fundamentals of electromagnetics by the author. It is not to be sold, reproduced
for any purpose other than stated here, or generally distributed. Any use other.
Fundamentals of Electromagnetics for Electrical and Computer Engineering in 108 Slides: A Tutorial
Nannapaneni Narayana Rao Edward C. Jordan Professor of Electrical and Computer Engineering University of Illinois at Urbana-Champaign, Urbana, Illinois, USA Distinguished Amrita Professor of Engineering Amrita Vishwa Vidyapeetham, Ettimadai, Coimbatore, Tamil Nadu, India
All Rights Reserved, January 2007 This publication is prepared for use with the presentation of tutorials on the fundamentals of electromagnetics by the author. It is not to be sold, reproduced for any purpose other than stated here, or generally distributed. Any use other than as stated here is not authorized and requires the written permission of the author.
Om Amriteswaryai Namaha! Om Namah Sivaya!
Inscription on wristband from Retreat with Amma Mata Amritanandamayi Devi: “Fill your heart with love and then express it in everything you do.” In this publication, I have attempted it through 108 slides on the fundamentals of engineering electromagnetics.
About the Author Nannapaneni Narayana Rao was born in Kakumanu, Guntur District, Andhra Pradesh, India. Prior to going to the United States in 1958, he attended high schools in Pedanandipadu and Nidubrolu; the Presidency College, Madras (now known as Chennai); and the Madras Institute of Technology. He completed high school from Nidubrolu in 1947, received the BSc. Degree in Physics from the University of Madras in 1952, and the Diploma in Electronics from the Madras Institute of Technology in 1955. In the United States, he attended the University of Washington, receiving the MS and PhD degrees in Electrical Engineering in 1960 and 1965, respectively. In 1965, he joined the faculty of the Department of Electrical Engineering, now the Department of Electrical and Computer Engineering, of the University of Illinois at Urbana-Champaign, Illinois, serving as Associate Head of the Department from 1987 to 2006. At the University of Illinois at Urbana-Champaign, he carried out research in the general area of radiowave propagation in the ionosphere and authored the undergraduate textbooks, Basic Electromagnetics with Applications (Prentice Hall, 1972), and six editions of Elements of Engineering Electromagnetics (Prentice Hall, 1977, 1987, 1991, 1994, 2000, and 2004). The fifth edition of Elements of Engineering Electromagnetics was translated into Bahasa Indonesia, the language of Indonesia, by a professor of physics at the Bandung Institute of Technology, Bandung. Professor Rao has received numerous awards and honors for his teaching and curricular activities, and contributions to engineering education in India, the USA, and Indonesia. He received the IEEE Technical Field Award in Undergraduate Teaching in
1994 with the citation, “For inspirational teaching of undergraduate students and the development of innovative instructional materials for teaching courses in electromagnetics.” In Fall 2003, Professor Rao was named to be the first recipient of the Edward C. Jordan Professorship in Electrical and Computer Engineering, created to honor the memory of Professor Jordan, who served as department head for 25 years, and to be held by a “member of the faculty of the department who has demonstrated the qualities of Professor Jordan and whose work would best honor the legacy of Professor Jordan.” In October 2006, Professor Rao was named the first “Distinguished Amrita Professor of Engineering” by Amrita Vishwa Vidyapeetham, Ettimadai, Coimbatore, Tamil Nadu, India, where, in July – August, 2006, he taught the inaugural course on the Indo-US Inter-University Collaborative Initiative on the Amrita e-learning network. A special Indian Edition of “Elements of Engineering Electromagnetics, Sixth Edition,” was published prior to this course offering.
Preface “… I am talking about the areas of science and learning that have been at the heart of what we know and what we do, that which has supported and guided us and which is fundamental to our thinking. It is electromagnetism (EM) in all its many forms that has been so basic, that haunts us and guides us…” – Nick Holonyak, Jr., John Bardeen Endowed Chair Professor of Electrical and Computer Engineering and of Physics, University of Illinois at Urbana-Champaign, and the inventor of the semiconductor visible LED, laser, and quantum-well laser, in a Foreword in the Indian Edition of “Elements of Engineering Electromagnetics, Sixth Edition,” by the author. “The electromagnetic theory, as we know it, is surely one of the supreme accomplishments of the human intellect, reason enough to study it. But its usefulness in science and engineering makes it an indispensable tool in virtually any area of technology or physical research.” – George W. Swenson, Jr., Professor Emeritus of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, and a pioneer in the field of radio astronomy, in the “Why Study Electromagnetics? Section in the Indian Edition of “Elements of Engineering Electromagnetics, Sixth Edition,” by the author. In this presentation, I present in a nutshell the fundamental aspects of engineering electromagnetics from the view of looking back in a reflective fashion what has already been learnt in undergraduate electromagnetics courses as a novice. The first question that comes to mind in this context is: What constitutes the fundamentals of engineering electromagnetics? If the question is posed to several individuals, it is certain that they will come up with sets of topics, not necessarily the same or in the same order, but all
containing the topic, “Maxwell’s Equations,” at some point in the list, ranging from the beginning to the end of the list. In most cases, the response is bound to depend on the manner in which the individual was first exposed to the subject. Judging from the contents of the vast collection of undergraduate textbooks on electromagnetics, there is definitely a heavy tilt toward the traditional, or historical, approach of beginning with statics and culminating in Maxwell’s equations, with perhaps an introduction to waves. Primarily to provide a more rewarding understanding and appreciation of the subject matter, and secondarily owing to my own fascination resulting from my own experience as a student, a teacher, and an author over a few decades, I have employed here the approach of beginning with Maxwell’s equations and treating the different categories of fields as solutions to Maxwell’s equations. In doing so, instead of presenting the topics in an unconnected manner, I have used the thread of staticsquasistatics-waves to cover the fundamentals and bring out the frequency behavior of physical structures at the same time. The material in this presentation is based on Chapter 1, entitled, “Fundamentals of Engineering Electromagnetics Revisited,” by the author, in Handbook of Engineering Electromagnetics, Marcel Dekker, 2004. I wish to express my appreciation to the following staff members of the Department of Electrical and Computer Engineering at the University of Illinois at Urbana-Champaign: Teresa Peterson for preparing the PowerPoint slides, and Kelly Collier for typing the text material. N. Narayana Rao
Fundamentals of Electromagnetics for Electrical and Computer Engineering in 108 Slides: A Tutorial
Nannapaneni Narayana Rao Edward C. Jordan Professor of Electrical and Computer Engineering University of Illinois at Urbana-Champaign, Urbana, Illinois, USA Distinguished Amrita Professor of Engineering Amrita Vishwa Vidyapeetham, Ettimadai, Coimbatore, Tamil Nadu, India
Note that this publication does not have page numbers. The right side pages contain slides numbered 1 through 108 in sequence. The left side pages contain explanations for the slides.
Slide Nos. 1-3 Electromagnetics is the subject having to do with electromagnetic fields. An electromagnetic field is made up of interdependent electric and magnetic fields, which is the case when the fields are varying with time, that is, they are dynamic. An electric field is a force field that acts upon material bodies by virtue of their property of charge, just as a gravitational field is a force field that acts upon them by virtue of their property of mass. A magnetic field is a force field that acts upon charges in motion. The subject of electromagnetics is traditionally taught using the historical approach of beginning with static fields and culminating in Maxwell’s equations, with perhaps an introduction to waves. Here, we employ the approach of beginning with Maxwell’s equations and treating the different categories of fields as solutions to Maxwell’s equations. In doing so, the thread of statics-quasistatics-waves is used to cover the fundamentals and bring out the frequency behavior of physical structures at the same time.
What is Electromagnetics? Electromagnetics Electromagnetics is is the the subject subject having having to to do do with with electromagnetic electromagnetic fields. fields. An An electromagnetic electromagnetic field field is is made made up up of of interdependent interdependent electric electric and and magnetic magnetic fields, fields, which which is is the the case case when when the the fields fields are are varying varying with with time, time, that that is, is, they they are are dynamic. dynamic. An An electric electric field field is is aa force force field field that that acts acts upon upon material material bodies bodies by by virtue virtue of of their their property property of of charge, charge, just just as as aa gravitational gravitational field field is is aa force force field field that that acts acts upon upon them them by by virtue virtue of of their their property property of of mass. mass. A A magnetic magnetic field field is is aa force force field field that that acts acts upon upon charges charges in in motion. motion.
11
The Approach Used Here The The subject subject of of electromagnetics electromagnetics is is traditionally traditionally taught taught using using the the historical historical approach approach of of beginning beginning with with static static fields fields and and culminating culminating in in Maxwell’s Maxwell’s equations, equations, with with perhaps perhaps an an introduction introduction to to waves. waves. Here, Here, we we employ employ the the approach approach of of beginning beginning with with Maxwell’s Maxwell’s equations equations and and treating treating the the different different categories categories of of fields fields as as solutions solutions to to Maxwell’s Maxwell’s equations. equations. In In doing doing so, so, the the thread thread of of staticsstaticsquasistatics-waves quasistatics-waves is is used used to to cover cover the the fundamentals fundamentals and and bring bring out out the the frequency frequency behavior behavior of of physical physical structures structures at at the the same same time. time.
22
Maxwell’s Equations dS == ∫ ρρ dv dv ∫SSDD⋅ dS VV
d E dl == –– d B E dl B⋅ dS dS ⋅ ∫CC ∫ dt dt SS
Charge Charge density density
Magnetic Magnetic flux flux density density
Electric Electric field field intensity intensity
((CC mm33))
22 ((Wb Wb m m ))
((VV mm))
dd D dS H dl JJ⋅ dS = + H dl dS D⋅ dS = + ⋅ ∫CC ∫SS dt dt ∫SS Current Magnetic Current Magnetic field density field intensity intensity density ((AA mm)) (AA mm22)
(
)
Displacement Displacement flux flux density density 22 C Cm m
(
))
dS == 00 ∫SSBB⋅ dS
33
Slide No. 4
A region is said to be characterized by an electric field if a particle of charge q moving with a velocity v experiences a force Fe, independent of v. The force, Fe, is given by Fe = qE
where E is the electric field intensity. We note that the units of E are newtons per coulomb (N/C). Alternate and more commonly used units are volts per meter (V/m), where a volt is a newton-meter per coulomb. The line integral of E between two points A and B in an electric field region,
B
∫A E • dl , has the meaning of voltage between A and B.
It is the work per unit charge done by the field in the movement of the charge from A to B. The line integral of E around a closed path C is also known as the electromotive force
or emf around C.
(1)
Electric Force on a Charge F Fee == qq E E N N
E E
FFee
C C V Vm m
E E
V V == (N-m)/C (N-m)/C
qq
cos αα))dl dl ∫AA EE⋅ ddll ==∫AA ((EE cos BB
BB
E E
== Voltage Voltage between between AA and and BB
AA
BB αα
dl dl 44
Slide No. 5
If the charged particle experiences a force which depends on v, then the region is said to be characterized by a magnetic field. The force, Fm, is given by Fm = qv x B
where B is the magnetic flux density. We note that the units of B are newtons/(coulombmeter per second), or, (newton-meter per coulomb) × (seconds per square meter), or, volt-seconds per square meter. Alternate and more commonly used units are webers per square meter (Wb/m2) or tesla (T), where a weber is a volt-second. The surface integral of B over a surface S,
∫
S
B • dS , is the magnetic flux (Wb) crossing the surface.
Equation (2) tells us that the magnetic force is proportional to the magnitude of v and orthogonal to both v and B in the right-hand sense. The magnitude of the force is qvB sin δ, where δ is the angle between v and B. Since the force is normal to v, there is
no acceleration along the direction of motion. Thus the magnetic field changes only the direction of motion of the charge, and does not alter the kinetic energy associated with it.
(2)
Magnetic Force on a Moving Charge FFmm == qq vv xx B B N N
B B 2
2 CC m m ss Wb Wb m m
B B
FFmm
Wb Wb == V-s V-s FFmm == qvB qvB sin sin δδ dS ∫∫SSBB⋅⋅dS == Magnetic Magnetic flux flux crossing crossing SS
dS dS
αα
qq
δδ vv
BB
dS dS
55
Slide No. 6
Since current flow in a wire results from motion of charges in the wire, a wire of current placed in a magnetic field experiences a magnetic force. For a differential length dl of a wire of current I placed in a magnetic field B, this force is given by dFm = I dl x B
(3)
Magnetic Force on a Current Element dFmm = I N dl xx BB N N
A A m m
22
Wb Wb m m
dF dFmm
B B B B
dl dl II
66
Slide No. 7
Combining (1) and (2), we obtain the expression for the total force F = Fe + Fm, experienced by a particle of charge q moving with a velocity v in a region of electric and magnetic fields, E and B, respectively, as F = qE + qv x B
= q(E + v x B) Equation (4) is known as the Lorentz force equation.
(4)
Lorentz Force Equation F = qE + qv x B = q(E + v x B) EE vv BB qq
77
Slide No. 8
The vectors E and B are the fundamental field vectors which define the force acting on a charge moving in an electromagnetic field, as given by the Lorentz force equation (4). Two associated field vectors D and H, known as the electric flux density (or the displacement flux density) and the magnetic field intensity, respectively, take into account the dielectric and magnetic properties, respectively, of material media. Materials contain charged particles which, under the application of external fields, respond giving rise to three basic phenomena known as conduction, polarization, and magnetization. Although a material may exhibit all three properties, it is classified as a conductor, a dielectric, or a magnetic material, depending upon whether conduction, polarization, or magnetization is the predominant phenomenon. While these phenomena occur on the atomic or “microscopic” scale, it is sufficient for our purpose to characterize the material based on “macroscopic” scale observations, that is, observations averaged over volumes large compared with atomic dimensions.
Materials Materials Materials contain contain charged charged particles particles that that under under the the application application of of external external fields fields respond respond giving giving rise rise to to three three basic basic phenomena phenomena known known as as conduction, conduction, polarization, polarization, and and magnetization. magnetization. While While these these phenomena phenomena occur occur on on the the atomic atomic or or “microscopic” “microscopic” scale, scale, itit is is sufficient sufficient for for our our purpose purpose to to characterize characterize the the material material based based on on “macroscopic” “macroscopic” scale scale observations, observations, that that is, is, observations observations averaged averaged over over volumes volumes large large compared compared with with atomic atomic dimensions. dimensions. 88
Slide No. 9
In the case of conductors, the effect of conduction is to produce a current in the material known as the conduction current. Conduction is the phenomenon whereby the free electrons inside the material move under the influence of the externally applied electric field with an average velocity proportional in magnitude to the applied electric field, instead of accelerating, due to the frictional mechanism provided by collisions with the atomic lattice. For linear isotropic conductors, the conduction current density, having the units of amperes per square meter (A/m2) is related to the electric field intensity in the manner Jc = σE
where σ is the conductivity of the material, having the units siemens per meter (S/m). In semiconductors, the conductivity is governed by not only electrons but also holes.
(5)
Conductors (conduction) JJcc == σσ E E 2 JJcc == Conduction Conduction current current density, density, A Am m2
σσ == Conductivity, Conductivity, SS m m
Ohm’s Ohm’s Law Law in in Circuit Circuit Theory Theory ll
VV == El El
σV JJcc == σσ EE == σ V ll σA II == JJcc AA == σ A VV ll
⎛ ⎞ VV == II⎜⎛⎜ ll ⎟⎞⎟ == IR ⎝⎝σσ AA⎠⎠ IR
AA EE II VV
l RR == l ,, resistance (ohms) σσ AA resistance (ohms)
99
Slide Nos. 10-11
While the effect of conduction is taken into account explicitly in the electromagnetic field equations through (5), the effect of polarization is taken into account implicitly through the relationship between D and E, which is given by D = εE
(6)
for linear isotropic dielectrics, where ε is the permittivity of the material having the units (coulombs)2 per (newton-meter2), commonly known as farads per meter (F/m), where a farad is a (coulomb)2 per newton-meter. Polarization is the phenomenon of creation and net alignment of electric dipoles, formed by the displacements of the centroids of the electron clouds of the nuclei of the atoms within the material, along the direction of an applied electric field. The effect of polarization is to produce a secondary field which acts in superposition with the applied field to cause the polarization. To implicitly take this into account, leading to (6), we begin with D = ε0E + P
(7)
where ε0 is the permittivity of free space, having the numerical value 8.854 × 10–12, or approximately 10–9/36π, and P is the polarization vector, or the dipole moment per unit volume, having the units (coulomb-meters) per meter3 or coulombs per square meter. Note that this gives the units of coulombs per square meter for D. The term ε0E accounts for the relationship between D and E if the medium were free space, and the quantity P represents the effect of polarization. For linear isotropic dielectrics, P is proportional to E in the manner P = ε0χeE
(8)
Dielectrics (polarization) Electric Electric Dipole Dipole Electric Electric Dipole Dipole Moment: Moment:
++ Q Q dd
EE ++ ––
pp == Q Qdd
–– –Q –Q
Polarization Polarization vector: vector: PP == 11 ∆v ∆v
NN ∆v ∆v
QE QE
ppjj == Np ∑ Np((CC m )) ∑ j =1 m 22
QQ ++
j =1
PP ==εε 00χχeeEE
E E
εε 00== Permittivity Permittivity of offree freespace space((FF m m)) χχee == Electric Electric susceptibility susceptibility
–QE –QE
–– –Q –Q
10 10
where χe, a dimensionless quantity, is the electric susceptibility, a parameter that signifies the ability of the material to get polarized. Combining (7) and (8), we have D = ε0(1 + χe)E
= ε 0 ε rE = εE where εr (= 1 + χe) is the relative permittivity of the material.
(9)
The Permittivity Concept Applied Applied Field, Field,EEaa
++ ++
Total TotalField Field EE == EEaa ++EEss
Dielectric Dielectric
Secondary SecondaryField, Field,EEss Polarization Polarization
D D == εε00EE ++ PP == εε00EE++εε00χχeeEE
D D == εε00(11 ++ χχee))E E
== εε00εεrrE E == εεE, E, Displacement Displacement Flux Flux Density Density
11 11
Slide Nos. 12-13
In a similar manner, the effect of magnetization is taken into account implicitly through the relationship between H and B, which is given by H= B µ
(10)
for linear isotropic magnetic materials, where µ is the permeability of the material, having the units newtons per (ampere)2, commonly known as henrys per meter (H/m), where a henry is a (newton-meter) per (ampere)2. Magnetization is the phenomenon of net alignment of the axes of the magnetic dipoles, formed by the electron orbital and spin motion around the nuclei of the atoms in the material, along the direction of the applied magnetic field. The effect of magnetization is to produce a secondary field which acts in superposition with the applied field to cause the magnetization. To implicitly take this into account, we begin with B = µ0H + µ0M
(11)
where µ0 is the permeability of free space, having the numerical value 4π × 10–7, and M is the magnetization vector or the magnetic dipole moment per unit volume, having the units (ampere-square meters) per meter3 or amperes per meter. Note that this gives the units of amperes per meter for H. The term µ0H accounts for the relationship between H and B if the medium were free space, and the quantity µ0M represents the effect of magnetization. For linear isotropic magnetic materials, M is proportional to H in the manner M = χmH
(12)
Magnetic Materials (Magnetization) Magnetic Magnetic Dipole Dipole Magnetic Magnetic Dipole Dipole Moment: Moment:
xx aann
m m == IIAa Aann
II
IIinin ××
IIout out
Magnetization Magnetization Vector: Vector: NN∆v ∆v
11 M m M == ∆v ∑ mjj == Nm Nm((AA m m)) ∆v j∑ =1
IIdl dl xx BB
j =1
χχm BB M M ==1 + χm µ 1 + χmm µ00
I ××I
IIdl dl xx BB
BB II
µµ00 == Permeability Permeability of of free free space space ((HH m m))
χχmm == Magnetic Magnetic susceptibility susceptibility
12 12
where χm, a dimensionless quantity, is the magnetic susceptibility, a parameter that signifies the ability of the material to get magnetized. Combining (11) and 12), we have H = µ (1B+ χ ) 0 m = µ Bµ 0 r =B
µ
where µr (= 1 + χm) is the relative permeability of the material.
(13)
The Permeability Concept Applied Applied Field, Baa Field,B
++ ++
Total Total Field Field BB == BBaa ++BBss
Magnetic MagneticMaterial Material
Secondary SecondaryField, Field, BBss Magnetization Magnetization
χχm B B B == µµ00H H ++ µµ00M M == µµ00H H ++ µµ001+ χm µB 1+ χmm µ00 B BB BB B H Magnetic Field Field Intensity Intensity H == µ (1+ χ ) == µ µ == µ ,, Magnetic µ00 (1+ χmm ) µ00 µrr µ
13 13
Slide No. 14
Equations (5), (6), and (10) are familiarly known as the constitutive relations, where σ, ε, and µ are the material parameters. The parameter σ takes into account explicitly the phenomenon of conduction, whereas the parameters ε and µ take into account implicitly the phenomena of polarization and magnetization, respectively.
Materials and Constitutive Relations Summarizing, Summarizing, JJcc == σσE Conductors E Conductors
D D == εεEE B H = H = µB
µ
Dielectrics Dielectrics Magnetic Magnetic materials materials
E E and and B B are are the the fundamental fundamental field field vectors. vectors. D D and and H H are are mixed mixed vectors vectors taking taking into into account account the the dielectric dielectric and and magnetic magnetic properties properties of of the the material material implicity and µµ,, respectively. respectively. implicity through through εε and
14 14
Slide No. 15
The constitutive relations (5), (6), and (10) tell us that Jc is parallel to E, D is parallel to E, and H is parallel to B, independent of the directions of the field vectors. For anisotropic materials, the behavior depends upon the directions of the field vectors. The constitutive relations have then to be written in matrix form. For example, in an anisotropic dielectric, each component of P and hence of D is in general dependent upon each component of E. Thus, in terms of components in the Cartesian coordinate system, the constitutive relation is given by
⎡ D x ⎤ ⎡ε 11 ε 12 ⎢ D ⎥ = ⎢ε ⎢ y ⎥ ⎢ 21 ε 22 ⎢⎣ D z ⎥⎦ ⎢⎣ε 31 ε 32
ε 13 ⎤ ⎡ E x ⎤ ε 23 ⎥⎥ ⎢⎢ E y ⎥⎥ ε 33 ⎥⎦ ⎢⎣ E z ⎥⎦
(14)
or, simply by [D] = [ε] [E] where [D] and [E] are the column matrices consisting of the components of D and E, respectively, and [ε] is the permittivity matrix containing the elements εij, i = 1, 2, 3 and j = 1, 2, 3. Similar relationships hold for anisotropic conductors and anisotropic magnetic materials.
(15)
Anisotropic Materials Example: Example: Anisotropic Anisotropic Dielectrics Dielectrics ⎡⎡D Exx⎤⎤ Dxx ⎤⎤ ⎡⎡εε11 11 εε12 12 εε13 13⎤⎤ ⎡⎡E ⎢⎢ ⎥⎥ ⎢⎢ ⎥⎥⎢⎢ ⎥⎥ = D ε ε ε E = D ε ε ε 22 23 ⎢⎢ yy ⎥⎥ ⎢⎢ 21 21 22 23⎥⎥ ⎢⎢ Eyy⎥⎥ ⎢⎣⎢D ⎥⎦⎥ ⎢⎣⎢εε31 εε32 εε33 ⎥⎦⎥⎢⎣⎢EEz ⎦⎥⎥ D z ⎣ z ⎦ ⎣ 31 32 33 ⎦ ⎣ z ⎦
[D] [D] == [[ε]] [E] [E] 15 15
Slide No. 16
Since the permittivity matrix is symmetric, that is, εij = εji, from considerations of energy conservation, an appropriate choice of the coordinate system can be made such that some or all of the nondiagonal elements are zero. For a particular choice, all of the nondiagonal elements can be made zero so that
⎡ε1 0 ⎢ [ε ] = ⎢ 0 ε 2 ⎢ ⎢⎣ 0 0
0⎤ ⎥ 0⎥ ⎥ ε 3 ⎥⎦
(16)
Then D x′ = ε 1 E x′
(17a)
D y ′ = ε 2 E y′
(17b)
D z ′ = ε 3 E z′
(17c)
so that D and E are parallel when they are directed along the coordinate axes, although with different values of “effective permittivity,” that is, ratio of D to E, for each such direction. The axes of the coordinate system are then said to be the “principal axes” of the medium. Thus when the field is directed along a principal axis, the anisotropic medium can be treated as an isotropic medium of permittivity equal to the corresponding effective permittivity.
Specific Example of Anisotropic Dielectrics ⎡⎡εε11 00 00 ⎤⎤ ⎢ ⎥ [ε ]] == ⎢⎢⎢ 00 εε22 00 ⎥⎥⎥ ⎢⎣⎢ 00 00 εε3 ⎦⎥⎥ ⎣ 3⎦
Dxx′′ == εε11EExx′′
Dyy′′ == εε22EEyy′′
Dzz′′ == εε33EEzz′′
Characteristic Characteristic Polarizations Polarizations –– xx′′,, yy′′,, zz′′ Effective Effective Permittivities Permittivities –– εε11,, εε22,, εε33 16 16
Slide Nos. 17-18
The electric and magnetic fields are governed by a set of four laws, known as the Maxwell’s equations, resulting from several experimental findings and a purely mathematical contribution. Together with the constitutive relations, Maxwell’s equations form the basis for the entire electromagnetic field theory. We shall consider the time variations of the fields to be arbitrary and introduce these equations and an auxiliary equation in the time domain form. In view of their experimental origin, the fundamental form of Maxwell’s equations is the integral form. We shall first present all four Maxwell’s equations in integral form and the auxiliary equation, the law of conservation of charge, and then discuss each equation and several points of interest pertinent to them. It is understood that all field quantities are real functions of position and time, that is, E = E(r, t) = E(x, y, z, t), etc.
Maxwell’s Equations in Integral Form and the Law of Conservation of Charge d
∫CCE ⋅ dl = – dt ∫SS B ⋅ dS
Faraday’s Faraday’s Law Law
dd D dS H dl J dS = + H dl J dS D⋅ dS = + ⋅ ⋅ ∫CC ∫SS dt dt ∫SS
Ampere’s Ampere’s Circuital Circuital Law Law 17 17
Maxwell’s Equations in Integral Form and the Law of Conservation of Charge
∫SS D ⋅dS = ∫VV ρ dv
Gauss’ Gauss’ Law Law for for the the Electric Electric Field Field
∫SS B ⋅ dS = 0
Gauss’ Gauss’ Law Law for for the the Magnetic Magnetic Field Field
dd J dS = − ρ dv J dS dv = − ⋅ ∫SS ∫ dt V dt V
Law Law of of Conservation Conservation of of Charge Charge
d J dS + ρ dv = 0 ∫SS ⋅ dt ∫VV
18 18
Slide No. 19
Faraday’s Law Faraday’s law is a consequence of the experimental finding by Michael Faraday in 1831 that a time varying magnetic field gives rise to an electric field. Specifically, the electromotive force around a closed path C is equal to the negative of the time rate of increase of the magnetic flux enclosed by that path, that is,
d
∫CE • dl = − dt ∫SB • dS where S is any surface bounded by C.
(18)
Faraday’s Law d B ⋅ dS ∫CCEE⋅ dldl == −− dtdt ∫∫SS B ⋅ dS d
B B
SS CC dS dS
Electromotive Electromotive Force Force (emf) (emf) or or voltage voltage around around C C = = Negative Negative of of the the time time rate rate of of increase increase of of the the magnetic magnetic flux flux enclosed enclosed by by C. C.
19 19
Slide No. 20
There are certain procedures and observations of interest pertinent to (18): 1. The direction of the infinitesimal surface vector dS denotes that the magnetic
flux is to be evaluated in accordance with the right hand screw rule (R.H.S. rule), that is, in the sense of advance of a right hand screw as it is turned around C in the sense of C. The R.H.S. rule is a convention that is applied consistently for all electromagnetic field laws involving integration over surfaces bounded by closed paths. 2. In evaluating the surface integrals in (18), any surface S bounded by C can be
employed. This implies that the time derivative of the magnetic flux through all possible surfaces bounded by C is the same in order for the emf around C to be unique. 3. The minus sign on the right side of (18) tells us that when the magnetic flux
enclosed by C is increasing with time, the induced voltage is in the sense opposite to that of C. If the path C is imagined to be occupied by a wire, then a current would flow in the wire which produces a magnetic field so as to oppose the increasing flux. Similar considerations apply for the case of the magnetic flux enclosed by C decreasing with time. These are in accordance with Lenz’ law which states that the sense of the induced emf is such that any current it produces tends to oppose the change in the magnetic flux producing it. 4. If loop C contains more than one turn, such as in an N-turn coil, then the
surface S bounded by C takes the shape of a spiral ramp. For a tightly wound coil, this is equivalent to the situation in which N separate, identical, single-turn loops are stacked so that the emf induced in the N-turn coil is N times the emf induced in one turn. Thus, for an N-turn coil,
Some Considerations RHS RHS rule rule
CC CC
N-turn N-turn loop loop
ddψ ψ emf == –N –N dt dt ψ == Magnetic Magnetic flux flux enclosed enclosed by by one one turn turn 20 20
emf = − N
dψ dt
(19)
where ψ is the magnetic flux computed as though the coil is a one-turn coil. Slide No. 21
Ampere’s Circuital Law Ampere’s circuital law is a combination of an experimental finding of Oersted that electric currents generate magnetic fields and a mathematical contribution of Maxwell that time-varying electric fields give rise to magnetic fields. Specifically, the magnetomotive force, or mmf, around a closed path C is equal to the sum of the current enclosed by that path due to actual flow of charges and the displacement current due to the time rate of increase of the electric flux (or displacement flux) enclosed by that path, that is, d
∫CH • dl = ∫S J • dS + dt ∫SD • dS where S is any surface bounded by C. Since magnetic force acts perpendicular to the motion of a charge, the
∫C
magnetomotive force, that is, H • dl , does not have a physical meaning similar to that of the electromotive force. The terminology arises purely from analogy with
∫C
electromotive force for E • dl . In evaluating the surface integrals in (20), any surface S bounded by C can be employed. However, the same surface S must be employed for both surface integrals. This implies that the sum of the current due to flow of charges and the displacement current through all possible surfaces bounded C is the same in order for the mmf around
C to be unique.
(20)
Ampere’s Circuital Law dd
dS ++ dt ∫S D D⋅ dS dS ∫CCHH⋅dldl == ∫SSJJ⋅ dS dt S
SS
J, J,DD
CC
dS dS
Magnetomotive Magnetomotive force force (mmf) (mmf) around around C C == Current due to charges crossing S Current due to charges crossing S bounded bounded by by C C ++ Time Time rate rate of of increase increase of of electric electric (or (or displacement) displacement) flux flux 21 21 crossing crossing SS
The current density J in (20) pertains to true currents due to motion of true charges. It does not pertain to currents resulting from the polarization and magnetization phenomena, since these are implicitly taken into account by D and H.
Slide No. 22 d D • dS is not a true current, that is, it is not a The displacement current, dt S
∫
current due to actual flow of charges, such as in the case of the conduction current in wires or a convection current due to motion of a charged cloud in space. Mathematically, d it has the units of dt [(C/m2) × m2] or amperes, the same as the units for a true current, as
it should be. Physically, it leads to the same phenomenon as a true current does, even in free space for which P is zero, and D is simply equal to ε0E. Without it, the uniqueness of the mmf around a given closed path C is not ensured. In fact, Ampere’s circuital law in its original form did not contain the displacement current term, thereby making it valid only for the static field case. It was the mathematical contribution of Maxwell that led to the modification of the original Ampere’s circuital law by the inclusion of the displacement current term. Together with Faraday’s law, this modification in turn led to the theoretical prediction by Maxwell of the phenomenon of electromagnetic wave propagation in 1864 even before it was confirmed experimentally 23 years later in 1887 by Hertz.
The Displacement Current Concept dS dS 11
dd H dl D == JJ⋅⋅ dS ⋅ H dl dS D⋅⋅dS dS11 11 + + ⋅ ∫∫CC11 ∫∫SS11 dt dt ∫∫SS11
d H dl == JJ⋅⋅dS ++ d D ⋅ H dl dS D⋅⋅dS dS22 2 ⋅ 2 ∫∫CC22 ∫∫SS22 dt dt∫∫SS22
dS dS22
CC1 CC2 1 2 SS22
SS1 1
d 00 == ∫ dS++ dtd ∫∫S + S D D⋅⋅dS dS ∫SS11++SS22 JJ⋅⋅dS dt S11 + S22 dd
SS
dS++ dt ∫∫S D D⋅⋅dS dS ==00 ∫∫SSJJ⋅⋅dS dt S
dd D D⋅⋅dS dS== ––∫∫S JJ⋅⋅dS dS dt dt ∫∫SS S
Capacitor Capacitor Plates Plates
DD I(t )
I(It()t )
Displacement Displacement current current out out of of SS == Current Current due due to to flow flow of of charges charges into into S. S.
++––
VV((tt))
22 22
Slide No. 23 Gauss’ Law for the Electric Field: Gauss’ law for the electric field states that electric charges give rise to electric field. Specifically, the electric flux emanating from a closed surface S is equal to the charge enclosed by that surface, that is,
∫SD • dS = ∫V ρ dv where V is the volume bounded by S. In (21), the quantity ρ is the volume charge density having the units coulombs per cubic meter (C/m3). The charge density ρ in (21) pertains to true charges. It does not pertain to charges resulting from the polarization phenomena, since these are implicitly taken into account by the definition of D. The cut view in the figure indicates that electric field lines are discontinuous wherever there are charges, diverging from positive charges and converging on negative charges.
(21)
Gauss’ Law for the Electric Field dS == ∫V ρρ dv dv ∫SS DD⋅dS V D D
ρρ VV
SS
dS dS
Electric Electric (or (or displacement) displacement) flux flux emanating emanating from from aa closed closed surface surface SS == Charge Charge contained contained in in the the volume volume VV bounded bounded by by SS..
23 23
Slide No. 24 Gauss’ Law for the Magnetic Field: Gauss’ law for the magnetic field states that the magnetic flux emanating from a closed surface S is equal to zero, that is,
∫SB • dS = 0
(22)
Thus, whatever magnetic flux enters (or leaves) a certain part of the closed surface must leave (or enter) through the remainder of the closed surface. The observation concerning the time derivative of the magnetic flux crossing all possible surfaces bounded by a given closed path C in connection with the discussion of Faraday’s law implies that the time derivative of the magnetic flux emanating from a closed surface S is zero, that is,
d dt SB • dS = 0
∫
One can argue then that the magnetic flux emanating from a closed surface is zero, since at an instant of time when no sources are present the magnetic field vanishes. Thus, Gauss’ law for the magnetic field is not independent of Faraday’s law. The cut view in the figure indicates that magnetic field lines are continuous having no beginnings or endings.
(23)
Gauss’ Law for the Magnetic Field dS == 00 ∫SS BB⋅ dS BB
dS dS SS
Magnetic Magnetic flux flux emanating emanating from from aa closed closed surface surface is is equal equal to to zero. zero. 24 24
Slide No. 25 Law of Conservation of Charge An auxiliary equation known as the law of conservation of charge states that the current due to flow of charges emanating from a closed surface S is equal to the time rate of decrease of the charge inside the volume V bounded by that surface, that is,
d
∫SJ • dS = − dt ∫V ρ dv or
d
∫SJ • dS + dt ∫V ρ dv = 0
(24)
Combining the observation concerning the sum of the current due to flow of charges and the displacement current through all possible surfaces bounded by a given closed path C in connection with the discussion of Ampere’s circuital law with the law of conservation of charge, we obtain for any closed surface S,
d ⎜⎛ D • dS − dt ⎝ S
∫
⎞
∫V ρ dν ⎟⎠ = 0
where V is the volume bounded by S. One can then argue that the quantity inside the parentheses is zero, since at an instant of time when no sources are present, it vanishes. Thus, Gauss’ law for the electric field is not independent of Ampere’s circuital law in view of the law of conservation of charge.
(25)
The Law of Conservation of Charge d JJ⋅⋅dS == –– d ρρ dv dS dv ∫SS dt dt ∫VV J
ρρ(t) (t)
VV
SS
dS
Current Current due due to to flow flow of of charges charges emanating emanating from from aa closed closed surface surface SS == Time Time rate rate of of decrease decrease of of charge charge within within the the volume volume VV bounded bounded by by S. S.
25 25
Slide Nos. 26-27 From the integral forms of Maxwell’s equations, one can obtain the corresponding differential forms through the use of Stoke’s and divergence theorems in vector calculus, given, respectively, by
∫CA • dl = ∫S(∇ x A) • dS
(26a)
∫SA • dS = ∫V (∇ • A)dν
(26b)
where in (26a), S is any surface bounded by C, and in (26b), V is the volume bounded by S.
Curl and Divergence ∇ A == ∇ xx A
Lim A⋅⋅ddll⎤⎤ Lim ⎡⎡∫ A ⎢⎢ ⎥⎥ aan Curl n Curl ∆∆SS → → 00 ⎢⎢ ∆∆SS ⎥⎥ ⎣⎣ ⎦⎦max max
Lim dS A⋅⋅dS Lim ∫∫SSA ∇ ∇ ⋅⋅A A == ∆v → 0 ∆v ∆v → 0 ∆v
In In Cartesian Cartesian coordinates, coordinates, aaxx ∂∂ ∇ A == ∂x ∇ xx A ∂x AAxx
aayy ∂∂ ∂y ∂y AAyy
aazz ∂∂ ∂z ∂z AAzz
Divergence Divergence ∂Ayy ∂A ∂A ∂Axx + ∂A ∂Az ∇ ⋅ A = ∇ ⋅ A = ∂x + ∂y ++ ∂z z ∂x ∂y ∂z ∇ A ≡≡ 00 Identity Identity ∇⋅⋅∇ ∇ xx A
26 26
Two Theorems Stokes’ Stokes’ Theorem Theorem
∫CCA ⋅ dl = ∫SS(∇ x A)⋅ dS Divergence Divergence Theorem Theorem
∫SSA ⋅ dS = ∫VV (∇ ⋅A) dν 27 27
Slide No. 28 Thus, Maxwell’s equations in differential form are given by ∇xE=−
∂B ∂t
∇ x H=J+
∂D ∂t
(27)
(28)
∇ •D = ρ
(29)
∇ •B = 0
(30)
corresponding to the integral forms (18), (20), (21), and (22), respectively. These differential equations state that at any point in a given medium, the curl of the electric field intensity is equal to the time rate of decrease of the magnetic flux density, and the curl of the magnetic field intensity is equal to the sum of the current density due to flow of charges and the displacement current density (time derivative of the displacement flux density), whereas the divergence of the displacement flux density is equal to the volume charge density, and the divergence of the magnetic flux density is equal to zero. Auxiliary to the Maxwell’s equations in differential form is the differential equation following from the law of conservation of charge (24) through the use of (26b). Familiarly known as the continuity equation, this is given by ∇•J +
∂ρ =0 ∂t
It states that at any point in a given medium, the divergence of the current density due to flow of charges plus the time rate of increase of the volume charge density is equal to zero.
(31)
Maxwell’s Equations in Differential Form and the Continuity Equation B ∇xE= –∂ ∂t ∇ x H = J + ∂D ∂t
∇ ∇⋅ D D == ρρ
⋅ ∇ ∇⋅ BB== 00
Faraday’s Faraday’s Law Law Ampere’s Ampere’s Circuital Circuital Law Law Gauss’ Gauss’ Law Law for for the the Electric Electric Field Field
Gauss’ Gauss’ Law Law for for the the Magnetic Magnetic Field Field
∂ρ ∂ρ ∇ ⋅ JJ ++ == 00 ∂∂ tt
Continuity Continuity Equation Equation
28 28
From the interdependence of the integral laws discussed in Slides 24 and 25, it follows that (30) is not independent of (27), and (29) is not independent of (28) in view of (31).
Slide No. 29 Maxwell’s equations in differential form lend themselves well for a qualitative discussion of the interdependence of time-varying electric and magnetic fields giving rise to the phenomenon of electromagnetic wave propagation. Recognizing that the operations of curl and divergence involve partial derivatives with respect to space coordinates, we observe that time-varying electric and magnetic fields coexist in space, with the spatial variation of the electric field governed by the temporal variation of the magnetic field in accordance with (27), and the spatial variation of the magnetic field governed by the temporal variation of the electric field in addition to the current density in accordance with (28). Thus, if in (28) we begin with a time-varying current source represented by J, or a time-varying electric field represented by ∂D/dt, or a combination of the two, then one can visualize that a magnetic field is generated in accordance with (28), which in turn generates an electric field in accordance with (27), which in turn contributes to the generation of the magnetic field in accordance with (28), and so on, as depicted on Slide 29. Note that J and ρ are coupled, since they must satisfy (31). Also, the magnetic field automatically satisfies (30), since (30) is not independent of (27). The process depicted is exactly the phenomenon of electromagnetic waves propagating with a velocity (and other characteristics) determined by the parameters of the medium. In free space, the waves propagate unattenuated with the velocity 1 / µ 0 ε 0 , familiarly represented by the symbol c. If either the term ∂B/∂t in (27) or the
Qualitative Qualitative Explanation Explanation of of Electromagnetic Electromagnetic Wave Wave Phenomenon Phenomenon from from Maxwell’s Maxwell’s Equations Equations ++
JJ
++
Ampere’s Ampere’s Circuital Circuital Law Law
Continuity Continuity Equation Equation
ρρ
Gauss’ Gauss’ Law Law for for EE
H H,B ,B
Faraday’s Faraday’s Law Law
D, E D,E
29 29
term ∂D/∂t in (28) is not present, then wave propagation would not occur. As already stated in Slide 22, it was through the addition of the term ∂D/∂t in (28) that Maxwell predicted electromagnetic wave propagation before it was confirmed experimentally.
Slide Nos. 30-31 Of particular importance is the case of time variations of the fields in the sinusoidal steady state, that is, the frequency domain case. In this connection, the frequency domain forms of Maxwell’s equations are of interest. Using the phasor notation based on
[
]
[
A cos (ωt + φ ) = Re Ae jφ e jωt = Re A e jωt
]
(32)
where A = Ae jφ is the phasor corresponding to the time function, we obtain these equations by replacing all field quantities in the time domain form of the equations by the corresponding phasor quantities and ∂/∂t by jω. Thus with the understanding that all phasor field quantities are functions of space coordinates, that is, E = E(r) , etc., we write the Maxwell’s equations in frequency domain as
∇ x E = − jω B
(33)
∇ x H = J + jω D
(34)
∇•D = ρ
(35)
∇•B = 0
(36)
Also, the continuity equation (31) transforms to the frequency domain form
∇ • J + jω ρ = 0 Note that since ∇ • ∇ x E = 0 , (36) follows from (33) and since ∇ • ∇ x H = 0 , (35) follows from (34) with the aid of (37).
(37)
Sinusoidal Case
[
jφ jω t
] [
jω t
]
j φ j ω t = Re A e jω t A cos ((ω ω tt ++φφ)) == Re Re Ae Ae ee = Re A e
Maxwell’s Equations ∇ x E = − jω B
∇ ∇⋅ D D == ρρ
∇ x H = J + jω D
∇ ∇⋅ B B == 00
⋅
⋅
30 30
Now the constitutive relations in phasor form are
D = εE
(38a)
B
(38b)
H=
µ
Jc = σ E
(38c)
Sinusoidal Case Continuity Equation ∇ ⋅ J + jω ρ = 0
Constitutive Relations D D == εεEE
B B H H == µ
µ
Jcc == σσE E 31 31
Slide No. 32 Substituting (38a)-(38c) into (33)-(36), we obtain for a material medium characterized by the parameters ε, µ, and σ, ∇ x E = − jω µ H
(39)
∇ x H = (σ + jωε ) E
(40)
∇•H = 0
(41)
ρ ε
(42)
∇•E =
Note however that if the medium is homogeneous, that is, if the material parameters are independent of the space coordinates, (40) gives
∇•E =
1 ∇•∇ x H = 0 σ + jωε
so that ρ = 0 in such a medium. A point of importance in connection with the frequency domain form of Maxwell‘s equations is that in these equations, the parameters ε, µ, and σ can be allowed to be functions of ω. In fact, for many dielectrics, the conductivity increases with frequency in such a manner that the quantity σ/ωε is more constant than is the conductivity. This quantity is the ratio of the magnitudes of the two terms on the right side of (40), that is, the conduction current density term σ E and the displacement current density term jωε E .
(43)
Maxwell’s Equations for the Sinusoidal Case for a Material Medium
∇ x E = – j ω µH
∇ ∇⋅ H H == 00
∇ x H = (σ + jωε )E
∇ ∇⋅ E E == ε
⋅
ρρ ε
1 ∇⋅E = ∇ ∇x H =0 σ + j ωε ⋅ 32 32
Slide Nos. 33-35 Maxwell’s equations in differential form govern the interrelationships between the field vectors and the associated source densities at points in a given medium. For a problem involving two or more different media, the differential equations pertaining to each medium provide solutions for the fields that satisfy the characteristics of that medium. These solutions need to be matched at the boundaries between the media by employing “boundary conditions,” which relate the field components at points adjacent to and on one side of a boundary to the field components at points adjacent to and on the other side of that boundary. The boundary conditions arise from the fact that the integral equations involve closed paths and surfaces and they must be satisfied for all possible closed paths and surfaces whether they lie entirely in one medium or encompass a portion of the boundary. The boundary conditions are obtained by considering one integral equation at a time and applying it to a closed path or a closed surface encompassing the boundary, and in the limit that the area enclosed by the closed path, or the volume bounded by the closed surface goes to zero. Let the quantities pertinent to medium 1 be denoted by subscript 1 and the quantities pertinent to medium 2 be denoted by subscript 2, an be the unit normal vector to the surface and directed into medium 1. Let all normal components at the boundary in both media be directed along an and denoted by an additional subscript
n, and all tangential components at the boundary in both media be denoted by an additional subscript t. Let the surface charge density (C/m2) and the surface current density (A/m) on the boundary be ρS and JS, respectively. Then, the boundary conditions corresponding to the Maxwell’s equations in integral form can be summarized as
Boundary Conditions JJn1 n1 JJS S
H Htt22
H Htt11 aan n
JJn2 n2
Medium Medium 1,1, zz >>00 σσ11,, εε11,, µµ11 BBn1 n1
D Dn1n1
ρρss EEt1 t1
BBn2 D Dn2n2 n2
EEt2 t2
zz== 00 zz Medium Medium 2, 2, zz , is given by
< P > = Re[P ]
(60)
where P is the complex Poynting vector given by
P=
1 E x H* 2
(61)
where the star denotes complex conjugate. The Poynting theorem for the frequency domain case, known as the complex Poynting’s theorem, is given by ⎛1 ⎞ − ∫ ⎜ E • J *0 ⎟ dv = ∫ < p d > dv + j 2ω ∫ (< wm > − < we > )dv + ∫ P • dS 2 ⎠ V⎝ V V S
(62)
where 1 * < pd > = σ E • E 2 < we > = 1 ε E • E* 4 1 * < wm > = µ H • H 4
are the time-average power dissipation density, the time-average electric stored energy density, and the time-average magnetic stored energy density, respectively. Equation (62) states that the time-average, or real, power delivered to the volume V by the current source is accounted for by the time-average power dissipated in the volume plus the timeaverage power carried by the electromagnetic field out of the volume through the surface S bounding the volume, and that the reactive power delivered to the volume V by the current source is equal to the reactive power carried by the electromagnetic field out of
(63a) (63b) (63c)
Complex Poynting Vector and Complex Poynting’s Theorem PP == Re Re[PP]
[]
1 ∗∗ PP == 1 E xx H E H 22
1 −−∫ ⎛⎝⎛ 1E ⋅ J **⎞⎞ dv = < p > dv + j2ω (< w > − < w >) dv + ⎝ 22 E ⋅ J00⎠⎠ dv = ∫∫ < pdd > dv + j2ω ∫∫ (< wmm > − < wee > ) dv + V V V V
V
> == 11σσEE⋅⋅EE** 22
V
* 1 == 41εεEE⋅⋅EE* 4
dS ∫∫PP⋅⋅dS S S
* 1 == 41µµH H⋅⋅H H* 4
The The time-average, time-average, or or real, real, power power delivered delivered to to the the volume volume VV by by the the current current source source isis accounted accounted for for by by the the time-average time-average power power dissipated dissipated in in the the volume volume plus plus the the time-average time-average power power carried carried by by the the electromagnetic electromagnetic field field out out of of the the volume volume through through the surface S bounding the volume. The reactive power delivered to the surface S bounding the volume. The reactive power delivered to the the volume volume VV by by the the current current source source isis equal equal to to the the reactive reactive power power carried carried by by the the electromagnetic electromagnetic V through the surface S plus a quantity that field out of the volume ω times times the the field out of the volume V through the surface S plus a quantity that isis 22ω difference difference between between the the time-average time-average magnetic magnetic and and electric electric stored stored energies energies in in the the 44 44 volume. volume.
the volume V through the surface S plus a quantity which is 2ω times the difference between the time-average magnetic and electric stored energies in the volume.
Slide No. 45 While every macroscopic field obeys Maxwell’s equations in their entirety, depending on their most dominant properties it is sufficient to consider a subset of, or certain terms only, in the equations. The primary classification of fields is based on their time dependence. Fields which do not change with time are called static. Field which change with time are called dynamic. Static fields are the simplest kind of fields, because for them ∂/∂t = 0 and all terms involving differentiation with respect to time go to zero. Dynamic fields are the most complex, since for them Maxwell’s equations in their entirety must be satisfied, resulting in wave type solutions, as provided by the qualitative explanation earlier. However, if certain features of the dynamic field can be analyzed as though the field were static, then the field is called quasistatic. If the important features of the field are not amenable to static type field analysis, they are generally referred to as time-varying, although in fact, quasistatic fields are also time-varying. Since in the most general case, time-varying fields give rise to wave phenomena, involving velocity of propagation and time delay that cannot be neglected, it can be said that quasistatic fields are those time-varying fields for which wave propagation effects can be neglected.
Classification of Fields Static Static Fields Fields (( No No time time variation; variation; ∂∂ ⁄⁄ ∂∂ tt == 0) 0) Static Static electric, electric, or or electrostatic electrostatic fields fields Static Static magnetic, magnetic, or or magnetostatic magnetostatic fields fields Electromagnetostatic Electromagnetostatic fields fields Dynamic Dynamic Fields Fields (Time-varying) (Time-varying) Quasistatic Quasistatic Fields Fields (Dynamic (Dynamic fields fields that that can can be be analyzed analyzed as as though though the the fields fields are are static) static) Electroquasistatic Electroquasistatic fields fields Magnetoquasistatic Magnetoquasistatic fields fields
45 45
Slide No. 46 Static Fields For static fields, ∂/∂t = 0. Maxwell’s equations in integral form and the law of conservation of charge become
∫ E • dl = 0
(64a)
∫ H • dl = ∫ J • dS
(64b)
∫ D • dS = ∫ ρ dv
(64c)
∫ B • dS = 0
(64d)
∫ J • dS = 0
(64e)
C
C
S
S
V
S
S
whereas Maxwell’s equations in differential form and the continuity equation reduce to
∇xE=0
(65a)
∇xH=J
(65b)
∇•D=ρ
(65c)
∇•B=0
(65d)
∇•J=0
(65e)
Immediately, one can see that, unless J includes a component due to conduction current, the equations involving the electric field are completely independent of those involving the magnetic field. Thus the fields can be subdivided into static electric fields, or electrostatic fields, governed by (64a) and (64c), or (65a) and (65c), and static magnetic fields, or magnetostatic fields, governed by (64b) and (64d), or (65b) and (65d).
Static Fields For For static static fields, fields, ∂∂ ⁄⁄ ∂∂ tt == 0, 0, and and the the equations equations reduce reduce to to
∫CCE ⋅ dl = 0
dS ∫CCHH⋅ dldl ==∫SSJJ ⋅dS dS == ∫ ρ dv dv ∫SSDD⋅ dS vv dS == 00 ∫SSBB⋅ dS ∫SSJJ ⋅ddSS == 00
∇x E=0 ∇xH = J ∇ ∇⋅ D D == ρ ∇ ∇⋅ B B == 00 ∇ ∇⋅ JJ == 00
46 46
The source of a static electric field is ρ, whereas the source of a static magnetic field is J. One can also see from (64e) or (65e) that no relationship exists between J and ρ. If J includes a component due to conduction current, then, since Jc = σE, a coupling between the electric and magnetic fields exists for that part of the total field associated with Jc. However, the coupling is only one way, since the right side of (64a) or (65a) is still zero. The field is then referred to as electromagnetostatic field. It can also be seen then that for consistency, the right sides of (64c) and (65c) must be zero, since the right sides of (64e) and (65e) are zero. We shall now consider each of the three types of static fields separately and discuss some fundamental aspects.
Slide No. 47 Electrostatic Fields The equations of interest are (64a) and (64c), or (65a) and (65c). The first of each pair of these equations simply tells us that the electrostatic field is a conservative field, and the second of each pair of these equations enables us, in principle, to determine the electrostatic field for a given charge distribution. Alternatively, the potential function equation (53), which reduces to
ρ
∇2 Φ = − ε
(66)
can be used to find the electric scalar potential, Φ, from which the electrostatic field can be determined by using (48), which reduces to
E = –∇Φ Equation (66) is known as the Poisson’s equation, which automatically includes the condition that the field be conservative. It is worth noting that the potential difference
(67)
Electrostatic Fields
∫CCE ⋅ dl = 0 ∫SSD ⋅ dS = ∫vv ρ dv ρρ ε E E == –∇Φ –∇Φ 2
2 =− ∇ ∇Φ Φ = −ε
∇x E=0
∇ ∇⋅ D D == ρρ
⋅
Poission’s Poission’s Equation Equation
[−∇Φ]⋅ dl dl == Φ ΦAA −− Φ ΦBB ∫AA EE ⋅ dldl == ∫AA [−∇Φ] BB
BB
47 47
between two points A and B in the static electric field, which is independent of the path followed from A to B because of the conservative nature of the field is B
∫A
E • dl =
B
∫A [−∇Φ]• dl
(68)
= ΦA − ΦB the difference between the value of Φ at A and the value of Φ at B. The choice of minus sign associated with ∇Φ in (48) is now evident. Slide No. 48 The solution to Poisson’s equation (66) for a given charge density distribution
ρ(r) is given by Φ (r ) =
1
ρ (r ′)
∫ r − r ′ dv′
4πε V ′
(69)
where the prime denotes source point and no prime denotes field point. Although cast in terms of volume charge density, (69) can be formulated in terms of a surface charge distribution, a line charge distribution, or a collection of point charges. In particular, for a point charge Q(r′), the solution is given by Φ (r ) =
Q(r ′) 4πε r − r ′
(70)
It follows from (67) that the electric field intensity due to the point charge is given by E(r ) =
Q(r ′)(r − r ′) 4πε r − r ′
3
which is exactly the expression that results from Coulomb’s law for the electric force between two point charges.
(71)
Solution for Potential and Field 11 Φ(r) = Φ(r) = 4πε 4πε
∫
VV′′
ρρ((rr′′)) dv′ rr−− rr′′ dv′
Solution Solution for for charge charge distribution distribution
Q( Q(rr′′)) Φ(r) = Φ(r) = 4πε r − r 4πε r − r′′
Solution Solution for for point charge point charge
Q( Q(rr′′)(r )(r −− rr′′)) E(r) = E(r) = 33 r − r ′ 44πε πε r − r ′
Electric Electric field field due due to point charge to point charge 48 48
Equation (69) or its alternate forms can be used to solve two types of problems: (a) finding the electrostatic potential for a specified charge distribution by evaluating the integral on the right side, which is a straightforward process with the help of a computer but can be considerably difficult analytically except for a few examples, and (b) finding the surface charge distribution on the surfaces of an arrangement of conductors raised to specified potentials, by inversion of the equation, which is the basis for numerical solution by the well-known method of moments. In the case of (a), the electric field can then be found by using (67). Slide No. 49 In a charge-free region, ρ = 0, and Poisson’s equation (66) reduces to ∇2 Φ = 0 which is known as the Laplace’s equation. The field is then due to charges outside the region, such as surface charge on conductors bounding the region. The situation is then one of solving a boundary value problem. In general, for arbitrarily-shaped boundaries, a numerical technique, such as the method of finite differences, is employed for solving the problem. Here, we consider analytical solution involving one-dimensional variation of Φ.
(72)
Laplace’s Equation and One-Dimensional Solution For ρ = 0, Poission’s equation reduces to 2
2 = 0 Laplace’s equation ∇ ∇Φ Φ=0
dd22Φ Φ =0 22 = 0 dx dx
Φ Φ == Ax Ax ++ BB 49 49
Slide Nos. 50-51 A simple example is that of the parallel-plate arrangement in which two parallel, perfectly conducting plates (σ = ∞, E = 0) of dimensions w along the y-direction and l along the z-direction lie in the x = 0 and x = d planes. The region between the plates is a perfect dielectric (σ = 0) of material parameters ε and µ. The thickness of the plates is shown exaggerated for convenience in illustration. A potential difference of V0 is maintained between the plates by connecting a direct voltage source at the end z = –l. If fringing of the field due to the finite dimensions of the structure normal to the x-direction is neglected, or, if it is assumed that the structure is part of one which is infinite in extent normal to the x-direction, then the problem can be treated as one-dimensional with x as the variable, and (72) reduces to d 2Φ =0 dx 2
(73)
The solution for the potential in the charge-free region between the plates is given by Φ ( x) =
V0 (d − x) d
(74)
which satisfies (73), as well as the boundary conditions of Φ = 0 at x = d and Φ = V0 at x = 0. The electric field intensity between the plates is then given by E = −∇Φ =
V0 ax d
(75)
as depicted in the cross-sectional view in Fig. (b) on Slide 50, and resulting in displacement flux density D=
ε V0 d
ax
(76)
Example of Parallel-Plate Arrangement; Capacitance ll
w w
yy
dd
VV0 0
zz (a) (a)
xx= = 00
ε,ε, µµ
xx
xx==dd
zz
zz==––l l
zz== 00
ρρSS VV 00
++ ++ ++ ++ ++ ++ ++ ++ x = 0, Φ = V x = 0, Φ = V00 E, D E, D d, Φ Φ== 00 –– –– –– –– –– –– –– –– xx ==d, zz zz==––l l ρ zz==00
ρSS
VV0 Φ(x) Φ(x)== d0 (d (d −− x) x) d yy
zz (b) x
50 50
Then, using the boundary condition for the normal component of D given by (44c) and noting that there is no field inside the conductor, we obtain the magnitude of the charge on either plate to be
εwl ⎛ εV ⎞ Q = ⎜ 0 ⎟ ( wl ) = V0 d ⎝ d ⎠
(77)
We can now find the familiar circuit parameter, the capacitance, C, of the parallelplate arrangement, which is defined as the ratio of the magnitude of the charge on either plate to the potential difference V0. Thus C=
Q ε wl = V0 d
(78)
Note that the units of C are the units of ε times meter, that is, farads. The phenomenon associated with the arrangement is that energy is stored in the capacitor in the form of electric field energy between the plates, as given by ⎛1 ⎞ We = ⎜ εE x2 ⎟ ( wld ) ⎝2 ⎠ 1 ⎛ εwl ⎞ 2 = ⎜ ⎟V0 2⎝ d ⎠ 1 = CV02 2 the familiar expression for energy stored in a capacitor.
(79)
Electrostatic Analysis of Parallel-Plate Arrangement εεVV0 D D == d 0 aaxx d
VV00 E = −∇Φ = E = −∇Φ = d aaxx d ⎛⎛ εεVV0 ⎞⎞ εεwl wl Q Q == ⎝⎝ d 0 ⎠⎠(wl) (wl) == d VV00 d d
Q Q εwl C C == V == εdwl V00 d
εεwwll C C == d d
Capacitance Capacitance of of the the arrangement, arrangement, FF
11⎛ εεwl ⎛ 11 22⎞ wl⎞ 22 11 22 W Wee == ⎝⎛⎝ 2 εεEExx ⎠⎞⎠(wld) CV (wld) == 2 ⎝⎛⎝ d ⎠⎞⎠VV00 == 2 CV 2 2 00 2 d 51 51
Slide No. 52 Magnetostatic Fields The equations of interest are (64b) and (64d), or (65b) and (65d). The second of each pair of these equations simply tells us that the magnetostatic field is solenoidal, which as we know holds for any magnetic field, and the first of each pair of these equations enables us, in principle, to determine the magnetostatic field for a given current distribution. Alternatively, the potential function equation (54), which reduces to ∇2A = –µJ can be used to find the magnetic vector potential, A, from which the magnetostatic field can be determined by using (47). Equation (80) is the Poisson’s equation for the magnetic vector potential, which automatically includes the condition that the field be solenoidal.
(80)
Magnetostatic Fields dS ∫CCHH⋅ dldl == ∫SSJJ ⋅ dS
∇ H == JJ ∇ xx H
dS == 00 ∫SSBB⋅ dS
∇ ∇⋅ BB== 00
2
∇ ∇ 2A A == ––µµJJ
⋅
Poisson’s Poisson’s equation equation for for magnetic magnetic vector vector potential potential
52 52
Slide No. 53 The solution to (80) for a given current density distribution J(r) is, purely from analogy with the solution (69) to (66), given by A(r ) =
µ J (r ′) dv ′ 4π V∫′ r − r ′
(81)
Although cast in terms of volume current density, (81) can be formulated in terms of a surface current density, a line current, or a collection of infinitesimal current elements. In particular, for an infinitesimal current element I dl(r′), the solution is given by A (r ) =
µI dl (r ′) 4π r − r ′
(82)
It follows from (47) that the magnetic flux density due to the infinitesimal current element is given by B(r ) =
µI dl (r ′) x (r − r ′) 4π r − r ′
3
(83)
which is exactly the law of Biot-Savart that results from Ampere’s force law for the magnetic force between two current elements. Similar to that in the case of (69), (81) or its alternate forms can be used to find the magnetic vector potential and then the magnetic field by using (47) for a specified current distribution. In a current-free region, J = 0, and (80) reduces to ∇2A = 0 The field is then due to currents outside the region, such as surface currents on conductors bounding the region. The situation is then one of solving a boundary value problem as in the case of (72). However, since the boundary condition (44b) relates the
(84)
Solution for Vector Potential and Field µµ J( J(rr′′)) A(r) A(r) == 4π ∫∫ r − r ′ ddvv′′ 4π V ′ r − r ′
Solution Solution for for current current distribution distribution
µµII dl( dl(rr′′)) A(r) A(r) == 4π r − r ′ 4π r − r ′
Solution Solution for for current current element element
µµII ddll((rr′′))xx ((rr−− rr′′)) B B((rr))== 3 44ππ rr −− rr′′ 3
Magnetic Magnetic field field due due to current element to current element
∇ ∇22A A == 00
For For current-free current-free region region
V′
53 53
magnetic field directly to the surface current density, it is straightforward and more convenient to determine the magnetic field directly by using (65b) and (65d). Slide Nos. 54-56 A simple example is that of the parallel-plate arrangement of Fig. (a) on Slide 50 with the plates connected by another conductor at the end z = 0 and driven by a source of direct current I0 at the end z = –l, as shown in Fig. (a) on Slide 54. If fringing of the field due to the finite dimensions of the structure normal to the x-direction is neglected, or, if it is assumed that the structure is part of one which is infinite in extent normal to the xdirection, then the problem can be treated as one-dimensional with x as the variable and we can write the current density on the plates to be
⎧( I 0 /w)a z ⎪ J S = ⎨( I 0 /w)a x ⎪− ( I /w)a 0 z ⎩
on the plate x = 0 on the plate z = 0
(85)
on the plate x = d
In the current-free region between the plates, (65b) reduces to
ax ∂ ∂x Hx
ay 0 Hy
az 0 =0
(86)
Hz
and (65d) reduces to ∂B x =0 ∂x
so that each component of the field, if it exists, has to be uniform. This automatically forces Hx and Hz to be zero since nonzero value of these components do not satisfy the boundary conditions (44b) and (44d) on the plates, keeping in mind that the field is
(87)
Example of Parallel-Plate Arrangement; Inductance ll yy ww
dd
z
I0I0 xx==00
(a)
xx
ε,ε,µµ xx==dd
zz==––l l
zz
zz==00
xx == 00 I0I0
yy
zz
HH, , BB
JJS S zz==––l l
xx ==dd zz
xx
(b)
zz==00
54 54
entirely in the region between the conductors. Thus, as depicted in the cross-sectional view in the figure, I H = w0 a y
(88)
which satisfies the boundary condition (44b) on all three plates, and results in magnetic flux density
µI B = w0 a y
(89)
The magnetic flux, ψ, linking the current I0, is then given by ⎛ µI ⎞ ⎛ µdl ⎞ ψ = ⎜ w0 ⎟ (dl) = ⎜ w ⎟ I0 ⎝ ⎠ ⎝ ⎠
(90)
We can now find the familiar circuit parameter, the inductance, L, of the parallelplate arrangement, which is defined as the ratio of the magnetic flux linking the current to the current. Thus
ψ µdl L= I = w 0
(91)
Note that the units of L are the units of µ times meter, that is, henrys. The phenomenon associated with the arrangement is that energy is stored in the inductor in the form of magnetic field energy between the plates, as given by ⎞ ⎛1 Wm = ⎜ µH 2 ⎟ ( wld ) ⎠ ⎝2 1 ⎛ µdl ⎞ 2 = ⎜ ⎟ I0 2⎝ w ⎠ 1 = LI 02 2 the familiar expression for energy stored in an inductor.
(92)
Magnetostatic Analysis of Parallel-Plate Arrangement on ⎧⎧(I (I0 w)a w)azz on the theplate platexx == 00 ⎪⎪ 0 JJSS == ⎨⎨(I w)axx on on the theplate platezz ==00 (I00 w)a ⎪⎪ w)azz on on the theplate platexx == dd −(I00 w)a ⎩⎩−(I
aaxx
aayy
aazz
∂∂ 0 00 ==00 ∂∂xx 0 H Hxx H Hyy H Hzz
∂B ∂Bxx == 00 ∂x ∂x II00 H = H = w aayy w 55 55
Magnetostatic Analysis of Parallel-Plate Arrangement µµII0 B B == w0 aayy w
⎛⎛ µII00⎞⎞
µdl (dl) = ⎛⎛ dl⎞⎞I
ψ == ⎝ (dl) = ⎝ w ⎠ I00 ⎝ w ⎝ w ⎠ w ⎠⎠ ψ µdl LL == == dl w II00 w
µ dl LL == µ dl w w
Inductance Inductance of of the the arrangement, arrangement, H H
2⎞ 1 ⎛ µµdl dl⎞ 2 1 22 ⎛1 W Wmm == ⎝⎛⎝ 21 µµH H2⎠⎞⎠((wld wld)) == 21 ⎝⎛⎝ w ⎠⎞⎠ II002 == 21 LI LI 2 2 w 2 00 56 56
Slide No. 57 Electromagnetostatic Fields The equations of interest are
∫ E • dl = 0
(93a)
∫ H • dl = ∫ Jc • dS = σ ∫ E • dS
(93b)
∫ D • dS = 0
(93c)
C
C
S
S
S
∫ B • dS = 0
(93d)
∇xE=0
(94a)
∇ x H = Jc = σE
(94b)
∇•D=0
(94c)
∇•B=0
(94d)
S
or, in differential form
From (94a) and (94c), we note that Laplace’s equation (72) for the electrostatic potential is satisfied, so that, for a given problem, the electric field can be found in the same manner as in the case of the example of the capacitor arrangement with perfect dielectric. The magnetic field is then found by using (94b), and making sure that (94d) is also satisfied.
Electromagnetostatic Fields (J (J == JJcc == σσE) E)
dl == 00 ∫CCEE⋅ dl dS == σσ ∫ E E⋅ dS dS ∫CCHH⋅ dldl ==∫SSJJcc ⋅ dS SS
dS == 00 ∫SSDD⋅ dS dS == 00 ∫SSBB⋅ dS
∇ E == 00 ∇ xx E ∇ E H == JJcc == σσE ∇ xx H
∇ ∇⋅ D D == 00
⋅ ∇ ∇⋅ BB== 00
57 57
Slide Nos. 58-60 A simple example is that of the parallel-plate arrangement of Fig. (a) on Slide 50, but with an imperfect dielectric material of parameters σ, ε, and µ, between the plates, as shown in Fig (a) on Slide 58. Then, the electric field between the plates is the same as that given by (75), that is, V E = d0 a x
(95)
σV J c = d0 a x
(96)
resulting in a conduction current of density
from the top plate to the bottom plate, as depicted in the cross-sectional view of Fig. (b) on Slide 58. Since ∂ρ/∂t = 0 at the boundaries between the plates and the slab, continuity of current is satisfied by the flow of surface current on the plates. At the input z = –l, this surface current, which is the current drawn from the source, must be equal to the total current flowing from the top to the bottom plate. It is given by
σwl ⎛ σV ⎞ I c = ⎜ 0 ⎟( wl ) = V0 d ⎝ d ⎠
(97)
We can now find the familiar circuit parameter, the conductance, G, of the parallel-plate arrangement, which is defined as the ratio of the current drawn from the source to the source voltage V0. Thus G=
I c σwl = V0 d
Note that the units of G are the units of σ times meter, that is, siemens(S). The reciprocal quantity, R, the resistance of the parallel-plate arrangement, is given by
(98)
Example of Parallel-Plate Arrangement ll yy ww
dd zz
VV 00 xx==00
σ, σ,ε,ε,µµ
xx (a) (a)
x x==dd
zz
zz==––l l
JJS S IcIc VV 00
zz==00
ρρSS
++ ++ ++ ++ ++ ++ ++ ++ HH, , BB
xx ==0, 0, Φ Φ== VV00 yy
zz
E, E, JJcc,, DD
–– –– –– –– –– –– –– –– xx ==d,d, ΦΦ== 00 zz z z==––l l ρρSS zz==00 JJ
xx
(b)
SS
58 58
V d R = I 0 = σ wl c
(99)
The unit of R is ohms. The phenomenon associated with the arrangement is that power is dissipated in the material between the plates, as given by Pd = (σE 2 )( wld ) ⎛ σwl ⎞ 2 =⎜ ⎟ V0 ⎝ d ⎠ = GV02 =
V02 R
the familiar expression for power dissipated in a resistor.
(100)
Electromagnetostatic Analysis of Parallel-Plate Arrangement VV00 E E == d aaxx d σσVV JJcc == 00 aaxx dd
σσVV σσwl wl V Icc == ⎛⎛ 00 ⎞⎞(wl) = (wl) = ⎝⎝ dd ⎠⎠ dd V00 59 59
Electromagnetostatic Analysis of Parallel-Plate Arrangement IIc σσwl wl G G == V c == d V00 d
Conductance, Conductance, SS
VV00 dd R = R = I == σ wl Icc σ wl
Resistance, Resistance, ohms ohms
σwl 2 PPdd == ((σσEE22)(wld) )(wld)== ⎛⎝⎛⎝ σwl⎞⎠⎞⎠VV002 dd 2 == G GVV002
VV0022 == RR 60 60
Slide No. 61 Proceeding further, we find the magnetic field between the plates by using (94b), and noting that the geometry of the situation requires a y-component of H, dependent on z, to satisfy the equation. Thus
H = Hy ( z) a y ∂H y ∂z H=−
=−
σV0
σ V0 d
d za y
where the constant of integration is set to zero, since the boundary condition at z = 0 requires Hy to be zero for z equal to zero. Note that the magnetic field is directed in the positive y-direction (since z is negative) and increases linearly from z = 0 to z = –l, as depicted in Fig. (b) on Slide 58. It also satisfies the boundary condition at z = –l by being consistent with the current drawn from the source to be w ⎡⎣ Hy ⎤⎦ = (σ V0 /d )( wl ) = Ic . z =− l Because of the existence of the magnetic field, the arrangement is characterized by an inductance, which can be found either by using the flux linkage concept, or by the energy method. To use the flux linkage concept, we recognize that a differential amount of magnetic flux dψ ′ = µHyd (dz ′) between z equal to (z′ – dz′) and z equal to z′, where –l < z′ < 0, links only that part of the current that flows from the top plate to the bottom plate between z = z′ and z = 0, thereby giving a value of (–z′/l) for the fraction, N, of the total current linked. Thus, the inductance, familiarly known as the internal inductance, denoted Li, since it is due to magnetic field internal to the current distribution, as
(101a) (101b)
(101c)
Electromagnetostatic Analysis of Parallel-Plate Arrangement H H == H Hyy(z) (z)aayy
σσVV00 H = − H = − d zz aayy d
∂∂H Hyy σσVV == −− 00 ∂∂zz dd
00 11 00 ⎛⎛ ––zz′′⎞⎞ 1 1 = LLii == ∫ N dψ ′ = [µ H d(dz′)] IIcc zz′′==−l−l N dψ ′ IIcc ∫zz′′==––ll⎝⎝ ll ⎠⎠ [µ Hyy d(dz′)]
1 µ dl == 1 µ dl 33 ww
Internal Internal Inductance Inductance 61 61
compared to that in (91) for which the magnetic field is external to the current distribution, is given by Li = I1 c
0
∫z ′= − l N dψ ′
(102)
µdl = 13 w or, 1/3 times the inductance of the structure if σ = 0 and the plates are joined at z = 0, as in the case of the arrangement of parallel-plates connected at the end.
Slide No. 62 Alternatively, if the energy method is used by computing the energy stored in the magnetic field and setting it equal to 12 Li Ic2 , then we have Li = 12 (dw) Ic
0
∫z =− l µHy
2
dz
µdl =1 3 w same as in (102). Finally, recognizing that there is energy storage associated with the electric field between the plates, we note that the arrangement has also associated with it a capacitance C, equal to εwl/d. Thus, all three properties of conductance, capacitance, and inductance are associated with the structure. Since for σ = 0 the situation reduces to that of the single capacitor arrangement, we can represent the arrangement to be equivalent to the circuit shown. Note that the capacitor is charged to the voltage V0 and the current through it is zero (open circuit condition). The voltage across the inductor is zero (short circuit condition) and the current through it is V0/R. Thus, the current drawn from the voltage source is V0/R and the voltage source views a single resistor R, as far as the current drawn from it is concerned.
(103)
Electromagnetostatic Analysis of Parallel-Plate Arrangement
Alternatively, 00 22 11 11 µµdl dl LLii == 22 (dw) µµH (dw)∫ Hyy dz dz == 3 w 3 w zz==−−ll IIcc
Equivalent Circuit VV 00
εεwl wl C C == d d
d RR == d σσwl wl
µ dl LLii == 11 µ dl 33 ww 62 62
Slide No. 63 Quasistatic Fields As already mentioned, quasistatic fields are a class of dynamic fields for which certain features can be analyzed as though the fields were static. In terms of behavior in the frequency domain, they are low-frequency extensions of static fields present in a physical structure, when the frequency of the source driving the structure is zero, or lowfrequency approximations of time-varying fields in the structure that are complete solutions to Maxwell’s equations. Here, we use the approach of low-frequency extensions of static fields. Thus, for a given structure, we begin with a time-varying field having the same spatial characteristics as that of the static field solution for the structure, and obtain field solutions containing terms up to and including the first power (which is the lowest power) in ω for their amplitudes. Depending on whether the predominant static field is electric or magnetic, quasistatic fields are called electroquasistatic fields or magnetoquasistatic fields. We shall now consider these separately.
Quasistatic Fields For For quasistatic quasistatic fields, fields, certain certain features features can can be be analyzed analyzed as as though though the the fields fields were were static. static. In In terms terms of of behavior behavior in in the the frequency frequency domain, domain, they they are are low-frequency low-frequency extensions extensions of of static static fields fields present present in in aa physical physical structure, structure, when when the the frequency frequency of of the the source source driving driving the the structure structure is is zero, zero, or or low-frequency low-frequency approximations approximations of of time-varying time-varying fields fields in in the the structure structure that that are are complete complete solutions solutions to to Maxwell’s Maxwell’s equations. equations. Here, Here, we we use use the the approach approach of of low-frequency low-frequency extensions extensions of of static static fields. fields. Thus, Thus, for for aa given given structure, structure, we we begin begin with with aa timetimevarying varying field field having having the the same same spatial spatial characteristics characteristics as as that that of of the the static static field field solution solution for for the the structure structure and and obtain obtain field field solutions solutions containing containing terms terms up up to to and and including including the the first first power power (which (which is is the the lowest lowest power) power) in in ω ω for for their their amplitudes. amplitudes. 63 63
Slide Nos. 64-65 Electroquasistatic Fields For electroquasistatic fields, we begin with the electric field having the spatial dependence of the static field solution for the given arrangement. An example is provided by the parallel-plate arrangement with perfect dielectric (Slide 50), excited by a sinusoidally time-varying voltage source Vg(t) = V0 cos ωt, instead of a direct voltage source. Then, E0 =
V0 cos ωt a x d
(104)
where the subscript 0 denotes that the amplitude of the field is of the zeroth power in ω. This results in a magnetic field in accordance with Maxwell’s equation for the curl of H, given by (28). Thus, noting that J = 0 in view of the perfect dielectric medium, we have for the geometry of the arrangement, ∂H y1 ∂z
=−
H1 =
∂Dx 0 ωεV0 = sin ωt ∂t d
ωεV0 z d
sin ωt a y
where we have also satisfied the boundary condition at z = 0 by choosing the constant of
[ ]z =0 is zero, and the subscript 1 denotes that the amplitude of
integration such that Hy1
the field is of the first power in ω. Note that the amplitude of Hy1 varies linearly with z, from zero at z = 0 to a maximum at z = –l.
(105)
(106)
Electroquasistatic Fields
IIg (t) (t) g
VVg ((tt))==VV0 cos ωt g 0 cos ω t
JJS ρρSS S
H H11
++ ++ ++ ++ ++ ++ ++ ε,ε, µµ
+ –+ –
yy
zz
EE00
–– –– –– –– –– –– –– zz== –l –l
++ x = 0 x =0
zz
x d –– x == d
xx
zz== 00
64 64
Electroquasistatic Analysis of Parallel-Plate Arrangement V V0 E = E00 = d0 cos cos ω tt aaxx d
∂∂H Hy1 ∂∂D VV00 Dxx00 ωε ωε y1 =− = sin ω t ∂∂ zz = − ∂∂tt = dd sin ω t H H11 ==
ωε ωεVV00zz dd
sin sin ω ω tt aayy 65 65
Slide No. 66 We stop the solution here, because continuing the process by substituting (106) into Maxwell’s curl equation for E, (27), to obtain the resulting electric field will yield a term having amplitude proportional to the second power in ω. This simply means that the fields given as a pair by (104) and (106) do not satisfy (27), and hence are not complete solutions to Maxwell’s equations. The complete solutions are obtained by solving Maxwell’s equations simultaneously and subject to the boundary conditions for the given problem. Proceeding further, we obtain the current drawn from the voltage source to be
[ ]z =−l
Ig (t ) = w H y1
⎛ εwl ⎞ = −ω ⎜ ⎟ V0 sin ω t ⎝ d ⎠ dVg (t ) =C dt
(107)
or, Ig = jω CVg
where C = (εwl/d) is the capacitance of the arrangement obtained from static field considerations. Thus, the input admittance of the structure is jωC so that its low frequency input behavior is essentially that of a single capacitor of value same as that found from static field analysis of the structure.
(108)
Electroquasistatic Analysis of Parallel-Plate Arrangement
[ ]zz==−−ll
Igg(t ) = w H yy11
ε wl ⎞ V sin ω t = − ω ⎛⎝ d ⎠ 00 dVgg(t ) =C dt
εεwwll C C == d d
εεwl wl IIgg == jjω CV C = where where ω CVgg C= d d 66 66
Slide No. 67 Indeed, from considerations of power flow, using Poynting’s theorem, we obtain the power flowing into the structure to be
[
Pin = wd Ex 0 H y1
]z=0
⎛ εwl ⎞ 2 = −⎜ ⎟ ωV0 sin ω t cos ω t ⎝ d ⎠ d ⎛1 ⎞ = ⎜ CVg2 ⎟ dt ⎝ 2 ⎠
which is consistent with the electric energy stored in the structure for the static case, as given by (79).
(109)
Electroquasistatic Analysis of Parallel-Plate Arrangement
[
]zz==00
Pinin = wd Exx00Hy1 y1
ε wl ⎞ 22 = −⎛⎝ ωV sin ω t cos ω t d ⎠ 00 22 d 1 = ⎛⎝ CVgg ⎞⎠ dt 2 67 67
Slide Nos. 68-69 Magnetoquasistatic Fields For magnetoquasistatic fields, we begin with the magnetic field having the spatial dependence of the static field solution for the given arrangement. An example is provided by the parallel-plate arrangement with perfect dielectric and connected at the end (Slide 54), excited by a sinusoidally time-varying current source Ig(t) = I0 cos ωt, instead of a direct current source. Then, I H 0 = w0 cos ω t a y
(110)
where the subscript 0 again denotes that the amplitude of the field is of the zeroth power in ω. This results in an electric field in accordance with Maxwell’s curl equation for E, given by (27). Thus, we have for the geometry of the arrangement, ∂B y 0 ωµI 0 ∂E x1 =− = sin ωt ∂z ∂t w E1 =
ωµI 0 z w
sin ωt a x
where we have also satisfied the boundary condition at z = 0 by choosing the constant of integration such that [E x1 ]z =0 = 0 is zero, and again the subscript 1 denotes that the
amplitude of the field is of the first power in ω. Note that the amplitude of Ex1 varies linearly with z, from zero at z = 0 to a maximum at z = –l. As is the case of electroquasistatic fields, we stop the process here, because continuing it by substituting (112) into Maxwell’s curl equation for H, (28), to obtain the resulting magnetic field will yield a term having amplitude proportional to the second power in ω. This simply means that the fields given as a pair by (110) and (112) do not
(111)
(112)
Magnetoquasistatic Fields
++ IIg ((tt))== II0 cos cosωωtt g
0
++ VVgg(t) (t) –– JJSS –– zz== –– ll
++
++
ρρSS
++
EE11
ε,ε, µµ ––
––
+
x=0
y
z
H 00 ––
x =d
– zz
x
zz == 00
68 68
satisfy (28), and hence are not complete solutions to Maxwell’s equations. The complete solutions are obtained by solving Maxwell’s equations simultaneously and subject to the boundary conditions for the given problem.
Magnetoquasistatic Analysis of Parallel-Plate Arrangement II00 H = H00 = w cos cosωωtt aayy w
∂∂BBy0 ∂∂EEx1 ωµII00 sin ω t y0 ωµ x1 = − = ∂∂zz = − ∂∂tt = ww sin ω t E E11 ==
ωµ ωµII00zz sin ω t a sin ω t axx w w 69 69
Slide No. 70 Proceeding further, we obtain the voltage across the current source to be Vg (t ) = d ⎣⎡ Ex1 ⎦⎤ z =− l ⎛ µ dl ⎞ = −ω ⎜ ⎟ I0 sin ω t ⎝ w ⎠ dIg (t ) =L dt or
Vg = jωL Ig
where L = (µdl/w) is the inductance of the arrangement obtained from static field considerations. Thus, the input impedance of the structure is jωL, such that its low frequency input behavior is essentially that of a single inductor of value same as that found from static field analysis of the structure.
(113)
(114)
Magnetoquasistatic Analysis of Parallel-Plate Arrangement V Vgg(t) (t) == dd[E Ex1 x1] z
z==−−ll
⎛ µdl ⎞ == −−ω ⎛ dl ⎞ II00 sin sin ω tt ⎝⎝ w w ⎠⎠ dI (t) dI (t) == LL gg dt dt
µ dl LL == µ dl ww
µµdl dl VVgg == jjω where L = ωLL IIgg where L= w w 70 70
Slide No. 71 Indeed, from considerations of power flow, using Poynting’s theorem, we obtain the power flowing into the structure to be
[
Pin = wd Ex1 H y 0
]z=−l
⎛ µdl ⎞ 2 = −⎜ ⎟ ω I 0 sin ωt cos ωt ⎝ w ⎠ d ⎛1 ⎞ = ⎜ LI g2 ⎟ dt ⎝ 2 ⎠
which is consistent with the magnetic energy stored in the structure for the static case, as given by (92).
(115)
Magnetoquasistatic Analysis of Parallel-Plate Arrangement
[
Pinin = wd Exx11Hyy00
]zz==−−ll
⎛ µ dl ⎞ I 22 sin t cos t = −⎝ ω ω ω w ⎠ 00 d 1 2 = ⎛⎝ LIgg2⎞⎠ dt 2 71 71
Slide No. 72 Quasistatic Fields in a Conductor If the dielectric slab in the arrangement of Slide 64 is conductive, then both electric and magnetic fields exist in the static case, because of the conduction current, as discussed under electromagnetostatic fields. Furthermore, the electric field of amplitude proportional to the first power in ω contributes to the creation of magnetic field of amplitude proportional to the first power in ω, in addition to that from electric field of amplitude proportional to the zeroth power in ω.
Quasistatic Fields in a Conductor E E00,, JJcc00 H H00
IIgg
VVg ((tt))==VV0 cos ωt g 0 cos ω t
xx == 00
σ, σ, ε,ε,µµ
++ ––
xx == dd zz == 00
zz
zz==––ll
yy
zz xx
(a) (a)
EExx00 H Hyy00 zz
zz==––ll H Hyd1 yd1
zz==––ll
H Hyc1 yc1
zz == 00
(b) (b)
E Ex1x1
zz
zz == 00
(c) (c)
72 72
Slide No. 73 Thus, using the results from the static field analysis from the arrangement of Slide 58, we have for this arrangement, E0 =
V0 cosωt a x d
J c 0 = σE 0 = H0 = −
σV0 d
σV0 z d
cosωt a x
cosωt a y
(116) (117) (118)
as depicted in the figure. Also, the variations with z of the amplitudes of Ex 0 and Hy 0 are shown in Fig. (b) on Slide 72. The magnetic field given by (118) gives rise to an electric field having amplitude proportional to the first power in ω, in accordance with Maxwell’s curl equation for E, (27). Thus ∂B y 0 ωµσ V0 z ∂E x1 =− =− sin ωt ∂z ∂t d E x1 = −
ωµσ V0 2d
(z
2
)
− l 2 sin ωt
where we have also made sure that the boundary condition at z = –l is satisfied. This boundary condition requires that Ex be equal to Vg /d at z = –l. Since this is satisfied by Ex 0 alone, it follows that Ex1 must be zero at z = –l.
(119) (120)
Quasistatic Analysis of Parallel-Plate Arrangement with Conductor VV00 E = E00 = d cos cosωωtt aaxx d σσVV00 JJcc00 == σσE = E00 = d cos cosωωtt aaxx d σσVV00zz H = − H00 = − d cos cosωωtt aayy d
∂∂BBy0 ∂∂EEx1 ωµσ ωµσ VV00zz sin ω t y0 x1 = − == −− = − sin ω t ∂∂zz ∂∂tt dd
ωµσ ωµσ VV00 z22 − l 22 sin ω t z − l sin ω t 2d 2d
EEx1 =− x1 = −
(
)
73 73
Slide No. 74 The electric field given by (116) and that given by (120) together give rise to a magnetic field having terms with amplitudes proportional to the first power in ω, in accordance with Maxwell’s curl equation for H, (28). Thus ∂H y1 ∂z
= −σE x1 − ε =
H y1 =
ωµσ 2V0 2d
(
∂E x 0 ∂t
(z
2
ωµσ 2V0 z 3 − 3zl 2 6d
)
− l 2 sin ωt +
ωεV0 d
(121) sin ωt
) sin ωt + ωεV z sin ωt 0
d
where we have also made sure that the boundary condition at z = 0 is satisfied. This boundary condition requires that Hy be equal to zero at z = 0, which means that all of its terms must be zero at z = 0. Note that the first term on the right side of (122) is the contribution from the conduction current in the material resulting from Ex1 and the second term is the contribution from the displacement current resulting from Ex 0 . Denoting these to be Hyc1 and Hyd1 , respectively, we show the variations with z of the amplitudes of all the field components having amplitudes proportional to the first power in ω, in Fig. (c) on Slide 72.
(122)
Quasistatic Analysis of Parallel-Plate Arrangement with Conductor ∂∂H Hyy11 ∂∂EEx0 x0 == −−σσEEx1 − ε − ε x1 ∂∂ zz ∂∂ tt ==
2 ωµσ ωµσ 2VV00
2d 2d
(
(
2
2
)
zz 2 −− ll 2 sin sinω ωtt ++
ωε ωεVV00 dd
sin sin ω ω tt
)
2 3 2 ωµσ ωµσ 2VV00 zz3 −− 3zl 3zl2 ωε ωεVV00zz sin ω t H = sin ω t + Hy1 sin ω t + d sin ω t y1 = 6d d 6d 74 74
Slide No. 75 Now, adding up the contributions to each field, we obtain the total electric and magnetic fields up to and including the terms with amplitudes proportional to the first power in ω to be Ex =
Hy = −
σV0 z d
(
cos ωt +
Ex =
or
Hy = −
σz d
)
ωµσV0 2 2 V0 cos ωt − z − l sin ωt d 2d ωεV0 z d Vg d
V g − jω
sin ωt +
+ jω
εz d
µσ 2d
(z
V g − jω
(123a)
(
ωµσ 2V0 z 3 − 3zl 2 6d 2
)
− l 2 Vg
(
µσ 2 z 3 − 3zl 2 6d
) sin ωt
(123b)
(124a)
)V
g
(124b)
Quasistatic Analysis of Parallel-Plate Arrangement with Conductor VV0 ωµσ ωµσVV00 z22 − l 22 sin ω t EExx == 0 cos tt −− ω cos z − l sin ω t ω dd 2d 2d
(
)
2
((3
2
))
ωµσ ωµσ 2VV00 zz3 −−3zl 3zl 2 σσVV00zz ωε VV00zz ωε H sin Hyy == −− d cos cos ω ωtt ++ d sin sinωωtt ++ sinωωtt 6d 6d d d VVg µσ µσ z22 − l 22 V EExx == g ++ jjω ω z − l Vgg dd 2d 2d
((
))
((
))
2 3 2 µσ 3zl 2 µσ 2 zz3 −−3zl εεzz σσ zz H VVgg Hyy == −− d VVgg −− jjω ω d VVgg −− jjωω 6d d d 6d
75 75
Slide No. 76 Finally, the current drawn from the voltage source is given by
[ ]z =−l
Ig = w H y
⎛ σwl εwl µσ 2 wl 3 ⎞⎟ ⎜ = + jω − jω Vg ⎜ d 3d ⎟⎠ d ⎝
(125)
The input admittance of the structure is given by Yin =
2 Ig σwl ⎜⎛ 1− jω µσ l ⎟⎞ = j ω εwl + d d ⎜⎝ 3 ⎟⎠ Vg
≈ jω εwl + d
d ⎛ σwl ⎜⎝ 1 +
1 jω
(126) µσl 2 ⎞ 3
⎟ ⎠
where we have used the approximation [1 + jω(µσl2/3)]–1 ≈ [1 – jω(µσl2/3)].
Quasistatic Analysis of Parallel-Plate Arrangement with Conductor
[ ]zz==−−ll
IIgg == w wH Hyy
22 33 ⎛⎛σσwl ⎞⎞ wl µσ wl ε wl µσ wl wl ε == ⎜⎜ ++ jjω −− jjω ω ω ⎟ VV dd 3d ⎝⎝ dd 3d ⎠⎟⎠ gg 2 IIgg ⎛⎛ ll 2⎞⎞ σσwl µσ wl ε wl µσ wl ε YYinin == == jjω ω d ++ d ⎜⎜11−− jjω ω 3 ⎟⎟ ⎝ d 3 ⎠⎠ d V ⎝ Vgg
εwl ≈≈ jjω ω εwl ++ dd
11 µσ µσl 11 ++ jjω ω 33 l
(
dd σσwl wl
22
)
76 76
Slide No. 77 Proceeding further, we have Yin = jω εwl + d
d σwl
1 + jω
µdl 3w
1 = jω C + R + j ωLi
where C = εwl/d is the capacitance of the structure if the material is a perfect dielectric, R = d/σwl is the resistance of the structure, and Li = µdl/3w is the internal inductance of the structure, all computed from static field analysis of the structure. The equivalent circuit corresponding to (127) consists of capacitance C in parallel with the series combination of resistance R and internal inductance Li, same as that on Slide 62. Thus, the low-frequency input behavior of the structure is essentially the same as that of the equivalent circuit on Slide 62, with the understanding that its input admittance must also be approximated to first-order terms. Note that for σ = 0, the input admittance of the structure is purely capacitive. For nonzero σ, a critical value of σ equal to
3ε/µl 2 exists for which the input admittance is purely conductive. For values of σ
smaller than the critical value, the input admittance is complex and capacitive, and for values of σ larger than the critical value, the input admittance is complex and inductive.
(127)
Quasistatic Analysis of Parallel-Plate Arrangement with Conductor εεwl 1 1 wl YYinin == jjω ω d ++ dd 1 µµdldl == jjωωCC++ R + 1jωL R + jωLii d ω 3w σ wl ++ jjω σ wl
3w
Equivalent Circuit εεwl wl C C == d d
RR == dd σσwl wl
µ dl LLii == 11 µ dl 33 ww 77 77
Slide No. 78 We have seen that quasistatic field analysis of a physical structure provides information concerning the low-frequency input behavior of the structure. As the frequency is increased beyond that for which the quasistatic approximation is valid, terms in the infinite series solutions for the fields beyond the first-order terms need to be included. While one can obtain equivalent circuits for frequencies beyond the range of validity of the quasistatic approximation by evaluating the higher order terms, no further insight is gained through that process and it is more straightforward to obtain the exact solution by resorting to simultaneous solution of Maxwell’s equations, when a closed form solution is possible.
Waves and the Distributed Circuit Concept We We have have seen seen that that quasistatic quasistatic field field analysis analysis of of aa physical physical structure structure Provides Provides information information concerning concerning the the low-frequency low-frequency input input behavior behavior of of the the structure. structure. As As the the frequency frequency is is increased increased beyond beyond that that for for which which the the quasistatic quasistatic approximation approximation is is valid, valid, terms terms in in the the infinite infinite series series solutions solutions for for the the fields fields beyond beyond the the first-order first-order terms terms need need to to be be included. included. While While one one can can obtain obtain equivalent equivalent circuits circuits for for frequencies frequencies beyond beyond the the range range of of validity validity of of the the quasistatic quasistatic approximation approximation by by evaluating evaluating the the higher higher order order terms, terms, no no further further insight insight is is gained gained through through that that process, process, and and itit is is more more straightstraightforward forward to to obtain obtain the the exact exact solution solution by by resorting resorting to to simultaneous simultaneous solution solution of of Maxwell’s Maxwell’s equations equations when when aa closed closed form form solution solution is is possible. possible. 78 78
Slide No. 79 Wave Equation Let us, for simplicity, consider the structures on Slides 64 and 68, for which the material between the plates is a perfect dielectric (σ = 0). Then, regardless of the termination at z = 0, the equations to be solved are ∇ x E=−
∇xH=
∂B ∂H = −µ ∂t ∂t
(128a)
∂D ∂E =ε ∂t ∂t
(128b)
For the geometry of the arrangements, E = Ex(z, t)ax and H = Hy(z, t)ay, so that (128a) and (128b) simplify to ∂H y ∂E x = −µ ∂z ∂t ∂H y ∂z
= −ε
∂E x ∂t
(129a)
(129b)
Combining the two equations by eliminating Hy, we obtain ∂ 2Ex ∂z 2 which is the wave equation.
= µε
∂ 2 Ex ∂t 2
(130)
Wave Equation ∂∂ H ∂∂ B B H ∇ E == −− ∂ t == −−µµ ∂ t ∇ xx E ∂t ∂t
∂∂ D ∂∂E D E ∇ H == ∂ t == εε ∂ t ∇ xx H ∂t ∂t
For For the the one-dimensional one-dimensional case case of of
E E == EExx((z,t z,t))aaxx and and H H == H Hyy((z,t z,t))aayy,,
∂∂H Hy ∂∂EExx == −−µµ y ∂∂ zz ∂∂tt
∂∂ 22EExx ∂∂ 22EExx = µε µε 22 22 = ∂∂zz ∂∂ tt
∂∂ H Hyy ∂∂EE == −−εε xx ∂∂zz ∂∂tt One-dimensional One-dimensional wave wave equation equation 79 79
Slide Nos. 80-81 The wave equation has solutions of the form
(
)
(
E x ( z ,t ) = A cos ω t − µε z + φ + + B cos ω t + µε z + φ −
)
(131)
The terms on the right side correspond to traveling waves propagating in the +z and –z directions, which we shall call the (+) and (–) waves, respectively, with the velocity 1 / µε , or c / µ r ε r , where c = 1 / µ 0ε 0 is the velocity of light in free space. This
can be seen by setting the derivative of the argument of the cosine function in each term equal to zero, or by plotting each term versus z for a few values of t, as on Slides 80 and 81, for the (+) and (–) waves, respectively. The corresponding solution for Hy is given by H y ( z ,t ) =
[A cos ω (t − µ/ε 1
)
(
µε z + φ + − B cos ω t + µε z + φ −
)]
(132)
Solution to the One-Dimensional Wave Equation
(
))
((
))
+ − EExx(z, (z,t)t) == AA cos cos ω ω tt −− µε µε zz++φφ + ++BB cos cosωω tt++ µε µεzz ++φφ −
Traveling Traveling wave wave propagating propagating in in the the +z +z direction direction
[ ((
))
]
+ cos cos ω ω tt –– µε µε zz ++φφ +
tt == 00
11
tt == ππ t = ππ 44ωω t = 2ω 2ω
11 22 ff µε µε
00
11 ff µε µε
zz
–1 –1
80 80
Solution to the One-Dimensional Wave Equation
[
((
))
))]]
((
+ − 1 H Hyy(z, (z, t)t) == µ1 ε AA cos cosωω tt −− µε µε zz ++φφ + −−BBcos cosωω tt++ µε µε zz ++φφ − µε
Traveling Traveling wave wave propagating propagating in in the the –z –z direction direction tt == ππ 22ωω
tt == ππ 44ω ω
[ ((
))
–
]
– cos cos ω ω tt ++ µε µε zz ++φφ
tt ==00
11 11 22 ff µε µε
–z –z
11 ff µε µε
00
–1 –1
81 81
Slide No. 82 For sinusoidal waves, which is the case at present, the velocity of propagation is known as the phase velocity, denoted by vp, since it is the velocity with which a constant phase surface moves in the direction of propagation. The quantity ω µε is the magnitude of the rate of change of phase at a fixed time t, for either wave. It is known as the phase constant and is denoted by the symbol β. The quantity
µ/ε , which is the
ratio of the electric field intensity to the magnetic field intensity for the (+) wave, and the negative of such ratio for the (–) wave, is known as the intrinsic impedance of the medium. It is denoted by the symbol η. Thus, the phasor electric and magnetic fields can be written as E x = A e − jβz + B e jβz Hy =
(A e η 1
− jβz
− B e jβz
(133)
)
(134)
General Solution in Phasor Form EExx == AAee−−jjββzz ++ BBeejjββzz
(
))
1 − jβ z jβ z H Hyy == η1 AAee − j β z −− BBee jβ z
η
jφ jφ AA == Ae Ae jφ ,, BB == Be Be jφ ++
––
ω ββ == ωω µε µε == vω ,, Phase Phase constant constant vpp vvpp == 11 ,, µε
Phase Phase velocity velocity
µ ηη == µ,, εε
Intrinsic Intrinsic impedance impedance
µε
82 82
Slide Nos. 83-85 We may now use the boundary conditions for a given problem and obtain the specific solution for that problem. For the arrangement on Slide 64, the boundary conditions are H y = 0 at z = 0 and Ex = Vg /d at z = –l. We thus obtain the particular solution for that arrangement to be Ex =
Hy =
Vg d cos β l − jV g
ηd cos β l
cos β z
(135)
sin β z
(136)
which correspond to complete standing waves, resulting from the superposition of (+) and (–) waves of equal amplitude. Complete standing waves are characterized by pure half-sinusoidal variations for the amplitudes of the fields, as shown on Slide 84. For values of z at which the electric field amplitude is a maximum, the magnetic field amplitude is zero, and for values of z at which the electric field amplitude is zero, the magnetic field amplitude is a maximum. The fields are also out of phase in time, such that at any value of z, the magnetic field and the electric field differ in phase by t = π/2ω.
Example of Parallel-Plate Structure Open-Circuited at the Far End IIgg(t) (t) VVg ((tt))==VV0 cos cosωωtt g
0
JJS ρρss S
H H
++ ++ + ++ ++ ++ + ε,ε, µµ
++ ––
xx = 00
yy
zz
E E
– – –– –– –– –– – zz== ––ll
+
zz
–
xx = dd
xx
zz == 00
H Hyy == 00 at at zz == 00 ⎫⎫ ⎪⎪ VVgg ⎬⎬ B.C. B.C. EExx == at at zz == −−ll⎪⎪ dd ⎭⎭ VVgg EExx == cos cos ββzz dd cos cos ββll
−−jV jVgg H Hyy == ηd cos β l sin sin ββzz ηd cos β l
83 83
Standing Wave Patterns (Complete Standing Waves) EExx
–z –z
77ππ 22ββ
55ππ 22ββ
ππ 00 22ββ
33ππ 22ββ
HHyy
–z –z
33ππ
ββ
22ππ
ββ
ππ ββ
00
84 84
Complete Standing Waves Complete Complete standing standing waves waves are are characterized characterized by by pure pure half-sinusoidal half-sinusoidal variations variations for for the the amplitudes amplitudes of of the the fields. fields. For For values values of of zz at at which which the the electric electric field field amplitude amplitude is is aa maximum, maximum, the the magnetic magnetic field field amplitude amplitude is is zero, zero, and and for for values values of of zz at at which which the the electric electric field field amplitude amplitude is is zero, zero, the the magnetic magnetic field field amplitude amplitude is is aa maximum. maximum. The The fields fields are are also also out out of of phase phase in in time, time, such such that that at at any any value value of of z, z, the the magnetic magnetic field field and and the the electric ω.. electric field field differ differ in in phase phase by by tt == ππ // 22ω 85 85
Slide No. 86 Now, the current drawn from the voltage source is given by
[ ]z =−l
Ig = w H y
jwV g
=
ηd
tan β l
(137)
so that the input impedance of the structure is
Yin =
Ig Vg
= j
w tan β l ηd
(138)
which can be expressed as a power series in βl. In particular, for βl < π/2, Yin = j
w ηd
⎡ ⎤ ( β l ) 3 2( β l ) 5 + + + "⎥ β l ⎢ 3 15 ⎢⎣ ⎥⎦
(139)
The first term on the right side can be identified as belonging to the quasistatic approximation. Indeed for βl
![[PDF] Fundamentals of Electromagnetics with ... - Google Sites](https://m.moam.info/img/260x300/pdf-fundamentals-of-electromagnetics-with-google-s_6477b2c7097c474e708bfe1d.jpg)
![FUNDAMENTALS OF APPLIED ELECTROMAGNETICS 6/e [PDF]](https://m.moam.info/img/260x300/fundamentals-of-applied-electromagnetics-6-e-pdf_6479c3cc098a9e3a068b4665.jpg)
![[PDF] Download Fundamentals of Electromagnetics ... - Google Sites](https://m.moam.info/img/260x300/pdf-download-fundamentals-of-electromagnetics-goog_6477de26097c474e708c2e44.jpg)
![[PDF] Fundamentals of Electromagnetics with ... - Google Sites](https://m.moam.info/img/260x300/pdf-fundamentals-of-electromagnetics-with-google-s_647720e4097c4737708b5bc6.jpg)
![[PDF] Download Fundamentals of Electromagnetics ... - Google Sites](https://m.moam.info/img/260x300/pdf-download-fundamentals-of-electromagnetics-goog_6476f369097c474c228b497a.jpg)