Section 3 5 TRƯỜNG ĐIỆN TỪ

3 5 Sinusoidally Time Varying Uniform Plane Waves in Free Space Slide Presentations for ECE 329, Introduction to Electromagnetic Fields, to supplement “Elements of Engineering Electromagnetics, Sixth[.]

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

Sinusoidally Time-Varying Uniform Plane Waves in

Free Space

Sinusoidal Traveling Waves

cos cos

p

p

f t z v

t z v







cos cos

p

p

g t z v

t z v

t z







0 0

where   v p   

  , cos  

f z t   t   z

1

 1



 4

t 





0

t 





2 



  , cos  

g z t   t   z

2

t 





4

t 





0

t 





2 



g 1

0

 1

 z

The solution for the electromagnetic field is

0 0

where   w vp  w  

For JS t  J S cos t ax for z 0,

0 cos for 0



0

2

S

p x



0 cos for 0

Parameters and Properties

1  t   z    Phase, 

2 radian frequency =

rate of change of phase with time for a fixed value of (movie)

t z







frequency 2

= number of 2 radians of phase change per sec





3 phase constant =

= magnitude of rate of change of phase with

distance for a fixed value of (still photograph)

z







4 phase velocity =

velocity with which a constant phase progresses along the direction of propagation

p

v











5 = wavelength =

distance in which the phase changes by 2 for a fixed t









0

7

= Ratio of the amplitude of to the amplitude

of for either wave

    

E H

6 Note that

2

2

in in MHz = 300



8 (Poynting Vector, )

for (+) wave

for ( ) wave

is in the direction of propagation.







x

H P

E

z y

x

E

y

z

Direction of propagation is –z.

Consider E 37.7 cos 6 10 t  2 z ay V m

Then

8

8

2 2

2

p

v















Array of Two Infinite Plane Current Sheets

1

S

J JS2

0

z  z   4

1 For ,JS

cos for 0 sin for 4



0 0

1

0 0

2

2

S

x

S

x







a E

a

For JS ,

2

2

S

x

S

x

S

x

S

x

S

x











a E

a a a

a

4









For both sheets,

No radiation to one side of the array

“Endfire” radiation pattern

   

   

   

1 2

0 0

0 0

sin sin for 0 4

0 for 0

z

z z

z



























E = E E

a a