electromagnetic waves theory - hanyang
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Electromagnetic Waves
Theory
Advanced Engineering Electromagnetics
by Constantine A. Balanis
Chapter 3.1 – 3.3
Changgon Han
2020.03.30
Antennas & RF Devices Lab.
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Contents
3. Wave Equation and its solutions
3.1 Introduction
3.2 Time-Varying Electromagnetic Fields
3.3 Time-Harmonic Electromagnetic Fields
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The EM fields of boundary-value problems are obtained as solution of
Maxwell’s Equations, which are first-order partial differential equations.
However, Maxwell’s equations are coupled partial differential equations;
each has both E- and H-fields. These equations can be uncoupled at the
Expense of raising the order to second order.
The new set of uncoupled second-order PDEs are known as the Wave Equation
for either the electric or magnetic fields, or any other fields or current densities.
Therefore an EM boundary-value problem can solved using either
Maxwell’s Equations or the Wave Equation.
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(3-1)
(3-2)en.wikipedia.org
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Figure 1-4 Geometry for boundary conditions of tangential
components.
(1-25)
0y
(1-26)
0y
(1-27)
(1-25a)
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Figure 1-4 Geometry for boundary conditions of normal
components.
(1-28)
0y
(1-29)
(1-30)
(1-31)
(1-32)
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Figure 1-4 Geometry for boundary conditions of tangential
components.
(1-33)
0y
(1-33a)
(1-33b)
(1-33c)
(1-33d)
(1-35)
(1-39)
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(1-39)
(1-43)
(1-47)
(1-40)
(1-26)
Figure 1-4 Geometry for boundary conditions of normal
components.
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(3-1)
(3-2)
(3 1) :
(3-3)
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(3-5)2( )F F F
(3-6)
(3-7)
22
2 2
1( )
iev i
t t t
J E Eq E = M
(3-8)
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(3 2) :
(3-4)
(3-1)
(3-2)
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2( )F F F
(3-9)
(3-10)
(3-11)
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(3-8)
(3-11)
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0, 0i ev i mv J q M q
22
2 2t t
E EE =
22
2 20
t t
E EE (3-12)
22
2 2t t
H HH =
22
2 20
t t
H HH (3-13)
0, 0i ev i mv J q M q
22
2 2t
EE =
22
2 2t
HH =
22
2 20
t
EE
22
2 20
t
HH
(3-14)
(3-15)
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2 2 22
2 2 2
f f ff
x y z
𝜕𝑓
𝜕𝑥=𝜕𝑓
𝜕𝑦= 0
2 2 22
2 2 2 2 2
1 10
ff f
t z t
22
2 20
t
EE
22
2 20
t
HH
(3-14)
(3-15)
Assume the solution 𝑓 𝑥, 𝑦, 𝑧 = 𝑓(𝑧 ± 𝑣𝑡)
t
z
0l z vt
l z vt
0t
l z vt
(a)
(b)
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BE
t
0B (1) (2)
B A
( ) 0A
Et
AE
t
(3)
E
A
t
(4) (5)
22 2
2E A
t t
22
2t
(6)
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, (divergence nullity)B A (3)A
Et
EB J
t
(7) 2( ) ( )B A A A 2
2
E AJ J
t t t
At
(8)
22
2A J
t
(9)
vector identity
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For time-harmonic fields, the wave equations can be derived
using similar procedure as in Section 3.2.
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iE M j H
iH J E j E
( )iE M j H
Taking curl left side (∇ ×)
Using Maxwell’s equation from above and vector identity 2( )F F F
2( ) [ ]i iE E M j J E j E
2 ]i iM j J j E E
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2( ) [ ]i iE E M j J E j E
2 ]i iM j J j E E
( ) evev
qD E E q E
We can write that
2 21i i evE M j J q j E E
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iE M j H
iH J E j E
( )iH J E j E
Taking curl left side (∇ ×)
Using Maxwell’s equation from above and vector identity 2( )F F F
2( ) ( )( )i iH H J j M j H
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2( ) ( )( )i iH H J j M j H
( ) mvmv
qB H H q H
2 21i i mv iH J M q j M j H H
We can write that
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However, instead of going through this process by comparing
Maxwell’s equations for the general time-varying fields with
those for time-harmonic fields, that one set can be obtained
from the other replacing 𝝏
𝝏𝒕≡ 𝒋𝝎,
𝝏𝟐
𝝏𝟐𝒕𝟐≡ 𝒋𝝎 𝟐 = −𝝎𝟐, and
the instantaneous fields respectively, with the
complex field (E, H, D, B) and vice versa.
(E,H,D,B),
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22
2 2Re( ); Re( ); , ( )j t j tEe He j j
t t
E H
(3-16a)
(3-16b)
(3-8)
(3-11)
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22
2 2Re( ); Re( ); , ( )j t j tEe He j j
t t
E H
(3-17a)
(3-17b)
(3-17c)
(3-12)
(3-13)
22
2 2t t
E EE =
22
2 2t t
H HH =
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(3-17d)
(3-17e)
(3-17f)
(3-18a)
(3-18b)
(3-18c)
(3-17c)
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22
fk
1f
en.wikipedia.org
f = frequency = 𝟏
𝑻
T = Cycle
7 2 12 2 2 31/[4 10 / ][8.854 10 / ]v mkg C C s kg m
8 72.9979 10 10 /m s
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2 2 0E E
2 2 0H H
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