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Electromagnetism INEL 4151 Ch 10 Waves


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Title: Electromagnetism INEL 4151 Ch 10 Waves

ElectromagnetismINEL 4151Ch 10 Waves
  • Sandra Cruz-Pol, Ph. D.
  • Mayagüez, PR

(No Transcript)
Electromagnetic Spectrum
Maxwell Equations in General Form
Differential form Integral Form
Gausss Law for E field.
Gausss Law for H field. Nonexistence of monopole
Faradays Law
Amperes Circuit Law
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
  • The equation of a wave!

Phasors complex s
  • Working with harmonic fields is easier, but
    requires knowledge of phasor, lets review
  • complex numbers and
  • phasors

  • Given a complex number z
  • where

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

For a time varying phase
  • Real and imaginary parts are

  • 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

To change back to time domain
  • The phasor is multiplied by the time factor,
    ejwt, and taken the real part.

Advantages of phasors
  • Time derivative is equivalent to multiplying its
    phasor by jw
  • Time integral is equivalent to dividing by the
    same term.

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
Gausss Law for E field.
Gausss Law for H field. No monopole
Faradays Law
Amperes Circuit Law
(substituting and
A wave
  • Start taking the curl of Faradays law
  • Then apply the vectorial identity
  • And youre left with

A Wave
  • Lets look at a special case for simplicity
  • without loosing generality
  • The electric field has only an x-component
  • The field travels in z direction
  • Then we have

To change back to time domain
  • From phasor
  • to time domain

Several Cases of Media
  1. Free space
  2. Lossless dielectric
  3. Lossy dielectric
  4. Good Conductor

eo8.854 x 10-12 F/m mo 4p x 10-7 H/m
1. Free space
  • There are no losses, e.g.
  • Lets 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?

3. Lossy Dielectrics(General Case)
  • In general, we had
  • From this we obtain
  • So , for a known material and frequency, we can
    find gajb

Intrinsic Impedance, h
  • 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.

Not in-phase for a lossy medium
  • E and H are perpendicular to one another
  • Travel is perpendicular to the direction of
  • The amplitude is related to the impedance
  • And so is the phase

Loss Tangent
  • If we divide the conduction current by the
    displacement current

Relation between tanq and ec
2. Lossless dielectric
  • Substituting in the general equations

Review 1. Free Space
  • Substituting in the general equations

4. Good Conductors
  • Substituting in the general equations

Is water a good conductor???
Any medium Lossless medium (s0) Low-loss medium (e/elt.01) Good conductor (e/egt100) Units
a 0 Np/m
b rad/m
h ohm
uc l w/b 2p/bup/f m/s m
In free space eo 8.85 x 10-12 F/m mo4p x 10-7 H/m In free space eo 8.85 x 10-12 F/m mo4p x 10-7 H/m In free space eo 8.85 x 10-12 F/m mo4p x 10-7 H/m In free space eo 8.85 x 10-12 F/m mo4p x 10-7 H/m In free space eo 8.85 x 10-12 F/m mo4p x 10-7 H/m In free space eo 8.85 x 10-12 F/m mo4p x 10-7 H/m
Skin depth, d
  • Is defined as the depth at which the electric
    amplitude is decreased to 37

Short Cut
  • You can use Maxwells 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).

  • Static charges gt static electric field, E
  • Steady current gt static magnetic field, H
  • Static magnet gt static magnetic field, H
  • Time-varying current gt time varying E(t) H(t)
    that are interdependent gt electromagnetic wave
  • Time-varying magnet gt time varying E(t) H(t)
    that are interdependent gt electromagnetic wave

EM waves dont need a medium to propagate
  • Sound waves need a medium like air or water to
  • EM wave dont. 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
  • direction of wave propagation,
  • wavelength in the material
  • phase velocity
  • Relative permittivity of material
  • Electric field phasor
  • Answer y, up 2x108 m/s, 1.26m, 2.25,

Power in a wave
  • A wave carries power and transmits it wherever it

The power density per area carried by a wave is
given by the Poynting vector.
See Applet by Daniel Roth at http//fipsgold.physi
Poynting Vector Derivation
  • Start with E dot Ampere
  • Apply vectorial identity
  • And end up with

Poynting Vector Derivation
  • Substitute Faraday in 1rst term

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.

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.

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

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)

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 er9 is characterized by
    Determine the
    direction of wave travel and the average power
    density carried by the wave
  • Answer

TEM wave
  • Transverse ElectroMagnetic plane wave
  • There are no fields parallel to the direction of
  • only perpendicular (transverse).
  • If have an electric field Ex(z)
  • then must have a corresponding magnetic field
  • The direction of propagation is
  • aE x aH ak

PE 10.7
  • In free space, H0.2 cos (wt-bx) z A/m. Find the
    total power passing through a
  • square plate of side 10cm on plane xz1
  • square plate at x1, 0

Answer Ptot 53mW
Answer Ptot 0mW!
Polarization of a wave
  • IEEE 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

Dual-Pol in Weather Radars
  • Dual polarization radars can estimate several
    return signal properties beyond those available
    from conventional, single polarization Doppler
  • Hydrometeors Shape, Direction, Behavior, Type,
  • Events Development, identification, extinction
  • Lineal Typical
  • Horizontal
  • Vertical

Dra. Leyda León
  • 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.

  • In general, plane wave has 2 components in x y
  • And y-component might be out of phase wrt to
    x-component, d is the phase difference between x
    and y.

Front View
Several Cases
  • Linear polarization ddy-dx 0o or 180on
  • Circular polarization dy-dx 90o EoxEoy
  • Elliptical polarization dy-dx90o Eox?Eoy, or
    d?0o or ?180on even if EoxEoy
  • Unpolarized- natural radiation

Linear polarization
Front View
  • d 0
  • _at_z0 in time domain

Back View
Circular polarization
  • Both components have same amplitude EoxEoy,
  • d d y-d x -90o Right circular polarized (RCP)
  • d 90o LCP

Elliptical polarization
  • X and Y components have different amplitudes
    Eox?Eoy, and d 90o
  • Or d ?90o and EoxEoy,

Polarization example
  • Determine the polarization state of a plane wave
    with electric field
  • a.
  • b.
  • c.
  • d.
  1. Elliptic
  2. -90, RHEP
  3. 90, LHCP
  4. -90, RHCP

Cell phone brain
  • Computer model for Cell phone Radiation inside
    the Human Brain

Radar bands
Band 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 GHz2.7-3.7gt Weather observations Cellular phones
C 4-8 GHz (4-8 cm) 5.25-5.925 GHz TV stations, short range Weather
X 8-12 GHz (2.54 cm) 8.5-10.68 GHz Cloud, 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 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!

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.

Attenuation rate, A
  • Represents the rate of decrease of the magnitude
    of Pave(z) as a function of propagation distance

Submarine 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
  • At 1kHz, sea water has er81, s4.
  • At what depth the amplitude of E has decreased to
    1 its initial value at z0 (sea surface)?

Exercise Lossy media propagation
  • For each of the following determine if the
    material is low-loss dielectric, good conductor,
  • Glass with mr1, er5 and s10-12 S/m at 10 GHZ
  • Animal tissue with mr1, er12 and s0.3 S/m at
    100 MHZ
  • Wood with mr1, er3 and s10-4 S/m at 1 kHZ
  • Answer
  • low-loss, a 8.4x10-11 Np/m, b 468 r/m, l 1.34
    cm, up1.34x108, hc168 W
  • general, a 9.75, b12, l52 cm, up0.5x108 m/s,
    hc39.5j31.7 W
  • Good conductor, a 6.3x10-4, b 6.3x10-4, l
    10km, up0.1x108, hc6.28(1j) W
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