is a pattern of values in space that appear to move as time evolves.

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4㨰㵜 Wave Equations & Time-Harmonic Fields

Z

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A

wave

is a pattern of values in space that appear to move as time evolves.

A

wave

is a solution to a wave equation.

Examples of waves include water waves, sound waves, seismic waves, and voltage and current waves on transmission lines.

Wave phenomena result from an exchange between two

different forms of energy

such that the time rate of change in one

form

leads to a spatial change in the other .

Overview of Wave

EElectromagnetic Fields

Z

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Maxwell’s equations give a complete description of the relation between electromagnetic fields and charge and current distributions.

For given charge and current distributions, ȡ and d J, we first solve the inhomogeneous wave equations for potentials,

and _A, ²_A ^2A_{2 J}

PH

t

P

§’  w  ·

¨ w ¸

© ¹

&

& & *

2 2

2

V V V

t

PH U

H

§’  w  ·

¨ w ¸

© ¹

With V and d A determined, d, E and d B can be found from

B & ’ u A &

E V A

t ’  w

w

&

&

V

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Except at the origin,

2 2

2 2

1 ( V ) V 0

R ^wR R ^wR PH ^wt

w w w

2 2

2

V V

t PH U

H

’  w 

w U( ) 't v'

Y+U/w,@B

Spherical coordinates !!

First, solve for the homogeneous equation. Next, particular solution.

Because of spherical symmetry, V depends only on R and t.

( , ) 1 ( , ) V R t U R t

Assume . R ^{2 2}

2 2 0

U U

R PH t

w  w

w w

sin 0 sin 1

sin 1 1

2 2 2

2 2 2 2

2

2 w

 w w

 w

¸¹

¨ ·

©

§

w w w

 w

¸¹

¨ ·

©

§ w w w

w

t V V

R V

R R

R V R

R PH

I T T T

T T

=0 =0

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V

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2 2

2 2 0

U U

R PH t

w  w

w w

(t r R PH )

It can be verified by direct substitution that any twice-differentiable function of is a solution, such as

( , ) ( )

U R t f t R

PH

From we get _{V R t( , )} ¹_{U R t( , ),} R

( , ) 1 ( / )

V R t f t R u

R 

( ) '

( ) 4

t v

V R R

U SH ' '

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'

1 ( / )

( , ) '

4 ^V

t R u

V R t dv

R

U

SH

³



R V q

SHo

4 for static E-field.

V

Vroxwlrq#ri#Zdyh#Htxdwlrqv#iru#Srwhqwldov This equation indicates that the V at a distance R from the source at time t depends on the value of the charge density at an earlier time (t - R/u).

( R)

t u

U ^_x^q

R

V t( )

'

( / )

( , ) '

4 ^V

J t R u

A R t dv

R P

S

³

^& 

&

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E and d B are also functions of (t-R/u) and retarded in time.

In the quasi-static approximation, we ignore this retardation effect and assume instant response.

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V

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t₁=0 sec xq





t 0sec



U

s m u 3u10⁸ /

m R 3u10⁸

t₂=1 sec

^{t 1sec}

V

   ₁

8 8

2 0sec

/ 10 3

10 sec 3

1 t

s m

m u

t R U U U

U ¸¸¹

·

¨¨©

§

u

 u

¸¹

¨ ·

©§ 

).

( determines

Therefore, ₂ V t₂

u t R¸

¹

¨ ·

©§  U

This equation indicates that the V at a distance R from the source at time t depends on the value of the charge density at an earlier time (t - R/u).

V

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0 0 Ÿ J &

V

Vrxufh#Iuhh=

˜ 0

’

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’

w

 w u

’

w

 w u

’

B D

t E B

t J D

H

*

&

&

&

&

&

&

U

Maxwell’s Equations

0 0

˜

’

˜

’

w

 w u

’

w u w

’

B D

t E B

t H D

*

&

&

&

&

&

Maxwell’s Equations

EElectromagnetic Fields

V

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t E B

w uw

’

u

’ u

’

&

&

2 2

t E E

w

 w u

’ u

’

&

&

H P

t E B

w

w u

’

&

&

 

^B

E& t &

u w ’

 w u

’ u

’



^H



E& t &

u w ’

 w u

’ u

’ P

¸¸¹

¨¨ ·

©

§ w w w

 w u

’ u

’ t

D E t

&

&

P

 

^{2 2}₂

t E E

E w

 w

’



˜

’

’

&

&

&

H P

2 0

2 2

w

 w

’ t

E E

&

&

H P

: Taking curl of both sides

 

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E& & ₂&

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’

’ u

’ u

’

H B& P &

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&

&

&

&

t D t

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H w

w w

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’

E D& H &

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0 

&

˜

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V

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2

0

2 2

w

 w

’ t

E E

&

&

H

P 1 0

2 2

2 2

w

 w

’ t

E E u

&

&

PH u 1

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1 0

2 2

2 2

w

 w

’ t

H H u

&

&

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Sinusoidal steady-state (or time-harmonic) analysis is very useful in electrical engineering because an arbitrary waveform can be represented by a superposition of sinusoids of different frequencies using Fourier analysis.

If the waveform is periodic, it can be represented using a Fourier series.

If the waveform is not periodic, it can be represented using a Fourier transform.

^ ^{E x y z e}

^{j t}

`

t z y x

E & ( , , , ) Re & ( , , )

^Z

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For time-harmonic (steady-state sinusoidal) fields, it is convenient to use a phasor notation.

Phasor: It contains amplitude and phase information.

Time-Independent Time-dependent

 

^{t E}



^ZtI



E ₀cos

Instantaneous(time-dependent) expression

Phasor is useful to analyze a linear system with time-harmonic excitations.

1

 

t e C idt

dt Ri

Ldi  

³

s s s

s E

j I RI C

I

Lj  

Z ¹ Z

^

ESe^jZt

`

^

^E0^ejZtjI

`

Re

Re Re

^

^E0^ejIejZt

`

I j

S E e

E ₀

Example:

> @ > @ >

s ^{j t}

@

t s j t

j s t

j

s e E e

j I e C

I R e

I j

L ^{Z Z Z Z}

Z 1 Re Z Re

Re

Re »

¼

« º

¬

 ª



s

s E

C I L j

j

R »

¼

« º

¬

ª  

Ÿ Z Z¹

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^ ^{E x y z e}

^{j t}

`

t z y x

E & ( , , , ) Re & ( , , )

^Z

E(x,y,z) is a vector phasor which contains information on direction, magnitude, and phase, but not time.

H j

B j

E & & &

ZP

Z 

 u

’

˜ 0

’ B &

U

˜

’ D &

E j

J D j J

H & & & & &

ZH

Z 

 u

’

Time-Harmonic

t

E B

w

 w u

’

&

&

˜ 0

’ B &

U

˜

’ D &

t

J D

H

w

 w u

’

&

&

&

General

S

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Given that in air, find ^E&(^x,^y,^z,^t) ^aˆy0.1sin



10S^x



cos



6S10^9tE^z



d H and ȕ.

Solution: It can be solved using time-harmonic analysis.

 

_»¼^º

«¬ªa_y x e^{j z}˜e^{j t} E^& Re ˆ 0.1sin10S ^{E 6S109}

¸¸

¹

·

¨¨

©

§

¸¸¹

¨¨ ·

©

§

w

w w

 w

¸¹

¨ ·

©

§

w

w w

 w

¸¸¹

¨¨ ·

©

§

w

w w

 w

y E x

a E x E z

a E z E y a E j

y x z z x

y z y

x ˆ ˆ

1 ˆ ZP0

¸¸¹

¨¨ ·

©

§

w

 w w

 w

 x

a E z a E j

y z y

x ˆ

1 ˆ ZP0

 

^{j z}

 

^{j z}

y e j x e

x z z

E _{E E}

S E

S ^  ^

w w w

w 0.1sin10 0.1sin10

Phasor form: E& aˆ_y0.1sin



10Sx



e^jEz j E

H& &

u

’



0

1 Phasor form: ZP



^x



^e



^x



e z x

E_{y j z j z}

S S

S ^E

E sin10 0.1 10 cos10 1

.

0 ^{ }

w w w

w

GHz 3 10 3

10 6 2

10 6

9 9 9

u u

Ÿ u

f

f S

S S

Z

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S

Sureohp#:058

¸¸¹

¨¨ ·

©

§

w

 w w

 w

 u

’

 x

a E z a E E j

H j 1 1 ˆ_x ^y ˆ_z ^y

0

0 ZP

ZP

&

&

 

     



^{a j x e a e x}



j

z j z

z j

x E S S S

ZP

E

E ˆ 0.1 10 cos10 10

sin 1 . ˆ 0

1

0



 







j H

E& &

u

’

0

1

ZH ^¸¸_¹

·

¨¨

©

§

¸¸¹

¨¨ ·

©

§

w

 w w

 w

¸¹

¨ ·

©

§

w

 w w

 w

¸¸¹

¨¨ ·

©

§

w

 w w w

y H x

a H x

H z

a H z

H y

a H j

y x z z x

y z y

x ˆ ˆ

1 ˆ ZH0

¸¸¹

¨¨ ·

©

§

w

 w

¸¹

¨ ·

©

§

w

w w

 w w w

y a H x

H z

a H y a H j

x z z x

y z

x ˆ ˆ

1 ˆ

ZH0 ^¨_©^§ _w ^¸_¹^·

w w w

x H z

H

a_y j ^{x z}

0

ˆ 1 ZH

 

 

^{j z}

z

z j x

e j x

H

e x H

E E

ZP S S ZP S

E









10 cos

10 1 sin

. 0

0 0

 

 

^{j z}

z

z x j

e j x

x H

e x z j

H

E E

ZP S S

ZP S E





w w

w w

10 10 sin

10 1 sin

. 0

0 2 0

2

In order to find ȕ,

¸¹

¨ ·

©

§

w

 w w u w

’ x

H z

H a j

j H

E _y ^{x z}

0 0

ˆ 1 1

ZH ZH

&

&

   

_¸^¸

¹

·

¨¨

©

§ ^{j z} ^{j z}

y x e

e j x j j

a ^E S ^E

ZP S S

ZP E

ZH ^sin10

10 10 1 sin

. 0 ˆ 1

0 2

0 2

0

   

^{j z}

y x e

a E S S ^E

P H Z

10 sin10 

1 . 1 0

ˆ ^{2 2}

0 2 0



2

 

2



2

 

2 2 0 0 0

20.01 E 10S 0.1 E 10S Z H P

P H

Z ^{ Ÿ }



⁸



²

0 2 0

9

10 3

1 1

10 6

u u

P c H

S Z

2 16

18 2

0

2 0 400

10 9

10

36S S

P H

Z u

u

S S

S

E 400 ² 100 ² 300

?

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^{j z}

y x e

a

E& ˆ 0.1sin10S ^{ E} Because ,



^{E S}



P H

Z ¹



^{0.1 2 10 2}

P0 2 0

.1 0

EElectromagnetic Fields

 

     



^{a j x e a e x}



H j _x E S ^{j z} _z ^{j z} S S

ZP

E

E ˆ 0.1 10 cos10 10

sin 1 . ˆ 0

1

0



 





& 

     



^x

 

^{t z}



a

z t

x a

t z y x H

z x

˜

 u

u



˜

 u

u







S S

S

S S

S

300 10

6 sin 10

cos 10 33 . ˆ 1

300 10

6 cos 10

sin 10 3 . ˆ 2

; , ,

9 4

9

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Sureohp#:058



^{x y z t}

 > @

^{Hej t}

H& & ^Z Re

; , ,

㷟Practice Problem 7-25 and 26.

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Nonhomogeneous Helmholtz Equations for Electromagnetic Waves in a Simple Medium

2 2

2 2 2

2

V V

t

A A J

t PH U

H

PH P

’  w 

w

’  w 

w

&

& *

w 0

 w

˜

’ t

A ^& PH V ’ ˜ A ^&  j ZPH V ₀

Lorentz Condition

 

 

j A J A

V j V

&

&

&

P Z

PH

H Z U

PH





’





’

2 2

2 2

ௗ

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2 2

2 2

V k V

A k A J

U H P

’  

’ & &  &

J A

A

V V

&

&

&

P PHZ

H PHZ U





’

Ÿ





’

Ÿ

2 2

2 2

Zdyhqxpehu=#n

PH Z PH Z

2

Ÿk2

k

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( ) '

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jkR

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jkR

V

V R e dv

R

A R Je dv

R U SH

P S





³

³

^&

&

'

'

1 ( / )

( , ) '

4

( / )

( , ) '

4

V

V

t R u

V R t dv

R J t R u

A R t dv

R U SH

P S





³

³

^&

&

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  kR  jkR

j t

j ¸ 

¹

¨ ·

©

§

 exp exp

exp ₀

Z Z Z

u t₀ R

k Z PH Zu Z 

kR

  ¸

¹

¨ ·

©§ 

 u

t R f t t

f ₀

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if 2 R 1, or

kR

S

R

O

O



'

'

( ) 1 '

4

( ) '

4

jkR

V

jkR

V

V R e dv

R

A R Je dv

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P S





³

³

^&

&

2 1 1

2

2  |





 k R 

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>#vlpsolilhg#wr#txdvl0vwdwlf#ilhog Condition for Quasis -i-Static Approx.

³

'

4 ' ) 1 (

V

R dv R

V U

SH

³

'

4 ' ) (

V

Rdv R J

A

&

&

S P

:Taylor-series expansion

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( ) ' ' ( / )

4

jkR

V

A R Je dv Wb m

R P

S

³

^& 

&

'

( ) 1 ' ( )

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V R e dv V

R U SH

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B R& ’ u A&

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E R& ’ V j AZ &

61#Ilqg#lqvwdqwdqhrxv E R t&( , ) Re[ ( )E R e& ^{j tZ} ] ( , ) Re[ ( ) ^{j t}] B R t& B R e& ^Z Procedure for determining E and B

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Time-Harmonic Maxwell’s Equations in Differential Form for a Simple and Source-FreeMedium

H j

E & &

ZP

 u

’

˜ W

’ H & H

˜ U

’ E &

E j

J

H & & &

ZH

 u

’

H j

E & &

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 u

’

˜ W

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Y Y

Y Y

Y



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’

u

’

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’

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E E

E E

E j j

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H j

E

H j

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&

&

&

&

&

&

&

&

&

&

&

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PH Z

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W

W ’u ’˜

˜

’

 u

’ E& j H& E& H& j E& H&

ZH ZP

   

W W

Y Y

Y Y

Y Y

Y



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’

’





’



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’

’

u

’ u

’ u

’

u

’

H k H

H H

H H

H j

j H H

E j

H

E j H

&

&

&

&

&

&

&

&

&

&

&

&

&

PH Z

PH Z

ZP ZH

ZH ZH

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• The Helmholtz equations are not independent.

• Usually, we solve the electric field equation and determine H from E using Faraday’s law.

0

2

2



’ E & k E &

0

2

2



’ H & k H &

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E j

J

H & & &

ZH

 u

’

( )

H j E j E

j

V ZH Z H V

Z

§ ·

’u  ¨  ¸

© ¹

& & &

j ZH

c

E ^&

Maxwell’s Equations in a lossy medium:

( / )

c jV F m

H H

 Z 0

0

˜

’ u

’

˜

’

 u

’

H E

j H

E H

j E

c

&

&

&

&

&

&

ZH ZP

If the simple medium is a lossy medium with finite conductivity,

Loss Tangent

( ' '')

c c

k

j Z PH

Z P H  H

2

0

2



’ E & k E &

Complex wavenumber:

in lossy medium

Loss tangent:

Î Measure of the power loss in the medium

Gc

Loss angle:

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ZH V H

G H #

c cc tan c

Z H cc V

;

2

0

2



’ E & k

_c

E &