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# ELECTROMAGNETICS1 ECE1259

Pitt

GPA 3.74

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This 6 page Class Notes was uploaded by Macy Lowe on Monday October 26, 2015. The Class Notes belongs to ECE1259 at University of Pittsburgh taught by Staff in Fall. Since its upload, it has received 35 views. For similar materials see /class/229455/ece1259-university-of-pittsburgh in Electronics and Computer Technology at University of Pittsburgh.

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Date Created: 10/26/15

Course Notes for ECE1266 Applications of fields and waves NOTES for Antennas I This lecture covers 81 and 82 Review of vector potential A Short dipole antenna Farfield approximation Antenna radiation characteristics Pattern dimensions directivity gain and resistance 759 Chapter 8 Antennas Lecture 1 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 0 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 elds 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 Course Notes for ECE1266 Applications of elds and Waves J a Tmndipalz by Eicumcaldipulz c7 mp 17 Helix 97 Lagrpenadic Phasesm zrs hung W Cuzxmlteed 352 M mumm Vector potential and current density We have learned in ECE 1259 that currents cause magnetic elds H or magnetic ux density B The magnetic ux density can by calculated from current density see gure below from z U y Figure 811 p 400 The vector magnetic ux density at the observation point 0 results from a current density distributed about the volume vd Chapter 8 Antennas Lecture 1 a gt u gt ada gt BU TIL devd adu 1s a umt vector from the source to 7 do the point of observation It is simpler to talk about magnetic ux density in terms of a vector potential A where the magnetic ux intensity is derived from A via BVXA Thus the determination of magnetic ux density B is in effect a determination of vector potential A Following our usual approach we will write phasor descriptions of all elds 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 0 in the previous gureFig 811 This delay leads to a phase factor exp jBRdo 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 Jd Jd eXP j Rda radian frequencyphase velocity u P This leads to an expression for the phasor expression for the vector potential l r ya Jr 7d A 47 Rdn Vd Course Notes for ECE1266 Applications of fields and waves Thus the problem of nding magnetic ux density B and magnetic eld HB40 becomes a problem of nding the phasor form of the vector potential A This vector potential is calculated from the delayed version or retarded phasor quantity st Short Dipole Antenna The simplest the antenna is a shortdipole antenna or Hertzian dipole antenna The following gure shows a shortdipole antenna text Figure 812 Figure 812 p 402 The vector magnetic potential is sought at point 0 from a z directed Hertzian dipole at the origin Chapter 8 Antennas Lecture 1 The short antenna referred to here means the length of dipole lltlt 2 roughly lltl50 so we can consider the AC current it on dipole to be uniform along the length of the dipole The text points out page 402 that To maintain constant current over its entire length it is helpful to imagine a pair of plates at the ends of the line that can store charge The stored charge at its ends resembles an electric dipole The short Hertzian dipole is most often analyzed as a building block that allows us to describe more complicated antennas as a sum of Hertzian dipoles We note however that a Hertzian dipole ie an antenna that is short with respect to a wavelength is a real antenna one that can be physically constructed If this antenna is oriented to zaxis and we supply an AC current to the dipole it 10 cosat a or a phasor current Is Joe 1 The current retarded density on the line will be a I e Jag ExeXlX J Rdg where the exponential faction accounts for the propagation phase delay and S is the crosssection area of the antenna If we recall previous page A0 Jag dva 47139 R010 We can calculate some mathematics is skipped the phasor form of the vector potential as A 0151 e w cos ar sin ag 47139 r Course Notes for ECE1266 Applications of fields and waves We can then nd the magnetic eld at any point away from the currentcarrying antenna as n H ivXZ 0 0 After some algebra we nd a 2 i119 gt H05k LZ sin6a 4 3 3 Far away from the current source the second term becomes small with respect to the rst We speak of this region as the far eld of the antenna In that region sin aj and the electric eld is just found by as gt 19 H Isl e 4711 from concepts developed earlier in the course the electric eld is perpendicular to the direction of propagation and to the direction of magnetic eld and is 110 times the magnetic eld or formally a I l e39w E0 770 0 H0 1770 5 made 4717 Finally the time average Poynting vector is found in general from Pr6 12ReEm gtlt H55 For the Hertzian dipole this becomes no 12071 and IS 10 em a 21212 a 12 2 a Pr 639 Tyo i ljsin2 639 a 157139 igiizsinz 639 a r 327W 2 Chapter 8 Antennas Lecture 1 General Properties of Antennas We will de ne some general properties of antennas and illustrate these de nitions by applying them to the Hertzian dipole Radiation Patterns The magnitude of the electric and magnetic elds in general decay as Jr when the point of point of observation is far from the antenna this ensures that the total power crossing a sphere located in the far eld is constant It is customary to describe the variation of Poynting vector powerarea with angle as a normalized power function Pn6 where Pr6 130945 Pm This function is also referred to as the normalized radiation intensity For an isotropic antenna one where the radiation pattern is independent of angle Pn6 1 For a Hertzian dipole Pn6 sm26 Radiation patterns are most often shown as polar plot The radiation pattern for a Hertzian dipole is shown in the text gure 813 copied below CourseNotes for ECE1266 Applications of elds and waves Directivity It is o en desirable to beam the antennas radiation pattern in a articular direction A measure ofhow well this is accomplished is called directivity D The directivity refers to the radiation power per angle where angle refers to a twodimension angle or solidangle Solid angle is measured in steradians One steradian is de ned as the angle subtended by an area r2 located at a distance r from the origin We recall that a onedimensional angle is de ned such that one radian subtends and arc of length r on a circle located a distance r from the origin Text gure 85 illustrates the concept of radians and steradians Chapter 8 Antennas Lecture 1 x r r i 1sr 1 radian a b Figure 85 p 393 a An arc with length equal to a circle39s radius de nes a radian b An area equal to the square of a sphere39s radius de nes a steradian An incremental angle d6 de ned by the differential length of arc traversed along the circumference of a circle divided by radius r similarly an incremental solid angle d0 is de ned by the differential area dA divided by r2 It is not hard to show that dfk sin L9 d6 d43 Total angle is 2739 and total solid angle is found from errr Q jjsin9d9d 47 An antenna s pattem solid angle is de ned by QF n L9 dQ and the normalized power s average value is De ned b Pquot aveng n d 4n Qph and directive gain D9CD is de ned from D8 P 8 m R g 0 47 n avarage p Course Notes for ECE1266 Applications of fields and waves For a Hertizian dipole the antenna s pattern solid angle is found Z from Qp Usinz 6619 JJsin2 6sin 6d6d 87r3 algebra 0 0 skipped and the directive gain is D09 sin2 639 Figure 86 of the text illustrates the signi cance of pattern solid angle It describes the equivalent solid angular spread of a beam that has uniform intensity within the solid angle Qp It describes an equivalent width of the main lobe of the antenna pattern Figure 86 p 394 The pattern solid angle in steradians for a typical antenna radiation pattern Chapter 8 Antennas Lecture 1 Directivity is defined as the maximum value of D9I For a Hertzian dipole the directivity is thus 15 Finally the radiation resistance of antenna is defined so that the total power radiated Pmd 12 10 Rmd where 10 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 eld is Mimi 2 P r6 s1n 6a 327Wquot2 r We find the total power radiated from PM Hu3lrzdojjli3lrzsin6d6d For a Hertzian dipole Z Z Z 2 PM Mlsinz 6er sin 6d6d 40321 102 K 327139 r j 1 Hence Rmd the radiation resistance Pm10 102 Rmd of the Hertzian dipole is 2 Rm 80521 1 Update November 06 2007

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