1 Dipolo Infinitesimal Sandra Cruz-Pol, Ph. D. INEL 5305 ECE UPRM Mayagüez, PR

2 Outline Maxwell’s equations Maxwell’s equations Wave equations for A and for  Wave equations for A and for  Power: Poynting Vector Power: Poynting Vector Dipole antenna Dipole antenna

3 Maxwell Equations Ampere: Faraday: Gauss:

4 Relaciones de Continuidad

5 Potencial Magnético Vectorial A Si lo sustituimos en la Ley de Faraday: Si lo sustituimos en la Ley de Faraday:  is the Electric Scalar Potential

6 Usando Ley de Ampere:

7 Lorentz’ condition Si escogemos Si escogemos Queda la Ecuacion de Onda Queda la Ecuacion de Onda donde J es la densidad de una fuente de corriente, si hay.

8 de la Ec. de Gauss Eléctrica

9 Wave equation For sinusoidal fields (harmonics): For sinusoidal fields (harmonics):where

10 Outline Maxwell’s equations Maxwell’s equations Wave equations for A and for  Wave equations for A and for  Power: Poynting Vector Power: Poynting Vector Dipole antenna Dipole antenna

11 Poynting Vector

12 Average S Para medios sin pérdidas:

13 Outline Maxwell’s equations Maxwell’s equations Wave equations for A and for  Wave equations for A and for  Power: Poynting Vector Power: Poynting Vector Infinitesimal Dipole antenna Infinitesimal Dipole antenna

14 Find A from Dipole with current J Line charge w/uniform charge density,  L Line charge w/uniform charge density,  L x z  r r 0 JzJz AzAz Assume the simplest solution A z (r):

15 To find…. Assume the simplest solution A z (r):

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

17 Apply B.C. If radiated wave travels outwards from the source: If radiated wave travels outwards from the source: To find C2, let’s examine what happens near the source. (in that case k tends to 0) To find C2, let’s examine what happens near the source. (in that case k tends to 0) So the wave equation reduces to So the wave equation reduces to

18 Now we integrate the volume around the dipole: And using the Divergence Theorem And using the Divergence Theorem

19 Comparing both, we get:

20 Now from A we can find E & H Using the Victoria IDENTITY: Using the Victoria IDENTITY: And And Substitute: Substitute:

21 The magnetic filed intensity from the dipole is:

22 Now the E field:

23 The electric field from infinitesimal dipole:

24 General @Far field r> 2D 2 / General @Far field r> 2D 2 / Note that the ratio of E/H is the intrinsic impedance of the medium.

25 Power :Hertzian Dipole

26 Resistencia de Radiacion La potencia tiene parte real y parte reactiva La potencia tiene parte real y parte reactiva La potencia irradiada es: La potencia irradiada es: Comparando con la P total, se halla la Impedancia Comparando con la P total, se halla la Impedancia COmparando cno la P rad, halla la R rad. COmparando cno la P rad, halla la R rad.