electromagnetism inel 4152 sandra cruz-pol, ph. d. ece uprm mayag ü ez, pr

ElectromagnetismElectromagnetismINEL 4152INEL 4152

Sandra Cruz-Pol, Ph. D.Sandra Cruz-Pol, Ph. D.ECE UPRMECE UPRM

MayagMayagüüez, PRez, PR

Cruz-Pol, Electromagnetics Cruz-Pol, Electromagnetics UPRMUPRM

Electricity => MagnetismElectricity => MagnetismElectricity => MagnetismElectricity => Magnetism

In 1820 Oersted discovered that a steady In 1820 Oersted discovered that a steady current produces a magnetic field while current produces a magnetic field while teaching a physics class. teaching a physics class.


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 in Mike Faraday in London and Joe London and Joe Henry in New York Henry 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:


Electromagnetic SpectrumElectromagnetic Spectrum


Some termsSome terms

E = electric field intensity [V/m]E = electric field intensity [V/m] D = electric field densityD = electric field density H = magnetic field intensity, [A/m]H = magnetic field intensity, [A/m] B = magnetic field density, [Teslas]B = magnetic field density, [Teslas]


Maxwell Equations Maxwell Equations in General Form in General Form

Differential formDifferential form Integral FormIntegral FormGauss’sGauss’s Law for Law for EE field.field.

Gauss’sGauss’s Law for Law for HH field. Nonexistence field. Nonexistence of monopole of monopole

Faraday’sFaraday’s Law Law

Ampere’sAmpere’s Circuit Circuit LawLaw

vD

0 B

t

BE

t

DJH

v

v

s

dvdSD

0s

dSB

sL

dSBt

dlE

sL

dSt

DJdlH


Maxwell’s Eqs.Maxwell’s Eqs.

Also the equation of continuity Also the equation of continuity Maxwell addedMaxwell added the term to Ampere’s the term to Ampere’s

Law so that it not only works for Law so that it not only works for staticstatic conditions but also for conditions but also for time-varyingtime-varying situations. situations.

This added term is called the This added term is called the displacement displacement current densitycurrent density, while , while JJ is the conduction is the conduction current.current.

tJ v

t

D


Maxwell put them togetherMaxwell put them together And added And added JJdd, the , the

displacement currentdisplacement current

IIdSJdlH enc

SL

1

02

SL

dSJdlHI

S2

S1

L

Idt

dQdSD

dt

ddSJdlH

SS

d

L

22

At low frequencies J>>Jd, but at radio frequencies both terms are comparable in magnitude.


Moving loop in static fieldMoving loop in static field

When a conducting loop is moving inside a When a conducting loop is moving inside a magnet (static magnet (static BB field, in Teslas), the force on a field, in Teslas), the force on a charge ischarge is

BlIF

BuQF

Encarta®


Who was NikolaTesla?Who was NikolaTesla?

Find out what inventions he madeFind out what inventions he made His relation to Thomas EdisonHis relation to Thomas Edison Why is he not well know?Why is he not well know?


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 & complex #’sPhasors & complex #’s

Working with Working with harmonic fieldsharmonic fields is easier, but is easier, but requires knowledge of requires knowledge of phasorphasor, let’s review , let’s review

complex numberscomplex numbers and and phasorsphasors


COMPLEX NUMBERS:COMPLEX NUMBERS:

Given a complex number Given a complex number zz

wherewhere

sincos jrrrrejyxz j

magnitude theis || 22 yxzr

angle theis tan 1

x

y


Review:Review:

Addition, Addition, Subtraction, Subtraction, Multiplication, Multiplication, Division, Division, Square Root, Square Root, Complex ConjugateComplex Conjugate


For a time varying phaseFor a time varying phase

Real and imaginary parts are:Real and imaginary parts are:

t

)cos(}Re{ trre j

)sin(}Im{ trre j


PHASORSPHASORS

For a sinusoidal current For a sinusoidal current

equals the real part of equals the real part of

The complex term which results from The complex term which results from dropping the time factor is called the dropping the time factor is called the phasor current, denoted by (s comes phasor current, denoted by (s comes from sinusoidal) from sinusoidal)

)cos()( tItI otjj

o eeI

joeI

tje

sI


To To changechange back to back to time time domaindomain

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.

}Re{ tjseAA


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

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


t

DJH

t

BE

0 B

vD

Maxwell Equations Maxwell Equations for for Harmonic fieldsHarmonic fields

Differential form* Differential form*

Gauss’sGauss’s Law for E field. Law for E field.

Gauss’sGauss’s Law for H field. Law for H field. No monopoleNo monopole

Faraday’sFaraday’s Law Law

Ampere’sAmpere’s Circuit Law Circuit Law

vE

0 H

HjE

EjJH

* (substituting and )ED BH


A waveA wave

Start taking the curl of Faraday’s lawStart taking the curl of Faraday’s law

Then apply the vectorial identityThen apply the vectorial identity

And you’re left withAnd you’re left with

AAA 2)(

ss HjE

s

sss

E

EjjEE2

2

)()(


A WaveA Wave

022 EE

Let’s look at a special case for simplicity Let’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(),(

Ejemplo 9.23Ejemplo 9.23

In free space, In free space,

Find Find k, Jk, Jdd and and H using phasors and H using phasors and

maxwells eqs.maxwells eqs.


mVkztE /)10cos(50 8


Several Cases of MediaSeveral Cases of Media

1.1. Free space Free space

2.2. Lossless dielectricLossless dielectric

3.3. Lossy dielectricLossy dielectric

4.4. Good ConductorGood Conductor )or ,,(

),,0(

)or ,,0(

),,0(

oro

oror

oror

oo

Recall: Permittivity

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

Permeabilityo= 4 x 10-7 [H/m]


1. Free space1. Free space

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

Let’s defineLet’s 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

112

2

)(2 jj

xzteEtzE zo

)cos(),(

j

2

2Re

22222

222

11

2

2


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

H

E

yzteE

tzH

xzteEtzE

zo

zo

ˆ)cos(),(

)cos(),(

*Not in-phase for a lossy medium

HFind

xeEEgiven zo


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

yzteE

tzH

xzteEtzE

zo

zo

ˆ)cos(),(

)cos(),(

yeeE

zH

xeeEzE

zjzo

zjzo

ˆ)(

)(

)(

)(


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

http://fipsgold.physik.uni-kl.de/software/java/polarisation


Relation between Relation between tantan and and cc

EjjEjEH

1

Ej c

'''1

isty permittivicomplex The

jjc

'

"tanas also defined becan tangent loss The


2. Lossless dielectric2. Lossless dielectric

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

)or ,,0( oror

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???


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; o =8.85 x 10-12 F/m o=4 x 10-7 H/m

11

2

2

j

j

f

u p

1

2

f

f

f

u p

1

f

u

f

p

4

)1( j

11

2

2

rr

c


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

We know that a wave attenuates in a lossy medium until it vanishes, but how deep does it go?

]/[)cos(),( mVxzteEtzE zo


Short Cut …Short Cut … You can use Maxwell’s or useYou can use Maxwell’s 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


WavesWaves

Static charges > Static charges > static electric field, static electric field, EE

Steady current > Steady current > static magnetic field, static magnetic field, HH

Static magnet > Static magnet > static magnetic field, static magnetic field, HH

Time-varying current > Time-varying current > time varying time varying E(t)E(t) & & H(t)H(t) that are that are interdependent > interdependent > electromagnetic waveelectromagnetic wave

Time-varying magnet > Time-varying magnet > time varying time varying E(t)E(t) & & H(t)H(t) that are that are interdependent > interdependent > electromagnetic waveelectromagnetic wave


EM waves don’t need a EM waves don’t need a medium to propagatemedium to propagate

Sound waves need a Sound waves need a medium like air or water medium like air or water to propagateto propagate

EM wave don’t. They can EM wave don’t. They can travel in free space in the travel in free space in the complete absence of complete absence of matter.matter.

Look at a “wind wave”; Look at a “wind wave”; the energy moves, the the energy moves, the plants stay at the same plants stay at the same place. place.


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://fipsgold.physik.uni-kl.de/software/java/oszillator/index.html

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


Poynting Vector DerivationPoynting Vector Derivation

Start with Start with EE dot Ampere’s dot Ampere’s

Apply Apply vector identityvector identity

And end up with:And end up with:

EHHEEH

BAABBA

:case in thisor

t

EEEEHE

t

EEHE

t

EEEHEH

2

2

2

1


Poynting Vector Poynting Vector Derivation…Derivation…

Substitute Faraday in 1Substitute Faraday in 1rst rst termterm

t

EEEH

t

HH

2

2

2

1

t

HH

t

HH

2

:function square of derivativein As

t

EEHE

t

H

2

22

22

HEEH

(-) sit' order, invert the if and

222

22E

t

H

t

EHE

Rearrange


Poynting Vector Poynting Vector Derivation…Derivation…

Taking the integral Taking the integral wrtwrt volume volume

Applying Theorem of Divergence Applying Theorem of Divergence

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

dvEdvHEt

dvHEvvv

222

22

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 Poynting says that the net power flowing out net power flowing out of a of a

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

][W/m 2HE

PRepresents the instantaneous power density 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:

*00

Re2

111ss

TT

ave HEtdHET

tdT

PP

]/[ˆ

]/[ˆ

mAyeeE

H

mVxeeEE

zjzos

zjzos

][W/m ˆcos

222

2

zeE zo

ave

P


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 power nomenclature, Note that power nomenclature, PP is not cursive. is not cursive. 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 |E(R)|=3000/R [V/m], where with distance as |E(R)|=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: 34.64 mAnswer: 34.64 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: ][W/m 05.0ˆˆcos2

22

1

2

xaeE

ko

ave

P

[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 aE x aH = ak

z

x

y

z

x


PE 10.7PE 10.7

In free space, In free space, H=0.2 cos (H=0.2 cos (t-t-x) x) zz A/m. Find A/m. Find the total power passing through a the total power passing through a

square plate of side 10cm on square plate of side 10cm on plane plane x+zx+z=1 =1

square plate at z=3square plate at z=3

Answer; Ptot = 53mWHz

Ey

x

Answer; Ptot = 0mW!


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) Round 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 more information and improve results.

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.


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 function of timetime seen from seen from behindbehind..

Simple casesSimple cases VerticalVertical, E, Exx

HorizontalHorizontal, , EEyy

x

x

y

y

zjoxs

ox

eEzE

xztEzE

)(

ˆ)cos()(

http://fipsgold.physik.uni-kl.de/software/java/polarisation/


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

oyy

zjoxx

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

yoxo

yoy

xox


Elliptical polarizationElliptical polarization

X and Y components have different amplitudes EX and Y components have different amplitudes Eoxox≠≠EEoy, oy, andand ==±±9090oo or or ≠≠±±9090o o and Eand Eoxox==EEoyoy

Or Or ≠0,180≠0,180oo,,

Or any other phase difference, for exampleOr any other phase difference, for example =56 =56oo


Polarization examplePolarization example

Polarizing glasses

Unpolarizedradiation enters

Nothing comes out this time.

All light comes out


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-t10sin(y)45z-tcos(5ˆ),( oo xtzE

)45z-t4sin(y-)45z-tcos(4ˆ),( oo xtzE

zs y-jxzE -j)eˆˆ(14)(

. =105, Elliptic

. =0, linear a 30o

c. +180, LP a 45o

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 Cell phone Computer model for Cell phone

Radiation inside the Human BrainRadiation inside the Human Brain SAR SAR Specific Absorption Rate Specific Absorption Rate

[W/Kg] FCC limit 1.6W/kg, [W/Kg] FCC limit 1.6W/kg, ~.2mW/cm~.2mW/cm22 for 30mins for 30mins

http://www.ewg.org/cellphoneradiation/Get-a-Safer-Phone/Samsung/Impression+%28SGH-a877%29/



Human absorptionHuman absorption

30-300 MHz is 30-300 MHz is where the human where the human body absorbs RF body absorbs RF energy most energy most efficientlyefficiently

http://handheld-safety.com/SAR.aspx http://www.fcc.gov/Bureaus/Engineering_Technology/Docume

nts/bulletins/oet56/oet56e4.pdf

* The FCC limit in the US for public exposure * The FCC limit in the US for public exposure from cellular telephones at the ear level is a from cellular telephones at the ear level is a SAR level of 1.6 watts per kilogram (1.6 W/kg) SAR level of 1.6 watts per kilogram (1.6 W/kg) as averaged over one gram of tissue.as averaged over one gram of tissue.

**The ICNIRP limit in Europe for public **The ICNIRP limit in Europe for public exposure from cellular telephones at the ear exposure from cellular telephones at the ear level is a SAR level of 2.0 watts per kilogram level is a SAR level of 2.0 watts per kilogram (2.0 W/kg) as averaged over ten grams of (2.0 W/kg) as averaged over ten grams of tissue.tissue.


http://handheld-safety.com/SAR.aspx

http://www.fcc.gov/Bureaus/Engineering_Technology/Documents/bulletins/oet56/oet56e4.pdf

http://www.fcc.gov/Bureaus/Engineering_Technology/Documents/bulletins/oet56/oet56e4.pdf


Radar bandsRadar bandsBand Name

Nominal FreqRange

Specific Bands Application

HF, VHF, UHF 3-30 MHz0, 30-300 MHz, 300-1000MHz

138-144 MHz216-225, 420-450 MHz890-942

TV, Radio,

L 1-2 GHz (15-30 cm) 1.215-1.4 GHz Clear air, soil moist

S 2-4 GHz (8-15 cm) 2.3-2.5 GHz2.7-3.7>

Weather observationsCellular phones

C 4-8 GHz (4-8 cm) 5.25-5.925 GHzTV stations, short range

Weather

X 8-12 GHz (2.5–4 cm) 8.5-10.68 GHzCloud, light rain, airplane

weather. Police radar.

Ku 12-18 GHz 13.4-14.0 GHz, 15.7-17.7 Weather studies

K 18-27 GHz 24.05-24.25 GHz Water vapor content

Ka 27-40 GHz 33.4-36.0 GHz Cloud, rain

V 40-75 GHz 59-64 GHz Intra-building comm.

W 75-110 GHz 76-81 GH, 92-100 GHz Rain, tornadoes

millimeter 110-300 GHz Tornado chasers


Microwave OvenMicrowave OvenMost food is lossy media at Most food is lossy media at

microwave frequencies, microwave frequencies, therefore EM power is lost therefore EM power is lost in the food as heat.in the food as heat.

Find depth of penetration if Find depth of penetration if meat which at 2.45 GHz has meat which at 2.45 GHz has the complex permittivity the complex permittivity given.given.

The power reaches the inside The power reaches the inside as soon as the oven in as soon as the oven in turned on!turned on!

[/m] 2817.4

)30(2

j

jc

f

jj co

cm 3.21/1

)130( joc


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, fields, and electric Note that for voltages, fields, and electric currents, the log is multiplied by 20 instead of 10.currents, the log is multiplied by 20 instead of 10.

2

12

2

21

2

1

2

1

log20log10log10][V

V

/RV

/RV

P

PdBG

P

P G


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


Submarine antennaSubmarine antenna

A submarine at a depth of 200m uses a wire antenna to receive signal transmissions at 1kHz.

Determine the power density incident upon the submarine antenna due to the EM wave with |Eo|= 10V/m.

[At 1kHz, sea water has r=81, =4].

At what depth the amplitude of E has decreased to 1% its initial value at z=0 (sea surface)?

][W/m ˆcos2

222

zeE zo

ave

P


Exercise: Lossy media Exercise: Lossy media propagationpropagation

For each of the following determine if the material is low-loss dielectric, good conductor, etc.

(a) Glass with r=1, r=5 and =10-12 S/m at 10 GHZ

(b) Animal tissue with r=1, r=12 and =0.3 S/m at 100 MHZ

(c) Wood with r=1, r=3 and =10-4 S/m at 1 kHZ

Answer:(a) low-loss,

.xNp/mr/m.cmup.xc

(b) general, .cmup.xm/sc.j31.7

(c) Good conductor, .x.xkm up.xc.j

electromagnetism inel 4152 sandra cruz-pol, ph. d. ece uprm mayag ü ez, pr

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