Lect10

1

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


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


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


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


Domain (Cont’d)Domain (Cont’d) Here we must have Here we must have

tzHaH yy ,ˆwhere

tvzptvzptzH ppy 21

1,

16Lecture 10


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


Domain (Cont’d)Domain (Cont’d) In free space (vacuum):In free space (vacuum):

377120

m/s 103 8

cvp

18Lecture 10


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


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


Domain (Cont’d)Domain (Cont’d) The electric and magnetic fields The electric and magnetic fields are given byare given by

21Lecture 10


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


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


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


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


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


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


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


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


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


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


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


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


Domain (Cont’d)Domain (Cont’d)

40Lecture 10

zteC

zteC

ezHtzH

z

z

tjyy

cos

cos

Re,

2

1


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


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


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


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



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



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


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


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


The phase velocity is given byThe phase velocity is given by

54Lecture 10

2

tan00

2/1

2

tan1

tan1

j

rr

ej

j


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


The phase velocity is given byThe phase velocity is given by

60Lecture 10

45

2

1 jej

j

j


The intrinsic impedance is given The intrinsic impedance is given byby

61Lecture 10


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


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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