Electromagnetism INEL 4151 Ch 10 Waves

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ElectromagnetismINEL 4151Ch 10 Waves

• Sandra Cruz-Pol, Ph. D.
• ECE UPRM
• Mayagüez, PR

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(No Transcript)
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Electromagnetic Spectrum
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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
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Who was NikolaTesla?

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

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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!

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Phasors complex s

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

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COMPLEX NUMBERS

• Given a complex number z
• where

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Review

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

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For a time varying phase

• Real and imaginary parts are

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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)

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To change back to time domain

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

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Advantages of phasors

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

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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.

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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
)
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A wave

• Start taking the curl of Faradays law
• Then apply the vectorial identity
• And youre left with

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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

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To change back to time domain

• From phasor
• to time domain

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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
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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?

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3. Lossy Dielectrics(General Case)

• In general, we had
• From this we obtain
• So , for a known material and frequency, we can
  find gajb

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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
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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

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Loss Tangent

• If we divide the conduction current by the
  displacement current

http//fipsgold.physik.uni-kl.de/software/java/pol
arisation
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Relation between tanq and ec
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2. Lossless dielectric

• Substituting in the general equations

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Review 1. Free Space

• Substituting in the general equations

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4. Good Conductors

• Substituting in the general equations

Is water a good conductor???
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Summary
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
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Skin depth, d

• Is defined as the depth at which the electric
  amplitude is decreased to 37

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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).

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Waves

• 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

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EM waves dont need a medium to propagate

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

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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,

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Power in a wave

• A wave carries power and transmits it wherever it
  goes

The power density per area carried by a wave is
given by the Poynting vector.
See Applet by Daniel Roth at http//fipsgold.physi
k.uni-kl.de/software/java/polarisation
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Poynting Vector Derivation

• Start with E dot Ampere
• Apply vectorial identity
• And end up with

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Poynting Vector Derivation

• Substitute Faraday in 1rst term

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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.

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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.
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Time Average Power

• The Poynting vector averaged in time is
• For the general case wave

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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)

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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

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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

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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

x
Answer Ptot 53mW
Hz
Ey
Answer Ptot 0mW!
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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

x
y
x
y
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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

• Lineal Typical
• Horizontal
• Vertical

ZHH ZVV ZHV ZVH
Dra. Leyda León
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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.

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Polarization

• 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
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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

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Linear polarization
Front View

• d 0
• _at_z0 in time domain

Back View
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Circular polarization

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

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Elliptical polarization

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

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Polarization example
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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

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Cell phone brain

• Computer model for Cell phone Radiation inside
  the Human Brain

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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
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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!

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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.

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Attenuation rate, A

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

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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
  10V/m.
• 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)?

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Exercise Lossy media propagation

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