Section 4 4 TRƯỜNG ĐIỆN TỪ

4 4 Wave Equation and Solution for Material Medium Slide Presentations for ECE 329, Introduction to Electromagnetic Fields, to supplement “Elements of Engineering Electromagnetics, Sixth Edition” by N[.]

Introduction to Electromagnetic Fields,

to supplement “Elements of Engineering

Electromagnetics, Sixth Edition”

by

Nannapaneni Narayana Rao Edward C Jordan Professor of Electrical and Computer Engineering

University of Illinois at Urbana-Champaign, Urbana, Illinois, USA

Distinguished Amrita Professor of Engineering Amrita Vishwa Vidyapeetham, Coimbatore, Tamil Nadu, India

Wave Equation and Solution for Material

Medium

Waves in Material Media

, ,

,

y x

x

x

y

y

H z t

E z t

E z t

E j H z

H

E j E j E z











 







Combining, we get

2

z   



         

2

2

 



Define

Then

Wave equation

 

Re Re

cos cos

x

j t

z z j t

j z j z j t j z j z j t

z

z

Ae e e e Be e e e















 = attenuation constant, Np/m

 = phase constant, rad/m

  wave

cos





        

attenuation

  wave

cos

z

B e   t z 





       

1

= propagation constant, m

attenuation

  , zcos  

f z t  e   t  z

2



 4



 0

t 

z

2 







f

1

 1

0

  , zcos  

g z t  e   t  z

0

t 

g

4

t 



 2

t 





 z

1

 1

0

2 







1 1 1

x

y

x y

z

E H

 

 













 











j







j

j e

j





 



1

Re

1 Im

j

 













 







For dry earth,  10 s/m,   5 , and   

Let us compute , , , , and for   v p   f 100 kHz

Solution:

5 8

1

1

2

3 10

j

f



 





 

 



 





0.004683 1.0628 19.8

j

0.004683 1.0309 9.9

j

0.004683 1.0155 0.1772

0.00083 j 0.004756

0.00083 Np/m



 

0.004756 rad/m

8

2 10 1.321 10 m/s 0.004756

p

0.004756





j j









1 1

j









1 j j















161.1 j28.1

1 168.6



163.55 9.9