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Dipolo Infinitesimal. Sandra Cruz-Pol, Ph. D. INEL 5305 ECE UPRM Mayagüez, PR. Outline. Maxwell’s equations Wave equations for A and for F Power: Poynting Vector Dipole antenna. Maxwell Equations. Ampere:. Faraday:. Gauss:. Relaciones de Continuidad. Potencial Magnético Vectorial A. - PowerPoint PPT Presentation

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  • Dipolo Infinitesimal

    Sandra Cruz-Pol, Ph. D.INEL 5305ECE UPRMMayagez, PR

  • OutlineMaxwells equationsWave equations for A and for FPower: Poynting VectorDipole antenna

  • Maxwell EquationsAmpere: Faraday:Gauss:

  • Relaciones de Continuidad

  • Potencial Magntico Vectorial A

    Si lo sustituimos en la Ley de Faraday:F is the Electric Scalar Potential

  • Usando Ley de Ampere:

  • Lorentz conditionSi escogemos

    Queda la Ecuacion de Onda

    donde J es la densidad de una fuente de corriente, si hay.

  • de la Ec. de Gauss Elctrica

  • Wave equationFor sinusoidal fields (harmonics):

    where

  • OutlineMaxwells equationsWave equations for A and for FPower: Poynting VectorDipole antenna

  • Poynting Vector

  • Average SPara medios sin prdidas:

  • OutlineMaxwells equationsWave equations for A and for FPower: Poynting VectorInfinitesimal Dipole antenna

  • Find A from Dipole with current JLine charge w/uniform charge density, rL xzqrr0JzAzAssume the simplest solution Az(r):

  • To find.Assume the simplest solution Az(r):

  • Fuera de la Fuente (J=0)Which has general solution of:

  • Apply B.C.If radiated wave travels outwards from the source:

    To find C2, lets examine what happens near the source. (in that case k tends to 0)

    So the wave equation reduces to

  • Now we integrate the volume around the dipole:

    And using the Divergence Theorem

  • Comparing both, we get:

  • Now from A we can find E & H

    Using the Victoria IDENTITY:

    And

    Substitute:

  • The magnetic filed intensity from the dipole is:

  • Now the E field:

  • The electric field from infinitesimal dipole:

  • General @Far field r> 2D2/lNote that the ratio of E/H is the intrinsic impedance of the medium.

  • Power :Hertzian Dipole

  • Resistencia de RadiacionLa potencia tiene parte real y parte reactiva

    La potencia irradiada es:

    Comparando con la P total, se halla la Impedancia

    COmparando cno la Prad, halla la Rrad.

    Dr. S. Cruz-Pol, INEL 4151-Electromagnetics IElectrical Engineering, UPRM (please print on BOTH sides of paper)*Dr. S. Cruz-Pol, INEL 4151-Electromagnetics IElectrical Engineering, UPRM (please print on BOTH sides of paper)*Dr. S. Cruz-Pol, INEL 4151-Electromagnetics IElectrical Engineering, UPRM (please print on BOTH sides of paper)*Dr. S. Cruz-Pol, INEL 4151-Electromagnetics IElectrical Engineering, UPRM (please print on BOTH sides of paper)*Dr. S. Cruz-Pol, INEL 4151-Electromagnetics IElectrical Engineering, UPRM (please print on BOTH sides of paper)*Dr. S. Cruz-Pol, INEL 4151-Electromagnetics IElectrical Engineering, UPRM (please print on BOTH sides of paper)*Dr. S. Cruz-Pol, INEL 4151-Electromagnetics IElectrical Engineering, UPRM (please print on BOTH sides of paper)*Dr. S. Cruz-Pol, INEL 4151-Electromagnetics IElectrical Engineering, UPRM (please print on BOTH sides of paper)*Dr. S. Cruz-Pol, INEL 4151-Electromagnetics IElectrical Engineering, UPRM (please print on BOTH sides of paper)*Dr. S. Cruz-Pol, INEL 4151-Electromagnetics IElectrical Engineering, UPRM (please print on BOTH sides of paper)*Dr. S. Cruz-Pol, INEL 4151-Electromagnetics IElectrical Engineering, UPRM (please print on BOTH sides of paper)*Dr. S. Cruz-Pol, INEL 4151-Electromagnetics IElectrical Engineering, UPRM (please print on BOTH sides of paper)*Dr. S. Cruz-Pol, INEL 4151-Electromagnetics IElectrical Engineering, UPRM (please print on BOTH sides of paper)*Dr. S. Cruz-Pol, INEL 4151-Electromagnetics IElectrical Engineering, UPRM (please print on BOTH sides of paper)*Dr. S. Cruz-Pol, INEL 4151-Electromagnetics IElectrical Engineering, UPRM (please print on BOTH sides of paper)*Dr. S. Cruz-Pol, INEL 4151-Electromagnetics IElectrical Engineering, UPRM (please print on BOTH sides of paper)*Dr. S. Cruz-Pol, INEL 4151-Electromagnetics IElectrical Engineering, UPRM (please print on BOTH sides of paper)*Dr. S. Cruz-Pol, INEL 4151-Electromagnetics IElectrical Engineering, UPRM (please print on BOTH sides of paper)*Dr. S. Cruz-Pol, INEL 4151-Electromagnetics IElectrical Engineering, UPRM (please print on BOTH sides of paper)*Dr. S. Cruz-Pol, INEL 4151-Electromagnetics IElectrical Engineering, UPRM (please print on BOTH sides of paper)*Dr. S. Cruz-Pol, INEL 4151-Electromagnetics IElectrical Engineering, UPRM (please print on BOTH sides of paper)*Dr. S. Cruz-Pol, INEL 4151-Electromagnetics IElectrical Engineering, UPRM (please print on BOTH sides of paper)*Dr. S. Cruz-Pol, INEL 4151-Electromagnetics IElectrical Engineering, UPRM (please print on BOTH sides of paper)*Dr. S. Cruz-Pol, INEL 4151-Electromagnetics IElectrical Engineering, UPRM (please print on BOTH sides of paper)*Dr. S. Cruz-Pol, INEL 4151-Electromagnetics IElectrical Engineering, UPRM (please print on BOTH sides of paper)*Dr. S. Cruz-Pol, INEL 4151-Electromagnetics IElectrical Engineering, UPRM (please print on BOTH sides of paper)*