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Antennas • Hertzian Dipole – Current Density – Vector Magnetic Potential – Electric and Magnetic Fields – Antenna Characteristics Hertzian Dipole Step 1: Current Density Let us consider a short line of current placed along the z-axis. i(t) I o cos t j Where the phasor I o I s e The stored charge at the ends resembles an electric dipole, and the short line of oscillating current is then referred to as a Hertzian Dipole. The current density at the origin seen by the observation point is J ds Is S e j Rdo a z A differential volume of this current element is dvd Sdz J ds dvd I s e j Rdo dza z Hertzian Dipole Step 2: Vector Magnetic Potential The vector magnetic potential equation is A os o 4 l 2 l 2 I s dza z e j R do Rdo A key assumption for the Hertzian dipole is that it is very short so Rdo r o I s l e j r A os az 4 r The unit vector az can be converted to its equivalent direction in spherical coordinates using the transformation equations in Appendix B. az cos ar sin a o I s l e j r A os cos ar sin a 4 r This is the retarded vector magnetic potential at the observation point resulting from the Hertzian dipole element oriented in the +az direction at the origin. Hertzian Dipole Step 3: Electric and Magnetic Fields The magnetic field is given by Bos A os H os = B os o 1 o A os I s l e j r 1 Hos j sin a 4 r r It is useful to group and r together H os I s l 2 e j r j 4 The electric field is given by sin a 2 r r Eos oar Hos . 1 Eos jo In the far-field, we can neglect the second term. Far-field condition: Hos j I s l e j r 4 r r 2 sin a 1 1 r r 2 I sl e j r 4 r sin a . Hertzian Dipole Step 4: Antenna Parameters Power Density: P r, , 1 2 Re Eos H*os o 2 I o2 l 2 2 P r, sin a r 2 2 32 r Maximum Power Density: Pmax o 2 I o2 l 2 32 2 r 2 Antenna Pattern Solid Angle: p sin 2 d sin 2 sin d d p 8 3 Directivity: Dmax 4 p 1.5 Hertzian Dipole Step 4: Antenna Parameters Total Radiated Power and Radiation Resistance : The total power radiated by a Hertzian dipole can be calculated by Prad r 2 Pmax p o 2 I o2 l 2 Prad r 2 2 32 r 2 The power radiated by the antenna is 2 l 2 40 Io P 2 Prad I o2 Rrad Circuit Analysis Field Analysis Rrad l 80 2 2 Hertzian Dipole - Example Example Electric Field: Power density: Maximum Power density: Normalized Power density Example Antenna Pattern Solid Angle: p Pn , sin d d d sin 3 cos 2 d d p sin d cos d 3 Radiated Power: Radiated Resistance: 2