ENE 325 Electromagnetic Fields and Waves

Lecture 11 Uniform Plane Waves 1

Introduction

http://www.phy.ntnu.edu.tw/ntnujava/viewtopic.php?t=52  From Maxwell’s equations, if the electric field

E

is changing with time, then the magnetic field

H

varies spatially in a direction normal to its orientation direction  A uniform plane wave, both electric and magnetic fields lie in the transverse plane, the plane whose normal is the direction of propagation  Both fields are of constant magnitude in the transverse plane, such a wave is sometimes called a transverse electromagnetic (TEM) wave.

2

Maxwell’s equations

 

H

 

D

 

v B

 0  

t B



D



t

(1) (2) (3) (4) 3

Maxwell’s equations in free space

  = 0, 

r

= 1, 

r

= 1 

H

  0  0 

E



t



H



t

Ampère’s law Faraday’s law 4

General wave equations

  Consider medium free of charge where For linear, isotropic, homogeneous, and time invariant medium, 

E

   

t E

  

H t

(1) (2) 5

General wave equations

Take curl of (2), we yield   ( 

E

  

t

 

t H

)  

E t

)    

E



t

   2

E



t

2 From

A A

2

A

then 2

E

   

E



t

   2

E



t

2 For charge free medium 0 6

Helmholtz wave equation

For electric field For magnetic field  2

E

  

E



t

   2

E



t

2  2

H

  

H



t

   2

H



t

2 7

Time-harmonic wave equations

 Transformation from time to frequency domain Therefore  

t



j

  2

E s



j

  

j

 )

E s

 2

E s



j

  

j

 )

E s

 0  2

E s

  2

E s

 0 8

or

Time-harmonic wave equations

 2

H s

  2

H s

 0 where  

j

  

j

 ) This  term is called propagation constant or we can write 

=



+j

 where   = attenuation constant (Np/m) = phase constant (rad/m) 9

Solutions of Helmholtz equations

 Assuming the electric field is in x-direction and the wave is propagating in z- direction  The instantaneous form of the solutions

E



E e

0 

z

cos(  

x



E e

0   

z

cos(    Consider only the forward-propagating wave, we have

E



E e

0  

z

cos(  

x

 Use Maxwell’s equation, we get

H



H e

 

z

cos(  

y x

10

Solutions of Helmholtz equations in phasor form

 Showing the forward-propagating fields without time-harmonic terms.

E s

  

z e



a x H s

  

z e



a y

 Conversion between instantaneous and phasor form Instantaneous field = Re(e j  t  phasor field) 11

Intrinsic impedance

 For any medium,  

E x H y

 

j

 

j

  For free space  

E x H y



E

0

H

0   0  0  120   12

Propagating fields relation

H s

 1 

a

 

E s E s

  

a

 

H s a

13

Propagation in lossless-charge free media

 Attenuation constant  = 0, conductivity  = 0  Propagation constant  Propagation velocity

u p

    1   for free space u

p

= 3  10 8 m/s (speed of light)  for non-magnetic lossless dielectric ( 

r u p



c



r

= 1), 14

Propagation in lossless-charge free media

 intrinsic impedance      Wavelength   2   15

Ex1 A 9.375 GHz uniform plane wave is propagating in polyethelene (



r = 2.26). If the amplitude of the electric field intensity is 500 V/m and the material is assumed to be lossless, find

a) phase constant b) wavelength in the polyethelene 16

c) propagation velocity d) intrinsic impedance e) amplitude of the magnetic field intensity 17

Propagation in dielectrics

 Cause   finite conductivity polarization loss ( 

=



’ -j



”

)  Assume homogeneous and isotropic medium 

H

 

E



j

  ' 

j

 " )

E



H

 [(   " ) 

j

 ' ]

E

18

Define from and

Propagation in dielectrics



eff

" ,  2 

j

  

j

 )  2  (  

j

 ) 2 19

Propagation in dielectrics

We can derive and  ( 1  2   2  1)  ( 1  2   2  1)     1  1

j

(   ) .

20

Loss tangent

 A standard measure of lossiness, used to classify a material as a good dielectric or a good conductor tan    ' "  

eff

 ' 21

Low loss material or a good dielectric (tan



« 1)

 and     2      (1 

j

2   ).

22

Low loss material or a good dielectric (tan



« 1)

 propagation velocity  wavelength

u p

    1    2   

f

1  23

High loss material or a good conductor (tan



» 1)

 1    2   therefore and    2

j

     

e j

45 .

24

High loss material or a good conductor (tan



» 1)

 depth of penetration or skin depth,  is a distance where the field decreases to e

-1

or 0.368 times of the initial field   1  1  1   m  propagation velocity  wavelength

u p

   2    2  25

Ex2 Given a nonmagnetic material having



r 3.2 and



= 1.5



10 -4 S/m, at f = 3 MHz, find =

a) loss tangent  b) attenuation constant  26

c) phase constant  d) intrinsic impedance 27

Ex3 Calculate the followings for the wave with the frequency f = 60 Hz propagating in a copper with the conductivity,



= 5.8



10 7 S/m:

a) wavelength b) propagation velocity 28

c) compare these answers with the same wave propagating in a free space 29