EE451_Lec6_ Solving the four fundamental EM equationsPolarization_Power.pdf

Maxwell Equations

1. ةبيط ةعماج
Omar Siddiqui
Department of Electrical Engineering
College of Engineering
Taiba University
Madinah
Email:[email protected]
EE 451 –Electromagnetic Waves
Lecture 6: Wave Polarization and Poynting Vector

2. Polarization of a Wave
Example
Locus of electric field at
point z
1
Linear (Vertical)
polarization
College of Engineering, Taibah University, MadinahEE451 Electromagnetics
Polarization of a wave is given by the locus of the tip of the electric field (in a plane
perpendicular to the direction of propagation) at a given point as a function of time.
z
E
y
Direction of travel
E
y
z=z
1
??????(�,�)=??????
�cos(��−��)
Polarization type:
Types of Polarization
Linear: when tip of E
makes linear pattern
Circular: when tip of E
rotates in a circular
pattern
Elliptical: when tip of E
rotates in an elliptical pattern

3. Types of Polarization: 1. Linear Polarization
??????�=ො�??????
�������−��+??????
�+ො�??????
�������−��+??????
�
A transverse electromagnetic (TEM) wave is linearly polarized if either the electric
field has one component or both orthogonal components of electric field are in
phase
(ෝ??????
�)
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Assuming +z propagation of a plane wave,
??????
�
??????
�
෡??????
Linear polarization conditions:
??????�=ො�??????
�������−��
??????�=ො�??????
�������−��+
??????
4
??????�=ො�??????
�������−��+ො�??????
�������−��
1.Either??????
�or??????
�=0
2.If??????
�≠0??????��??????
�≠0
x or Horizontally polarized
y or Vertically polarized
diagonally polarized
??????
thenΔ??????=??????
�−??????
�=0��±??????
Examples
??????
�=0
??????
�=0
Δ??????=0
Electric Field Condition Polarization Sub-Type

4. Example of Linear Polarization (Horizontal)
College of Engineering, Taibah University, MadinahEE451 Electromagnetics
??????�=ො�??????
�������−��(x or Horizontally polarized)

5. Example of Linear Polarization (Vertical)
College of Engineering, Taibah University, MadinahEE451 Electromagnetics
??????�=ො�??????
�������−��(y or Vertically polarized)

6. Example of Linear Polarization (Diagonal)
College of Engineering, Taibah University, MadinahEE451 Electromagnetics
??????�=ො�??????
�������−��+ො�??????
�������−��(diagonally polarized)

7. Types of Polarization: 2. Circular Polarization
??????�=ො�??????
�������−��+??????
�+ො�??????
�������−��+??????
�
A transverse electromagnetic (TEM) wave is circularly polarized if the two components
of electric field have equal magnitude and a phase difference of (2n+1)p/2
College of Engineering, Taibah University, MadinahEE451 Electromagnetics
Assuming +z propagation of a plane wave
Circular polarization conditions:
??????�=ො�??????
������−��+ො�??????
������−��−
??????
2
1.??????
��=??????
��=??????
�
2.Δ??????=??????
�−??????
�=±
(2�+1)??????
2
(ෝ??????
�)
??????
�
??????
�
෡??????
??????
Examples
Δ??????=−
??????
2
Electric Field Condition Polarization Sub-Type
Circular Counter Clockwise
??????�=ො�??????
������−��+ො�??????
������−��+
??????
2
Δ??????=+
??????
2
Circular Clockwise

8. Example of Circular Polarization (CW)
College of Engineering, Taibah University, MadinahEE451 Electromagnetics
(Circular CW polarized)??????�=ො�??????
������−��+ො�??????
������−��+
??????
2

9. Example of Circular Polarization (CCW)
College of Engineering, Taibah University, MadinahEE451 Electromagnetics
(Circular CCW polarized)??????�=ො�??????
������−��+ො�??????
������−��−
??????
2

10. Types of Polarization: 3. Elliptical Polarization
??????�=ො�??????
�������−��+??????
�+ො�??????
�������−��+??????
�
A transverse electromagnetic (TEM) wave is circularly polarized if the two components
of electric field have non-equal magnitude and a phase difference of (2n+1)p/2
College of Engineering, Taibah University, MadinahEE451 Electromagnetics
Assuming +z propagation of a plane wave
Circular polarization conditions:
??????�=ො�??????
�������−��+ො�??????
�������−��−
??????
2
1.??????
�≠??????
�
2.Δ??????=??????
�−??????
�=±
(2�+1)??????
2
(ෝ??????
�)
??????
�
??????
�
෡??????
??????
Examples
Δ??????=−
??????
2
Electric Field Condition Polarization Sub-Type
Elliptical Counter Clockwise
??????�=ො�??????
�������−��+ො�??????
�������−��+
??????
2
Δ??????=+
??????
2
Elliptical Clockwise

11. Example of Elliptical Polarization (CCW)
College of Engineering, Taibah University, MadinahEE451 Electromagnetics
(Elliptical CCW polarized)??????�=ො�??????
�������−��+ො�??????
�������−��+
??????
2

12. Comparing Different Polarizations
College of Engineering, Taibah University, MadinahEE451 Electromagnetics
elliptical CCW polarizedCircular CCW polarizedCircular CW polarized
Linear (diagonal) polarizedLinear (vertical polarizedLinear (horizontal polarized

13. How polarized waves are produced
College of Engineering, Taibah University, MadinahEE451 Electromagnetics
Vertical polarized (dipole based LPDA antenna)
Horizontal polarized (dipole based Yagi antenna)
Vertical polarized (dish fed with horn antenna)
Circular polarized (helical antenna)

14. Poynting Theorem
College of Engineering, Taibah University, MadinahEE451 Electromagnetics
∇×??????=−??????
�??????
��
and�×??????=????????????+??????
�??????
��
⇒??????.�×??????=−??????.??????
�??????
��
⇒??????.�×??????=????????????.??????+??????.??????
�??????
��
⇒??????.�×??????=−
??????
2
�??????
2
��
⇒??????.�×??????=????????????
2
+
??????
2
�??????
2
��
�.??????×??????=??????.�×??????)−??????.(�×??????
⇒−�.??????×??????=??????.�×??????−??????.(�×??????)
⇒�.??????×??????=−??????.�×??????+??????.(�×??????)
−????????????
2
−
??????
2
�??????
2
��
⇒�.??????×??????=
Lets dot the Faraday Law eq with H and Amp Law with E
Using Vector Identity
−
??????
2
�??????
2
��
Take the volume integral of both sides
න
??????
�.??????×??????�??????=−
�
��
න
??????
1
2
????????????
2
+
1
2
????????????
2
�??????−න
??????
????????????
2
�??????
⇒න
??????
(�.??????)�??????=
Apply divergence theorem
න
�
??????.��
⇒න
??????
�.??????×??????�??????=න
�
??????×??????.��
⇒න
�
??????×??????.��
=−
�
��
න
??????
1
2
????????????
2
+
1
2
????????????
2
�??????−න
??????
????????????
2
�??????
From Faraday and Ampere’s Laws:
Stored magnetic energy in H-field
Ohmic loss
Total power leaving the volume v
Poynting theorem provides the budget of the power which is flowing with the electromagnetic
wave. It is derived from Maxwell equations as follows:
=
1
2
????????????
2
Stored electric energy in E-field=
1
2
????????????
2
=න
??????
????????????
2
�??????
=න
�
??????×??????.��

15. Explanation of the Powers in Poynting theorem
College of Engineering, Taibah University, MadinahEE451 Electromagnetics
න
�
??????×??????.��=−
�
��
න
??????
1
2
????????????
2
+
1
2
????????????
2
�??????
Equation of Poynting Theorem
The law of conservation of energy can
be applied. The power leaving the
volume is equal to the decrease in the
electromagnetic energies and the
power dissipated.
Total power leaving the
volume v
Rate of decrease of the electrical
and mechanical energies
−න
??????
????????????
2
�??????
Power dissipated in ohmic
losses

16. Poynting Vector: Power Flow with the Wave propagation
The average power density is obtained by integrating it
over a time period. Equally:
The term on the left side of the Poynting Theorem equation is called the Poynting vector.
It also gives the direction of power flow
ሜ�=ሜ??????×ሜ??????
Total Power incident on an area
??????=න
??????
ሜ??????
????????????.????????????S nˆ k
ˆ E H HES = k
ˆ
ሜ�
????????????=
1
2
Reሜ??????×ሜ??????
∗
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dS

17. Calculation of The Poynting Vector (ሜ??????×??????) for a TEM Wave
⇒??????(�)=ො??????
�??????
??????��
−????????????�
�
−��
+ො??????
�??????
??????��
−????????????�
�
−��
??????=
1
�
෠�×??????
College of Engineering, Taibah University, MadinahEE451 Electromagnetics
⇒??????(�)=ො??????
�??????
���
−��
+ො??????
�??????
���
−���=�+�
??????=ሜ??????×??????
Assuming z directed propagation, the general electric field can have x and y components
??????=−
1
��??????
(�×??????)
For a TEM wave
⇒??????=
�
−????????????�
�
−��
�
ො??????
�×ො??????
�??????
??????�+ො??????
�??????
??????�
Here෠�=ො??????
�
1.
1
�
ො??????
�×ത??????=
1
�
ො??????
�ො??????
�ො??????
�
001
??????
�??????
�??????
�
⇒
1
�
ො??????
�×ത??????=
�
−??????��
�
−��
�
ො??????
�ො??????
�ො??????
�
001
??????
��??????
��1
⇒??????=
1
�
ො??????
�×??????
⇒
1
�
ො??????
�×ത??????=
�
−??????��
�
−��
�
−ො??????
�??????
??????�+ො??????
�??????
??????�
⇒??????(�)=
�
−��
�
−ො??????
�??????
??????�+ො??????
�??????
??????��
−??????��
⇒??????(�)=�
−��
ො??????
�??????
��+ො??????
�??????
���
−??????��
Poynting Vector
=
1
�
ො??????
�×[�
−????????????�
�
−��
ො??????
�??????
??????�+ො??????
�??????
??????�]⇒??????
Cross Product
⇒??????(�,�)=�
−��
ො??????
�??????
��+ො??????
�??????
��cos(��−��)
Magnetic field
Forlossymedia�=�∠�
�
=
�
−��
��
??????�??????
−ො??????
�??????
??????�+ො??????
�??????
??????��
−??????��
=
�
−��
�
−ො??????
�??????
??????�+ො??????
�??????
??????��
−??????(��−�??????)
In time domain⇒??????(�,�)=
�
−��
�
−ො??????
�??????
??????�+ො??????
�??????
??????����(��−��−�
�)
⇒??????(�)⇒??????(�)

18. Calculation of The Poynting Vector (ሜ??????×??????) for a TEM Wave
College of Engineering, Taibah University, MadinahEE451 Electromagnetics
??????=ሜ??????×??????
⇒??????×??????==
??????
−2????????????
�
ො??????
�??????
��
2
cos(��−��)���(��−��−�
�)+??????
��
2
cos(��−��)���(��−��−�
�)
Poynting Vector
??????(�,�)=�
−��
ො??????
�??????
��+ො??????
�??????
��cos(��−��)
??????(�,�)=
�
−��
�
−ො??????
�??????
??????�+ො??????
�??????
??????����(��−��−�
�)
From previous slide:
⇒??????×??????=
??????
−2????????????
�
ො??????
� ො??????
�ො??????
�
??????
��cos(��−��) ??????
��cos(��−��) 0
−??????
�����(��−��−�
�)??????
�����(��−��−�
�)0
⇒??????×??????=
??????
−2????????????
�
ො??????
�??????
��
2
+??????
��
2
cos(��−��)���(��−��−�
�) ??????
�
2
=??????
��
2
+??????
��
2
Assuming
⇒??????×??????=
�
−2��
�
ො??????
�??????
�
2
cos(��−��)���(��−��−�
�)
⇒??????(�,�)=??????×??????=ො??????
�
??????
�
2
2�
�
−2��
����
�+cos(2��−2��−�
�)
⇒�
????????????(�)=
1
�
න
�
??????(�,�)��
⇒�
????????????�=ො??????
�
??????
??????
2
2�
�
−2��
����
�

19. Calculation of Vector (ሜ??????×ሜ??????
∗
) for a TEM Wave
⇒??????(�)=ො??????
�??????
??????��
−????????????�
�
−��
+ො??????
�??????
??????��
−????????????�
�
−��
College of Engineering, Taibah University, MadinahEE451 Electromagnetics
�=�+�
ሜ??????×ሜ??????
∗
=�
−????????????�
�
−��
ො??????
�??????
??????�+ො??????
�??????
??????�×
�
+??????��
�
−��
�
∗
−ො??????
�??????
??????�
∗
+ො??????
�??????
??????�
∗
⇒??????
∗
=
�
+??????��
�
−��
�
∗
−ො??????
�??????
��
∗
+ො??????
�??????
��
∗
??????=
�
−??????��
�
−��
�
−ො??????
�??????
??????�+ො??????
�??????
??????�
??????(�)=�
−��
ො??????
�??????
��+ො??????
�??????
���
−??????��
ሜ??????×ሜ??????
∗
=
??????
−2????????????
�
∗
ො??????
�ො??????
�ො??????
�
??????
��??????
��0
−??????
��
∗
??????
��
∗
0
⇒ሜ??????×ሜ??????
∗
=
??????
−2????????????
�
∗
ො??????
�??????
��
∗
??????
??????�+??????
��
∗
??????
??????�
Cross Products
??????
��
∗
??????
??????�=??????
�??????
??????
??????
��
∗
??????
��=??????
��
2
⇒ሜ??????×ሜ??????
∗
=
??????
−2????????????
�
∗
ො??????
�??????
��
2
+??????
��
2
⇒ሜ??????×ሜ??????
∗
=
??????
−2????????????
�
∗
ො??????
�??????
??????�+ො??????
�??????
??????�×−ො??????
�??????
��
∗
+ො??????
�??????
��
∗
We have
??????
??????
2
=??????
??????�
2
+??????
??????�
2
Assuming
⇒ሜ??????×ሜ??????
∗
=
??????
−2????????????
�
∗
ො??????
�??????
??????
2
⇒ሜ??????×ሜ??????
∗
=
??????
−2????????????
�
∗
ො??????
�??????
??????
2

20. Calculation of Average Power Flow (S
av) for a TEM Wave
College of Engineering, Taibah University, MadinahEE451 Electromagnetics
ሜ�
????????????=
1
2
Reሜ??????×ሜ??????
∗
=?
From previous slide:??????(�)=�
−��
ො??????
�??????
��+ො??????
�??????
���
−??????��
??????=
??????
−????????????
�
−ො??????
�??????
??????�+ො??????
�??????
??????��
−??????��
⇒ሜ�
????????????=
1
2
Reሜ??????×ሜ??????
∗
Average power flow:
⇒ሜ�
????????????
=
�
−2��
2
ො??????
�??????
??????
2
��
1
�
∗
??????
�
2
Where �=
??????
??????
1−�
??????
�??????
−
1
2
�
∗
=
??????
??????
1+�
??????
�??????
−
1
2
�
∗
=
??????
??????
1+
??????
2
�
2
??????
2
1
2
�
????????????�
−1??????
????????????
−
1
2
⇒ሜ�
????????????=ො??????
�
??????
??????
2
2
�
−2��
��
1
�
∗
??????
�
2
�
∗
=
??????
??????
1+
??????
2
�
2
??????
2
−
1
4
�
??????
2
????????????�
−1??????
????????????
⇒�
∗
=�∠−�
�
⇒ሜ�
????????????=ො??????
�
??????
??????
2
2
�
−2��
��
1
��
−??????�
??????
??????
�
2
⇒ሜ�
????????????=ො??????
�
??????
??????
2
2�
�
−2��
������
�−�����
�
??????
�
2
⇒ሜ�
????????????=ො??????
�
??????
??????
2
2�
�
−2��
����
�
??????
�
2
Lossless media (s=0, a=0):⇒�=
??????
??????
⇒ሜ�
????????????=ො??????
�
??????
??????
2
2�
??????
�
2
??????
�
2
⇒ሜ??????×ሜ??????
∗
=
??????
−2????????????
�
∗
ො??????
�??????
??????
2

21. Reading from book
College of Engineering, Taibah University, MadinahEE451 Electromagnetics
✓10.7 WAVE POLARIZATION pages. 498-501
✓10.8 POWER AND THE POYNTING VECTOR pages 501-506
Sadiku, Elements of Electromagnetics 7
th
edition: Chapter 10