Uniform Plane Waves (1)

8/11/2019 Uniform Plane Waves (1)

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EMG2016

ELECTROMAGNETIC THEORY

UNIFORM PLANE WAVES


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Learning Outcomes


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Outline Plane Wave Propagation

Time Harmonic Fields

Wave Equations Plane wave propagation in lossless media

Polarization of Waves

Plane wave propagation in lossy media

Electromagnetic Power density


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Plane wave propagation

a time-varying electric field E(t) produces a magnetic fieldH(t)and, conversely, a timevarying magnetic field

produces an electric field.

This cyclic pattern generates electromagnetic (EM) waves

capable of propagating through free space and in material

media.

When its propagation is guided by a material structure, the

EM wave is said to be traveling in a guided medium (e.g.

a transmission line).

EM waves also can travel in unbounded media. (e.g. lightwaves emitted by the sun and radio transmissions by

antennas).


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For a transmission-line circuit, we can model wave

propagation on such a transmission line either in terms ofthe voltages across the line and the currents through its

conductors or in terms of the electric and magnetic fields in

the dielectric medium between the conductors (bounded

case).

In this chapter we focus our attention on wave propagation

in unbounded media that can be categorized into two

classes lossless & lossy

The medium are assumed to be homogeneous &

isotropic


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When energy isemitted by a

source, such as an

antenna, it

expands outwardly

from the source inthe form of

spher ical waves.

Plane Wave


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To an observer very far way from the source, the

wavefront of the spherical wave appears

approximately planar, as if it were part of auni form planewave with uniform properties at all

points in the plane tangent to the wavefront

To analyse the unbounded EM wave, Maxwellsequations are used


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MAXWELL EQUATIONS. D = v

E= -B/t

. B= 0

H = J+ D/t

where E= electric field intensity D= electric flux density

H= magnetic field intensity B= magnetic flux density

v= electric charge volume density

J= conduction current density


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Time-Harmonic Fields D, E, B, H, Jand v depend on spatial coordinates (x,y,z)

and the time variable, t.

If their time variation is sinusoidalwith angular frequency, then these quantities can be represented by a phasor

that depends on (x,y,z) only.

Why phasor form?


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For a linear, isotropic, and homogeneous medium, the

MaxwellsEquations in phasor form is given as:

For time-harmonic quantities, differentiation in timedomain corresponds to multiplication by j in phasor

domain.

/v E

j E H

0 H

j H J E

Converted using

( , , , ) ( , , ) j tx y z t e x y z e E E


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j H J E

j j -j

H E E

By introducing the complex permittivity c which is

defined as

c= (-j/)

c= (-j/) = -j

with= , and = /

For a lossless medium (= 0), it follows that = 0 and c

= = .


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0 E

-j E H0 H

cj H E


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HjE ~~

EEjjE cc~~~ 2

EEE ~~~ 2

To the derive the wave equation from Maxwell

equation, we do a curl operation on both sides of the

second and fourth Maxwellsequations

Wave Equations


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2 2- 0 E E

2 2- 0 H H


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THE WAVE EQUATIONS FORLOSSLESS MEDIUM

0~~ 22 EE

0

~~ 22 HH

Homogeneous Vector Helmhotzs equations

0~

E~ 22 Ek

kSince

therefore,


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a plane wave has no electric-or magnetic-field components along its

direction of propagation

Wave equation:

For the phasor quantity x , the general solution of the ordinary

differential equation of the wave equation is

0~

E~

z zH

jkz

xo

jkz

xoxxx eEeEzEzEE )(

~)(

~~

Uniform Plane Wave

A uniform plane wave is characterized by electric and magnetic

fields that have uniform properties at all points across an infinite

plane and if this is the x-y plane so Eand Hdo not vary with x

andy

0~

~2

2

2

xx Ek

dz

Ed(x component ofE)


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Assume thatEhas only a component inxand that xconsists of a

wave travelling in the +z direction

( ) ( ) jkzx xoz E z E e E x x

0 0

x y z

x

j H H Hx y z

E

x y z

E x y z

0

~1~

~1~

0~

y

E

j

H

z

E

jH

H

xz

xy

x

jkz

yo

jkz

xoy eHeEkzH ~~)(~

yo xo

kH E


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I ntr insic impedance,h,of a lossless medium is defined as

h

k

We can summarize our results as

The electric and magnetic fields are perpendicular to each other, and

both are perpendicular to the direction of wave travel. These

directional properties characterize a transverse electromagnetic

(TEM) wave

jkz

xoeExz )(E

~x(z)E

~x

jkzxo eEyz hh //)(E~

y(z)H~

x


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TEM Wave [1]

[1] http://www.inchem.org/documents/ehc/ehc/ehc137.htm

E(z,t)andH(z,t) are in-phase

Phase velocity

wavelength

1pu

k

2 pu

k f
http://www.inchem.org/documents/ehc/ehc/ehc137.htmhttp://www.inchem.org/documents/ehc/ehc/ehc137.htmhttp://www.inchem.org/documents/ehc/ehc/ehc137.htmhttp://www.inchem.org/documents/ehc/ehc/ehc137.htmhttp://www.inchem.org/documents/ehc/ehc/ehc137.htmhttp://www.inchem.org/documents/ehc/ehc/ehc137.htmhttp://www.inchem.org/documents/ehc/ehc/ehc137.htm


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In lossless dielectric:

thus

rr 00 ,,0 then

0, k

1u 0

h


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Exercise 1


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General Relation between E and H

It can be shown that, for any uniform plane wave traveling

in an arbitrary direction denoted by the unit vector , themagnetic field phasor is interrelated to the electric field

phasor by

Right hand rule applies: when we rotate the four fingers of

the right hand from the direction of Etoward that of H, the

thumb points in the direction of wave travel,

1/h H k E

-h E k H

(7.39a)

k

k

E

Hk


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The wave may be considered the sum of two waves, one with (E+x,H+

y)

components and another with (E+y, H+

x) components.

In general, a TEM wave may have an electric field in any direction in the

plane orthogonal to the direction of wave travel, and the associated magnetic

field is also in the same plane and its direction is dictated by Eq.(7.39a,

Ulaby).


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Exercise 2


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POLARIZATION OF PLANE WAVES

EEy

Ex

cos cosx ya t kz a t kz E x y

The polarization of a uniform plane wave describes the locus traced by

the tip of the E vector (in the plane orthogonal to the direction ofpropagation) at a given point in space as a function of time


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SENSE OF POLARIZATION

Linear circularelliptical elliptical


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Answer:


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To examine wave propagation in a conducting medium we

return to the wave equation

with2= -2c= -

2(- j)

where = and = /. Since is complex, we express it

as

= +j,

where is the attenuation constantof the medium and is

its phase constant.

2 2E 0E

Plane Wave Propagation inLossy media (0)


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By replacing with (+j), we have

(+j)2= (2- 2) +j2= -2+j2.

Hence,

2- 2= -2,

2= 2

.Solving these two equations for and gives

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1

'

"1

2

'

2/12

1

'

"1

2

'

or2/1

2

112

2/12

112


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zAssuming that the wave propagates along direction

and E

~

has only an x-component

x ( )E E zx

0~)-( 22 ESubstitute into

yields solution ' z0 0( )

z

x x xE z E e E e

Inserting the time factor


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( , ) Re[ ( ) ]j t

xz t E z e

E x ( )0

Re[ ]z j t zE e e

x

0( , ) cos( )zz t E e t z E x

For magnetic field,

( )

0( , ) Re[ ]z j t zz t H e e H y

where 00 c

EHh

y

nj| | | |ec c n cjj

h h h

and

intrinsic impedance


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thus0

( , ) cos( )

| |

z

c

Ez t e t z

h

h

H y

hc= intrinsic impedance for lossy medium= attenuation constant(factor)

= phase constant

h = phase angle of the intrinsic impedance for lossy medium

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1

/||

hc

h2tan

E(z,t) & H(z,t) not in-phase

Magnitude ofEx(z) decrease exponentially.


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For a lossy medium, the ratio /= /appears in

all these expressions and plays an important role in

determining how lossy a medium is.

When /1, the medium is characterized as a

good conductor. (/>102)


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Low-Loss Dielectric

The general expression for is given by

2'2

"

'

h c


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thus

then

In good conductors:

r 00 ,,

f2

2p

u

45c

h

1 (1 )cj f

j j

h


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0

( , ) cos( )zz t E e t z E x

thus

0( , ) cos( 45 )zE

z t e t z

H y


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Electromagnetic Power Density

The Poynting vector S is defined as

represents the power density (power per unit area) carried

by the wave. Its direction is along the propagation direction of

the wave

The instantaneous Poynting vector or power density vector is

given by

na

HE

P

P

tjtj etzHetzEetzHtzEtz

),(~

),(~

),(),(),(

P


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The total power that flows through or is intercepted by

the aperture is

(W)

Sd.

SaveP aveP


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In practice, the quantity of greater interest is the average

power density of the wave, Pave, which is the time-averaged value of

where and are in phasor formE~ H~

]~~

Re[)2/1( *ave HEP

aveP


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A plane wave is propagating in the +z direction in a lossy

medium

For a lossless medium

The attenuation raterepresents the rate of decrease of the

magnitude of Pave (z) as a function of propagation distance

A= 10 log10[Pave(z)/Pave(0) ] = -8.68 z (dB)

)cos(2

2

2

ave nzo e

Ez

hP

hh 2

2

22

ave

Ez

Ez

c

o P


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Decibel Scale for power ratios

G = P1 / P2

G(dB) = 10 log(G) = 10 log(P1/P2) (dB)

For voltage or current ratio, it can be written as

G(dB) = 20 log(V1/V2) (dB)


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Uniform Plane Waves (1)