Vector Potential and Antenna Radiation in ECE1266 Applications of Fields and Waves Course, Study notes of Electrical and Electronics Engineering

Download Vector Potential and Antenna Radiation in ECE1266 Applications of Fields and Waves Course and more Study notes Electrical and Electronics Engineering in PDF only on Docsity! Course Notes for ECE1266 Applications of fields and waves NOTES for Antennas I This lecture covers 8.1 and 8.2 1. Review of vector potential A 2. Short dipole antenna 3. Far-field approximation 4. Antenna radiation characteristics: Pattern, dimensions, directivity, gain, and resistance An antenna is a device that transducer a guided wave into an unbounded medium or vice versa. Antennas are made in various shapes and size and used in radio, V, communication, cell phone, radar, etc, etc. Here are the summary of some fundamental terms and properties of antenna.  Reciprocity: Most linear antennas are reciprocal, meaning that antennas exhibit the same radiation pattern for transmission as for reception.  Polarization: antenna polarization describes the direction of E- or H- radiation fields that are transmitted by the antenna.  Antenna impedance: pertains to the transfer of power from a generator to the antenna when the antenna is used as transmitter and, conversely, the transfer of power from the antenna to a load when the antenna is used as a receiver. Chapter 8: Antennas Lecture 1 1 Course Notes for ECE1266 Applications of fields and waves Vector potential and current density We have learned in ECE 1259 that currents cause magnetic fields H or magnetic flux density B. The magnetic flux density can by calculated from current density (see figure below) from Figure 8.11 (p. 400) The vector magnetic flux density at the observation point (o) results from a current density distributed about the volume vd.     d do do d dv R a xJB 2 0 0 4   doa is a unit vector from the source to the point of observation. It is simpler to talk about magnetic flux density in terms of a vector potential A where the magnetic flux intensity is derived from  A via   AB Thus the determination of magnetic flux density B is in effect a determination of vector potential A. Following our usual approach we will write phasor descriptions of all fields (and thus suppress any time variation from the resulting equations.) We will also note that there is a time delay or a phase shift for the result of any variation in current to reach an observation point (point o in the previous figure-Fig 8.11). This delay leads to a phase factor exp (-jRdo). (This accounts for the time for a disturbance to propagate from its source to the point of observation.) We will write this delay is terms of a delayed version of the phasor expression for the current density via )exp( dodds RjJJ   velocityphasefrequency/radian  pu   This leads to an expression for the phasor expression for the vector potential     d do ds os dvR J A   4 0 Thus the problem of finding magnetic flux density B (and magnetic field H=B/μμ0 becomes a problem of finding the phasor form of the vector potential A. This vector potential is calculated from the delayed version (or retarded phasor quantity) Jds. Chapter 8: Antennas Lecture 1 2 Course Notes for ECE1266 Applications of fields and waves called directivity D. The directivity refers to the radiation power per “angle” where “angle” refers to a two-dimension angle or solid-angle. Solid angle is measured in steradians. One steradian is defined as the “angle” subtended by an area r2 located at a distance r from the origin. We recall that a one-dimensional angle is defined such that one radian subtends and arc of length r on a circle located a distance r from the origin. Text figure 8.5 illustrates the concept of radians and steradians . Figure 8.5 (p. 393) (a) An arc with length equal to a circle's radius defines a radian. (b) An area equal to the square of a sphere's radius defines a steradian. An incremental angle dθ defined by the differential length of arc traversed along the circumference of a circle divided by radius r; similarly an incremental solid angle d is defined by the differential area dA divided by r2. It is not hard to show that d= sin θ dθ dФ Total angle is 2π and total solid angle is found from    4)(sin 2 0 0   dd An antenna’s pattern solid angle is defined by p =Pn(θ,Ф)d and the normalized power’s average value is Defined by Pn average =Pn(θ,Ф)d/(4π)=π)=)=p/4π)=π)= and directive gain D(,Ф) is) is defined from    4 ),(),( ),D( average p n n n P P P   For a Hertizian dipole the antenna’s pattern solid angle is found from 3/8)(sinsinsin 2 0 0 22      dddp (algebra skipped) and the directive gain is 2 3 sin),( 2  D . Figure 8.6 of the text illustrates the significance of pattern solid angle. It describes the equivalent solid angular spread of a beam that has uniform intensity within the solid angle p. It describes an equivalent width of the main lobe of the antenna pattern. Chapter 8: Antennas Lecture 1 5 Course Notes for ECE1266 Applications of fields and waves Figure 8.6 (p. 394) The pattern solid angle, in steradians, for a typical antenna radiation pattern. Directivity is defined as the maximum value of D(,Ф) is). For a Hertzian dipole the directivity is thus 1.5. Finally the radiation resistance of antenna is defined so that the total power radiated Prad = ½ I02 Rrad where I0 is the magnitude of the phase current driving the antenna. For the Hertzian dipole we showed earlier that the Poynting vector radiated into the far field is           rar lI rP     2 22 22 0 2 0 sin 32 ),( We find the total power radiated from     ddrPdrPPrad sin|||| 22 For a Hertzian dipole 2 0 2 222 22 22 0 2 0 40sinsin 32 I l ddr r lI Prad                  Hence Rrad, the radiation resistance (Prad=1/2 I02 Rrad) of the Hertzian dipole is 2 280          l Rrad Update: November 06, 2007 Chapter 8: Antennas Lecture 1 6