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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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Thank you for your

attention