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ECE 546 Lecture 02 Review of Electromagnetics

ECE 546 Lecture 02 Review of Electromagnetics. Spring 2014. Jose E. Schutt-Aine Electrical & Computer Engineering University of Illinois jschutt@emlab.uiuc.edu. Electromagnetic Quantities. Electric field (Volts/m). Electric flux density (Coulombs/m 2 ). Magnetic field (Amperes/m).

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ECE 546 Lecture 02 Review of Electromagnetics

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  1. ECE 546 Lecture 02 Review of Electromagnetics Spring 2014 Jose E. Schutt-Aine Electrical & Computer Engineering University of Illinois jschutt@emlab.uiuc.edu

  2. Electromagnetic Quantities Electric field (Volts/m) Electric flux density (Coulombs/m2) Magnetic field (Amperes/m) Magnetic flux density (Webers/m2) Current density (Amperes/m2) Charge density (Coulombs/m2)

  3. Maxwell’s Equations Faraday’s Law of Induction Ampère’s Law Gauss’ Law for electric field Gauss’ Law for magnetic field

  4. Constitutive Relations Permittivity e: Farads/m Permeability m: Henries/m Free Space

  5. Continuity Equation

  6. Electrostatics Assume no time dependence  Poisson’s Equation if no charge is present  Laplace’s Equation

  7. Integral Form of ME

  8. Boundary Conditions

  9. Free Space Solution Faraday’s Law of Induction Ampère’s Law Gauss’ Law for electric field Gauss’ Law for magnetic field

  10. Wave Equation  Wave Equation can show that

  11. Wave Equation separating the components

  12. Wave Equation Plane Wave (a) Assume that only Ex exists  Ey=Ez=0 (b) Only z spatial dependence  This situation leads to the plane wave solution In addition, assume a time-harmonic dependence then

  13. Plane Wave Solution solution where propagation constant In the time domain solution

  14. Plane Wave Characteristics where propagation constant In free space

  15. Solution for Magnetic Field then If we assume that then If we assume that intrinsic impedance of medium

  16. Time-Average Poynting Vector Poynting vector W/m2 time-average Poynting vector W/m2 We can show that

  17. Material Medium or s: conductivity of material medium (W-1m-1) since then

  18. Wave in Material Medium g is complex propagation constant a: associated with attenuation of wave b: associated with propagation of wave

  19. Wave in Material Medium Solution: decaying exponential Complex intrinsic impedance Magnetic field

  20. Wave in Material Medium Phase Velocity: Wavelength: Special Cases 1. Perfect dielectric air, free space and

  21. Wave in Material Medium 2. Lossy dielectric Loss tangent:

  22. Wave in Material Medium 3. Good conductors Loss tangent:

  23. Material Medium

  24. Radiation - Vector Potential Assume time harmonicity ~ (1) (2) (3) (4)

  25. Radiation - Vector Potential Using the property: : vector potential

  26. Vector Potential Since a vector is uniquely defined by its curl and its divergence, we can choose the divergence of A Lorentz condition

  27. Vector Potential D’Alembert’s equation

  28. Vector Potential Three-dimensional free-space Green’s function Vector potential From A, get E and H using Maxwell’s equations

  29. Vector Potential For infinitesimal antenna, the current density is: Calculating the vector potential, In spherical coordinates,

  30. Vector Potential Resolving into components,

  31. E and H Fields Calculate E and H fields

  32. E and H Fields

  33. E and H Fields

  34. E and H Fields

  35. Far Field Approximation Note that:

  36. Far Field Approximation Characteristics of plane waves • Uniform constant phase locus is a plane • Constant magnitude • Independent of q • Does not decay Similarities between infinitesimal antenna far field radiated and plane wave • E and H are perpendicular • E and H are related by h • E is perpendicular to H

  37. Poynting Vector Time-average Poynting vector or TA power density E and H here are PHASORS

  38. Time-Average Power Total power radiated (time-average)

  39. Time-Average Power

  40. Directivity For infinitesimal antenna,

  41. Directivity Directivity: gain in direction of maximum value Radiation resistance: From we have: For infinitesimal antenna:

  42. Radiation Resistance For free space, (for Hertzian dipole) The radiation resistance of an antenna is the value of a fictitious resistance that would dissipate an amount of power equal to the radiated power Pr when the current in the resistance is equal to the maximum current along the antenna A high radiation resistance is a desirable property for an antenna

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