Gradient coil with cancelled net thrust force

Superconducting magnets (10) generate a temporally constant magnetic field through a bore (12). The bore, hence the superconducting magnets, have a length which is relatively short compared to its diameter, a length to diameter ratio of less than 1.0 to 1.5 and preferably 1:1. A gradient coil assembly (30) is disposed around the bore for generating gradient magnetic fields across the bore. With such a relatively short bore magnet, in the region in which the gradient field coil is disposed, the main magnetic field suffers non-uniformities including radial magnetic field components. When current pulses are fed through the windings of a primary gradient coil (32) and a secondary gradient coil (34), the currents interact in an unbalanced manner with the non-uniformities and radial components of the temporally constant magnetic field, causing a net force in axial and/or transverse directions. Additional windings (62, 72) are mounted adjacent the ends of the gradient coil, carrying current in an opposite direction to z and transverse gradient windings in order to produce a counterbalancing force such that the net and counterbalancing forces substantially cancel. In this manner, the gradient coil assembly is freed from the necessity of a substantial mechanical mountings and the vibration and distortion which large net forces can cause.

BACKGROUND OF THE INVENTION 
The present invention relates to the magnetic resonance imaging arts. It 
finds particular application in conjunction with gradient coils for 
"short" bore magnets and will be described with particular reference 
thereto. It is to be appreciated, however, that the present invention will 
also find application in conjunction with coils for other magnets, 
particularly magnets in which a gradient coil is disposed in a magnetic 
field such that Lorentzian force components are not balanced. 
Magnetic resonance imagers commonly include a bore having a diameter of 90 
centimeters or more for receiving the body of an examined patient. The 
bore is surrounded by a series of annular superconducting magnets for 
creating a substantially uniform magnetic field longitudinally along the 
patient receiving bore. The more axially spaced the annular magnets, the 
more uniform the primary magnetic field within the patient receiving bore 
tends to be and the longer the axial dimension over which such magnetic 
field uniformity exists. Typically, the bore is at least 1.6 meters long 
and often longer. 
One of the drawbacks to such "long" bore magnets is that the region of 
interest is often inaccessible to medical personnel. If a procedure is to 
be performed based on the image, the patient must be removed from the bore 
before the procedure can be performed. Moving the patient risks potential 
misregistration problems between the image and the patient. 
One way to improve access to the patient is to shorten the length of the 
magnet and the patient receiving bore. If the magnet and the bore were 
shortened to about 1 meter or roughly the diameter of the bore, patient 
access is much improved. Although the size of the uniform magnetic field 
area compresses to a more disk-like shape, the area of substantial 
uniformity is still sufficient for a series of 10 to 20 contiguous slice 
images. NMR helical or continuous scanning methods can also be employed. 
Although an adequate imaging volume remains, the magnetic field in the 
volume around the periphery of the bore which receives the gradient coil 
tends to become relatively non-uniform and has both axial z-components and 
radial x,y-components. The gradient coil generally includes windings for 
generating three linear and orthogonal magnetic field gradients for 
providing spatial resolution and discrimination of nuclear magnetic 
resonance signals. Gradient coils are typically designed and constructed 
to optimize strength and linearity over the imaging volume and stored 
energy and inductance in the gradient coil. See, for example, U.S. Pat. 
No. 5,296,810 of Morich. To create the magnetic field gradients, current 
pulses are applied to the x, y, and/or z-gradient coils. These currents 
interact with the main magnetic field to generate Lorentz forces on the 
gradient coil. Due to the symmetries included in gradient coil currents, 
the Lorentz forces across the entire coil cancel when the gradient coil is 
disposed in a uniform magnetic field. However, when the main magnetic 
field is less uniform, particularly when there are significant radial and 
non-uniform axial components in the neighborhood of the gradient coils, a 
net thrust can be developed. Typically, pulsing the z-gradient coil causes 
a net thrust in the z-direction, pulsing the x-gradient coil develops a 
net thrust in the x-direction, and pulsing the y-gradient coil causes a 
net thrust in the y-direction. In the case of the z-gradient coil, the net 
axial force can be on the order of a few hundred pounds. These net thrusts 
tend to push or urge the gradient coil axially out of the bore. Although 
the gradient coil can be anchored mechanically, these large forces still 
tend to cause acoustic noise and increased vibrations to the magnet and 
the patient. Such vibration has deleterious effects on imaging, such as a 
loss of resolution. 
The present invention contemplates a new and improved gradient coil which 
overcomes the above-referenced problems and others. 
SUMMARY OF THE INVENTION 
In accordance with the present invention, a magnetic resonance imaging 
apparatus is provided. A gradient magnetic field coil is disposed around a 
patient receiving bore which is surrounded by a main magnetic field 
magnet. The main magnetic field magnet generates a magnetic field which is 
less than ideally uniform and has radial components in the region of the 
gradient field coil. The gradient field coil is designed to optimize 
strength, linearity over the imaging volume, and stored energy and 
inductance, as well as to minimize forces due to interaction of the 
current pulses applied to the gradient coil and the main magnetic field. 
In accordance with a more limited aspect of the present invention, 
additional windings are provided at least at one end of the gradient coil 
winding. The additional windings receive current pulses which interact 
with the main magnetic field in such a manner that any net force or thrust 
attributable to the current being pulsed through the other windings is 
substantially cancelled. 
In accordance with a more limited aspect of the present invention, the 
gradient coil is a self-shielded gradient coil. The primary gradient 
magnetic field coil is wound adding coils as necessary in order to zero 
the net force on the primary and shield gradient coils. 
In accordance with another aspect of the present invention, the gradient 
coil includes a z-gradient coil which is a plurality of loops extending 
circumferentially around the bore. On one side of isocenter, most of the 
windings carry current in a first direction and on the other side of 
isocenter, most of the windings carry current in a second direction 
opposite to the first direction. Windings are included on the first end of 
the coil having current flowing in the second direction, and windings are 
added on at the end of the second end having current flowing in the first 
direction. The amounts of opposite direction current flowing in these 
windings are selected such that there is substantially zero net axial 
thrust in the z-direction on the z-gradient coil windings. 
In accordance with another aspect of the present invention, a new and 
improved method of calculating gradient current windings is provided. An 
additional constraint is added such that the net thrust forces on all 
windings is zero. 
One advantage of the present invention is that it facilitates access to 
portions of the patient in the examination region. 
Another advantage of the present invention is that it improves image 
quality. 
Another advantage of the present invention is that it simplifies mounting 
and construction of gradient coils. 
Other advantages of the present invention include reduced vibrations and 
facilitating the construction of magnetic resonance imaging systems with 
short bores or main magnets. 
Still further advantages of the present invention will become apparent to 
those of ordinary skill in the art upon reading and understanding the 
following detailed description of the preferred embodiments.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
With reference to FIG. 1, a plurality of primary magnet coils 10 generate a 
temporally constant magnetic field along a longitudinal or z-axis of a 
central bore 12. In a preferred superconducting embodiment, the primary 
magnet coils are supported by a former 14 and received in a toroidal 
helium vessel or can 16. The vessel is filled with liquid helium to 
maintain the primary magnet coils at superconducting temperatures. The can 
is surrounded by one or more cold shields 18 which are supported in a 
vacuum dewar 20. 
A whole body gradient coil assembly 30 includes x, y, and z-coils mounted 
around the bore 12. Preferably, the gradient coil assembly is a 
self-shielded gradient coil assembly that includes primary x, y, and 
z-coil assemblies 32 potted in a dielectric former 34 and a secondary or 
shield gradient coil assembly 36 that is supported on a bore defining 
cylinder 38 of the vacuum dewar 20. The dielectric former 34 with potted 
gradient coils can function as a bore liner or a cosmetic bore liner can 
be inserted to line it. Preferably, shims (not shown) for adjusting the 
magnetic field are positioned as needed between the primary and shield 
coil dielectric formers as taught by DeMeester, et al. in U.S. Pat. No. 
5,349,297. A whole body RF coil 40 is mounted inside the gradient coil 
assembly 30. A whole body RF shield 42, e.g. a layer of copper mesh, is 
mounted between RF coil 40 and the gradient coil assembly 30. 
An operator interface and control station 50 includes a human-readable 
display such as a video monitor 52 and an operator input means including a 
keyboard 54 and a mouse 56. Track balls, light pens, and other operator 
input devices are also contemplated. Computer racks 58 hold a magnetic 
resonance sequence memory and controller, a reconstruction processor, and 
other computer hardware and software for controlling the radio frequency 
coil 38 and the gradient coil 30 to implement any of a multiplicity of 
conventional magnetic resonance imaging sequences, including echo-planar, 
echo-volume, spin echo, and other imaging sequences. Echo-planar and 
echo-volume imaging sequences are characterized by short data acquisition 
times and high gradient strengths and slew rates. The racks 58 also hold a 
digital transmitter for providing RF excitation and resonance manipulation 
signals to the RF coil and a digital receiver for receiving and 
demodulating magnetic resonance signals. An array processor and associated 
software reconstruct the received magnetic resonance signals into an image 
representation which is stored in computer memory, on disk, or in other 
recording media. A video processor selectively extracts portions of the 
stored reconstructed image representation and formats the data for display 
by the video monitor 52. An image printer can also be provided for making 
paper copies of selected images. 
In the preferred embodiment, the diameter and length of the bore 12 have a 
size ratio of about 1:1. However, it is to be appreciated that the present 
invention is also applicable to other magnets, particularly magnets with 
limited main magnetic field uniformity. Typically, one might expect 
magnets with a bore length to diameter ratio of up to 1.5:1 to be 
candidates for the present invention. However, the present invention will 
also be applicable to magnetic resonance imagers with longer bores in 
which there is a sufficient main magnetic field non-uniformity, 
particularly radial field components, in the area of the gradient coils to 
cause image degradation due to vibration or in which there are sufficient 
net forces that mechanical mounting of the gradient coils becomes 
difficult. 
The main magnetic field magnets 10 produce a main magnetic field B.sub.0 
over the imaging volume. In the short bore magnets, the z-component of the 
main magnetic field B.sub.z0 remains constant over a central imaging 
region of about 40 to 45 centimeters in diameter in a x,y plane at the 
magnetic isocenter. Along the z-axis, the imaging region is shorter than 
40 centimeters. Outside the ellipsoidal, uniform, imaging region, the B 
field changes significantly at larger axial or z-displacements from 
isocenter and with radial or .rho. displacement from the central axis of 
the bore. The dependence of B.sub.z0 on the axial and radial positions 
becomes more significant around the gradient and RF coil locations, 
typically the region bounded by 30 cm..ltoreq..rho..ltoreq.40 cm. and -60 
cm..ltoreq.z.ltoreq.60 cm and encompassing all angular positions. Although 
inside the imaging region, the contribution of the radial component of the 
main magnetic field B.sub..rho.0 to the total value of the B.sub.0 field 
is negligible, it is much more significant at the gradient coil sets' 
location, particularly in short bore magnet designs. 
The interaction between the spatially varying components of the magnetic 
fields and the current densities of the coil set creates two distinct 
problems. First, there is an interaction between the azimuthally directed 
current of the z-gradient coil with B.sub..rho.0. Because both quantities 
are odd symmetric functions around the geometric center of the magnet plus 
gradient system, they generate a z-directed thrust force according to the 
Lorentz force equation. Depending on the axial or z-behavior of 
B.sub..rho.0, in particular its value at the location of the conductors of 
the z-gradient coil, the value of the thrust force can reach several 
hundred pounds or a few thousand Newtons. 
The second problem deals with the interaction between the current density 
for a transverse x or y-gradient coil and both the B.sub..rho.0 and 
B.sub.z0 components of the main magnetic field. The result of such an 
interaction is again determined from the Lorentz force equation, but is a 
radially directed net force. Due to the azimuthal dependence of the 
transverse gradient's current density, the radially directed force is 
along the x or y-direction for an x or y-gradient coil, respectively. 
Again, the magnitude of this force is dependent on the value of the static 
magnetic fields B.sub..rho.0 and B.sub.z0 components at the locations of 
the current lines for the x and y-gradient coils. Because the radius of 
the x and y-gradient coils is different, the value of the corresponding 
force will also be different. The value of the components of B.sub.0 
change with radial position. 
Looking first to the analytical evaluation of the z-directed thrust force 
which is the result of the interaction between the z-gradient coil and the 
B.sub..rho.0 component, a minimization technique is presented for 
designing z-gradient coils with a zero net thrust force. For a 
conventional z-gradient coil, the current is odd-symmetric around the 
geometric center of the coil and the magnetic field B.sub..rho.0 is also 
odd-symmetric around the geometric center of the magnet. Because these two 
centers normally coincide, the result of the interaction between the 
current and the B.sub..rho.0 field component is a z-directed thrust force 
on the gradient coil. 
With reference to FIGS. 2A and 2B, the z-gradient coil is self-shielded, 
having an inner or primary winding 60 of radius a and an outer or 
secondary shield coil 62 of radius b. For simplicity of construction, the 
primary z-gradient coil is preferably circular loops wound on the 
dielectric former 34 and potted in epoxy or other resin. The shield 
z-gradient coil is preferably circularly wound on the dielectric former 38 
that is incorporated into the vacuum dewar. Of course, alternate 
embodiments where it is not an integral part of the vacuum dewar are also 
possible. The current density for the inner gradient coil 
j.sub..phi..sup.a (z) is azimuthally directed and varies only along the 
axial direction of the coil. Similarly, the current density for the outer 
gradient coil J.sub..phi..sup.b (z) is azimuthally directed and also 
varies along the axial direction of the coil. The interaction between 
these two current densities with the radial component of the main magnetic 
field at the radial locations of the coil, i.e., B.sub..rho.0.sup.a and 
B.sub..rho.0.sup.b, respectively generates a Lorentz force which is 
directed along the z-direction. 
In general, the Lorentz equation describes the elemental force experienced 
by a current element J.sub..phi..sup.a,b (z)dz(a,b)d.phi..alpha..sub..phi. 
in the presence of the magnetic field B.sub..rho.0.sup.a,b as: 
EQU dF=J.sub..phi..sup.a (z)dzad.phi..alpha..sub..phi. 
.times.B.sub..rho.0.sup.a .alpha..sub..rho. +j.sub..phi..sup.b 
(z)dzbd.phi..alpha..sub..phi. .times.B.sub..rho.0.sup.b .alpha..sub..rho.( 
1). 
The net force which is generated from these two coils is along the 
z-direction and has the form: 
EQU F.sup.net =F.sub.z.sup.net .alpha..sub.z 
with 
##EQU1## 
where B.sub..rho.0.sup.a,b (z) are known real functions of (z). The 
Fourier transform pairs for j.sub..phi..sup.a,b and B.sub..rho.0.sup.a,b 
are: 
##EQU2## 
These equations can be used to obtain the Fourier or k-space 
representation of the net thrust force as: 
##EQU3## 
Because B.sub..rho.0.sup.a,b (z) is a pure real, odd-symmetric function of 
z: 
##EQU4## 
For a set of z-gradient coils with radius a,b, respectively, the expression 
of the axial z-component of the magnetic field due to the gradient coil 
B.sub.z, in the three regions which the two coils define can be 
simplified, due to the behavior of the current density. More specifically, 
no angular dependence is considered for a current which is uniform around 
the circumference of a cylindrical gradient coil. The current in the 
z-gradient coil is restricted to vary along the z-direction. With these 
constraints, the expression of B.sub.z in the three regions is: 
##EQU5## 
where j.sub..phi..sup.a (k), j.sub..phi..sup.b (k) are the Fourier 
transforms of the current densities of the inner coil of radius a and the 
outer coil of radius b, respectively. Ideally, the z-gradient coil 
generated magnetic field B.sub.z is linear in the internal region of both 
coils and zero outside the two coils. In order to satisfy the constraint 
that the field is zero outside of the two coils, B.sub.z in the region 
b&lt;.rho. is set to zero, i.e., Equation (6c) is set to zero. One way to set 
Equation (6c) to zero is by relating the current densities of the inner 
and outer coils such that their Fourier components sum to zero, i.e.: 
EQU aj.sub..phi..sup.a (k)K.sub.0 (k.rho.)I.sub.1 (ka)+bj.sub..phi..sup.b 
(k)K.sub.0 (k.rho.)I.sub.1 (kb)=0 (7). 
or: 
##EQU6## 
Substituting Equation (8) into Equation (6a), the expression for the 
magnetic field for the region inside of the two coils .rho.&lt;a becomes: 
##EQU7## 
From Equation (8), the net force expression can also be simplified: 
##EQU8## 
The stored energy in the coil W in terms of Equation (8) is: 
##EQU9## 
For a gradient coil of length L, the Fourier expansion around the 
geometric center for the current of the inner coil in a self-shielded 
design is: 
##EQU10## 
where j.sub.n.sup.a are the Fourier coefficients of the expansion and sin 
k.sub.n z represents the antisymmetry condition of the current around the 
origin. For a coil of length L, the current is preferably restricted to 
become zero at the ends of the coil which restricts the values that 
k.sub.n can take. Thus, the allowable values for k.sub.n are: 
##EQU11## 
The Fourier transform of j.sub..phi..sup.a (z) is: 
##EQU12## 
with 
##EQU13## 
Due to the symmetry requirements, the dependence of .psi..sub.n (k) in the 
variable k is: 
EQU .psi..sub.n (-k)=-.psi..sub.n (k) (17). 
From the expression of the Fourier component of the current, the 
expressions for the gradient magnetic field B.sub.z, the stored magnetic 
energy W in the gradient coil, and the z-directed thrust force F.sub.z 
are: 
##EQU14## 
Because B.sub..rho.0.sup.a,b (z) is an odd-symmetric function with respect 
to z, its Fourier transform is defined as: 
##EQU15## 
with: 
##EQU16## 
Equation (20) then has the form: 
##EQU17## 
From the expressions for the magnetic field, the stored energy, and the 
z-directed thrust, the functional E can be defined as: 
##EQU18## 
where B.sub.zSC and F.sub.zSC.sup.net are the prespecified constraint 
values of the magnetic field at the constraint points and the z-directed 
thrust force, respectively. 
Minimizing E with respect to j.sub.n.sup.a, one obtains a matrix equation 
for the j.sub.n'.sup.a as follows: 
##EQU19## 
Depending on the value of j, the D.sub.jn matrix can correspond to either 
the magnetic field expression or to the z-directed thrust force 
expression. 
Specifically, the conversion for the D.sub.jn matrix is: 
##EQU20## 
For the expression of B.sub..rho.0.sup.a,b (z), data is obtained from the 
actual main magnet 10. Due to the antisymmetry of the B.sub..rho.0.sup.a,b 
(Z), only positive z-values are generated. In order to incorporate the 
generated data for the radial component of the magnetic field into 
Equation (24a), consider the following: 
##EQU21## 
where .OMEGA. corresponds to the total number of points along z, and 
B.sub..gamma..sup.a,b is the corresponding value of B.sub..rho.0.sup.a,b 
(z) at the location z.sub..gamma.. The Fourier transform 
B.sub..rho.0.sup.a,b (k), is: 
##EQU22## 
Furthermore, the generated values for B.sub..rho.0.sup.a,b (z) are for the 
axial distances which are larger than the half length of the gradient coil 
and up to the point where there is no significant action between the 
current density of the gradient coils and the main magnet's radial 
components. Truncating the infinite summations at M terms, the matrix 
representation for Equation (24b) becomes: 
EQU J.sup.a C=.lambda.D or J.sup.a =.lambda.DC.sup.-1 (28) 
where J.sup.a is a 1.times.M matrix, C is a M.times.M matrix, .lambda. is a 
1.times.N.sub.1 +1 matrix, and D is an N.sub.1 +1.times.M matrix. 
Using the constraint equation for the gradient magnetic field B.sub.z and 
the z-directed thrust force, the Lagrangian multipliers are determined. 
The Fourier components of the current density for the primary or inner 
gradient coil in matrix form are: 
EQU J.sup.a =B.sub.z [DC.sup.-1 D.sup.t ].sup.-1 DC.sup.-1 (29). 
Finding the expression for the Fourier components of the current for the 
inner coil and with the help of Equation (14), the continuous distribution 
of current density of the inner coil is generated. To determine the 
current for the outer coil, the shielding relationship between the Fourier 
transform for both current densities of Equation (8) is used. An inverse 
Fourier transform is then used to obtain the continuous current 
distribution for the outer coil. 
The next step is the process for discretization of the continuous current 
distribution for both coils. The continuous current distribution is 
divided into positive and negative current regions. Integrating the area 
underneath each region, the total current contained in each region is 
obtained. When the current for all of the regions of the cylinder is 
calculated, discrete current loops are positioned on a dielectric former 
to mimic the behavior of the continuous current pattern. Each region is 
filled with discrete wires carrying the prescribed amount of current. The 
amount of current is the same for each wire loop, in the preferred 
embodiment. In each region, the continuous current density is divided into 
smaller segments which correspond to the selected equal amount of current. 
The selected current amount may be iteratively adjusted in order to match 
the continuous current densities in the selected current regions more 
closely. Each wire is placed at the midpoint of the corresponding segment 
in order to obtain an equal distribution from both sides of the segment. 
This current distribution is then analyzed to calculate the generated 
magnetic field to assure that the desired magnetic field is, in fact, 
obtained. Alternately, one can use the center of mass scheme taught in 
previously referenced U.S. Pat. No. 5,296,810. 
With continuing reference to FIG. 2, in the preferred embodiment, the inner 
or primary coil has a radius of about 340 millimeters and a length of 
about 700 to 900 millimeters. The outside or shield coil is about 385 
millimeters in diameter. For the Fourier series expansion, ten points 
provide a reasonable definition of the current density, although larger or 
smaller numbers may be chosen. Five constraint points are preferably 
chosen to define the characteristics of the field and the thrust force. 
The field is to be constrained inside a 40 cm. to 50 cm. generally 
ellipsoidal imaging volume. The first constraint point establishes a 
gradient field strength, e.g., about 13.5 mT/m. The second constraint 
point defines the linearity of the field along the gradient axis. 
Preferably, at the edges of the 25 cm. dimension of the volume, the 
magnetic field is confined to vary not more than 5%. The remaining two 
constraints define the uniformity of the field across a plane 
perpendicular to the gradient axis. Preferably, the magnetic field is 
constrained to stay within 7% of its actual value at a radial distance of 
about 22 cm. from the center of the coils. 
The last constraint point defines the value of the z-directed thrust force 
which is preferably less than -1.0 e.sup.-08 Newtons. A suitable primary 
gradient coil meeting these constraints is illustrated in FIG. 2A and a 
suitable outer or shield gradient coil meeting these conditions is 
illustrated in FIG. 3B. It will be noted that the primary gradient coil 
has several force correcting windings 64 of reversed polarity at its ends 
or extremes. If these reverse polarity windings were removed, the gradient 
coil would suffer a net longitudinal force component in excess of the 
above-discussed constraints. 
Although the above-described method calculates an ideal current 
distribution, the force correcting current for cancelling the axial force 
can be determined iteratively. More specifically, several coil windings 
are disposed at the ends of one of the primary and secondary coils, 
preferably the primary coils. The net force on the gradient coil assembly 
during a gradient current pulse is measured. The current flow through the 
force correcting windings (or the number of force correcting loops) is 
iteratively adjusted until the axial force is substantially zeroed. 
With reference to FIGS. 3A and 3B, the gradient coil assembly further 
includes an x-gradient coil and a y-gradient coil. The x and y-gradient 
coils are of substantially the same construction, but rotated 90.degree.. 
Of course, because one is laminated over the other, they will have a small 
difference in radius. The x and y-gradient coils each include four 
substantially identical windings arranged symmetrically on either side of 
the isocenter. Each of the quadrants of the primary x or y-gradient coil 
contains a winding substantially as illustrated in FIG. 3A. Each of the 
quadrants of the shield x or y-gradient coil contains a winding 
substantially as illustrated in FIG. 3B. A current distribution is again 
calculated to produce the selected gradient strength, minimize stored 
energy, and zero the net lateral Lorentz forces. 
For the transverse x or y-gradient coils, the total current density can be 
represented as: 
EQU J.sup.a (r)=[j.sub..phi..sup.a (.phi.,z).alpha..sub..phi. +j.sub.z.sup.a 
(.phi.,z).alpha..sub.z ].delta.(.rho.-.rho..sub.0) (30), 
where .delta.(.rho.-.rho..sub.0) is the restriction that the current is 
confined on a cylindrical surface of radius .rho..sub.0. Again, the x and 
y-gradient coils are self-shielded. That is, they have an inner or primary 
coil of radius a and an outer secondary or shield coil of radius b. The 
current density for the outer coil of radius b is analogous to Equation 
(30). The interaction between the components of the current density and 
the corresponding components of the main magnetic field yields an 
x-directed Lorentz force for self-shielded x-gradient coils and a 
y-direction force for self-shielded y-gradient coils. The following 
discussion focuses on x-gradient coils. However, it is to be appreciated 
that the same discussion is equally applicable to y-gradient coils which 
will be of the same construction but rotated 90.degree. and of slightly 
different radius. 
The Lorentz equation describes the elemental force experienced by a current 
element Idl in the presence of the magnetic field B.sub.0 : 
EQU dF=Idl.times.B.sub.0 (31a) , 
or 
EQU dF=(j.sub..phi..sup.a,b (z)cos (.phi.).alpha..sub..phi. +j.sub.z.sup.a,b 
(z)sin(.phi.).alpha..sub.z)dz(a,b)d.phi..times.(B.sub..rho.o.sup.a,b 
(z).alpha..sub.r +B.sub.z0.sup.a,b (z).alpha..sub.z) (31b), 
where B.sub..rho.0.sup.a,b (z) and B.sub.z0.sup.a,b (z) are radial and 
axial components of the main magnetic field B.sub.0 at radii locations 
(a,b). Integrating Equation (31b) results in a net force along the x-axis: 
EQU F.sup.net =F.sub.x.sup.net .alpha..sub.x (32a), 
with: 
##EQU23## 
where B.sub..rho.0.sup.a,b (z) and B.sub.z0.sup.a,b (z) are known real 
functions of z. The Fourier transform pairs for j.sub..phi..sup.a,b, 
j.sub.z.sup.a,b, B.sub..rho.0.sup.a,b, and B.sub.z0.sup.a,b are: 
##EQU24## 
Using the constraint: 
EQU .gradient..multidot.J=0 
and the shielding conditions, the following relationships are derived: 
##EQU25## 
The expression for F.sub.x.sup.net from Equation (32b) is rewritten as: 
##EQU26## 
where * represents the complex conjugate of the Fourier transforms of the 
two components of the main magnetic field. 
For a real magnet design, there is no useful analytic expression for the 
B.sub.z0.sup.a,b (z) and B.sub..rho.0.sup.a,b (z). In order to incorporate 
analytically the numerically generated data for both components of the 
magnetic field along the central z-axis of the magnet, consider the 
following representations: 
##EQU27## 
where B.sub.z0.sup.a,b (z) is symmetric around z and B.sub..rho.0.sup.a,b 
(z) is antisymmetric around z. 
The restriction to the inner coil length, the confinement of the current 
density to a cylindrical surface, the azimuthal and axial symmetries for 
the j.sub..phi..sup.a and j.sub.z.sup.a and the demand that the current 
density obeys the continuity equation, provides a Fourier series expansion 
for both components around the geometric center of the coil: 
##EQU28## 
where j.sub..phi.n.sup.a are the Fourier coefficients, L represents the 
total length of the inner coil, and k.sub.n =2n.pi./L because the current 
cannot flow off of the ends of the cylinder. Furthermore, both current 
components are zero for .vertline.z.vertline.&gt;L/2. 
The general expression for the magnetic field for a self-shielded gradient 
coil in terms of the Fourier transform of the current density is: 
##EQU29## 
where j.sub..phi..sup.a (m,k) is the double Fourier transform of 
J.sub..phi..sup.a (.phi.,z). Since the azimuthal dependence of 
j.sub..phi..sup.a is proportional to cos(.phi.), the Fourier transform of 
j.sub..phi..sup.a is non-zero when m=.+-.1. In this case the 
two-dimensional Fourier transform of the current density is: 
##EQU30## 
with: 
##EQU31## 
where .psi..sub.n (k) is an even function of k and j.sub..phi..sup.a 
(+1,k)=j.sub..phi..sup.a (-1,k). Therefore, the expression of the gradient 
field has the form: 
##EQU32## 
In a similar fashion, the stored magnetic energy in the system is: 
##EQU33## 
and the expression for the x-directed Lorentz force is written as: 
##EQU34## 
The functional E is constructed and has the identical form of Equation 
(22). However, in the present case, the dependence of E is over 
j.sub..phi.n.sup.a and the force is directed along the x-axis instead of 
the z-axis. Thus, minimizing E with respect to j.sub..phi.n.sup.a yields a 
matrix equation for j.sub..phi.n'.sup.a as follows: 
##EQU35## 
Truncating the infinite summations at M terms, and using compact notation, 
Equation (38) simplifies to: 
##EQU36## 
or: 
EQU J.sup.a C=.lambda.D or J.sup.a =.lambda.DC.sup.-1 (46), 
where J.sup.a is a 1.times.M matrix, C is a M.times.M matrix, .lambda. is a 
1.times.N.sub.1 +1 matrix, and D is a N.sub.1 +1.times.M matrix with: 
##EQU37## 
Because Equation (46) is the same as Equation (28), the expressions of the 
continuous current distribution for both coils is found by following the 
steps described above. To discretize both current densities, one first 
considers the continuity equation for the current density: 
EQU .gradient..multidot.J=0 (48). 
In analogy with the magnetic field, where a vector potential is introduced, 
the current density is expressed as a curl of the stream function s as: 
EQU J=.gradient..times.S (49). 
Because the current is restricted to flow on the surface of a cylinder of 
radius a=.rho..sub.a and has only angular and axial dependence, the 
relation between the current density and the stream function in 
cylindrical coordinates is: 
##EQU38## 
and S.sub..rho. is found from: 
##EQU39## 
The contour plots of the current density are determined by: 
##EQU40## 
where N is the number of the current contours, S.sub.min is the minimum 
value of the current density, and S.sub.inc represents the amount of the 
current between two contour lines. The determination of S.sub.inc is: 
##EQU41## 
with S.sub.max representing the maximum value of the current density. The 
contours which are generated by this method follow the flow of the current 
and the distance between them corresponds to a current equal to an amount 
of S.sub.inc in amps. The discrete wires are positioned to coincide with 
these contour lines. 
In the preferred embodiment, the self-shielded force free x-gradient coil 
has a radius of the inner coil of about 1/3 meter and a length which is 
about 700 to 900 mm. The radius of the outer coil is about 0.4 meters. For 
the design of this self-shielded coil, the number of Fourier coefficients 
which are used is equal to the total number of constraint points increased 
by 1. In addition, four constraint points are chosen to specify the 
quality of the magnetic field inside a 40-50 cm. by 25 cm. ellipsoidal 
volume, and to eliminate the net Lorentz force for the whole gradient coil 
system. The first constraint sets the strength of the gradient field, e.g. 
13.5 mT/m. The second constraint specifies the linearity of the gradient 
field along the x-axis and up to the distance of 25 cm. from the isocenter 
of the gradient field, e.g., 5%. The third constraint specifies the 
uniformity of the gradient field along the z-axis at a distance up to 
.+-.6 cm. from the isocenter of the gradient field, e.g., 12%. The fourth 
constraint specifies the value of the net x-directed force on the gradient 
sets location, e.g., 1.times.10.sup.-7 Newtons. 
The presence of this set of constraints generate a continuous current 
distribution for both the inner and outer coils which, when discretized, 
takes the form illustrated in FIGS. 3A and 3B. More specifically, applying 
these design requirements generates a continuous current distribution of 
j.sub.za and j.sub.zb. Only the z-component of the current density is 
necessary for the creation of a discrete current pattern. The constraints 
are also chosen such that an integer number of turns for the inner and 
outer coils with a constant amount of current per loop can be obtained. 
Using the stream function technique, the discrete current distribution of 
the inner coil for the lateral force free configuration is obtained as 
illustrated in FIG. 3A. It will be noted that the inner or primary coil 
has a thumbprint winding 70 and an additional or force offsetting winding 
72. The current in the force correcting winding effectively flows in an 
opposite spiral direction to the primary coil portion 70. The current 
distribution and number of turns in the force cancelling coil portion 72 
is selected to zero the net x or y-direction force generated by the 
primary coil section 70 of FIG. 3A. Depending on the nature of the B.sub.0 
field components in a particular design, similar force free results can 
also be obtained by placing a winding having an opposite sense than the 
secondary x or y-gradient coil near an outer edge of that coil. The 
amperage through the force offsetting windings or the number of windings 
are adjusted until a current density is achieved which produces an x or 
y-force component which is equal and opposite to the force component 
generated by the other x or y-gradient coils. 
Although force balancing coils are applied to only the primary winding of 
shielded gradient coils in the preferred embodiment, it is to be 
appreciated that the primary and secondary coils can each be individually 
force balanced. Further, it is to be appreciated that the thrust force 
balancing is also applicable to gradient coils which are not 
self-shielded, i.e., which have only a primary winding and no secondary 
shield winding. 
The invention has been described with reference to the preferred 
embodiment. Obviously, modifications and alterations will occur to others 
upon reading and understanding the preceding detailed description. It is 
intended that the invention be construed as including all such 
modifications and alterations insofar as they come within the scope of the 
appended claims or the equivalents thereof.