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Slides for the chapter 10 of Electromagnetics engieering 8th Edition.
Typology: Lecture notes
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Department of Electricaland Computer Engineering
ECSE 352 Electromagnetic Waves
Module 2: The uniform plane wave
Reference: H&B, Section 11.
1-
In this class we will investigate the way that electromagnetic wavesarise from Maxwell's equations. All waves require a source and wewill see that when electric and magnetic dipoles oscillate at highfrequencies, waves are radiated. We will then follow the derivationof the source-free wave equations from Maxwell's equations. Inmany cases of practical interest, the frequency of a wave-sourceremains constant and this leads to the concept of harmonic wavesthat have a single well-defined frequency. This assumption greatlysimplifies the mathematical treatment of EM waves and results in atime-independent formulation of the wave equations called thevector Helmholtz equation. We will investigate the concept ofphasor representation of EM waves and will see that simple cosinefunctions are solutions of this equation. Finally we will examine therelationship between the electric and the magnetic fields.
1-
-^ Review of Maxwell’s equations•^ Origins of EM waves•^ Derivation of the source-free wave equation•^ Transverse electromagnetic waves
1-
transmission line: Equivalent lumped element model:Charging speed goes as
dIV L dt
dVI C dt
H-field
E-field
(^1) v LC
V^0
RL
L C
L
L^ C
L C
++ V =V^0 -
V^0
RL
z
1-
1-
in time involves a change of
H^ in
space E.g. Dipole with
E = E a x
x^ t y and/or z
E^ x H^ z H^ y
ˆ x^ y^
z x^ y^
z x^ y^
z H^ H^
H ^ ^
^
^ ^
a^ a^
a H
1-
1-
Now using the E
and Hx^
polarizedy^
fields into eq. 2 lead to two equations analog totelegrapher’s equations
0^ y 0 x y^
t H y E z^
E a
a y 0 0 x y^
y y^
x x^
H x E z^
t H^
z^
a^
a a^
a
ˆ^ ˆ^
ˆ x^ y^
z x^ y^
z x^ y^
z E^ E^
E ^ ^
^
^ ^
a^ a^
a E
1-
z
on first equation, and derivativewith respect to
t^ on second, combine equations and concludeThis has a known solution
y 0 0 x y^
y y^
x x^
H x E z^
t H^
z^
a^
a a^
a
2
(^20 ) 2
2 x^
x E^
z^
t
^ ^
1
2 , x
z^
z
E^ z t^
f^ t^
f^ t v v ^ ^
^
^
^ ^
^ ^
^
1 0 0 0 v^
c ^
^
^
^
^
A plane wave with E and H orthogonal and enclosed within aplane orthogonal to propagation direction is a solution offree-space Maxwell’s equations
This is a TEM wave
1-
^ ^
(^010)
0
0
0 1 1 02 1 0
0
,^ cos^212
Re Re
x^
x
j^ t^ k zj x j^ t^ k z
j^ t^ k z
x^
x j^ t^
j^ t
xs^
xs
E^ z t^
t^ k z E^ e^
e^
cc E^ e^
cc^
E^ e
E e^
cc^
E e
^
'
0
0 1
0
0 2
,^
cos^
cos
x^
x^
x
E^ z t
E^
t^ k z^
E^
t^ k z
^
^
^
^
0 0 1 0 0
jk z xs^ x
j E^ E^ x^ x
e E^ E
^ e ^
Time-independent form^ E-field amplitude
1-
0 y^
x x^
x H^
E z^
t ^
^
a^
a^
0 0 1 j^ 0 0 t x^ xs
jk z xs^ x
j E^ E eE^ E^ e E^ E^ x^ x
e ^
^
^ 0^ t 0 t 0 0 ^
^ ^ ^
^ ^
^ ^
E H
H E E H
0 ys x^
xs^ x H^
j^ E z
^ a^ ^
a
(^0000) s^
s s^
s j^ j s s ^ ^ ^ H^ E E^ H E ^ H Time-independentMaxwell’s equations
1-
The H-field is provided from the E-field solutioninto and providesWhere the ratio of electric field to magnetic field is theintrinsic impedance
(^0000) s^
s s^
s j^ j s s ^ ^ ^ H^ E E^ H E ^ H
0 '^0 0 jk z^ jk z^0 xs^ x^
x E^ E^
e^ E ^ e ^
^0 s s^0 xsy^
j ys^ y dE^
j^ H dz
^ ^ E^
H a^
a 0
0 0 '^0 0
0 0
0 0
(^0) ' 0
0 0
0 1 jk z^ jk z^1 ys^ x^
x jk z jk z x^
x H^ E^
e^ E
e E^ e^ E^ e ^
^
^0377 ^ 0 ^0
ˆ^ ˆ^
ˆ x^ y^
z x^ y^
z x^ y^
z E^ E^
E ^ ^
^
^ ^
a^ a^
a E
1-
^ ^^ are always orthogonal following the RH rule
^
^ ^
^
0
0 0
0 0 ,^
cos ,^ x cos y z t^
E^
t^ k z E z t^
t^ k z ^
^
^
E^
a H^
a