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Scattering Parameters
in RF and Microwave Circuit
Analysis and Design
6631 Book_R1.indb 1 4/21/16 3:17 PM

For a listing of recent titles in the
Artech House Microwave Library,
turn to the back of this book.
6631 Book_R1.indb 2 4/21/16 3:17 PM

Scattering Parameters
in RF and Microwave Circuit
Analysis and Design
Janusz A. Dobrowolski
artechhouse.com
6631 Book_R1.indb 3 4/21/16 3:17 PM

Library of Congress Cataloging-in-Publication Data
A catalog record for this book is available from the U.S. Library of Congress
British Library Cataloguing in Publication Data
A catalog record for this book is available from the British Library.
ISBN-13: 978-1-63081-093-1
Cover design by John Gomes
© 2016 Artech House
685 Canton St.
Norwood, MA
All rights reserved. Printed and bound in the United States of America. No part of this
book may be reproduced or utilized in any form or by any means, electronic or mechanical,
including photocopying, recording, or by any information storage and retrieval system,
without permission in writing from the publisher.
All terms mentioned in this book that are known to be trademarks or service marks
have been appropriately capitalized. Artech House cannot attest to the accuracy of this
information. Use of a term in this book should not be regarded as affecting the validity
of any trademark or service mark.
10 9 8 7 6 5 4 3 2 1
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v
Contents
Preface xv
1 Introduction 1
References 5
2 Theory of Uniform Waveguides 7
2.1 Modal Electromagnetic Fields 8
2.2 Power Transmitted in a Waveguide 9
2.3 Characteristic Impedance 11
2.4 Normalization of Waveguide Voltage and
Current 13
2.5 Transmission Line Equivalent Circuit of a
Waveguide 14
References 16
3 Theory of Transmission Lines 17
3.1 Lumped Element Circuit Model of a
Transmission Line 18
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vi Scattering Parameters in RF and Microwave Circuit Analysis and Design
3.2 Voltage and Current Wave Propagation in a
Transmission Line 18
3.3 Terminated Transmission Line 22
3.4 Terminated Transmission Line Special Cases 25
References 26
4 Wave Variables and the Scattering Matrix 27
4.1 Voltage Traveling Waves and the Scattering
Matrix 28
4.1.1 Physical Interpretation of Scattering
Parameters 29
4.1.2 A Shift in Reference Plane 31
4.1.3 Scattering Matrix Properties 33
4.1.4 Example 4.1 34
4.1.5 Conversions Between the Scattering Matrix
and Other Matrix Descriptions
of Microwave Networks 35
4.2 Normalized Voltage Traveling Waves and the
Generalized Scattering Matrix 35
4.2.1 Physical Interpretation of Generalized
Scattering Parameters 36
4.2.2 Example 4.2 37
4.3 Traveling Wave Intensities and the True
Scattering Matrix 38
4.3.1 Example 4.3 40
4.4 Pseudowaves and the Pseudoscattering Matrix 41
4.4.1 Pseudoscattering Matrix Properties 43
4.4.2 Example 4.4 45
4.4.3 Conversions Between the Pseudoscattering
Matrix and Other Matrix Descriptions of
Microwave Networks 47
4.4.4 Change of Reference Impedances 50
4.4.5 One-Port Reference Impedance
Transformation 52
4.4.6 Multiport Network Reference Impedance
Transformation 53
4.4.7 Two-Port Reference Impedance
Transformation 54
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Contents vii
4.4.8 Three-Port to Two-Port Network
Scattering Matrix Transformation 55
4.4.9 Scattering Matrix of the Cascade of
Two-Port Networks 59
4.4.10 Scattering Matrix of an Embedded
Multiport Network 61
4.5 Generalized Multiport Network Cascade Matrix 63
4.5.1 T-Matrix to S-Matrix and S-Matrix to
T-Matrix Transformation for Multiport
Networks with the Same Number of
Input and Output Ports (Balanced
Networks) 66
4.5.2 T-Matrix to S-Matrix and S-Matrix to
T-Matrix Transformation for Multiport
Networks with Different Numbers of
Input and Output Ports (Unbalanced
Networks) 68
4.6 Load Impedance 70
4.7 Power Waves and the Power Scattering Matrix 71
4.7.1 Physical Interpretation of Power Waves 73
4.7.2 Physical Interpretation of Power
Scattering Parameters 76
4.7.3 Example 4.5 78
4.7.4 Conversions Between Power Wave
Scattering Matrix and Other Matrix
Descriptions of Microwave Networks 80
4.7.5 Power Scattering Matrix Properties 82
4.7.6 Port Connections 83
References 85
5 Signal Analysis of Multiport Networks 87
5.1 Wave Relations for Basic Elements of Multiport
Networks 88
5.1.1 Signal Source 88
5.1.2 Load 90
5.1.3 Signal Source Available Power 92
5.2 Microwave Network Analysis Using Scattering
Parameters and Signal Flow Graphs 94
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viii Scattering Parameters in RF and Microwave Circuit Analysis and Design
5.3 Signal Analysis of Two-Port Networks 97
5.3.1 Transducer Power Gain 98
5.3.2 Example 5.1 99
5.3.3 Power Gain 100
5.3.4 Available Power Gain 101
5.3.5 Stability Consideration for Active
5.3.6 Maximum Power Gain 105
5.3.7 Constant Power Gain Circles 107
5.3.8 Constant Available Power Gain Circles 108
5.3.9 Insertion Loss 108
5.3.10 Voltage Gain 110
5.3.11 Voltage Transfer Gain 111
5.4 Multiport Network Analysis 112
5.4.1 Transducer Power Gain of Multiport
Networks 113
5.4.2 Power Gain of Multiport Networks 116
5.5 Multielement Multiport Network Analysis
Using Connection Scattering Matrix Approach 117
5.5.1 Transducer Power Gain of Multielement
Multiport Networks 120
5.5.2 Power Gain of Multielement
References 122
6 Mode Wave Variables and Mixed-Mode
Scattering Matrix of Differential Networks 123
6.1 Differential-Mode and Common-Mode
Definitions 124
6.2 Mode-Specific Waves and Impedances 126
6.3 Mixed-Mode Scattering Parameters 127
6.4 Transformation Between Standard-Mode and
Mixed-Mode Scattering Parameters 130
6.5 Generalized Mixed-Mode Pseudoscattering
Matrix 135
6.5.1 Example 6.1 146
6.5.2 Example 6.2 150
6.5.3 Example 6.3 152
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Contents ix
6.5.4 Example 6.4 155
6.5.5 Example 6.5 156
6.6 Mixed-Mode Cascade Matrix 160
References 166
7 Signal Analysis of Differential Multiport
Networks 169
7.1 Wave Relations for Basic Elements of
Differential Multiport Networks 170
7.1.1 Differential Signal Source 170
7.1.2 Differential Load 181
7.1.3 Differential Signal Source Available
Power 190
7.2 Signal Analysis of Differential Two-Port
Networks 192
7.2.1 Transducer Power Gain of Differential
Two-Ports 193
7.2.2 Power Gain of Differential Two-Ports 196
7.2.3 Available Power Gain of Differential
Two-Ports 197
7.2.4 Differential Amplifier Maximum
Power Gain 198
7.2.5 Differential Insertion Loss 199
7.2.6 Differential Voltage Gain 200
7.2.7 Differential Voltage Transfer Gain 200
7.3 Differential Multiport Network Analysis 201
7.3.1 Differential Transducer Power Gain of
7.3.2 Differential Power Gain of Multiport
Networks 209
7.4 Differential Multielement Multiport Network
Analysis Using Connection Scattering Matrix
Approach 211
7.4.1 Transducer Power Gain of Differential
Multielement Multiport Networks 215
7.4.2 Power Gain of Differential Multielement
References 217
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x Scattering Parameters in RF and Microwave Circuit Analysis and Design
8 Noise Wave Variables and the Scattering
Matrix 219
8.1 Noise Waves 220
8.1.1 Noise Power Waves 220
8.1.2 Noise Pseudowaves 221
8.2 Noise Wave Representation of Microwave
Networks 222
8.3 Other Noise Representations of Noisy Networks
and Their Transformations to Noise Wave
Parameters 225
8.3.1 Chain Matrix Noise Representation 225
8.3.2 Cascade Matrix Noise Representation 229
8.3.3 Impedance Matrix and Admittance
Matrix Noise Representations 233
8.4 Noise Models of Microwave Network Elements 236
8.4.1 Noise Wave Correlation Matrices of
Passive Multiport Networks 236
8.4.2 Example 8.1 238
8.4.3 Noise Correlation Matrices of Passive
Multiport Networks Embedded in
Lossy Waveguides 239
8.4.4 Noise Wave Correlation Matrices of
Active Two-Port Networks 241
8.4.5 Example 8.2 241
8.5 Two-Port-to-Three-Port Network Noise Wave
Transformation 245
8.6 Noise Wave Correlation Matrices of Embedded
8.6.1 Example 8.3 253
8.7 Deembedding Noise Wave Parameters of
Cascaded Noisy Two-Port Networks 256
References 258
9 Noise Analysis of Multiport Networks 261
9.1 Basic Relationships for Noisy Multiport
Networks 262
9.2 Classical Two-Port Network Noise Theory 263
9.3 Noise Factor of Two-Port Networks 266
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Contents xi
9.3.1 Constant Noise Factor Circles 268
9.4 Two-Port Network Noise Analysis Using Noise
Waves and Scattering Matrix 269
9.4.1 Example 9.1 273
9.5 Noise Analysis of Two-Port Networks Using
Noise Waves and Cascade (Transfer Scattering)
Matrix 274
9.5.1 Noise Wave Parameters of Cascade
Connected Two-Port Networks 277
9.6 Noise Analysis of Multielement Multiport
Networks Using Connection Scattering Matrix
Approach 278
9.6.1 Noise Factor of Multielement Multiport
Networks 281
9.6.2 Signal-to-Noise Ratio of Multielement
9.7 Noise Analysis of Multiport Networks 286
9.7.1 Noise Factor of Multiport Networks 288
9.7.2 Signal-to-Noise Ratio of Multiport
Networks 289
9.7.3 Example 9.2 290
9.7.4 Example 9.3 293
9.7.5 Example 9.4 294
References 296
10 Differential- and Common-Mode Noise Waves
and Correlation Matrices 297
10.1 Differential- and Common-Mode Noise Waves 298
10.2 Generalized Mixed-Mode Noise Wave
Correlation Matrix 306
10.3 Mixed-Mode Noise Wave Correlation Matrices
of Passive Networks 320
10.3.1 Mixed-Mode Noise Wave Correlation
Matrix of Passive Two-Port Networks 321
Matrix of a Balun 323
Matrix of Passive Four-Port Networks 325
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xii Scattering Parameters in RF and Microwave Circuit Analysis and Design
10.4 Mixed-Mode Noise Wave Correlation Matrices
of Active Differential Networks 327
References 329
11 Noise Analysis of Differential Networks 331
11.1 Noise Analysis of Differential Two-Port Networks
332
11.1.1 Noise Figure of Balanced Two-Port
Networks 332
11.1.2 Noise Figure of a Cascade of Balanced
11.1.3 Noise Figure of Fully Differential Two-
Port Networks 336
11.1.4 Noise Figure of Power-Splitting Balun 338
11.1.5 Noise Figure of Power Combine Balun
Excited by Differential Signal Source 340
11.1.6 Single-Ended Noise Source and a Balun
as a Source of Differential- and Common-
Mode Noise Waves 342
11.1.7 Noise Figure of Differential Amplifier in
Single-Ended Environment 344
11.2 Differential Two-Port Network Noise Analysis
Using Mixed-Mode Scattering Matrix 350
11.2.1 Mixed-Mode Noise of Differential
Signal Source 350
11.2.2 Differential Noise Figure of a Differential
Two-Port Network in Terms of Mixed-
Mode Scattering Parameters 353
11.3 Noise Analysis of Mixed-Mode Multielement
Multiport Networks Using Connection
Scattering Matrix Approach 355
11.3.1 Differential Noise Figure of Mixed-Mode
Multielement Multiport Networks 360
11.3.2 Differential Signal-to-Noise Ratio of
Mixed-Mode Multielement Multiport
Networks 365
11.4 Noise Analysis of Mixed-Mode Multiport
Networks 367
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Contents xiii
11.4.1 Differential Noise Figure of Mixed-Mode
11.4.2 Differential Signal-to-Noise Ratio of
Mixed-Mode Multiport Networks 372
References 374
12 Scattering Functions in Nonlinear Modeling of
Microwave Devices 375
12.1 Large-Signal Scattering Functions 376
12.2 Linearization of Scattering Functions 379
12.3 The Time Reference 384
12.4 Application of the Response Coefficients
Matrices S and S′ to Predict Nonlinear Device
Performance 386
12.5 Experimental Determination of the Response
Coefficients Matrices S and S′ 387
References 391
Appendix 393
Basics of Fourier Transform in Application-to-
Noise Waves 393
About the Author 397
Index 399
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6631 Book_R1.indb 14 4/21/16 3:17 PM

xv
Preface
The first edition of Microwave Network Design Using the Scattering Matrix was
published in 2010. Since the publication of the first edition, I have worked
extensively to prepare and write new material concerning full theory for appli-
cation of mode-specific signal waves, noise waves, generalized mixed-mode scat-
tering matrices, and generalized mixed-mode noise wave correlation matrices
to the analysis and design of microwave networks with any topology. Three
new chapters: Chapter 7—Signal Analysis of Differential Multiport Networks,
Chapter 10—Differential- and Common-Mode Noise Waves and Correlation
Matrices, and Chapter 11—Noise Analysis of Differential Networks cover this
material. The text from the first edition has been thoroughly revised.
The new edition presents complete and detailed presentation of the
wave approach to microwave network characterization, analysis, and design
using signal and noise wave variables, scattering parameters, and noise wave
parameters in application to the standard, single-ended multiport networks,
as well as to the differential multiport networks.
I believe that this book will continue serving the needs of many students
and microwave engineers.
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6631 Book_R1.indb 16 4/21/16 3:17 PM

1
1
Introduction
For the past 60 years, scattering parameters have been used very extensively
by the microwave community for characterization, modeling, and design of
microwave devices and networks. Because at RF and microwave frequen-
cies, impedance or admittance description of networks is in many aspects
abstraction since the voltages, currents, impedances, or admittances cannot
be measured in direct manner, and then using equipment called vector net-
work analyzer (VNA), we measure the wave reflection coefficient at particular
network ports or wave transmission coefficient through a network [1–3]. The
directly measurable quantities are magnitudes and phase angles of the waves
reflected or scattered from the junction relative to the incident wave magni-
tude and phase angle. Thanks to the linearity of the field equations and the
assumed linearity of such devices as transistors and diodes, the amplitudes
of scattered waves are linearly dependent on the amplitudes of the incident
waves. The matrix describing the linear relationship between the incident and
scattered waves at ports of a network is called the scattering matrix.
A vector network analyzer (VNA) measures the magnitude and phase
characteristics of microwave devices including passive and active components
of microwave networks, as well as multifunctional monolithic microwave
integrated circuits (MMICs) and radio frequency integrated circuits (RFICs).
The scattering parameters and the network analyzer have equipped microwave
engineers and researchers with very valuable device information, transcending
6631 Book_R1.indb 1 4/21/16 3:17 PM

2 Scattering Parameters in RF and Microwave Circuit Analysis and Design
the role of test equipment and data to become vital component in the design
process.
The first introduction of the scattering representation of microwave net-
works took place many years ago [4, 5]. Since then, definitions of wave vari-
ables have been modified many times. In relation to this fact, it is important
to know, understand, and realize the consequences of different approaches to
the microwave networks scattering representation that are presented in the
literature of the subject [4, 6–12]. In particular, an understanding of scatter-
ing description based on wave variables referenced to complex impedances is
very important, because incorrect use of this tool may lead to unacceptable
results and mistakes.
This book presents, in detail, the theoretical foundation for the wave
approach to microwave network characterization, analysis, and design using
scattering parameters.
Chapter 1 is an updated introductory chapter.
Chapter 2 briefly presents theory of uniform waveguides helpful to
understand the microwave network concept of incident wave and reflected or
scattered wave, which is the basis for the scattering parameter description of
microwave networks. Modal electromagnetic field representation discussed
here will be used in Chapter 4 to define the wave variables and the scattering
matrix. Properly normalized modal fields are used to determine electromag-
netic power transmitted in a waveguide, to define characteristic impedance
of a waveguide, and to introduce normalized waveguide voltage and current.
At the end of this chapter, the reader will find considerations on transmission
line equivalent circuit of a single-mode waveguide.
Theory of transmission lines given briefly in Chapter 3 is of significant
importance in microwave network theory. As a bridge between field analy-
sis and microwave network theory, it is very helpful in many considerations
and developments of some basic relations applied to scattering parameters.
This will be demonstrated with examples given in Chapter 4. Starting from
lumped element circuit model of a transmission line, we present and discuss
voltage and current wave propagation, and impedance relations for terminated
transmission lines.
In Chapter 4, we introduce wave variables and the scattering matrix.
There are presented definitions of voltage traveling waves and scattering matrix,
normalized voltage scattering waves and generalized scattering matrix, traveling
wave intensities and true scattering matrix, pseudowaves and pseudoscattering
matrix, and finally power scattering waves and power scattering matrix. The
physical interpretation of these quantities is presented and compared. These
considerations are illustrated by examples, showing different approaches to
6631 Book_R1.indb 2 4/21/16 3:17 PM

Introduction3
wave variables and the scattering matrix. In addition, the consequences of
the change of the reference impedance in the scattering matrix definition
and the relations among the scattering matrix, the pseudoscattering matrix,
the power scattering matrix, and other matrix representations of microwave
networks are discussed.
This book guides the reader through various applications of the scat-
tering matrix. The remaining chapters of this book are devoted to scattering
matrix–based methods of microwave network analysis and design.
Microwave network signal analysis methods based on scattering matrix
description of network elements are discussed in Chapter 5. Three approaches
to this problem are presented: (1) multiport network analysis based on signal
flow graph, (2) multiport network analysis, and (3) multielement multiport
network analysis using connection scattering matrix approach. Multiport
network analysis approach may be applied to networks considered as a single
multiport excited by one-port signal sources and loaded by one-port termi-
nations. The connection scattering matrix approach may be used to analyze
multiport multielement networks considered as a connection of many indi-
vidual multiport elements characterized by their individual scattering matrices.
Microwave networks with any arbitrary topology may be analyzed using this
matrix formalism. Algorithms for different network function calculations are
presented in detail.
In Chapter 6, we present the theory related to the mixed mode scatter-
ing matrix of differential networks. Based on differential mode and common
mode definitions, mode specific wave variables and mixed-mode scattering
parameters are introduced here. Also, discussed here are transformation rela-
tions between standard single-ended and mixed-mode scattering parameters
of multiport networks. A generalized mixed mode scattering matrix for mul-
tiport networks with single-ended and differential inputs and outputs is also
discussed. At the end of this chapter, the presented theory is applied to the
analysis of the differential amplifier.
Chapter 7 develops the complete theory for the signal analysis of dif-
ferential microwave networks, based on mixed-mode wave variables and
mixed-mode scattering matrices. There are presented three approaches to this
problem: (1) differential two-port network analysis based on mixed-mode scat-
tering matrix, (2) differential multiport network analysis and (3) differential

multielement multiport network analysis using connection scattering matrix
approach. Differential multiport network analysis approach may be applied
to networks considered as single mixed-mode multiport excited by differential
signal sources and loaded by differential or single-ended terminations. The
connection scattering matrix approach may be used to analyze differential
6631 Book_R1.indb 3 4/21/16 3:17 PM

multiport networks considered as connections of many individual differential
multiport elements characterized by their individual generalized mixed-mode
scattering matrices. Microwave networks with any arbitrary topology may be
analyzed using this matrix formalism. Algorithms for different network func-
tions calculations are presented in detail.
In Chapter 8, we introduce the definition of the noise wave variables and
the noise wave representation of noisy microwave networks in the form of noise
wave correlation matrix. Chain matrix, impedance matrix, and admittance
matrix noise representations of noisy multiport networks are also presented.
Two-port to three-port noise wave correlation matrix transformation and the
noise wave correlation matrix of embedded multiport networks are discussed.
We also present noise wave modeling of passive multiport networks and of
active two-ports. These relations are essential in noise wave modeling problems
of microwave semiconductor devices such as MESFETs, HEMTs, or HJTs.
In Chapter 9, a scattering matrix and a noise wave correlation matrix
are used for the noise analysis of multiport networks. First, we define and
discuss the noise figure for two-port networks and extend the definition of
this parameter to noisy multiports. Then, two methods of multiport network
noise analysis are given. The first method, applicable to multielement multiport
networks, is based on the connection scattering matrix approach. The second
method may be applied to networks considered as a single multiport termi-
nated at its ports by signal sources and loads. In both cases, the algorithms
of noise figure and signal-to-noise ratio calculations are discussed in detail.
Chapter 10 is on the basis of differential mode and common-mode noise
wave variables and on generalized mixed-mode noise wave correlation matri-
ces. There are, discussed here, mixed-mode noise wave correlation matrices of
passive networks. We discuss, in detail, mixed-mode correlation matrices of
two-port networks, baluns, and differential four-port networks, and present
their relation to standard, single-ended scattering parameters, as well as to
the mixed-mode scattering parameters. There are also discussed mixed-mode
correlation matrices of active two-ports.
Chapter 11 treats a number of topics related to the noise analysis of
differential networks. First, it presents and discusses the noise analysis of dif-
ferential two-port networks.
There are derived relations for the noise figure of balanced two-ports,
fully differential two-ports, power splitting baluns, and power combine baluns.
There are presented and discussed in detail the relation for the noise figure
of differential amplifiers in single-ended environment. Then, two methods
of differential multiport network noise analysis based on generalized mixed-
mode noise wave correlation matrix are given. The first method, applicable
6631 Book_R1.indb 4 4/21/16 3:17 PM

Introduction5
to multielement multiport networks with differential ports and single-ended
ports, is based on the generalized connection scattering matrix approach. The
second method may be applied to networks considered as a single mixed-mode
multiport terminated at its ports by differential signal sources and differential
or single-ended loads. In both cases, algorithms of the noise figure and signal-
to-noise ratio calculations are derived and discussed.
The scattering parameters discussed in this book can only accurately
represent linear networks such as filters, directional couplers, waveguides and
transmission lines, and in approximation, small signal devices, and amplifiers.
They are based on the superposition principle and may represent semiconduc-
tor devices, such as transistors, diodes, and amplifiers, when the applied signal
is assumed to be small enough to justify the superposition principle. To break
the limitations of the small-signal scattering parameters, a large signal scat-
tering function theory has been invented. This theory, which has been very
extensively studied in last years, extends small signal theory by considering
the scattering wave variables not only at one fundamental frequency but also
at harmonic and non-harmonic frequencies. The contribution of all such
spectral components is formulated into nonlinear scattering functions that
allow the characterization of nonlinear devices and networks. In Chapter 12,
we introduce scattering functions, their linearization, and an application for
the modeling of nonlinear microwave devices.
In the Appendix, the reader can find basics of Fourier transform in
application-to-noise waves.
This book is an excellent source of theoretical as well as practical infor-
mation on the wave variables and scattering matrix, and their application
to microwave network characterization, modeling, analysis, and design. It
is suitable for beginners and students, as well as experienced engineers and
researchers working in the field of microwaves.
References
[1] Adam, S. F., “A New Precision Automatic Microwave Measurement System,” IEEE
Trans. on Instrumentation and Measurements, Vol. 17, No. 4, 1968, pp. 308–313.
[2] Howell, K., and K. Wong, “DC to 110 GHz Measurements in Coax Using 1mm Con-
nectors,” Microwave Journal, Vol. 42, July 1999, pp. 22–34.
[3] Rumiantsev, A., and N. Ridler, “VNA Calibration,” IEEE Microwave Magazine, Vol. 9,
No. 3, June 2008, pp. 86–99.
[4] Montgomery, C. G., R. H. Dicke, and E. M. Purcell, Eds., Principles of Microwave
Circuits, McGraw Hill, 1948.
6631 Book_R1.indb 5 4/21/16 3:17 PM

[5] Marcuvitz, N., Waveguide Handbook, McGraw Hill, 1951.
[6] Harrington, R. F., Time Harmonic Electromagnetic Waves, New York: McGraw Hill,
1961.
[7] Kurokawa, K., “Power Waves and the Scattering Matrix,” IEEE Trans. on Microwave
Theory and Techniques, Vol. MTT-13, No. 2, 1965, pp. 194–202.
[8] Kurokawa, K., An Introduction to the Theory of Microwave Circuits, New York: Academic
Press, 1969.
[9] Collin, R. E., Foundations for Microwave Engineering, Tokyo: McGraw Hill Kogakusha,
Ltd., 1966.
[10] Marks, R. B., and D. F. Williams, “A General Waveguide Circuit Theory,” Jour-
nal of Research of the National Institute of Standards and Technology, Vol. 97, 1992,
pp. 533–562.
[11] Altman, J. L., Microwave Circuits, Princeton, New Jersey: D. Van Nostrand Company,
Inc., 1964.
[12] Pozar, D. M., Microwave Engineering, New York, USA: Addison-Wesley Publishing
Company, 1990.
6631 Book_R1.indb 6 4/21/16 3:17 PM

7
2
Theory of Uniform Waveguides
This chapter is based on a fundamental work of Roger B. Marks and Dylan F.
Williams published in [1], where it presents a theory of uniform waveguides
required to understand a concept of waveguide voltage and current waves that
propagate in a waveguide. Voltage and current waves defined here are properly
normalized to simplify further derivations and results. There are derived and
discussed relations for transmitted power and the modal characteristic imped-
ance of the uniform waveguide. This chapter ends with the definitions of the
equivalent circuit parameters in terms of the characteristic impedance of the
uniform waveguide. There are also given explicit expressions for L, R, C, and
G parameters in terms of the modal fields in a lossy waveguide. They are used
for the theoretical determination of the L, R, C, and G network parameters
of any lossy waveguide [2].
According to the relations presented and discussed in this chapter, trans-
verse electric (TE) and transverse magnetic (TM) fields of waves propagating in
the waveguide may be expressed in terms of waveguide voltage and waveguide
current. It can be assumed that the characteristic impedance of the mode equals
the ratio of complex amplitudes of waveguide voltage and waveguide current
waves propagating in forward direction. These quantities, waveguide voltage
and waveguide current, and mode characteristic impedance are the basis for
the definition of traveling wave intensities (or shortly, traveling waves) and the
true scattering parameters that will be introduced and discussed in Chapter 4.
6631 Book_R1.indb 7 4/21/16 3:17 PM

Relations for transmitted power and the waveguide characteristic imped-
ance presented here on the basis of field theory are going to be exploited and
discussed in detail in Chapter 4 in connection with traveling wave intensities
on the basis of microwave network theory.
The L,R,C,G equivalent circuit of the uniform lossy waveguide will be
exploited in Chapter 3 in order to derive transmission line impedance equa-
tions very useful in scattering matrix theory.
2.1 Modal Electromagnetic Fields
In a uniform waveguide, propagating a single mode, the transverse components
of the total fields E and H are given as [1]
Et
= c+
e−gz
et
+ c−
e+gz
et
≡
v(z)
v0
et
(2.1)
and
Ht
= c+
e−gz
ht
− c−
e+gz
ht
≡
i(z)
i0
ht
(2.2)
where γ is the modal propagation constant having real and imaginary parts
α and β
g ≡ a + jb (2.3)
It is assumed that z-axis is oriented along the waveguide axis.
The fields in the waveguide are linear combinations of the forward and
backward mode fields. The first terms in (2.1) and (2.2) are transverse nor-
malized fields, which correspond to a mode propagating in the +z direction
with the propagating constant +γ, while the second terms are transverse
normalized fields of the same mode propagating in the −z direction with the
propagating constant −γ.
In (2.1)
v(z) = c+
v0
e−gz
+ c−
v0
e+gz
= V0
(+)
e−gz
+V0
(−)
e+gz
= v(+)
(z) + v(−)
(z) (2.4)
is the waveguide voltage associated with the transverse components of the
total electric field in the waveguide.
6631 Book_R1.indb 8 4/21/16 3:17 PM

Theory of Uniform Waveguides9
Similarly, in (2.2)
i(z) = c+
i0
e−gz
− c−
i0
e+gz
= I0
(+)
e−gz
+ I0
(−)
e+gz
= i(+)
(z) + i(−)
(z) (2.5)
is the waveguide current associated with the transverse components of the
total magnetic field in the waveguide.
In (2.4) and (2.5), the waveguide voltage
v+
(z) = c+
v0
e−gz
= V0
(+)
e−gz
(2.6)
and the waveguide current
i+
(z) = c+
i0
e−gz
= I0
(+)
e−gz
(2.7)
are voltage and current waves that propagate in forward direction, while the
waveguide voltage
v−
(z) = c−
v0
e+gz
= V0
(−)
e+gz
(2.8)
and the waveguide current
i−
(z) = c−
i0
e+gz
= I0
(−)
e+gz
(2.9)
are voltage and current waves that propagate in backward direction.
v0 and i0 are normalizing constants. Units of v0 are volts and units of
i0 are amperes. Thanks to such normalization, Et and et have units appropri-
ate to electric fields, and Ht and ht have units appropriate to magnetic fields,
while v has units of voltage and i has units of current.
2.2 Power Transmitted in a Waveguide
The integral of the Poyinting vector over the cross section S of the waveguide
equals the net complex power p(z), crossing a given transverse plane in the
waveguide [1, 3, 4]
p(z) = Et
× Ht
∗
( )⋅z dS
S
∫ =
v(z)i∗
(z)
v0
i0
∗ p0
(2.10)
6631 Book_R1.indb 9 4/21/16 3:17 PM

where
p0
≡ et
× ht
∗
( )⋅z dS
S
∫ (2.11)
is the complex power, which is carried by the normalized mode across the
surface S.
In (2.10), there is no one-half factor, which means that the magnitudes
of the complex time dependent fields are assumed to be the root mean square,
not the peak values.
Because, as in electrical circuit theory, it is convenient to require that
p(z) = v(z)i∗
(z) (2.12)
than the normalizing constants v0 and i0 cannot be arbitrarily chosen. They
have to satisfy the constraint
p0
= v0
i0
∗
(2.13)
The forward mode is defined as that in which the power flows in the
+z direction; that is
P = Re p0
{ }≥ 0 (2.14)
The average power flow P(z) across the cross section S of a waveguide
equals the real part of p(z)
P(z) = Re p(z)
{ }= Re Et
× Ht
∗
( )⋅z dS
S
∫
⎧
⎨
⎪
⎩
⎪
⎫
⎬
⎪
⎭
⎪
= Re v(z)i∗
(z)
{ } (2.15)
In a case when only forward mode propagates in the waveguide, the
complex power is
p(z) = p0
e−2az
(2.16)
Similarly, when only backward mode is present in the waveguide, the
complex power is
6631 Book_R1.indb 10 4/21/16 3:17 PM

p(z) = − p0
e+2az (2.17)
The associated average powers are, respectively
P(z) = Re p0
{ }e−2az
(2.18)
and
P(z) = −Re p0
{ }e+2az
(2.19)
As it comes from nonlinear relation in (2.15), in general, the net real
power is not a simple difference of the forward and backward mode powers
[5]. This will be presented and discussed in Chapter 4 of this book.
2.3 Characteristic Impedance
The forward mode characteristic impedance is defined as the ratio of complex
amplitudes of waveguide voltage and waveguide current waves propagating
in forward direction or as the ratio with minus sign of complex amplitudes
of waveguide voltage and waveguide current waves propagating in backward
direction [1]
Z0
=
v0
i0
(2.20)
However, because (2.13) is imposed, we can also write
Z0
=
v0
i0
=
v0
2
p0
∗ =
p0
i0
2 (2.21)
The three definitions of Z0, namely, “voltage-current,” “voltage-power,”
and “current-power,” are consistent. If p0, v0, and i0 were defined indepen-
dently, as for example, in terms of some power, voltage drop, and current in
the waveguide, the three definitions of Z0 would be inconsistent and (2.21)
6631 Book_R1.indb 11 4/21/16 3:17 PM

would not hold. Such an approach to the characteristic impedance definition
can be found in the literature.
The phase of the characteristic impedance Z0 is identical to that of p0.
The phase of Z0 is a fixed, inherent, and unambiguous property of the mode.
Equations (2.14) and (2.20) constrain the sign of Z0, such that
Re Z0
{ }≥ 0 (2.22)
The characteristic impedance Z0 of any propagating mode of a lossless
line is real and positive.
When only a single forward propagating mode exists in a waveguide, then
v(z)/i(z) = v0/i0 = Z0 for all z values. Likewise, when only a single backward
mode exists in a waveguide, then v(z)/i(z) = −v0/i0 = Z0. In a case, when both
forward and backward modes exist, v(z)/i(z) depends on z due to the interfer-
ence between the two waves propagating the waveguide in opposite directions.
Let us consider now the correspondence between the above presented def-
inition of Z0 and the conventional definitions of the characteristic impedance.
In a homogeneous waveguide, the fields of TE, TM, and transverse
electromagnetic (TEM) modes satisfy
z × et
= hht
(2.23)
where η is the wave impedance, which is a constant over the cross section S
of the waveguide. In such a case
Z0
=
v0
2
et
2
dS
S
∫
h (2.24)
Because the modal field et is normalized and the value of the denomina-
tor is fixed, the magnitude of Z0 depends only on v0. From (2.24), it is also
seen that the phase of the characteristic impedance Z0 is equal to that of the
wave impedance η.
These results indicate that the definition of mode characteristic imped-
ance given by (2.21) is in coincidence with the most conventional definitions
of Z0.
In particular, for the TEM modes, η equals to the intrinsic wave imped-
ance of the medium fulfilling the line
6631 Book_R1.indb 12 4/21/16 3:17 PM

h =
m
e
(2.25)
and from this
arg Z0
{ }=
1
2
arg{m} − arg{e}
( ) (2.26)
In the case when the medium of the line is lossy dielectric and μ is real,
then
arg Z0
{ }= −
1
2
d (2.27)
where tgδ ≡ Im{ε}/Re{ε} is the dielectric loss tangent.
If in (2.24) v0 is taken to be the voltage between two conductors of the
TEM line than the mode characteristic impedance Z0 defined by (2.21) equals
the conventional characteristic impedance of the TEM transmission line.
For TE and TM modes
h =
m
e
1−
kc
2
w2
me
⎛
⎝
⎜
⎞
⎠
⎟
±1/2
(2.28)
where “+” sign corresponds to TM mode, “−” sign corresponds to TE mode,
and kc = 2π/λc is the cutoff wave number in which λc is the cutoff wavelength
of a mode.
2.4 Normalization of Waveguide Voltage and Current
The magnitude of Z0 depends on the choice of v0 and i0. Because of modal
field normalization [(2.1) and (2.2)] and the constraint defined by (2.13), only
one of these quantities may be assigned independently. One useful normaliza-
tion defines the constant v0 as the path integral
v0
= − et
⋅ dl
path
∫ (2.29)
6631 Book_R1.indb 13 4/21/16 3:17 PM

The path is restricted to a single transverse plane, and the integral, in
general, depends on the path between two given endpoints. When the mode
in a waveguide is TEM or TM, this integral depends only on the endpoints,
not on the path between them.
The voltage v0 defined by (2.29) is analogous to the voltage given by
v(z) = − Et
(z)⋅ dl
path
∫ (2.30)
Normalization given by (2.29) together with (2.21) leads to “voltage-
power” definition of the characteristic impedance Z0.
It is also possible to use “current-power” definition by choosing i0 to be a
current. However, because of (2.21), it is not possible to use “voltage-current”
definition of the characteristic impedance. It is so, because the phases of v0
and i0 may not satisfy the relation v0/i0 = Z0.
2.5 Transmission Line Equivalent Circuit of a Waveguide
Figure 2.1 presents the equivalent circuit of a uniform waveguide with a dis-
tributed shunt capacitance C, a conductance G, a series resistance R, and an
inductance L, all per unit length. These parameters are defined as [1, 2]
G + jwC ≡
g
Z0
(2.31)
and
R + jwL ≡ gZ0
(2.32)
The derivations presented in [1, 2] lead to the following relations for the
circuit parameters:
C =
1
v0
2
′
e et
2
ds
S
∫ − ′
m hz
S
∫
2
dS
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
(2.33)
L =
1
i0
2
′
m ht
2
ds
S
∫ − ′
e ez
S
∫
2
dS
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
(2.34)
6631 Book_R1.indb 14 4/21/16 3:18 PM

G =
1
v0
2
′′
ε et
2
ds
S
∫ + ′′
m hz
S
∫
2
dS
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
(2.35)
R =
1
i0
2
′′
m ht
2
ds
S
∫ + ′′
e ez
S
∫
2
dS
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
(2.36)
where ε ≡ ε′ − jε″ and μ ≡ μ′ − jμ″. In passive media, all these parameters
are nonnegative.
A transmission line with circuit parameters, given above, models a lossy
waveguide, which is characterized by a complex propagation constant γ and a
complex power flow P(z). This fact is very important when one wants to model
properties of lines over lossy semiconductor substrates.
As it comes from these relations, the parameters C, L, G, and R are
normalized as Z0, with respect to v0 and i0. In a case of a lossless TEM line,
when v0 is taken as the voltage between two active conductors, then L and C
are the conventional inductance and capacitance per unit length of the line.
From (2.31) and (2.32), the characteristic impedance of the waveguide
Z0
=
jwL + R
jwC + G
(2.37)
and the propagation constant
g = ( jwL + R)( jwC + G) (2.38)
The above relations are identical to those derived from the conventional
circuit theory for a transmission line with the distributed shunt admittance
Y = jωC + G and the series impedance Z = jωL + R, in which C, L, G,
and R are the capacitance, inductance, conductance, and resistance per unit
length of the line. In the steady-state sinusoidal, voltage v and current i in a
Figure 2.1 Uniform waveguide equivalent circuit.
6631 Book_R1.indb 15 4/21/16 3:18 PM

transmission line are described by the equations presented and discussed in
Chapter 3 of this book.
References
[1] Marks, R. B., and D. F. Williams, “A General Waveguide Circuit Theory,” Jour-
nal of Research of the National Institute of Standards and Technology, Vol. 97, 1992,
pp. 533–562.
[2] Brews, J. R., “Transmission Line Models for Lossy Waveguide Interconnections in
VLSI,” IEEE Trans. on Electron Devices, Vol. ED-33, 1986, pp. 1356–1365.
[3] Harrington, R. F., Time Harmonic Electromagnetic Waves, New York: McGraw Hill,
1961.
[4] Montgomery, C. G., R. H. Dicke, and E. M. Purcell, Eds., Principles of Microwave
Circuits, McGraw Hill, 1948.
[5] Marcuvitz, N., Waveguide Handbook, McGraw Hill, 1951.
[6] Collin, R. E., Foundations for Microwave Engineering, Tokyo: McGraw Hill Kogakusha,
Ltd., 1966.
6631 Book_R1.indb 16 4/21/16 3:18 PM

17
3
Theory of Transmission Lines
The use of transmission lines to model waveguides is very common. The theory
of transmission lines plays a very important role in the analysis and design of
microwave networks composed of waveguides. The analogy between wave-
guides and transmission lines comes from the fact that both structures propa-
gate waves. A waveguide propagates waves of electric and magnetic fields, while
voltage and current waves propagate in a transmission line. If the direction of
wave propagation is chosen as the z-direction, then the z-dependence of the
waves of transverse electric field in the waveguide and the z-dependence of the
voltage in the equivalent transmission line are the same. The same statement
is true for the waves of transverse magnetic field in the waveguide and of the
current waves in a transmission line.
In this chapter, we present voltage and current wave propagation in
transmission lines, define transmission line characteristic impedance, and
discuss the total power flow. The relations for total voltage and total current
are discussed and, resulting from this, the input impedance of terminated
transmission line is given.
Relations presented here are helpful in deriving scattering parameters
of many microwave elements composed of waveguides, coupled waveguides,
terminated waveguides, and so forth. They will be used in Chapter 4 in exam-
ples, illustrating theoretical derivation of transmission line segment scattering
parameters.
6631 Book_R1.indb 17 4/21/16 3:18 PM

3.1 Lumped Element Circuit Model of a Transmission Line
The transmission line can be described as a distributed-parameter electric net-
work [1–5]. The equivalent circuit of a section of transmission line of differential
length is presented in Figure 2.1. Parameters R[Ω/m], L[H/m], G[S/m], and
C[F/m] of this circuit are, respectively, resistance, inductance, conductance,
and capacitance per unit length of the line. They are given by (2.33) to (2.36).
Resulting from Kirchhof’s laws, the equations of this equivalent circuit
are [1–5]
∂v(z,t)
∂z
= −Ri(z,t) − L
∂i(z,t)
∂t
(3.1)
and
∂i(z,t)
∂z
= −Gv(z,t) − C
∂v(z,t)
∂t
(3.2)
For sinusoidal steady state, when v(z) and i(z) represent the voltage and
current without the time dependence ejωt
, the basic equations for the circuit are
dv(z)
dz
= −( jwL + R)i(z) (3.3)
di(z)
dz
= −( jwC + G)v(z) (3.4)
where v(z,t) = v(z)ejωt
and i(z,t) = i(z)ejωt
are the complex voltage and current
in the line.
3.2 Voltage and Current Wave Propagation in a
Transmission Line
From (3.3) and (3.4), the wave equations for v(z) and i(z) in a transmission
line have the form
d2
v(z)
dz2 − g 2
v(z) = 0 (3.5)
6631 Book_R1.indb 18 4/21/16 3:18 PM

Theory of Transmission Lines19
d2
i(z)
dz2 − g 2
i(z) = 0 (3.6)
where
g = a + jb = (R + jwL)(G + jwC)
= −w2
LC + RG + jw(RC + LG)
(3.7)
is the complex propagation constant.
The general solution to (3.5) is
v(z) = V0
(+)
e−gz
+V0
(−)
egz
= V (+)
(z) +V (−)
(z) (3.8)
The solution for the currrent i from (3.6) is
i(z) = I0
(+)
e−gz
− I0
(−)
egz
= I(+)
(z) − I(−)
(z) (3.9)
or
i(z) =
V0
(+)
Z0
e−gz
−
V0
(−)
Z0
egz
(3.10)
where
Z0
=
R + jwL
g
=
R + jwL
G + jwC
(3.11)
is the characteristic impedance of the line.
In both (3.8) and (3.9), the first term represents the wave propagating
in the +z direction, and the second term represents the wave propagating in
the −z direction of the z-axis. Quantities V0
(+)
, I0
(+)
, V0
(–)
, and I0
(–)
are the com-
plex amplitudes of voltages and currents of voltage waves and current waves
at point z = 0 of the transmission line.
The characteristic impedance
Z0
=
V0
(+)
I0
(+) = −
V0
(−)
I0
(−) (3.12)
6631 Book_R1.indb 19 4/21/16 3:18 PM

of the line is equal to the ratio of the complex voltage amplitude V0
(+)
(or V0
(–)
)
of the voltage wave, and to the complex current amplitude I0
(+)
(or −I0
(–)
) of the
current wave propagating in the +z direction (or in the −z direction) of the line.
A ratio of reflected and incident voltage wave amplitudes at any point
l = −z of the line
Γ(l) =
V (−)
V (+) =
V0
(−)
V0
(+) e−2gl
= ΓL
e−2gl
(3.13)
is called the voltage reflection coefficient. In (3.13)
ΓL
=
V0
(−)
V0
(+) =
ZL
− Z0
ZL
+ Z0
(3.14)
is the load reflection coefficient.
The total power flow at z is given by
P(z) = Re vi∗
( )= Re V (+)
+V (−)
( ) I(+)
− I(−)
( )∗
{ }
= Re V (+)
+V (−)
( )Y0
∗
V (+)
−V (−)
( )∗
{ }
= P+
1− Γ
2
− 2Im{Γ}
Im Y0
{ }
Re Y0
{ }
⎛
⎝
⎜
⎞
⎠
⎟
(3.15)
where
P+
(z) = Re Y0
{ }V (+) 2
= Re Y0
{ }V0
(+) 2
e−2az
(3.16)
is the power transmitted by the wave propagating in the +z direction and
Y0 = 1/Z0 is the complex characteristic admittance of the transmission line.
In the case of dissipative transmission line, ⎪Γ⎪2
cannot be regarded as
the power reflection coefficient. The value of ⎪Γ⎪ may exceed unity if Z0 is
not real.
In the case of a lossless line, R = 0 and G = 0, (3.7) describing propaga-
tion coefficient has a form
6631 Book_R1.indb 20 4/21/16 3:18 PM

g = a + jb = jw LC (3.17)
what means that
a = 0 oraz b = w LC (3.18)
and the characteristic impedance of a transmission line is real and equal
Z0
=
L
C
(3.19)
The resulting traveling wave solutions are now represented as
v(z) = V0
(+)
e− jbz
+V0
(−)
e jbz
= V (+)
(z) +V (−)
(z) (3.20)
and
i(z) =
V0
(+)
Z0
e− jbz
−
V0
(−)
Z0
e jbz
= I(+)
(z) − I(−)
(z) (3.21)
The voltage reflection coefficient is now
Γ(z) =
V0
(−)
V0
(+) e j2b(z−z0
)
= Γ z0
( )e j2b(z−z0
)
(3.22)
In a nondissipative uniform transmission line, the total average power
flow at any point z becomes
P = Re vi∗
( )=
V0
(+) 2
Z0
−
V0
(−) 2
Z0
=
V0
(+) 2
Z0
1− Γ
2
( )= P+
1− Γ
2
( ) (3.23)
Equation (3.23) is simple to interpret. This power equals the difference
between the incident and the reflected power flowing down the transmission
line. ⎪Γ⎪2
is just the power reflection coefficient.
6631 Book_R1.indb 21 4/21/16 3:18 PM

3.3 Terminated Transmission Line
We assume now that the transmission line presented in Figure 3.1 is excited
by a signal source located on the left-hand side of the origin of the coordinate
system (z 0). Because at the end of the line, the ratio v/i is equal to ZL, and
simultaneously, V(+)
/I(+)
= Z0, which means that the wave transmitted in the
line must be reflected from the load in such a way that the ratio of the resul-
tant voltage v and the resultant current i at the end of the line is equal to ZL.
The total voltage and the total current in the line can be expressed as
v(l) = V0
(+)
egl
+ ΓL
e−gl
⎡
⎣ ⎤
⎦ (3.24)
and
i(l) =
V0
(+)
Z0
egl
− ΓL
e−gl
⎡
⎣ ⎤
⎦ (3.25)
where l = z is the positive distance measured from the load toward the generator.
For a lossless transmission line, (3.24) and (3.25) are
v(l) = V0
(+)
e jbl
+ ΓL
e− jbl
⎡
⎣ ⎤
⎦ (3.26)
and
i(l) =
V0
(+)
Z0
e jbl
− ΓL
e− jbl
⎡
⎣ ⎤
⎦ (3.27)
Figure 3.1 Terminated transmission line.
6631 Book_R1.indb 22 4/21/16 3:18 PM

where
ΓL
=
V0
(−)
V0
(+) =
ZL
− Z0
ZL
+ Z0
(3.28)
When the load impedance equals the characteristic impedance, ZL =
Z0, then ΓL = 0, what means that there is no reflected wave. When ΓL = 0,
the whole power transmitted in the wave propagating in the +z direction dis-
sipates in the load.
Using (3.24) and (3.25), we can determine the input impedance at a
distance l = −z from the end of the line (from the load ZL), looking toward
the load. This impedance is defined as
ZIN
(l) =
V (l)
I(l)
= Z0
V (+)
egl
+ ΓL
e−gl
⎡
⎣ ⎤
⎦
V (+)
egl
− ΓL
e−gl
⎡
⎣ ⎤
⎦
= Z0
1+ ΓL
e−2gl
1− ΓL
e−2gl
(3.29)
Using (3.29) in (3.28), we also have
ZIN
(l) = Z0
ZL
+ Z0
( )egl
+ ZL
− Z0
( )e−gl
ZL
+ Z0
( )egl
− ZL
− Z0
( )e−gl
= Z0
ZL
coshgl + Z0
sinhgl
Z0
coshgl + ZL
sinhgbl
= Z0
ZL
+ Z0
tanhgl
Z0
+ ZL
tanhgl
(3.30)
For a lossless transmission line
ZIN
(l)= Z0
ZL
+ Z0
( )e jbl
+ ZL
− Z0
( )e− jbl
ZL
+ Z0
( )e jbl
− ZL
− Z
( )e− jbl
= Z0
ZL
cosbl + jZ0
sinbl
Z0
cosbl + jZL
sinbl
= Z0
ZL
+ jZ0
tanbl
Z0
+ jZL
tanbl
(3.31)
6631 Book_R1.indb 23 4/21/16 3:18 PM

It comes from (3.31) that the lossless transmission line input impedance
is a periodic function of the line lenght l, with a period equal to λg/2.
In the same way, we can determine the input admittance of the line. It
is defined as
YIN
(l) = Y0
YL
+ Y0
( )egl
+ YL
− Y0
( )e−gl
YL
+ Y0
( )egl
− YL
− Y0
( )e−gl
= Y0
YL
coshgl + Y0
sinhgl
Y0
coshgl + YL
sinhgl
= Y0
YL
+ Y0
tanhgl
Y0
+ YL
tanhgl
(3.32)
And for a lossless transmission line
YIN
(l) = Y0
YL
+ Y0
( ) jbl
e + YL
− Y0
( ) -jbl
e
YL
+ Y0
( ) jbl
e − YL
− Y0
( ) -jbl
e
= Y0
YL
cosbl + jY0
sinbl
Y0
cosbl + jYL
sinbl
= Y0
YL
+ jY0
tanbl
Y0
+ jYL
tanbl
(3.33)
where Y0 = 1/Z0 and YL = 1/ZL.
For lossless transmission lines, both the functions ZIN(βl) and YIN(βl)
are periodic and the period equals π. Because βl = 2πl/λg, the same value of
the input impedance ZIN (or the input admittance YIN) repeats along the line
every distance equal to λg/2, what means that
ZIN
(l) = ZIN
l + klg
/2
( ), k = 1,2,3,… (3.34)
and
YIN
(l) = YIN
l + klg
/2
( ), k = 1,2,3,… (3.35)
6631 Book_R1.indb 24 4/21/16 3:18 PM

3.4 Terminated Transmission Line Special Cases
The special cases of terminated transmission lines considered now are short
circuited (ZL = 0) and open circuited (ZL = ∞) lossless transmission lines.
In the case of a short circuited at the end of the transmission line, the
load reflection coefficient ΓL = −1, the load voltage v(0) = 0, the load current
i(0) reaches maximum value, and according to (3.28), the input impedance
of such a line is
ZIN
(l) = jZ0
tanbl = jXIN
(3.36)
The input impedance ZIN of the short-circuited transmission line is
purely imaginary, and its value depends on the line length l. For l = 0 + k λg/2,
k = 1,2,3,…, ZIN = 0, and for l = λg/4 + k λg/2, k = 1,2,3,…, ZIN → ∞.
In a case of open circuited at the end of the transmission line, the
load reflection coefficient ΓL = 1, the load current i(0) = 0, the load voltage
reaches maximum, and according to (3.28), the input impedance of such a
line is
ZIN
(l) = − jZ0
cotbl = jXIN
(3.37)
The input impedance ZIN of the open-circuited transmission line is
purely imaginary, and its value depends on the line length l. For l = 0 + k λg/2,
k = 1,2,3,…, ZIN → ∞ (open circuit), and for l = λg/4 + k λg/2, k = 1,2,3,…,
ZIN = 0 (short circuit).
Other two very important cases of terminated transmission lines are a
line with length l = λg/2 and a line with lenght l = λg/4.
If l = λg/2, then according to (3.31)
ZIN
= ZL
(3.38)
what means that the transmision line with length equal to a half of wave-
length or a multiple of a half of wavelength does not change the impedance
value. This property does not depend on the value of the transmission line
characteristic impedance Z0.
6631 Book_R1.indb 25 4/21/16 3:18 PM

When the transmission line length equals λg/4 (a quarter of the wave-
length), or more generally, when l = λg/4 + k λg/2, k = 1,2,3,…, then accord-
ing to (3.31), its input impedance satisfies the relation
ZIN
=
Z0
2
ZL
(3.39)
Such a line is called a quarter wavelength transformer.
In a case when a transmission line is terminated by a load with imped-
ance ZL equal to the transmission line characteristic impedance Z0, then
independently on line length l
ZIN
= Z0
(3.40)
Such a line load is called a matching load.
References
[1] Collin, R. E., Foundations for Microwave Engineering, Tokyo: McGraw-Hill, Ltd., 1992.
[2] Pozar, D. M., Microwave Engineering, New York: Addison-Wesley Publishing Company,
1990.
[3] Russer, P., Electromagnetics, Microwave Circuit and Antenna Design for Communications
Engineering, Norwood, Massachusetts: Artech House, 2006.
[4] Nibler, F., High-Frequency Circuit Engineering, IEE Circuits and Systems, Series 6,
London, U.K., 1990.
[5] Ramo, S., J. R. Whinnery, and T. Van Duzer, Field and Waves in Communication
Electronics, New York: J. Wiley Sons, Inc., 1965.
6631 Book_R1.indb 26 4/21/16 3:18 PM

27
4
Wave Variables and the
Scattering Matrix
In this chapter, we review different definitions of wave variables that are pre-
sented in [1–6, 8–10]. Definitions of voltage traveling waves and the scatter-
ing matrix, normalized voltage traveling waves and the generalized scattering
matrix, traveling wave intensities and the true scattering matrix, pseudowaves
and the pseudoscattering matrix, and the power scattering waves and the power
scattering matrix are discussed and compared. We also discuss the physical
interpretation of these wave variables, properties of different scattering param-
eters, conversions between the scattering matrices and the impedance and
admittance matrices, and the relations for reference impedance renormalization.
Some examples illustrate differences between different approaches to the wave
variables and the scattering parameters. Because many microwave networks
and systems may be considered as cascade connections of two-port networks,
we discuss an efficient method for the computation of the resultant scattering
matrix of a cascade of two two-port networks described by their individual
scattering matrices. We present a generalized multiport cascade (transfer scat-
tering) matrix, very useful and convenient in network description of cascaded
multiport networks, applicable in analysis of multiline interconnects. We also
present problems encountered in characterization of active devices, and how
to transform three-port scattering parameters of three-terminal devices into
equivalent two-port scattering parameters and vice versa. These procedures
6631 Book_R1.indb 27 4/21/16 3:18 PM

have their place in transistor scattering parameter transformation for differ-
ent common terminal configurations. Finally, we present a procedure for the
resultant scattering matrix computations of embedded multiport networks. This
procedure is used in active device modeling and characterization problems. An
embedded network represents an intrinsic chip of an active device, while passive
parasitic elements, such as resistances, inductances, and capacitances, create
an embedding network. Similar procedures are used in deembedding a real
device under test (DUT) data from vector network analyzer measurement data.
4.1 Voltage Traveling Waves and the Scattering Matrix
Let us consider a linear microwave network with an arbitrary number of uni-
form transmission lines (waveguides) that create ports of the network. To define
a port in each waveguide, a cross-sectional reference is chosen. It is assumed that,
at the reference plane, only a single mode exist. This condition will be fulfilled
by choosing the reference plane sufficiently far from waveguide connection that
it can be assumed that higher-order modes have decayed and are negligible.
As discussed in Chapter 3, the incident and reflected voltage waves travel-
ing in the transmission lines creating ports of a multiport can be considered as
wave variables at these ports of the network. This definition of wave variables
was introduced by R. E. Collin [1]
a ≡ v(+) (4.1)
b ≡ v(−)
(4.2)
Voltage traveling waves are discussed in Chapter 3, and they are defined as
(3.8). Figure 4.1 illustrates a multiport network with incident and reflected
waves at its ports.
The scattering matrix, or S-matrix, is defined in relation to these incident
and reflected voltage waves as
b = Sa (4.3)
where
a =
a1
a2
!
aN
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
= V(+)
=
v1
(+)
v2
(+)
!
vN
(+)
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
(4.4)
6631 Book_R1.indb 28 4/21/16 3:18 PM

Wave Variables and the Scattering Matrix 29
and
b =
b1
b2
!
bN
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
= V(−)
=
v1
(−)
v2
(−)
!
vN
(−)
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
(4.5)
are the vectors of incident and reflected voltage waves. S is the complex squared
matrix.
4.1.1 Physical Interpretation of Scattering Parameters
A specific element of the scattering matrix from the main diagonal can be
determined as
Sii
=
vi
(−)
vi
(+)
vk
(+)
=0 for k≠i
(4.6)
Thus, Sii is the voltage reflection coefficient of the ith port when all other
ports are terminated in matched loads. Matched loads requirement comes
from the condition: vk
(+)
= 0 for k ≠ i.
The off-diagonal element of the S-matrix can be determined as
Sij
=
vi
(−)
vj
(+)
vk
(+)
=0 for k≠ j
(4.7)
Figure 4.1 N-port network with scattered waves at its ports.
6631 Book_R1.indb 29 4/21/16 3:18 PM

what means that Sij is the ratio of the outgoing wave voltage amplitude vi
(–)
,
coming out of port i, to the incident wave voltage amplitude vj
(+)
at the port
j, when all other ports are terminated in matched loads. Both the voltage
amplitudes are referenced to the reference planes of the considered ports.
At the same time, the incident waves at all ports except the driven port j are
equal to zero, what means that all ports must be terminated in matched loads.
Sij parameter is the voltage transmission coefficient from port j to port i,
with the assumption that all other ports are loaded by matched loads.
Because scattering parameters relate amplitudes (both magnitude and
phase) of traveling wave voltages, reference planes must be specified at each
port of the network.
If someone wants to relate, defined as (4.1) and (4.2), wave variables with
a power transmitted in all ports, it is possible to assume that transmitted in
the +z direction power in ith port is given by [1]
Pi
(+)
=
1
2
vi
(+) 2
(4.8)
while a power transmitted in a port in the −z direction is
Pi
(−)
=
1
2
vi
(−) 2
(4.9)
This corresponds to choosing, for all ports, the equivalent characteristic
impedances Z0 equal to unity. Of course, any value different from Z0 = 1 Ω
would also be possible and suitable. In such a case, transmitted powers will
always be equal to ⎪vi
(+)
⎪2
and ⎪v(–)
⎪2
multiplied by some constant.
With the assumed normalization (Z0 = 1 Ω), the total voltage v and the
total current i at the reference plane of the port are
v = v(+)
+ v(−)
(4.10)
and
i = i(+)
− i(−)
= v(+)
− v(−)
(4.11)
Thus, the wave variables are the linear combinations of variables v and i
v(+)
=
v + i
2
(4.12)
6631 Book_R1.indb 30 4/21/16 3:18 PM

and
v(−)
=
v − i
2
(4.13)
The real power that flows across the reference plane of a port is

P(z) =
1
2
Re vi∗
{ }=
1
2
Re v(+) 2
− v(−) 2
+ v(−)
v(+)∗
− v(+)
v(−)∗
( )
{ }
=
1
2
v(+) 2
− v(−) 2
( )=
1
2
a
2
− b
2
( )
(4.14)
since the quantity (v(–)
v(+)∗
− v(+)
v(–)∗
) is purely imaginary. The net real power
P crossing the reference plane is equal to the difference of the power carried
by the forward and backward waves acting independently. This result comes
from the fact that the uniform transmission line propagating voltage waves
is assumed to be lossless.
The reflection coefficient Γ of the voltage waves is defined as
Γ(z) =
b(z)
a(z)
=
v(−)
v(+) =
v − i
v + i
=
Z −1
Z +1
(4.15)
In (4.15), Z = v/i is the impedance seen at the reference plane looking
forward the line.
4.1.2 A Shift in Reference Plane
The scattering matrix elements (scattering parameters) are functions of fre-
quency and of locations of reference planes that define the network ports.
The values of scattering matrix elements change with frequency in a manner,
which is not generally deduced analytically. At a fixed frequency, changes
of locations of reference planes specified for each port and related with this
transformation of the scattering matrix is easy to establish. To do this, let us
consider the N-port presented in Figure 4.2.
The original terminal plane of the ith port is located at zi = 0 (zi is the
length coordinate measured along the transmission line forming the ith port).
The scattering matrix of the original network is S. Let us assume now that
reference planes have been shifted outward to the locations zi = li, i = 1,2,…,N,
and the new scattering matrix is denoted as S′.
6631 Book_R1.indb 31 4/21/16 3:18 PM

For the network defined by a set of original reference planes located at
zi = 0, we can write
V(−)
= SV(+)
(4.16)
while for the network referenced to the new set of reference planes located
at zi = li
′
V (−)
= ′
S ′
V (+)
(4.17)
Using the theory of lossless transmission lines, we can write relations
between voltage wave amplitudes at original reference planes (zi = 0) and at
new reference planes (zi = li)
′
vi
(+)
= vi
(+)
e jqi
(4.18)
′
vi
(−)
= vi
(−)
e jqi
(4.19)
where θi = βli is the electrical length of the outward shift of the reference
plane of port i.
Writing (4.18) and (4.19) in matrix form gives
′
V (+)
= DV(+) (4.20)
and
′
V (−)
= D∗
V(−)
(4.21)
where V′(+)
and V(+)
are the vectors of voltage amplitudes of incident waves and
V′(–)
and V(–)
voltage amplitudes of outgoing waves at all circuit ports of the
original network and of the network with new reference planes, respectively.
Figure 4.2 N-port with shifted reference planes at its ports.
6631 Book_R1.indb 32 4/21/16 3:18 PM

D is the diagonal matrix whose ith diagonal component is given by ejθi. D∗
indicates the complex conjugate of D.
Substituting (4.20) and (4.21) into (4.17) gives
D∗
V(−)
= ′
S DV(+) (4.22)
Then, using (4.16), we get
D∗
SV(+)
= ′
S DV(+)
(4.23)
and finally multiplying (4.23) (from the right-hand side) by the inverse of
DV(+)
, we find the desired result
′
S = D∗
SD−1
= D∗
SD∗
(4.24)
because D–1
= D∗
. It is important to note that Sii′ = e–2jθiSii, which means that
the change in the phase of Sii is twice the electrical length of the shift in the
ith reference plane. This is because the wave travels the reference shift distance
twice, forth and back.
4.1.3 Scattering Matrix Properties
In special cases of networks, their scattering matrices have to satisfy some
conditions.
4.1.3.1 Reciprocal Network
For a reciprocal network, the scattering matrix defined as (4.1) to (4.5) is sym-
metrical, that is [1]
S = ST
(4.25)
provided that the equivalent voltages are chosen, so that the transmitted power
is given by 1/2⎪vi⎪2
for all ports. Equation (4.25) comes from the fact that the
reciprocal network is assumed to be lossless.
4.1.3.2 Lossless Network
In a case of lossless network, the total power entering the N-ports must be
equal to the total leaving power. This is power conservation condition, and
mathematically it is represented as
vi
(+)
i=1
N
∑
2
− vi
(−)
i=1
N
∑
2
= 0 (4.26)
6631 Book_R1.indb 33 4/21/16 3:18 PM

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from himself in reading; and, indeed, he read an astonishing
multitude of books upon an astonishing multitude of subjects. But
now and then, in spite of his efforts to be blind, the actual Elias
Bacharach would loom up big before him, in all his ghastly
demoralization; and sick with self-loathing, he would bury his face in
his hands, and demand bitterly, impotently, why he had ever been
born? what single earthly purpose he was good for? why he could
not be abolished utterly forthwith? But these dark moods, or lucid
intervals, were commonly of short duration. He was generally able to
forget them in a novel. He watched his wedding-day draw near and
nearer, without the slightest quickening of the pulse. As I have said,
he took a certain insipid pleasure in the thought of his marriage. He
fancied it would be rather agreeable than otherwise to have Tillie a
constant inmate of his house. She would brighten it up, put a little
electricity into its atmosphere, relieve the excessive tedium of life in
it. But this pleasure was very mild indeed; the languid pleasure that
one might experience at the prospect of becoming the owner of a
languidly admired vase or piece of furniture. Yes, he was glad
enough that it was going to be his; but he did not care a great deal
one way or the other; and as the day approached which was to
inaugurate his proprietorship, he felt no flutter of the heart, no
accession of eagerness or interest. Tillie's excitement, on the
contrary, intensified perceptibly. It had the effect of beautifying her,
and of civilizing her. With heightened color and brightened eyes, she
was an exceedingly pretty girl, one that any man might have been
proud of for his bride. Then, she did not talk half so loudly as she
had used to do; and her choice of words, phrases, and figures,
underwent a notable modification for the better. The adjectives,
grand, ideal, elegant, fearful, and such like, for example, dropped
almost entirely out of her daily speech.
Of course, before long, the wedding-presents began to come in.
Tillie's delight knew no bounds. Every evening Elias discovered her in
an ecstasy over the things that had arrived that day, and joyfully
anticipating those that would arrive to-morrow. Some of these
presents made the poor fellow groan inwardly. Mr. Blum, for

instance, sent an enormous worsted-work picture of Ruth and Boaz,
with a charming, though misapplied, inscription cunningly
embroidered in gold thread: “Whither thou goest, I will go,” etc.
Elias knew that this would have to be hung in a conspicuous place in
his house; for, of course, when Mr. Blum came to see them, he
would look for it, and, if it wasn't visible, would feel hurt and
slighted. Mrs. Blum sent a pair of diamond ear-rings. Tillie at once
put them on; and she never afterward appeared without them; so
that, from this point, whenever she figures upon these pages, the
reader will kindly imagine a lustrous solitaire pendent from each of
her tiny ears. They were large and handsome; and Mr. Blum
confidentially informed Elias that he had got them at a bargain, but
that they had coast him a heap of money all the same.
Neither Mr. Sternberg's parlors, nor Mr. Koch's, were spacious
enough to accommodate a tithe of the people who would have to be
invited to the wedding; and therefore it was decided to follow the
common Jewish practice, and engage for the occasion a public hall.
Mr. Koch engaged the hall of the Advance Club.
There, accordingly, in the afternoon of Monday, the seventh of
January, 1884, and in the presence of rather more than three
hundred witnesses, Mr. Elias Bacharach and Miss Matilda Morgenthau
were pronounced irrevocably man and wife; the Reverend Dr.
Gedaza, assisted by the Reverend Mr. Lewis, as cantor, officiating.
The ceremonies were conducted in the strictest orthodox style. The
happy couple stood beneath a silken canopy, supported by four
young gentlemen designated by the groom; all the men present
covered their heads, some with hats, some with handkerchiefs; the
cantor intoned an invocation, a prayer, a benediction; the rabbi put
the requisite questions, and got the regulation responses, both in
Hebrew; after which, he made a very pretty and touching speech,
kissed the bride, and said, “Mrs. Bacharach, accept my heartiest
congratulations.” The wine, meanwhile, had been spilled and
drunken, and the goblet crushed under the bridegroom's heel. For
upwards of an hour afterward, there was a wild clamor of talk; and

every body shook hands with Elias, and gave Tillie a kiss. Then they
all sat down to dinner. The chazzan chanted a grace. The banqueters
fell to. By and by toasts were proposed, and harangues delivered.
The dancing began at eleven o'clock, and held out until five the next
morning.
So they were married.

F
XIX
IRST of all, weakened in body and mind by an epileptic stroke;
then scared literally out of his wits, terrified into a mental and
emotional stupor, by the belief that that which we know to
have been an epileptic stroke was a visitation from an angry God; a
victim, rather than a villain; the creature of disease and superstition,
of heredity and education; Elias Bacharach had deserted and
forgotten the woman whom he loved, and had allowed himself to be
seduced into a marriage with a woman whom he did not love. That
a reawakening, accompanied by all the horrors of despair and
remorse, should come sooner or later, was, of course, inevitable. It
did not come, however, till some nine months after his separation
from Christine Redwood, which was some nine months too late.
I have in my possession a quantity of manuscript, in Elias's
crabbed handwriting, which gives a deep and clear, though
fragmentary, insight into the life he led after his marriage. It is in the
form of a long, turbulent, and often hysterical letter, addressed by
him, under circumstances which will in due time be explained, to
Christine—a letter, however, which was never sent—and it bears date
February, 1885. I have already made one or two quotations from it. I
shall avail myself freely of it in the present chapter.
About the relations between himself and Tillie, Elias writes, “there
is not much to be said. Our relations were perfectly amicable, but
perfectly superficial. Man and wife in name, in reality we were simply
good friends; scarcely that, indeed; scarcely more than friendly
acquaintances. She was invariably bright, cheerful, amiable,
unselfish. I tried to do my duty by her, as I conceived it; to be
always kind to her, and to seize every opportunity that I saw to
afford her pleasure, or to spare her annoyance. I dare say this was
not enough. I dare say she deserved better of me than she got; that

I ought to have striven to be her husband in a more genuine and
vital sense of the word. But I did not; and if, in this way, I sinned
against her, it was at least an unintentional sin, a sin of omission,
and one which she remained unaware of. I was egotistical and self-
centered, as it is my nature to be. She was not at all exacting. If I
would listen to her when she talked, and admire her dresses, and
enjoy her playing, and take her to the theater or to parties, she was
quite contented. She neither asked, nor appeared to expect, any
thing further. So that, though we saw each other every day, and
were together a good deal of the time, we were as far as possible
from being intimate. Our real, innermost selves never approached
each other. In fact, she and my uncle were much more intimate than
she and I. He was always having her to sit with him in his study,
where he would talk to her of the subjects that interested him, or
get her to read aloud to him, or to act as his amanuensis, and write
under his dictation. She thought my uncle was a 'perfectly adorable
old man'; and he called her 'the light of his declining years.'
“I, meanwhile, lived my own life, such as it was, in silence. But it
was not much of a life. It was not especially enjoyable, and it was
altogether valueless. I produced nothing, accomplished nothing, was
of no earthly use or benefit to anybody in the world—except a sort
of convenient appendage to my wife. My favorite occupation—the
only one that I cared any thing about—consisted in getting away by
myself, and reading. My studio was my castle. Once inside it, with
the door closed behind me, I was sure of not being disturbed. I, had
forsworn my painting, as I fancied, for good and all. I had got utterly
discouraged about it, had lost all zest in it, had vowed never to
return to it. But up here in my studio I had a lot of books; and here
for hours I would sit at the window, reading. My appetite for reading
had recently become voracious, insatiable. I can't convey to you an
idea of how dependent I was upon my books. They were the world
in which I lived, moved, had my being. Away from them, I kept
thinking about them, longing to get back to them. Not that I derived
so much pleasure from them, but simply that I was unhappy unless I
had them. They were to me, I suppose, in my dead-and-alive

condition, something like what his drug is to an opium-eater—not so
harmful, of course, but just as indispensable: a stimulant, which I
could not do without. What the books were, doesn't matter. All sorts,
from the latest sensational novel, or wildest exposition of
spiritualism, up to Milton and the Bible. Yet, perhaps, I ought to give
you the names of some of these books, for some of them produced
a very deep and vivid impression upon me, and no doubt contributed
more or less to my subsequent state of mind—helped, I mean, to
bring it on. Well, I reread Wilhelm Meister; and I read for the first
time Rousseau's Confessions, de Musset's La Confession d'un Enfant
du Siècle, and Browning's Inn Album and The Ring and the Book,
besides many of his shorter poems. I mention these five particularly,
because they were the ones that had really strong effects. They
stirred me; pierced to my heart, and hurt me; where other books
merely interested or amused me. What I mean is, they appealed to
my emotions, where other books merely appealed to my intelligence.
Especially Browning. When I read Browning, the exhilaration was
almost physical. It was like breathing some vivifying atmosphere, like
drinking some powerful elixir. It made me glow and tingle through
and through. It was as though the very inmost quick of my spirit had
been touched, and made to throb and thrill. I had never supposed, I
would never have believed, that any book could possibly have
exerted such a profound and irresistible influence over the reader.
My sensation was like an acute pain, that yet somehow verged
toward—not pleasure—something deeper and better than pleasure.
No music, not even Beethoven's or Wagner's, ever moved me, ever
carried me away, as these poems of Browning's did. They literally
transfixed me, magnetized me, like the spell of a magician. The
reason was, of course, partly because the poetry is in itself so great;
so intense, so penetrating, so vibrant with the living truth, so warm
with human blood and passion; and I don't believe that any man
could read it understandingly without being affected by it very much
as I was. But the reason was also partly personal. In The Ring and
the Book I found expressed, in clear, straightforward language, all
those deep, strenuous emotions which I myself had experienced in
my love of you, which had always groped and struggled for

expression, but which to me had always been inexpressible—
yearnings which I had felt with all their force and ardor, which I had
labored hard to speak, but which I had never been able to speak,
any more than as if I had been dumb; which, pent up in my heart,
and straining for an outlet, had sought one by means of broken
syllables, glances, caresses. In The Ring and the Book I found them
expressed; found my own unutterable secrets uttered. Oh, if only
when you and I were together I had had The Ring and the Book to
read aloud to you from! Then, perhaps, I could have made you feel
how deeply, utterly, I loved you. In the Inn Album, too, another
chapter of my own story was told, more of my own secrets were laid
bare. The material conditions, the circumstances, the accidentals, to
be sure, were totally different; but the essentials seemed to me the
same. A man had irretrievably wronged a woman—a noble, beautiful
woman, who loved him and trusted him. A lover had acted basely
toward his sweetheart. And there, also, I found an expression for my
remorse and my despair. But now I am anticipating. For the present
these thoughts had not come to me—the thought of you, and of
what had been between you and me, and of how I had wronged
you. I mean to say, they had come to me after a fashion; now and
then, spasmodically, by fits and starts; but they had not pierced
more than skin-deep, and they had not taken fast hold. They had
come and gone. Later on, they came and staid—like coals burning in
my heart. For the present, I did a great deal of reading and scarcely
any thinking. Sometimes, it is true, instead of reading, I would sit
still, looking out of the window, and carrying on a certain mental
process which might perhaps have been called thinking: but it was
the sort of thinking known as mooning. I mean it was vague, listless,
purposeless; it had no vigor, no point; and it bore no result. You, and
our love, and the misery I had caused you, were the subjects of it,
yes; but it was like thinking in a fog. It had not grown intense and
clear. It had not crystallized. It awoke in my breast a sort of
sluggish, languid melancholy, instead of the pain that I ought to
have felt, and by and by did feel—and feel now, and so long as I live
shall feel. Whatever there is in me that is not wholly bad and callous,
what I suppose would be called my better nature, was just preparing

to wake up; and these were the dull, premonitory throes. I was just
beginning to come to myself, out of a long lethargy. My remorse was
just beginning to kindle. It had not yet sprung into the white-hot
continuous fire that it has since become.”
In another place he says: “As I write to you now, what I am trying
hard to do, is to get at close quarters with the real, bare truth; to
look straight and steadily at it; and to tell you, as clearly and as
calmly as I can, what I see. But the truth is so deep and subtle,
though so unmistakable; and I am so unused to writing; and it is so
hard for me to keep down my feelings, that I can't seem to find the
right words. After I have written a sentence, when I come to read it
over, it seems almost as though I might as well not have written at
all. What I write does not express half clearly, or fully, or forcibly
enough what is in my mind. So I can't help fearing that you may not
understand. Yet my desire that you shall understand is so strong, I
am so serious, so much in earnest, I can hardly believe it possible
that my words can entirely fail to show you what I mean. If they
should do so, if in this letter I do fail to make you understand, then I
will say this: the only purpose that I have left in life will be defeated.
That is the only object that I care to live for: to make you
understand. Oh, I beg of you, try to understand. I have no right to
ask you to do any thing, to expect any kindness, any common mercy
even, from you: and yet I do ask, I implore you to read this letter
through, and to try to understand what I am trying to express. Not a
single line is written which I do not feel in the bottom of my heart. I
am striving honestly, with all my might, to strip my soul naked
before you. And when what I write seems feeble or obscure, please
endeavor to pierce through to the meaning and the feeling of it. You
have a kind and pitiful heart; and if a human being, no matter how
low or base, called out to you in great pain to stoop and do a little
thing—a little, easy thing—to soothe and relieve him, I know you
would do it. Well, that is the way I call out to you now, and beg you
to read and try to understand my letter. As I write, I feel like a dumb
man, his heart big and sore with something that presses desperately
to be spoken, laboring to speak. Well, what I want to make you

understand is this. Very slowly and gradually, by imperceptible
degrees, a great change was coming over me, was being wrought in
me. This change was really nothing but a return to health, mental
and moral health. Ever since that night on which we were to have
been married, I had been mentally and morally sick—in an
unhealthy, unnatural state. My moral nature, and many of my mental
faculties, had lain torpid and inactive, as if deadened—had not
performed their functions. Well, health was now slowly returning to
them, health and vitality. The depths of my spirit—it is a canting
phrase, but it expresses exactly what I mean—the depths of my
spirit, which had long lain stagnant, were being stirred. I had always
comprehended, as a mere intellectual proposition, how much you
must have suffered. It was obvious. Dull and half stupefied as I was,
I could not help comprehending that. It was like two-and-two-make-
four. But the comprehension had got no further than my brain. It
had not touched my heart, and made it shudder with horror, and
burn with remorse, for my own baseness, and for the agony that I
had inflicted upon you, as it has done since. I had comprehended,
but I had not felt it. My love of you had been struck dead; and my
imagination—or whatever the faculty is, which causes us to
sympathize with another's pain—was failing to act. So I had gone
about the daily affairs of my life, in no wise troubled or affected by
the fact, which I was perfectly aware of, that you, at the same time,
in solitude, were suffering the worst sorrow possible in the world—
yes, absolutely the worst; I know it. I had gone about, and got what
apology for enjoyment, what vulgar amusement, I could, out of life;
had eaten, drunken, talked, laughed, read, smoked, paid calls,
listened to music, all precisely as though you did not exist, never
had existed; and finally I had become engaged and married; and all
the while I knew what hopeless, speechless anguish you were
enduring, thanks to me; I knew it, but did not care. Now and then I
would think of it; but so dead was my heart, the thought never
aroused a single throe of pain in it. I thought of it on the night of my
wedding. In the midst of the dancing, in the midst of the loud,
romping merriment, I thought: 'What is she doing at this moment?'
But it was nothing like sympathy or self-reproach, that prompted

me. It was a sense of the curious incongruity. I shrugged my
shoulders, said to myself that I could not help it, and went on
dancing. This will show you how low I had sunken, how callous I
had become; and you may imagine how I despise myself, how I hate
and abhor myself, as I recall it now. Oh, my God! my God!—
Christine, for God's sake, when you read this, don't harden against
me, because of it, and refuse to read any more. Don't stop reading.
For God's sake, in mercy to me, go on reading to the end. Don't
close your ears against me, and refuse to listen. The only alleviation
of my torments that I have, is the hope that you will read this letter
through, and understand how I have repented.... Well, as I say, this
state of being was now slowly, gradually, changing. Not a day
passed now but I would think of you, and of every thing that had
been between you and me, from the beginning to the end; and now
these thoughts did arouse pains in my heart—vague pains, that I did
not understand—dull pains, such as one feels in sleep, or while
under the influence of an opiate—but still, certainly, pain. As I said
before, I was only just beginning to come to myself. My realization
of what I had done, of what you had suffered, of what I had made
you suffer, had not yet crystallized. My love had not yet waked up.
My remorse had not yet got really afire. But all of a sudden, one day,
the complete change came. The change was precipitated.
“It was a Friday afternoon late in February, a year ago—dark,
rainy, warmish. My wife had gone to the rehearsal at Steinway Hall. I
had agreed to meet her in the lobby, at the end, and bring her
home. All day long, that day, I had done nothing but mope. I had
sat at my studio window looking out into the gray, wet park, or up
into the heavy, inky clouds, and giving myself over to the blues—
thinking that there was the world, full of interests and activities, the
same world that I had used to find so pleasant, and in which I had
hoped to work and to be of service, the same world quite unaltered;
and that yet, somehow, unchanged as it appeared to be, it had
changed totally for me, had lost all its flavor for me, all its attraction
for me; the light, the spirit, had died out of it. I got no pleasure from
it. I was of no use in it. I was so much inert, obstructive stuff and

lumber. Then, why did I continue to exist? Neither useful nor happy,
what excuse for being had I? Why should I not at once be
annihilated and done away with? etc., etc. This was the strain that
my mind had been running in all day long. Then, toward five o'clock,
I put on my hat and walked around to Steinway Hall to wait for
Tillie. It was singular, and even now I can not account for it by any
ordinary theory, that, as I stood there in the lobby waiting, while the
audience, mostly women, passed out, I was conscious of a strange
trembling of the heart, such as one feels in anticipation of some
momentous event, such as usually accompanies what we call a
presentiment—a presentiment that something portentous for our
good or for our evil is about to happen. I could not understand it at
all. I could not imagine what it was caused by. And yet,
notwithstanding, I could not subdue it. It went on from moment to
moment getting more intense; troubling me, perplexing me. I
concluded that it must be the wind-up and climax of my blues, just
as a dull, dark day sometimes winds up and reaches its climax in a
thunder-storm. I said to myself, 'You have not felt any thing like this
for nearly a year. This is the sort of thing you used to feel when you
were in love—after you had rung Christine's door-bell, while you
were waiting and chafing for the door to be opened.' Meantime the
audience were pouring out past me, laughing, chatting, greeting
their acquaintances, putting up their umbrellas; and I was keeping a
look-out for my wife. When, all of a sudden, my heart, which had
been trembling in the way I have described, all of a sudden it gave a
great, terrible leap, and then stood stock still; and I could not
breathe nor move, but was literally petrified, rooted to the spot, and
felt a fearful pain begin to burn in my breast. For I saw—I saw you.
Oh, my God! I saw you come out of the hall, and move slowly
through the lobby, passing within almost a yard of me, so that I
could have stretched out my hand and touched you, so that, if I had
whispered your name, you would have heard me, and saw you go
down the stairs and disappear in the street. I stood there with wide,
staring eyes and parted lips, like a man turned to stone. How shall I
ever disentangle, and put before you in some sort of consecutive
order, the great crowd of thoughts and emotions that suddenly, and

all at the same time, broke loose in my heart and brain? In that brief
interval—it could not have been more than a minute altogether—I
lived through almost every thing that I have lived through since. It
was all compressed into that minute. I shall try hard to give you
some sort of an account of it, to make it as clear and as
comprehensible as I can. But I know that, however hard I try, I shall
only be able to give you a very meager and faint conception. If I
could only see you, and speak to you—if for one moment I could
kneel down at your feet, and touch your hand, and look into your
face, and utter one long, deep sigh—oh, I should feel then as
though I had in some degree expressed what was, and has been
ever since, in my heart and mind. Sometimes, when I have listened
to certain pieces of music, I have felt that in them was the
expression for my unspeakable emotions. I have felt this about some
of Chopin's impromptus and nocturnes—that if I could somehow
make you hear them, you would somehow understand. Do you know
the Impromptu in C-sharp minor? That sometimes seems to express
almost perfectly my grief and passion and remorse and hopeless
longing. But—but to touch your hand, and look into your eyes, and
sob at your feet—I would be willing to die at the end of one minute
spent that way. But see—see how I am compelled to sit here, away
from you, and realize that never, never, so long as I live, shall I be
allowed to approach you, or speak to you. Can you imagine the
agony it is, to yearn with your whole soul to speak one word to a
woman; to have your whole soul and heart and mind burdened with
something that burns like fire, and will never cease burning until you
have emptied soul and heart and mind at her feet; and to know that
she is scarcely a mile distant from you, in the same city with you;
and yet to know that if she were dead she would not be further
removed from you, it could not be more impossible for you ever to
approach her, ever to speak with her? Can you imagine that? Oh,
sometimes I can not believe it—believe that facts can be so
inexorable. Sometimes it seems against nature that a man's whole
strength, whole life, can be concentrated in one single wish, and yet
the fulfillment of that wish be absolutely beyond hope. It is too
stupendous, too monstrous. Oh, to think! To think that at this very

moment you, your own living self, are almost within reach of my
voice! It would not take half an hour to bring me to your side. And
once there, once in your actual presence—Oh, my God! This
unceasing agony would be ended, this unutterable agony would be
uttered. We two should be together once again—you and I. Oh, the
joy, the joy, to sob out all our grief together, and soothe each other's
pain! And yet, if I were at the other extremity of the earth, or if you
were dead, it could not be more impossible, I could not be more
hopeless. Christine!
“But there! I am losing control of myself, crying out and raving in
my despair. But what I have set myself to do, is to keep perfectly
calm, and, by the aid of all my forces, to try to give you a clear
statement of what I have been through. If I ever succeed in making
you realize how thoroughly I have understood your pain, how
completely I have appreciated the enormity of my own conduct, and
how bitterly I have repented it, I shall be almost happy, and I shall
have discharged a duty toward you—the only duty that I have a
right any more to owe you.
“Well, now, I tell you that in that one minute—in the time that
elapsed from the instant I first caught sight of you, down to the
instant when you disappeared in the street below—in that minute,
with intensity proportionate to the rapidity, I lived through nearly
every thing that I have lived through since. All my vivid realization of
how utterly base I myself had been, and of your unspeakable agony,
caused by me, your despair, your humiliation; all my remorse, my
yearning to atone for what could never be atoned for, to repair the
irreparable wrong that I had done; all my sense of what I had
wantonly flung away, and lost beyond recovery; all my despair; in a
word, all my love—love that had lain stunned, as I supposed dead,
but now suddenly had come to, never to let me rest any more:
these, and much else that I shall not attempt to reduce to words,
these were what sprang upon me all at once, shaking my soul to its
foundations, and holding me rigid, horrified, in their grasp. Oh, help
me to find an expression for what strains so hard to be spoken. I

have just read over what I have written. It sounds vague, cold,
formal. If I had left the paper blank, it would have done about as
well. What I have written conveys only the weak echo of what I
want to say, of what I feel. I stood there in the lobby of Steinway
Hall; and I watched you pass under my eyes; and I saw how pale
you were, how large and dark and sorrowful your eyes were; and
suddenly I knew, I understood, how I, my very self, had made you
suffer, you whom I loved, and how never, never, no matter how long
I might live, could I in any way do any thing to soothe you, to
comfort you, to make up to you for the suffering I had caused you; I
knew and understood all this; and my heart went out to you,
bounding and burning with a thousand fierce emotions, with an
anguish of remorse and love—oh, my sweet, injured lady beautiful,
frail Christine!—and now, now when I try to give you some faint idea
of it, I am as helpless to do so, as if I were trying to scream out in a
nightmare, and my voice failed me, and my tongue clove to the roof
of my mouth. What if I had trampled down all conventional
restraints, and then and there, in spite of the crowd, in spite of
every thing, had rushed forward and stopped you, and thrown
myself upon the ground before you, abasing myself at your feet, and
just moaned out loud—letting it all burst forth in one good, deep,
satisfying sob? My heart throbs hard at the thought. Yet, of course, I
had no right to do it. If I had done it, I should only have relieved
myself, at the cost of paining you—you whom, God knows, I have
already pained enough.. . . Oh, well, I must try to do my best with
pen and ink. Well, as I say, I stood there, breathing heavily, at last,
after many months of death, at last alive, I stood there like that,
when—when my wife came up, and took my arm, and demanded,
startled by my appearance, what the matter was. My wife! And I had
just seen you; and my soul was full of you, you whom I had
wronged and lost! And here was my wife, taking my arm, speaking
to me, emphasizing the antithesis. The past and the present! What I
had given up, and what I had got in place of it! After my glimpse of
you, the reality—Tillie! Oh, it was as though a starving man had just
seen bread, smelled meat, and then, looking into his own hand, had
found a stone there. She took my arm; and I turned her question as

best I could; and I led her home. Conceive how, as I walked home
from Steinway Hall this Friday afternoon, the ghost of a certain other
Friday afternoon bore me company. One Friday afternoon, only a
little more than a year earlier, in December, 1882, you had gone with
me there, to hear the Damnation of Faust. Do you remember? You
had sat at my side, close at my side. You had looked into my eyes,
had touched my arm, had spoken to me. The sweetness of the rose
that you wore in your bosom, had filled my nostrils. For one instant,
one delirious instant, your breath, your very breath, had fallen upon
my cheek! You had allowed me to wrap you in your cloak, when you
felt a draught—in the fur circular you used to wear; I remember the
faint perfume that always clung to it. We were so intimate, so
confidential, you and I! You were happy. And I loved you; and I had
the possibility of winning your love open before me. And now! God,
to think that the possibility which that afternoon held safe in store
for me, had been used and wasted! To think that by no remaining
possibility it could ever be won back! Every thing was destroyed. I
myself, by my own voluntary act, had destroyed every thing—even
hope. Well, well, my wife and I walked home. My brain and my heart
were burning. Chaos was let loose in them. I wanted to scream out,
to beat my breast, to rend my garments. But I had, instead, to put
on an indifferent face, exchange commonplaces with her, take her
home; and, it being Sabbath by this time, had to join in the praying
and the Scripture-reading, and all that. Of course, I was eager, wild,
to get away, by myself. But I had to sit it out with the family—my
wife, her mother, my uncle—till ten o'clock that night. I was pretty
nearly beside myself. But at last I escaped, and got into my studio.
There is no use my writing about that night, the night I passed alone
up here in my studio—alone with you; for, so intense was my
thought of you, you were all but palpable at my side. I had given
you back, as I supposed, all your letters—every keepsake I had to
connect me with the past. But this night, as the reward of much
ransacking, I found in the drawer of my desk the very first note you
had ever written me, the one in which you said you would go with
me to the exhibition. Do you remember? How we walked up and
down the galleries? And how you leaned upon my arm? And the little

red bonnet that you wore? And how, afterward, we went to
Delmonico's? That little note, ever since, has been the most precious
of all my possessions. Your own hand traced these letters! Your own
breath fell upon this paper! What effect it had upon me that night, I
shall not attempt to tell you. Think of this: it still kept a faint trace of
its fragrance—of the sweet smell it had had, when you first sent it to
me. That that should have remained, that immaterial, evanescent
perfume! That that should have outlasted the rest! No; there is no
use of my writing a line about that night. I should only be
incoherent, if I tried. All I will say is this: if you had cared about
revenge, and had witnessed my suffering that night, you would have
been satisfied.”
Still elsewhere, he goes on as follows: “Christine, what I want to
say to you is very simple. I don't understand why I should have so
much difficulty in saying it, why every attempt I make at saying it
should be such a wretched failure. I suppose it is because, when I
bring my mind to bear upon it, when I look it squarely in the face, it
appalls me so, I get so excited, my feelings get so wrought up, that
I lose the self-command which a man must retain, in order to
express himself clearly and fully with his pen. It is as if, instead of
saying what I have to say, fluently and directly, I were to falter, and
stammer, and gasp forth inarticulate, unmeaning sounds. If only the
impossible were not impossible; if only the hopeless were not
hopeless; if for one minute I could stand in your presence; alone
with you, and look into your eyes, and touch your hand, and speak
one word to you—just call you by your name, Christine!—or, no, not
even do that, not even speak, but simply stand there silent, and look
at you: then, I feel sure that somehow you would understand, and
then I could find something like peace. You would understand by
instinct, by intuition, what my mind and heart are full of. If such a
meeting might only come to pass! But I do not delude myself. I
know that it never can come to pass—never, not if we go on living in
the same city for fifty years. Constant and intense as my longing to
see you is, fiercely as my heart beats at the thought of meeting you,
I know that I might as well long to see, think of meeting, one who is

dead. I am a married man, and have no right to seek to see you. But
even if I were not a married man, you, whose scorn and hatred of
me must be bottomless, you would spurn me, you would refuse,
shuddering, to look at me, or to listen to me. I know it. Even if you
ever, in your holy goodness and mercy, can forgive me in some
degree for what I have done, I know you never can forgive me
enough to let me approach you, to let me speak to you by word of
mouth. The mere idea of meeting me, I suppose, must always be
full of horror for you. I can never atone for the wrong I have done
you. I can never even tell you of my remorse, and beseech your
forgiveness, except by writing. So I write, begging you, in charity, to
read and to try and get my meaning. If it were not for the hope that
you will read this letter through, I believe my agony would drive me
mad. This hope is the only thing that mitigates it, and makes it
bearable.
“Well, then, here is the simple truth, told as simply as, by my
utmost effort, I can tell it. For a period of some months, I had been
in a condition which you must let me compare roughly to
somnambulism—a sort of daze, a dull, half-waking trance. While in
that condition, a great number of my mental and moral faculties had
lain absolutely dormant—just as much so, as if I had not possessed
them. From that unconscious fit into which I fell on the night of our
wedding, I had never perfectly recovered. My body had recovered,
yes, and a part of my mind—the every-day, working part. But the
rest of my mind, the better part of it, had never emerged from the
coma which it sank into then. And during this period, I want to say, I
do not think I was, in the ordinary sense, responsible for what I did.
I was mentally responsible: that is, I knew what I was doing, and I
chose to do it. But I was not exactly morally responsible, because
morally I was blind. My moral sense—my heart and conscience, I
mean, were in a state of suspended animation; and I acted without
their guidance. I don't say this with a view to excusing myself. I say
it, because I honestly believe that it is true, and because, to some
extent, it accounts for my otherwise unaccountable way of acting.
Well, let me call it somnambulism. Then, on that Friday afternoon,

when I so unexpectedly caught sight of you in the lobby of Steinway
Hall, there, at that instant, all of a sudden, I woke up; I came to my
senses, in heart and mind was my complete self again. And awaking
in this way, getting my moral eyes opened, my moral faculties into
running order, I then for the first time, saw, realized, understood,
what, while in that irresponsible, somnambulistic state, I had done.
Dumfoundered, aghast, I saw the ruin I had wrought—ruin of your
life, your world, and of mine—total, hopeless ruin. I have read of a
man who dearly loved his wife, and who, one night, in his sleep, got
up and murdered her. When he awoke next morning, and found her
lying dead beside him, and made the horrible discovery that he
himself had done it—well, he must have felt a little as I felt after I
had seen you that day at Steinway Hall. And the worst of it—the
aspect of it which was most unbearable, most infuriating—was this
knowledge, that loomed up before me, as big and as unalterable as
a mountain of granite: the knowledge that what I had done could
never be undone; that the desolation to which I had reduced our
world, could never be repaired; that, no matter how bitter my
remorse was, no matter how poignant my regret, I could never
atone for the wrong I had committed, never could win back again
the treasure I had thrown away. It was a mountain of granite, I say,
against which, frantically, with all my puny strength, I dashed
myself; thereby making no impression, but falling back, bruised,
stunned, disheartened. My knowledge now of your suffering, my
knowledge of how I had made you suffer, and that, though my
whole life yearned toward you with tenderness, love, contrition,
unutterable, I never in all my life could do the slightest, smallest
thing toward making amends to you, toward soothing the pain,
healing the wounds, that I had inflicted upon you—upon you, my
pale, sweet lady—oh, I ask you to imagine how heavily that
knowledge weighed upon my spirit, how sharp its clutch was, how it
would never let me rest, never allow me a moment of forgetfulness,
but clung constantly and grimly, a monster with which it would be
futile for me to hope to struggle. That last meeting between us,
when you came here to my studio, to this very room, to the room I
am writing in now, and I here, in my uncle's presence, threw you

down and trampled upon you, and allowed him to lead you away,
crushed and bleeding—that last meeting, when I still had it in my
power to spare you all that shame and sorrow, to take you in my
arms, and quiet all your pain, and kiss away all your fear, and to
keep you—keep you for myself—oh, you may imagine how my
memory of that meeting, my realization of how I had hurt and
humiliated you, my recognition of the wasted possibilities it had
held, would not out of my heart, but abode there all the time, eating
into it like acid. The walls and ceiling of the room, which had been
witnesses of that last meeting, seemed eternally to be crying it out
at me. When I looked at the floor, it was as if I saw a blood-stain
there where you had stood. Oh, to think that there for one long
minute you did really stand, you yourself, within arm's-reach of me;
and I might have put out my hand, and touched you, and taken hold
of you, and kept you to me forever, but did not! To think that I let
you go; and you went; and I did not call you back! Oh, God, if I had
only come to my senses soon enough to have called you back! But
no, no; you went; and there was an end of it all. Love, happiness,
hope, all went out with you. I drove you out. I drove them out.
Christine, for every single pain that I inflicted upon you at that
meeting, I ask you to believe, I have never ceased to pay with the
acutest anguish that I am capable of feeling. That spot on my floor
where you stood—ah, God, how many thousand times have I kissed
it since! Ah, God, if there were only some power in earth or heaven
that could bring you back there, make you stand there, again, for
just one minute more! And it was I—I, whose soul goes out to you
with an immensity of love that I can not find words for—— I, who
would give all the rest of my life for the privilege of caressing and
comforting you for a single instant—I, whose place it was to shield
you and protect you—I myself, who drove you awray from here,
heart-broken, never to return. Oh, my beautiful, pale darling!
Christine, lost, lost forever! Here am I, my heart bursting with the
desire to be, in some way, of some sort of service to you; and there
are you, needing perhaps some little service: and yet if we were
upon different planets, it could not be more impossible for me ever
to lift my finger in your aid! Oh, I say, it is infuriating. It is too much.

Oh, if I could tear open my breast, and let you look in, and see!—
see the love, the remorse, the despair, that are stirring in perpetual
fever there.. . . Oh, the misery I caused you! The long, hateful days
that you had to drag through afterward, while I was amusing myself,
dining out, learning to dance, getting engaged and married! Far and
wide, as far as your eye could see, the world, which had been a fair
and fragrant garden in your sight, had crumbled suddenly to a bleak
waste of dust and ashes. The hand that you loved had dealt you a
blow worse than a death-blow. You had entrusted your happiness to
me, and I had betrayed my trust; had taken it, and deliberately
dashed it to the ground, and shattered it beyond possibility of
mending. My frail, beautiful lady. Yes, if I had stabbed you with a
knife, I should not have been so brutal, so base, so cruel; your pain
would not have been so great; I should have less to reproach myself
with to-day. Yes, I know it.”
But, the reader may curiously ask, how about his theology? his
belief that it had been the act of heaven? This question he touches
upon only incidentally, and disposes of briefly: “In the light of my
resuscitated love, the mere remembrance of that blasphemous
delusion filled me with loathing for myself—made me shudder, and
draw back, sickened. It was a monstrous lie. I can not bring myself
to write about it.” And on another page, he says: “My superstition
was the dragon, whose breath poisoned our joy, withered our world,
burned out our hearts. The dragon was killed at last, but too late—
after its ravages had been accomplished, after it had done its worst.”
I may seize this opportunity, also, to request that if Elias is not
always so scrupulous about his syntax and rhetoric as one might

wish, the reader will charitably pardon him, in view of the high
degree of mental excitement under which he is manifestly laboring.
“Well,” he continues, “after this reawakening, what of my life?
Externally my life went on precisely as before. I was married. I had
married of my own free will. I knew that, however detestable my
marriage might now have become to me, I was bound in all honor
and decency not to do any thing that could make my wife unhappy. I
had already done mischief enough in the world. I must not, if I could
help it, do any more. I must keep my secret. Though all the forces of
my body and soul were sucked up and concentrated in that one
fierce secret, as they were, I must not let it appear. So, the relations
between my wife and myself went on precisely as before; and I tried
to be a good husband to her, and to give her what pleasure, and
spare her what pain, I could. The same theaters, dinners, parties;
the same talk about dresses, the same piano playing. Sometimes,
even while, with as much nonchalance of manner as I could master,
I was listening to her prattle, my secret would be burning so hot in
my breast, it was a wonder to me that she did not guess it, or
suspect it—that she did not feel it. Sometimes, even while I was
directly speaking to her, answering some question that she had
asked me, or what not, my heart was being wrung by such strong
emotions, it seemed as though she could not help but divine them.
It was hard work, keeping this constant guard over myself, wearing
this mask. But, of course, I was in duty bound to wear it. The relief
was immense when I could get away by myself, and let it drop off.
Away by myself, I could, any how, be myself—lead my own life,
without dissembling.

“My own life—what was it like? Well, outwardly it was a life of
silence and inaction. My real life was an inward life—lived in my own
heart. My heart was like a furnace. Shut up there, my love, my
remorse, my despair at the past, my hopelessness of the future, a
hundred nameless, restless, futile fears and longings, burned
steadily all day long from day to day. Sometimes one emotion would
be paramount, sometimes another. Sometimes memory would take
possession of me; and, seated at my studio window, with my one
relic of you clasped in my hand, I would go back, and live over again
all that had passed between us, from the day when I first saw you,
down to the day when, in this same room, I had put you from me.
Do you remember that day—the day I first saw you? Do you
remember our first speech together? And how awkward I was? and
embarrassed? Do you remember the night of the party—New Year's
Eve— when the heel of your slipper broke off? And how jealous I
was? And how angry you got with me? And how you scolded me?
And then—in the carriage, going home? Do you remember your
birthday? and mine? The silk handkerchief you embroidered for me
with my initials? The concerts we used to go to together? and the
little suppers afterward? The books we read together? Detmold? The
Portrait of a Lady? The poems you were so fond of? The letters we
used to write to each other, even when we were going to see each
other the very same day?... Or, perhaps, instead of sitting still here
at my studio window, I would leave the house, and go for a walk in
the old places—the places that were associated with our love, and
now for me were sorrowfully consecrated by it. I would walk up
Eighth Avenue, over the ground that I had used to cover every time
I went to see you; would cross the great circle at Fifty-ninth Street;
would come within eye-shot of your door, look up at your window,
recall the time when I had had right of entrance, wonder what you
were doing now; would enter the park, and even seek out our pine-
trees, and stay for a while there in their shadow—there, where—! Do
you remember? You may imagine whether this was bitter-sweet. To
go back to the time when you had been mine, wholly mine, and live
over all the rapture of that time, in all its minute, intimate details;
and then, with an infinite hunger for you gnawing in my heart, to

return to the present, look into the future, and realize that I, by my
own act, had let you go, had lost you forever! You may imagine with
what woe and fury, deep and frantic, and yet dumb, I would recall
and repeat to myself that verse of Rossetti's poetry: 'Could we be so
now?' And there was the truth, the relentless truth, for me to
confront, and reconcile myself to, if I could: 'Not if all beneath
heaven's pall lay dead but I and thou, could we be so now!' The
truth which, as I said, was like a mountain of granite, separating you
and me. Oh, but at other times I could not believe that the truth was
the truth. It was too cruel. It was incredible. It must be some
hideous hallucination—some nightmare, that I should sooner or later
wake up from. I could not believe that it was in the possible order of
nature for a man and a woman to have loved each other as you and
I had loved each other, and yet to have become so utterly lost to
each other as it now seemed that we were; for two human lives to
have been so perfectly fused together, blended together like two
colors upon my palette, and yet afterward to have become so
completely rent asunder. I could not believe it possible for my soul to
yearn toward you and thirst for you constantly, as it did, and yet be
debarred forever from any sort of communion with you. It seemed
as though somehow, sometime, somewhere, we must come
together—you and I once more!—and all our sorrow be swept away
by the great joy of our reunion. Oh, Christine, if it might be so! If
only it might be so! At these moments my imagination would break
the bonds of reason and fly off in daydreams, long, delicious flights
of fancy, visiting wondrous air castles where you and I dwelt
together—only shortly to drop back upon the awful reality. The
reality: I married, and all your love for me, your priceless love for
me, by my fault, turned to horror and hatred. And yet, in spite of the
reality, in the very teeth of it, I would think: 'Well, what if my wife
should die?' As long as I am telling you the truth, I may as well tell
you the whole truth, no matter how bad it may make you think I am.
Yes, I would say: 'What if my wife should die?' And then I would
repeat to myself what you had once said about that very same verse
of Rossetti's poetry: 'I can't understand why it should be so
absolutely hopeless. If they really were all alone together, and she

saw how dreadfully he had suffered, I don't understand how she
could help forgiving him and loving him again.' And then, for an
instant my heart would bound with something like hope. But only for
an instant. As soon as my reason could make itself heard, I would
acknowledge that I had sinned too much ever to expect forgiveness
from you. No, it would be past human nature.... At still other times
my uppermost feeling would be simply an intense desire to see you
—not for any special purpose, not with a view to speaking to you—
simply a craving for the sight of your face. I felt that if I could only
look upon you for an instant, catch one brief glimpse of you, I
should have something to remember and cherish, something for my
heart to feed upon, which was feeding upon itself. It would be an
agony.
“I knew that. The mere thought of it was that. But it would also
be the nearest approach to a joy that I could expect. So, in the hope
that I might see you, I would stand for hours on the corner of your
street, in the snow, in the rain, in the hot sun or cold wind, watching
the door of your house, waiting for you to pass in or out—very much
as, in the old times, I would watch the door of a house where I
knew that you were visiting, and wait to join you at your exit. (Do
you remember? And how surprised you always used to be?) But I
was always disappointed. I never once saw you. I would walk, also,
in those quarters of the city where ladies throng to do their
shopping; always searching for one face in the crowd, but never
finding it. And I haunted regularly the rehearsals at Steinway Hall
and at the Academy of Music, closely watching the audience as it
passed out, always hoping that my experience of that afternoon in
February might be repeated, invariably getting my labor for my
pains. Where did you keep yourself? Oh, sometimes I felt that I
positively could not live without a sight of you. I was starving for a
sight of you. Only to see you for one little moment! Only to feed my
heart with one brief glimpse of you! That did not seem such a
greedy or unreasonable desire. It could do you no harm, provided I
were careful not to be seen, as well as to see; and I meant to be
careful about that. It could do no living creature harm; and to me—

oh, to me it would be like a drop of water to a man consumed by
thirst. Then my wish would become the father of my thought. I
would say:
“'Surely, if I go out now, and scour the city, visiting every spot that
in any possibility she may visit—the shops, the park, Fourteenth
Street, Twenty-third Street—surely, at some point our paths will
cross each other, and I shall see her.' Well, I would go out. I would
give my thought a trial. I would walk the streets till I was fagged out
and foot-sore. I would come back home, with a heart sick for hope
deferred.... What fears tormented me all this time, you will surely be
able to conceive for yourself. How could I know but that you might
have died? One morning at the breakfast-table my uncle glanced up
from his newspaper, and, looking very queerly at me, said, 'Here,
Elias, here's news for you. An old friend of yours is dead.' With a
horrible, sick heart-leap, I thought: 'Ah, she is dead.' With as
indifferent an air as I could put on, I asked, 'Who?' He handed me
the paper, pointing to the death notices. It cost me all my strength
to look; but I looked. Yes; there I saw your name, Redwood. With
the courage of despair, I read the notice. 'No; it was not you; it was
your father. But how could I know—what assurance had I—that you
had not died, too, without my chancing to learn of it? The thought
that you might have, got to be a fixed idea in my brain. There was
no way by which I could find out. I knew nobody to whom I could
apply for information. But at last, one day, by accident, in looking
through a newspaper, I again caught sight of your name, Redwood.
Ah, how the sight of it made my temples throb! I read that you had
been appointed a teacher in the Normal College. So, my doubts on
the score of your death were set at rest. It may seem strange to you
that I should care so much whether you live or die, since already
you are as far and as hopelessly removed from me, as if you were
dead; yet the thought that you may die is the blackest of all
thoughts to me. I don't know why it is, but I feel that so long as you
remain in it, the world will not be quite a blank wilderness to me.
There is still some warmth, some beauty, in the light of day, which
would go out utterly if you were to die. So long as you live, I want to

live. It seems as though there were something to live for; though I
can't tell what. But if you were to die—oh, God! if she were to die! I
pray God to put an end to my life at once. Oh, don't die, Christine.
Oh, to think that if you were to die, I might not hear of it, and might
go on living! To think that I can do nothing to make life worth living
for you! Nothing to protect you from the danger of death! To think
that if you were lying on a sick bed, and I knew it, I could do
nothing to soothe you, to nurse you back to health! Oh, Christine!
Oh, God grant that at least we may both live until I have finished
this letter, and you have read it! I must not die, you must not die,
until I have finished, and you have read, this letter.... Once in a
great while, once in six or eight weeks, or even seldomer, I would
dream about you. These dreams were the one luxury of my life,
being, as they were, the one means of escape from my life;
reversing, as they did, the real truth of my life. Every night, when I
lay down to sleep, I would think to myself:
“'Perhaps to-night I shall dream of her. She will come to me in my
dream.' These dreams always annihilated the recent past, and
carried me back to our happy days. You were mine again, with me
again. All was as it had been. My lost treasure was for a brief space
restored to me. The great joy that I experienced in these dreams, I
can not describe. It was boundless, unspeakable. Of course, to wake
up in the morning, and realize that it had only been a dream, was
hard. To wake up, and look around me, and see the walls of mv
bedroom, the view from my window, and breathe the air, and listen
to the sounds, of the morning, all quite unchanged, just as they had
used to be in the old time; and then to think how completely all the
rest was changed—changed beyond possibility of retrieval—you and
your love lost to me forever—that was hard enough. It was like a
famished man dreaming of food, and waking up to find a stone in
his hand. And yet—and yet, so great was the rapture of them, while
they lasted, my dreams were worth purchasing at almost any price;
certainly, at the price of the pain of waking. To see you, to speak to
you, to touch you; to be spoken to, and touched, by you; to hold
your little, soft, warm hand in mine, to hear the music of your

laughter, to breath the fragrance that the air caught from your
presence, to gaze into the depths of your eyes, even though in a
dream—it was better than nothing, wasn't it? Better than never,
dreaming or waking, to see you at all. So, as I say, every night I
would hope to dream of you—notwithstanding the thought that
perhaps I had no right to dream of you, that you perhaps would
begrudge me the possession of you, even in my dreams; but, as I
say, my hope was rewarded very seldom—not oftener than once in
every six or eight weeks. This was strange, seeing that you absorbed
my mind constantly, all day long, every day.
“I believe I called my life purposeless and hopeless; but it was not
exactly this. One purpose and one hope, each forlorn enough, I
clung to. They furnished the only light that I could see, as I looked
forward into the future. The same hope and purpose that animate
me now, as I write. I purposed and I hoped, sometime, by some
means, to let you know—to let you know what I have been trying to
let you know by all this writing; how thoroughly I had appreciated
my own brutality and baseness, how intensely I had realized your
suffering, and how my heart was devoured by remorse, despair, and
love. This desire to let you know, was the one constant desire that
never left me. It was like an extreme thirst, that would not let me
rest till I had satisfied it. I could not understand it. Even now I do
not understand it. What good could it do either you or me? No good
to you, surely; for the most that you can possibly care about, in
regard to me, is to be let alone, and allowed to forget me. And what
good to me? Would it give you back to me? Would it allay my
remorse? Not unless it could undo the past, and blot out the pain I
had caused you. Would it rekindle your love? I might as well expect,
by my touch, to raise the dead, as ever, by any means, to rekindle
your love. Would it even win for me your forgiveness? I knew that it
was not within the capacity of human nature, ever really, from the
bottom of the heart, without a reservation, to forgive such wrong as
I had done to you. This was what my reason said; and yet, despite
all this, I felt—and still feel, and can not help feeling—that somehow
I ought to let you know, that it was only right to let you know. I

longed to let you know. That is the substance of it. I longed to let
you know; and my longing defied my reason, just as hunger defies
reason. If I could only let you know, it seemed as though both you
and I should then be able to find something like peace and repose.
My soul ached to unbosom itself before you; and all reasoning to the
contrary notwithstanding, my instincts told me that you, as well as I
myself, would be happier—at least, less unhappy—afterward. It was
as though I had something big and heavy in my heart, that pressed
to be got out; that would strain and rack my heart until it was got
out; and that could only be got out by letting you know. I suppose
this is always the way, when a man's heart is full of conscious guilt.
But how to let you know? Oh, my impulses answered at once. They
said: 'Seek her out. Kneel down before her. Look into her face. Touch
her hand. Give it vent—let it all burst forth—in one good, long,
satisfying sob! Then, she will understand. She will understand what
is too deep, too passionate, for any speech. Her heart and yours will
be at rest. This anguish will be relieved.' Oh, how my temples
throbbed, how my breath quickened, how my whole spirit thrilled, as
I allowed myself to shape that thought. You, my frail darling, whom
I had hurt so! You, my sweet rose-lady, whom I had torn, and
crushed, and made to bleed! Christine, pale, sad Christine! To spend
one moment weeping at your feet, trying a little to soothe and
comfort and console you, to atone a little for the sorrow I had
caused you, to pour out my love and my remorse before you! Oh,
good God! But of course, of course, I knew that I might as well hope
to speak with one who was dead. I, a married man, had no right,
even in my own secret thoughts, to wish for such a meeting
between you and me. And you, despising me, you would fly from
me, you would never permit me to draw near to you. And yet, it is
so hard to reconcile one's self to the truth, even when one can have
no doubt about it, I would go on hoping, in spite of the
hopelessness, in spite of the fact that I had no right to hope—hoping
that somehow the impossible might come to pass. But at the same
time, I would think: 'How else? Is there any other way?' Necessarily,
it occurred to me to write. But the idea of writing was repugnant. I
never could tell the half of what I had to tell by writing; and then,

what assurance had I that you would read my letter? (What
assurance have I, even now?) So, for the time being, I put the plan
of writing out of my head; and went back, and asked again: 'How
else?' Was there no possible method by which I could let you know
what weighed so heavily, so heavily, upon my mind? Sometimes the
most absurd notions would seize hold of me, with all the force of
realities. For a little while, this would become not merely a theory, as
of a thing conceivable, but a conviction, as of a thing actual; that,
thinking of you as constantly and as intently as I did, by some occult
means in nature, my spirit was enabled to transcend the limitations
of space and matter, and to reach yours, and to communicate with
it. For hours at a stretch, I would sit here at my studio window,
harboring this delicious fancy: that now, at this very moment, by the
operation of some subtle psychic force, you were receiving the
message which my heart was sending you. I had read of such things
in wonder-tales, even in serious pseudoscientific treatises. Why
might there not be something in them? But, as I have said, only for
a little while could a fancy like this hold its place. In a little while my
common-sense would assert itself, and bring the dismal truth
looming up again stark before me. All of a sudden, one day, I
thought of my painting. It made my pulse leap. It seemed like an
inspiration. I would paint a picture which—if you saw it; and if I sent
it to the exhibition, you would very likely see it—which would tell you
the whole story. In a fever of impatience to get the picture begun,
and without having stopped to determine what the picture was to
be, I procured canvas, paints, brushes. Then I paused, and asked:
'But what shall I paint?' It did not require much thinking, to make
the futility of the whole design clear to me. Unless I could tear my
heart out, and paint it, with all the fierce passions fermenting in it, I
might as well not paint any thing at all. Now, at last, you see, I have
returned to my former plan of writing. I have done so, in despair of
any other means, and because it is no longer possible for me to hold
back. I have held back until I am tired out, worn out. I have been
writing at this letter, from time to time, during the past fortnight. To-
day is Friday, February 13th. I have much left to say. As soon as it is
finished, I shall send it to you.”

“As soon as it is finished!” It was never finished. Events now
supervened, which interrupted it, and prevented its completion.
Those events, it will be my business, in the concluding chapters of
this story, to relate.

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