1ECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityLecture 29The Hertzian Dipole AntennaIn this lecture you will learn:• Hertzian dipole antenna• Gain and radiation pattern of an antennaECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityHertzian Dipole Antenna - Review• A Hertzian dipole is represented by an arrow whose direction indicates the positive direction of the current and also the orientation of the dipole in space• By assumption, the size of the dipole is much smaller than the wavelength of the emitted radiation, i.e. d << λxzdq(t)-q(t)xzI(t)In phasor notation• Because d << λ, the current density associated with the dipole is represented mathematically as a delta function with an appropriate weight () ( )αω+= tItIocos()()redIzrJjorrr3ˆδα=2ECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityRadiation Emitted by a Hertzian Dipole - ReviewNeed to solve for the radiated fields:() ()()rJrAkrAorrrrrrµ−=+∇22Use the superposition integral form of the solution:()()()rkjorkjorrkjoerIdzerIdzrAdverrrJrA−−−−==⇒∫∫∫−=πµπµπµ4ˆ4ˆ ''4' 'rrrrrrrrrrrrFind the H-field:()() ()[]rkjoerIdrrA−−=⇒πµθθθ4sinˆcosˆ rr() ()()()θπφµsin114ˆ o⎥⎦⎤⎢⎣⎡+=⇒×∇=−rkjerIdkjrHrArHrkjrrrrrrxzθ()()rdIzrJrrr3ˆδ=Working in spherical coordinatesyφECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityRadiation Emitted by a Hertzian Dipole - ReviewH-field was:()()θπφsin114ˆ⎥⎦⎤⎢⎣⎡+=−rkjerIdkjrHrkjrrFind the E-field:Use Ampere’s Law:()()()rEjrJrHorrrrrrεω+=×∇ Away from the dipole the current density is zero, therefore:() ()rHjrEorrrr×∇=εω1 ()()()⎪⎭⎪⎬⎫⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛++⎪⎩⎪⎨⎧+⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛+=−θθθπηsin111ˆ cos211ˆ422rkjrkjrkjrkjrerIdkjrErkjorrxzθ()()rdIzrJrrr3ˆδ=Working in spherical coordinatesyφ3ECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityRadiation Emitted by a Hertzian Dipole - Review()() ()⎪⎭⎪⎬⎫⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛++⎪⎩⎪⎨⎧+⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛+=−θθθπηsin111ˆcos211ˆ422rkjrkjrkjrkjrerIdkjrErkjorrNear field region (r << λ / 2π)()trE ,rrECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityFar-Fields of a Hertzian Dipolexzθ()()rdIzrJrrr3ˆδ=yφ()()θπφsin114ˆ⎥⎦⎤⎢⎣⎡+=−rkjerIdkjrHrkjrr()()()⎪⎭⎪⎬⎫⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛++⎪⎩⎪⎨⎧+⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛+=−θθθπηsin111ˆ cos211ˆ422rkjrkjrkjrkjrerIdkjrErkjorrFar-field is the field far away from the dipole where: kr >> 1(or more accurately where: d << λ/2π<< r )()()θπφsin4ˆrkjfferIdkjrH−=rr()()θπηθsin4ˆrkjofferIdkjrE−=rrE-field and H-field are in phase in the far-field4ECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityPower Emitted by a Hertzian Dipole - Ixzθ()()rdIzrJrrr3ˆδ=Working in spherical coordinatesyφThe time average power per unit area going in the (θ , φ ) direction radiated by the Hertzian dipole is given by the Poynting vector:() (){}() (){}() (){}()θπη22**sin42ˆ Re21 Re21Re21,rIdkrrHrErHrErStrSoffff=×=×==rrrrrrrrrrrrNo radiation is emitted in directions given by θ = 0 or θ = 180 degreesCan simply use the field expressions in the far-field:ECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityPower Emitted by a Hertzian Dipole - IIThe total time average power radiated by the Hertzian dipole is given by integrating the Poynting vector over any closed surface surrounding the dipole:Assume the closed surface to be a sphere for simplicity:()()()2200212 sinˆ., .,IdkddrrtrSadtrSPoradπηφθθππ=∫∫=∫∫=rrrrrxzysurrounding sphererNote:For the same current, more power is radiated if d is larger, i.e. if the size of the dipole is larger.5ECE 303 – Fall 2005 – Farhan Rana – Cornell UniversitySome Useful Characteristics of AntennasAntenna Gain:The gain G(θ,φ) of an antenna is defined as the ratio of the power density (i.e. power per unit area) emitted radially outward in the (θ,φ) direction to the power density in the same direction radiated by an isotropic source that emits the same total power()()24ˆ.,,rPrtrSGradπφθrr=Example: For a Hertzian dipole the gain is:()()()θπφθ22sin234ˆ.,, ==rPrtrSGradrr⇒() ()πφθθφθππ4sin,200=∫∫ddGAntenna Radiation Pattern:The radiation pattern p(θ,φ) of an antenna is defined as the ratio of the gain G(θ,φ) to the maximum value of the gain()()max,,GGpφθφθ=Example: For a Hertzian dipole the radiation pattern is:()()()θφθφθ2maxsin,, ==GGpECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityCharacteristics of a Hertzian Dipole AntennaAntenna Gain:For a Hertzian dipole the gain is:()()()θπφθ22sin234ˆ.,, ==rPrtrSGradrrAntenna Radiation Pattern:For a Hertzian dipole the radiation pattern is:()()()θφθφθ2maxsin,, ==GGp()0, =φθpθ (degrees)0180903060120150()0,=φθpθ6ECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityCharacteristics of a Hertzian Dipole Antenna - IIAntenna Radiation Pattern in 3D:()()()θφθφθ2maxsin,, ==GGp()φθ,pECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityRadiation Resistance of Antennas212IdkPoradπη=Antenna Radiation Resistance:The radiation resistance of an antenna is found by equating the power radiated by an antenna to the power dissipated in a resistor carrying the same current as the antennaFor a Hertzian dipole:xz()()rdIzrJrrr3ˆδ=RIIFor a Resistor:()2262kdIPRoradradπη==RIPdissipated221=Radiation resistance of a Hertzian dipole:Circuit Model of a Hertzian DipoleradR7ECE 303 – Fall 2005 – Farhan Rana – Cornell Universityxz() ()rdIzrJrrr3ˆδ=A Single Hertzian Dipole Antenna()()()θπφθ22sin234ˆ.,, ==rPrtrSGrr()()()θφθφθ2maxsin,, ==GGp()()rkjfferIdkjrH−=θπφsin4ˆrr()()rkjofferIdkjrE−=θπηθsin4ˆrrOne is usually interested in only radiation far-fields:0180903060120150()0,=φθpθEHHEECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityA Single Hertzian Dipole Antenna Not at Origin - Iyz()()hrdIzrJrrrr−=3ˆδWhat if one has a Hertzian dipole sitting at some arbitrary point?If one is interested in radiation far-fields only, then