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Magnetic Resonance in Med - 1999 - Morgan - Resistive homogeneous MRI magnet design by matrix subset selection

Magnetic Resonance in Medicine 41:1221–1229 (1999)
Resistive Homogeneous MRI Magnet Design by Matrix
Subset Selection
Patrick N. Morgan,1* Steven M. Conolly,2 and Albert Macovski2
A new technique for designing resistive homogeneous multicoil magnets for magnetic resonance imaging (MRI) is presented. A linearly independent subset of coils is chosen from a
user-defined feasible set using an efficient numerical algorithm.
The coil currents are calculated using a linear least squares
algorithm to minimize the deviation of the actual magnetic field
from the target field. The solutions are converted to practical
coils by rounding the currents to integer ratios, selecting the
wire gauge, and optimizing the coil cross-sections. To illustrate
the technique, a new design of a short, homogeneous MRI
magnet suitable for low-field human torso imaging is presented.
Magnets that satisfy other constraints on access and field
uniformity can also be designed. Compared with conventional
techniques that employ harmonic expansions, this technique is
flexible, simple to implement, and numerically efficient. Magn
Reson Med 41:1221–1229, 1999. r 1999 Wiley-Liss, Inc.
Key words: resistive; homogeneous; magnet design
For the decade following the first demonstration of MRI in
1973 (1), resistive magnets were most often used to create
the main magnetic field, B0 (2–4). Technical advances over
the last 20 years have contributed to the proliferation of
superconducting and permanent magnet systems in MRI.
Today MRI scanners based on resistive main magnets are
relatively scarce in the marketplace.
However, cost pressures have encouraged the reconsideration of resistive B0 magnets in MRI scanners. Even with
the additional costs of a power supply with precision
regulation and a cooling system, resistive magnets are
generally much cheaper and simpler to build than superconductors (5). Precision-regulated power supplies with ratings up to 50 kW can be purchased commercially (6) or
created by externally stabilizing a standard power supply
with feedback circuitry and a current sensor (7,8) or a field
probe (4,9–11). If the supply cost is high, efficiency may be
improved by incorporating ferrite materials into the electromagnet (12).
Some examples of clinical whole-body scanners successfully operated with resistive magnets are the 0.15 T scanner at the University of Cincinnati (13), the 0.02 T scanner
made by Instrumentarium (14), and the 0.08 T scanner at
the University of Aberdeen (15). Resistive main magnets
1Department of Electrical Engineering, Texas A&M University, College Station,
2Department of Electrical Engineering, Stanford University, Stanford, California.
Grant sponsors: National Science Foundation, Air Defense Systems Department of the Johns Hopkins University Applied Physics Laboratory, Whitaker
Foundation, and GE Medical Systems.
*Correspondence to: Patrick N. Morgan, 208A Zachry Engineering Center,
Department of Electrical Engineering, Texas A&M University, College Station,
TX 77843-3128. E-mail: morgan@ee.tamu.edu
Received 10 July 1998; revised 3 February 1999; accepted 5 February 1999.
r 1999 Wiley-Liss, Inc.
have also been employed by researchers in low-field MRI
(16), Overhauser imaging (8), and prepolarized MRI (17,18).
Various techniques have been used to design a resistive
B0 magnet. The most venerable of these relies on a spherical harmonic expansion of the magnetic field (19). For
magnets consisting of multiple coils, the coil currents,
radii, and locations are determined by setting the harmonic
coefficients to zero. These equations have a linear dependence on the coil currents and a much more complicated
nonlinear dependence on the coil radii and locations.
Historically, the equations have been solved by making
assumptions such as equal coil currents or radii (20,21).
Hand calculation or nonlinear optimization may also be
used (22,23). Although they permit a solution to be obtained, these assumptions limit the ability of this design
approach to satisfy geometric constraints such as maximum or minimum coil sizes or gaps. For example, the
equal current assumption creates solutions where the radii
and locations of the coils approximate a sphere around the
sample. If access is required between the inner-most coils
in the arrangement, the design must either be scaled in size
or redesigned using a different number of coils. Either way,
the harmonic design technique does not guarantee an
optimal solution that satisfies constraints on both access
and field homogeneity.
Another technique is to employ a ‘‘target field’’ approach
whereby the design is optimized based on a desired
magnetic field profile within a user-defined region. This
approach has been previously used to design MRI gradient
coils (24), NMR field cycling magnets (25), and shim coils
(26). In those approaches, no attempt was made to optimize
coil radii or locations, so a fixed magnet geometry is
required. Pissanetzky (27) presented a target field approach
that does optimize these parameters along with the coil
currents using quadratic programming. The optimization
causes most, if not all, of the currents to attain their upper
or lower bounds, resulting in clustering of similar current
values. In the case of relatively few magnet coils, the
clustering may be desirable, but in other cases it may prove
overly restrictive. Simulated annealing is another numerical approach that has recently been applied to open access
superconducting magnet design (28,29). Linear programming has also been used to optimize coil locations and
radii (30).
Here we present an alternative technique to design a
resistive B0 magnet. Ferrite materials have not been modeled, so the technique is suitable for designing air core
electromagnets only. Our approach involves selecting an
optimal subset of coils from a user-defined feasible set
prior to determining the coil currents using the target field.
All computations require only standard matrix decompositions and linear matrix operations. Once the optimal coil
locations and currents have been calculated, the solutions
Morgan et al.
are converted to practical coils by rounding the currents to
integer ratios, selecting the wire gauge, and optimizing the
coil cross sections. Compared with previous techniques,
this technique involves relatively few calculations, often
less than 10 million floating point operations (flops),
which can typically be executed in less than 1 sec on a SUN
Using this technique, the magnet designer is required to
specify the bounds of the feasible set as well as the
location, shape, and size of the target field. This approach
has the advantage of enabling the design to meet constraints on both access and field homogeneity. Furthermore, the designer can trade field homogeneity for power
dissipation by changing the number of selected coils. This
flexibility is particularly valuable since closely spaced
coils in the feasible set often produce largely redundant
magnetic fields. By changing the bounds of the feasible set
or other design parameters, variation of power dissipation,
field homogeneity, access, or other quantities can be explored interactively.
Following presentation and discussion of the theory, we
illustrate use of the technique with a new design of a short
B0 magnet. This magnet may be integrated into a low-cost
MRI scanner designed to image the human torso with
minimal patient claustrophobia.
Our magnet design technique is based on minimizing the
␦ ⫽ 0A I ⫺ b0,
where 0 · · · 0 denotes vector length (sometimes called ‘‘2norm’’), A denotes the magnetic field created by the set of
feasible coils, I denotes the coil currents, and b denotes the
target field points. The residual can be minimized by
calculating the coil currents using the ‘‘pseudo-inverse’’ of
A, as shown previously (26). If the target field constraints
can be satisfied exactly, so ␦ ⫽ 0, then the pseudo-inverse
provides the solution that minimizes the sum of the
squares of the coil currents (31). In either case, this
approach generally requires all coils in the feasible set to
carry a nonzero amount of current.
Rather than using all coils in the feasible set, we select an
optimal subset of coils prior to minimizing the residual
using the pseudo-inverse. Our implementation of this
technique and some of the relevant tradeoffs are discussed
in this section.
FIG. 1. A feasible set of N coaxial coils. Multiple coils may exist at the
same location or with the same radius. When connected in series,
any subset of the coils forms a multiple coil magnet.
matrix, A, written as
A ⫽ (A1A2A3 . . . AN).
If the cross-sectional dimensions of the coils are small
compared with the coil radii, then the magnetic field can
be accurately calculated by modeling the coils as infinitesimal current loops. For the coaxial coils shown in Fig. 1, the
magnetic field at the location (r, z) is given by (32, p. 237)
Br(an, zn; r, z) ⫽
␮(z ⫺ zn)
2␲r (an ⫹ r)2 ⫹ (z ⫺ zn)2
⭈ ⫺K(␬) ⫹
Bz(an, zn; r, z) ⫽
an2 ⫹ r 2 ⫹ (z ⫺ zn)2
(an ⫺ r)2 ⫹ (z ⫺ zn)2
2␲ (an ⫹ r)2 ⫹ (z ⫺ zn)2
⭈ K(␬) ⫹
an2 ⫺ r 2 ⫺ (z ⫺ zn)2
(an ⫺ r)2 ⫹ (z ⫺ zn)2
where Br and Bz are the radial and axial components of the
field and r ⫽ 冑x2⫹ y 2. The functions E( ) and K( ) are
elliptic integrals where ␬ is defined as
Feasible Set
The feasible set consists of coils that are sampled over a
range of allowable coil radii and locations. In this paper,
we consider feasible sets consisting of N coaxial coils, as
shown in Fig. 1. Multiple coils may exist at the same
location or with the same radius. When connected in
series, any subset of the coils forms a multiple-coil magnet.
The total current carried by the nth coil, in amp-turns, is
denoted by In, the average radius of the coil is denoted by
an, and the average location is denoted by zn.
The magnetic fields produced by the feasible coils
carrying unit current are collected as columns of the
(an ⫹ r)2 ⫹ (z ⫺ zn)2
We model the elliptic integrals using polynomial approximations that are rapid to calculate and accurate to 2 ⫻ 10⫺8
(33, pp. 591–592).
In MRI, the magnetic moments respond to the magnitude
of the field, 0 B 0 ⫽ 冑Br2 ⫹ B2z . In the vicinity of the uniform
magnetic field created by coaxial current loops, the radial
component is much smaller than the axial component.
Hence 0 B 0 ⬇ Bz and the columns of the feasible matrix can
MRI Magnet Design by Subset Selection
be calculated by
An ⫽ Bz(an, zn; r, z).
For magnets consisting of loops with even symmetry in zn,
the columns may be calculated by
An ⫽
Bz(an, zn; r, z) ⫹ Bz(an, ⫺zn; r, z)
Bz(an, 0; r, z)
zn ⫽ 0
zn ⫽ 0.
Designs producing fields that have odd symmetry in zn
would have the fields subtracted since the currents are
equal and opposite on either side of zn ⫽ 0. Since the
feasible set of coils is restricted to zn ⱖ 0, this approach is
desirable to maximize computational efficiency, but it is
not required to create symmetric magnet designs. Our
experience indicates that nearly symmetric coils are always selected when symmetric target points are specified.
Designing magnets that produce other field profiles, such
as a shim coil or gradient coil, may require consideration of
field components other than Bz.
ment since the series is convergent for R ⬍ Rn for all n. Note
that as the order increases the magnetic field becomes more
uniform around the origin.
Constant target field values can be specified by locating
the points at the zeros of the Legendre function, PL(cos ␪). A
good approximation to the zeros is given by (33, p. 787)
␪l ⫽
(4l ⫺ 1)␲
4L ⫹ 2
2 12
(4l ⫺ 1)␲
4L ⫹ 2
for 1 ⱕ l ⱕ L. The error in the approximation is on the order
of 1/L3. For symmetric arrangements that calculate the field
using Eq. [6], the number of target points is M ⫽ L/2
arranged over a quarter circle for 0 ⬍ ␪ ⬍ ␲/2.
Subset Selection
We apply an algorithm suggested by Golub that selects a
subset of columns of A based on linear independence
according to the following steps (35, pp. 590–595).
1. Calculate the singular value decomposition (SVD) of
A, defined as (31, p. 442)
Target Field
We denote the target field by a vector, b, that consists of M
discrete points. All the magnetic fields in the matrix, A,
are sampled at the target locations, so the dimension of A
is therefore M ⫻ N. The matrix element occupying the mth
row and the nth column can be written as
A mn ⫽ Bz(an, zn; rm, zm)
1 ⱕ m ⱕ M, 1 ⱕ n ⱕ N.
Assuming the target volume contains no current sources,
both the divergence and the curl of the field are zero, i.e.,
ⵜ · B ⫽ 0 and ⵜ ⫻ B ⫽ 0. Hence specification of a uniform
target field along a boundary is sufficient to guarantee a
uniform field inside. For the coaxial set of loops shown in
Fig. 1, the target field can be specified along a semicircular
boundary in the r ⫺ z plane. For symmetric arrangements
that calculate the field using Eq. [6], the points at z ⬍ 0 are
redundant, reducing the boundary to a quarter of a circle.
Other target field shapes, such as an ellipsoid, can be
specified using an appropriate boundary.
For a spherical boundary around a uniform field, the
exact location of the target field points can be obtained by
expanding the axial component using spherical harmonics,
Bz(R, ␪) ⫽
A ⫽ U S V T,
where U is M ⫻ M and orthogonal, S is M ⫻ N and
diagonal, and V is N ⫻ N and orthogonal. The
diagonal elements of S are the singular values of A.
The SVD is calculated so the singular values are
ordered in decreasing magnitude.
2. Calculate the QR decomposition with column pivoting of the first N̂ columns of V, defined as (35, p. 271),
V (:, 1 : N̂)T ⫽ Q R P T,
where N̂ denotes the actual number of coils ultimately used to construct the magnet. The matrix, Q, is
N̂ ⫻ N̂ and orthogonal, R is N̂ ⫻ N and upper
triangular, and P is the N ⫻ N square permutation
matrix that orders the diagonal elements of R in
decreasing magnitude.
3. Calculate the columns of A to be selected by collectˆ,
ing them into a matrix, A
ˆ ⫽ A P (:, 1 : N̂),
C R P (cos ␪)
⬇ B(0) ⫹
CLRLPL(cos ␪),
where Cl are the spherical harmonic coefficients, B(0) is the
field strength at the origin, and (R, ␪) are spherical coordinates. The functions Pl( ) are Legendre functions of degree
l and Pl1( ) are associated Legendre functions of the first
kind of degree l and order 1 (34, p. 1014). The approximation is valid for magnets that produce a field of order L over
regions extending to the innermost coils in the arrange-
ˆ is M ⫻ N̂. The columns of A are sorted by
where A
multiplying it with the permutation matrix, P, obˆ,
tained in step 2. The matrix of selected columns, A
contains the first N̂ sorted columns of A, corresponding to the subset of coils to be retained. The remaining
coils are discarded.
To obtain useful solutions with the algorithm, some
relations among the matrix dimensions are required. To
approximate closely the target field typically requires
defining more feasible coils than target points, so
M ⱕ N.
Morgan et al.
For all matrices, the number of linearly independent
columns of A, known as the rank, must be less or equal to
the smallest dimension,
R ⫽ rank( A ) ⱕ M.
If R ⫽ M, the matrix is termed ‘‘full rank’’; otherwise it is
termed ‘‘rank deficient.’’ Since there are a total of R
independent columns, the minimum residual approximation of the target field requires no more than R coils
selected from the feasible set. Hence the rank provides an
upper bound to the number of selected coils,
N̂ ⱕ R.
Provided Eq. [15] is satisfied, Golub’s algorithm selects a
subset of columns that is guaranteed to be linearly independent since the QR decomposition performs Gram-Schmidt
orthogonalization. Heuristically, the subset tends to be
well conditioned. However, the conditioning comes at the
expense of discarding some nearly dependent columns
that may be required to minimize the residual. Alternative
algorithms, such as performing a QR decomposition directly on A, may lead to poorly conditioned subsets
containing nearly dependent columns that create homogeneous, high-power magnet designs. Later in this paper, we
explore the tradeoff between power and homogeneity by
varying the number of selected coils, N̂.
constraints. These tradeoffs will be illustrated with some
design examples later in this paper.
Power Dissipation and Coil Mass
For resistive magnets, the total power dissipation can be
written as the sum of the power dissipated by the individual coils,
Since the magnetic field depends linearly on the coil
currents, the optimum currents can be calculated as the
solution to a linear least squares (LLS) problem (31, p. 156),
ˆ )⫺1A
ˆ T b,
Î ⫽ (Â TA
which minimizes the residual between the magnetic field
produced by the subset of coils and the target field values,
as given by
ˆ Î ⫺ b 0 .
␦ˆ ⫽ 0 A
ˆ are linearly independent, the
Since the columns of A
inversion can always be calculated.
If the number of selected coils equals the rank of A, N̂ ⫽
ˆ spans the range
R, the residual is minimized since A
of A. All other subsets can produce no smaller residual.
␦ˆ ⱖ ␦,
where ␦ is obtained with no subset selection as given by Eq.
[1]. Although reducing the number of selected coils,
N̂, may produce a larger residual, the magnet design is
simplified since fewer coils are used. In some cases, the
magnet power dissipation decreases. Hence this technique
enables the magnet designer to trade field homogeneity for
lower power by changing N̂. This enables the designer to
obtain a low-power design that just satisfies the target field
兺 a I /A ,
n n
where ␴ is the electrical conductivity of the wire material,
an is the average radius of the nth coil, In is the nth element
of the vector of currents, I, in amp-turns, and An is the coil
cross-sectional area. The total mass of the coils is given by
m ⫽ 2␲␳
where ␳ is the density of the wire material.
Given the coil currents, equations [19] and [20] indicate
that the power and the mass can be traded off for each other
by changing the wire gauge. Our approach to this tradeoff is
to specify the wire current, Iw, based on the maximum
rating of the power supply. The wire area, Aw, can be
determined by
Aw ⬇ Iw
Linear Least Squares
␳cp␴ dT/dt
where cp is the specific heat, and dT/dt is an approximation for the uncooled heating rate of the coils (18). Typically we specify a slow heating rate of dT/dt ⫽ 67C/min to
permit minimal air cooling. Higher values could be specified if a more sophisticated cooling system is to be designed to manage the heating. The appropriate wire gauge
is selected by comparing Aw with a specification list
available from the wire manufacturer.
We implemented this technique in the mathematical analysis software package, MATLAB. Interactivity is achieved in
the MATLAB environment through a text-based user interface and multiple plotting windows. Computationally intensive functions, such as those for field calculation, were
developed in C and linked into MATLAB as object files.
Magnets were defined as arrays of coil structure elements
each containing the wire currents, locations, radii, number
of wires, and wire sizes. The magnet structure was passed
among custom functions developed for subset selection,
coil optimization, field calculation, and display.
After subset selection, the solutions were converted to
practical coils by quantizing the loop currents to integer
ratios based on the desired wire current, Iw. We implemented a constrained nonlinear optimization procedure to
minimize the RMS inhomogeneity by varying the coil
locations and radii over a small range. The optimization
was implemented using a sequential quadratic program-
MRI Magnet Design by Subset Selection
ming method with the MATLAB function ‘‘constr( ).’’ A
similar method was implemented to optimize the coil
aspect ratios, defined as the ratio of the build of the coil in r
to the width of the coil in z. In our experience, these
optimization steps converge quickly since the starting
values determined by subset selection are close to the
optimum values.
To minimize calculation time, we perform ‘‘economy’’
decompositions that return the first R columns of V and
return the permutation matrix as a vector. For numerical
stability, we calculate the pseudo-inverse based on the
ˆ . For M ⫽ 20 target field points arranged on a
SVD of A
semicircle of radius 10 cm and N ⫽ 1000 feasible coils of
radii 25 cm uniformly sampled along a 2 m long solenoid,
the total calculation time is about 350 msec, corresponding
to about 5.5 million floating point operations (Mflops).
Although not fundamentally important to the magnet
design problem, rapid computation enables the designer to
explore variation of power dissipation, field homogeneity,
access, or other quantities by changing the bounds of the
feasible set or other design parameters interactively.
We are most interested in low-field MRI of the human torso
with 10–20 ppm field inhomogeneity over a 20 cm DSV
and a field strength of 0.07 T, corresponding to a Larmor
frequency of 3 MHz for protons in water. The minimum
bore size is 45 cm.
Solenoid Magnet
Figure 2 shows the feasible set and target field points for a
solenoidal magnet. The feasible set consists of 40 coils
uniformly sampled down the length of the magnet. To
ensure a sufficiently large bore, all feasible coils have equal
radii set to 25 cm. The magnet length is judiciously set to 3
m to minimize the resulting field inhomogeneity. The
target field points, denoted by ‘‘o,’’ are set at the zeros of the
Legendre function corresponding to a 12th order field and
mapped to a semicircle of diameter 20 cm in the r ⫺ z
The currents and magnetic field profile obtained after
subset selection are shown in Fig. 3. The currents obtained
after subset selection are denoted by ‘‘x.’’ For the sake of
comparison, the currents and fields for a magnet designed
without subset selection, using the pseudo-inverse of A
ˆ , are shown by the dashed lines in the figure.
instead of A
The negative currents correspond to counter-wound coils
leading to an ‘‘overshoot’’ of the magnetic field. When
subset selection is used, none of the coils are counter-
FIG. 3. Coil currents (a) and on-axis magnetic fields (b) for 3 MHz
solenoidal magnets designed by subset selection (x and solid line)
and designed by pseudo-inverse without subset selection (dashed
wound, and the entire magnet is much shorter. In this case,
the magnetic field falls off gradually at the edges. Both
magnetic fields are comparable in field strength and uniformity over the target DSV.
Table 1 shows a more detailed comparison of this magnet
design with similar designs obtained using other techniques. For equivalent access, the inner radius was set to
25 cm for all designs. The RMS inhomogeneity was calculated over a 20 cm DSV. The power dissipation and coil
mass were calculated assuming an uncooled temperature
rise of 67C/min. The pseudo-inverse technique involves no
subset selection, so the magnet consists of 40 coils. This
approach achieves minimum inhomogeneity but very high
power and mass because of the counter-wound coils. The
peak current technique, which involves calculating the
pseudo-inverse only at the six locations of peak positive
current, has lower power dissipation and mass but proTable 1
Comparison of Magnet Design Techniques
FIG. 2. Feasible set and target field points for a solenoidal magnet.
Peak currents
Lee-Whiting solenoid
Garrett solenoid
Golub subset selection
No. of
inhomogeneity dissipation mass
Morgan et al.
FIG. 4. Coil cross sections and simulated magnetic field for a six-coil,
3 MHz solenoidal magnet. Contours are drawn for 1, 10, and 100
ppm field inhomogeneity.
duces an excessively inhomogeneous field. The four-coil
Lee-Whiting solenoid produces an 8th order field that is
also excessively inhomogeneous (20). The six-coil Garrett
solenoid produces a 12th order field with moderate power
(21). However, some improvement can be gained using the
subset selection technique that converges approximately to
the Garrett solenoid but achieves somewhat lower power
by relaxing the field inhomogeneity closer to our 10 ppm
Figure 4 shows the results of this design when converted
to practical coils. Contours are drawn around the coil cross
sections and for field inhomogeneity values of 1, 10, and
100 ppm. The magnet and coil design parameters are
shown in Table 2. Minimal air cooling is required since the
coils heat slowly. Designing the magnet to use a more
sophisticated cooling system could allow less massive
coils at the expense of higher power dissipation. The mass
of the support material is neglected. The coil axial locations and radii refer to the centers-of-mass of the turns.
Note that some extra turns are required to be placed on top
of the others.
Figure 5 shows a plot of the power-mass product versus
RMS field inhomogeneity for subsets consisting of 1–12
coils selected using the feasible set and target points shown
in Fig. 2. The number above the ‘‘x’’ denotes the number of
selected coils in the subset. To improve the field homogeneity, the number of feasible coils was increased by 1 for odd
numbers of selected coils. The general trend of the plot
indicates that the field becomes more homogeneous at the
expense of higher power-mass as the number of coils
increases. In this case, there are two exceptions to this rule.
First, the power-mass rapidly increases for N̂ ⫽ 12 because
this design requires counter-wound coils that substantially
cancel the field. The field inhomogeneity is slightly worse
(less than 0.1 ppm) than for N̂ ⫽ 11, perhaps because of
numerical roundoff error. Second, two sets of designs
(N̂ ⫽ 7, 9) and (N̂ ⫽ 8, 10) appear nearly identical on the
plot. In this case, the design with the fewer number of coils
is preferable to minimize manufacturing cost. Other criteria such as sensitivity to coil positions could also be
Short Magnet
Figure 6 shows the feasible set and target field points for a
solenoidal magnet. Each intersection point defines the
Table 2
Design Parameters for the Six-Coil, 3 MHz Solenoidal Magnet
Shown in Fig. 4
Magnet parameters
No. of coils
Wire current
Field strength
Power dissipation
Uncooled heat rate
Coil mass
Wire gauge
RMS inhomogeneity
100 A
0.07 T
6.665 kW
171 kg
0.441 m
0.608 m
AWG #5
11 ppm
Coil parameters
Axial locations
No. of turns
⫾0.273215 m
0.251312 m
13 ⫻ 13 ⫹ 2
⫾0.116104 m
0.249898 m
⫾0.036940 m
0.252810 m
FIG. 5. Plot of power-mass product versus RMS field inhomogeneity
for subsets consisting of 1–12 coils. The number above the x denotes
the number of selected coils in the subset.
MRI Magnet Design by Subset Selection
FIG. 6. Feasible set and target field points for a short solenoidal
magnet. Each intersection point defines the radius and location of
each coil in the feasible set.
radius and location of each coil in the feasible set. In this
case the maximum length is restricted to 30 cm. To include
the possibility of larger coils at the same locations as
smaller coils, both dimensions are included in the feasible
set. To ensure a sufficiently large bore, the minimum coil
radius is set to 25 cm. The number of feasible coil locations
was set to 40, and the number of feasible coil radii was set
to 3. In this case, the best results were obtained when the
number of target field points was increased to 20, as
denoted by ‘‘o.’’ As before, the target DSV is set to 20 cm.
Increasing the target points beyond the desired field order
may be required for magnets designed with restrictive
length constraints.
Figure 7 shows the results of this design when converted
to practical coils. Contours are drawn around the coil cross
sections and for field inhomogeneity values of 1, 10, and
100 ppm. The magnet and coil design parameters are
shown in Table 3. For comparable heating and field
homogeneity, this design requires 3.7 times more power
than the previous design. The top coils carry most of the
current and are counter-wound relative to all the other
points or less are sufficient to obtain useful magnet designs. To strictly enforce constraints such as minimum
power, mass, or inductance, other subset selection or
optimization algorithms may be developed.
Although not analyzed here, successful magnet design
often requires consideration of other factors such as the
extent of fringe fields, Lorentz forces, insulation, mechanical stresses, or use of water cooling. These factors become
most significant at a high field strength or with numerous
counter-wound turns. The sensitivity of field homogeneity
to inevitable coil winding or positioning errors is also a
consideration. However, for a realistic position tolerance of
100 ␮m, the field degradation for the magnets considered
here was simulated to be less than 100 ppm, so homogeneity could be restored by low-power shim coils. Ferrite
materials have not been modeled, so the technique is
suitable for designing air core electromagnets only. Shielded
magnets may be designed with our technique by incorporating remote target field points. It may also be desirable to
design the coils with equal inner radii to reduce manufacturing costs or to improve mechanical stability. To avoid
the nonlinear optimization steps when converting the
designs to practical coils, the columns of A could be
calculated using expressions derived for finite coils, such
as those provided by Urankar (36).
All our calculations have assumed resistive copper wire.
Alternatively, the coils could be wound with aluminum
wire, but because of differences in the specific heat and
electric conductivity, approximately 1.5 times more coil
area would be required for the same heating rate. Even with
the additional volume the aluminum coils may be cheaper
and would weigh less than the copper coils. Even with the
slightly greater expansion with temperature, aluminum
coils could be more cost effective than copper coils. Other
wire materials could also be considered.
Design Technique
Selecting the optimal subset of coils based on linear
independence has the advantage of generating sparse solutions. The disadvantage is that the coils are selected
independently of the target field values, so for some
combinations of feasible sets and target fields, or if the
feasible set is undersampled, the technique may yield
designs that are excessively inhomogeneous. In this case,
the feasible set or the target points must be redefined to
obtain a solution. In our experience, feasible sets consisting of 1000 coils or less, and target fields consisting of 20
FIG. 7. Coil cross sections and simulated magnetic field for an
eight-coil, 3 MHz short solenoidal magnet. Contours are drawn for 1,
10, and 100 ppm field inhomogeneity. The two large coils are
counter-wound relative to the others.
Morgan et al.
Table 3
Design Parameters for the Eight-Coil, 3 MHz Short Solenoidal Magnet Shown in Fig. 7
Magnet parameters
No. of coils
Wire current
Field strength
Power dissipation
Uncooled heat rate
Coil mass
Wire gauge
RMS inhomogeneity
100 A
0.07 T
24.911 kW
640 kg
0.460 m
0.416 m
AWG #5
16 ppm
Coil parameters
Axial locations
No. of turns
⫾0.153777 m
0.426109 m
22 ⫻ 23 ⫹ 9
⫾0.154710 m
0.248902 m
⫾0.086846 m
0.250157 m
⫾0.030587 m
0.249711 m
Short MRI Magnets
In our experience, counter-wound coils appear to be required for a homogeneous magnetic field when setting
aggressive physical constraints such as a short magnet
length or small maximum coil radius. Similar counterwound results have recently been shown by other researchers (12,28). In this regime, the magnet efficiency is low
because the opposing currents cancel out the magnetic
field. Because of the low efficiency, designs requiring
counter-wound coils are likely to be more expensive to
manufacture and operate than unconstrained designs producing a comparable magnetic field. For very inefficient
designs, permanent material or superconducting wire may
be required.
Magnets of various other geometries could be considered. For example, the magnet could be designed for the
patient to lie within the gap between the innermost coils in
the arrangement. In this case the feasible set would consist
of a constrained set of coils outside the region of the gap.
Asymmetric magnets with offset target fields or projected
fields may also be designed.
Subset selection is a new design technique based on using
matrix manipulations to select an optimal subset of coils
from a user-defined feasible set. These manipulations
efficiently optimize coil radii and locations over a wide
range of geometries based on desired access to the sample,
maximum or minimum magnet length or radius, or other
factors. The residual obtained after subset selection is
always greater than or equal to that obtained without
subset selection, but requires fewer coils than if all coils
were selected. Using fewer coils can reduce magnet power
dissipation, mass, or manufacturing cost. For short magnets, large counter-wound coils may be required. Our
results indicate that a 33% reduction in magnet length
requires 3.7 times more power for comparable heating and
field homogeneity with sufficient access for a human torso.
Patrick Morgan gratefully acknowledges the support of a
National Science Foundation Graduate Research Fellow-
ship and the Air Defense Systems Department of the Johns
Hopkins University Applied Physics Laboratory. Steven
Conolly gratefully acknowledges the support of a Whitaker
Foundation Biomedical Research Grant. The authors also
acknowledge the support of the Whitaker Cost Effective
Health Care Technology Program and thank Hao Xu for
helpful discussions and GE Medical Systems for help and
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