Elements of Microwave Networks Basics of Microwave Engineering pdf pdf
ELEMENTS
OF MICROWAVE
NETWORKS
z
Basics of Microwave Engineering
Nor-/lwcls/rmUniv., USA
Y
zy
zy
CARMINE VITTORIA
zyxwvu
$World Scientific
Singapore*New
'
Jersey*London*HongKong
zyxwvutsrq
zyxwvut
zyxwvutsrq
zyxw
Published by
World Scientific Publishing Co. Re. Ltd.
P 0 Box 128. Famr Road. Singapore 912805
USA oflce: Suite 1B. 1060 Main Street. River Edge. NJ 07661
UK ofice: 57 Shelton Street. Coven1Garden, London WQH 9HE
Library of Congress Cataloging-in-Publication Data
Vinoria C.
Wemenu of microwave networks : basics of microwave engineering /
Carmine Vittoria.
p. cm.
Includes bibliographical references(p. ) and index.
ISBN 9810234244
I. Microwave circuits -- Mathematical models. 2. Microwaves - Mathematical models. 3. Microwave transmission lines -- Mathematical
models. 1. Title.
TK7876.VS8 1998
98-10764
621.381'32 dc21
CIP
--
British Library Cataloguing-in-Publication Data
A catalogue record for this book i s available from the British Library.
zyxwvuts
Copyright Q 1998 by World Scientific PublishingCo PIC. Lid.
All rights resenwd. This book or parts thereo/: may not be reproduced in any form or by any meam.
electronic or mechanical. includingphofocopying. recording or any information storage and retrieval
system now known or to be invented. without written permissionfrom the Publisher.
For photocopying of material in this volume. please pay a copying fee through the Copyright
Clearance Center. Inc., 222 Rosewood Drive. Danven. M A 01923. USA. I n this case permission to
photocopy is not required from the publisher.
This book i s printed on acid-free paper.
Printed in Singapore by Uto-Pnni
zyx
zyxwv
zy
Preface
The purpose of this book is to bridge the knowledge gap between fundamental
textbooks used in required courses and the rapid advances made in the
microwave technology. Today, microwave systems and/or requirements are
too complex or advanced to be analyzed by simple graphical approaches as
the Smith chart, for example. As such, this book is to be used in an elective
course at the undergraduate level. In putting together this book, I chose topics
which (1) did not duplicate topics covered in textbooks used in required
courses, (2) were a logical extension of required courses, and (3) reached out
to a larger audience of students majoring in science and engineering.
zyxwvu
Using criteria (1) as a basis I have purposely left out all formal
derivations of Maxwell's equations and subsequent material related to it. I
introduced Maxwell's equations as a given fact only to make some
connections in later chapters. From criteria (2) the following topics were
chosen: (a) lossy medium, (b) scattering S parameters, (c) matrix
representation of wave propagation, and (d) microwave properties of ferrite
networks. All microwave systems contain some element of losses. As such,
lossy medium as well as absorbing networks were extensively covered. The
S matrix and its properties is introduced for the sole purposes of preparing
students for the demands of the microwave industry. The matrix
representation of wave propagation has been used for many years in other
fields of science. We have re-introduced it here in relation to microwave
networks. Finally, ferrite usage represents the largest industry in technology.
We should introduce topics in ferrite networks in a way that is complementary
to the core material in undergraduate education. We need to go beyond the
description of a scalar permeability at the undergraduate level.
The book consists of eight chapters. The first two chapters are
introductory material for students who do not have an adequate background
in electromagnetic theory, such as students in other engineering disciplines
and science majors. These two chapters are not to be covered in a regular
course. A semester course covers six chapters starting with chapter 3. It is not
necessary to cover every single topic in each chapter. It is possible to adapt
or re-arrange the topics in a way to meet the needs of students depending on
the makeup of the class. For example, as an elective course we covered
chapters 3-8 minus topics in ferrite networks and specialized topics in a
quarter which is shorter than a semester. Although it was an electrive course
V
vi
zyxwvuts
zyxwvutsrq
zyxwvu
zy
Elemcnb o j Micrmwve Nctmonkr
zyxwv
zyxwv
for juniors/seniors, the book could be used as a springboard to a more
advanced course at the graduate level bu including advanced topics in ferrite
networks and matrix representation of microwave networks in general.
I wish to thank Professor George Alexandrakis for his hospitality while
I was preparing the manuscript at the University of Miami. Also, the able
assistance of Drs. H. How and W.Hu, and Y. Biaz is acknowledged.
zyx
zyxwvut
Contents
Preface
zyxwv
z
V
Review of Maxwell Equations
Chapter I
A. Maxwell's Equations in MKS System of Units
B. General Constitutive Relations
C. External, Surface and Internal Electromagnetic Fields
D. Practical Example
E. Electric Polarization of Microwave Signal
F. Microwave Response and Polarization
References
Examples
i
1
2
5
6
10
13
14
zyxwvu
14
Common Waveguide Structures
Chapter II
A. Parallel - Plate Waveguides
B. Coaxial Line
C. Rectangular Waveguide
References
Examples
21
21
26
29
33
34
Chapter 111 Telegraph's Equations
A. Types of Transmission Lines
B. Wave Equation
C. Connection to Circuit Parameters
D. Formal Solution
E. Electrical Quantities
F. Bounce Diagrams
References
Examples
39
39
39
Chapter IV Analytical Solution
A. Lossy Case
B. Real Time Solutions
C. Lossless Case
48
52
57
68
79
79
90
91
96
97
vii
viii
zyxwvutsrq
zyxwvu
zyxwv
zyxwvu
Elements of Micrwwovc Networks
D. Determination of I, and I,
E. Coupling Between Source and Transmission Line
References
Examples
102
109
112
113
zyxwv
Chapter V
Graphical Solution
A. Mathematical Terminologies
B. Plot of r ( - I )in Complex Plane
C. Projection of Z(-I) onto the r(-I)Complex Plane-Smith Chart
D. Projection of V(4) on Smith Chart
E. Projection of I(-() on Smith Chart
F. Graphical Methods for Lossy Lines
References
Examples
127
127
128
131
141
146
147
150
151
Chapter V1 Special Topics in 'hansmission Lines
A. Stub Tuners
B. Graphical Solution
C. Quarter and Half Wavelength Transmission Lines
D. Microwave Absorbers
References
Examples
160
1 60
Chapter VII Electromagnetic Scattering Parameters
A. Definitions of S-Parameters
A. 1 . Algebraic Properties of S Matrix
B. Relationship Between Measurements and Network
Electrical Parameters
C. Example of Inverse Scattering Problem
D. General Comments
References
Examples
199
199
199
Chapter VIlI Matrix Representation of Microwave Networks
A. Transfer Function Matrix of Two Port Network
163
169
176
188
189
210
214
215
216
217
224
224
z
zyxwv
zyxwvu
B. Transmission Line Analysis Using Matrix Representation
C. Connection Between Scattering Parameters and Matrix
Representation
D. General Properties of Matrix Representation
D. 1. Single Networks
D.2. Cascaded Networks
D.3. Periodic Network Systems
D.4. Application
E. Some Special Properties of Ferrite Networks
F. Relationship Between Scattering S-Parameters and
Matrix Elements
G. Example of Inverse Scattering
H. Special Applications of Matrix Representation
1. Three and Four Port Networks
I. 1. Three Port Network
1.2. Four Port Network
J. Equivalent Circuit of Ferrite Circulator
References
Examples
Index
226
229
232
232
236
237
24 1
244
247
252
254
256
257
26 1
262
264
265
283
zy
zyxw
zyx
Chapter I
Review of Maxwell Equations
The foundations of electromagnetic wave propagation in a medium are
Maxwell’s equations. In this book we introduce measurement quantities such
as voltage, V, and current, I, to solve for electromagnetic wave propagation
characteristics of specialized guided structures. As a result of introducing
measurement variables V and I. it is mathematically convenient to introduce
electrical circuit parameters, such as R, L, C and G. Before we go much into
details of the analysis of wave propagation let’s re-acquaint ourselves with
Maxwell equations written in terms of field variables E and f i , the electric
and magnetic field variables.
zyx
zyx
A. Maxwell’s Equations in MKS System of Units
Maxwell’s equations in MKS maybe written in the differential form as
follows:
where
zyxwvu
zyxw
j = d
The two constitutive equations are usually written as
B=h(fi+@=pfi
Gi=&$+jLEE
mi
,
The units of each field quantity are defined as follows
J = Current Density (amplm’)
B = Magnetic Flux Density (webedm’)
H = Magnetic Field Intensity (amp/m)
1
2
zyxwvutsr
zyxwvuts
zyxwvu
zyxwv
Elmentr of Macroawvc Networh
zyx
D = Electric Displacement (coulomb/m’)
p = Charge Density (coulomb/m’)
u
= Conductivity (mhoslm)
M = Magnetization (amp/m2)
P = Electric Polarization (coulomb/m*)
p,, = Permeability of free space (henrys/m)
q,= Dielectric constant of free space (faradsh)
E and p may be defined in terms of their respective susceptibilities x
P=N(l+X,)
E=Eo(l+X,)
or
.
where x, = magnetic susceptibility and x, = electric susceptibility.
B. General Constitutive Relations
We have assumed in the above relations that both p and E are scalar
quantities. This implies for example that the differential permeability is
isotropic.
and that
Similarly,
zyxwv
ax 6Dy a!
4 4- 4
&=-=---
and
zyx
zyxw
zyx
z
zyx
zyx
zyxwvuts
Review of M w w c l l Equotioru 3
For most practical microwave materials p and E are not scalar, but
tensors. The above relations state that E and p are proportional to induced
incremental polarizations (upon application of external E or g ) . The
polarization may be in the form of induced magnetization or electric
polarization. The more one is able to polarize a material the higher the values
of p or E . However, if we allow the possibility that the induced polarization
direction (magnetic or electric) may be orthogonal (in direction) to the
application of an external field, we find that we can no longer describe p and
E in terms of a scalar quantity. In general a change in polarization along a
given direction may be due to fields applied in an arbitrary direction.
Mathematically, an incremental 6B along a given direction may be expressed
as follows
zyxwvu
We have implied in the above equation that a change in 6B,, can be
induced by a small change in field along the x, y and z directions. In general,
one may write
c
In matrix form one may represent eq.(8) symbolically as follows
zyx
=z
MI
We define a differential susceptibility tensor element X,
, where
i, j = x, y, z. is the unit matrix. Hence, pii = (1 +x,J p,, and pIj’, for i j.
Finally, a tensor for [p] may be defined as follows
m
4
zyxwvutsrqp
zyxwvu
zyxwvuts
zyxwv
Elements of Micnwow Ncturwlks
or equivalently
zyxwvut
The tensor for [E] can be written in a similar fashion.
[&I= 4 ~ l + [ r , l ]
or equivalently
Here now
(&)q
1B:
=--
z
zyxw
for i, j = x, y and z.
4 a,
In summary, the value of each matrix element of [E] and [p] is
proportional to the induced electric or magnetic polarization. The existence
of the off-diagonal matrix elements of [E] and [p] is due to anisotropic
inductions of polarization in a magneto-dielectric medium. If changes in
external fields are defined with respect to a zero field reference, we may write
the constitutive equations simply as follows
zyx
zy
z
zyxwvu
zyx
zyx
Review of Manuel1 Equotiona 5
where 5, is a unit vector in the i direction (x, y,
2.).
C. External, Surface and Internal Electromagnetic Fields
Maxwell's equations represent a set of equations which relate the internal
electromagnetic fields in a magneto-dielectric medium to the polarization
fields of that medium. It establishes fundamental relationships between these
set of fields. Polarization fields are the result of local interactions between
the electromagnetic fields and the medium. A convenient way to relate
internal fields with surface electromagnetic fields is via the so-called Poynting
relation or integral
where
cf?
is a vector normal to a surface containing the medium of interest
and is in units of m2. The left hand side of the equation is a surface integral,
where the surface is enclosed and the fields are determined on the surface.
The right hand side of the equation is a volume integral in which the fields are
determined within the enclosed surface. They are internal and polarizing
fields. The minus sign is important, if we want to conserve energy as the
electromagnetic wave propagates in and out of the enclosed surface. In Eq.
(16) p and E may be either scalars or tensors. The total stored energy density
is usually written as
where U, and U, are the stored electric and magnetic energies, respectively.
The Poynting relation may be written in terms of U
The above equation states that the net time rate of change of
electromagnetic energy within an enclosed volume is equal to the negative of
the total work done by the fields on the medium. It is understood that the left
hand side of the above equation contains fields only at the surface of the
medium. The beauty of the above relationship is that any changes in the
stored energy or the potential energy within a medium manifest itself as a
change in the surface fields. In fact it is exactly this principle that allows
6
zyxwvutsrq
zyxwvu
zyxwvuts
zyxwv
Elements of M i c m w v c Networks
experimentalists to be able to characterize the properties of microwave
materials. Interaction between internal fields and the medium are included in
U. Finally, surface fields may be related to fields far removed from the
medium by simple application of electromagnetic boundary conditions at the
surface, see figure 1.1.
External
or Incident
Wave
z
p
z
zyxwvu
I
dg
Fig. 1 . 1 Internal, external and surface
elctromagnetic fields.
D. Practical Example
If there is no absorption of microwave energy within the enclosed surface, the
steady state power entering and exiting the enclosed surface is the same.
Mathematically, it means that if we integrate over the enclosed volume and
average the power over one period of time, the average power enclosed is
zero, assuming the dissipative term
is zero. Let's consider conceptually
the application of eq.(16) by assuming wave propagation in one direction, zaxis, and l? and fi perpendicular to each other but transverse to the z-axis.
Thus, surface integration over the x-z and y-z planes is zero, since E and A
are in the x-y plane, see figure 1.2.
zyxwvu
z
The two surfaces of integration are the surfaces in which
electromagnetic energy is impinged upon and in which the energy is exiting
the volume in question. At the input surface one can always adjust the
relative phase of E and fi to be zero so that one can write at the input
surface the following fields
zyx
zy
zy
Revicw of M o w e l l Equatim 7
(19)
zyxwvu
zyxw
zyxwvuts
zyxwvu
Fig. I .2 Field configuration relative to enclosed
volume.
At the other surface E, and fi, are again assumed to be in the x-y plane but
have now some phase relative to the input fields at surface (1). Write at the
output surface the following
OF MICROWAVE
NETWORKS
z
Basics of Microwave Engineering
Nor-/lwcls/rmUniv., USA
Y
zy
zy
CARMINE VITTORIA
zyxwvu
$World Scientific
Singapore*New
'
Jersey*London*HongKong
zyxwvutsrq
zyxwvut
zyxwvutsrq
zyxw
Published by
World Scientific Publishing Co. Re. Ltd.
P 0 Box 128. Famr Road. Singapore 912805
USA oflce: Suite 1B. 1060 Main Street. River Edge. NJ 07661
UK ofice: 57 Shelton Street. Coven1Garden, London WQH 9HE
Library of Congress Cataloging-in-Publication Data
Vinoria C.
Wemenu of microwave networks : basics of microwave engineering /
Carmine Vittoria.
p. cm.
Includes bibliographical references(p. ) and index.
ISBN 9810234244
I. Microwave circuits -- Mathematical models. 2. Microwaves - Mathematical models. 3. Microwave transmission lines -- Mathematical
models. 1. Title.
TK7876.VS8 1998
98-10764
621.381'32 dc21
CIP
--
British Library Cataloguing-in-Publication Data
A catalogue record for this book i s available from the British Library.
zyxwvuts
Copyright Q 1998 by World Scientific PublishingCo PIC. Lid.
All rights resenwd. This book or parts thereo/: may not be reproduced in any form or by any meam.
electronic or mechanical. includingphofocopying. recording or any information storage and retrieval
system now known or to be invented. without written permissionfrom the Publisher.
For photocopying of material in this volume. please pay a copying fee through the Copyright
Clearance Center. Inc., 222 Rosewood Drive. Danven. M A 01923. USA. I n this case permission to
photocopy is not required from the publisher.
This book i s printed on acid-free paper.
Printed in Singapore by Uto-Pnni
zyx
zyxwv
zy
Preface
The purpose of this book is to bridge the knowledge gap between fundamental
textbooks used in required courses and the rapid advances made in the
microwave technology. Today, microwave systems and/or requirements are
too complex or advanced to be analyzed by simple graphical approaches as
the Smith chart, for example. As such, this book is to be used in an elective
course at the undergraduate level. In putting together this book, I chose topics
which (1) did not duplicate topics covered in textbooks used in required
courses, (2) were a logical extension of required courses, and (3) reached out
to a larger audience of students majoring in science and engineering.
zyxwvu
Using criteria (1) as a basis I have purposely left out all formal
derivations of Maxwell's equations and subsequent material related to it. I
introduced Maxwell's equations as a given fact only to make some
connections in later chapters. From criteria (2) the following topics were
chosen: (a) lossy medium, (b) scattering S parameters, (c) matrix
representation of wave propagation, and (d) microwave properties of ferrite
networks. All microwave systems contain some element of losses. As such,
lossy medium as well as absorbing networks were extensively covered. The
S matrix and its properties is introduced for the sole purposes of preparing
students for the demands of the microwave industry. The matrix
representation of wave propagation has been used for many years in other
fields of science. We have re-introduced it here in relation to microwave
networks. Finally, ferrite usage represents the largest industry in technology.
We should introduce topics in ferrite networks in a way that is complementary
to the core material in undergraduate education. We need to go beyond the
description of a scalar permeability at the undergraduate level.
The book consists of eight chapters. The first two chapters are
introductory material for students who do not have an adequate background
in electromagnetic theory, such as students in other engineering disciplines
and science majors. These two chapters are not to be covered in a regular
course. A semester course covers six chapters starting with chapter 3. It is not
necessary to cover every single topic in each chapter. It is possible to adapt
or re-arrange the topics in a way to meet the needs of students depending on
the makeup of the class. For example, as an elective course we covered
chapters 3-8 minus topics in ferrite networks and specialized topics in a
quarter which is shorter than a semester. Although it was an electrive course
V
vi
zyxwvuts
zyxwvutsrq
zyxwvu
zy
Elemcnb o j Micrmwve Nctmonkr
zyxwv
zyxwv
for juniors/seniors, the book could be used as a springboard to a more
advanced course at the graduate level bu including advanced topics in ferrite
networks and matrix representation of microwave networks in general.
I wish to thank Professor George Alexandrakis for his hospitality while
I was preparing the manuscript at the University of Miami. Also, the able
assistance of Drs. H. How and W.Hu, and Y. Biaz is acknowledged.
zyx
zyxwvut
Contents
Preface
zyxwv
z
V
Review of Maxwell Equations
Chapter I
A. Maxwell's Equations in MKS System of Units
B. General Constitutive Relations
C. External, Surface and Internal Electromagnetic Fields
D. Practical Example
E. Electric Polarization of Microwave Signal
F. Microwave Response and Polarization
References
Examples
i
1
2
5
6
10
13
14
zyxwvu
14
Common Waveguide Structures
Chapter II
A. Parallel - Plate Waveguides
B. Coaxial Line
C. Rectangular Waveguide
References
Examples
21
21
26
29
33
34
Chapter 111 Telegraph's Equations
A. Types of Transmission Lines
B. Wave Equation
C. Connection to Circuit Parameters
D. Formal Solution
E. Electrical Quantities
F. Bounce Diagrams
References
Examples
39
39
39
Chapter IV Analytical Solution
A. Lossy Case
B. Real Time Solutions
C. Lossless Case
48
52
57
68
79
79
90
91
96
97
vii
viii
zyxwvutsrq
zyxwvu
zyxwv
zyxwvu
Elements of Micrwwovc Networks
D. Determination of I, and I,
E. Coupling Between Source and Transmission Line
References
Examples
102
109
112
113
zyxwv
Chapter V
Graphical Solution
A. Mathematical Terminologies
B. Plot of r ( - I )in Complex Plane
C. Projection of Z(-I) onto the r(-I)Complex Plane-Smith Chart
D. Projection of V(4) on Smith Chart
E. Projection of I(-() on Smith Chart
F. Graphical Methods for Lossy Lines
References
Examples
127
127
128
131
141
146
147
150
151
Chapter V1 Special Topics in 'hansmission Lines
A. Stub Tuners
B. Graphical Solution
C. Quarter and Half Wavelength Transmission Lines
D. Microwave Absorbers
References
Examples
160
1 60
Chapter VII Electromagnetic Scattering Parameters
A. Definitions of S-Parameters
A. 1 . Algebraic Properties of S Matrix
B. Relationship Between Measurements and Network
Electrical Parameters
C. Example of Inverse Scattering Problem
D. General Comments
References
Examples
199
199
199
Chapter VIlI Matrix Representation of Microwave Networks
A. Transfer Function Matrix of Two Port Network
163
169
176
188
189
210
214
215
216
217
224
224
z
zyxwv
zyxwvu
B. Transmission Line Analysis Using Matrix Representation
C. Connection Between Scattering Parameters and Matrix
Representation
D. General Properties of Matrix Representation
D. 1. Single Networks
D.2. Cascaded Networks
D.3. Periodic Network Systems
D.4. Application
E. Some Special Properties of Ferrite Networks
F. Relationship Between Scattering S-Parameters and
Matrix Elements
G. Example of Inverse Scattering
H. Special Applications of Matrix Representation
1. Three and Four Port Networks
I. 1. Three Port Network
1.2. Four Port Network
J. Equivalent Circuit of Ferrite Circulator
References
Examples
Index
226
229
232
232
236
237
24 1
244
247
252
254
256
257
26 1
262
264
265
283
zy
zyxw
zyx
Chapter I
Review of Maxwell Equations
The foundations of electromagnetic wave propagation in a medium are
Maxwell’s equations. In this book we introduce measurement quantities such
as voltage, V, and current, I, to solve for electromagnetic wave propagation
characteristics of specialized guided structures. As a result of introducing
measurement variables V and I. it is mathematically convenient to introduce
electrical circuit parameters, such as R, L, C and G. Before we go much into
details of the analysis of wave propagation let’s re-acquaint ourselves with
Maxwell equations written in terms of field variables E and f i , the electric
and magnetic field variables.
zyx
zyx
A. Maxwell’s Equations in MKS System of Units
Maxwell’s equations in MKS maybe written in the differential form as
follows:
where
zyxwvu
zyxw
j = d
The two constitutive equations are usually written as
B=h(fi+@=pfi
Gi=&$+jLEE
mi
,
The units of each field quantity are defined as follows
J = Current Density (amplm’)
B = Magnetic Flux Density (webedm’)
H = Magnetic Field Intensity (amp/m)
1
2
zyxwvutsr
zyxwvuts
zyxwvu
zyxwv
Elmentr of Macroawvc Networh
zyx
D = Electric Displacement (coulomb/m’)
p = Charge Density (coulomb/m’)
u
= Conductivity (mhoslm)
M = Magnetization (amp/m2)
P = Electric Polarization (coulomb/m*)
p,, = Permeability of free space (henrys/m)
q,= Dielectric constant of free space (faradsh)
E and p may be defined in terms of their respective susceptibilities x
P=N(l+X,)
E=Eo(l+X,)
or
.
where x, = magnetic susceptibility and x, = electric susceptibility.
B. General Constitutive Relations
We have assumed in the above relations that both p and E are scalar
quantities. This implies for example that the differential permeability is
isotropic.
and that
Similarly,
zyxwv
ax 6Dy a!
4 4- 4
&=-=---
and
zyx
zyxw
zyx
z
zyx
zyx
zyxwvuts
Review of M w w c l l Equotioru 3
For most practical microwave materials p and E are not scalar, but
tensors. The above relations state that E and p are proportional to induced
incremental polarizations (upon application of external E or g ) . The
polarization may be in the form of induced magnetization or electric
polarization. The more one is able to polarize a material the higher the values
of p or E . However, if we allow the possibility that the induced polarization
direction (magnetic or electric) may be orthogonal (in direction) to the
application of an external field, we find that we can no longer describe p and
E in terms of a scalar quantity. In general a change in polarization along a
given direction may be due to fields applied in an arbitrary direction.
Mathematically, an incremental 6B along a given direction may be expressed
as follows
zyxwvu
We have implied in the above equation that a change in 6B,, can be
induced by a small change in field along the x, y and z directions. In general,
one may write
c
In matrix form one may represent eq.(8) symbolically as follows
zyx
=z
MI
We define a differential susceptibility tensor element X,
, where
i, j = x, y, z. is the unit matrix. Hence, pii = (1 +x,J p,, and pIj’, for i j.
Finally, a tensor for [p] may be defined as follows
m
4
zyxwvutsrqp
zyxwvu
zyxwvuts
zyxwv
Elements of Micnwow Ncturwlks
or equivalently
zyxwvut
The tensor for [E] can be written in a similar fashion.
[&I= 4 ~ l + [ r , l ]
or equivalently
Here now
(&)q
1B:
=--
z
zyxw
for i, j = x, y and z.
4 a,
In summary, the value of each matrix element of [E] and [p] is
proportional to the induced electric or magnetic polarization. The existence
of the off-diagonal matrix elements of [E] and [p] is due to anisotropic
inductions of polarization in a magneto-dielectric medium. If changes in
external fields are defined with respect to a zero field reference, we may write
the constitutive equations simply as follows
zyx
zy
z
zyxwvu
zyx
zyx
Review of Manuel1 Equotiona 5
where 5, is a unit vector in the i direction (x, y,
2.).
C. External, Surface and Internal Electromagnetic Fields
Maxwell's equations represent a set of equations which relate the internal
electromagnetic fields in a magneto-dielectric medium to the polarization
fields of that medium. It establishes fundamental relationships between these
set of fields. Polarization fields are the result of local interactions between
the electromagnetic fields and the medium. A convenient way to relate
internal fields with surface electromagnetic fields is via the so-called Poynting
relation or integral
where
cf?
is a vector normal to a surface containing the medium of interest
and is in units of m2. The left hand side of the equation is a surface integral,
where the surface is enclosed and the fields are determined on the surface.
The right hand side of the equation is a volume integral in which the fields are
determined within the enclosed surface. They are internal and polarizing
fields. The minus sign is important, if we want to conserve energy as the
electromagnetic wave propagates in and out of the enclosed surface. In Eq.
(16) p and E may be either scalars or tensors. The total stored energy density
is usually written as
where U, and U, are the stored electric and magnetic energies, respectively.
The Poynting relation may be written in terms of U
The above equation states that the net time rate of change of
electromagnetic energy within an enclosed volume is equal to the negative of
the total work done by the fields on the medium. It is understood that the left
hand side of the above equation contains fields only at the surface of the
medium. The beauty of the above relationship is that any changes in the
stored energy or the potential energy within a medium manifest itself as a
change in the surface fields. In fact it is exactly this principle that allows
6
zyxwvutsrq
zyxwvu
zyxwvuts
zyxwv
Elements of M i c m w v c Networks
experimentalists to be able to characterize the properties of microwave
materials. Interaction between internal fields and the medium are included in
U. Finally, surface fields may be related to fields far removed from the
medium by simple application of electromagnetic boundary conditions at the
surface, see figure 1.1.
External
or Incident
Wave
z
p
z
zyxwvu
I
dg
Fig. 1 . 1 Internal, external and surface
elctromagnetic fields.
D. Practical Example
If there is no absorption of microwave energy within the enclosed surface, the
steady state power entering and exiting the enclosed surface is the same.
Mathematically, it means that if we integrate over the enclosed volume and
average the power over one period of time, the average power enclosed is
zero, assuming the dissipative term
is zero. Let's consider conceptually
the application of eq.(16) by assuming wave propagation in one direction, zaxis, and l? and fi perpendicular to each other but transverse to the z-axis.
Thus, surface integration over the x-z and y-z planes is zero, since E and A
are in the x-y plane, see figure 1.2.
zyxwvu
z
The two surfaces of integration are the surfaces in which
electromagnetic energy is impinged upon and in which the energy is exiting
the volume in question. At the input surface one can always adjust the
relative phase of E and fi to be zero so that one can write at the input
surface the following fields
zyx
zy
zy
Revicw of M o w e l l Equatim 7
(19)
zyxwvu
zyxw
zyxwvuts
zyxwvu
Fig. I .2 Field configuration relative to enclosed
volume.
At the other surface E, and fi, are again assumed to be in the x-y plane but
have now some phase relative to the input fields at surface (1). Write at the
output surface the following