Lect10

EEE 498/598

EEE 498/598

Overview of Electrical

Overview of Electrical

Engineering

Engineering

Lecture 10:

Uniform Plane Wave

Uniform Plane Wave

Solutions to Maxwell’s

Lecture 10 Objectives

Lecture 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.

Overview of Waves

Overview of Waves

 A A wavewave is a pattern of values in space is a pattern of values in space

that appear to move as time 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, waves, sound waves, seismic waves,

and voltage and current waves on and voltage and current waves on

transmission lines. transmission lines.

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

Time-Domain Maxwell’s

Time-Domain Maxwell’s

Equations in Differential

Equations in Differential

Form

Form

mv

ev

q

B

t

D

J

H

q

D

t

B

K

E







































J

J



i c

K

K



Time-Domain Maxwell’s

Time-Domain Maxwell’s

Equations in Differential Form

Equations in Differential Form

for a Simple Medium

for a Simple Medium

      mv i ev i m q H t E J E H q E t H K H E                      H K E J H B E

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 Medium and Lossless Medium

0 0  

 

 

 

 

 

 

 



H t

E H

E t

H E

 

0 0

0    



 _{i ev mv m}

i K q q

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 Medium and 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.

Derivation of Wave Equations for Derivation of Wave Equations for

Electromagnetic Waves in a Simple, Electromagnetic Waves in a Simple,

Source-Free, Lossless Medium Free, Lossless Medium

















Wave Equations for

Wave Equations for

Electromagnetic Waves in a

Electromagnetic Waves in a

Simple, Source-Free, Lossless

Simple, Source-Free, Lossless

Medium

Medium

0

2 2

2 _

  



t H H 

0 2

2

2 _

  



t E

E 

 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 law.

Uniform Plane Wave

Uniform Plane Wave

Solutions in the Time

Solutions in the Time

Domain

Domain

 A A uniform plane wave_{uniform plane wave} is an electromagnetic wave in which is an electromagnetic wave in which the electric and magnetic fields and the direction of

the electric and magnetic fields and the direction of

propagation are mutually orthogonal, and their

propagation are mutually orthogonal, and their

amplitudes and phases are constant over planes

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 byLet us examine a possible plane wave solution given by

 

z

t

E

a

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}

0

2 2 2

2

 

 

 

t E z

E_{x  x}

 

z

t

p



z

v

t



p



z

v

t



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-v_ppt)t) and and pp_{2 2} (z+v(z+v_ppt)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.

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



p

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

 

z

t

H

a

H



ˆ

,

where

 

z

t



p



z

v

t



p



z

v

t





H

,



1







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 impedance_{intrinsic impedance} of the medium of the medium

given by given 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.

   

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):

 



 



377 120

m/s 10

3 8





c v_p

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 of waves is very important in the study of

electromagnetics. electromagnetics.

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

























 









 













H z t C z t C t z E y x                  cos cos 1 , cos cos , 2 1 2 1

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 by

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 phase

instantaneous phase of the field: _{of the field:}

 

z

t



t



z

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

0 0

 



 

 



dz v

t z

z t

  



  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

2

2 radians for a fixed value of radians for a fixed value of time is the

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

wavelength

wavelength is is also the

also the distance distance between between

every other every other

zero zero

crossing of crossing of

the sinusoid. the sinusoid.



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





v



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

by

by

p

v













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):

0 0

0 0

2





















k



c

free space wavenumber (rad/m)

Time-Harmonic

Time-Harmonic

Analysis

Analysis

 Sinusoidal steady-stateSinusoidal steady-state (or (or time-harmonictime-harmonic) analysis ) analysis

is very useful in electrical engineering

is very useful in electrical engineering

because an arbitrary waveform can be

because an arbitrary waveform can be

represented by a superposition of sinusoids

represented by a superposition of sinusoids

of different frequencies using

of different 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

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 Medium Possibly Lossy Medium

0

0



























H

E

j

H

E

H

j

E

























j

j

j

j





















Derivation of Helmholtz Equations for Derivation of Helmholtz Equations for Electromagnetic Waves in a Simple, Electromagnetic Waves in a Simple,

Source-Free, Possibly Lossy Medium Free, Possibly Lossy Medium









Helmholtz Equations for

Helmholtz Equations for

Electromagnetic Waves in a Simple,

Electromagnetic Waves in a Simple,

Source-Free, Possibly Lossy Medium

Source-Free, Possibly Lossy Medium

0

2

2

_

_



E



E

 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 law.

using Faraday’s law.

0

2

2

_

_

Uniform Plane Wave

Uniform Plane Wave

Solutions in the Frequency

Solutions in the Frequency

Domain

Domain

 Assuming a plane wave solution of Assuming a plane wave solution of

the form the form

 The Helmholtz equation simplifies toThe Helmholtz equation simplifies to

 

z

E

a

E



ˆ

0

2 2

2



 _x

x _E

dz E

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 as written as





















j





j

attenuation constant

(Np/m)

phase constant

(rad/m) (m-1)

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

medium

 The general solution to this wave The general solution to this wave equation is equation is

 

e

e

C

e

e

C

e

C

e

C

z

E









Uniform Plane Wave

Uniform Plane Wave

Solutions in the

Solutions in the

Frequency Domain

Frequency Domain

(Cont’d)

(Cont’d)

 

z

E

E

 

z

 • wave traveling in

the +z-direction

• wave traveling in the -z-direction

 



 











e C

e z E

t z E

z z

t j x

x

 



 





 

 



 _{cos cos}

Re ,

2 1

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 have

time domain, we have

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

E

H

H

j

E















 





 

 



E

z

E

z



e

C

e

C

z

H

x x

z z

y

 















 

1

1

2 1

Uniform Plane Wave

Uniform Plane Wave

Solutions in the

Solutions in the

Frequency Domain

Frequency Domain

(Cont’d)

(Cont’d)

 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.









e



Uniform Plane Wave

Uniform Plane Wave

Solutions in the Frequency

Solutions in the Frequency

Domain (Cont’d)

 



 











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 have

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

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

have have

 These form a These form a

right-handed right-handed

coordinate coordinate

system system

z

a H

E   ˆ

E

aˆ

aˆ aˆ

 Uniform plane Uniform plane

waves are a

waves are a

type of

type of transverse transverse

electromagnetic

electromagnetic

(

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:

H

a

E

E

a

H

p p











ˆ

ˆ

1





unit vector in direction of propagation

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 191 1 S/m 01 . 0 5 . 2 m 300 . 0 Hz 10 1 0 0 0 9                      . α f

 





Uniform Plane Wave

Uniform Plane Wave

Solutions in the

Solutions in the

Frequency Domain

Frequency Domain

(Cont’d)

(Cont’d)

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

E x

+ (z

,t)

z

e

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

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 (vv_pp))  intrinsic impedance (intrinsic impedance ())

 propagation constant constant propagation constant constant

(

(==jj)₎

 wavelength (wavelength ())

   vp  2

• also depend on frequency

Dispersion

Dispersion

_{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 5 10 15 20 25

input signal

Plane Wave Propagation in

Plane Wave Propagation in

Lossy Media

Lossy 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.

 

z

t

E

e



t

z



Plane Waves in a Low-Loss

Plane Waves in a Low-Loss

Dielectric

Dielectric

 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









 0

  _r  

Plane Waves in a

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 tangent

tangent of the material is small: _{of the material is small:}

1

tan















Plane Waves in a

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.





r p

c

k

v















Plane Waves in a

Plane Waves in a

Low-Loss Dielectric (Cont’d)

Loss Dielectric (Cont’d)





Plane Waves in a

Plane Waves in a

Low-Loss Dielectric (Cont’d)

Loss Dielectric (Cont’d)

 The intrinsic impedance is given byThe intrinsic impedance is given by

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.

Plane Waves in a Good

Plane Waves in a Good

Conductor

Conductor

 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 are The properties of a good conductor are

determined primarily by its

determined primarily by its

conductivity.

Plane Waves in a Good

Plane Waves in a Good

Conductor

Conductor

 For a good conductor,For a good conductor,

 Hence, Hence,

1















Plane Waves in a Good

Plane Waves in a Good

Conductor (Cont’d)

Conductor (Cont’d)





2

2 1

 

 



 

 

 

 

 

 

    

  



j

j j

j j

c

v















2

Plane Waves in a Good

Plane Waves in a Good

Conductor (Cont’d)

Conductor (Cont’d)



 



 

 

45

2

1 _j

e j

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

by

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



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





Wave Equations for

Wave Equations for

Time-Harmonic Fields in Simple

Harmonic Fields in Simple

Medium

Medium

0    