Electromagnetic waves Unifrom Plane Slides, Lecture notes of Classical Mechanics

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Department of Electricaland Computer Engineering

ECSE 352 Electromagnetic Waves

Module 2: The uniform plane wave

2.1 EM Wave propagation in free

space

Reference: H&B, Section 11.

1-

Overview

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.

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Contents

-^ Review of Maxwell’s equations•^ Origins of EM waves•^ Derivation of the source-free wave equation•^ Transverse electromagnetic waves

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Rem: Transmission line propagation Lossless

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

• I

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Electromagnetic wave spectrum



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Free-space WE: InterpretationChange of^ E

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

  0^ t  0 t 0 0

 ^ 

 ^  

EH H E     E    H ˆ ˆ^

ˆ x^ y^

z x^ y^

z x^ y^

z H^ H^

H ^ ^

  ^ 

^ ^

 a^ a^

a H

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Plane wave

1-

  0^ t  0 t 0 0

 ^ 

 ^  

EH H E     E    H

Fields forming a plane wave

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^

   t

 ^  

^

H 

E a

a y 0 0 x y^

y y^

x x^

H x E z^

t H^

E

z^

   t

^

^

^

a^

a a^

a

ˆ^ ˆ^

ˆ x^ y^

z x^ y^

z x^ y^

z E^ E^

E ^ ^

  ^ 

^ ^

 a^ a^

a E

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Transverse electromagnetic wave - TEMTake derivative with respect to

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^

E

z^

   t

^

^

^

a^

a a^

a

2

(^20 ) 2

2 x^

x E^

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

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Complex solutions

Keeping in mind that there are 2 solutions, backwardand forward, we generally work with only the forwardsolution to simplify the analysis.As well, the complex notation is used^ ^

^

^

^   ^

^   

^

(^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^

E^

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

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Maxwell’s equations in cpx formThe time-independent form is most often used. Forexample with previous equationbecomesand more generallyvalid for any E, H fields (even beyond planar waves)

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

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…and magnetic field solution

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

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Transverse ElectroMagnetic (TEM) wave E^ and^ H

^ ^^ 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

E

y

z

H

x Right hand rule ExHst 1  End 2  H Thumb^ ^ Direction of propagation