Electrical Engineering Hand Book by ramesh kumar

“Frontmatter” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000

Product Manager: Project Editor: Packaging design:

Maureen Aller Ibrey Woodall Jonathan Pennell

These files shall remain the sole and exclusive property of CRC Press LLC, 2000 Corporate Blvd., N.W., Boca Raton, FL 33431. The contents are protected by copyright law and international treaty. No part of the Electrical Engineering Handbook CRCnetBASE CD-ROM product may be duplicated in hard copy or machine-readable form without prior written authorization from CRC Press LLC, except that the licensee is granted a limited, non-exclusive license to reproduce limited portions of the context for the licensee’s internal use provided that a suitable notice of copyright is included on all copies. This CD-ROM incorporates materials from other sources reproduced with the kind permission of the copyright holder. Credit to the original sources and copyright notices are given with the figure or table. No materials in this CD-ROM credited to these copyright holders may be reproduced without their written permission. WARRANTY The information in this product was obtained from authentic and highly regarded sources. Every reasonable effort has been made to give reliable data and information, but the publisher cannot assume responsibility for the validity of all materials or the consequences of their uses. © 2000 by CRC Press LLC No claim to original U.S. Government works International Standard Book Number 0-8493-2177-8 International Standard Series Number 1097-9409

Preface

Purpose The purpose of The Electrical Engineering Handbook, 2nd Edition is to provide in a single volume a ready reference for the practicing engineer in industry, government, and academia. The book in its comprehensive format is divided into twelve sections which encompass the field of electrical engineering. The goal is to provide the most up-to-date information in the classical fields of circuits, signal processing, electronics, electromagnetic fields, energy devices, systems, and electrical effects and devices, while covering the emerging fields of communications, digital devices, computer engineering, systems, and biomedical engineering. In addition, the final section provides a complete compendium of information regarding physical, chemical, and materials data, as well as widely inclusive information on mathematics.

Organization The fundamentals of electrical engineering have evolved to include a wide range of knowledge, substantial empirical data, and a broad range of practice. The focus of the handbook is on the key concepts, models, and equations that enable the electrical engineer to analyze, design, and predict the behavior of complex electrical devices, circuits, instruments, and systems. While data and formulae are summarized, the main focus is the provision of the underlying theories and concepts and the appropriate application of these theories to the field of electrical engineering. Thus, the reader will find the key concepts defined, described, and illustrated in order to serve the needs of the engineer over many years. With equal emphasis on electronics, circuits, power systems, instruments, materials, effects and devices, systems, and control, the engineer should encounter a wide range of concepts and considerable depth of exploration of these concepts as they lead to application and design. The level of conceptual development of each topic is challenging, but tutorial and relatively fundamental. Each article, of which there are more than 200, is written to enlighten the expert, refresh the knowledge of the mature engineer, and educate the novice. The information is organized into twelve major sections. The first eleven sections encompass 118 chapters and the last section summarizes the applicable mathematics, symbols, and physical constants. Each section contains one or more historical vignettes that serve to enliven and illuminate the history of the subject of that section. Furthermore, each section is preceded by a photo of a device, circuit, or system that demonstrates an application illustrative of the material in the section. Each article includes four important and useful categories: defining terms, related topics, references, and further information. Defining terms are key definitions and the first occurrence of each term defined is indicated in boldface in the text. The definitions of these terms are summarized as a list at the end of each chapter or article. Related Topics are cross-references to related articles. The related topics are provided at the end of each chapter or article. The references provide a list of useful books and articles for follow-up reading. Finally, further information provides some general and useful sources of additional information on the topic.

Locating Your Topic Numerous avenues of access to information contained in the handbook are provided. A complete table of contents is presented at the front of the book. In addition, an individual table of contents precedes each of the twelve sections. Finally, each chapter begins with its own table of contents. The reader should look over these tables of contents to become familiar with the structure, organization, and content of the book. For example, see Section II: Signal Processing, then Chapter 17: Multidimensional Signal Processing, and then Chapter 17.2: Video Signal Processing. This tree-and-branch table of contents enables the reader to move up the tree to locate information on the topic of interest. Five indexes have been compiled to provide multiple means of accessing information. Three indexes are listed in alphabetical order: (1) subject index, (2) index of basic equations by title or name, and (3) index of contributing authors. The subject index can also be used to locate key definitions. The page on which the definition appears for each key (defining) term is clearly identified in the subject index. Two additional indexes are sequenced by order of appearance: (1) index of key tables of data or information and (2) index of key figures. The Electrical Engineering Handbook, 2nd Edition is designed to provide answers to most inquiries and direct the inquirer to further sources and references. We hope that this handbook will be referred to often and that informational requirements will be satisfied effectively.

Acknowledgments This handbook is testimony to the dedication of the Board of Advisors, the publishers, and my editorial associates. I particularly wish to acknowledge at CRC Press Ron Powers, Publisher; Kristen Peterson, Developmental Editor; and Susan Fox, Senior Project Editor. Finally, I am indebted to the assistance of Sara Hare, who served as editorial assistant.

Richard C. Dorf Editor-in-Chief

Editor-in-Chief

Richard C. Dorf, professor of electrical and computer engineering at the University of California, Davis, teaches graduate and undergraduate courses in electrical engineering in the fields of circuits and control systems. He earned a Ph.D. in electrical engineering from the U.S. Naval Postgraduate School, an M.S. from the University of Colorado, and a B.S. from Clarkson University. Highly concerned with the discipline of electrical engineering and its wide value to social and economic needs, he has written and lectured internationally on the contributions and advances in electrical engineering. Professor Dorf has extensive experience with education and industry and is professionally active in the fields of robotics, automation, electric circuits, and communications. He has served as a visiting professor at the University of Edinburgh, Scotland; the Massachusetts Institute of Technology; Stanford University; and the University of California, Berkeley. A Fellow of The Institute of Electrical and Electronics Engineers, Dr. Dorf is widely known to the profession for his Modern Control Systems, 7th edition (Addison-Wesley, 1995) and The International Encyclopedia of Robotics (Wiley, 1988). Dr. Dorf is also the co-author of Circuits, Devices and Systems (with Ralph Smith), 5th edition (Wiley, 1992) and Electric Currents, 3rd edition (Wiley, 1996).

Advisory Board

Frank Barnes

William Kersting

Richard S. Sandige

University of Colorado Boulder, Colorado

New Mexico State University Las Cruces, New Mexico

University of Wyoming Laramie, Wyoming

Joseph Bronzino

Vojin Oklobdzia

Leonard Shaw

Trinity College Hartford, Connecticut

University of California, Davis Davis, California

Polytechnic University Brooklyn, New York

Wai-Kai Chen

John V. Oldfield

John W. Steadman

University of Illinois Chicago, Illinois

Syracuse University Syracuse, New York

University of Wyoming Laramie, Wyoming

Delores Etter

Banmali Rawat

R. Lal Tummala

University of Colorado Boulder, Colorado

University of Nevada Reno, Nevada

Michigan State University East Lansing, Michigan

Lyle Feisel State University of New York Binghamton, New York

Contributors M. Abdelguerfi

R.A. Backer

R. Bartnikas

University of New Orleans New Orleans, Louisiana

A. Terry Bahill

Institut de Recherche d’Hydro-Québec Varennes, Canada

Samuel O. Agbo

The University of Arizona Tucson, Arizona

Stella N. Batalama

Inder J. Bahl

SUNY at Buffalo Buffalo, New York

California Polytechnic State University San Luis Obispo, California

Ahmed Amin Texas Instruments

ITT-Gallium Arsenide Technology Center Roanoke, Virginia

Norman Balabanian

Consultant in Information Storage Technology Los Altos Hills, California

University of Florida Gainesville, Florida

William T. Baumann

N. Alemadi Michigan State University East Lansing, Michigan

Jan P. Allebach Purdue University West Lafayette, Indiana Braden Allenby Lawrence Livermore National Lab Livermore, California

Glen Ballou Ballou Associates Guilford, Connecticut

Brian R. Bannister

Kristinn Andersen

University of Hull Hull, United Kingdom

Marel Corporation Hafnarfjoudur, Iceland

Joseph Bannister

Nick Angelopoulos Gould Shawmut Toronto, Canada

Carl A. Argila Software Engineering Consultant Pico Rivera, California

C.P. Arnold

Geoffrey Bate

University of Southern California Information Sciences Institute Marina Del Ray, California

Avram Bar-Cohen

Virginia Polytechnic Institute and State University Blacksburg, Virginia

John S. Bay Virginia Polytechnic Institute and State University Blacksburg, Virginia

R.A. Becker Integrated Optical Circuit Consultants Cupertino, California

Melvin L. Belcher

University of Minnesota Minneapolis, Minnesota

Georgia Tech Research Institute Atlanta, Georgia

Matthew F. Baretich

Peter Bendix

University of Canterbury Christchurch, New Zealand

University Hospital University of Colorado Denver, Colorado

Jos Arrillaga

Allen M. Barnett

LSI Logic Corp. Milpitas, California

John P. Benner

University of Canterbury Christchurch, New Zealand

AstroPower, Inc. Newark, Delaware

National Renewable Energy Laboratory Golden, Colorado

Derrick Atherton

Robert Joel Barnett

Edward J. Berbari

University of Sussex Falmer, Brighton, England

Vanderbilt University Nashville, Tennessee

Purdue University Indianapolis, Indiana

Ashoka K.S. Bhat

John R. Brews

Allen H. Cherin

University of Victoria Victoria, Canada

The University of Arizona Tucson, Arizona

Lucent Technologies Norcross, Georgia

Imran A. Bhutta

William L. Brogan

John Choma, Jr.

University of Nevada Las Vegas, Nevada

University of Southern California Los Angeles, California

Joseph D. Bronzino

Mo-Yuen Chow

RFPP

Theodore A. Bickart Michigan State University East Lansing, Michigan

Bill Bitler InfiMed Liverpool, New York

Glenn R. Blackwell Purdue University West Lafayette, Indiana

Karen Blades Lawrence Livermore National Lab Livermore, California

Linda Sue Boehmer LSB Technology Clairton, Pennsylvania

Theodore F. Bogart, Jr. University of Southern Mississippi Hattiesburg, Mississippi

Martin Bolton Ensigma Ltd. Chepstow, United Kingdom

Bruce W. Bomar University of Tennessee Space Institute Tullahmoa, Tennessee

Anjan Bose Washington State University Pullman, Washington

Bimal K. Bose University of Tennessee Knoxville, Tennessee

N.K. Bose The Pennsylvania State University University Park, Pennsylvania

Charles A. Bouman Purdue University West Lafayette, Indiana

Joseph Boykin Clarion Advanced Storage South Borough, Massachusetts

Trinity College Hartford, Connecticut

North Carolina State University Raleigh, North Carolina

Marcia A. Bush

Michael D. Ciletti

Xerox Palo Alto Research Center Palo Alto, California

University of Colorado Colorado Springs, Colorado

Doug Burger

Gerlad A. Clapp

University of Wisconsin-Madison Madison, Wisconsin

James A. Cadzow Vanderbilt University Nashville, Tennessee

Gordon L. Carpenter California State University Long Beach, California

Bill D. Carroll The University of Texas at Arlington Arlington, Texas

G. Clifforn Carter NUWC Detachment New London, Connecticut

Shu-Park Chan International Technological University Santa Clara, California

Rulph Chassaing Roger Williams University Bristol, Rhode Island

Willam Check GE Spacenet McLean, Virginia

Kao Chen Carlson Consulting Engineers San Diego, California

Mo-Shing Chen

Almon H. Clegg Cci Rockaway, New Jersey

J.R. Cogdell The University of Texas at Austin Austin, Texas

Richard C. Compton Cornell University Ithaca, New York

George E. Cook Vanderbilt University Nashville, Tennessee

J. Arlin Cooper Sandia National Laboratories Albuquerque, New Mexico

Timothy J. Coutts National Renewable Energy Laboratory Golden, Colorado

Thomas M. Cover Stanford University Stanford, California

Edward W. Czeck Northeastern University Boston, Massachusetts

University of Texas at Arlington Arlington, Texas

John N. Daigle

Sue-Ling Chen

Thomas E. Darcie

Allied Signal Falls Church, Virginia

Joe E. Brewer

Wai-Kai Chen

Northrop Grumman Corp. Baltimore, Maryland

University of Illinois at Chicago Chicago, Illinois

Naval Command Control and Ocean Surveillance Center San Diego, California

University of Mississippi At&T Bell Laboratories Holmdel, New Jersey

Larry S. Darken Oxford Instruments Oak Ridge, Tennessee

Fred Daum

Gerald W. Farnell

Gennady Sh. Gildenblat

Raytheon Company Marlborough, Massachusetts

McGill University Montreal, Canada

The Pennsylvania State University University Park, Pennsylvaina

Kevin A. Delin

James M. Feldman

Jay C. Giri

Jet Propulsion Lab Pasadena, California

Northeastern University Boston, Massachusetts

Cegelec ESCA Corporation Bellevue, Washington

Edward J. Delp

Tse-yun Feng

Purdue University West Lafayette, Indiana

The Pennsylvania State University State College, Pennsylvania

J. Duncan Glover

Kenneth Demarest

J. Patrick Fitch

University of Kansas Lawrence, Kansas

Lawrence Livermore National Laboratory Livermore, California

Bülent I. Dervisoglu Hewlett Packard Mountain View, California

Allan Dewey Duke University Durham, North Carolina

Daniel F. DiFonzo Planar Communications Corporation Rockville, Maryland

Dennis F. Doelitzsch 3-D Communications Corporation Marion, Illinois

Robert C. Durbeck IBM Corporation (retired) San Jose, California

Alexander C. Ehrlich U.S. Naval Research Laboratory Washington, D.C.

Mohamed E. El-Hawary Technical University of Nova Scotia Halifax, Nova Scotia

Aicha Elshabini-Riad

Martin D. Fox University of Connecticut Storrs, Connecticut

Larry A. Franks Sandia National Laboratory Livermore, California

James F. Frenzel University of Idaho Moscow, Idaho

Leon A. Frizzell University of Illinois Urbana, Illinois

Jesse W. Fussell Department of Defense Fort Meade, Maryland

Yuqing Gao IBM, T.J. Watson Research Center Yorktown Heights, New York

Donald Galler MIT Cambridge, Massachusettts

Failure Analysis Associates Framingham, Massachusetts

Oded Goldreich Weizmann Institute of Science Rehovot, Israel

James Richard Goodman University of Wisconsin-Madison Madison, Wisconsin

Peter J. Graham Florida Atlantic University Boca Raton, Florida

L. L. Grigsby Auburn University Auburn, Alabama

Charles A. Gross Auburn University Auburn, Alabama

R.B. Gungor University of South Alabama Mobile, Alabama

Chris G. Guy University of Reading Reading, United Kingdom

V. Carl Hamacher Queen’s University Kingston, Ontario, Canada

A.P. Hanson

Viginia Polytechnic Institute and State University Blacksburg, Virginia

Susan A.R. Garrod Purdue University West Lafayette, ndiana

Auburn University Auburn, Alabama

Yariv Ephraim

L.A. Geddes

Royce D. Harbor

Purdue University Lafayette, Indiana

University of West Florida Pensacola, Florida

Boris Gelmont

Gary L. Hawke

George Mason University Fairfax, Virgnia

R. Eskicioglu University of New Olreans New Orleans, Lousiana

Delores M. Etter University of Colorado Boulder, Colorado

K.F. Etzold IBM T.J. Watson Research Center Yorktown Heights, New York

University of Virginia Charlottesville, Virginia

Mario Gerla University of California, Los Angeles Los Angeles, California

Jerry D. Gibson Texas A&M University College Station, Texas

Unversity of Kansas Lawrence, Kansas

Jeff Hecht Laser Focus World Auburndale, Massachusetts

Lealand H. Hemming McDonnell Douglas Helicopter Systems Mesa, Arizona

H. Scott Hinton

Randy H. Katz

Dhammika Kurumbalapitiya

University of Colorado Boulder, Colorado

University of California Berkeley, California

Harvey Mudd College Claremont, California

Conrad H. Hoeppner

Myron Kayton

The Johns Hopkins University Baltimore, Maryland

Juang-Chung Lai

Kayton Engineering Co. Santa Monical, California

University of Texas at Arlington Arlington, Texas

S. Ratnajeevan H. Hoole

Dmimitri Kazakos

Harvey Mudd College Claremont, California

University of Southwestern Louisiana Lafayette, Louisiana

Pradeep Lall

Manfred N. Huber

Lawrence L. Kazmerski

Siemens Munich, Germany

National Renewable Energy Laboratory Newark, Delaware

Jeffy L. Hudgins

E.J. Kennedy

University of South Carolina Columbia, South Carolina

J. David Irwin Auburn University Auburn, Alabama

Raymond G. Jacquot University of Wyoming Laramie, Wyoming

University of Tennessee Knoxville, Tennessee

Michael Peter Kennedy University College Dublin Dublin, Ireland

William H. Kersting New Mexico State University Las Cruces, new Mexico

William J. Kerwin Ralph B. James Sandia National Laboratories Livermore, California

W. Kenneth Jenkins University of Illinois Urbana, Illinois

Christopher P. Jobling University of Wales Swansea, Wales

Barry W. Johnson

The University of Arizona Tucson, Arizona

Nicholas J. Kolias Raytheon Company Andover, Massachusetts

Géza Kolumbán Technical University of Budapest Budapest, Hungary

Jin Au Kong Massachusetts Institute of Technology Cambridge, Massachusetts

Motorola, Inc. Plantation, Florida

Ty A. Lasky University of California, Davis Davis, California

Gordon K.F. Lee North Carloina State University Raleigh, North Carolina

Peter A. Lee Department of Trade and Industry London, Elngland

William C.Y. Lee AirTouch Communications Walnut Creek, California

Cornelius T. Leondes University of California, San Diego La Jolla, California

Ted G. Lewis Naval Postgraduate School Montery, California

Jay Liebowitz George Washington University Washington, D.C.

Jefferson F. Lindsey III

Lurt L. Kosbar

Southern Illinois University at Carbondale Carbondale, Illinois

David E. Johnson

University of Missouri-Rolla Rolla, Missouri

Birmingham-Southern College Birmingham, Alabama

Chen-Ching Liu

Allan D. Kraus

University of Washington Seattle, Washington

Philip L. Johnson

Allan D. Kraus Associates Pacific Grove, California

University of Alabama at Birmingham Birmingham, Alabama

Mark H. Kryder

Capers Jones

Carnegie Mellon University Pittsburgh, Pennsylvania

Software Productivity Research, Inc. Burlington, Massachusetts

Luis G. Kun

University of Nevada Reno, Nevada

Center for Information Technology/AHCPR Rockville, Maryland

Raymond Luebbers

University of Virginia Charlottesville, Virginia

Ivan P. Kaminow AT&T Bell Laboratories Holmdel, New Jersey

George G. Karady Arizona State University Tempe, Arizona

Benjamin C. Kuo University of Illinois at Urbana-Champaign Urbana, Illinois

David C. Look Wright State University Dayton, Ohio

Carl Grant Looney

Penn State University University Park, Pennsylvania

Steven L. Maddy SpectraLink Corp. Boulder, Colorado

Donald C. Malocha

James E. Morris

Keshab K. Parhi

University of Central Florida Orlando, Florida

State University of New York Binghamton, New York

University of Minnesota Minneapolis, Minnesota

Giridhar D. Mandyam

Gregory L. Moss

Nokia Research Center Irving, Texas

Purdue University West Lafayette, Indiana

Harold G. Parks

Masud Mansuripur

Wayne Needham

University of Arizona Tucson, Arizona

Intel Corporation Chandler, Arizona

Robert J. Marks II

P.S. Neelakanta

University of Washington Seattle, Washington

Florida Atlantic University Boca Raton, Florida

André Martin

Josh T. Nessmith

Hughes Display Products Lexington, Kentucky

Georgia Research Institute Atlanta, Georgia

Johannes J. Martin

Paul Neudorfer

Virginia Polytechnic Institute and State University Blacksburg, Virginia

University of New Orleans New Orleans, Louisana

Seattle University Seattle, Washington

Michael Picheny

Daniel A. Martinec

Michael R. Neuman

IBM, T.J. Watson Research Center Yorktown Heights, New York

Aeronautical Radio, Inc. Annapolis, Maryland

Case Western Reserve University Cleveland, Ohio

Robert E. Massara

Robert E. Newnham

University of Essex Colchester, Essex, United Kingdon

Penn State University University Park, Pennsylvania

Kartikeya Mayaram

Norman S. Nise

Washington State University Pullman, Washington

California State Polytech University Pomona, California

Stan McClellan

Nicholas G. Odrey

University of Alabama Birmingham, Alabama

Lehigh University Bethlehem, Pennsylvania

John E. McInroy

Vojin G. Oklobdzija

Unviersity of Wyoming Laramie, Wyoming

University of California, Davis Davis, California

Duane McRuer

John V. Oldfield

Systems Technology, Inc. Hawthorne, California

Syracuse University Syracuse, New York

Sanjay K. Mehta

Chee-Mun Ong

NUWE Detachment New London, Connecticut

Purdue University West Lafayette, Indiana

Miran Milkovic

Terry P. Orlando

Analog Technology Consultants Alpharetta, Georgia

Massachusetts Institute of Technology Cambridge, Massachusetts

Brown University Providence, Rhode Island

Edmund K. Miller

Mil Ovan

W. David Pricer

Los Alamos National Laboratory Los Alamos, New Mexico

Motorola, Inc. Arlington Heights, Illinois

IBM Essex Junction, Vermont

L.B. Milstein

Joseph C. Palais

Jose C. Principe

University of California San Diego, California

Arizona State University Tempe, Arizona

University of Florida Gainesville, Florida

The University of Arizona, Tucson Tucson, Arizona

Clayton R. Paul University of Kentucky Lexington, Kentucky

Michael Pecht University of Maryland College Park, Maryland

Arun G. Phadke

Charles L. Phillips Auburn University Auburn, Alabama

S. Unnikrishna Pillai Polytechnic University Brooklyn, New York

Charles Polk University of Rhode Island Kingston, Rhode Island

H. Vincent Poor Princeton University Princeton, New Jersey

Alexander D. Poularikas University of Alabama in Huntsville Huntsville, Alabama

David M. Pozar University of Massachusetts Amherst, Massachusetts

Franco P. Preparata

Yuan Pu

Peter H. Rogers

Solomon Sherr

Applied Materials

Georgia Institute of Technology Atlanta, Georgia

Westland Electronics Old Chatham, New York

J. Gregory Rollins

Theodore I. Shim

Technology Modeling Associates Inc. Sunnyvale, California

Polytechnic University Brooklyn, New York

John M. Roman

L.H. Sibul

Sarah A. Rajala North Carolina State University Raleigh, North Carolina

Conor Rafferty Bell Laboratories, Lucent Technology Murry Hill, New Jersey

S. Rajaram Lucent Technologies Piscataway, New Jersey

Kaushik Rajashekara

Telematics Ft. Lauderdale, Florida

Evelyn P. Rozanski Rocherster Institute of Technology Rochester, New York

Delphi Energy & Engineering Management System Indianapolis, Indiana

Marcos Rubinstein

Bhuvana Ramabhadran

Matthew N.O. Sadiku

IBM, T.J. Watson Research Center Yorktown Heights, New York

Temple University Philadelphia, Pennsylvania

R. Ramakumar

Andrew P. Sage

Oklahoma State University Stillwater, Oklahoma

George Mason University Fairfax, Virginia

Abdul Hamid Rana

Stanley Salek

GE LogistiCom

Hammett & Edison, Inc. San Francisco, California

Banmali S. Rawat University of Nevada, Reno Reno, Nevada

Jacques Raymond Ottawa University Ottawa, Canada

J. Patrick Reilly Metatec Associates Silver Spring, Maryland

C. Magnus Rimvall F.L. Smith & Co. Valby, Copenhagen, Denmark

Thomas G. Robertazzi

Swiss PTT Bern, Switzerland

Ricardo S. Sánchez Peña University of Buenos Aires Argentina

Richard S. Sandige University of Wyoming Laramie, Wyoming

C. Sankaran Electro-Test Seattle, Washington

R.A. Schlueter Michigan State University East Lansing, Michigan

State University of New York at Stony Brook Stony Brook, New York

John L. Schmalzel

Charles J. Robinson

Juergen Schroeter

Louisiana Tech University Overton Brooks VA Medical Center Ruston, Louisiana

Richard B. Robrock II Bell Communications Research Piscataway, New Jersey

Martin S. Roden California State University Los Angeles, California

Rowan University Glassboro, New Jersey Acoustics Research Department AT&T Bell Laboratories Murray Hill, New Jersey

Micaela Serra University of Victoria Victoria, Canada

Leonard Shaw Polytechnic University Brooklyn, New York

The Pennsylvania State University University Park, Pennsylvania

M.K. Simon Jet Propulsion Laboratory

L. Montgomery Smith University of Tennessee Space Institute Tullahoma, Tennessee

Rosemary L. Smith University of California, Davis Davis, California

William M. Smith University of Alabama at Birmingham Birmingham, Alabama

Sidney Soclof California Stte University Los Angeles, California

Gurindar S. Sohi University of Wisconsin-Madison Madison, Wisconsin

René Speé Oregon State University Corvallis, Oregon

Cary R. Spitzer AvioniCon Inc. Williamsburg, Virginia

K. Neil Stanton Stanton Associates Bellevue, Washington

John Staudhammer University of Florida Gainesville, Florida

J.W. Steadman University of Wyoming Laramie, Wyoming

Michael B. Steer North Carolina State University Raleigh, North Carolina

F.W. Stephenson Virginia Polytechnic Institute and State University Blacksburg, Virginia

Elias G. Strangas

Robert J. Trew

Roger W. Whatmore

Michigan State University East Lansing, Michigan

Case Western Reserve University Cleveland, Ohio

V. Sundar

C.W. Trowbridge

Cranfield University Cranfield, Bedfordshire, United Kingdon

The Pennsylvania State University University Park, Pennsylvania

Vector Fields, Ltd. Oxford, England

Witold Suryn

R. Lal Tummala

Gemplus Quebec, Canada

James A. Svoboda Clarkson University Pottsdam, New York

David Sworder University of California, San Diego San Diego, California

Mario Sznaier Pennsylvania State University — University Park University Park, Pennsylvania

Michigan State University East Lansing, Michigan University of Florida Gainesville, Florida

B.M. Wilamowski

Vichate Ungvichian

University of Wyoming laramie, Wyoming

Florida Atlantic University Boca Raton, Florida

Michel Veillette Gemplus Quebec, Canada

Sergio Verdu

The University of Arizona Tucson, Arizona

Zvonko G. Vranesic

Ronald J. Tallarida

University of Toronto Toronto, Ontario, Canada

Temple University Philadelphia, Pennsylvania

Khoi Tien Vu

Stuart K. Tewksbury West Virginia University Morgantown, West Virginia

Rao S. Thallam Salt River Project Phoenix, Arizona

Joy A. Thomas IBM T.J. Watson Research Center Hawthorne, New York

Richard F. Tinder Washington State University Pullman, Washington

Spyros Tragoudas Southern Illinois University at Carbondale Carbondale, Illinois

William H. Tranter University of Missouri-Rolla Rolla, Missouri

Donald G. Whitehead University of Hull Hull, United Kingdom

Ferenc Szidarovszky

Lucent Technologies India Pvt. Ltd. New Delhi, India

Technical Press Morgan Hill, California

Martin A. Uman

Princeton University Princeton, New Jersey

Basant K. Tariyal

Jerry Whitaker

Lynn D. Wilcox FX Palo Alto Lab Palo Alto, California

James C. Wiltse Georgia Tech Institute Spartanburg, South Carolina

Phillip J. Windley Brigham Young University Provo, Utah

Wayne Wolf

ABB Transmission Technical Institute Raleigh, North Carolina

Princeton University Princeton, New Jersey

John V. Wait

David Young

The University of Arizona (retired) Tucson, Arizona

Alan K. Wallace Oregon State University Corvallis, Oregon

Zhen Wan University of California, Davis Davis, California

Rockwell Semiconductor Systems Newport Beach, California

Yixin Yu Tianjin University Tianjin, People’s Republic of China

Safwat G. Zaky University of Toronto Toronto, Ontario, Canada

Laurence S. Watkins Lucent Technologies Princeton, New Jersey

Joseph Watson University of Wales, Swansea Swansea, United Kingdom

N.R. Watson University of Canterbury Christchurch, New Zealand

Larry F. Weber Plasmaco, subsidiary of Matsushita Electric Industrial Co., Ltd. Highland, New York

Mehdi R. Zargham Southern Illinois University at Carbondale Carbondale, Illinois

Ya-Qin Zhang Microsoft Research, China Cranbury, New Jersey

Rodger E. Ziemer University of Colorado at Colorado Springs Colorado Springs, Colorado

Contents

SECTION I Circuits Introduction

Passive Components 1.1 1.2 1.3 1.4

Resistors Michael Pecht and Pradeep Lall Capacitors and Inductors Glen Ballou Transformers C. Sankaran Electrical Fuses Nick Angelopoulos

Voltage and Current Sources 2.1 2.2 2.3

Shu-Park Chan

Step, Impulse, Ramp, Sinusoidal, Exponential, and DC Signals Richard C. Dorf and Zhen Wan Ideal and Practical Sources Clayton R. Paul Controlled Sources J. R. Cogdell

Linear Circuit Analysis 3.1 3.2 3.3 3.4 3.5 3.6 3.7

Voltage and Current Laws Michael D. Ciletti Node and Mesh Analysis J. David Irwin Network Theorems Allan D. Kraus Power and Energy Norman Balabanian and Theodore A. Bickart Three-Phase Circuits Norman Balabanian Graph Theory Shu-Park Chan Two-Port Parameters and Transformations Norman S. Nise

Passive Signal Processing William J. Kerwin

Nonlinear Circuits 5.1 5.2 5.3 5.4

Diodes and Rectifiers Jerry L. Hudgins Limiters Theodore F. Bogart, Jr. Distortion Kartikeya Mayaram Communicating with Chaos Michael Peter Kennedy and Géza Kolumbán

Laplace Transform 6.1 6.2

Definitions and Properties Richard C. Dorf and Zhen Wan Applications David E. Johnson

7 State Variables: Concept and Formulation Wai-Kai Chen 8 The z-Transform Richard C. Dorf and Zhen Wan 9 T-Π Equivalent Networks Zhen Wan and Richard C. Dorf 10 Transfer Functions of Filters Richard C. Dorf and Zhen Wan 11 Frequency Response Paul Neudorfer 12 Stability Analysis Ferenc Szidarovsky and A. Terry Bahill 13 Computer Software for Circuit Analysis and Design 13.1 Analog Circuit Simulation J. Gregory Rollins 13.2 Parameter Extraction for Analog Circuit Simulation Peter Bendix

SECTION II Signal Processing Introduction

Delores M. Etter

14 Digital Signal Processing 14.1 Fourier Transforms W. Kenneth Jenkins 14.2 Fourier Transforms and Fast Fourier Transform Alexander D. Poularikas 14.3 Design and Implementation of Digital Filters Bruce W. Bomar and L. Montgomery Smith 14.4 Signal Restoration James A. Cadzow

15 Speech Signal Processing 15.1 15.2 15.3 15.4 15.5

Coding, Transmission, and Storage Stan McClellan and Jerry D. Gibson Speech Enhancement and Noise Reduction Yariv Epharim Analysis and Synthesis Jesse W. Fussell Speech Recognition Lynn D. Wilcox and Marcia A. Bush Large Vocabulary Continuous Speech Recognition Yuqing Gao, Bhuvana Ramabhadran, Michael Picheny

16 Spectral Estimation and Modeling 16.1 Spectral Analysis S. Unnikrishna Pillai and Theodore I. Shim 16.2 Parameter Estimation Stella N. Batalama and Dimitri Kazakos 16.3 Kalman Filtering Fred Daum

17 Multidimensional Signal Processing 17.1 17.2 17.3 17.4 17.5

Digital Image Processing Edward J. Delp, Jan Allebach, and Charles A. Bouman Video Signal Processing Sarah A. Rajala Sensor Array Processing N. K. Bose and L. H. Sibul Video Processing Architectures Wayne Wolf MPEG-4 Based Multimedia Information System Ya-Qin Zhang

18 VLSI for Signal Processing 18.1 Special Architectures Keshab K. Parhi 18.2 Signal Processing Chips and Applications Rulph Chassaing and Bill Bitler

19 Acoustic Signal Processing 18.1 Digital Signal Processing in Audio and Electroacoustics Juergen Schroeter 18.2 Underwater Acoustical Signal Processing Sanjay K. Mehta and G. Clifford Carter

20 Artificial Neural Networks

Jose C. Principe

21 Computing Environments for Digital Signal Processing

Delores M. Etter

SECTION III Electronics Introduction

John W. Steadman

22 Semiconductors 22.1 Physical Properties Gennady Sh. Gildenblat and Boris Gelmont 22.2 Diodes Miram Milkovic 22.3 Electrical Equivalent Circuit Models and Device Simulators for Semiconductor Devices Aicha Elshabini-Riad, F. W. Stephenson, and Imran A. Bhutta 22.4 Electrical Characterization of Semiconductors David C. Look

23 Semiconductor Manufacturing 23.1 23.2 23.3 23.4

Processes Harold G. Parks Testing Wayne Needham Electrical Characterization of Interconnections S. Rajaram Process Modeling and Simulation Conor Rafferty

24 Transistors 24.1 Junction Field-Effect Transistors Sidney Soclof 24.2 Bipolar Transistors Joseph Watson 24.3 The Metal-Oxide Semiconductor Field-Effect Transistor (MOSFET) John R. Brews

25 Integrated Circuits 25.1 Integrated Circuit Technology Joe E. Brewer 25.2 Layout, Placement, and Routing Mehdi R. Zargham and Spyros Tragoudas 25.3 Application-Specific Integrated Circuits Stuart Tewksbury

26 Surface Mount Technology

Glenn R. Blackwell

27 Operational Amplifiers 27.1 Ideal and Practical Models E. J. Kennedy 27.2 Applications John V. Wait

28 Amplifiers 28.1 Large Signal Analysis Gordon L. Carpenter 28.2 Small Signal Analysis John Choma, Jr.

29 Active Filters 29.1 Synthesis of Low-Pass Forms Robert E. Massara 29.2 Realization J. W. Steadman and B. M. Wilamowski 29.3 Generalized Impedance Convertors and Simulated Impedances James A. Svoboda

30 Power Electronics 30.1 30.2 30.3 30.4

Power Semiconductor Devices Kaushik Rajashekara Power Conversion Kaushik Rajashekara Power Supplies Ashoka K. S. Bhat Converter Control of Machines Bimal K. Bose

31 Optoelectronics 31.1 Lasers Jeff Hecht 31.2 Sources and Detectors Laurence S. Watkins 31.3 Circuits R. A. Becker

32 D/A and A/D Converters Susan A. R. Garrod 33 Thermal Management of Electronics Avram Bar-Cohen 34 Digital and Analog Electronic Design Automation Allen Dewey SECTION IV Electromagnetics Introduction

Banmali S. Rawat

35 Electromagnetic Fields Jin Au Kong 36 Magnetism and Magnetic Fields 36.1 Magnetism Geoffrey Bate 36.2 Magnetic Recording Mark H. Kryder

37 Wave Propagation 37.1 Space Propagation Matthew N. O. Sadiku 37.2 Waveguides Kenneth Demarest

38 Antennas 38.1 Wire N. J. Kolias and R. C. Compton 38.2 Aperture J. Patrick Fitch 38.3 Microstrip Antennas David M. Pozar

39 Microwave Devices 39.1 Passive Microwave Devices Michael B. Steer 39.2 Active Microwave Devices Robert J. Trew

40 Compatibility 40.1 Grounding, Shielding, and Filtering Leland H. Hemming 40.2 Spectrum, Specifications, and Measurement Techniques Vichate Ungvichian and John M. Roman 40.3 Lightning Martin Uman and Marcos Rubinstein

41 Radar 41.1 Pulse Radar Melvin L. Belcher and Josh T. Nessmith 41.2 Continuous Wave Radar James C. Wiltse

42 Lightwave 42.1 Lightwave Waveguides Samuel O. Agbo 42.2 Optical Fibers and Cables Allen H. Cherin and Basant K. Tariyal

43 Solid State Circuits I. J. Bahl 44 Three-Dimensional Analysis C. W. Trowbridge 45 Computational Electromagnetics E. K. Miller

SECTION V Electrical Effects and Devices Introduction

Lyle D. Feisel

46 Electroacoustic Devices

Peter H. Rogers

47 Surface Acoustic Wave Filters 48 Ultrasound

Donald C. Malocha

Gerald W. Farnell

49 Ferroelectric and Piezoelectric Materials

K. F. Etzold

50 Electrostriction

V. Sundar and R. E. Newnham

51 Piezoresistivity

Ahmed Amin

52 The Hall Effect

Alexander C. Ehrlich

53 Superconductivity

Kevin A. Delin and Terry P. Orlando

54 Pyroelectric Materials and Devices 55 Dielectrics and Insulators 56 Sensors

Roger W. Whatmore

R. Bartnikas

Rosemary L. Smith

57 Magnetooptics

David Young and Yuan Pu

58 Smart Materials

P. S. Neelakanta

SECTION VI Energy Introduction

William H. Kersting

59 Conventional Power Generation

George G. Karady

60 Power Systems and Generation 60.1 Distributed Power Generation R. Ramakumar 60.2 Photovoltaic Solar Cells Allen M. Barnett and Lawrence L. Kazmerski 60.3 Thermophotovoltaics John P. Benner and Timothy J. Coutts

61 Transmission 61.1 Alternating Current Overhead: Line Parameters, Models, Standard Voltages, Insulators Mo-Shing Chen 61.2 Alternating Current Underground: Line Parameters, Models, Standard Voltages, Cables Mo-Shing Chen and K. C. Lai 61.3 High-Voltage Direct-Current Transmission Rao S. Thallam 61.4 Compensation Mohamed E. El-Hawary 61.5 Fault Analysis in Power Systems Charles Gross 61.6 Protection Arun G. Phadke

61.7 Transient Operation of Power Systems R. B. Gungor 61.8 Planning J. Duncan Glover

62 Power Quality Jos Arrillaga 63 Power Systems 63.1 Power System Analysis L.L. Grigsby and A.P. Hanson 63.2 Voltage Instability R.A. Schlueter and N. Alemadi

64 Power Transformers 65 Energy Distribution 66 Electrical Machines 66.1 66.2 66.3 66.4

Charles A. Gross George G. Karady

Generators Chen-Ching Liu, Khoi Tien Vu, and Yixin Yu Motors Donald Galler Small Electric Motors Elias G. Strangas Simulation of Electric Machinery Chee-Mun Ong

67 Energy Management K. Neil Stanton, Jay C. Giri, and Anjan Bose 68 Power System Analysis Software C. P. Arnold and N. R. Watson SECTION VII Communications Introduction

Leonard Shaw

69 Broadcasting 69.1 69.2 69.3 69.4 69.5

Modulationa nd Demodulation Richard C. Dorf and Zhen Wan Radio Broadcasting Jefferson F. Lindsey III and Dennis F. Doelitzsch Television Systems Jerry Whitaker High-Definition Television Martin S. Roden Digital Audio Broadcasting Stanley Salek and Almon H. Clegg

70 Digital Communication 70.1 Error Control Coding Richard C. Dorf and Zhen Wan 70.2 Equalization Richard C. Dorf and Zhen Wan 70.3 Spread Spectrum Communications L. B. Milstein and M. K. Simon

71 Optical Communication 71.1 Lightwave Technology for Video Transmission T. E. Darcie 71.2 Long Distance Fiber Optic Communications Joseph C. Palais 71.3 Photonic Networks Ivan P. Kaminow

72 Networks 72.1 72.2 72.3 72.4

B-ISDN Manfred N. Huber Computer Communications Networks J. N. Daigle Local-Area Networks Joseph Bannister and Mario Gerla The Intelligent Network Richard B. Robrock II

73 Information Theory 73.1 Signal Detection H. Vincent Poor 73.2 Noise Carl G. Looney

73.3 73.4 73.5 73.6

Stochastic Processes Carl G. Looney The Sampling Theorem R. J. Marks II Channel Capacity Sergio Verdú Data Compression Joy A. Thomas and Thomas M. Cover

74 Satellites and Aerospace 75 Personal and Office 75.1 75.2 75.3 75.4

Daniel F. DiFonzo

Mobile Radio and Cellular Communications William C. Y. Lee Facsimile Rodger E. Ziemer Wireless Local-Area Networks for the 1990s Mil Ovan Wireless PCS Giridhar D. Mandyam

76 Phase-Locked Loop

Steven L. Maddy

77 Telemetry Conrad H. Hoeppner 78 Computer-Aided Design and Analysis of Communication Systems William H. Tranter and Kurt L. Kosbar

SECTION VIII Digital Devices Introduction

Richard S. Sandige

79 Logic Elements 79.1 79.2 79.3 79.4

IC Logic Family Operation and Characteristics Gregory L. Moss Logic Gates (IC) Peter Graham Bistable Devices Richard S. Sandige Optical Devices H. S. Hinton

80 Memory Devices 80.1 80.2 80.3 80.4

Integrated Circuits (RAM, ROM) W. David Pricer Basic Disk System Architectures Randy H. Katz Magnetic Tape Peter A. Lee Magneto-Optical Disk Data Storage M. Mansuripur

81 Logical Devices 81.1 81.2 81.3 81.4 81.5

Combinational Networks and Switching Algebra Franco P. Preparata Logic Circuits Richard S. Sandige Registers and Their Applications B. R. Bannister and D. G. Whitehead Programmable Arrays Martin Bolton Arithmetic Logic Units Bill D. Carroll

82 Microprocessors 82.1 Practical Microprocessors John Staudhammer and Sue-Ling Chen 82.2 Applications Phillip J. Windley and James F. Frenzel

83 Displays 83.1 83.2 83.3 83.4

Light-Emitting Diodes James E. Morris Liquid-Crystal Diodes James E. Morris The Cathode Ray Tube André Martin Color Plasma Displays Larry F. Weber

84 Data Acquisition 85 Testing

Dhammika Kurumbalaptitiya and S. Ratnajeevan H. Hoole

85.1 Digital IC Testing Micaela Serra 85.2 Design for Test Bulent I. Dervisoglu

SECTION IX Computer Engineering Introduction

Organization 86.1 86.2 86.3 86.4

John V. Oldfield and Vojin G. Oklobdzija

Number Systems Richard F. Tinder Computer Arithmetic Vojin G. Oklobdzija Architecture V. Carl Hamacher, Zvonko G. Vranesic, and Safwat G. Zaky Microprogramming Jacques Raymond

Programming 87.1 87.2 87.3

Assembly Language James M. Feldman and Edward W. Czeck High-Level Languages Ted G. Lewis Data Types and Data Structures Johannes J. Martin

Memory Systems

Input and Output 89.1 89.2 89.3

Doug Burger, James R. Goodman, and Gurindar S. Sohi

Input Devices Solomon Sherr Computer Output Printer Technologies Robert C. Durbeck Smart Cards Witold Suryn and Michel Veillette

Software Engineering 90.1 90.2 90.3

Tools and Techniques Carl A. Argila Testing, Debugging, and Verification Capers Jones Programming Methodology Johannes J. Martin

Computer Graphics

Evelyn P. Rozanski

Computer Networks

Thomas G. Robertzaai

Fault Tolerance

Knowledge Engineering 94.1 94.2

Barry W. Johnson

Databases M. Abdelguerfi and R. Eskicioglu Rule-Based Expert Systems Jay Liebowitz

Parallel Processors

Tse-yun Feng

Operating Systems

Joseph Boykin

Computer Security and Cryptography 97.1 97.2

Computer and Communications Security J. Arlin Cooper Fundamentals of Cryptography Oded Goldreich

Computer Reliability

The Internet and Its Role in the Future

Chris G. Guy Gary L. Hawke

SECTION X Systems Introduction

100

Control Systems 100.1 100.2 100.3 100.4 100.5 100.6 100.7 100.8 100.9

101

R. Lal Tummala

Models William L. Brogan Dynamic Response Gordon K. F. Lee Frequency Response Methods: Bode Diagram Approach Andrew P. Sage Root Locus Benjamin C. Kuo Compensation Charles L. Phillips and Royce D. Harbor Digital Control Systems Raymond G. Jacquot and John E. McInroy Nonlinear Control Systems Derek P. Atherton Optimal Control and Estimation John S. Bay and William T. Baumann Neural Control Mo-Yuen Chow

Robotics 101.1 Robot Configuration Ty A. Lasky and Tien C. Hsia 101.2 Dynamics and Control R. Lal Tummala 101.3 Applications Nicholas Odrey

102

Aerospace Systems 102.1 Avionics Systems Cary R. Spitzer, Daniel A. Martinec, and Cornelius T. Leondes 102.2 Communications Satellite Systems: Applications Abdul Hamid Rana and William Check

103

Command, Control, and Communications (C 3) G. Clapp and D. Sworder

104

Industrial Systems 104.1 Welding and Bonding George E. Cook, Kristinn Andersen, and Robert Joel Barnett 104.2 Large Drives Alan K. Wallace and René Spée 104.3 Robust Systems Mario Sznaier and Ricardo S. Sánchez Peña

105

Man-Machine Systems

106

Vehicular Systems

107

Industrial Illuminating Systems

108

Instruments

109

Navigation Systems

110

Reliability Engineering

Duane McRuer

Linda Sue Boehmer Kao Chen

John L. Schmalzel Myron Kayton R. Rama Kumar

111

Environmental Effects

112

Computer-Aided Control Systems Design

Karen Blades and Braden Allenby

C. Magnus Rimvall and Christopher P. Jobling

SECTION XI Biomedical Systems Introduction

113

Joseph D. Bronzino

Bioelectricity 113.1 Neuroelectric Principles J. Patrick Reilly 113.2 Bioelectric Events L. A. Geddes 113.3 Application of Electric and Magnetic Fields in Bone and Soft Tissue Repair C. Polk

114

Biomedical Sensors

115

Bioelectronics and Instruments

Michael R. Neuman

115.1 The Electroencephalogram Joseph D. Bronzino 115.2 The Electrocardiograph Edward J. Berbari 115.3 Pacemakers/Implantable Defibrillators Philip L. Johnson and William M. Smith

116

Medical Imaging 116.1 Tomography M. D. Fox 116.2 Ultrasound Leon A. Frizzell 116.3 Semiconductor Detectors for Radiation Measurements Larry S. Darken and Ralph B. James

117

Biocomputing 117.1 Clinical Information Systems Luis Kun 117.2 Hospital Information Systems Matthew F. Baretich

118

Computer Design for Biomedical Applications

119

Rehabilitation Engineering, Science, and Technology 119.1 119.2 119.3 119.4 119.5 119.6

Raymond Luebbers

Rehabilitation Concepts Engineering Concepts in Sensory Rehabilitation Engineering Concepts in Motor Rehabilitation Engineering Concepts in Communications Disorders Appropriate Technology The Future of Engineering in Rehabilitation

SECTION XII Mathematics, Symbols, and Physical Constants Introduction

Ronald J. Tallarida

Greek Alphabet International System of Units (SI) Conversion Constants and Multipliers

Physical Constants Symbols and Terminology for Physical and Chemical Quantities

Associations and Societies

Chan, Shu-Park “Section I – Circuits” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000

The Intel Pentium® processor, introduced at speeds of up to 300 MHz, combines the architectural advances in the Pentium Pro processor with the instruction set extensions of Intel MMX™ media enhancement technology. This combination delivers new levels of performance and the fastest Intel processor to workstations. The Pentium II processor core, with 7.5 million transistors, is based on Intel’s advanced P6 architecture and is manufactured on .35-micron process technology. First implemented in the Pentium Pro processor, the Dual Independent Bus architecture is made up of the L2 cache bus and the processor-to-main-memory system bus. The latter enables simultaneous parallel transactions instead of single, sequential transactions of previous generation processors. The types of applications that will benefit from the speed of the Pentium II processor and the media enhancement of MMX technology include scanning, image manipulation, video conferencing, Internet browsers and plug-ins, video editing and playback, printing, faxing, compression, and encryption. The Pentium II processor is the newest member of the P6 processor family, but certainly not the last in the line of high performance processors. (Courtesy of Intel Corporation.)

I Circuits 1 Passive Components M. Pecht, P. Lall, G. Ballou, C. Sankaran, N. Angelopoulos Resistors • Capacitors and Inductors • Transformers • Electrical Fuses

2 Voltage and Current Sources R.C. Dorf, Z. Wan, C.R. Paul, J.R. Cogdell Step, Impulse, Ramp, Sinusoidal, Exponential, and DC Signals • Ideal and Practical Sources • Controlled Sources

3 Linear Circuit Analysis M.D. Ciletti, J.D. Irwin, A.D. Kraus, N. Balabanian, T.A. Bickart, S.P. Chan, N.S. Nise Voltage and Current Laws • Node and Mesh Analysis • Network Theorems • Power and Energy • Three-Phase Circuits • Graph Theory • Two Port Parameters and Transformations

4 Passive Signal Processing W.J. Kerwin Low-Pass Filter Functions • Low-Pass Filters • Filter Design

5 Nonlinear Circuits J.L. Hudgins, T.F. Bogart, Jr., K. Mayaram, M.P. Kennedy, G. Kolumbán Diodes and Rectifiers • Limiters • Distortion • Communicating with Chaos

6 Laplace Transform

R.C. Dorf, Z. Wan, D.E. Johnson

Definitions and Properties • Applications

7 State Variables: Concept and Formulation W.K. Chen State Equations in Normal Form • The Concept of State and State Variables and Normal Tree • Systematic Procedure in Writing State Equations • State Equations for Networks Described by Scalar Differential Equations • Extension to Time-Varying and Nonlinear Networks

8 The z-Transform

R.C. Dorf, Z. Wan

Properties of the z-Transform • Unilateral z-Transform • z-Transform Inversion • Sampled Data

9 T-P Equivalent Networks

Z. Wan, R.C. Dorf

Three-Phase Connections • Wye ⇔ Delta Transformations

10 Transfer Functions of Filters

R.C. Dorf, Z. Wan

Ideal Filters • The Ideal Linear-Phase Low-Pass Filter • Ideal Linear-Phase Bandpass Filters • Causal Filters • Butterworth Filters • Chebyshev Filters

11 Frequency Response P. Neudorfer Linear Frequency Response Plotting • Bode Diagrams • A Comparison of Methods

12 Stability Analysis F. Szidarovszky, A.T. Bahill Using the State of the System to Determine Stability • Lyapunov Stability Theory • Stability of Time-Invariant Linear Systems • BIBO Stability • Physical Examples

13 Computer Software for Circuit Analysis and Design J.G. Rollins, P. Bendix Analog Circuit Simulation • Parameter Extraction for Analog Circuit Simulation

Shu-Park Chan International Technological University

HIS SECTION PROVIDES A BRIEF REVIEW of the definitions and fundamental concepts used in the study of linear circuits and systems. We can describe a circuit or system, in a broad sense, as a collection of objects called elements (components, parts, or subsystems) which form an entity governed by certain laws or constraints. Thus, a physical system is an entity made up of physical objects as its elements or components. A subsystem of a given system can also be considered as a system itself. A mathematical model describes the behavior of a physical system or device in terms of a set of equations, together with a schematic diagram of the device containing the symbols of its elements, their connections, and numerical values. As an example, a physical electrical system can be represented graphically by a network which includes resistors, inductors, and capacitors, etc. as its components. Such an illustration, together with a set of linear differential equations, is referred to as a model system. Electrical circuits may be classified into various categories. Four of the more familiar classifications are (a) linear and nonlinear circuits, (b) time-invariant and time-varying circuits, (c) passive and active circuits, and (d) lumped and distributed circuits. A linear circuit can be described by a set of linear (differential) equations; otherwise it is a nonlinear circuit. A time-invariant circuit or system implies that none of the components of the circuit have parameters that vary with time; otherwise it is a time-variant system. If the total energy delivered to a given circuit is nonnegative at any instant of time, the circuit is said to be passive; otherwise it is active. Finally, if the dimensions of the components of the circuit are small compared to the wavelength of the highest of the signal frequencies applied to the circuit, it is called a lumped circuit; otherwise it is referred to as a distributed circuit. There are, of course, other ways of classifying circuits. For example, one might wish to classify circuits according to the number of accessible terminals or terminal pairs (ports). Thus, terms such as n-terminal circuit and n-port are commonly used in circuit theory. Another method of classification is based on circuit configurations (topology),1 which gives rise to such terms as ladders, lattices, bridged-T circuits, etc. As indicated earlier, although the words circuit and system are synonymous and will be used interchangeably throughout the text, the terms circuit theory and system theory sometimes denote different points of view in the study of circuits or systems. Roughly speaking, circuit theory is mainly concerned with interconnections of components (circuit topology) within a given system, whereas system theory attempts to attain generality by means of abstraction through a generalized (input-output state) model. One of the goals of this section is to present a unified treatment on the study of linear circuits and systems. That is, while the study of linear circuits with regard to their topological properties is treated as an important phase of the entire development of the theory, a generality can be attained from such a study. The subject of circuit theory can be divided into two main parts, namely, analysis and synthesis. In a broad sense, analysis may be defined as “the separating of any material or abstract entity [system] into its constituent elements;” on the other hand, synthesis is “the combining of the constituent elements of separate materials or abstract entities into a single or unified entity [system].”2 It is worth noting that in an analysis problem, the solution is always unique no matter how difficult it may be, whereas in a synthesis problem there might exist an infinite number of solutions or, sometimes, none at all! It should also be noted that in some network theory texts the words synthesis and design might be used interchangeably throughout the entire discussion of the subject. However, the term synthesis is generally used to describe analytical procedures that can usually be carried out step by step, whereas the term design includes practical (design) procedures (such as trial-and-error techniques which are based, to a great extent, on the experience of the designer) as well as analytical methods. In analyzing the behavior of a given physical system, the first step is to establish a mathematical model. This model is usually in the form of a set of either differential or difference equations (or a combination of them), 1

Circuit topology or graph theory deals with the way in which the circuit elements are interconnected. A detailed discussion on elementary applied graph theory is given in Chapter 3.6. 2The definitions of analysis and synthesis are quoted directly from The Random House Dictionary of the English Language, 2nd ed., Unabridged, New York: Random House, 1987. © 2000 by CRC Press LLC

the solution of which accurately describes the motion of the physical systems. There is, of course, no exception to this in the field of electrical engineering. A physical electrical system such as an amplifier circuit, for example, is first represented by a circuit drawn on paper. The circuit is composed of resistors, capacitors, inductors, and voltage and/or current sources,1 and each of these circuit elements is given a symbol together with a mathematical expression (i.e., the voltage-current or simply v-i relation) relating its terminal voltage and current at every instant of time. Once the network and the v-i relation for each element is specified, Kirchhoff ’s voltage and current laws can be applied, possibly together with the physical principles to be introduced in Chapter 3.1, to establish the mathematical model in the form of differential equations. In Section I, focus is on analysis only (leaving coverage of synthesis and design to Section III, “Electronics”). Specifically, the passive circuit elements—resistors, capacitors, inductors, transformers, and fuses—as well as voltage and current sources (active elements) are discussed. This is followed by a brief discussion on the elements of linear circuit analysis. Next, some popularly used passive filters and nonlinear circuits are introduced. Then, Laplace transform, state variables, z-transform, and T and p configurations are covered. Finally, transfer functions, frequency response, and stability analysis are discussed.

Nomenclature Symbol

Quantity

Unit

Symbol

Quantity

Unit

A B C e e e f F f I J k k K L l M n n

area magnetic flux density capacitance induced voltage dielectric constant ripple factor frequency force magnetic flux current Jacobian Boltzmann constant dielectric coefficient coupling coefficient inductance eigenvalue mutual inductance turns ratio filter order

m2 Tesla F V F/m

w P PF q Q R R(T)

angular frequency power power factor charge selectivity resistance temperature coefficient of resistance resistivity Laplace operator damping factor phase angle velocity voltage energy reactance admittance impedance

rad/s W

Hz Newton weber A 1.38 ´ 10–23 J/K

H H

r s t q v V W X Y Z

C W W / °C Wm

degree m/s V J W S W

1Here, of course, active elements such as transistors are represented by their equivalent circuits as combinations of resistors and dependent sources.

Pecht, M., Lall, P., Ballou, G., Sankaran, C., Angelopoulos, N. “Passive Components” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000

1 Passive Components Michael Pecht

1.1

Resistors

1.2

Capacitors and Inductors

1.3

Transformers

Resistor Characteristics • Resistor Types

University of Maryland

Pradeep Lall Motorola

Capacitors • Types of Capacitors • Inductors Types of Transformers • Principle of Transformation • Electromagnetic Equation • Transformer Core • Transformer Losses • Transformer Connections • Transformer Impedance

Glen Ballou Ballou Associates

C. Sankaran Electro-Test

Nick Angelopoulos Gould Shawmut Company

1.1

1.4

Electrical Fuses Ratings • Fuse Performance • Selective Coordination • Standards • Products • Standard— Class H • HRC • Trends

Resistors

Michael Pecht and Pradeep Lall The resistor is an electrical device whose primary function is to introduce resistance to the flow of electric current. The magnitude of opposition to the flow of current is called the resistance of the resistor. A larger resistance value indicates a greater opposition to current flow. The resistance is measured in ohms. An ohm is the resistance that arises when a current of one ampere is passed through a resistor subjected to one volt across its terminals. The various uses of resistors include setting biases, controlling gain, fixing time constants, matching and loading circuits, voltage division, and heat generation. The following sections discuss resistor characteristics and various resistor types.

Resistor Characteristics Voltage and Current Characteristics of Resistors The resistance of a resistor is directly proportional to the resistivity of the material and the length of the resistor and inversely proportional to the cross-sectional area perpendicular to the direction of current flow. The resistance R of a resistor is given by

R =

rl A

(1.1)

where r is the resistivity of the resistor material (W · cm), l is the length of the resistor along direction of current flow (cm), and A is the cross-sectional area perpendicular to current flow (cm2) (Fig. 1.1). Resistivity is an inherent property of materials. Good resistor materials typically have resistivities between 2 ´ 10–6 and 200 ´ 10–6 W · cm. © 2000 by CRC Press LLC

The resistance can also be defined in terms of sheet resistivity. If the sheet resistivity is used, a standard sheet thickness is assumed and factored into resistivity. Typically, resistors are rectangular in shape; therefore the length l divided by the width w gives the number of squares within the resistor (Fig. 1.2). The number of squares multiplied by the resistivity is the resistance.

l Rsheet = rsheet w

(1.2)

FIGURE 1.1 Resistance of a rectangular cross-section resistor with cross-sectional area A and length L.

where rsheet is the sheet resistivity (W/square), l is the length of resistor (cm), w is the width of the resistor (cm), and Rsheet is the sheet resistance (W). The resistance of a resistor can be defined in terms of the voltage drop across the resistor and current through the resistor related by Ohm’s law,

R =

V I

(1.3)

where R is the resistance (W), V is the voltage across the resistor (V), and I is the current through the resistor (A). Whenever a current is passed through a resistor, a voltage is dropped across the ends of the resistor. Figure 1.3 depicts the symbol of the resistor with the Ohm’s law relation. All resistors dissipate power when a voltage is applied. The power dissipated by the resistor is represented by

P =

V2 R

(1.4)

where P is the power dissipated (W), V is the voltage across the resistor (V), and R is the resistance (W). An ideal resistor dissipates electric energy without storing electric or magnetic energy. Resistor Networks Resistors may be joined to form networks. If resistors are joined in series, the effective resistance (RT) is the sum of the individual resistances (Fig. 1.4). n

RT =

åR

(1.5)

i =1

FIGURE 1.3 A resistor with resistance R having a current I flowing through it will have a voltage drop of IR across it.

FIGURE 1.4 Resistors connected in series.

If resistors are joined in parallel, the effective resistance (RT) is the reciprocal of the sum of the reciprocals of individual resistances (Fig. 1.5). n 1 1 = RT i =1 R i

(1.6)

Temperature Coefficient of Electrical Resistance The resistance for most resistors changes with temperature. The temperature coefficient of electrical resistance is the change in electrical resistance of a resistor per unit change in temperature. The temperature coefficient of resistance is measured in W/°C. The temperature coefficient of resistors may be either positive or negative. A positive temperature coefficient denotes a rise in resistance with a rise in temperature; a negative temperature coefficient of resistance denotes a decrease in resistance with a rise in temperature. Pure metals typically have a positive temperature coefficient of resistance, while some metal alloys such as constantin and manganin have a zero temperature coefficient of resistance. Carbon and graphite mixed with binders usually exhibit negative temperature coefficients, although certain choices of binders and process variations may yield positive temperature coefficients. The temperature coefficient of resistance is given by

FIGURE 1.5 parallel.

R ( T 2) = R ( T 1)[1 + aT1( T 2 – T 1)]

Resistors connected in

(1.7)

where aT1 is the temperature coefficient of electrical resistance at reference temperature T1, R(T2) is the resistance at temperature T2 (W), and R(T1) is the resistance at temperature T1 (W). The reference temperature is usually taken to be 20°C. Because the variation in resistance between any two temperatures is usually not linear as predicted by Eq. (1.7), common practice is to apply the equation between temperature increments and then to plot the resistance change versus temperature for a number of incremental temperatures. High-Frequency Effects Resistors show a change in their resistance value when subjected to ac voltages. The change in resistance with voltage frequency is known as the Boella effect. The effect occurs because all resistors have some inductance and capacitance along with the resistive component and thus can be approximated by an equivalent circuit shown in Fig. 1.6. Even though the definition of useful frequency FIGURE 1.6 Equivalent circuit for a resistor. range is application dependent, typically, the useful range of the resistor is the highest frequency at which the impedance differs from the resistance by more than the tolerance of the resistor. The frequency effect on resistance varies with the resistor construction. Wire-wound resistors typically exhibit an increase in their impedance with frequency. In composition resistors the capacitances are formed by the many conducting particles which are held in contact by a dielectric binder. The ac impedance for film resistors remains constant until 100 MHz (1 MHz = 106 Hz) and then decreases at higher frequencies (Fig. 1.7). For film resistors, the decrease in dc resistance at higher frequencies decreases with increase in resistance. Film resistors have the most stable high-frequency performance. © 2000 by CRC Press LLC

FIGURE 1.7 Typical graph of impedance as a percentage of dc resistance versus frequency for film resistors.

The smaller the diameter of the resistor the better is its frequency response. Most high-frequency resistors have a length to diameter ratio between 4:1 to 10:1. Dielectric losses are kept to a minimum by proper choice of base material. Voltage Coefficient of Resistance Resistance is not always independent of the applied voltage. The voltage coefficient of resistance is the change in resistance per unit change in voltage, expressed as a percentage of the resistance at 10% of rated voltage. The voltage coefficient is given by the relationship

Voltage coefficient =

100(R1 – R2 ) R2 (V1 – V2 )

(1.8)

where R1 is the resistance at the rated voltage V1 and R2 is the resistance at 10% of rated voltage V2. Noise Resistors exhibit electrical noise in the form of small ac voltage fluctuations when dc voltage is applied. Noise in a resistor is a function of the applied voltage, physical dimensions, and materials. The total noise is a sum of Johnson noise, current flow noise, noise due to cracked bodies, and loose end caps and leads. For variable resistors the noise can also be caused by the jumping of a moving contact over turns and by an imperfect electrical path between the contact and resistance element. The Johnson noise is temperature-dependent thermal noise (Fig. 1.8). Thermal noise is also called “white noise” because the noise level is the same at all frequencies. The magnitude of thermal noise, ERMS (V), is dependent on the resistance value and the temperature of the resistance due to thermal agitation.

ERMS =

4kRTDf

(1.9)

where ERMS is the root-mean-square value of the noise voltage (V), R is the resistance (W), K is the Boltzmann constant (1.38 ´ 10–23 J/K), T is the temperature (K), and Df is the bandwidth (Hz) over which the noise energy is measured. Figure 1.8 shows the variation in current noise versus voltage frequency. Current noise varies inversely with frequency and is a function of the current flowing through the resistor and the value of the resistor. The magnitude of current noise is directly proportional to the square root of current. The current noise magnitude is usually expressed by a noise index given as the ratio of the root-mean-square current noise voltage (ERMS) © 2000 by CRC Press LLC

FIGURE 1.8 The total resistor noise is the sum of current noise and thermal noise. The current noise approaches the thermal noise at higher frequencies. (Source: Phillips Components, Discrete Products Division, 1990–91 Resistor/Capacitor Data Book, 1991. With permission.)

over one decade bandwidth to the average voltage caused by a specified constant current passed through the resistor at a specified hot-spot temperature [Phillips, 1991].

æ Noise voltage ö N.I. = 20 log10 ç ÷ è dc voltage ø

(1.10)

æf ö E RMS = Vdc ´ 10N.I. / 20 log ç 2 ÷ è f1 ø

(1.11)

where N.I. is the noise index, Vdc is the dc voltage drop across the resistor, and f1 and f2 represent the frequency range over which the noise is being computed. Units of noise index are mV/V. At higher frequencies, the current noise becomes less dominant compared to Johnson noise. Precision film resistors have extremely low noise. Composition resistors show some degree of noise due to internal electrical contacts between the conducting particles held together with the binder. Wire-wound resistors are essentially free of electrical noise unless resistor terminations are faulty. Power Rating and Derating Curves Resistors must be operated within specified temperature limits to avoid permanent damage to the materials. The temperature limit is defined in terms of the maximum power, called the power rating, and derating curve. The power rating of a resistor is the maximum power in watts which the resistor can dissipate. The maximum power rating is a function of resistor material, maximum voltage rating, resistor dimensions, and maximum allowable hot-spot temperature. The maximum hot-spot temperature is the temperature of the hottest part on the resistor when dissipating full-rated power at rated ambient temperature. The maximum allowable power rating as a function of the ambient temperature is given by the derating curve. Figure 1.9 shows a typical power rating curve for a resistor. The derating curve is usually linearly drawn from the full-rated load temperature to the maximum allowable no-load temperature. A resistor may be operated at ambient temperatures above the maximum full-load ambient temperature if operating at lower than full-rated power capacity. The maximum allowable no-load temperature is also the maximum storage temperature for the resistor. © 2000 by CRC Press LLC

FIGURE 1.9 Typical derating curve for resistors.

Voltage Rating of Resistors The maximum voltage that may be applied to the resistor is called the voltage rating and is related to the power rating by

V =

(1.12)

where V is the voltage rating (V), P is the power rating (W), and R is the resistance (W). For a given value of voltage and power rating, a critical value of resistance can be calculated. For values of resistance below the critical value, the maximum voltage is never reached; for values of resistance above the critical value, the power dissipated is lower than the rated power (Fig. 1.10). Color Coding of Resistors Resistors are generally identified by color coding or direct digital marking. The color code is given in Table 1.1. The color code is commonly used in composition resistors and film resistors. The color code essentially consists of four bands of different colors. The first band is the most significant figure, the second band is the second significant figure, the third band is the multiplier or the number of zeros that have to be added after the first two significant figures, and the fourth band is the tolerance on the resistance value. If the fourth band is not present, the resistor tolerance is the standard 20% above and below the rated value. When the color code is used on fixed wire-wound resistors, the first band is applied in double width.

TABLE 1.1 Color Code Table for Resistors Color

First Band

Second Band

Third Band

Black Brown Red Orange Yellow Green Blue Violet Gray White Gold Silver No band

0 1 2 3 4 5 6 7 8 9

1 10 100 1,000 10,000 100,000 1,000,000 10,000,000 100,000,000 1,000,000,000 0.1 0.01

Fourth Band Tolerance, %

5% 10% 20%

Blanks in the table represent situations which do not exist in the color code.

Resistor Types Resistors can be broadly categorized as fixed, variable, and special-purpose. Each of these resistor types is discussed in detail with typical ranges of their characteristics. Fixed Resistors The fixed resistors are those whose value cannot be varied after manufacture. Fixed resistors are classified into composition resistors, wire-wound resistors, and metal-film resistors. Table 1.2 outlines the characteristics of some typical fixed resistors. Wire-Wound Resistors. Wire-wound resistors are made by winding wire of nickel-chromium alloy on a ceramic tube covering with a vitreous coating. The spiral winding has inductive and capacitive characteristics that make it unsuitable for operation above 50 kHz. The frequency limit can be raised by noninductive winding so that the magnetic fields produced by the two parts of the winding cancel. Composition Resistors. Composition resistors are composed of carbon particles mixed with a binder. This mixture is molded into a cylindrical shape and hardened by baking. Leads are attached axially to each end, and the assembly is encapsulated in a protective encapsulation coating. Color bands on the outer surface indicate the resistance value and tolerance. Composition resistors are economical and exhibit low noise levels for resistances above 1 MW. Composition resistors are usually rated for temperatures in the neighborhood of 70°C for power ranging from 1/8 to 2 W. Composition resistors have end-to-end shunted capacitance that may be noticed at frequencies in the neighborhood of 100 kHz, especially for resistance values above 0.3 MW. Metal-Film Resistors. Metal-film resistors are commonly made of nichrome, tin-oxide, or tantalum nitride, either hermetically sealed or using molded-phenolic cases. Metal-film resistors are not as stable as the TABLE 1.2 Characteristics of Typical Fixed Resistors Resistor Types Wire-wound resistor Precision Power Metal-film resistor Precision Power Composition resistor General purpose

Resistance Range

Watt Range

Operating Temp. Range

a, ppm/°C

0.1 to 1.2 MW 0.1 to 180 kW

1/8 to 1/4 1 to 210

–55 to 145 –55 to 275

10 260

1 to 250 MW 5 to 100 kW

1/20 to 1 1 to 5

–55 to 125 –55 to 155

50–100 20–100

2.7 to 100 MW

1/8 to 2

–55 to 130

1500

wire-wound resistors. Depending on the application, fixed resistors are manufactured as precision resistors, semiprecision resistors, standard general-purpose resistors, or power resistors. Precision resistors have low voltage and power coefficients, excellent temperature and time stabilities, low noise, and very low reactance. These resistors are available in metal-film or wire constructions and are typically designed for circuits having very close resistance tolerances on values. Semiprecision resistors are smaller than precision resistors and are primarily used for current-limiting or voltage-dropping functions in circuit applications. Semiprecision resistors have long-term temperature stability. General-purpose resistors are used in circuits that do not require tight resistance tolerances or long-term stability. For general-purpose resistors, initial resistance variation may be in the neighborhood of 5% and the variation in resistance under full-rated power may approach 20%. Typically, general-purpose resistors have a high coefficient of resistance and high noise levels. Power resistors are used for power supplies, control circuits, and voltage dividers where operational stability of 5% is acceptable. Power resistors are available in wire-wound and film constructions. Film-type power resistors have the advantage of stability at high frequencies and have higher resistance values than wire-wound resistors for a given size. Variable Resistors Potentiometers. The potentiometer is a special form of variable resistor with three terminals. Two terminals are connected to the opposite sides of the resistive element, and the third connects to a sliding contact that can be adjusted as a voltage divider. Potentiometers are usually circular in form with the movable contact attached to a shaft that rotates. Potentiometers are manufactured as carbon composition, metallic film, and wire-wound resistors available in single-turn or multiturn units. The movable contact does not go all the way toward the end of the resistive element, and a small resistance called the hop-off resistance is present to prevent accidental burning of the resistive element. Rheostat. The rheostat is a current-setting device in which one terminal is connected to the resistive element and the second terminal is connected to a movable contact to place a selected section of the resistive element into the circuit. Typically, rheostats are wire-wound resistors used as speed controls for motors, ovens, and heater controls and in applications where adjustments on the voltage and current levels are required, such as voltage dividers and bleeder circuits. Special-Purpose Resistors Integrated Circuit Resistors. Integrated circuit resistors are classified into two general categories: semiconductor resistors and deposited film resistors. Semiconductor resistors use the bulk resistivity of doped semiconductor regions to obtain the desired resistance value. Deposited film resistors are formed by depositing resistance films on an insulating substrate which are etched and patterned to form the desired resistive network. Depending on the thickness and dimensions of the deposited films, the resistors are classified into thick-film and thin-film resistors. Semiconductor resistors can be divided into four types: diffused, bulk, pinched, and ion-implanted. Table 1.3 shows some typical resistor properties for semiconductor resistors. Diffused semiconductor resistors use resistivity of the diffused region in the semiconductor substrate to introduce a resistance in the circuit. Both n-type and p-type diffusions are used to form the diffused resistor. A bulk resistor uses the bulk resistivity of the semiconductor to introduce a resistance into the circuit. Mathematically the sheet resistance of a bulk resistor is given by

Rsheet =

re d

(1.13)

where Rsheet is the sheet resistance in (W/square), re is the sheet resistivity (W/square), and d is the depth of the n-type epitaxial layer. Pinched resistors are formed by reducing the effective cross-sectional area of diffused resistors. The reduced cross section of the diffused length results in extremely high sheet resistivities from ordinary diffused resistors.

TABLE 1.3 Typical Characteristics of Integrated Circuit Resistors

Resistor Type Semiconductor Diffused Bulk Pinched Ion-implanted Deposited resistors Thin-film Tantalum SnO2 Ni-Cr Cermet (Cr-SiO) Thick-film Ruthenium-silver Palladium-silver

Sheet Resistivity (per square)

Temperature Coefficient (ppm/°C)

0.8 to 260 W 0.003 to 10 kW 0.001 to 10 kW 0.5 to 20 kW

1100 to 2000 2900 to 5000 3000 to 6000 100 to 1300

0.01 to 1 kW 0.08 to 4 kW 40 to 450 W 0.03 to 2.5 kW

m100 –1500 to 0 m100 m150

10 W to 10 MW 0.01 to 100 kW

m200 –500 to 150

Ion-implanted resistors are formed by implanting ions on the semiconductor surface by bombarding the silicon lattice with high-energy ions. The implanted ions lie in a very shallow layer along the surface (0.1 to 0.8 mm). For similar thicknesses ion-implanted resistors yield sheet resistivities 20 times greater than diffused resistors. Table 1.3 shows typical properties of diffused, bulk, pinched, and ion-implanted resistors. Typical sheet resistance values range from 80 to 250 W/square. Varistors. Varistors are voltage-dependent resistors that show a high degree of nonlinearity between their resistance value and applied voltage. They are composed of a nonhomogeneous material that provides a rectifying action. Varistors are used for protection of electronic circuits, semiconductor components, collectors of motors, and relay contacts against overvoltage. The relationship between the voltage and current of a varistor is given by

V = kI b

(1.14)

where V is the voltage (V), I is the current (A), and k and b are constants that depend on the materials and manufacturing process. The electrical characteristics of a varistor are specified by its b and k values. Varistors in Series. The resultant k value of n varistors connected in series is nk. This can be derived by considering n varistors connected in series and a voltage nV applied across the ends. The current through each varistor remains the same as for V volts over one varistor. Mathematically, the voltage and current are expressed as

nV = k 1 I b

(1.15)

Equating the expressions (1.14) and (1.15), the equivalent constant k1 for the series combination of varistors is given as

k 1 = nk

(1.16)

Varistors in Parallel. The equivalent k value for a parallel combination of varistors can be obtained by connecting n varistors in parallel and applying a voltage V across the terminals. The current through the varistors will still be n times the current through a single varistor with a voltage V across it. Mathematically the current and voltage are related as

(1.17)

From Eqs. (1.14) and (1.17) the equivalent constant k2 for the series combination of varistors is given as

k2 =

k nb

(1.18)

Thermistors. Thermistors are resistors that change their resistance exponentially with changes in temperature. If the resistance decreases with increase in temperature, the resistor is called a negative temperature coefficient (NTC) resistor. If the resistance increases with temperature, the resistor is called a positive temperature coefficient (PTC) resistor. NTC thermistors are ceramic semiconductors made by sintering mixtures of heavy metal oxides such as manganese, nickel, cobalt, copper, and iron. The resistance temperature relationship for NTC thermistors is

RT = Ae B/T

(1.19)

where T is temperature (K), RT is the resistance (W), and A, B are constants whose values are determined by conducting experiments at two temperatures and solving the equations simultaneously. PTC thermistors are prepared from BaTiO3 or solid solutions of PbTiO3 or SrTiO3. The resistance temperature relationship for PTC thermistors is

RT = A + Ce BT

(1.20)

where T is temperature (K), RT is the resistance (W), and A, B are constants determined by conducting experiments at two temperatures and solving the equations simultaneously. Positive thermistors have a PTC only between certain temperature ranges. Outside this range the temperature is either zero or negative. Typically, the absolute value of the temperature coefficient of resistance for PTC resistors is much higher than for NTC resistors.

Defining Terms Doping: The intrinsic carrier concentration of semiconductors (e.g., Si) is too low to allow controlled charge transport. For this reason some impurities called dopants are purposely added to the semiconductor. The process of adding dopants is called doping. Dopants may belong to group IIIA (e.g., boron) or group VA (e.g., phosphorus) in the periodic table. If the elements belong to the group IIIA, the resulting semiconductor is called a p-type semiconductor. On the other hand, if the elements belong to the group VA, the resulting semiconductor is called an n-type semiconductor. Epitaxial layer: Epitaxy refers to processes used to grow a thin crystalline layer on a crystalline substrate. In the epitaxial process the wafer acts as a seed crystal. The layer grown by this process is called an epitaxial layer. Resistivity: The resistance of a conductor with unit length and unit cross-sectional area. Temperature coefficient of resistance: The change in electrical resistance of a resistor per unit change in temperature. Time stability: The degree to which the initial value of resistance is maintained to a stated degree of certainty under stated conditions of use over a stated period of time. Time stability is usually expressed as a percent or parts per million change in resistance per 1000 hours of continuous use. Voltage coefficient of resistance: The change in resistance per unit change in voltage, expressed as a percentage of the resistance at 10% of rated voltage. Voltage drop: The difference in potential between the two ends of the resistor measured in the direction of flow of current. The voltage drop is V = IR, where V is the voltage across the resistor, I is the current through the resistor, and R is the resistance. Voltage rating: The maximum voltage that may be applied to the resistor.

Related Topics 22.1 Physical Properties • 25.1 Integrated Circuit Technology • 51.1 Introduction

References Phillips Components, Discrete Products Division, 1990–91 Resistor/Capacitor Data Book, 1991. C.C. Wellard, Resistance and Resistors, New York: McGraw-Hill, 1960.

Further Information IEEE Transactions on Electron Devices and IEEE Electron Device Letters: Published monthly by the Institute of Electrical and Electronics Engineers. IEEE Components, Hybrids and Manufacturing Technology: Published quarterly by the Institute of Electrical and Electronics Engineers. G.W.A. Dummer, Materials for Conductive and Resistive Functions, New York: Hayden Book Co., 1970. H.F. Littlejohn and C.E. Burckel, Handbook of Power Resistors, Mount Vernon, N.Y.: Ward Leonard Electric Company, 1951. I.R. Sinclair, Passive Components: A User’s Guide, Oxford: Heinmann Newnes, 1990.

1.2

Capacitors and Inductors

Glen Ballou Capacitors If a potential difference is found between two points, an electric field exists that is the result of the separation of unlike charges. The strength of the field will depend on the amount the charges have been separated. Capacitance is the concept of energy storage in an electric field and is restricted to the area, shape, and spacing of the capacitor plates and the property of the material separating them. When electrical current flows into a capacitor, a force is established between two parallel plates separated by a dielectric. This energy is stored and remains even after the input is removed. By connecting a conductor (a resistor, hard wire, or even air) across the capacitor, the charged capacitor can regain electron balance, that is, discharge its stored energy. The value of a parallel-plate capacitor can be found with the equation

C =

x e[(N – 1)A] ´ 10 –13 d

(1.21)

where C = capacitance, F; e = dielectric constant of insulation; d = spacing between plates; N = number of plates; A = area of plates; and x = 0.0885 when A and d are in centimeters, and x = 0.225 when A and d are in inches. The work necessary to transport a unit charge from one plate to the other is

e = kg

(1.22)

where e = volts expressing energy per unit charge, g = coulombs of charge already transported, and k = proportionality factor between work necessary to carry a unit charge between the two plates and charge already transported. It is equal to 1/C, where C is the capacitance, F. The value of a capacitor can now be calculated from the equation

q C = e

(1.23)

where q = charge (C) and e is found with Eq. (1.22). The energy stored in a capacitor is

W =

CV 2 2

(1.24)

where W = energy, J; C = capacitance, F; and V = applied voltage, V. The dielectric constant of a material determines the electrostatic energy which may be stored in that material per unit volume for a given voltage. The value of the dielectric constant expresses the ratio of a capacitor in a vacuum to one using a given dielectric. The dielectric of air is 1, the reference unit employed for expressing the dielectric constant. As the dielectric constant is increased or decreased, the capacitance will increase or decrease, respectively. Table 1.4 lists the dielectric constants of various materials. The dielectric constant of most materials is affected by both temperature and frequency, except for quartz, Styrofoam, and Teflon, whose dielectric constants remain essentially constant. The equation for calculating the force of attraction between two plates is

F =

AV 2

(1.25)

k (1504S )2

TABLE 1.4 Comparison of Capacitor Dielectric Constants Dielectric Air or vacuum Paper Plastic Mineral oil Silicone oil Quartz Glass Porcelain Mica Aluminum oxide Tantalum pentoxide Ceramic

K (Dielectric Constant) 1.0 2.0–6.0 2.1–6.0 2.2–2.3 2.7–2.8 3.8–4.4 4.8–8.0 5.1–5.9 5.4–8.7 8.4 26 12–400,000

Source: G. Ballou, Handbook for Sound Engineers, The New Audio Cyclopedia, Carmel, Ind.: Macmillan Computer Publishing Company, 1991. With permission.

where F = attraction force, dyn; A = area of one plate, cm2; V = potential energy difference, V; k = dielectric coefficient; and S = separation between plates, cm. The Q for a capacitor when the resistance and capacitance is in series is

Q =

1 2p f RC

(1.26)

where Q = ratio expressing the factor of merit; f = frequency, Hz; R = resistance, W; and C = capacitance, F. When capacitors are connected in series, the total capacitance is

CT =

1 1/C1 + 1/C 2 + × × × + 1/Cn

(1.27)

and is always less than the value of the smallest capacitor. When capacitors are connected in parallel, the total capacitance is

CT = C1 + C2 + · · · + Cn

(1.28)

and is always larger than the largest capacitor. When a voltage is applied across a group of capacitors connected in series, the voltage drop across the combination is equal to the applied voltage. The drop across each individual capacitor is inversely proportional to its capacitance.

VC =

V AC X CT

(1.29)

where VC = voltage across the individual capacitor in the series (C1, C 2, ...,Cn ), V; VA = applied voltage, V; CT = total capacitance of the series combination, F; and CX = capacitance of individual capacitor under consideration, F. In an ac circuit, the capacitive reactance, or the impedance, of the capacitor is

XC =

1 2pfC

(1.30)

where XC = capacitive reactance, W ; f = frequency, Hz; and C = capacitance, F. The current will lead the voltage by 90° in a circuit with a pure capacitor. When a dc voltage is connected across a capacitor, a time t is required to charge the capacitor to the applied voltage. This is called a time constant and is calculated with the equation

t = RC

(1.31)

where t = time, s; R = resistance, W; and C = capacitance, F. In a circuit consisting of pure resistance and capacitance, the time constant t is defined as the time required to charge the capacitor to 63.2% of the applied voltage. During the next time constant, the capacitor charges to 63.2% of the remaining difference of full value, or to 86.5% of the full value. The charge on a capacitor can never actually reach 100% but is considered to be 100% after five time constants. When the voltage is removed, the capacitor discharges to 63.2% of the full value. Capacitance is expressed in microfarads (mF, or 10–6 F) or picofarads (pF, or 10–12 F) with a stated accuracy or tolerance. Tolerance may also be stated as GMV (guaranteed minimum value), sometimes referred to as MRV (minimum rated value). All capacitors have a maximum working voltage that must not be exceeded and is a combination of the dc value plus the peak ac value which may be applied during operation. Quality Factor (Q) Quality factor is the ratio of the capacitor’s reactance to its resistance at a specified frequency and is found by the equation

1 2 pfCR 1 = PF

Q =

(1.32)

where Q = quality factor; f = frequency, Hz; C = value of capacitance, F; R = internal resistance, W; and PF = power factor Power Factor (PF) Power factor is the preferred measurement in describing capacitive losses in ac circuits. It is the fraction of input volt-amperes (or power) dissipated in the capacitor dielectric and is virtually independent of the capacitance, applied voltage, and frequency. Equivalent Series Resistance (ESR) Equivalent series resistance is expressed in ohms or milliohms (W , mW) and is derived from lead resistance, termination losses, and dissipation in the dielectric material. Equivalent Series Inductance (ESL) The equivalent series inductance can be useful or detrimental. It reduces high-frequency performance; however, it can be used in conjunction with the internal capacitance to form a resonant circuit.

Dissipation Factor (DF) The dissipation factor in percentage is the ratio of the effective series resistance of a capacitor to its reactance at a specified frequency. It is the reciprocal of quality factor (Q) and an indication of power loss within the capacitor. It should be as low as possible. Insulation Resistance Insulation resistance is the resistance of the dielectric material and determines the time a capacitor, once charged, will hold its charge. A discharged capacitor has a low insulation resistance; however once charged to its rated value, it increases to megohms. The leakage in electrolytic capacitors should not exceed

I L = 0.04C + 0.30

(1.33)

where IL = leakage current, mA, and C = capacitance, mF. Dielectric Absorption (DA) The dielectric absorption is a reluctance of the dielectric to give up stored electrons when the capacitor is discharged. This is often called “memory” because if a capacitor is discharged through a resistance and the resistance is removed, the electrons that remained in the dielectric will reconvene on the electrode, causing a voltage to appear across the capacitor. DA is tested by charging the capacitor for 5 min, discharging it for 5 s, then having an open circuit for 1 min after which the recovery voltage is read. The percentage of DA is defined as the ratio of recovery to charging voltage times 100.

Types of Capacitors Capacitors are used to filter, couple, tune, block dc, pass ac, bypass, shift phase, compensate, feed through, isolate, store energy, suppress noise, and start motors. They must also be small, lightweight, reliable, and withstand adverse conditions. Capacitors are grouped according to their dielectric material and mechanical configuration. Ceramic Capacitors Ceramic capacitors are used most often for bypass and coupling applications (Fig. 1.11). Ceramic capacitors can be produced with a variety of K values (dielectric constant). A high K value translates to small size and less stability. High-K capacitors with a dielectric constant >3000 are physically small and have values between 0.001 to several microfarads.

FIGURE 1.12 Film-wrapped film capacitors. (Courtesy of Sprague Electric Company.)

Good temperature stability requires capacitors to have a K value between 10 and 200. If high Q is also required, the capacitor will be physically larger. Ceramic capacitors with a zero temperature change are called negative-positive-zero (NPO) and come in a capacitance range of 1.0 pF to 0.033 mF. An N750 temperature-compensated capacitor is used when accurate capacitance is required over a large temperature range. The 750 indicates a 750-ppm decrease in capacitance with a 1°C increase in temperature (750 ppm/°C). This equates to a 1.5% decrease in capacitance for a 20°C temperature increase. N750 capacitors come in values between 4.0 and 680 pF. Film Capacitors Film capacitors consist of alternate layers of metal foil and one or more layers of a flexible plastic insulating material (dielectric) in ribbon form rolled and encapsulated (see Fig. 1.12). Mica Capacitors Mica capacitors have small capacitance values and are usually used in high-frequency circuits. They are constructed as alternate layers of metal foil and mica insulation, which are stacked and encapsulated, or are silvered mica, where a silver electrode is screened on the mica insulators. Paper-Foil-Filled Capacitors Paper-foil-filled capacitors are often used as motor capacitors and are rated at 60 Hz. They are made of alternate layers of aluminum and paper saturated with oil that are rolled together. The assembly is mounted in an oilfilled, hermetically sealed metal case. Electrolytic Capacitors Electrolytic capacitors provide high capacitance in a tolerable size; however, they do have drawbacks. Low temperatures reduce performance, while high temperatures dry them out. The electrolytes themselves can leak and corrode the equipment. Repeated surges above the rated working voltage, excessive ripple currents, and high operating temperature reduce performance and shorten capacitor life. Electrolytic capacitors are manufactured by an electrochemical formation of an oxide film on a metal surface. The metal on which the oxide film is formed serves as the anode or positive terminal of the capacitor; the oxide film is the dielectric, and the cathode or negative terminal is either a conducting liquid or a gel. The equivalent circuit of an electrolytic capacitor is shown in Fig. 1.13, where A and B are the capacitor terminals, C is the effective capacitance, and L is the self-inductance of the capacitor caused by terminals, electrodes, and geometry. © 2000 by CRC Press LLC

The shunt resistance (insulation resistance) Rs accounts for the dc leakage current. Heat is generated in the ESR from ripple current and in the shunt resistance by voltage. The ESR is due to the spacer-electrolyte-oxide system and varies only slightly except at low temperature, where it increases greatly. FIGURE 1.13 Simplified equivalent circuit of The impedance of a capacitor (Fig. 1.14) is frequency-depen- an electrolytic capacitor. dent. The initial downward slope is caused by the capacitive reactance XC . The trough (lowest impedance) is almost totally resistive, and the upward slope is due to the capacitor’s self-inductance XL . An ESR plot would show an ESR decrease to about 5–10 kHz, remaining relatively constant thereafter. Leakage current is the direct current that passes through a capacitor when a correctly polarized dc voltage is applied to its terminals. It is proportional to temperature, becoming increasingly important at elevated ambient temperatures. Leakage current decreases slowly after voltage is applied, reaching steady-state conditions in about 10 min. If a capacitor is connected with reverse polarity, the oxide film is forward-biased, offering very little resistance to current flow. This causes overheating and self-destruction of the capacitor. The total heat generated within a capacitor is the sum of the heat created by the I leakage ´ Vapplied and the I 2R losses in the ESR. The ac ripple current rating is very important in filter applications because excessive current produces temperature rise, shortening capacitor life. The maximum permissible rms ripple current is limited by the internal temperature and the rate of heat dissipation from the capacitor. Lower ESR and longer enclosures increase the ripple current rating. Capacitor life expectancy is doubled for each 10°C decrease in operating temperature, so a capacitor operating at room temperature will have a life expectancy 64 times that of the same capacitor operating at 85°C (185°F). The surge voltage specification of a capacitor determines its ability to withstand high transient voltages that generally occur during the starting up period of equipment. Standard tests generally specify a short on and long off period for an interval of 24 h or more, and the allowable surge voltage levels are generally 10% above the rated voltage of the capacitor. Figure 1.15 shows how temperature, frequency, time, and applied voltage affect electrolytic capacitors. Aluminum Electrolytic Capacitors. Aluminum electrolytic capacitors use aluminum as the base material (Fig. 1.16). The surface is often etched to increase the surface area as much as 100 times that of unetched foil, resulting in higher capacitance in the same volume. Aluminum electrolytic capacitors can withstand up to 1.5 V of reverse voltage without detriment. Higher reverse voltages, when applied over extended periods, lead to loss of capacitance. Excess reverse voltages applied for short periods cause some change in capacitance but not to capacitor failure. Large-value capacitors are often used to filter dc power supplies. After a capacitor is charged, the rectifier stops conducting and the capacitor discharges into the load, as shown in Fig. 1.17, until the next cycle. Then the capacitor recharges again to the peak voltage. The De is equal to the total peak-to-peak ripple voltage and is a complex wave containing many harmonics of the fundamental ripple frequency, causing the noticeable heating of the capacitor. Tantalum Capacitors. Tantalum electrolytics are the preferred type where high reliability and long service life are paramount considerations. Tantalum capacitors have as much as three times better capacitance per volume efficiency than aluminum electrolytic capacitors, because tantalum pentoxide has a dielectric constant three times greater than that of aluminum oxide (see Table 1.4). The capacitance of any capacitor is determined by the surface area of the two conducting plates, the Figure 1.14 Impedance characteristics of a capacitor. © 2000 by CRC Press LLC

FIGURE 1.15 Variations in aluminum electrolytic characteristics caused by temperature, frequency, time, and applied voltage. (Courtesy of Sprague Electric Company.)

FIGURE 1.16 Company.)

Verti-lytic® miniature single-ended aluminum electrolytic capacitor. (Courtesy of Sprague Electric

distance between the plates, and the dielectric constant of the insulating material between the plates [see Eq. (1.21)]. In tantalum electrolytics, the distance between the plates is the thickness of the tantalum pentoxide film, and since the dielectric constant of the tantalum pentoxide is high, the capacitance of a tantalum capacitor is high.

Tantalum capacitors contain either liquid or solid electrolytes. The liquid electrolyte in wet-slug and foil capacitors, generally sulfuric acid, forms the cathode (negative) plate. In solid-electrolyte capacitors, a dry material, manganese dioxide, forms the cathode plate. Foil Tantalum Capacitors. Foil tantalum capacitors can be designed to voltage values up to 300 V dc. Of the three types of tantalum electrolytic capacitors, the foil design has the lowest capacitance per unit volume and is best suited for the higher voltages primarily found in older designs of equipment. It is expensive and used only where neither a solid-electrolyte (Fig. 1.18) nor a wet-slug (Fig. 1.19) tantalum capacitor can be employed.

FIGURE 1.17 Full-wave capacitor charge and discharge.

FIGURE 1.18 Tantalex® solid electrolyte tantalum capacitor. (Courtesy of Sprague Electric Company.)

FIGURE 1.19 Hermetically sealed sintered-anode tantalum capacitor. (Courtesy of Sprague Electric Company.)

Foil tantalum capacitors are generally designed for operation over the temperature range of –55 to +125°C (–67 to +257°F) and are found primarily in industrial and military electronics equipment. Solid-electrolyte sintered-anode tantalum capacitors differ from the wet versions in their electrolyte, which is manganese dioxide. Another variation of the solid-electrolyte tantalum capacitor encases the element in plastic resins, such as epoxy materials offering excellent reliability and high stability for consumer and commercial electronics with the added feature of low cost. Still other designs of “solid tantalum” capacitors use plastic film or sleeving as the encasing material, and others use metal shells that are backfilled with an epoxy resin. Finally, there are small tubular and rectangular molded plastic encasements. Wet-electrolyte sintered-anode tantalum capacitors, often called “wet-slug” tantalum capacitors, use a pellet of sintered tantalum powder to which a lead has been attached, as shown in Fig. 1.19. This anode has an enormous surface area for its size. Wet-slug tantalum capacitors are manufactured in a voltage range to 125 V dc. Use Considerations. Foil tantalum capacitors are used only where high-voltage constructions are required or where there is substantial reverse voltage applied to a capacitor during circuit operation. Wet sintered-anode capacitors, or “wet-slug” tantalum capacitors, are used where low dc leakage is required. The conventional “silver can” design will not tolerate reverse voltage. In military or aerospace applications where utmost reliability is desired, tantalum cases are used instead of silver cases. The tantalum-cased wet-slug units withstand up to 3 V reverse voltage and operate under higher ripple currents and at temperatures up to 200°C (392°F). Solid-electrolyte designs are the least expensive for a given rating and are used where their very small size is important. They will typically withstand a reverse voltage up to 15% of the rated dc working voltage. They also have good low-temperature performance characteristics and freedom from corrosive electrolytes.

Inductors Inductance is used for the storage of magnetic energy. Magnetic energy is stored as long as current keeps flowing through the inductor. In a perfect inductor, the current of a sine wave lags the voltage by 90°. Impedance Inductive reactance X L , the impedance of an inductor to an ac signal, is found by the equation

X L = 2p f L

(1.34)

where XL = inductive reactance, W; f = frequency, Hz; and L = inductance, H. The type of wire used for its construction does not affect the inductance of a coil. Q of the coil will be governed by the resistance of the wire. Therefore coils wound with silver or gold wire have the highest Q for a given design. To increase inductance, inductors are connected in series. The total inductance will always be greater than the largest inductor.

L T = L1 + L 2 + · · · + L n

(1.35)

To reduce inductance, inductors are connected in parallel.

LT =

1 1 /L 1 + 1 /L 2 + × × × + 1 /L n

The total inductance will always be less than the value of the lowest inductor.

(1.36)

Mutual Inductance Mutual inductance is the property that exists between two conductors carrying current when their magnetic lines of force link together. The mutual inductance of two coils with fields interacting can be determined by the equation

M =

LA – LB 4

(1.37)

where M = mutual inductance of LA and L B , H; LA = total inductance, H, of coils L1 and L2 with fields aiding; and LB = total inductance, H, of coils L1 and L2 with fields opposing. The coupled inductance can be determined by the following equations. In parallel with fields aiding,

LT =

1 1 1 + L1 + M L2 + M

(1.38)

In parallel with fields opposing,

LT =

1 1 1 – L1 – M L2 – M

(1.39)

In series with fields aiding,

L T = L 1 + L 2 + 2M

(1.40)

L T = L 1 + L 2 – 2M

(1.41)

In series with fields opposing,

where L T = total inductance, H; L1 and L 2 = inductances of the individual coils, H; and M = mutual inductance, H. When two coils are inductively coupled to give transformer action, the coupling coefficient is determined by

K =

(1.42)

L 1L 2 where K = coupling coefficient; M = mutual inductance, H; and L1 and L2 = inductances of the two coils, H. An inductor in a circuit has a reactance equal to j2pf L W. Mutual inductance in a circuit has a reactance equal to j2pf L W. The operator j denotes that the reactance dissipates no energy; however, it does oppose current flow. The energy stored in an inductor can be determined by the equation

W =

LI 2 2

(1.43)

Coil Inductance Inductance is related to the turns in a coil as follows: 1. The inductance is proportional to the square of the turns. 2. The inductance increases as the length of the winding is increased. 3. A shorted turn decreases the inductance, affects the frequency response, and increases the insertion loss. 4. The inductance increases as the permeability of the core material increases. 5. The inductance increases with an increase in the cross-sectional area of the core material. 6. Inductance is increased by inserting an iron core into the coil. 7. Introducing an air gap into a choke reduces the inductance. A conductor moving at any angle to the lines of force cuts a number of lines of force proportional to the sine of the angles. Thus,

V = bLv sin q ´ 10 –8

(1.44)

where b = flux density; L = length of the conductor, cm; and v = velocity, cm/s, of conductor moving at an angle q. The maximum voltage induced in a conductor moving in a magnetic field is proportional to the number of magnetic lines of force cut by that conductor. When a conductor moves parallel to the lines of force, it cuts no lines of force; therefore, no current is generated in the conductor. A conductor that moves at right angles to the lines of force cuts the maximum number of lines per inch per second, therefore creating a maximum voltage. The right-hand rule determines direction of the induced electromotive force (emf). The emf is in the direction in which the axis of a right-hand screw, when turned with the velocity vector, moves through the smallest angle toward the flux density vector. The magnetomotive force (mmf) in ampere-turns produced by a coil is found by multiplying the number of turns of wire in the coil by the current flowing through it.

æV ö Ampere-turns = T ç ÷ èRø

(1.45)

= TI where T = number of turns; V = voltage, V; and R = resistance, W. The inductance of a single layer, a spiral, and multilayer coils can be calculated by using either Wheeler’s or Nagaoka’s equations. The accuracy of the calculation will vary between 1 and 5%. The inductance of a singlelayer coil can be calculated using Wheeler’s equation:

L =

B2N 2 9B + 10 A

(1.46)

For the multilayer coil,

L =

0.8B 2 N 2 6B + 9 A + 10C

(1.47)

For the spiral coil,

L =

B2N 2 8B + 11C

(1.48)

where B = radius of the winding, N = number of turns in the coil, A = length of the winding, and C = thickness of the winding. Q Q is the ratio of the inductive reactance to the internal resistance of the coil and is affected by frequency, inductance, dc resistance, inductive reactance, the type of winding, the core losses, the distributed capacity, and the permeability of the core material. The Q for a coil where R and L are in series is

Q =

2pf L R

(1.49)

where f = frequency, Hz; L = inductance, H; and R = resistance, W. The Q of the coil can be measured using the circuit of Fig. 1.20 for frequencies up to 1 MHz. The voltage across the inductance (L) at resonance equals Q(V) (where V is the voltage developed by the oscillator); therefore, it is only necessary to measure the output voltage from the oscillator and the voltage across the inductance. The oscillator voltage is driven across a low value of resistance, R, about 1/100 of the anticipated rf resistance of the LC combination, to FIGURE 1.20 Circuit for measuring assure that the measurement will not be in error by more than 1%. For the Q of a coil. most measurements, R will be about 0.10 W and should have a voltage of 0.1 V. Most oscillators cannot be operated into this low impedance, so a step-down matching transformer must be employed. Make C as large as convenient to minimize the ratio of the impedance looking from the voltmeter to the impedance of the test circuit. The LC circuit is then tuned to resonate and the resultant voltage measured. The value of Q may then be equated

Q =

resonant voltage across C voltage across R

(1.50)

The Q of any coil may be approximated by the equation

2pf L R XL = R

Q =

(1.51)

where f = the frequency, Hz; L = the inductance, H; R = the dc resistance, W (as measured by an ohmmeter); and XL = the inductive reactance of the coil. iTme Constant When a dc voltage is applied to an RL circuit, a certain amount of time is required to change the circuit [see text with Eq. (1.31)]. The time constant can be determined with the equation

T =

L R

(1.52)

The right-hand rule is used to determine the direction of a magnetic field around a conductor carrying a direct current. Grasp the conductor in the right hand with the thumb extending along the conductor pointing in the direction of the current. With the fingers partly closed, the finger tips will point in the direction of the magnetic field. Maxwell’s rule states, “If the direction of travel of a right-handed corkscrew represents the direction of the current in a straight conductor, the direction of rotation of the corkscrew will represent the direction of the magnetic lines of force.” Impedance The total impedance created by resistors, capacitors, and inductors in circuits can be determined with the following equations. For resistance and capacitance in series,

Z =

R 2 + XC2

(1.53)

XC R

(1.54)

R 2 + X L2

(1.55)

XL R

(1.56)

when X L > XC when XC > X L

(1.57) (1.58)

q = arctan For resistance and inductance in series,

Z =

q = arctan For inductance and capacitance in series,

ïìX – XC Z = í L ïîXC – X L For resistance, inductance, and capacitance in series,

Z =

R 2 + (X L - XC )2

(1.59)

X L – XC R

(1.60)

q = arctan For capacitance and resistance in parallel,

Z =

RXC

(1.61)

R 2 + XC2 For resistance and inductance in parallel,

Z =

RX L R 2 + X L2

(1.62)

For capacitance and inductance in parallel,

ì X L XC ï ï X – XC Z = í L ï XC X L ïî XC – X L

when X L > XC

(1.63)

when XC > X L

(1.64)

For inductance, capacitance, and resistance in parallel,

RX L XC

Z =

X L2 XC2

q = arctan

+ R 2 (X L – XC )2 R ( X L – XC ) X L XC

(1.65)

(1.66)

For inductance and series resistance in parallel with resistance,

Z = R2

R12 + X L2 (R1 + R2 )2 + X L2

q = arctan

X L R2 R12

+ X L2 + R1R2

(1.67)

(1.68)

For inductance and series resistance in parallel with capacitance,

R 2 + X L2

Z = XC

R 2 + (X L - XC )2

q = arctan

X L ( XC – X L ) - R 2 RXC

(1.69)

(1.70)

For capacitance and series resistance in parallel with inductance and series resistance,

Z =

(R12 + X L2 )(R22 + XC2 ) (R1 + R2 )2 + (X L – XC )2

q = arctan

X L (R22 + XC2 ) - XC (R12 + X L2 ) R1(R22 + XC2 ) + R2 (R12 + X L2 )

(1.71)

(1.72)

where Z = impedance, W; R = resistance, W; L = inductance, H; XL = inductive reactance, W; XC = capacitive reactance, W; and q = phase angle, degrees, by which current leads voltage in a capacitive circuit or lags voltage in an inductive circuit (0° indicates an in-phase condition). Resonant Frequency When an inductor and capacitor are connected in series or parallel, they form a resonant circuit. The resonant frequency can be determined from the equation

f =

2 p LC 1 = 2 pCXC =

(1.73)

XL 2 pL

where f = frequency, Hz; L = inductance, H; C = capacitance, F; and XL, XC = impedance, W. The resonant frequency can also be determined through the use of a reactance chart developed by the Bell Telephone Laboratories (Fig. 1.21). This chart can be used for solving problems of inductance, capacitance, frequency, and impedance. If two of the values are known, the third and fourth values may be found with its use.

Defining Terms Air capacitor: A fixed or variable capacitor in which air is the dielectric material between the capacitor’s plates. Ambient temperature: The temperature of the air or liquid surrounding any electrical part or device. Usually refers to the effect of such temperature in aiding or retarding removal of heat by radiation and convection from the part or device in question. Ampere-turns: The magnetomotive force produced by a coil, derived by multiplying the number of turns of wire in a coil by the current (A) flowing through it. Anode: The positive electrode of a capacitor. Capacitive reactance: The opposition offered to the flow of an alternating or pulsating current by capacitance measured in ohms. Capacitor: An electrical device capable of storing electrical energy and releasing it at some predetermined rate at some predetermined time. It consists essentially of two conducting surfaces (electrodes) separated by an insulating material or dielectric. A capacitor stores electrical energy, blocks the flow of direct current, and permits the flow of alternating current to a degree dependent essentially upon capacitance and frequency. The amount of energy stored, E = 0.5 CV 2. Cathode: The capacitor’s negative electrode. Coil: Anumber of turns of wire in the form of a spiral. The spiral may be wrapped around an iron core or an insulating form, or it may be self-supporting. A coil offers considerable opposition to ac current but very little to dc current. Conductor: Abare or insulated wire or combination of wires not insulated from one another, suitable for carrying an electric current. Dielectric: The insulating (nonconducting) medium between the two electrodes (plates) of a capacitor. Dielectric constant: The ratio of the capacitance of a capacitor with a given dielectric to that of the same capacitor having a vacuum dielectric. Disk capacitor: A small single-layer ceramic capacitor with a dielectric insulator consisting of conductively silvered opposing surfaces. Dissipation factor (DF): The ratio of the effective series resistance of a capacitor to its reactance at a specified frequency measured in percent. Electrolyte: Current-conducting solution between two electrodes or plates of a capacitor, at least one of which is covered by a dielectric. © 2000 by CRC Press LLC

FIGURE 1.21 Reactance chart. (Courtesy AT&T Bell Laboratories.)

Electrolytic capacitor: A capacitor solution between two electrodes or plates of a capacitor, at least one of which is covered by a dielectric. Equivalent series resistance (ESR): All internal series resistance of a capacitor concentrated or “lumped” at one point and treated as one resistance of a capacitor regardless of source, i.e., lead resistance, termination losses, or dissipation in the dielectric material. Farad: The basic unit of measure in capacitors. Acapacitor charged to 1 volt with a charge of 1 coulomb (1 ampere flowing for 1 second) has a capacitance of 1 farad. Field: Ageneral term referring to the region under the influence of a physical agency such as electricity, magnetism, or a combination produced by an electrical charged object. Impedance (Z): Total opposition offered to the flow of an alternating or pulsating current measured in ohms. (Impedance is the vector sum of the resistance and the capacitive and inductive reactance, i.e., the ratio of voltage to current.) Inductance: The property which opposes any change in the existing current. Inductance is present only when the current is changing. Inductive reactance (XL ): The opposition to the flow of alternating or pulsating current by the inductance of a circuit. © 2000 by CRC Press LLC

Inductor: Aconductor used to introduce inductance into a circuit. Leakage current: Stray direct current of relatively small value which flows through a capacitor when voltage is impressed across it. Magnetomotive force: The force by which the magnetic field is produced, either by a current flowing through a coil of wire or by the proximity of a magnetized body. The amount of magnetism produced in the first method is proportional to the current through the coil and the number of turns in it. Mutual inductance: The property that exists between two current-carrying conductors when the magnetic lines of force from one link with those from another. Negative-positive-zero (NPO): An ultrastable temperature coefficient (±30 ppm/°C from –55 to 125°C) temperature-compensating capacitor. Phase: The angular relationship between current and voltage in an ac circuit. The fraction of the period which has elapsed in a periodic function or wave measured from some fixed origin. If the time for one period is represented as 360° along a time axis, the phase position is called phase angle. Polarized capacitor: An electrolytic capacitor in which the dielectric film is formed on only one metal electrode. The impedance to the flow of current is then greater in one direction than in the other. Reversed polarity can damage the part if excessive current flow occurs. Power factor (PF): The ratio of effective series resistance to impedance of a capacitor, expressed as a percentage. Quality factor (Q): The ratio of the reactance to its equivalent series resistance. Reactance (X): Opposition to the flow of alternating current. Capacitive reactance (Xc) is the opposition offered by capacitors at a specified frequency and is measured in ohms. Resonant frequency: The frequency at which a given system or object will respond with maximum amplitude when driven by an external sinusoidal force of constant amplitude. Reverse leakage current: Anondestructive current flowing through a capacitor subjected to a voltage of polarity opposite to that normally specified. Ripple current: The total amount of alternating and direct current that may be applied to an electrolytic capacitor under stated conditions. Temperature coefficient (TC): A capacitor’s change in capacitance per degree change in temperature. May be positive, negative, or zero and is usually expressed in parts per million per degree Celsius (ppm/°C) if the characteristics are linear. For nonlinear types, TC is expressed as a percentage of room temperature (25°C)capacitance. Time constant: In a capacitor-resistor circuit, the number of seconds required for the capacitor to reach 63.2% of its full charge after a voltage is applied. The time constant of a capacitor with a capacitance (C) in farads in series with a resistance (R) in ohms is equal to R ´ C seconds. Winding: Aconductive path, usually wire, inductively coupled to a magnetic core or cell.

Related Topic 35.1 Maxwell Equations

References P. Beckman and A. Spizzichino, The Scattering of Electromagnetic Waves from Random Surfaces, New York: Macmillan, 1963. L.V. Blake, Radar Range-Performance Analysis, Norwood, Mass.: Artech House, 1986, pp. 253–271. R.E. Collin, Antennas and Radiowave Propagation, New York: McGraw-Hill, 1985, pp. 339–456. G. Hyde, “Microwave propagation,” in Antenna Engineering Handbook, 2nd ed., R.C. Johnson and H. Jasik, Eds., New York: McGraw-Hill, 1984, pp. 45.1–45.17. D.E. Kerr, Propagation of Short Radio Waves, New York: McGraw-Hill (republished by Peter Peregrinus, London, 1987), 1951, pp. 396–444. C.H. Liu and D.J. Fang, “Propagation,” in Antenna Handbook: Theory, Applications, and Design, Y.T. Lo and S.W. Lee, Eds., New York: Van Nostrand Reinhold, 1988, pp. 29.1–29.56. T. Oguchi, “Electromagnetic wave propagation and scattering in rain and other hydrometeors,” Proc. IEEE, vol. 71, pp. 1029–1078, 1983. M.N.O. Sadiku, Numerical Techniques in Electromagnetics, Boca Raton, Fla.: CRC Press, 1992, pp. 96–116. B.G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Ant. Prog., vol. 15, pp. 668–671, 1967.

Further Information There are several sources of information dealing with the theory and practice of wave propagation in space. Some of these are in the reference section. Journals such as Radio Science, IEE Proceedings Part H, and IEEE Transactions on Antennas and Propagation are devoted to EM wave propagation. Radio Science is available from the American Geophysical Union, 2000 Florida Avenue NW, Washington DC 20009; IEE Proceedings Part H from IEE Publishing Department, Michael Faraday House, 6 Hills Way, Stevenage, Herts, SG1 2AY, U.K.; and IEEE Transactions on Antennas and Propagation from IEEE, 445 Hoes Lane, P.O. Box 1331, Piscataway, NJ 08855-1331. Other mechanisms that can affect EM wave propagation in space, not discussed in this section, include clouds, dust, and the ionosphere. The effect of the ionosphere is discussed in detail in standard texts.

37.2

Waveguides

Kenneth Demarest Waveguide Modes Any structure that guides electromagnetic waves can be considered a waveguide. Most often, however, this term refers to closed metal cylinders that maintain the same cross-sectional dimensions over long distances. Such a structure is shown in Fig. 37.7, which consists of a metal cylinder filled with a dielectric. When filled with low-loss dielectrics (such as air), waveguides typically exhibit lower losses than transmission lines, which makes them useful for transporting RF energy over relatively long distances. They are most often used for frequencies ranging from 1 to 150 GHz. Every type of waveguide has an infinite number of distinct electromagnetic field configurations that can exist inside it. Each of these configurations is called a waveguide mode. The characteristics of these modes depend upon the cross-sectional dimensions of the conducting cylinder, the type of dielectric material inside the waveguide, and the frequency of operation. Waveguide modes are typically classed according to the nature of the electric and magnetic field components Ez and H z . These components are called the longitudinal components of the fields. Several types of modes are possible in waveguides: TE modes:

Transverse-electric modes, sometimes called H modes. These modes have Ez = 0 at all points within the waveguide, which means that the electric field vector is always perpendicular (i.e., transverse) to the waveguide axis. These modes are always possible in waveguides with uniform dielectrics.

TM modes:

Transverse-magnetic modes, sometimes called E modes. These modes have Hz = 0 at all points within the waveguide, which means that the magnetic field vector is perpendicular to the waveguide axis. Like TE modes, they are always possible in waveguides with uniform dielectrics.

EH modes:

EH modes are hybrid modes in which neither Ez nor Hz are zero, but the characteristics of the transverse fields are controlled more by Ez than H z . These modes are often possible in waveguides with inhomogeneous dielectrics.

HE modes:

HE modes are hybrid modes in which neither Ez nor Hz are zero, but the characteristics of the transverse fields are controlled more by Hz than E z . Like EH modes, these modes are often possible in waveguides with inhomogeneous dielectrics.

TEM modes:

Transverse-electromagnetic modes, often called transmission line modes. These modes can exist only when a second conductor exists within the waveguide, such as a center conductor on a coaxial cable. Because these modes cannot exist in single, closed conductor structures, they are not waveguide modes.

FIGURE 37.7 A uniform waveguide with arbitrary cross section.

Waveguide modes are most easily determined by first computing the longitudinal field components, Ez and Hz , that can be supported by the waveguide. From these, the transverse components (such as Ex and Ey) can easily be found simply by taking spatial derivatives of the longitudinal fields [Collin, 1992]. When the waveguide properties are constant along the z axis, Ez and Hz vary in the longitudinal direction as Ez , Hz µ exp(wt – gz), where w = 2pf is the radian frequency of operation and g is a complex number of the form

g = a + jb

(37.59)

The parameters g, a, and b are called the propagation, attenuation, and phase constants, respectively, and j = – 1 . When there are no metal or dielectric losses, g is always either purely real or imaginary. When g is real, Ez and Hz have constant phase and decay exponentially with increasing z. When g is imaginary, Ez and Hz vary in phase with increasing z but do not decay in amplitude. When this occurs, the fields are said to be propagating. When the dielectric is uniform (i.e., homogeneous), Ez and Hz satisfy the scalar wave equation at all points within the waveguide:

Ñt 2Ez + h2Ez = 0

(37.60)

Ñt 2Hz + h2Hz = 0

(37.61)

h2 = (2p f)2me + g 2 = k2 + g 2

(37.62)

and

where

Here, m and e are the permeability and permittivity of the dielectric media, respectively, and k = 2pf me is the wavenumber of the dielectric. The operator Ñt2 is called the transverse Laplacian operator. In Cartesian coordinates,

Ñt2 =

¶2

¶x

¶y 2

+ 2

Most of the properties of the allowed modes in real waveguides can usually be found by assuming that the walls are perfectly conducting. Under this condition, Ez = 0 and ¶H z /¶p = 0 at the waveguide walls, where p is the direction perpendicular to the waveguide wall. When these conditions are imposed upon the general solutions of Eqs. (37.60) and (37.61), it is found that only certain values of h are allowed. These values are called the modal eigenvalues and are determined by the cross-sectional shape of the waveguide. Using Eq. (37.62), the propagation constant g for each mode varies with frequency according to

æ f ö g = a + jb = h 1 - ç ÷ è fc ø

(37.63)

where

fc =

h 2p me

(37.64)

The modal parameter fc has units hertz and is called the cut-off frequency of the mode it is associated with. According to Eq. (37.63), when f > fc , the propagation constant g is imaginary and thus the mode is propagating. On the other hand, when f < fc , g is real, which means that the fields decay exponentially with increasing values of z. Modes operated at frequencies below their cut-off frequency are not able to propagate energy over long distances and are called evanescent modes. The dominant mode of a waveguide is the one with the lowest cut-off frequency. Although higher-order modes are often useful for a variety of specialized uses of waveguides, signal distortion is usually minimized when a waveguide is operated in the frequency range where only the dominant mode exists. This range of frequencies is called the dominant range of the waveguide. The distance over which the fields of propagating modes repeat themselves is called the guide wavelength l g . From Eq. (37.63), it can be shown that lg always varies with frequency according to

lg =

lo æf ö 1- ç c÷ è f ø

(37.65)

where lo = 1/( f me) is the wavelength of a plane wave of the same frequency in an infinite sample of the waveguide dielectric. For f >> f c , lg » lo . Also, lg ® ¥ as f ® fc , which is one reason why it is usually undesirable to operate a waveguide mode near modal cut-off frequencies. Although waveguide modes are not plane waves, the ratio of their transverse electric and magnetic field magnitudes is constant throughout the cross section of the waveguide, just as for plane waves. This ratio is called the modal wave impedance and has the following values for TE and TM modes:

Z TE =

ET j wm = HT g

(37.66)

Z TM =

ET g = HT j we

(37.67)

and

where E T and HT are the magnitudes of the transverse electric and magnetic fields, respectively. In the limit as f ® ¥, both ZT E and ZTM approach me, which is the intrinsic impedance of the dielectric medium. On the other hand, as f ® f c , ZTE ® ¥ and ZTM ® 0, which means that the transverse electric fields are dominant in TE modes near cut-off and vice versa for TM modes.

Rectangular Waveguides A rectangular waveguide is shown in Fig. 37.8. The conducting walls are formed such that the inner surfaces form a rectangular cross section, with dimensions a and b along the x and y coordinate axes, respectively. If the walls are perfectly conducting and the dielectric material is lossless, the field components for the TEmn modes are given by

Ex = H0

æ np ö æ mp ö j wm æ n p ö x ÷ sin ç y ÷ exp( j wt - g mn z ) ç ÷ cos ç 2 è b ø è a ø hmn è b ø

(37.68)

FIGURE 37.8 A rectangular waveguide.

æ np ö æ mp ö j wm æ m p ö sin cos x y ÷ exp( j wt - g mn z ) ç ÷ ç ÷ ç 2 è b ø è a ø è a ø hmn

(37.69)

H x = H0

æ np ö æ mp ö g mn æ m p ö x ÷ cos ç y ÷ exp( j wt - g mn z ) ÷ sin ç 2 ç è b ø è a ø hmn è a ø

(37.70)

H y = H0

æ np ö æ mp ö g mn æ n p ö cos sin x y ÷ exp( j wt - g mn z ) ç ÷ ç ÷ ç 2 è b ø è a ø è b ø hmn

(37.71)

E y = -H 0 Ez = 0

æ mp ö æ np ö H z = H 0 cos ç x ÷ cos ç y ÷ exp( j wt - g mn z ) è a ø è b ø

(37.72)

where 2

hmn =

æ mp ö æ npö ç ÷ +ç ÷ è a ø è b ø

= 2 pf c mn me

(37.73)

For the TMmn modes, m and n can be any positive integer value, including zero, as long as both are not zero. The field components for the TMmn modes are

E x = -E 0

æ np ö æ mp ö g mn æ m p ö x ÷ sin ç y ÷ exp( j wt - g mn z ) ÷ cos ç 2 ç è b ø è a ø hmn è a ø

(37.74)

E y = -E 0

æ np ö æ mp ö g mn æ n p ö sin cos x y ÷ exp( j wt - g mn z ) ç ÷ ç ÷ ç 2 è b ø è a ø è b ø hmn

(37.75)

æ np ö æ mp ö E z = E 0 sin ç x ÷ sin ç y ÷ exp( j wt - g mn z ) è b ø è a ø

(37.76)

FIGURE 37.9 Field configurations for the TE10 (dominant) mode of a rectangular waveguide. Solid lines, E; dashed lines, H. (Source: Adapted from N. Marcuvitz, Waveguide Handbook, 2nd ed., London: Peter Peregrinus Ltd., and New York: McGrawHill, 1986, p. 63. With permission.) TABLE 37.3 Cut-off Frequencies of the Lowest-Order Rectangular Waveguide Modes (Referenced to the Cut-off Frequency of the Dominant Mode) for a Rectangular Waveguide with a/b = 2.1

H x = E0

fc /fc10

Modes

1.0 2.0 2.1 2.326 2.9 3.0 3.662 4.0

TE10 TE20 TE01 TE11, TM11 TE21, TM21 TE30 TE31, TM31 TE40

æ np ö æ mp ö j we æ n p ö sin cos x y ÷ exp( j wt - g mn z ) ç ÷ ç ÷ ç 2 è b ø è a ø è b ø hmn

(37.77)

æ np ö æ mp ö j we æ m p ö x ÷ sin ç y ÷ exp( j wt - g mn z ) ÷ cos ç 2 ç è b ø è a ø hmn è a ø

(37.78)

H y = -E 0 Hz = 0

(37.79)

where the values of hmn and fcmn are given by Eq. (37.73). For the TMmn modes, m and n can be any positive integer value except zero. The dominant mode in a rectangular waveguide is the TE10 mode, which has a cut-off frequency

f c 10 =

1 2a me

c 2a

(37.80)

where c is the speed of light in the dielectric media. The modal field patterns for this mode are shown in Fig. 37.9. Table 37.3 shows the cut-off frequencies of the lowest-order rectangular waveguide modes (as referenced to the cut-off frequency of the dominant mode) when a/b = 2.1. The modal field patterns for several lower-order modes are shown in Fig. 37.10.

Circular Waveguides A circular waveguide with inner radius a is shown in Fig. 37.11. Here the axis of the waveguide is aligned with the z axis of a circular-cylindrical coordinate system, where r and f are the radial and azimuthal coordinates, © 2000 by CRC Press LLC

FIGURE 37.10 Field configurations for the TE11, TM11, and the TE21 modes. Solid lines, E; dashed lines, H. (Source: Adapted from N. Marcuvitz, Waveguide Handbook, 2nd. ed., London: Peter Peregrinus Ltd., and New York: McGraw-Hill, 1986, p. 59. With permission.)

FIGURE 37.11 A circular waveguide.

respectively. If the walls are perfectly conducting and the dielectric material is lossless, the equations for the TEnm modes are

Er = H 0

Ef = H 0

j wmn

J n (hnm r) sin(n f) exp( j wt - g nm z )

(37.81)

j wm J n¢ (hnm r) cos (n f) exp( j wt - g nm z ) hnm

(37.82)

2 r hnm

Ez = 0

(37.83)

H r = -H 0

Hf = H0

g nm J ¢ (h r) cos (n f) exp( j wt - g nm z ) hnm n nm g nm

2 r hnm

J n (hnm r) sin (n f) exp( j wt - g nm z )

H z = H 0 J n (hnm r) cos (n f) exp( j wt - g nm z )

(37.84)

(37.85)

(37.86)

where n is any positive valued integer, including zero, and Jn(x) and J ¢n(x) are the regular Bessel function of order n and its first derivative, respectively. The allowed values of the modal eigenvalues hnm satisfy

(37.87)

where m signifies the root number of Eq. (37.87). By convention, 1 < m < ¥, where m = 1 indicates the smallest root. The equations that define the TMnm modes in circular waveguides are

g nm J n¢ (hnm r) cos(n f) exp( j wt - g nm z ) hnm

E r = -E 0

E f = E0

g nm 2 r hnm

J n (hnm r) sin (n f) exp( j wt - g nm z )

E z = E 0 J n (hnm r) cos (n f) exp( j wt - g nm z ) H r = -E 0

H f = -E 0

j wen

(37.88)

(37.89)

(37.90)

J n (hnm r) sin (n f) exp( j wt - g nm z )

(37.91)

j we J n¢ (hnm r) cos (n f) exp( j wt - g nm z ) hnm

(37.92)

2 r hnm

Hz = 0

(37.93)

where n is any positive valued integer, including zero. For the TMnm modes, the values of the modal eigenvalues are solutions of

Jn(hnma) = 0

(37.94)

where m signifies the root number of Eq. (37.94). As in the case of the TE modes, 1 < m < ¥. The dominant mode in a circular waveguide is the TE11 mode, which has a cut-off frequency given by

f c 11 =

0.293

(37.95)

a me

The configuration of the electric and magnetic fields of this mode is shown in Fig. 37.12.

FIGURE 37.12 Field configuration for the TE11 (dominant) mode of a circular waveguide. Solid lines, E; dashed lines, H. (Source: Adapted from N. Marcuvitz, Waveguide Handbook, 2nd ed., London: Peter Peregrinus Ltd., and New York: McGrawHill, 1986, p. 68. With permission.)

TABLE 37.4 Cut-off Frequencies of the Lowest-Order Circular Waveguide Modes, Referenced to the Cut-off Frequency of the Dominant Mode fc /fc11

Modes

1.0 1.307 1.66 2.083 2.283 2.791 2.89 3.0

TE11 TM01 TE21 TE01, TM11 TE31 TM21 TE41 TE12

FIGURE 37.13 Field configurations for the TM01, TE21, and TE01 circular waveguide modes. Solid lines, E; dashed lines, H. (Source: Adapted from N. Marcuvitz, Waveguide Handbook, 2nd ed., London: Peter Peregrinus Ltd., and New York: McGrawHill, 1986, p. 71. With permission.)

Table 37.4 shows the cut-off frequencies of the lowest-order modes for circular waveguides, referenced to the cut-off frequency of the dominant mode. The modal field patterns for several lower-order modes are shown in Fig. 37.13.

Commercially Available Waveguides The dimensions of standard rectangular waveguides are given in Table 37.5. In addition to rectangular and circular waveguides, several other waveguide types are commonly used in microwave applications. Among these are ridge waveguides and elliptical waveguides. The modes of elliptical waveguides can be expressed in terms of Mathieu functions [Kretzschmar, 1970] and are similar to those of circular waveguides but are less perturbed by minor twists and bends of the waveguide. This property makes them attractive for coupling to antennas. Single-ridge and double-ridge waveguides are shown in Fig. 37.14. The modes of these waveguides bear similarities to those of rectangular guides, but can only be derived numerically [Montgomery, 1971]. Ridge waveguides are useful because their dominant ranges exceed those of rectangular waveguides. However, this range increase is obtained at the expense of higher losses. Waveguides are also available in a number of constructions, including rigid, semirigid, and flexible. In applications where it is not necessary for the waveguide to bend, rigid construction is always the best since it exhibits the lowest loss. In general, the more flexible the waveguide construction, the higher the loss.

Waveguide Losses There are two mechanisms that cause losses in waveguides: dielectric losses and metal losses. In both cases, these losses cause the amplitudes of the propagating modes to decay as exp(- a z), where a is the attenuation constant, measured in units of nepers/meter. Typically, the attenuation constant is considered as the sum of

TABLE 37.5

Standard Rectangular Waveguides Cut-off Frequency for

EIAa Designation WRb( ) 2300 2100 1800 1500 1150 975 770 650 510 430 340 284 229 187 159 137 112 90 75 62 51 42 34 28 22 19 15

Physical Dimensions Inside, cm (in.)

Outside, cm (in.)

Width

Height

Width

Height

58.420 (23.000) 53.340 (21.000) 45.720 (18.000) 38.100 (15.000) 29.210 (11.500) 24.765 (9.750) 19.550 (7.700) 16.510 (6.500) 12.954 (5.100) 10.922 (4.300) 8.636 (3.400) 7.214 (2.840) 5.817 (2.290) 4.755 (1.872) 4.039 (1.590) 3.485 (1.372) 2.850 (1.122) 2.286 (0.900) 1.905 (0.750) 1.580 (0.622) 1.295 (0.510) 1.067 (0.420) 0.864 (0.340) 0.711 (0.280) 0.569 (0.224) 0.478 (0.188) 0.376 (0.148)

29.210 (11.500) 26.670 (10.500) 22.860 (9.000) 19.050 (7.500) 14.605 (5.750) 12.383 (4.875) 9.779 (3.850) 8.255 (3.250) 6.477 (2.500) 5.461 (2.150) 4.318 (1.700) 3.404 (1.340) 2.908 (1.145) 2.215 (0.872) 2.019 (0.795) 1.580 (0.622) 1.262 (0.497) 1.016 (0.400) 0.953 (0.375) 0.790 (0.311) 0.648 (0.255) 0.432 (0.170) 0.432 (0.170) 0.356 (0.140) 0.284 (0.112) 0.239 (0.094) 0.188 (0.074)

59.055 (23.250) 53.973 (21.250) 46.350 (18.250) 38.735 (15.250) 29.845 (11.750) 25.400 (10.000) 20.244 (7.970) 16.916 (6.660) 13.360 (5.260) 11.328 (4.460) 9.042 (3.560) 7.620 (3.000) 6.142 (2.418) 5.080 (2.000) 4.364 (1.718) 3.810 (1.500) 3.175 (1.250) 2.540 (1.000) 2.159 (0.850) 1.783 (0.702) 1.499 (0.590) 1.270 (0.500) 1.067 (0.420) 0.914 (0.360) 0.772 (0.304) 0.681 (0.268) 0.579 (0.228)

29.845 (11.750) 27.305 (10.750) 23.495 (9.250) 19.685 (7.750) 15.240 (6.000) 13.018 (5.125) 10.414 (4.100) 8.661 (3.410) 6.883 (2.710) 5.867 (2.310) 4.724 (1.860) 3.810 (1.500) 3.233 (1.273) 2.540 (1.000) 2.344 (0.923) 1.905 (0.750) 1.588 (0.625) 1.270 (0.500) 1.207 (0.475) 0.993 (0.391) 0.851 (0.335) 0.635 (0.250) 0.635 (0.250) 0.559 (0.220) 0.488 (0.192) 0.442 (0.174) 0.391 (0.154)

Recommended Frequency

Air-filled

Range for

Waveguide, GHz

TE10 Mode, GHZ

0.257

0.32–0.49

0.281

0.35–0.53

0.328

0.41–0.62

0.394

0.49–0.75

0.514

0.64–0.98

0.606

0.76–1.15

0.767

0.96–1.46

0.909

1.14–1.73

1.158

1.45–2.20

1.373

1.72–2.61

1.737

2.17–3.30

2.079

2.60–3.95

2.579

3.22–4.90

3.155

3.94–5.99

3.714

4.64–7.05

4.304

5.38–8.17

5.263

6.57–9.99

6.562

8.20–12.50

7.874

9.84–15.00

9.494

11.90–18.00

11.583

14.50–22.00

14.058

17.60–26.70

17.361

21.70–33.00

21.097

26.40–40.00

26.362

32.90–50.10

31.381

39.20–59.60

39.894

49.80–75.80

TABLE 37.5 (continued)

Standard Rectangular Waveguides Cut-off Frequency for

EIAa Designation WRb( ) 12 10 8 7 5 4 3

Physical Dimensions Inside, cm (in.)

Outside, cm (in.)

Width

Height

Width

Height

0.310 (0.122) 0.254 (0.100) 0.203 (0.080) 0.165 (0.065) 0.130 (0.051) 0.109 (0.043) 0.086 (0.034)

0.155 (0.061) 0.127 (0.050) 0.102 (0.040) 0.084 (0.033) 0.066 (0.026) 0.056 (0.022) 0.043 (0.017)

0.513 (0.202) 0.457 (0.180) 0.406 (0.160) 0.343 (0.135) 0.257 (0.101) 0.211 (0.083) 0.163 (0.064)

0.358 (0.141) 0.330 (0.130) 0.305 (0.120) 0.262 (0.103) 0.193 (0.076) 0.157 (0.062) 0.119 (0.047)

Recommended Frequency

Air-filled

Range for

Waveguide, GHz

TE10 Mode, GHZ

48.387

60.50–91.90

59.055

73.80–112.00

73.892

92.20–140.00

90.909

114.00–173.00

115.385

145.00–220.00

137.615

172.00–261.00

174.419

217.00–333.00

Electronic Industry Association. Rectangular waveguide.

FIGURE 37.14 Single- and double-ridged waveguides.

two components: a = a die + amet, where adie and amet are the dielectric and metal attenuation constants, respectively. The attenuation constant adie can be found directly from Eq. (37.63) simply by generalizing the dielectric wavenumber k to include the effect of the dielectric conductivity s. For a lossy dielectric, the wavenumber is given by k2 = w2me[1 + (s/jwe)]. Thus, from Eqs. (37.62) and (37.63) the attenuation constant adie due to dielectric losses is given by

é æ s öù a die = real ê h 2 - w 2 me ç 1 + ÷ú êë j we ø úû è

(37.96)

where the allowed values of h are given by Eq. (37.73) for rectangular modes and Eqs. (37.87) and (37.94) for circular modes.

FIGURE 37.15 Values of metallic attenuation constant a for the first few waveguide modes in a circular waveguide of diameter d, plotted against normalized wavelength. (Source: A.J. Baden Fuller, Microwaves, 2nd ed., New York: Pergamon Press, 1979, p. 138. With permission.)

The metal loss constant amet is usually obtained by assuming that the wall conductivity is high enough to have only a negligible effect on the transverse properties of the modal field patterns. Using this assumption, the power loss in the walls per unit distance along the waveguide can then be calculated to obtain amet [Marcuvitz, 1986]. Figure 37.15 shows the metal attenuation constants for several circular waveguide modes, each normalized to the resistivity Rs of the walls, where Rs = ( pfm ¤ s ) and where m and s are the permeability and conductivity of the metal walls, respectively. As can be seen from this figure, the TE0m modes exhibit particularly low loss at frequencies significantly above their cut-off frequencies, making them useful for transporting microwave energy over large distances.

Mode Launching When coupling electromagnetic energy into a waveguide, it is important to ensure that the desired modes are excited and that reflections back to the source are minimized. Similar concerns must be considered when

Antenna probe

l 2 TE10 mode

FIGURE 37.16 Coaxial to rectangular waveguide transition that couples the transmission line mode to the dominant waveguide mode

Short-circuited end l 2 Antenna probe TM11 mode

TM21 mode

FIGURE 37.17 Coaxial to rectangular waveguide transitions that couple the transmission line mode to the TM11 and TM21 waveguide modes.

coupling energy from a waveguide to a transmission line or circuit element. This is achieved by using launching (or coupling) structures that allow strong coupling between the desired modes on both structures. Figure 37.16 shows a mode launching structure for coaxial cable to rectangular waveguide transitions. This structure provides good coupling between the TEM (transmission line) mode on a coaxial cable and the TE10 mode in the waveguide because the antenna probe excites a strong transverse electric field in the center of the waveguide, directed between the broad walls. The distance between the probe and the short circuit back wall is chosen to be approximately l/4, which allows the TE10 mode launched in this direction to reflect off the short circuit and arrive in phase with the mode launched towards the right. Launching structures can also be devised to launch higher-order modes. Mode launchers that couple the transmission line mode on a coaxial cable to the TM11 and TM21 waveguide mode are shown in Fig. 37.17.

Defining Terms Cut-off frequency: The minimum frequency at which a waveguide mode will propagate energy with little or no attenuation. Guide wavelength: The distance over which the fields of propagating modes repeat themselves in a waveguide. Waveguide: A closed metal cylinder, filled with a dielectric, used to transport electromagnetic energy over short or long distances. Waveguide modes: Unique electromagnetic field configurations supported by a waveguide that have distinct electrical characteristics. Wave impedance: The ratio of the transverse electric and magnetic fields inside a waveguide.

Related Topics 35.1 Maxwell Equations • 39.1 Passive Microwave Devices • 42.1 Lightwave Waveguides

References A. J. Baden Fuller, Microwaves, 2nd ed., New York: Pergamon Press, 1979. R. E. Collin, Foundations for Microwave Engineering, 2nd ed., New York: McGraw-Hill, 1992. J. Kretzschmar, “Wave propagation in hollow conducting elliptical waveguides,” IEEE Transactions on Microwave Theory and Techniques, vol. MTT-18, no. 9, pp. 547–554, Sept. 1970. S. Y. Liao, Microwave Devices and Circuits, 3rd ed., Englewood Cliffs, N.J.: Prentice-Hall, 1990. N. Marcuvitz, Waveguide Handbook, 2nd ed., London: Peter Peregrinus Ltd., 1986. J. Montgomery, “On the complete eigenvalue solution of ridged waveguide,” IEEE Transactions on Microwave Theory and Techniques, vol. MTT-19, no. 6, pp. 457–555, June 1971.

Further Information There are many textbooks and handbooks that cover the subject of waveguides in great detail. In addition to the references cited above, others include L. Lewin, Theory of Waveguides, New York: John Wiley, 1975. Reference Data for Radio Engineers, Howard W. Sams Co., 1975. R. E. Collin, Field Theory of Guided Waves, 2nd ed., Piscataway, N.J.: IEEE Press, 1991. F. Gardiol, Introduction to Microwaves, Dedham, Mass.: Artech House, 1984. S. Ramo, J. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics, New York: John Wiley, 1965.

Kolias, N.J., Compton, R.C., Fitch, J.P., Pozar, D.M. “Antennas” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000

38 Antennas 38.1 Wire

N.J. Kolias Raytheon Company

R.C. Compton Cornell University

J. Patrick Fitch Lawrence Livermore Laboratory

David M. Pozar University of Massachusetts at Amherst

38.1

Short Dipole • Directivity • Magnetic Dipole • Input Impedance • Arbitrary Wire Antennas • Resonant Half-Wavelength Antenna • End Loading • Arrays of Wire Antennas • Analysis of General Arrays • Arrays of Identical Elements • Equally Spaced Linear Arrays • Planar (2-D) Arrays • Yagi–Uda Arrays • LogPeriodic Dipole Arrays

38.2 Aperture The Oscillator or Discrete Radiator • Synthetic Apertures • Geometric Designs • Continuous Current Distributions (Fourier Transform) • Antenna Parameters

38.3 Microstrip Antennas Introduction • Basic Microstrip Antenna Element • Feeding Techniques for Microstrip Antennas • Microstrip Antenna Arrays • Computer-Aided Design for Microstrip Antennas

Wire

N.J. Kolias and R.C. Compton Antennas have been widely used in communication systems since the early 1900s. Over this span of time scientists and engineers have developed a vast number of different antennas. The radiative properties of each of these antennas are described by an antenna pattern. This is a plot, as a function of direction, of the power Pr per unit solid angle W radiated by the antenna. The antenna pattern, also called the radiation pattern, is usually plotted in spherical coordinates q and j. Often two orthogonal cross sections are plotted, one where the E-field lies in the plane of the slice (called the E-plane) and one where the H-field lies in the plane of the slice (called the H-plane).

Short Dipole Antenna patterns for a short dipole are plotted in Fig. 38.1. In these plots the radial distance from the origin to the curve is proportional to the radiated power. Antenna plots are usually either on linear scales or decibel scales (10 log power). The antenna pattern for a short dipole may be determined by first calculating the vector potential A [Collin, 1985; Balanis, 1982; Harrington, 1961; Lorrain and Corson, 1970]. Using Collin’s notation, the vector potential in spherical coordinates is given by

A = m 0I dl

e - jk 0r (a r cos q - a q sin q) 4 pr

(38.1)

FIGURE 38.1 Radiation pattern for a short dipole of length dl (dl << l0). These are plots of power density on linear scales. (a) E-plane; (b) H-plane; (c) three-dimensional view with cutout.

where k0 = 2p /l0, and I is the current, assumed uniform, in the short dipole of length dl (dl << l0). Here the assumed time dependence e j w t has not been explicitly shown. The electric and magnetic fields may then be determined using

E = - j wA +

ÑÑ × A j wm 0 e 0

H =

1 Ñ´A m0

(38.2)

The radiated fields are obtained by calculating these fields in the so-called far-field region where r >> l. Doing this for the short dipole yields

E = jZ 0Idl k 0 sin q

e - jk 0r aq 4 pr

e - jk 0r H = jIdl k 0 sin q aj 4 pr

(38.3)

where Z0 = m 0 ¤ e 0. The average radiated power per unit solid angle W can then be found to be

DPr (q, j) 1 sin2 q = r 2 Âe{E ´ H * ×a r } = * I * 2 Z 0 (dl )2 k 02 2 DW 32 p2

(38.4)

Directivity The directivity D(q,j) and gain G(q,j) of an antenna are defined as

D(q, j) =

Radiated power per solid angle DPr (q, j)/DW = Total radiated power/4p Pr /4 p

(38.5)

DPr (q, j)/DW Radiated power per solid angle G (q, j) = = Pin /4 p Total input power/4p Antenna efficiency, h, is given by

h º

Pr G (q, j) = Pin D(q, j)

(38.6)

For many antennas h »1 and so the words gain and directivity can be used interchangeably. For the short dipole

D(q, j) =

3 2 sin q 2

(38.7)

The maximum directivity of the short dipole is 3/2. This single number is often abbreviated as the antenna directivity. By comparison, for an imaginary isotropic antenna which radiates equally in all directions, D(q,j) = 1. The product of the maximum directivity with the total radiated power is called the effective isotropic radiated power (EIRP). It is the total radiated power that would be required for an isotropic radiator to produce the same signal as the original antenna in the direction of maximum directivity.

Magnetic Dipole A small loop of current produces a magnetic dipole. The far fields for the magnetic dipole are dual to those of the electric dipole. They have the same angular dependence as the fields of the electric dipole, but the polarization orientations of E and H are interchanged.

H = - Mk 02 sin q E =

e - jk 0r aq 4 pr

e MZ 0 k 02 sin q

(38.8)

- jk 0r

4 pr

where M = p r02 I for a loop with radius r0 and uniform current I.

Input Impedance At a given frequency the impedance at the feedpoint of an antenna can be represented as Za = Ra + jXa. The real part of Za (known as the input resistance) corresponds to radiated fields plus losses, while the imaginary part (known as the input reactance) arises from stored evanescent fields. The radiation resistance is obtained from Ra = 2Pr /|I|2 where Pr is the total radiated power and I is the input current at the antenna terminals. For electrically small electric and magnetic dipoles with uniform currents

æ dl ö Ra = 80 p ç ÷ è l0 ø

electric dipole

ær ö Ra = 320 p6 ç 0 ÷ è l0 ø

(38.9)

magnetic dipole

The reactive component of Za can be determined from Xa = 4w(Wm-We)/|I|2 where Wm is the average magnetic energy and We is the average electric energy stored in the near-zone evanescent fields. The reflection coefficient, G, of the antenna is just

G =

Za - Z0 Za + Z0

(38.10)

where Z0 is the characteristic impedance of the system used to measure the reflection coefficient.

Arbitrary Wire Antennas An arbitrary wire antenna can be considered as a sum of small current dipole elements. The vector potential for each of these elements can be determined in the same way as for the short dipole. The total vector potential is then the sum over all these infinitesimal contributions and the resulting E in the far field can be found to be

E(r ) = jk 0Z 0

e - jk 0r 4pr

ò [(a × a )a - a]I (l ¢)e c

jk 0 a r × r ¢

dl ¢

(38.11)

where the integral is over the contour C of the wire, a is a unit vector tangential to the wire, and r¢ is the radial vector to the infinitesimal current element.

Resonant Half-Wavelength Antenna The resonant half-wavelength antenna (commonly called the half-wave dipole) is used widely in antenna systems. Factors contributing to its popularity are its well-understood radiation pattern, its simple construction, its high efficiency, and its capability for easy impedance matching. The electric and magnetic fields for the half-wave dipole can be calculated by substituting its current distribution, I = I0 cos(k0z), into Eq. (38.11) to obtain

E = jZ 0I 0

H= j I 0

ö æp cos ç cos q÷ ø e - jk 0r è2 sin q

2 pr

ö æp cos ç cos q÷ ø e - jk 0r è2 sin q

2 pr

aq (38.12)

The total radiated power, Pr , can be determined from the electric and magnetic fields by integrating the expression 1/2 Re {E ´ H* · a r} over a surface of radius r. Carrying out this integration yields Pr = 36.565 |I0|2. The radiation resistance of the half-wave dipole can then be determined from

Ra =

2Pr * I0 *2

» 73 W

(38.13)

This radiation resistance is considerably higher than the radiation resistance of a short dipole. For example, if we have a dipole of length 0.01l, its radiation resistance will be approximately 0.08 W (from Eq. 38.9). This resistance is probably comparable to the ohmic resistance of the dipole, thereby resulting in a low efficiency. The half-wave dipole, having a much higher radiation resistance, will have much higher efficiency. The higher resistance of the half-wave dipole also makes impedance matching easier.

End Loading At many frequencies of interest, for example, the broadcast band, a half-wavelength becomes unreasonably long. Figure 38.2 shows a way of increasing the effective length of the dipole without making it longer. Here, additional wires have been added to the ends of the dipoles. These wires increase the end capacitance of the dipole, thereby increasing the effective electrical length.

Arrays of Wire Antennas Often it is advantageous to have several antennas operating together in an array. Arrays of antennas can be made to produce highly directional radiation patterns. Also, small antennas can be used in an array to obtain the level of performance of FIGURE 38.2 Using end loading to increase the a large antenna at a fraction of the area. The radiation pattern of an array depends on the number and type of antennas effective electrical length of used, the spacing in the array, and the relative phase and magnitude of the excitation an electric dipole. currents. The ability to control the phase of the exciting currents in each element of the array allows one to electronically scan the main radiated beam. An array that varies the phases of the exciting currents to scan the radiation pattern through space is called an electronically scanned phased array. Phased arrays are used extensively in radar applications.

Analysis of General Arrays To obtain analytical expressions for the radiation fields due to an array one must first look at the fields produced by a single array element. For an isolated radiating element positioned as in Fig. 38.3, the electric field at a farfield point P is given by

Ei = a i K i (q, j)e j [k 0 (R i × i p ) - a i ]

(38.14)

where Ki (q,j) is the electric field pattern of the individual element, ai e –jai is the excitation of the individual element, Ri is the position vector from the phase reference point to the element, i p is a unit vector pointing toward the far-field point P, and k0 is the free space wave vector. Now, for an array of N of these arbitrary radiating elements the total E-field at position P is given by the vector sum N -1

E tot =

å i =0

N -1

Ei =

å a K (q, j)e i

j [k 0 ( R i × i p ) - a i ]

(38.15)

i =0

This equation may be used to calculate the total field for an array of antennas where the mutual coupling between the array elements can be neglected. For most practical antennas, however, there is mutual coupling,

FIGURE 38.3 Diagram for determining the far field due to radiation from a single array element. (Source: Reference Data for Radio Engineers, Indianapolis: Howard W. Sams & Co., 1975, chap. 27–22. With permission.)

and the individual patterns will change when the element is placed in the array. Thus, Eq. (38.15) should be used with care.

Arrays of Identical Elements If all the radiating elements of an array are identical, then K i (q,j) will be the same for each element and Eq. (38.15) can be rewritten as N -1

åae

E tot = K(q, j)

j [k 0 ( R i × i p ) - a i ]

(38.16)

i =0

This can also be written as N -1

E tot = K(q, j) f (q, j)

where f (q, j) =

åae i

j [k 0 ( R i × i p ) - a i ]

(38.17)

i =0

The function f(q,j) is normally called the array factor or the array polynomial. Thus, one can find E tot by just multiplying the individual element’s electric field pattern, K(q,j), by the array factor, f (q,j). This process is often referred to as pattern multiplication. The average radiated power per unit solid angle is proportional to the square of E tot . Thus, for an array of identical elements

DPr (q, j) ~ * K(q, j) *2 * f (q, j) *2 DW

(38.18)

Equally Spaced Linear Arrays An important special case occurs when the array elements are identical and are arranged on a straight line with equal element spacing, d, as shown in Fig. 38.4. If a linear phase progression, a, is assumed for the excitation currents of the elements, then the total field at position P in Fig. 38.4 will be

FIGURE 38.4 A linear array of equally spaced elements.

N -1

åa e

E tot = K(q, j)

jn (k 0d cos q - a )

n =0

(38.19)

N -1

åa e

= K(q, j)

jn y

= K(q, j) f (y )

n =0

where y = k0d cos q – a. Broadside Arrays Suppose that, in the linear array of Fig. 38.4, all the excitation currents are equal in magnitude and phase (a0 = a1 = . . . = aN – 1 and a = 0). The array factor, f(y), then becomes N -1

f (y ) = a 0

e jn y = a 0

1 - e jN y

n =0

1 - ejy

(38.20)

This can be simplified to obtain the normalized form

Ny f (y ) 2 f ¢(y ) = = a 0N y N sin 2 sin

(38.21)

Note that f '(y) is maximum when y = 0. For our case, with a = 0, we have y = k0d cosq. Thus f '(y) will be maximized when q = p/2. This direction is perpendicular to the axis of the array (see Fig. 38.4), and so the resulting array is called a broadside array.

Phased Arrays By adjusting the phase of the elements of the array it is possible to vary the direction of the maximum of the array’s radiation pattern. For arrays where all the excitation currents are equal in magnitude but not necessarily phase, the array factor is a maximum when y = 0. From the definition of y, one can see that at the pattern maximum

k 0d cos q = a Thus, the direction of the array factor maximum is given by

æ a ö q = cos -1 ç ÷ è k 0d ø

(38.21b)

Note that if one is able to control the phase delay, a, the direction of the maximum can be scanned without physically moving the antenna.

Planar (2-D) Arrays Suppose there are M linear arrays, all identical to the one pictured in Fig. 38.4, lying in the yz-plane with element spacing d in both the y and the z direction. Using the origin as the phase reference point, the array factor can be determined to be N -1 M -1

f (q, j) =

å åa e mn

[ jn (k 0 d cos q - a z ) + jm (k 0 d sin q sin j - a y )]

(38.22)

n =0 m =0

where ay and az are the phase differences between the adjacent elements in the y and z directions, respectively. The formula can be derived by considering the 2-D array to be a 1-D array of subarrays, where each subarray has an antenna pattern given by Eq. (38.19). If all the elements of the 2-D array have excitation currents equal in magnitude and phase (all the amn are equal and az = ay = 0), then the array will be a broadside array and will have a normalized array factor given by

f ¢(q, j) =

ö æ Nk d sin ç 0 cos q÷ ø è 2

ö æ Mk 0d sin ç sin q sin j÷ ø è 2

ö æk d ö æk d N sin ç 0 cos q÷ M sin ç 0 sin q sin j÷ ø è 2 ø è 2

(38.23)

Yagi–Uda Arrays The Yagi–Uda array can be found on rooftops all over the world—the standard TV antenna is a Yagi–Uda array. The Yagi–Uda array avoids the problem of needing to control the feeding currents to all of the array elements by driving only one element. The other elements in the Yagi–Uda array are excited by near-field coupling from the driven element. The basic three-element Yagi–Uda array is shown in Fig. 38.5. The array consists of a driven antenna of length l1, a reflector element of length l2, and a director element of length l3. Typically, the director element is shorter than the driven element by 5% or more, while the reflector element is longer than the driven element by 5% or more [Stutzman and Thiele, 1981]. The radiation pattern for the array in Fig. 38.5 will have a maximum in the +z direction. © 2000 by CRC Press LLC

One can increase the gain of the Yagi–Uda array by adding additional director elements. Adding additional reflector elements, however, has little effect because the field behind the first reflector element is small. Yagi–Uda arrays typically have directivities between 10 and 100, depending on the number of directors [Ramo et al., 1984]. TV antennas usually have several directors.

Log-Periodic Dipole Arrays Another variation of wire antenna arrays is the log-periFIGURE 38.5 Three-element Yagi–Uda antenna. odic dipole array. The log-periodic is popular in applica- (Source: Shintaro Uda and Yasuto Mushiake, Yagi–Uda tions that require a broadband, frequency-independent Antenna, Sendai, Japan: Sasaki Printing and Publishing antenna. An antenna will be independent of frequency if Company, 1954, p. 100. With permission.) its dimensions, when measured in wavelengths, remain constant for all frequencies. If, however, an antenna is designed so that its characteristic dimensions are periodic with the logarithm of the frequency, and if the characteristic dimensions do not vary too much over a period of time, then the antenna will be essentially frequency independent. This is the basis for the log-periodic dipole array, shown in Fig. 38.6. In Fig 38.6, the ratio of successive element positions equals the ratio of successive dipole lengths. This ratio is often called the scaling factor of the log-periodic array and is denoted by

t =

z n +1 L = n +1 zn Ln

(38.24)

Also note that there is a mechanical phase reversal between successive elements in the array caused by the crossing over of the interconnecting feed lines. This phase reversal is necessary to obtain the proper phasing between adjacent array elements. To get an idea of the operating range of the log-periodic antenna, note that for a given frequency within the operating range of the antenna, there will be one dipole in the array that is half-wave resonant or is nearly so. This half-wave resonant dipole and its immediate neighbors are called the active region of the log-periodic array. As the operating frequency changes, the active region shifts to a different part of the log-periodic. Hence, the frequency range for the log-periodic array is roughly given by the frequencies at which the longest and shortest dipoles in the array are half-wave resonant (wavelengths such that 2LN < l < 2L1 ) [Stutzman and Thiele, 1981].

FIGURE 38.6 The log-periodic dipole array. (Source: D.G. Isbell, “Log periodic dipole arrays,” IRE Transactions on Antennas and Propagation, vol. AP-8, p. 262, 1960. With permission.) © 2000 by CRC Press LLC

Defining Terms Antenna gain: The ratio of the actual radiated power per solid angle to the radiated power per solid angle that would result if the total input power were radiated isotropically. Array: Several antennas arranged together in space and interconnected to produce a desired radiation pattern. Directivity: The ratio of the actual radiated power per solid angle to the radiated power per solid angle that would result if the radiated power was radiated isotropically. Oftentimes the word directivity is used to refer to the maximum directivity. Phased array: An array in which the phases of the exciting currents are varied to scan the radiation pattern through space. Radiation pattern: A plot as a function of direction of the power per unit solid angle radiated in a given polarization by an antenna. The terms radiation pattern and antenna pattern can be used interchangeably.

Related Topics 37.1 Space Propagation • 69.2 Radio

References C.A. Balanis, Antenna Theory Analysis and Design, New York: Harper and Row, 1982. R. Carrel, “The design of log-periodic dipole antennas,” IRE International Convention Record (part 1), 1961, pp. 61–75. R.E. Collin, Antennas and Radiowave Propagation, New York: McGraw-Hill, 1985. R.F. Harrington, Time Harmonic Electromagnetic Fields, New York: McGraw-Hill, 1961. D.E. Isbell, “Log periodic dipole arrays,” IRE Transactions on Antennas and Propagation, vol. AP-8, pp. 260–267, 1960. P. Lorrain and D.R. Corson, Electromagnetic Fields and Waves, San Francisco: W.H. Freeman, 1970. S. Ramo, J.R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics, New York: John Wiley & Sons, 1984. W.L. Stutzman and G.A. Thiele, Antenna Theory and Design, New York: John Wiley & Sons, 1981. S. Uda and Y. Mushiake, Yagi–Uda Antenna, Sendai, Japan: Sasaki Printing and Publishing Company, 1954.

Further Information For general-interest articles on antennas the reader is directed to the IEEE Antennas and Propagation Magazine. In addition to providing up-to-date articles on current issues in the antenna field, this magazine also provides easy-to-read tutorials. For the latest research advances in the antenna field the reader is referred to the IEEE Transactions on Antennas and Propagation. In addition, a number of very good textbooks are devoted to antennas. The books by Collin and by Stutzman and Thiele were especially useful in the preparation of this section.

38.2

Aperture

J. Patrick Fitch The main purpose of an antenna is to control a wave front at the boundary between two media: a source (or receiver) and the medium of propagation. The source can be a fiber, cable, waveguide, or other transmission line. The medium of propagation may be air, vacuum, water, concrete, metal, or tissue, depending on the application. Antenna aperture design is used in acoustic, optic, and electromagnetic systems for imaging, communications, radar, and spectroscopy applications. There are many classes of antennas: wire, horn, slot, notch, reflector, lens, and array, to name a few (see Fig. 38.7). Within each class is a variety of subclasses. For instance, the horn antenna can be pyramidal or conical. The horn can also have flaring in only one direction (sectoral horn), asymmetric components, shaped

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edges, or a compound design of sectoral and pyramidal combined. For all antennas, the relevant design and analysis will depend on antenna aperture size and shape, the center wavelength l, and the distance from the aperture to a point of interest (the range, R). This section covers discrete oscillators, arrays of oscillators, synthetic apertures, geometric design, Fourier analysis, and parameters of some typical antennas. The emphasis is on microwave-type designs.

FIGURE 38.7 Examples of several types of antennas: (a) pyramidal horn, (b) conical horn, (c) axial slot on a cylinder, and (d) parabolic reflector.

The Oscillator or Discrete Radiator The basic building block for antenna analysis is a linear conductor. Movement of electrons (current) in the conductor induces an electromagnetic field. When the electron motion is oscillatory—e.g., a dipole with periodic electron motion, the induced electric field, E, is proportional to cos(wt – kx + f), where w is radian frequency of oscillation, t is time, k is wave number, x is distance from the oscillator, and f is the phase associated with this oscillator (relative to the time and spatial coordinate origins). When the analysis is restricted to a fixed position x, the electric field can be expressed as

E (t ) = A cos(wt + f)

(38.25)

where the phase term f now includes the kx term, and all of the constants of proportionality are included in the amplitude A. Basically, the assumption is that oscillating currents produce oscillating fields. The description of a receiving antenna is analogous: an oscillating field induces a periodic current in the conductor. The field from a pair of oscillators separated in phase by d radians is

E d (t ) = A1 cos (wt + f) + A2 cos (wt + f + d)

(38.26)

Using phasor notation, Ẽd, the cosines are converted to complex exponentials and the radial frequency term, wt, is suppressed,

E˜ d (t ) = A1e i f + A2e i (f + d)

(38.27)

The amplitude of the sinusoidal modulation E d(t) can be calculated as êẼdï. The intensity is

I = * E˜ d * 2 = * A1 * 2 + * A2 * 2 + 2 A1A2 cos(d)

(38.28)

When the oscillators are of the same amplitude, A = A1 = A2, then

E d (t ) = A cos(wt + f) + A cos(wt + f + d) æ æ dö dö = 2 A cos ç ÷ cos ç wt + f + ÷ 2ø è è 2ø

(38.29)

For a series of n equal amplitude oscillators with equal phase spacing n -1

En d (t ) =

å A cos(wt + f + j d) j =0

(38.30)

FIGURE 38.8 A two-element and an n-element array with equal spacing between elements. The propagation length difference between elements is d sinq, which corresponds to a phase difference of kd sinq, where k is the wave number 2p/l. The length L corresponds to a continuous aperture of length nd with the sample positions beginning d/2 from the ends.

By using phasor arithmetic the intensity is given as

I n d (t ) = *E˜n d * 2 = Ae i f

n -1

ij d

= A2

j =0

= I0

1 - e in d

1 - eid

= I0

1 - cos(n d) 1 - cos(d)

(38.31)

sin2 (n d/2) sin2 (d/2)

where I0 =n–2 to normalize the intensity pattern at d = 0. For an incoming plane wave which is tilted at an angle q from the normal, the relative phase difference between two oscillators is kd sinq, where d is the distance between oscillators and k is the wave number 2p/l (see Fig. 38.8). For three evenly spaced oscillators, the phase difference between the end oscillators is 2kd sinq. In general, the end-to-end phase difference for n evenly spaced oscillators is (n – 1)kd sinq. This formulation is identical to the phase representation in Eq. (38.30) with d = kd sinq. Therefore, the intensity as a function of incidence angle q for an evenly spaced array of n elements is

I nL (q) = I 0

ö æ1 sin2 ç knd sin q÷ ø è2 ö æ1 sin ç kd sin q÷ ø è2 2

= I0

ö æ1 sin2 ç kL sin q÷ ø è2 ö æ 1 sin ç kL sin q÷ ø è 2n 2

= I0

ö æ pL sin2 ç sin q÷ ø è l ö æ pL sin ç sin q÷ ø è nl

(38.32)

where L = nd corresponds to the physical dimension (length) of the aperture of oscillators. The zeros of this function occur at kL sinq = 2mp, for any nonzero integer m. Equivalently, the zeros occur when sinq = ml/L. When the element spacing d is less than a wavelength, the number of zeros for 0 < q < p/2 is given by the largest integer M such that M £ L/l. Therefore, the ratio of wavelength to largest dimension, l/L, determines both the location (in q space) and the number of zeros in the intensity pattern when d £ l. The number of oscillators controls the amplitude of the side lobes. For n = 1, the intensity is constant—i.e., independent of angle. For l > L, both the numerator and denominator of Eq. (38.32) have no zeros and as the length of an array shortens (relative to a wavelength), the intensity pattern converges to a constant (n = 1 case). As shown in Fig. 38.9, a separation of l/4 has an intensity rolloff less than 1 dB over p/2 radians (a l/2 separation rolls off 3 dB). This implies that placing antenna elements closer than l/4 does not significantly change the intensity pattern. Many microwave antennas exploit this and use a mesh or parallel wire (for polarization sensitivity) design rather than covering the entire aperture with conductor. This reduces both weight and sensitivity to wind loading. Note that the analysis has not accounted for phase variations from position errors in the element placement where the required accuracy is typically better than l/10. © 2000 by CRC Press LLC

FIGURE 38.9 Normalized intensity pattern in decibels (10 log(I)) for a two-element antenna with spacing 4l, 2l, l, l/2, and l/4 between the elements.

FIGURE 38.10 Normalized intensity pattern in decibels (10 log(I)) for a length 4l array with 2, 3, 4, 5, and 8 elements.

For L > > l, sinq » q, which implies that the first zero is at q = l/L. The location of the first zero is known as the Rayleigh resolution criteria. That is, two plane waves separated by at least l/L radians can be discriminated. For imaging applications, this corresponds roughly to the smallest detectable feature size. As shown in Fig. 38.10, the first zero occurs at approximately l/L = 0.25 radians (the Rayleigh resolution). Note that there is no side lobe suppression until d £ l, when the location of the zeros becomes fixed. Having more than eight array elements (separation of less than a quarter wavelength) only moderately reduces the height of the maximum side lobe.

Synthetic Apertures In applications such as air- and space-based radar, size and weight constraints prohibit the use of very large antennas. For instance, if the L-band (23.5-cm wavelength) radar imaging system on the Seasat satellite (800-km altitude, launched in 1978) had a minimum resolution specification of 23.5 m, then, using the Rayleigh resolution criteria, the aperture would need to be 8 km long. In order to attain the desired resolution, an aperture is “synthesized” from data collected with a physically small (10 m) antenna traversing an 8-km flight © 2000 by CRC Press LLC

FIGURE 38.11 Synthetic aperture radar geometry and nearly orthogonal partitioning of the footprint by range (circular) and Doppler frequency (hyperbolic) contours.

path. Basically, by using a stable oscillator on the spacecraft, both amplitude and phase are recorded, which allows postprocessing algorithms to combine the individual echoes in a manner analogous to an antenna array. From an antenna perspective, an individual scattering element produces a different round trip propagation path based on the position of the physical antenna—a synthetic antenna array. Using the geometry described in Fig. 38.11, the phase is

f(x ) =

2p 2p 2R ( x ) = 2 x2 + y 2 + z2 l l

(38.33)

It is convenient to assume a straight-line flight path along the x-axis, a planar earth (x, y plane), and a constant velocity, v, with range and cross-range components vr(x) and vc( x), respectively. In many radar applications the broad side distance to the center of the footprint, R, is much larger than the size of the footprint. This allows the distance R(x) to be expanded about R resulting in

f(t ) =

ìï 2R 2p 2v v 2 üï 2R(vt ) = 2 p í + r t + c t2 ý l l lR þï îï l

(38.34)

The first term in Eq. (38.34) is a constant phase offset corresponding to the center of beam range bin and can be ignored from a resolution viewpoint. The second term, 2v r/l, is the Doppler frequency shift due to the relative (radial) velocity between antenna and scattering element. The third term represents a quadratic correction of the linear flight path to approximate the constant range sphere from a scattering element. It is worth noting that synthetic aperture systems do not require the assumptions used here, but accurate position and motion compensation is required. For an antenna with cross range dimension D and a scattering element at range R, the largest synthetic aperture that can be formed is of dimension lR/D (the width of the footprint). Because this data collection scenario is for round trip propagation, the phase shift at each collecting location is twice the shift at the edges of a single physical antenna. Therefore at a range R, the synthetic aperture resolution is

lR lR D = = DSA 2 lR /D 2

(38.35)

The standard radar interpretation for synthetic apertures is that information coded in the Doppler frequency shift can be decoded to produce high-resolution images. It is worth noting that the synthetic aperture can be formed even with no motion (zero Doppler shift). For the no-motion case the antenna array interpretation is appropriate. This approach has been used for acoustic signal processing in nondestructive evaluation systems as © 2000 by CRC Press LLC

FIGURE 38.12 Parabolic reflector systems: (a) geometry for determining the function with a constant path length and specular reflection, (b) single-bounce parabolic reflector, (c) two-bounce reflector with a parabolic primary and hyperbolic secondary (Cassegrain).

well as wave migration codes for seismic signal processing. When there is motion, the Doppler term in the expansion of the range dominates the phase shift and therefore becomes the useful metric for predicting resolution.

Geometric Designs The phase difference in a linear array was caused by the spatial separation and allowed the discrimination of plane waves arriving at different angles. Desired phase patterns can be determined by using analytic geometry to position the elements. For example, if coherent superposition across a wave front is desired, the wave front can be directed (reflected, refracted, or diffracted) to the receiver in phase. For a planar wave front, this corresponds to a constant path length from any point on the reference plane to the receiver. Using the geometry in Fig. 38.12, the sum of the two lengths (x, R + h) to (x, y) and (x, y) to (0, h) must be a constant independent of x—which is R + 2h for this geometry. This constraint on the length is

R +h-y +

x 2 + (h - y )2 = R + 2h

x 2 = 4hy

(38.36)

This is the equation for a parabola. Losses would be minimized if the wave front were specularly reflected to the transceiver. Specular reflection occurs when the angles between the normal vector N [or equivalently the tangent vector T = (x,f¢ ¢(x))] = (1, x/2h) and the vectors A = (0, –1) and B = (–x, h – y) are equal. This is the same as equality of the inner products of the normalized vectors, which is shown by

Tˆ × Aˆ = Tˆ × Bˆ =

(2h, x ) x 2 + 4h 2 (2h, x ) x 2 + 4h 2

× (0, -1) =

-x

(38.37)

x 2 + 4h 2

(- x , h - y ) x 2 + (h - y )2

- x ( x 2 + 4h 2 ) 2

2 3 /2

( x + 4h )

-x

(38.38)

x 2 + 4h 2

The constant path length and high gain make the parabolic antenna popular at many wavelengths including microwave and visible. More than one reflecting surface is allowed in the design. The surfaces are typically conical sections and may be designed to reduce a particular distortion or to provide better functionality. Compound designs often allow the active elements to be more accessible and eliminate long transmission lines. A two-bounce reflector with a parabolic primary and a hyperbolic secondary is known as a Cassegrain system. In all reflector systems it is important to account for the blockage (“shadow” of the feed, secondary reflector, and support structures) as well as the spillover (radiation propagating past the intended reflecting surface).

Continuous Current Distributions (Fourier Transform) Ideally, antennas would be designed using solutions to Maxwell’s equations. Unfortunately, in most cases exact analytic and numerical solutions to Maxwell’s equations are difficult to obtain. Under certain conditions, © 2000 by CRC Press LLC

approximations can be introduced that allow solution to the wave equations. Approximating spherical wave fronts as quadratics has been shown for the synthetic aperture application and is valid when the propagation distance is greater than (pL2/4l)1/3, where L is the aperture size. In general, this is known as the Fresnel or near-field approximation. When the propagation distance is at least 2L2/l, the angular radiation pattern can be approximated as independent of distance from the aperture. This pattern is known as the normalized farfield or Fraunhofer distribution, E(q), and is related to the normalized current distributed across an antenna aperture, i(x), by a Fourier transform:

ò i(x ¢)e

E (u ) =

i 2 pux ¢

dx ¢

(38.39)

where u = sinq and x ¢ = x/l. Applying the Fraunhofer approximation to a line source of length L

L /2 l

E L (u = sin q) =

ò e

i 2 pux ¢

dx ¢ =

- L /2 l

æ pL ö sin ç u÷ è l ø pL u l

ö æ pL sin ç sin q÷ ø è l pL sin q l

(38.40)

which is Eq. (38.32) when n >> L/l. As with discrete arrays, the ratio L/l is the important design parameter: sinq = l/L is the first zero (no zeros for l > L) and the number of zeros is the largest integer M such that M £ L/l. In two dimensions, a rectangular aperture with uniform current distribution produces

E R (u1 , u 2 ) =

ö æp ö æp sin ç u1L1 ÷ sin ç u 2 L2 ÷ ø èl ø èl p u1L1 l

p u 2 L2 l

and I R (u1 , u 2 ) = * E L (u1 ) * 2 * E L (u 2 ) * 2 (38.41)

The field and intensity given in Eq. (38.41) are normalized. In practice, the field is proportional to the aperture area and inversely proportional to the wavelength and propagation distance. The normalized far-field intensity distribution for a uniform current on a circular aperture is a circularly symmetric function given by

é æp öù ê 2 J 1 ç uL ÷ ú ê èl øú I C (u ) = ê ú p ê uL ú úû êë l

(38.42)

where J1 is the Bessel function of the first kind, order one. This far-field intensity is called the Airy pattern. As with the rectangular aperture, the far-field intensity is proportional to the square of the area and inversely proportional to the square of the wavelength and the propagation distance. The first zero (Rayleigh resolution criteria) of the Airy pattern occurs for uL/l = 1.22 or sinq = 1.22l/L. As with linear and rectangular apertures, the resolution scales with l/L. Figure 38.13 shows a slice through the normalized far-field intensity of both a rectangular aperture and a circular aperture. The linearity of the Fourier transform allows apertures to be represented as the superposition of subapertures. The primary reflector, the obscurations from the support structures, and the secondary reflector © 2000 by CRC Press LLC

FIGURE 38.13 Normalized intensity pattern in decibels (10 log[I(v = uL/l)]) for a rectangular and a circular antenna aperture with uniform current distributions.

TABLE 38.1 Directivity and Gain of Some Higher Frequency Antennas Antenna Type

Directivitya

Uniform rectangular aperture

4p L L l2 x y

Large square aperture

æ Lö 12.6 ç ÷ è lø

Large circular aperture (parabolic reflector)

æ Dö 9.87 ç ÷ è lø

4p L L l2 x y

æ Lö 7.7 ç ÷ è lø

æ 4p ö ç 2 ÷ Lx L y èl ø

Pyramidal horn

Gaina

æ Dö 7ç ÷ è lø

æ 4p ö 0.5 ç 2 ÷ Lx L y èl ø

Directivity and gain are relative to a half-wave dipole.

of a Cassegrain-type antenna can be modeled. Numerical evaluation of the Fourier transform permits straightforward calculation of the intensity patterns, even for nonuniform current distributions.

Antenna Parameters Direct solutions to Maxwell’s equations or solutions dependent on approximations provide the analytic tools for designing antennas. Ultimately, the analysis must be confirmed with experiment. Increasingly sensitive radar and other antenna applications have resulted in much more attention to edge effects (from the primary aperture, secondary, and/or support structures). The geometric theory of diffraction as well as direct Maxwell solvers are making important contributions. With the diversity of possible antenna designs, a collection of design rules of thumb are useful. The directivity and gain for a few popular antenna designs are given in Table 38.1. Directivity is the ratio of the maximum to average radiation intensity. The gain is defined as the ratio of the maximum radiation intensity from the subject antenna to the maximum radiation intensity from a reference antenna with the same power input. The directivity, D, and gain, G, of an antenna can be expressed as

and

æ 4p ö G = ç 2 ÷ Ae èl ø

(38.43)

where Aem is the maximum effective aperture and Ae is the actual effective aperture of the antenna. Because of losses in the system, Ae = kAem, where k is the radiation efficiency factor. The gain equals the directivity when there are no losses (k = 1), but is less than the directivity if there are any losses in the antenna (k < 1), that is, G = kD. As an example, consider the parabolic reflector antenna where efficiency degradation includes • Ohmic losses are small (k = 1) • Aperture taper efficiency (k = 0.975) • Spillover (feed) efficiency (k = 0.8) • Phase errors in aperture field (k = 0.996 to 1) • Antenna blockage efficiency (k = 0.99) • Spar blockage efficiency (k = 0.994) Each antenna system requires a customized analysis of the system losses in order to accurately model performance.

Defining Terms Antenna: A physical device for transmitting or receiving propagating waves. Aperture antenna: An antenna with a physical opening, hole, or slit. Contrast with a wire antenna. Array antenna: An antenna system performing as a single aperture but composed of antenna subsystems. Directivity: The ratio of the maximum to average radiation intensity. Fraunhofer or far field: The propagation region where the normalized angular radiation pattern is independent of distance from the source. This typically occurs when the distance from the source is at least 2L2/l, where L is the largest dimension of the antenna. Fresnel or near field: The propagation region where the normalized radiation pattern can be calculated using quadratic approximations to the spherical Huygens’ wavelet surfaces. The pattern can depend on distance from the source and is usually valid for distances greater than (p/4l)1/3L2/3, where L is the largest dimension of the antenna. Gain: The ratio of the maximum radiation intensity from the subject antenna to the maximum radiation intensity from a reference antenna with the same power input. Typical references are a lossless isotropic source and a lossless half-wave dipole. Oscillator: A physical device that uses the periodic motion within the material to create propagating waves. In electromagnetics, an oscillator can be a conductor with a periodic current distribution. Reactive near field: The region close to an antenna where the reactive components of the electromagnetic fields from charges on the antenna structure are very large compared to the radiating fields. Considered negligible at distances greater than a wavelength from the source (decay as the square or cube of distance). Reactive field is important at antenna edges and for electrically small antennas.

Related Topic 35.1 Maxwell Equations

References A. Barone and G. Paterno, Physics and Applications of the Josephson Effect, New York: Wiley, 1982. R. J. Donnelly, “Cryogenics,” in Physics Vade Mecum, H.L. Anderson, Ed., New York: American Institute of Physics, 1981. S. Foner and B. B. Schwartz, Superconducting Machines and Devices, New York: Plenum Press, 1974. S. Foner and B. B. Schwartz, Superconducting Materials Science, New York: Plenum Press, 1981. J. Knuutila, M. Kajola, H. Seppä, R. Mutikainen, and J. Salmi, Design, optimization, and construction of a DC SQUID with complete flux transformer circuits, J. Low. Temp. Phys., 71, 369–392, 1988. K. K. Likharev, Dynamics of Josephson Junctions and Circuits, Philadelphia, Pa.: Gordon and Breach Science Publishers, 1986. T. P. Orlando and K. A. Delin, Foundations of Applied Superconductivity, Reading, Mass.: Addison-Wesley, 1991. S. T. Ruggiero and D. A. Rudman, Superconducting Devices, Boston: Academic Press, 1990. B. B. Schwartz and S. Foner, Superconducting Applications: SQUIDs and Machines, New York: Plenum Press, 1977. T. Van Duzer and C. W. Turner, Principles of Superconductive Devices and Circuits, New York: Elsevier North Holland, 1981.

H. Weinstock and R. W. Ralston, The New Superconducting Electronics, Boston, Mass: Kluwer Academic Publishers, 1993. J. P. Wikswo, SQUID magnetometers for biomagnetism and non-destructive testing: important questions and initial answers, IEEE Trans. Appl. Supercond., 5, 74–120, 1995. M. N. Wilson, Superconducting Magnets, Oxford: Oxford University Press, 1983.

Further Information Every two years an Applied Superconductivity Conference is held devoted to practical technological issues. The proceedings of these conferences have been published every other year from 1977 to 1991 in the IEEE Transactions on Magnetics. In 1991, the IEEE Transactions on Applied Superconductivity began publication. This quarterly journal focuses on both the science and the technology of superconductors and their applications, including materials issues, analog and digital circuits, and power systems. The proceedings of the Applied Superconductivity Conference now appear in this journal.

Whatmore, R.W. “Pyroelectric Materials and Devices” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000

54 Pyroelectric Materials and Devices Roger W. Whatmore Cranfield University

54.1

54.1 54.2 54.3 54.4

Introduction Polar Dielectrics The Pyroelectric Effect Pyroelectric Materials and Their Selection

Introduction

It was known over 2000 years ago that certain minerals such as tourmaline would attract small objects when heated. It was understood over 200 years ago that this attraction was a manifestation of the appearance of electrical charges on the surface as a consequence of the temperature change. This is called the pyroelectric effect and over the last 15 years has become the basis of a major worldwide industry manufacturing detectors of infrared radiation. These are exploited in such devices as “people detectors” for intruder alarms and energy conservation systems, fire and flame detectors, spectroscopic gas analyzers—especially looking for pollutants from car exhausts—and, more recently, devices for thermal imaging. Such thermal imagers can be used for night vision and, by exploiting the smoke-penetrating properties of long-wavelength infrared radiation, in devices to assist firefighters in smoke-filled spaces. The major advantages of the devices in comparison with the competing infrared detectors that exploit narrow bandgap semiconductors are that no cooling is necessary and that they are cheap and consume little power. The pyroelectric effect appears in any material which possesses a polar symmetry axis. This chapter describes the basic effect, gives a brief account of how it can be used in radiation detection, and discusses the criteria by which materials can be selected for use in this application, concluding with a comparison of the properties of several of the most commonly used materials.

54.2

Polar Dielectrics

A polar material is one whose crystal structure contains a unique axis, along which an electric dipole moment will exist. There are 10 polar crystal classes: • Triclinic • Tetragonal • Hexagonal

1 4, 4mm 6, 6mm

• Monoclinic • Orthorhombic • Trigonal

2, m mm2 3, 3m

All crystals whose structures possess one of these symmetry groups will exhibit both pyroelectric and piezoelectric characteristics. In ferroelectrics, which are a subset of the set of pyroelectrics, the orientation of the polar axis can be changed by application of an electric field of sufficient magnitude. The original and final states of the crystal are symmetrically related. It is important to note that

TABLE 54.1 Spontaneous Polarizations and Curie Temperatures for a Range of Ferroelectrics Material

Tc(k)

Ps(cm-2)

T(k)

KH2PO4 (KDP) Triglycine sulphate Polyvinylidene fluoride (PVDF) DOBAMBC (liquid crystal) PbTiO3 BaTiO3

123 322 > 453 359 763 393

0.053 0.028 0.060 ~3 ´ 10–5 0.760 0.260

96 293 293 354 293 296

1. Not all polar materials are ferroelectric. 2. There is a set of point groups which lack a center of symmetry, without possessing a polar axis. The – – – — — crystals belonging to these groups (222, 4, 422, 42m , 32, 6, 6m2, 23, and 43m) are piezoelectric without being pyroelectric. (432 is a noncentrosymmetric, nonpiezoelectric class.) Typical values of spontaneous polarizations (Ps) and Curie temperatures (Tc) for a range of ferroelectrics are given in Table 54.1. A very wide range of materials exhibit ferroelectric, and thus pyroelectric, behavior. These range from crystals, such as potassium dihydrogen phosphate and triglycine sulphate, to polymers, such as polyvinylidene fluoride, and liquid crystals such as DOBAMBC and ceramics, such as barium titanate and lead zirconate titanate. Most ferroelectrics exhibit a Curie temperature (Tc) at which the spontaneous plarization goes to zero. (A few ferroelectrics, such as the polymer polyvinylidene fluoride [PVDF] melt before this temperature is reached.) The fact that the orientation of the polar axis in ferroelectrics can be changed by the application of a field has a very important consequence for ceramic materials. If a polycrystalline body is made of a polar material, then the crystal axes will, in general, be randomly oriented. It cannot therefore show pyroelectricity. However, if an electric field greater than the coercive field (Ec) is applied to a ferroelectric ceramic, then the polar axes within the grains will tend to be reoriented so that they each give a component along the direction of the applied field. This process is called “poling.” The resulting ceramic is polar (with a point symmetry ¥m) and will show both piezoelectricity and pyroelectricity.

54.3

The Pyroelectric Effect

The pyroelectric effect is described by:

Pi = pi DT

(54.1)

where Pi is the change in the coefficient of the polarization vector due to a change in temperature DT and pi is the pyroelectric coefficient, which is a vector. The effect and its applications have been extensively reviewed in Whatmore [1986]. The effect of a temperature change on a pyroelectric material is to cause a current, ip , to flow in an external circuit, such that

i p = ApdT/dt

(54.2)

where A is the electroded area of the material, p the component of the pyroelectric coefficient normal to the electrodes, and dT/dt the rate of change of temperature with time. Pyroelectric devices detect changes in temperature in the sensitive material and as such are detectors of supplied energy. It can be seen that the pyroelectric current is proportional to the rate of change of the material with time and that in order to obtain a measurable signal, it is necessary to modulate the source of energy. As energy detectors, they are most frequently applied to the detection of incident electromagnetic energy, particularly in the infrared wavebands.

FIGURE 54.1 Pyroelectric detector with FET amplifier.

Typically, a pyroelectric detector element will consist of a thin chip of the pyroelectric material cut perpendicular to the polar axis of the material, electroded with a conducting material such as an evaporated metal and connected to a low-noise, high-input impedance amplifier—for example, a junction field-effect transistor (JFET) or metal-oxide gate transistor (MOSFET)—as shown in Fig. 54.1. In some devices, the radiation is absorbed directly in the element. In this case the front electrode will be a thin metal layer matched to the permittivity of free space with an electrical surface resistivity of 367 W/square. However, in most high-performance devices, the element is coated with a layer designed to absorb the radiation of interest. The element itself must be thin to minimize the thermal mass and, in most cases, well isolated thermally from its environment. These measures are designed to increase the temperature change for a given amount of energy absorbed and thus the electrical signal generated. The necessary modulation of the radiation flux can be achieved either by deliberately only “looking” for moving objects or other radiation sources (e.g., flickering flames for a flame detector) or by interposing a mechanical radiation “chopper” such as a rotating blade. The voltage responsivity of a device such as this is defined as R v = Vo / W, where Vo is the output voltage and W is the input radiation power. For radiation sinusoidally modulated at a frequency w, Rv is given by

Rv =

RG hpAw 2 2 1/ 2 GT (1 + w tT ) (1 + w 2 t2E )1/2

(54.3)

where G T is the thermal conductance from the element to the environment, t T is the thermal time constant of the element, t E is the electrical time constant of the element, R G is the electrical resistance across the element, h is the emissivity of the element for the radiation being detected, and A is the sensitive area of the element. It is easy to show that the response of a pyroelectric device maximizes at a frequency equal to the inverse of the geometric mean of the two time constants and that above and below the two frequencies given by t T–1 and tE –1, R v falls as w–1. The consequence of this is that pyroelectric detectors have their sensitivities maximized by having fairly long electrical time constants (0.1 to 10 s) and that such detectors thus work best at low frequencies (0.1 to 100 Hz). However, if high sensitivity is not required, extremely large bandwidths with little sensitivity variation can be obtained by shortening these time constants (making R G and C E low and G T high). In this way, detectors have been made which give picosecond time responses for tracking fast laser pulses. There are several noise sources in a pyroelectric device. These are discussed in detail in Whatmore [1986]. In many cases of interest, the dominant noise source is the Johnson noise generated by the ac conductance in the capacitance of the detector element. This noise is given by DVj , where 1/ 2

ìï tand üï -1/2 DV j = í4kT ý w C E þï îï

for C E >> C A

(54.4)

where k is Boltzmann’s constant, T is the absolute temperature, tand is the dielectric loss tangent of the detector material, C E is the electrical capacitance of the element, and C A is the input capacitance of the detector element amplifier. The input radiation power required to give an output equal to the noise at a given frequency in unity bandwidth is known as the noise equivalent power (NEP). This is given by

NEP = Vn / R v

(54.5)

where Vn is the total RMS voltage noise from all sources. A performance figure of merit frequently used when discussing infrared detectors is the detectivity, usually designated as D*. This is given by

D* = A 1/2/ NEP

(54.6)

Thus, the detectivity of a pyroelectric detector can be derived from Eqs. (54.3) to (54.6) and is given by

D* =

hd 1/ 2

(4kT )

p c ¢(eeo tand)

1/ 2

1 w1/2

(54.7)

where c ¢ is the volume specific heat, e is the dielectric constant of the pyroelectric, and d is the thickness of the pyroelectric element. The roll-off in D* at high frequencies is thus 1/w1/2. Pyroelectric single-element IR detectors come in many different varieties. A typical commercial device will have the sensitive element made from a material of the type discussed in the next section, such as a piece of lithium tantalate crystal or a ferroelectric ceramic. The element size will be a few millimeters square. Typical performance figures at about 10 Hz would be a responsivity of a few hundred volts per watt of input radiation, a noise equivalent power of about 8 ´ 10–9 W/Hz1/2, and a detectivity of about 2 ´ 108 cm Hz1/2 W–1 for unity bandwidth. The detector can be fitted with a wide variety of windows, depending upon the wavelength of the radiation to be detected. As noted above, pyroelectric devices have also been used for thermal imaging. In this application, their main advantage when compared with photon detector materials such as mercury cadmium telluride (CMT) (which are more sensitive) is that they can be used at room temperature. All the photon detectors require cooling, typically to 77 K. A very successful device for pyroelectric thermal imaging is the pyroelectric vidicon which uses a thin plate of pyroelectric material contained in a vacuum tube. The thermal image is focused onto the surface of the material using a germanium lens. This causes the formation of a pattern of pyroelectric charges, which are “read” by means of an electron beam. Typical sensitivities for such devices are between 0.50 and 1 K temperature differences in the scene for an f/1 lens. This compares with <0.10 K for a cooled CMT detectorbased imager. Recently, a solid-state approach to pyroelectric thermal imaging has been developed. In this, an array of many thousands of very small identical detectors, each between 50 and 100 mm square, depending on the array design, are linked to a silicon amplifier/multiplexer circuit which allows the signals from all the elements to be read onto a single output line. These devices have been primarily developed for thermal imaging applications and excellent sensitivities (close to those achieved by many cooled systems) have been demonstrated.

54.4

Pyroelectric Materials and Their Selection

There are many different types of pyroelectrics and the selection of a material depends strongly upon the application. It is possible to formulate from the given equations a number of figures of merit which describe the contribution of the physical properties of a material to the performance of a device. For example, the current responsivity is proportional to Fi :

(54.8)

TABLE 54.2 Pyroelectric Properties of Selected Materials

Material (Temperature)

Dielectric Properties (1 kHz)

Pyroelectric Coefficient P 10–4 cm–2 K–1

5.5 5.5 0.27 2.3 3.8 3.8

55 43 12 47 290 220

TGS (35°C) DTGS (40°C) PVDF polymer LiTaO3 crystal Modified PZ ceramic Modified PT ceramic

tand

Volume-Specific Heat c¢ 106 Jm–3 K–1

Thermal Conductivity K 10–7 m2 s–1

0.025 0.020 0.015 0.005 0.003 0.011

2.6 2.4 2.43 3.2 2.5 2.5

3.3 3.3 0.62 13.0

Fv m2 C–1

FD 10– 5 Pa–1/2

Fvid 106 sC–1

0.43 0.60 0.10 0.17 0.06 0.08

6.1 8.3 0.88 4.9 5.8 3.3

1.3 1.8 1.6 0.13

PZ = PbZrO3, PT = PbTiO 3 .

The voltage response for a pyroelectric element feeding into a high-input impedance, unity gain amplifier (such as a source follower FET) as shown in Fig. 54.1 is proportional to Fv :

F v = p/c ¢ eeo

(54.9)

F D = p/{c ¢ (ee otand) 1/2}

(54.10)

The detectivity is proportional to FD :

For the pyroelectric vidicon, thermal spreading of the pattern on the target is important and the relevant figure of merit is Fvi d :

F vid = Fv /K

(54.11)

where K is the thermal conductivity of the pyroelectric. It should be noted that the use of these merit figures must be tempered with a knowledge of the type of detector the material is to be used in. It is necessary, if possible, to match the capacitance of the detector to the input capacitance of the amplifier. Hence, lowpermittivity materials are better suited to large-area detectors, and conversely arrays of small-area detectors are better served by materials with high permittivities. Table 54.2 lists the pyroelectric properties of several different materials, single crystals, ceramics, and polymers. It can be seen that triglycine sulphate (TGS) and its deuterated isomorph (DTGS) exhibit the highest value of Fv and are frequently used for high-performance single-element detectors. These are the preferred materials for pyroelectric vidicon targets. However, they are water soluble, difficult to handle, and show poor long-term stability, both chemically and electrically, because of their low Curie temperatures. Furthermore, their dielectric loss is rather high, so that the F D figures are not so favorable. Lithium tantalate, on the other hand, is an oxide single-crystal material which possesses a relatively low value of F v but a very low loss so that F D is favorable. The material is very stable and is now widely used for single-element detectors. Its thermal conductivity is quite high so that it is not a good material for the pyroelectric vidicon. The ferroelectric polymers possess relatively low pyroelectric coefficients and low dielectric constants with high losses, so that their figures of merit are also quite low. Their low thermal conductivities make them quite favorable for use in the pyroelectric vidicon and the fact that they are commercially available in thin sections (down to 6 mm) at low cost, removing any requirement for expensive lapping and polishing, makes them attractive for some low-cost detectors. Their low permittivities make them particularly well suited to large-area detectors. The ceramic materials modified lead zirconate and modified lead titanate are interesting in that they possess high pyroelectric coefficients with relatively high permittivities and low losses. The modified lead zirconate is a solid solution of lead zirconate with lead iron niobate and lead titanate, with small additions of uranium as a stabilizing dopant. The use of uranium in this material minimizes the dielectric constant and loss (thus maximizing FD) while also permitting control over the electrical resistivity, allowing the gate bias resistor in © 2000 by CRC Press LLC

Fig. 54.1 to be designed into the sensor element. The modified lead titanate is doped with calcium titanate and lead cobalt tungstate. The use of hot pressing in ceramic manufacture permits the fabrication of very low porosity material, which can be lapped and polished to very thin sections (as low as 20 mm) while being mechanically strong enough to be placed on a mount which provides support only over a small area, permitting the fabrication of detectors with maximum sensitivity. While the Fv values are relatively small in these materials, the FD values are as good as most of the single-crystal materials. They are very well suited to small-area detectors because their high dielectric constants enable the element capacitance to be matched to that of the amplifier. Pyroelectric ceramics are now finding use in a wide range of the infrared detector market, from low-cost intruder alarms to high-value imaging arrays. Recently, a new class of pyroelectric materials which use the effect in the region of Tc have been developed [Whatmore, 1991]. In these materials a bias field must be applied to stabilize the effect, but F D values as high as 10 to 15 ´ 10–5 Pa–1/2 have been recorded in such materials as barium strontium titanate or lead scandium tantalate, both perovskite ceramics. This mode of operation is usually called “dielectric bolometer.”

Defining Terms Coercive field (Ec ): The field required to invert a sufficient proportion of the polarization of a body of a poled ferroelectric such that the net measurable external dipole moment is zero. Curie temperature (Tc ): The temperature at which the spontaneous polarization of a ferroelectric goes to zero. Ferroelectric: A polar dielectric in which the crystallographic orientation of the internal dipole moment can be changed by the application of an electric field. Paraelectric: The nonpolar phase into which the ferroelectric transforms above Tc , frequently called the paraelectric phase. Piezoelectric: A material which possesses a noncentrosymmetric crystal structure which will generate charge on the application of a mechanical stress. As in the case of a pyroelectric, this can be detected as either a potential difference or as a charge flowing in an external circuit. Pyroelectric: A polar dielectric material in which the internal dipole moment is temperature dependent. This leads to a change in the charge balance at the surface of the material which can be detected as either a potential difference or as a charge flowing in an external circuit. Remanent polarization: The value to which the externally measured polarization of a ferroelectric body relaxes after it has been subjected to an electric field much greater than the coercive field, which is then removed. Saturation polarization: The value to which the externally measured electrical dipole moment of a ferroelectric body tends when subjected to an external electrical field greater than the coercive field. Spontaneous polarization: The value of the electrical dipole moment of a ferroelectric crystal due to the separation of positive and negative charges within the unit cell.

Related Topic 35.1 Maxwell Equations

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Further Information W.R. Adey, “Electromagnetic fields, cell membrane amplification, and cancer promotion,” in Extremely Low Frequency Electromagnetic Fields: The Question of Cancer, B.W. Wilson, R.G. Stevens, and L. E. Anderson (eds.), Columbus, Ohio: Battelle Press, 1990, pp. 211–249. C.A.L. Bassett, “Bioelectromagnetics in service of medicine,” Bioelectromagnetics, 13:7–17, 1992. Bioelectromagnetics, the bi-monthly (formerly quarterly) journal of the Bioelectromagnetics Society. Published by Wiley-Liss, New York, since 1980. J. Black, Electrical Stimulation, New York: Praeger, 1987. M. Blank (ed.), Electricity and Magnetism in Biology and Medicine, San Francisco: San Francisco Press, 1993. C. Branden and J. Tooze, Introduction to Protein Structure, New York: Garland Publishing, 1991. C.T. Brighton, J. Black, and S. Pollack (eds.), Electrical Properties of Bone and Cartilage, New York: Grune and Stratton, 1979. C.T. Brighton and S.R. Pollack (eds.), Electromagnetics in Medicine and Biology, San Francisco: San Francisco Press, 1991. C. Polk and E. Postow, CRC Handbook of Biological Effects of Electromagnetic Fields, (second edition) Boca Raton, Fla.: CRC Press, 1996. Transactions of the Bioelectric Repair and Growth Society. Vols. I through XI. Published annually since 1980 by the Bioelectric Repair and Growth Society, Dresher, Pennsylvania.

Neuman, M.R. “Biomedical Sensors” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000

114 Biomedical Sensors

Michael R. Neuman Case Western Reserve University

114.1

114.1 114.2 114.3 114.4 114.5 114.6

Introduction Physical Sensors Chemical Sensors Bioanalytical Sensors Applications Summary

Introduction

Any instrumentation system can be described as having three fundamental components: a sensor, a signal processor, and a display and/or storage device. Although all these components of the instrumentation system are important, the sensor serves a special function in that it interfaces the instrument with the system being measured. In the case of biomedical instrumentation a biomedical sensor (which in some cases may be referred to as a biosensor) is the interface between the electronic instrument and the biologic system. There are some general concerns that are very important for any sensor in an instrumentation system regarding its ability to effectively carry out the interface function. These concerns are especially important for biomedical sensors, since the sensor can affect the system being measured and the system can affect the sensor performance. Sensors must be designed so that they minimize their interaction with the biologic host. It is important that the presence of the sensor does not affect the variable being measured in the vicinity of the sensor as a result of the interaction between the sensor and the biologic system. If the sensor is placed in a living organism, that organism will probably recognize the sensor as a foreign body and react to it. This may in fact change the quantity being sensed in the vicinity of the sensor so that the measurement reflects the foreign body reaction rather than a central characteristic of the host. Similarly, the biological system can affect the performance of the sensor. The foreign body reaction might cause the host to attempt to break down the materials of the sensor as a way to remove it. This may, in fact, degrade the sensor package so that the sensor can no longer perform in an adequate manner. Even if the foreign body reaction is not strong enough to affect the measurement, just the fact that the sensor is placed in a warm, aqueous environment may cause water to eventually invade the package and degrade the function of the sensor. Finally, as will be described below, sensors that are implanted in the body are not accessible for calibration. Thus, such sensors must be extremely stable so that frequent calibrations are not necessary. Biomedical sensors can be classified according to how they are used with respect to the biologic system. Table 114.1 shows that sensors can range from noninvasive to invasive as far as the biologic host is concerned. The most noninvasive of biomedical sensors do not even contact the biological system being measured. Sensors of radiant heat or sound energy coming from an organism are examples of noncontacting sensors. Noninvasive sensors can also be placed on the body surface. Skin surface thermometers, biopotential electrodes, and strain gauges placed on the skin are examples of noninvasive sensors. Indwelling sensors are those which can be placed into a natural body cavity that communicates with the outside. These are sometimes referred to as minimally invasive sensors and include such familiar sensors as oral-rectal thermometers, intrauterine pressure transducers, and stomach pH sensors. The most invasive sensors are those that need to be surgically placed and that

CARDIAC MONITOR

he basic method of assessing heart function is thermodilution, a procedure that involves insertion of a catheter into the pulmonary artery and is demanding in terms of cost, equipment, and skilled personnel time. For monitoring astronauts in flight, NASA needed a system that was non-invasive and considerably less complex. In 1965, Johnson Space Center contracted with the University of Minnesota to explore the then-known but little-developed concept of impedance cardiography (ICG) as a means of astronaut monitoring. A five-year program led to the development of the Minnesota Impedance Cardiograph (MIC), an electronic system for measuring impedance changes across the thorax that would be reflective of cardiac function and blood flow from the heart’s left ventricle into the aorta. ICG clearly had broad potential for hospital applications but further development and refinement was needed. A number of research institutions and medical equipment companies launched development of their own ICGs, using MIC technology as a departure point. Among them were Renaissance Technologies, Inc., Newtown, Pennsylvania, and Drexel University of Philadelphia, who jointly developed the IQ System. The system provides a simple, repeatable, non-invasive way of assessing cardiac function at dramatically reduced cost. The IQ System is in wide use in hospital intensive care units, emergency rooms, operating rooms, and laboratories in the U.S. and abroad. IQ has two basic elements: the non-invasive, disposable patient interface known as IQ-Connect and the touch screen monitor, which calculates and displays cardiac output values and trends. The hardware design of the original MIC was retained but IQ has advanced automated software that features the signal processing technology known as TFD (Time Frequency Distribution). TFD provides three-dimensional distribution of the hemodynamic signals being measured, enabling visualization of the changes in power, frequency, and time. This clinically proven capability allows IQ to measure all cardiac events without using estimation techniques required in some earlier systems. (Courtesy of National Aeronautics and Space Administration.)

The IQ-Connect interface electronically measures impedance changes across the thorax to reflect heart function. (Photo courtesy of National Aeronautics and Space Administration.)

TABLE 114.1 Classification of Biomedical Sensors According to Their Interface with the Biologic Host Noninvasive Invasive

Noncontacting Body surface Indwelling Implanted

TABLE 114.2 Physical Variables Sensed by Biomedical Sensors Displacement, velocity, acceleration (linear and angular) Temperature Force (weight and mass) Pressure Flow Radiant energy (optical)

involve some tissue damage associated with their installation. For example, a needle electrode for picking up electromyographic signals directly from muscles; a blood pressure sensor placed in an artery, vein, or the heart itself; or a blood flow transducer positioned on a major artery are all examples of invasive sensors. We can also classify sensors in terms of the quantities that they measure. Physical sensors are used in measuring physical quantities such as displacement, pressure, and flow, while chemical sensors are used to determine the concentration of chemical substances within the host. A subgroup of the chemical sensors that are concerned with sensing the presence and the concentration of biochemical materials in the host are known as bioanalytical sensors, or sometimes they are referred to as biosensors. In the following paragraphs we will look at each type of sensor and present some examples as well as describe some of the important issues surrounding these types of sensors.

114.2

Physical Sensors

Physical variables associated with biomedical systems are measured by a group of sensors known as physical sensors. A list of typical variables that are frequently measured by these devices is given in Table 114.2. These quantities are similar to physical quantities measured by sensors for nonbiomedical applications, and the devices used for biomedical and nonbiomedical sensing are, therefore, quite similar. There are, however, two principal exceptions: pressure and flow sensors. The measurement of blood pressure and blood flow in humans and other animals remains a difficult problem in biomedical sensing. Direct blood pressure measurement refers to evaluation of the blood pressure using a sensor that is in contact with the blood being measured or contacts it through an intermediate fluid such as a physiologic saline solution. Direct blood pressure sensors are invasive. Indirect blood pressure measurement involves a sensor that does not actually contact the blood. The most familiar indirect blood pressure measurement is the sphygmomanometer cuff that is usually used in most medical examinations. It is a noninvasive instrument. Until recently, the primary sensor used for direct blood pressure measurement was the unbonded strain gauge pressure transducer shown in Fig. 114.1. The basic principle of this device is that a differential pressure seen across a diaphragm will cause that diaphragm to deflect. This deflection is then measured by a displacement transducer. In the unbonded strain gauge sensor a closed chamber is covered by a flexible diaphragm. This diaphragm is attached to a structure that has four fine gauge wires drawn between it and the chamber walls. A dome with the appropriate hardware for coupling to a pressure source covers the diaphragm on the side opposite the chamber such that when the pressure in the dome exceeds the pressure in the chamber, the diaphragm is deflected into the chamber. This causes two of the fine wires to stretch by a small amount while the other two wires contract by the same amount. The electrical resistance of the wires that are stretched increases while that of the wires that contract decreases. By connecting these wires, or more correctly these unbonded strain gauges, into a Wheatstone bridge circuit, a voltage proportional to the deflection of the diaphragm can be obtained. In recent years semiconductor technology has been applied to the design of pressure transducers. Silicon strain gauges that are much more sensitive than their wire counterparts are formed on a silicon chip, and micromachining technology is used to form this portion of the chip into a diaphragm with the strain gauges integrated into its surface. This structure is then incorporated into a plastic housing and dome assembly. The entire sensor can be fabricated and sold inexpensively so that disposable, single-use devices can be made. These

FIGURE 114.1 An unbonded strain gauge pressure transducer.

have the advantage that they are only used on one patient and they do not have to be cleaned and sterilized between patients. By using them on only one patient, the risk of transmitting blood-borne infections is eliminated. In biomedical applications pressure is generally referenced to atmospheric pressure. Therefore, the pressure in the chamber of the pressure transducer must be maintained at atmospheric pressure. This is done by means of a vent in the chamber wall or a fine bore, flexible capillary tube that couples the chamber to the atmosphere. This tube is usually included in the electrical cable connecting the pressure transducer to the external instrumentation such that the tube is open to the atmosphere at the cable connecter. In using this sensor to measure blood pressure the dome is coupled to a flexible plastic tube, and the dome and tube are filled with a physiological saline solution.1 As described by Pascal’s Law, the pressure in the dome, and hence against the diaphragm, will be the same as that at the tip of the tube provided the tip of the tube is at the same horizontal level as the dome. Thus by threading the tube into a blood vessel, an invasive procedure, the blood pressure in that vessel can be transmitted to the dome and hence the diaphragm of the pressure transducer. The pressure transducer will, therefore, sense the pressure in the vessel. This technique is known as external direct blood pressure measurement, and the flexible plastic tube that enters the blood vessel is known as a catheter. It is important to remember that the horizontal level of the blood pressure transducer dome must be the same as that of the tip of the catheter in the blood vessel to accurately measure the pressure in that vessel without adding an error due to the hydrostatic pressure in the catheter. In addition to problems due to hydrostatic pressure differences between the chamber and the dome, catheters introduce pressure errors as a result of the dynamic properties of the catheter, fluid, dome, and diaphragm. These properties as well as air bubbles in the catheter, or obstructions due to clotted blood or other materials, introduce resonances and damping. These problems can be minimized by utilizing miniature pressure transducers fabricated using microelectronic semiconductor technology that are located at the tip of a catheter FIGURE 114.2 A catheter tip pressure transducer. rather than at the end that is external to the body. A general arrangement for such a pressure transducer is shown in Fig. 114.2. As with the disposable sensors, strain gauges are integrated into the diaphragm of the transducer such that they detect very small deflections of this diaphragm. Because of the small size, small diaphragm displacement, and lack of a catheter with a fluid column, these sensors have a much broader frequency response, give a clearer signal, and do not have any hydrostatic pressure error.

1It must be pointed out that the use of such a sensor is not limited to blood pressure measurement. The strain gauge pressure sensor can be used to measure the pressure of any fluid to which it is appropriately coupled.

FIGURE 114.3 The Clark electrode, an amperometric electrochemical sensor of oxygen.

Although the indwelling catheter tip pressure transducer appears to solve many of the problems associated with the external pressure transducer, there are still important problems in pressure transducer design that need to be addressed. Long-term stability of pressure transducers is not very good. This is especially problematic for venous pressure measurements which are carried out at relatively low pressure. Long-term changes in baseline pressure require pressure transducers to be frequently adjusted to be certain of zero pressure. While this can be done relatively easily for external and indwelling pressure transducers, there is no way to carry out this procedure for implanted transducers, since there is not a way to establish zero pressure at the sensor. Thus devices that have very low long-term baseline drift are essential for implantable applications. The packaging of the pressure transducer also represents a problem that needs to be addressed. Packaging must both protect the transducer and be biocompatible. It also must allow the appropriate pressure to be transmitted from the biologic fluid to the diaphragm. The amount of packaging material required should be kept at a minimum so as not to substantially increase the size of implantable or indwelling sensors. Furthermore, the material must be mechanically stable so that it does not swell or contract, since this will most likely change the baseline pressure seen by the sensor. These problems need to be overcome before miniature pressure transducers can be used reliably in implantable applications.

114.3

Chemical Sensors

There are many biomedical problems where it is TABLE 114.3 Classifications of Chemical Biomedical Sensors necessary to know the concentration of a particular 1. Electrochemical a. Amperometric substance in a biological sample. Chemical sensors b. Potentiometric provide the interface between an instrument and c. Coulometric the specimen to allow one to determine this con2. Optical centration. These sensors can be used on a biologa. Colorimetric ical specimen taken from the host and tested in a b. Emission and absorption spectroscopy c. Fluorescence laboratory, or they can be used for in vivo measured. Chemiluminescence ments either as noninvasive or invasive sensors, the 3. Thermal methods latter being the most frequently used. a. Calorimetry There are many types of chemical sensors used b. Thermoconductivity in biomedical instrumentation. Table 114.3 lists 4. Nuclear magnetic resonance some general categories of sensors. Electrochemical and optical sensors are most frequently used for biomedical measurements both in vivo and in vitro. An example of an electrochemical sensor is the Clark electrode illustrated in Fig. 114.3. This consists of an electrochemical cell separated from the specimen being measured by an oxygen-permeable membrane. The cell is driven at a fixed potential of 600 mV, and under these conditions the following reaction occurs at the noble metal cathode:

O 2 + 4e – + H 2O ® 4OH –

FIGURE 114.4 A glass electrode pH sensor.

This reaction involves the reduction of molecular oxygen that diffuses into the cell through the oxygenpermeable membrane. Since the other components of the reaction are in abundance, the rate of the reaction is limited by the amount of oxygen available. Thus, the rate of electrons used at the cathode is directly related to the available oxygen. In other words, the cathode current is proportional to the partial pressure of oxygen in the specimen being measured. The electrochemical cell is completed by the silver anode. The reaction at the anode involves forming the low-solubility salt, silver-chloride, from the anode material itself and the chloride ion contained in the electrolyte. The cell is designed so that these materials are also in abundance so that their concentration does not affect the sensor performance. This type of sensor is an example of an amperometric electrochemical sensor. Another type of electrochemical sensor that is frequently used in biomedical laboratories is the glass pH electrode illustrated in Fig. 114.4. The acidity or alkalinity of a solution is characterized by its pH. This quantity is defined as

pH = – log 10 [H +] where [H+] is the activity of the hydrogen ions in solution, a quantity that is related to the concentration of the hydrogen ions. This sensor only works in an aqueous environment. It consists of an inner chamber containing an electrolytic solution of known pH and an outer solution with an unknown pH that is to be measured. The membrane consists of a specially formulated glass that will in essence allow hydrogen ions to pass in either direction but will not pass other chemical species. If the concentration of hydrogen ions in the external solution is greater than that in the internal solution, there will be a gradient forcing hydrogen ions to diffuse through the membrane into the internal solution. This will cause the internal solution to have a greater positive charge than the external solution so that an electrical potential and, hence, an electric field will exist across the membrane. This field will counteract the diffusion of hydrogen ions due to the concentration difference and so an equilibrium will be eventually established. The potential across the membrane at this equilibrium condition will be related to the hydrogen ion concentration difference (or more accurately the activity difference) between the inner and outer solutions. This potential is given by the Nernst equation

E = -

RT æ a1 ö ln ç ÷ nF è a2 ø

where E is the potential measured, R is the universal gas constant, T is the absolute temperature, n is the valence of the ion, and a1 and a2 are the activities of the ions on each side of the membrane. Thus the potential measured across the glass membrane will be proportional to the pH of the solution being studied. At room temperature © 2000 by CRC Press LLC

FIGURE 114.5 A generalized bioanalytical sensor.

the theoretical sensitivity of the electrode is approximately 60 mV/pH. It is not practical to measure the potential across the membrane directly and so reference electrodes, sensors that can be used to measure electrical potential of an electrolytic solution, are used to contact the solution on either side of the membrane to measure the potential difference across it. The reference electrodes and the glass membrane are incorporated into the structure shown in Fig. 114.4 known as a glass pH electrode. This is an example of a potentiometric measurement made using an ion-selective membrane. There are other types of ion-selective membrane potentiometric chemical sensors that are used for biomedical applications. The membranes of these sensors determine the ion being sensed. The membrane can be based upon glass or a polymeric material such as polyvinyl chloride, but the key component is the substance that is added to the membrane that allows it to selectively pass a single ion. Important problems in the development of chemical biomedical sensors are similar to those discussed above for the pressure sensor. Issues of long-term stability and packaging are critical to the success of a chemical sensor. The package is even more critical in chemical sensors than it was in pressure sensors in that the package must protect portions of the sensor that require isolation from the solutions being measured while it provides direct contact of the chemically sensitive portions of the sensor to the solution. The maintenance of a window through the package for this contact represents a critical aspect of sensor development. Frequent calibration is also necessary for chemical sensors. Just about every type of chemical sensor requires some sort of calibration using a standard solution with known concentration of the analyte being sensed. The best calibration method is a two-point procedure where two standards are used to establish the slope and the intercept of the calibration line. Some chemical sensors have stable slopes but need to be calibrated in terms of the baseline or intercept. In this case a single-point calibration can be used.

114.4

Bioanalytical Sensors

A special class of sensors of biological molecules has evolved in recent years. These bioanalytical sensors take advantage of one of the following biochemical reactions: (1) enzyme-substrate, (2) antigen-antibody, or (3) ligand-receptor. The advantage of using these reactions in a sensor is that they are highly specific for a particular biological molecule, and sensors with high sensitivity can be developed based upon these reactions. The basic structure of a bioanalytical sensor is shown in Fig. 114.5. There are two principal portions of the sensor. The first contains one component of the biological sensing reaction such as the enzyme or the antibody, and the second component involves a means of detecting whether the biological reaction has taken place. This second portion of a bioanalytical sensor is made up of either a physical or chemical sensor that serves as the detector © 2000 by CRC Press LLC

of the biological reaction. As illustrated in Fig. 114.5, this detector can consist of an electrical sensor such as used in electrochemical sensors, a thermal sensor, a sensor of changes in capacitance, a sensor of changes in mass, or a sensor of optical properties. An example of a bioanalytical sensor is a glucose sensor. The first portion of the sensor contains the enzyme glucose oxidase. This enzyme promotes the oxidation of glucose to glucuronic acid and consumes oxygen in the process. Thus, by placing an oxygen sensor along with the glucose oxidase in the bioanalytical sensor, one can determine the amount of glucose oxidized by measuring the amount of oxygen consumed. An even better approach is to have two identical sensor structures in the same package. The only difference is that only one of the sensors contains the enzyme. When there is no glucose present, both sensors will measure the same oxygen partial pressure. The presence of glucose, however, will cause the sensor with the glucose oxidase to have a reduced partial pressure of oxygen due to the oxygen consumption of the reaction. By making a differential measurement of oxygen partial pressure with both sensors, other factors that can cause an apparent change in oxygen partial pressure such as temperature will have a much lower effect than if a single sensor was used. Stability problems are important for bioanalytical sensors, especially those that are used for long-term measurements. Not only are the stability issues the same as for the physical and chemical sensors, but they are also related to preservation of the biological molecules used in the first stage of the sensor. These molecules can often be degraded or destroyed by heat or exposure to light. Even aging can degrade some of these molecules. Thus, an important issue in dealing with bioanalytical sensors is the preservation of the biochemical components of the sensor. Not all biochemical reactions are entirely reversible, and so the bioanalytical sensors based on them will not be reversible as well. This may be acceptable for some applications but not for others and must be taken into consideration in choosing a bioanalytical sensor.

114.5

Applications

Biomedical sensors and instrumentation are used in biomedical research and patient care applications. In terms of patient care, sensors are used as a part of instruments that carry out patient screening by making measurements such as blood pressure using automated apparatus. Specimen analysis is another important application of biomedical sensors in patient care. This can include analyses that can be carried out by the patients themselves in their homes such as is done with home blood glucose analyzers. Instrumentation based upon biomedical sensors can be used in the physician’s office for carrying out some chemical analyses of patient specimens such as urinalysis or elementary blood chemistries such as serum glucose and electrolytes. Sensors also are a part of large multicomponent automatic blood analyzers used in the central clinical laboratory of major medical centers. Another application for biomedical sensors is in patient monitoring. Sensors represent the front end of critical care monitors used in the intensive care unit and in the operating and recovery rooms. Measurements cover a wide range of biomedical variables such as continuous recordings of blood pressure and transcutaneous measurement of the partial pressure of carbon dioxide in the blood. The performance of these instruments is strongly dependent on biomedical sensors. Patient monitoring can also be carried out in the various clinical units of the hospital. Devices such as ambulatory cardiac monitors that allow patients to be observed while they are free to move around if they desire are becoming important in clinical care in “step-down” units for patients who have completed their stay in the intensive care unit. Patient monitoring has even made its way into the home. Home cardiorespiratory monitors are thought to have some potential value in identifying infants at risk of sudden infant death.

114.6

Summary

Sensors serve an important function in biomedical instrumentation systems in that they provide the interface between the electronic instrument and the biologic system being measured. Very often the quality of the instrument is based upon the quality of the sensor at the instrument’s front end. Although electronic signal processing has been developed to a high level, the signals are no better than the quality of the sensors that provide them. Although there have been many advances in biomedical sensor technology, many problems remain. Biomedical sensors will continue to be an import area for research and development in biomedical engineering.

Defining Terms Amperometric sensor: An electrochemical sensor that determines the amount of a substance by means of an oxidation-reduction reaction involving that substance. Electrons are transferred as a part of the reaction, so that the electrical current through the sensor is related to the amount of the substance seen by the sensor. Analyte: The substance being measured by a chemical or bioanalytical sensor and instrumentation system. Bioanalytical sensor: A special case of a chemical sensor for determining the amount of a biochemical substance. This type of sensor usually makes use of one of the following types of biochemical reactions: enzyme-substrate, antigen-antibody, or ligand-receptor. Biomedical sensor: A device for interfacing an instrumentation system with a biological system such as a biological specimen or an entire organism. The device serves the function of detecting and measuring in a quantitative fashion a physiological property of the biologic system. Chemical sensor: The interface device for an instrumentation system that determines the concentration of a chemical substance. Noninvasive sensor: The interface device of an instrumentation system that measures a physiologic variable from an organism without interrupting the integrity of that organism. This device can be in direct contact with the surface of the organism or it can measure the physiologic quantity while remaining remote from the organism. Physical sensor: An interface device at the input of an instrumentation system that quantitatively measures a physical quantity such as pressure or temperature. Potentiometric sensor: A chemical sensor that measures the concentration of a substance by determining the electrical potential between a specially prepared surface and a solution containing the substance being measured.

Related Topics 56.1 Introduction • 56.2 Physical Sensors • 56.3 Chemical Sensors • 56.4 Biosensors • 56.5 Microsensors

References R.S.C. Cobbold, Transducers for Biomedical Measurements: Principles and Applications, New York: John Wiley, 1974. B.R. Eggins, Biosensors: An Introduction, Chichester; New York: John Wiley, 1996. D.G. Fleming, W.H. Ko, and M.R. Neuman, Eds., Indwelling and Implantable Pressure Transducers, Cleveland: CRC Press, 1977. L.A. Geddes, The Direct and Indirect Measurement of Blood Pressure, Chicago: Year Book Medical Publishers, 1970. L.A. Geddes, Electrodes and the Measurement of Bioelectric Events, New York: John Wiley, 1972. W. Göpel, J. Hesse and J.N. Zemel, Sensors; A Comprehensive Survey, Weinheim, Germany: VCH Verlagsgesellschaft, 1989. A.H. Hall, Biosensors, Englewood Cliffs, N.J.: Prentice Hall, 1991. J. Janata, Principles of Chemical Sensors, New York: Plenum Press, 1989. M.R. Neuman, R.P. Buck, V.V. Cosofret, E. Lindner, and C.C. Liu, “Fabricating biomedical sensors with thinfilm technoogy, IEEE Engineering in Medicine and Biology Magazine, 13, 409–419, 1994. R. Pallas-Areny and J.G. Webster, Sensors and Signal Conditioning, New York: John Wiley, 1991. J.I. Peterson and G.G. Vurek, “Fiber-optic sensors for biomedical applications,” Science, vol. 224, pp. 123–127, 1984. P. Rolfe, “Review of chemical sensors for physiological measurement,” J. Biomed. Eng., vol. 10, pp. 138–145, 1988. J.G. Webster, Ed., Encyclopedia of Medical Devices and Instrumentation, New York: John Wiley, 1988. O.S. Wolfbeis, Ed., Fiber Optic Chemical Sensors and Biosensors, Boca Raton, Fla: CRC Press, 1991.

Further Information Research reports on biomedical sensors appear in many different journals ranging from those that are concerned with clinical medicine through those that are engineering and chemistry oriented. Three journals, however, represent major sources of biomedical sensor papers. These are listed as follows: The IEEE Transactions on Biomedical Engineering is a monthly journal devoted to research papers on biomedical engineering. Papers on biomedical sensors frequently appear, and the February 1986 issue was devoted entirely to the topic of biomedical sensors. For more information or subscriptions, contact IEEE Service Center, 445 Hoes Lane, P.O. Box 1331, Piscataway, NJ 08855-1331. The international journal Medical and Biological Engineering and Computing is published bi-monthly by the International Federation for Medical and Biological Engineering. This journal also contains frequent reports on biomedical sensors and related topics. Subscription information can be obtained from Peter Peregrinus Ltd., P.O. Box 96, Stevenage, Herts SG12SD, United Kingdom. The journal Biomedical Instrumentation and Technology is published by the Association for the Advancement of Medical Instrumentation. This bimonthly journal has reports on biomedical instrumentation for clinical applications, and these include papers on biomedical sensors. Subscription information can be obtained from Hanley and Belfus, 210 S. 13th Street, Philadelphia, PA 19107. There are also several scientific meetings that include biomedical sensors. The major meeting in the area is the international conference of the IEEE Engineering in Medicine and Biology Society. An extensive book or CD ROM of extended abstracts for this meeting is published each year by the IEEE. Further information can be obtained by contacting the IEEE at the address listed above.

Bronzino, J.D., Berbari, E.J., Johnson, P.L., Smith, W.M. “Bioelectronics and Instruments” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000

115 Bioelectronics and Instruments

Joseph D. Bronzino Trinity College/Biomedical Allience for Central Connecticut (BOACON)

Edward J. Berbari Purdue University

115.1

Philip L. Johnson University of Alabama at Birmingham

115.2

William M. Smith University of Alabama at Birmingham

The Electroencephalogram The Language of the Brain • Historical Perspective • EEG Recording Techniques • Frequency Analysis of the EEG • Nonlinear Analysis of the EEG • Topographic Mapping

The Electrocardiograph Physiology • Instrumentation • Conclusions

115.3

Pacemakers/Implantable Defibrillators Pacemakers • Implantable Cardioverter Defibrillators

115.1 The Electroencephalogram Joseph D. Bronzino Electroencephalograms (EEGs) are recordings of the minute (generally less than 300 µV) electrical potentials produced by the brain. Since 1924, when Hans Berger reported the measurements of rhythmic electrical activity on the human scalp, it has been suggested that these patterns of bioelectrical origin may provide clues regarding the neuronal bases for specific behaviors and has offered great promise to reveal correlations between pathological processes and the electrical activity of specific regions of the brain. Over the years, EEG analyses have been conducted primarily in clinical settings, to detect gross organic pathologies and the epilepsies, and in research facilities to quantify the central effect of new pharmacological agents. As a result of these efforts, cortical EEG patterns have been shown to be modified by a wide variety of variables including biochemical, metabolic, circulatory, hormonal, neuroelectric, and behavioral factors. In the past, interpretation of the EEG was limited to visual inspection by a trained electroencephalographer capable of distinguishing normal activity from localized or generalized abnormalities of particular types from relatively long EEG records. This approach has left clinicians and researchers alike lost in a sea of EEG paper records. Computer technology has permitted the application of a host of methods to quantify EEG changes. With this in mind, this section provides an introduction to some of the basic concepts underlying the generation of the EEG, a review of the basic approaches used in quantifying alterations in the EEG, and some insights regarding quantitative electrophysiology techniques.

The Language of the Brain The mass of brain tissue is composed of bundles of nerve cells (neurons) which constitute the fundamental building blocks of the nervous system. Figure 115.1 is a schematic drawing of just such a cell. It consists of three major components: the cell body (or soma), the receptor zone (or dendrites), and the axon, which carries electrical signals from the soma to target sites such as muscles, glands, or other neurons. Numbering approximately 20 billion in each human being, these tiny cells come in a variety of sizes and shapes. Although neurons are anatomically distinct units having no physical continuity between their processes, the axon ends on the soma and the dendrites of other cells in what is called a synapse. Under the microscope this often stands out as a spherical enlargement at the end of the axon to which various names have been given, for example, boutons,

end-plate, or synaptic terminals. This ending does not actually make physical contact with the soma or dendrite but is separated by a narrow cleft (gap) of approximately 100 to 200 Å (10–9 m) wide. This is known as the synaptic cleft. Each of these synaptic endings contains a large number of submicroscopic spherical structures (synaptic vesicles) that can be detected only under an electron microscope. These synaptic vesicles, in turn, are essentially “chemical carriers” containing transmitter substance that is released into the synaptic cleft on excitation. When an individual neuron is excited, an electrical signal is transmitted along its axon to many tiny branching, diverging fibers near its far end. These axonal terminals end as synapse on a large number of other neurons. When an electrical pulse arrives at the synapse, it triggers the FIGURE 115.1 Basic structure of the neuron. release of a tiny amount of transmitter substance which crosses the synaptic cleft thereby altering the membrane potential of the receiving neuron. If the change is above a certain threshold value, the neuron is activated and generates an action potential of its own which is propagated along its axon, and the process is repeated. Neurons are involved in every conceivable action taken by the body, whether it is to control its own internal environment or to respond to changes in the external world. As a result, they are responsible for such essential functions as: • Accepting and converting sensory information into a form that can be processed within the nervous system by other neurons. • Processing and analyzing this information so that an “integrated portrait” of the incoming data can be obtained. • Translating the final outcome or “decision” of this analysis process into appropriate electrical or chemical form needed to stimulate glands or activate muscles. Evolution has played a role in the development of these unique neurons and in the arrangement and development of interconnections between nerve cells in the various parts of the brain. Since the brain is a most complex organ, it contains numerous regions designed for specific tasks. One might, in fact, consider it to be a collection of organs arranged together to act in the harmony of activity we recognize as the individual’s state of consciousness or as life itself. Over the years, anatomists and physiologists have identified and named most pathways (tracts), most groups of neurons (nuclei), and most of the major parts of the human brain. Such attention to detail is certainly not necessary here. It will serve our purpose to simply provide a broad overview of the organization of the brain and speak of three general regions: the brainstem, cerebellum, and the cerebral cortex. The brainstem, or old brain, is really an extension and elaboration of the spinal chord. This section of the brain evolved first and is the location of all the centers that control the regulatory systems, such as respiration, necessary for physical survival of the organism. In addition, all sensory pathways find their way into the brainstem, thereby permitting the integration of complex input patterns to take place within its domain. Above the brainstem is a spherical mass of neuronal tissue called the cerebellum. This remarkable structure is a complex monitor and modifier of body movements. The cerebellum does not initiate movements, but only modifies motor control activated in other areas. Cerebellar operation is not only dependent on evolutionary development, but relies heavily on actual use and patterns of learned motor behavior acquired throughout life. It is for this reason that the movements of a gymnast are smooth and seemingly effortless. The most conspicuous part of all in the human brain is the cerebral cortex. Compared to most mammals, it is so large in man that it becomes a covering that surrounds and hides most of the other regions of the brain. Wrinkled and folded, the cerebral tissue is literally pressed into the limited space allocated to it. Although it has been possible to ascertain that certain cortical areas such as visual cortex, the sensory projection area, and the motor strip are associated with specific functions, the overall operation of this complex structure is still

FIGURE 115.2 Major divisions of the cerebral cortex.

not completely understood. However, for the sake of convenience, it has been arbitrarily divided (based primarily on anatomical considerations) into the following areas: frontal lobe, parietal lobe, temporal lobe, and occipital lobe (Fig. 115.2). Each of these segments of the cortex, which is the source of intellectual and imaginative capacities, includes millions of neurons and a host of interconnections. It is generally agreed that brain function is based on the organization of the activity of large numbers of neurons into coherent patterns. Since the primary mode of activity of these nerve cells is electrical in nature, it is not surprising that a composite of this activity can be detected in the form of electrical signals. Of extreme interest, then, are the actual oscillations, rhythms, and patterns seen in the cryptic flow of electrical energy coming from the brain itself, i.e., in the EEG.

Historical Perspective In 1875, Caton published the initial account of the recording of the spontaneous electrical activity of the brain from the cerebral cortex of an experimental animal. The amplitude of these electrical oscillations was so low, that is, on the order of microvolts, that Caton’s discovery is all the more amazing because it was made 50 years before suitable electronic amplifiers became available. In 1924, Hans Berger, of the University of Jena in Austria, carried out the first human EEG recordings using electrical metal strips pasted to the scalps of his subjects as electrodes and a sensitive galvanometer as the recording instrument. Berger was able to measure the irregular, relatively small electrical potentials (i.e., 50 to 100 mV) coming from the brain. By studying the successive positions of the moving element of the galvanometer recorded on a continuous roll of paper, he was able to observe the resultant patterns in these brain waves as they varied with time. From 1924 to 1938, Berger laid the foundation for many of the present applications of electroencephalography. He was the first to use the word electroencephalogram in describing these brain potentials in man. Berger noted that these brain waves were not entirely random, but instead displayed certain periodicities and regularities. For example, he observed that although these brain waves were slow (i.e., exhibited a synchronized patter of high amplitude and low frequency, <3 Hz) in sleep and states of depressed function, they were faster (i.e., exhibited a desynchronized pattern of low amplitude and high frequency, 15–25 Hz) during waking behavior. He suggested, quite correctly, that the brain’s activity changed in a consistent and recognizable fashion when the general status of the subject changed, as from relaxation to alertness. Berger also concluded that these brain waves could be greatly affected by certain pathological conditions after noting the marked increase in the amplitude of these brain waves brought about by convulsive seizures. However, in spite of the insights provided by these studies, Berger’s original paper published in 1929 did not excite much attention. In essence, the efforts of this most remarkable pioneer were largely ignored until similar investigations were carried out and verified by British investigators. It was not until 1934 when Adrian and Matthews published their classic paper verifying Berger’s findings that the reality of human brain waves was accepted and EEG studies were put on a firmly established basis. One of their primary contributions was the identification of certain rhythms in the EEG, regular oscillations at approximately 10–12 Hz in the occipital lobes of the cerebral cortex. They found that this alpha rhythm in the EEG would disappear when the brain displayed any type of attention or alertness or focused on objects in the visual field. The physiological basis for these results, the “arousing influence” of external stimuli on the

cortex, was not formulated until 1949 when Moruzzi and Magoun demonstrated the existence of widely spread pathways through the central reticular core of the brainstem capable of exerting a diffuse activating influence on the cerebral cortex. This reticular activating system has been called the brain’s response selector because it alerts the cortex to focus on certain incoming information while ignoring other. It is for this reason that a sleeping mother will immediately be awakened by her crying baby or the smell of smoke, and yet ignore the traffic outside her window or the television still playing in the next room. An in-depth discussion of these early studies is beyond the scope of this presentation; however, for the interested reader an excellent historical review of this early era in brain research has been recorded in a fascinating text by Brazier [1968].

EEG Recording Techniques Scalp recordings of spontaneous neuronal activity in the brain, identified as the EEG, allow measurement of potential changes over time between a signal electrode and a reference electrode [Kondraski, 1986]. Compared to other biopotentials, such as the electrocardiogram, the EEG is extremely difficult for an untrained observer to interpret. As might be expected, partially as a result of the spatial mapping of functions onto different regions of the brain, correspondingly different waveforms are visible, depending on electrode placement. Recognizing that some standardization was necessary for comparison of research as well as clinical EEG records, the International Federation in Electroencephalography and Clinical Neurophysiology adopted the 10–20 electrode placement system, [Jasper, 1958]. Additional electrodes to monitor extracerebral contaminants of the EEG such as eye movement, EKG, and muscle activity are essential. The acquisition of EEG for quantitative analysis should also require the ability to view the EEG during collection on a polygraph or high-resolution video display. Since amplification, filtering, and digitization determine the frequency characteristics of the EEG and the source of potential artifacts, the acquisition parameters must be chosen with an understanding of their effects on signal acquisition and subsequent analysis. Amplification, for example, increases the amplitude range (volts) of the analog-to-digital (A/D) converter. The resolution of the A/D converter is determined by the smallest amplitude of steps that can be sampled. This is calculated by dividing the voltage range of the A/D converter by 2 to the power of the number of bits of the A/D converter. For example, an A/D converter with a range of ± 5 V with 12-bit resolution can resolve samples as small as ± 2.4 mV. Appropriate matching of amplification and A/D converter sensitivity permits resolution of the smallest signal while preventing clipping of the largest signal amplitudes. The bandwidth of the filters and the rate of digitization determine the frequency components of interest that are passed, while other frequencies outside the band of interest that may represent potential artifacts, such as aliasing, are rejected. A filter’s characteristics are determined by the rate of the amplitude decrease at the bandwidth’s upper and lower edges. Proper digital representation of the analog signal depends on the rate of data sampling, which is governed by the Nyquist theorem that states that data sampling should be at least twice the highest frequency of interest. In addition to the information available from spontaneous electrical activity of the EEG, the brain’s electrical response to sensory stimulation can contribute data as to the status of cortical and subcortical regions activated by sensory input. Due to the relatively small amplitude of a stimulus-evoked potential as compared to the spontaneous EEG potentials, the technique of signal averaging is used to enhance the stimulus-evoked response. Stimulus averaging takes advantage of the fact that the brain’s electrical response is time-locked to the onset of the stimulus and the nonevoked background potentials are randomly distributed in time. Consequently, the average of multiple stimulus responses will result in the enhancement of the time-locked activity, while the averaged random background activity will approach zero. The result is an evoked response that consists of a number of discrete and replicable peaks that occur, depending upon the stimulus and the recording parameters, at predicted latencies from the onset of stimulation. The spatial localization of maximum peak amplitudes has been associated with cortical generators in primary sensory cortex. Instrumentation required for EEG recordings can be simple or elaborate [Kondraski, 1986]. (Note: Although the discussion presented in this section is for a single-channel system it can be extended to simultaneous multichannel recordings simply by multiplying the hardware by the number of channels required. In cases that do not require true simultaneous recordings, special electrode selector panels can minimize hardware requirements.) Any EEG system consists of electrodes, amplifiers (with appropriate filters) and a recording device.

Commonly used scalp electrodes consist of Ag-AgCl disks, 1 to 3 mm in diameter, with a very flexible long lead that can be plugged into an amplifier. Although it is desirable to obtain a low-impedance contact at the electrode ski interface (less than 10 kW), this objective is confounded by hair and the difficulty of mechanically stabilizing the electrodes. Conductive electrode paste helps obtain low impedance and keep the electrodes in place. A type of cement (collodion) is used to fix small patches of gauze over electrodes for mechanical stability, and leads are usually taped to the subject to provide some strain relief. Slight abrasion of the skin is sometimes used to obtain better electrode impedances, but this can cause irritation and sometimes infection (as well as pain in sensitive subjects). For long-term recordings, as in seizure monitoring, electrodes present major problems. Needle electrodes, which must be inserted into the tissue between the surface of the scalp and skull, are sometimes useful. However, the danger of infection increases significantly. Electrodes with self-contained miniature amplifiers are somewhat more tolerant because they provide a low-impedance source to interconnecting leads, but they are expensive. Despite numerous attempts to simplify the electrode application process and to guarantee long-term stability, none has been widely accepted. Instruments are available for measuring impedance between electrode pairs. The procedure is recommended strongly as good practice, since high impedance leads to distortions that may be difficult to separate from actual EEG signals. In fact, electrode impedance monitors are built into some commercial devices for recording EEGs. Standard dc ohmmeters should not be used, since they apply a polarizing current that causes build-up of noisy electrode potential at the skin-electrode interface. Commercial devices apply a known-amplitude sinusoidal voltage (typically 1 kHz) to an electrode pair circuit and measure root mean square (rms) current, which is directly related to the magnitude of the impedance. From carefully applied electrodes, signal amplitudes of 1 to 10 mV can be obtained. Considerable amplification (gain = 106) is required to bring these levels up to an acceptable level for input to recording devices. Because of long electrode leads and the common electrically noisy environment where recordings take place, differential amplifiers with inherently high input impedance and high common mode rejection ratios are essential for highquality EEG recordings. In some facilities, special electrically shielded rooms minimize environmental electrical noise, particularly 60-Hz alternating current (ac) line noise. Since much of the information of interest in the EEG lies in the frequency bands less than 40 Hz, low-pass filters in the amplifier can be switched into attenuate 60-Hz noise sharply. For attenuating ac noise when the low-pass cutoff is greater than 60 Hz, many EEG amplifiers have notch filters that attenuate only frequencies in a narrow band centered around 60 Hz. Since important signal information may also be attenuated, notch filtering should be used as a last resort; one should try to identify and eliminate the source of interference instead. In trying to identify 60-Hz sources to eliminate or minimize their effect, it is sometimes useful to use a dummy source, such as a fixed 100-kW resistor attached to the electrodes. An amplifier output represents only contributions from interfering sources. If noise can be reduced to an acceptable level (at least by a factor of 10 less than EEG signals) under this condition, one is likely to obtain uncontaminated EEG records. Different types of recording instruments obtain a temporary or permanent record of the EEG. The most common recording device is a pen or chart recorder (usually multichannel) that is an integral part of most commercially available EEG instruments. The bandwidth of clinical EEGs is relatively low (less than 40 Hz) and therefore within the frequency response capabilities of these devices. Recordings are on a long sheet of continuous paper (from a folded stack), fed past the moving pen at one of several selectable constant speeds. The paper speed translates into distance per unit time or cycles per unit time, to allow EEG interpreters to identify different frequency components or patterns within the EEG. Paper speed is selected according to the monitoring situation at hand: slow speeds (10 mm/s) for observing the spiking characteristically associated with seizures and faster speeds (up to 120 mm/s) for the presence of individual frequency bands in the EEG. In addition to (or instead of) a pen recorder, the EEG may be recorded on a multichannel frequency modulated (FM) analog tape recorder. During such recordings, a visual output device such as an oscilloscope or video display is necessary to allow visual monitoring of signals, so that corrective action (reapplying the electrodes and so on) can take place immediately if necessary.

Sophisticated FM cassette recording and playback systems allow clinicians to review long EEG recordings over a greatly reduced time, compared to that required to flip through stacks of paper or observe recordings as they occur in real time. Such systems take advantage of time compensation schemes, whereby a signal recorded at one speed (speed of the tape moving past the recording head of the cassette drive) is played back at a different, faster speed. The ratio of playback to recording speed is known, so the appropriate correction factor can be applied to played-back data to generate a properly scaled video display. A standard ratio of 60:1 is often used. Thus, a trained clinician can review each minute of real-time EEG in 1 s. The display appears to be scrolled at a high rate horizontally across the display screen. Features of these instruments allow the clinician to freeze a segment of EEG on the display and to slow down or accelerate tape speed from the standard playback as needed. A time mark channel is usually displayed as one of the traces as a convenient reference (vertical “tick” mark displayed at periodic intervals across the screen). Computers can also be recording devices, digitizing (converting to digital form) one or several amplified EEG channels at a fixed rate. In such sampled data systems, each channel is repeatedly sampled at a fixed time interval (sample interval) and this sample is converted into a binary number representation by an A/D converter. The A/D converter is interfaced to a computer system so that each sample can be saved in the computer’s memory. A set of such samples, acquired at a sufficient sampling rate (at least two times the highest frequency component in the sampled signal), is sufficient to represent all the information in the waveform. To ensure that the signal is band-limited, a low-pass filter with a cutoff frequency equal to the highest frequency of interest is used. Since physically realizable filters do not have the ideal characteristics, the sampling rate is usually greater than two times the filter’s cutoff frequency. Furthermore, once converted to a digital format, digital filtering techniques can be used. On-line computer recordings are only practical for short-term recordings or for situations in which the EEG is immediately processed. This limitation is primarily due to storage requirements. For example, a typical sampling rate of 128 Hz yields 128 new samples per second that require storage. For an 8-channel recording, 1,024 samples are acquired per second. A 10-minute recording period yields 614,400 data points. Assuming 8bit resolution per sample, over 0.5 megabyte (MB) of storage is required to save the 10-minute recording. Processing can consist of compression for more efficient storage (with associated loss of total information content), as in data record or epoch averaging associated with evoked responses, or feature extraction and subsequent pattern recognition, as in automated spike detection in seizure monitoring.

Frequency Analysis of the EEG In general, the EEG contains information regarding changes in the electrical potential of the brain obtained from a given set of recording electrodes. These data include the characteristic waveform with its variation in amplitude, frequency, phase, etc. and the occurrence of brief electrical patterns, such as spindles. Any analysis procedure cannot simultaneously provide information regarding all of these variables. Consequently, the selection of any analytic technique will emphasize changes in one particular variable at the expense of the others. This observation is extremely important if one is to properly interpret the results obtained by any analytic technique. In this chapter, special attention is given to frequency analysis of the EEG. In early attempts to correlate the EEG with behavior, analog frequency analyzers were used to examine single channels of EEG data. Although disappointing, these initial efforts did introduce the utilization of frequency analysis to study gross brain wave activity. Although, power spectral analysis, i.e., the magnitude square of Fourier transform, provides a quantitative measure of the frequency distribution of the EEG, it does so as mentioned above, at the expense of other details in the EEG such as the amplitude distribution, as well as the presence of specific patterns in the EEG. The first systematic application of power spectral analysis by general-purpose computers was reported in 1963 by Walter; however, it was not until the introduction of the fast Fourier transform (FFT) by Cooley and Tukey in the early 1970s that machine computation of the EEG became commonplace. Although an individual FFT is ordinarily calculated for a short section of EEG data (e.g., from 1 to 8 s epoch), such segmentation of a signal with subsequent averaging over individual modified periodograms has been shown to provide a consistent estimator of the power spectrum, and an extension of this technique, the compressed spectral array, has been particularly useful for computing EEG spectra over long periods of time. A detailed review of the

development and use of various methods to analyze the EEG is provided by Givens and Redmond [1987]. Figure 115.3 provides an overview of the computational processes involved in performing spectral analysis of the EEG, i.e., including computation of auto and cross spectra [Bronzino, 1984]. It is to be noted that the power spectrum is the autocorrellogram, i.e., the correlation of the signal with itself. As a result, the power spectrum provides only magnitude information in the frequency domain; it does not provide any data regarding phase. The power spectrum is computed by:

P ( f ) = Re 2[ X ( f )] + Im 2[ X ( f )]

(115.1)

where X(f ) is the Fourier transform of the EEG. Power spectral analysis not only provides a summary of the EEG in a convenient graphic form, but also facilitates statistical analysis of EEG changes which may not be evident on simple inspection of the records. In addition to absolute power derived directly from the power spectrum, other measures calculated from absolute power have been demonstrated to be of value in quantifying various aspects of the EEG. Relative power expresses the percent contribution of each frequency band to the total power and is calculated by dividing the power within a band by the total power across all bands. Relative power has the benefit of reducing the intersubject variance associated with absolute power that arises from intersubject differences in skull and scalp conductance. The disadvantage of relative power is that an increase in one frequency band will be reflected in the calculation by a decrease in other bands; for example, it has been reported that directional shifts between high and low frequencies are associated with changes in cerebral blood flow and metabolism. Power ratios between low (0–7 Hz) and high (10–20 Hz) frequency bands have been demonstrated to be an accurate estimator of changes in cerebral activity during these metabolic changes. Although the power spectrum quantifies activity at each electrode, other vari- FIGURE 115.3 Block diaables derivable from FFT offer a measure of the relationship between activity gram of measures determined recorded at distinct electrode sites. Coherence (which is a complex number), cal- from spectral analysis. culated from the cross-spectrum analysis of two signals, is similar to cross-correlation in the time domain. The magnitude squared coherence (MSC) values range from 1 to 0, indicating maximum or no synchrony, respectively, and are independent of power. The temporal relationship between two signals is expressed by phase, which is a measure of the lag between two signals for common frequency components or bands. Phase is expressed in units of degrees, 0° indicating no time lag between signals or 180° if the signals are of opposite polarity. Phase can also be transformed into the time domain, giving a measure of the time difference between two frequencies. Cross spectrum is computed by:

Cross spectrum = X ( f ) Y *( f )

(115.2)

where X(f ), Y(f ) are Fourier transforms and * indicates complex conjugates and coherence is calculated by

Coherence =

Cross spectrum

(115.3)

PX ( f ) - PY ( f ) Since coherence is a complex number, the phase is simply the angle associated with the polar expression of that number. MSC and phase represent measures that can be employed to investigate the cortical interactions of cerebral activity. For example, short (intracortical) and long (cortico-cortical) pathways have been proposed

as the anatomic substrate underlying the spatial frequency and patterns of coherence. Therefore, discrete cortical regions linked by such fiber systems should demonstrate a relatively high degree of synchrony, whereas the time lag between signals, as represented by phase, quantifies the extent to which one signal leads another.

Nonlinear Analysis of the EEG As mentioned earlier, the EEG has been studied extensively using signal-processing schemes, most of which are based on the assumption that the EEG is a linear, gaussian process. Although linear analysis schemes are computationally efficient and useful, they only utilize information retained in the autocorrelation function (i.e., the second-order cumulant). Additional information stored in higher-order cumulants is therefore ignored by linear analysis of the EEG. Thus, while the power spectrum provides the energy distribution of a stationary process in the frequency domain, it cannot distinguish nonlinearly coupled frequencies from spontaneously generated signals with the same resonance condition [Nikias and Raghvveer, 1987]. There is evidence showing that the amplitude distribution of the EEG often deviates from gaussian behavior. It has been reported, for example, that the EEG of humans involved in the performance of mental arithmetic task exhibits significant nongaussian behavior. In addition, the degree of deviation from gaussian behavior of the EEG has been shown to depend to the behavioral state, with the state of slow-wave sleep showing less gaussian behavior than quiet waking, which is less gaussian than rapid eye movement (REM) sleep [Ning and Bronzino, 1989a,b]. Nonlinear signal-processing algorithms such as bispectral analysis are therefore necessary to address nongaussian and nonlinear behavior of the EEG in order to better describe it in the frequency domain. But what exactly is the bispectrum? For a zero-mean, stationary process {X(k)}, the bispectrum, by definition, is the Fourier transform of its third-order cumulant (TOC) sequence: a

B (w1 , w 2 ) =

å å C (m, n)e (

- j w1m + w 2n )

(115.4)

m = -a n = -a

The TOC sequence {C(m, n)} is defined as the expected value of the triple product

{

}

C (m, n ) = E X (k )X (k + m )X (k + n )

(115.5)

If process X(k) is purely gaussian, then its third-order cumulant C(m, n) is zero for each (m, n), and consequently, its Fourier transform, the bispectrum, B(w1, w2) is also zero. This property makes the estimated bispectrum an immediate measure describing the degree of deviation from gaussian behavior. In our studies [Ning and Bronzino, 1989a,b], the sum of magnitude of the estimated bispectrum was used as a measure to describe the EEG’s deviation from gaussian behavior, that is,

D =

å B(w , w ) 1

(115.6)

(w1w2 )

Using bispectral analysis, the existence of significant quadratic phase coupling (QPC) in the hippocampal EEG obtained during REM sleep in the adult rat was demonstrated [Ning and Bronzino, 1989a,b, 1990]. The result of this nonlinear coupling is the appearance, in the frequency spectrum, of a small peck centered at approximately 13 to 14 Hz (beta range) that reflects the summation of the two theta frequency (i.e., in the 6to 7-Hz range) waves. Conventional power spectral (linear) approaches are incapable of distinguishing the fact that this peak results from the interaction of these two generators and is not intrinsic to either. To examine the phase relationship between nonlinear signals collected at different sites, the cross-bispectrum is also a useful tool. For example, given three zero-mean, stationary processes ”xj(n)j = 1, 2, 3}, there are two conventional methods for determining the cross-bispectral relationship, direct and indirect. Both methods first divide these three processes into M segments of shorter but equal length. The direct method computes the

Fourier transform of each segment for all three processes and then estimates the cross-bispectrum by taking the average of triple products of Fourier coefficients over M segments, that is, M

1 Bx1x 2x 3 (w1 , w 2 ) = X m (w )X m (w )X m * (w1 + w 2 ) M m =1 1 1 2 2 3

(115.7)

where Xmj (w) is the Fourier transform of the mth segment of {xj(n)}, and * indicates the complex conjugate. The indirect method computes the third-order cross-cumulant sequence for all segments:

C xm1x 2x 3 (k , l ) =

å x (n)x (n + k )x (n + l ) m 1

m 2

m 3

(115.8)

n et

where t is the admissible set for argument n. The cross-cumulant sequences of all segments will be averaged to give a resultant estimate: M

1 C x1x 2x 3 (k , l ) = C m (k , l ) M m = 1 x1x 2x 3

(115.9)

The cross-bispectrum is then estimated by taking the Fourier transform of the third-order cross-cumulant sequence: a

Bx1x 2x 3 (w1 , w 2 ) =

å åC

k = -a

- j w1k + w 2l )

x1x 2x 3

(k , l ) (

(115.10)

l = -a

Since the variance of the estimated cross-bispectrum is inversely proportional to the length of each segment, computation of the cross-bispectrum for processes of finite data length requires careful consideration of both the length of individual segments and the total number of segments to be used. The cross-bispectrum can be applied to determine the level of cross-QPC occurring between {x1(n)} and {x2(n)} and its effects on {x3(n)}. For example, a peak at Bx1x2x3(w1, w2) suggests that the energy component at frequency w1 + w2 of {x3(n)} is generated due to the QPC between frequency w1 of {x1(n)} and frequency w2 of {x2(n)}. In theory, the absence of QPC will generate a flat cross-bispectrum. However, due to the finite data length encountered in practice, peaks may appear in the cross-bispectrum at locations where there is no significant cross-QPC. To avoid improper interpretation, the cross-bicoherence index, which indicates the significance level of cross-QPC, can be computed as follows:

bic x1x 2x 3 (w1 , w 2 ) =

Bx1x 2x 3 (w1 , w 2 ) Px1 (w1 )Px 2 (w 2 )Px 3 (w1 + w 2 )

(115.11)

where Pxj(w) is the power spectrum of process {xj(n)}. The theoretical value of the bicoherence index ranges between 0 and 1, i.e., from nonsignificant to highly significant. In situations where the interest is the presence of QPC and its effects on {x(n)}, the cross-bispectrum equations can be modified by replacing {x1(n)} and {x3(n)} with {x(n)} and {x2(n)} with {y(n)}, that is, M

Bxyz (w1 , w 2 ) =

1 X m (w1 ) Y m (w 2 ) X m * (w1 + w 2 ) M m =1

(115.12)

FIGURE 115.4 A and B represent the averaged power spectra of 80 4-s epochs of REM sleep (sampling rate = 128 Hz) obtained from hippocampal CA1 and the dentate gyrus, respectively. Note that both spectra exhibit clear power peaks at 7Hz (theta) and 14-Hz (beta) frequencies. C and D represent the bispectrum of these same epochs from CA1 and the dentate gyrus, respectively. Computation of the bicoherence index at 7 Hz shows significant quadratic phase coupling at this frequency, indicating that the 14-Hz peak is not spontaneously generated, but results from quadratic phase coupling.

In theory, both methods will lead to the same cross-bispectrum when data length is infinite. However, with finite data records, direct and indirect methods generally lead to cross-bispectrum estimates with different shapes (Fig. 115.4). Therefore, like power spectrum estimation, users have to choose an appropriate method to extract the information desired.

Topographic Mapping Computerized tomography (CT) and magnetic resonance imaging (MRI) have demonstrated the impact of spatial displays on data interpretation and analysis. Similarly, mapping techniques have been applied to electrophysiologic data to depict the spatial information available from multielectrode recordings. This effort has been assisted by the development and implementation of low-cost, high-resolution graphic displays on microcomputer systems. The data are frequently presented as two-dimensional topographic color maps [Zappulla, 1991]. In the time domain, color values depict the changes in potential across the scalp at each time point. This is exemplified by mapping peaks of an evoked potential or the spatial distribution of an epileptic spike. Temporal changes in the spatial distribution of voltage can be presented graphically as a series of maps constructed at adjacent time points or by cartooning the topographic maps over the time interval of interest. In the frequency domain, color coding can be used to spatially map power, covariance, and phase values. These maps may be constructed for the broadband activity or for selective frequency components. Unlike CT and MRI displays where each picture element or pixel value represents real data, most of the pixels comprising an EEG and ER topographic map consist of interpolated values. This is because the activity

from a finite number of electrodes represents a sampling of the spatial activity over the scalp. Consequently, the remaining values of the map located outside the electrode positions must be estimated from this sampled activity. One technique for deriving these values is linear interpolation. In the case of a four-point interpolation, the map is divided into boxes whose corners are defined by real data. The interpolated points within the boxes are calculated by the weighted sum of the four real data points, based on their distance from the interpolated point. Although linear interpolation is the most popular technique, polynomial regression and surface spline interpolation have been employed as alternative procedures. These methods reduce the discontinuities inherent in linear interpolation and offer better estimates of extreme values. Polynomial regression has the additional advantage of permitting quantitative comparisons between maps by taking into account the topographic information represented in the map. Maps can be presented in any of several projections to assist in interpretation [Zappulla, 1991]. The most common projection is the top view which presents the spatial distribution of variables from all leads simultaneously. Lateral, posterior, and anterior projections highlight focal areas of interest. Although mapping presents a method by which spatial information can be efficiently communicated, it is important to be alert to the artifacts that can arise from map construction and manipulation. Topographic spatial artifacts that can lead to misinterpretation include ring enhancement around a spike using source-derivation references, spatial aliasing arising from linear interpolation which causes maximal activity to be mapped at electrode sites, the enhancement of activity away from the midline, and the attenuation of midline activity on amplitude asymmetry maps (centrifugal effect). The quality of the spatial information derivable from EEG recordings depends upon the number of recording electrodes, the choice of the reference electrode, and the conductive properties of intracranial and extracranial structures. The localization of cortical activity from scalp recordings assumes that the potentials recorded from the scalp reflect cortical activity generated in proximity to the recording electrode. Therefore, the greater the density of recording electrodes, the more accurate the estimate of the spatial distribution of scalp potentials and the localization of cortical generators. However, since the distance between the cortical source and recording electrode, as well as the low conductivity of the skull, results in a selective attenuation of small dipole fields, most available EEG information can be obtained with an average scalp-electrode spacing of 2 cm. Topographic maps are constructed from monopolar electrodes referenced to a common cephalic (linked ears or mandible, chin and nose) or noncephalic (linked clavicles or a balanced sternum-vertebra) electrode. Although the reference electrode should be free of any EEG activity, in practice most cephalic electrodes contain some EEG activity, while noncephalic electrodes are a potential source of EKG or muscle activity. Differential amplification of an EEG-contaminated reference electrode can decrease or cancel similar activity in neighboring electrodes, while at electrodes distant from the reference, the injected activity will be present as a potential of opposite polarity. Similarly, noncerebral potentials can be injected into scalp electrodes and misinterpreted as cerebral activity. Therefore, a nonneutral reference electrode can result in misleading map configurations. Several techniques have been applied to circumvent this problem. The construction of multiple maps using several different references can sometimes assist in differentiating active and reference electrode activity. This can be accomplished by acquiring serial EEG records using different references. Alternatively, various references can be acquired simultaneously during acquisition, and various montages can be digitally reconstructed, post hoc. A more computationally intensive method for localizing a source at an electrode involves calculating the local source activity at any one electrode based on the average activity of its neighbors, weighted by their distance from the source. The technique has the advantage of suppressing potentials that originate outside the measurement area and weighing factors for implementing source deviation techniques for each of the electrodes in the 10–20 system are available. Another reference technique, the average head reference, uses the average activity of all active electrodes as the common reference. In this approach, the activity at any one electrode will vary depending upon the activity at the site of the reference electrode, which can be anywhere on the recording montage. Therefore, for N number of recording electrodes, each being a potential reference, there are N – 1 possible voltage measurements at each instant of time for each electrode. Maps constructed using the average head reference represent a unique solution to the problem of active reference electrodes in that the average reference produces an amplitude-weighted reference-free map of maximal and minimal field potentials. Power maps constructed from the average reference

best depict the spatial orientation of the generating field, and the areas with extreme values are closest to the generating processes [Zappulla, 1991]. Topographical maps represent an efficient format for displaying the extensive amount of data generated by quantitative analysis. However, for reasons discussed above, the researcher and clinician must be cautious in deriving spatial and functional conclusions from mapped data. Although the replicability of map configurations across subjects or experimental conditions may represent a useful basis for experimental and diagnostic classification, judgments concerning the localization of cortical generators or functional localization of cerebral activity are less certain and more controversial. Research continues on defining models and validating assumptions that relate scalp potentials to cortical generators in an attempt to arrive at accurate mathematical solutions that can be applied to mapping functions.

Defining Terms Bispectra: Computation of the frequency distribution of the EEG exhibiting nonlinear behavior. Cross spectra: Computation of the energy in the frequency distribution of two different electrical signals. Electroencephalogram (EEG): Recordings of the electrical potentials produced by the brain. Fast Fourier transform (FFT): Algorithms that permit rapid computation of the Fourier transform of an electrical signal, thereby representing it in the frequency domain. Magnitude squared coherence (MSC): A measure of the degree of synchrony between two electrical signals at specific frequencies. Power spectral analysis: Computation of the energy in the frequency distribution of an electrical signal. Quadratic phase coupling: A measure of the degree to which specific frequencies interact to produce a third frequency.

Related Topic 35.1 Maxwell Equations

References L. Axel, P.H. Arger, and R. Zimmerman, “Applications of computerized tomography to diagnostic radiology,” Proceedings of the IEEE, vol. 71, no. 3, p. 293, March 1983. K.R. Castleman, Digital Image Processing, Englewood Cliffs, N.J.: Prentice-Hall, 1979. R.E. Coleman, “Single photon emission computed tomography and positron emission tomography,” Cancer, vol. 67 (4 Suppl.), pp. 1261–1270, Feb. 1991.

P.M. Hurculak, “Positron emission tomography,” Canadian Journal of Medical Radiation Technology, vol. 18, no. 1, March 1987. G.F. Knoll, “Single-photon emission computed tomography,” Proceedings of the IEEE, vol. 71, no. 3, p. 320, March 1983. A. Macovski, Medical Imaging Systems, Englewood Cliffs, N.J.: Prentice-Hall, 1983. H.R. Schelbert, “Future perspectives: Diagnostic possibilities with positron emission tomography,” Roentgen Blaetter, vol. 43, no. 9, pp. 384–390, Sept. 1990. G.L. Wolf and C. Popp, NMR, A Primer for Medical Imaging, Thorofare, N.J.: Slack, Inc., 1984.

Further Information The journal IEEE Transactions on Medical Imaging describes advances in imaging techniques and image processing. Investigative Radiology, published by the Association of University Radiologists, emphasizes research carried out by hospital-based physicists and engineers. Radiology, published by the North American Society of Radiologists, contains articles which emphasize clinical applications of imaging technology. Diagnostic Imaging, publishing by Miller Freeman, Inc., is a good source of review articles and information on the imaging marketplace.

116.2

Ultrasound

Leon A. Frizzell Ultrasound, acoustic waves at frequencies higher than those audible by humans, has developed over the past 35 years into an indispensable clinical diagnostic tool. Currently, ultrasound is used to image most parts of the body. More than half of all pregnant women in the United States are examined with ultrasound. This widespread utilization has resulted from ultrasound’s proven clinical utility for imaging soft tissues compared to more expensive imaging techniques. The development of ultrasound, particularly for fetal examinations, has also been fostered by its safety record; no case of an adverse biological effect induced by diagnostic ultrasound has ever been reported in humans [AIUM, 1988]. Diagnostic ultrasound systems are used primarily for soft tissue imaging, motion detection, and flow measurement. Except for some Doppler instruments, these systems operate in a pulse-echo mode. A brief summary of some of the fundamentals of acoustic wave propagation and the principles of ultrasound imaging follows.

Fundamentals of Acoustics Unlike electromagnetic waves, acoustic waves require a medium for propagation. The acoustic wave phenomenon causes displacement of particles (consisting of many molecules), which results in pressure and density changes within the medium. For a traveling sinusoidal wave, the variation in acoustic pressure (the difference between the total and ambient pressure), excess density, particle displacement, particle velocity, and particle acceleration can all be represented by the form

p = P e –ax cos(wt – kx)

(116.1)

for a wave propagating in the positive x direction, where p is the pressure (or one of the other parameters listed above), P is its amplitude, w is the angular frequency, and w = 2pf where f is the frequency in hertz, k is the propagation constant and k = w/c where c is the propagation speed, a is the attenuation coefficient, and t is the time. The wave can experience significant attenuation, as represented by the exponential decay of amplitude with distance, during propagation in tissues. The attenuation coefficient varies greatly among tissues [Goss et al., 1978, 1980; Haney and O’Brien, 1986] but is low for most body fluids, much higher for solid tissues, and very high for bone and lung (see Table 116.1). The skin depth is the distance that the wave can propagate before being attenuated to e–1 of its original amplitude and is thus simply the inverse of the attenuation coefficient.

TABLE 116.1 Approximate Ultrasonic Attenuation Coefficient, Speed, and Characteristic Impedance for Water and Selected Tissues at 3.5 MHz Tissue Water Amniotic fluid Blood Liver Muscle Bone Lung

Attenuation Coefficient (m–1)

Speed (m/s)

Characteristic Impedance (106 Pa s/m)

0.2 0.7 7 35 50 800 1000

1520 1510 1550 1580 1560 3360 340

1.50 1.51 1.60 1.74 1.72 5.70 0.25

Ultrasound is typically used to image soft body tissues such as liver, but the sound beam often travels through fluids, for example, through amniotic fluid when imaging the fetus. Generally, bone and lung are not imaged with ultrasound. The attenuation processes include absorption, which is the conversion of acoustic energy to heat, and scattering, which will be addressed later. The attenuation increases roughly linearly with frequency in the 2- to 10-MHz range typically used for medical imaging. This range represents a compromise between increased penetration at lower frequencies (because of decreased attenuation) and improved resolution associated with higher frequencies as discussed below. Thus, the lower frequencies are used when greater penetration is required, such as for fetal imaging in the obese patient, and higher frequencies for lesser penetration, such as the examination of peripheral vascular flow. When an acoustic wave impinges on an interface between two media of different specific acoustic impedance, a portion of the incident energy is reflected. For normal incidence on an infinite plane interface, the pressure reflection coefficient is given by [Kinsler et al., 1982]

R =

z 2 - z1 z 2 + z1

(116.2)

where z1 and z2 are the specific acoustic impedance of the incident and transmitting media, respectively. For a plane wave the specific acoustic impedance is equal to the characteristic impedance which is the product of the density and acoustic speed in the medium (see Table 116.1). The speed is dependent upon the density and the elastic properties of the medium. Thus, at an interface between media exhibiting different densities or elastic properties, i.e., compressibility, some acoustic energy will be reflected. Although the reflection coefficient at an interface between muscle and bone is large (approximately 0.54) the reflection coefficient between two soft tissues such as liver and muscle is quite small (approximately 0.006). Reflection at oblique incidence obeys Snell’s law in the same way it applies to electromagnetic waves. In addition to the specular reflection that occurs at an interface between two media of different specific acoustic impedance as described above (where any curvature along the interface is negligible over distances comparable to a wavelength), energy may also be scattered in all directions by inhomogeneities in the medium. An acoustic image is formed by using this scattered energy as well as specular reflections. The fraction of the incident energy reflected or scattered is very small for soft tissues. Although it is convenient to consider plane waves of infinite lateral extent, as was done above, real sources generate finite beams of ultrasound. These sources may be unfocused, but for the typical diagnostic system they are focused. Figure 116.4 shows the acoustic field from a typical focused source. The source consists of a piezoelectric transducer which converts electrical to acoustic energy and vice versa. Most transducers for medical applications are made from ceramic materials such as a lead zirconate titanate (PZT) mixture. For a circular aperture these may be circular disks with a plano-concave lens mounted in front to produce spherical focusing. Alternatively, the transducer itself may be a spherical segment that produces a focused field without a lens. Some probes utilize electronic focusing methods. Such a phased array probe consists of many individual elements which can be excited with signals having a controlled delay with respect to one another such that the

FIGURE 116.4 Cross section of a typical focused circular ultrasound source of aperture diameter D and focal length F, showing the focused beam of lateral beam width at the focus DL.

signals constructively interfere at the desired focal region. At a receiver the signals are combined with delays associated with various elements to provide reinforcement of the signals from a receiving focal region. The 3 db lateral beam width DL is directly dependent upon the wavelength l and focal length F and inversely related to the aperture diameter (diameter of the transducer) D [Kino, 1987]:

DL = 1.02

Fl D

(116.3)

Because f l = c, the higher the frequency the smaller is l and DL. The smaller DL, the better the lateral resolution near the focus, but the beam spread is greater with distance from the focus. Thus, the strength of the focusing varies among transducers so that the user may choose very good resolution over a short region or somewhat poorer resolution that is maintained over a greater depth. With phased array transducers, the focal region can be varied dynamically to optimize lateral resolution at all distances.

Principles of Pulse-Echo Ultrasound Ultrasound imaging usually employs frequencies in the 2- to 10-MHz range, though some of the new intravascular probes use higher frequencies. Images are formed by using a transducer within a probe to generate a short pulse (typically on the order of 1 ms in duration) of ultrasound which is propagated through the tissue. A portion of the energy in this pulse is reflected back toward the transducer from specular reflectors and from scatterers in the tissue. These acoustic echoes, with amplitudes much lower than the transmitted pulse, are converted by the transducer to electrical signals which are converted to a (rectified) video signal, amplified by a time gain controlled amplifier, and displayed. The A-mode display is rarely used but simply involves display of the received echoes as amplitude versus time of arrival. The time of arrival is related by the wave speed to the tissue depth from which the echo returns, i.e., d = ct/2. Figure 116.5 provides a very simple representation of this process where the A-mode display associated with specular reflection from three different interfaces is illustrated. For clinical imaging the interfaces would not necessarily be perpendicular to the axis of the sound beam, and there would be a continuum of echoes, a continuous received signal, due to energy backscattered from within the tissues. Since the ultrasound pulse is attenuated as it propagates, all ultrasonic imaging systems use a logarithmic variation of amplifier gain with time to compensate the exponential attenuation of the tissue. Thus, echoes from structures reflecting or backscattering the same fraction of the incident signal will have the same amplitude after passing through the time gain controlled amplifier. A B-mode display is typically used for ultrasound imaging. It involves display of the echoes at various brightness or gray levels corresponding to their amplitude. A two-dimensional B-mode display involves movement of the transducer (manually or automatically), movement of a mirror to change the direction of the field (automatically), or movement of the ultrasound beam directly (electrically) such that it scans a plane through the body. Figure 116.6 provides a simplified representation (again, echoes are shown as arising from interfaces only) of the formation of a B-mode image. The direction of the beam is monitored so that the received signals along each path are placed in their correct location on the display. Typically, the orientation information and echoes are processed by a digital scan converter for appropriate display of the two-dimensional image on © 2000 by CRC Press LLC

FIGURE 116.5 (a) The transmitted pulse (heavy wave) and echoes from reflecting structures; (b) the resulting A-mode display.

FIGURE 116.6 (a) The transmitted pulse paths for a rotating transducer probe; (b) the resulting two-dimensional B-mode display of echoes from the interfaces only.

a cathode ray tube in the standard format used for television picture display. Most B-mode systems in use today create an image in 0.1 s or less, so that the image is displayed in real-time for viewing of moving structures, such as structures in the heart or the fetus moving within the womb. This is not possible with the typical magnetic resonance or computed tomography system.

Many systems now use digital processing to enhance portions of the image. For example, it is possible to emphasize the large amplitude, small amplitude, or midrange signals. It is also possible to perform a more sophisticated analysis to enhance edges. Many specialty probes have been designed for intracavitary examination. Examples include examination of the fetus with a vaginal probe, the prostate with a rectal probe, and blood vessel walls with intravascular probes. The intracavitary probe offers the advantage of decreasing the distance from the transducer to the tissue of interest and thus decreasing attenuation such that higher frequencies can be used for greater resolution. The lateral resolution of a focused probe is improved with frequency as discussed in the preceding subsection, but the axial (along the ultrasound beam) resolution also improves with frequency. The shorter the transmitted pulse, the better the axial resolution. Shorter pulses are generated by sources with a larger bandwidth, which corresponds to a higher center frequency when the sources have bandwidths which are approximately the same fraction of the center frequency. The use of ultrasound for motion detection and measurement has increased tremendously in recent years. Most of these systems use the Doppler principle, but some use time domain detection. In Doppler detection, if the ultrasound is reflected from a target moving at some speed vt toward (away from) the source at an angle q with respect to the beam axis, the frequency of the transmitted signal f is shifted up (down) by an amount fD, the Doppler shift, according to the following relation:

fD =

2 fv t cos q c

(116.4)

In principle a measurement of fD, when f, c, and q are known, will yield the speed of the target vt. However, it is often difficult to determine q because the angle the transducer axis makes with a blood vessel, for example, is often unknown. Even when that angle is known, the flow is not necessarily along the direction of the vessel at every location and for all times. Time domain detection of motion, by measuring the movement of specific echoes from one pulse to another, is a recently developed alternative to Doppler detection that is not currently widely used. For many years duplex systems, which provide both a two-dimensional image and a Doppler signal, showing the change of target speed with time, from a particular selected target area, have been in wide use. More recently, color flow imaging has been employed, which provides a two-dimensional color (typically red or blue) image of flow toward and away from the transducer superimposed on the gray scale image of stationary tissue structures. For these systems the speed, whether from Doppler or time domain detection schemes, is indicated by color saturation, hue, or luminance. These systems have proven very valuable for detecting the existence of flow in a region, detecting obstructions to flow and the turbulence associated with this, detecting reduced flow, and so on. Some other systems add to the color flow image a display of speed versus time for a region that is defined by a user-movable box.

Future Developments It seems clear that the continuing development of intracavitary transducers, particularly for intravascular imaging, and the use of ultrasound intraoperatively will lead to more high-frequency commercially available probes that will produce better resolution images for these applications. The development of useful threedimensional ultrasonic imaging is progressing rapidly and should immediately improve the measurement of tissue volumes [Gilja et al., 1995].

Defining Terms A-mode display: Returned ultrasound echoes displayed as amplitude versus depth into the body. B-mode display: Returned ultrasound echoes displayed as brightness or gray scale levels corresponding to the amplitude versus depth into the body.

Color flow imaging or color Doppler: Two-dimensional image showing color-coded flow toward and away from the transducer displayed with the two-dimensional gray scale image of stationary targets. Duplex ultrasound: Simultaneous display of speed versus time for a chosen region and the two-dimensional B-mode image. Pulse-echo ultrasound: Using a probe containing a transducer to generate a short ultrasound pulse and receive echoes of that pulse, associated with specular reflection from interfaces between tissues or scattering from inhomogeneities within the tissue, to form a display of the tissue backscatter properties. Two-dimensional B-mode display: Echoes from a transducer, or beam, scanned in one plane displayed as brightness (or gray scale) versus location for the returned echo to produce a two-dimensional image.