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Electromagnetism INEL 4151 Ch 10 Waves

Electromagnetism INEL 4151 Ch 10 Waves. Sandra Cruz-Pol, Ph. D. ECE UPRM Mayag ü ez, PR. Electromagnetic Spectrum. Maxwell Equations in General Form. Who was NikolaTesla?. Find out what inventions he made His relation to Thomas Edison Why is he not well know?. Special case.

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Electromagnetism INEL 4151 Ch 10 Waves

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  1. ElectromagnetismINEL 4151Ch 10 Waves Sandra Cruz-Pol, Ph. D. ECE UPRM Mayagüez, PR

  2. Cruz-Pol, Electromagnetics UPRM

  3. Electromagnetic Spectrum Cruz-Pol, Electromagnetics UPRM

  4. Maxwell Equations in General Form Cruz-Pol, Electromagnetics UPRM

  5. Who was NikolaTesla? • Find out what inventions he made • His relation to Thomas Edison • Why is he not well know? Cruz-Pol, Electromagnetics UPRM

  6. 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! Cruz-Pol, Electromagnetics UPRM

  7. Phasors & complex #’s Working with harmonic fields is easier, but requires knowledge of phasor, let’s review • complex numbers and • phasors Cruz-Pol, Electromagnetics UPRM

  8. COMPLEX NUMBERS: • Given a complex number z where Cruz-Pol, Electromagnetics UPRM

  9. Review: • Addition, • Subtraction, • Multiplication, • Division, • Square Root, • Complex Conjugate Cruz-Pol, Electromagnetics UPRM

  10. For a time varying phase • Real and imaginary parts are: Cruz-Pol, Electromagnetics UPRM

  11. 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) Cruz-Pol, Electromagnetics UPRM

  12. To change back to time domain • The phasor is multiplied by the time factor, ejwt, and taken the real part. Cruz-Pol, Electromagnetics UPRM

  13. Advantages of phasors • Timederivative is equivalent to multiplying its phasor by jw • Timeintegral is equivalent to dividing by the same term. Cruz-Pol, Electromagnetics UPRM

  14. 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. Cruz-Pol, Electromagnetics UPRM

  15. Maxwell Equations for Harmonic fields Cruz-Pol, Electromagnetics UPRM * (substituting and )

  16. A wave • Start taking the curl of Faraday’s law • Then apply the vectorial identity • And you’re left with Cruz-Pol, Electromagnetics UPRM

  17. A Wave • Let’s 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 Cruz-Pol, Electromagnetics UPRM

  18. To change back to time domain • From phasor • …to time domain Cruz-Pol, Electromagnetics UPRM

  19. Several Cases of Media • Free space • Lossless dielectric • Lossy dielectric • Good Conductor eo=8.854 x 10-12[ F/m] mo= 4px 10-7 [H/m] Cruz-Pol, Electromagnetics UPRM

  20. 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? Cruz-Pol, Electromagnetics UPRM

  21. 3. Lossy Dielectrics(General Case) • In general, we had • From this we obtain • So , for a known material and frequency, we can find g=a+jb Cruz-Pol, Electromagnetics UPRM

  22. 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 Cruz-Pol, Electromagnetics UPRM

  23. Note… • E and H areperpendicular to one another • Travel is perpendicular to the direction of propagation • The amplitude is related to the impedance • And so is the phase Cruz-Pol, Electromagnetics UPRM

  24. Loss Tangent • If we divide the conduction current by the displacement current http://fipsgold.physik.uni-kl.de/software/java/polarisation Cruz-Pol, Electromagnetics UPRM

  25. Relation between tanq and ec Cruz-Pol, Electromagnetics UPRM

  26. Substituting in the general equations: 2. Lossless dielectric Cruz-Pol, Electromagnetics UPRM

  27. Review: 1. Free Space • Substituting in the general equations: Cruz-Pol, Electromagnetics UPRM

  28. 4. Good Conductors • Substituting in the general equations: Is water a good conductor??? Cruz-Pol, Electromagnetics UPRM

  29. Summary Cruz-Pol, Electromagnetics UPRM

  30. Skin depth, d • Is defined as the depth at which the electric amplitude is decreased to 37% Cruz-Pol, Electromagnetics UPRM

  31. 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). Cruz-Pol, Electromagnetics UPRM

  32. 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 Cruz-Pol, Electromagnetics UPRM

  33. 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. Cruz-Pol, Electromagnetics UPRM

  34. 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, Cruz-Pol, Electromagnetics UPRM

  35. 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.physik.uni-kl.de/software/java/polarisation Cruz-Pol, Electromagnetics UPRM

  36. Poynting Vector Derivation • Start with E dot Ampere • Apply vectorial identity • And end up with Cruz-Pol, Electromagnetics UPRM

  37. Rearrange Poynting Vector Derivation… • Substitute Faraday in 1rst term Cruz-Pol, Electromagnetics UPRM

  38. Rate of change of stored energy in E or H Ohmic losses due to conduction current Total power across surface of volume 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. Cruz-Pol, Electromagnetics UPRM

  39. 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. Cruz-Pol, Electromagnetics UPRM

  40. Time Average Power • The Poynting vector averaged in time is • For the general case wave: Cruz-Pol, Electromagnetics UPRM

  41. 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) Cruz-Pol, Electromagnetics UPRM

  42. 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 er=9 is characterized by Determine the direction of wave travel and the average power density carried by the wave • Answer: Cruz-Pol, Electromagnetics UPRM

  43. x x z z y 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 • aEx aH = ak Cruz-Pol, Electromagnetics UPRM

  44. PE 10.7 In free space, H=0.2 cos (wt-bx) 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 x Answer; Ptot = 53mW Hz Ey Answer; Ptot = 0mW! Cruz-Pol, Electromagnetics UPRM

  45. x z y x z y 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 Cruz-Pol, Electromagnetics UPRM

  46. 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

  47. 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. Cruz-Pol, Electromagnetics UPRM

  48. x Ex x y Ey y 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 Cruz-Pol, Electromagnetics UPRM

  49. Several Cases • Linear polarization: d=dy-dx =0oor ±180on • Circular polarization: dy-dx=±90o & Eox=Eoy • Elliptical polarization: dy-dx=±90o & Eox≠Eoy, or d=≠0o or ≠180on even if Eox=Eoy • Unpolarized- natural radiation Cruz-Pol, Electromagnetics UPRM

  50. x Ex x y Ey y Linear polarization Front View • d =0 • @z=0 in time domain Back View: Cruz-Pol, Electromagnetics UPRM

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