said windings is used to cancel
harmonics from -M to M said error magnetic field.
15. The gradient coil set of claim 14 wherein said gradient coil
does is a parallel loop gradient coil and 7 I k ( z ) = 0 z z ' 0 (
z ' ) + ( k - 1 ) / N for k=1 to k=N.
16. The gradient coil set of claim 14 wherein said gradient coil
does not consist of parallel loops and the winding equation of said
windings is given by simultaneously solving the following
equations: 8 0 z z ' 0 ( z ' ) = ( 1 / N ) k = 1 N ( I k ( z ) - I
k ( 0 ) ) for m=0, and 9 0 z z ' m ( z ' ) = ( - 1 / 2 im N ) k = 1
N [ exp ( - 2 im I k ( z ) ) - exp ( - 2 imI k ( 0 ) ) ] for
non-zero m from -M to M.
17. The gradient coil set of claim 12 wherein said windings of said
shielding coil are displaced 360.degree./N with respect to each
other.
18. A gradient coil set of claim 13 wherein said windings of said
shielding coil are displaced 360.degree./N with respect to each
other.
19. The gradient coil set of claim 10 further comprising: a second
coil support structure coaxial with said first coil support
structure, said first coil support structure located within said
second coil support structure, wherein at least one of the said N
windings is wound around said first coil support structure and at
least one of the said N windings is wound around said second coil
support structure.
20. A magnetic resonance imaging system comprising: a main magnetic
component; a detection component; a cylindrical gradient coil; and
a cylindrical shielding coil comprising: a non-magnetic
electrically insulating cylindrical coil support having an internal
cavity which forms a volume for accepting said gradient coil; and N
electrically conductive cylindrical windings wound in a plurality
of turns in a substantially helical path about a surface of said
coil support, each turn of each winding being electrically spaced
from each other turn of each said winding such that spacing between
respective turns of each said winding decreases in approximate
proportion to the distance of said respective turns from a center
of each of said windings in a direction parallel to said axis of
said coil, each of said N windings being interleavingly wound in
the same direction about said surface of said coil support and
separated from each other winding of said N windings in an angular
orientation of approximately 360.degree./N about said coil support,
whereby N is determined so as to cancel harmonics of an error
magnetic field gradient created within an imaging volume of said
gradient coil by eddy currents generated by said gradient coil in a
volume outside of said gradient coil up to and including the
M.sup.th harmonic.
21. The magnetic resonance imaging system of claim 20 wherein said
gradient coil is a parallel loop gradient coil and N=M+1.
22. The magnetic resonance imaging system of claim 20 wherein said
gradient coil does not consist of parallel loops and N is no
greater than 2M+1.
23. The magnetic resonance imaging system of claim 20 wherein a
winding equation of .theta.=2.pi. I.sub.k(z) of said windings is
used to cancel harmonics from -M to M said error magnetic
field.
24. The magnetic resonance imaging system of claim 23 wherein said
gradient coil is a parallel loop gradient coil and 10 I k ( z ) = 0
z z ' 0 ( z ' ) + ( k - 1 ) / N for k=1 to k=N.
25. The magnetic resonance imaging system of claim 23 wherein said
gradient coil does not consist of parallel loops and the winding
equation of said windings is given by simultaneously solving the
following equations: 11 0 z z ' 0 ( z ' ) = ( 1 / N ) k = 1 N ( I k
( z ) - I k ( 0 ) ) for m=0 and 12 0 z z ' m ( z ' ) = ( - 1 / 2 im
N ) k = 1 N [ exp ( - 2 im I k ( z ) ) - exp ( - 2 im I k ( 0 ) ) ]
for non-zero m from -M to M.
26. The magnetic resonance imaging system of claim 21 wherein said
windings of said shielding coil are displaced 360.degree./N with
respect to each other.
27. A magnetic resonance imaging system of claim 22 wherein said
windings of said shielding coil are displaced 360.degree./N with
respect to each other.
28. The magnetic resonance imaging system of claim 20 wherein said
shielding coil includes at least one electrical connection that
provides current to said shielding coil.
29. The magnetic resonance imaging system of claim 20 further
comprising: a second coil support structure coaxial with said first
coil support structure, said first coil support structure located
within said second coil support structure, wherein at least one of
the said N windings is wound around said first coil support
structure and at least one of the said N windings is wound around
said second coil support structure.
30. A method of determining a number of windings, and a winding
equation of said windings, in a z-gradient shielding coil required
to cancel harmonics of an error magnetic field created within an
imaging volume by eddy currents generated by a gradient coil in a
volume outside of said gradient coil comprising: selecting a
maximum M.sup.th harmonic to be cancelled in said imaging volume;
determining the type of gradient coil; if said gradient coil is a
parallel loop gradient coil, then determining the number of
windings, N, according to N=M+1, and determining the winding
equation, .theta.=2.pi. I.sub.k(z), of said windings, according to
13 I k ( z ) = 0 z z ' 0 ( z ' ) + ( k - 1 ) / N for k=1 to k=N;
and if said gradient does not consist of parallel loops, then
determining the number of windings, N, where N is no greater than
2M+1, and determining the winding equation, .theta.=2.pi.
I.sub.k(z), of said windings, where the winding equation of said
windings is given by simultaneously solving the following
equations: 14 0 z z ' 0 ( z ' ) = ( 1 / N ) k = 1 N ( I k ( z ) - I
k ( 0 ) ) for m=0 and 15 0 z z ' m ( z ' ) = ( - 1 / 2 im N ) k = 1
N [ exp ( - 2 im I k ( z ) ) - exp ( - 2 im I k ( 0 ) ) ] for
non-zero m from -M to M.
31. A gradient coil set comprising: a gradient coil comprising a
plurality of gradient windings, each said gradient winding having
an inductance and resistance; a shielding coil comprising a
plurality of shield windings; and at least one external coil having
an inductance and resistance substantially equivalent to the
inductance and resistance of said shielding coil, wherein said
plurality of shield windings are connected in parallel, one of said
plurality of gradient windings is connected in series to said
plurality of shield windings, the remaining gradient windings are
connected in series with an external coil, and the combination of
said one of said plurality of gradient windings connected in series
to said plurality of shield windings is connected in parallel to
the combination of said remaining gradient windings connected in
series with said external coil.
32. The gradient coil set of claim 31 further comprising a
plurality of external coils each of said plurality of external
coils connected in series with each remaining gradient windings.
Description
BACKGROUND OF THE INVENTION
[0001] 1. Field of the Invention
[0002] The present invention generally relates to magnetic
resonance imaging coils and, more particularly, to z-gradient
shielding coils.
[0003] 2. Background of the Invention
[0004] Magnetic resonance imaging (MRI) systems are currently
employed in forming images of the internal human anatomy. In such
systems, a patient is placed in a magnetic field and is subjected
to radio-frequency electromagnetic pulses. The magnetic resonance
of the atomic nuclei of the patient are detected with a radio
frequency receiver to provide information from which an image of
that portion of the patient containing these nuclei may be formed.
The magnetic field includes a main magnetic field and three
additional fields with linear spatial gradients in the x, y, and z
directions
[0005] The main magnetic field is a very strong magnetic field,
which may be created by a super-conducting coil, a resistive coil,
or a permanent magnet. Normally, the z-axis is parallel to the axis
of the main magnetic field for systems in which the magnet has
cylindrical geometry, such as for whole body imaging. The linear
gradient magnetic fields are typically created by resistive coils
and are referred to as gradient coils. The resistive coils create a
magnetic field within the coil with a linear spatial gradient, also
referred to as a magnetic gradient. Typically, there is one
gradient coil for each of the x, y and z-axes, which create x, y,
and z magnetic gradients, respectively. Two different types of
gradient coils are typically used to produce the magnetic gradients
for MRI, one which creates a magnetic gradient along the z (or
longitudinal) axis of the coil, and two others which create
magnetic gradients along either the x or y (transverse) axes.
[0006] In operation, for imaging purposes, it is necessary to
rapidly pulse electrical current through the three gradient coils.
When this is done, a problem commonly encountered is the induction
of eddy currents in various metallic parts of the MRI system. The
MRI system typically contains a metallic cylinder called a bore
tube. The inside of the bore tube is an image volume; however, most
imaging occurs only in the central portion of the bore tube. The
current in the gradient coils induce eddy currents in the bore tube
of the MRI system that, in turn, induce a magnetic field within the
image volume, referred to as the error field. The magnetic field
created by the eddy currents is undesirable in the image volume. In
many medically useful imaging procedures, it is highly desirable to
reduce or eliminate these eddy currents.
[0007] Typically, eddy currents are reduced by surrounding each
gradient coil, also referred to as the inner coil, with another
similar coil, an outer coil or a shielding coil, to cancel the
magnetic and induced electric fields in the region outside of the
outer coil. A set of a gradient coil and its associated shielding
coil is referred to as a shielded gradient coil set. Ideally, the
shielding coil is designed to exactly cancel the electric and
magnetic field outside of the coil set. If no field exists outside
of the shielded gradient coil set, then no eddy currents can be
induced in the metallic parts of the MRI system, and therefore, no
error field will be produced in the image volume.
[0008] Not all eddy currents affect the imaging volume equally, in
particular, the induced eddy currents and magnetic fields can be
analyzed in terms of the azimuthal harmonic number, m. The
azimuthal harmonic number m means that the field or gradient varies
in azimuthal angle like cosine(m .phi.), sin(m .phi.), or a linear
combination of the two, where .phi. is the azimuthal angle as shown
in FIG. 5B. That is, the field goes through exactly m full cycles
as the angle varies from 0 to 360 degrees. The worst effects are
seen from eddy currents with m=0. These harmonics also have the
longest lifetime, which can be as long as several seconds. In
general, the lower the m number, the worse the effects on an MRI
system.
[0009] Existing attempts to reduce the eddy current effects have
only been partially successful, especially for the z-gradient coil
set. One common technique for making z-gradient coils is using
circular parallel loops of wire, all of which lie in planes that
are perpendicular to the z-axis. The loops are interconnected by
straight wires that lie on the outer cylindrical surface of a
support structure and are parallel with the z-axis. This design has
the advantage that it creates no x or y magnetic gradient. This is
important because it is undesirable to use a z-gradient coil that
creates x or y magnetic gradients. However, the z-gradient that is
created is not exactly homogenous, but varies with the radius from
the z-axis.
[0010] The problem with this conventional shielding is that it is
impossible to exactly cancel the field outside of the z-gradient
coil set. A continuous surface current distribution would be
required on the surface of the shielding coil to exactly cancel the
field outside of the gradient coil set. Conventional shielding
simulates a continuous surface current distribution by winding
several discrete circular loops around a support structure.
However, these discrete circular loops cannot exactly simulate a
continuous surface distribution, and therefore, never exactly
cancel the field outside of the gradient coil set.
[0011] While is not possible to exactly cancel the entire field
outside of the gradient coil set, it would be desirable to cancel
the specific harmonics that are most troublesome to the MRI system.
Therefore, a shielding coil for a z-gradient coil that exactly
cancels the magnetic fields of low azimuthal harmonic number, m,
outside of the z-gradient coil set would be very desirable.
[0012] FIG. 1 illustrates an exemplary prior art MRI system 10 as
disclosed in U.S. Pat. No. 4,733,189. As shown in FIG. 1, the MRI
system 10 includes a main magnetic component 20, gradient coils 30,
shielding coils 40, and a detection component 50.
[0013] The main magnetic component 20 can be a permanent magnet, a
resistive electromagnet, or a superconducting system as shown, in
which a solenoidal electromagnet 22 is encased within a cryogenic
vessel 26. Bore tube 28 supports the solenoidal electromagnet 22.
Image volume 24 is located centrally to the main magnetic component
20.
[0014] Gradient coils 30 include an x-gradient coil 32, a
y-gradient coil 34 and a z-gradient coil 36, disposed to create
gradient fields orthogonal to each other. X and y
gradient-producing coils are preferably implemented by
saddle-shaped coil elements disposed about the main magnetic field
axis and rotated ninety degrees from each other in orientation. As
shown, the z-gradient coil 36 is implemented by a parallel loop
gradient coil coaxial with the main magnetic field axis.
[0015] Detection component 50 includes a radio frequency (RF) coil
52 and an RF interrogator 56 and receiver 58. The interrogator 56
produces a pulse of radio frequency excitation and the energy
emitted as the atoms return to an aligned state is captured via
coil 52 and used to obtain an image signal. In use, a patient or
other object is positioned within the image volume 24 of the system
10.
[0016] Shield component 40 includes an x-shielding coil 42, a
y-shielding coil 44, and a z-shielding coil 46 disposed to
counteract the eddy currents induced by the gradient-producing
coils 32, 34 and 36, respectively. The x and y shielding coils, 42
and 44 may be implemented by saddle-shaped coils cut from flat
copper sheets and rolled into the appropriate saddle shapes. As
shown, the z-shielding coil 46 is implemented by a parallel loop
shielding coil coaxial with the main magnetic field axis.
[0017] FIG. 2 illustrates an exemplary prior art parallel loop
gradient coil of a type which may be used as a z-gradient coil 36
in FIG. 1. As shown in FIG. 2, parallel loop gradient coil 80
includes loops 81 interconnected by straight wires 82 that lie on
the outer cylindrical surface of a support structure 84 and
parallel with the z-axis. The loops 81 and straight wires 82 are
formed from a single wire 86. Terminal connections 88 are connected
to both ends of the single wire 86. This design has the advantage
that it creates no x or y magnetic gradient. This is important
because it is undesirable to use a z-gradient coil that creates x
or y magnetic gradients. However, the z-gradient that is created is
not exactly homogenous but varies with the radius from the z-axis.
The number of loops is determined by the current available and the
gradient desired. Typically, two pairs of loops are used, called a
Maxwell pair.
[0018] FIG. 3 illustrates an exemplary prior art multiple winding
gradient coil as described in U.S. Pat. No. 5,289,129 to Joseph,
which may be used as a z-gradient coil 36 in the exemplary MRI
system of FIG. 1. As shown in FIG. 3, multiple winding gradient
coil 100 includes two windings. The first electrically conductive
winding 112 is wound about the surface of cylindrical coil support
structure 110. The first electrically conductive winding 112 is
wound helically in a symmetric manner about the center 114 of the
coil support structure 110. Terminal connections 116 are connected
to the first winding 112. A second electrically conductive winding
118 is also wound in an interleaved manner with respect to the
first winding 112. In accordance with the invention, the second
winding 118 is offset in azimuthal angle by 180.degree.
(360.degree./2 windings) with respect to the first winding 112. For
ease of illustration, the wire diameter of the second winding 118
has been illustrated to have a smaller diameter than the wire
diameter of first winding 112. As with the first winding 112,
second winding 118 also has a terminal connection 120 to which
current is applied from a power supply (not shown) for generating a
magnetic field. In another embodiment, the winding gradient coil
may include a plurality of windings, offset in azimuthal angle by
360.degree./X windings. Any number of windings can be used, but for
simplicity, only two are shown. The contents of U.S. Pat. No.
5,289,129, are hereby incorporated by reference for ease of
description.
[0019] FIG. 4 illustrates an exemplary prior art parallel loop
shielding coil, for use as a shielding coil 46 as shown in FIG. 1.
As shown in FIG. 4, loops 130 are interconnected by straight wires
132 that lie on the outer cylindrical surface of a support
structure 134 and are parallel with the z-axis. The loops 130 and
straight wires 132 are formed from a single wire 136. Terminal
connections 138 are connected to both ends of the single wire 136.
Experience indicates that existing art is only partially successful
in reducing eddy current effects, especially for the z gradient
coil set.
[0020] A z-gradient shielding coil and a z-gradient coil set is
desired that improves upon the coils and coil sets of the prior art
to reduce eddy currents induced in the MRI system, particularly
eddy currents with low azimuthal harmonic number, m. The present
invention has been developed to address these needs in the art.
SUMMARY OF THE INVENTION
[0021] The above mentioned needs are met by a cylindrical whole
body magnetic resonance imaging system gradient shielding coil
having multiple windings which are individually wound about a
cylindrical support structure such that the windings are evenly
distributed in azimuthal angle and interleaved with one another.
The windings are preferably spaced in azimuthal angle by
360.degree./N, where N is the number of windings. In such a design,
m azimuthal harmonic components of the error field caused by the
effects of eddy currents within the imaging volume can be canceled.
Canceling an increasing number of m components requires increasing
the N number of windings.
[0022] The cylindrical shielding coil of the present invention
provides a canceling magnetic field gradient to cancel an error
magnetic field gradient created within the imaging volume by eddy
currents generated by a gradient coil in a volume outside of the
gradient coil. The shielding coil includes a non-magnetic
electrically insulating cylindrical coil support having an internal
cavity which forms a volume for accepting the gradient coil and the
imaging volume. N electrically conductive cylindrical windings are
wound in a plurality of turns in a substantially helical path about
a surface of the coil support, each turn of each winding being
electrically spaced from each other turn of each the winding such
that spacing between respective turns of each the winding decreases
in approximate proportion to the distance of the respective turns
from a center of each of the windings in a direction parallel to
the axis of the coil. Each of the N windings is interleavingly
wound in the same direction about the surface of the coil support
and separated from each other winding of the N windings in an
angular orientation of approximately 360.degree./N about the coil
support. N is determined so as to cancel all harmonics up to and
including the M.sup.th harmonic of the error magnetic field
gradient.
[0023] The present invention also includes a gradient coil set for
use in a magnetic resonance imaging system. The gradient coil set
includes a cylindrical gradient coil and a cylindrical shielding
coil. The shielding coil is as described above. The gradient coil
may be any cylindrical gradient coil.
[0024] The present invention also includes a gradient coil set
electrically connected together to provide a fraction of the
current to the shielding coil. The gradient coil may include a
plurality of gradient windings with an inductance and a resistance.
The shielding coil may include a plurality of shield windings with
an inductance and a resistance. The gradient coil set may include
at least one external coil having an inductance and resistance
substantially equivalent to the inductance and resistance of the
gradient winding of the shielding coil. The plurality of shield
windings are connected in parallel. One of the plurality of
gradient windings are connected in series to the plurality of
shield windings. The remaining gradient windings are connected in
series with an external coil. The combination of the plurality of
gradient windings connected in series to the plurality of shield
windings is connect in parallel to the combination of the remaining
gradient windings connected in series with the external coil.
BRIEF DESCRIPTION OF THE DRAWINGS
[0025] The above and other aspects and advantages of the present
invention will become more apparent and more readily appreciated
from the following detailed description of the presently preferred
exemplary embodiments of the invention taken in conjunction with
the accompanying drawings, of which:
[0026] FIG. 1 illustrates an exemplary prior art magnetic resonance
imaging system;
[0027] FIG. 2 illustrates an exemplary prior art parallel loop
gradient coil;
[0028] FIG. 3 illustrates an exemplary prior art multiple winding
gradient coil;
[0029] FIG. 4 illustrates an exemplary prior art parallel loop
shielding coil;
[0030] FIGS. 5A and 5B illustrate a coordinate system for analyzing
coil magnetic fields;
[0031] FIG. 6 illustrates exemplary eddy currents in a
cylinder;
[0032] FIG. 7 illustrates an exemplary continuous surface current
distribution required for exact cancellation of magnetic fields
outside of a shielding coil;
[0033] FIG. 8 illustrates error fields for various azimuthal
harmonic numbers, m;
[0034] FIG. 9 illustrates a preferred embodiment of a multiple
winding shielding coil, in accordance with the present
invention;
[0035] FIG. 10 is a cross sectional view of one embodiment of a
multiple winding shielding coil, in accordance with the present
invention;
[0036] FIG. 11 illustrates an embodiment of a multiple winding
shielding coil in a z-coil gradient set, in accordance with the
present invention;
[0037] FIG. 12 illustrates a preferred embodiment of a multiple
winding shielding coil in a z-coil gradient set, in accordance with
the present invention;
[0038] FIG. 13 is a schematic diagram of an embodiment of the
present invention;
[0039] FIG. 14 is a graph showing calculated shielding efficiencies
at z=0 meters of a multiple loop shielding coil and a multiple
winding shielding coil in accordance with the present invention;
and
[0040] FIG. 15 is a graph showing calculated shielding efficiencies
at z=0.25 meters of a multiple loop shielding coil and a multiple
winding shielding coil in accordance with the present
invention.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
[0041] The main advantage of the multiple winding shielding coil
technique of the present invention over the parallel loop shielding
coil technique of the prior art is that the eddy current
distributions with low azimuthal harmonic number, m, can
theoretically be canceled exactly. Due to practical constraints,
including the placement of coil wires and the movement of coil
wires over time, exact cancellation of the eddy currents by the
multiple winding shielding coil of the present invention is not
probable. However, the present invention may perform better than
conventional shielding coils with regard to low number m
harmonics.
[0042] In accordance with the present invention, cancellation of
the m=0 component may be achieved with a single wire, but at the
expense of creating eddy currents and resultant magnetic fields
with higher m numbers. However, these higher m fields make no
contribution to the magnetic field in the z direction in the center
of the image volume and also decay much faster than the m=0
components. Therefore, these higher m number eddy currents and
resultant magnetic fields are less troublesome in MRI applications.
Furthermore, by increasing the number of windings, higher m errors
can be reduced to arbitrarily low levels. With a parallel loop
gradient coil and N spiral windings distributed equally in angle on
a shield coil, all harmonic components, m, are cancelled except
those that are integer multiples of N. It is not necessary that N
be a power of 2 or an even number. For example, with N=3, all
components will be cancelled except for .vertline.m.vertline.=3, 6,
9, 12, etc.
[0043] The cause of the eddy current problem addressed by the
present invention can be understood if the induced currents and
magnetic fields are analyzed in terms of the azimuthal harmonic
number, m. FIGS. 5A and 5B illustrate a coordinate system for
analyzing harmonics of induced currents and magnetic fields from a
coil. FIG. 5A is a perspective view of a coil. The coil 140 may be
a gradient coil, a shielding coil, or a main magnet coil as all
these coils are typically coaxial in an MRI full body imaging
system. As shown in FIG. 5A, the z-axis is co-axial with the main
axis of the coil 140. FIG. 5B is a cross sectional view of the coil
of FIG. 5A. As shown in FIG. 5B, angles .phi. and .theta. represent
the angle between the x-axis and a point on the coil. The angle
.phi. runs from 0.degree. to 360.degree. and then begins again at
0.degree. after one complete rotation around the coil. The angle
.theta. increments by 360.degree. upon every complete rotation
around the coil.
[0044] FIG. 6, shows the eddy currents produced for different
azimuthal harmonic numbers, m. As shown in FIG. 6, only the m=0
component 150 of the current circulates fully around the cylinder
154. The cylinder 154 may be the cryogenic vessel as shown in FIG.
1, specifically the inside diameter of the cryogenic vessel also
referred to as the bore tube 28, or any electrically conducting
object outside of the gradient coil. All higher values of
.vertline.m.vertline. have current distributions which pass part
way around the cylinder 154 and then, while flowing also along the
z direction, turn around and make a complete loop without
encircling the z-axis. For example, as shown in FIG. 6, the m=1
component 152 flows partially along the x-axis, then partially
along the y-axis, but never encircles the z-axis. The m=0 component
is the only component in which there is no motion of the current
along the z-axis, as it is a purely transverse current
distribution.
[0045] To fully cancel all harmonics of all eddy currents requires
that the shielding coil have a two dimensional continuous
distribution of surface current density .lambda.(.phi.,z) which is
determined by the design of the underlying main gradient coil.
There are established ways to calculate the desired surface current
density .lambda.(.phi.,z); for example, see Carlson and Zha, Magn.
Resn. Med. 1996; 36:950-54. However, there are no practical methods
of implementing a system of this type. Existing techniques use a
series of circular loop coils arranged perpendicular to the z-axis
as shown in FIG. 4 to approximate a continuous current density. The
parallel loop shielding coil cancels the error field along the
.theta. axis well, but only approximates the cancellation of the
error field along the z-axis.
[0046] The reason that the parallel loop shielding coil fails to
completely shield the gradient coil is illustrated in FIG. 7, which
shows an illustration of the form of the needed surface current
distribution, .lambda..sub.0(z), 161 as a smooth function of z. Any
design for the z-gradient shielding coil that uses parallel loops
is equivalent to sampling that distribution at a finite number of z
values, shown as 160 in FIG. 7. Such finite number of samples can
never exactly reproduce the continuous function of z that is
required for exact cancellation of fields external to the shielding
coil.
[0047] The present invention provides a shielding coil for exact
cancellation of error fields along the z-axis, at least for one
azimuthal harmonic number (m=0), and for cancellation of up to a
maximum azimuthal harmonic number, M. The invention, however, does
introduce errors of higher number m components, and does not
exactly cancel the error fields along the .theta. axis. Even though
these errors are not completely cancelled, they are less
troublesome for MRI systems.
[0048] Analysis of the azimuthal harmonic number is equivalent to
expanding the angular dependence in a Fourier series; for example,
the transverse surface current density, .lambda., in amps/meter can
be expressed as:
.lambda.(.phi.,z)=.SIGMA..sub.m.lambda..sub.m(z)exp(i m .phi.)
Equation 1
[0049] where .phi. is the azimuthal angle of a point on the surface
of the cylinder of the shielding coil and m is an integer running
from -.infin. to .infin..
[0050] The Fourier coefficients .lambda..sub.m (z) are functions of
z and can be computed from a known current density using:
.lambda..sub.m(z)=(1/2.pi.) .intg.d.phi. .lambda.(.phi.,z)exp(-i m
.phi.) Equation 2
[0051] and the integral is over a range of 2.pi. for the angle
.phi..
[0052] At issue is which m components are most troublesome for the
MRI system. This can by understood by looking at the radial
dependence of the magnetic fields generated by a current density of
a specific m value. The fall analysis of this problem involves
another Fourier transformation over the z coordinate and is beyond
the scope of this invention. The important conclusion is that near
the center of the cylinder, which is where the person or object to
the imaged is placed, the magnetic field strength will be
approximately proportional to the radius raised to the m-th
power.
Field .varies. r.sup.m Equation 3
[0053] Thus, the worst effects are seen from eddy currents with m=0
since they create magnetic fields which are approximately
transversely uniform within the imaged volume. They also have the
longest lifetime, which can be as long as several seconds. If
.vertline.m.vertline.>0, then the field will vanish at the
center, r=0. If .vertline.m.vertline.=1, then one gets an
approximately linear gradient in the magnetic field near the
center, which is also very undesirable. In general, it is most
desirable to suppress as many low values of m as possible.
[0054] FIG. 8 illustrates the error field induced by eddy currents
versus the radius from the z-axis, or center of the coil for
different values of the azimuthal harmonic number, m. As shown in
FIG. 8, the m=0 component is approximately constant within the
coil. This approximately constant error creates the largest
contribution of all the azimuthal harmonics induced in the MRI
system. The m=1 component approximates a linear function versus the
radius from the z-axis. Therefore, there is no m=1 error field in
the center of the coil, but the m=1 error field increases
approximately linearly with increasing radius. This error field is
zero at the center of the coil (where most of the imaging occurs).
The m=2 component approximates a quadratic function versus the
radius from the z-axis. Again, there is no m=2 field at the center
of the coil, but the m=2 error field increases with increasing
radius. However, the increase is slower than that of the m=1 error
field. These curves illustrate that the m=0 component is the most
critical component to eliminate in the MRI system. Also, these
curves illustrate that the lower the m azimuthal harmonic number,
the more important it becomes to reduce that component of the error
field.
[0055] FIG. 9 illustrates one embodiment of a multiple winding
shielding coil in accordance with the present invention. As shown
in FIG. 9, the multiple winding shielding coil 200 includes at
least two windings. A first electrically conductive winding 212 is
wound about the surface of cylindrical coil support structure 210.
The first electrically conductive winding 212 is wound helically in
a symmetric manner about the center 214 of the coil support
structure 210. Terminal connections 216 are connected to the first
winding 212. A second electrically conductive winding 218 is also
wound in an interleaved manner with respect to the first winding
212. In accordance with the invention, the second winding 218 is
offset in azimuthal angle by 180.degree. (360.degree./2 windings)
with respect to the first winding 212. For ease of illustration,
the wire diameter of the second winding 218 has been illustrated to
have a smaller diameter than the wire diameter of first winding
212. In this embodiment, the multiple winding shielding coil has
two windings or N=2, where N=the number of windings. As with the
first winding 212, second winding 218 also has a terminal
connection 220 to which current may be applied. The multiple
winding shielding coil is displaced substantially coaxial and
outside of a gradient coil 230, for example, the parallel loop
gradient coil 80 as shown in FIG. 2 or the multiple loop gradient
coil 100 as shown in FIG. 3.
[0056] In another embodiment, four interleaved windings are offset
in azimuthal angle by 90.degree. (360.degree./4 windings) from each
other. The multiple winding shielding coil thus corresponds to that
of FIG. 9 except that a third winding (not shown) and a fourth
winding (not shown) are also wound in an interleaved manner with
the first and second windings such that the four windings are
offset by 90.degree. with respect to each other. This multiple
winding shielding coil has N=4.
[0057] Increasing the number of windings increases the performance
of the multiple winding shielding coil. However, this requires more
wires be placed on the coil support structure 210. One way to fit
more wires is to decrease the wire gauge, however, this increases
the resistance of the wires, causing increased heat to be
generated. Placing the windings in layers, allows more space for
windings, while still allowing wires that won't cause excessive
heat generation.
[0058] FIG. 10 shows an alternative embodiment of the present
invention, where the multiple winding shielding coil includes more
than one layer. As shown in FIG. 10, a first layer of windings 236
is wrapped around a first coil support structure 235. An individual
winding in the first layer of windings is shown as element 237. The
first layer of windings 236 may include a plurality of windings. A
second layer of windings 239 is wrapped around a second coil
support structure 238. An individual winding in the first layer of
windings is shown as element 240. The second layer of windings 240
may include a plurality of windings. The first coil support
structure 235 is coaxial with and located within the second coil
support structure 238. Alternatively, there may be a plurality of
coil support structures.
[0059] In one embodiment, the same winding equation for each spiral
winding is used and the spiral windings are placed on the different
diameter support structures. This will create some error because
the ideal winding equation is dependent on the diameter of the
support structure. However, such a scheme would still provide
approximate shielding.
[0060] In another embodiment, separate winding equations for each
spiral are calculated, assuming that each spiral is located on its
own cylinder with its own diameter. This would mean that each
spiral would exactly cancel the m=0 azimuthal component, but it
would also mean that the each layer would have its own winding
equation and so the various windings would not maintain the exact
angular distribution needed to cancel the higher m angular
harmonics. This embodiment would also provide only approximate
shielding.
[0061] In a preferred embodiment, N spiral winding would be wound
on each of P coil support structures. In this manner, each cylinder
would have exactly the optimum distribution of wires for shielding.
As before, all of the spiral windings would be connected in
parallel and so the total resistance of the coil set would be
reduced to reduce power dissipation and heat. In this case, the
formulas used to derive the winding equation would be changed so
that each wire was carrying 1/(P*N) amps instead of 1/(N) amps.
[0062] Preferably, the windings are placed onto the coil support
structure 210 very precisely. To achieve this precision, it is
preferred to machine the coil support structure with a computer
numerically controlled (CNC) lathe or other equivalent device to
produce grooves in the coil support structure. The grooves are
placed according to the winding equation and number of windings
required to achieve a desired performance level. For example, a
desired performance level may be the cancellation of the m=0
component only. A method is described below, with formulas, for
determining the number of windings required and the winding
equation for each winding. Preferably, the windings are aligned
with the grooves and are displaced within the grooves.
[0063] FIG. 11 illustrates a preferred embodiment of a gradient
coil set in accordance with the present invention. As shown in FIG.
11, gradient coil set 250 comprises a parallel loop gradient coil
80 and a multiple winding shielding coil 200. The parallel loop
gradient coil 80 is located coaxially and within the multiple
winding shielding coil 200.
[0064] Although the described embodiments include only a parallel
winding gradient coil and a multiple winding gradient coil, the
present invention may be used to shield any type of gradient coil.
The following describes a general method of calculating the number
of windings, N, and the winding equation required for canceling a
particular number of azimuthal harmonics. First, a desired surface
current density .lambda.(.phi.,z) is calculated using known means.
The maximum harmonic, M, desired to be cancelled is selected. Using
the desired surface current density .lambda.(.phi.,z) and m, the
number of windings required, N, and the winding equation are
calculated. Examples are given below for both a parallel loop
gradient coil, as shown in FIG. 11 and a multiple winding gradient
coil (i.e., a gradient coil not consisting of parallel loops), as
shown in FIG. 12.
Parallel Loop Gradient Coil
[0065] In the embodiment of FIG. 11, the gradient coil includes
only circular loops parallel to the z-axis. It is assumed that the
desired current distribution, .lambda.(.phi.,z), has been
determined using known means. If the underlying gradient coil
consists solely of circular loops parallel to the z-axis, then
.lambda.(.phi.,z) has no .phi. dependence and has only the m=0
component, referred to as .lambda..sub.0(z). The k-th winding in
the shielding coil is represented by the winding equation:
.theta.=2.pi. I.sub.k(z) Equation 4
[0066] where .theta. is the total number of radians of angle in the
winding starting from z=0 meters. That is, .theta. increases by
2.pi. radians each time the wire encircles the cylinder, and
I.sub.k(z)-I.sub.k(0) can be thought of as the number of complete
encirclements that the winding undergoes from the center up to
position z. Thus, at any point on the winding, the azimuthal angle
coordinate .phi. is given by
.phi.=.theta. mod(2.pi.) Equation 5.A
and
exp(i .phi.)=exp(i .theta.) Equation 5.B
[0067] The function I.sub.k are computed from: 1 I k ( z ) = 0 z z
' 0 ( z ' ) + ( k - 1 ) / N Equation 6
[0068] for k=1 to k=N, and N is the number of windings in the
multiple winding shielding coil.
[0069] This algorithm gives N windings of similar form uniformly
distributed over the azimuthal angle on the cylinder. This method
will exactly cancel, in theory, the m=0 component of the field from
the parallel loop gradient coil, and will not introduce any other
field components with .vertline.m.vertline.<N. In other words,
N=M+1, and k runs from k=1 to K=N. For example, using N=2 windings
will produce zero for the m=-1, 0, and 1 components of the fields
and eddy currents, so that both the field and the gradient due to
eddy currents will vanish at all points on the central axis of the
cylinder.
Multiple Winding Gradient Coil/Other Than Parallel Loops
[0070] FIG. 12 illustrates a preferred embodiment of a gradient
coil set in accordance with the present invention. As shown in FIG.
12, gradient coil set 270 comprises a multiple winding gradient
coil 100 and a multiple winding shielding coil 200. The multiple
winding gradient coil 100 is located coaxially and within the
multiple winding shielding coil 200.
[0071] If the underlying gradient coil is not composed of circular
loops, or for some reason has Fourier components with
.vertline.m.vertline.>0- , then the preceding algorithm will not
work. A gradient coil of this type is shown in FIG. 3. While FIG. 3
illustrates a multiple winding gradient coil, the present invention
applies to any gradient coil not consisting of parallel loops. In
this case, using N=2M+1 windings will match all the Fourier
components of .lambda.(.phi.,z) up to .vertline.m.vertline.=M. M is
the maximum azimuthal harmonic number, m, desired to be cancelled
and N is the number of windings in the multiple winding shielding
coil. For example, using N=3 winding will cancel both the m=0 and
.vertline.m.vertline.=1 (M=1) components. There may exist gradient
coils that require less than 2M +1 windings to cancel m harmonics,
however, using 2M+1 windings will ensure cancellation of up to and
including the M.sup.th harmonic.
[0072] As in Equation 2, the Fourier coefficient functions for the
N=2M+1 Fourier components are calculated to be matched by the
spiral windings. It will be necessary to calculate the N functions
I.sub.k(z), indexed as k=1 to k=N. To do this, the following N
simultaneous equations are set up according to: 2 0 z z ' 0 ( z ' )
= ( 1 / N ) k = 1 N ( I k ( z ) - I k ( 0 ) ) Equation 7.A
[0073] for m=0, and 3 0 z z ' m ( z ' ) = ( - 1 / 2 im N ) k = 1 N
[ exp ( - 2 im I k ( z ) ) - exp ( - 2 imI k ( 0 ) ) ] for non -
zero m from - M to M Equation 7.B
[0074] Equation (7.B) applies only to the cases
.vertline.m.vertline.>0- . Equations (7.A) and (7.B) constitute
a set of N coupled non-linear equations that can be solved
numerically using well-established iterative algorithms. The
left-hand sides are known, computable, functions of z. The
right-hand sides contain the N unknown functions I.sub.k(z). To
compute the solutions, it is necessary to specify the values
I.sub.k(0) as well as approximate values of I.sub.k(z) to start the
iteration. The former should be specified as:
I.sub.k(0)=(k-1)/N for k=1 to N Equation 8
[0075] and the starting value of all the I.sub.k(z) can be taken
from Equation 6, e.g., one can assume that the components with
.vertline.m.vertline.>0 are small compared with the m=0
component for any reasonable z-gradient coil.
[0076] With the algorithm described, solutions for the N functions
are easily obtained with digital computers. For example, a
simulated example with N=3 was solved using the commercial software
Mathematica (Wolfram Research, Champaign, Ill.) in 13 milliseconds
per z point. Thus the computations involved are neither excessively
complex nor slow when used in practical cases.
[0077] The shielding coil current density will be less than that
used in the gradient coil. This is because the function of the
shielding coil is to cancel the external field from the gradient
coil, and this external field becomes weaker as the gap between the
two coils is increased. The current density needed on the shielding
coil is computable as described above. It is customary to force the
same current to flow through the shielding coil as through the main
coil. This has the advantage that the two currents are always
directly linked and proportional, which is effective for shielding.
However, the reduced current density in the shielding coil means
that the winding density will be reduced, and this will tend to
reduce the efficiency of shielding. One way to alleviate this
problem without increasing N, the number of loops in the shielding
coil, is to allow only a fraction of the main coil current to flow
in the shielding coil. This could be done, for example, by
employing two independent windings in the gradient coil, as
illustrated in FIG. 12. Then the shielding coil is fed with the
current from one of the gradient coil windings, thus reducing the
current to the shielding coil by a factor of 2. The factor by which
the current in the shielding coil is reduced would be exactly equal
to the number, N', of such independent windings in the gradient
coil. In this case, as explained in Equations 6, 7A, and 7B, the
right hand side of the equations would have an additional factor of
N', with the result that the winding density of the shielding coil
spirals would be increased, thus giving more effective
shielding.
[0078] A technical problem with this approach may be maintaining an
exact and precise division of current by factor N' between the
gradient coil and shielding coils. This could be accomplished by
inserting a compensating inductive and resistive component in
series with that part of the gradient current that does not flow
into the shielding coil. The purpose of that external component
would be to provide to those windings in the gradient coil an
inductive and resistive component that matches that of the
shielding coil, so that proportional currents would at all times
flow in the gradient and shielding coils. An embodiment is
illustrated in FIG. 13.
[0079] FIG. 13 shows one embodiment of the present invention, with
external coils used to control the current between the gradient
coil and the shielding coil. In the embodiment of FIG. 13, the
gradient coil set includes one gradient coil and one shielding
coil, for example, the gradient coil set of FIG. 12. The gradient
coil includes two gradient windings (elements 300a and 300b) and
the shielding coil includes two shield windings (elements 302a and
302b). As shown in FIG. 13, the shield windings 302a, 302b are
connected in parallel. This parallel connection is then connected
in series to the a gradient winding 300a. This passive connection
provides that the current through the shielding coil will be a
fraction of the current of the gradient coil. This is important,
because the current is the shielding coil 200 must be a particular
fraction of the current in the gradient coil to properly cancel the
magnetic field of the gradient coil. This fraction depends on the
difference in radius between the gradient coil and the shielding
coil.
[0080] The remaining windings of the gradient coil, which are not
connected to the shielding coil, may be connected to an external
coil to balance the current between the windings of the gradient
coil. As shown in FIG. 13, a gradient winding 300b, which is not
connected to the shielding coil, is connected in series to an
external coil 304. The series combination of the gradient winding
300b and the external coil are connected in parallel to the series
combination of the gradient winding 300a and the shield windings
(302a, 302b). The external coil 304 should have inductance and
resistance substantially equivalent to those of the shielding coil.
This provides an economical and highly reliable way to apply only a
fraction of the current in the gradient coil 100 to the shielding
coil 200. If the same current were applied to both the gradient
coil 100 and shielding coil 200, the winding density of the
shielding coil 200 would have to be decreased. This would, in turn,
increase the eddy current inhomogeneity of higher m harmonic
numbers. Hence, applying the multiple lead principle to both the
gradient coil 100 and the shielding coil 200 can contribute to
better shielding performance than would be possible if the gradient
coil contained only a single current path. Alternately, a plurality
of external coils may each be connected to in series with each of
the remaining gradient windings.
[0081] FIG. 14 shows calculated shielding efficiencies at z=0
meters of a multiple loop shielding coil and a multiple winding
shielding coil in accordance with the present invention. As shown
in FIG. 14, the shielding efficiency of the multiple winding
shielding coil with three spirals shields significantly better than
a multiple loop shielding coil with twenty-four loops.
[0082] FIG. 15 shows calculated shielding efficiencies at z=0.25
meters of a multiple loop shielding coil and a multiple winding
shielding coil in accordance with the present invention. As shown
in FIG. 15, the shielding efficiency of the multiple winding
shielding coil with three spirals shields significantly better than
a multiple loop shielding coil with twenty-four loops. In both
FIGS. 14 and 15, results are plotted versus the x-coordinate out to
a radius of 0.25 meters.
[0083] To perform the calculations, the gradient coil was given a
diameter of 1.0 meter. The shielding coil diameter was 1.1 meter,
and the bore tube of the magnet was taken to be 1.2 meters. The
length of the gradient coil was 2.0 meters. These numbers are
similar to what are used in clinical whole body MRI scanners.
[0084] The gradient coil had the multiple loop design, with a total
of 60 loops arranged with 30 loops on each side of the z=0 plane.
The spacing of the loops decreased in proportion to the distance
from the xy-plane so as to create an approximately linear magnetic
field gradient. No effort was made to further decrease the loop
spacing at the ends so as to improve the homogeneity of the
gradient in the middle region.
[0085] No gradient coil is expected to be useful throughout its
entire volume; one always uses a central region where the gradient
is constant enough for good imaging. In this case, it was assumed
that the useful region would extend to
.vertline.z.vertline.<0.25 meters in the z-direction and out to
a radius of 0.25 meters. The latter represents the maximum
transverse diameter of most adult humans (50 cm).
[0086] In all cases the distribution of current in the shielding
coil was calculated using well known Fourier Transform methods.
Once that is done, the number and position of the loops in the
multiple loop design is fixed and can not be changed. However, in
the multiple winding shielding coil technique one has the freedom
to choose the number of winding used in the spirals.
[0087] Once the shielding coil parameters were determined, it is
possible to calculate the eddy currents that flow in the bore tube.
The magnetic field generated by these eddy currents is then
calculated for the region inside the gradient coil, (i.e., the
region in which the patient's body is present for imaging). The
goal of the shielding coil is to reduce these eddy current fields
to a minimum in the central portion of the coil.
[0088] The multiple winding shielding coil provides, theoretically,
zero eddy current field at all points along the z-axis. It will
show progressively worse performance as one moves out in radius
away from the z axis. However, by increasing the number of windings
one achieves better and better suppression of the eddy current
fields near the z-axis. It was found that choosing three spirals
gave excellent shielding efficiency out to a radius of 0.25 meters.
It was found that this coil showed its worst performance in the
central plane, z=0 meters, and got progressively better moving out
along the z axis. At z=0.25 meters, the largest eddy current field
was about 0.02% of the main gradient field.
[0089] The z dependence of the multiple loop coil was the opposite
of the multiple winding shielding coil. The multiple loop shield
coil gave its best performance in the central plane at z=0 meters,
and was progressively worse as one moves out along the z axis to
z=0.25 meters. Its worst performance at x=y=0 meters and z=0.25
meters showed 3.5% contribution of the eddy currents as compared to
the gradient field.
* * * * *