electromagnetic waves -review- sandra cruz-pol, ph. d. ece uprm mayag ü ez, pr

Electromagnetic Electromagnetic waveswaves

--ReviewReview--Sandra Cruz-Pol, Ph. D.Sandra Cruz-Pol, Ph. D.

ECE UPRMECE UPRM

MayagMayagüüez, PRez, PR

Cruz-Pol, Electromagnetics Cruz-Pol, Electromagnetics UPRMUPRM


Electromagnetic SpectrumElectromagnetic Spectrum


Uniform plane em waveapproximation


Maxwell Equations Maxwell Equations in General Form in General Form

Differential formDifferential form Integral FormIntegral FormGaussGauss’’ss Law for Law for EE field.field.

GaussGauss’’ss Law for Law for HH field. Nonexistence field. Nonexistence of monopole of monopole

FaradayFaraday’’ss Law Law

AmpereAmpere’’ss Circuit Circuit LawLaw

vD

0 B

t

BE

t

DJH

v

v

s

dvdSD

0s

dSB

sL

dSBt

dlE

sL

dSt

DJdlH


Would magnetism would Would magnetism would produce electricity?produce electricity?

Eleven years later, Eleven years later, and at the same time, and at the same time, Mike Faraday Mike Faraday in in London and London and Joe Joe Henry Henry in New York in New York discovered that a discovered that a time-varying time-varying magnetic magnetic field would produce field would produce an electric current! an electric current!

dt

dNVemf

sL

dSBt

dlE


Electromagnetics was born!Electromagnetics was born!

This is the principle of This is the principle of motors, hydro-electric motors, hydro-electric generators and generators and transformers operation.transformers operation.

sL

dSt

DJdlH

*Mention some examples of em waves

This is what Oersted discovered This is what Oersted discovered accidentally:accidentally:


Special caseSpecial case Consider the case of a Consider the case of a lossless mediumlossless medium

with no charges, i.e. . with no charges, i.e. .

The wave equation can be derived from Maxwell The wave equation can be derived from Maxwell equations asequations as

What is the solution for this differential equation? What is the solution for this differential equation? The equation of a wave!The equation of a wave!

00v

022 EE c


Phasors for harmonic fieldsPhasors for harmonic fields

Working with Working with harmonic fieldsharmonic fields is easier, but is easier, but requires knowledge of requires knowledge of phasorphasor..

The phasor is multiplied by the time factor, The phasor is multiplied by the time factor, eejjtt, , and taken the real part.and taken the real part.

t

)cos(}Re{ trre j

)sin(}Im{ trre j


Advantages of phasorsAdvantages of phasors

TimeTime derivativederivative is equivalent to is equivalent to multiplying its phasor by multiplying its phasor by jj

TimeTime integralintegral is equivalent to dividing by is equivalent to dividing by the same term.the same term.

sAjt

A

jA

tA s


Time-Harmonic fields Time-Harmonic fields (sines and cosines)(sines and cosines)

The wave equation can be derived from Maxwell The wave equation can be derived from Maxwell equations, indicating that the changes in the equations, indicating that the changes in the fields behave as a wave, called an fields behave as a wave, called an electromagneticelectromagnetic field. field.

Since any periodic wave can be represented Since any periodic wave can be represented as as a suma sum of sines and cosines (using Fourier), then of sines and cosines (using Fourier), then we can deal only with harmonic fields to simplify we can deal only with harmonic fields to simplify the equations.the equations.


t

DJH

t

BE

0 B

vD

Maxwell Equations Maxwell Equations for for Harmonic fieldsHarmonic fields

Differential form* Differential form*

GaussGauss’’ss Law for E field. Law for E field.

GaussGauss’’ss Law for H field. Law for H field. No monopoleNo monopole

FaradayFaraday’’ss Law Law

AmpereAmpere’’ss Circuit Law Circuit Law

vE

0 H

HjE

EjJH

* (substituting and )ED BH


A waveA wave

Start taking the curl of FaradayStart taking the curl of Faraday’’s laws law

Then apply the vectorial identityThen apply the vectorial identity

And youAnd you’’re left withre left with

AAA 2)(

ss HjE

s

sss

E

EjjEE2

2

)()(


A WaveA Wave

022 EE

LetLet’’s look at a special case for simplicity s look at a special case for simplicity without loosing generality:without loosing generality:

•The electric field has only an The electric field has only an xx-component-component•The field travels in The field travels in zz direction directionThen we haveThen we have

zo

zo eEe EE(z)

tzE

'

issolution general whose

),(


To change back to time To change back to time domaindomain

From phasor From phasor

……to time domainto time domain

)()( jzo

zoxs eEeEzE

xzteEtzE zo

)cos(),(


Several Cases of MediaSeveral Cases of Media

1.1. Free space Free space

2.2. Lossless dielectricLossless dielectric

3.3. Low-lossLow-loss

4.4. Lossy dielectricLossy dielectric

5.5. Good ConductorGood Conductor

Permitivity: o=8.854 x 10-12[ F/m]

Permeability: o= 4 x 10-7 [H/m]


1. Free space1. Free space

There are no losses, e.g.There are no losses, e.g.

LetLet’’s defines define The phase of the waveThe phase of the wave The angular frequencyThe angular frequency Phase constantPhase constant The phase velocity of the waveThe phase velocity of the wave The period and wavelengthThe period and wavelength How does it moves?How does it moves?

xztAtzE

)sin(),(


3. Lossy Dielectrics3. Lossy Dielectrics(General Case)(General Case)

In general, we hadIn general, we had

From this we obtainFrom this we obtain

So , for a known material and frequency, we can find So , for a known material and frequency, we can find jj

11

2 and 11

2

22

)(2 jj

xzteEtzE zo

)cos(),(

j

222222

2222Re


SummarySummaryAny medium Lossless

medium (=0)

Low-loss medium

(”/’<.01)

Good conductor

(”/’>100)Units

0 [Np/m]

[rad/m]

[ohm]

uucc

up/f

[m/s]

[m]

**In free space; **In free space; oo =8.85 10 =8.85 10-12-12 F/m F/m oo=4=4 10 10-7-7 H/m H/m =120=120

j

j

f

u p

1

2

f

f

f

u p

1

f

u

f

p

4

)1( j

11

2

2


For lossless media,The wavenumber, k, is equal toThe phase constant. This is not so inside waveguides.

(Relative) Complex Permittivity


Intrinsic Impedance, Intrinsic Impedance, If we divide If we divide EE by by HH, we get units of ohms and , we get units of ohms and

the definition of the intrinsic impedance of a the definition of the intrinsic impedance of a medium at a given frequency.medium at a given frequency.

][

j

j

yzteE

tzH

xzteEtzE

zo

zo

ˆ)cos(),(

)cos(),(

*Not in-phase for a lossy medium


Note…Note…

EE and and HH are are perpendicularperpendicular to one another to one another TravelTravel is is perpendicularperpendicular to the direction of to the direction of

propagationpropagation The The amplitudeamplitude is related to the impedance is related to the impedance And so is the And so is the phasephase H lags EH lags E

yzteE

tzH

xzteEtzE

zo

zo

ˆ)cos(),(

)cos(),(


Loss TangentLoss Tangent

If we divide the conduction current by the If we divide the conduction current by the displacement current displacement current

tangentosstan lEj

E

J

J

s

s

ds

cs


Relation between Relation between tantan and and cc

EjjEjEH

1

Ej c

'

"tanas also defined becan tangent loss The


2. Lossless dielectric2. Lossless dielectric

Substituting in the general equations:Substituting in the general equations:

o

u

0

21

,0


Review: 1. Free SpaceReview: 1. Free Space


mAyztE

tzH

mVxztEtzE

o

o

o

/ˆ)cos(),(

/)cos(),(

) ,,0( oo

3771200

21

/,0

o

o

o

oo

cu

c


4. Good Conductors4. Good Conductors


]/[ˆ)45cos(),(

]/[)cos(),(

mAyzteE

tzH

mVxzteEtzE

oz

o

o

zo

) ,,( oro

o

u

45

22

2

Is water a good conductor???


Skin depth, Skin depth, Is defined as the Is defined as the

depth at which the depth at which the electric amplitude is electric amplitude is decreased to 37%decreased to 37%

/1at

%)37(37.01

1

zee

ez

[m] /1


Short Cut …Short Cut … You can use MaxwellYou can use Maxwell’’s or uses or use

where where kk is the direction of propagation of the wave, is the direction of propagation of the wave, i.e., the direction in which the EM wave is i.e., the direction in which the EM wave is traveling (a unitary vector).traveling (a unitary vector).

HkE

EkH

ˆ

ˆ1


Exercises: Wave Propagation in Exercises: Wave Propagation in Lossless materialsLossless materials

A wave in a nonmagnetic material is given byA wave in a nonmagnetic material is given by

Find:Find:

(a)(a) direction of wave propagation,direction of wave propagation,

(b)(b) wavelength in the materialwavelength in the material

(c)(c) phase velocityphase velocity

(d)(d) Relative permittivity of materialRelative permittivity of material

(e)(e) Electric field phasor Electric field phasor

Answer: +y, up= 2x108 m/s, 1.26m, 2.25,2.25,

[mA/m])510cos(50ˆ 9 ytzH

[V/m]57.12ˆ 5 yjexE


Power in a wavePower in a wave

A wave A wave carries powercarries power and and transmitstransmits it it wherever it goeswherever it goes

See Applet by Daniel Roth at

http://www.netzmedien.de/software/download/java/oszillator/

The power density per area carried by a wave is given by the Poynting vector.


Poynting Vector Poynting Vector Derivation…Derivation…

Which means that the total power coming out of Which means that the total power coming out of a volume is either due to the electric or a volume is either due to the electric or magnetic field energy variations or is lost as magnetic field energy variations or is lost as ohmic losses.ohmic losses.

dvEdvHEt

dSHEvvS

222

22

Total power across surface of volume

Rate of change of stored energy in E or H

Ohmic losses due to conduction current


Power: Poynting VectorPower: Poynting Vector

Waves carry Waves carry energyenergy and and informationinformation Poynting says that the net power flowing out of a Poynting says that the net power flowing out of a

given volume is = to the decrease in time in given volume is = to the decrease in time in energy stored minus the conduction losses.energy stored minus the conduction losses.

Represents the instantaneous power vector associated to the electromagnetic wave.


Time AverageTime Average Power Power

The Poynting vector The Poynting vector averaged in timeaveraged in time is is

For the general case wave:For the general case wave:

]/[ˆ

]/[ˆ

mAyeeE

H

mVxeeEE

zjzos

zjzos

For general lossy media


Total Power in WTotal Power in W

The The total powertotal power through a surface through a surface SS is is

Note that the units now are in Note that the units now are in WattsWatts

Note that the dot product indicates that the Note that the dot product indicates that the surface surface area needs to be area needs to be perpendicularperpendicular to the Poynting to the Poynting vector so that all the power will go thru. (give example vector so that all the power will go thru. (give example of receiver antenna)of receiver antenna)

][WdSPS

aveave P


Exercises: PowerExercises: Power

1. At microwave frequencies, the power density considered 1. At microwave frequencies, the power density considered safe for human exposure is 1 mW/cmsafe for human exposure is 1 mW/cm22. A radar radiates . A radar radiates a wave with an electric field amplitude E that decays a wave with an electric field amplitude E that decays with distance as with distance as E(R)E(R)=3000/R [V/m], where =3000/R [V/m], where RR is the is the distance in meters. What is the radius of the unsafe distance in meters. What is the radius of the unsafe region?region?

Answer: 34.6 mAnswer: 34.6 m

2. A 5GHz wave traveling in a nonmagnetic medium with 2. A 5GHz wave traveling in a nonmagnetic medium with rr=9 is characterized by =9 is characterized by

Determine the direction of wave travel and the average Determine the direction of wave travel and the average power density carried by the wavepower density carried by the wave

Answer: Answer:

[V/m])cos(2ˆ)cos(3ˆ xtzxtyE


TEM waveTEM wave

TTransverse ransverse EElectrolectroMMagnetic = plane waveagnetic = plane wave There are no fields parallel to the direction of

propagation, only perpendicular (transverse). If have an electric field Ex(z)

…then must have a corresponding magnetic field Hx(z)

The direction of propagation is

z

x

y

z

x

kHE aaa ˆˆˆ


x

x

y

y

z

Polarization of a wavePolarization of a wave

IEEE Definition: IEEE Definition: The trace of the tip of the The trace of the tip of the E-field vector as a E-field vector as a function of time seen from function of time seen from behindbehind..

Simple casesSimple cases Vertical, EVertical, Exx

Horizontal, Horizontal, EEyy

x

x

y

y

xztEzE

eEzE

ox

zjoxs

ˆ)cos()(

)(


PolarizationPolarization::

Why do we care?? Why do we care?? Antenna applications – Antenna applications –

Antenna can only TX or RX a polarization it is designed to support. Straight wires, square waveguides, and similar rectangular systems support linear waves (polarized in one direction, often) Circular waveguides, helical or flat spiral antennas produce circular or elliptical waves.

Remote Sensing and Radar Applications – Remote Sensing and Radar Applications – Many targets will reflect or absorb EM waves differently for different

polarizations. Using multiple polarizations can give different information and improve results. Rain attenuation effect.

Absorption applications – Absorption applications – Human body, for instance, will absorb waves with E oriented from

head to toe better than side-to-side, esp. in grounded cases. Also, the frequency at which maximum absorption occurs is different for these two polarizations. This has ramifications in safety guidelines and studies.


PolarizationPolarization In general, plane wave has 2 components; in In general, plane wave has 2 components; in xx & & yy

And y-component might be out of phase wrt to x-And y-component might be out of phase wrt to x-component, component, is the phase difference between x and y. is the phase difference between x and y.

Ey ExzE yx ˆˆ)(

zj

oy

zjox

e E E

e E E x

yEy

Ex

y

x

Front View


Several CasesSeveral Cases

Linear polarization: Linear polarization: yy--xx =0 =0oo or or ±±180180oonn

Circular polarization: Circular polarization: yy--xx ==±±9090oo & & EEoxox=E=Eoyoy

Elliptical polarization: Elliptical polarization: yy--xx==±±9090oo & & EEoxox≠≠EEoyoy, ,

or or ≠≠00oo or or ≠≠180180oon even if n even if EEoxox=E=Eoyoy

Unpolarized-Unpolarized- natural radiation natural radiation


Linear polarizationLinear polarization

=0=0

@z=0 in time domain@z=0 in time domain

zjoy

zjox

e E E

e E E

x

yEy

Ex

Front View

y

x

Back View:

t)cos(

t)cos(

yoy

xox

E E

E E


Circular polarizationCircular polarization Both components have Both components have

same amplitude Esame amplitude Eoxox=E=Eoy, oy,

== yy-- xx= -90= -90oo = Right = Right

circular polarized (RCP)circular polarized (RCP) =+90=+90oo = LCP = LCP

ˆˆˆˆ

:phasorin

)90tcos(

t)cos(

90

o

yjEExe EyExE

E E

E E

yoxoj

yoxox

yoy

xox

x

y


Elliptical polarizationElliptical polarization

X and Y components have different amplitudes X and Y components have different amplitudes EEoxox≠≠EEoy, oy, andand ==±±9090oo

Or Or ≠≠±±9090o o and Eand Eoxox==EEoy, oy,


Polarization examplePolarization example

Polarizing glasses

Unpolarizedradiation enters

Nothing comes out this time.

All light comes out

Polarization ParametersPolarization Parameters Ellipticy angle, Ellipticy angle,

Rotation angle,Rotation angle,

Axial ratioAxial ratio


Polarization StatesPolarization States



ExampleExample

Determine the polarization state of a plane wave Determine the polarization state of a plane wave with electric field:with electric field:

a.a.

b.b.

c.c.

d.d.

)45z-t4sin(y-)30z-tcos(3ˆ),( oo xtzE

)45z-t8sin(y)45z-tcos(3ˆ),( oo xtzE

)45z-t4sin(y-)45z-tcos(4ˆ),( oo xtzE

ys z-jxyE -j)eˆˆ(14)(

a. Elliptic

b. -90, RHEP

c. LP<135

d. -90, RHCP

)(cos)180(c

)sin()180sin(o

o

os)(s)90(c

)cos()90sin(o

o

inos


Cell phone & brainCell phone & brain

Computer model for Computer model for Cell phone Radiation Cell phone Radiation inside the Human inside the Human BrainBrain


Decibel ScaleDecibel Scale In many applications need comparison of two In many applications need comparison of two

powers, a powers, a power ratiopower ratio, e.g. reflected power, , e.g. reflected power, attenuated power, gain,… attenuated power, gain,…

The decibel (dB) scale is logarithmicThe decibel (dB) scale is logarithmic

Note that for voltages, the log is multiplied by 20 Note that for voltages, the log is multiplied by 20 instead of 10.instead of 10.

i

o

i

o

in

out

in

out

V

V

/RV

/RV

P

PdBG

P

P G

log20log10log10][ 2

2

Power RatiosPower RatiosG G [dB]

10x 10x dB

100 20 dB

4 6 dB

2 3 dB

1 0 dB

.5 -3 dB

.25 -6 dB

.1 -10 dB

.001 -30 dB


Attenuation rate, AAttenuation rate, A

Represents the rate of decrease of the magnitude Represents the rate of decrease of the magnitude of of PPaveave(z)(z) as a function of propagation distance as a function of propagation distance

]Np/m[68.8]dB/m[

where

[dB] -z8.68- log20

log100

log10

dB

dB

2

zez

e)(P

(z)PA z

ave

ave

Assigned problems ch 2 1-4,7-9, 11-13, 15-22, 28-30,32-34, 36-42

quizquiz

Based on wave attenuation and reflection Based on wave attenuation and reflection measurements conducted at 1MHz, it was measurements conducted at 1MHz, it was determined that the intrinsic impedance of a determined that the intrinsic impedance of a certain medium is 28.1 ⁄45certain medium is 28.1 ⁄45oo and the skin and the skin depth is 2m. Find: depth is 2m. Find:

1.1.the conductivity of the materialthe conductivity of the material

2.2.The wavelength in the mediumThe wavelength in the medium

3.3.And phase velocityAnd phase velocity



SummarySummaryAny medium Lossless

medium (=0)

Low-loss medium

(”/’<.01)

Good conductor

(”/’>100)Units

0 [Np/m]

[rad/m]

[ohm]

uucc

up/f

[m/s]

[m]

**In free space; **In free space; oo =8.85 10 =8.85 10-12-12 F/m F/m oo=4=4 10 10-7-7 H/m H/m

j

j

f

u p

1

2

f

f

f

u p

1

f

u

f

p

4

)1( j

11

2

2


Reflection and TransmissionReflection and Transmission

Wave incidenceWave incidence Wave arrives at an angle

SnellSnell’’s Law and Critical angles Law and Critical angle ParallelParallel or or PerpendicularPerpendicular BrewsterBrewster angle angle


EM Waves EM Waves

Normal , Normal , aann Plane of Plane of

incidenceincidence Angle of Angle of

incidenceincidence

zy

z=0

Medium 1 : 1 , 1 Medium 2 : 2, 2

i

rt

ki

kr

kt

kix

kiz


PropertyProperty Normal Normal IncidenceIncidence

Perpendicular Perpendicular ParallelParallel

Reflection Reflection coefficientcoefficient

Transmission Transmission coefficientcoefficient

RelationRelation

Power Power ReflectivityReflectivity

Power Power TransmissivityTransmissivity

SnellSnell’’s Law: s Law:

it

it

coscos

coscos

12

12||

ti

i

coscos

cos2

12

2

ti

ti

coscos

coscos

12

12

t

i

12

22

it

i

coscos

cos2

12

2||

1 1 t

i

cos

cos1 ||||

2

|||| R

RT 1 RT 1|||| 1 RT

2222

1 wheresinsin rrit nn

n


Critical angle, Critical angle, cc

…All is reflected …All is reflected

When When tt =90 =90oo, the refracted , the refracted

wave flows along the surface wave flows along the surface and and no energy is transmittedno energy is transmitted into medium 2. into medium 2.

The value of the angle of The value of the angle of incidence corresponding to incidence corresponding to this is called this is called critical anglecritical angle, , cc..

If If ii > > cc, the incident wave is , the incident wave is

totally reflectedtotally reflected..

)(for

]90[ sinsin

21

1

2

1

2

o

1

2

r

r

ttc

n

n

n

n


Brewster angle, Brewster angle, BB

Is defined as the incidence angle at which the Is defined as the incidence angle at which the reflection coefficient is 0 (reflection coefficient is 0 (totaltotal transmission). transmission).

The Brewster angle The Brewster angle does not existdoes not exist for for perpendicularperpendicular polarization for polarization for nonmagneticnonmagnetic materials. materials.

221

1221||

12

12

12||

)/(1

)/(1sin

0coscos

0coscos

coscos

B

Bt

Bt

Bt

* BB is

known as the polarizing angle

http://www.amanogawa.com/archive/Oblique/Oblique-2.html


Reflection vs. Incidence Reflection vs. Incidence angle.angle.

Reflection vs. incidence angle for different types of soil and parallel or perpendicular polarization.

Dielectric Slab:Dielectric Slab:2 layers2 layers

Medium 1: AirMedium 1: Air Medium 2: layer of thickness d, low-loss Medium 2: layer of thickness d, low-loss

(ice, oil, snow) (ice, oil, snow) Medium 3: LossyMedium 3: Lossy


Snell’s Law Phase matching condition at interphase:

Reflections at interfacesReflections at interfaces

At the At the toptop boundary, boundary, 1212,,

At the At the bottombottom boundary, boundary, 2323


For H polarization:

For V polarization:

Multi-reflection MethodMulti-reflection Method Propagation factor: Propagation factor:


1

2

3

d

Cont… for H PolarizationCont… for H Polarization


Substituting the geometric series:

And then Substituting and


Antennas Antennas

Now letNow let’’s review antenna theorys review antenna theory

electromagnetic waves -review- sandra cruz-pol, ph. d. ece uprm mayag ü ez, pr

Documents