2_Electromagnetic waves and propagation Lectures_part1_1.pdf

Wave

1. LECTURES ON
ELECTROMAGNETIC WAVES
AND WAVE PROPAGATION
Assoc. Prof. Yasser Mahmoud Madany
Senior Member, IEEE
URSI Senior Member
Founder and Chair of the IEEE Egypt AP-S/MTT-S Joint Chapter
Founder and Counselor of the IEEE AL Ryada Student Branch

2. PART ONE
Maxwell’s Equations and
Electromagnetic Waves

3. INTRODUCTION
❑Staticelectricfieldhasapplicationsin:
❖Cathode rayoscilloscopesfordeflecting
chargedparticles.
❖Inkjetprinterstoincreasethespeedofprinting
andimproveprintquality.
❑Staticmagneticfieldhasapplicationsin:
❖Magneticseparatorstoseparatemagnetic
materialsfromnon-magneticmaterials.
❖Cyclotronsforimpartinghighenergytocharged
particles.

4. INTRODUCTION
❑Time-varyingfieldsconstituteelectromagnetic(EM)waves
whichhavewideapplicationsinallthecommunications,
radarsandalsoinBio-medicalengineering.
❑EMwavesareproducedbytime-varyingcurrents.
❑Inbrief,itmaybenotedthat:
❖Staticchargesproduceelectrostaticfields.
❖Steadycurrents(DCcurrents)producemagneto-static
fields.
❖Staticmagneticcharges(magneticdipoles)also
producemagneto-staticfields.
❖Time-varyingcurrentsproduceEMwavesorEMfields.

5. THE FIELDS ഥ�,ഥ�,ഥ�AND ഥ??????IN STATIC FORM
Coordinate
/ Field
CartesianCylindricalSpherical
Cartesian
Time-
varying
Electric
Field,ഥ�
ഥ�(�,�,�) ഥ�(�,∅,�) ഥ�(�,�,∅) ഥ�(�,�,�,�)
Magnetic
Field, ഥ�
ഥ�(�,�,�) ഥ�(�,∅,�) ഥ�(�,�,∅) ഥ�(�,�,�,�)
Electric Flux
Density, ഥ�
ഥ�(�,�,�) ഥ�(�,∅,�) ഥ�(�,�,∅) ഥ�(�,�,�,�)
Magnetic
Flux Density,
ഥ??????
ഥ??????(�,�,�) ഥ??????(�,∅,�) ഥ??????(�,�,∅) ഥ??????(�,�,�,�)

6. EQUATION OF CONTINUITY FOR TIME-VARYING
FIELDS
❑Consideraclosedsurfaceenclosingacharge�and
thereexistsanoutwardflowofcurrentgivenby
�=ර
�
ҧ�.�ത�
❑Thepreviousequationknownasequationofcontinuityin
integralform,where
�isthecurrentflowingawaythroughaclosedsurface
(A).
ҧ�istheconductioncurrentdensity(A/m
2
).
�ത�isthedifferentialareaonthesurfacewhosedirection
isalwaysoutwardandnormaltothesurface.

7. ❑Asthereisoutwardflowofcurrent,therewillbea
decreasetherateofchargeofwhichisgivenby
�=
−��
��
where�istheenclosedcharge(C).
❑Fromtheprincipleofconservativeofcharge,wehave
�=ර
�
ҧ�.�ത�=
−��
��
❑Fromdivergencetheorem,wehave
ර
�
ҧ�.�ത�=න
�
??????.ҧ���

8. ❑Hence,
න
�
??????.ҧ���=
−��
��
❑Bydefinition,
�=න
�
�
���
where�
�isthevolumechargedensity(C/m
3
).
❑So,
න
�
??????.ҧ���=න
�
−��
�
��
��=න
�
−ሶ�
���
❑Twovolumeintegralsareequalonlyiftheirintegrandsare
equal.So,theequationofcontinuityinpointformis
??????∙ҧ�=
−��
�
��
=−ሶ�
�

9. MAXWELL’S EQUATIONS FOR TIME-VARYING
FIELDS
❑Maxwell’sequationsindifferentialformaregivenby
??????×ഥ�=
�ഥ�
��
+ҧ??????=
ሶഥ�+ҧ?????? (1)
??????×ഥ�=
−�ഥ??????
��
=−
ሶഥ?????? (2)
??????∙ഥ�=�
� (3)
??????∙ഥ??????=� (4)
where
ഥ�isthemagneticfieldstrength(A/m).
ഥ�istheelectricfluxdensity(C/m
2
).

10. MAXWELL’S EQUATIONS FOR TIME-VARYING
FIELDS
where:
ሶഥ�=
�ഥ�
��
isthedisplacementelectriccurrentdensity(A/m
2
).
ҧ�istheconductioncurrentdensity(A/m
2
).
ഥ�istheelectricfieldstrength(V/m).
ഥ??????isthemagneticfluxdensity(wb/m
2
)or(Tesla).
ሶഥ??????=
�ഥ??????
��
isthetime-derivativeofmagneticfluxdensity(wb/m
2
.sec).Also,
ሶഥ??????isthemagneticcurrentdensity(V/m
2
)or(Tesla/sec).
�
�isthevolumechargedensity(C/m
3
)

11. MAXWELL’S EQUATIONS FOR TIME-VARYING
FIELDS
❑Maxwell’sequationsinintegralformaregivenby
ׯ
??????
ഥ�.�ഥ??????=ׯ
�
ሶഥ�+ҧ??????.�ത�(1)
ׯ
??????
ഥ�.�ഥ??????=−ׯ
�
ሶഥ??????.�ത�(2)
ׯ
�
ഥ�.�ത�=׬
�
�
���(3)
ׯ
�
ഥ??????.�ത�=� (4)
where
�ഥ??????isthedifferentiallength.
�ത�isthedifferentialsurfaceareawhosedirectionisalways
outwardandnormaltothesurface.

12. CONVERSION OF DIFFERENTIAL FORM OF
MAXWELL’S EQUATIONS TO INTEGRAL FORM
❑ConsiderthefirstMaxwell’sequation
??????×ഥ�=
ሶഥ�+ҧ??????
❑Takesurface integral on both sides
ׯ
�
??????×ഥ�.�ത�=ׯ
�
ሶഥ�+ҧ??????.�ത�
❑ApplyingStokes’ theorem, we can write
ර
�
??????×ഥ�.�ത�=ර
??????
ഥ�.�ത??????
❑Therefore, the first low
ර
??????
ഥ�.�ത??????=ර
�
ሶഥ�+ҧ??????.�ത�

13. ❑ConsiderthesecondMaxwell’sequation
??????×ഥ�=−
ሶഥ??????
❑Takesurface integral on both sides
ׯ
�
??????×ഥ�.�ത�=−ׯ
�
ሶഥ??????.�ത�
❑ApplyingStokes’ theorem, we can write
ර
�
??????×ഥ�.�ത�=ර
??????
ഥ�.�ത??????
❑Therefore, the second low
ර
??????
ഥ�.�ത??????=−ර
�
ሶഥ??????.�ത�

14. ❑ConsiderthethirdMaxwell’sequation
??????∙ഥ�=�
�
❑Takevolume integral on both sides
න
�
??????∙ഥ���=න
�
�
���
❑Applyingdivergence theorem, we can write
න
�
??????∙ഥ���=ර
�
ഥ�.�ത�
❑Therefore, the third low
ර
�
ഥ�.�ത�=න
�
�
���

15. ❑ConsiderthefourthMaxwell’sequation
??????∙ഥ??????=�
❑Takevolume integral on both sides
න
�
??????∙ഥ??????��=�
❑Applyingdivergence theorem, we can write
න
�
??????∙ഥ??????��=ර
�
ഥ??????.�ത�
❑Therefore, the fourth low
ර
�
ഥ??????.�ത�=�

16. MAXWELL’S EQUATIONS FOR STATIC FIELDS
??????×ഥ�=ҧ??????⇔
ර
??????
ഥ�.�ത??????=ර
�
ҧ??????.�ത�
??????×ഥ�=�⇔
ර
??????
ഥ�.�ത??????=�
??????∙ഥ�=�
�⇔
ර
�
ഥ�.�ത�=න
�
�
���
??????∙ഥ??????=�⇔
ර
�
ഥ??????.�ത�=�

17. CHARACTERISTICS OF FREE SPACE
Parameter Symbol Value
Relative permittivity ??????
� 1
Relative permeability ??????
� 1
Conductivity ?????? 0
Conduction current density ҧ�0
Volumecharge density �
� 0
Intrinsic or characteristic
impedance
�����??????��????????????

18. MAXWELL’S EQUATIONS FOR FREE SPACE
??????×ഥ�=
ሶഥ�⇔
ර
??????
ഥ�.�ത??????=ර
�
ሶഥ�.�ത�
??????×ഥ�=−
ሶഥ??????⇔
ර
??????
ഥ�.�ത??????=−ර
�
ሶഥ??????.�ത�
??????∙ഥ�=�⇔
ර
�
ഥ�.�ത�=�
??????∙ഥ??????=�⇔
ර
�
ഥ??????.�ത�=�

19. MAXWELL’S EQUATIONS FOR STATIC FIELDS IN
FREE SPACE
??????×ഥ�=�⇔
ර
??????
ഥ�.�ത??????=�
??????×ഥ�=�⇔
ර
??????
ഥ�.�ത??????=�
??????∙ഥ�=�⇔
ර
�
ഥ�.�ത�=�
??????∙ഥ??????=�⇔
ර
�
ഥ??????.�ത�=�

20. PROOF OF MAXWELL’S EQUATIONS
❑First,fromAmpere’scircuitallaw,wehave
??????×ഥ�=ҧ??????
❑Takedot product on both sides
??????.??????×ഥ�=??????.ҧ??????
❑As the divergence of curlof a vector is zero,
??????.ҧ??????=�
❑But the equation of continuity in point form
??????∙ҧ�=
−��
�
��
=−ሶ�
�
❑This means that if ??????×ഥ�=ҧ??????istrue, we get ??????.ҧ??????=�.
❑Astheequationofcontinuityismorefundamental,Ampere’s
circuitallawshouldbemodified.

21. ❑ModifyingAmpere’scircuitallaw,wehave
??????×ഥ�=ҧ??????+ഥ�
❑Takedot product on both sides
??????.??????×ഥ�=??????.ҧ??????+??????.ഥ�
❑As the divergence of curlof a vector is zero,
??????.ҧ??????+??????.ഥ�=�
❑Substitutingthevalueof??????.ҧ??????fromtheequationof
continuityinthepreviousequation,
??????∙ഥ�+−ሶ�
�=�OR ??????∙ഥ�=ሶ�
�

22. ❑ButthepointformofGauss’slaw
??????.ഥ�=�
�OR??????.
ሶഥ�=ሶ�
�
❑Then,
??????.ഥ�=??????.
ሶഥ�
❑Thedivergenceoftwovectorsareequalonlyifthe
vectorsareidentical.
ഥ�=
ሶഥ�
❑Substitutingthevalueofഥ�,
??????×ഥ�=
ሶഥ�+ҧ??????

23. ❑Second,accordingtoFaraday’slaw
���=−
�∅
��
,∅∶����������������
❑By definition,
���=ර
??????
ഥ�.�ഥ??????
❑Then,
ර
??????
ഥ�.�ഥ??????=−
�∅
��
❑But,
∅=ׯ
�
ഥ??????.�ത�OR −
�∅
��
=−ׯ
�
�ഥ??????
��
.�ത�
❑So,
ර
??????
ഥ�.�ഥ??????=−ර
�
ሶഥ??????.�ത�

24. ❑ApplyingStokes’ theorem, we get
ර
�
??????×ഥ�.�ത�=ර
??????
ഥ�.�ഥ??????
❑Hence,
ර
�
??????×ഥ�.�ത�=−ර
�
ሶഥ??????.�ത�
❑Twosurfaceintegralsareequalonlyiftheirintegrandsare
equal,
??????×ഥ�=−
ሶഥ??????

25. ❑Third,fromGauss’slawinelectricfield
ׯ
�
ഥ�.�ത�=�=׬
�
�
���
❑Applyingdivergence theorem, we get
ර
�
ഥ�.�ത�=න
�
??????∙ഥ���=න
�
�
���
❑Twovolumeintegralsareequalonlyiftheirintegrandsare
equal,
??????∙ഥ�=�
�
❑Fourth,fromGauss’slawformagneticfield[themagneticflux
linesareaclosedloop]
ර
�
ഥ??????.�ത�=�
❑Applyingdivergence theorem, we get
׬
�
??????∙ഥ??????��=�OR ??????∙ഥ??????=�

26. SINUSOIDAL TIME-VARYING FIELDS
❑Inpractice,electricandmagneticfieldsvary
sinusoidally.
❑Anyperiodicvariationcanbedescribedintermsof
sinusoidalvariations.
❑Thefieldscanberepresentedby
෩ഥ�=�
�??????????????????��OR
෩ഥ�=�
������
where
�: Angular frequency, �=���
�: Frequency variation of the field.
�
�: The maximum field strength.

27. ❑Itisalsopossibletorepresentthefieldsusingthe
phasornotation.
❑Thetime-varyingfield
෩ഥ�(�,�)isrelatedtophasorfield
ഥ�(�)as
෩ഥ�(�,�)=��ഥ���
���
OR
෩ഥ�(�,�)=��ഥ���
���
where
�: Angular frequency.

28. MAXWELL’S EQUATIONS IN PHASOR FORM
❑ConsiderthefirstMaxwell’sequation
??????×
෩ഥ�=
෩ሶഥ�+ሚҧ??????
❑If
෩ഥ�=��ഥ��
���
෩ഥ�=��ഥ��
���
෨ҧ�=��ҧ��
���
Then
??????×��ഥ��
���
=
�
��
��ഥ��
���
+��ҧ��
���
❑Interchangingtheoperationoftakingtherealpart,weget
��??????×ഥ�−��ഥ�−ҧ��
���
=�
∴??????×ഥ�=��ഥ�+ҧ�

29. ❑ConsiderthesecondMaxwell’sequation
??????×
෩ഥ�=−
෩ሶഥ??????
❑If
෩ഥ�=��ഥ��
���
෩ഥ??????=��ഥ??????�
���
❑Then
??????×��ഥ��
���
=−
�
��
��ഥ??????�
���
❑Interchangingtheoperationoftakingtherealpart,we
get
��??????×ഥ�+��ഥ??????�
���
=�
∴??????×ഥ�=−��ഥ??????

30. ❑ConsiderthethirdMaxwell’sequation
??????∙
෩ഥ�=�
�
❑If
෩ഥ�=��ഥ��
���
❑Then
??????∙��ഥ��
���
=�
�
❑Interchangingtheoperationoftakingtherealpart,we
get
��??????∙ഥ��
���
=�
�
∴??????∙ഥ�=�
�

31. ❑ConsiderthefourthMaxwell’sequation
??????∙
෩ഥ??????=�
❑If
෩ഥ??????=��ഥ??????�
���
❑Then
??????∙��ഥ??????�
���
=�
❑Interchangingtheoperationoftakingtherealpart,we
get
��??????∙ഥ??????�
���
=�
∴??????∙ഥ??????=�

32. ❑Insummary,Maxwell’sequationsinphasorformare
asfollows:
??????×ഥ�=��ഥ�+ҧ�
??????×ഥ�=−��ഥ??????
??????∙ഥ�=�
�
??????∙ഥ??????=�

33. The Part (1) will be
continued …
Thanks