CARTESIAN FEEDBACK CONTROL FOR MRI

CARTESIAN FEEDBACK CONTROL

FOR MRI TRANSMITTER ARRAY SYSTEMS

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF ELECTRICAL

ENGINEERING

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Marta Gaia Zanchi

May 2010

http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/jd326wm8459

© 2010 by Marta Gaia Zanchi. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.

ii



http://purl.stanford.edu/jd326wm8459

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

John Pauly, Primary Adviser


Thomas Lee


Greig Cameron Scott

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

iii

Preface

Accurate control of the radio-frequency (RF) electromagnetic fields in Magnetic Res-

onance Imaging (MRI) is necessary to ensure patient safety and provide high-quality

diagnostic capabilities. Precise control is however becoming increasingly difficult to

achieve, given the recent trends toward high fields and transmitter array systems. At

high fields, imaging is performed in a frequency regime where the wavelength is on the

order of, or smaller than, the dimensions of the human body. This leads to prominent

wave behavior, non-uniform field patterns, and increased power deposition. Multi-

element transmitter array systems with independent phase and amplitude control of

their elements support methods that can mitigate these problems. However, in turn,

they demand high fidelity RF reproduction and may lead to undesired electromagnetic

interactions between elements of the arrays and with interventional devices.

Frequency-offset Cartesian feedback can be used to address all of these issues. In

combination with the use of polyphase error amplifiers—to implement a low-IF control

bandwidth—Cartesian feedback can be used with MRI power amplifiers and transmit

coils to increase the fidelity of RF reproduction, without the in-bandwidth DC-offsets

and quadrature mismatches that may lead to imaging artifacts such as bright spots

and ghosting. In addition, the control system—which includes autotuning circuitry

for stability and vector multipliers circuitry for feedback manipulation—can be used

to tune the series output impedance of these amplifiers, thereby reducing the like-

lihood of interactions between elements of transmit array systems. Furthermore, a

miniaturized variation of the control system (called Active Cable Trap) can be used

on guidewires to suppress undesired currents elicited by coupling with the RF fields

of the transmit coils.

iv

In an era of rapid progress in high field MRI for clinical applications, the frequency-

offset Cartesian feedback method and system thus promises to address many of the

challenges faced by designers of multi-element transmitter array systems.

v

Acknowledgements

Had I not been fortunate enough to meet and work with Dr. Greig Scott during the

past four years, this dissertation would look very different. It would use an excessive

number of words (Dr. Scott would tactfully describe it as “verbose”) to explain far

fewer interesting concepts and results. He is one of the most intellectually honest,

creative, and enthusiastic scientists I have ever met.

I am equally fortunate to have been guided along the Ph.D. path by Prof. John

Pauly. He has always been encouraging, sympathetic, and available whenever I was

in need of help or advice. I am particularly grateful that he has always stood by

me when I was planning one of my new summer adventures, whether I wanted to

join a start-up company in Fremont, California, or work in the government offices in

Rockville, Maryland. These great opportunities would not have come true without

his ongoing support.

Prof. Thomas Lee is one of the most unexpected surprises that Stanford has

thrown at me. The little time I was privileged to spend in conversations with him has

made my Ph.D. journey much more enjoyable and has given me strength and hope

for what would come after. Of the 2,000 people who populate the world, he shines as

one of my favorites. Thinking of him, I am reminded of Friedrich Nietzsche’s famous

quote, ”One must have chaos [in his office] to give birth to a dancing star.” Indeed!

To Prof. Michael McConnell, I want to express my deepest appreciation for join-

ing Dr. Scott, Prof. Pauly, and Prof. Lee in my oral dissertation committee. I am

particularly happy to have had Prof. Jelena Vuckovic chair the committee. Prof.

Vuckovic taught the very first class—Applied Quantum Mechanics— I took at Stan-

ford in the fall of 2006. With her strength, independence, and hard work, she has

vi

always been a role model for me.

Evelin Sullivan has been my great friend and writing tutor at the Stanford School

of Engineering since 2007. In the past three years, she read almost every single page

I have written (God knows I wrote many!), and patiently educated me in the art of

clear and logical writing... even when this meant correcting my mistakes seven times

in a row. With her, I share the passion for a well written document and the joy of

scooter riding, to which I am proud and happy to have introduced her.

I am indebted to many people at the Magnetic Resonance Systems Research Labo-

ratory (MRSRL). All of them have been great friends, helpful colleagues, and patient

teachers. Prof. Albert Macovski is, to say the least, an inspiration to build great

things that last generations and beyond. Prof. Dwight Nishimura, Dr. Adam Kerr,

Prof. Steven Conolly, and Dr. William Overall have been always supportive, en-

couraging, and generous with their time and consideration. Joelle Barral and Pascal

Stang are destined to amazing careers and I am privileged to have received their help

and enjoyed their friendship. I am grateful and happy to have been at MRSRL with

Hattie Dong, Thomas Grafendorfer, Okai Addy, Joseph Cheng, Emine Saritas, Kim

Shultz, and many others. I thank Ross Venook for the research he has done before

me, for the interest he has demonstrated in what I built on his legacy, and for always

greeting me with a smile and a word of encouragement. I am grateful to Lily Shuye

Huan for her work and assistance throughout the years. My deepest thanks go to

Maryam Etezadi-Amoli, the kindest, sweetest, most selfless, as well as one of the most

intelligent young women I have ever met. I am proud she gave me her friendship and

I hope it will continue for many, many years ahead.

A few more remarkable men at and around Stanford have made a strong impression

in my life. I want to thank in particular Prof. Robert Dawson at the Department of

Art and Art History in Stanford University. In 2006, he opened the doors of his office

(and his analog photography darkroom) to me, and later helped me being admitted

to the M.F.A. program in Photography at the Academy of Art University in San

Francisco. Here, I want to thank Prof. Will Mosgrove for seeing me as an artist with

an engineering degree, rather than as an engineer with a digital camera.

The summer of 2007 in Volterra Semiconductors, Fremont, California, was beyond

vii

my expectations and the opportunity to meet a number of wonderful people including

Som Chakraborty, Milovan Glogovac, Alex Ikriannikov, Michael McJimsey, and many

others. I am very grateful to Ognjen Djekic for welcoming me in his System Design

group and for being a great boss then, and a great friend after.

I want to acknowledge the support and great work of the organizers and teachers

at the Summer Institute for Entrepreneurship at the Stanford Graduate School of

Business, with whom I was privileged to spend the summer of 2008.

Dr. Sunder Rajan mentored me during the summer of 2009 at the Office of Device

Evaluation, Food & Drug Administration, in Rockville, Maryland. I am very grateful

for he was the kindest and most understanding mentor I could have ever hoped for.

I am fortunate to have worked with him and to have had the opportunity to know

him and his lovely wife outside of work. He is an example of generosity and, with his

work for Habitat for Humanity, he inspired me to dedicate more time volunteering

for the Peninsula Humane Society.

In and outside Stanford, I was blessed with the friendship of some truly special

people. I want to thank Wei Wu and Forrest Foust for the great time spent together

in conversations about comics, art, Chinese anecdotes, dreams, Wisconsin cheese,

traveling, and so much more. I thank Maryam Fathi for the great work we did (and

all the fun we had while doing it) in the radiofrequency classes we took together as

a team during the past four years. Of Alex Tung, I cherish the memories of the

time spent together in the basement of Packard. He is not with us anymore, but his

spirit is, and so are the beautiful fruits of his humanitarian work. From all over the

world, my dear friends Alessandro Rossi, Lara Gherardi, Alessandro Restelli, Alberto

Carrera, and Ivan Labanca have always encouraged me to be strong and helped me

to find happiness in life. I hope they know just how much I treasure their beautiful

friendship.

To my parents, Paolo Zanchi and Claudia Modesti, I owe a tremendous amount of

gratitude. They have always loved and supported me unconditionally, even when my

pursuit of happiness meant taking away a piece of theirs. There is not a single day in

which I do not feel fortunate and proud to say that I am their daughter and biggest

fan. I am grateful to my grandma Natalina Nervi, to my cousins, nephews, aunties

viii

and uncles, and to all the rest of my family for their encouragement and affection. I

miss them all, dearly. I thank my new extended Garcea family as well, in particular

Bruno, Maura, Rosetta, Teresa, Dino, and their kind sisters, for welcoming me in

their homes as if I had always been a part of them.

Finally, I thank my husband, Giovanni Garcea. He is my best friend, my playmate,

my anchor to sanity, my dream maker. With his caring presence, he always reminds

me of the truly important things in life. With his example, he teaches me that there

are no boundaries to what we can pursue. With his passion, he makes every day

worth living to the fullest. This thesis is dedicated to him, with love.

E quidi uscimmo a riveder le stelle.

ix

Contents

Preface iv

Acknowledgements vi

1 Introduction 1

1.1 MR Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 MR Trends and Challenges . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.1 Towards higher RF bandwidth and frequency . . . . . . . . . 6

1.2.2 Towards Arrays of Transmitters . . . . . . . . . . . . . . . . . 7

1.2.3 Towards Interventional MRI . . . . . . . . . . . . . . . . . . . 8

1.3 Translating Challenges into Goals . . . . . . . . . . . . . . . . . . . . 10

1.4 Dissertation Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Cartesian Feedback 13

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Cartesian Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.1 Brief History . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.2 Classic Cartesian Feedback . . . . . . . . . . . . . . . . . . . . 15

2.2.3 Problems of Classic Cartesian Feedback . . . . . . . . . . . . . 19

2.2.4 Towards a Modified Cartesian Feedback Architecture . . . . . 21

2.2.5 Adapting Cartesian Feedback to Application in MRI . . . . . 23

2.3 Alternatives to Cartesian Feedback . . . . . . . . . . . . . . . . . . . 24

2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

x

3 Active Polyphase Amplifiers 29

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Brief History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3 Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.5 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.5.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4 System Design 46

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2 Motivations, Requirements and Objectives . . . . . . . . . . . . . . . 46

4.3 High-Level System Preview . . . . . . . . . . . . . . . . . . . . . . . 49

4.4 The Transmitter: Genie . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.4.1 Image Reject Down-Converter . . . . . . . . . . . . . . . . . . 53

4.4.2 DC Management Circuitry . . . . . . . . . . . . . . . . . . . . 56

4.4.3 Polyphase Amplifier Loop Filter . . . . . . . . . . . . . . . . . 57

4.4.4 Mixers and Phase Shift Control . . . . . . . . . . . . . . . . . 57

4.4.5 Additional Genie Components . . . . . . . . . . . . . . . . . . 59

4.5 Closing the Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.5.1 Power Amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.5.2 Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.5.3 Coupling devices . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.5.4 Auto-Calibration Network . . . . . . . . . . . . . . . . . . . . 65

4.5.5 Medusa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.5.6 PE001 Card and GUI Interface . . . . . . . . . . . . . . . . . 69

4.5.7 AVRmini and Matlab Interface . . . . . . . . . . . . . . . . . 69

4.6 Analysis of Performance . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.7 Analysis of Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

xi

5 Improving the Fidelity of RF Reproduction 75

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.2 Nature of Amplifier Distortion . . . . . . . . . . . . . . . . . . . . . . 76

5.3 Reduced AM-AM, AM-PM Distortion . . . . . . . . . . . . . . . . . 78

5.3.1 Voltage-Mode Amplitude Test . . . . . . . . . . . . . . . . . . 79

5.3.2 Current-Mode Amplitude Test . . . . . . . . . . . . . . . . . . 80

5.4 Reduced Two-Tone and QAM Distortion . . . . . . . . . . . . . . . . 82

5.4.1 Two-Tone Test . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.4.2 QAM Constellation Test . . . . . . . . . . . . . . . . . . . . . 84

5.5 Reduced MRI Pulse Distortion . . . . . . . . . . . . . . . . . . . . . . 86

5.5.1 Sinc Pulse Linearization Test . . . . . . . . . . . . . . . . . . 87

5.5.2 VSS Pulse Linearization Test . . . . . . . . . . . . . . . . . . 89

5.6 Effect of Linearization on Magnetization . . . . . . . . . . . . . . . . 90

5.7 Closed Loop Image Rejection Performance . . . . . . . . . . . . . . . 93

5.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6 Manipulating the Amplifier Impedance 95

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.2 The problem of Coil Interactions . . . . . . . . . . . . . . . . . . . . 96

6.2.1 Available Methods . . . . . . . . . . . . . . . . . . . . . . . . 97

6.3 Theory of Impedance Manipulation . . . . . . . . . . . . . . . . . . . 101

6.4 Load Pull Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.5 Impedance Control System Configuration . . . . . . . . . . . . . . . . 105

6.6 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.6.1 With Single Power Amplifier . . . . . . . . . . . . . . . . . . . 109

6.6.2 With Balanced Power Amplifier . . . . . . . . . . . . . . . . . 110

6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

7 Conclusion 114

A Active Cable Trap 117

A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

xii

A.2 Theory and Previous Work . . . . . . . . . . . . . . . . . . . . . . . . 118

A.3 Feedback Method for Current Attenuation . . . . . . . . . . . . . . . 119

A.3.1 Toroidal Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . 121

A.3.2 Quadrature Demodulator . . . . . . . . . . . . . . . . . . . . . 122

A.3.3 Polyphase Loop Error Amplifiers . . . . . . . . . . . . . . . . 123

A.3.4 Quadrature Modulator . . . . . . . . . . . . . . . . . . . . . . 124

A.3.5 Toroidal actuator . . . . . . . . . . . . . . . . . . . . . . . . . 124

A.4 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Bibliography 128

xiii

List of Tables

3.1 Simulated and Measured Sideband Rejection at Different Center Fre-

quencies of the Polyphase Passband . . . . . . . . . . . . . . . . . . . 45

4.1 S-parameters of custom-made coupler. Port 1 = Input; Port 2 = Out-

put; Port 3 = Voltage Sample; Port 4 = Current Sample. . . . . . . . 65

4.2 S-parameters of C7149 coupler by Werlatone. Port 1 = Input; Port 2 =

Output; Port 3 = Forward Voltage Sample; Port 4 = Reverse Voltage

Sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3 Gain and maximum input and output levels of the main loop com-

ponents. The maximum up-mixer output is taken after the on-board

filters. The maximum down-mixer input is valid at minimum gain set-

ting; typically, the gain is 0±3 dB and the corresponding maximum

input is +2∓3 dBm. The power amplifier is the custom-made ampli-

fier built using an AN779H 20 W predriver and an AR313 amplifier by

Communication Concepts, Inc. . . . . . . . . . . . . . . . . . . . . . . 74

5.1 Measured Sideband Rejection of the Closed Loop FOCF System . . . 94

xiv

List of Figures

1.1 MRI system overview. The high field magnet is responsible for the

magnetization of the imaging volume. Transmit RF coils and RF

power amplifiers (PA) create pulses of energy that perturb (excite) the

original magnetization. Gradient coils and amplifiers introduce linear

variations in the static field for phase and frequency encoding. Receive

RF coils and preamplifiers measure the voltages induced by the pre-

cessing transverse magnetization. Additional electronics are used for

post-processing and image reconstruction. . . . . . . . . . . . . . . . 2

1.2 Desired (reference) VSS pulse (top, left) compared to the actual VSS

pulse (top, right), measured at the output of an RF power amplifier.

The effect of each pulse on the magnetization of the nuclei, calculated

using the Bloch equations, is shown in the plots below. Clearly, the

effect on the magnetization of the actual VSS pulse is substantially

altered by the distortion of the power amplifier. As a consequence of

this distortion, the quality of the MR image as well as the image’s

diagnostic potential can be drastically compromised. . . . . . . . . . 6

xv

1.3 Measurement setup (top) and measured currents and heating (bot-

tom) induced by the MRI RF field in a guidewire. The measurement

setup shows a previously developed optically-coupled current moni-

toring device, which consists of toroidal sensor, transmitter, receiver,

and display of the measured signals (in this case, an oscilloscope). The

guidewire was fed in the cavity of the toroidal sensor. The temperature

rise at the guidewire tip was measured with a commercial temperature

sensor. At increasing body coil excitation, the measured current (bot-

tom, right) increases linearly and the measured temperature increase

(bottom, left) increases quadratically. . . . . . . . . . . . . . . . . . . 9

2.1 Simplified schematic of a Cartesian feedback control system for power

amplifier linearization. The basic Cartesian loop consists of two identi-

cal feedback circuits operating independently on the quadrature (I/Q)

channels. Each of the quadrature baseband inputs is applied to a dif-

ferential amplifier, with the resulting difference (error) signals being

modulated (up-converted) onto quadrature carriers at the local oscil-

lator frequency and then combined to drive the power amplifier. A

sampled version of the power amplifier output is quadrature-down-

converted (synchronously with the up-conversion process). The result-

ing quadrature feedback signals form the second inputs to the differ-

ential integrators, completing the two feedback loops. . . . . . . . . . 15

2.2 Classic loop amplifiers. In a Cartesian feedback system, these ampli-

fiers subtract the reference and feedback signal, amplify the resulting

difference, and are responsible for the loop compensation. . . . . . . . 16

2.3 Classic loop amplifiers response. Reference signal, feedback signal, and

error amplification are at DC. . . . . . . . . . . . . . . . . . . . . . . 17

2.4 Complex bandpass amplifiers response. A reference signal, feedback

signal, and error amplification shifted at a complex IF frequency moves

the control bandwidth away from the frequencies where DC offset and

quadrature mismatches exist. . . . . . . . . . . . . . . . . . . . . . . 22

xvi

3.1 Control bandwidth options. In a classic pair of lowpass amplifiers,

amplification and subtraction between the reference input signals and

feedback signals occur at DC (top). A pair of bandpass amplifier cre-

ates two separate bandwidths at both positive and negative frequencies

(middle). Polyphase amplifiers, instead, create a single complex band-

pass control bandwidth (bottom); hence, they selectively amplify the

desired signals over quadrature mismatches and DC offsets. . . . . . . 31

3.2 Simplified schematic (left) and frequency response (right) of the fully-

differential polyphase difference amplifiers that have been used as the

loop error amplifiers of the frequency-offset Cartesian feedback system

described in this dissertation. . . . . . . . . . . . . . . . . . . . . . . 33

3.3 Actual polyphase amplifier transfer function with varying operational

amplifier gain (left, pole frequency is constant) and pole frequency

(right, gain is constant). If the gain-bandwidth product of the opera-

tional amplifier is the same, the effects of these non-idealities on the

desired polyphase transfer function are virtually indistinguishable. . . 38

3.4 Picture and simplified schematic of PCB for testing of polyphase am-

plifiers. The polyphase amplifiers can be tested with either two fully-

differential input signals or two positive input signals (the negative

inputs being AC-grounded). . . . . . . . . . . . . . . . . . . . . . . . 40

3.5 Normalized real input (i, q) to real output (I, Q) simulated trans-

fer functions. The latter can be also measured by driving the PCB

polyphase amplifiers with only one non-zero quadrature input signal (i

or q) at a time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.6 Up, Um, Vp, Vm simulated transfer functions obtained by combining the

functions in Figure 3.5. . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.7 Overall simulated transfer functions of the polyphase amplifiers ob-

tained by combining the functions in Figure 3.6. The desired frequency

response is obtained by merging Up with the mirrored Vm. The mirror

frequency response is obtained by merging Vp with the mirrored Um. . 43

xvii

3.8 Experimental transfer functions of the polyphase amplifiers. These

functions were constructed from the normalized real input (i, q) to

real output (I, Q) transfer functions measured by driving the amplifiers

with only one non-zero quadrature input signal (i or q) at a time. . . 44

4.1 Simplified hardware diagram of the frequency-offset Cartesian feedback

system. In addition to the RF power amplifier, the system includes

the transmitter Genie, a power amplifier load, and an RF coupler.

The components of an auto-calibration network (RF switches) and the

Medusa console are not shown here and will be described later. . . . . 50

4.2 Top: Printed Circuit Board (PCB) of the frequency-offset Cartesian

transmitter, Genie. Bottom: Simplified block diagram showing the

position on the board and relationship between the reference generation

circuitry, polyphase amplifiers, CMX998, and local oscillator in Genie. 52

4.3 Reference Generation Circuitry. R1 is 649 Ω, R2 is 680 Ω, C1 and C2

are 470 pF. All the passive components have 0.1% tolerance. The fully-

differential amplifiers driven by the ADL5387 quadrature demodulator

are THS4131 devices by Texas Instruments. (The THS4131 devices of

the DC management circuitry are also shown.) The LO frequency is

the same reference sent to the down/up-mixers of the feedback loop. . 54

4.4 Simulated complex response of the passive polyphase filter. These

passive filters create two complex notches near -500 kHz. Over 40

dB attenuation is obtained in the negative frequency band opposite

the desired positive frequency bandwidth defined by the polyphase

amplifiers in the frequency-offset Cartesian feedback loop. . . . . . . . 56

4.5 Genie polyphase amplifier. The frequency response of the amplifier has

peak gain of 70 VV

(36.9 dB), center frequency of about 500 kHz, and

passband half-width of about 140 kHz. The final design values of the

passive components, all of which have 0.1% tolerance, were Ri = 750 Ω,

RC = 16 kΩ,RF = 50 kΩ, C = 22 pF (nominal). The fully-differential

amplifiers are THS4131 devices by Texas Instruments. . . . . . . . . . 58

xviii

4.6 Phase shift deviation from the desired value at 64 MHz, 128 MHz, and

300 MHz. Although the CMX998 is specified for operation above 100

MHz RF, the phase shift control circuitry operates with good linearity

(less than ± 4 deg error) and can thus be used for 1.5 T MRI amplifiers

feedback control. The CMX998 fails at RF frequencies below 40 MHz. 59

4.7 Simplified schematic of a 6” by 3” surface transmit coil. The capaci-

tance is distributed to minimize the e-field. The integrated coil current

sensor is also shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.8 Photo (left) and simplified schematic (right) of custom-made slotted-

line style sensing circuit. A tapped RC is used for voltage sensing and

a pick-up loop for current sensing. . . . . . . . . . . . . . . . . . . . . 64

4.9 Schematic and photo of ZASWA-2-50DR switch by Mini-circuits. The

switch provides internal 50 Ω termination. . . . . . . . . . . . . . . . 66

4.10 Schematic of the feedback system with auto-calibration network. When

the stability conditions of the system must be investigated, Medusa

toggles the switches in position A. In this state, the loop is open and

the reference signal is sent to the feedback down-mixer. A sample of the

output signal is used to measure the loop phase rotation and calculate

the phase shift setting that compensates for it. Once the stability

conditions are known, Medusa toggles the switches to position B. In

this state, the loop is closed and the reference signal is sent to the

input of Genie. A sample of the output signal is used to measure, for

example, the linearization performance of the system. . . . . . . . . . 67

4.11 Simplified schematic of the feedback system for loop analysis. In the

frequency-offset Cartesian feedback system, block A includes the cou-

pler attenuation coefficient, the down-mixer conversion gain, and the

loss of combiners and additional pads. H(ω) includes the polyphase

amplifier gain and the up-mixer conversion gain. . . . . . . . . . . . . 70

xix

5.1 Output Voltage Amplitude (left) and Phase Error (right) of the power

amplifier without (red traces) and after addition (blue traces) of the

frequency-offset Cartesian feedback system. Both AM-AM and AM-

PM distortions are reduced by a factor of at least 14 (23 dB), which

approximates the loop gain of the system in its chosen configuration

during the experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.2 Coil Current Amplitude (left) and Phase Error (right) before and after

addition of the frequency-offset Cartesian feedback system (shown in

red and blue, respectively). Both AM-AM and AM-PM distortions

are reduced by a factor of about 10 (20 dB) in the range between

10% and 90% of the total output current. This value approximates the

minimum loop gain of the system in its chosen configuration during the

experiment. The reduced linearization performance at the extremes of

the range can be explained by the reduced phase margin, due to the

variation in the load impedance with varying frequency. . . . . . . . . 81

5.3 Two-tone Test. The output spectrum of the power amplifier driven di-

rectly (top) with two tones closely spaced in frequency shows odd-order

inter-modulation products, which are reduced to the noise floor after

addition of the frequency-offset Cartesian feedback system (bottom).

Some increase in noise level is evident with closed-loop operation, es-

pecially near the main tones. . . . . . . . . . . . . . . . . . . . . . . . 83

5.4 Two-tone Test. The output spectrum of the power amplifier driven di-

rectly (top) with two tones closely spaced in frequency shows odd-order

inter-modulation products, which are reduced to the noise floor after

addition of the classic Cartesian feedback system (bottom) obtained by

removing the coupling between the quadrature error signals amplified

by the loop error amplifiers. The ”spike” at the center of the control

bandwidth is the LO leakage created by DC offsets and self-mixing

of the LO frequency at the down-mixer. The LO phase noise is also

present near the center frequency. . . . . . . . . . . . . . . . . . . . . 84

xx

5.5 QAM Test. The QAM diagram of the power amplifier driven directly

(top) shows gain compression effects. After addition of the frequency-

offset Cartesian feedback system, the compression effects are virtually

eliminated (middle); higher noise is evident as in the two tone test

results. Removing the coupling between quadrature error signals at the

loop error amplifiers results in an appreciably distorted constellation

(bottom), in which low power symbols especially suffer from DC/LO

leakage and quadrature errors within the loop. . . . . . . . . . . . . . 85

5.6 Sinc pulse test. The measured sinc pulse at the output of the power

amplifier driven directly is overlaid on the reference sinc pulse in the

two upper panels at the right (showing real and imaginary parts, re-

spectively). The two bottom panels below show that amplitude and

phase errors are ±5% and ±20, respectively. Memory effects are

also evident, especially in the phase behavior. After addition of the

frequency-offset Cartesian feedback system, the four plots at the left

are obtained. The amplitude and phase errors are reduced to less than

±1% and ±2, respectively, even if the power amplifier behavior is not

memory-less. Simple pre-distortion techniques based on look-up tables

are not able to compensate for memory effects, hence, would not have

been able to demonstrate the same result. . . . . . . . . . . . . . . . 88

5.7 VSS pulse test. The measured VSS pulse at the output of the power

amplifier (driven directly) is overlaid on the reference VSS pulse in

the two right upper panels (showing real and imaginary parts, respec-

tively). The two right bottom panels show that amplitude and phase

error are ±5% and ±20, respectively. Memory effects are also evident,

especially in the phase behavior. After addition of the frequency-offset

Cartesian feedback system, the four plots at the left are obtained. The

amplitude and phase error are reduced to less than ±1% and ±2,

respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

xxi

5.8 Magnetization profile of VSS pulse. While the time envelope of the VSS

pulse at the output of the power amplifier driven directly (top, first

plot) does not appear appreciably different from the reference signal

(top, second plot), the effect of the distorted and reference pulses on

the magnetization does (bottom second and first plot, respectively).

The suppression band is altered from about 1% (desired) to over 20%

the unaltered magnetization. After the addition of the frequency-offset

Cartesian feedback system, the desired suppression band is faithfully

reproduced. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.9 Magnetization profile of 5 kHz-modulated VSS pulse. Despite the in-

creased bandwidth, the system shows performance similar to the case of

the un-modulated VSS pulse. The power amplifier alters the two sup-

pression bands from about 1% (desired) to over 30% of the unaltered

magnetization. After the addition of the frequency-offset Cartesian

feedback system, the desired suppression bands are again faithfully

reproduced. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.1 Equivalent representation of different impedance manipulation tech-

niques. Depending on the length of transmission line between the

power amplifier and coil, some techniques attempt to decrease the

power amplifier impedance, others, to increase. . . . . . . . . . . . . . 97

6.2 The case of the distributed-C surface transmit coil. The coil is designed

so that the impedance of the inductance XL matches the impedance

of the capacitance XC . If the power amplifier drives the inductance

directly, then a power amplifier output impedance Zout equal to zero

is desirable, since by doing so XL and XC are in parallel. Conversely,

if the power amplifier is separated from the coil by a length of trans-

mission line equal to λ4, the same result is obtained with a very high

amplifier output impedance. . . . . . . . . . . . . . . . . . . . . . . . 98

xxii

6.3 MRI receive technologies. A PIN diode (bottom, during the transmit

interval) or a properly-matched preamplifier input impedance (top,

during the receive interval) presents a short-circuit to the surface re-

ceive coil. If XL = XC , this short-circuit allows the tank circuit in-

ductor and capacitor to create a very high input impedance, which

open-circuits the coil. Methods of dealing with transmit coil interac-

tions based on power amplifier impedance manipulation attempt to

emulate these “Q-spoiling” techniques. . . . . . . . . . . . . . . . . . 99

6.4 Simplified frequency-offset Cartesian feedback system with power am-

plifier loaded by a transmit coil, separated from it by an arbitrary

length of transmission line. The loop forces a precise relationship be-

tween Vr and Vf , hence, a precise value of the reflection coefficient ΓA

at the output of the power amplifier. The desired value of the reflec-

tion coefficient depends on the length of transmission line. The goal is

to obtain a transmit coil that presents a very high impedance to the

other coils in the array. . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.5 Simplified schematic of load-pull setup. By switching between two

different known output loads and measuring output voltage and cur-

rent for each load, the internal amplifier impedance responsible for the

output level change can be calculated. . . . . . . . . . . . . . . . . . 104

6.6 Simplified frequency-offset Cartesian feedback setup with a single com-

biner to create a very high (or very low) output impedance of the power

amplifier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.7 Simplified schematic of a vector multiplier consisting of a two-stage

polyphase filter, four-quadrant multiplier AD835 (here shown as two

mixers), DACS LT1655, and output buffer LT1395. . . . . . . . . . . 106

6.8 Simplified frequency-offset Cartesian feedback setup with a pair of vec-

tor multipliers—each one weighting Vf or Vr to create any arbitrary

output impedance of the power amplifier. . . . . . . . . . . . . . . . . 107

xxiii

6.9 Simplified schematic of a balanced amplifier. A balanced amplifier has

two matched amplifying devices that are run in quadrature. A 200

W RF 3 dB hybrid on the input creates two quadrature signals from

the single RF signal; a second, identical 3 dB hybrid on the output

recombines in phase the two quadrature signals. If the two amplifying

devices are well matched, the balanced amplifier has excellent input

and output return loss (input and output impedance are approximately

50 Ω). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.10 Simplified schematic of 3 dB quadrature hybrid and phase truth table. 109

6.11 Experimental plot of the output impedance, obtained by summing

(blue trace) or subtracting (green trace) Vf and Vr, at increasing RF

output voltage. The plot also shows the output impedance obtained

by disconnecting Vr (red trace). . . . . . . . . . . . . . . . . . . . . . 110

6.12 Smith chart of the experimental power amplifier output reflection coef-

ficient obtained with constant magnitude of the ration of α and β and

varying phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.13 Experimental plot of the series real and imaginary output impedance,

obtained with a balanced amplifier and a single amplifier, at increasing

RF output voltage. The feedback control variable was Vr. . . . . . . . 112

A.1 Simplified schematic of active cable trap. The feedback loop input is

the RF current flowing in a potentially dangerous conductor, which is

detected by a toroidal sensor. After down-conversion, the loop com-

pares the detected signal to a DC reference, and amplifies the difference

by the polyphase amplifier gain. After up-conversion, the amplified er-

ror signal drives a toroidal actuator that induces in the conductor itself

an RF current that opposes the one induced by the B1 field. . . . . . 120

A.2 A toroid-cavity senses the RF currents in a wire fed through it. The

toroid consists of copper tape wrapped around a toroidal Teflon core

with 1.55 mm inner diameter and 5.50 mm outer diameter. . . . . . . 122

xxiv

A.3 Active cable trap polyphase amplifier. The frequency response of the

amplifier has peak gain of 70 (36.9 dB), center frequency of about

330 kHz, and passband half-width of about 140 kHz. The final design

values of the passive components, all of which have 0.1% tolerance, were

Ri = 750 Ω, RC = 20 kΩ, RF = 50 kΩ, C = 22 pF (nominal). The two

fully-differential amplifiers are model THS4140 by Texas Instruments. 123

A.4 Setup for experimental validation on the bench. Experiments were con-

ducted with the toroidal sensor and the actuator, and, by interrupting

the continuity of a looped wire as shown here. In the latter case, the

wire current is fed at the input of the demodulator directly; similarly,

the modulator’s output stage drives the wire current directly. . . . . . 125

A.5 Effect of the active cable trap on the currents induced in a looped wire.

When the loop phase rotation is accurately compensated, the suppres-

sion of the wire current is the highest and mirrors the amplification

of the polyphase loop error amplifiers. A small phase misalignment

between the up- and down-conversion reduces the effective suppression

of the wire current. Increasing the phase misalignment causes instabil-

ity: positive feedback amplification is obtained, and the wire current

is higher than that originally induced. . . . . . . . . . . . . . . . . . . 126

xxv

Chapter 1

Introduction

Accurate control of the radio-frequency (RF) field in Magnetic Resonance Imaging

(MRI) is necessary to ensure patient safety and provide high-quality diagnostic ca-

pabilities. Precise control is however becoming increasingly difficult to achieve, given

the recent trends toward high fields and high bandwidth, as well as toward the use

of transmitter array systems. In addition, the increasing use of interventional MRI

poses concerns for the safety of the patient.

At high fields, imaging is performed in a frequency regime where the wavelength

is on the order of, or smaller than, the dimensions of the human body. This leads to

prominent wave behavior, non-uniform field patterns, and increased power deposition.

Multi-element transmitter array systems with independent phase and amplitude con-

trol of their elements support methods that can mitigate these problems. However,

they demand high fidelity RF reproduction and may lead to undesired electromag-

netic interactions between elements of the arrays. Interactions with interventional

devices can also occur, and the number of unsafe events has been increasing steadily

with the use of interventional devices.

In this chapter, I introduce MRI and describe the trends that are changing the

face of this relatively young imaging modality. I explain the challenges that these

trends introduce and discuss how they translate into a precise set of goals that must

be addressed to enable MRI moving forward. Finally, I introduce modified Cartesian

feedback methods, which are proposed as a solution to reach all of these goals.

1

CHAPTER 1. INTRODUCTION 2

Figure 1.1: MRI system overview. The high field magnet is responsible for the mag-netization of the imaging volume. Transmit RF coils and RF power amplifiers (PA)create pulses of energy that perturb (excite) the original magnetization. Gradientcoils and amplifiers introduce linear variations in the static field for phase and fre-quency encoding. Receive RF coils and preamplifiers measure the voltages inducedby the precessing transverse magnetization. Additional electronics are used for post-processing and image reconstruction.

1.1 MR Imaging

Magnetic Resonance Imaging (MRI) is a non-toxic imaging modality that offers ar-

bitrary imaging planes and a high flexibility of applications for the diagnosis and

staging of diseases. The key steps necessary to obtain the final MR image are:

1. magnetization (or, polarization)

2. radio-frequency (RF) excitation and slice selection,

3. frequency/phase encoding,


4. RF detection,

5. post-processing and image reconstruction.

A specific hardware sub-system is associated with each of these steps; these five

sub-systems are, respectively,

1. a high-field coil magnet,

2. transmit RF coil and RF power amplifier,

3. gradient coils and gradient amplifiers,

4. receive RF coil and preamplifier,

5. a processing unit (a PC).

Figure 1.1 shows where these hardware sub-systems are located in a (simplified)

architecture of an MRI system; for ease of representation, transmit and receive RF

coils are shown as a combined device, though the use of two separate sets of coils is

common.

In the polarization phase, the high field magnet generates a strong static magnetic

field B0 that causes the hydrogen nuclei in the body to preferably align in the direction

of the field, creating a net magnetization.

In the RF excitation phase, the RF transmit coil and power amplifier create pulses

of RF energy, which are obtained when an alternating current is passed through the

transmit coil, at the characteristic Larmor frequency

ω = γB0 (1.1)

which perturbs the original spin magnetization. On an atomic level, ω is equivalent

to the quantum of energy required for the spins of the nuclei to make the transition to

a higher energy state. On a macroscopic level, the effect is that of a perturbation in

the direction of the net magnetization proportional to the duration and magnitude,

B1, of the RF pulse.


In the frequency/phase encoding steps, the gradient coils and amplifiers create

linear variations of the static field, B0, in the x, y, and z directions (the gradient

fields Gx, Gy, Gz), thereby affecting the magnetization of the nuclei with spins in a

fashion that is a function of their exact location within the volume.

In the RF detection phase, the precessing transverse magnetization induces a

voltage in the receive coil and detector.

In the post-processing phase, the information regarding duration and intensity of

the gradient fields and RF fields, together with the received RF signal, is used to

obtain the desired final spatial map of the distribution of the nuclei in the patient’s

body.

While the role of all of the above hardware sub-systems is critical to obtain the

desired image, it is certainly true that much of the flexibility of the MRI modality

relies on the ability of the RF transmit coil and power amplifier to faithfully reproduce

complex RF envelope and phase modulations. These modulations are employed by

the RF pulse designer to physically manipulate the magnetization. A simple example

is the sinc pulse. The sinc pulse has a square frequency distribution; hence, applied

in conjunction with a one-dimensional, linear magnetic field gradient, it will rotate

spins which are located in a slice or plane through the object. This principle is known

as “slice selection” and is commonly employed in MRI. More complex modulations

can be created to target more specific applications. For example, a Very Selective

Saturation (VSS) pulse envelope selectively suppresses the magnetization of the spins

in a well-defined frequency band, as it can be shown by the Bloch equations (a set of

coupled differential equations used to describe the behavior of a magnetization vector

under any condition), and finds application in brain imaging and prostate imaging.

In theory, the capabilities of the MRI modality are limited only by the creativity

of the RF pulse designers. In reality, three critical trends in the MRI field have been

identified that are pushing the available RF hardware transmit paths to the limits of

their ability to faithfully and safely reproduce the desired RF pulses. These trends

are the increasing bandwidth and frequency of the RF fields, the increasing use of

arrays of transmitters, and the increasing use of interventional MRI.


MRI is a relatively new and expanding technique where new developments con-

stantly emerge to address some of MRI’s most serious limitations, most notably in

terms of sensitivity and speed. The enhancement of the overall sensitivity and speed

of MRI by the transition to ever higher magnetic field strength (and thus Larmor

frequencies) and increasing RF bandwidth may be viewed as the response to these

limitations. In particular, moving toward higher RF bandwidths collides with the lim-

itations of the transmit coil RF power amplifier, whose non-linear behavior at rapidly

varying frequencies and amplitudes of the RF pulse distorts the desired envelope.

Simultaneously, moving towards higher fields poses challenges such as how to over-

come wave effects and create uniform fields as the Larmor frequency increases. The

trend toward the use of parallel transmission promises to solve the latter problem, by

offering new versatility in high field imaging. Similar to parallel reception, which was

developed to increase signal-to-noise ratio and speed of MRI, it is possible to drive

several transmit coil elements not only with independent amplitudes and phases but

also with independent RF pulse shapes. Similar to the shimming technique, which is

performed for the static magnetic field, parallel transmission makes possible a much

more uniform field distribution in vivo if multiple ports (channels) are driven with

RF energy, where the amplitude and the phase of the RF pulse varies independently

for each port. The problem here is that of RF coupling between the elements of the

array.

Contemporaneously to the increase of field strength and the development of par-

allel transmission techniques, interventional MRI has also gained increased attention.

Broadly defined, interventional MRI makes use of devices simultaneously with imag-

ing, for example, to guide minimally-invasive interventions or monitor the patient’s

vitals using a diagnostic procedure. The problem here is that of interactions between

the devices and the RF field, which can cause substantial RF currents and heating

at the points where the device is in contact with the patient’s tissue. The result of

these currents and heating is accidental RF ablation.

These three trends and the technological challenges created by them are described

in the next section.


0 0.5 1 1.5 2 2.5 3time (ms)

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0.15

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A A

Figure 1.2: Desired (reference) VSS pulse (top, left) compared to the actual VSSpulse (top, right), measured at the output of an RF power amplifier. The effect ofeach pulse on the magnetization of the nuclei, calculated using the Bloch equations,is shown in the plots below. Clearly, the effect on the magnetization of the actualVSS pulse is substantially altered by the distortion of the power amplifier. As aconsequence of this distortion, the quality of the MR image as well as the image’sdiagnostic potential can be drastically compromised.

1.2 MR Trends and Challenges

1.2.1 Towards higher RF bandwidth and frequency

The key asset of imaging at high field is increased baseline SNR [59, 64]. This in-

crease results from larger equilibrium polarization and higher resonance frequency.

These beneficial effects are only partly negated by increased thermal noise, resulting

in a significant net SNR gain. The downside of high fields is closely related to these

mechanisms. As magnetic field strengths continue to increase in human MRI, the

bandwidth and electrical power required to flip magnetization also increase, and the

wavelength of lossy propagation in the human body becomes shorter than the body’s


size [66, 78]. Higher resonance frequency leads to increased specific absorption rates

(SAR), since the energy deposition caused by radiofrequency irradiation grows as the

square of the frequency. Field perturbations caused by varying magnetic suscepti-

bility scale with the external field strength. A higher main field, therefore, causes a

stronger local field inhomogeneity; field inhomogeneity, in turn, causes artifacts and

blurring in sequences with long acquisition intervals [26, 49]. Furthermore, at very

high frequencies, the effective wavelength of the RF field is comparable to the size

of the anatomy under investigation and to the length of the coil elements, and wave

effects (i.e., the phase of the wave) can no longer be ignored. As a result, the technical

challenges associated with control over the RF transmission field become more com-

plex: high power components of widely-employed class AB power amplifiers rapidly

heat and cause drift in output impedance, gain and phase, which in turn causes

distortion. Simultaneously, the complexity (bandwidth) of the RF pulses increases

because it is desirable to increase the complexity (bandwidth) of the modulation of

the magnetization of the nuclei. The latter, in fact, opens the door to more sophis-

ticated applications, in less time. However, as the bandwidth requirement increases,

the distortion introduced by the RF power amplifiers can also increase.

If neglected, the distortion introduced by the RF power amplifiers can result in

degradation of the image quality. As an example, Figure 1.2 (top) compares the

desired (reference) VSS pulse to the actual VSS pulse, measured at the output of an

RF power amplifier. The same figure (bottom) compares the effects of the two pulses

on the magnetization of the nuclei, calculated using the Bloch equations. Clearly,

the latter is substantially altered by the distortion of the power amplifier. As a

consequence of this distortion, the quality of the MR image as well as the image’s

diagnostic potential can be drastically compromised.

1.2.2 Towards Arrays of Transmitters

Parallel transmission offers new versatility in high field imaging, integration of trans-

mit mode interventional devices, and improved RF safety and SAR control [72, 82].

As with B0 shimming [54], which is performed for the static magnetic field, parallel


transmission makes possible a much more uniform field distribution in vivo if multiple

ports (channels) are driven with RF energy, where the amplitude and the phase of

the RF pulse varies independently for each port. As with parallel reception [65], it is

possible to drive several transmit coil elements not only with independent amplitudes

and phases but also with independent RF pulse forms, an approach that has been

named both parallel transmission and Transmit SENSE (SENSitivity Encoding). Fi-

nally, parallel transmission may compensate for some of the drawbacks of higher field

strengths, such as increased SAR and field inhomogeneity [1, 17].

These are considerable advantages. However, a major engineering challenge posed

by them is the minimization of the unwanted sources of error in real parallel transmit

systems. One source of error is the coil-to-coil coupling at different power levels,

which creates interference patterns and, hence, an inhomogeneous B1 field. Another

is RF leakage causing unwanted bulk excitation. Yet another source of error is non-

linear behavior and memory effects of the coil-driving amplifier, which can also affect

performance in many ways, for example by causing inaccurate pulse reproduction and

spectral spreading as well as poor selectivity [2, 46]. Non-linear behavior is mainly

described as static non-linearity, which can take the form of amplitude-to-amplitude

or amplitude-to-phase distortion, and memory effects, which include amplifier heating

and aging (responsible for bias drifts), as well as power supply droop and bandwidth.

1.2.3 Towards Interventional MRI

The radiofrequency pulse created by the transmit coil will not only be absorbed by

the nuclear spins in our bodies, but may also couple with devices that are attached

to the body during the imaging procedure, thus creating a RF current in the device

itself.

Three different physical phenomena can occur that explain the onset of these RF

currents: electromagnetic (EM) induction with a non-resonant looped device (such

as a looped guidewire), EM induction in a resonant looped device, and coupling with

a resonant elongated device whose length is a multiple of the half-wavelength of the

coupled RF field. The latter is essentially identical to that of an antenna that couples


T fiber optic

temperaturesensor

guidewire

receivertransmitter

sensor

scopeRF whole body coil

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

dT [C

]

Body Coil Excitation [%]0 10 20 30 40 50 60 70 80 90 100

0.10.20.30.40.50.60.70.80.91.0

I WIR

E / I PE

AK [A

/A]

Body Coil Excitation [%]

Ipeak = 390 mArms

Figure 1.3: Measurement setup (top) and measured currents and heating (bottom) in-duced by the MRI RF field in a guidewire. The measurement setup shows a previouslydeveloped optically-coupled current monitoring device, which consists of toroidal sen-sor, transmitter, receiver, and display of the measured signals (in this case, an oscillo-scope). The guidewire was fed in the cavity of the toroidal sensor. The temperaturerise at the guidewire tip was measured with a commercial temperature sensor. At in-creasing body coil excitation, the measured current (bottom, right) increases linearlyand the measured temperature increase (bottom, left) increases quadratically.

to a wireless field and it is associated with the highest currents [42–44].

The problem with the induced currents is that at the sharp end of the device, the

local electric field is high and can induce currents in a body in its close vicinity. Under

these conditions, dissipation of heat in the body will occur, possibly high enough to

create burns [21, 45, 61]. Figure 1.3 (bottom) shows the current and temperature

increase measured with an optically-coupled current monitoring device that has been

developed in the context of these studies. A simplified measurement setup is shown

in the same Figure 1.3 (top). As shown, a temperature increase of up to 80 degrees

celsius can be measured.


1.3 Translating Challenges into Goals

For each of the technical challenges posed by the trends in the MRI field, a solution

is needed that will enable the continuous progress of this imaging modality.

To deal with the challenge of increasing RF distortion created by the increasing

frequency and bandwidth of the RF transmit signal, a solution is to increase the

fidelity of RF reproduction by reducing the non-linearity and memory effects of the

RF power amplifiers driving the transmit coils.

To reduce interactions between coils of transmit arrays, a possible approach con-

sists of controlling the RF power amplifier output impedance. Indeed, if a high-

impedance can be created at the input of each transmit coil, the currents induced

by the time-varying magnetic fields created by the neighboring transmitters will be

greatly reduced.

To attenuate the risk for safety posed by the presence of interventional devices

in the MRI RF field, new methods and systems can be implemented that reduce the

currents induced in these devices.

In this dissertation, I propose and describe a modified Cartesian feedback control

method and system that promises to solve these problems. Cartesian feedback is

a negative-feedback technique that has been demonstrated to increase the linearity

and mitigate memory effects of existing power amplifiers in the field of mobile com-

munications. Similar linearization performance can be achieved in the field of MRI,

especially at high field strength and near the limit of the amplifier’s power-handling

capability. Also, Cartesian feedback can be used to manipulate the power amplifier

output impedance, in order to reduce the interactions between elements of transmit-

ter array systems. Finally, a miniaturized variation of this system can be used to

substantially attenuate currents induced in guidewires by the MRI fields.

Applying this promising technology to the field of MRI is, however, a challenging

task for two reasons. First of all, Cartesian feedback suffers from the presence of

undesired spurious frequencies within the feedback control bandwidth, which can

create undesired artifacts in the MR image if they are not suppressed before they are

sent to the transmit coil. To address this problem, a modified architecture based on


polyphase amplifiers in place of the classic amplifiers for the loop error amplification is

proposed. Second, Cartesian feedback is traditionally adapted to the communications

field; hence, new solutions are needed to address issues that are specific to the MRI

environment. To address these issues, a polyphase Cartesian feedback (also known

as frequency-offset Cartesian feedback, FOCF) system has been designed specifically

for application in MRI.

1.4 Dissertation Overview

The central contribution of this dissertation is the introduction of a modified Cartesian

feedback method and system, named frequency-offset Cartesian feedback or polyphase

Cartesian feedback, intended specifically for applications in MRI. The following chap-

ters present this contribution:

• Chapter 2 describes the classic Cartesian feedback method, starting with an

overview of its development in the field of communication. It presents an

overview of alternative methods used in communications to deal with the prob-

lem of distortion introduced by the RF power amplifiers, which is the first of

the three challenges addressed by the work described in this dissertation, and

motivates the choice of Cartesian feedback for application to MRI. Chapter

2 also introduces the problems of the traditional Cartesian feedback method

and the solution identified in this dissertation, namely, the implementation of a

low-frequency complex baseband amplification of the feedback loop error made

possible by the use of polyphase amplifiers. Finally, Chapter 2 introduces the

main issues specific to the MRI environment that need to be addressed for the

successful implementation of Cartesian feedback in this field.

• Chapter 3 describes the polyphase amplifiers used in the frequency-offset Carte-

sian feedback system. A theoretical analysis of their ideal behavior and practical

limitations, including component mismatching and limited gain-bandwidth ca-

pability, is presented. The insight on the polyphase amplifier behavior offered


by this mathematical analysis is compared to the experimental results obtained

with printed circuit board polyphase amplifiers.

• Chapter 4 motivates and presents the discrete design of the frequency-offset

Cartesian feedback system and of its components. Particular attention is de-

voted to the parts of the system that have been designed to address the issues

specific to the MRI environment. The chapter includes a theoretical analysis of

expected linearization performance and stability needs of the system, in partic-

ular in the presence of multiple feedback loops with coupled loads. The latter is

of particular interest in the use of the system in parallel transmit applications,

where interactions of transmit coils may occur.

• Chapter 5 demonstrates the ability of the modified Cartesian feedback system

to improve the linearity of the transmit path, thereby addressing the challenge

of increasing RF distortion in MRI. The chapter presents the characterization of

the open-loop behavior of the FOCF system and the demonstrated linearization

performance of its closed-loop operation in a variety of situations, for example

situations where output voltage control, output current control, or coil current

control is desired.

• Chapter 6 describes how the impedance manipulation ability is a solution to the

problem of the interactions between elements of MRI transmitter array systems,

and it demonstrates the ability of the modified Cartesian feedback system to

electronically manipulate the output impedance of power amplifiers.

• Chapter 7 summarizes lessons learned and suggests future directions.

• Appendix 1 presents the Active Cable Trap concept and prototype based on a

miniaturized version of the frequency-offset Cartesian feedback for attenuation

of the currents induced in interventional devices. By substantially attenuating

these currents, the Active Cable Trap could virtually eliminate the increase in

temperature that may occur at the points where the device is in contact with

the patient.

Chapter 2

Cartesian Feedback

2.1 Introduction

This chapter describes the classic Cartesian feedback method. It motivates the choice

of this particular linearization technique for application in MRI over alternative meth-

ods, which are used in communications to deal with the problem of distortion intro-

duced by the RF power amplifiers.

The problems of the traditional Cartesian feedback method are also described.

These problems are the sensitivity of this technique to LO-leakage and quadrature

mismatches. The implementation of a low-frequency complex baseband amplification

of the feedback loop error is then presented as a solution to the latter.

Finally, this chapter introduces the main issues specific to the MRI environment

that need to be addressed for the successful application of Cartesian feedback in this

field.

2.2 Cartesian Feedback

2.2.1 Brief History

Cartesian feedback control was invented in the early 1980s by Petrovic [63] to address

the problem of distortion introduced by RF-power amplifiers used in high-frequency

13

CHAPTER 2. CARTESIAN FEEDBACK 14

transmitters for communications. In communications, RF power amplifiers are used

in a variety of applications including radio and TV transmitters, wireless communi-

cations, and satellite communication systems. While high-linearity power amplifiers

are generally available, these are also characterized by low efficiency, which poses

problems such as high RF dissipation and thus heating, high costs, and difficulty of

integration of these devices. In communications, the application of Cartesian feedback

to a higher-efficiency amplifier allows one to relax the trade-off between linearization

and efficiency and to obtain acceptably low distortion during transmission.

Despite its potential for application in the field, the popularity of the invention was

held back by complexities associated with the actual implementation of the system.

Only in the mid 1990s, thanks to the progress made in the field of analog integrated

electronics, did Cartesian feedback become a topic of intense study and development.

From 1991 to 1994, Johansson and Mattsson demonstrated the flexibility of the

technique with applications in the linearization of RF power amplifiers for personal

communication networks, [35], linear TDMA modulation [36], and multi-carrier com-

munication systems [37, 38]. During those same years, Briffa and Faulkner focused

most of their studies and implementation efforts on solving the problems of stability of

Cartesian feedback, taking in consideration the non-idealities of the actual implemen-

tation of the system [9–12, 24, 25]. In 1996, Boolorian and McGeehan published new

solutions for maximizing the linearization bandwidth [6], new compensation strate-

gies [7], and new applications [4, 5, 8]. In 1997, Kenington for the first time studied

the noise performance of a Cartesian loop transmitter [41]. More recently, Dawson

and Lee implemented the first fully integrated Cartesian feedback system for power

amplifier linearization [20]. Integrated solutions for Cartesian feedback linearization

were then developed by two manufacturers, namely CML Microelectronics (in 2007)

and Motorola (expected in 2010).

In the context of MRI, the Cartesian feedback technique first appeared in 2004

thanks to the work by Hoult, who proposed this technique as a solution to the problem

of coil interactions in an array of coils for the purpose of transmission in MRI [31–33].


Figure 2.1: Simplified schematic of a Cartesian feedback control system for poweramplifier linearization. The basic Cartesian loop consists of two identical feedback cir-cuits operating independently on the quadrature (I/Q) channels. Each of the quadra-ture baseband inputs is applied to a differential amplifier, with the resulting difference(error) signals being modulated (up-converted) onto quadrature carriers at the localoscillator frequency and then combined to drive the power amplifier. A sampledversion of the power amplifier output is quadrature-down-converted (synchronouslywith the up-conversion process). The resulting quadrature feedback signals form thesecond inputs to the differential integrators, completing the two feedback loops.

2.2.2 Classic Cartesian Feedback

Cartesian feedback is a negative feedback technique that includes a frequency down-

conversion step in the feedback path, so that the loop is closed at baseband instead

of at the carrier frequency, and is based on the Cartesian coordinates of the baseband

signal.

The basic Cartesian loop consists of two identical feedback circuits operating in-

dependently on the two channels, known as the I and Q channels, as shown in the

simplified schematic of a Cartesian feedback control system for linearization of power

amplifiers in Figure 2.1. Each of the quadrature baseband inputs I and Q is applied to

a differential amplifier, with the resulting difference (error) signals being modulated

(up-converted) onto quadrature carriers at the local oscillator (LO) frequency and


Vcm

Vcm

ir

Q

I

if

qr

qf

Figure 2.2: Classic loop amplifiers. In a Cartesian feedback system, these amplifierssubtract the reference and feedback signal, amplify the resulting difference, and areresponsible for the loop compensation.

then combined to drive the power amplifier. A sampled version of the power am-

plifier output is quadrature-down-converted (synchronously with the up-conversion

process). The resulting quadrature feedback signals form the second inputs to the

differential integrators, completing the two feedback loops.

The forward path consists of the differential amplifiers, the synchronous up-mixer,

the non-linear power amplifier, and the output load (an antenna in communications,

a transmit coil in MRI). The differential amplifiers are characterized, to first-order

approximation, by the transfer function HC(ω), which describes the relationship be-

tween the complex output I+ jQ and the complex input i+ jq. Dawson and Lee [18]

emphasize the importance of choosing HC(s) = k/sx, where 0 < x < 1, as a compen-

sation strategy for robustness to phase misalignments that impact stability. However,

these “slow-rolloff” functions are not truly realizable with a lumped-element network

and are usually approximated by alternating poles and zeros that ensure that the


-500 0 5000.20

0.32

0.50

0.79

1.26

2

Frequency (KHz)

Freq

uenc

y R

espo

nse,

Mag

nitu

de (V

/V)

-90

-45

0

45

90

Frequency Response, Phase (degree)

Freq. Response, AmplitudeFreq. Response, Phase

Figure 2.3: Classic loop amplifiers response. Reference signal, feedback signal, anderror amplification are at DC.

average slope of HC(s) has the appropriate roll-off. In practice, it is not uncommon

to find Cartesian feedback systems in which the difference amplifiers are characterized

by as few as one single pole at a frequency other than DC and one single zero at higher

frequency, such that the transfer function near DC can be roughly approximated by

H(ω) = (K

1 + j( ωωo

)). (2.1)

Such an example of a classic loop difference amplifier is shown in 2.2, where i = ir−ifand q = qr − qf . The normalized amplitude of the amplifier frequency response is

shown in 2.3; in this example, fo = ωo

2πis 300 kHz.

The feedback path consists of a coupler that sends a sample of the power am-

plifier output voltage (or current) to the synchronous down-mixer. The quadrature

baseband components resulting from this down-conversion are used as feedback sig-

nals and subtracted from the baseband reference signals at the input of the difference

amplifiers. Since both the reference signals and feedback signals are quadrature base-

band signals, the resulting amplified quadrature error signal is also at baseband. The

quadrature error signal is often known as the loop error signal. After amplification


and up-conversion, the two paths in quadrature are summed to form the control sig-

nal driving the power amplifier. Once the loop is closed and if the conditions for

stability are met, then the control signal is the pre-distorted version of the desired

reference signal needed to compensate for the distortion of the power amplifier. As

will be shown in Chapter 4,

errorbaseband ∝1

loopgain. (2.2)

Hence, for very high loop gain (ideally, infinite) the control law of the system is,

simply

feedbackbaseband ≈ referencebaseband (2.3)

and, correspondingly

outputRF ≈ referenceRF/C (2.4)

where C is the attenuation coefficient of the coupler sampling the output of the power

amplifier. In words, once the loop is closed and if the conditions for stability are met,

then for very high loop gain the output signal will be an exact replica of the reference

signal, translated to the RF band and amplified by 1/C.

The last indispensable component of a Cartesian feedback system is the phase

shifter. Synchronism between the up- and down-mixers is obtained by splitting a

common RF carrier (the local oscillator, or LO, frequency), however the different

phase rotation (phase shift) through the feedback and forward paths cause the refer-

ence and feedback signals to be phase misaligned, a situation that compromises the

stability of the system. The phase shifter is thus necessary to compensate for the

phase shift and maintain the relationship that guarantees the loop stability. In ad-

dition to phase shift between the up-converted and down-converted signals, a second

effect—time delay in the loop—limits stability. This effect is typical of any feedback

system and defines the bandwidth allowed within the loop and thus the amount of

linearization that can be applied over a given bandwidth [29].

The system comprising the loop error amplifiers, up- and down-conversion mixers,

and phase shifter is known as the transmitter of the Cartesian feedback system.


Cartesian feedback has received a great deal of attention in communications thanks

to its advantages over alternative methods: it does not require a detailed knowledge

of the power amplifier behavior and is immune to changes such as those due to

temperature and aging. Moreover, it is suitable for almost any type of modulation

of the reference signal, including those characterized by a substantial variation in the

signal envelope as measured by the peak-to-average ratio (PAR). However, the classic

Cartesian feedback architecture is not immune to problems. Since both the reference

and feedback signals and the amplification of the loop error signals occur at baseband,

the transmitter of the system must be designed with tight specifications regarding the

matching between the two quadrature paths. Mismatches between these two paths

can create undesired frequencies that also are within the bandwidth of amplification

of the loop error signal (aka, the control loop bandwidth) and thus appear at the

output of the power amplifier.

2.2.3 Problems of Classic Cartesian Feedback

Designers of Cartesian feedback systems face two practical challenges when imple-

menting the system transmitter: minimizing the mismatches between amplitude and

phase of the two quadrature signals, and minimizing the differences between the volt-

age and current settings of their two paths at the zero frequency (DC).

Mismatches between amplitude and phase of the two quadrature signals will cre-

ate an image of the desired signal, that is, an undesired frequency component of the

signal spectrum at the opposite side of the carrier (LO). This is known as “quadrature

mismatch.” For a classic Cartesian feedback transmitter, whose loop error amplifica-

tion is at baseband (symmetric around DC), the image will always be in the control

loop bandwidth. Image suppression of 40 dB is desirable but very difficult to achieve.

One of the two limits to obtaining good image suppression is in producing accurate

quadrature reference baseband signals. Petrovic has shown that considerable effort

was expended on finding solutions to this problem. Although excellent results were

achieved, this was at the expense of increased cost and complexity. The availability


of low cost, high performance Digital Signal Processors (DSPs) and Digital to Ana-

logue Converters (DACs) provides a solution to this problem. The second limit is in

producing accurate quadrature feedback baseband signals at the output of the Carte-

sian feedback down-converter. The down-converter inevitably suffers from amplitude

and phase inaccuracies for which, again, only expensive and complex solutions can

compensate.

To understand how undesired frequencies are created in the output spectrum by

quadrature errors, an open-loop analysis of the circuit is useful. Consider the case of

the loop open at the output of the down-mixer. Let the power amplifier be a perfect,

ideal amplification block of gain G. The baseband feedback complex signals, if (t) and

qf (t), appear at the down-mixer output as a result of input reference signals ir(t) and

qr(t) of time-varying envelope A(t) and modulation frequency ωB. If the up-mixer is

error-free and the down-mixer introduces a phase quadrature error φ, then

if (t) = KGA(t) cos(ωBt−φ

2) (2.5)

qf (t) = KGA(t) sin(ωBt+φ

2) (2.6)

that is, the overall feedback signal Sf is

Sf (t) = if (t) + jqf (t) = KGA(t)

(ej(ωB)t cos(

φ

2) + jej(−ωB)t sin(

φ

2)

). (2.7)

If φ is zero, then Sf contains only the desired frequency +ωB; otherwise, a so-called

image (or, ghost) frequency −ωB is also produced before the subtraction node. Since

the non-ideality responsible for this unwanted behavior is in the feedback path, it

will not be compensated by the loop operation. It can be shown that for the image

rejection ratio (the ratio of the amplitudes of the desired and image frequencies) to

be at least 40 dB, the maximum value of the phase error φ must be 1.15. Similarly,

the maximum value of amplitude imbalance to obtain the same image rejection ratio

is 2%.

As with the case of amplitude and quadrature mismatches, differences between

the DC settings of the quadrature path (known as DC offset) will create an undesired


frequency component of the signal spectrum exactly at the carrier (LO) known as

“LO leakage.” Operational amplifiers within the system have DC offsets that drift

with temperature, and the down-converter mixers produce a frequency dependent

DC offset. In addition, 1f

noise drops below thermal noise above 100 Hz to 100 kHz

(depending on the technological process) and is thus a problem near DC. For these

reasons, at least 40 dB of DC offset correction is desirable. The typical method of

carrier suppression is to use sample-and-hold devices to null out the closed loop DC

offset. This solution, once again, adds complexity and cost to the implementation.

It is also possible to use a DSP to measure the DC offsets and to digitally predistort

the I-Q drive to the transmitter. However, this method eliminates fewer DC offsets.

Anticipating the application of Cartesian feedback in MRI, the undesired quadra-

ture mismatches and LO leakage in the RF transmit field would create artifacts to

appear in the final image, which would compromise the diagnostic information con-

tained in the image. The classic approach to Cartesian feedback would require that

the transmitter meet the very tight specification on amplitude and phase mismatches

and DC offset. A better solution is to offset the reference signal, feedback signal, and

loop error amplification of the classic Cartesian feedback system away from baseband,

to only one side of the carrier, so that both the quadrature mismatches and LO leak-

age would be outside the control bandwidth. This solution is accessible, thanks to

the use of polyphase amplifiers.

2.2.4 Towards a Modified Cartesian Feedback Architecture

While there is a strong theoretical motivation to pursue Cartesian feedback, its adop-

tion has been held back by the complexities associated with the actual implementation

of the system as explained in Section 2.2.3. Issues such as the impact of phase mis-

alignment on stability, phase and amplitude quadrature errors (particularly in the

down-converter of the feedback path), and DC offsets (particularly at the output of

the multipliers and at the input of the loop error amplifiers) have been and still are the

subject of many studies. The limit imposed by the accuracy of the down-conversion

is fundamental to linearization strategies, as errors in the feedback path cannot be


-500 0 500Frequency (KHz)

Freq

uenc

y R

espo

nse,

Mag

nitu

de (V

/V)

0.20

0.32

0.50

0.79

1.26

2

-90

-45

0

45

90

Frequency Response, Phase (degree)Freq. Response, Amplitude

Freq. Response, Phase

Figure 2.4: Complex bandpass amplifiers response. A reference signal, feedback sig-nal, and error amplification shifted at a complex IF frequency moves the controlbandwidth away from the frequencies where DC offset and quadrature mismatchesexist.

compensated by the loop operation and further complicate the analysis of the phase

alignment control problem. DC offsets also lessen the quality of the output baseband

spectrum.

While classic Cartesian feedback would require that the transmitter meet the very

tight specifications on amplitude and phase mismatches and DC offset, the solution

proposed in this dissertation consists of using a complex reference input signal cen-

tered at a low positive intermediate frequency (IF) band. The intended modulation

bandwidth also occupies only positive frequencies. The sample of the power amplifier

output signal is quadrature down-converted as feedback to this low IF band instead

of to DC. Hence, the loop error amplifiers perform the subtraction between the refer-

ence input and feedback signals at the IF instead of at DC. The classic (matched-pair)

difference amplifiers employed by Cartesian feedback cannot be used in this scenario

because their control bandwidth and peak gain are centered at DC. A matched pair of

bandpass differential amplifiers centered at the IF are also problematic. The bandpass

amplifiers would certainly prevent LO leakage by rejecting DC but would create two


control bands for complex signals centered at the positive and negative IF frequen-

cies. Quadrature image errors would remain. More importantly, because two high

gain control bands are generated, the system is potentially unstable as the desired sig-

nal and its quadrature mismatches experience different loop phase rotation and hence

demand different compensation strategies. The optimal solution to the problem of

subtracting the reference input and the feedback signal would be a complex bandpass

difference amplifier, which would create a single control bandwidth centered at the

positive IF only. In the realm of quadrature signals, a complex passband amplifier

does exist and can be synthesized with “active polyphase amplifiers.” The key mod-

ification to the classic Cartesian feedback control loop thus consists of substituting

active polyphase difference amplifiers for the classic matched difference amplifiers of

the Cartesian feedback system. The net result of this change is to move the loop con-

trol bandwidth away from DC (at baseband) and from the local oscillator frequency

(at RF), as shown in Figure 2.4, so that the undesired frequencies that would be

created by both quadrature errors and DC offsets are outside this bandwidth. With

this solution, even if quadrature errors and offsets within the loop are not stringently

minimized, they do not impair the performance of the Cartesian feedback system.

2.2.5 Adapting Cartesian Feedback to Application in MRI

In addition to using polyphase amplifiers in place of the classic baseband amplifiers

for the loop error amplification, developing a Cartesian feedback-controlled power

amplifier system for applications in MRI involves meeting the power, frequency, lin-

earity, and safety requirements of the latter, as well as creating solutions to address

issues that are not found in communications, where this technique was invented, and

that are specific to the MRI environment.

In particular, the power and frequency requirements of the system applied to

the MRI RF power amplifier are substantially different from those of the system

applied to the RF amplifier in communications. Moreover, the robustness of the

system is imperative for application in MRI, since the system controls the powerful

RF amplifiers driving the transmit coils.


In addition to meeting the above requirements, the implementation of the Carte-

sian feedback transmitter must include circuits to adapt the real signals employed

in MRI to the quadrature representation of both reference and feedback signals. For

example, the reference generation circuitry must convert the MRI signal into a pair of

(very well matched) reference quadrature signals. Similarly, the feedback path must

contain the circuitry necessary to generate the feedback quadrature signals from a

sample of the real signal at the output of the power amplifier.

2.3 Alternatives to Cartesian Feedback

The choice of Cartesian feedback for application to the field of MRI is warranted for a

number of reasons, which can be best appreciated when the technique is compared to

the available alternative methods of linearization of RF power amplifiers [40]. These

methods include (1) power back-off, (2) polar feedback, (3) feed-forward, and (4)

predistortion or iterative predistortion.

1. Power Back-Off. When high-linearity is needed but sophisticated linearization

techniques are not available, the simplest approach is to use the amplifier in

its most linear region, that is, to operate the amplifier at output powers much

lower than the maximum available. This approach is known as power back-off,

and the basic principle can be seen by looking at the characteristic equation

that describes the output envelope of any power amplifier as a function of the

input amplitude Vin and a series of coefficients:

Vout = a1Vin + a2V2in + a3V

3in + a4V

4in + . . . (2.8)

If Vin is small, the linear term with coefficient a1 is dominant. The main problem

with this approach is that the efficiency of an amplifier with power back-off will

be considerably lower than its maximum efficiency, which is obtained at full

power. Underutilizing the technology is one very expensive solution to the

problem at hand.


2. Polar Feedback. A polar feedback loop transmitter uses two feedback loops

from the output of the power amplifier, one for the amplitude and the other

for the phase of the transmitted signal. Amplitude-to-amplitude distortion is

corrected by the amplitude loop; amplitude-to-phase distortion, by the phase

feedback loop. The input signal to the power amplifier is the output of a Voltage

Controlled Oscillator (VCO), phase modulated and with constant amplitude.

The amplitude modulation is added at the output by varying the RF signal gain

of the amplifier. Because the polar-loop transmitter does not use up-conversion

in the RF chain, the need to suppress conversion images is obviated [80]. This

intrinsic advantage has drawbacks, however: the architecture needs a high-

quality dynamic VCO and a high-bandwidth envelope detector in the feedback

path, and it is not applicable to all modulation schemes.

3. Feed-Forward. A feed-forward amplifier consists of two loops: a signal cancel-

lation loop followed by an error cancellation loop. In the signal cancellation

loop a sample of the main amplifier output, attenuated by a factor matching

precisely the gain of the main amplifier, is subtracted from the properly delayed

version of the input. By accurately choosing the attenuation factor and input

delay, one ensures that the resulting signal contains only the distortion informa-

tion from the main amplifier; ideally, none of the input energy remains. In the

error-cancellation loop, this signal is linearly amplified by a secondary amplifier

and injected in anti-phase to a delayed version of the main amplifier output. At

the output of the feed-forward amplifier, only the ideally amplified version of

the input appears. The major advantage of a feed-forward amplifier is the lack

of a feedback loop, which means stability is not an issue: wideband operation

is guaranteed with significant linearity improvement. Drawbacks of this archi-

tecture are the need to accurately align the delays of the signal-cancellation

and error-cancellation loops, the sensitivity to gain and phase variations of the

secondary amplifier, and the generally poor efficiency.

4. Predistortion. A look-up table (LUT) that captures the inverse of the amplifier

nonlinear behavior precedes the amplifier itself, to ensure that the cascade of the


two is linear. Training of the LUT relies on feedback and adaptative algorithms,

based on accurate modeling of the amplifier. Because the system is based on

knowing and mathematically inverting the amplifier non-linearity, complexity

and cost are not trivial. The architecture is also limited by the inability to

account for changes in the amplifier behavior, such as thermal drift, unless the

LUT is updated often (Iterative Predistortion). Incidentally, Cartesian feedback

has been proposed for LUT training of digital predistortion systems [16].

In comparison to all of the above techniques, a frequency-offset Cartesian feedback

has a number of advantages that make it ideal for application to MRI:

1. No need for a dynamic, fast VCO/PLL to track the rapid phase changes which

can occur in some high-bandwidth RF envelopes found in MRI. In addition,

the PLL arrangement (such as in a polar loop) can have problems tracking or

locking at low-envelope levels, such as those occurring when the MRI signal

envelope passes through zero. A sinc pulse is a classic example.

2. The modulation signal is reduced to a simple mixer. The need for a separate

modulator at the final output RF is eliminated.

3. Simplicity of implementation. (In particular with the use of polyphase ampli-

fiers, which substantially relax the specifications of the transmitter compared

to the classic architecture.)

4. Applicable to any type of RF pulse envelope and any type of envelope modula-

tion.

5. A flexible approach to the choice of signal to be sampled at the output of the

RF power amplifier. We will see that, for example, sampling of the output

voltage, output current, or combination of the two, can be obtained without

substantially altering the hardware configuration.

Instead of linearizing efficient amplifiers, an alternative view is to boost the effi-

ciency of conventional linear amplifiers. A number of these efficiency-boosting power


amplification techniques exist and are often described as performing amplifier lin-

earization, when in fact they simply overcome the efficiency obstacle to the utiliza-

tion of already linear amplifiers. Examples of these techniques are (1) Doherty archi-

tecture, (2) envelope elimination and restoration, and (3) linear amplification with

non-linear components.

1. A very popular example is the Doherty technique. The basic idea behind the

Doherty technique is to allow one amplifier to operate at its peak envelope

power level, where its efficiency is maximum but its capability to deal with the

modulation peaks is very poor, while a second linear amplifier works in parallel

to faithfully reproduce the modulation peaks. When the input power is low, the

first amplifier acts alone and the second is shut off by drive control circuitry.

Other methods track the signal envelope level, to vary the standing DC bias on

a low-efficiency, high-linearity amplifier as a way to compensate for the loss of

efficiency caused by the use of back-off (Adaptative Bias), or manipulate the

reference signal to reduce its peak to r.m.s. amplitude (Crest Factor Reduction).

2. Envelope Elimination and Restoration (EER). EER involves signal processing

to radically alter the original signal. It is essentially a high-level modulation

technique, and as such may be implemented as either a complete linear trans-

mitter or as an RF amplifier. In this technique, the input signal, which may

contain both amplitude and phase modulation, is split to form a baseband

path containing the envelope of the input signal and an RF path containing a

constant-envelope modulated carrier signal. The latter is then amplified by a

high-efficiency RF amplifier, which will transmit the phase modulation infor-

mation to the output of the system. The baseband envelope signal is amplified

by an audio amplifier which modulates the power supply of the RF power stage.

If the delays of the two paths are properly matched, this modulation process

restores the signal envelope and results in a high-power replica of the input

being produced at the output. In EER, phase matching between the two sig-

nal paths is critical, and restoring the envelope in a power-efficient way is very

challenging.


3. Linear Amplification with Non-linear Components. The input signal is split

into two constant-envelope, phase-modulated signals by the signal separation

or generation process, and each is fed to its own non-linear RF power amplifier.

The two amplifiers increase the power of each signal by an identical amount. The

two signals are then summed in a quadrature combiner resulting in an amplified

version of the original signal that has, ideally, no added distortion. The key

barrier to the acceptance of this technique, also called “outphasing amplifier”

is obtaining good power combining with both low loss and high isolation.

Since they are not, strictly speaking, linearization techniques, these efficiency-

boosting techniques are not alternatives to the use of frequency-offset Cartesian feed-

back; they may, however, be used in conjunction with the latter, as a way to further

increase the efficiency of the Cartesian feedback-linearized RF power amplifier.

2.4 Summary

In Chapter 2, the architecture of classic Cartesian feedback has been introduced.

Alternatives to the use of this RF power amplifier linearization technique have also

been presented, and the choice of this particular method for application to MRI has

been motivated. As a solution to the problem of the high sensitivity of this method to

quadrature mismatches and DC offsets, which can create undesired frequencies at the

output of the power amplifier, the use of low-IF reference signals, feedback signals,

and loop error amplification bandwidth is proposed. In particular, a complex low-IF

control bandwidth can be obtained with the use of polyphase amplifiers. Chapter 3

is entirely dedicated to this particular and little-known class of amplifiers.

Chapter 3

Active Polyphase Amplifiers

3.1 Introduction

The successful implementation of a frequency-offset Cartesian feedback system is

based on the availability of complex-passband gain to selectively amplify the fre-

quencies of the feedback signal while rejecting the carrier and image frequencies. An

example of an electrical circuit with this characteristic gain is an active polyphase

amplifier. After a brief historical introduction and description of the requirements

of polyphase amplifiers for Cartesian feedback applications in MRI, this chapter in-

troduces the architecture and qualitative behavior of active polyphase amplifiers. A

mathematical analysis of the polyphase amplifier operation is then presented, with

the goal of demonstrating that satisfactory performance (in terms of rejection of un-

desired frequencies, bandpass center frequency, and bandwidth) can be achieved for

application of these circuits as the loop error amplifiers of frequency-offset Cartesian

feedback in MRI. The results of this analysis support those obtained in a series of

experiments, also described here.

3.2 Brief History

Active polyphase amplifiers, also known as active polyphase filters, are a recent in-

vention by Christopher Marshall of U.S. Philips Corporation [55]. In his U.S. patent

29

CHAPTER 3. ACTIVE POLYPHASE AMPLIFIERS 30

number 4,723,318 published in 1988, Marshall provides the first available description

of these circuits:

An active polyphase filter [is an] arrangement in which asymmetric

poles and zeros are obtained using feedback.

Active polyphase amplifiers did not receive any significant interest until the mid-

1990’s, when they were first used in the design of low-intermediate frequency (IF)

receivers in communications by Crols [18, 19], Chou [14], Linggajaya [53], and Not-

ten [60]. Used in an image-reject down-converter, the active polyphase filter imple-

mentation eliminated the needs or relaxed the requirements for the traditionally used

low-pass filter after quadrature down-conversion while providing gain.

In this dissertation, for the first time, active polyphase amplifiers support the key

modification to classic Cartesian feedback architecture that significantly relaxes the

specifications of the transmitter in terms of both quadrature mismatches and DC

offset.

3.3 Requirements

In the modified Cartesian feedback system, the complex reference input signal and

down-converted feedback signal are at a low positive IF band. The intended modula-

tion bandwidth also occupies only positive frequencies. Polyphase amplifiers replace

the classic amplifiers for the loop error amplification, in which amplification and sub-

traction between the reference input signals and feedback signals occur at this IF

instead of at DC. Contrary to a pair of bandpass amplifiers, they do so while creating

a single complex bandpass control bandwidth instead of two separate bandwidths at

both positive and negative frequencies (see Figure 3.1). The net result is to selectively

amplify the desired signals over quadrature mismatches and DC offsets, which relaxes

the specifications of the transmitter substantially.

The main characteristics of the complex bandpass control band are its bandwidth

and center frequency. The requirements regarding the latter depend on the partic-

ular application of the frequency-offset Cartesian feedback system; specifically, they


wanted signalmirror signal

LO

control BW

DC

DC

DC

RF

RF-RF

DC IF

DC RF-RF

-RF

IF

DC IFIF

LO leakage

CL

ASS

IC C

TR

L B

WPA

SSB

AN

D C

TR

L B

WC

OM

PLE

X C

TR

L B

W

mirror LO

Figure 3.1: Control bandwidth options. In a classic pair of lowpass amplifiers, am-plification and subtraction between the reference input signals and feedback signalsoccur at DC (top). A pair of bandpass amplifier creates two separate bandwidths atboth positive and negative frequencies (middle). Polyphase amplifiers, instead, cre-ate a single complex bandpass control bandwidth (bottom); hence, they selectivelyamplify the desired signals over quadrature mismatches and DC offsets.

depend on the necessary signal modulation bandwidth. In general,

• the control bandwidth must be wide enough to accommodate the desired mod-

ulation bandwidth, hence,

• the center frequency of the polyphase passband must be at least half the de-

sired modulation bandwidth, so that the entire control band exists only in the

positive-frequency band (or, conversely, only in the negative-frequency band) of

the spectrum.

In the research presented in this dissertation, the desired application for the

frequency-offset Cartesian feedback is MRI; hence, the desired signal modulation


bandwidth is that of the MRI RF pulse envelope. Typically, the latter is rarely above

20 kHz, though some very demanding MR strategies may require up to 100 kHz mod-

ulation. As a result, a polyphase amplification bandwidth at a center frequency of at

least 50 kHz must be demonstrated.

While perfect complex gain amplifiers would have zero overall response to the fre-

quency opposite (mirror) the desired frequency, in reality, even the ideal architecture

has a gain with finite roll-off with frequency, as will be shown below; therefore, it has

non-zero amplification at both DC and the mirror frequency of the desired signals.

For this reason, it is desirable to move the signal IF frequency band and thus the

center frequency of the polyphase amplifiers farther away from DC, so that the gain

will be significantly lower at both DC and the mirror frequency band than the gain

at the desired frequency band. In MRI, this consideration translates into a center

frequency ideally a decade higher than the minimum desired 50 kHz, that is, a 500

kHz or even higher center frequency should be demonstrated.

Besides the finite roll-off of the gain with frequency, other factors may cause the

polyphase amplifiers to respond to DC and opposite frequency. Imperfections of the

polyphase architecture cause both positive and negative frequencies to be amplified

as well as components of opposite frequency to be originated at the output. Whatever

the particular application may be, this consideration dictates an additional require-

ment for the practical polyphase amplifiers used in Cartesian feedback systems:

• the amplification of the desired positive frequencies (aka, the reference and

feedback signal) must be much higher than the amplification of the opposite

negative (image) frequencies (aka, the quadrature mismatches) created by the

loop imperfections.

The worst case scenario is that the quadrature mismatch image signal is as high

as the desired signal itself. In this catastrophic scenario, and for the specific case

of the application of the feedback system in MRI, a 40 dB differential amplification

would be considered adequate and values of 60 dB and over would be ideal.

To demonstrate that these performance requirements can be met in practical

implementations of polyphase amplifiers, a mathematical analysis of the polyphase


Vcm

Vcm

ir

Q

I

if

qr

qf

Ri RF

C

RC

+

−

+

−0

Frequency (kHz)

Freq

uenc

y R

espo

nse,

Mag

nitu

de (a

.u.)

0.7·K

K

-45

0

45

Frequency Response, Phase (degree)

Freq. Response, AmplitudeFreq. Response, Phase

ωc

ωc = RCC 1 ωo = RFC

1

ωo

K = Ri

RF

Figure 3.2: Simplified schematic (left) and frequency response (right) of the fully-differential polyphase difference amplifiers that have been used as the loop erroramplifiers of the frequency-offset Cartesian feedback system described in this disser-tation.

amplifier behavior is presented in the next section.

3.4 Theory

Figure 3.2 shows a simplified schematic of the (fully-differential) polyphase difference

amplifiers that have been used as the loop error amplifiers of the frequency-offset

Cartesian feedback system that will be described in Chapter 4. In this application, the

polyphase amplifiers take the difference between the quadrature differential reference

signals (ir, qr) and the quadrature differential feedback signals (if , qf ). As in any

classic amplifier architecture, the transfer function H(ω) can be used to describe the

relationship between the complex output signal I + jQ and the complex input signal

i+ jq, where i = ir − if and q = qr − qf , of the polyphase architecture:

H(ω) =

(K

1 + j(ω+ωc

ωo)

)(3.1)


where K, ωc, and ωo are the peak gain, center frequency, and half width of the

baseband signal band, respectively. H(ω) is the response of a single pole low pass

filter shifted by ωc away from 0 frequency as shown in Figure 3.2. It is as described

by 3.1 only if its active elements (such as the fully differential operational amplifiers

in the discrete implementation) are ideal blocks of infinite gain and bandwidth, its

passive components are perfectly matched, and the i and q input signals have the

same amplitude and are in perfect quadrature.

Qualitatively, the polyphase amplifier acts as two asymmetrically cross-coupled

amplifiers. The coupling from Q− i is the opposite sign of I − q. A quadrature ±90

degree phase relationship representing positive or negative input frequencies leads to

constructive or destructive interference in the outputs and to enhanced selectivity of

positive frequencies.

An equivalent representation of the operation of polyphase amplifiers can be ob-

tained if one considers the four real input (i, q) to real output (I, Q) transfer functions

and obtains the overall complex response by appropriately combining these functions.

For a unity-gain polyphase difference amplifier, these four equations are

i2I =I

i=

(ωojω− ω2

o

ω2

1

D

)(3.2)

q2I =I

q=

(+ωoωcω2

1

D

)(3.3)

q2Q =Q

q=

(ωojω− ω2

o

ω2

1

D

)(3.4)

i2Q =Q

i=

(−ωoωc

ω2

1

D

)(3.5)

where

D = 1− ω2o + ω2

c

ω2+

2ωojω

. (3.6)

This approach to polyphase amplifier analysis allows us to derive the equations that

separate the desired and undesired components of the output response, namely Up,

Vp, Um, Vm.


Up is the transfer function of the positive-frequency output to a positive-frequency

input complex signal,

Up = (i2I − jq2I) + j (i2Q− jq2Q) , (3.7)

Vp is the transfer function of the negative-frequency output to a positive-frequency

input (aka the “mirror” of the desired signal),

Vp = [(i2I − jq2I)∗ + j (i2Q− jq2Q)∗]∗, (3.8)

Vm is the transfer function of the negative-frequency output to a negative-frequency

input,

Vm = (i2I + jq2I)∗ + j (i2Q+ jq2Q)∗ , (3.9)

and finally Um is the transfer function of the positive-frequency output to a negative-

frequency input signal

Um = [(i2I + jq2I) + j (i2Q+ jq2Q)]∗ . (3.10)

When the ideal polyphase amplifier described by 3.1 is used as the difference amplifier

of a Cartesian feedback control system, the control bandwidth is frequency offset to

the positive axis; moreover in this ideal case, i2I = q2Q, and q2I = −i2Q such that

Vp = 0 and Um = 0.

Given a positive reference input frequency and an error-free down-mixer in the

control loop, the feedback input will always be composed of only positive frequencies,

and the only component of interest in the overall amplifier response is Up. However,

if the down-mixer (and other circuitry in the feedback path) introduces quadrature

mismatches and DC offsets, negative (mirror) frequencies and DC components will

also be generated at the feedback input of the polyphase difference amplifiers. In the

ideal architecture, where Vp = 0 and Um = 0, the amplification of these unwanted

frequencies originates from the spontaneous roll-off only of 3.1 in the negative fre-

quency band, described by Vm. To reduce this amplification (without affecting K),

the solution is simple and consists of increasing the center frequency ωc.


In the practical architecture, as will be shown later, Vp and Um are not null. In

the presence of quadrature mismatches and DC offsets created by the down-mixer,

the analysis of all four components Up, Vp, Um, and Vm becomes important, and the

ability of the polyphase amplifiers to reject the undesired mirror-frequency inputs

becomes a figure of merit.

Effects of Component Mismatching

Ideal polyphase architectures are obtained when their active elements (such as the

fully differential operational amplifiers in the discrete implementation) are ideal blocks

of infinite gain and bandwidth and their passive components are perfectly matched.

In this case, these amplifiers have zero overall mirror response (Vp = 0 and Um =

0). In reality, imperfections of practical polyphase architectures are inevitable and

cause both positive and negative frequencies to be amplified as well as components

of opposite frequency to be generated at the output.

In a discrete implementation, which is particularly appealing in the context of

our application of frequency-offset Cartesian feedback to MRI power amplifiers, the

major deviation from the ideal case is the mismatch of the capacitors. Indeed, 0.1%

surface mount technology (SMT) resistors are readily available (Panasonic ECG, ERA

series), while only 1% SMT capacitors are available (AVX Corporation, C0G/NP0

ceramics).

Building on the contribution by [19], this dissertation presents a novel analysis of

the effects of capacitor mismatching on the transfer functions of the ideal polyphase

architecture. Let us consider a mismatch dC that affects the capacitors C of one fully

differential amplifier relative to those of the other. In this case, the mismatch of the

half-bandwidth ωo between the two channels is

dωoωo

=

(1 +

dC

C

)−1

− 1 (3.11)

and the transfer functions of the mismatched architecture become

i2Iε =I

i=

(ωo + dωo

jω− ω2

o − dω2o

ω2

1

Dε

)(3.12)


q2Iε =I

q= +

(ωoωcω2− ωcdω

2o

ωoω2

)1

Dε

(3.13)

q2Qε =Q

q=

(ωo − dωo

jω− ω2

o − dω2o

ω2

1

Dε

)(3.14)

i2Qε =Q

i= −

(ωoωcω2− ωcdω

2o

ωoω2

)1

Dε

(3.15)

where

Dε = 1−ω2o + ω2

c − dω2o

(1 + ω2

c

ω2o

)ω2

+2ωojω

. (3.16)

It can be shown that, to first order approximation in case dC/C 1, the errors ε are

ε[i2I] = i2Iε − i2I ≈ +dωojω

1

D(3.17)

ε[q2I] = q2Iε − q2I ≈ 0 (3.18)

ε[q2Q] = q2Qε − q2Q ≈ −dωojω

1

D(3.19)

ε[i2Q] = i2Qε − i2Q ≈ 0 (3.20)

and thus the first-order errors to the desired and undesired baseband components of

the output response are

ε[Up] ≈ 0 (3.21)

ε[Vp] ≈(−2

dωojω

1

D

)∗(3.22)

ε[Vm] ≈ 0 (3.23)

ε[Um] ≈ −2dωojω

1

D(3.24)

which tells us that the polyphase amplifier with a small capacitive mismatch has

a desired response very close to that of the perfect amplifier. However, its mirror

response is not null anymore; moreover, since ε[Vp] is the complex conjugate of ε[Um],

the mirror response has even amplitude and odd phase.

This conclusion is validated by both simulations and experiments. As an example,


Frequency (KHz)

Freq

uenc

y R

espo

nse,

Mag

nitu

de (V

/V)

4002000-200-400

10

50Red : A=100M, p=10HzGreen : A=10M, p=10HzBlue : A=1M, p=10HzBrown : A=0.1M , p=10Hz

Freq

uenc

y R

espo

nse,

Mag

nitu

de (V

/V)

Frequency (KHz)

Red: A=100M, p=10HzGreen: A=100M, p=1HzBlue : A=100M, p=0.1HzBrown: A=100M , p=0.01Hz

4002000-200-400

10

50

Figure 3.3: Actual polyphase amplifier transfer function with varying operationalamplifier gain (left, pole frequency is constant) and pole frequency (right, gain isconstant). If the gain-bandwidth product of the operational amplifier is the same, theeffects of these non-idealities on the desired polyphase transfer function are virtuallyindistinguishable.

the simulated and measured response of the polyphase amplifier with a capacitive

mismatch of 0.85% is presented later.

Effects of Limited Gain-Bandwidth Product

As previously discussed, in some applications of the frequency-offset Cartesian method

it may be desirable to move the mirror frequency band away from the LO frequency by

an amount equal to at least half the control bandwidth (so that the mirror frequency

band is outside the MRI transmitter bandwidth), and ideally as high as possible in

order to increase the separation between the desired frequency band and the mirror

frequency band.

There are, however, limits to the maximum value of the low-IF passband center

frequency that can be obtained. One of these limits is the finite gain-bandwidth

product of the amplifiers implementing the loop driver polyphase amplifiers.

To derive the effective transfer function He(ω) of the polyphase architecture em-

ploying fully-differential (FD) amplifiers with finite gain-bandwidth product, GBP,

(and with perfectly matched components), each FD amplifier can be characterized


with analog behavior of finite DC gain, A, and finite bandwidth defined by the fre-

quency of a single pole, ωp; the transfer function of this behavior is shown in 3.25.

The GBP is, by definition, given by the product of A and ωp.

G(ω) =A

1 + j ωωp

. (3.25)

The next step is to solve the linear equation for the outputs Q, I as a function of

the inputs q, i. The following two new quantities can be defined in the process, the

magnitude factor M(ω) and shift factor S:

M(ω) = 1 +1 +K

A+ωcωp − ω2

Aωoωp= 1 +

(1 +K)A

GBP+ωcωp − ω2

GBPωo(3.26)

S(ω) = 1 +(1 +K)ωo + ωp + ωc

Aωp= 1 +

(1 +K)ωo + ωp + ωcGBP

. (3.27)

The effective transfer function He(ω) of the polyphase architecture employing FD

amplifier with finite GBP is

He(ω) =K

M(ω)

1

1 + jω−ωc

S

ωoM(ω)

S

(3.28)

in which ωec = ωc

Sis the effective passband center frequency, ωoc = ωo

M(ω)S

is the

effective half passband width, and Ke = KM(ω)

is the effective peak gain.

Figure 3.3 (left) shows the amplitude of the effective transfer function of a poly-

phase architecture of desired passband center frequency 200 kHz, desired half pass-

band width 80 kHz, and desired peak gain 50, with varying DC gain of the FD

amplifiers (A varies from 108 to 105): the finite bandwidth of the FD amplifiers was

held fixed at 10 Hz (ωp = 204π rads

). He(ω) in the case of A = 108 approximates the

ideal transfer function very well.

Similarly, Figure 3.3 (right) shows the amplitude of the effective transfer function

of the same polyphase architecture as a function, this time, of the pole frequency of

the FD amplifiers (from 10 to 0.01 Hz) while A is fixed and equal to 108.

The comparison of the two figures illustrates one of the key findings of this analysis:


Vcm

Vcm

i

Q

Ii

q

q

Ri RF

C

RC

Figure 3.4: Picture and simplified schematic of PCB for testing of polyphase ampli-fiers. The polyphase amplifiers can be tested with either two fully-differential inputsignals or two positive input signals (the negative inputs being AC-grounded).

given the desired transfer function and the chosen FD amplifiers, and considering

that ωp is typically small (less than 500 Hz) compared to ωo and ωc, then the effective

transfer function is almost entirely determined by the product GBP, not by A and ωp

independently.

3.5 Experiments

To validate the mathematical model described above and to demonstrate that discrete

polyphase amplifiers can have the performance required to be used as components of

Cartesian feedback systems, a printed circuit board (PCB) polyphase amplifier was

built that could be driven in one of three different ways:

• with four single-ended independent inputs

• with two fully-differential independent inputs

• with two independent (positive) inputs, the other two (negative) inputs being

AC grounded.


Figure 3.5: Normalized real input (i, q) to real output (I, Q) simulated transferfunctions. The latter can be also measured by driving the PCB polyphase amplifierswith only one non-zero quadrature input signal (i or q) at a time.

A picture and a simplified schematic of this board is shown in Figure 3.4. The

board allowed bench testing of polyphase amplifiers built using discrete passive com-

ponents of known tolerance and discrete fully-differential amplifiers of known (nomi-

nal) GBP.

3.5.1 Results

The experiments described below were obtained by driving the polyphase amplifiers

with four single-ended independent inputs (i+, i−, q+, q−) and measuring the four

single-ended outputs elicited by each input (I+, I−, Q+, Q−). Appropriately com-

bining these 16 measurements allowed one to obtain the transfer functions i2I, i2Q,

q2I and q2Q, the output components Up, Um, Vp, and Vm, and their combination into

the overall desired and undesired responses.

The response of the discrete polyphase amplifiers was measured on the bench for

different values of capacitive matching as well as at different values of the center

frequency of the polyphase passband, and compared with the results of simulations

obtained using the mathematical model described in the previous sections.

In a first experiment, the components of the discrete polyphase circuit were sorted

in order to obtain 0.85% mismatch dCC

of the capacitors, peak gain of 40 dB, and

center frequency of nearly 200 kHz. Initially, the LT1994 fully differential discrete

operational amplifier by Linear Technology (with GBP of 70 MHz) was selected to


0

2

4

6

8x 10-3

0Fr

eque

ncy

Res

pons

e, M

agni

tude

(V/V

)

0 100 200 300 400 5000.2

0.6

1

1.4

1.8

Frequency (kHz)

x 10

Simulated VpSimulated Um

Simulated UpSimulated Vm

Figure 3.6: Up, Um, Vp, Vm simulated transfer functions obtained by combining thefunctions in Figure 3.5.

build polyphase amplifiers with the desired amplitude and shape of the polyphase

passband. The passive resistors had 0.01% matching, obtained after measuring and

cherry picking from a batch of 0.1% resistors. The simulated transfer functions and

output response were calculated using the mathematical model with a 0.85% mis-

match dCC

between capacitors and the same values of peak gain and center frequency.

Then, the normalized results of these simulations were compared with the normalized

results of the experiments.

Figure 3.5 shows the simulated magnitude transfer functions i2Q, i2I, q2I, q2Q

while Figure 3.6 shows the simulated desired response from Up and Vm (bottom) and

the simulated undesired response from Vp and Um (top). In these plots, the negative

frequencies are folded in with the positive frequencies.

From Up, Vm, and Vp, Um, the complete amplitude and phase for the desired and

mirror responses can be obtained as shown in Figure 3.7 over negative and positive

frequencies. Figure 3.8 show the experimental amplitude and phase response, created

by the same construction procedure. The measured functions i2Q, i2I, q2I, q2Q

and the measured responses Up, Vm, Vp, Um, that were used to obtain Figure 3.8


10-3

10-2

10-1

100

Mag

nitu

de (V

/V)

-90

0

90

-500 0 500Frequency (kHz)

Phas

e (d

egre

e)

-180

180

Desired Frequency Response

Mirror Frequency Response

Figure 3.7: Overall simulated transfer functions of the polyphase amplifiers obtainedby combining the functions in Figure 3.6. The desired frequency response is obtainedby merging Up with the mirrored Vm. The mirror frequency response is obtained bymerging Vp with the mirrored Um.

are not shown, as they were virtually indistinguishable from those of their respective

simulations.

One important measure of non-ideality of the polyphase amplifiers is the mirror

(sideband) rejection, defined as the ratio between the peak gain at center frequency

ωc of the desired frequency response (Up, Vm) and the gain at the same frequency

of the mirror frequency response (Um, Vp). Both model and measurement show that

a 0.85% capacitor matching provides about 47 dB mirror rejection, which meets the

desired minimum rejection requirement of 40 dB. Moreover, this performance has

been met with a polyphase amplification centered at 200 kHz, four times higher than

the minimum requirement of 50 kHz.

In an effort to increase both the rejection of the mirror frequency band and the

center frequency of the polyphase passband, experiments were conducted with capac-

itor matching between 0.05% and 0.1% and center frequencies up to 1.65 MHz. At


10-3

10-2

10-1

100

Mag

nitu

de (V

/V)

-500 0 500Frequency (kHz)

Phas

e (d

egre

e)

0

90

180

-90

-180

Desired Frequency Response

Mirror Frequency Response

Figure 3.8: Experimental transfer functions of the polyphase amplifiers. These func-tions were constructed from the normalized real input (i, q) to real output (I, Q)transfer functions measured by driving the amplifiers with only one non-zero quadra-ture input signal (i or q) at a time.

center offset frequencies over 650 kHz, the LT1994 (GBP 70 MHz) by Linear Tech-

nologies was substituted with the THS4141 (GBP 200 MHz) by Texas Instruments,

to faithfully reproduce the desired amplitude and shape of the polyphase passband.

The peak gain was again 100, the resistor matching 0.01%. All of these experiments

validate the mathematical model. The results of some of these experiments are sum-

marized in Table I, and compared to the results of the simulations.

These simulations and experiments demonstrate that it is possible to build discrete

polyphase amplifiers that meet and exceed not only the minimum requirements but

also the desired requirements (≥ 500 kHz center frequency, ≥ 60 dB mirror rejection)

for their use in Cartesian feedback systems applied to linearization and control of RF

power amplifiers in MRI. The next step is to use these amplifiers as the loop error

amplifiers of a complete frequency-offset Cartesian feedback system.


Table 3.1: Simulated and Measured Sideband Rejection at Different Center Frequen-cies of the Polyphase Passband

C Mismatch Center Frequency Simulated SR Measured SR% kHz dB dB

0.065 158 69.76 68.520.050 640 72.04 74.430.100 1490 66.02 65.19

3.6 Summary

In this chapter, a mathematical formulation of the behavior of the polyphase ampli-

fiers has been presented, which includes the effect of the mismatches that are present

in any practical polyphase circuit. The insights provided by this analysis support

the experimental results obtained with a discrete implementation of polyphase am-

plifiers. In particular, both simulations and experiments support the conclusion that

these circuits can be built to meet and exceed the requirements for their use as the

loop error amplifiers of the modified Cartesian feedback system proposed in this dis-

sertation. In fact, they offer very high rejection (in excess of 60 dB) of the undesired

frequencies that can be created by mismatches and DC offsets in the feedback path

of the Cartesian transmitter, and they do so at a center frequency (higher than 500

kHz) able to accommodate the entire MRI bandwidth of signal modulation in only

the positive-frequency part of the spectrum.

In the next chapter, the high-level overview and details of the frequency-offset

Cartesian feedback system including polyphase amplifiers for the loop error amplifi-

cation are discussed.

Chapter 4

System Design

4.1 Introduction

To improve the fidelity of the MRI RF transmit path and to electronically manipulate

the output impedance of the amplifiers driving transmit array coils, a frequency-offset

Cartesian feedback system has been designed and built.

This chapter discusses the design of the system. The motivations and high-level

objectives of the work, introduced in Chapter 1 and 2, are first revisited. Then, the

high-level system architecture and all its various parts are described in detail. Finally,

the system operation and linearization performance are analyzed and the conditions

necessary for the stability of the loop are discussed.

4.2 Motivations, Requirements and Objectives

As discussed in Chapter 1 and Chapter 2, a frequency-offset Cartesian feedback sys-

tem promises to address the challenges created by the recent trends in MRI, including

the trend toward higher frequencies and bandwidth, toward increasing use of trans-

mitter array systems, and toward increasing use of interventional devices. The main

challenges are distortion of the RF pulse envelope created by the MRI power amplifier,

interactions between the coils of transmitter array systems, and induced RF currents

in interventional devices, respectively. To address these challenges, frequency-offset

46

CHAPTER 4. SYSTEM DESIGN 47

Cartesian feedback methods must improve the fidelity of RF reproduction, manipulate

the output impedance of the amplifiers driving the transmit coils, and significantly

attenuate the current induced in interventional devices.

The first two goals (improved RF fidelity of reproduction and impedance ma-

nipulation) call for a very similar implementation of the frequency-offset Cartesian

feedback system: in both cases, the Cartesian feedback transmitter drives the input

of the MRI RF power amplifiers, receives the MRI signal at its reference input, and

receives a sample of the power amplifier output at its feedback input. This chapter

describes the system implementation; the experimental results will be presented in

the next two chapters. The third goal (to attenuate current in devices) requires a

dedicated implementation and will be discussed in Appendix A.

To improve the RF fidelity of reproduction and to manipulate the power amplifier

impedance, the system must automatically control a particular combination of the

RF output voltage and current generated by the RF power amplifier. In the first

case, the load to the amplifier can be a tuned RF transmit coil or, more generally,

the tuned RF body coil of a traditional MRI system; in the second case, the load is

always a tuned RF transmit coil. In any case, the power and frequency requirements

that the frequency-offset Cartesian feedback system must satisfy are similar and have

been in part introduced in Chapter 2. In particular,

• the feedback system must demonstrate control of power amplifiers with output

power of 50 dBm (100 W) or higher.

• the feedback system should work with a stable control bandwidth large enough

to accommodate the modulation bandwidth of most, if not all, MRI signals.

Though the great majority of MRI pulses are signals with modulation bandwidths

of less than 20 kHz, very demanding applications may require up to 100 kHz modu-

lation. For this reason, the goal is to demonstrate at least 50 kHz control bandwidth,

with 100 kHz considered ideal.

Given the available test equipment at the Magnetic Resonance Systems Research

laboratory (MRSRL), the frequency-offset Cartesian feedback system has been de-

signed to demonstrate reduced RF distortion and the ability to manipulate amplifier


output impedance with various RF power amplifiers for 1.5 T MRI. However, the

system should work in any MRI system or Nuclear Magnetic Resonance (NMR) spec-

troscopy system of field strength from 1.5 T (64 MHz RF) up to 23 T (1 GHz RF)

with only minor modifications. This requirement on the Larmor frequency of the

MRI system translates into a technical specification for the operating frequency of

the control architecture:

• the feedback system must control any RF power amplifier between 64 MHz

(Larmor frequency at 1.5 T field strength) and 300 MHz (at 7 T), with only

minor modifications to the system architecture.

Finally, it is important that the phase shift that compensates the misalignment

between the up- and down-converted signals in the loop (see Chapter 2) can be studied

accurately and implemented rapidly before each use of the system. In fact, the correct

phase shift value depends on the behavior of the power amplifier and on the load,

and can differ greatly with the particular imaging system and imaging methodology.

Not only different amplifiers and loads could be used, but even given the same power

amplifier and load, the characteristic of the object being imaged (the patient) can

change the loaded Q and, consequently, the conditions for stability. In summary,

• the feedback system must support methods for the automatic measurement of

the phase loop rotation and automatic calibration of the phase shift necessary

for the system stability.

More specific objectives that motivated both the actual implementation choices

and the approach to its theoretical analysis have been:

1. to demonstrate a discrete prototype of a frequency-offset Cartesian feedback

power amplifier control system based on polyphase loop error amplifiers;

2. to demonstrate that both DC-coupling and AC-coupling of the control loop are

possible, and explore the trade-offs involved in choosing one approach versus

the other;


3. to demonstrate the behavior of the control loop with different types of loads at

the power amplifier output, such as a 50 Ω dummy load, a loaded or unloaded

transmit coil element, and a bird cage body coil;

4. to demonstrate the behavior of the control loop with different gains for the

forward and feedback path;

5. to demonstrate the frequency range of stable operation at varying values of the

phase shift between the up- and down-conversion mixers;

6. to be able to change gain and phase characteristics of the loop using a PC

interface;

7. to demonstrate that frequency-offset Cartesian feedback works with any choice

of feedback signal obtained by sampling an arbitrary combination of the output

voltage and current of the power amplifier;

8. to demonstrate that the frequency-offset Cartesian feedback works with a feed-

back signal obtained by sampling the current in a transmit coil directly.

In the next sections, the frequency-offset Cartesian feedback architecture will be

described, with emphasis on the choices that enabled the above requirements and

objectives to be met.

4.3 High-Level System Preview

Figure 4.1 shows the basic hardware diagram of the frequency-offset Cartesian feed-

back system. In addition to the RF power amplifier, the basic blocks of the system

are:

1. a printed-circuit board (PCB) frequency-offset Cartesian feedback transmitter,

which will be referred to as “Genie”;

2. a load to the RF power amplifier, which can be a 50 Ω dummy load, a tuned

transmit coil, or an RF body coil;


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3. RF probes (couplers) that sample the output voltage or output current from the

power amplifier. An alternative approach is to sample the forward voltage and

reflected voltage of the transmission line between the power amplifier and the

load, from which the output voltage and output current can be easily derived;

4. devices that combine the samples from the RF probes to obtain the desired

feedback variable;

5. RF switches to toggle the system between the open-loop and closed-loop con-

figuration;

6. a PC software interface;

7. a Medusa console, developed by Pascal Stang at MRSRL [73], which offers

network-analyzer like capabilities to transmit the reference signal, gate the RF

switches, and perform analysis of the RF feedback variable.

Genie, the frequency-offset Cartesian feedback transmitter, is the core of the con-

trol system and is described in Section 4.4. The remaining components will be dis-

cussed in Section 4.5.

4.4 The Transmitter: Genie

Genie is a second generation prototype PCB frequency-offset Cartesian feedback

transmitter that consists of several functional blocks. The main blocks are an image-

reject down-converter, the circuitry for DC-level management, a polyphase amplifier

loop filter, up/down quadrature mixers, and phase shift control circuitry. Auxiliary

blocks include circuitry for the local oscillator frequency generation, power supply

management, and more. A picture of Genie is shown in Figure 4.2.

The image-reject down-converter, driven by an external RF source, is an optional

module that provides the quadrature reference signals to the Cartesian loop. (Alter-

natively, a direct digital synthesizer (DDS) can generate a direct low-IF quadrature

reference.)


PhaseShifter

Figure 4.2: Top: Printed Circuit Board (PCB) of the frequency-offset Cartesiantransmitter, Genie. Bottom: Simplified block diagram showing the position on theboard and relationship between the reference generation circuitry, polyphase ampli-fiers, CMX998, and local oscillator in Genie.

The DC setting circuitry allows matching of the DC levels of the quadrature ref-

erence signals to the DC levels of the feedback reference signal. While the classic

Cartesian feedback transmitter would require very complex and sophisticated cir-

cuitry to minimize DC offsets and thus the presence of the undesired LO-leakage

frequency, frequency-offset Cartesian feedback makes use of a very simple alternative.

The polyphase amplifier loop filter is constructed with discrete devices: they re-

ceive the reference signals (from the reference demodulator) and feedback signals

(from the feedback down-mixer) at their input, amplify the difference between the


two (i.e., the error signal), and provide this amplified signal to the input of the up-

mixer.

The mixers and phase shift control circuitry are provided in the commercially-

available classic Cartesian feedback transmitter model CMX998 by CML Microcir-

cuits. The CMX998 is an integrated solution for classic Cartesian feedback loop based

linear transmitters, which offers the possibility to interface the phase shift control cir-

cuitry, as well as to vary the up- and down-converter gain, via a user-friendly software

interface.

The polyphase amplifiers in Genie were designed to create a control band with

a center frequency offset of about 500 kHz. The peak gain and bandwidth of the

polyphase passband are about 70 and 140 kHz, respectively. The polyphase amplifier

alone does not completely determine the loop gain and stable control bandwidth of

the overall feedback system: other circuitry in and outside Genie will contribute extra

gain and phase behavior. In the next subsections, each of the above main functional

blocks is described in detail.

4.4.1 Image Reject Down-Converter

The MRI RF signal is a real signal, undergoing both amplitude and phase modulation.

Cartesian feedback, instead, is based on a quadrature representation of both reference

and feedback signals. Hence, for Cartesian feedback techniques to be used in MRI,

a quadrature reference generation circuit is necessary to convert the MRI RF signals

to a demodulated pair of quadrature signals. This approach is ideal for insertion into

an existing transmit chain. Alternatively, the MRI scanner should provide the low-IF

quadrature signals directly by DDS.

The reference generation circuitry in Genie is a fully differential complex baseband

quadrature demodulator acting as an image-reject down-converter. Its first two main

functional blocks are:

1. a commercially available broadband 50 MHz to 2 GHz single-ended quadrature

demodulator (model ADL5387 by Analog Devices). The ADL5387 outputs are

fully-differential with maximum amplitude of 2 Vpp and a DC level of 2.3 V;


Phase

Splitter

RFREFERENCE

INPUT

ADL5387

Reference generation circuitry

LO a b

a b

THS4131

R1 R2

C1 C2

THS4131

THS4131THS4131

Figure 4.3: Reference Generation Circuitry. R1 is 649 Ω, R2 is 680 Ω, C1 and C2are 470 pF. All the passive components have 0.1% tolerance. The fully-differentialamplifiers driven by the ADL5387 quadrature demodulator are THS4131 devices byTexas Instruments. (The THS4131 devices of the DC management circuitry are alsoshown.) The LO frequency is the same reference sent to the down/up-mixers of thefeedback loop.

typically, these outputs have a 0.4 phase error and 1.16 % amplitude imbal-

ance (equivalent to a 36 dB image rejection ratio, dominated by the amplitude

imbalance);

2. circuitry obtained with a pair of unity-gain fully-differential amplifiers (model

THS4131 by Texas Instruments). These amplifiers provide the recommended

loading conditions to the demodulator, filter the eventual high-frequency com-

ponents (such as residual LO frequency leaking at the output of the demodula-

tor), and shift the DC-level from the 2.3 VDC provided by the demodulator to

0 VDC.

In principle, the frequency-offset Cartesian feedback system will linearize a positive

IF band and reject negative frequencies. Even so, good matching of the quadrature

reference signals is still advantageous to avoid the generation of a negative frequency.

The reference signals are compared to the feedback signals; under closed-loop condi-

tions, the two will be approximately equal (as explained in Chapter 2). In practice,

the frequency-offset Cartesian feedback system is designed so that the loop gain is


well below unity in the negative frequency image band opposite the desired positive

frequency band; however, should this condition change during the investigation of

the behavior of the system, a non-negligible signal in this frequency band could be

reproduced at the output of the power amplifier.

The main requirement for the image-reject down-converter reference circuit is thus

to minimize the phase and amplitude mismatch and to ensure that the polyphase

loop driver amplifiers receive good-quality reference signals. To do so, a passive

polyphase filter is cascaded at the output of the unity-gain fully-differential amplifiers

buffering the ADL5387 quadrature demodulator. The passive polyphase block of the

reference generation circuitry acts as an image-reject filter. It increases the amplitude

and quadrature phase matching of the reference signals by providing two complex

notches within the mirror (negative IF) bandwidth opposite the desired (positive IF)

bandwidth provided by the loop. The minimum rejection inside a 200 kHz mirror

bandwidth is over 40 dB. The result is a nominal image rejection ratio of the reference

generation circuitry, including the passive polyphase filter, of over 70 dB, equivalent

to less than 0.1 phase error and less than 0.1% amplitude imbalance. The detailed

schematic of this filter is explained below and the complete schematic of the complex

baseband demodulator is shown in 4.3.

Polyphase Filter for High I/Q Matching

The two-stage passive polyphase filtering of the image-reject down-converter circuitry

improves the quadrature matching (both amplitude and phase) of the positive fre-

quency reference components by substantially reducing the negative frequency com-

ponents created by the imperfect behavior of the ADL5387 [3, 39, 51, 68, 75]. It does

so thanks to two complex notches near -500 kHz, as shown in the circuit complex

response in Figure 4.4, which reduce the baseband image by more than 40 dB in the

range from -400 kHz to -600 kHz.

While polyphase filter architectures for single-ended quadrature signals have been

presented in the literature, optimum image suppression can be obtained only with

fully-differential signals. This is a very fortunate outcome considering that the signals

used in frequency-offset Cartesian feedback must be fully-differential, since it allows


Frequency [MHz]-0.8

Gai

n [d

B]

-80

-60

-40

-20

0

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

Figure 4.4: Simulated complex response of the passive polyphase filter. These passivefilters create two complex notches near -500 kHz. Over 40 dB attenuation is obtainedin the negative frequency band opposite the desired positive frequency bandwidthdefined by the polyphase amplifiers in the frequency-offset Cartesian feedback loop.

the implementation of the polyphase passive filter to be very close to the loop error

amplifiers.

Genie includes mini-connectors before and after the passive polyphase filters, to

allow by-passing of these circuits to test the response of the loop error amplifiers to

a negative frequency input, if desired.

4.4.2 DC Management Circuitry

At the output of the reference generation circuitry, the quadrature reference signals

are 0 VDC (and fully differential). The feedback signals down-converted by the

CMX998 are, instead, 1.6 VDC (and single-ended). Since the gain of the polyphase

loop error amplifier, which amplifies the difference between the reference and feedback

signals, is not null at DC, it is necessary to offset the DC level. Since the output of

the polyphase loop error amplifier must be 1.6 VDC to drive the up-converter of the

CMX998 (see Section 4.4.3), the decision was to shift the 0 VDC level of the reference


signals to the latter.

In a classic Cartesian feedback loop, complex DC management strategies would

be necessary for the minimization of the DC offset between all the signal lines at the

input of the loop error amplifiers. Here, the modification implemented with the use

of the polyphase loop error amplifiers relaxes this requirement substantially.

For this reason, to shift the DC level, simple circuitry is sufficient that consists

of a pair of fully differential amplifiers with the common mode control pin receiving

the buffered reference of 1.6 VDC provided by the CMX998; similarly, to compensate

for DC offsets, a simple balanced passive network is sufficient where each quadrature

differential signal line at the input of the polyphase loop error amplifier receives a

manually trimmed constant voltage from a buffer.

4.4.3 Polyphase Amplifier Loop Filter

The polyphase amplifier in Genie has the same fully-differential architecture shown

and discussed in Chapter 3. The simplified schematic is also reproduced here, in

Figure 4.5. The polyphase amplifier was implemented using a pair of fully-differential

amplifiers, model THS4131 by Texas Instruments. The mismatch of the passive com-

ponents, both resistors and capacitors, is 0.1% to guarantee at least 60 dB rejection

to quadrature mismatches created in the feedback path.

The actual values of these components were chosen to obtain a control band with

center frequency ωc of about 500 kHz. The peak gain and passband half-width ωo of

the polyphase amplifiers are about 70 and 140 kHz, respectively.

4.4.4 Mixers and Phase Shift Control

The mixers and phase shift control circuitry of Genie were those of the CMX998

by CML Microcircuits, a commercially available integrated solution for a Cartesian

feedback loop based linear transmitter. This chip also includes an instability detector

and uncommitted op-amps (for input signal conditioning), input amplifiers, filters,

single-ended differential integrators for loop error amplification, and feedback base-

band amplifiers. These circuits were designed for implementing a classic Cartesian


Vcm

Vcm

ir

Q

I

if

qr

qf

Ri RF

C

RC

+

−

+

−

Figure 4.5: Genie polyphase amplifier. The frequency response of the amplifier haspeak gain of 70 V

V(36.9 dB), center frequency of about 500 kHz, and passband

half-width of about 140 kHz. The final design values of the passive components,all of which have 0.1% tolerance, were Ri = 750 Ω, RC = 16 kΩ,RF = 50 kΩ, C= 22 pF (nominal). The fully-differential amplifiers are THS4131 devices by TexasInstruments.

feedback control system but they are unused for the frequency-offset Cartesian feed-

back application in MRI. In particular, the loop error integrator cannot be modified

to obtain the complex loop error amplification offered by a polyphase implementation.

For this reason, only the mixers and loop phase control circuitry are used.

The CMX998 chip is specified for the RF range from 100 MHz to 1 GHz, which

allows it to be used for application in MRI at 3 T and above; on the other hand,

successful operation at 64 MHz (for 1.5 T MRI) was questionable. Initial evaluation

of the chip performance—using the CML Microcircuits evaluation board EV998—

demonstrated that the circuit can operate at carrier frequencies well below 100 MHz.

The only drawback to operation below 100 MHz is that the performance of the phase

control circuit starts dropping, until the circuit eventually fails at 40 MHz. However

as shown in Figure 4.6, at 64 MHz the actual phase deviation from linearity (plotted


0 45 90 135 180 225 270 315

-5

0

5

Desired Phase Shift (deg)

Phas

e Er

ror (

deg)

(63.75+0.25) MHz

-5

0

5

Phas

e Er

ror (

deg)

-5

0

5

Phas

e Er

ror (

deg)

at 64 MHz

at 128 MHz

at 300 MHz

Figure 4.6: Phase shift deviation from the desired value at 64 MHz, 128 MHz, and300 MHz. Although the CMX998 is specified for operation above 100 MHz RF, thephase shift control circuitry operates with good linearity (less than ± 4 deg error)and can thus be used for 1.5 T MRI amplifiers feedback control. The CMX998 failsat RF frequencies below 40 MHz.

versus the value of nominal phase shift) is still a remarkably low ±4.

4.4.5 Additional Genie Components

In addition to reference generation circuitry, DC management circuitry, polyphase

amplifiers, mixers, and phase shift control circuitry, Genie includes circuits for the

carrier frequency (LO) generation, conversion of the (single-ended) output signals of

the down-mixer to differential signals, and the power supply.

For the carrier frequency generation, a programmable surface-mount device (SMD)

oscillator by ECS Inc., International is used that drives both the down-converter

for the image-reject reference generation circuitry and the up/down mixers of the


CMX998. This off-the-shelf commercial component can be programmed by the man-

ufacturer or vendor with accuracy of 0.1% (± 6 kHz deviation from the desired 64

MHz frequency, for example). In principle, more accurate devices (such as a direct

digital synthesizer (DDS)) would be desirable to precisely set the center frequency of

the RF control bandwidth; however, in a frequency-offset Cartesian feedback trans-

mitter, variations in the carrier frequency can be compensated for by tuning the

center frequency of the polyphase amplifier passband. (In a classic Cartesian feed-

back transmitter, a more accurate carrier frequency generator would be necessary

since the center frequency of the loop error amplifiers is fixed at DC.) Even more im-

portant, if one wishes to by-pass the reference down-converter of Genie and generate

the low-IF quadrature reference signals using circuitry external to Genie, frequency

errors in the carrier generator would cause phase misalignments between the MRI

reference and the transmitted RF signal. Future generations of Genie should address

this limitation. Here, by using the same LO for the image-reject down-converter and

for the feedback loop, phase errors are cancelled and the output power is phase-locked

to the MRI reference.

To convert the single-ended feedback signals at the output of the CMX998 down-

mixers to differential, a pair of matched unity-gain low-pass amplifiers has been used.

The output signal lines of these amplifiers include 470 nF series capacitors to allow

optional AC-coupling of the feedback signals at the input of the polyphase loop driver

amplifiers.

The bias to all the components on board is obtained by means of four low dropout

(LDO) linear regulators (one LM2990, one LM1086, and two LP2992 devices by

National Instruments) that convert the input +-7 V to +5 V, -5 V, +3.3 V (digital),

and +3.3 V (analog). Genie’s current consumption is 350 mA with zero reference

input signal, and up to 700 mA at maximum control output signal.

4.5 Closing the Loop

At various times during development, the system included different types of loads and

RF power amplifiers. Different coupling devices have also been investigated to sample


the output voltage and current of the power amplifier, or, the forward voltage and

reverse voltage of the transmission line between the amplifier and load. RF switches

have also been added to the system to support auto-calibration, that is, for setting of

the stability conditions of the feedback system before each use. Finally, the network

analyzer-like capabilities of a Medusa console developed by Pascal Stang have been

used to provide the reference RF signal, analyze the feedback variable, and gate the

RF switches for auto-calibration.

The above list does not include the components of a load-pull set that has been

used to demonstrate the electronic manipulation of the output impedance of the RF

power amplifier. These components will be described in Chapter 6. Also in this

chapter the use of balanced amplifiers and of vector multipliers in the feedback path

will be presented.

4.5.1 Power Amplifiers

Typical MRI power amplifiers are expensive and low-efficiency class A or A/B RF

amplifiers with output power level ranging from 100 W to even 30 kW. An example of

this class of devices is the A/B wideband power amplifier KAA2040 by AR Modular

RF. At a cost of $10,000, it delivers in excess of 200 W power into a 50 Ω load over

the frequency range of 500 kHz to 100 MHz with nominal power gain of 53 dB. The

KAA2040 has been used in the frequency-offset Cartesian feedback system especially

during the initial stages of development.

The limited output power of Genie combined with the power gain of the KAA2040,

however, is not sufficient to drive the power amplifier up to its full power capabilities.

For this reason, as an alternative to the KAA2040, a set of custom-made 200 W

to 250 W (depending on supply voltage) class A/B power amplifiers with power

gain in excess of 60 dB have been preferred to demonstrate the linearization and

impedance manipulation of the feedback system up to the full output power range.

These amplifiers were built using an AN779H 20 W predriver and an AR313 amplifier

by Communication Concepts, Inc. and corresponding application notes by Motorola,

Inc. [27, 28, 74]. These kits were modified to incorporate RF gating, to perform


Figure 4.7: Simplified schematic of a 6” by 3” surface transmit coil. The capacitanceis distributed to minimize the e-field. The integrated coil current sensor is also shown.

programmable shutdown. (Although specified for 2 to 30 MHz, the AN779H can

provide 20 W drive at 64 MHz.)

4.5.2 Loads

In addition to a 150 W RF dummy load model 8141 by Bird, different types of RF

coils for MRI have been used. RF coils create the B1 field that rotates the net

magnetization in a pulse sequence (and also detects the magnetization as it precesses

transverse to Bo). They can be divided into three general categories:

1. transmit and receive coils. They serve as the transmitter of the B1 fields and

receiver of RF energy from the imaged object;

2. transmit only coils. They serve only as the transmitter of the B1 fields;

3. receive only coils. Used in conjunction with separate transmit coils, they detect

the signal from the spins in the imaged object.

Several varieties of each category exist; nevertheless, they are very similar in that

they must resonate at the Larmor frequency and they are composed of inductive and

capacitive elements. The resonant frequency, ν, of an RF coil is determined by the

inductance L and capacitance C of the inductor capacitor circuit:


ν =1

2π√LC

. (4.1)

The capacitance is usually distributed to minimize the electric field. The dissipative

nature of the copper coil is described by its resistance R and depends on the con-

ductivity of the material and on the geometry of the coil. However, the actual losses

are dominated by the sample [34]. A lossy sample when immersed in the B1 fields

created by the coil (such as a saline load, in most of the experiments described later)

will dissipate power because of RF eddy currents. This loss effectively adds a series

resistance to the coil, which depends not only on the nature of the sample but also

on its proximity to the coil.

Surface coils and bird cage coils are the two most common designs and both have

been used during the experiments described in Chapters 5 and 6. The schematic of

a 6” by 3” transmit-only surface coil, used in these experiments, is shown in Figure

4.7 [70]. They have been tuned to match the center of the complex control bandwidth

of the frequency-offset Cartesian feedback system, which of course must be equal to

the MRI Larmor frequency.

4.5.3 Coupling devices

To sample the output of the RF power amplifier, two options are available: a custom-

made slotted-line style coupler, sampling both the total voltage and total current,

and a C7149 made-to-order Werlatone coupler, sampling both forward and reverse

voltage. When the attenuation coefficient of these two devices was not sufficient, fixed

coaxial attenuators by Minicircuits were used in series.

The custom-made sensing circuit, whose schematic is shown in Figure 4.8 [69],

is a variation of the Bruene directional coupler [13] to simultaneously measure the

line voltage and current. It can be inserted in series with a transmitter coaxial cable

and is made with transmission lines and discrete components. The circuit consists

of two sections, one of which inductively samples the total current in the output

line of the amplifier, while the other simultaneously senses the total voltage. The


f (I)

f(V)

1:0.01

1:1

1:11 pF

50 Ω

25 Ω

25 Ω

Figure 4.8: Photo (left) and simplified schematic (right) of custom-made slotted-linestyle sensing circuit. A tapped RC is used for voltage sensing and a pick-up loop forcurrent sensing.

coupling coefficient of both sections is approximately 40 dB at 64 MHz. It is essen-

tially a narrowband device, since its coupling coefficients exhibit an approximately

linear dependence with frequency, increasing approximately by 20 dB per decade of

frequency.

The C7149 coupler by Werlatone is a high-power (200 W), wide bandwidth (60-600

MHz) bidirectional coupler that samples both forward voltage and reverse voltage,

with attenuation coefficient equal to 20±0.5 dB over the entire range.

The relationships between the output voltage, output current, and forward or

reverse transmission line voltages are described by the following equations:

Itot =VFWD − VREV

Z0

(4.2)

Vtot = VFWD + VREV . (4.3)

The s-parameters of the custom-made coupler and of the C7149 coupler are presented

in Table 4.1 and Table 4.2, respectively.


Table 4.1: S-parameters of custom-made coupler. Port 1 = Input; Port 2 = Output;Port 3 = Voltage Sample; Port 4 = Current Sample.

InputPort 1 Port 2 Port 3 Port 4

Outp

ut Port 1 0.04406 73.0 0.9940 6 -8.1 0.0104 6 -112.6 0.08906 57.8

Port 2 0.99406 -8.1 0.0447 6 73.1 0.01056 -112.4 0.00886 -122Port 3 0.01046 -112.6 0.01056 -112.4 0.11206 69.1 0.00016 100Port 4 0.08906 57.8 0.00886 -122 0.00016 100 0.28006 58.5

Table 4.2: S-parameters of C7149 coupler by Werlatone. Port 1 = Input; Port 2 =Output; Port 3 = Forward Voltage Sample; Port 4 = Reverse Voltage Sample.

InputPort 1 Port 2 Port 3 Port 4

Outp

ut Port 1 0.02606 -33.0 0.99006 -29.3 0.09566 -16.2 0.00146 60.0

Port 2 0.99006 -29.3 0.02706 -30.0 0.00266 16.0 0.09626 15.0Port 3 0.09566 -16.2 0.0026 6 16.0 0.02166 -143 0.97306 -151.7Port 4 0.0014 6 60.0 0.0962 6 15.0 0.97306 -151.7 0.01736 -138.0

Combiners

Commercially available, 2-way power combiners by Mini-circuits are used to com-

bine the outputs of the couplers (or, of the vector multipliers) to obtain the desired

feedback signal at the input of Genie’s down-converter. Both 0 combiners (model

ZFSC-2-4+) and 180 combiners (model ZFSCJ-2-4+) are available, to perform sum

and subtraction, respectively, of two samples.

4.5.4 Auto-Calibration Network

An automated switching network has been developed for the calibration of the feed-

back system in which RF switches toggle the system from the open-loop to the closed-

loop configuration. At the beginning of each experiment, the system is in open-loop

configuration and the phase shift setting that ensures the stability of the closed system

is measured and implemented by controlling the phase shift circuitry of the CMX998


Figure 4.9: Schematic and photo of ZASWA-2-50DR switch by Mini-circuits. Theswitch provides internal 50 Ω termination.

via a software interface (described later). After this operation, the switches are tog-

gled and the system works stably in closed-loop configuration. In the earlier stages

of the system development, this operation was performed by manually routing the

cable connections between the source of the reference signal, Genie, and the coupler

(or combiner).

Figure 4.9 shows the schematic and photo of the single-port, double-throw (SPDT)

RF switch model number ZASWA-2-50DR by Mini-circuits. A PCB was built hosting

two of these devices and a PTK10-Q24-D5 DC-DC converter by CUI provides the

required ±5 V dual supply voltage to the switches from a 24 V single supply (shared

with the power amplifier).

The ZASWA-2-50DR is a wideband (DC to 5 GHz), TTL-driven high-isolation

switch with one gate port. In addition to its exceptionally high isolation, this par-

ticular device provides a 50 Ω termination at the output port when “off”. By doing

so, the switch ensures that a 50 Ω termination is presented at the input of Genie’s

reference down-conversion circuitry when the feedback system is in open-loop config-

uration. The drawback of this approach is that the source of the RF reference signal

is loaded by two parallel resistances, each approximately 50 Ω, during both open-loop

analysis and closed-loop operation. Because of the non-zero output impedance of

the RF source, the amplitude of the signal from this source is reduced at the input

of the reference down-conversion circuitry. The benefits of the automated switching


FWD REV

Figure 4.10: Schematic of the feedback system with auto-calibration network. Whenthe stability conditions of the system must be investigated, Medusa toggles theswitches in position A. In this state, the loop is open and the reference signal issent to the feedback down-mixer. A sample of the output signal is used to measurethe loop phase rotation and calculate the phase shift setting that compensates for it.Once the stability conditions are known, Medusa toggles the switches to position B.In this state, the loop is closed and the reference signal is sent to the input of Ge-nie. A sample of the output signal is used to measure, for example, the linearizationperformance of the system.

network, however, more than compensate for the latter drawback; moreover, the ref-

erence down-conversion circuitry includes active components that can be used if the

available source output range is limited and amplification of the reference signals is

necessary.

Figure 4.10 is a simplified schematic of the auto-calibration circuitry showing

the port/throw connections of the switches. To make the switching network fully

automatic, the status of the switches must be computer controllable. This is possible

thanks to the Medusa console, which is also the source of the RF reference signal and

contains the circuitry for the analysis of the feedback signal.


4.5.5 Medusa

Medusa was developed by Pascal Stang at MRSRL for parallel imaging applications,

as well as for vector modulation of an array system for parallel transmit and array

coil decoupling [73]. It is built from an extensible set of intelligent RF and gradient

modules, including local synchronization logic, a Direct Memory Access (DMA) en-

gine, and a 2-megabyte waveform buffer, which together provide the core functionality

required by MRI.

Medusa’s network analyzer-like capabilities supported the system development

described in this chapter by providing an RF reference signal to Genie, receiving and

analyzing a copy of the feedback signal sent to the CMX998 down-mixer, gating the

power amplifier synchronously with the onset and duration of the reference signal (in

order to avoid excessive dissipation and thus heating, which could irreversibly damage

its functionality), and controlling the RF switches to toggle the system configuration

between open- and closed-loop.

Medusa can be controlled entirely via PC, using a Matlab interface. In addition

to the routines necessary for the automatic calibration of the loop, several pulse se-

quence routines were developed to investigate the performance of the frequency-offset

Cartesian feedback system in a variety of experiments. Some routines, for example,

allow the generation of the reference signals and the measurement of the linearization

performance of the system at varying signal amplitudes (at fixed frequency) or fre-

quencies (at fixed amplitude). Other routines create MRI-like modulated reference

signals such as sinc pulses, Gaussian pulses, rectangular pulses, or more complicated

Very Selective Saturation pulses [77]. Other routines control the auxiliary circuitry

necessary for measuring the output impedance of the power amplifier (the load pull

setup, see Chapter 6) or for changing phase and amplitude of the sampled forward

and reverse voltages from the Werlatone coupler with the vector multipliers. Higher-

level routines combine these capabilities to analyze the performance of the system

when multiple functionalities are needed at once.

In addition to controlling Medusa, it is also possible to control the CMX998 mixers’

gain and phase shift settings via PC. To do so, two options have been demonstrated.

One consists of using the commercially available PE001 card and GUI interface by


CML Microcircuits. The other is based on a development board hosting an AVR

processor by Atmel controlled via a Matlab interface.

4.5.6 PE001 Card and GUI Interface

The PE0001 interface card is a global interface system designed for use with evaluation

kits for new generation ICs manufactured by CML. Supplied with a PC GUI, the

PE0001 provides a graphical method of addressing all the CMX998 on-chip registers

via the C-BUS interface. The information generated by the GUI is formatted, timed

and delivered to the target IC via the C-BUS serial interface hosted by Genie. Power

to the PE0001 is also obtained from Genie via the same serial connector.

The PE0001 card and GUI interface offer many capabilities; however, given that

only the mixers and phase shift circuitry of the CMX998 are used, the card and

interface are used only to address the registers that enable the CMX998 bias, enable

the forward and feedback circuitry, select the gain of the mixers (to modify the forward

and feedback loop gain), and control the phase shift circuitry.

4.5.7 AVRmini and Matlab Interface

As an alternative to the PE0001 card, the use of an AVRmini V4.0 development board

(by P. Stang) with Ethernet and USB, hosting an ATmega644P AVR processor by

Atmel, has also been demonstrated. The board is connected to Genie via a second

serial connector, and to a PC via Ethernet. The AVRmini is programmed by Matlab

and fully replaces all functions supported by the PE0001, that is, it enables the

CMX998 bias, forward and feedback circuitry, selects the gain of the mixers, and

controls the phase shift circuitry. It does so with the added advantage that the number

of conductive pathways between Genie and the PC is reduced to that necessary for

the Medusa console only. One drawback of this option is that, without the PE0001

GUI interface, the values stored in the CMX998 registers are not all immediately

visible at any time, which makes testing more cumbersome.

This section concludes the description of the components in the frequency-offset

Cartesian feedback system. In the next section, a theoretical analysis of the loop


+ control

error

GENIE TRANSMITTER

COUPLER

-H(ω)

POWER AMPLIFIER

k·e j(θ-ωτ)

A

Z

X(ω) Y(ω)+

d(ω)

+

n(ω)

+

+

Figure 4.11: Simplified schematic of the feedback system for loop analysis. In thefrequency-offset Cartesian feedback system, block A includes the coupler attenuationcoefficient, the down-mixer conversion gain, and the loss of combiners and additionalpads. H(ω) includes the polyphase amplifier gain and the up-mixer conversion gain.

performance and stability considerations are presented.

4.6 Analysis of Performance

As in the case of any other negative feedback method, the analysis of the frequency-

offset Cartesian feedback loop starts with a calculation of the loop gain.

Figure 4.11 shows a simplified block diagram of the feedback system. For analysis

purposes, the quadrature modulators and demodulators can be assumed ideal: the

frequency conversion allows the loop gain analysis to be referred to the low-IF range;

any distortion can be considered as an additive error; the mixer harmonics as well

as image aliasing can also be considered as additive errors. With this assumption,

the forward gain of the loop is comprised of the gain in the low-frequency error

amplifiers, H(ω), the gain and group delay at the RF frequency of the power amplifier,

ke−jωτ , and a loop phase rotation, ejθ. The feedback gain instead consists of only

the attenuation A of the coupler, which is obtained with passive components and

thus is usually constant over a wide bandwidth and over a wide range of operating

conditions.

The loop gain, which ultimately controls the level of linearization of the RF power


amplifier, is thus

G(ω) = Ak|H(ω)|ejθ−jωτ+j 6 H(ω) (4.4)

and the output control variable is

Y (ω) = X(ω)k|H(ω)|ejθ−jωτ+j 6 H(ω)

1 + Ak|H(ω)|ejθ−jωτ+6 H(ω). (4.5)

If the conditions for stability are met, and Ak|H(ω)| is greater than unity within the

control bandwidth, then

Y (ω) ≈ X(ω)1

A(4.6)

that is, the gain of the system is determined by the attenuation of the coupler only

and the output signal is free, to first order approximation, from the distortion of the

power amplifier.

To understand the effect of the loop on noise and distortion created by the blocks

of the system, two summing junctions can be included in the forward path and in

the feedback path, respectively, into which distortion and noise (d(ω) and n(ω)) are

added. In this case,

Y (ω) = X(ω)k|H(ω)|ejθ−jωτ+j 6 H(ω)

1 + Ak|H(ω)|ejθ−jωτ+6 H(ω)

+d(ω)1

1 + Ak|H(ω)|ejθ−jωτ+6 H(ω)

+n(ω)Ak|H(ω)|ejθ−jωτ+j 6 H(ω)

1 + Ak|H(ω)|ejθ−jωτ+ 6 H(ω).

(4.7)

Assuming, for the moment, that the loop is stable and Ak|H(ω)| is greater than unity,

then

Y (ω) ≈ X(ω)1

A+ d(ω)

1

G(ω)+ n(ω). (4.8)

The noise and distortion d(ω) produced by the blocks in the forward path are reduced

by a factor equal to the loop gain by the inclusion of the feedback loop. Thus, for

example, if the loop gain is 10, the distortion of the closed loop system will be 20


dB lower than the distortion introduced by the blocks in the forward path, which is

usually dominated by the power amplifier.

The noise and distortion n(ω) produced by the blocks in the feedback path, in-

stead, directly corrupt the output of the closed system. For this reason, minimizing

any source of noise and distortion in the feedback path is extremely important.

4.7 Analysis of Stability

For the closed loop transmission to be determined (mostly) by the coupler coefficient

only, the conditions for stability of the loop must be met. As in the classic Cartesian

feedback system, the conditions for stability are as follows:

1. the phase adjuster must be set to cancel any phase shift (phase rotation) around

the loop (i.e., force θ equal to 0);

2. the loop phase margin pm should be at least 45.

The phase margin (pm) is the difference between the phase of G(ω) and 180 (or

-180) when the loop gain is unity (i.e., ω = ωpm, the unity gain crossover frequency):

pm = π − (θ − ωpmτ + 6 H(ωpm)). (4.9)

When the loop gain approaches unity, the ability of the loop to reduce the distortion

of the power amplifier is effectively nil. In a frequency-offset Cartesian feedback

system, the separation between ωpm and ωc (the center frequency of the polyphase

complex band) is thus the minimum desired half-width of the control bandwidth.

In other words, ωpm must be at least larger than half the modulation bandwidth

of the desired signal, and should include the intermodulation bandwidth (since only

intermodulation products within the control bandwidth can be linearized). With this

requirement on ωpm and assuming that the phase rotation is compensated for, then

the specification on the minimum phase margin drives the design of H(ωpm), that

is, it dictates the choice of peak gain and pole frequency of the polyphase loop error

amplifiers described in Chapter 3.


If the phase rotation is not compensated, the phase margin can become less than

45 and stability can be compromised. For example, positive phase error increases

the phase margin for positive offset frequencies and reduces it for negative offset

frequencies. When the phase error is negative, the opposite happens.

The loop gain of the frequency-offset Cartesian feedback loop has changed sig-

nificantly during the design, since different power amplifiers and couplers have been

used. In general, the system was characterized by loop gain of magnitude in the range

10 to 20 and with a stable control band 50 kHz to 100 kHz wide.

The contributions (and limitations) to the loop gain of the frequency-offset Carte-

sian feedback system are shown in Table 4.3. In particular, gain and dynamic range

of all the main components in Genie, including not only the polyphase amplifiers but

also the variable gain amplifiers following the mixers in the forward and feedback

path, dictated the choice of having an overall feedback attenuation (including the

attenuation coefficient A and down-mixer gain) usually equal or slightly higher (by

10 dB to 14 dB) than the amplification k of the RF power amplifier. The magnitude

of the loop gain is thus, generally, equal to or lower (by the same amount) than the

peak gain of the polyphase amplifiers (including the up-mixer gain, always kept at its

highest value of -2dB). With a polyphase gain of about 70, the loop gain magnitude

ranged between 10 (17 dB below 70) and 20 (11 dB below 70).

While traditional frequency response techniques can be applied to a single Carte-

sian feedback control loop and offer a useful starting point for further stability anal-

ysis, the situation where multiple loops control different resonant loads inductively-

coupled to one another is of particular interest in MRI and the possibility to study the

stability of the latter with traditional methods of analysis is debated. The application

of the control theory for Multiple-Input, Multiple-Output (MIMO) systems might be

a solution to this problem. However, this approach to the study of stability can be

extremely challenging. Simpler approaches would be desirable. The application of

the Middlebrook criterion [56–58] to arrays of control loops could be a solution to

this problem. While the application of this method has not been verified in practice,

it might represent a starting point for the understanding of the effects of coil-to-coil

interactions on stability. One of the most interesting results of this analysis is that


Table 4.3: Gain and maximum input and output levels of the main loop components.The maximum up-mixer output is taken after the on-board filters. The maximumdown-mixer input is valid at minimum gain setting; typically, the gain is 0±3 dB andthe corresponding maximum input is +2∓3 dBm. The power amplifier is the custom-made amplifier built using an AN779H 20 W predriver and an AR313 amplifier byCommunication Concepts, Inc.

Component Gain Max Input Max OutputdB dBm dBm

Polyphase Amplifier 36 – 13.5Up-Mixer -32 to -2 +4 +0

Down-Mixer -5 to +24 +7 +4BP Filter -3 – –

Power Amplifier +60 – +53Coupler -40 to -20 – –

Combiner -3 – –Pad -40 to -10 – –

identical systems, if stable uncoupled, will be unconditionally stable even when cou-

pled. Mismatches between the phase and gain characteristics of the different systems

may cause instability of the array, even if each uncoupled system is stable.

4.8 Summary

In this chapter, the requirements and desired specifications that have determined the

design choices of the frequency-offset Cartesian feedback system have been presented.

Then, the fundamental blocks and circuit details of the system have been described,

including the components providing autotuning capabilities for the analysis of the

phase-shift settings that compensate for the loop phase rotation. Finally, a mathe-

matical formulation to the analysis of the loop stability and linearization performance

has been provided. In the next chapter, the latter will be demonstrated.

Chapter 5

Improving the Fidelity of RF

Reproduction

5.1 Introduction

As the frequency and bandwidth of the MRI signal increase, the problem of the

RF distortion in the transmit path becomes non-negligible. The frequency-offset

Cartesian feedback method and system employing power amplifiers for MRI promises

to address this problem by providing increased fidelity of RF reproduction.

In this chapter, the nature of the distortion created by the behavior of any RF

power amplifier is briefly discussed. Then, the ability of the feedback system to

improve the fidelity of RF reproduction is demonstrated in a number of situations:

not only have different types of tests been performed, but also different variables—

such as the output voltage of the power amplifier, or the output current, or even the

transmit coil current—have been chosen as the feedback control loop variable.

The conventional linearity tests used in the field of communications to demonstrate

the performance of a classic Cartesian feedback system have also been performed and

are described here. These tests show that the frequency-offset Cartesian feedback

system offers state of the art performance equal to the classic implementation. In

addition, experiments have been conducted to show that excellent performance is

obtained with much relaxed specifications for the transmitter, thanks to the use

75

CHAPTER 5. IMPROVING THE FIDELITY OF RF REPRODUCTION 76

of polyphase loop error amplifiers. Indeed, if the polyphase loop error amplifiers

are modified to obtain the classic loop error amplifiers, the system performance is

significantly hindered.

To demonstrate the significance of the application of the frequency-offset Carte-

sian feedback system in MRI, the effects of using the system have been demonstrated

with a variety of typical MRI pulses, such as sinc pulses, Gaussian pulses, rectangu-

lar pulses, and many more. Some of the results obtained in these experiments are

described here.

Finally, the ability to reject the mirror frequency of the desired reference signal

at the carrier has been measured. This test shows that the system is capable of a

significant rejection of the undesired frequencies created by the down-mixer.

5.2 Nature of Amplifier Distortion

Amplifier distortion has been a concern for many years in virtually every field of ampli-

fier design. The nature of such distortion is usually described in terms of amplitude-

to-amplitude (AM-AM) distortion, amplitude-to-phase (AM-PM) distortion, phase

dispersion, and memory effects [40].

A perfect amplifier would have a linear transfer characteristic, where the output

signal Sout would be a scalar multiple of the input signal Sin, that is,

Sout(t) = kSin(t) (5.1)

where k is the gain of the amplifier and

Sin = A(t) cos(ωCt). (5.2)

In reality, all practical amplifiers are characterized by a certain degree of amplitude

non-linearity, that is, a relationship between input and output that is a non-linear

function fAM() of the amplitude modulation of the input signal:

Sout(t) = kfAM(A(t)) cos(ωCt). (5.3)


This non-linear relationship is often termed AM-AM conversion since it is a conversion

between the amplitude modulation present on the input signal and the distorted

amplitude modulation present on the output signal.

Another effect is a conversion from amplitude modulation of the input signal to

phase modulation on the output signal, and is known as AM-PM conversion. In this

case,

Sout(t) = kA(t) cos(ωCt+ fPM(A(t))) (5.4)

that is, the resulting output spectrum is that of a carrier whose phase is modulated by

a non-linear function fPM() of the amplitude modulation of the input signal. Usually,

the undesired AM-PM modulation at the output of the amplifier is similar to that

resulting from AM-AM non-linearity; since the two forms of distortion often coexist,

separating their effects is generally very difficult.

To complicate the picture even further, a practical amplifier does not delay all

frequency components within the input signal by the same amount when they reach

the output (regardless of the amplitude of each of these components). The relationship

between time delay, τ , and phase shift, ϕ (in radians), is

τ =ϕ

2πf(5.5)

where f is the frequency dispersion. If the time delay is not identical for all frequency

components of the input signal, the phase shift does not increase in proportion to

the frequency and the output waveshape will be distorted. This non-linear phase

characteristic is known as the phase dispersion of the amplifier.

Finally, in all practical amplifiers, the AM/AM, AM/PM and phase dispersion are

not immutable but vary with changes in the environment and with the operating fre-

quency of the amplifier. Typical changes that influence the power amplifier behavior

are fluctuations in the biasing conditions and temperature. The distortions created

by these changes generally fall under the category of memory effects.

In the field of communication the distortion of a power amplifier is usually charac-

terized through standardized tests such as 1 dB compression testing, two-tone testing,


and QAM constellation testing. In the field of MRI, while these non-linearity mea-

sures are still very useful, it is easier to appreciate the extent to which the amplifier

distortion affects the quality of an image by solving the Bloch equations and com-

paring the effects on the spin magnetization of a desired (reference input) waveshape

and the experimental (output) MR pulse, distorted by the amplifier.

In the remaining sections of this chapter, simple tests are presented that show the

AM-AM, AM-PM, and phase linearization enhancement obtained with the frequency-

offset Cartesian system over stand-alone MRI power amplifiers. Then, both standard-

ized communication-style test results and MRI-like test results are presented to show

the positive effects of this enhancement on the fidelity of reproduction of the signals

used in both fields.

5.3 Reduced AM-AM, AM-PM Distortion

In a first set of experiments, the AM-AM and AM-PM distortion of the power ampli-

fier is studied and compared with the distortion of the closed-loop feedback system.

To study AM-AM and AM-PM distortion and feedback linearization, a series

of pulses was used as the input reference signal, each pulse consisting of a sinusoidal

signal of fixed frequency and fixed amplitude; all the pulses in the series had the same

duration and frequency, but their amplitude increased in the series. The duration of

the pulse was made much longer than the settling time of the power amplifier (with

or without control feedback loop), so that the distortion could be evaluated when

the amplifier was at equilibrium. Initially, the reference signal was applied directly

at the input of the power amplifier; then, it was applied at the input of the reference

down-converter of the transmitter in the complete feedback system.

Enhanced linearization can be demonstrated in the voltage-to-voltage character-

istic of the power amplifier or in the voltage-to-current characteristic by choosing the

output voltage or output current, respectively, of the amplifier as the feedback vari-

able of the control system. For application in MRI, both situations are of interest,

depending on the particular design of the coil loading the amplifier.


0

10

20

30

40

50

60

70

80

90

100

0.0250.0

520.0

790.1

050.1

320.1

590.1

860.2

120.2

390.2

660.2

930.3

200.3

460.3

730.4

00

Normalized Input Amplitude [V/V]

Out

put V

olta

ge [V

]

-25

-20

-15

-10

-5

0

5

10

15

20

0.0250.0

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050.1

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730.4

00


Out

put P

hase

Dev

iatio

n [d

eg]

closed loop

no loop

closed loop

no loop

frequency-offset

Cartesian feedback

transmitter

V50 Ω

Figure 5.1: Output Voltage Amplitude (left) and Phase Error (right) of the poweramplifier without (red traces) and after addition (blue traces) of the frequency-offsetCartesian feedback system. Both AM-AM and AM-PM distortions are reduced by afactor of at least 14 (23 dB), which approximates the loop gain of the system in itschosen configuration during the experiment.

5.3.1 Voltage-Mode Amplitude Test

The first fundamental test was to characterize the ability of the system to significantly

reduce AM-AM and AM-PM distortion when the feedback variable is a sample of the

total voltage at the output of the RF power amplifier obtained with the custom-

made Bruene-style directional coupler of coupling coefficient equal to -40 dB. In this

experiment, the load to the 200 W output, 60±2 dB gain (depending on the output

power) power amplifier built with the AN779H 20 W predriver and an AR313 amplifier

was a 50 Ω dummy load by Bird. A series 20 dB pad was added at the coupler’s output.

The down-mixer gain was 0 dB and the up-mixer gain -2 dB. A coaxial bandpass filter

(at the input of the amplifier) and a combiner (splitting the voltage sample into a

monitoring signal for Medusa and a feedback signal for Genie) added a loss of about


3 dB each. The autotuning switches accounted for 2 dB additional path loss. The

nominal total loop gain, excluding cable losses, was about 26±2 dB.

Figure 5.1 compares the amplitude and phase distortion obtained with the power

amplifier driven directly and after addition of the feedback system, up to the regime

of full compression of the power amplifier. Both amplitude and phase distortions are

reduced by over 23 dB (i.e. the AM-AM, AM-PM distortion after linearization is

about 7% the original amplifier distortion). This value approximates well the mini-

mum loop gain magnitude of the system in the configuration used for this particular

experiment.

5.3.2 Current-Mode Amplitude Test

In MRI, direct control of transmit coil loop current is of particular interest, since these

currents generate the B1 field that excites the magnetization of the imaging sample.

The AM-AM and AM-PM enhancement was thus measured when the feedback signal

was a sample of the coil current obtained with a current sensor integrated in a 6” by

3” transmit surface coil loading the power amplifier.

The experimental setup was very similar to that of the previous experiment. The

200 W output, 60±2 dB gain power amplifier built with the AN779H 20 W predriver

and an AR313 amplifier was chosen to drive the transmit coil. A series 24 dB pad was

added at the coupler’s output. The down-mixer gain was 0 dB, the up-mixer gain -2

dB. A coaxial bandpass filter (at the input of the amplifier) and a combiner (splitting

the voltage sample into a monitoring signal for Medusa and a feedback signal for

Genie) added a loss of about 3 dB each. The autotuning switches accounted for 2 dB

additional path loss. The nominal total loop gain, excluding cable losses, was about

22±2 dB.

Figure 5.2 compares the amplitude and phase distortion obtained with the power

amplifier driven directly and with the feedback system in place, up to the regime of

full compression of the power amplifier. The maximum current that could be driven

in the coil by the power amplifier was about 3.5 A. After linearization, the amplitude

distortion was reduced by about 20 dB (i.e. the AM-AM and AM-PM distortion


0.0250.0

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050.1

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Coi

l Cur

rent

Pha

se D

evia

tion

[I]

-15

-10

-5

0

5

10

15

0

0.5

1

1.5

2

2.5

3

3.5

4

0.0250.0

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460.3

730.4

00


Coi

l Cur

rent

[I]

closed loop

no loop

closed loop

no loop

frequency-offset

Cartesian feedback

transmitter

Figure 5.2: Coil Current Amplitude (left) and Phase Error (right) before and afteraddition of the frequency-offset Cartesian feedback system (shown in red and blue,respectively). Both AM-AM and AM-PM distortions are reduced by a factor of about10 (20 dB) in the range between 10% and 90% of the total output current. This valueapproximates the minimum loop gain of the system in its chosen configuration duringthe experiment. The reduced linearization performance at the extremes of the rangecan be explained by the reduced phase margin, due to the variation in the loadimpedance with varying frequency.

after linearization was about 10% of the original amplifier distortion). Also the phase

distortion was reduced by the same amount, but only within 20% and 90% of the

output range: the plot of phase distortion shows a reduced linearization performance

near the minimum and maximum limits of the power amplifier operating range.

The latter observation, which was not made in the previous experiment, can be

explained by the different type of loading to the power amplifier: in the previous

experiment, the broadband 50 Ω dummy load was presenting a constant impedance

to the amplifier; in this experiment, the transmit coil is a tuned (narrowband) device.

The narrowband behavior of the coil affects the phase margin of the feedback system,


whose conditions for stability are fixed and equal to those ideal for operation in

the middle of the output range. The decrease in phase margin is not sufficient to

compromise the stability of the system; however, it does spoil its ability to reduce

distortion.

5.4 Reduced Two-Tone and QAM Distortion

In the field of communications, a useful, directly applicable measure of the amplifier

dispersion is usually obtained by performing two-tone, multi-tone, and QAM constel-

lation tests.

All these tests have been performed and demonstrate that the frequency-offset

Cartesian feedback system offers linearization performance as effective as the classic

Cartesian feedback system. In addition, experiments have been conducted to show

that this performance is obtained with much relaxed specifications for the transmit-

ter, thanks to the use of polyphase loop error amplifiers. When the polyphase loop

error amplifiers are modified to obtain the classic loop error amplifiers, the system

performance is significantly compromised.

5.4.1 Two-Tone Test

Fig. 5.3 (top) shows the output spectrum of the 200 W power power amplifier—built

with an AN779H 20 W predriver and an AR313 amplifier—when driven directly by the

two tones closely spaced in frequency, at 64.455 MHz and 64.475 MHz, respectively.

These match the +10 kHz and -10 kHz offset from the frequency-offset Cartesian

feedback RF control center frequency of 64.465 MHz.

Fig. 5.3 (bottom) shows the output spectrum of the power amplifier after addition

of the frequency-offset Cartesian closed loop control. The two tones are fed at the

input of the transmitter’s reference down-converter. Here the distortion products are

attenuated down to the noise level. Some increase in noise level is also apparent

with closed-loop operation, especially near the main tones. This noise arose because

of signal level and attenuation requirements for using the CMX998. Future designs


Figure 5.3: Two-tone Test. The output spectrum of the power amplifier drivendirectly (top) with two tones closely spaced in frequency shows odd-order inter-modulation products, which are reduced to the noise floor after addition of thefrequency-offset Cartesian feedback system (bottom). Some increase in noise levelis evident with closed-loop operation, especially near the main tones.

of the system transmitter can address this limitation, and optimization strategies

are available since the noise analysis of a frequency-offset Cartesian feedback system

should be equivalent to the noise analysis of a classic Cartesian feedback loop.

The two-tone distortion results are repeated in Figure 5.4 for the classic Cartesian

feedback configuration, obtained by removing the crossing resistors of the polyphase

loop error amplifiers (and thus, the coupling between I/Q error signals) to obtain a

pair of classic low-pass amplifiers. Figure 5.4(top) shows the output spectrum of the

power amplifier when directly driven by two tones at 64.01 MHz and 63.99 MHz (these

tones are +10 kHz and -10 kHz offset from the LO and control center frequency of 64

MHz). Figure 5.4(bottom) shows the output spectrum of the power amplifier when

controlled by the classic transmitter configuration. The system reduces the distortion

products down to the noise level, just as the frequency-offset Cartesian feedback

system does, but some LO-leakage and LO-noise is evident. More precise trimming of

DC-offsets would have suppressed this effect. The frequency-offset Cartesian feedback


Figure 5.4: Two-tone Test. The output spectrum of the power amplifier drivendirectly (top) with two tones closely spaced in frequency shows odd-order inter-modulation products, which are reduced to the noise floor after addition of the classicCartesian feedback system (bottom) obtained by removing the coupling between thequadrature error signals amplified by the loop error amplifiers. The ”spike” at thecenter of the control bandwidth is the LO leakage created by DC offsets and self-mixing of the LO frequency at the down-mixer. The LO phase noise is also presentnear the center frequency.

system is immune to this effect, because the loop error amplification takes place at

a low IF. The classic Cartesian feedback system, however, is not: the loop error

amplification band includes DC, where LO-leakage, DC offset, and LO-noise now

create the undesirable “spike” at the center of the control bandwidth.

5.4.2 QAM Constellation Test

Quadrature Amplitude Modulation (QAM) constellation diagrams are used to graph-

ically represent the quality and distortion of a digital signal. In a QAM test the

amplitude of two waves, 90 degrees out-of-phase with each other (in quadrature) is

changed to represent the data signal. Amplitude modulating two carriers in quadra-

ture can be equivalently viewed as both amplitude modulating and phase modulating


-0.3

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0

0.1

0.2

0.3

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

Figure 5.5: QAM Test. The QAM diagram of the power amplifier driven directly(top) shows gain compression effects. After addition of the frequency-offset Cartesianfeedback system, the compression effects are virtually eliminated (middle); highernoise is evident as in the two tone test results. Removing the coupling betweenquadrature error signals at the loop error amplifiers results in an appreciably distortedconstellation (bottom), in which low power symbols especially suffer from DC/LOleakage and quadrature errors within the loop.


a single carrier.

A 9x9 QAM constellation test was used to study the ability of the frequency-offset

Cartesian feedback to enhance the quality of this type of digital modulation format

by feeding the amplitude and phase-modulated carrier first directly at the input of

the power amplifier, and then at the input of Genie’s reference down-converter in the

closed loop system configuration.

Figure 5.5 (top) shows the QAM constellation of the power amplifier driven di-

rectly with 64.5 MHz symbols, where gain compression effects are evident at the

highest carrier amplitudes. This amplifier was then linearized by frequency-offset

Cartesian feedback. The linearized QAM grid is shown in Figure 5.5 (middle), where

gain compression effects have been removed. The noise level is slightly increased, for

the same reason discussed previously.

A second amplifier driven to similar gain compression levels (QAM grid not shown)

was linearized by classic Cartesian feedback configuration, obtained by removing the

crossing resistors of the polyphase loop error amplifiers. In Figure 5.5 (bottom), the

resulting QAM constellation of 64 MHz symbols (matching the LO) also shows lin-

earization of high power symbols, but low power symbols suffer from DC/LO leakage

and quadrature errors within the CF loop. Moreover, since the reference down-

converter image reject filter was ineffective at DC, the down-converter quadrature

and LO leakage errors also distort the baseband reference signals, prior to the CF

loop. Higher noise levels at baseband are again evident.

5.5 Reduced MRI Pulse Distortion

In the field of communications, the popularity of two-tone and QAM constellation

tests is explained by the immediate applicability of the information that these tests

provide to the understanding of the effects of distortion on the modulation formats

employed to transmit information via antennas. In some applications, the modulation

format is such that only the AM-AM distortion of the amplifier poses concerns; in

others, the phase characteristic is also important.

Similarly, in the field of MRI, tests must be designed that provide information


about the effect of distortion on the signals of interest, that is, on the shape of the

RF pulse employed during excitation of the magnetization.

To achieve this goal, the ability of the frequency-offset Cartesian feedback sys-

tem to significantly reduce the power amplifier distortion has been investigated with

MRI-like pulses such as sinc pulses (both windowed and non-windowed), rectangular

pulses, skewed triangular pulses, and Very Selective Saturation (VSS) pulses. In this

section, the linearity enhancement obtained with frequency-offset Cartesian feedback

is demonstrated with a sinc pulse and with a VSS pulse. Similar improvements have

been demonstrated with the other pulse shapes as well.

5.5.1 Sinc Pulse Linearization Test

A sinc pulse is used in MRI to perform slice selection, that is, to select spins in a plane

through the object. Slice selection is achieved by applying a one-dimensional, linear

magnetic field gradient during the sinc pulse interval. The gradient creates a Larmor

frequency proportional to the position of the spins. Only the spins at spatial locations

with Larmor frequency that falls within the rectangular frequency bandwidth of the

sinc pulse will be excited. An accurate fidelity of reproduction of the desired sinc

shape is thus important in order to avoid out-of-slice artifacts reducing the quality of

the image.

The two upper panels of Figure 5.6 show real and imaginary parts, respectively,

of the measured output sinc pulse produced by only the power amplifier (right) and

by the amplifier after feedback linearization had been applied (left) in response to a

reference input equal to a sinc pulse of bandwidth equal to 400 Hz. Both plots are

overlaid on the reference (desired) signal. The two lower panels of the same Figure

5.6 show the amplitude and phase distortion before and after linearization.

Over a factor of 10 attenuation of both amplitude and phase distortion is demon-

strated by these plots. The fact that both types of distortion can be successfully

reduced is of particular significance in MRI. In fact, contrary to some applications

in communications, MRI uses linear modulation schemes, that is, the information is


WITH CONTROL WITHOUT CONTROL

ReferenceExperimental

-5

0

5 Am

pl. Error [%]

0 2 4 6 8 10Time, ms

-10

0

10

Phase Error [deg]

Time, ms

Norm

. Real Part

Norm

. Imag. Part


Am

pl. E

rror

[%]

Phas

e Er

ror [

deg]

Nor

m. R

eal P

art

Nor

m. I

mag

. Par

t

0

0.5

-0.2

-0.1

0

-5

0

5

-10

0

10

1.0

0

0.5

-0.2

-0.1

0

1.0

0 2 4 6 8 10

Figure 5.6: Sinc pulse test. The measured sinc pulse at the output of the power am-plifier driven directly is overlaid on the reference sinc pulse in the two upper panelsat the right (showing real and imaginary parts, respectively). The two bottom panelsbelow show that amplitude and phase errors are ±5% and ±20, respectively. Mem-ory effects are also evident, especially in the phase behavior. After addition of thefrequency-offset Cartesian feedback system, the four plots at the left are obtained.The amplitude and phase errors are reduced to less than ±1% and ±2, respectively,even if the power amplifier behavior is not memory-less. Simple pre-distortion tech-niques based on look-up tables are not able to compensate for memory effects, hence,would not have been able to demonstrate the same result.

transmitted in both the amplitude and phase of the RF signal. It is thus very impor-

tant that the MRI power amplifier system has not only low AM-AM distortion, but

also low AM-PM distortion and phase distortion.

In addition to reducing the AM-AM and AM-PM distortion of the power am-

plifier, the frequency-offset Cartesian feedback system also reduces memory effects.

Memory effects are evident in the two lower panels of Figure 5.6, before the feedback

linearization is applied. Here amplitude and phase distortions are not functions of

the amplitude and phase of the signal only: instead, the two halves of the waveform


Abs(TX) Error

-0.5

0

0.5


-0.5

0

0.5

-5

0

5

Am

pl. Error [%]

0 0.5 1 1.5 2 2.5 3Time, ms

0.5 1 1.5 2 2.5 3

-10

0

10

Phase Error [deg]

Time, ms

Ph(RX)-Ph(TX) Error

Norm

. Real Part

Norm

. Imag. Part


Am

pl. E

rror

[%]

Phas

e Er

ror [

deg]

Nor

m. R

eal P

art

Nor

m. I

mag

. Par

t

-0.5

0

0.5

-0.5

0

0.5

-5

0

5

-10

0

10

0

WITH CONTROL WITHOUT CONTROL

Figure 5.7: VSS pulse test. The measured VSS pulse at the output of the poweramplifier (driven directly) is overlaid on the reference VSS pulse in the two rightupper panels (showing real and imaginary parts, respectively). The two right bottompanels show that amplitude and phase error are ±5% and ±20, respectively. Mem-ory effects are also evident, especially in the phase behavior. After addition of thefrequency-offset Cartesian feedback system, the four plots at the left are obtained.The amplitude and phase error are reduced to less than ±1% and ±2, respectively.

before and after its peak value differ, for example, by both maximum amplitude error

(5% vs. 6%) and maximum phase error (+10 and -10). After linearization, these

errors are equally reduced; types of linearization other than frequency-offset Carte-

sian feedback (such as pre-distortion with simple look-up tables) would not have been

able to demonstrate this result.

5.5.2 VSS Pulse Linearization Test

While a sinc pulse is a widely used MRI shape, its modulation bandwidth is usually

very limited. To challenge the ability of the frequency-offset Cartesian feedback to

improve the fidelity of reproduction for pulse shapes of higher modulation bandwidth,


a similar test was performed with a Very Selective Saturation (VSS) pulse as the

reference signal [67, 77].

Unlike conventional Shinnar-Le Roux (SLR) pulses [62], VSS pulses are quadratic

phase modulated (linear frequency sweep) as well as amplitude modulated to spread

the energy evenly throughout the entire pulse duration. Despite their short dura-

tion and small in-band equi-ripple (equal amplitude), they possess large excitation

bandwidths and narrow transition bands (sharper edge profiles) useful, for exam-

ple, to overcome the geometric restrictions in volume prescription and chemical shift

registration errors in MRI spectroscopy.

The VSS pulse used in this test had 5 kHz excitation bandwidth. Figure 5.7

compares the results of driving the power amplifier directly and adding the frequency-

offset Cartesian feedback system to the same amplifier. Despite the larger modulation

bandwidth, the system shows performance similar to that demonstrated for the case

of a sinc pulse. The amplitude distortion is reduced from over ±5% (power amplifier

only) to less than ±1% (after addition of the feedback system). The phase error is

reduced from ±10 to less than ±2. Also in this case, memory effects are evident in

the behavior of the power amplifier and are significantly attenuated by the action of

the loop.

5.6 Effect of Linearization on Magnetization

Once the waveshape of a pulse is known, it is possible to calculate the effect that the

pulse has on the magnetization using the Bloch equations [76].

The Bloch equations are a set of coupled differential equations that can be used

to describe the behavior of the magnetization vector under any conditions. Let ~M(t)

be the nuclear magnetization, γ the gyromagnetic ratio, and ~B(t) the total magnetic

field. Then, the Bloch equations are

dMx(t)

dt= γ( ~M(t)× ~B(t))x −

Mx(t)

T2

(5.6)


00.5

1

MZ

10-410-2100

||

0 0.5 1 1.5 2 2.5 3-0.15

-0.09

-0.03

0.03

0.09

0.15

time (ms)


-0.15

-0.09

-0.03

0.03

0.09

0.15

MZ

A

00.51

10-410-210 0

0 0.5 1 1.5 2 2.5 3time (ms)

0 0.5 1 1.5 2 2.5 3time (ms)



WITH CONTROLWITHOUT CONTROLREFERENCE

Figure 5.8: Magnetization profile of VSS pulse. While the time envelope of the VSSpulse at the output of the power amplifier driven directly (top, first plot) does notappear appreciably different from the reference signal (top, second plot), the effectof the distorted and reference pulses on the magnetization does (bottom second andfirst plot, respectively). The suppression band is altered from about 1% (desired)to over 20% the unaltered magnetization. After the addition of the frequency-offsetCartesian feedback system, the desired suppression band is faithfully reproduced.

dMy(t)

dt= γ( ~M(t)× ~B(t))y −

My(t)

T2

(5.7)

dMz(t)

dt= γ( ~M(t)× ~B(t))z −

Mz(t)−Mo

T1

. (5.8)

Here, Mo is the steady state nuclear magnetization, T1 is the time constant that

describes the length of time it takes Mz to return to its equilibrium value (i.e. the

spin lattice relaxation time), and T2 is the time constant that describes the return to

equilibrium of the transverse magnetization, Mxy (aka the spin-spin relaxation time).

Applying the Bloch equations to the 5 kHz bandwidth VSS pulse results of Figure

5.7, the magnetization profiles in 5.8 are obtained. The figure shows that the desired

suppression bandwidth, severely distorted by the power amplifier non-linear behav-

ior, is faithfully reproduced with the addition of the frequency-offset Cartesian loop


-0.25

-0.15

-0.05

0.05

0.15

0.25

-0.25

-0.15

-0.05

0.05

0.15

0.25

00.5

1

MZ

10-410-2100

||

0 0.5 1 1.5 2 2.5 3time (ms)


MZ

A

00.51

10-410-210 0

0 0.5 1 1.5 2 2.5 3time (ms)

0 0.5 1 1.5 2 2.5 3time (ms)



WITH CONTROLWITHOUT CONTROLREFERENCE

Figure 5.9: Magnetization profile of 5 kHz-modulated VSS pulse. Despite the in-creased bandwidth, the system shows performance similar to the case of the un-modulated VSS pulse. The power amplifier alters the two suppression bands fromabout 1% (desired) to over 30% of the unaltered magnetization. After the additionof the frequency-offset Cartesian feedback system, the desired suppression bands areagain faithfully reproduced.

system.

The bandwidth of the VSS pulse can be easily increased by applying a sinusoidal

modulation at the frequency f0. A cosine modulation produces two suppression bands

of equal thickness that are spatially located, and displaced, using

FT (cos 2πf0t) =δ(f − f0) + δ(f + f0)

2(5.9)

where FT is the symbol of a Fourier transform. An increasing modulation at fre-

quency f0 has been applied to the VSS pulse to study the frequency regime in which

the frequency-offset Cartesian feedback system operates without loss of linearization

performance.

Figure 5.9, as in Figure 5.8, compares the magnetization profile for the particular

case of f0 = 5 kHz (making the total bandwidth of the modulated signal 15 kHz). Up

to 70 kHz modulation was applied without loss of performance; at higher modulation


bandwidth, the linearization effect of the loop is lower, consistent with the slow roll-off

of the error amplification created by the polyphase amplifiers.

5.7 Closed Loop Image Rejection Performance

The ability of the frequency-offset Cartesian feedback system to reject negative fre-

quencies is an important figure of merit, as it describes its robustness to the undesired

quadrature mismatches created by the down-mixer and other components of the feed-

back path that may corrupt the desired signal. For this reason, the lower sideband

rejection of the closed-loop system in a 400 kHz span at the center frequency of the

polyphase difference amplifiers (64.5 MHz) has been measured.

Table 5.1 shows the results of this measurement. The sideband rejection of the

closed-loop frequency-offset Cartesian feedback system is highest at 64.5 MHz and,

as expected, is close to the sideband rejection of the polyphase difference amplifiers

measured in the previous experiment (described in Chapter 3). The difference be-

tween the values in this and the previous experiment results from image (negative

frequency) generation by the error and reference input quadrature down-converters,

which are then amplified unequally by the desired polyphase response to negative in-

put frequencies. In this manner, besides mismatches in the quadrature up-converter,

image rejection limits by quadrature down-conversion can translate to sideband gen-

eration. The same argument can be made regarding the observation that the sideband

rejection decreases more rapidly at frequencies lower than 64.5 MHz.

5.8 Summary

In this chapter, the nature of the distortion created by the behavior of any RF power

amplifier has been discussed and the ability of the feedback system to improve the

fidelity of RF reproduction has been demonstrated with different feedback control

loop variables and power amplifiers. Notably, excellent performance is obtained with

much relaxed specifications for the transmitter with respect to conventional Cartesian

feedback systems, thanks to the use of polyphase loop error amplifiers.


Table 5.1: Measured Sideband Rejection of the Closed Loop FOCF System

Desired Frequency Mirror Frequency Measured RejectionkHz MHz dB

63.35 63.65 40.9464.40 63.60 48.8264.45 63.55 57.4964.50 63.50 61.1664.55 63.45 58.6464.60 63.40 56.8464.65 63.35 55.10

The significance of the application of the frequency-offset Cartesian feedback sys-

tem in MRI has been, in particular, substantiated by showing the linearization effects

of using the system with a variety of typical MRI pulses, such as sinc pulses, Gaussian

pulses, VSS pulses, and more.

Improving the fidelity of RF reproduction in MRI is a solution to the challenge

of increasing RF distortion created by the trend of this imaging modality towards

higher RF bandwidth and frequency. Higher fidelity of RF reproduction is beneficial

also for the minimization of the unwanted sources of error in another recent techno-

logical advancement, namely, parallel transmit systems. Here, poor fidelity can cause

inaccurate pulse reproduction, spectral spreading, and poor selectivity if neglected.

However, the issue remains of coil-to-coil coupling at different power levels, which

creates interference patterns and, hence, an inhomogeneous B1 field. In the next

chapter, manipulating the power amplifier output impedance with frequency-offset

Cartesian feedback is presented as a potential solution to this issue.

Chapter 6

Manipulating the Amplifier

Impedance

6.1 Introduction

Coupling between coil elements of a transmit array is one of the key challenges faced

by designers of MRI transmitter arrays. As discussed in this chapter, different meth-

ods of dealing with the challenge have been presented in the literature. The alter-

native method proposed and motivated in this dissertation consists of manipulating

the output impedance of the RF power amplifier using frequency-offset Cartesian

feedback.

The chapter describes the theory behind impedance manipulation, providing an

understanding of how this manipulation can be obtained electronically using the

feedback method and system described in the previous chapters. The results of sim-

ulations based on this mathematical analysis are also presented. Then, the “load

pull” method and setup is also presented, which was added to the frequency-offset

Cartesian feedback system described in Chapter 4 to measure the output impedance

of the power amplifier under transmit power conditions.

Finally, the results of electronically manipulating the output impedance of both

single amplifiers (with and without terminating circulator) and balanced amplifiers

are presented, which demonstrate the ability to predictably manipulate the real and

95

CHAPTER 6. MANIPULATING THE AMPLIFIER IMPEDANCE 96

imaginary parts of the series output impedance of the power amplifier from very low

to very high values.

6.2 The problem of Coil Interactions

The principal difficulty in the use of a transmitter array of coils is one of electro-

magnetic coupling interaction. This interaction can occur not only directly between

the coils themselves, but also via the sample; hence, quantifying its effect on the coil

currents and electromagnetic field a priori is a very challenging undertaking. Even

if the equations were available for an array of known geometry and frequency char-

acteristics, a new set of equations would be necessary for each imaging procedure,

since the nature of the sample and its relative distance to each array element will be

different each time.

A practical way to describe the problem of coil interactions makes use of the

concept of mutual impedance between the coils. The mutual impedance is defined

by:

Zij =ViIj

= Rij + jXij (6.1)

where Ik = 0 for each value of k other than j. Zij is the ratio of the voltage at port

i due to an excitation current at port j, with all other ports open-circuited. It is

important to note that this includes the effects of other coils, which may still affect

the mutual impedances although open-circuited (Ik = 0). Rij is the mutual resistance

of Zij, and Xij is the mutual inductance. When the mutual inductance is not null,

the currents driven in each coil by the power amplifiers of the array (or, induced in

each coil by the sample magnetization) will in turn induce currents in the other coils.

During the past years, dealing with the problems of coil interaction has been

an active topic of discussion in the MRI community; different approaches have been

proposed in the literature, some of which are briefly discussed below. As an alternative

to these methods, the possibility of reducing coil interactions by manipulating the

output impedance of the RF power amplifier using Cartesian feedback methods is

described and motivated.


=

== 0

XL

XC I(t)

XL

XCZout

λ/4

Z@λ/4

XL

XC

XL

XC

λ/4

V(t)

Figure 6.1: Equivalent representation of different impedance manipulation techniques.Depending on the length of transmission line between the power amplifier and coil,some techniques attempt to decrease the power amplifier impedance, others, to in-crease.

6.2.1 Available Methods

Since the early 2000s, several diverse methods have been proposed in an attempt to

suppress inter-element coupling effects in transmitter arrays.

In 2002, Lee developed a current source design to present the amplifier load with a

very high impedance [52]. In 2004, similar to the work of Lee, Kurpad presented the

implementation of a coil-integrated RF power MOSFET acting as a voltage controlled

current source in a transmit phased array coil [47], which eventually led to the so

called active rung concept [48]. Here, one rung of a volume coil connected across

the output terminals of an RF power MOSFET turns the amplifier into a very high

output impedance device. In 2009, Chu [15] proposed a drastically different solu-

tion, consisting of an output-matching network, to drastically lower the drain-source

impedance of the power amplifier output transistor.

What these methods have in common is the attempt to perform some kind of

manipulation of the output impedance of the amplifier: some methods attempt to

increase it; others, to decrease it. The rationale behind these various solutions lies


XL

XCZout

XL

XCZout

λ/4

=

= 0

Z@λ/4 = 0

Figure 6.2: The case of the distributed-C surface transmit coil. The coil is designed sothat the impedance of the inductance XL matches the impedance of the capacitanceXC . If the power amplifier drives the inductance directly, then a power amplifieroutput impedance Zout equal to zero is desirable, since by doing so XL and XC arein parallel. Conversely, if the power amplifier is separated from the coil by a lengthof transmission line equal to λ

4, the same result is obtained with a very high amplifier

output impedance.

in the variety of designs of the system consisting of the power amplifier and loading

coil. If the power transistor is in close proximity to a series LC transmit coil, such

as in Kurpad’s design, a very high impedance ensures that the electromotive force

induced by the neighboring coils will create a negligible current. Conversely, if the

power amplifier is separated from the same transmit coil by a λ4

stub of transmission

line, the same result can be obtained if the output impedance of the power amplifier

is very low: at the input of the transmit coil, this impedance will be transformed into

a very high impedance by the transmission line. The transmission line equations for

the impedance transformation are

Zs(l) = Z0Zout + jZ0 tan βl

Z0 + jZout tan βl(6.2)

where β = 2πλ

is the wavenumber, Zout is the output impedance of the power amplifier,


XL

XC

Zin

XL

XC

is a short circuit

(ideally)

PIN diode

Figure 6.3: MRI receive technologies. A PIN diode (bottom, during the transmitinterval) or a properly-matched preamplifier input impedance (top, during the re-ceive interval) presents a short-circuit to the surface receive coil. If XL = XC , thisshort-circuit allows the tank circuit inductor and capacitor to create a very high in-put impedance, which open-circuits the coil. Methods of dealing with transmit coilinteractions based on power amplifier impedance manipulation attempt to emulatethese “Q-spoiling” techniques.

and Zs is the impedance seen by the transmit coil separated from the power amplifier

by a transmission line of length l and characteristic impedance Z0. Z0 is defined as

the coaxial characteristic impedance. If l is null, then

Zs(l) = Zout (6.3)

and to minimize induced currents, Zout should be very high. If l is λ4, then

Zs(l) =Z2

0

Zout(6.4)

and to minimize induced currents, Zout should be very low (Zs should be high). See

Figure 6.1.

If the transmit coil is as shown in Chapter 4 (a distributed-C surface transmit


coil with input series inductance), then the design requirements would be opposite

from the ones described above. Let us consider the two situations in Figure 6.2.

The distributed-C surface transmit coil is designed so that the impedance of the

inductance XL matches the impedance of the capacitance XC . If the power amplifier

drives the inductance directly, then a power amplifier output impedance Zout equal

to zero is desirable, since by doing so XL and XC are in parallel and thus present

a very high impedance to any current induced in the coil by the neighboring array

elements. Conversely, if the power amplifier is separated from the coil by a length of

transmission line equal to λ4, the same result is obtained with a very high amplifier

output impedance (the transmission line transforms the latter into a short-circuit at

the input of the coil inductor).

For all intents and purposes, methods of dealing with transmit coil interactions

based on power amplifier impedance manipulation attempt to emulate Q-spoiling

techniques based on PIN diodes or preamplifier decoupling. As shown in Figure 6.3,

these techniques—commonly used in standard MRI receive technologies— present

a short-circuit at the input of the receive coil series inductor. If XL = XC , the

short-circuit will allow the inductor and capacitor to resonate and create a very

high impedance, which open-circuits the coil. Q-spoiling based on PIN diodes is

useful during the transmit interval, to avoid measuring the very high signals that may

otherwise damage the receive circuitry. Q-spoiling based on preamplifier decoupling

is useful during the receive interval, to create a virtual short so that an adjacent

coil is not detuned by resonant coupling when both coils are activated in the receive

interval.

While promising, all of the methods of power amplifier impedance manipulation

described above have drawbacks and limitations: they assume linear behavior of the

amplifier, lessen the power amplifier efficiency, and are unable to accurately predict

the exact value of the output impedance of the amplifier as well as to maintain the

latter constant over a wide range of output power. In addition, they are designed to

work only for particular configurations of the system including the power amplifier

and loading coil, which limits their applicability significantly.

Given these considerations, a method that allows one to predictably obtain any


arbitrary value of the amplifier output impedance is very appealing, as it would be

applicable not only to both the situations without transmission line and with a λ4

stub of transmission line but also to any other situation with any arbitrary length of

transmission line.

In addition, if impedance manipulation were obtained with a feedback method,

the drawbacks and limitations of the methods described above—such as the linearity

assumption and the lessening of the amplifier efficiency—could be overcome. As ex-

plained in the previous chapter, some of the most significant advantages of a feedback

technique are that the non-linearity of the power amplifier needs not to be character-

ized a priori and the power amplifier efficiency is not compromised. Moreover, any

changes to the power amplifier behavior can be predicted with the use of the loop

gain equations, and these changes are usually constant over a wide range of output

power and frequency (within the control bandwidth of the loop).

In the next section, the method of arbitrarily manipulating the output impedance

of RF power amplifiers based on frequency-offset Cartesian feedback is described.

6.3 Theory of Impedance Manipulation

To explain how impedance manipulation can be achieved with Cartesian feedback

methods, the simplified representation of the frequency-offset Cartesian feedback in

Figure 6.4 can be used. Here the power amplifier is loaded by a transmit coil, sepa-

rated from it by an arbitrary length of transmission line. The assumption is that the

coupler can provide, as the loop feedback variable, a sample Vc of the transmission

line forward voltage (Vf ), or reflected voltage (Vf ), or any combination of the two:

Vc = αVf + βVr. (6.5)

Vc can be obtained, for example, by using a coupler model C7149 by Werlatone and

two vector multipliers described later; α and β are, in this case, the complex weighting

coefficients of the multipliers.

For simplicity, let us also assume that the reference input signal (x(t)) to the


out

x(t)

feedback

Transmitter

COUPLER

ph. ctlr

AMPLIFIER

i1(t) i2(t)

dΦ2(t)

αVF(t)+βVR(t)

VF(t)VR(t)

l

di1(t)

s

Figure 6.4: Simplified frequency-offset Cartesian feedback system with power ampli-fier loaded by a transmit coil, separated from it by an arbitrary length of transmissionline. The loop forces a precise relationship between Vr and Vf , hence, a precise valueof the reflection coefficient ΓA at the output of the power amplifier. The desired valueof the reflection coefficient depends on the length of transmission line. The goal is toobtain a transmit coil that presents a very high impedance to the other coils in thearray.

transmitter is zero; hence, the desired coil current i1(t) should also be zero. Without

any measure in place to avoid it, the electromagnetic interaction between the coil

and its neighbor— carrying a current i2(t)—will cause a non-zero current i1(t) to

circulate. Manipulating the amplifier impedance Zout in order to obtain a very high

source impedance Zs is the key to minimize this induced current. The necessary value

of Zout will depend on the length of the transmission line l.

As explained in Chapter 4, with x(t)=0, the effect of the feedback loop is to force

the feedback variable Vc equal to zero, that is,

Vc = αVf + βVr = 0. (6.6)

Hence, the loop forces a precise relationship between Vr and Vf :

VfVr

= −βα. (6.7)

The ratioVf

Vris, by definition, the reflection coefficient ΓA at the output of the power


amplifier, which is also described as

ΓA =VfVr

=Zout − Z0

Zout + Z0

(6.8)

where Z0 is the impedance of the coupler (approximately 50 Ω). In summary, weight-

ing the transmission line voltages with coefficients α and β synthesizes an impedance

at the output of the power amplifier described by

Zout = Z01 + ΓA1− ΓA

= Z0

1− βα

1 + βα

(6.9)

which can be tuned in order to obtain a very high impedance Zs(l) using the trans-

mission line equation.

If, for example, l is zero, then Zout should be very high—ideally, infinite; hence,

the denominator of Eq. 6.9 should be zero and

ΓA = −βα

= 1. (6.10)

If instead l is a quarter wave length, then Zout should be very low—ideally, zero;

hence,

ΓA = −βα

= −1. (6.11)

In summary, choosing the weighted combination αVf + βVr as the feedback variable

of the frequency-offset Cartesian feedback allows manipulating the output impedance

of the power amplifier in order to obtain a value that depends only on the ratio of

the two weighting coefficients.

6.4 Load Pull Setup

To characterize the power amplifier impedance behavior under transmit power condi-

tions, the automated load-pull and hot S22 described by Scott [69] was used. Mea-

surements are based on the concept of a nonlinear Thevenin circuit in which equiva-

lent open circuit output voltage and output series impedance can be assigned at each


Figure 6.5: Simplified schematic of load-pull setup. By switching between two differ-ent known output loads and measuring output voltage and current for each load, theinternal amplifier impedance responsible for the output level change can be calculated.

power level. By switching between two different known output loads, ZL1 and ZL2,

and measuring output voltage and current, the internal amplifier impedance respon-

sible for the output level change can be calculated. The amplifier output impedance

is estimated by

Zout = −V1

I1

V2

V1− 1

I2I1− 1

(6.12)

where V1 (V2) and I1 (I2) are the total voltage and total current, respectively, at the

output of the power amplifier loaded by ZL1 (ZL2). To provide a sample of these volt-

ages and currents, the variation of the Bruene directional coupler described in Chapter

4 was included in the frequency-offset Cartesian Feedback system. The dummy loads

ZL1 and ZL2 had a nominal impedance equal to 50 Ω and 43 Ω, respectively. To

switch between the two loads, a high power coaxial single-pole double-throw (SPDT)

relay model CX-230 by Tohtsu was also included in the loop. The total voltage and

total current samples were multiplexed by a SPDT switch model ZX80-DR230-S+ by

Mini-Circuits to the MEDUSA console described in Chapter 4.


Figure 6.6: Simplified frequency-offset Cartesian feedback setup with a single com-biner to create a very high (or very low) output impedance of the power amplifier.

6.5 Impedance Control System Configuration

The case of a power amplifier separated from the transmit coil by a length of transmis-

sion line equal to an even or odd multiple “n” of a quarter wavelength is very common.

For this reason, synthesizing a very high (when n is even) or very low (when “n” is

odd) output impedance of the RF power amplifier is particularly important.

As demonstrated above, to obtain an open or short-circuited output impedance,

the weighting coefficients α and β of the transmission line voltages must be equal

in magnitude and equal or opposite in sign. In these simple scenarios, the use of

simple combiners at the Vr and Vf ports of the directional coupler is sufficient: a 180

combiner allows one to fix α = -β, hence, ΓA = 1 (high Zout); a 0 combiner allows

one to fix α = β, hence, ΓA = -1 (low Zout).

Figure 6.6 shows the simplified frequency-offset Cartesian feedback system includ-

ing only the directional coupler by Werlatone, in which both samples of the forward

and reverse voltages are summed (or, subtracted) by a 0 (180) combiner model


Figure 6.7: Simplified schematic of a vector multiplier consisting of a two-stagepolyphase filter, four-quadrant multiplier AD835 (here shown as two mixers), DACSLT1655, and output buffer LT1395.

ZFSC-2-1+ (ZFSCJ-2-1+) by Mini-Circuits before they are sent to the down-mixer

of the feedback transmitter. This setup also allows one to simply disconnect the sam-

ple Vr from the coupler (that is, β is zero); in this case, the power amplifier output

reflection coefficient is forced to be equal zero, hence,

Zout = Z0 ≈ 50Ω. (6.13)

The output impedance of the power amplifier will be equal to the coupler character-

istic impedance Z0.

Almost any other arbitrary value of output impedance can be obtained when two

vector multipliers are integrated in the frequency-offset Cartesian feedback system,

between the directional coupler and a 0 combiner.

Vector multipliers [71] are circuits that can be used to individually shape the

waveforms provided by the coupler sampling the forward and reverse voltages at the

output of the power amplifier.

Each vector multiplier used in this research work incorporates a two-stage passive

polyphase filter to generate differential precision phase shifts in 90 increments. The


Figure 6.8: Simplified frequency-offset Cartesian feedback setup with a pair of vectormultipliers—each one weighting Vf or Vr to create any arbitrary output impedanceof the power amplifier.

RC time cross-over frequencies are chosen to bracket 64 MHz ± 2 MHz to provide

broadband phase shift precision. The quadrature signals are then multiplied by 250

MHz, voltage output four-quadrant multipliers, model AD835 by Analog Devices,

with the output of two 16 bit serial Digital-to-Analog Converters (DACs), model

LT1655 by Linear Technology. The latter represent the cosine and sine weightings to

synthesize the signal A sin(ωτ+ψ) as A sin(ψ) cos(ωτ)+A cos(ψ) sin(ωτ). The output

of the vector multiplier is driven by a buffer made with an LT1395 current-feedback

amplifier by Linear Technology.

The input waveform amplitude can be attenuated by a minimum of 10 dB to a

maximum of 60 dB, and the phase can be shifted continuously over the entire 360

range. Waveform generator and control are provided by Matlab via a serial (RS-232)

connector and Serial-to-USB converter interfacing the vector modulator with a PC.

A simplified schematic of the vector multipliers is shown in Figure 6.7. The

frequency-offset Cartesian feedback system with vector multipliers is shown in Figure


error

3 dBQUAD

HYBRID

3 dBQUAD

HYBRID

50Ω

50Ω

QUAD HYBRID BALANCED POWER AMPLIFIER

in

Z

PA

PA

Figure 6.9: Simplified schematic of a balanced amplifier. A balanced amplifier has twomatched amplifying devices that are run in quadrature. A 200 W RF 3 dB hybrid onthe input creates two quadrature signals from the single RF signal; a second, identical3 dB hybrid on the output recombines in phase the two quadrature signals. If thetwo amplifying devices are well matched, the balanced amplifier has excellent inputand output return loss (input and output impedance are approximately 50 Ω).

6.8. Here one vector multiplier receives the Vf sample, the other, the Vr sample.

Experiments have been conducted initially with the custom-made amplifier de-

scribed in Chapter 4. The output impedance of the latter was measured and was

found to vary from 35 Ω to 72 Ω in the range of output power up to 150 W.

Then, experiments were conducted with a balanced amplifier obtained by assem-

bling custom-made amplifiers in pairs. A balanced amplifier architecture, whose sim-

plified schematic is shown in Figure 6.9 [22,23], is created when a well-matched pair

of power amplifiers is combined with a pair of 3 dB quadrature hybrids (schematic

and phase truth table shown in Figure 6.10). In Figure 6.9, the hybrid on the input

splits a signal into two equal amplitude outputs that are 90 apart; the second hybrid

on the output shifts the signals by the opposite amount at the amplifier outputs so

they combine in phase at the load port.

The main advantage of a balanced architecture is that, in principle and if the

amplifying devices are well-matched, the input and output return loss is very low

(the input and output impedance of the balanced amplifier is approximately 50 Ω)

and does not vary with the output power or biasing conditions. The output impedance

of the balanced amplifier built during this research work was measured and found to


-90°

-90°0°

0°

A

D

C

B

A

ISO

0

-90

B

ISO

-90

0

C

0

-90

ISO

D

-90

0

ISO

A

B

C

D

ISO = Isolation

Figure 6.10: Simplified schematic of 3 dB quadrature hybrid and phase truth table.

vary from 40 Ω to 44 Ω in the range of output power up to 150 W.

6.6 Experiments

6.6.1 With Single Power Amplifier

Figure 6.11 shows the experimental results obtained with the setup in Figure 6.6

while the reference input of the transmitter was driven with a sinusoid of increasing

amplitude (up to 150 W RF output power) and of frequency equal to the center

frequency of the RF control bandwidth. The minimum value of the impedance (α =

β) is 2 ± 0.3 Ω; the maximum (α = –β), 432.7 ± 39.0 Ω. The output impedance

obtained with β = 0 is 49.1 ± 1.1 Ω. Importantly, these values deviate by less than

10% over a wide range of output power.

The Smith chart in Figure 6.12 shows the measured values of power amplifier out-

put reflection coefficients obtained with the setup in Figure 6.8. The input reference

signal was, this time, constant in both amplitude and frequency (once again, at the

center frequency of the RF control bandwidth). Each of the closed trajectories in

the figure was obtained by keeping the ratio of α and β constant in magnitude and

varying in phase. Ideally, these trajectories would be perfect circles on the Smith

chart, that is, regions of the reflection coefficient of constant magnitude and varying


1

10

100

1000

1.7 10.2 18.7 27.2 35.7 44.2 52.7 61.2 69.7 78.2Vout [Vrms]

Zou

t [oh

m]

86.5

Figure 6.11: Experimental plot of the output impedance, obtained by summing (bluetrace) or subtracting (green trace) Vf and Vr, at increasing RF output voltage. Theplot also shows the output impedance obtained by disconnecting Vr (red trace).

phase. Nevertheless, simulations and measurements match well, especially for lower

values of |ΓA| (|ΓA| ≤ 0.5); at higher values, deviations from theory result from the

open-loop impedance of the amplifier, load-pull coupler losses, and the finite value of

the loop gain.

6.6.2 With Balanced Power Amplifier

Figure 6.13 shows the experimental results obtained with the setup in Figure 6.6

while the reference input of the transmitter was driven with a sinusoid of increasing

amplitude (up to 150 W RF output power) and of frequency equal to the center

frequency of the RF control bandwidth. The feedback control variable was Vf . Here

the measured series impedance (both real and imaginary parts) obtained with a single

power amplifier is compared to that obtained with a balanced amplifier. Similar

results are obtained, as expected; however, some differences are worth mentioning.

First of all, the imaginary series impedance obtained with a balanced amplifier is

approximately null over the entire range, while about -10 Ω imaginary impedance


|β/α| = 0.8

|β/α| = 0.6

|β/α| = 0.4

|β/α| = 0.2

Figure 6.12: Smith chart of the experimental power amplifier output reflection coef-ficient obtained with constant magnitude of the ration of α and β and varying phase.


1.7 10.2 18.7 27.2 35.7 44.2 52.7 61.2 69.7 78.2Vout [Vrms]

Zou

t [oh

m]

86.5-30

-20

-10

0

10

20

30

40

50

60

Real Zout, single amplifier

Real Zout, balanced amplifier

Imag Zout, single amplifier

Imag Zout, balanced amplifier

Figure 6.13: Experimental plot of the series real and imaginary output impedance,obtained with a balanced amplifier and a single amplifier, at increasing RF outputvoltage. The feedback control variable was Vr.

was measured with a single power amplifier. In addition, both magnitude and phase

of the series impedance obtained with a balanced amplifier vary by about ±1% over

the entire range of output power, while those of the series impedance obtained with

a single amplifier vary by about ±5% over almost the entire range. These differences

are explained by the different output impedance behavior of the two power amplifiers.

While the single power amplifier output impedance is 53.5±18.5Ω (53.5Ω±34.5%) in

the 150 W output power range, the balanced power amplifier impedance is 42±2Ω

(42Ω±4.8%) in the same range. The feedback loop attenuates both deviations from

the desired value of output impedance (50 Ω) by the same amount; however, the

initial ratio between the two stays approximately the same.

6.7 Summary

In this chapter, power amplifier impedance manipulation with frequency-offset Carte-

sian feedback has been presented as a possible solution to one of the key challenges


faced by designers of MRI transmitter arrays, namely, the problem of coupling be-

tween coil elements.

The setup and results of experiments conducted with both single and balanced

power amplifiers have been shown and demonstrate that the output reflection coef-

ficient of these amplifiers can be manipulated to obtain almost any value within a

large area of the Smith chart, and is stable over the power range.

In principle, since Zout is tunable, any length of coaxial cable can be used between

the power amplifier and transmit coil and the system can be used with a variety of

different amplifier and coil designs.

Chapter 7

Conclusion

The research in this dissertation grew from the study of different methods and tech-

nologies, and from their successful integration and application to the field of MRI.

Petrovic’s publications on Cartesian feedback (1984), Marshall’s patent of active

polyphase amplifiers (1986), Kurpad’s work on increasing the impedance of power

amplifiers for arrays of MRI coils (2006), and several other concepts and resources

were merged and applied in this research work, which led to the original solution to

the problems of MRI RF transmit power generation described in this dissertation.

The main lesson that I have learned through my research is that we tend to

approach the science of MRI from a very limited angle, looking for knowledge and

inspiration within the community of scientists to which we belong. The result is that

“new” technologies are invented 20 years after they were first proposed in a different

field. Opportunities to accelerate growth are missed. Conversely, great ideas and

advancements are confined and do not reach the potential they truly deserve. We

should stop looking at scientific fields as “compartments,” and be open to discussing

problems and resources outside our comfort zone. The new kinds of solutions that

can be created with this approach will benefit everyone.

MRI RF systems are quickly reaching a level of complexity that calls for a deeper

relationship between this field and the field of communications. As the MRI frequency

and bandwidth increases, the overlap with the operating frequency of current radio

systems and devices increases. As MRI transmitter array systems gain acceptance,

114

CHAPTER 7. CONCLUSION 115

the use of multiple antennas at both the transmitter and receiver will also contribute

to attenuating the differences between the two fields.

Obviously, some elements will certainly remain to differentiate MRI from commu-

nications substantially. While the concepts and ideas can be translated from one field

to another, the requirements of their particular implementation will perhaps always

be very different. The frequency-offset Cartesian feedback system presented in this

dissertation would look profoundly different in a smart phone.

This is a very exciting time to be working in MRI. There is a lot of work to be done

to make the MRI RF systems more linear, more efficient, safer, and faster. The same

architecture presented in this dissertation offers many opportunities for continuous

advancement and further investigation:

• Efficiency boosting methods, such as the Doherty technique, could be applied

with frequency-offset Cartesian feedback to lower the RF dissipation in MRI.

• Digital predistortion could be also merged with the present solution, to develop

the frequency-offset variation of the two-point architecture described by [16].

• Different compensation methods could be explored that will relax the require-

ments for stability and allow one to obtain higher modulation bandwidths. (For

example, polyphase loop error amplifiers with lead-lag compensation can be im-

plemented.)

• A frequency-offset Cartesian feedback with tunable center frequency of the con-

trol bandwidth could be obtained using voltage-controlled digital resistors (in

a discrete design) or switched-capacitors (in an analog design) in place of the

fixed-value crossing resistors that couple the quadrature signals in the polyphase

architecture. This architecture would make frequency-hopping for multi-slice

applications possible, simply by tuning the center frequency of the polyphase

amplifier’s complex IF bandwidth.

• Similarly, a tunable-RF-frequency Cartesian feedback based on polyphase am-

plifiers could be demonstrated by using a voltage-controlled oscillator (VCO) or

CHAPTER 7. CONCLUSION 116

direct digital synthesizer (DDS) in place of the fixed local oscillator currently im-

plemented in the Cartesian transmitter. The system would thus be adaptable to

image multiple nuclei, such as carbon-13 (gyromagnetic ratio 10.71 MHz/T) and

phosphorus-31(gyromagnetic ratio 17.25 MHz/T), at different field strengths .

• A re-designed Cartesian feedback transmitter could be useful that integrates on-

board the main discrete components of the system. For example, two on-board

vector modulators could be obtained using two feedback signal inputs, two vari-

able gain feedback amplifiers, and two phase shifters (one for each path). Also,

better matching of the chip’s input/output dynamic range to the typical MRI

power amplifier’s range would be desirable, to eliminate the need for significant

attenuation in the feedback path that currently makes the system susceptible

to the noise generated by the down-mixer.

We are in the middle of a revolution in new kinds of MRI RF systems to solve

some of the crucial problems that limit the applicability of the most recent MRI

technologies. I feel that frequency-offset Cartesian feedback methods are an oppor-

tunity to deal with these problems successfully and I look forward to seeing what the

beginnings described in this dissertation will set in motion.

Appendix A

Active Cable Trap

A.1 Introduction

This appendix presents the “active cable trap” concept and prototype for attenuation

of the currents induced in interventional devices. It is, in essence, a miniaturized and

simplified version of the frequency-offset Cartesian feedback for MRI power amplifiers

described in this dissertation.

The idea and prototyping of the active cable trap actually preceded in time the

development of the frequency-offset Cartesian feedback system, and served as a learn-

ing tool that inspired much of the design choices regarding the latter (in particular,

the auto-calibration setup).

The active cable trap concept was invented to deal with the problem of induced

RF currents in interventional devices by the RF fields of MRI. At the sharp end of

the device, if the conditions for resonance are established, the local electric field is

high and can induce currents in a body in its close vicinity. These currents can, in

turn, create significant heating and thus burns.

In this appendix, the theory and previous work on RF-induced currents and heat-

ing in interventional devices is revisited. The architecture of the active cable trap is

then discussed. Finally, the results of experiments on looped wire current attenuation

are presented.

117

APPENDIX A. ACTIVE CABLE TRAP 118

A.2 Theory and Previous Work

The high-power RF fields employed by MRI can induce RF currents in long conductors

found in interventional devices such as guidewires, causing RF heating effects and

posing safety risks during the clinical use of MRI in the presence of these devices [79].

These currents can be dangerously high especially if the conditions for resonance

and the generation of a transverse electromagnetic (TEM) field are established. In

the simple case of a straight, bare wire parallel to the symmetry axis of the MRI

scanner, these conditions exist when the wire length is an integer number of half the

wavelength of the dissipative medium.

Embedded in air, the bare, straight wire has a 230 cm minimum resonant length

at 64.0 MHz; embedded in tissue, it has a minimum resonant length of 21 cm at that

same frequency.

In reality, the situation of a straight, bare wire in a completely homogenous,

isotropic medium is not one commonly encountered in an MRI scanner. Typically,

the wire construction and the nature of its environment can change the value of the

resonant length significantly, making the task to predict potentially dangerous reso-

nant conditions very difficult. These uncertainties of RF safety lead to unpredicted

risks of adverse events in unsafe settings, to the disqualification of entire patient pop-

ulations from receiving any form of MRI scan (even in situations where it might be

safe), and to limitations in the development and use of interventional MRI devices.

Studies conducted with an optically-coupled system—which was designed to de-

tect RF currents in long conductors external to the patient [81]— revealed that cur-

rents up to several hundred mA can be induced. Moreover, the time constants associ-

ated with the onset of these currents are not long enough to allow the MRI operator

to react promptly once the dangerous resonant conditions are established. Therefore,

reliable methods and sub-systems are needed to attenuate the undesired currents

automatically, without any external intervention.

Many devices and wire constructions have been proposed in the literature to at-

tenuate the risk of wire resonance and heating. The devices most often used are

cable traps. Cable trap devices prevent heating conditions by targeting regions of


high RF current, where energy storage is inductive. Toroidal cable traps inductively

couple a high resistance in series with the wire to spoil the resonant Q or modify the

resonant frequency of the wire. Here, the wire itself acts as the primary coil, and a

resonant toroidal cavity enfolding the wire acts as the secondary coil. In 2000, Ladd

and Quick [50] added coaxial chokes with length λ4

to coaxial cables in order to reduce

the amplitude of the MRI-induced currents on the cable shields. In 2005, Hillenbrand

[30] described the “bazooka coil,” a balun-style RF trap, and demonstrated its ca-

pability to reduce the formation of resonating RF waves on long conductors during

transmission. Hillenbrand also interfaced the cable trap to a preamplifier circuit, used

in this case as a dual-mode receiver.

As an alternative to cable traps, the use of lossy dielectrics (also called Q-spoilers)

has been proposed [79]. The principle of these devices is that of coupling of the

free-end of a wire–where the E field is maximum and energy storage is primarily

capacitive—with a lossy dielectric: by inserting the wire into the dielectric, the quality

factor of the resonant wire is reduced, thereby reducing the dangerous resonant effects.

All of the above are passive methods and thus share one important limitation,

that is, their ability to attenuate wire currents is significantly restricted by component

loss and tuning interactions. In experiments with the optically-coupled monitoring

system, it was demonstrated that a Q-spoiling lossy dielectric reduces the RF power

of a resonant guidewire current to 14% of its value in the free wire, and a properly

tuned balun-style RF trap reduces it to 23%; however, variations in the guidewire

environment or in the devices inevitably changed the resonant characteristics and

can drastically impact these outcomes.

A.3 Feedback Method for Current Attenuation

A potential solution to the problem of RF heating in guidewires is to create an auto-

matic negative feedback control system to sense and attenuate the RF wire current. A

device capable of doing so is a much simplified version of the frequency-offset Carte-

sian feedback system described in the previous chapters: equipped with the same

current sensor used in the optically-coupled monitoring system, the system would, in


Sensor

LO

Actuator

QUADRATUREUPMIXER

OPENLOOP

CLOSEDLOOP

WIRE

TEST

RF‐INDUCED CURRENT

/2 p

hase

sp

litte

r

QUADRATURE DOWNMIXER

LO

balun balun

/2 p

hase

sp

litte

r

Vcm

Vcm

Sensor

LO

Actuator

AD5385UP-MIXER

WIRERF-INDUCED CURRENT

AD8348DOWN-MIXER

LO

balun balun

phase splitter

phase splitter

POLYPHASE AMPLIFIERS

Figure A.1: Simplified schematic of active cable trap. The feedback loop input isthe RF current flowing in a potentially dangerous conductor, which is detected bya toroidal sensor. After down-conversion, the loop compares the detected signal toa DC reference, and amplifies the difference by the polyphase amplifier gain. Afterup-conversion, the amplified error signal drives a toroidal actuator that induces inthe conductor itself an RF current that opposes the one induced by the B1 field.

theory, offer a wire current attenuation equal to the system loop gain over a range

of frequencies designed to match the known bandwidth of RF excitation. To test

these ideas, a miniaturized version of the frequency-offset Cartesian feedback system,

acting as an “active cable trap,” was designed.

Figure A.1 shows a simplified schematic of the active cable trap. The feedback

loop input of the active cable trap is the RF current flowing in a potentially dangerous

conductor, which is detected by a toroidal sensor. After down-conversion, the loop

compares the detected signal to a DC reference, and amplifies the difference (i.e.,

the error signal) by the polyphase amplifier gain H(ω). After up-conversion, the

amplified error signal drives a toroidal actuator that induces in the conductor itself

an RF current that opposes the one induced by the B1 field. As in the frequency-

offset Cartesian control system for MRI power amplifiers, the active cable trap uses a

Cartesian down/up-conversion scheme, so that amplification is obtained at a low-IF


band, where values of loop gain up to 100 (40 dB) and bandwidth of hundreds of

kHz can be obtained; in addition, the polyphase loop error amplifiers eliminate the

possibility of quadrature errors caused by phase and amplitude mismatches, which

in this application could lead to the generation of an “image” or quadrature ghost

excitation by the wire.

As previously discussed, the ability of the control loop to preserve the phase in-

tegrity of the signal is crucial to the stability of the loop. While in the MRI power

amplifier system, phase alignment is obtained with the shift control circuitry inte-

grated in the CMX998 IC, with the active cable trap the phase alignment is obtained

by phase-locking two external signals providing the LO frequencies to the demodula-

tor and modulator, respectively.

The architecture of the active cable trap can be conceptually divided into five

separate elements: a toroidal sensor, a quadrature demodulator, the polyphase loop

error amplifiers, a quadrature modulator, and a toroidal actuator. These elements

are discussed in detail below.

A.3.1 Toroidal Sensor

.

The long conductor, whose RF current must be monitored and reduced, is fed

through a toroid-cavity sensor shown in Figure A.2. The toroid-cavity acts as a

volume-rotated rectangular single-turn transformer secondary and can be thought of

as a self-shielded pick-up loop. It consists of copper tape wrapped around a toroidal

Teflon core with 1.55 mm inner diameter and 5.50 mm outer diameter. Any long

conductor fed through the core of the toroid-cavity acts as the transformer primary.

MRI-induced currents on the wire will couple a magnetic flux to the toroid secondary

given by

Φ = µ0

ln ba

2πg · I = M · I (A.1)

where a, b are the inner and outer radius of the toroid, respectively; g is the length of

the toroidal cavity; I is the wire current, and M is the mutual inductance. Faraday’s


Figure A.2: A toroid-cavity senses the RF currents in a wire fed through it. Thetoroid consists of copper tape wrapped around a toroidal Teflon core with 1.55 mminner diameter and 5.50 mm outer diameter.

law yields the toroid voltage produced by the RF current I,

V = MdI

dt. (A.2)

The sensor outputs a voltage that is proportional to the undesired RF current. During

the MR procedure, this current is induced by the B1 field of the imaging procedure.

The frequency of this signal is the characteristic Larmor frequency of the B0 field,

modulated by the bandwidth of the B1 field.

A.3.2 Quadrature Demodulator

The signal detected by the toroid-cavity sensor is down-converted to occupy the

IF bandwidth in the frequency range 50 kHz – 2 MHz. To implement the down-

conversion of the signal from the Larmor frequency to the IF band, the device model

AD8348 by Analog Devices was selected. The AD8348 is a 50 MHz to 1000 MHz

quadrature demodulator with integrated variable gain amplifier, Gilbert cell mixers,

LO quadrature phase splitter, and integrated baseband amplifiers. The AD8348 offers

a quadrature phase error of less than 0.5 and an I/Q amplitude imbalance of less


Vcm

Vcm

ir

Q

I

if

qr

qf

Ri RF

C

RC

+

−

+

−

Figure A.3: Active cable trap polyphase amplifier. The frequency response of theamplifier has peak gain of 70 (36.9 dB), center frequency of about 330 kHz, and pass-band half-width of about 140 kHz. The final design values of the passive components,all of which have 0.1% tolerance, were Ri = 750 Ω, RC = 20 kΩ, RF = 50 kΩ, C =22 pF (nominal). The two fully-differential amplifiers are model THS4140 by TexasInstruments.

than 0.25 dB, as well as a gain control range of 44 dB to drive the polyphase loop

error amplifiers with up to 3.5 Vpp when a 5 V single-ended supply is adopted.

The power output capabilities of the AD8348 are, however, limited; at the peak

output current of only 1 mA, the demodulator requires a resistive output load of at

least 1 kΩ (single-ended to ground) and capacitive loads of no more than 1 nF series.

Future designs of the active cable-trap should address this limitation.

A.3.3 Polyphase Loop Error Amplifiers

The polyphase loop error amplifier provides the negative amplification H(ω) of the

down-converted signal and shapes the loop signal in order to guarantee stability within

the desired bandwidth.

The architecture is equivalent to the one described in Chapter 3. The amplifier


is realized with two cross-connected fully-differential amplifiers (model THS4140 by

Texas instruments). Each amplifier, arranged in a single-supply fully differential

configuration, is DC-coupled to the demodulator’s differential output.

The passive components in the loop error amplifiers, whose values are described in

Figure A.3, have 0.1% tolerance to guarantee 60 dB sideband rejection to quadrature

mismatches created by the AD8348.

A.3.4 Quadrature Modulator

The output of the polyphase loop error amplifiers drives a quadrature modulator

(model AD5385 50 MHz to 2200 MHz by Analog Devices). At the output frequency

of 65 MHz, the AD5385 offers a quadrature phase error of 0.17 degrees only and a

very low quadrature amplitude imbalance of 0.03 dB.

As in the AD8348, the power output capabilities of the AD5385 are modest (a

maximum power of 8 dBm can be delivered), which makes it possible to induce

wire currents of only about 10 mARMS using the toroidal actuator. While adequate

for testing the proposed application on the bench, future designs should consider

replacing the AD5385 or the toroidal actuator with devices that offer higher output

capabilities. This will allow the active cable trap to work with much higher wire

currents. The AD5385 is powered by a 5V single supply, as is the AD8348.

A.3.5 Toroidal actuator

The toroidal actuator is an exact replica of the toroidal sensor. Driven by the AD5385

modulator output signal, the actuator operates on the long conductor. When the

feedback loop is closed and stable, the toroidal actuator induces in the conductor a

current which is, ideally, the negated and amplified version of the current originally

caused by interactions with the B1 field. The net result is that a substantially smaller

current, inversely proportional to the value of the loop gain, is present on the wire.


Figure A.4: Setup for experimental validation on the bench. Experiments were con-ducted with the toroidal sensor and the actuator, and, by interrupting the continuityof a looped wire as shown here. In the latter case, the wire current is fed at the inputof the demodulator directly; similarly, the modulator’s output stage drives the wirecurrent directly.

A.4 Validation

For feasibility bench testing, currents near 64 MHz were induced in a looped-wire

using a 3 in. diameter transmit coil controlled by a network analyzer. The looped-wire

was transformer-coupled to the control loop. The wire currents were independently

measured using a resistor in series with the wire. Using the setup shown in Figure

A.4, the closed feedback network was characterized while varying the relative phase

shift of the LO frequencies (64 MHz) of the demodulator and modulator.

As shown in Figure A.5, when the loop phase rotation is accurately compensated,

the suppression of the wire current is the highest and mirrors the amplification of

the polyphase loop error amplifiers. A small phase misalignment between the up-

and down-conversion processes reduces the effective suppression of the wire current.

Increasing the phase misalignment further causes instability: positive feedback am-

plification is obtained, and the wire current is higher than that originally induced.

While the results obtained on the bench show that frequency-offset Cartesian feed-

back is a promising technology for attenuating RF-induced currents, much remains


63.4 63.6 63.8 64 64.2 64.4 64.6

10-1

100

63.4 63.6 63.8 64 64.2 64.4 64.610

-2

10-1

100

Frequency [MHz]

Nor

m. L

oop

Gai

nN

orm

aliz

ed W

ire

Cur

rent

1) accurate compensation

2) imperfect compensation

3) positive feedback

wire current without feedback

Figure A.5: Effect of the active cable trap on the currents induced in a looped wire.When the loop phase rotation is accurately compensated, the suppression of thewire current is the highest and mirrors the amplification of the polyphase loop erroramplifiers. A small phase misalignment between the up- and down-conversion reducesthe effective suppression of the wire current. Increasing the phase misalignment causesinstability: positive feedback amplification is obtained, and the wire current is higherthan that originally induced.

to be done to turn this device into one that can be used in an interventional setting.

As mentioned, the input/output dynamic range needs to be improved to deal with

currents that are at least an order of magnitude higher than those of the above exper-

iments. Secondly, circuitry should be added to the prototype to guarantee that the

conditions for stability (the compensation of the loop phase rotation) can be found

and set automatically before each use of the device. (This concern was addressed

during the design of the frequency-offset Cartesian feedback system for MRI power

amplifiers.) In addition, issues such as the effect of the distance between actuator and

sensor are not obvious, because they are in some way related to the characteristics of

the standing waves on the conductor, which are unknown a priori. These effects could


not only ultimately hinder the effectiveness of the negative feedback attenuation, but

also change the characteristics of the loop and thus compromise its stability.

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CARTESIAN FEEDBACK CONTROL FOR MRI