Section 3 6 TRƯỜNG ĐIỆN TỪ

No Slide Title Slide Presentations for ECE 329, Introduction to Electromagnetic Fields, to supplement “Elements of Engineering Electromagnetics, Sixth Edition” by Nannapaneni Narayana Rao Edward C Jor[.]

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

3.6 Polarization of Sinusoidally

Time-Varying Fields

Polarization is the characteristic which describes how the position of the tip of the vector varies with time

Linear Polarization:

Tip of the vector describes a line

Circular Polarization:

Tip of the vector describes a circle

Elliptical Polarization:

Tip of the vector describes an ellipse

(i) Linear Polarization

Linearly polarized in the x direction.



F1 F1cos ( at ) x

Direction remains

along the x axis

 Magnitude varies sinusoidally with time

F2 F2cos ( at ) y

Direction remains

along the y axis

Magnitude varies sinusoidally with time

Linearly polarized in the y direction.



If two (or more) component linearly polarized vectors are in phase, (or in phase opposition), then their sum vector is also linearly polarized.

Ex: F F1 cos (t  ) ax  F2 cos (t  ) ay

(ii) Circular Polarization

If two component linearly polarized vectors are

(b) differ in direction by 90˚

then their sum vector is circularly polarized

 tan–1 F2 cos (t  )

F1 cos (t  )

tan–1 F2

F1

constant

y

x



F1

   

1

1 1 1 1

, constant

sin tan

cos tan tan

F















F

1

F

2

F

F

x y



(iii) Elliptical Polarization

In the general case in which either of (i) or (ii) is not satisfied, then the sum of the two component

linearly polarized vectors is an elliptically polarized vector.

Ex: F  F1 cos t a x  F2 sin t a y

1

F

2

F

F

x y

–F0

–F0

F0

F0

F1

/4

y

D3.17

F1 and F2 are equal in amplitude (= F0 ) and differ in

direction by 90˚ The phase difference (say  ) depends

on z in the manner –2z – (–3z) = z.

(a) At (3, 4, 0),  = (0) = 0.

(b) At (3, –2, 0.5),  = (0.5) = 0.5 .

8

8

x

y

F F1  2  is linearly polarized.

F F1  2 is circularly polarized

(c) At (–2, 1, 1),  = (1) = .

(d) At (–1, –3, 0.2) =  = (0.2) = 0.2.

F F1  2  is linearly polarized.

F F1  2  is elliptically polarized