Lect11

(1)

EEE 498/598

Overview of Electrical

Engineering

Lecture 11:

Electromagnetic Power Flow;

Reflection And Transmission

Of Normally and Obliquely

(2)

Lecture 11 Objectives

 To study electromagnetic power To study electromagnetic power

flow; reflection and transmission flow; reflection and transmission

of normally and obliquely of normally and obliquely

incident plane waves; and some incident plane waves; and some

useful theorems. useful theorems.

(3)

Flow of

Electromagnetic Power

 Electromagnetic waves transport throughout Electromagnetic waves transport throughout

space the energy and momentum arising from a

set of charges and currents (the sources).

 If the electromagnetic waves interact with If the electromagnetic waves interact with another set of charges and currents in a

another set of charges and currents in a

receiver, information (energy) can be delivered

from the sources to another location in space.

 The energy and momentum exchange between The energy and momentum exchange between waves and charges and currents is described by

waves and charges and currents is described by

the Lorentz force equation.

(4)

Derivation of Poynting’s

Theorem

 Poynting’s theorem concerns the Poynting’s theorem concerns the

conservation of energy for a conservation of energy for a

given volume in space. given volume in space.

 Poynting’s theorem is a Poynting’s theorem is a

consequence of Maxwell’s consequence of Maxwell’s

equations. equations.

(5)

Derivation of Poynting’s

Theorem in the Time

Domain (Cont’d)

 Time-Domain Maxwell’s curl Time-Domain Maxwell’s curl

equations in differential form equations in differential form

t D J

J H

t B K

K E

c i

   

 



   

  

(6)

Derivation of Poynting’s

Theorem in the Time

Domain (Cont’d)

 Recall a vector identityRecall a vector identity

 Furthermore,Furthermore,



E



H





H







E



E







H





B H

K H

E H

t D E

J E

H

E _i _c

  

 

 

 

   

 

 

  

(7)

Derivation of Poynting’s

Theorem in the Time

Domain (Cont’d)





t D E

J E

t B H

K H

H E

E H

H E

c i

   

 

   

 



   

   

 

(8)

Derivation of Poynting’s

Theorem in the Time

Domain (Cont’d)

 Integrating over a volume Integrating over a volume V_V bounded by bounded by a closed surface

a closed surface SS, we have_{, we have}

   



                          V V c V c V V i i dv H E dv M H dv J E dv t B H t D E dv K H J E

(9)

Derivation of Poynting’s

Theorem in the Time

Domain (Cont’d)

 Using the divergence theorem, we Using the divergence theorem, we

obtain the general form of Poynting’s

theorem theorem    



                         S V c V c V V i i s d H E dv M H dv J E dv t B H t D E dv K H J E

(10)

Derivation of Poynting’s

Theorem in the Time

Domain (Cont’d)

 For simple, lossless media, we haveFor simple, lossless media, we have

 Note thatNote that

 

2 1

A A

A     

   



                     S V V i i s d H E dv t H H t E E dv K H J

(11)

Derivation of Poynting’s

Theorem in the Time

Domain (Cont’d)

 Hence, we have the form of Hence, we have the form of

Poynting’s theorem valid in simple,

lossless media:

 



 



   



 _

    

 

S

V V

i i

s d H

E

dv H

E t

dv K

H J

E 2 2

2 1 2

1

 

(12)

Derivation of Poynting’s

Theorem in the Frequency

Domain (Cont’d)

 Time-Harmonic Maxwell’s curl equations Time-Harmonic Maxwell’s curl equations

in differential form for a simple medium

i i

J

E

j

H

K

H

j

E





















   



(13)

Derivation of Poynting’s

Theorem in the Frequency

Domain (Cont’d)

 Poynting’s theorem for a simple Poynting’s theorem for a simple

medium medium  





 



               _ _ _      S V m V V V V i i s d H E dv H dv E dv H E dv H E j dv K H J E 2 2 2 2 2 2 2 1 2 1        

(14)

Physical Interpretation

of the Terms in

Poynting’s Theorem

 The termsThe terms

represent the

represent the instantaneous power instantaneous power dissipated

dissipated in the electric and _{in the electric and}

magnetic conductivity losses, magnetic conductivity losses,

respectively, in volume respectively, in volume VV._.





V

m V

dv

H

dv

E

2



2

(15)

Physical Interpretation of

the Terms in Poynting’s

Theorem (Cont’d)

 The termsThe terms

represent the

represent the instantaneous power instantaneous power dissipated

dissipated in the polarization and _{in the polarization and} magnetization losses,

magnetization losses, respectively, in volume respectively, in volume VV._.





_



V V

dv

H

dv

E

2





2





(16)

Physical Interpretation of

the Terms in Poynting’s

Theorem (Cont’d)

 Recall that the electric energy Recall that the electric energy

density is given by

 Recall that the magnetic energy Recall that the magnetic energy

density is given by

2

2 1

E w_e   

2

2 1

H w_m  

(17)

Physical Interpretation of

the Terms in Poynting’s

Theorem (Cont’d)

 Hence, the terms Hence, the terms

represent the

represent the total electromagnetic total electromagnetic energy stored

energy stored in the volume _{in the volume}VV. _.





  



 _ _ _

V

dv H

E2 2

2 1 2

1

 

(18)

Physical Interpretation of

the Terms in Poynting’s

Theorem (Cont’d)

 The termThe term

represents

represents the flow of instantaneous the flow of instantaneous power

power out of the volume _{out of the volume}VV

through the surface through the surface SS._.







 

S

s d H

(19)

Physical Interpretation of

the Terms in Poynting’s

Theorem (Cont’d)

 The term The term

represents the

represents the total electromagnetic total electromagnetic energy generated by the sources

energy generated by the sources in the _{in the} volume

volume VV. _.













V

i

H

K

dv

J

E

(20)

Physical Interpretation of

the Terms in Poynting’s

Theorem (Cont’d)

 In words the Poynting vector can be In words the Poynting vector can be

stated as

stated as “The sum of the power generated by “The sum of the power generated by

the sources, the imaginary power (representing

the time-rate of increase) of the stored electric

and magnetic energies, the power leaving, and

the power dissipated in the enclosed volume is

equal to zero.”

                            _ _ _     V V V i i s d H E dv H dv E dv H E dv H E j dv K H J E 0 2 1 2 1 2 2 2 2 2 2        

(21)

Poynting Vector in the

Time Domain

 We define a new vector called the We define a new vector called the

(instantaneous)

(instantaneous) Poynting vectorPoynting vector as as

 The Poynting vector has the same direction as The Poynting vector has the same direction as

the direction of propagation.

 The Poynting vector at a point is equivalent to The Poynting vector at a point is equivalent to

the power density of the wave at that point.

H

E

S





•vector has units of The Poynting W/m2.

(22)

Time-Average Poynting

Vector

 The time-average Poynting The time-average Poynting

vector can be computed from the

instantaneous Poynting vector as

 

S

 

r t dt T

r S

p

T

p

av 



0

, 1

(23)

Time-Average Poynting

Vector (Cont’d)

vector can also be computed as vector can also be computed as

 



*



Re 2

1

H E

r

S _av  

(24)

Time-Average Poynting

Vector for a Uniform

Plane Wave

 Consider a uniform plane wave Consider a uniform plane wave

traveling in the +

traveling in the +zz-direction in a -direction in a lossy medium:

lossy medium:

 

z j z c

y

z j z

x

e e

E z

H

e e

E z

E

 



 

 

0 0

(25)

Time-Average Poynting

Vector for a Uniform Plane

Wave (Cont’d)

vector is vector is





 

         cos 2 ˆ Re 2 ˆ 1 Re 2 ˆ Re 2 1 2 2 0 2 2 2 0 * 2 2 0 * z z z z z z av e E a e E a e E a H E S              

(26)

Time-Average Poynting

Vector for a Uniform Plane

Wave (Cont’d)

 For a lossless medium, we haveFor a lossless medium, we have

 





2 ˆ

0 0

2 0

E a

S _av  _z

 

(27)

Reflection and

Transmission of Waves at

Planar Interfaces

medium 2 medium 1

incident wave

reflected wave

(28)

Normal Incidence on a

Lossless Dielectric

 Consider both medium 1 and medium Consider both medium 1 and medium

2 to be lossless dielectrics.

 Let us place the boundary between the Let us place the boundary between the

two media in the

two media in the z z = 0 plane, and _{= 0 plane, and}

consider an incident plane wave which

is traveling in the +

is traveling in the +zz-direction._-direction.

 No loss of generality is suffered if we No loss of generality is suffered if we

assume that the electric field of the

(29)

Normal Incidence on a

Lossless Dielectric

(Cont’d)

z x

1 1, H

E E 2, H 2

0 ,

, ₁ ₁

1   

(30)

Normal Incidence on a

Lossless Dielectric

(Cont’d)

 Incident waveIncident wave

z j i

y i

z i

z j i

x i

e E

a E

a H

e E

a E

1 1

1 0 1

0

ˆ ˆ

1 ˆ

 

 

 

 

 

known

1 1

1 _

 

  

(31)

Normal Incidence on a

Lossless Dielectric

(Cont’d)

 Reflected waveReflected wave





_r j z

y r

z r

z j r

x r

e E

a E

a H

e E

a E

1 1

1 0 1

0

ˆ ˆ

1 ˆ

 

 

 

  

 



(32)

Normal Incidence on a

Lossless Dielectric

(Cont’d)

 Transmitted waveTransmitted wave

z j t

y t

z t

z j t

x t

e E

a E

a H

e E

a E

2 2

2 0 2

0

ˆ ˆ

1 ˆ

 

 

 

 

 

unknown

2 2

2 _

 

 



(33)

Normal Incidence on a

Lossless Dielectric

(Cont’d)

 The total electric and magnetic The total electric and magnetic

fields in medium 1 are





                  z j r z j i y r i z j r z j i x r i e E e E a H H H e E e E a E E E 1 1 1 1 1 0 1 0 1 0 0 1 ˆ ˆ      

(34)

Normal Incidence on a

Lossless Dielectric

(Cont’d)

 The total electric and magnetic The total electric and magnetic

fields in medium 2 are

z j t

y t

z j t

x t

e E

a H

H

e E

a E

E

2 2

2 0 2

0 2

ˆ ˆ

 



 

 

(35)

Normal Incidence on a

Lossless Dielectric

(Cont’d)

 To determine the unknowns To determine the unknowns EE_r0_r0 and and

E

E_t0_t0, we must enforce the BCs at _{, we must enforce the BCs at}zz = 0 = 0:_:











0





0



0 0

2 1

 



 



z H

z E

(36)

Normal Incidence on a

Lossless Dielectric

(Cont’d)

 From the BCs we haveFrom the BCs we have

2 0 1

0 1

0

0 0

0

 

i r t

t r

i

E E

E

E E

E

 

 

or

0 1

2

2 0

0 1

2

1 2

0

2

, _t _i

i

r E E E

E

 



 

 

  

(37)

Reflection and

Transmission

Coefficients

 Define the Define the reflection coefficientreflection coefficient as as

 Define the Define the transmission coefficient_{transmission coefficient} as as

1 2

0 0

 

  

 

i r

E E

2

0 2

 

 

 

 t

E E

(38)

Reflection and

Transmission

Coefficients (Cont’d)

 Note also thatNote also that

 The definitions of the reflection and The definitions of the reflection and

transmission coefficients do generalize

to the case of lossy media.

 For lossless media, For lossless media,  and and  are real. are real.

 For lossy media, For lossy media,  and and  are complex. are complex.



   1

2 0

, 1

1     

(39)

Traveling Waves and

Standing Waves

 The total field in medium 1 is The total field in medium 1 is

partially a

partially a traveling wavetraveling wave and and partially a

partially a standing wavestanding wave._.

 The total field in medium 2 is a The total field in medium 2 is a

pure

(40)

Traveling Waves and

Standing Waves

(Cont’d)

 The total electric field in medium The total electric field in medium

1 is given by

















 



e j z



E a e e e E a e e E a E E E z j i x z j z j z j i x z j z j i x r i 1 0 0 0 1 sin 2 1 ˆ 1 ˆ ˆ 1 1 1 1 1 1                               standing

(41)

Traveling Waves and

Standing Waves:

Example

z x

0 ,

, ₁ ₀ ₁

0

1       

 ₂  4₀, ₂  ₀, ₂  0

0

1 

  0 ₂

2

  

3 1

  

3 2



(42)

Traveling Waves and

Standing Waves:

Example (Cont’d)

0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

N

o

rm

a

liz

e

d

E

f

ie

(43)

Standing Wave Ratio

 The The standing wave ratiostanding wave ratio is defined as is defined as

 In this example, we haveIn this example, we have

 

 

  



1 1 min

1

max 1

z E

S

2 1

1

3 1 1

 

 

(44)

Time-Average Poynting

Vectors

 





 





     



2



0 2

1 1 2 0 2 * 1 2 0 * 2 1 ˆ 2 ˆ Re 2 1 2 ˆ Re 2 1    i z r av i av av i z r r r av i z i i i av E a S S S E a H E S E a H E S             

(45)

Time-Average Poynting

Vectors (Cont’d)

   





2 2 0 2 * 2 2 ˆ Re 2 1   i z t t t av av E a H E S

S    

We note that

        2 2 2 2 1 2 1 2 2 1 2 2 1 2 1 2 1 2 1 2 1 1 2 2 1 4 1 1 1 1 1                                                                 

(46)

Time-Average Poynting

Vectors (Cont’d)

 Hence, Hence,

   

     

av i av r av t av

av

S S

S

S S

 



or

2 1

(47)

Oblique Incidence at a

Dielectric Interface

1 1,

 _₂_,_₂

0



z

E E

E   _E _ _E

i



r

(48)

Oblique Incidence at a

Dielectric Interface: Parallel

Polarization (TM to z)





 





 





 _t _t 

r r i i z x jk t t t z x jk r r r z x jk i i i e z x E E e z x E E e z x E E             cos sin 0 cos sin 0 cos sin 0 2 1 1 sin ˆ cos ˆ sin ˆ cos ˆ sin ˆ cos ˆ              

(49)

Dielectric Interface: Parallel

Polarization (TM to z)

i t

i

i t

 

cos cos

cos 2

cos cos

1 2

2

1 2

 



  

(50)

Dielectric Interface:

Perpendicular Polarization (TE

to z)

 

 _t _t 

r r i i z x jk t z x jk r z x jk i

e

y

E

e

y

E

e

y

E

      cos sin 0 cos sin 0 cos sin 0 2 1 1

ˆ

     









(51)

Dielectric Interface:

Perpenidcular Polarization

(TM to z)

t i

i

t i

 

cos cos

cos 2

cos cos

1 2

2

1 2

 



  

(52)

Brewster Angle

 The Brewster angle is a special The Brewster angle is a special

angle of incidence for which angle of incidence for which



=0._=0.

 For dielectric media, a Brewster For dielectric media, a Brewster

angle can occur only for parallel

polarization.

(53)

Critical Angle

 The critical angle is the largest The critical angle is the largest

angle of incidence for which

angle of incidence for which kk22 is is

real (i.e., a propagating wave real (i.e., a propagating wave

exists in the second medium). exists in the second medium).

 For dielectric media, a critical For dielectric media, a critical

angle can exist only if

(54)

Some Useful Theorems

 The reciprocity theoremThe reciprocity theorem  Image theoryImage theory

(55)

Image Theory for Current

Elements above a Infinite, Flat,

Perfect Electric Conductor

 



actual sources

images electric _magnetic

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Image Theory for Current

Elements above a Infinite, Flat,

Perfect Magnetic Conductor

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actual sources

images electric _magnetic

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