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Applications and Advanced Features Gradient Linearity Definitions
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GRADIENTS WITH INTEGRATED SHIMS - BFG-S SERIES GRADIENT DESIGN PRINCIPLES Eddy
currents are generated by the interaction of the fields external to the gradient
assembly due to the pulsed gradient coils with the external electrically
conductive structures. The
conventional method for dealing with the eddy current problem relies on compensating
for the eddy fields. The
second, preferred, method deals with the problem by eliminating
the eddy currents that generate the eddy fields.
. This technique reduces the
fields outside the pulsed gradient coils substantially to zero. Since the surrounding structure is not exposed to changing
magnetic fields, no eddy currents are induced and no eddy fields are generated. Pulsed
gradient coils employing this technique are referred to as actively shielded
gradient coils, (ASGCs), because the field that they generate is shielded from
their exterior at all times. This
is achieved by winding an additional set of coils, (the shields), at a radius
larger than the primary windings, (the coils).
The current in the shields is of polarity opposite to that in the coils
and the winding distribution is such that beyond the radius of the shields, at
all points in space, the sum of the field due to the coils and the field due to
the shields is zero. This means
that all of the magnetic flux generated by the coil/shield system that
penetrates the coil returns in the annular space between the coil and the
shield. It
should be noted that the opposite polarity current in the shield that is
required for cancellation of the coil's external field, creates an internal
field opposite to that of the coil. Thus
the act of shielding the outside makes the ASGC less efficient than an
unshielded pulsed gradient coil, and the closer the shield and coil approach
then the less efficient will be the ASGC. If
they are infinitesimally close to touching then almost equal and opposite
currents in almost identical winding patterns will cancel the outside field but
of course will also almost cancel the internal field.
Then an almost infinite current will be required to produce a finite
field gradient strength. At the
other extreme, as the shield gets farther from the coil, less and less shield
current is required and in the limit it goes to zero leaving the situation as
for an unshielded coil. COIL
EFFICIENCY The
most important requirements of an MR experiment that are relevant to the
performance of pulsed gradient coils are: gradient strength, pulse rise time,
gradient linearity and duty cycle. The
specific requirements vary widely from experiment to experiment but there is
always a need to maximize gradient strength, linearity and duty cycle, and
minimize rise time within the constraints of available gradient amplifier power
and cooling capability. High
gradient strength is required in many applications, including situations where
increased spatial resolution is necessary as in MR microscopy or when very short
imaging times are desired, as is the situation for echo-planar imaging
techniques, or when it is necessary to move through k space in unusual patterns.
High gradients are necessary even for more mundane situations when
minimization of imaging time is desired. For
example, in spin echo imaging the time for a pulse sequence may be shortened by
modifying the phase compensation pulses that occur after the slice select pulse
and before the phase encode pulse. In
such pulses the flat top time may be reduced provided that the gradient strength
is increased to the extent that the integral of gradient strength with respect
to time remains constant. This
means that whatever the specifications imposed by an MR imaging experiment on
the design of an ASGC, the design must be as efficient as possible.
ASGC efficiency can be expressed in many ways but one of the most
relevant is in terms of the relationship between the energy stored in the
magnetic field, the gradient strength, the diameter of the system and the ratio
of the diameters of the coil and shield. The
reason that the magnetic stored energy is important is that it is related to the
peak power requirement. This occurs
at the end of the ramp up to flat top. For
a linear ramp with respect to time we can write; Ppeak
= PL
+ PR
(1) PL
= 2 E / tr
(2) where: Ppeak[VA]
= peak power demand PL[VA] = maximum reactive power demand PR[W]
= maximum resistive power demand E[J]
= magnetic stored energy tr[s]
= gradient rise time It can be shown that the magnetic stored energy is related to the gradient strength and system diameter by the relation:
where: G[T/m]
= gradient strength s[m] = radius of shield 0[H/m] = permeability of free space
The stored energy factor
0[m] = radius of coil ze[m] = half length of coil
then:
It can further be shown that:
where the approximation improves as
the relative system length
Equation
3 shows that for geometrically similar systems the magnetic stored energy of the
system is proportional to the square of the gradient strength, e.g. four times
the stored energy for twice the gradient strength and to the fifth power of the
shield radius, e.g. twice the stored energy for a 15 percent increase in the
shield radius. Equation 4 shows
that the stored energy factor
The
confined space that is usually available for pulsed gradient coil occupation
means that relatively high radius ratios
COIL DESIGN OPTIMIZATION Once
the pulsed gradient coil envelope dimensions have been defined, i.e. the maximum
and minimum OD and ID respectively, three coils and three shields corresponding
to the three pulsed gradient coil axes are then determined according to
tentative conductor thickness dimensions. The
conductor thickness dimensions are supposed to make resistive power dissipation
consistent with duty cycle specifications.
The coils and shields are then maximally separated to minimize the coil
to shield radius ratio
Given a coil length and radius and a coil turns distribution it is possible to obtain the magnetic scalar potential corresponding to this distribution everywhere inside an infinitely long cylinder of radius equal to the shield radius subject, to the boundary condition that the radial field on the cylinder radius is zero, i.e., there is no penetration of magnetic flux. This involves solving Laplace's equation in cylindrical coordinates. The
resulting analytical solution is in terms of Bessel functions.
This solution corresponds to the state of ideal shielding.
Having obtained that magnetic scalar potential solution it is possible to
derive the shield turns distribution from the magnetic scalar potential
distribution at the shield radius. Analytic
expressions involving Bessel functions are also obtained for the source
coefficients corresponding to a spherical harmonic expansion of the magnetic
scalar potential in the central region. This
is used in the process of meeting field gradient linearity specifications. Although this distribution is consistent with the magnetic field shielding constraint it is not in general consistent with the minimum stored energy requirement nor with the linearity constraints. To achieve minimum stored energy consistent with constrained gradient linearity requires the ability to manipulate the distribution of turns within the coil. To this end the coil turns distribution is modeled as a Fourier series of appropriate symmetry with unknown amplitudes. The solution depends upon finding a set of amplitudes that are consistent with minimizing stored energy and satisfying the gradient linearity constraint. The problem is highly non-linear and objective function gradient evaluations are impossible analytically. This precludes the use of many efficient function minimization methods. The algorithm used to find the minimum energy then is a variant, due to Brent1, of the conjugate direction method invented by Powell2. This method does not require any information concerning the derivatives with respect to the optimization variables, of the objective function, in this case the system stored energy; only function evaluations are required. This alone, only allows a minimum energy solution to be found; to impose constraints on gradient linearity requires additional treatment. This is achieved by using a modified objective function. The
objective function i.e., the magnetic stored energy, is modified by a penalty
term that is a function of the extent by which a particular configuration
violates the linearity constraint. The
particular method used is Bertsekis's3
augmented Lagrangian multiplier modification of the exterior penalty method.
The solution then proceeds by a series of unconstrained minimizations of
a modified objective function4. After
the completion of such a series a solution is obtained that is both of minimum
stored energy and which satisfies the linearity constraints. Constraints other than gradient linearity are also applied
during the solution process. These
include constraints on such parameters as maximum current density and resistive
power dissipation. Having
obtained such a solution the design is further investigated to determine its
thermal and mechanical characteristics. Adjustments
to coil length, conductor thickness, etc., are made and the process is repeated
for as many times as is necessary to obtain an optimal design meeting all design
specifications. REFERENCES
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