Copyright 2006 Regents of University of California1EE 210 Applied EM Fall 2006, Neureuther Lecture #21 Ver 11/12/06EE243 Advanced Electromagnetic TheoryLec # 21 Radiation• Vector Potential Formulation for Finding Fields• Near and Far Field Limits• Multipole Components• Far-Field ApplicationsReading: Jackson Chapter 9Fixed Slides 2, 3, 5, 9Copyright 2006 Regents of University of California2EE 210 Applied EM Fall 2006, Neureuther Lecture #21 Ver 11/12/06OverviewCharges in motion radiate. The radiation can be found by evaluating the vector potential and then taking the curl to get H and then another curl to get E. The radiating fields have E and H transverse to the outward direction and E/H is proportional to the impedance of free space and decrease as 1/r. Various oscillating charge moments create electric and magnetic dipoles and multipoles and each has a characteristic radiation pattern. These moments help characterize radion from small holes and slots. In the far-field the radiation pattern is the Fourier transform of the current distribution.Copyright 2006 Regents of University of California3EE 210 Applied EM Fall 2006, Neureuther Lecture #21 Ver 11/12/06Localized Oscillating Source• Electric monopole part of the potential (and fields) of a localized source is of necessity static.• Hence the vector potential is sufficient to describe the radiating field.() ()() ()() ()()()xdxxxxHkiZEAHxdxxexJxAexJtxJextxxxiktiti′′−′=ΦΦ∇−×∇=×∇=′′−′===∫∫′−−−3003014,,ρµπµρρωωCopyright 2006 Regents of University of California4EE 210 Applied EM Fall 2006, Neureuther Lecture #21 Ver 11/12/06Radiating Zones• Near (Static) Zone d << r << λ– Exponential is unity, => static and no radiation• Intermediate (Induction) Zone d << r ~ λ– General expansion required• Far (Radiation) Zone d << λ << r– Approximate denominator as 1/r– Approximate exponential as quadratic => Fresnel – Or Approximate exponential as linear => Fraunhoffer() ()()xdexJrexAxnrxxxnikikr′′=′⋅−≈′−′⋅∫304|πµApproximated by projection parallel to nIn Fourier transformCopyright 2006 Regents of University of California5EE 210 Applied EM Fall 2006, Neureuther Lecture #21 Ver 11/12/06Electric Dipole Fields and Radiation• Approximate exponent as constant• Apply iωρ = Div J• Integrate by parts• Fields are perpendicular to n and perpendicular to each other• Both E and H decrease as 1/r() ()()()nHZErepnckHxdxxprepixdxxexJxAikrikrxxik×=×≈′′′=−=′′−′=∫∫′−023030444πρπωµπµCopyright 2006 Regents of University of California6EE 210 Applied EM Fall 2006, Neureuther Lecture #21 Ver 11/12/06Poynting Vector for Electric Dipole• Poynting vector gives power density per unit solid angle• Substitute for fields• Sin squared polar angle• Integrate over azimuthal and polar angles to get net power radiated.[]()240222420224202*212sin3232Re21pkZcPpkZcddPnpnkZcddPHEnrddPπθππ==Ω××=Ω×⋅=ΩCopyright 2006 Regents of University of California7EE 210 Applied EM Fall 2006, Neureuther Lecture #21 Ver 11/12/06Radiation for Short Wire Antenna• Assume current is triangular –d/2 to d/2• Evaluate dipole moment•Evaluate Power• Interpret coefficient of maximum current squared over 2 as resistance• Cell phone at 2 GHz and 5 cm high has impedance of about 15 ohms()()()ohmskdRkdIZPkdIZddPdiIpRAD2222002222000548sin1282≈==Ω=πθπωCopyright 2006 Regents of University of California8EE 210 Applied EM Fall 2006, Neureuther Lecture #21 Ver 11/12/06Magnetic Dipole and Electric Quadrapole Fields• Approximate exponential phase in integral for A over source by more terms in a Taylor series• Constant => electric dipole p• First => magnetic dipole m (circulating current) plus quadrapole Qαβ• Second = further monopolesCopyright 2006 Regents of University of California9EE 210 Applied EM Fall 2006, Neureuther Lecture #21 Ver 11/12/06Characterizing Small Sources• Many types of sources small sources– Probes, current loops, holes in metal screens,• When sources are smaller than a wavelength they can be approximated by their electric and magnetic dipole moments• The source contributions to producing fields can be evaluated using the reaction theorem• Above is an example for waveguide apertures of radius R()()()() ()[]...00438ˆ2234030tan030tan+⋅−⋅==⋅=×=−=⋅=±±±∫∫∫λλλλωµωεεBmEpZiAHRdaHnxdaEnimERdaExnpeffeffeffeffCopyright 2006 Regents of University of California10EE 210 Applied EM Fall 2006, Neureuther Lecture #21 Ver 11/12/06Aperture Radiation• Rectangular current patch flowing in x direction over-a/2 < x < a/2-b/2 < y < b/2• Plug in Fraunhoffer approximation for A• Factor to F(x)G(y)• View as product of two Fourier Transforms() ()()()()()()⎥⎦⎤⎢⎣⎡=′′=′′=′⋅−≈′−∫∫∫−′−′′⋅rkayrkayrkaxrkaxabrexAydexderexAxdexJrexAxnrxxikrbbyikaaxikikrxnikikr2/2/sin2/2/sin444|02/2/2/2/030πµπµπµxyzCopyright 2006 Regents of University of California11EE 210 Applied EM Fall 2006, Neureuther Lecture #21 Ver 11/12/06Aperture Radiation Beamwidth• Look for first null• Occurs when aperture is one wavelength wide and ful cycle integrates to zero• FWHM = 60 degrees/size in wavelengths()()()()arxrkaxrkaxrkaxrkayrkayrkaxrkaxabrexAikr22/2/02/2/sin2/2/sin2/2/sin40λππµ===⎥⎦⎤⎢⎣⎡=Copyright 2006 Regents of University of California12EE 210 Applied EM Fall 2006, Neureuther Lecture #21 Ver 11/12/06Antenna Array Patterns• Composite Array id built from a element instantiated at array positions (convolution of element with space array factor)• FT of convolution is product of FT’s• Composite pattern is the array pattern times element pattern.CompositeArray