electromagnetism inel 4151 ch 10 waves sandra cruz-pol, ph. d. ece uprm mayag ü ez, pr

ElectromagnetismINEL 4151

Ch 10 Waves

Sandra Cruz-Pol, Ph. D.ECE UPRM

Mayagüez, PR

Cruz-Pol, Electromagnetics UPRM

Electromagnetic Spectrum


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

Who was NikolaTesla?

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


Special case• Consider the case of a lossless medium

• with no charges, i.e. .

The wave equation can be derived from Maxwell equations as

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


00v

022 EE c

Phasors & complex #’s

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

• complex numbers and• phasors


COMPLEX NUMBERS:

• Given a complex number z

where


sincos jrrrrejyxz j

magnitude theis || 22 yxzr

angle theis tan 1

x

y

Review:

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


For a time varying phase

• Real and imaginary parts are:


t

)cos(}Re{ trre j

)sin(}Im{ trre j

PHASORS

• For a sinusoidal current equals the real part of

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


)cos()( tItI otjj

o eeI

joeI

tje

sI

To change back to time domain

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


}Re{ tjseAA

Advantages of phasors

• Time derivative is equivalent to multiplying its phasor by jj

• Time integral is equivalent to dividing by the same term.


sAjt

A

jA

tA s

Time-Harmonic fields (sines and cosines)

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

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


Maxwell Equations for Harmonic 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


t

DJH

t

BE

0 B

vD vE

0 H

HjE

EjJH

* (substituting and )ED BH

A wave

• Start taking the curl of Faraday’s law

• Then apply the vectorial identity

• And you’re left withAAA 2)(

s

sss

E

EjjEE2

2

)()(


ss HjE

A Wave

022 EE

zo

zo eEe EE(z)

tzE

'

issolution general whose

),(


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

To change back to time domain

• From phasor

• …to time domain

)()( jzo

zoxs eEeEzE

xzteEtzE zo

)cos(),(


Several Cases of Media

1. Free space 2. Lossless dielectric3. Lossy dielectric4. Good Conductor )or ,,(

),,0(

)or ,,0(

),,0(

oro

oror

oror

oo


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

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

1. Free space

There are no losses, e.g.

Let’s define• The phase of the wave• The angular frequency• Phase constant• The phase velocity of the wave• The period and wavelength• How does it moves?

xztAtzE

)sin(),(


3. Lossy Dielectrics(General Case)

• In general, we had

• From this we obtain

• So , for a known material and frequency, we can find j

11

2 and 11

2

22

)(2 jj


xzteEtzE zo

)cos(),(

j

222222

2222Re

Intrinsic Impedance, • If we divide E by H, we get units of ohms and the

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

][

j

j


yzteE

tzH

xzteEtzE

zo

zo

ˆ)cos(),(

)cos(),(

*Not in-phase for a lossy medium

Note…

• E and H are perpendicular to one another• Travel is perpendicular to the direction of

propagation• The amplitude is related to the impedance• And so is the phase

yzteE

tzH

xzteEtzE

zo

zo

ˆ)cos(),(

)cos(),(


Loss Tangent

• If we divide the conduction current by the displacement current

tangentosstan lEj

E

J

J

s

s

ds

cs


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

Relation between tan and c

EjjEjEH

1

Ej c

'''1

isty permittivicomplex The

jjc

'

"tanas also defined becan tangent loss The


2. Lossless dielectric)or ,,0( oror


• Substituting in the general equations:

o

u

0

21

,0

Review: 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 Conductors


]/[ˆ)45cos(),(

]/[)cos(),(

mAyzteE

tzH

mVxzteEtzE

oz

o

o

zo


) ,,( oro

o

u

45

22

2

Is water a good conductor???

SummaryAny 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

j

j


f

u p

1

2

f

f

f

u p

1

f

u

f

p

4

)1( j

11

2

2

Skin depth, • Is defined as the depth

at which the electric amplitude is decreased to 37%

/1at

%)37(37.01

1

zee

ez

[m] /1


Short Cut …

• You can use Maxwell’s or use

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

HkE

EkH

ˆ

ˆ1


Waves

• Static charges > static electric field, E

• Steady current > static magnetic field, H

• Static magnet > static magnetic field, H

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

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


EM waves don’t need a medium to propagate

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

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

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


Exercises: Wave Propagation in Lossless materials

• A wave in a nonmagnetic material is given by

Find:

(a) direction of wave propagation,

(b) wavelength in the material

(c) phase velocity

(d) Relative permittivity of material

(e) 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 wave

• A wave carries power and transmits it wherever it goes

Cruz-Pol, Electromagnetics UPRMSee Applet by Daniel Roth at

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

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

Poynting Vector Derivation

• Start with E dot Ampere

• Apply vectorial identity

• And end up with

EHHEEH

BAABBA

:case in thisor

t

EEEEHE

t

EEHE


t

EEEHEH

2

2

2

1

Poynting Vector Derivation…

• Substitute Faraday in 1rst term

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 Derivation…

• Taking the integral wrt volume

• Applying theory of divergence

• Which simply means that the total power coming out of a volume is either due to the electric or magnetic field 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 Vector

• Waves carry energy and information• Poynting says that the net power flowing out of a

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

][W/m 2HE

P


Represents the instantaneous power vector associated to the electromagnetic wave.

Time Average Power

• The Poynting vector averaged in time is

• For the general case wave:

*

00

Re2

111ss

TT

ave HEtdHET

tdT

PP

]/[ˆ

]/[ˆ

mAyeeE

H

mVxeeEE

zjzos

zjzos


][W/m ˆcos2

222

zeE zo

ave

P

Total Power in W

The total power through a surface S is

• Note that the units now are in Watts• Note that power nomenclature, P is not cursive.• Note that the dot product indicates that the surface area

needs to be perpendicular to the Poynting vector so that all the power will go thru. (give example of receiver antenna)

][WdSPS

aveave P


Exercises: Power

1. At microwave frequencies, the power density considered safe for human exposure is 1 mW/cm2. A radar radiates a wave with an electric field amplitude E that decays with distance as E(R)=3000/R [V/m], where R is the distance in meters. What is the radius of the unsafe region?

• Answer: 34.64 m

2. A 5GHz wave traveling In a nonmagnetic medium with r=9 is characterized by Determine the direction of wave travel and the average power density carried by the wave

• Answer: ][W/m 05.0ˆ 2xave P

[V/m])cos(2ˆ)cos(3ˆ xtzxtyE


TEM wave

Transverse ElectroMagnetic = 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 10.7

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

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

• square plate at x=1, 0


Answer; Ptot = 53mWHz

Ey

x

Answer; Ptot = 0mW!

Polarization of a waveIEEE Definition:

The trace of the tip of the E-field vector as a function of time seen from behind.

Simple cases• Vertical, Ex

• Horizontal, Ey

xztEzE

eEzE

ox

zjoxs

ˆ)cos()(

)(


x

x

y

y

z

x

x

y

y

Dual-Pol in Weather Radars• Dual polarization radars can

estimate several return signal properties beyond those available from conventional, single polarization Doppler systems.

• Hydrometeors: Shape, Direction, Behavior, Type, etc…

• Events: Development, identification, extinction

ZHH ZVV

ZHV ZVH

Lineal Typical•Horizontal•Vertical

Dra. Leyda León

Polarization:Why do we care?? • 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) Round waveguides, helical or flat spiral antennas produce circular or elliptical waves.

• 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.

• 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.


Polarization• In general, plane wave has 2 components; in x & y

• And y-component might be out of phase wrt to x-component, 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 Cases

• Linear polarization: y-x =0o or ±180on

• Circular polarization: y-x =±90o & Eox=Eoy

• Elliptical polarization: y-x=±90o & Eox≠Eoy, or ≠0o or ≠180on even if Eox=Eoy

• Unpolarized- natural radiation


Linear polarization

• =0

• @z=0 in time domain

t)cos(

t)cos(

yoy

xox

E E

E E


zjoy

zjox

e E E

e E E

x

yEy

Ex

Front View

y

x

Back View:

Circular polarization• Both components have

same amplitude Eox=Eoy,

• = y- x= -90o = Right circular polarized (RCP)

• =+90o = LCP ˆˆˆˆ

:phasorin

)90tcos(

t)cos(

90

o

yjEExe EyExE

E E

E E

yoxoj

yoxo

yoy

xox


Elliptical polarization

• X and Y components have different amplitudes Eox≠Eoy,

and =±90o

• Or ≠±90o and Eox=Eoy,


Polarization example


Polarizing glasses

Unpolarizedradiation enters

Nothing comes out this time.

All light comes out

Example

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

a. b.

c.

d.

)45z-t4sin(y-)30z-tcos(3ˆ),( oo xtzE

)45z-t8sin(y)45z-tcos(3ˆ),( oo xtzE


)45z-t4sin(y-)45z-tcos(4ˆ),( oo xtzEa. Elliptic

b. -90, RHEP

c. +90, LHCP

d. -90, RHCP

)(cos)180(c

)sin()180sin(o

o

os )(s)90(c

)cos()90sin(o

o

inos

Cell phone & brain

• Computer model for Cell phone Radiation inside the Human Brain


Radar 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 chasersCruz-Pol, Electromagnetics UPRM

Microwave Oven

Most food is lossy media at microwave frequencies, therefore EM power is lost in the food as heat.

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

The power reaches the inside as soon as the oven in turned on! [/m] 2817.4

)30(2

j

jc

f

jj co

cm 3.21/1 Cruz-Pol, Electromagnetics UPRM

)130( joc

Decibel Scale

• In many applications need comparison of two powers, a power ratio, e.g. reflected power, attenuated power, gain,…

• The decibel (dB) scale is logarithmic

• Note that for voltages, 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, A

• Represents the rate of decrease of the magnitude of Pave(z) 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 antennaA 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 propagation

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/mcmupxc

(b) general, cmupxm/scj31.7

(c) Good conductor, xxkm upxcj


electromagnetism inel 4151 ch 10 waves sandra cruz-pol, ph. d. ece uprm mayag ü ez, pr

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