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Novel techniques for pulsed field gradient NMR measurements

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Novel techniques for pulsed field gradient NMR measurements
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NOVEL TECHNIQUES FOR
PULSED FIELD GRADIENT NMR MEASUREMENTS

















By

WILLIAM W. BREY


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA


1994














ACKNOWLEDGMENTS


I would like to thank Janel LeBelle, Igor Friedman, and

Don Sanford for construction of the gradient coil

prototypes, and Jerry Dougherty for performing the

simulations of the coils they all helped to construct.

Stanislav Sagnovski, Eugene Sczezniak, Doug Wilken, and

Randy Duensing participated in many helpful discussions

concerning gradient coils. Debra Neill-Mareci provided the

excellent illustration of a gradient coil in Figure 22. For

their part in the microscopy project, thanks go to Barbara

Beck, Michael Cockman, and Dawei Zhou. Ed Wirth and Louis

Guillette provided the samples. For help measuring eddy

current fields I am grateful to Wenhua Xu, and to Steve Patt

for help with the software. Thanks go to my parents, Mary

Louise and Wallace Brey, and my brother, Paul Brey, for

encouragement and help with red tape. Paige Brey has my

special thanks for her extensive help preparing the thesis.

Katherine Scott, Richard Briggs, Jeff Fitzsimmons, and Neil

Sullivan enriched my graduate experience with their wide

knowledge and diverse interests. Thanks go to them for

their enthusiasm and for reading this thesis. Raymond

Andrew served as supervisory committee chairman. Thanks go

to Thomas Mareci for directing the research, for providing









financial and moral support, and for encouraging me to

pursue this work to its conclusion.


iii















TABLE OF CONTENTS



ACKNOWLEDGMENTS .......................................... ii

ABSTRACT ................................................. v

GENERAL INTRODUCTION ....................................... 1

MEASUREMENT OF EDDY CURRENT FIELDS ....................... 5
Introduction ........................................ 5
Literature Review ................................... 11
Spin-Echo Techniques ................................ 23
Stimulated Echo Techniques .......................... 28
Results ............................................. 34
Conclusion .......................................... 41

GRADIENT COIL DESIGN ..................................... 46
Introduction and Theory ............................. 46
Literature Review ................................... 50
Field Linearity ..................................... 61
Efficiency .......................................... 62
Eddy Currents ....................................... 68
Coil Projects ....................................... 72
Amplifiers ....................................... 72
16 mm Coil for NMR Microscopy .................. 76
9 cm Coil for Small Animals .................... 79
15 cm Coil for Small Animals ................... 87
Concentric Return Path Coil .................... 98

SYSTEM DEVELOPMENT FOR NMR MICROSCOPY .................... 126
Introduction ........................................ 126
Literature Review ................................... 128
Instrument Development .............................. 133
Results ............................................. 149
Conclusion .......................................... 153

CONCLUSION ............................................... 155

REFERENCES ............................................... 156

BIOGRAPHICAL SKETCH ...................................... 162










Abstract of Dissertation Presented to the Graduate School of
the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy



NOVEL TECHNIQUES FOR PULSED FIELD GRADIENT NMR MEASUREMENTS

By

William W. Brey

December, 1994


Chairman: E. Raymond Andrew
Major Department: Physics

Pulsed field gradient (PFG) techniques now find

application in multiple quantum filtering and diffusion

experiments as well as in magnetic resonance imaging and

spatially selective spectroscopy. Conventionally, the

gradient fields are produced by azimuthal and longitudinal

currents on the surfaces of one or two cylinders. Using a

series of planar units consisting of azimuthal and radial

current elements spaced along the longitudinal axis, we have

designed gradient coils having linear regions that extend

axially nearly to the ends of the coil and to more than 80%

of the inner radius. These designs locate the current

return paths on a concentric cylinder, so the coils are

called Concentric Return Path (CRP) coils. Coils having

extended linear regions can be made smaller for a given

sample size. Among the advantages that can accrue from

using smaller coils are improved gradient strength and









switching time, reduced eddy currents in the absence of

shielding, and improved use of bore space.

We used an approximation technique to predict the

remaining eddy currents and a time-domain model of coil

performance to simulate the electrical performance of the

CRP coil and several reduced volume coils of more

conventional design. One of the conventional coils was

designed based on the time-domain performance model.

A single-point acquisition technique was developed to

measure the remaining eddy currents of the reduced volume

coils. Adaptive sampling increases the dynamic range of the

measurement. Measuring only the center of the stimulated

echo removes chemical shift and B0 inhomogeneity effects.

The technique was also used to design an inverse filter to

remove the eddy current effects in a larger coil set.

We added pulsed field gradient and imaging capability

to a 7 T commercial spectrometer to perform neuroscience and

embryology research and used it in preliminary studies of

binary liquid mixtures separating near a critical point.

These techniques and coil designs will find application

in research areas ranging from functional imaging to NMR

microscopy.














GENERAL INTRODUCTION


As pulsed field gradient technology for NMR matures,

new and diverse applications develop. Pulsed Gradient Spin

Echo techniques allow the measurement not only of the bulk

diffusion tensor, but of the structure factor of the

sample.1 Editing techniques use pulsed field gradients to

simplify the complex spectra of biomolecules.2 Local

gradient coils allow functional imaging in the human head.3

NMR microscopy can require field gradients much larger and

switched more rapidly than conventional imaging

experiments.4 Localized spectroscopy allows chemical shift

information to be collected from specific voxels in a living

animal.5 This paper will address some approaches for

producing and evaluating pulsed field gradients.

A technique was developed to measure the eddy current

field that persists after a field gradient is switched off

and, based on the measurement, a filter to correct for the

eddy current field was designed. The technique, which

employs a series of experiments based on the stimulated

echo, was then used to evaluate the performance of the



1D. G. Cory and A. N. Garroway, Magn. Reson. Med. 14, 435, 1990.
2D. Brihwiler and G. Wagner, J. Magn. Reson. 69, 546, 1986.
3K. K. Kwong et al., Proc. Natl. Acad. Sci. 89, 5675, 1992.
4Z. H. Cho et al., Med. Phys. 15, 815, 1988.
5H. R. Brooker et al., Macn. Reson. Med. 5, 417, 1987.









filter. If an eddy current field persists during the period

when the NMR signal is detected, distortions in the spectrum

or image will result. The distortions are particularly

severe when chemical shift information is obtained in the

same experiment as spatial localization by encoding spatial

information in the phase of the NMR signal. It is important

to be able to measure the residual gradient field, which is

usually due to eddy currents in the metal structures of the

magnet, so that it can be corrected by changing the drive to

the gradient amplifier, or by whatever other technique is

available, and to evaluate the remaining uncorrected field

to estimate the distortion that will result in a desired

experiment.

One way to avoid eddy currents for experiments such as

spatially selective spectroscopy is to employ actively

shielded gradient coils. Another, much simpler, approach is

to reduce the size of the gradient coil so that it is widely

separated from the eddy-current-producing structures in the

magnet. This approach is only possible when the clear bore

of the magnet is much larger than the volume of interest,

which is often the case. To make possible experiments, such

as spatially selective spectroscopy, that require rapidly

switched high intensity field gradients, I developed pulsed

field gradient systems based on reduced volume gradient

coils for a 2 T, 31 cm bore magnet used for small animal

studies. This magnet was replaced with a 4.7 T, 33 cm bore

magnet, and the gradient systems were adapted accordingly.









These pulsed field gradient systems offer much better

performance than the large and unshielded gradient system

supplied with the magnet, given their limitation on sample

size. I also developed a pulsed field gradient coil for a 7

T, 51 mm bore magnet used for NMR microscopy and

spectroscopy.

Another experiment which requires gradient coils to

perform exceptionally well is functional imaging of the

human brain. The head is much smaller than the whole-body

magnets in general use. A smaller coil can allow faster

switching to higher gradient fields, as well as reduce eddy

current fields. In order to get a gradient coil that is

matched to the size of the head, some provision must be made

to allow for the shoulders. Conventional designs, even

existing designs with a large linear volume, have current

return paths arrayed on both sides of the linear volume. A

coil matched to the size of the head would not fit over the

shoulders. A coil that trades radial linear region for

increased axial linear region is more appropriate. A design

utilizing concentric return paths was developed that

significantly improved the axial region of linearity. A

prototype was constructed and tested.

In order to perform NMR microscopy and pulsed field

gradient experiments, we adapted an NMR spectrometer and

probe for a 7 T, 51 mm bore magnet. The instrument included

a simple amplitude modulator to carry out slice selection.

A probe that allowed sample loading from above was






4


constructed. Artifacts were eliminated from the images. A

software interface that allows the user to set up an

experiment by entering values in a spreadsheet was

developed. Useful contrast was obtained on fixed biological

samples. Preliminary imaging experiments on both biological

and nonbiological systems were carried out.














MEASUREMENT OF EDDY CURRENT FIELDS


Introduction

It is well known that, when a current pulse is passed

through a field gradient coil in a superconducting magnet,

eddy currents are produced in the conducting structures of

the magnet. Experiments such as diffusion-weighted imaging6

and multiple-quantum spectroscopy7 require that the eddy

current field be a much smaller fraction of the applied

field than do conventional spin-echo magnetic-resonance

imaging experiments. Strategies to reduce the eddy current

field consequently become increasingly important. The two

effective strategies are signal processing of the gradient

demand, known as preemphasis, and self-shielding of gradient

coils, which greatly reduces the interaction of the coil

with the metal structures of the magnet. Often, the two

techniques are used together. When the sample or subject is

substantially smaller than the magnet, another approach is

to minimize the size of the gradient coil. In order to

evaluate and improve the effectiveness of these three

strategies, it is desirable to have a technique to measure

eddy current fields. To implement the preemphasis, it is

necessary to measure the eddy current field in order to


6D. G. Cory and A. N. Garroway, Magn. Reson. Med. 14, 435, 1990.
7C. Boesch et al., Magn. Reson. Med. 20, 268, 1991.








cancel it. An eddy current measurement technique is also

useful in order to evaluate the possibility of performing a

given experiment with available hardware. In this chapter,

a technique for measuring and analyzing the time behavior of

eddy current fields is developed and experimental results

are presented. Some general physical considerations of eddy

currents are discussed, and existing techniques for eddy

current field measurement are reviewed.

An introduction to the Bloch equations will be

preliminary to a discussion of the effect of the eddy

current field on the nuclear magnetization. The Bloch

equations provide a phenomenological description of some

aspects of the behavior of spins in a magnetic field. Let M

be the bulk nuclear magnetization, y the gyromagnetic ratio,

By the polarizing magnetic field, and B1 the amplitude of

the radio frequency excitation field which has rotational

frequency o. T1 and T2 are the time constants associated

with longitudinal and transverse relaxation, respectively.

Mx = Y(BoMy + BlMz sin (ot) Mx/T2 [1]

My = Y(BiMZ cos Ct BoMx) My/T2 [2]

Mz = -y(BlMx sin Cot + BiMy cos Cot) (Mz Mo)/IT [3]

Instead of T2, the symbol T2* is used to denote the time

constant of apparent transverse relaxation when

inhomogeneity in BO is present. Neglecting the effects of

T1 and T2 and assuming B1 = 0, the equations can be

simplified.









MX = TMyBo [4]

My = -'YMBO [5]

Mz = 0 [6]

We can introduce a complex transverse magnetization M = Mx +

iMy so that

M = -iyMB0. [7]

Assume that Bo consists of a constant and a component

linearly dependent on position: B0 = BO+gx. BO is

independent of time and space, while g is quasi-static. If

we define m, the magnetization in the rotating frame, by

M = me-iBOt [8]

then

M = -iyBOM + me-iyBot [9]

Substituting back into Equation [7] gives

-iyBOM + ine-iyBot = -iyM(BO + gx). [10]

Simplifying Equation [10] yields

ne- iyBO t = -iyMgx. [11]

Combining Equation [11] with Equation [8] yields

m = -iygxm, [12]

which has the immediate solution

-iyxft gdt'
m(t) = m(to)e t [13]

If the magnetization has been prepared to a non-zero m(to)

by a radio frequency pulse, the evolution described by

Equation [13] is called a free induction decay (FID).









Consider the characteristics of a general eddy current

field. The eddy currents give rise to a magnetic field that

roughly tends to cancel the applied field of the gradient

coil. The spatial dependence of the eddy current field is

not exactly the same as the applied gradient field.8

The time behavior of the eddy current field is a

multiexponential decay, which can be seen by considering the

form of the solution to the differential equation governing

the decay of magnetic induction due to current flow in the

conductor. Maxwell's equations9 in a vacuum in SI units are

V B = 0 [14]

V *E = [15]
aB
V X E + -- = 0 [16]
at

V X B oF00 DE = o0J [17]

where B is the magnetic induction and E is the electric

field, p is the charge density, e0 and go are the

permittivity and permeability of free space, and J is the

current density. We also assume Ohm's law, J = GE, where y

is the conductivity, assumed to be isotropic and

homogeneous. Taking the curl of both sides of Ampere's law,

Equation [17], neglecting the displacement current, and

using the identity



8R. Turner and R.M. Bowley, J. Phys E: Sci. Instrum. 19, 876, 1986.
9J. D. Jackson, Classical Electrodynamics, John Wiley & Sons, New York,
1975.









V x (V x A) = V(V .A) V2A [18]

gives

-V2B = g0V X J. [19]

Using Ohm's law to eliminate J for E, neglecting the

displacement current, yields

-V2B = 0CFV X E, [20]

so Equation [16] allows this to be expressed as

rB
V2B = 0a [21]
at
The decay of the magnetic induction must be a solution to

this diffusion equation. Separation of variables gives

solutions for the time part having an exponential time

dependence. This makes it possible to correct for the

linear spatial term in the eddy current field with a linear

filter network. Such a network is known as a preemphasis

circuit.

In a superconducting magnet, the conducting structures

involved are often at very low temperatures and hence have

much greater conductivity than might otherwise be expected.

For example, pure aluminum at 10 K has a resistivity of

1.93 x 10-12 J2-m, while at a room temperature of 293 K its

resistivity10 is 2.65 x 10-8 Q-m. The time scale of the

eddy current decay is directly proportional to its

conductivity, as can be inferred from Equation [21], so eddy

currents will persist 13,700 times longer in an aluminum


10D. R. Lide, (Ed.), CRC Handbook of Chemistry and Physics, 72nd
Edition, CRC Press, Boca Raton, 1991.









structure at 10 K than one at 293 K. In a commercial

aluminum alloy the conductivity will vary from that of the

pure metal, especially at low temperature, so the effect may

not be as great. In practice, the principal source of eddy

current fields is generally the innermost low temperature

aluminum cylinder, which is at approximately the boiling

point of liquid nitrogen, 77 K. The resistivity of aluminum

at 80 K is 2.45 x 10-9 Q-m, so the time constant is about

11 times greater than it would be at room temperature.

We consider the desirable characteristics for an eddy

current measurement technique. Our primary goal will be to

measure the eddy current field in order to evaluate the

feasibility of performing a given experiment, not to

compensate for the eddy current field. Therefore dynamic

range is more important than absolute accuracy. It must be

possible to measure eddy currents produced by specific pulse

sequences, probably by appending the eddy current

measurement experiment to the end of the sequence under

evaluation. It is also preferable to have a technique that

is insensitive to inhomogeneity so that no swimming is

necessary. Since the shim coil power supply may respond

dynamically to the gradient pulse and distort the measured

eddy current field, it is useful to be able to turn the shim

supply off. Experiments based on Selective Fourier

Transform11 and other chemical shift imaging techniques rely


11H. R. Brooker et al., Maan. Reson. Med. 5, 417, 1987.









for spatial localization on the integral of the eddy current

field, so it is desirable to have a measurement technique

that is based upon the integral of the eddy current field.

If possible, the technique should have no special hardware

requirements.


Literature Review

Many workers have addressed the problem of eddy current

measurement and compensation in the literature. The two

aspects of the eddy current field to measure are the spatial

and time behaviors. We review publications that include

descriptions of eddy current measurements, although in most

cases the emphasis is placed upon the preemphasis

compensation process and its effectiveness, not the

measurement. The measurement process can be divided into

techniques that detect the derivative of the eddy current

field, those that detect the eddy current field itself, and

those that detect the integral of the eddy current field.

The derivative of the field is sensed by a pickup coil

consisting of turns of wire through which the changing flux

of the eddy current field produces an electromotive force

that is proportional to the rate of change of the field.12

A high impedance preamplifier boosts the signal. An analog

integrator is usually used to convert the measured voltage

into a quantity proportional to the field, although it is

possible to use digital integration. When used in a magnet


12D. J. Jensen et al., Med. Phys. 14, 859, 1987.









at field, the pickup coil is sensitive also to any change in

flux resulting from mechanical motion, which can contaminate

the measurement. Since the field of the main magnet is

generally about four orders of magnitude larger than the

eddy current field and the time scale of mechanical modes is

smaller than that of the longer time-constant eddy currents,

mechanical stability of the coil is crucial. Drift in the

analog electronics is another potential difficulty with the

pickup coil technique. Even with digital integration, the

preamplifier can experience thermal drift on time scales not

too different from the eddy current field. In spite of

these difficulties, pickup coils are simple to use and can

be used effectively to adjust preemphasis compensation.

They are used routinely to correct for eddy currents in

commercial, clinical MRI installations.13

A different approach to measuring the eddy current

field is through its effect on the NMR resonance. One

advantage here is that a pickup coil and its associated

hardware are not needed. These proportional techniques

measure a frequency shift in the NMR resonance that is

directly related to the eddy current field.14 From Equation

[13], the phase of freely-precessing magnetization in the

rotating frame at time t with respect to to can be written

as




13Personal communication, Dye Jensen.
14Ch. Boesch et al., Magn. Reson. Med. 20, 268, 1991.






13

= yxJ gdt'. [22]

The instantaneous frequency o(t), which can be defined as

the rate of change of the phase of 0 by (0(t) = d)/dt is

related to the eddy current field through the Larmor

equation (0 = yB. Magnetic field homogeneity is important

when using this approach, so that the FID will persist long

enough to obtain a meaningful measurement.

In another approach based on the NMR experiment, the

phase of the magnetization 0 is measured at a single point

in time. The phase at that point reflects the integral of

the eddy current field over certain intervals in the

experiment. Since only one point is sampled in each

experiment, many more experiments are required to map the

decay of the eddy current field than with the proportional

techniques. However, T2* and off-resonance effects do not

affect the usefulness of the technique. The experiment

proposed later is a single-point acquisition technique.

All the techniques surveyed were implemented for

unshielded gradient units, although preemphasis is typically

used on systems with shielded gradient sets as well.15

Boesch, Gruetter and Martin of the University Children's

Hospital in Zurich16,17 measure and correct eddy currents on

a 2.35 T, 40 cm Bruker magnet. The unshielded gradient set

has an inner diameter of 35 cm and a maximum gradient of 1


15R. Turner, Magn. Reson. Imaq. 11, 903, 1993.
16Ch. Boesch et al., Magn. Reson. Med. 20, 268, 1991.
17Ch. Boesch et al., SMRM 1989, 965.









G/cm. They use two NMR techniques to measure the eddy

current field. They interactively correct, using a 12 cm

diameter glass sphere filled with distilled water, and they

use no spatial discrimination in order to get all spatial

components. The experiment consists of a 2.5 s gradient

pulse of 0.6 G/cm followed by a train of 8 FIDs. There is a

20 ms delay between the time the gradient is switched off

and the first radiofrequency (RF) pulse. The RF pulses have

a 20 flip angle in order to reduce echo signals. The total

eight FID acquisition time is 200 ms. They solve the Bloch

equation for a sample with a single resonance frequency and

decay constant and extract

yABz(t) = (MydMx / dt MxdMy / dt) / (M2 + My) [23]

as an estimate of time-dependent Bo shift. They claim this

gives enough information for interactive preemphasis

adjustment. The one measurement they publish is of an

already corrected system and shows 7ABz(t) decaying from 2

to 0 ppm as time t increases from 20 to 200 ms. Glitches

are apparent at the ends of the FIDs.

To map the spatial variations, they place a stimulated

echo (STE) imaging experiment following the gradient pattern

of the experiment they want to analyze. The STE sequence is

applied with and without the preceding gradient pattern.

The difference in phase is considered to be due to the time

integral of the eddy current field in the interval between

the first two pulses of the STE sequence. A series of









slices tilted by multiples of 22.50 is obtained from the

same 12 cm diameter phantom. The images were phase

corrected. The phase of points along the z axis and on

circles around the z axis was measured and used as data for

a polynomial regression analysis to determine the

coefficients of the various spatial harmonics. A table of

the harmonic components following a 2.5 second x gradient

pulse of 0.3 G/cm is presented. The delay between the end

of the gradient pulse and the first RF pulse in the three

pulse STE experiment is 20 ms, and the delay between the

first and second RF pulse is 15 ms. The experiment was

conducted following adjustment of the preemphasis unit. In

decreasing order of magnitude, x, z, y, z2, xz2, xz, and x2-

y2 terms were present. The value of the B0 term was not

reported. Note that after x, the dominant terms should be

eliminated by the symmetry of the coil/cylinder system.

Only the x and xz terms would appear in an ideal system.

The presence of terms having even-order in x can be due to

two reasons. First, the terms may really exist due to

asymmetries in the magnet and gradient coil, crosstalk

between amplifiers, etc. Second, the spherical harmonic

analysis is highly sensitive to the point chosen to be the

origin, and the most favorable origin may not have been

employed.

A series of the phase-modulated images is presented as

well, with delays of 5, 20, 50, and 100 ms between the x









gradient and the STE imaging sequence. The images are all

from an already compensated system.

Van Vaals and Bergman of Philips Research Laboratories

in Eindhoven, the Netherlands,18,19 have a 6.3 T, 20 cm

horizontal bore Oxford magnet with 2 G/cm non shielded

gradients leaving a 13.5 cm clear bore. To measure the

eddy currents, they use a 4 cm diameter spherical phantom.

After swimming, they perform a simple "long gradient pulse,

delay 8, RF pulse, acquire" sequence. The gradient is

switched on for typically 3 s, but at least 5 times the

largest eddy current time constant. For various values of

8, the magnet is re-shimmed to maximize the signal during

the first 10 ms of the FID. The difference in shim values

with and without the gradient pulse is interpreted to be a

spherical harmonic expansion of the eddy current field.

Exact values of 8 are not listed, nor are tables of shim

values. Instead, the amplitudes and time constants of the

eddy current fields, as derived by a Laplace transform

technique, are given. Only the B0 and linear terms are

given; presumably only these terms were shimmed.

Jehenson, Westphal and Schuff of the Service

Hospitalier Frederic Joliot, Orsay, France, and Bruker,20

corrected eddy currents on a 3 T, 60 cm Bruker magnet. The

0.5 G/cm unshielded gradient coils had a clear bore of 50



18J. J. van Vaals and A. H. Bergman, J. Magn. Reson. 90, 52, 1990.
19J. J. van Vaals et al., SMRM 1989, 183.
20P. Jehenson et al., J. Macrn. Reson. 90, 264, 1990.









cm. They use the same type of multiple FID sequence as

Boesch, Gruetter and Martin, with an exponentially

increasing sampling interval and 30 sampling points. The

gradient prepulse is 10 s in length. The first FID is

sampled at 1.5 ms after switching off the gradient, and

sampling continues for 4 s using multiple FIDs. They plot

the measured field vs. the time with and without

compensation. They use a 1 mm by 3 mm water-filled

capillary positioned at +/- 5 cm to discriminate Bo and

linear terms. They do not consider crosstalk or higher-

order terms. They use the same Laplace transform technique

as van Vaals and Bergman, but they apply it iteratively to

get better correction.

Heinz Egloff at SISCO (Spectroscopy and Imaging

Systems, Sunnyvale, CA)21 used a pickup coil to measure eddy

current fields. To correct the B0 component of the eddy

current fields, he moved the gradient coils until the field

shift was eliminated.

Riddle, Wilcott, Gibbs and Price22 considered the

performance of a Siemens 1.5 T Magnetom. They measured the

instantaneous frequency do/dt of a 100 ml round flask

(presumably filled with water) following a 256 ms, 0.8 G/cm

gradient pulse. They present plots for imaging and

spectroscopy shims as well as for the gradient pulse. They

endorse do/dt as an indication of shim. It would seem to


21H. Egloff, SMRM 1989, 969.
22W. R. Riddle et al., SMRM 1991, 453.









work only for single-line samples, however. Following the

gradient pulse, the plot of d'/dt contains peaks that are

not explained. They may be an indication of the true do/dt,

or they may be artifacts from beginnings and ends of FIDs.

The sensitivity of the technique as presented here seems to

be about 1 Hz.

Hughes, Liu and Allen23 of the Departments of Physics

and Applied Sciences in Medicine at the University of

Alberta measured the eddy current fields of their 2.35 T, 40

cm bore Bruker magnet. After 57 delays ranging between 500

is and 2.5 s following a 0.2 G/cm gradient pulse the FID was

measured and the offset frequency of the line determined.

They placed a 13 mm diameter spherical water sample at +/-

1, 2, 4 cm along the axes of the radial gradients under

test. A four-exponential fit was applied to all six

locations simultaneously. The shortest time constant was

associated with the amplifier rise time. An interesting

plot shows that the field associated with each time constant

is essentially linear. The Bo fields associated with the

various time constants are different, however, suggesting a

unique isocenter for each time constant.

Zur, Stokar, and Morad24 of Elscint in Israel place a

doped water sample at +/- 5 cm from the center in the

direction of the gradient of the field. A train of 256 FIDs

is acquired after switching off the gradient. Each FID is


23D. G. Hughes et al., SMRM 1992, 362.
24Y. Zur et al., SMRM 1992, 363.









Fourier transformed, bandpass filtered, then inverse

transformed. The instantaneous magnetic field is obtained

from d)/dt. The digital filtering points to a problem with

phase measurements. The low-pass filters required to

eliminate Nyquist aliasing and to improve the signal-to-

noise ratio (SNR) distort the phase of the received signal.

Digital filtering enables one to recover the SNR ratio of a

small bandwidth without significant phase distortion.

Wysong and Lowe25 at Carnegie Mellon and the University

of Pittsburgh measured eddy current fields on a Magnex 2.35

T 31 cm magnet with unshielded gradient coils. A 1 cm

diameter sphere containing water doped to T-T2-~1 ms is

used. A 0.9 G/cm gradient is applied for 1.0 s, then ramped

down in 128 ps. A train of pulses of flip angle n/2 set 1

ms apart is applied for 1 second. One point is sampled for

each FID. With the system adjusted so the FID is in-phase

in the absence of a gradient field, the out-of-phase

component is proportional to sin(yABte-t/2) = yABte-t/2 for

small values of time and gradients.

Keen, Novak, Judson, Ellis, Vennart and Summers26 of

the Department of Physics, University of Exeter, propose

using a phantom slightly smaller than the imaging volume.

Having switched off the gradient, they delay a variable

time, then pulse and acquire the FID. The Fourier transform



25R. E. Wysong and I. J. Lowe, SMRM 1991, 712.
26M. Keen et al., SMRM 1992, 4029.









of the FID represents a projection of the phantom in the

quasi-steady eddy current field. Measuring the distance

between the peaks that appear as edge artifacts gives the

eddy current field.

Teodorescu, Badea, Herrick, and Huson27 at the Texas

Accelerator Center and Baylor College of Medicine measured

eddy current fields in their 4 T, 30 cm superferric self-

shielded magnet. The magnet was operated at 2.19 T. They

follow Riddle et al.28 in their measurement. A small

phantom is placed at various off-center locations. They use

a 0.8 G/cm gradient pulse of 15 ms and a 750 gs rise/fall

time. This is followed by an FID (or a series of them) that

is acquired for 20 ms. They compare this to the result

obtained from a pickup coil.

The eddy current field was measured with a sense coil

and analog integrator by Morich, Lampman, Dannels, and

Goldie.29 They used a Laplace transform approach to derive

correct parameter values for an analog inverse filter to

compensate for the eddy currents. The analog inverse filter

was of conventional design,30 placed at the input of the

gradient power supply. The theory was tested on an Oxford

Magnet Technology whole body superconducting magnet.

The approach is based on the ease with which a linear

system can be analyzed in the reciprocal space s defined by


27M. R. Teodorescu et al., SMRM 1992, 364.
28W. R. Riddle et al., SMRM 1991, 453.
29M. A. Morich et al., IEEE Trans. Med. Imac. 7, 247, 1988.
30D. J. Jensen et al., Med. Phys. 14, 859, 1987.









the Laplace transform. We can understand the calculation as

follows. Assume the gradient field for t>0 in response to a

unit step function is

N
g(t) = 1- ae-tw i i [24]
i=1

The amplitudes ai and time constants Ti can be determined

through a best-fit to experimental data. To determine the

inverse filter, the first step is to deconvolve the step

function to find the impulse response h(t), which can more

conveniently be accomplished by a multiplication in the

complex frequency space, s. The equivalent function G(s) is

obtained by a Laplace transform

1 N a
G(s) = _- N- ai [25]
S is + Wi

Then the impulse response in the s domain, H(s), is found

through the relation

G(s) = H(s)/s, [26]

so that
N
H(s) = sG(s) = 1 N ais [27]
s + wi
i=1 1
is the impulse response. The inverse filter's impulse

response is just the reciprocal of the impulse response of

the eddy currents,


YH(s) = N [28]
s + Wi









The step response of the inverse filter, F(s), is the

convolution of a step function and the impulse response:

1 1
F(s) =- [29]
sH(s) N ais2
S -
S + wi
i=1 i

The amplitudes bi and time constants vi of the inverse

filter can be read directly from the inverse Laplace

transform, f(t), of F(s):

N
f(t) = 1 + bie-t/vi [30]
i=l

The inverse Laplace transform was performed by matrix

inversion for a four-time-constant case using Gaussian

elimination.

Now the appropriateness of these techniques to the

project of following the time evolution of the eddy current

field can be considered. Two of the techniques, those of

Egloff and Morich, involve the use of a pickup coil,

preamplifier, and integrator. We choose to confine

ourselves to NMR techniques. The procedures of Boesch, van

Vaals, Jehenson, Riddle, Keen, Hughes and Teodorescu require

swimming to correct for the inhomogeneity of Bo. The fact

that T2* must be reasonably long also limits the region

where eddy current fields can be measured to well inside the

active imaging volume. Wysong and Zur propose similar NMR

techniques that do not require swimming. In general,

however, it is samples with long relaxation times that are

most sensitive to small eddy current fields, and the use of









a sample with especially short (T~-T2-1 ms) relaxation times

is not an obvious way to detect low-level fields. The T2 of

the sample limits the duration of the interval in which

phase can be sampled. Another drawback is that the trains

of 7/2 pulses will produce stimulated echoes, even if T1 is

on the order of the interpulse separation. However, this

may be the most promising of the techniques surveyed.


Spin-Echo Techniques

Distortions in the phase of spectra spatially localized

with a two-pulse Selective Fourier Transform technique31

were observed by Mareci.32 He observed that the distortions

were reduced by lengthening the echo time, consistent with

the known behavior of field distortions due to eddy currents

induced in the metal structures of the magnet by the pulsed

gradient fields used for spatial localization. We consider

how a series of spin echo experiments identical except for

n/2y it.

RF nA


g


I I I
0 TE/2 TE

Figure 1. Two-pulse experiment with pulsed field gradient.
The long trailing edge of the gradient pulse indicates
distortion due to the eddy current field.


31H. R. Brooker et al., Magn. Reson. Med. 5, 417, 1987.
32T. H. Mareci, Personal communication.









increasing echo time (TE) gives an indication of the eddy

current field distortion as a function of time. Consider

the evolution of the rotating-frame magnetization m in the

presence of the gradient field g illustrated in the pulse

sequence in Figure 1. For O
Equation [13] so

-iyx f gdt'
m(t) = m(O)e 0 < t < TE / 2. [31]

We can also apply the result directly to describe the

magnetization's evolution following the n pulse. Let TE/2+

be the time just after the n pulse. Then


m(t) = m(TE / 2+)e TE / 2 5 t. [32]

The t pulse along x inverts the sign of the imaginary part

of m(t), equivalent to taking the complex conjugate:

iyx JTE/2 gdt'
m(TE / 2+) = m*(0)e 0 [33]

Putting it together gives

iyxITE/2 gdt' -iE/2 gdt
m(t) = m (0)e e TE / 2 < t [34]
[ (TE/2 : ,
iyx gdt- gdt'
m(t) = m*(O)e J JTE/2 J TE / 2 < t. [35]

Measurement of the phase exactly at the center of the Hahn

echo should remove off-resonance effects, whether due to

chemical shift or field inhomogeneity. Now it remains to be

shown that measurements of the phase at a series of echo

times can be used to find g(t). If 00 is the phase without

a gradient pulse applied, then








(TE) 40 = yx E/2 gdt' J/ gdt] = yx2fTE/2 gdt' JTEgdt].

[36]

We define a function G(t) by

G(t) = yxJ gdt' [37]

which simplifies the expression above for ):

O(TE) 40 = yx[2G(TE / 2) G(TE)] [38]

By measuring 00 and measuring ) at a number of echo times,

we hope to be able to extrapolate the function G(TE), whose

rate of change gives the eddy current field. By performing

a series of experiments in which the values of TE are

related by successive powers of two (TEi1j = 2TEi), we can

obtain a series of coupled equations. Using the shorthand

0 (TEi) = 0j,

Oi+1 = y [2G +i] i = 1, 2, [39]

Inverting for G, yields

GI = [(4i+l 0o)/yx + Gi+1]/2 i = 1, 2, ... [40]

For large enough i, Gi = Gi+I, and the equation has an

immediate solution. The remaining Gi can be determined

recursively. The rate of change of G(TE) is the eddy

current field.

Experiments and subsequent data analysis have pointed

to several drawbacks in this approach. The first is that

the echo time TE limits the maximum length of the gradient

pulse. A gradient pulse long in comparison to the eddy

current decay time approximates a step function, which









simplifies the analysis of the eddy current response.33

However, lengthening the TE reduces the time resolution of

the experiment. Placing the gradient pulse before the

excitation pulse as in Figure 2 eliminates the problem and

decouples the length of the gradient pulse from the echo

time.

n/2y nx

RFnA






Id2 0 d3 TE/2 TE/2

Figure 2. Gradtest v.1.2 is a spin-echo experiment for
measuring the eddy current field following a pulsed field
gradient.


The above analysis assumes a point sample. Any real

sample has finite extent and will experience some dephasing,

and associated signal loss, as its phase evolves in the

gradient field. By not subjecting the transverse

magnetization to the gradient pulse but only to the eddy

current field, the dephasing effect is reduced. Another

drawback proved to be that the signal decayed due to T2

relaxation before Gi stabilized. With the gradient pulse

before the excitation, the condition Gi = Gi, could be met

for small values of TE. However, for large TE we could


33M. A. Morich et al., IEEE Trans. Med. Imaq. 7, 247, 1988.









assume that g = 0 while for small TE, g # 0. To solve for

the Gi it is necessary to know one of them in advance, so to

determine Gi for large TE, another experiment was performed.

TE was held fixed at a large value and d3, the interval

between the end of the gradient pulse and the RF excitation

pulse, was varied in steps of TE/2. A system of

simultaneous equations describes the phase obtained by

varying d3 in steps of TE/2, starting with d3 = 0:

O(TE + d3) 00 = yx[2G(TE / 2 + d3) G(TE + d3) G(d3)]

[41]

The problem of signal decay due to T2 is thus circumvented.

This technique could be used by itself or, as we used it,

only to obtain a starting point for varying TE.

A remaining difficulty is the ambiguity of phase

measurement. Phase can be directly measured only modulo

3600, but the accumulated phase in our experiment may be

much greater. One way around this difficulty is to reduce

the applied gradient so that we can be sure that our sample

rate is above the Nyquist limit, so that 4j+1 i < 1800

To get an upper bound that guarantees no phase ambiguity,

assume that the eddy current field has the same amplitude as

the applied field before the n pulse and zero amplitude

following the 7 pulse. Protons process at 4258 Hz/G. To get

a measurement for TE/2 = 512 ms without phase ambiguity

would, for a sample 1 cm from the center, require a gradient

pulse no greater than 0.000229 G/cm. Such a small gradient









pulse would result in no detectable phase accumulation in

practical cases. Experimental experience showed that it was

not simple to choose in advance a gradient amplitude that

would result in measurable phase accumulation, but no phase

ambiguity, at all echo times. Instead, we repeated the

experiment for a series of increasing gradient amplitudes.

For phase changes of less than 3600, the phase doubles as

the gradient doubles. We could keep track of phase

accumulations greater than 3600, thereby decreasing the

minimum detectable eddy current field.


Stimulated Echo Techniques

The stimulated echo (STE) has advantages over the spin

echo as the basis of an eddy current field measurement

experiment. Consider the stimulated echo sequence Gradtste

in Figure 3. The "e" at the end of the pulse sequence name

indicates that this is a stimulated echo experiment. A third

pulse is required to excite a stimulated echo. The

magnetization of interest is flipped into the transverse

plane by the first RF pulse, where it accumulates phase


RF n n





I I I I I
t T t
grad decay 1

Figure 3. Diagram for Gradtste, a three pulse stimulated
echo experiment for measurement of the eddy current field.









shift due to static field inhomogeneity and eddy current

fields. Then, stored by the second RF pulse along the z

axis, the magnetization accumulates no more phase until the

final RF pulse tips it back into the transverse plane. The

phase accumulation due to static inhomogeneity now unwraps,

resulting in the stimulated echo. If tj is long enough,

there is essentially zero eddy current field in the second

T, so the phase accumulated due to eddy current fields in

the first T is preserved.

It is possible to follow the eddy current decay by

incrementing either tdecay or T between experiments. If T is

incremented, the procedure for determining the eddy current

field is similar to that for spin echo experiments. The

phase shift for two experiments with different T is

subtracted to get the integral of the eddy current field in

the time between the earlier and later T. A more direct

approach is to increment tdecay between experiments, keeping

T small. Using this approach, each experiment yields the

integral of the eddy current field over a short interval T.

Dividing by T yields the average eddy current field in the

interval.

Two advantages of the STE are immediately evident. A

single STE experiment can be directly related to phase

accumulation in a single interval, eliminating the need for

the recursive data analysis or simultaneous equations

associated with the spin echo technique. This would also

seem to make the choice of gradient pulse amplitude more









straightforward. Since tI is limited by T1, which is

generally longer than T2, it is possible to sample with

smaller residual gradient field than in the spin echo

experiment.

The eddy current field is subject to a multiexponential

decay. The integral of a multiexponential decay is another

multiexponential decay. We can expect these functions to be

reasonably smooth. That is, if we notice that the phase is

not changing much between delay increments, we could either

increase the delay increment or increase the amplitude of

the gradient pulse. This is a form of adaptive sampling,

since the sampling strategy for the gradient field depends

upon its behavior. The sampling technique should be capable

of following the residual field decay when preemphasis is

used, and in this situation the field will not in general

decay monotonically, since some of the decay components may

be overcompensated. Therefore the adaptive sampling must

also be able to decrease sensitivity when needed.

Since the eddy current field generally changes most

rapidly at short times, varying T to keep the measured phase

shift approximately constant for each value of tdecay yields

less densely spaced measurements when the field is changing

slowly. We have implemented such an adaptive sampling

technique by writing a recursive macro Adgrad in the Varian

MAGICAL language to perform a series of measurements in

which T is varied to "lock" the phase shift to 45. The

macro functions as a command to the Varian program "VNMR"









through which the spectrometer is controlled. Adgrad allows

the automatic measurement of the eddy current field over a

large dynamic range. Forty-five degrees is large enough to

measure with enough precision and yet small enough to

minimize the possibility of aliasing. The values of phase


'user enters
" Adgrad ( t max) "


Figure 4. Flow chart of the macro Adgrad, which executes
adaptive sampling of the eddy current field. The dotted
portion is not part of the macro.









shift A0 and T are easily reduced to a plot of the eddy

current field vs. time. A flow chart of Adgrad is found in

Figure 4. It is most easily explained in the context of the

whole experimental procedure. The user notes the phase of

the STE for an experiment with gph, the value of the

gradient pulse, set to zero. He then selects a combination

of T, tdecay, and gph that results in phase accumulation of

about 450 and acquires an FID. He also removes the file

"phase.out" if it remains from a previous session. Then he

executes the macro Adgrad(tmax, o0), where tmax is the value

of tdecay at which the macro will stop and 00 is the phase

with gph = 0.

Adgrad first calls the macro Calcphase to compute the

phase A0 at the center of the acquisition window (that is

also the center of the FID) for the data already in memory.

Adgrad then stores the values of A0 and T as the first line

in the output file "phase.out." Next, Adgrad tests to find

if tdecay > tmax. If so, it ends the experiment. This

should not occur on the first pass through the test. In the

following two steps, Adgrad sets up the timing for the next

experiment. The new tdecay is set to be greater than the

old by T, to provide for a contiguous series of intervals t.

The new T is set so that if the eddy current field remains

constant, the next measurement will yield a phase A0 of 45.

Now the measurement is started. Following the measurement,

the macro calls itself and the process repeats. When the









tdecay > tmax test is passed, Adgrad returns control to the
operator.

Two other adaptive sampling macros have been developed

for eddy current testing. Adgrad2 changes both T and gph to

lock the phase to 45. The resulting series of experiments

are more closely spaced in time than Adgrad. Using Adgrad2,

it is possible to follow the eddy current field over a wider

range of values than with Adgrad. However, linearity error

in the digital-to-analog converter or nonlinear amplifier

response will be reflected in error in the eddy current

field. Adgradl80 locks the phase to 1800. It can only work

when the phase accumulation is monotonically decreasing yet

never changes sign, which is true for the uncompensated

gradients. Otherwise, Adgradl80 may lose its lock. If the

error in the measured angle is constant, the accuracy of the

technique, when applicable, should be about 5 times better

than for Adgrad or Adgrad2.

In the preliminary data analysis, we assumed that the

eddy current field was essentially constant over the

sampling interval T, so that g = AO/yxT. An Excel

spreadsheet was used to reduce and analyze the data and plot

the results. An example is shown in Figure 5. It is a plot

of the average field in each of the measurement intervals.

Since the field is dropping exponentially, not linearly,

fits to the mean value will have systematic errors. A

better way is to assume a multiexponential decay of the eddy

current field, and then calculate what phase will be








measured in the STE experiment. If we define OI(t) as the

total phase shift from t = 0 to tn for gph = 1, then
n
(D(t) = [42]
i=l
This phase shift is just, for a single experiment,

*((t) = yx g(t'dt' [43]

Now we assume that the eddy current field can be described

by a three-time-constant decay,

g(t) = Ae- tta + Be-t/tb + Ce- c. [44]

Integration gives a function to which the measured phase can

be fit:

D (t) = yx[taA(l e-tta) + tbB(l e-t/tb) + tc( etc)].

[45]

Results

Eddy current measurements were made on several gradient

coils of practical interest. Tests of the Oxford gradient

coils in the 2 and 4.7 T magnets were conducted. For the

4.7 T magnet, the eddy current measurements were used to

adjust the preemphasis network. Measurements of the eddy

current fields associated with home-built gradient coils

were also made. The detailed design and construction of the

coils, on 9 and 15 cm former, is described in the following

chapter.

Initial measurements were made using the spin-echo

technique of Gradtest vl.2. A 5 mm NMR tube with about 5 mm










of H20 trapped by a vortex plug was used as a sample and

placed 1.7 cm from the center of a 2 T, 31 cm horizontal-

bore magnet (as measured from an image). The Oxford Z

gradient in the Oxford 2 T magnet was pulsed to a value of

1000 units or 1 G/cm. The manufacturer-installed

preemphasis filter was in place. A d3 array with four

elements was used to establish the phase value for large

echo times via matrix inversion of simultaneous equations.

An echo time array resulting in a series of coupled

equations was used to work back to 1 ms. The resulting plot

is shown in Figure 5. The bumpiness of the plot may be due

to the preemphasis. Data points are plotted in the center

of the interval for which they represent the average

gradient.



0.045

0.04

0.035

0.03
U
0.025

0.02

m 0.015

0.01

0.005

0 --- I I-Illi'- -
0 100 200 300 400 500 600 700 800 900
t (ms)


Figure 5. Eddy current field as a fraction of applied field
for Oxford gradient coil.









Stimulated echo measurements using the pulse sequence

Gradtste and the macro Adgrad were conducted for the Oxford

gradients as well as for the 9 cm home-built gradient coil

in the 2 T, 31 cm diameter magnet. The eddy currents for

the Oxford gradients were measured with the manufacturer-

installed preemphasis filter in place. The 9 cm coils had

no preemphasis. A 5 mm NMR tube with about 5 mm of H20

trapped by a vortex plug was used as a sample and placed

between 1 and 2 cm from the center of the magnet. The

center of the sample was determined from an image. In all

cases, tgrad = 2 s, dl = 2 s, tj = 0.5 s, and two averages

were acquired. The parameter T was set to 4 ms and tdecay was

1 ms for the initial experiment. The data were analyzed in

Excel spreadsheets. In the plots of field vs. time given in

Figures 6 and 7 for the Oxford and 9 cm coils respectively,

the average of the eddy current field over the sampling

interval is plotted against the middle of each sampling

interval. The eddy current field is represented as a

percentage of the applied gradient. Note that without

preemphasis, the eddy current field due to the 9 cm coil

declines monotonically, while the preemphasis filters

contribute to the measured field of the Oxford coils. For

the 15 cm coil tested in the 2.0 T magnet, eddy current

measurement was used to calculate values for an inverse

filter. The coils and samples were removed between the

experiments before and after preemphasis. The Techron 7540

amplifiers were used to drive the coils in current mode.


























Hm r in

(%) (^)6


_ --,- I


I~ 11 I -


C,, u-


H4 u-
0


S L

I


(%) (4)B


a) 00 r- w


in m (N H-


t


0 H-
I


(%) (4))


S Al 0Co


m in


-mm "m **.


4

*-)



0






,3-
co
0)H












-H




0 -H

0)-


rd
d4
01






o *
O u




44
0) -H






x eo
0
r-
r- O
0 m













00
p H
: 4-4


a) -H
*H (I
(0) 0




rx.4 1


* -


















m
U







.m m
*









U *

W ..


(%) (q)6


r- 0 0 0 0

(%) ( )B


M *
, -- I .


(%) (1)


r-1
-H
-r 0
(s u






S-H
0u
u
( >4



E-H -H


0
0
-A

0



U
-H o
ro
.,-0 )
U2




O rd
a-,-




0(U





Oi -



0(1)



S-H





., --I



>H
-H U

rX :


* N











The same sample and RF coil were used as in the Oxford and 9

cm tests. In all cases, tgrad = 0.5 s, tL = 0.5 s, the

sample position was between 1 and 2 cm from the center, and

the position was measured in an image. The data were


g(t) (%)
5

4

3

2

1


t (s)


0.1 0.2 0.3 0.4


0.5


g(t) (%)
6


4
3
2
1


t (s)


0.2


0.3 0.4 0.5


= t (s)
0.5


0.1 0.2 0.3 0.4


Figure 8. Eddy current field of 15 cm gradient coil set in
2.0 T magnet system before (upper curve) and after
compensation (lower curve). a) X coil; b) Y coil; c) Z coil.


g(t) (%)

6
5
4
3
2
1
0









acquired with Adgrad2, that changes gph as well as T to keep

the phase locked. Eight averages were acquired. The eddy

current field was measured out to 1 s, although the plots in

Figure 8 only show 0.5 s. The data were analyzed both with

the average-field technique used in the Excel (Microsoft,

Inc.) spreadsheet and with a multiexponential curve fit in

Mathematica (Wolfram Research, Inc.). The curve fits seemed

more satisfactory, and are shown in Figure 8. The lower

curves represent the eddy current field after compensation.

The curves plotted are the derivatives of the exponential

curves that were fitted to the raw data. The preemphasis

filter amplitudes and time constants were taken to be those

of the eddy current field itself. This procedure should


0(t) (degrees) Echo Phase Shift



400



300



200



100



t (s)
0.2 0.4 0.6 0.8 1

Figure 9. Fit to raw data of eddy current field of Oxford Z
gradient field for 4.7 T magnet system before compensation.









tend to underestimate the preemphasis required, but since

the unshielded eddy current fields were already less than 5%

of the applied field, the error is not severe.

The 4.7 T magnet that replaced the 2 T 31 cm magnet did

not have manufacturer-installed preemphasis, so the eddy

current measurement techniques were applied to design an

appropriate preemphasis filter. Since the uncompensated

eddy currents were on the order of 50% of the applied field,

the approximation used to compensate the 15 cm coil would

not be effective. An inverse Laplace transform technique

was used to design the filters. The technique was

implemented through the symbolic inverse Laplace transform

capability of Mathematica. An example of a multiexponential

fit to the raw phase accumulation performed with Mathematica

is shown in Figure 9. Eddy current fields before and after

compensation are presented in Figure 10. The upper curves

represent the field before, and the lower curves after,

preemphasis. For the Y coil, the procedure was repeated a

second time to obtain an additional reduction of the eddy

current field. The lowest curve in Figure 10 (b) represents

the eddy current field after the second pass of eddy current

correction.


Conclusion

A technique to measure the eddy current field of a

pulsed field gradient based on the phase of the stimulated-

echo NMR signal has been proposed. Experimental







42

g(t) (%)
50
40
30

20


0 t (s)
a) 0.2 0.4 0.6 0.8 1


g(t) (%)
40
35
30
25
20
15
10
5
0 t (s)
b) 0.2 0.4 0.6 0.8 1

g(t) (%)
40

30

20

10

0 t (s)

C) 0.2 0.4 0.6 0.8 1




Figure 10. Eddy current field of Oxford gradient coil in
4.7 T magnet system before (upper curve) and after
compensation (lower curve). a) X coil; b) Y coil. Lowest
curve was acquired after second-pass preemphasis; c) Z coil.


verification consists of measurements of the eddy current

field before and after preemphasis. The level of the eddy

current field after preemphasis can be interpreted as an

upper limit on the error bar of the measurement. It is only

an upper limit, since other errors also contribute to the









residual eddy current field. Any error in the values of

timing components in the preemphasis filter will add to the

residual eddy current field. Also, any distortion in the

amplifier will reduce the effectiveness of the compensation,

since the filter was designed based on the assumption that

the amplifier is linear. Error in the eddy current

measurement technique itself might be due to other echo

terms than the stimulated echo contributing to the signal.

However, experiments have shown that other echoes are

essentially negligible due to a combination of favorable

timing and phase cycling. In the case of Adgrad2, which

scales gph as well as T, it is clear that some error is due

to inaccuracies in the digital-to-analog converter (DAC)

output level. The applied gradient is then not proportional

to the DAC code, and so there is an error in normalizing to

the applied gradient. Error in the curve fits may be

significant, since in a multiple-exponential fit it is

difficult to get an accurate fit if the time constants are

not widely separated. Note that second-pass adjustment of

the preemphasis was more effective in reducing the residual

eddy current field.

The technique came out of a need to quantify phase

distortions in localized spectroscopy. It is therefore

better-suited to measuring the time integral of the eddy

current field than the field itself, and it is the integral

of the field that gives rise to errors in phase-sensitive

techniques such as SFT. It is often useful to employ the









basic stimulated echo experiment without adaptive sampling

to quantify the integral of the eddy current field over an

interval, and allow one to predict the resulting phase

distortion directly.

The adaptive sampling algorithm is able to follow eddy

current fields that are not simply monotonically decreasing.

I found experimentally that if the angle became much

different than 450, the values for T would bounce around a

lot before stabilizing. This is probably due to the control

being purely proportional. Introducing an integral term

might help.

The relatively large eddy current field produced by the

15 cm Z gradient compared to the X and Y channels is due to

its extended-linearity design, which locates the currents

farther from the region of interest than a Maxwell pair.

The relatively large eddy current field produced by the 9 cm

Y gradient compared to the X and Y channels may be due to a

problem with centering the gradient coils in the bore. The

measured field gradient would then depend strongly on the

position of the sample.34

The contrast in eddy current field between the large

and small coils is clear. There is a factor of about 10 in

eddy current field between the 15 cm coil and the larger

Oxford coil. There is a factor of about 180 in eddy current

field between the 9 cm coil and the Oxford gradient coil set


34D. J. Jensen et al., Med. Phys. 14, 859, 1987.






45


in the X and Z channels. Experimental evidence demonstrates

the advantage in eddy current field obtainable with reduced

size gradient coils.














GRADIENT COIL DESIGN


Introduction and Theory

Although virtually all NMR measurements rely on

auxiliary field coils, there has been comparatively little

published work on the design and analysis of shim and field

gradient coils compared to that for radio frequency coils.

However, high levels of performance have become increasingly

important for these low-frequency room-temperature coils on

several frontiers of the NMR technique. Three of these

areas are gradient coils for NMR microscopy, coils for

spatial localization of spectra, and local gradient coils

for functional imaging of the human brain.

The simple forms of discrete element coil designs have

linear regions that are about 1/3 of the coil radius.35

Therefore the gradient coil must be considerably larger than

the sample. Increasing the linear region would allow

smaller coils to be used, generally improving efficiency and

decreasing eddy current fields. Several approaches are

available to increase the region of linearity. Adding

discrete elements to cancel more high-order terms in the

harmonic expansion has been done successfully by Suits and

Wilken.36 Continuous current density coils have also been


35F. Romeo and D. I. Hoult, Magn. Reson. Med. 1, 44, 1984.
36B. H. Suits and D. E. Wilken, J. Phys. E: Sci. Instrum. 23, 565, 1989.









designed with linear regions that are a large fraction of

the radius.37 We have tried to take a fresh approach,

combining aspects of both continuous and discrete designs.

For a solenoidal main magnet, available radial gradient coil

designs are longer and less efficient than axial designs, so

we have chosen to concentrate on the radial case.


z

z


e
/ 0 y



X

Figure 11. The coordinate system used in the text.


An appropriate starting point to find a new radial

gradient coil design might be: what current distribution on

the surface of an infinitely long cylinder would produce a

field in which the axial component is linearly proportional

to the radial position, Bzo x? To describe surface

currents and fields, we introduce the three coordinate

systems described by Figure 11. Any point can be described

in any of three orthogonal coordinate systems. In the

Cartesian system a point is described by its location along

the three axes (x, y, z). In the spherical system, it is

described by two angles and the distance from the origin:


37R. Turner, J. Phvs. D: ADpp. Phys. 19, L147, 1986.









(r, 0, )) In the cylindrical system, the point is

described by (p, ), z). It can be easily shown that an
azimuthal component of the surface current, J0, proportional

to cos) and independent of z produces the desired spatial

dependence. Neglecting for the moment the problem of

current continuity, there are two possible approaches to

achieving the cos) angular dependence. First, it can be

approximated by superimposing azimuthal currents with no

axial component. The solutions are exactly the same as for

discrete filamentary currents. The first approximation, the

1200 arc familiar from the so-called Golay double-saddle

design,38,39 is shown in Figure 12(a). This class of

designs has been called the "Golay Cage" because of its

correspondence to the double-saddle design. Higher-order

approximations utilizing superimposed arcs are derived by

Suits and Wilken.40 The other approach is to use our

freedom to choose any axial current to meet J4o cos# by

varying the current direction, for example, J4Oc cos#), Jzo

sin). This approach leads to the Cosine Coil shown in

Figure 12(b). Note that in Figure 12 the return paths are

located away from the active volume of the coil. For a coil

of practical length, the current return paths can

significantly reduce and distort the gradient field.




38F. Romeo and D. I. Hoult, Magn. Reson. Med. 1, 44, 1984.
39M. J. E. Golay, Rev. Sci. Inst. 29, 313, 1958.
40B. H. Suits and D. E. Wilken, J. Phys. E: Sci. Instrum. 23, 565, 1989.






















x x

(a) (b)


Figure 12. Two radial gradient coils, a) The Golay Cage
Coil; b) Cosine Coil.


Another approach to current return paths is possible if

we relax the requirement that the current is confined to

the surfaces of cylinders. The current return paths can be

located in the same plane as the azimuthal current paths. A

gradient coil can be constructed of a stack of planes

approximating a current sheet, such as shown in Figure 34

(a) on page 111. The planes include radial as well as

azimuthal current elements. The radial currents do

contribute to the axial magnetic field. It happens that the

third-order harmonic terms eliminated by using 1200 arcs are

independently zero for the radial currents connecting the

arcs. These Concentric Return Path (CRP) Coils can have a

linear region that can be increased in length by stacking

more planes together. The overall combined coil structure

can also be very short, since the return paths do not

require extra length. In order to improve the linear region









beyond that produced by a constant current density along z,

we can adjust the relative current or position of each

planar unit.


Literature Review

This literature review will be focused on efforts to

increase the useful volume of a gradient coil, to optimize

its performance, and to understand the eddy current field

associated with a switched gradient it produces. The

specific requirements of coils of interest for functional

imaging of the human head are discussed, along with several

approaches to meeting those requirements.

Gradient coils can be grouped into two broad

categories: those made up of discrete current elements as in

Figure 13, and those approximating a continuous current

density. The former include the original NMR shim coil

designs,41'42 while the latter approach has been used to

make possible actively shielded gradient coils.43

Anderson described a set of electrical current shims

for an NMR system based on an electromagnet.44 The coils

were located in two parallel planes, one against each

poleface, to allow access to the sample. Each coil was

designed to produce principally one term in the spherical

harmonic expansion of the field. The orthogonality of the



41W. A. Anderson, Rev. Sci. Inst. 32, 241, 1961.
42M. J. E. Golay, Rev. Sci. Inst. 29, 313, 1958.
43P. Mansfield and B. Chapman, J. Maan. Reson. 66, 573, 1986.
44W. A. Anderson, Rev. Sci. Inst. 32: 241 1961.









expansion ensured relatively independent adjustment of the

current in the various coils.

Techniques for designing higher-order shim coils for

solenoidal magnets were set forth by Romeo and Hoult.45

Coils are designed by expanding the Biot-Savart integral for

Bz, the axial component of the field, in a spherical

harmonic series about the center of the coil for simple

filamentary building block currents.
1
Bz(r,9,()) = XAl,mPi(cos )ei. [46]
1=0m=-1
The functions Pmf(cos ) are the associated Legendre

functions. As building blocks are added in the form of arcs

on the surface of a cylinder, more terms in a spherical

harmonic expansion of the field can be set to zero. The

designer connects the building blocks in such a way as to

satisfy the requirement of current continuity, which is not

built into the Biot-Savart law. By setting each undesired

term in the harmonic series to zero, a system of equations

results. The solutions are the current, length, and

position parameters of the coil designs. A Maxwell pair, as

shown in Figure 13(b), is composed of a loop placed at 0 =

600 and another having opposite current direction placed at

8 = 120. This separation is required to cancel the (1, m)

= (3, 0) term, while the odd symmetry cancels the (2, m) and

(1, m # 0) terms. The desired (1,0) term remains. The


45F. Romeo and D. I. Hoult, Maan. Reson. Med. 1, 44, 1984.









simplest coil producing a gradient perpendicular to the axis

of the cylinder is the double saddle or "Golay" coil

illustrated in Figure 13(a). The arcs all subtend 1200 and

are placed at the four angles 01 = 68.70, 02 = 21.30, 1800-1,

and 1800-2, where they produce an (1, m) = (1, 0) term but

no (1, m) = (3, 0) term. The relative current directions

are shown in Figure 13(a). A family of solutions exists for

which the sum of the (1, m) = (3, 0) terms produced by the

arcs cancels, but the (1, m) = (3, 0) terms produced by each

arc are not necessarily zero. We designate such coil

designs by the two angles 01 and 02, so that the design

above would be described as 68.70/21.30. Adding additional

current elements to the coils adds degrees of freedom to

the system of simultaneous equations, and makes it possible

to cancel more terms.

Adding another pair of loops adds two more degrees of

freedom (the current and position of the new loops), and

makes it possible to cancel higher-order terms including (5,

0). Note that the equations are not linear, so for a large

number of current elements the procedure becomes unwieldy.

For shim coils, it is less important to improve the linear

region of a first-order or gradient coil than to design

additional coils whose lowest-order terms are of

increasingly high order. The simple saddle coil in Figure

13(a) designed by this technique has a useful volume with a

radius of about 1/3 that of the cylinder.





















x x

(a) (b)


Figure 13. Field gradient coils that use discrete
filamentary current elements. a) Double-saddle 68.70/21.30
coil to produce the radial field gradient x or y; b)
Maxwell pair produces the axial, or z, field gradient.


The approach is best suited to cases where the gradient

coil is much larger than the sample, since the harmonic

series approximation to the field converges more rapidly

near the center of the coil. Although in theory the current

elements are lines, in practice they do have finite

dimensions, especially where large field intensity is

required. Including the wire diameter would greatly

complicate the design procedure. It is natural to use this

technique to design the shim coils mentioned above, where it

is conventional to have separate adjustments for as many as

twelve or more terms in the harmonic series. The coils

designed this way have the advantage of simplicity of

construction.









The building block approach was successfully extended

by Suits and Wilken46 to use discrete wires to produce a

constant field gradient over an extended region. They

evaluated designs for cylinders with the polarizing field

both parallel and perpendicular to the axis. To improve the

useful volume of the radial gradient coil, they superimposed

four saddle coils. The available degrees of freedom then

included the number of turns in each of the four coils, the

angular width of the four arcs, and the axial positions of

the arcs. Systems of nonlinear equations result that were

solved to null desired terms in an expansion of the field in

orthogonal functions. Numerical plots demonstrate that the

useful volume was extended to about eight times that of the

simple saddle coil. In each case, the volume was

nonspherical. The problems of extending this approach

further are that larger systems of nonlinear equations are

increasingly difficult to solve, and that the orthogonal

expansions do not converge rapidly away from the center of

the coil.

Bangert and Mansfield47 designed a gradient coil in

which the wires were included in two intersecting planes.

The wires in each plane formed two trapezoids symmetrically

placed about the z axis. By setting the angle between the

planes to 450, the third-order terms in the magnetic field

are canceled.


46B. H. Suits and D. E. Wilken, J. Phys. E: Sci. Instrum. 23, 565, 1989.
47V. Bangert and P. Mansfield, J. Phys. E: Sci. Instrum. 15, 235, 1982.









Approaches based on a continuous current density are

most often used for gradient coils, where performance and

linear volume are more important than simplicity. The

impetus for these coils was the echo-planar imaging

technique of Mansfield,48 which requires high intensity

field gradients switched about an order of magnitude faster

than conventional Fourier imaging. Also, shielded coils are

useful in other imaging experiments that require rapidly

switched gradient fields, and in volume localized

spectroscopy. Without shielding to cancel the external

field, the higher frequency and intensity lead to greater

eddy currents in the cryostat and magnet, that in turn

distort the linearity and time response of the field. Using

current on two concentric cylinders, it is possible to

produce a linear field inside the inner cylinder and zero

field outside the outer cylinder. Continuous current

density coils can be designed to have a large linear region,

and, since current flows on the surface of the whole

cylinder, high efficiency.

A Fourier transform technique was applied by Turner to

design gradient coils that approximate a continuous current

distribution. The approach arose from consideration of the

eddy currents induced on cylindrical shields concentric to

gradient coils made up of discrete arcs.49 Expansion of the

Green's function in cylindrical coordinates was a natural


48p. Mansfield and I. L. Pykett, J. Macn. Reson. 29, 355, 1978.
49R. Turner and R. M. Bowley, J. Phys. E: Sci. Instrum. 19, 876, 1986.









approach to calculating the eddy current distribution. It

was then possible to write the field produced by a general

current distribution on the surface of a cylinder as a

Fourier-Bessel series.50 An inverse Fourier transform of

the Fourier-Bessel series allowed the current to be

expressed in terms of the desired field on the surface of an

imaginary cylinder. The field must satisfy Laplace's

equation to allow the existence of the inverse Fourier

transform. So by specifying the inverse Fourier transform

of the desired field, the current distribution required to

generate that field could be calculated. The continuous

distribution of current is approximated by discrete wires.

The wires are placed along the contour lines of integrated

current. Although the principal application of the

technique was shielded gradients, unshielded coils having

extended linearity were also designed. For example, a

radial gradient coil is reported to have a gradient uniform

to within 5% over 80% of the radius and a length of twice

the radius. The overall length of the coil is about 9 times

the radius.

It was pointed out by Engelsberg et al. for the case of

a uniform solenoid that the homogeneity of the coil depends

strongly on the radius of the target cylinder.51 They note

that the field has the target value only on the surface of

the target cylinder. For example, in order to achieve a


50R. Turner, J. Phvs. D: Appl. Phys. 19, L147, 1986.
51M. Engelsberg et al., J. Phys. D. 21, 1062, 1988.









homogeneous field along the axis of the solenoid, the target

cylinder should be as narrow as possible. The effect is

especially pronounced at the ends of the target cylinder.

The importance of functional imaging of the human brain

and its reliance on the echo planar imaging technique puts

special demands on the rise time and field of the gradient

coil. The fact that smaller coils will be more efficient

and less affected by eddy currents has motivated several

workers to design gradient coils that will fit closely over

the head. To use a small gradient coil it is necessary to

have extended linearity in the radial and axial directions.

For a head coil, extended axial linearity is especially

important to allow the diameter of the coil to be smaller

than the width of the shoulders.

Wong applied conjugate gradient descent optimization to

the design of gradient coils with extended linearity.52 He

allows the position of current elements to vary to minimize

an error function. It is possible to define the error

function as desired, so it is simple to optimize over

regions of any shape, or for coil former of any shape. It

is also simple to include parameters such as coil length.

Repeated numerical evaluation of the Biot-Savart law for the

test wire positions would limit the application to coils

with a fairly small number of elements. Wong applied the

technique to the design of a local gradient coil for the


52E. C. Wong et al., Maan. Reson. Med. 21, 39, 1991.









human head.53 Its overall length was 37 cm, diameter 30 cm.

The region of interest is a cylinder 18.75 cm in diameter

and 16.5 cm long, over which the RMS (root mean square)

error in the field was less than 3% for all three axes. The

gradient coil was symmetric to avoid torque. In order to

make still shorter coils, Wong placed the return paths on a

larger cylinder.54 The wires on the inner cylinder were

connected to the return paths on the outer cylinder over

both endcaps. A coil was designed of 30 cm length, 30 cm

inner diameter, and 50 cm outer diameter. The optimization

region was a cylinder 24 cm long and 20 cm in diameter, and

the RMS error over the cylinder was 7.2%. The symmetry of

the coil eliminated the torque that arises in other short

designs. Additional points on a cylinder 70 cm in diameter

were added to the region of interest to force some partial

shielding.

Another approach to the design of gradient coils that

will fit over the head is to design a coil that has its

linear region at one end. Myers and Roemer55 used only half

of a conventional coil to move the linear region to the end.

A target field approach was used by Petropoulos et al. to

design an asymmetric coil with low stored energy.56 The

coil simulated was 60 cm long and 36.4 cm in diameter. The

"center" of the coil was 14.5 cm from one end. The stored


53E. C. Wong et al., SMRM 1992, 105.
54E. C. Wong and J. S. Hyde, SMRM 1992, 583.
55C. C. Myers and P. B. Roemer, SMRM 1991, 711.
56L. S. Petropoulos et al., SMRM 1992, 4032.









energy for a gradient of 4 G/cm was calculated to be 7.93 J.

Since these coils can be made much smaller than the bore of

the magnet, eddy currents are not a serious problem and

neither of these coils is shielded. Unlike symmetric

designs, these coils experience a net torque in the magnetic

field that is potentially dangerous.

Another coil at a larger radius can be used to cancel

the torque experienced by an asymmetric gradient coil.

Petropoulos et al.57 took this approach to design a head

coil with an inner diameter of 36.4 cm, the same as their

single-layer coil described above, and an outer diameter of

48 cm. The length of both inner and outer coils was 60 cm.

The coil was designed to have a useful region that is a

sphere of 25 cm diameter. There is a price to pay in

increased stored energy, which increases over the single

layer coil value of 7.93 J to 19.2 J. Torque-compensating

windings can be added to the same cylinder as the primary

coil, resulting in a long structure one end of which is

placed over the head of the patient. Abduljalil et al.58

developed such a coil set for echo-planar imaging. The

diameter of the two radial coils was 27.2 cm and 31.2 cm.

The center of the linear region was 17 cm from the end. The

overall length was not reported, but based on artwork for

the wire pattern, it seems to be about 116 cm.




57L. S. Petropoulos et al., SMRM 1993 1305.
58A. M. Abduljalil et al., SMRM 1993, 1306.









Turner has suggested that the best approach to a

compact gradient head coil design is that of Wong, in which

the return paths are placed on a larger cylinder.59 He

points to the trapezoidal gradient coil designed by Bangert

and Mansfield,60 and discussed above, as a starting point

for this approach. The concept for such a gradient coil is

described in a patent by Frese, for a cylindrical

geometry.61 It can be thought of as a Bangert and Mansfield

coil in which the inner and outer wires have been stretched

into arcs on concentric cylinders. This is the design

independently developed by Brey and Andrew and dubbed the

Concentric Return Path Coil (CRPC). Frese suggested using a

stack of the planar CRPC units with spacing along the

cylinder's axis varied to improve size of the linear region.

He also suggested that the angle of the arcs could be varied

from plane to plane. No specific information on the spacing

or angle of the arcs is provided.

A survey of the literature suggests that it is

desirable to design a short gradient coil using the basic

concentric return path geometry to be used for the human

head. A direct error-minimization technique is appropriate

for two reasons. First, the Fourier-Bessel transform

technique, although computationally efficient, limits the

shape of the region of optimization to the surface of a



59R. Turner, Maan. Reson. Imaa. 11, 903, 1993.
60V. Bangert and P. Mansfield, J. Phys. E.: Sci. Instrum. 15, 235, 1982.
61G. Frese and E. Stetter, U. S. Patent 5,198,769, 1993.









cylinder, and axial linearity is important for the head

coil. Second, the currents are not confined to the surface

of a cylinder, and the transform technique in its present

form allows only for current on the surface of a cylinder.


Field Linearity

An extended linear region is one of the goals of a

reduced-size gradient coil. In order to evaluate a coil

design in terms of its linear region, it is necessary to

define the boundary of the linear region. An appropriate

definition for the error associated with a field gradient

reflects the purpose of the gradient coil. In an error

minimization technique, the error definition is central to

the coil design. A reasonable parameter to use is the error

in the field, B.E.= Bz() where the desired gradient, G, is
G x

measured at the center of the coil. Another error parameter

is the error in the gradient, G.E., defined by

1 dBz(x)
G.E.= In an NMR image, error due to the
IGI dx

gradient coil simply produces an error in the mapping

between the sample and the image. The absolute mapping

error is simply the error in the field, B.E. In practice,

samples are usually centered in the gradient coil, so we may

want to weight the error toward the center of the coil. We

use an error parameter that corresponds to the mapping shift

relative to the component of the distance to the center in









the direction of the gradient, the relative error

B (x)- G x
R.E., defined by R.E.= Bz)- G
G x


Efficiency

In order to make use of the efficiency the reduced size

of an extended-linearity gradient coil design can provide,

it is necessary to construct the coil in such a way that it

can be driven efficiently by an amplifier. By adjusting the

number of turns, it is possible to trade maximum gradient

for switching time. We will show that to obtain optimal

switching time, the amplifier should be current-controlled

to compensate for the inductance of the coil. To reduce

switching time with such an amplifier, the coil resistance

per turn should be as low as possible, even though the time

constant of the coil will be lengthened. A time-domain

model for the coil and amplifier will be used to explore the

tradeoff between maximum gradient and switching time.




L

c
R,



Figure 14. Time-dependent voltage source v(t) drives
inductive load.


We show that a current-controlled amplifier gives

better switching performance into an inductive load than a









voltage source. The amplifier, modeled by a time-dependent

voltage source, v(t), is connected to a load with

resistance, Rc, and inductance, Lc, as shown in Figure 14.

When a demand is applied to a current-controlled amplifier

for some current, io, it will by definition change its

output voltage, v(t), as much and as rapidly as possible to

change the current through an inductance across the output.

If the maximum output voltage of the amplifier is Vo, and we
define the steady state output voltage v0 = Vo / Rc, where

Vo>v>O, then the amplifier output voltage and current as a

function of time will be

0 t < 0 0 t < 0
v(t) Vo 0 < t < to i(t) = I. 1- e-t/ 0 < t < to
Vo to < t < t Rc
i to < t<


[47]

where to is the time at which the output current reaches the

desired current io, and T = Lc/Rc. It is straightforward to

calculate that

to = TIn Vo [48]

The smaller the ratio of V0 to vo, the greater the switching

time to will become. If the amplifier is a voltage source,

the desired current will never be exactly reached. It is

more desirable to use a current-controlled amplifier for

which o>>v0.









It will be shown that, with a current-controlled

amplifier, additional series resistance per turn always

decreases performance. Therefore, the series resistance

should be reduced as much as possible, for example, by using

a larger cross-sectional area for the winding in a coil of

fixed radius. Consider again the time response of a

current-controlled amplifier from Equation [47]. Any

internal amplifier resistance can be included in Rc to avoid

any loss of generality. Assume there is some finite,

positive Rc that maximizes i(t). Then for that
di
R, = 0. Solving for Rc:
dRc

di V + Vo [491
di =_ i-e- L-+ -e- = 0 [49]
dRc Rc2 c c L

and assuming that Rc and Vo do not vanish,


1 e + R/c L e = 0. [50]


This can be written as

(1 + x)e-x = 1 where t/T = x, [51]

therefore

ex = + x. [52]

There is no positive value of x that satisfies Equation
di
[52]. Hence < 0 for all t>O, Rc>O. A lower Rc is
dRc

always an advantage when using a current-controlled

amplifier, although the time constant T = Lc/Rc of the

gradient system increases as Rc decreases. It is then









appropriate to maximize the cross-sectional area of the

windings subject to considerations of linearity and

available space.

Next we consider how the number of turns of wire in a

coil of fixed cross-sectional area can be varied to achieve

the desired performance. It is important to note that the

time constant of a coil, for a fixed area, can properly be

considered to be independent of the number of turns. This

result follows from consideration of a gradient coil at low

frequencies, as described by the equivalent circuit of a

series resistor Rc and inductor Lc as shown in Figure 15.




LC


Re



Figure 15. Equivalent circuit of a gradient coil in the
low-frequency limit.


Let R1 be the resistance, and let L1 be the inductance of a

single turn coil. It is well known that the inductance of a

coil increases as the square of the number of turns of wire

N.62 The resistance also increases as the square of the

number of turns if the area is held constant, since as the

number of turns increases, the area of each turn diminishes.



62T. N. Trick, Introduction to Circuit Analysis, p. 256, John Wiley and
Sons, New York, 1977.









The total resistance and inductance are then

Rc = R1N2 Lc = L1N2. [53]

The time constant T of the coil is just the ratio

S= Lc/ R [54]

Perhaps surprisingly, T is independent of N. The result

does not apply when additional turns of wire are added to an

existing coil, thus increasing the area. However, since the

area should already be as large as possible to maximize the

performance, it will not be possible to increase N without

decreasing the size of the wire.

To determine how many turns of wire N should be used in

the gradient coil, we consider how rapidly and to what value

the current rises for various N, holding the area constant.

It will be shown that with a current-controlled amplifier,

the coil is optimized to switch to the field at which it

reaches a saturation current, IO, which is the maximum that

the amplifier can supply. Consider an amplifier with

negligible output impedance switching at time t = 0 from

zero current to maximum current, I0, through a gradient

coil, reaching I0 at to. Assume that Rc < VO/IO. The

current as a function of time is:


1-e1 0 t 5 to
i(t) = Rc [55]
TO t > to

We define a current efficiency k for a single turn so that

the gradient field G(t) = kNi(t). Rc varies as N2, and the









magnetic field G varies as N, so

kVo -/
f Wo1 e- 0 < t < to
G(t) = RIN [56]
kNIo t 2 to

A plot of G(t) for various values of N is shown in Figure

16. All three curves have the same time constant, so


G(t)


0.175

0.15

0.125

0.1

0.075

0.05

0.025


0.1 0.2 0.3


Figure 16. Magnetic field produced by a current controlled
linear amplifier coupled to a coil of fixed dimensions.
Each curve represents a different number of turns.


the difference in slope is due to the relative amplitude of

the maximum gradient. The dotted line connects all the

current-limit points. Since at the current-limit point to

the amplifier is both a voltage and a current source, we can

eliminate G(to) in favor of N and to, yielding an optimum

number of turns for a given switching time.








Nopt(to) = [1[- e [57

By substituting [57] back into [56], we obtain an
expression for Gmax(to), the maximum field attainable at a
given switching time for a class of coils having the same
design except for the number of turns.


Gmax(t0) = k I 1 e-t [58]
YR,

Gmax(to) is just the dotted line in Figure 15. The
tradeoff between switching time and field strength is
described by the plot of Gmax(to).
In summary, a design procedure has been developed for
optimizing the switching performance of a gradient coil.
Use of a current-controlled amplifier reduces switching
time. The cross-sectional area of the winding is maximized
subject to constraints that include linearity and available
space. Then the number of turns is computed from Equation
[57], given the desired switching time to. The resulting
coil will give the largest possible gradient for the desired
switching time.

Eddy Currents
Shielding efficiency of self-shielded gradient coils is
typically evaluated using a screening factor, a ratio of the
magnetic field outside the unshielded coil to the field
outside the shielded coil.63 It is possible to take this


63R. Turner, Maan. Reson. Imaa. 11, 903, 1993.









type of approach further and evaluate the ratio of the

gradient at the center of the coil with and without the

shield. This would seem to be a useful approach when

evaluating reduced-size gradient coils and comparing them to

shielded coils. For small eddy current fields, as in the

case of reduced-size coils, an iterative approximation

technique described below can be used to solve the integral

equation for the eddy current field. This technique is best

suited to situations where the eddy current field is much

smaller than applied field, so that a first-order

approximation can be used. However, it is simple and

flexible.

To estimate the eddy current field due to a gradient

coil, we assume that there is a passive shield surrounding

the coil. The shield is typically part of the cryostat. The

boundary conditions at the shield will be

(B2 B1) n = 0 [59]

SX (H2 H1) = K [60]

where B1 and H1 are the magnetic induction and field inside

the shield, B2 and H2 outside the shield, K is the surface

current on the shield, and n is an outwardly directed unit

vector normal to the surface of the shield.64 We assume the

shield is perfectly conducting, so that with H2 = 0,

H1 x n = K, [61]



64J. D. Jackson, Classical Electrodynamics, p. 1.5, John Wiley & Sons,
New York, 1975.









or more conveniently,
1
K = -B x p [62]
go
since B = o0H and n = p at the cylinder. Recall the

Biot-Savart law:

B(x) = J(x') x d3x'. [63]
4r Ix x'13

Let BO(x) be the free-space field from the gradient coil.

Then

B(x) = B0(x) + K(x') X dx [64]
4 x x'13

and substituting Equation [62] for the surface current, for

9 = Io,
1 X X'
B(x) = Bo(x) + -- B(x') x p'] x d x [65]
4 |ix x'3

where p' = p(x'). This is an integral equation for B. We

can solve it iteratively. If we define Bn(x) as the field

to nth order, then the first-order solution is

B1(x) = Bo(x) + f[Bo(x') x p'] x d x. [66]
4x Ix x'|3

The first-order solution does not take into account

eddy currents induced by eddy currents. When the coil and

shield are in close proximity, not only are the eddy

currents larger but they are also closer to the coil, so the

second-order effect can be important. To second-order,

1 x -X d2,
B2(x) = BO(x) + [Bl(x) x '] d2x. [67]
4t Ix X'1










The expressions to first- and second-order for the eddy

current field will be used to evaluate numerically the eddy

current field of several coil designs. Although the result

is not exact, the expressions can easily be integrated for

coils and shields of totally arbitrary shape, assuming they

are not too close together.

The first-order calculated eddy current field of a

68.70/21.30 radial gradient coil is plotted in Figure 17.

The first-order approximation breaks down for ratios of

shield-to-coil radius of less than about 1.5.




shield radius / coil radius

1.5 2 2. 3.5 4


-20



-40



-60
eddy
current
field
(%)


Figure 17. Eddy current field of 68.70/21.30 double-saddle
radial gradient coil. The field as a percentage of applied
field is plotted against the ratio of shield radius to coil
radius.









Coil Projects

Coil projects were intended to meet experimental needs

while exploring some aspect of coil design. The 15 cm, 9 cm

and 16 mm NMR microscopy coils are well separated from any

sources of eddy currents, and demonstrate the results that

can be achieved with simple filamentary designs and without

shielding. The CRP coil development was begun to produce a

coil with good axial linearity for NMR microscopy, so that

long, narrow samples could be observed. It seemed to be

well-suited for use as a head coil for echo planar imaging,

and we turned the development toward that possible

application.


Amplifiers

Three Techron 7540 dual-channel amplifier units (Crown

International, Elkhorn, Illinois) are used to drive the

three-axis gradient coil sets. Each axis of the gradient

coil set is split into two halves, and one channel of each

amplifier unit is wired to each half. The plane in which

the field is always zero can be shifted slightly by varying

the relative gain in the amplifiers. This is particularly

useful in the 51 mm, 7 T magnet, since the sample is

inaccessible once loaded, and mechanical centering is

difficult. The amplifiers are rated to produce 23.8 A at 42

V direct current output. The maximum slew rate is 16 V/gs.

The output impedance is less than 7 mnQ in series with less

than 3 gH, which is negligible. The power response into a 4






73


0 load is +/- 1 dB up to 25 kHz for 265 W. The noise is

rated to be 112 dB below the maximum output from 20 Hz to 20

kHz.65

Tests of the Techron 7540 were conducted into six

loads consisting of wire-wound resistors between 1 and 9 Q.

The amplifiers were pulsed to saturation at low duty cycle.

10-90% rise times were between 4 and 6.5 9s, and so are

essentially independent of load. Thus the amplifier was

bandwidth limited, not slew-rate limited, and it is

appropriate to use a linear model. The voltage and current



60 16 T
50 14
12
40 1
4 U 10
o 30 U 8
20 6 4
> H
4
10 2
0 I I 0
0 2 4 6 8 10 0 2 4 6 8 10
R (ohms) R (ohms)


(a) (b)

Figure 18. Output of Techron 7540 measured into load. a)
Measured voltage; b) Calculated current.


produced are shown in Figure 18. For load resistance of

four ohms or more, the amplifier at saturation can be

modeled by a 56 V voltage source. For a load resistance of


65Crown International, Techron 7540, Elkhorn, Illinois.









four ohms or less, it can be modeled as a 15 A current

source.

The Techron amplifiers are equipped with optional

current-control modules. With current control switched on,

an amplifier behaves like a voltage-controlled current

source. With current control switched off, an amplifier

behaves like a voltage-controlled voltage source. Current

control serves two functions when driving gradient coils.

It compensates for any variation in temperature of the

gradient coil due to resistive heating. More importantly,

it enables the coil to be switched to low fields much more

rapidly than the coil's time constant would otherwise allow.

The current-control module compares the demand (or input

signal) to the voltage across a small shunt resistor in the

output circuit. With a highly inductive load such as a

gradient coil, at high frequencies the amplifier's output

voltage is shifted almost n/2 with respect to the coil

current, and the amplifier is unstable and will oscillate.

The voltage and current response of one of the Techron

amplifiers in current mode is shown in Figure 19. The

controlled voltage overshoot reduces the current switching

time. Approximately 5 A is being switched into a 7 Q load.

An adjustable resistor-capacitor (RC) network in parallel

with the coil rolls off the high frequency gain to

compensate for the instability. The values of the RC

network are determined by the inductance of the coil. Since

the 7540 amplifiers are used with more than one coil, the









current-control units were modified so the RC networks can

be plugged in and out when gradient coils are changed.


time


1.79 ms


Figure 19. Output voltage and current of Techron 7540
amplifier with current-control module. The load is the
highly inductive 9 cm field gradient coil.



The amplifier rack was equipped with wheels and shared

between the NMR microscopy and small-animal spectrometers.

It was used in voltage-control mode with the NMR microscopy

system, and current-control mode with the small animal

system where the coil inductance was much higher. Input and

output connectors were standardized to facilitate quick

conversion. A fully-shielded output cable terminated in a

fuse-and-filter chassis eliminated interference from the RF

coils.









16 mm Coil for NMR Microscopy

The 16 mm gradient coil was developed as part of the

NMR microscope development project described below. Earlier

NMR microscopy gradient coils described in the literature

were located outside of the RF probe insert, as part of the

shim coil set. A simple and straightforward approach to

improving the coil switching time, increasing the field

strength, and decreasing the eddy current field is to

integrate the gradient coils with the RF probe. This also

allows the use of a narrowbore (51 mm) magnet. Drawbacks to

this approach include a lack of flexibility. If the

gradient coil is outside the RF probe, then any RF probe can

be used. In our approach, one gradient coil is required for

each RF probe. Also, since one of the dewars associated

with the variable-temperature (VT) control system is

replaced by an acrylic tube, the range of the VT system is

reduced. Our probe did not contain any VT control

capability. The fact that the coil former was so small

encouraged us to choose a simple design to ease the

assembly.

Since the sample-tube inner diameter was 4.5 mm and the

first metal tube, or shield, had an inner diameter of 33 mm,

this was a favorable case for using a reduced-size gradient

coil. A simple 68.70/21.30 radial gradient coil as

described above has a useful volume with a diameter of about









1/3 that of the coil,66 so the gradient former was chosen to

have a diameter of about 15 mm, or 5/8". A factor of two

remains in the ratio of the coil to the shield diameter.

This results in an eddy current field for the 68.70/21.30

Golay radial gradient coil, based on Figure 17, of about 20%

of the applied field.

The NMR microscope gradient coil set is of the

conventional Maxwell and Golay design described above. It

was constructed to accommodate standard 5 mm tubes used in

analytical NMR work. The 10 turns of 36 AWG enameled magnet

wire are wound on a 5/8" nominal outer diameter acrylic tube

(15.9 mm). Using a value of 1.36 Q/m for the wire67 and a

length of 0.135 m per turn, the resistance of each side of

the coil is 1.84 Q. The coil inductance can be estimated68

to be about 8 LH. The time constant of the coil is then

about 4 Js. The current efficiency of a 68.70/21.30 Golay

radial gradient coil is 0.918/a2 G/cm-A, where a is the coil

radius,69 so the coil has a current efficiency of 14.1 G/cm-

A. A Maxwell pair has a current efficiency of 0.808/a2

G/cm-A, so the coil has a current efficiency of 15.3 G/cm-A.

The typical figure for the linear region of 1/3 the diameter

of the coil is then enough to accommodate a sample. The

coil is driven by the Techron 7540 amplifier set. The coil



66F. Romeo and D. I. Hoult, Maan. Reson. Med. 1, 44, 1984.
67D. Lide,(Ed.), CRC Handbook of Chemistry and Physics, 51st Edition,
CRC Press, Boca Raton, 1970, p. 15-29.
68F. E. Terman, Radio Engineers' Handbook, McGraw-Hill, New York, 1943.
69F. Romeo and D.I. Hoult, Maan. Reson. Med. 1, 44, 1984.









has a small time constant, so using the Techron in voltage

mode does not limit the switching time. The two halves of

each gradient coil are driven separately. Since the voltage

gain of the amplifiers can be adjusted manually, it is

convenient to vary the relative gain in the coils to shift

the zero point of the magnetic field to make up for sample

misregistration.

The details of the coil construction are visible in

Figure 47 in the following chapter. The radial coils were

wound on a flat winding former, then removed and attached to

the acrylic tube with epoxy. To eliminate any solder

connectors within the coil, the winding former allowed two

loops to be wound at once, held apart at the correct

distance. General Electric #7031 varnish was used to hold

the wires together while the coil was being clamped to the

former. No attempt was made to arrange the wire into a

packed structure. The Maxwell pair was wound around the

radial coils. The whole assembly was potted in epoxy to

secure the coils to the former, and the 36 AWG wires were

run down to a small printed circuit board mounted to the

structure of the probe. It was necessary to pot the fine

wires to keep them from moving in the magnetic field when a

current pulse is applied.

An example of the results obtained with the coil is

reproduced in Figure 60. Although the coil is capable of

about 150 G/cm, in routine operation, the coil was operated

at a full-scale field gradient of 5 G/cm for the radial









gradients, and 10 G/cm for the axial gradient, to allow

sufficient resolution.


9 cm Coil for Small Animals

An NMR magnet is frequently used for samples or animals

significantly smaller than the available bore size. It is

possible to take advantage of this fact and scale the size

of the gradient coil to match the size of the sample. One

advantage that accrues is reduced eddy current fields, since

the coil and the source of eddy currents are better

separated. Another is the increased efficiency possible

with smaller coils, since efficiency scales as the fifth

power of the diameter.70 Many applications require more

rapidly switched and more intense gradient fields than are

generally available. Diffusion-weighted imaging and

localized spectroscopy are two examples. Also, to achieve

the same bandwidth per pixel, small samples require larger

gradient fields.

The 31 cm 2 T small-animal imaging spectrometer was

supplied with a gradient coil set manufactured by Oxford

instruments that has a clear bore of 22.5 cm, and is capable

of producing a maximum gradient of 2 G/cm with a switching

time of 1 ms. Although rat, mouse, and lizard studies, do

not require the full 22.5 cm bore, they benefit from the

horizontal orientation and will not fit into other available

magnets. Additionally, localization techniques such as


70R. Turner, Macn. Reson. Imaq. 11, 903, 1993.









selective Fourier transform71 typically require better

gradient performance than is available with a large,

unshielded gradient coil set.

To meet some of these needs, a conventional Maxwell and

68.70/21.30 Golay radial gradient coil set was designed and

constructed with a clear bore of 8.3 cm in diameter. The

useful region is a sphere of about 1/3 the diameter of the

coil, or about 3 cm. The coil was designed to accommodate

rats up to 150 g, and was able to achieve 12 G/cm with a 200

ps switching time. The intense gradients are needed for

imaging experiments on smaller samples. The coil was used

for projects involving lizards, and for development of

techniques to produce diffusion images of the spinal cord of

a rat model.

We can consider the application of the time-domain

model to the 9 cm coil. As wound, the coil will produce the

field shown in Figure 20 when driven to saturation. The

Maxwell pair, Z-axis coil, is the most efficient, followed

by the inner radial, or X-axis coil, which has better

performance than the outer radial, or Y-axis coil, because

of its smaller radius. The Maxwell pair reaches the 16 A

current limit of the amplifier, and does not increase in

field beyond that point. The radial gradient coils never

reach the current limit.


71T. H. Mareci and H. R. Brooker, J. Maan. Reson. 57, 157, 1984.









G(t) (G/cm)


-f t (ms)
0.2 0.4 0.6 0.8 1

Figure 20. The gradient produced by the 9 cm gradient coil
set following a demand that saturates the amplifier.


Figure 21 describes the maximum field Gmax(to) possible

for switching time to for each of the three coils. The


Gx (G/cm)

25

20

15

10

5


I .. .' to (m s )
0.2 0.4 0.6 0.8 1

Figure 21. The maximum gradient that could be achieved by a
coil identical to the 9 cm coil, with the same cross-
sectional area, but with varying number of turns.



inherently lower inductance and resistance of the Maxwell

pair are reflected in its greater field. The actual and

optimal fields obtainable at 200 ps are compared in Table 1.

The Maxwell pair has about the optimal number of turns, and

its field is about the same as the optimum. The Golay coils









have about twice as many turns as optimum, and yield fields

about 80% of optimum level.


Table 1. Gradient fields for 9 cm coil set.

Gradient Actual Gradient Optimal Gradient
channel no. of after 200 no. of after 200
turns ps (G/cm) turns Rs (G/cm)


X 52 15.2 27.9 19.0

Y 52 12.3 26.9 15.9

Z 52 24.0 53.6 24.8


The radial and axial gradient coils consist of 52 turns

of AWG 27 enameled magnet wire. The wire was wound in a 7-

6-7-... close-pack configuration to minimize the cross-

sectional area of the winding, which is reduced by a factor

of 0.866 from a square winding pattern. The resulting

winding cross-section is about 2.6 mm on a side, only about

6% of the coil radius, so the winding approximates a

filament. The mean radius of the coils is 4.6 cm, 4.8 cm,

and 5.1 cm. The Maxwell pair is wound on the outside

because it is inherently more efficient and will hold down

the other coils. The two halves of each coil are wound

separately, and one channel of a stereo amplifier is wired

to each. It is driven in current mode from the Techron 7540

amplifiers. The current-control circuit helps to buck the

inductance of the coil. The coil resistance for the radial

windings is predicted to be about 6.7 Q for the inner set.

The measured resistances and time constants including the









leads and filters are given in Table 2. The time constant

of the shorted power cable with filters and fuses was too

short to measure with the amplifier, so it can be assumed to

be negligible. The last column is the inductance estimated

from the Bowtell and Mansfield formulation for coils on the

surface of cylinders.72 To adapt for the thick winding, the

height is added to the width of the winding. Calculations

for loops of square cross section compare closely to

heuristic formulas.73


Table
for 9

Coil




X1

X2

Y1

Y2

Zl

Z2


2. Comparison of measured and predicted inductance
cm gradient coil set.

Measured Measured Time Experimental Theoretica
Resistance Constant Inductance Inductanc
(Q) (9s) (gH) (UH)


7.02

7.08

7.57

7.61

3.50

3.53


175

175

195

190

150

150


1229

1239

1476

1446

525

530


l
e


1727

1727

1866

1866

330

330


The difference between the theoretical and experimental

inductance is not due to the inductance of the power cable,

which was measured by the same technique to be about 18 pH.

It is primarily due to poor control of the cross-sectional



72R. Bowtell and P. Mansfield, Meas. Sci. Technol. 1, 431, 1990.
73F. E. Terman, Radio Engineers' Handbook, McGraw-Hill, New York, 1943.









dimensions of the winding, which expands when it is removed

from the winding former. The maximum operating field as the

system has been installed is 12.83 G/cm for X, 12.28 G/cm

for Y and 8.80 G/cm for Z. By limiting the gradient field

to a value below that corresponding to the steady state

current, the switching time is better controlled.

A drawing of the coil that illustrates how the Faraday

shield and RF coil fit together is shown in Figure 22.


Golay windings











RF coil Faraday Centering Maxwell Clamping
shield disk pair cam

Figure 22. Drawing of 9 cm gradient coil set with Faraday
shield and RF coil.


Figure 23 is a photograph of the assembly. The former

is an acrylic tube of 3.5" (89 mm) nominal outer diameter

and 1/8" (3.2 mm) wall thickness. The radial gradient coils

are wound on rectangular bobbins made from three flat pieces

of acrylic. The wires are held together in a hexagonal

matrix by General Electric varnish #7031 diluted in acetone.

After winding, the bobbin is disassembled and the coil is

glued onto the cylindrical former with epoxy. A variety of
































Figure 23. Photograph of 9 cm gradient coil set. The power
cable and water supply cables are visible at left. The
axial and radial gradient coils are visible through the
cooling tubing.


RF coils were developed as inserts for the probe, including

19F, 1H birdcage, 31P/1H double-tuned saddle coil, and my

own 1H saddle coil. A pair of cams connected by a rod and

mounted on the edge of the mounting flanges served to lock

the probe into the magnet. A Faraday shield was used in

addition to the filter/fuse box to isolate the RF coils from

the gradient coils. The shield consisted of strips of

Reynolds heavy-duty aluminum foil approximately 2" wide,

overlapped by about 1/2", and insulated from the other

strips by masking tape that also secured the strips to

manila card stock. At one end, all strips contacted a

header strip. Provision was made to ground the shield, but









in practice it was not used. The interdigitated geometry

reduced eddy currents from the gradient coil, but allowed

the shield to serve as a barrier to the RF field. The

Maxwell pair is on the outside, which helps to hold the

radial gradient coils in place. Epoxy was initially used to

hold the windings to the former and pot the windings, but

the epoxy did not withstand the temperatures developed by

the coil and depolymerized. Polyester was selected as a

casting resin that would outperform the acrylic former under

warm conditions, and the coils were potted in polyester.

In order to cool the unit, approximately 25 m of 1/8"

O. D. by 1/64" wall polypropylene tubing was wound around

the coils. It was connected to a Neslab circulating system

that is capable of producing a pressure head of 40 psi.

Supply tubing consisted of about 30 m of 1/4" 0. D. by

1/32" wall polypropylene. Assuming laminar flow, the water

flux through a tube74 of radius r (cm) is

flux = r 4Ap
8 l

where Ap is the pressure in dyne/cm2, 1 is the length of the

tube in cm, and g is the viscosity in poise. The resulting

flux for a 40 psi drop is 8.6 cm3/s. One KW of power

transferred to the water in the tube will raise its

temperature by about 280 C. Measurement of the motion of

bubbles in the tubing reveals a flow of about 3.5 cm3/s.


74D. Lide,(Ed.), CRC Handbook of Chemistry and Physics, 51st Edition,
CRC Press, Boca Raton, 1970, p. F-34.









The difference is almost certainly due to quick-release

connectors that allow the probe to be removed or inserted at

operating pressure.


15 cm Coil for Small Animals

The 15 cm coil was designed to accommodate larger rats

and other medium-sized laboratory animals and still produce

a higher field and faster switching time than the Oxford

gradient set. Like the 9 cm and the NMR microscope coil

set, it is based on filamentary winding design. Since it is

also driven by the Techron 7540 amplifier set, which is

under-powered for a coil of this diameter, switching

performance was at a premium. So, in contrast to the 9 cm

coil, the 15 cm coil was designed to have optimal switching

performance for the chosen switching time. In order to

provide more flexibility in choosing either high field

intensity or fast switching time, the windings were split

and could be driven either in series or parallel. Since the

coil would tend to become somewhat unwieldy as an insertable

unit, its length was the shortest that would give

essentially undiminished field intensity. Plots of the

relative error in Figures 26 and 27 illustrate the linear

region of the radial coil design. In order to avoid

aliasing signals from long animals, the axial coil was based

on an extended linearity design by Suits and Wilken.75 The


75B. H. Suits and D. E. Wilken, J. Phys. E: Sci. Instrum. 23, 565, 1989.









coil and amplifiers were capable of developing 9 G/cm in a

100 gs switching time on X, Y and Z channels.

The arcs in the 15 cm coil were arranged to have the

minimum length without losing a significant amount of

efficiency in the static limit. The standard solution of

68.70/21.30 for the arc positions arcl/arc2 leads to no

third order component from either arc, but a family of

solutions for which the third order components cancel is

available. These solutions are graphed in Figure 24.


arc2 (deg)


60

50

40

30

arcl (deg)
30 40 50 60

Figure 24. The solutions to the arc position of the double-
saddle radial gradient coil.


Each solution is graphed twice, since exchanging arcl and

arc2 results in the same coil design. To improve the

relative size of the linear region to the coil, one would

like to make the coil shorter than the 68.70/21.30 solution.

The current efficiency decreases with the length, since the

return arcs tend to cancel the desired field, and moving

them closer increases the effect. However, in order to










include the effect of reducing resistive loss in the coil,

one must divide the current efficiency by the square root of

the length. The resulting measure is an indication of the

relative field that can be produced with an amplifier of a

given power. Figure 25 illustrates the relative power

efficiency as a function of the position of the return arc.


rel. power eff.

0.3

0.25

0.2
0.15

0.1

0.05
angle (deg)
20 25 30 35 40 45 50

Figure 25. The relative power efficiency of the double-
saddle radial gradient coil as a function of the angle
between the z-axis and the current return path.


The peak efficiency is achieved at 260, but efficiency is a

weak function of angle and much shorter coils can be used

with little loss of performance. Based on this curve, the

arcs of the 15 cm radial coil were placed at 30.20 and 66.10

from the axis of the coil, compared to 21.30 and 68.70 for

the Golay coil. The overall length of the coil is reduced

by 33%. The current efficiency is 0.819/a2 G/cm-A compared

to 0.808/a2 G/cm-A for the 68.70/21.30 Golay coil. It would

be even better to use a variant of the field-versus-

switching-time approach to determine the change in

















0.4

-1%
-3%
0.2
-1%



Z 0 +1%+3%+5%



-0.2




-0.4





-0.6 -0.4 -0.2 0 0.2 0.4 0.6

a) Y







0.4 +5%

+3%



+1%
0.2






Y 0






-0.2





-0.4

-0.4 -0.2 0 0.2 0.4

b) x




Figure 26. Relative error plots of 30.20/66.10 radial
gradient coil. Radius of coil corresponds to 1 on scale.
a) YZ plane; b) XY plane.


1












0.61
+5%
+3%
0.4 +1%
-1%
-5%
-3%
0.2


Z 0 +1%+3%+5%


-0.2


-0.4


-0.6
-1 -0.5 0 0.5 1


X




Figure 26--continued. Relative error plots of 30.20/66.10
radial gradient coil. Radius of coil corresponds to 1 on
scale. c) XZ plane.



performance for coils of different lengths, taking the

changing inductance into account.

The size and shape of the linear region of the 66.10

/30.20 radial gradient coil design is described by the plots

of Figures 26 and 27. It is not greatly different from the

longer 68.70/21.30 design. Note from Figure 26 that the

regions of equal relative error are not simply connected. A

three-dimensional plot of a region as large as that in the

two dimensional plots would give a false impression of the

size of the linear region, since the apparently solid

volumes would contain large holes. To avoid these "bubbles"

of linearity and give a true picture of the useful volume,


r










the region displayed in Figure 27 is truncated. As a result

of the truncation, it is possible to see through the linear

region. All plots were produced by direct evaluation of the

Biot-Savart law.


-2

z
0

1





0.5

x 0


-0.5


-1
0.5
0
-0.
Y -1


Figure 27. Perspective rendering of the 5% relative error
region of 30.20/66.10 radial gradient coil.


The axial coil was built to an extended-linearity

design proposed by Suits and Wilken.76 It consists of loops

at both 40.00 and 66.30 from the z axis. The outer loops

carry 7.5 times more current than the inner loops. Using

the additional degrees of freedom of the second-loop

position and the ratio of current in the loops, the fifth


76B. H. Suits and D. E. Wilken, J. Phys. E: Sci. Instrum. 23, 565, 1989.









and seventh order terms are canceled, resulting in about

eight times the useful volume of a Maxwell pair. For a coil

with more turns in some loops than others, the current

efficiency does not have an unambiguous definition. With

respect to the current in the outer loops, the current

efficiency of the coil is 0.635/a2 G/cm-A.

The 15 cm coil was matched to the Techron 7540

amplifiers in our laboratory using the time-domain model of

gradient performance described above. The height and width

of the windings was set to be 1 by 1 cm for the radial and 1

by 2 cm for the outer axial, to ensure that the assumption

of filamentary wires in the calculation of angle position

would be valid. The width of the outer axial winding was

increased from 1 to 2 cm, since it is farther from the

center than the others, and it was necessary to increase it

to match the radial performance. Inductance of the radial

and axial coils was calculated using a Fourier-Bessel

approach.77 The available combinations of switching time

and maximum gradient are shown in Figure 28.

In order to improve the switching time to smaller

fields, the 15 cm coil is constructed from split windings.

All the gradient coils described have the two sides driven

by separate amplifiers so that the magnetic center of the

coil can be moved. The 15 cm coil has each side split

further into two identical but electrically separate



77R. Bowtell and P. Mansfield, Meas. Sci. Technol. 1, 431, 1990.










windings. Placing the windings in parallel trades field

intensity for switching time; placing them in series reduces

switching time at the expense of lower field intensity.


Gma (G/cm)

14 X
12
10
8
6
4
2

to (ms)
0.5 1 1.5 2

Figure 28. Maximum field Gmax (G/cm) vs. switching time to
(ms) for the 15 cm field gradient coil set as driven by the
Techron 7540. Points along the curves represent designs
with different numbers of turns, increasing from left to
right. The top curve represents the inner radial coil. The
lower curves represent the outer radial coil and the axial
coil. A "slower" coil has a larger maximum field.


All three coils are optimized to approximately 10 G/cm

for the series mode. The exact number of turns for the

desired field is not used, due to the limited number of

standard wire sizes and the use of rectangular winding cross

sections. For the number of turns actually used, to and

Gmax based on the time domain model are tabulated in Table

3.

A photograph of the 15 cm coil assembly is shown in

Figure 29. The coil is wound on an acrylic former having a

nominal O. D. of 6" (152 mm) and a wall thickness of 1/4"

(6.4 mm). The overall length was 38 mm. The radial coils




Full Text
CONCLUSION
These developments in pulse field gradient technology
may find application in research areas ranging from
functional imaging to NMR microscopy. Stimulated echo
techniques allow measurement of the eddy current field over
a wide dynamic range without any special hardware and can be
used to adjust the eddy current compensation and assess the
effectiveness of reduced-size gradient coils. This
technique is now being used to adjust the compensation for
new gradient coils for a 4.7 T small-animal spectrometer.
Reduced-size gradient coils produce reduced eddy currents
and short switching times and can be applied to small-animal
imaging and NMR microscopy. The CRP coil is a good
candidate for application to clinical imaging. Moving the
region of sensitivity to one end, for example, will allow
better coverage of the human neck.
155


133
250 flm. The cell nucleus was easily visible in the dark
background of the cytoplasm.
Magnetization transfer was used to show that the
shortened T2 in cell cytoplasm comes from a broad component
with restricted mobility, not from "microsusceptibility"
broadening at surfaces.93 The T2 weighted images suggested
the presence of bound water.
Images of blood flowing through a beating, perfused rat
heart were obtained using a modified Nicolet NT 360 (8.45 T)
spectrometer by Delayre et al.94 23Na was imaged, and
contrast depended on the fact that there is more Na in the
blood than in the heart tissue. Imaging gradients were
produced by the x, y, and z shim coils that were controlled
from the computer through digital-to-analog converters. An
oscillating z gradient defined the 1.5 mm thick slice.
Projection-reconstruction was used to reconstruct the image
matrix of 64 by 64 pixels. The heart was placed in a 20 mm
diameter tube. The NMR signal was detected with a saddle
coil. The NMR imaging experiment was gated to the cardiac
cycle to obtain images at systole and diastole.
Instrument Development
To enable an existing, commercially available NMR
spectrometer to perform NMR imaging, capability was added to
produce and control pulsed field gradients and amplitude-
93E. W. Hsu et al., SMRM 1992, 974.
94J. L. Delayre et al., Science 212, 935, 1981.


GRADIENT COIL DESIGN
Introduction and Theory
Although virtually all NMR measurements rely on
auxiliary field coils, there has been comparatively little
published work on the design and analysis of shim and field
gradient coils compared to that for radio frequency coils.
However, high levels of performance have become increasingly
important for these low-frequency room-temperature coils on
several frontiers of the NMR technique. Three of these
areas are gradient coils for NMR microscopy, coils for
spatial localization of spectra, and local gradient coils
for functional imaging of the human brain.
The simple forms of discrete element coil designs have
linear regions that are about 1/3 of the coil radius.35
Therefore the gradient coil must be considerably larger than
the sample. Increasing the linear region would allow
smaller coils to be used, generally improving efficiency and
decreasing eddy current fields. Several approaches are
available to increase the region of linearity. Adding
discrete elements to cancel more high-order terms in the
harmonic expansion has been done successfully by Suits and
Wilken.36 Continuous current density coils have also been
35F. Romeo and D. I. Hoult, Maan. Reson. Med. 1, 44, 1984.
36B. H. Suits and D. E. Wilken, J. Phvs. E: Sci. Instrum. 23, 565, 1989.
46


108
interior expansion does not converge rapidly unless r << f ,
a target field approach was tried.
Turner defined a target consisting of points evenly
spaced on a cylinder, so that he could perforin the coil
optimization in a reciprocal space.83 We perform a
minimization of the mean square relative error of the field
at a list of points freely chosen in the region of interest,
similar to a method used by Wong.84 The points are chosen
opportunistically to push out the region of linearity in the
desired directions. Typically, points are evenly spaced in
two lines. One line is close to the z axis and one is
slightly inside the edge of the desired linear region. As
in the harmonic term nulling procedure, only the currents
are varied. In principle, if the number of degrees of
freedom is equal to the number of constraints, it is
possible to find a solution with zero error at the target
points. In practice, there are drawbacks to this approach.
Since the target points define the periphery of the linear
region, the field will naturally have a small error there,
so the constraints are not well related to the parameters of
interest. Solutions having large currents of opposite sign
in adjacent planes are apt to occur, which in practice would
give very poor power efficiency. A better approach is to
use more constraints (points in the target field) than
degrees of freedom (number of planar units). The
83R. Turner, Maan. Reson. Imaa. 11, 903, 1993.
84E. C. Wong et al., Maan. Reson. Med. 21, 39, 1991.


117
-1 -0.5 0 0.5 1
Y
Figure 35--continued. e) Contour plot of the relative error
of the 30 unit CRP Coil in the X = 0 plane.


44
basic stimulated echo experiment without adaptive sampling
to quantify the integral of the eddy current field over an
interval, and allow one to predict the resulting phase
distortion directly.
The adaptive sampling algorithm is able to follow eddy
current fields that are not simply monotonically decreasing.
I found experimentally that if the angle became much
different than 45, the values for x would bounce around a
lot before stabilizing. This is probably due to the control
being purely proportional. Introducing an integral term
might help.
The relatively large eddy current field produced by the
15 cm Z gradient compared to the X and Y channels is due to
its extended-linearity design, which locates the currents
farther from the region of interest than a Maxwell pair.
The relatively large eddy current field produced by the 9 cm
Y gradient compared to the X and Y channels may be due to a
problem with centering the gradient coils in the bore. The
measured field gradient would then depend strongly on the
position of the sample.34
The contrast in eddy current field between the large
and small coils is clear. There is a factor of about 10 in
eddy current field between the 15 cm coil and the larger
Oxford coil. There is a factor of about 180 in eddy current
field between the 9 cm coil and the Oxford gradient coil set
34D. J. Jensen et al., Med. Phvs. 14, 859, 1987.


103
C ) length
Figure 33. Relative Error of CRP coils with uniform current
density. Length and distance are in units of inner radius.
Outer radius is 1.5 times inner radius, a) R. E. along x
axis; b) R. E. along x = y; c) R. E. along y axis.


130
gradient coils were wound on a 4.5 cm diameter cylinder.
Sixteen AWG wire was used. The coils produced fields of 0.8
for x, 1.0 for y and 0.6 G/cm-A for z. A 100 A, 5V power
supply was used to drive the coils, with the current
switched on and off by a transistor switch controlled by the
pulse sequencer. Only a single level of gradient field was
possible with this arrangement. The gradient switching rate
was 0.4 G/cm-ms. Solenoidal RF coils were used, with 10 and
2 mm diameter versions for the 1H coil. A sample in the 2
mm proton coil could be shimmed to 10 Hz. Images of Rana
pipiens egg cells were acquired with 15 (Jm resolution and
200 |im slice thickness.
Cho et al. predict and measure the SNR obtainable at
7.1 T.89 They derive a diffusion-limited resolution of
which yields 1 |lm for D = 10~5 cm2/s at Tacq
1.5 ms. They selected a 5-turn solenoid with a diameter of
1 mm, length of 0.8 mm, and wire diameter of 0.14 mm, as an
RF coil. The loaded Q was 83; unloaded Q was 86. Other
solenoid designs were tried and rejected as being less
sensitive. They built unshielded double-saddle gradient
coils that provided 800 G/cm with a rise time of 100 |is.
Images of human hair in glycerin with 4 |lm resolution in
plane and 3 00 Jim slice thickness were acquired in one hour.
They use a gradient echo pulse sequence in which the half-
89Z. H. Cho et al., Med. Phvs. 15, 815, 1988.


27
assume that g = 0 while for small TE, g 0. To solve for
the Gi it is necessary to know one of them in advance, so to
determine Gi for large TE, another experiment was performed.
TE was held fixed at a large value and d3, the interval
between the end of the gradient pulse and the RF excitation
pulse, was varied in steps of TE/2. A system of
simultaneous equations describes the phase obtained by
varying d3 in steps of TE/2, starting with d3 = 0:
0(TE + d 3) 0O = 7X[2 G(TE / 2 + d 3) G(TE + d 3) G(d3)]
[41]
The problem of signal decay due to T2 is thus circumvented.
This technique could be used by itself or, as we used it,
only to obtain a starting point for varying TE.
A remaining difficulty is the ambiguity of phase
measurement. Phase can be directly measured only modulo
360, but the accumulated phase in our experiment may be
much greater. One way around this difficulty is to reduce
the applied gradient so that we can be sure that our sample
rate is above the Nyquist limit, so that 0i + 1 0. < 180.
To get an upper bound that guarantees no phase ambiguity,
assume that the eddy current field has the same amplitude as
the applied field before the n pulse and zero amplitude
following the K pulse. Protons precess at 4258 Hz/G. To get
a measurement for TE/2 = 512 ms without phase ambiguity
would, for a sample 1 cm from the center, require a gradient
pulse no greater than 0.000229 G/cm. Such a small gradient


132
reduces attenuation due to diffusion. Long, cylindrical air
spaces result in local B0 gradients that dominate the
appearance of the images. First, the local inhomogeneity
results in distortion in the readout direction, as spins at
(y, z) are mapped to (y, z + AB0(y,z)/G). More importantly,
the gradient due to the air spaces is as much as 2000 G/cm,
which greatly increases attenuation associated with
diffusion. One simple solution to the problem might be to
turn the sample so the long dimension of the air spaces is
parallel to B0 and the susceptibility effect is minimized.
Another solution would be to use two phase-encode
directions.
However, a sequence of 180 pulses in the readout
period was used to reduce the effects of susceptibility and
diffusion.91 The magnetization is sampled once at the peak
of each of a train of echoes. This CPMG (Carr-Purcell-
Meiboom-Gill) approach removes the effect of the B0
inhomogeneity and reduces the attenuation loss due to
diffusion.
The first images of a single cell were obtained by
Aguayo et al. in a 9.5 T, 89 mm bore spectrometer modified
for NMR microscopy.92 The gradient coil system was capable
of developing gradients of 20 G/cm. Using a 5 mm solenoid
RF coil, an image of a Xenopus Laevis ovum was obtained with
spatial resolution of 10 by 13 (lm and a slice thickness of
91J. C. Sharp et al., SMRM 1992, 688.
92J. B. Aguayo et al., Nature 322, 190, 1986.


17
cm. They use the same type of multiple FID sequence as
Boesch, Gruetter and Martin, with an exponentially
increasing sampling interval and 30 sampling points. The
gradient prepulse is 10 s in length. The first FID is
sampled at 1.5 ms after switching off the gradient, and
sampling continues for 4 s using multiple FIDs. They plot
the measured field vs. the time with and without
compensation. They use a 1 mm by 3 mm water-filled
capillary positioned at + /- 5 cm to discriminate B0 and
linear terms. They do not consider crosstalk or higher-
order terms. They use the same Laplace transform technique
as van Vaals and Bergman, but they apply it iteratively to
get better correction.
Heinz Egloff at SISCO (Spectroscopy and Imaging
Systems, Sunnyvale, CA)21 used a pickup coil to measure eddy
current fields. To correct the B0 component of the eddy
current fields, he moved the gradient coils until the field
shift was eliminated.
Riddle, Wilcott, Gibbs and Price22 considered the
performance of a Siemens 1.5 T Magnetom. They measured the
instantaneous frequency d^/dt of a 100 ml round flask
(presumably filled with water) following a 256 ms, 0.8 G/cm
gradient pulse. They present plots for imaging and
spectroscopy shims as well as for the gradient pulse. They
endorse d<^>/dt as an indication of shim. It would seem to
21H. Egloff, SMRM 1989, 969.
22W. R. Riddle et al., SMRM 1991, 453.


152
Figure 60. Spin-echo NMR image at 7.1 T of excised rat
spine in buffer solution. The parameters are: TE/TR:
13/2000 ms, FOV = 5 mm by 5 mm, slice thickness = 1.0 mm,
matrix = 128 by 128.
by varying the TR to provide T1 contrast. In these gradient
echo images, TE = 5.45 ms, the field-of-view (FOV) is 4 by 4
mm, and the slice thickness is 0.5 mm. The repetition times
TR range from 50 to 500 ms. The number of acquisitions is
varied along with TR to keep the intensity approximately
constant. The bright rings in (b) and (c) of Figure 59
appear to indicate the medullary cytoplasm. The bright
central spot is the germinal vesicle. The dark area is the
cortical cytoplasm.
The image of Figure 60 illustrates the contrast still
present in fixed tissues. The transverse section through


15
slices tilted by multiples of 22.5 is obtained from the
same 12 cm diameter phantom. The images were phase
corrected. The phase of points along the z axis and on
circles around the z axis was measured and used as data for
a polynomial regression analysis to determine the
coefficients of the various spatial harmonics. A table of
the harmonic components following a 2.5 second x gradient
pulse of 0.3 G/cm is presented. The delay between the end
of the gradient pulse and the first RF pulse in the three
pulse STE experiment is 20 ms, and the delay between the
first and second RF pulse is 15 ms. The experiment was
conducted following adjustment of the preemphasis unit. In
decreasing order of magnitude, x, z, y, z^, xz^, xz, and x^-
y2 terms were present. The value of the B0 term was not
reported. Note that after x, the dominant terms should be
eliminated by the symmetry of the coil/cylinder system.
Only the x and xz terms would appear in an ideal system.
The presence of terms having even-order in x can be due to
two reasons. First, the terms may really exist due to
asymmetries in the magnet and gradient coil, crosstalk
between amplifiers, etc. Second, the spherical harmonic
analysis is highly sensitive to the point chosen to be the
origin, and the most favorable origin may not have been
employed.
A series of the phase-modulated images is presented as
well, with delays of 5, 20, 50, and 100 ms between the x


12
at field, the pickup coil is sensitive also to any change in
flux resulting from mechanical motion, which can contaminate
the measurement. Since the field of the main magnet is
generally about four orders of magnitude larger than the
eddy current field and the time scale of mechanical modes is
smaller than that of the longer time-constant eddy currents,
mechanical stability of the coil is crucial. Drift in the
analog electronics is another potential difficulty with the
pickup coil technique. Even with digital integration, the
preamplifier can experience thermal drift on time scales not
too different from the eddy current field. In spite of
these difficulties, pickup coils are simple to use and can
be used effectively to adjust preemphasis compensation.
They are used routinely to correct for eddy currents in
commercial, clinical MRI installations.13
A different approach to measuring the eddy current
field is through its effect on the NMR resonance. One
advantage here is that a pickup coil and its associated
hardware are not needed. These proportional techniques
measure a frequency shift in the NMR resonance that is
directly related to the eddy current field.14 From Equation
[13], the phase of freely-precessing magnetization in the
rotating frame at time t with respect to t0 can be written
as
13Personal communication, Dye Jensen.
14Ch. Boesch et al., Maan. Reson. Med. 20, 268, 1991.


40
acquired with Adgrad2, that changes gph as well as T to keep
the phase locked. Eight averages were acquired. The eddy-
current field was measured out to 1 s, although the plots in
Figure 8 only show 0.5 s. The data were analyzed both with
the average-field technique used in the Excel (Microsoft,
Inc.) spreadsheet and with a multiexponential curve fit in
Mathematica (Wolfram Research, Inc.). The curve fits seemed
more satisfactory, and are shown in Figure 8. The lower
curves represent the eddy current field after compensation.
The curves plotted are the derivatives of the exponential
curves that were fitted to the raw data. The preemphasis
filter amplitudes and time constants were taken to be those
of the eddy current field itself. This procedure should
(t) (degrees) Echo Phase Shift
Figure 9. Fit to raw data of eddy current field of Oxford Z
gradient field for 4.7 T magnet system before compensation.


91
X
Figure 26--continued. Relative error plots of 30.2/66.1
radial gradient coil. Radius of coil corresponds to 1 on
scale, c) XZ plane.
performance for coils of different lengths, taking the
changing inductance into account.
The size and shape of the linear region of the 66.1
/30.2 radial gradient coil design is described by the plots
of Figures 26 and 27. It is not greatly different from the
longer 68.7/21.3 design. Note from Figure 26 that the
regions of equal relative error are not simply connected. A
three-dimensional plot of a region as large as that in the
two dimensional plots would give a false impression of the
size of the linear region, since the apparently solid
volumes would contain large holes. To avoid these "bubbles"
of linearity and give a true picture of the useful volume,


48
(r, 0, <]>) In the cylindrical system, the point is
described by (p, <\>, z) It can be easily shown that an
azimuthal component of the surface current, JI, proportional
to cost)) and independent of z produces the desired spatial
dependence. Neglecting for the moment the problem of
current continuity, there are two possible approaches to
achieving the cos<|) angular dependence. First, it can be
approximated by superimposing azimuthal currents with no
axial component. The solutions are exactly the same as for
discrete filamentary currents. The first approximation, the
120 arc familiar from the so-called Golay double-saddle
design,38'39 is shown in Figure 12(a). This class of
designs has been called the "Golay Cage" because of its
correspondence to the double-saddle design. Higher-order
approximations utilizing superimposed arcs are derived by
Suits and Wilken.40 The other approach is to use our
freedom to choose any axial current to meet cos([> by
varying the current direction, for example, cos sin<(). This approach leads to the Cosine Coil shown in
Figure 12(b). Note that in Figure 12 the return paths are
located away from the active volume of the coil. For a coil
of practical length, the current return paths can
significantly reduce and distort the gradient field.
38F
Romeo and D.
I.
Hoult,
Maan.
Reson. Med. 1.
44, 1984.
39m.
J.
E. Golay,
Rev. Sci.
Inst.
29, 313, 1958.
40b.
H.
Suits and
D.
E. Wilken, J,
. Phvs. E: Sci.
Instrum. 23. 565. 1989


92
the region displayed in Figure 27 is truncated. As a result
of the truncation, it is possible to see through the linear
region. All plots were produced by direct evaluation of the
Biot-Savart law.
-2
z
0
1
0.5
X 0
-0.5
-1
1
Figure 27. Perspective rendering of the 5% relative error
region of 30.2/66.1 radial gradient coil.
The axial coil was built to an extended-linearity
design proposed by Suits and Wilken.76 It consists of loops
at both 40.0 and 66.3 from the z axis. The outer loops
carry 7.5 times more current than the inner loops. Using
the additional degrees of freedom of the second-loop
position and the ratio of current in the loops, the fifth
76B. H. Suits and D. E. Wilken, J. Phvs. E: Sci. Instrum. 23, 565, 1989.


23
a sample with especially short (T1~T2~1 ms) relaxation times
is not an obvious way to detect low-level fields. The T2 of
the sample limits the duration of the interval in which
phase can be sampled. Another drawback is that the trains
of 71/2 pulses will produce stimulated echoes, even if is
on the order of the interpulse separation. However, this
may be the most promising of the techniques surveyed.
Soin-Echo Techniques
Distortions in the phase of spectra spatially localized
with a two-pulse Selective Fourier Transform technique31
were observed by Mareci.32 He observed that the distortions
were reduced by lengthening the echo time, consistent with
the known behavior of field distortions due to eddy currents
induced in the metal structures of the magnet by the pulsed
gradient fields used for spatial localization. We consider
how a series of spin echo experiments identical except for
Jt/2y 7tx
RF
9
Jl
P-
1 I 1
0 TE/2 TE
Figure 1. Two-pulse experiment with pulsed field gradient.
The long trailing edge of the gradient pulse indicates
distortion due to the eddy current field.
31H. R. Brooker et al., Maan. Reson. Med. 5, 417, 1987.
32T. H. Mareci, Personal communication.


14
G/cm. They use two NMR techniques to measure the eddy
current field. They interactively correct, using a 12 cm
diameter glass sphere filled with distilled water, and they
use no spatial discrimination in order to get all spatial
components. The experiment consists of a 2.5 s gradient
pulse of 0.6 G/cm followed by a train of 8 FIDs. There is a
20 ms delay between the time the gradient is switched off
and the first radiofrequency (RF) pulse. The RF pulses have
a 2 flip angle in order to reduce echo signals. The total
eight FID acquisition time is 200 ms. They solve the Bloch
equation for a sample with a single resonance frequency and
decay constant and extract
yABz(t) = (MydMx / dt MxdMy / dt) / (Mx + My) [23]
as an estimate of time-dependent B0 shift. They claim this
gives enough information for interactive preemphasis
adjustment. The one measurement they publish is of an
already corrected system and shows yABz(t) decaying from 2
to 0 ppm as time t increases from 20 to 200 ms. Glitches
are apparent at the ends of the FIDs.
To map the spatial variations, they place a stimulated
echo (STE) imaging experiment following the gradient pattern
of the experiment they want to analyze. The STE sequence is
applied with and without the preceding gradient pattern.
The difference in phase is considered to be due to the time
integral of the eddy current field in the interval between
the first two pulses of the STE sequence. A series of


151
(a)
(b)
Figure 59. Salamander (Taricha granulosa) follicles in 5 mm
NMR tube. Image series with various TR shows T2 variation.
In the gradient echo images, TE = 5.45 ms, FOV = 4 by 4 mm,
slice thickness = 0.5 mm. a) TR = 500, NA = 2; b) TR = 200
ms, NA = 20; c) TR = 100 ms, NA = 40 ms; d) TR = 50 ms, NA =
1000.


GENERAL INTRODUCTION
As pulsed field gradient technology for NMR matures,
new and diverse applications develop. Pulsed Gradient Spin
Echo techniques allow the measurement not only of the bulk
diffusion tensor, but of the structure factor of the
sample.1 Editing techniques use pulsed field gradients to
simplify the complex spectra of biomolecules.2 Local
gradient coils allow functional imaging in the human head.3
NMR microscopy can require field gradients much larger and
switched more rapidly than conventional imaging
experiments.4 Localized spectroscopy allows chemical shift
information to be collected from specific voxels in a living
animal.5 This paper will address some approaches for
producing and evaluating pulsed field gradients.
A technique was developed to measure the eddy current
field that persists after a field gradient is switched off
and, based on the measurement, a filter to correct for the
eddy current field was designed. The technique, which
employs a series of experiments based on the stimulated
echo, was then used to evaluate the performance of the
4D. G. Cory and A. N. Garroway, Maan. Reson. Med. 14, 435, 1990.
2D. Brhwiler and G. Wagner, J. Maan. Reson. 69, 546, 1986.
3K. K. Kwong et al., Proc. Natl. Acad. Sci. 89, 5675, 1992.
4Z. H. Cho et al., Med. Phvs. 15, 815, 1988.
5H. R. Brooker et al., Maan. Reson. Med. 5, 417, 1987.
1


119
Figure 36. Three-dimensional plot of useful volume of 10-
plane CRP coil. The shaded surface is the 5% relative error
contour.
current field is opposite to that for the single-cylinder
gradient coils.
A prototype of the 10-plane CRP coil was constructed.
It was designed to fit into the bore of our 31 cm 2 T magnet
(Oxford 85/310). The coil former consisted of 11 squares of
3/8" AB plywood 15.9 cm on a side. The square coil former
just fit into the 22.5 cm clear bore. A hole saw was used
to bore a 3" diameter hole for the sample. Grooves were
made with a 1/8" router bit to accommodate the wire in 10 of
the squares. The winding consists of 27 AWG enameled magnet
wire.


22
The step response of the inverse filter, F(s), is the
convolution of a step function and the impulse response:
sH (s)
s -
The amplitudes bj_ and time constants V- of the inverse
filter can be read directly from the inverse Laplace
transform, f(t), of F(s):
N
X
as
1 = 15 + wi
[29]
N
f(t) = 1 + £ bie~t/Vi [30]
i = 1
The inverse Laplace transform was performed by matrix
inversion for a four-time-constant case using Gaussian
elimination.
Now the appropriateness of these techniques to the
project of following the time evolution of the eddy current
field can be considered. Two of the techniques, those of
Egloff and Morich, involve the use of a pickup coil,
preamplifier, and integrator. We choose to confine
ourselves to NMR techniques. The procedures of Boesch, van
Vaals, Jehenson, Riddle, Keen, Hughes and Teodorescu require
shimming to correct for the inhomogeneity of B0. The fact
that T2* must be reasonably long also limits the region
where eddy current fields can be measured to well inside the
active imaging volume. Wysong and Zur propose similar NMR
techniques that do not require shimming. In general,
however, it is samples with long relaxation times that are
most sensitive to small eddy current fields, and the use of


ACKNOWLEDGMENTS
I would like to thank Janel LeBelle, Igor Friedman, and
Don Sanford for construction of the gradient coil
prototypes, and Jerry Dougherty for performing the
simulations of the coils they all helped to construct.
Stanislav Sagnovski, Eugene Sczezniak, Doug Wilken, and
Randy Duensing participated in many helpful discussions
concerning gradient coils. Debra Neill-Mareci provided the
excellent illustration of a gradient coil in Figure 22. For
their part in the microscopy project, thanks go to Barbara
Beck, Michael Cockman, and Dawei Zhou. Ed Wirth and Louis
Guillette provided the samples. For help measuring eddy
current fields I am grateful to Wenhua Xu, and to Steve Patt
for help with the software. Thanks go to my parents, Mary
Louise and Wallace Brey, and my brother, Paul Brey, for
encouragement and help with red tape. Paige Brey has my
special thanks for her extensive help preparing the thesis.
Katherine Scott, Richard Briggs, Jeff Fitzsimmons, and Neil
Sullivan enriched my graduate experience with their wide
knowledge and diverse interests. Thanks go to them for
their enthusiasm and for reading this thesis. Raymond
Andrew served as supervisory committee chairman. Thanks go
to Thomas Mareci for directing the research, for providing
11


89
include the effect of reducing resistive loss in the coil,
one must divide the current efficiency by the square root of
the length. The resulting measure is an indication of the
relative field that can be produced with an amplifier of a
given power. Figure 25 illustrates the relative power
efficiency as a function of the position of the return arc.
rel. power eff.
Figure 25. The relative power efficiency of the double
saddle radial gradient coil as a function of the angle
between the z-axis and the current return path.
The peak efficiency is achieved at 26, but efficiency is a
weak function of angle and much shorter coils can be used
with little loss of performance. Based on this curve, the
arcs of the 15 cm radial coil were placed at 30.2 and 66.1
from the axis of the coil, compared to 21.3 and 68.7 for
the Golay coil. The overall length of the coil is reduced
by 33%. The current efficiency is 0.819/a2 G/cm-A compared
to 0.808/a2 G/cm-A for the 68.7/21.3 Golay coil. It would
be even better to use a variant of the field-versus-
switching-time approach to determine the change in


70
or more conveniently,
K = B X p ,
^0
[62]
since B = H0H and h = p at the cylinder. Recall the
Biot-Savart law:
B ( x ) = J (x') x d3x' [ 63 ]
Air J
j X X |
Let B0(x) be the free-space field from the gradient coil.
Then
B ( X ) = B0(x) + -t- f K(x') X
0 4 K J
M- f w X X j2
dz /
x
[64]
x x
and substituting Equation [62] for the surface current, for
M- = Ho*
B ( x ) = B0 (x) + |[b(x') x p'] x p pp d2x' [65]
jx X
where p' = p(x'). This is an integral equation for B. We
can solve it iteratively. If we define Bn(x) as the field
to nth order, then the first-order solution is
i2
Bi ( x ) = Bn ( x
> + X p']
x x ,
x T d X
I '|3
X X I
[66]
The first-order solution does not take into account
eddy currents induced by eddy currents. When the coil and
shield are in close proximity, not only are the eddy
currents larger but they are also closer to the coil, so the
second-order effect can be important. To second-order,
B2(x) = B0(x) + ^ |[b1(x') x p'] x
X X 2 ,
rdY. [67]
lx x'P


4
constructed. Artifacts were eliminated from the images. A
software interface that allows the user to set up an
experiment by entering values in a spreadsheet was
developed. Useful contrast was obtained on fixed biological
samples. Preliminary imaging experiments on both biological
and nonbiological systems were carried out.


g(t)
0.25
0.2
0.15
dP
0.1
0.05
0
t (s)
0.6
t (s)
0.25 t
0.2
0.15
4J
01 0.1
0.05
i 1-
0 0.2 0.4 0.6
t (s)
(jO
oo
(a)
(b)
(c)
Figure 7. Eddy current field of home-built 9 cm gradient coil in 2 T 31 cm magnet as
measured with Adgrad. No preemphasis filter was used. a) X coil; b) Y coil; c) Z coil.


115
-1 -0.5 0 0.5 1
X
Figure 35--continued. c) Contour plot of the field of the 30
unit CRP Coil in the Y = 0 plane.


114
b) x
Figure 35. A Concentric Return Path Coil with 30 planar
units. The ratio of length:inner diameter:outer diameter is
6:2:3. a) Wire configuration, b) Contour plot of the field
in the Z = 0 plane.


49
(a) (b)
Figure 12. Two radial gradient coils, a) The Golay Cage
Coil; b) Cosine Coil.
Another approach to current return paths is possible if
we relax the requirement that the current is confined to
the surfaces of cylinders. The current return paths can be
located in the same plane as the azimuthal current paths. A
gradient coil can be constructed of a stack of planes
approximating a current sheet, such as shown in Figure 34
(a) on page 111. The planes include radial as well as
azimuthal current elements. The radial currents do
contribute to the axial magnetic field. It happens that the
third-order harmonic terms eliminated by using 120 arcs are
independently zero for the radial currents connecting the
arcs. These Concentric Return Path (CRP) Coils can have a
linear region that can be increased in length by stacking
more planes together. The overall combined coil structure
can also be very short, since the return paths do not
require extra length. In order to improve the linear region


146
of attenuation and back into the receiver. The magnitude of
the detected signal is proportional to the voltage produced
by the decoupler and can be used to construct a lookup
table. The transfer function of the modulator before and
after compensation is reproduced in Figure 53. A slice
profile of a frequency-selective pulse shaped in the time
domain is shown in Figure 54. Spin echo and gradient echo
imaging sequences were developed for use on the NMR
microscope. A timing diagram of a gradient echo experiment
is shown in Figure 55.
7t/2
RF
gx
gy
gz
r~ rIrn 1
d5 p2 d3 dl acq
Figure 55. Timing diagram for single slice gradient echo
imaging pulse sequence GESXYZ.
The code for this experiment is reproduced below.
GESXYZ
GRADIENT ECHO, TRANSVERSE VI.1
D5
P2/0,N,C@A+ 0,G
D3,F,U,W
Dl
A@A, E
D2,E J1
PHASE A = (2*S)+(2*#)


96
was potted in polyester resin to hold the coils in place.
Each of the windings is split so that it can be driven
either in series or in parallel. A simple adapter installed
between the power cable and the filter box enabled the coil
to be driven in series. The hardware and software have not
been designed to change configurations during an experiment.
Measurements have been made of the resistance and time
constant of the load that the 15 cm coil, cable, and filter
box present to the amplifier. Tables 4 and 5 allow the
measured values to be compared to predictions. The
experimental and theoretical inductance are reasonably
close. The inductance of coils constructed in this way is
not well controlled, since the cross section of the winding
can expand somewhat when the winding former is disassembled.
Table 4. Comparison of measured and predicted inductance
for 15 cm gradient coil set in series configuration.
Coil
Measured
Resistance
()
Measured Time
Constant
(M-s)
Experimental
Inductance
(flH)
Theoretical
Inductance
(JIH)
XI
2.48
740
1835
2000
X2
2.47
700
1729
2000
Yl
2.59
680
1761
2184
Y2
2.62
720
1886
2184
Z1
1.70
1120
1904
2120
Z2
1.50
1300
1950
2120


65
appropriate to maximize the cross-sectional area of the
windings subject to considerations of linearity and
available space.
Next we consider how the number of turns of wire in a
coil of fixed cross-sectional area can be varied to achieve
the desired performance. It is important to note that the
time constant of a coil, for a fixed area, can properly be
considered to be independent of the number of turns. This
result follows from consideration of a gradient coil at low
frequencies, as described by the equivalent circuit of a
series resistor Rc and inductor Lc as shown in Figure 15.
Figure 15. Equivalent circuit of a gradient coil in the
low-frequency limit.
Let R1 be the resistance, and let L2 be the inductance of a
single turn coil. It is well known that the inductance of a
coil increases as the square of the number of turns of wire
N.62 The resistance also increases as the square of the
number of turns if the area is held constant, since as the
number of turns increases, the area of each turn diminishes.
62T. N. Trick, Introduction to Circuit Analysis, p. 256, John Wiley and
Sons, New York, 1977.


47
designed with linear regions that are a large fraction of
the radius.37 We have tried to take a fresh approach,
combining aspects of both continuous and discrete designs.
For a solenoidal main magnet, available radial gradient coil
designs are longer and less efficient than axial designs, so
we have chosen to concentrate on the radial case.
Figure 11. The coordinate system used in the text.
An appropriate starting point to find a new radial
gradient coil design might be: what current distribution on
the surface of an infinitely long cylinder would produce a
field in which the axial component is linearly proportional
to the radial position, x? To describe surface
currents and fields, we introduce the three coordinate
systems described by Figure 11. Any point can be described
in any of three orthogonal coordinate systems. In the
Cartesian system a point is described by its location along
the three axes (x, y, z). In the spherical system, it is
described by two angles and the distance from the origin:
37R. Turner, J. Phvs. D: AppI. Phvs. 19, L147, 1986.


13
$ = yxj5 gdt'. [22]
L0
The instantaneous frequency (0(t), which can be defined as
the rate of change of the phase of related to the eddy current field through the Larmor
equation co = yB. Magnetic field homogeneity is important
when using this approach, so that the FID will persist long
enough to obtain a meaningful measurement.
In another approach based on the NMR experiment, the
phase of the magnetization is measured at a single point
in time. The phase at that point reflects the integral of
the eddy current field over certain intervals in the
experiment. Since only one point is sampled in each
experiment, many more experiments are required to map the
decay of the eddy current field than with the proportional
techniques. However, T2* and off-resonance effects do not
affect the usefulness of the technique. The experiment
proposed later is a single-point acquisition technique.
All the techniques surveyed were implemented for
unshielded gradient units, although preemphasis is typically
used on systems with shielded gradient sets as well.15
Boesch, Gruetter and Martin of the University Children's
Hospital in Zurich16-17 measure and correct eddy currents on
a 2.35 T, 40 cm Bruker magnet. The unshielded gradient set
has an inner diameter of 35 cm and a maximum gradient of 1
15R. Turner, Maan. Reson. Imaa. 11, 903, 1993.
16Ch. Boesch et al., Maan. Reson. Med. 20, 268, 1991.
17Ch. Boesch et al., SMRM 1989, 965.



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L E P 81,9(56,7< 2) )/25,'$


100
the ends of the coil. This is possible because the current
return paths are located on a larger concentric cylinder, as
shown in Figure 31. The coil is inherently torque free. In
fact, since each planar unit of the CRP coil is
independently torque free, axially asymmetric CRP coils will
also be torque free. Radial volume efficiency is traded for
axial volume efficiency. This new approach can lead to a
family of new designs; here we explore only two simple
possibilities.
Figure 31. A planar unit of a Concentric Return Path Coil.
Inner and outer arcs subtend 120. The ratio of the outer
radius to the inner radius is 1.5. Current direction is
indicated by arrows.
One way to understand the topology of the CRP coil is
to start with a double-saddle radial gradient coil as shown
in Figure 13(a). The only part of the coil that makes a
useful contribution to the field is the set of four inner
arcs. The longitudinal elements and the outer arcs make up
the current return paths that are needed to provide for


90
a)
-0.6 -0.4 -0.2 0 0.2 0.4 0.6
Y
b) X
Figure 26. Relative error plots of 30.2/66.1 radial
gradient coil. Radius of coil corresponds to 1 on scale,
a) YZ plane; b) XY plane.


157
Brooker, H. R., T. H. Mareci, and J. Mao. "Selective
Fourier Transform Localization." Maan. Reson. Med. 5: 417
(1987) .
Brhwiler, D., and G. Wagner. "Selective Excitation of 1H
Resonances Coupled to 13C. Hetero COSY and RELAY
Experiments with 1H Detection for a Protein." J. Maan.
Reson. 69: 546 (1986).
Cho, Z. H., C. B. Ahn, S. C. Juh, H. K. Lee, R. E. Jacobs,
S. Lee, J. H. Yi, and J. M. Jo. "Nuclear Magnetic
Resonance Microscopy with 4-|im Resolution: Theoretical
Study and Experimental Results." Med. Phvs. 15: 815
(1988) .
Cockman, M. D. "Novel Pulse Methods for Multidimensional NMR
Imaging and Spectroscopy." Dissertation. University of
Florida, 1988.
Cory, D. G., and A. N. Garroway. "Measurement of
Translational Displacement Probabilities by NMR: An
Indicator of Compartmentation." Maan. Reson. Med. 14: 435
(1990) .
Crown International. Techron 7540. Elkhorn, Illinois.
Delayre, J. L., J. S. Ingwall, C. Malloy, and E. T. Fossel.
"Gated Sodium-23 Nuclear Magnetic Resonance Images of an
Isolated Perfused Working Rat Heart." Science 212: 935
(1981) .
Edelstein, W. A., J. M. S. Hutchison, G. Johnson, and T.
Redpath. "Spin Warp NMR Imaging and Applications to Human
Whole Body Imaging." Phvs. Med. Biol. 25: 751 (1980).
Egloff, H. "Elimination of Base Field Shifts Due to Pulsed
Magnetic Field Gradients." Book of Abstracts, SMRM, 8th
Annual Meeting, 1989, p. 969.
Engelsberg, M., R. E. de Souza, and C. M. Dias Pazos. "The
Limitations of a Target Field Approach to Coil Design." J.
Phvs. D.: Appl. Phvs. 21: 1062 (1988).
Frahm, J., K. Merbolt, W., Hnicke. "Functional MRI of
Human Brain Activation at High Spatial Resolution." Maan.
Reson. Med. 29: 139 (1993).
Frese, G., and E. Stetter. "Tesseral Gradient Coil for a
Nuclear Nagnetic Resonance Tomography Apparatus." U. S.
Patent 5,198,769 (1993) .


1780
199 i
, b m
UNIVERSITY OF FLORIDA


94
windings. Placing the windings in parallel trades field
intensity for switching time; placing them in series reduces
switching time at the expense of lower field intensity.
Gmax (G/cm)
Figure 28. Maximum field Gmax (G/cm) vs. switching time t0
(ms) for the 15 cm field gradient coil set as driven by the
Techron 7540. Points along the curves represent designs
with different numbers of turns, increasing from left to
right. The top curve represents the inner radial coil. The
lower curves represent the outer radial coil and the axial
coil. A "slower" coil has a larger maximum field.
All three coils are optimized to approximately 10 G/cm
for the series mode. The exact number of turns for the
desired field is not used, due to the limited number of
standard wire sizes and the use of rectangular winding cross
sections. For the number of turns actually used, t0 and
Gmax based on the time domain model are tabulated in Table
3 .
A photograph of the 15 cm coil assembly is shown in
Figure 29. The coil is wound on an acrylic former having a
nominal O. D. of 6" (152 mm) and a wall thickness of 1/4"
(6.4 mm). The overall length was 38 mm. The radial coils


29
shift due to static field inhomogeneity and eddy current
fields. Then, stored by the second RF pulse along the z
axis, the magnetization accumulates no more phase until the
final RF pulse tips it back into the transverse plane. The
phase accumulation due to static inhomogeneity now unwraps,
resulting in the stimulated echo. If tj is long enough,
there is essentially zero eddy current field in the second
x, so the phase accumulated due to eddy current fields in
the first X is preserved.
It is possible to follow the eddy current decay by
incrementing either tdecay or x between experiments. If x is
incremented, the procedure for determining the eddy current
field is similar to that for spin echo experiments. The
phase shift for two experiments with different x is
subtracted to get the integral of the eddy current field in
the time between the earlier and later x. A more direct
approach is to increment tdecay between experiments, keeping
X small. Using this approach, each experiment yields the
integral of the eddy current field over a short interval x.
Dividing by x yields the average eddy current field in the
interval.
Two advantages of the STE are immediately evident. A
single STE experiment can be directly related to phase
accumulation in a single interval, eliminating the need for
the recursive data analysis or simultaneous equations
associated with the spin echo technique. This would also
seem to make the choice of gradient pulse amplitude more


128
Single slice gradient and spin echo images as well as
preliminary chemical shift images were obtained.
Literature Review
NMR microscopes have been created or adapted from
instruments designed for spectroscopy by a number of
research groups. Typically, these instruments are based on
89 mm bore magnets. Field strengths vary from 4.7 to 11.75
T.
L. D. Hall added imaging capability to a home-built
high-resolution spectrometer based on an Oxford Instruments
6.2 T magnet with a 54 mm bore.85 Early images of capillary
tubes containing water, benzene, acetone and methylene
chloride were made using the first-order shim coils to
produce field gradients for a projection-reconstruction
experiment. The shim coils provided a gradient of about
0.11 G/cm which yielded about 0.5 mm resolution for the
axially invariant capillary samples. Systematic
characterization of the shim coils as imaging gradient coils
was performed for the radial gradients using glass capillary
tubes filled with water and placed in a teflon plug and, for
the axial gradients, by a teflon plug into which was cut a
series of holes.86 It is much easier to use these phantoms
to characterize small bore systems than to make point-by
point measurements. To overcome the effects of magnetic
85L. D. Hall and S. Sukumar, J. Maan. Reson. 50, 161, 1982.
86L. D. Hall et al., J. Maan. Reson. 60, 199, 1984.


32
shift A<\> and x are easily reduced to a plot of the eddy
current field vs. time. A flow chart of Adgrad is found in
Figure 4. It is most easily explained in the context of the
whole experimental procedure. The user notes the phase of
the STE for an experiment with gp^, the value of the
gradient pulse, set to zero. He then selects a combination
of x, decay' an<^ 9ph that results in phase accumulation of
about 45 and acquires an FID. He also removes the file
"phase.out" if it remains from a previous session. Then he
executes the macro Adgrad(tmax, (|)0) where tmax is the value
of decay at which the macro will stop and <|)0 is the phase
with gph = 0.
Adgrad first calls the macro Calcphase to compute the
phase A(J) at the center of the acquisition window (that is
also the center of the FID) for the data already in memory.
Adgrad then stores the values of A in the output file "phase.out." Next, Adgrad tests to find
if decay > max If so, it ends the experiment. This
should not occur on the first pass through the test. In the
following two steps, Adgrad sets up the timing for the next
experiment. The new decay is set to greater than the
old by X, to provide for a contiguous series of intervals x.
The new x is set so that if the eddy current field remains
constant, the next measurement will yield a phase A Now the measurement is started. Following the measurement,
the macro calls itself and the process repeats. When the


93
and seventh order terms are canceled, resulting in about
eight times the useful volume of a Maxwell pair. For a coil
with more turns in some loops than others, the current
efficiency does not have an unambiguous definition. With
respect to the current in the outer loops, the current
efficiency of the coil is 0.635/a^ G/cm-A.
The 15 cm coil was matched to the Techron 7540
amplifiers in our laboratory using the time-domain model of
gradient performance described above. The height and width
of the windings was set to be 1 by 1 cm for the radial and 1
by 2 cm for the outer axial, to ensure that the assumption
of filamentary wires in the calculation of angle position
would be valid. The width of the outer axial winding was
increased from 1 to 2 cm, since it is farther from the
center than the others, and it was necessary to increase it
to match the radial performance. Inductance of the radial
and axial coils was calculated using a Fourier-Bessel
approach.77 The available combinations of switching time
and maximum gradient are shown in Figure 28.
In order to improve the switching time to smaller
fields, the 15 cm coil is constructed from split windings.
All the gradient coils described have the two sides driven
by separate amplifiers so that the magnetic center of the
coil can be moved. The 15 cm coil has each side split
further into two identical but electrically separate
77R. Bowtell and P. Mansfield, Meas. Sci. Technol. 1, 431, 1990.


143
written by David Brown. The internal RF trigger of the
spectrometer starts the free-running pattern generator. An
inverter/driver shown in Figure 50 converts the spectrometer
Figure 50. Inverter/Driver circuit for Pattern Generator
trigger. All resistors are 2 KQ. The transistor is type
2N3053.
trigger to the correct polarity and impedance. The pattern
generator output controls a four-quadrant double-balanced
diode mixer type SRA-1 (Mini-Circuits, Brooklyn, NY) as
shown in Figure 51. The mixer can be placed at almost any
low-power point in the transmitter chain, but the best
results were achieved with the mixer modulating the 160 MHz
Figure 51. Simple ring modulator to control the sign and
amplitude of the RF pulse.


55
Approaches based on a continuous current density are
most often used for gradient coils, where performance and
linear volume are more important than simplicity. The
impetus for these coils was the echo-planar imaging
technique of Mansfield,48 which requires high intensity
field gradients switched about an order of magnitude faster
than conventional Fourier imaging. Also, shielded coils are
useful in other imaging experiments that require rapidly
switched gradient fields, and in volume localized
spectroscopy. Without shielding to cancel the external
field, the higher frequency and intensity lead to greater
eddy currents in the cryostat and magnet, that in turn
distort the linearity and time response of the field. Using
current on two concentric cylinders, it is possible to
produce a linear field inside the inner cylinder and zero
field outside the outer cylinder. Continuous current
density coils can be designed to have a large linear region,
and, since current flows on the surface of the whole
cylinder, high efficiency.
A Fourier transform technique was applied by Turner to
design gradient coils that approximate a continuous current
distribution. The approach arose from consideration of the
eddy currents induced on cylindrical shields concentric to
gradient coils made up of discrete arcs.49 Expansion of the
Green's function in cylindrical coordinates was a natural
48P. Mansfield and I. L. Pykett, J. Maan. Reson. 29, 355, 1978.
49R. Turner and R. M. Bowley, J. Phvs. E: Sci. Instrum. 19, 876, 1986.


136
Twinax
Connector
Figure 43. One of the 3 channels of the pulsed field-
gradient controller.
as shown in Figure 43. Also, twinaxial cables were used to
couple the gradient demand to the gradient power amplifiers
Care was taken to place the PFG control unit as close as
possible to the source of the analog DAC levels to minimize
noise pickup. On the high-power side, the gradient coils
were not designed to dissipate the output of the amplifier
at a high duty cycle, so a fuse and filter unit as shown in
Figure 44 was incorporated between the power amplifiers and
Pi Filter
From Amplifier Fuse To Probe
Figure 44. One channel of the gradient filter unit. The Pi
filter is a Murata-Erie 1212-502.


142
MHz transcoupler are shown in Figures 48 and 49,
respectively. Bench tests with a network analyzer showed
the isolation between the RF IN and PREAMP ports to be 30.0
dB at 300 MHz. The loss between the PROBE and PREAMP ports
was 0.87 dB, and the loss between the RF IN and PROBE ports
was 1.3 dB. The Miteq preamp was used in conjunction with
the new transcoupler.
NMR spectrometers often possess a second RF transmitter
system known as a decoupler. In the Nicolet NT-300, the
decoupler is a single-frequency transmitter tuned to 300
MHz. Its control is more flexible than for the first, or
observe, transmitter system. The decoupler level can be
changed by means of a PIN diode switch between two power
levels set by rotary attenuators. The phase is under
quadrature control. The amplifier, although lower in power
at 20 W, is suited to RF pulses of longer duration.
Because of its flexibility, the decoupler was chosen to
replace the observe channel for experiments that require
modulated RF pulses. Dual attenuator control of the RF
level facilitated separate power adjustment of 71/2 and n
pulses. An additional advantage over the observe channel
that we did not exploit is that the decoupler frequency can
be moved independently of the receiver, so that off-center
slices can be excited.
In order to operate the selective pulse capability of
the spectrometer, the operator loads a pulse shape into the
pattern generator board described below using a program


145
A pulse sequence turns on both the decoupler, triggering a
full-scale linear ramp in the pattern generator, and the
ADC. The output of the decoupler is routed through 70 dB
(a) (b)
Figure 53. Response curve of modulator, a) Uncorrected; b)
corrected with lookup table.
10000 0 -10000 Hz
Figure 54. Slice profile of selective RF pulse.


25
()>(TE) <|>0
= yx
2%*" gdt'- Jf gdt'
We define a function G(t) by
G( t) = yx]q gdt'
which simplifies the expression above for 4>:
[36]
[37]
4> (TE) (j)0 = yx[2G(TE / 2) G(TE)] [38]
By measuring we hope to be able to extrapolate the function G(TE), whose
rate of change gives the eddy current field. By performing
a series of experiments in which the values of TE are
related by successive powers of two (TEi+1 = 2TE) we can
obtain a series of coupled equations. Using the shorthand
4> (TE) = i(
^i + 2 ~ §0 = Y*[2Gi Gi + 1] i = 1, 2, [39]
Inverting for Gi yields
Gi = [($i + i ~ 0 )/y* + Gi + i]/2 i = 1/ 2 [40]
For large enough i, Gi Gi+1, and the equation has an
immediate solution. The remaining can be determined
recursively. The rate of change of G(TE) is the eddy
current field.
Experiments and subsequent data analysis have pointed
to several drawbacks in this approach. The first is that
the echo time TE limits the maximum length of the gradient
pulse. A gradient pulse long in comparison to the eddy
current decay time approximates a step function, which


REFERENCES
Abduljalil, A. M., A. H. Aletras, J. C. Pruski, A. Ahmad, B.
S. Palmer, and P-M. L. Robitaille. "Torque Compensated
Asymmetric Gradient Coils for EPI." Book of Abstracts,
SMRM, 12th Annual Meeting, 1993, p. 1306.
Aguayo, J. B., S. J. Blackband, J. Schoeniger, M. A.
Mattingly, and M. Hinterman. "Nuclear Magnetic Resonance
Imaging of a Single Cell." Nature 322: 190 (1986).
Anderson, W. A. "Electrical Current Shims for Correcting
Magnetic Fields." Rev. Sci. Inst. 32: 241 (1961).
Bangert, V. and P. Mansfield. "Magnetic Field Gradient
Coils for NMR Imaging." J. Phvs. E.: Sci. Instrum. 15: 235
(1982) .
Barker, G. J., and T. H. Mareci. "Suppression of Artifacts
in Multiple-Echo Magnetic Resonance." J. Maan. Reson. 83:
11 (1989) .
Boesch, Ch., R. Gruetter, and E. Martin. "Temporal and
Spatial Analysis of Fields Generated by Eddy Currents in
Superconducting Magnets: Optimization of Corrections and
Quantitative Characterization of Magnet/Gradient Systems."
Maan. Reson. Med. 20: 268 (1991).
Boesch, Ch., R. Gruetter, and E. Martin. "Temporal and
Spatial Characterization of Eddy Current Fields in High
Field Superconducting Magnets." Book of Abstracts, SMRM,
8th Annual Meeting, 1989, p. 965.
Bowtell, R., G. D. Brown, P. M. Glover, M. McJury, and P.
Mansfield. "Resolution of Cellular Structures by NMR
Microscopy at 11.7 T." Phil. Trans. R. Soc. Lond. A 333:
457 (1990) .
Bowtell, R., and P. Mansfield. "Screened Coil Designs for
NMR Imaging in Magnets with Transverse Field Geometry."
Meas. Sci. Technol. 1: 431 (1990).
156


34
measured in the STE experiment. If we define (t) as the
total phase shift from t = 0 to tn for gph = 1, then
n
O(t) = [42]
2 1
This phase shift is just, for a single experiment,
'0
Now we assume that the eddy current field can be described
by a three-time-constant decay,
g(t) = Ae~t/ta + + Ce~^tc [44]
Integration gives a function to which the measured phase can
be fit:
0( t) = yx
taA
(l e ty,ta ) + ^(l e t/tb )
tcc(i -
s- fcc
[45]
Results
Eddy current measurements were made on several gradient
coils of practical interest. Tests of the Oxford gradient
coils in the 2 and 4.7 T magnets were conducted. For the
4.7 T magnet, the eddy current measurements were used to
adjust the preemphasis network. Measurements of the eddy
current fields associated with home-built gradient coils
were also made. The detailed design and construction of the
coils, on 9 and 15 cm formers, is described in the following
chapter.
Initial measurements were made using the spin-echo
technique of Gradtest vl.2. A 5 mm NMR tube with about 5 mm


102
Figure 32.
We apply Ampere's law | B dl = ^0Jenclosed >
where ^enclosed the current enclosed by an Amperian loop
in the YZ plane outside of the solenoid. Since
I
enclosed
= 0,
B dl = J B dr + j^B dz + j^B dr + j B dz = 0 [68]
The current density is independent of z, so the magnetic
field B must also be independent of z and the integrals over
1 and 3 must cancel leaving
$ B dl = J b dz + J B dz = 2BZL = 0, [69]
3
4
1
Figure 32. Ampere's law applied to a solenoid of infinite
length. The solenoid has an arbitrary cross-sectional
shape. The arrow indicates the direction of current flow.
An Amperian loop is drawn at right.
where L is the length of sides 2 and 4 of the loop. So
Bz = 0. Therefore, outside any infinitely long solenoid of
arbitrary shape, the component of magnetic field parallel to
the axis of the solenoid vanishes. An infinitely long CRP
coil would consist of two such solenoids and have zero field
parallel to its axis. Therefore, we cannot approximate a


63
voltage source. The amplifier, modeled by a time-dependent
voltage source, v(t), is connected to a load with
resistance, Rc, and inductance, Lc, as shown in Figure 14.
When a demand is applied to a current-controlled amplifier
for some current, i0, it will by definition change its
output voltage, v(t), as much and as rapidly as possible to
change the current through an inductance across the output.
If the maximum output voltage of the amplifier is V0, and we
define the steady state output voltage v0 = V0 / Rc, where
Vq>v0>0, then the amplifier output voltage and current as a
function of time will be
v(t)
0 t < 0
* Vq 0 t t Q
V0 t0 < t < OO
i(t) =
0
t <
0
Vo
t/T j
l1
- e~
Ao
fc0
A
rr
< OO
0 < t < tr
[47]
where t0 is the time at which the output current reaches the
desired current i0, and X = Lc/Rc. It is straightforward to
calculate that
tn = X In
Vn
\vo ~ vo J
[48]
The smaller the ratio of V0 to v0, the greater the switching
time t0 will become. If the amplifier is a voltage source,
the desired current will never be exactly reached. It is
more desirable to use a current-controlled amplifier for
which Vqv0.


54
The building block approach was successfully extended
by Suits and Wilken46 to use discrete wires to produce a
constant field gradient over an extended region. They
evaluated designs for cylinders with the polarizing field
both parallel and perpendicular to the axis. To improve the
useful volume of the radial gradient coil, they superimposed
four saddle coils. The available degrees of freedom then
included the number of turns in each of the four coils, the
angular width of the four arcs, and the axial positions of
the arcs. Systems of nonlinear equations result that were
solved to null desired terms in an expansion of the field in
orthogonal functions. Numerical plots demonstrate that the
useful volume was extended to about eight times that of the
simple saddle coil. In each case, the volume was
nonspherical. The problems of extending this approach
further are that larger systems of nonlinear equations are
increasingly difficult to solve, and that the orthogonal
expansions do not converge rapidly away from the center of
the coil.
Bangert and Mansfield47 designed a gradient coil in
which the wires were included in two intersecting planes.
The wires in each plane formed two trapezoids symmetrically
placed about the z axis. By setting the angle between the
planes to 45, the third-order terms in the magnetic field
are canceled.
46B. H. Suits and D. E. Wilken, J. Phvs. E: Sci. Instrum. 23, 565, 1989.
47V. Bangert and P. Mansfield, J. Phvs. E: Sci. Instrum. 15, 235, 1982.


18
work only for single-line samples, however. Following the
gradient pulse, the plot of d<|>/dt contains peaks that are
not explained. They may be an indication of the true d or they may be artifacts from beginnings and ends of FIDs.
The sensitivity of the technique as presented here seems to
be about 1 Hz.
Hughes, Liu and Allen23 of the Departments of Physics
and Applied Sciences in Medicine at the University of
Alberta measured the eddy current fields of their 2.35 T, 40
cm bore Bruker magnet. After 57 delays ranging between 500
(is and 2.5 s following a 0.2 G/cm gradient pulse the FID was
measured and the offset frequency of the line determined.
They placed a 13 mm diameter spherical water sample at +/-
1, 2, 4 cm along the axes of the radial gradients under
test. A four-exponential fit was applied to all six
locations simultaneously. The shortest time constant was
associated with the amplifier rise time. An interesting
plot shows that the field associated with each time constant
is essentially linear. The B0 fields associated with the
various time constants are different, however, suggesting a
unique isocenter for each time constant.
Zur, Stokar, and Morad24 of Elscint in Israel place a
doped water sample at +/- 5 cm from the center in the
direction of the gradient of the field. A train of 256 FIDs
is acquired after switching off the gradient. Each FID is
23D. G. Hughes et al., SMRM 1992, 362.
24Y. Zur et al., SMRM 1992, 363.


73
£2 load is + /- 1 dB up to 25 kHz for 265 W. The noise is
rated to be 112 dB below the maximum output from 20 Hz to 20
kHz .65
Tests of the Techron 7540 were conducted into six
loads consisting of wire-wound resistors between 1 and 9 Q.
The amplifiers were pulsed to saturation at low duty cycle.
10-90% rise times were between 4 and 6.5 |ls, and so are
essentially independent of load. Thus the amplifier was
bandwidth limited, not slew-rate limited, and it is
appropriate to use a linear model. The voltage and current
60
50
w 40
4J
o 30
20
>
10
0
4 6
R (ohms)
10
16
14
12
w
10
&
g
(d
8
6
H
4
4 6
R (ohms)
10
(a) (b)
Figure 18. Output of Techron 7540 measured into load, a)
Measured voltage; b) Calculated current.
produced are shown in Figure 18. For load resistance of
four ohms or more, the amplifier at saturation can be
modeled by a 56 V voltage source. For a load resistance of
65
Crown International, Techron 7540. Elkhorn, Illinois.


140
preamplifiers with a new preamp decreased the noise figure
to 5.5 dB.
The new preamp is a low-nose bipolar type model AU-
1054-1103 (Miteq, Inc., Hauppauge, NY). The gain as
measured at the factory is 30.7 +/- 0.25 dB. The noise
figure was measured by the factory to be 1.28 dB at 250 MHz,
1.46 dB at 500 MHz. The -1 dB gain compression point is +9
dBm.
Better results were obtained by replacing the entire
transcoupler circuit, which switches the probe between the
RF power amplifier for transmit and the preamplifier for
receive. The circuit and photographs of the home-built 300
Figure 48. The transcoupler circuit.


60
Turner has suggested that the best approach to a
compact gradient head coil design is that of Wong, in which
the return paths are placed on a larger cylinder.59 He
points to the trapezoidal gradient coil designed by Bangert
and Mansfield,60 and discussed above, as a starting point
for this approach. The concept for such a gradient coil is
described in a patent by Frese, for a cylindrical
geometry.61 It can be thought of as a Bangert and Mansfield
coil in which the inner and outer wires have been stretched
into arcs on concentric cylinders. This is the design
independently developed by Brey and Andrew and dubbed the
Concentric Return Path Coil (CRPC). Frese suggested using a
stack of the planar CRPC units with spacing along the
cylinder's axis varied to improve size of the linear region.
He also suggested that the angle of the arcs could be varied
from plane to plane. No specific information on the spacing
or angle of the arcs is provided.
A survey of the literature suggests that it is
desirable to design a short gradient coil using the basic
concentric return path geometry to be used for the human
head. A direct error-minimization technique is appropriate
for two reasons. First, the Fourier-Bessel transform
technique, although computationally efficient, limits the
shape of the region of optimization to the surface of a
59R. Turner, Maan. Reson. Imaa. 11, 903, 1993.
60V. Bangert and P. Mansfield, J. Phvs. E.: Sci. Instrum. 15, 235, 1982.
61G. Frese and E. Stetter, U. S. Patent 5,198,769, 1993.


69
type of approach further and evaluate the ratio of the
gradient at the center of the coil with and without the
shield. This would seem to be a useful approach when
evaluating reduced-size gradient coils and comparing them to
shielded coils. For small eddy current fields, as in the
case of reduced-size coils, an iterative approximation
technique described below can be used to solve the integral
equation for the eddy current field. This technique is best
suited to situations where the eddy current field is much
smaller than applied field, so that a first-order
approximation can be used. However, it is simple and
flexible.
To estimate the eddy current field due to a gradient
coil, we assume that there is a passive shield surrounding
the coil. The shield is typically part of the cryostat. The
boundary conditions at the shield will be
(B2 B-l ) n = 0
h x (h2 H-l) = K
[59]
[60]
where Bx and Hx are the magnetic induction and field inside
the shield, B2 and H2 outside the shield, K is the surface
current on the shield, and h is an outwardly directed unit
vector normal to the surface of the shield.64 We assume the
shield is perfectly conducting, so that with H2 = 0,
H-l x n K ,
[61]
64J. D. Jackson, Classical Electrodynamics, p. 1.5, John Wiley & Sons,
New York, 1975.


financial and moral support, and for encouraging me to
pursue this work to its conclusion.
in


9
V x (V x a) = V(V a) V2a
[18]
gives
-V2b = (i0V x j .
[19]
Using Ohm's law to eliminate J for E, neglecting the
displacement current, yields
-V2b = |i0aV x e ,
[20]
so Equation [16] allows this to be expressed as
[21]
The decay of the magnetic induction must be a solution to
this diffusion equation. Separation of variables gives
solutions for the time part having an exponential time
dependence. This makes it possible to correct for the
linear spatial term in the eddy current field with a linear
filter network. Such a network is known as a preemphasis
circuit.
In a superconducting magnet, the conducting structures
involved are often at very low temperatures and hence have
much greater conductivity than might otherwise be expected.
For example, pure aluminum at 10 K has a resistivity of
1.93 x 10-12 il-m, while at a room temperature of 293 K its
resistivity10 is 2.65 x 10-8 Q-m. The time scale of the
eddy current decay is directly proportional to its
conductivity, as can be inferred from Equation [21], so eddy
currents will persist 13,700 times longer in an aluminum
10D. R. Lide, (Ed.) CRC Handbook of Chemistry and Physics. 72nd
Edition, CRC Press, Boca Raton, 1991.


149
spreadsheet as shown in Figure 56 provided a convenient way
to make the necessary calculations. Command macros were
created to set up the spectrometer automatically for an
experiment, then to transfer the image to the PC for storage
and display. The macros were executed from a pull-down
menu. The parameters in the spreadsheet can be divided into
dependent and independent. The independent parameters are
those that would be set by the operator and are color-coded
red or blue. The dependent parameters are color-coded
black. Front-panel control of the gain of each of the
channels of the Techron amplifier facilitated manual
adjustment of the magnetic center of the gradient coils.
However, adjusting the gain resulted in small changes in
gradient calibration. In order to account for the changing
calibration factors in a convenient way, NMR.XLS shown in
Figure 57 provided a worksheet for calculating the current
calibration and storing it in the relevant worksheets.
Results
Phantoms consisting of melting-point capillary tubes
in a 5 mm NMR tube of water are used to test the NMR
microscope; an example is shown in Figure 58. The capillary
tubes are 5, 10, 25, and 50 |IL. The 5 [XL tubes have the
smallest I. D. of 270 |lm. The TE is 10 ms; the repetition
time TR is 1 s. The slice is 0.5 mm thick. Two averages
were used. A spin-echo sequence is employed. The field of
view is 5 mm by 5 mm, and the image matrix is 128 by 128.


57
homogeneous field along the axis of the solenoid, the target
cylinder should be as narrow as possible. The effect is
especially pronounced at the ends of the target cylinder.
The importance of functional imaging of the human brain
and its reliance on the echo planar imaging technique puts
special demands on the rise time and field of the gradient
coil. The fact that smaller coils will be more efficient
and less affected by eddy currents has motivated several
workers to design gradient coils that will fit closely over
the head. To use a small gradient coil it is necessary to
have extended linearity in the radial and axial directions.
For a head coil, extended axial linearity is especially
important to allow the diameter of the coil to be smaller
than the width of the shoulders.
Wong applied conjugate gradient descent optimization to
the design of gradient coils with extended linearity.52 He
allows the position of current elements to vary to minimize
an error function. It is possible to define the error
function as desired, so it is simple to optimize over
regions of any shape, or for coil formers of any shape. It
is also simple to include parameters such as coil length.
Repeated numerical evaluation of the Biot-Savart law for the
test wire positions would limit the application to coils
with a fairly small number of elements. Wong applied the
technique to the design of a local gradient coil for the
52E. C. Wong et al., Macm. Reson. Med. 21, 39, 1991.


35
of H20 trapped by a vortex plug was used as a sample and
placed 1.7 cm from the center of a 2 T, 31 cm horizontal-
bore magnet (as measured from an image). The Oxford Z
gradient in the Oxford 2 T magnet was pulsed to a value of
1000 units or 1 G/cm. The manufacturer-installed
preemphasis filter was in place. A d3 array with four
elements was used to establish the phase value for large
echo times via matrix inversion of simultaneous equations.
An echo time array resulting in a series of coupled
equations was used to work back to 1 ms. The resulting plot
is shown in Figure 5. The bumpiness of the plot may be due
to the preemphasis. Data points are plotted in the center
of the interval for which they represent the average
gradient.
Figure 5. Eddy current field as a fraction of applied field
for Oxford gradient coil.


86
in practice it was not used. The interdigitated geometry-
reduced eddy currents from the gradient coil, but allowed
the shield to serve as a barrier to the RF field. The
Maxwell pair is on the outside, which helps to hold the
radial gradient coils in place. Epoxy was initially used to
hold the windings to the former and pot the windings, but
the epoxy did not withstand the temperatures developed by
the coil and depolymerized. Polyester was selected as a
casting resin that would outperform the acrylic former under
warm conditions, and the coils were potted in polyester.
In order to cool the unit, approximately 25 m of 1/8"
0. D. by 1/64" wall polypropylene tubing was wound around
the coils. It was connected to a Neslab circulating system
that is capable of producing a pressure head of 40 psi.
Supply tubing consisted of about 30 m of 1/4" 0. D. by
1/32" wall polypropylene. Assuming laminar flow, the water
flux through a tube74 of radius r (cm) is
flux =
7tr4Ap
8|ll
where Ap is the pressure in dyne/cm2, 1 is the length of the
tube in cm, and |l is the viscosity in poise. The resulting
flux for a 40 psi drop is 8.6 cm3/s. One KW of power
transferred to the water in the tube will raise its
temperature by about 28 C. Measurement of the motion of
bubbles in the tubing reveals a flow of about 3.5 cm3/s.
74D. Lide,(Ed.), CRC Handbook of Chemistry and Physics. 51st Edition,
CRC Press, Boca Raton, 1970, p. F-34.


42
g(t) (%)
g(t) (%)
g(t) (%)
Figure 10. Eddy current field of Oxford gradient coil in
4.7 T magnet system before (upper curve) and after
compensation (lower curve), a) X coil; b) Y coil. Lowest
curve was acquired after second-pass preemphasis; c) Z coil.
verification consists of measurements of the eddy current
field before and after preemphasis. The level of the eddy
current field after preemphasis can be interpreted as an
upper limit on the error bar of the measurement. It is only
an upper limit, since other errors also contribute to the


84
dimensions of the winding, which expands when it is removed
from the winding former. The maximum operating field as the
system has been installed is 12.83 G/cm for X, 12.28 G/cm
for Y and 8.80 G/cm for Z. By limiting the gradient field
to a value below that corresponding to the steady state
current, the switching time is better controlled.
A drawing of the coil that illustrates how the Faraday
shield and RF coil fit together is shown in Figure 22.
Figure 22. Drawing of 9 cm gradient coil set with Faraday
shield and RF coil.
Figure 23 is a photograph of the assembly. The former
is an acrylic tube of 3.5" (89 mm) nominal outer diameter
and 1/8" (3.2 mm) wall thickness. The radial gradient coils
are wound on rectangular bobbins made from three flat pieces
of acrylic. The wires are held together in a hexagonal
matrix by General Electric varnish #7031 diluted in acetone.
After winding, the bobbin is disassembled and the coil is
glued onto the cylindrical former with epoxy. A variety of


66
The total resistance and inductance are then
Rc = R2N2 Lc = LtN2 .
The time constant X of the coil is just the ratio
w w
x =
C,
'Ri
[53]
[54]
Perhaps surprisingly, x is independent of N. The result
does not apply when additional turns of wire are added to an
existing coil, thus increasing the area. However, since the
area should already be as large as possible to maximize the
performance, it will not be possible to increase N without
decreasing the size of the wire.
To determine how many turns of wire N should be used in
the gradient coil, we consider how rapidly and to what value
the current rises for various N, holding the area constant.
It will be shown that with a current-controlled amplifier,
the coil is optimized to switch to the field at which it
reaches a saturation current, J0, which is the maximum that
the amplifier can supply. Consider an amplifier with
negligible output impedance switching at time t = 0 from
zero current to maximum current, J0, through a gradient
coil, reaching I0 at t0. Assume that Rc < V0/I0. The
current as a function of time is:
i(t) =
[ Vo
( -t/>
1 e A
0 < t < t0
V )

[55]
To
CT
IV
We define a current efficiency k for a single turn so that
the gradient field G(t) = kNi(t) Rc varies as N2, and the


30
straightforward. Since t1 is limited by Tlt which is
generally longer than T2, it is possible to sample with
smaller residual gradient field than in the spin echo
experiment.
The eddy current field is subject to a multiexponential
decay. The integral of a multiexponential decay is another
multiexponential decay. We can expect these functions to be
reasonably smooth. That is, if we notice that the phase is
not changing much between delay increments, we could either
increase the delay increment or increase the amplitude of
the gradient pulse. This is a form of adaptive sampling,
since the sampling strategy for the gradient field depends
upon its behavior. The sampling technique should be capable
of following the residual field decay when preemphasis is
used, and in this situation the field will not in general
decay monotonically, since some of the decay components may
be overcompensated. Therefore the adaptive sampling must
also be able to decrease sensitivity when needed.
Since the eddy current field generally changes most
rapidly at short times, varying x to keep the measured phase
shift approximately constant for each value of tdecay yields
less densely spaced measurements when the field is changing
slowly. We have implemented such an adaptive sampling
technique by writing a recursive macro Adgrad in the Varian
MAGICAL language to perform a series of measurements in
which x is varied to "lock" the phase shift to 45. The
macro functions as a command to the Varian program "VNMR"


6
cancel it. An eddy current measurement technique is also
useful in order to evaluate the possibility of performing a
given experiment with available hardware. In this chapter,
a technique for measuring and analyzing the time behavior of
eddy current fields is developed and experimental results
are presented. Some general physical considerations of eddy
currents are discussed, and existing techniques for eddy
current field measurement are reviewed.
An introduction to the Bloch equations will be
preliminary to a discussion of the effect of the eddy
current field on the nuclear magnetization. The Bloch
equations provide a phenomenological description of some
aspects of the behavior of spins in a magnetic field. Let M
be the bulk nuclear magnetization, 7 the gyromagnetic ratio,
B0 the polarizing magnetic field, and B^ the amplitude of
the radio frequency excitation field which has rotational
frequency to. T1 and T2 are the time constants associated
with longitudinal and transverse relaxation, respectively.
Mx j(bomy
+ B^My
sin COt)
- mx/t2
[1]
My = y(B1Mz
cos cot
- b0mx )
- My/T2
[2]
y(b1Mx sin cot
+ Bj-My
cos cot)
- [mz M0 )/t1
[3]
Instead of T2i the symbol T2* is used to denote the time
constant of apparent transverse relaxation when
inhomogeneity in B0 is present. Neglecting the effects of
T-j and T2 and assuming B1 = 0, the equations can be
simplified.


116
-1 -0.5 0 0.5 1
X
Figure 35continued, d) Contour plot of the relative error
of the 30 unit CRP Coil in the Y = 0 plane.


99
demand for gradient systems that produce a field that is
high in intensity and can be switched rapidly on and off.
Since smaller coils of a given design are more efficient,
and gradient coils designed specially for the head can be
made smaller than those that accommodate the whole body,
such coils are useful in high-performance applications.
Another benefit of smaller designs is the reduced
interaction with the cryostat and other metal structures in
the magnet, and the smaller resulting eddy currents. The
need for shielding is thereby reduced. Some smaller designs
incorporate shielding, but many do not. The principal
problem associated with designing gradient coils for the
human head is that the linear region of traditional designs
is a rather small fraction of the length of the assembly and
is located at the center of the assembly. A traditional
coil design will not fit over the human shoulders. A
successful alternative has been to design asymmetric coils
in which the linear region is pushed to one end of the
coil.80 In general such coils experience a torque in the
main magnetic field that is not experienced by symmetric
coils. These coils must be mechanically fixed in place.
Torque-free coils have been designed by imposing extra
constraints.81
We have taken an approach that results in a symmetric
coil that has a linear region that extends quite close to
80C. C. Myers and P. B. Roemer, SMRM 1991, 711.
8-'-L. S. Petropoulos et al. SMRM 1993, 13 05.


BIOGRAPHICAL SKETCH
William W. Brey was born in Gainesville, Florida, in
1961. He attended Rice University and graduated in 1983
with a B.A. in physics and a concentration in space physics.
After working for four years as a senior research assistant
in the labs of Jean Delayre and Ponnada Naryana at the
University of Texas Medical School in the University of
Texas Health Science Center in Houston, Mr. Brey returned to
Gainesville to pursue graduate studies with Raymond Andrew
and Tom Mareci. He has enjoyed the cooperative nature and
diverse interests of the NMR group at the University of
Florida.
Following completion of his degree, Mr. Brey will join
Conductus, Inc., in Sunnyvale, California, to develop
superconducting electronics technology for NMR applications.
162


112
c)
+ 1% ....
+3% _
+ 5%
-1%
-3%
-5%
d) x
Figure 34--continued. c) Contour plot of the field of the
10 unit CRP Coil in the Y = 0 plane, d) Contour plot of the
relative error of the 10 unit CRP Coil in the Z=0 plane.


125
nearly to each end of the coil. The coil is inherently
torque-free, a practical advantage that is particularly
important for safety in clinical applications. While the
coil is not self-shielded, the eddy current field has the
sign opposite to that of a coil on a single cylinder, which
reduces the dynamic range required of the amplifier.
Insignificant tuning shift or reduction of Q was observed
when an RF coil was placed inside, even in the absence of a
Faraday shield. Any RF field used for NMR would be in the
plane of the planar units, and so flux through the planes
should be minimized. This may make it possible to use an RF
coil that is an especially large fraction of the inner
radius of the CRP coil.
Many refinements of the design are possible. A more
efficient coil might be designed by including power
efficiency in the error function. It would be possible to
locate the useful volume off-center, which would improve the
usefulness of the coil for head imaging, since more of the
neck could be visualized. Dr. Raymond Andrew and Dr. Eugene
Sczezniak are refining the design for use on the human head.


106
ml
i(\|/2 VJ/-L K / 2) = (2n + 1)K / 2 [76]
for r¡ some nonnegative integer. The linear term, which is
needed in the field gradient coil, has n = 1, m = 1. The
arc widths that do not contribute m= 3 terms are V)/2 Vi =0,
K/3, 2k/3, 3l. The arc width 2 k / 3 is employed in the
classical radial gradient coil designs.
A similar analysis can be applied to the radial current
elements. The result is
dBz =
ZHo jr _jn_|Ln+l.-l(1 S-.oK1 + + Upnm(COs0)eijn(t,,-',')(
2 n=0m=-n U-n + i,m + l (n + m + 2)(n + m + 1)
[77]
where
Ids (n m)! .
^nm ~ £-m ~ £n+1 7 H 77 ^>nm cos ) [78]
47tf (n + m) !
To eliminate terms with a given m, the radial currents must
be placed at positions that make dBz completely imaginary:
tmf=nK for n a nonnegative integer. For m = 3 we obtain
V|/ = 0, K/3, -K/3, 271/3, -271/3, 71 as the only positions that
do not contribute term with m = 3. But these include
exactly the solutions for the arcs! Note that the solutions
for the radial currents are valid independent of length. We
can choose the lengths as desired, or use the length to
cancel higher order terms. We might also be able to find
solutions in which the arcs and the radial currents
contribute m = 3 terms of equal magnitude but opposite sign.
Perhaps this would allow arc widths of approximately 71/2,
which would be most convenient for coil construction.


20
of the FID represents a projection of the phantom in the
quasi-steady eddy current field. Measuring the distance
between the peaks that appear as edge artifacts gives the
eddy current field.
Teodorescu, Badea, Herrick, and Huson27 at the Texas
Accelerator Center and Baylor College of Medicine measured
eddy current fields in their 4 T, 30 cm superferric self-
shielded magnet. The magnet was operated at 2.19 T. They
follow Riddle et al.28 in their measurement. A small
phantom is placed at various off-center locations. They use
a 0.8 G/cm gradient pulse of 15 ms and a 750 (is rise/fall
time. This is followed by an FID (or a series of them) that
is acquired for 20 ms. They compare this to the result
obtained from a pickup coil.
The eddy current field was measured with a sense coil
and analog integrator by Morich, Lampman, Dannels, and
Goldie.29 They used a Laplace transform approach to derive
correct parameter values for an analog inverse filter to
compensate for the eddy currents. The analog inverse filter
was of conventional design,30 placed at the input of the
gradient power supply. The theory was tested on an Oxford
Magnet Technology whole body superconducting magnet.
The approach is based on the ease with which a linear
system can be analyzed in the reciprocal space s defined by
27M. R. Teodorescu et al., SMRM 1992, 364.
28W. R. Riddle et al., SMRM 1991, 453.
2%. A. Morich et al. IEEE Trans. Med. Imaa. 7, 247, 1988.
3D. J. Jensen et al., Med. Phvs. 14, 859, 1987.


129
field inhomogeneity and to observe resonances with larger
line widths, a high-resolution probe was subsequently
adapted to produce larger gradient fields.87 The glass
variable-temperature dewar was replaced with an acrylic tube
of 37 mm 0. D. and 5 mm thickness. Grooves 3 mm deep were
cut into the former, and 27 AWG copper wire was placed into
the grooves. For the radial gradients, the radius of the
coil was 14.5 mm and 8 turns were used. The axial gradient
consisted of a Maxwell pair, also with 8 turns. Using an
Amcron model M-600 audio amplifier to supply a 1 A pulse,
they obtained the fields shown in Table 6. Melting point
Table 6. Gradient field and inductance of Hall's microscopy
coils.
Direction Inductance (JIH) Field (G/cm)
X 35 3.71
Y 35 3.95
Z 12 2.5
capillaries were imaged without slice selection to obtain
images with 100 |lm resolution. A 10 ms delay between the
gradient pulse and the data acquisition period was required
to produce an undistorted spectrum.
An 8.5 T, 89 mm bore magnet at the Francis Bitter
National Magnet Laboratory served as the heart of a home-
built spectrometer imaging system.88 The Maxwell and Golay
87L. D. Hall et al., J. Maan. Reson. 66, 349, 1986.
88E. W. McFarland et al., Maan. Reson. Imaa. 6, 507, 1988.


138
sheets of 2 mil thickness are separated by 1 mil mylar film.
The capacitance density is about 115 pF/cm2. The angular
width of the foil strips is about 60. The foil and mylar
sheets are wrapped on a 6.5-PP quartz tube with 6.5 mm O. D.
(Wilmad Assoc., Buena, NJ). Balanced matching reduces
losses in conductive samples.98 The Q of the RF coil inside
the gradient set is about 100. To avoid eddy currents, no
Faraday shield was used between the RF coil and the gradient
coil. To reduce the effect of coupling between the RF and
Figure 46. Equivalent circuit for RF probe. Components on
the left are lumped; those on the right are formed from foil
sheets.
gradient coil leads, feed-through capacitors were placed at
the base of the probe. The standard commercial probe design
includes a glass dewar surrounding the RF coil to facilitate
variable temperature operation. An otherwise standard
design can be converted for PFG operation by replacing this
dewar with a cylindrical PFG coil set. The range of
variable-temperature (VT) operation would be restricted, but
98J. Murphy-Boesch and A. P. Koretsky, J. Maan. Reson. 54, 526, 1983.


139
some VT capability would remain. The details of the
gradient coils are set forth in the previous chapter. In
the prototype probe, no VT capability was provided. The
probe is shown in Figure 47.
Figure 47. Probe assembly with RF coil and PFG coils
visible.
The RF front end supplied with the NT-300 consisted of
a rack into which filters, Transmit/Receive switches,
filters and preamplifiers are plugged. The plug-in modules
select the frequency band and type of experiment. It was
possible to use the factory front end for NMR microscopy
experiments; however, the noise figure of the NT-300 was
measured to be 6.5 dB. Replacing the Avantek UTO-511


eddy current field /
applied field
120
Figure 37. Eddy current field of the 10-element CRPC,
expressed as a fraction of the applied gradient.
eddy current field/
applied field
Figure 38. Eddy current field of the 30-element CRPC,
expressed as a fraction of the applied gradient.
The coil was connected to a Techron 7540 power supply
amplifier. All the windings were placed in series, so that
only one of the two channels was used.
The phantom was integrated into the RF coil assembly.
It consisted of a five by nine cell section of 1/2" deep


127
linearity over the field of view, and interaction between
the gradient coils and the RF coil. Specimens consisted of
amphibian oocytes and excised tissue samples because of the
ease of handling these samples. The biological studies
guided the hardware and software design. The goal was to
begin to consider whether NMR can detect changes in quality
or maturation stage of oocytes in vivo. High-field systems
also show promise for pathology, since excised tissue tends
to be somewhat smaller than what is effectively imaged on
the 2 T system. Also, there is no other route to studying
very high-field imaging, since large-bore high-field magnets
do not exist.
Accordingly, a 7.1 T Nicolet NT-300 multinuclear pulsed
NMR spectrometer was modified to perform NMR microscopy and
other experiments requiring pulsed field gradients.
Previous work by faculty, graduate students and staff was
extended to produce a practical imaging system. Both
hardware and software enhancements to the instrument were
required. The additional capability included the production
and control of magnetic field gradients for imaging and
localized spectroscopy, the production of amplitude-
modulated RF pulses for slice selection, the ability to
display and store images, and improved sensitivity. Also, a
user interface was designed to facilitate the control of
imaging experiments. The spectrometer was used for
preliminary imaging experiments on phantoms, excised rodent
spinal cords, and follicles from Tarichula Granulosa.


131
echo is sampled in order to reduce echo time. Evidently
they reconstruct the images using a phase map to determine
the real part and avoid taking the magnitude.
An NMR microscope was created for an 11.74 T, 89 mm
bore magnet at the University of Nottingham.90 The
transverse gradient coils are of the primary/screen type,
with the spacing of the primary coils adjusted to correct
for the effect of the screen. The axial gradient coil was
designed with the target field method. The coils are wound
on cylinders of 30 mm and 65 mm inner diameter. Grooves for
the screens were cut into the formers with a computer-
controlled cutting machine. The gradient coils have the
characteristics given in Table 7.
Table 7. Gradient field and inductance of Nottingham
microscopy coils.
Direction Efficiency (G/cm-A) Inductance (JJ.H)
X 2.33 30
Y 2.04 40
Z 1.41 20
The instrument has been used to study stems of the
geranium (Peargonium graveoleus). The RF coil used is a 6-
turn solenoid with an inner diameter of 1.5 mm. The
resolution is as little as 4.5 Jim in-plane with a slice
thickness of 100 |im. Use of a half-Fourier imaging sequence
90R. Bowtell et al., Phil. Trans. R. Soc. Lond. A 333, 457, 1990.


107
Notice above that arc widths of n/3 also meet the criterion
of nulling m = 0 terms, and would have large advantages for
coil construction since the radial gradient coils would not
overlap.
The radial part of the linearity has been accomplished
by using a spherical harmonic expansion for the field and
choosing current elements that do not give undesired terms.
It seems reasonable to try the same approach with the axial
linear region. One can space the current on the z axis so
that undesired harmonic terms are canceled. Unfortunately,
the terms in the expansion of the field for the radial
currents are not as easily evaluated as those in the
expansion for azimuthal currents. However, coils have been
designed using a computational approach to null higher-order
harmonics terms. To eliminate the undesired terms, there
are two degrees of freedom available for each planar unit:
its position and its current. Since the field strength
depends linearly on the current, but nonlinearly on the
position, it is computationally more efficient to vary the
current. We chose to leave the units evenly spaced at 0.2
of the inner radius and vary only the current. Biot-Savart
simulations showed that this spacing reasonably well
approximated a continuous current distribution, so that the
field does not fluctuate with the period of the spacing.
However, the designs exhibited regions of linearity that
were disappointingly small. Since we expect to be able to
design coils with arbitrarily long linear regions, and the


56
approach to calculating the eddy current distribution. It
was then possible to write the field produced by a general
current distribution on the surface of a cylinder as a
Fourier-Bessel series.50 An inverse Fourier transform of
the Fourier-Bessel series allowed the current to be
expressed in terms of the desired field on the surface of an
imaginary cylinder. The field must satisfy Laplace's
equation to allow the existence of the inverse Fourier
transform. So by specifying the inverse Fourier transform
of the desired field, the current distribution required to
generate that field could be calculated. The continuous
distribution of current is approximated by discrete wires.
The wires are placed along the contour lines of integrated
current. Although the principal application of the
technique was shielded gradients, unshielded coils having
extended linearity were also designed. For example, a
radial gradient coil is reported to have a gradient uniform
to within 5% over 80% of the radius and a length of twice
the radius. The overall length of the coil is about 9 times
the radius.
It was pointed out by Engelsberg et al. for the case of
a uniform solenoid that the homogeneity of the coil depends
strongly on the radius of the target cylinder.51 They note
that the field has the target value only on the surface of
the target cylinder. For example, in order to achieve a
50R. Turner, J. Phvs. D: AppI. Phvs. 19, L147, 1986.
51M. Engelsberg et al., J. Phvs. D. 21, 1062, 1988.


36
Stimulated echo measurements using the pulse sequence
Gradtste and the macro Adgrad were conducted for the Oxford
gradients as well as for the 9 cm home-built gradient coil
in the 2 T, 31 cm diameter magnet. The eddy currents for
the Oxford gradients were measured with the manufacturer-
installed preemphasis filter in place. The 9 cm coils had
no preemphasis. A 5 mm NMR tube with about 5 mm of H20
trapped by a vortex plug was used as a sample and placed
between 1 and 2 cm from the center of the magnet. The
center of the sample was determined from an image. In all
cases, tgrad -2s, dl 2s, tq = 0.5 s, and two averages
were acquired. The parameter x was set to 4 ms and tdecay was
1 ms for the initial experiment. The data were analyzed in
Excel spreadsheets. In the plots of field vs. time given in
Figures 6 and 7 for the Oxford and 9 cm coils respectively,
the average of the eddy current field over the sampling
interval is plotted against the middle of each sampling
interval. The eddy current field is represented as a
percentage of the applied gradient. Note that without
preemphasis, the eddy current field due to the 9 cm coil
declines monotonically, while the preemphasis filters
contribute to the measured field of the Oxford coils. For
the 15 cm coil tested in the 2.0 T magnet, eddy current
measurement was used to calculate values for an inverse
filter. The coils and samples were removed between the
experiments before and after preemphasis. The Techron 7540
amplifiers were used to drive the coils in current mode.


2
filter. If an eddy current field persists during the period
when the NMR signal is detected, distortions in the spectrum
or image will result. The distortions are particularly
severe when chemical shift information is obtained in the
same experiment as spatial localization by encoding spatial
information in the phase of the NMR signal. It is important
to be able to measure the residual gradient field, which is
usually due to eddy currents in the metal structures of the
magnet, so that it can be corrected by changing the drive to
the gradient amplifier, or by whatever other technique is
available, and to evaluate the remaining uncorrected field
to estimate the distortion that will result in a desired
experiment.
One way to avoid eddy currents for experiments such as
spatially selective spectroscopy is to employ actively
shielded gradient coils. Another, much simpler, approach is
to reduce the size of the gradient coil so that it is widely
separated from the eddy-current-producing structures in the
magnet. This approach is only possible when the clear bore
of the magnet is much larger than the volume of interest,
which is often the case. To make possible experiments, such
as spatially selective spectroscopy, that require rapidly
switched high intensity field gradients, I developed pulsed
field gradient systems based on reduced volume gradient
coils for a 2 T, 31 cm bore magnet used for small animal
studies. This magnet was replaced with a 4.7 T, 33 cm bore
magnet, and the gradient systems were adapted accordingly.


64
It will be shown that, with a current-controlled
amplifier, additional series resistance per turn always
decreases performance. Therefore, the series resistance
should be reduced as much as possible, for example, by using
a larger cross-sectional area for the winding in a coil of
fixed radius. Consider again the time response of a
current-controlled amplifier from Equation [47]. Any
internal amplifier resistance can be included in Rc to avoid
any loss of generality. Assume there is some finite,
positive Rc that maximizes i(t). Then for that
di
Rc, = 0. Solving for Rc.
di
v0
, v0
f
t }
f
-t/\
O /T
dRc
Rc2
l1 e J
He
\
Lcj
0
>
and assuming that Rc and V0 do not vanish,
This can be written as
therefore
(1 + x)e x = 1 where t/x x,
[51]
ex = 1 + x [52]
There is no positive value of x that satisfies Equation
di
[52] Hence < 0 for all t >0, Rr>0. A lower Rn is
dRc
always an advantage when using a current-controlled
amplifier, although the time constant X = Lc/Rc of the
gradient system increases as Rc decreases. It is then


28
pulse would result in no detectable phase accumulation in
practical cases. Experimental experience showed that it was
not simple to choose in advance a gradient amplitude that
would result in measurable phase accumulation, but no phase
ambiguity, at all echo times. Instead, we repeated the
experiment for a series of increasing gradient amplitudes.
For phase changes of less than 360, the phase doubles as
the gradient doubles. We could keep track of phase
accumulations greater than 360, thereby decreasing the
minimum detectable eddy current field.
Stimulated Echo Techniques
The stimulated echo (STE) has advantages over the spin
echo as the basis of an eddy current field measurement
experiment. Consider the stimulated echo sequence Gradtste
in Figure 3. The "e" at the end of the pulse sequence name
indicates that this is a stimulated echo experiment. A third
pulse is required to excite a stimulated echo. The
magnetization of interest is flipped into the transverse
plane by the first RF pulse, where it accumulates phase
RF nn ru=-jW
n 1 1 1 1 1
t t T t T
grad decay 1
Figure 3. Diagram for Gradtste, a three pulse stimulated
echo experiment for measurement of the eddy current field.


72
Coil Projects
Coil projects were intended to meet experimental needs
while exploring some aspect of coil design. The 15 cm, 9 cm
and 16 mm NMR microscopy coils are well separated from any
sources of eddy currents, and demonstrate the results that
can be achieved with simple filamentary designs and without
shielding. The CRP coil development was begun to produce a
coil with good axial linearity for NMR microscopy, so that
long, narrow samples could be observed. It seemed to be
well-suited for use as a head coil for echo planar imaging,
and we turned the development toward that possible
application.
Amplifiers
Three Techron 7540 dual-channel amplifier units (Crown
International, Elkhorn, Illinois) are used to drive the
three-axis gradient coil sets. Each axis of the gradient
coil set is split into two halves, and one channel of each
amplifier unit is wired to each half. The plane in which
the field is always zero can be shifted slightly by varying
the relative gain in the amplifiers. This is particularly
useful in the 51 mm, 7 T magnet, since the sample is
inaccessible once loaded, and mechanical centering is
difficult. The amplifiers are rated to produce 23.8 A at 42
V direct current output. The maximum slew rate is 16 V/Jis.
The output impedance is less than 7 mil in series with less
than 3 |1H, which is negligible. The power response into a 4


122
An image of the eggcrate phantom was acquired with the
CRP coil, and can be compared to a reference image and a
simulated image. In a perfect image, the square grid of the
Figure 40. Photograph of the prototype CRP coil with the
eggcrate phantom inserted.
eggcrate will be reproduced without distortion. A
calculated "image" of the intersections of a grid with
approximately the same spacing and position as the eggcrate
phantom shown in Figure 41(a) illustrates the large size of
the linear region of the CRP coil, and the character of the
distortions that are observed at the edges of the field. A
Biot-Savart calculation of the field of the CRP coil was
used to produce the horizontal spacing, and the vertical


Figure 6. Eddy current field of Oxford gradient coil in 2 T 31 cm magnet as measured with
Adgrad. The manufacturer-installed preemphasis filter was in place. a) X coil; b) Y
coil; c) Z coil.


153
the excised spinal cord of an adult female Sprague-Dawley
rat shows a clear distinction between white and gray matter.
A spin-echo image sequence was used. The parameters are:
TE/TR: 13/2000 ms, FOV = 5 mm by 5 mm, slice thickness = 1.0
mm, matrix = 128 by 128. It is evident from the
characteristic butterfly shape of the gray matter that the
slice is at the level of the cervical enlargement,
approximately C5. The bright signal present around the cord
is the Sorenson's phosphate-buffered solution. The slight
hyperintensity in the dorsal horns indicates the substantia
gelatinosa, a myelin-free region that is involved in
transmission of pain signals.
Conclusion
An NMR spectrometer has been adapted to perform NMR
microscopy. A resolution of 40 by 40 by 500 |lm has been
obtained. Although a 7.1 T magnet with 51 mm bore was used,
the images compare favorably with those obtained on 9 T, 89
mm bore magnets. A fixed tissue sample and living amphibian
follicles were studied. Meaningful contrast was obtained in
both cases.
The high noise figure of the spectrometer and the
limited performance of the RF coil did not provide
especially high sensitivity. A 1H transcoupler and
preamplifier improved the noise figure and eliminated RF
interference. The RF coil Q was about half, and 90 pulse
length about twice, that obtained by spectrometer


33
tdecay > max test is passed, Adgrad returns control to the
operator.
Two other adaptive sampling macros have been developed
for eddy current testing. Adgrad2 changes both x and gph to
lock the phase to 45. The resulting series of experiments
are more closely spaced in time than Adgrad. Using Adgrad2,
it is possible to follow the eddy current field over a wider
range of values than with Adgrad. However, linearity error
in the digital-to-analog converter or nonlinear amplifier
response will be reflected in error in the eddy current
field. Adgradl80 locks the phase to 180. It can only work
when the phase accumulation is monotonically decreasing yet
never changes sign, which is true for the uncompensated
gradients. Otherwise, Adgradl80 may lose its lock. If the
error in the measured angle is constant, the accuracy of the
technique, when applicable, should be about 5 times better
than for Adgrad or Adgrad2.
In the preliminary data analysis, we assumed that the
eddy current field was essentially constant over the
sampling interval x, so that g = A^/yxi. An Excel
spreadsheet was used to reduce and analyze the data and plot
the results. An example is shown in Figure 5. It is a plot
of the average field in each of the measurement intervals.
Since the field is dropping exponentially, not linearly,
fits to the mean value will have systematic errors. A
better way is to assume a multiexponential decay of the eddy
current field, and then calculate what phase will be


I certify that; 2 hfvi? read this study and that in ray
opinin -if confirms to acceptable- standards of scholarly
presentation, and Is .fully adequate, in scope and quality as
i.* dissertation tor the degree of Doctor o Philosophy.
M~
Kathez i.ne N, Scott
Professor of Ondear ncjj
So:i cocos
nearing
'this dissertation *ms> oivbR ittsd to the viraduat-.e fc.cui:y
of t'hft r.vepr-phTent: of Phvsion in College of Ltbor&X Arts
and. hccrncen and to the 'tea-5vete fohool and va-> accepted as.
partial fulfillatent of ~ nt coqn >rojuents for fcite degree of
to-..: cor (j : Phx 1 os nphy.
f-:v'C:K<=b. 2 4
Dean
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113
e)
Figure 34--continued
the 10 unit CRP Coil
Contour plot of the relative error of
e) Y = 0 plane; f) X = 0 plane.


85
Figure 23. Photograph of 9 cm gradient coil set. The power
cable and water supply cables are visible at left. The
axial and radial gradient coils are visible through the
cooling tubing.
RF coils were developed as inserts for the probe, including
19F, 1H birdcage, 31P/1H double-tuned saddle coil, and my
own 1H saddle coil. A pair of cams connected by a rod and
mounted on the edge of the mounting flanges served to lock
the probe into the magnet. A Faraday shield was used in
addition to the filter/fuse box to isolate the RF coils from
the gradient coils. The shield consisted of strips of
Reynolds heavy-duty aluminum foil approximately 2" wide,
overlapped by about 1/2", and insulated from the other
strips by masking tape that also secured the strips to
manila card stock. At one end, all strips contacted a
header strip. Provision was made to ground the shield, but


62
the direction of the gradient, the relative error
, _. , (x) G x
R.E., defined by R.E.= .
G x
Efficiency
In order to make use of the efficiency the reduced size
of an extended-linearity gradient coil design can provide,
it is necessary to construct the coil in such a way that it
can be driven efficiently by an amplifier. By adjusting the
number of turns, it is possible to trade maximum gradient
for switching time. We will show that to obtain optimal
switching time, the amplifier should be current-controlled
to compensate for the inductance of the coil. To reduce
switching time with such an amplifier, the coil resistance
per turn should be as low as possible, even though the time
constant of the coil will be lengthened. A time-domain
model for the coil and amplifier will be used to explore the
tradeoff between maximum gradient and switching time.
Figure 14. Time-dependent voltage source v(t) drives
inductive load.
We show that a current-controlled amplifier gives
better switching performance into an inductive load than a


61
cylinder, and axial linearity is important for the head
coil. Second, the currents are not confined to the surface
of a cylinder, and the transform technique in its present
form allows only for current on the surface of a cylinder.
Field Linearity
An extended linear region is one of the goals of a
reduced-size gradient coil. In order to evaluate a coil
design in terms of its linear region, it is necessary to
define the boundary of the linear region. An appropriate
definition for the error associated with a field gradient
reflects the purpose of the gradient coil. In an error
minimization technique, the error definition is central to
the coil design. A reasonable parameter to use is the error
Bz(x)
in the field, B.E.=
where the desired gradient, G, is
G x
measured at the center of the coil. Another error parameter
is the error in the gradient, G.E., defined by
1 dBz ( x )
G. E. =
dx
In an NMR image, error due to the
gradient coil simply produces an error in the mapping
between the sample and the image. The absolute mapping
error is simply the error in the field, B.E. In practice,
samples are usually centered in the gradient coil, so we may
want to weight the error toward the center of the coil. We
use an error parameter that corresponds to the mapping shift
relative to the component of the distance to the center in


101
current continuity. Unfortunately, the current return paths
make the coil much longer. How could such a coil be
shortened? The quality of the field produced by the coil
might not be greatly affected by moving the current return
paths off the cylinder and bringing them into the planes of
the four inner arcs, but at a larger radius. This results
in a simple CRP coil. All the current lies in two planes,
each of which we call a planar unit.
Another approach to understanding the CRP coil is to
ask if there is a way to make a radial gradient coil that is
indefinitely long, as a linear field analog to a solenoid.
In such a coil the current cannot depend on the axial
coordinate. The obvious place to start is with the inner
arcs, that now can be thought of as surface currents
independent of z, not filamentary currents. The width of
the inner arcs must be the same as for the double-saddle
coil, since the analysis used for the width of the arcs in
the double-saddle coil did not consider their axial
position. In the double-saddle design, the outer arcs are
the same width as the inner arcs; carrying this over to the
indefinitely long coil leads to arcs with a larger radius
concentric with the inner arcs. Radial elements naturally
connect these nested arcs. The problem is that an
infinitely long CRP coil with uniform current distribution
has zero external field! This can be seen through a simple
application of Ampere's law. Consider a single solenoid of
arbitrary cross section with surface current K as shown in


78
has a small time constant, so using the Techron in voltage
mode does not limit the switching time. The two halves of
each gradient coil are driven separately. Since the voltage
gain of the amplifiers can be adjusted manually, it is
convenient to vary the relative gain in the coils to shift
the zero point of the magnetic field to make up for sample
misregistration.
The details of the coil construction are visible in
Figure 47 in the following chapter. The radial coils were
wound on a flat winding former, then removed and attached to
the acrylic tube with epoxy. To eliminate any solder
connectors within the coil, the winding former allowed two
loops to be wound at once, held apart at the correct
distance. General Electric #7031 varnish was used to hold
the wires together while the coil was being clamped to the
former. No attempt was made to arrange the wire into a
packed structure. The Maxwell pair was wound around the
radial coils. The whole assembly was potted in epoxy to
secure the coils to the former, and the 36 AWG wires were
run down to a small printed circuit board mounted to the
structure of the probe. It was necessary to pot the fine
wires to keep them from moving in the magnetic field when a
current pulse is applied.
An example of the results obtained with the coil is
reproduced in Figure 60. Although the coil is capable of
about 150 G/cm, in routine operation, the coil was operated
at a full-scale field gradient of 5 G/cm for the radial


144
reference to the decoupler. The intermediate frequency
allowed for better isolation between the mixer input and
output than the 300 MHz observe frequency. The complete
system configuration is illustrated in Figure 52.
Figure 52. Block diagram of RF transmitter chain.
Amplifier is a Minicircuits ZFL 500, with 20 dB of gain.
The pattern generator consisted of a WSB-10 (Quatech,
Inc., Akron, OH) arbitrary analog waveform synthesizer card
in a Gateway 386SX 25 MHz microcomputer. The WSB-10 allows
analog signals to be generated independent of the host
computer. The operator is then free to use the computer for
data analysis and display while the experiment executes.
The onboard clock allows data rates up to 200 ns/point. The
DAC has 12 bits of resolution.
The simple ring modulator mixer shown in Figure 51
suffers from nonlinearity. The distortion can be partly
alleviated by using a lookup table to filter the desired
pulse shape. The lookup table is simply the response curve
of the modulator to a full-scale ramp. For these
measurements, the spectrometer was used as a digital
oscilloscope to measure the output voltage of the decoupler.


121
white plastic eggcrate diffuser intended for fluorescent
lighting fixtures (Area Lighting Products A4524). The cells
are nominally 1/2" on a side, separated by 1/16" walls.
Acrylic sheets (K-S-H, Inc., MS CLR ACY 30 x 36) 1/8" thick
sealed the top and bottom of the grid. The acrylic was
welded into place with methylene chloride, then the joints
were coated with epoxy to seal against leaks. The assembly
was filled with de-ionized water during the assembly
process. The RF coil consisted of a single turn of 1/2"
copper tape around the outer edge of the plane defined by
the diffuser. Variable capacitors (Johanson Manufacturing
Corp., Boonton, NJ) and chip capacitors (American Technical
Ceramics Corp., Huntington Station, NY) were used to tune
and match the coil to 85.4 MHz. The circuit is shown in
Figure 39. The coil had a measured Q of 214. The phantom
just fit into the 3" aperture of the CRP coil, as shown in
Figure 40. Inside the gradient coil, the measured Q was
Figure 39. The RF coil. All capacitances are in pF.
169. The tuning shift when the RF coil was centered in the
gradient coil was 125 KHz, although the shift was much
larger when the coil was not centered.


19
Fourier transformed, bandpass filtered, then inverse
transformed. The instantaneous magnetic field is obtained
from dty/dt. The digital filtering points to a problem with
phase measurements. The low-pass filters required to
eliminate Nyquist aliasing and to improve the signal-to-
noise ratio (SNR) distort the phase of the received signal.
Digital filtering enables one to recover the SNR ratio of a
small bandwidth without significant phase distortion.
Wysong and Lowe25 at Carnegie Mellon and the University
of Pittsburgh measured eddy current fields on a Magnex 2.35
T 31 cm magnet with unshielded gradient coils. A 1 cm
diameter sphere containing water doped to T1~T2~1 ms is
used. A 0.9 G/cm gradient is applied for 1.0 s, then ramped
down in 128 (is. A train of pulses of flip angle k/2 set 1
ms apart is applied for 1 second. One point is sampled for
each FID. With the system adjusted so the FID is in-phase
in the absence of a gradient field, the out-of-phase
yABte 2 ) ~ yA.Bte 2 for
small values of time and gradients.
Keen, Novak, Judson, Ellis, Vennart and Summers26 of
the Department of Physics, University of Exeter, propose
using a phantom slightly smaller than the imaging volume.
Having switched off the gradient, they delay a variable
time, then pulse and acquire the FID. The Fourier transform
25R. E. Wysong and I. J. Lowe, SMRM 1991, 712.
26M. Keen et al., SMRM 1992, 4029.


135
control and processing capability of the system. The
original Nicolet NT-300 spectrometer and the modifications
are outlined in Figure 42.
The system is based on a 7.1 T vertical bore
superconducting magnet manufactured by Nicolet Nalorac
Corporation, Concord, CA. (now Nalorac Cryogenics
Corporation, Martinez, CA.) The room temperature bore of
the magnet has a diameter of 51 mm. The bore inside the
shim coil and spinner stack has a diameter of 37 mm. In
order to facilitate adjustment of the field homogeneity the
factory shim coils were left in place, at the cost of
considerable bore space. Since the gradient coil is much
smaller than the shim coils, coupling between the gradient
and shim amplifiers did not become a problem. Basic and
essential gradient control hardware and software originally
developed by Mareci, Cockman, and Thomas for another project
was adapted for the NMR microscope.
A pulsed field gradient (PFG) control unit designed and
constructed by Ray Thomas allowed programmed analog levels
produced by digital-to-analog converters (DACs) intended to
control the shim power supply to be switched to the gradient
amplifiers. Spare control lines from the experiment
controller were used to operate the PFG control unit. The
gradient demand could be switched between two gradient
levels and off during experiment execution for each of the
three channels. For the NMR microscope, buffer amplifiers
were added to lower the output impedance in the off state,


80
selective Fourier transform71 typically require better
gradient performance than is available with a large,
unshielded gradient coil set.
To meet some of these needs, a conventional Maxwell and
68.7/21.3 Golay radial gradient coil set was designed and
constructed with a clear bore of 8.3 cm in diameter. The
useful region is a sphere of about 1/3 the diameter of the
coil, or about 3 cm. The coil was designed to accommodate
rats up to 150 g, and was able to achieve 12 G/cm with a 200
(is switching time. The intense gradients are needed for
imaging experiments on smaller samples. The coil was used
for projects involving lizards, and for development of
techniques to produce diffusion images of the spinal cord of
a rat model.
We can consider the application of the time-domain
model to the 9 cm coil. As wound, the coil will produce the
field shown in Figure 20 when driven to saturation. The
Maxwell pair, Z-axis coil, is the most efficient, followed
by the inner radial, or X-axis coil, which has better
performance than the outer radial, or Y-axis coil, because
of its smaller radius. The Maxwell pair reaches the 16 A
current limit of the amplifier, and does not increase in
field beyond that point. The radial gradient coils never
reach the current limit.
71
T. H. Mareci and H. R. Brooker, J. Macrn. Reson. 57, 157, 1984.


SYSTEM DEVELOPMENT FOR NMR MICROSCOPY
Introduction
NMR imaging has the capability to reveal the structure
and hydration state of soft tissue in biological systems.
The technique has been successfully applied to routine
diagnostic procedures on human subjects with a typical in
plane resolution of 1 mm and slice thickness of 5 mm.
Imaging of embryos or pathological specimens requires higher
resolution, resulting in fewer spins per voxel and so less
signal. The NMR sensitivity scales as roughly the 3/2 power
of the polarizing field, so increasing the magnetic field
can in part make up for the loss of signal.
It is desirable to use as small a bore as possible to
image a given system. NMR microscope systems are typically
built around wide bore magnets, but narrow bore magnets are
much less expensive and more popular, and so it would be
desirable to use them instead.
The purpose of the NMR microscope project was to
develop an NMR imaging spectrometer for the study of
microscopic biological specimens in a narrow-bore
superconducting magnet. Technical issues included the
pulsed field gradients, the RF homogeneity, and the sample
positioning. Problems of using pulsed field gradients in a
narrow bore magnet include eddy current fields, gradient
126


TABLE OF CONTENTS
ACKNOWLEDGMENTS ii
ABSTRACT V
GENERAL INTRODUCTION 1
MEASUREMENT OF EDDY CURRENT FIELDS 5
Introduction 5
Literature Review 11
Spin-Echo Techniques 23
Stimulated Echo Techniques 28
Results 34
Conclusion 41
GRADIENT COIL DESIGN 46
Introduction and Theory 46
Literature Review 50
Field Linearity 61
Efficiency 62
Eddy Currents 68
Coil Projects 72
Amplifiers 72
16 mm Coil for NMR Microscopy 76
9 cm Coil for Small Animals 79
15 cm Coil for Small Animals 87
Concentric Return Path Coil 98
SYSTEM DEVELOPMENT FOR NMR MICROSCOPY 12 6
Introduction 126
Literature Review 12 8
Instrument Development 133
Results 149
Conclusion 153
CONCLUSION 155
REFERENCES 156
BIOGRAPHICAL SKETCH 162
IV


7
Mx = YMyB0 [4]
My = -yMXB0 [5]
Mz = 0 [6]
We can introduce a complex transverse magnetization M = Mx +
iMY so that
M = iyMB0 [ 7 ]
Assume that B0 consists of a constant and a component
linearly dependent on position: B0 = BO+gx. BO is
independent of time and space, while g is quasi-static. If
we define m, the magnetization in the rotating frame, by
M = me 01 [ 8 ]
then
M = -iyBOM + me 27B0t .
Substituting back into Equation [7] gives
-iyBOM + me~iyBOt = -iyM(B0 + gx).
Simplifying Equation [10] yields
me iyB 0t = iyMgx .
Combining Equation [11] with Equation [8] yields
m = iygxm ,
which has the immediate solution
[9]
[10]
[11]
[12]
~iyx¡t0 gdt'
m(t) = m(t0)e [13]
If the magnetization has been prepared to a non-zero m(t0)
by a radio frequency pulse, the evolution described by
Equation [13] is called a free induction decay (FID).


NOVEL TECHNIQUES FOR
PULSED FIELD GRADIENT NMR MEASUREMENTS
By
WILLIAM W. BREY
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1994


110
The difference in the coils is in the number of planar units
utilized, and in the relative current in the planes. A
short coil containing 10 planes was designed to have a
generally spherical linear region. A second design of 30
planar units illustrates the flexibility in length of the
CRP topology. Figures 34(b)-(f) and 35(b)-(f) exhibit the
field and relative error associated with the coils. The
linear volumes extend to a large fraction of the coil
length. The 1.8:2:3 coil has 10 planar units, with a turn
ratio of 20, -2, 11, 6, 8, 8, 6, 11, -2, 20. The turn ratio
rationalization procedure did not work satisfactorily with
the 6:2:3 design, so we present the exact solution: 50.45,
1.44, 3.94, 3.24, 3.86, 3.96, 4.28, 4.47, 4.60, 4.83, 4.99,
5.09, 5.09, . 50.45. The size of the linear region in
relation to the size of the coil can be most clearly
visualized from Figure 36, a three-dimensional rendering of
the 5% relative error contour.
For a coil constructed on a single cylinder, Lenz's law
guarantees that any eddy current field generated on a
concentric shield will oppose the field produced by the
coil. The CRP coil is a different case. The field outside
the coil is dominated by the outer arcs. The outer arcs
tend to cancel the field due to the inner arcs. The Lenz1s
law argument applied to the CRP coil results in an eddy
current field that actually reinforces the applied field.
The single-cylinder coil's transfer function is a low-pass
filter, whereas the CRP coil transfer function is that of a


switching time, reduced eddy currents in the absence of
shielding, and improved use of bore space.
We used an approximation technique to predict the
remaining eddy currents and a time-domain model of coil
performance to simulate the electrical performance of the
CRP coil and several reduced volume coils of more
conventional design. One of the conventional coils was
designed based on the time-domain performance model.
A single-point acquisition technique was developed to
measure the remaining eddy currents of the reduced volume
coils. Adaptive sampling increases the dynamic range of the
measurement. Measuring only the center of the stimulated
echo removes chemical shift and B0 inhomogeneity effects.
The technique was also used to design an inverse filter to
remove the eddy current effects in a larger coil set.
We added pulsed field gradient and imaging capability
to a 7 T commercial spectrometer to perform neuroscience and
embryology research and used it in preliminary studies of
binary liquid mixtures separating near a critical point.
These techniques and coil designs will find application
in research areas ranging from functional imaging to NMR
microscopy.
vi


16
gradient and the STE imaging sequence. The images are all
from an already compensated system.
Van Vaals and Bergman of Philips Research Laboratories
in Eindhoven, the Netherlands,18'19 have a 6.3 T, 20 cm
horizontal bore Oxford magnet with 2 G/cm non shielded
gradients leaving a 13.5 cm clear bore. To measure the
eddy currents, they use a 4 cm diameter spherical phantom.
After shimming, they perform a simple "long gradient pulse,
delay 6, RF pulse, acquire" sequence. The gradient is
switched on for typically 3 s, but at least 5 times the
largest eddy current time constant. For various values of
5, the magnet is re-shimmed to maximize the signal during
the first 10 ms of the FID. The difference in shim values
with and without the gradient pulse is interpreted to be a
spherical harmonic expansion of the eddy current field.
Exact values of 8 are not listed, nor are tables of shim
values. Instead, the amplitudes and time constants of the
eddy current fields, as derived by a Laplace transform
technique, are given. Only the B0 and linear terms are
given; presumably only these terms were shimmed.
Jehenson, Westphal and Schuff of the Service
Hospitalier Frederic Joliot, Orsay, France, and Bruker,20
corrected eddy currents on a 3 T, 60 cm Bruker magnet. The
0.5 G/cm unshielded gradient coils had a clear bore of 50
18J. J. van Vaals and A. H. Bergman, J. Maan. Reson. 90, 52, 1990.
19J. J. van Vaals et al., SMRM 1989, 183.
29P. Jehenson et al., J. Maan. Reson. 90, 264, 1990.


118
i
0.5
-0.5
-1
Figure 35--continued. f) Contour plot of the relative error
of the 30 unit CRP Coil in the Z = 0 plane.
high-pass filter. Rather than applying preemphasis, a high-
pass filter, to cancel the eddy current field, it would be
appropriate to apply a low-pass filter. Any estimate of the
amplifier specifications required to drive a CRP coil should
take the above distinction into account, since it is easier
for the amplifier to reproduce the low-pass current
waveform.
The results from a first-order calculation of the
gradient at the center of the coil due to the eddy current
field from a concentric cylinder are given in Figures 37 and
38. This shows that there is a large difference in eddy
current performance between the 30-plane and the 10-plane
versions of the coils. Note that the sign of the eddy
-1 -0.5 0 0.5 1
x


certify that X have read this study and that in ray
opinion it. donSoxTas to acceptable standards of scholarly
presentation -arid is fully adequate, i.r; scope and qualify
as a dissertation tor the degree of Doctor of Philosophy
E, Raymond Andrew, Cbfi.rrrr.en
G?. a do ate Research Prof essor
of Physics
1 certify that I h ive read this study and that in ray
opinion it conforms to acceptable standard? of scholarly
presentar, ion .and is fully adequate, in scope and qua lit:;/
as a-ciissortacion for the degree of Doctor of. 'Phiicsop.tr/
"'horas H. Mareei
Associate Professor of Physi.co
I certify that t have read thin study and that in my
opinicii it. coni 01 ms to acceptable standards of scholarly
psresfebtftion 'ant is fuJ ly adequate, in scope and quality
as a .dasserta tie*, far the degree of Doctor of: Phi losophy
No: 1 55.. Sullivan
Professor of Physics
1 certify thet 3 have read this study and that m a¡y
opinivi d.L contonos to acceptable standards of scholarly
presence;Lion- and is fully adequate,, in scope and quality
as a dssepfati a a for the degree of Doctor of Philosophy
"to hard W. Briggs
Associate Profosso
of
Biochemistry and Molecular
Biology


24
increasing echo time [TE) gives an indication of the eddy-
current field distortion as a function of time. Consider
the evolution of the rotating-frame magnetization m in the
presence of the gradient field g illustrated in the pulse
sequence in Figure 1. For 0 Equation [13] so
iyxjt gdt'
m( t) = m(0)e 0 < t < TE / 2.
We can also apply the result directly to describe the
magnetization's evolution following the n pulse. Let
be the time just after the n pulse. Then
~YxJte/2 9r<^t
m(t) = m\TE / 2+)e TE / 2 < t.
The 7l pulse along x inverts the sign of the imaginary
of m(t), equivalent to taking the complex conjugate:
iyxj/2 gdt'
m(TE / 2+) = m*(0)e
Putting it together gives
* iyxj^2 gdt' -iyx¡tE/2gdt
m(t) = m (0)e e TE / 2 < t
[31]
TE/ 2 +
[32]
part
[33]
[34]
m(t) = m (0)e
iyx
rTE/2
ft
[ gdt'-
[ gdt'
Jo
JTE/2
TE / 2 < t.
[35]
Measurement of the phase exactly at the center of the Hahn
echo should remove off-resonance effects, whether due to
chemical shift or field inhomogeneity. Now it remains to be
shown that measurements of the phase at a series of echo
times can be used to find g(t) If (|)0 is the phase without
a gradient pulse applied, then


51
expansion ensured relatively independent adjustment of the
current in the various coils.
Techniques for designing higher-order shim coils for
solenoidal magnets were set forth by Romeo and Hoult.45
Coils are designed by expanding the Biot-Savart integral for
Bz, the axial component of the field, in a spherical
harmonic series about the center of the coil for simple
filamentary building block currents.
Bz(r,9,(()) = X AlimP^{cosQ)ein^. [46]
l=0m=1
The functions P_f(cos 0) are the associated Legendre
functions. As building blocks are added in the form of arcs
on the surface of a cylinder, more terms in a spherical
harmonic expansion of the field can be set to zero. The
designer connects the building blocks in such a way as to
satisfy the requirement of current continuity, which is not
built into the Biot-Savart law. By setting each undesired
term in the harmonic series to zero, a system of equations
results. The solutions are the current, length, and
position parameters of the coil designs. A Maxwell pair, as
shown in Figure 13(b), is composed of a loop placed at 0 =
60 and another having opposite current direction placed at
0 = 120. This separation is required to cancel the (1, m)
- (3, 0) term, while the odd symmetry cancels the (2, m) and
(1, m 0) terms. The desired (1,0) term remains. The
45F. Romeo and D. I. Hoult, Macm. Reson. Med. 1, 44, 1984.


58
human head.53 Its overall length was 37 cm, diameter 30 cm.
The region of interest is a cylinder 18.75 cm in diameter
and 16.5 cm long, over which the RMS (root mean square)
error in the field was less than 3% for all three axes. The
gradient coil was symmetric to avoid torque. In order to
make still shorter coils, Wong placed the return paths on a
larger cylinder.54 The wires on the inner cylinder were
connected to the return paths on the outer cylinder over
both endcaps. A coil was designed of 30 cm length, 30 cm
inner diameter, and 50 cm outer diameter. The optimization
region was a cylinder 24 cm long and 20 cm in diameter, and
the RMS error over the cylinder was 7.2%. The symmetry of
the coil eliminated the torque that arises in other short
designs. Additional points on a cylinder 70 cm in diameter
were added to the region of interest to force some partial
shielding.
Another approach to the design of gradient coils that
will fit over the head is to design a coil that has its
linear region at one end. Myers and Roemer55 used only half
of a conventional coil to move the linear region to the end.
A target field approach was used by Petropoulos et al. to
design an asymmetric coil with low stored energy.56 The
coil simulated was 60 cm long and 36.4 cm in diameter. The
"center" of the coil was 14.5 cm from one end. The stored
53E. C. Wong et al., SMRM 1992, 105.
54E. C. Wong and J. S. Hyde, SMRM 1992, 583.
55C. C. Myers and P. B. Roemer, SMRM 1991, 711.
56L. S. Petropoulos et al., SMRM 1992, 4032.


53
Figure 13. Field gradient coils that use discrete
filamentary current elements, a) Double-saddle 68.7/21.3
coil to produce the radial field gradient x or y; b)
Maxwell pair produces the axial, or z, field gradient.
The approach is best suited to cases where the gradient
coil is much larger than the sample, since the harmonic
series approximation to the field converges more rapidly
near the center of the coil. Although in theory the current
elements are lines, in practice they do have finite
dimensions, especially where large field intensity is
required. Including the wire diameter would greatly
complicate the design procedure. It is natural to use this
technique to design the shim coils mentioned above, where it
is conventional to have separate adjustments for as many as
twelve or more terms in the harmonic series. The coils
designed this way have the advantage of simplicity of
construction.


98
field
(%)
Figure 30. Eddy current field of 30.2/66.1 radial
gradient coil, as calculated to first order.
The field produced by an optimal coil increases with
the cross-sectional area of the winding, as the current is
distributed over a larger area, so designing the coil in a
filamentary approximation is fundamentally not conducive to
high performance. A better approach that preserves some of
the simplicity of design and construction would be to
include the arc width in the coil design process. The
appropriate terms in the spherical harmonic expansion of the
field would be set to zero for arcs of desired, non-zero
width, on the surface of a cylinder.
Concentric Return Path Coil
The development of functional imaging78 and diffusion-
weighted imaging79 of the human brain in vivo has created a
78J. Frahm et al., Maan. Reson. Med. 29, 139, 1993.
79P. Gideon, et al., J. Maqn. Reson. Imaq. 4, 185, 1994.


150
* ¡1
H

'IT'*
Figure 58. Spin-echo image of capillary phantom. Tap water
fills the open melting point capillary tubes and the spaces
between them.
Note the absence of phase- encode artifacts that would be
due to inadequate filtering of alternate coherence
pathways.99
When using long capillary tubes, slice selection is
not essential if the shims are adjusted carefully and the
tubes are straight. With a known phantom, the RF power can
be adjusted to get even intensity in the image, compensating
for the inhomogeneity of the coil. It is possible to center
the image by adjusting the gain of the gradient amplifiers.
Then the height and width of the image are used to calibrate
the gradients.
A series of images of follicles of the salamander
Taricha granulosa illustrates the contrast effects possible
99G. J. Barker and T. H. Mareci, J. Maan. Reson. 83, 11, 1989.


68
f
v
Ac
( -t
Nopt(to) =
/x
[57]
J )
By substituting [57] back into [56], we obtain an
expression for Gmax(t0)i the maximum field attainable at a
given switching time for a class of coils having the same
design except for the number of turns.
[58]
Gmax(t0) is just the dotted line in Figure 15. The
tradeoff between switching time and field strength is
described by the plot of Gmax(t0) .
In summary, a design procedure has been developed for
optimizing the switching performance of a gradient coil.
Use of a current-controlled amplifier reduces switching
time. The cross-sectional area of the winding is maximized
subject to constraints that include linearity and available
space. Then the number of turns is computed from Equation
[57], given the desired switching time t0. The resulting
coil will give the largest possible gradient for the desired
switching time.
Eddv Currents
Shielding efficiency of self-shielded gradient coils is
typically evaluated using a screening factor, a ratio of the
magnetic field outside the unshielded coil to the field
outside the shielded coil.63 It is possible to take this
63R. Turner, Macrn. Reson. Imaa. 11, 903, 1993.


Ill
z
Figure 34. A Concentric Return Path Coil with 10 planar
units. The ratio of length:inner diameter:outer diameter is
1.8:2:3. a) Wire configuration, b) Contour plot of the field
in the Z = 0 plane.


109
minimization starts with equal current in each planar unit,
which is in some sense the most power-efficient solution.
The current in the planes is then varied to minimize the
error. We hope to find a local minimum of the relative
error that is as close as possible to the uniform current
starting point.
The target-field approach yields noninteger currents in
the various planes. In practice, one would like to vary the
current in the planes by varying the number of turns of
wire, which requires integer currents. Then all the planes
can be wired in series, so the current is precisely
controlled and independent of the quality of the connections
in the coil. A simple way to choose the smallest integral
currents that give essentially the same performance as the
minimal error design is described below. If Ji is the
nonintegral current of plane i in the solution, we generate
trial practical solutions = Roundialj^) where l' is the
integral current in the plane, a is an adjustable parameter,
and Round() is a function that rounds a number to the
nearest integer. By varying a and plotting the error that
results, we can choose the smallest value of a that yields
acceptable error.
We have designed CRP coils with length:inner
diameter:outer diameter ratios of 1.8:2:3 and 6:2:3, as
shown in Figure 34(a) and 35(a), respectively. Both utilize
an inner-radius to outer-radius ratio of 1:1.5. Both have a
spacing between planar units of 0.2 of the inner radius.


97
Table 5. Comparison of measured and predicted inductance
for 15 cm gradient coil set in parallel configuration.
Coil
Measured
Resistance
(fl)
Measured Time
Constant
(|!s)
Experimental
Inductance
(|IH)
Theoretical
Inductance
(|XH)
XI
0.95
440
418
500
X2
0.96
440
422
500
Yl
0.96
440
422
546
Y2
0.96
460
441
546
Z1
0.84
560
470
488
Z2
0.70
640
448
488
The first order calculation for the eddy current field
of the 66.1/30.2 radial gradient coil as plotted in Figure
30 is about the same as for a 68.7/21.3 Golay coil of the
same diameter shown in Figure 17.
This project demonstrates application of the time-
domain performance model to optimize the number of turns in
the coil. It also shows that is possible to reduce the
length of the coil without significantly reducing the size
of the linear region or the coil efficiency. A further
refinement of the optimization of performance would be to
include the coil length as a second parameter in the time-
domain optimization procedure. This project also
illustrates how using split windings allows the user to
trade off switching time for field strength as needed for a
given experiment.


45
in the X and Z channels. Experimental evidence demonstrates
the advantage in eddy current field obtainable with reduced
size gradient coils.


83
leads and filters are given in Table 2. The time constant
of the shorted power cable with filters and fuses was too
short to measure with the amplifier, so it can be assumed to
be negligible. The last column is the inductance estimated
from the Bowtell and Mansfield formulation for coils on the
surface of cylinders.72 To adapt for the thick winding, the
height is added to the width of the winding. Calculations
for loops of square cross section compare closely to
heuristic formulas.73
Table 2. Comparison of measured and predicted inductance
for 9 cm gradient coil set.
Coil
Measured
Resistance
(G)
Measured Time
Constant
(|is)
Experimental
Inductance
(|1H)
Theoretical
Inductance
(M-H)
XI
7.02
175
1229
1727
X2
7.08
175
1239
1727
Yl
7.57
195
1476
1866
Y2
7.61
190
1446
1866
Z1
3.50
150
525
330
Z2
3.53
150
530
330
The difference between the theoretical and experimental
inductance is not due to the inductance of the power cable,
which was measured by the same technique to be about 18 (IH.
It is primarily due to poor control of the cross-sectional
72R. Bowtell and P. Mansfield, Meas. Sci. Technol. 1, 431, 1990.
73F. E. Terman, Radio Engineers' Handbook. McGraw-Hill, New York, 1943.


31
through which the spectrometer is controlled. Adgrad allows
the automatic measurement of the eddy current field over a
large dynamic range. Forty-five degrees is large enough to
measure with enough precision and yet small enough to
minimize the possibility of aliasing. The values of phase
'user enters
"Adgrad (0, t max)
Figure 4. Flow chart of the macro Adgrad, which executes
adaptive sampling of the eddy current field. The dotted
portion is not part of the macro.


76
16 mm Coil for NMR Microscopy
The 16 mm gradient coil was developed as part of the
NMR microscope development project described below. Earlier
NMR microscopy gradient coils described in the literature
were located outside of the RF probe insert, as part of the
shim coil set. A simple and straightforward approach to
improving the coil switching time, increasing the field
strength, and decreasing the eddy current field is to
integrate the gradient coils with the RF probe. This also
allows the use of a narrowbore (51 mm) magnet. Drawbacks to
this approach include a lack of flexibility. If the
gradient coil is outside the RF probe, then any RF probe can
be used. In our approach, one gradient coil is required for
each RF probe. Also, since one of the dewars associated
with the variable-temperature (VT) control system is
replaced by an acrylic tube, the range of the VT system is
reduced. Our probe did not contain any VT control
capability. The fact that the coil former was so small
encouraged us to choose a simple design to ease the
assembly.
Since the sample-tube inner diameter was 4.5 mm and the
first metal tube, or shield, had an inner diameter of 33 mm,
this was a favorable case for using a reduced-size gradient
coil. A simple 68.7/21.3 radial gradient coil as
described above has a useful volume with a diameter of about


43
residual eddy current field. Any error in the values of
timing components in the preemphasis filter will add to the
residual eddy current field. Also, any distortion in the
amplifier will reduce the effectiveness of the compensation,
since the filter was designed based on the assumption that
the amplifier is linear. Error in the eddy current
measurement technique itself might be due to other echo
terms than the stimulated echo contributing to the signal.
However, experiments have shown that other echoes are
essentially negligible due to a combination of favorable
timing and phase cycling. In the case of Adgrad2, which
scales gph as well as x, it is clear that some error is due
to inaccuracies in the digital-to-analog converter (DAC)
output level. The applied gradient is then not proportional
to the DAC code, and so there is an error in normalizing to
the applied gradient. Error in the curve fits may be
significant, since in a multiple-exponential fit it is
difficult to get an accurate fit if the time constants are
not widely separated. Note that second-pass adjustment of
the preemphasis was more effective in reducing the residual
eddy current field.
The technique came out of a need to quantify phase
distortions in localized spectroscopy. It is therefore
better-suited to measuring the time integral of the eddy
current field than the field itself, and it is the integral
of the field that gives rise to errors in phase-sensitive
techniques such as SFT. It is often useful to employ the


11
for spatial localization on the integral of the eddy current
field, so it is desirable to have a measurement technique
that is based upon the integral of the eddy current field.
If possible, the technique should have no special hardware
requirements.
Literature Review
Many workers have addressed the problem of eddy current
measurement and compensation in the literature. The two
aspects of the eddy current field to measure are the spatial
and time behaviors. We review publications that include
descriptions of eddy current measurements, although in most
cases the emphasis is placed upon the preemphasis
compensation process and its effectiveness, not the
measurement. The measurement process can be divided into
techniques that detect the derivative of the eddy current
field, those that detect the eddy current field itself, and
those that detect the integral of the eddy current field.
The derivative of the field is sensed by a pickup coil
consisting of turns of wire through which the changing flux
of the eddy current field produces an electromotive force
that is proportional to the rate of change of the field.12
A high impedance preamplifier boosts the signal. An analog
integrator is usually used to convert the measured voltage
into a quantity proportional to the field, although it is
possible to use digital integration. When used in a magnet
12D. J. Jensen et al., Med. Phvs. 14, 859, 1987.


137
the gradient coils in order to protect the gradient coils.
Slow-blow fuses, typically 1 A, protected the gradient
coils. A set of power low-pass filters in the same chassis
ensured that RF pickup from the probe or cables to the probe
would not be coupled back into the gradient amplifier and
that the cable from the gradient amplifier would not serve
as an antenna coupling RF noise into the probe. Software
modifications by Mareci and Cockman95 enabled the amplitude
of the gradient pulse to be varied in a two dimensional
experiment, making possible experiments based on the spin-
warp modification96 to 2D Fourier imaging.97
Figure 45. Foil sheets for RF probe.
The RF probe is a slotted resonator of conventional
design. Copper foil sheets cut as shown in Figure 45 are
wrapped around a tube so that the tabs overlap to form
capacitors. Figure 46 is the equivalent circuit. The foil
95M. D. Cockman, Dissertation, 1988.
96W. A. Edelstein et al., Phvs. Med. Biol. 25, 751, 1980.
97A. Kumar et al., J. Maan. Reson. 18, 69, 1975.


41
tend to underestimate the preemphasis required, but since
the unshielded eddy current fields were already less than 5%
of the applied field, the error is not severe.
The 4.7 T magnet that replaced the 2 T 31 cm magnet did
not have manufacturer-installed preemphasis, so the eddy
current measurement techniques were applied to design an
appropriate preemphasis filter. Since the uncompensated
eddy currents were on the order of 50% of the applied field,
the approximation used to compensate the 15 cm coil would
not be effective. An inverse Laplace transform technique
was used to design the filters. The technique was
implemented through the symbolic inverse Laplace transform
capability of Mathematica. An example of a multiexponential
fit to the raw phase accumulation performed with Mathematica
is shown in Figure 9. Eddy current fields before and after
compensation are presented in Figure 10. The upper curves
represent the field before, and the lower curves after,
preemphasis. For the Y coil, the procedure was repeated a
second time to obtain an additional reduction of the eddy
current field. The lowest curve in Figure 10 (b) represents
the eddy current field after the second pass of eddy current
correction.
Conclusion
A technique to measure the eddy current field of a
pulsed field gradient based on the phase of the stimulated-
echo NMR signal has been proposed. Experimental


123
spacing is simply that of the undistorted grid. It can be
seen that the field is a little larger at the ends of the
coil, causing the phantom to appear to bulge. Just beyond
the end of the coil the field changes sign. The points in
the simulation are not connected, but the sign change would
be visible if they were.
The only difference between the phantom images is the
difference in the X gradient coil. The standard SISCO spin-
echo sequence "image" was used as the pulse sequence for
both images. The slice thickness was 2.3 mm. Two averages
were collected. The gain of the amplifiers was adjusted so
the size of the inner cells is the same; that is, the
calibration factor is the same for the Oxford and the CRP
coils. The RF coil is so close to the sample that the
periphery of the sample is liable to be much brighter than
the center. To compensate, the RF power was adjusted beyond
the power needed for maximum signal so that the intensity is
as homogeneous as possible.
A CRP coil image of the eggcrate phantom acquired in
the XZ plane is shown in Figure 41(b). The Oxford gradients
were used for the Y and Z localization, and the CRP coil
used only for X. X was chosen to be the phase-encode
direction so that there would be no position errors due to
shimming. The sign reversal of the field beyond the end of
the coil is clearly visible here. The grid distortion
generally matches that in the simulated image. The bulge in
the upper left corner is probably due to the poor machining


158
Gideon, P., C. Thomsen, and 0. Henriksen. "Increased Self-
Diffusion of Brain Water in Normal Aging." J. Macrn. Reson.
Imaa. 4: 185 (1994) .
Golay, M. J. E. "Field Homogenizing Coils for Nuclear Spin
Resonance Instrumentation." Rev. Sci. Inst. 29: 313
(1958) .
Hall, L. D., S. Luck, and V. Rajanayagam. "Construction of
a High-Resolution NMR Probe for Imaging with Submillimeter
Spatial Resolution" J. Maan. Reson. 66: 349 (1986).
Hall, L. D., V. Rajanayagam, and S. Sukumar. "Adaptation of
High-Resolution NMR Spectrometers for Chemical Microscopy.
Evaluation of Gradient Magnitudes and B1 Homogeneity." J.
Maan. Reson. 60: 199 (1984).
Hall, L. D., and S. Sukumar. "Chemical Microscopy Using a
High-Resolution NMR Spectrometer. A Combination of
Tomography/Spectroscopy Using Either 1H or 13C." J. Maan.
Reson. 50: 161 (1982).
Hsu, E. W., N. R. Aiken, J. S. Schoeniger, and S. J.
Blackband. "Magnetization Transfer Microimaging of Single
Neurons." Book of Abstracts, SMRM, 11th Annual Meeting,
1992, p. 974.
Hughes, D. G., Q. Liu, and P. S. Allen. "Spatial Dependence
of the Eddy Current Fields in a 40 cm Bore Magnet." Book
of Abstracts, SMRM, 11th Annual Meeting, 1992, p. 362.
Jackson, J. D. Classical Electrodynamics. John Wiley and
Sons: New York, 1975.
Jehenson, P., M. Westphal, and N. Schuff. "Analytical
Method for the Compensation of Eddy-Current Effects Induced
by Pulsed Magnetic Field Gradients in NMR Systems." J.
Maan. Reson. 90: 264 (1990).
Jensen, D. J., W. W. Brey, J. L. Delayre, and P. A.
Narayana. "Reduction of Pulsed Gradient Settling Time in
the Superconducting Magnet of a Magnetic Resonance
Instrument." Med. Phvs. 14: 859 (1987).
Keen, M., M. Novak, R. Judson, R. E. Ellis, W. Vennart and
I. R. Summers. "Characterization of Eddy Current
Compensation over a Large Volume." Book of Abstracts,
SMRM, 11th Annual Meeting, 1992, p. 4029.


105
infinitesimal volume element. For wires, we write J = Ids,
where I is the current and ds is a unit vector in the
direction of the current. Then the vector potential of an
|l0Jds
infinitesimal current element is dA =
4 nr
Now we expand
dA in associated Legendre polynomials to find that
Pnm (cos G)e^-V),
[tnXds v (n m) !
dA = X X : FT pnm(cos0
f r^n
4n ^ (n + m)\
n=0m=-n s '
\f )
[70]
where f is the distance between the current element and the
origin. The Neumann factor £m = 1 if m = 0, and otherwise
m 2. All variables are defined as in Romeo and Hoult, who
derive the magnetic field of an arc element to be
o n
dBZ = X X + Fn,m + lJn,m + l}rnPnm(csQ)eim{(p-'l')
n=0m=-n m
[71]
where
Fnm = I (sin a)em(n m + 1)! Pn+i/in(cos a)/4n(n + m + 1)! [72]
G = (i + 5,o)/2i" + 1
Jnm = (l 5,0)(n + m + 1 )(n + m)/2fn+1 .
A finite arc of constant radius and axial position can be
trivially integrated from V|/ = to V|/2 to give
[73]
[74]
oo n
n = 0m = n l+ un,m + l
n'm lGn'm 1 l.rnPnm(cos0)ieiMpeiffl(v2"Vl_,t/2).
[75]
m
To make a term of given m vanish, we only must make
im(y2 Vi -n/2) n
3 purely imaginary, i.e. we set


3
These pulsed field gradient systems offer much better
performance than the large and unshielded gradient system
supplied with the magnet, given their limitation on sample
size. I also developed a pulsed field gradient coil for a 7
T, 51 mm bore magnet used for NMR microscopy and
spectroscopy.
Another experiment which requires gradient coils to
perform exceptionally well is functional imaging of the
human brain. The head is much smaller than the whole-body
magnets in general use. A smaller coil can allow faster
switching to higher gradient fields, as well as reduce eddy
current fields. In order to get a gradient coil that is
matched to the size of the head, some provision must be made
to allow for the shoulders. Conventional designs, even
existing designs with a large linear volume, have current
return paths arrayed on both sides of the linear volume. A
coil matched to the size of the head would not fit over the
shoulders. A coil that trades radial linear region for
increased axial linear region is more appropriate. A design
utilizing concentric return paths was developed that
significantly improved the axial region of linearity. A
prototype was constructed and tested.
In order to perform NMR microscopy and pulsed field
gradient experiments, we adapted an NMR spectrometer and
probe for a 7 T, 51 mm bore magnet. The instrument included
a simple amplitude modulator to carry out slice selection.
A probe that allowed sample loading from above was


161
van Vaals, J. J., A. H. Bergman, and R. P. van Stapele.
"Optimization of Eddy Current Compensation: Competition for
Shielded Gradients." Book of Abstracts, SMRM, 8th Annual
Meeting, 1989, p. 183.
Wong, E. C., P. A. Bandettini, and J. S. Hyde. "Echo-Planar
Imaging of the Human Brain Using a Three Axis Local
Gradient Coil." Book of Abstracts, SMRM, 11th Annual
Meeting, 1992, p. 105.
Wong, E. C., and J. S. Hyde. "Short Cylindrical Transverse
Gradient Coils Using Remote Current Return." Book of
Abstracts, SMRM, 11th Annual Meeting, 1992, p. 583.
Wong, E. C., A. Jesmanowicz, and J. S. Hyde. "Coil
Optimization for MRI by Conjugate Gradient Descent." Maan.
Reson. Med. 21: 39 (1991).
Wysong, R. E., and I. J. Lowe. "A Simple Method of
Measuring Gradient Induced Eddy Currents to Set
Compensation Networks." Book of Abstracts, SMRM, 10th
Annual Meeting, 1991, p. 712.
Zur, Y., S. Stoker, and R. Morad. "An Automated Algorithm
for Eddy Current Compensation." Book of Abstracts, SMRM,
11th Annual Meeting, 1992, p. 363.


8
Consider the characteristics of a general eddy current
field. The eddy currents give rise to a magnetic field that
roughly tends to cancel the applied field of the gradient
coil. The spatial dependence of the eddy current field is
not exactly the same as the applied gradient field.8
The time behavior of the eddy current field is a
multiexponential decay, which can be seen by considering the
form of the solution to the differential equation governing
the decay of magnetic induction due to current flow in the
conductor. Maxwell's equations9 in a vacuum in SI units are
V B = 0
[14]
V E =
[15]
80
3b
V X E + = 0
[16]
3t
3e
V x b n0e0 = |i0J
at
[17]
where B is the magnetic induction and E is the electric
field, p is the charge density, e0 and \i0 are the
permittivity and permeability of free space, and J is the
current density. We also assume Ohm's law, J = aE, where a
is the conductivity, assumed to be isotropic and
homogeneous. Taking the curl of both sides of Ampere's law,
Equation [17], neglecting the displacement current, and
using the identity
8R. Turner and R.M. Bowley, J. Phvs E: Sci. Instrum. 19, 876, 1986.
9J. D. Jackson, Classical Electrodynamics. John Wiley & Sons, New York,
1975 .


39
The same sample and RF coil were used as in the Oxford and 9
cm tests. In all cases, tgrad = 0.5 s, fc] = 0.5 s, the
sample position was between 1 and 2 cm from the center, and
the position was measured in an image. The data were
g(t) (%)
g(t) (%)
g(t) (%)
Figure 8. Eddy current field of 15 cm gradient coil set in
2.0 T magnet system before (upper curve) and after
compensation (lower curve). a) X coil; b) Y coil; c) Z coil.


77
1/3 that of the coil,66 so the gradient former was chosen to
have a diameter of about 15 mm, or 5/8". A factor of two
remains in the ratio of the coil to the shield diameter.
This results in an eddy current field for the 68.7/21.3
Golay radial gradient coil, based on Figure 17, of about 20%
of the applied field.
The NMR microscope gradient coil set is of the
conventional Maxwell and Golay design described above. It
was constructed to accommodate standard 5 mm tubes used in
analytical NMR work. The 10 turns of 36 AWG enameled magnet
wire are wound on a 5/8" nominal outer diameter acrylic tube
(15.9 mm) Using a value of 1.3 6 Q/m for the wire67 and a
length of 0.135 m per turn, the resistance of each side of
the coil is 1.84 Q. The coil inductance can be estimated68
to be about 8 |1H. The time constant of the coil is then
about 4 |is. The current efficiency of a 68.7/21.3 Golay
radial gradient coil is 0.918/a2 G/cm-A, where a is the coil
radius,69 so the coil has a current efficiency of 14.1 G/cm-
A. A Maxwell pair has a current efficiency of 0.808/a2
G/cm-A, so the coil has a current efficiency of 15.3 G/cm-A.
The typical figure for the linear region of 1/3 the diameter
of the coil is then enough to accommodate a sample. The
coil is driven by the Techron 7540 amplifier set. The coil
66F. Romeo and D. I. Hoult, Maan. Reson. Med. 1, 44, 1984.
67D. Lide,(Ed.), CRC Handbook of Chemistry and Physics. 51st Edition,
CRC Press, Boca Raton, 1970, p. 15-29.
68F. E. Terman, Radio Engineers' Handbook. McGraw-Hill, New York, 1943.
69F. Romeo and D.I. Hoult, Maan. Reson. Med. 1, 44, 1984.


26
simplifies the analysis of the eddy current response.33
However, lengthening the TE reduces the time resolution of
the experiment. Placing the gradient pulse before the
excitation pulse as in Figure 2 eliminates the problem and
decouples the length of the gradient pulse from the echo
time.
RF
g
K/2 y kx
Jl
-T-1-
'd2 1 d3 1 TE/2 TE/2 '
Figure 2. Gradtest v.1.2 is a spin-echo experiment for
measuring the eddy current field following a pulsed field
gradient.
The above analysis assumes a point sample. Any real
sample has finite extent and will experience some dephasing,
and associated signal loss, as its phase evolves in the
gradient field. By not subjecting the transverse
magnetization to the gradient pulse but only to the eddy
current field, the dephasing effect is reduced. Another
drawback proved to be that the signal decayed due to T2
relaxation before Gj_ stabilized. With the gradient pulse
before the excitation, the condition Gi = Gi+1 could be met
for small values of TE. However, for large TE we could
33
M. A. Morich et al. IEEE Trans. Med. Imacr. 7, 247, 1988.


10
structure at 10 K than one at 293 K. In a commercial
aluminum alloy the conductivity will vary from that of the
pure metal, especially at low temperature, so the effect may
not be as great. In practice, the principal source of eddy
current fields is generally the innermost low temperature
aluminum cylinder, which is at approximately the boiling
point of liquid nitrogen, 77 K. The resistivity of aluminum
at 80 K is 2.45 x 10-9 Q-m, so the time constant is about
11 times greater than it would be at room temperature.
We consider the desirable characteristics for an eddy
current measurement technique. Our primary goal will be to
measure the eddy current field in order to evaluate the
feasibility of performing a given experiment, not to
compensate for the eddy current field. Therefore dynamic
range is more important than absolute accuracy. It must be
possible to measure eddy currents produced by specific pulse
sequences, probably by appending the eddy current
measurement experiment to the end of the sequence under
evaluation. It is also preferable to have a technique that
is insensitive to inhomogeneity so that no shimming is
necessary. Since the shim coil power supply may respond
dynamically to the gradient pulse and distort the measured
eddy current field, it is useful to be able to turn the shim
supply off. Experiments based on Selective Fourier
Transform11 and other chemical shift imaging techniques rely
11H. R. Brooker et al., Macrn. Reson. Med. 5, 417, 1987.


50
beyond that produced by a constant current density along z,
we can adjust the relative current or position of each
planar unit.
Literature Review
This literature review will be focused on efforts to
increase the useful volume of a gradient coil, to optimize
its performance, and to understand the eddy current field
associated with a switched gradient it produces. The
specific requirements of coils of interest for functional
imaging of the human head are discussed, along with several
approaches to meeting those requirements.
Gradient coils can be grouped into two broad
categories: those made up of discrete current elements as in
Figure 13, and those approximating a continuous current
density. The former include the original NMR shim coil
designs,41,42 while the latter approach has been used to
make possible actively shielded gradient coils.43
Anderson described a set of electrical current shims
for an NMR system based on an electromagnet.44 The coils
were located in two parallel planes, one against each
poleface, to allow access to the sample. Each coil was
designed to produce principally one term in the spherical
harmonic expansion of the field. The orthogonality of the
41W. A. Anderson, Rev. Sci, Inst. 32, 241, 1961.
42M. J. E. Golay, Rev. Sci. Inst. 29, 313, 1958.
43P. Mansfield and B. Chapman, J. Maan, Reson. 66, 573, 1986.
44W. A. Anderson, Rev. Sci. Inst. 32: 241 1961.


81
Figure 20. The gradient produced by the 9 cm gradient coil
set following a demand that saturates the amplifier.
Figure 21 describes the maximum field Gmax(t0) possible
for switching time t0 for each of the three coils. The
Gma* (G/cm)
Figure 21. The maximum gradient that could be achieved by a
coil identical to the 9 cm coil, with the same cross-
sectional area, but with varying number of turns.
inherently lower inductance and resistance of the Maxwell
pair are reflected in its greater field. The actual and
optimal fields obtainable at 200 |is are compared in Table 1.
The Maxwell pair has about the optimal number of turns, and
its field is about the same as the optimum. The Golay coils


71
The expressions to first- and second-order for the eddy-
current field will be used to evaluate numerically the eddy
current field of several coil designs. Although the result
is not exact, the expressions can easily be integrated for
coils and shields of totally arbitrary shape, assuming they
are not too close together.
The first-order calculated eddy current field of a
68.7/21.3 radial gradient coil is plotted in Figure 17.
The first-order approximation breaks down for ratios of
shield-to-coil radius of less than about 1.5.
field
(%)
Figure 17. Eddy current field of 68.7/21.3 double-saddle
radial gradient coil. The field as a percentage of applied
field is plotted against the ratio of shield radius to coil
radius.


87
The difference is almost certainly due to quick-release
connectors that allow the probe to be removed or inserted at
operating pressure.
15 cm Coil for Small Animals
The 15 cm coil was designed to accommodate larger rats
and other medium-sized laboratory animals and still produce
a higher field and faster switching time than the Oxford
gradient set. Like the 9 cm and the NMR microscope coil
set, it is based on filamentary winding design. Since it is
also driven by the Techron 7540 amplifier set, which is
under-powered for a coil of this diameter, switching
performance was at a premium. So, in contrast to the 9 cm
coil, the 15 cm coil was designed to have optimal switching
performance for the chosen switching time. In order to
provide more flexibility in choosing either high field
intensity or fast switching time, the windings were split
and could be driven either in series or parallel. Since the
coil would tend to become somewhat unwieldy as an insertable
unit, its length was the shortest that would give
essentially undiminished field intensity. Plots of the
relative error in Figures 26 and 27 illustrate the linear
region of the radial coil design. In order to avoid
aliasing signals from long animals, the axial coil was based
on an extended linearity design by Suits and Wilken.75 The
75B. H. Suits and D. E. Wilken, J. Phvs. E: Sci. Instrum. 23, 565, 1989.


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88
coil and amplifiers were capable of developing 9 G/cm in a
100 fls switching time on X, Y and Z channels.
The arcs in the 15 cm coil were arranged to have the
minimum length without losing a significant amount of
efficiency in the static limit. The standard solution of
68.7/21.3 for the arc positions arcl/arc2 leads to no
third order component from either arc, but a family of
solutions for which the third order components cancel is
available. These solutions are graphed in Figure 24.
arc2 (deg)
Figure 24. The solutions to the arc position of the double
saddle radial gradient coil.
Each solution is graphed twice, since exchanging arel and
arc2 results in the same coil design. To improve the
relative size of the linear region to the coil, one would
like to make the coil shorter than the 68.7/21.3 solution.
The current efficiency decreases with the length, since the
return arcs tend to cancel the desired field, and moving
them closer increases the effect. However, in order to


79
gradients, and 10 G/cm for the axial gradient, to allow
sufficient resolution.
9 cm Coil for Small Animals
An NMR magnet is frequently used for samples or animals
significantly smaller than the available bore size. It is
possible to take advantage of this fact and scale the size
of the gradient coil to match the size of the sample. One
advantage that accrues is reduced eddy current fields, since
the coil and the source of eddy currents are better
separated. Another is the increased efficiency possible
with smaller coils, since efficiency scales as the fifth
power of the diameter.70 Many applications require more
rapidly switched and more intense gradient fields than are
generally available. Diffusion-weighted imaging and
localized spectroscopy are two examples. Also, to achieve
the same bandwidth per pixel, small samples require larger
gradient fields.
The 31 cm 2 T small-animal imaging spectrometer was
supplied with a gradient coil set manufactured by Oxford
instruments that has a clear bore of 22.5 cm, and is capable
of producing a maximum gradient of 2 G/cm with a switching
time of 1 ms. Although rat, mouse, and lizard studies, do
not require the full 22.5 cm bore, they benefit from the
horizontal orientation and will not fit into other available
magnets. Additionally, localization techniques such as
70R. Turner, Macm. Re son. Imacr. 11, 903, 1993.


104
CRP coil as being infinitely long. We are led to consider
the effect of length on the linear volume of a CRP coil
having constant current density over its finite length.
Plots in Figure 33 of the linear region vs. coil length
demonstrate that the linear volume of such a coil is a very
strong function of length. Along both the x axis and x = y,
the linear region is strongly peaked at a length that is
twice the inner radius. This is in contradistinction to the
homogeneous region inside a solenoid, which would grow with
the length. For the CRP coil with uniform current density,
as the length grows both the field intensity and the linear
volume decline.
We consider the design of the CRP coil in two steps.
First is the design of the planar unit. Second is the
adjustment of current between planar units to optimize the
axial extent of the linear region. Since the magnetic field
in the coil must be a solution to Laplace's equation, which
is separable in cylindrical and spherical coordinates, we
can separate the axial and radial parts of the solution. To
design the planar unit we follow the spherical harmonic
approach of Romeo and Hoult.82 Starting with a definition
of the vector potential A by B = V x A where B is the
magnetic field, we use the well-known result that in free
Ho r j
space A = dv where J is the current density, r is the
4 K J r
distance between the current and the observer, and dv is an
82F. Romeo and D. I. Hoult, Maan. Reson. Med. 1, 44, 1984.


75
current-control units were modified so the RC networks can
be plugged in and out when gradient coils are changed.
Figure 19. Output voltage and current of Techron 7540
amplifier with current-control module. The load is the
highly inductive 9 cm field gradient coil.
The amplifier rack was equipped with wheels and shared
between the NMR microscopy and small-animal spectrometers.
It was used in voltage-control mode with the NMR microscopy
system, and current-control mode with the small animal
system where the coil inductance was much higher. Input and
output connectors were standardized to facilitate quick
conversion. A fully-shielded output cable terminated in a
fuse-and-filter chassis eliminated interference from the RF
coils.


74
four ohms or less, it can be modeled as a 15 A current
source.
The Techron amplifiers are equipped with optional
current-control modules. With current control switched on,
an amplifier behaves like a voltage-controlled current
source. With current control switched off, an amplifier
behaves like a voltage-controlled voltage source. Current
control serves two functions when driving gradient coils.
It compensates for any variation in temperature of the
gradient coil due to resistive heating. More importantly,
it enables the coil to be switched to low fields much more
rapidly than the coil's time constant would otherwise allow.
The current-control module compares the demand (or input
signal) to the voltage across a small shunt resistor in the
output circuit. With a highly inductive load such as a
gradient coil, at high frequencies the amplifier's output
voltage is shifted almost 71/2 with respect to the coil
current, and the amplifier is unstable and will oscillate.
The voltage and current response of one of the Techron
amplifiers in current mode is shown in Figure 19. The
controlled voltage overshoot reduces the current switching
time. Approximately 5 A is being switched into a 7 Q load.
An adjustable resistor-capacitor (RC) network in parallel
with the coil rolls off the high frequency gain to
compensate for the instability. The values of the RC
network are determined by the inductance of the coil. Since
the 7540 amplifiers are used with more than one coil, the


134
modulated RF pulses. An imaging display and storage
workstation was added, along with enhancements to the

Nicolet 1180
Computer
Nicolet Pulse
Sequencer
Synthesizer
H Modulator
Decoupler
Synthesizer
Control Unit
Observe
Unit
Gradient
Controller
Shim Current
Controller
data
control
r >\ r
Magnet
Shim Coils
Shim Power
Supply
Figure 42. Block diagram of Nicolet NT-300 Spectrometer as
modified for NMR microscopy. Modified or added sections are
shaded.


159
Kumar, A., D. Welti, and R. R. Ernst. "NMR Fourier
Zeugmatography." J. Maan. Reson. 18: 69 (1975).
Kwong, K. K., J. W. Belliveau, D. A. Chesler, I. E.
Goldberg, R. M. Weisskoff, B. P. Poncelet, D. N. Kennedy,
B. E. Hoppel, M. S. Cohen, R. Turner, H. M. Cheng, T. J.
Brady, and B. R. Rosen. "Dynamic Magnetic Resonance
Imaging of Human Brain Activity During Primary Sensory
Stimulation." Proc. Natl. Acad. Sci. 89: 5675 (1992).
Lide, D. R. (Ed.) CRC Handbook of Chemistry and Physics.
51st Edition. CRC Press: Boca Raton, 1970.
Lide, D. R. (Ed.) CRC Handbook of Chemistry and Physics.
72nd Edition. CRC Press: Boca Raton, 1991.
Mansfield, P., and B. Chapman. "Active Magnetic Screening
of Gradient Coils in NMR Imaging." J. Maan. Reson. 66: 573
(1986).
Mansfield, P., and I. L. Pykett. "Biological and Medical
Imaging by NMR." J. Maan, Reson. 29: 355 (1978).
Mareci, T. H., and H. R. Brooker. "High-Resolution Magnetic
Resonance Spectra from a Sensitive Region Defined with
Pulsed Field Gradients." J. Maan. Reson. 57: 157 (1984).
McFarland, E. W., L. J. Neuringer, and M. J. Kushmerik,
"Chemical Exchange Magnetic Resonance Imaging (CHEMI)."
Maqn. Reson. Imaq. 6: 507 (1988) .
Morich, M. A., D. A. Lampman, W. R. Dannels, and F. T.
Goldie. "Exact Temporal Eddy Current Compensation in
Magnetic Resonance Imaging Systems." IEEE Trans. Med.
Imaq. 7: 247 (1988).
Murphy-Boesch, J., and A. P. Koretsky. "An in Vivo NMR
Probe Circuit for Improved Sensitivity." J. Maqn. Reson.
54: 526 (1983).
Myers, C. C., and P. B. Roemer. "Highly Linear Asymmetric
Transverse Gradient Coil Design for Head Imaging." Book of
Abstracts, SMRM, 10th Annual Meeting, 1991," p. 711.
Petropoulos, L. S., J. L. Patrick, M. A. Morich, M. R.
Thompson, and R. W. Brown. "Torque-Free Asymmetric
Transverse Gradient Coil." Book of Abstracts, SMRM, 12th
Annual Meeting, 1993, p. 1305.


59
energy for a gradient of 4 G/cm was calculated to be 7.93 J.
Since these coils can be made much smaller than the bore of
the magnet, eddy currents are not a serious problem and
neither of these coils is shielded. Unlike symmetric
designs, these coils experience a net torque in the magnetic
field that is potentially dangerous.
Another coil at a larger radius can be used to cancel
the torque experienced by an asymmetric gradient coil.
Petropoulos et al.57 took this approach to design a head
coil with an inner diameter of 36.4 cm, the same as their
single-layer coil described above, and an outer diameter of
48 cm. The length of both inner and outer coils was 60 cm.
The coil was designed to have a useful region that is a
sphere of 25 cm diameter. There is a price to pay in
increased stored energy, which increases over the single
layer coil value of 7.93 J to 19.2 J. Torque-compensating
windings can be added to the same cylinder as the primary
coil, resulting in a long structure one end of which is
placed over the head of the patient. Abduljalil et al.58
developed such a coil set for echo-planar imaging. The
diameter of the two radial coils was 27.2 cm and 31.2 cm.
The center of the linear region was 17 cm from the end. The
overall length was not reported, but based on artwork for
the wire pattern, it seems to be about 116 cm.
57L. S. Petropoulos et al., SMRM 1993 1305.
58A. M. Abduljalil et al., SMRM 1993, 1306.


21
the Laplace transform. We can understand the calculation as
follows. Assume the gradient field for t>0 in response to a
unit step function is
N
g(t) = 1 aie~tw = yx, [24]
i = 1
The amplitudes and time constants can be determined
through a best-fit to experimental data. To determine the
inverse filter, the first step is to deconvolve the step
function to find the impulse response h(t), which can more
conveniently be accomplished by a multiplication in the
complex frequency space, s. The equivalent function G(s) is
obtained by a Laplace transform
G(s) = - Y ^ [25]
S .. S + Wi
1 = 1 2
Then the impulse response in the s domain, H(s), is found
through the relation
G(s) = H ( s )/s, [26]
so that
H(s) = sG{s) = 1- Y
i = ls + wi
[27]
is the impulse response. The inverse filter's impulse
response is just the reciprocal of the impulse response of
the eddy currents,
[28]


160
Petropoulos, L. S., M. A. Morich, J. L. Patrick, D. A.
Lampman, Haiying Liu, and R. W. Brown. "Insertable
Asymmetric Cylindrical Gradient Coils." Book of Abstracts,
SMRM, 11th Annual Meeting, 1992, p. 4032.
Riddle, W. R., M. R. Wilcott, S. J. Gibbs, and R. R.
Price. "Using the Phase of the Quadrature Signal in
Magnetic Resonance Spectroscopy to Evaluate Magnetic Field
Homogeneity and Temporal Stability." Book of Abstracts,
SMRM, 10th Annual Meeting, 1991, p. 453.
Romeo, F., and D. I. Hoult. "Magnet Field Profiling:
Analysis and Correcting Coil Design." Maan. Reson, Med.
1: 44 (1984).
Sharp, J. C., R. W. Bowtell, and P. Mansfield. "Elimination
of Susceptibility Distortions and Reduction of Diffusion
Attenuation in Microscopic NMR Imaging by Line Narrowed
2DFT." Book of Abstracts, SMRM, 11th Annual Meeting, 1992,
p. 688.
Suits, B. H., and D. E. Wilken. "Improving Magnetic Field
Gradient Coils for NMR Imaging." J. Phvs. E: Sci.
Instrum, 23: 565 (1989) .
Teodorescu, M. R., A. E. Badea, R. C. Herrick, and F. R.
Huson. "Experimental Analysis and Finite Element Solution
of Eddy Current Effects in Superconductive Magnets." Book
of Abstracts, SMRM, 11th Annual Meeting, 1992, p. 364.
Terman, F. E. Radio Engineers' Handbook. McGraw-Hill: New
York, 1943.
Trick, T. N. Introduction to Circuit Analysis. John Wiley
and Sons: New York, 1977.
Turner, R. "Gradient Coil Design: A Review of Methods."
Maan. Reson. Imaq. 11: 903 (1993).
Turner, R. "A Target Field Approach to Optimal Coil
Design." J. Phvs. D: Aool. Phvs. 19: L147 (1986).
Turner, R., and R. M. Bowley. "Passive Screening of
Switched Magnetic Field Gradients." J. Phvs E: Sci.
Instrum. 19: 876 (1986).
van Vaals, J. J., and A. H. Bergman. "Optimization of Eddy
Current Compensation." J. Maan. Reson. 90: 52 (1990).


52
simplest coil producing a gradient perpendicular to the axis
of the cylinder is the double saddle or "Golay" coil
illustrated in Figure 13(a). The arcs all subtend 120 and
are placed at the four angles 0X = 68.7, 02 = 21.3, 18O-01(
and 18O-02, where they produce an (1, m) = (1, 0) term but
no (I, m) = (3, 0) term. The relative current directions
are shown in Figure 13(a). A family of solutions exists for
which the sum of the (1, m) = (3, 0) terms produced by the
arcs cancels, but the (1, m) = (3, 0) terms produced by each
arc are not necessarily zero. We designate such coil
designs by the two angles 0-^ and 02, so that the design
above would be described as 68.7/21.3. Adding additional
current elements to the coils adds degrees of freedom to
the system of simultaneous equations, and makes it possible
to cancel more terms.
Adding another pair of loops adds two more degrees of
freedom (the current and position of the new loops), and
makes it possible to cancel higher-order terms including (5,
0). Note that the equations are not linear, so for a large
number of current elements the procedure becomes unwieldy.
For shim coils, it is less important to improve the linear
region of a first-order or gradient coil than to design
additional coils whose lowest-order terms are of
increasingly high order. The simple saddle coil in Figure
13(a) designed by this technique has a useful volume with a
radius of about 1/3 that of the cylinder.


147
The suffixes E, F, G, U, W, V turn on a field gradient pulse
in the interval. The suffix A defines the transmitter and
receiver RF phase. The suffix N enables the decoupler, and
C sets the output to continuous wave.
The instrument could be controlled directly from a
virtual terminal on the PC, which replaced the usual
Figure 56. Excel spreadsheet calculates dependent
parameters in spin-echo imaging experiment. The
spectrometer can be controlled from the pull down menu.


141
(b)
Figure 49. The transcoupler, a) Front view; b) Rear view.


Abstract of Dissertation Presented to the Graduate School of
the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
NOVEL TECHNIQUES FOR PULSED FIELD GRADIENT NMR MEASUREMENTS
By
William W. Brey
December, 1994
Chairman: E. Raymond Andrew
Major Department: Physics
Pulsed field gradient (PFG) techniques now find
application in multiple quantum filtering and diffusion
experiments as well as in magnetic resonance imaging and
spatially selective spectroscopy. Conventionally, the
gradient fields are produced by azimuthal and longitudinal
currents on the surfaces of one or two cylinders. Using a
series of planar units consisting of azimuthal and radial
current elements spaced along the longitudinal axis, we have
designed gradient coils having linear regions that extend
axially nearly to the ends of the coil and to more than 80%
of the inner radius. These designs locate the current
return paths on a concentric cylinder, so the coils are
called Concentric Return Path (CRP) coils. Coils having
extended linear regions can be made smaller for a given
sample size. Among the advantages that can accrue from
using smaller coils are improved gradient strength and
v


95
Table 3. Predictions for 15 cm coil set based on time-
domain model.
Series
Turns
t0 (|XS)
Gmax (G/
X
52
735
10.9
Y
52
816
9.5
Z
75
717
9.9
Parallel
Turns
t0 (M-s)
Gmax (G/cm)
26
150
5.5
26
164
00
37.5
157
5.7
Figure 29. Photograph of the 15 cm gradient coil assembly.
consist of 52 turns of 20 AWG enameled magnet wire in 7-6
close-pack matrix. The wire is first wound onto a
rectangular bobbin as described above, then cemented to the
former. The axial coil, consisting of 75 turns 18 AWG
magnet wire, is wound over the radial coils. The assembly


82
have about twice as many turns as optimum, and yield fields
about 80% of optimum level.
Table 1.
Gradient
fields for 9
cm coil set
Gradient
Actual
Gradient
Optimal
Gradient
channel
no. of
after 200
no. of
after 200
turns
(is (G/cm)
turns
(is (G/cm)
X
52
15.2
27.9
19.0
Y
52
12.3
26.9
15.9
Z
52
24.0
53.6
24.8
The radial and axial gradient coils consist of 52 turns
of AWG 27 enameled magnet wire. The wire was wound in a 7-
6-7-... close-pack configuration to minimize the cross-
sectional area of the winding, which is reduced by a factor
of 0.866 from a square winding pattern. The resulting
winding cross-section is about 2.6 mm on a side, only about
6% of the coil radius, so the winding approximates a
filament. The mean radius of the coils is 4.6 cm, 4.8 cm,
and 5.1 cm. The Maxwell pair is wound on the outside
because it is inherently more efficient and will hold down
the other coils. The two halves of each coil are wound
separately, and one channel of a stereo amplifier is wired
to each. It is driven in current mode from the Techron 7540
amplifiers. The current-control circuit helps to buck the
inductance of the coil. The coil resistance for the radial
windings is predicted to be about 6.7 £2 for the inner set.
The measured resistances and time constants including the


148
electromechanical input device. A second virtual terminal
is used to control the data transfer. Two 9600 baud serial
RS-232 lines connect the PC and the Nicolet 1180E computer,
one for control and one for data.
It is desirable to control an imaging experiment by
setting parameters such as the echo time, TE, that must be
translated into hardware delays. An Excel (Microsoft Corp.)
Figure 57. Spreadsheet to update gradient calibration
parameters.


67
magnetic field G varies as N, so
G(t)
'kV0 (
< RXN V
kNI0
t/
1 e A
t > t0
\
/
0 < t < t0
[56]
A plot of G(t) for various values of N is shown in Figure
16. All three curves have the same time constant, so
G(t)
Figure 16. Magnetic field produced by a current controlled
linear amplifier coupled to a coil of fixed dimensions.
Each curve represents a different number of turns.
the difference in slope is due to the relative amplitude of
the maximum gradient. The dotted line connects all the
current-limit points. Since at the current-limit point t0
the amplifier is both a voltage and a current source, we can
eliminate G(t0) in favor of N and t0, yielding an optimum
number of turns for a given switching time.


154
manufacturers. Reducing the stray inductance by routing
both leads from the same side of the coil should improve the
sensitivity. Some loss in Q was due to loading of the
gradient coil.
The vertical orientation of the coil and sample helped
to eliminate problems with susceptibility that have been
observed with horizontal samples. It also facilitated
changing the sample. The lack of control over the
rotational position provided by the pneumatic sample
insertion system was not a problem for the samples studied.
The unshielded three-axis gradient set provided
enough gradient strength and switching time to use for
diffusion or other more demanding experiments. There is
adequate gradient strength for higher-resolution imaging if
the sensitivity of the RF coil were improved. Some spatial
distortion in the images can be attributed to inaccuracy in
the wire position. Nothing observed in the images was
attributed to eddy currents. The eddy current measurement
technique described above should be applied to evaluate the
system experimentally.


124
a a a a
2
O
-2
-4-20 2
(a) (b) (C)
Figure 41. Simulated and real CRP coil images. The real
and simulated phantom have the same width. a) The image is
simulated from the Biot-Savart law; b) Image of egg crate
phantom obtained with CRP coil; c) Image obtained with
Oxford coil.
tolerance of the plywood. The length of the useful region
is about 7 cm. Since the distance between the outermost
planar units is 9.5 cm, the useful region extends to 74% of
the coil length.
An image obtained with the Oxford gradients used in all
three directions was made for comparison and shown in Figure
41(c). The Oxford gradients have a much larger linear
region due to their larger size. There is some distortion
visible in the Y direction at the ends of the phantom.
The CRP approach has yielded a radial gradient coil
design that has a useful volume that can be made to extend


MEASUREMENT OF EDDY CURRENT FIELDS
Introduction
It is well known that, when a current pulse is passed
through a field gradient coil in a superconducting magnet,
eddy currents are produced in the conducting structures of
the magnet. Experiments such as diffusion-weighted imaging6
and multiple-quantum spectroscopy7 require that the eddy
current field be a much smaller fraction of the applied
field than do conventional spin-echo magnetic-resonance
imaging experiments. Strategies to reduce the eddy current
field consequently become increasingly important. The two
effective strategies are signal processing of the gradient
demand, known as preemphasis, and self-shielding of gradient
coils, which greatly reduces the interaction of the coil
with the metal structures of the magnet. Often, the two
techniques are used together. When the sample or subject is
substantially smaller than the magnet, another approach is
to minimize the size of the gradient coil. In order to
evaluate and improve the effectiveness of these three
strategies, it is desirable to have a technique to measure
eddy current fields. To implement the preemphasis, it is
necessary to measure the eddy current field in order to
6D. G. Cory and A. N. Garroway, Maun. Reson. Med. 14, 435, 1990.
7C. Boesch et al., Maan. Reson. Med. 20, 268, 1991.
5