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EEE 498/598EEE 498/598Overview of Electrical Overview of Electrical
EngineeringEngineering
Lecture 10:Lecture 10:
Uniform Plane Wave Uniform Plane Wave Solutions to Maxwell’s Solutions to Maxwell’s
EquationsEquations
2Lecture 10
Lecture 10 ObjectivesLecture 10 Objectives
To study uniform plane wave To study uniform plane wave solutions to Maxwell’s solutions to Maxwell’s equations:equations: In the time domain for a lossless In the time domain for a lossless
medium.medium. In the frequency domain for a In the frequency domain for a
lossy medium.lossy medium.
3Lecture 10
Overview of WavesOverview of Waves
A A wavewave is a pattern of values in is a pattern of values in space that appear to move as time space that appear to move as time evolves.evolves.
A A wavewave is a solution to a is a solution to a wave equationwave equation.. Examples of waves include water Examples of waves include water
waves, sound waves, seismic waves, sound waves, seismic waves, and voltage and current waves, and voltage and current waves on transmission lines. waves on transmission lines.
4Lecture 10
Overview of Waves Overview of Waves (Cont’d)(Cont’d)
Wave phenomena result from an exchange Wave phenomena result from an exchange between two different forms of energy between two different forms of energy such that the time rate of change in one such that the time rate of change in one form leads to a spatial change in the other.form leads to a spatial change in the other.
Waves possessWaves possess no massno mass energyenergy momentummomentum velocityvelocity
5Lecture 10
Time-Domain Maxwell’s Time-Domain Maxwell’s Equations in Differential Equations in Differential
FormForm
mv
ev
qBt
DJH
qDt
BKE
ic JJ
ic KK
6Lecture 10
Time-Domain Maxwell’s Time-Domain Maxwell’s Equations in Differential Form Equations in Differential Form
for a Simple Mediumfor a Simple Medium
mvi
evim
qH
t
EJEH
qE
t
HKHE
HKEJHBED mcc
7Lecture 10
Time-Domain Maxwell’s Equations in Time-Domain Maxwell’s Equations in Differential Form for a Simple, Source-Free, Differential Form for a Simple, Source-Free,
and Lossless Mediumand Lossless Medium
0
0
Ht
EH
Et
HE
000 mmvevii qqKJ
8Lecture 10
Time-Domain Maxwell’s Equations in Time-Domain Maxwell’s Equations in Differential Form for a Simple, Source-Free, Differential Form for a Simple, Source-Free,
and Lossless Mediumand Lossless Medium
Obviously, there must be a Obviously, there must be a source for the field somewhere.source for the field somewhere.
However, we are looking at the However, we are looking at the properties of waves in a region properties of waves in a region far from the source.far from the source.
9Lecture 10
Derivation of Wave Equations for Derivation of Wave Equations for Electromagnetic Waves in a Simple, Source-Electromagnetic Waves in a Simple, Source-
Free, Lossless MediumFree, Lossless Medium
2
2
2
2
2
2
t
H
t
E
HHH
t
E
t
H
EEE
0
0
10Lecture 10
Wave Equations for Wave Equations for Electromagnetic Waves in a Electromagnetic Waves in a
Simple, Source-Free, Lossless Simple, Source-Free, Lossless MediumMedium
02
22
t
HH
02
22
t
EE
The wave equations The wave equations are not are not independent.independent.
Usually we solve Usually we solve the electric field the electric field wave equation and wave equation and determine determine HH from from EE using Faraday’s using Faraday’s law.law.
11Lecture 10
Uniform Plane Wave Uniform Plane Wave Solutions in the Time Solutions in the Time
DomainDomain A A uniform plane waveuniform plane wave is an electromagnetic wave in is an electromagnetic wave in which the electric and magnetic fields and the which the electric and magnetic fields and the direction of propagation are mutually orthogonal, and direction of propagation are mutually orthogonal, and their amplitudes and phases are constant over planes their amplitudes and phases are constant over planes perpendicular to the direction of propagation. perpendicular to the direction of propagation.
Let us examine a possible plane wave solution given Let us examine a possible plane wave solution given byby
tzEaE xx ,ˆ
12Lecture 10
Uniform Plane Wave Uniform Plane Wave Solutions in the Time Solutions in the Time
Domain (Cont’d)Domain (Cont’d) The wave equation for this field simplifies toThe wave equation for this field simplifies to
The general solution to this wave equation isThe general solution to this wave equation is
02
2
2
2
t
E
z
E xx
tvzptvzptzE ppx 21,
13Lecture 10
Uniform Plane Wave Uniform Plane Wave Solutions in the Time Solutions in the Time
Domain (Cont’d)Domain (Cont’d) The functionsThe functions pp11(z-v(z-vppt)t) and and pp2 2 (z+v(z+vppt)t)
represent uniform waves propagating represent uniform waves propagating in the in the +z+z and and -z-z directions respectively. directions respectively.
Once the electric field has been Once the electric field has been determined from the wave equation, determined from the wave equation, the magnetic field must follow from the magnetic field must follow from Maxwell’s equations.Maxwell’s equations.
14Lecture 10
Uniform Plane Wave Uniform Plane Wave Solutions in the Time Solutions in the Time
Domain (Cont’d)Domain (Cont’d) The The velocity of propagationvelocity of propagation is determined solely by the medium: is determined solely by the medium:
The functions The functions pp11 and and pp22 are determined by the source and the other boundary conditions. are determined by the source and the other boundary conditions.
1
pv
15Lecture 10
Uniform Plane Wave Uniform Plane Wave Solutions in the Time Solutions in the Time
Domain (Cont’d)Domain (Cont’d) Here we must have Here we must have
tzHaH yy ,ˆwhere
tvzptvzptzH ppy 21
1,
16Lecture 10
Uniform Plane Wave Uniform Plane Wave Solutions in the Time Solutions in the Time
Domain (Cont’d)Domain (Cont’d) is the is the intrinsic impedanceintrinsic impedance of the medium of the medium
given bygiven by
Like the velocity of propagation, the Like the velocity of propagation, the intrinsic impedance is independent of the intrinsic impedance is independent of the source and is determined only by the source and is determined only by the properties of the medium.properties of the medium.
17Lecture 10
Uniform Plane Wave Uniform Plane Wave Solutions in the Time Solutions in the Time
Domain (Cont’d)Domain (Cont’d) In free space (vacuum):In free space (vacuum):
377120
m/s 103 8
cvp
18Lecture 10
Uniform Plane Wave Uniform Plane Wave Solutions in the Time Solutions in the Time
Domain (Cont’d)Domain (Cont’d) Strictly speaking, uniform plane Strictly speaking, uniform plane
waves can be produced only by waves can be produced only by sources of infinite extent.sources of infinite extent.
However, point sources create However, point sources create spherical waves. Locally, a spherical spherical waves. Locally, a spherical wave looks like a plane wave.wave looks like a plane wave.
Thus, an understanding of plane Thus, an understanding of plane waves is very important in the study waves is very important in the study of electromagnetics.of electromagnetics.
19Lecture 10
Uniform Plane Wave Uniform Plane Wave Solutions in the Time Solutions in the Time
Domain (Cont’d)Domain (Cont’d) Assuming that the source is sinusoidal. We haveAssuming that the source is sinusoidal. We have
p
pp
p
pp
p
v
ztCtvzv
Ctvzp
ztCtvzv
Ctvzp
coscos
coscos
222
111
20Lecture 10
ztCztCtzH
ztCztCtzE
y
x
coscos1
,
coscos,
21
21
Uniform Plane Wave Uniform Plane Wave Solutions in the Time Solutions in the Time
Domain (Cont’d)Domain (Cont’d) The electric and magnetic fields The electric and magnetic fields are given byare given by
21Lecture 10
Uniform Plane Wave Uniform Plane Wave Solutions in the Time Solutions in the Time
Domain (Cont’d)Domain (Cont’d) The argument of the cosine The argument of the cosine
function is the called the function is the called the instantaneous phaseinstantaneous phase of the field: of the field:
zttz ,
22Lecture 10
Uniform Plane Wave Uniform Plane Wave Solutions in the Time Solutions in the Time
Domain (Cont’d)Domain (Cont’d) The speed with which a constant value The speed with which a constant value of instantaneous phase travels is called of instantaneous phase travels is called the the phase velocityphase velocity. For a . For a losslesslossless medium, medium, it is equal to and denoted by the same it is equal to and denoted by the same symbol as the symbol as the velocity of propagationvelocity of propagation..
1
00
dt
dzv
tzzt
p
23Lecture 10
2
2
Uniform Plane Wave Uniform Plane Wave Solutions in the Time Solutions in the Time
Domain (Cont’d)Domain (Cont’d) The distance along the direction The distance along the direction
of propagation over which the of propagation over which the instantaneous phase changes by instantaneous phase changes by 22 radians for a fixed value of radians for a fixed value of time is the time is the wavelengthwavelength..
24Lecture 10
0 2 4 6 8 10 12 14 16 18 20
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Uniform Plane Wave Uniform Plane Wave Solutions in the Time Solutions in the Time
Domain (Cont’d)Domain (Cont’d) The The
wavelengthwavelength is is also the also the distance distance between between every other every other zero zero crossing of crossing of the sinusoid.the sinusoid.
Function vs. position at a fixed time
25Lecture 10
Uniform Plane Wave Uniform Plane Wave Solutions in the Time Solutions in the Time
Domain (Cont’d)Domain (Cont’d) Relationship between Relationship between wavelengthwavelength
and frequency in free space:and frequency in free space:
Relationship between Relationship between wavelengthwavelength and frequency in a material and frequency in a material medium:medium:
f
c
f
vp
26Lecture 10
Uniform Plane Wave Uniform Plane Wave Solutions in the Time Solutions in the Time
Domain (Cont’d)Domain (Cont’d) is the is the phase constantphase constant and is given and is given
byby
pv
rad/m
27Lecture 10
Uniform Plane Wave Uniform Plane Wave Solutions in the Time Solutions in the Time
Domain (Cont’d)Domain (Cont’d) In free space (vacuum):In free space (vacuum):
0000
2
k
c
free space wavenumber (rad/m)
28Lecture 10
Time-Harmonic Time-Harmonic AnalysisAnalysis
Sinusoidal steady-stateSinusoidal steady-state (or (or time-harmonictime-harmonic) ) analysis is very useful in electrical analysis is very useful in electrical engineering because an arbitrary engineering because an arbitrary waveform can be represented by a waveform can be represented by a superposition of sinusoids of different superposition of sinusoids of different frequencies using frequencies using Fourier analysisFourier analysis..
If the waveform is periodic, it can be If the waveform is periodic, it can be represented using a represented using a Fourier seriesFourier series..
If the waveform is not periodic, it can be If the waveform is not periodic, it can be represented using a represented using a Fourier transformFourier transform..
29Lecture 10
Time-Harmonic Maxwell’s Equations in Time-Harmonic Maxwell’s Equations in Differential Form for a Simple, Source-Free, Differential Form for a Simple, Source-Free,
Possibly Lossy MediumPossibly Lossy Medium
0
0
HEjH
EHjE
mjj
jj
30Lecture 10
Derivation of Helmholtz Equations for Derivation of Helmholtz Equations for Electromagnetic Waves in a Simple, Source-Electromagnetic Waves in a Simple, Source-
Free, Possibly Lossy MediumFree, Possibly Lossy Medium
HEj
HHH
EHj
EEE
2
2
2
2
0
0 2
2
31Lecture 10
Helmholtz Equations for Helmholtz Equations for Electromagnetic Waves in a Simple, Electromagnetic Waves in a Simple, Source-Free, Possibly Lossy MediumSource-Free, Possibly Lossy Medium
022 EE
The Helmholtz The Helmholtz equations are not equations are not independent.independent.
Usually we solve Usually we solve the electric field the electric field equation and equation and determine determine HH from from EE using Faraday’s using Faraday’s law.law.
022 HH
32Lecture 10
Uniform Plane Wave Uniform Plane Wave Solutions in the Frequency Solutions in the Frequency
DomainDomain Assuming a plane wave solution of Assuming a plane wave solution of
the formthe form
The Helmholtz equation simplifies toThe Helmholtz equation simplifies to
zEaE xxˆ
022
2
xx E
dz
Ed
33Lecture 10
Uniform Plane Wave Uniform Plane Wave Solutions in the Solutions in the
Frequency Domain Frequency Domain (Cont’d)(Cont’d) The propagation constant is a The propagation constant is a
complex number that can be complex number that can be written aswritten as
jj 2
attenuation constant (Np/m)
phase constant (rad/m)(m-1)
34Lecture 10
Uniform Plane Wave Uniform Plane Wave Solutions in the Frequency Solutions in the Frequency
Domain (Cont’d)Domain (Cont’d) is the is the attenuation constantattenuation constant and has and has
units of nepers per meter (Np/m).units of nepers per meter (Np/m). is the is the phase constantphase constant and has and has
units of radians per meter units of radians per meter (rad/m).(rad/m).
Note that in general for a lossy Note that in general for a lossy mediummedium
35Lecture 10
The general solution to this wave The general solution to this wave equation isequation is
zjzzjz
zzx
eeCeeC
eCeCzE
21
21
Uniform Plane Wave Uniform Plane Wave Solutions in the Solutions in the
Frequency Domain Frequency Domain (Cont’d)(Cont’d)
zEx zEx
• wave traveling in the +z-direction
• wave traveling in the -z-direction
36Lecture 10
zteCzteC
ezEtzEzz
tjxx
coscos
Re,
21
Uniform Plane Wave Uniform Plane Wave Solutions in the Frequency Solutions in the Frequency
Domain (Cont’d)Domain (Cont’d) Converting the phasor Converting the phasor
representation of representation of EE back into the back into the time domain, we havetime domain, we have
• We have assumed that C1 and C2 are real.
37Lecture 10
Uniform Plane Wave Uniform Plane Wave Solutions in the Solutions in the
Frequency Domain Frequency Domain (Cont’d)(Cont’d) The corresponding magnetic field for The corresponding magnetic field for
the uniform plane wave is obtained the uniform plane wave is obtained using Faraday’s law:using Faraday’s law:
j
EHHjE
38Lecture 10
zEzE
eCeCzH
xx
zzy
1
121
Uniform Plane Wave Uniform Plane Wave Solutions in the Solutions in the
Frequency Domain Frequency Domain (Cont’d)(Cont’d) Evaluating Evaluating HH we have we have
39Lecture 10
We note that the intrinsic impedance We note that the intrinsic impedance is a complex number for lossy media.is a complex number for lossy media.
je
Uniform Plane Wave Uniform Plane Wave Solutions in the Frequency Solutions in the Frequency
Domain (Cont’d)Domain (Cont’d)
40Lecture 10
zteC
zteC
ezHtzH
z
z
tjyy
cos
cos
Re,
2
1
Uniform Plane Wave Uniform Plane Wave Solutions in the Solutions in the
Frequency Domain Frequency Domain (Cont’d)(Cont’d) Converting the phasor Converting the phasor
representation of representation of HH back into the back into the time domain, we havetime domain, we have
41Lecture 10
Uniform Plane Wave Uniform Plane Wave Solutions in the Frequency Solutions in the Frequency
Domain (Cont’d)Domain (Cont’d) We note that in a lossy medium, We note that in a lossy medium,
the electric field and the the electric field and the magnetic field are no longer in magnetic field are no longer in phase.phase.
The magnetic field lags the The magnetic field lags the electric field by an angle of electric field by an angle of ..
42Lecture 10
Uniform Plane Wave Uniform Plane Wave Solutions in the Solutions in the
Frequency Domain Frequency Domain (Cont’d)(Cont’d) Note that we Note that we
havehave
These form a These form a right-handed right-handed coordinate coordinate systemsystem
zaHE ˆ
Ea
Haza
Uniform plane Uniform plane waves are a waves are a type of type of transverse transverse electromagneticelectromagnetic ((TEMTEM)) wave. wave.
43Lecture 10
Uniform Plane Wave Uniform Plane Wave Solutions in the Solutions in the
Frequency Domain Frequency Domain (Cont’d)(Cont’d) Relationships between the phasor Relationships between the phasor
representations of electric and representations of electric and magnetic fields in uniform plane magnetic fields in uniform plane waves:waves:
HaE
EaH
p
p
ˆ
ˆ1
unit vector in direction of propagation
44Lecture 10
Uniform Plane Wave Uniform Plane Wave Solutions in the Solutions in the
Frequency Domain Frequency Domain (Cont’d)(Cont’d)
Example:Example:
ConsiderConsider
rad/m 33.16
Np/m 1911
S/m 01.0
5.2
m300.0Hz 101
0
0
09
.α
f
ztetzE zx cos,
45Lecture 10
Uniform Plane Wave Uniform Plane Wave Solutions in the Solutions in the
Frequency Domain Frequency Domain (Cont’d)(Cont’d)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
z/0
Ex+ (
z,t)
ze
Snapshot of Ex+(z,t) at t = 0
46Lecture 10
Uniform Plane Wave Uniform Plane Wave Solutions in the Frequency Solutions in the Frequency
Domain (Cont’d)Domain (Cont’d) Properties of the wave Properties of the wave
determined by the source:determined by the source: amplitudeamplitude phasephase frequencyfrequency
47Lecture 10
Uniform Plane Wave Uniform Plane Wave Solutions in the Solutions in the
Frequency Domain Frequency Domain (Cont’d)(Cont’d) Properties of the wave Properties of the wave
determined by the medium are:determined by the medium are: velocity of propagation (velocity of propagation (vvpp)) intrinsic impedance (intrinsic impedance ()) propagation constant constant propagation constant constant
((==jj)) wavelength (wavelength ())
2
f
vp
• also depend on frequency
48Lecture 10
DispersionDispersion For a signal (such as a pulse) comprising a band of frequencies, different frequency components For a signal (such as a pulse) comprising a band of frequencies, different frequency components
propagate with different velocities causing distortion of the signal. This phenomenon is called propagate with different velocities causing distortion of the signal. This phenomenon is called dispersiondispersion..
0 100 200 300 400 500 600-5
0
5
10
15
20
25
input signal
output signal
49Lecture 10
Plane Wave Propagation in Plane Wave Propagation in Lossy MediaLossy Media
Assume a wave propagating in Assume a wave propagating in the +the +zz-direction:-direction:
We consider two special cases:We consider two special cases: Low-loss dielectric.Low-loss dielectric. Good (but not perfect) conductor.Good (but not perfect) conductor.
zteEtzE zxx cos, 0
50Lecture 10
Plane Waves in a Low-Loss Plane Waves in a Low-Loss DielectricDielectric
A lossy dielectric exhibits loss A lossy dielectric exhibits loss due to molecular forces that the due to molecular forces that the electric field has to overcome in electric field has to overcome in polarizing the material.polarizing the material.
We shall assume thatWe shall assume that
tan1tan1
1
0 jj
jj
r
0
r
51Lecture 10
Plane Waves in a Low-Plane Waves in a Low-Loss Dielectric (Cont’d)Loss Dielectric (Cont’d)
Assume that the material is a Assume that the material is a low-loss dielectric, i.e, the low-loss dielectric, i.e, the loss loss tangenttangent of the material is small: of the material is small:
1tan
52Lecture 10
Plane Waves in a Low-Plane Waves in a Low-Loss Dielectric (Cont’d)Loss Dielectric (Cont’d) Assuming that the loss tangent is small, Assuming that the loss tangent is small,
approximate expressions for approximate expressions for and and can can be developed.be developed.
2
tan
2
tan
2
tan1
tan1
0
00
0
0
k
kk
jjj
jjj
r
wavenumber
2
11 2/1 xx
53Lecture 10
r
p
c
kv
Plane Waves in a Low-Plane Waves in a Low-Loss Dielectric (Cont’d)Loss Dielectric (Cont’d)
The phase velocity is given byThe phase velocity is given by
54Lecture 10
2
tan00
2/1
2
tan1
tan1
j
rr
ej
j
Plane Waves in a Low-Plane Waves in a Low-Loss Dielectric (Cont’d)Loss Dielectric (Cont’d)
The intrinsic impedance is given byThe intrinsic impedance is given by
2
11 2/1 xx xex 1
55Lecture 10
Plane Waves in a Low-Loss Plane Waves in a Low-Loss Dielectric (Cont’d)Dielectric (Cont’d)
In most low-loss dielectrics, In most low-loss dielectrics, rr is is more or less independent of more or less independent of frequency. Hence, dispersion frequency. Hence, dispersion can usually be neglected.can usually be neglected.
The approximate expression for The approximate expression for is used to accurately compute is used to accurately compute the loss per unit length.the loss per unit length.
56Lecture 10
Plane Waves in a Good Plane Waves in a Good ConductorConductor
In a perfect conductor, the In a perfect conductor, the electromagnetic field must vanish.electromagnetic field must vanish.
In a good conductor, the In a good conductor, the electromagnetic field experiences electromagnetic field experiences significant attenuation as it significant attenuation as it propagates.propagates.
The properties of a good conductor The properties of a good conductor are determined primarily by its are determined primarily by its conductivity.conductivity.
57Lecture 10
Plane Waves in a Good Plane Waves in a Good ConductorConductor
For a good conductor,For a good conductor,
Hence, Hence,
1
j
58Lecture 10
Plane Waves in a Good Plane Waves in a Good Conductor (Cont’d)Conductor (Cont’d)
2
2
21
j
jjjj
59Lecture 10
cvp
2
Plane Waves in a Good Plane Waves in a Good Conductor (Cont’d)Conductor (Cont’d)
The phase velocity is given byThe phase velocity is given by
60Lecture 10
45
2
1 jej
j
j
Plane Waves in a Good Plane Waves in a Good Conductor (Cont’d)Conductor (Cont’d)
The intrinsic impedance is given The intrinsic impedance is given byby
61Lecture 10
Plane Waves in a Good Plane Waves in a Good Conductor (Cont’d)Conductor (Cont’d)
The The skin depthskin depth of material is the of material is the depth to which a uniform plane depth to which a uniform plane wave can penetrate before it is wave can penetrate before it is attenuated by a factor of attenuated by a factor of 1/e1/e..
We haveWe have
1
1 e
62Lecture 10
Plane Waves in a Good Plane Waves in a Good Conductor (Cont’d)Conductor (Cont’d)
For a good conductor, we haveFor a good conductor, we have
21
63Lecture 10
Wave Equations for Time-Wave Equations for Time-Harmonic Fields in Simple Harmonic Fields in Simple
MediumMedium
ir
ir
r
ir
ir
r
KjJ
EkE
JjK
EkE
02
0
02
0
1
1
000 k
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