Copyright 2006 Regents of University of California1EE 210 Applied EM Fall 2006, Neureuther Lecture #10 Ver 09/26/06EE243 Advanced Electromagnetic TheoryLec # 10: Poynting’s Theorem, Time-Harmonic EM Fields• Poynting’s Theorem Conservation of energy and momentum• Poynting’s Theorem for Linear Dispersive Media• Poynting’s Theorem for Time-Harmonic Fields • Definition of Impedance and Admittance• Foster’s Reactance Theorem • Lorentz ReciprocityReading: Jackson Ch 6.7-6. 9 (skip 6.10)Collin pp 2.12, 4.3, 4.4Copyright 2006 Regents of University of California2EE 210 Applied EM Fall 2006, Neureuther Lecture #10 Ver 09/26/06Overview• Starting from the work done on a current source it is possible to develop a conservation of energy that includes the flow E cross H (Poynting’s Vector).• This approach generalizes to– Momentum using q(E + v cross B)– (Phasor notation and even/odd consequences )– Linear dispersive media– Time-harmonic fields– Reactance has positive slope– ReciprocityCopyright 2006 Regents of University of California3EE 210 Applied EM Fall 2006, Neureuther Lecture #10 Ver 09/26/06Work done on Source J by Field E• Work done by fields on sources•Replace J• Use integration by parts like vector idenity• Interpretation: Work done by fields on sources equals the energy flow into the voume plus the decrease in energy stored in the fields in the volume()()()())(21)(33333HBDEuHEStuSEJxdtDHtDEHExdEJHEEHHExdtDEHExdEJxdEJVVVVV⋅+⋅=×=∂∂−⋅−∇=⋅⎥⎦⎤⎢⎣⎡∂∂⋅+∂∂⋅+×⋅∇−=⋅×∇⋅−×∇⋅=×⋅∇⎥⎦⎤⎢⎣⎡∂∂⋅−×∇⋅=⋅⋅∫∫∫∫∫Copyright 2006 Regents of University of California4EE 210 Applied EM Fall 2006, Neureuther Lecture #10 Ver 09/26/06Linear Momentum• Force on particle• Momentum = force/time• Substitute• Many manipulations• Integration by parts and Div theorem• Define momentum g• Interpretation: Rate of change in mechanical momentum plus rate of change in linear momentum in volume is equal to flow of mentum across the surface into the volume.()boundaryacrossflowtermsHEcgdatermsxdBEdtddtdPtEBJExdBJEdtdPBvEqFVVmechVmech__1)(1)()(2300003=×==×+∂∂−×∇==×+=×+=∫∫∫∂εεµερρCopyright 2006 Regents of University of California5EE 210 Applied EM Fall 2006, Neureuther Lecture #10 Ver 09/26/06Fourier Representation Properties• Here f is any function• Fourier Representation• Fourier Spectrum• When f is real f(–ω) = f*(ω)),(),(),(),(),(),(21),(),,(),,(),,(),(*ωωωωπεωωωxfdwetxfxfdwetxfxfdwexftxfetctxtxHtxEtxftititi==−===−∞∞−∞∞−−∞∞−∫∫∫Copyright 2006 Regents of University of California6EE 210 Applied EM Fall 2006, Neureuther Lecture #10 Ver 09/26/06Fourier Representation Implications• Real nature of signals gives analytical properties to spectrum in the complex plane()())()()()()()()()(),(),(),(),(),(),(),(),(),(),(**********ωωωωωωωωωωωεωωωωωεωωωωZIVIVZIIVVxxExDxExDxxDxDxExE==−−=−=−=−==−−=−=−=−Copyright 2006 Regents of University of California7EE 210 Applied EM Fall 2006, Neureuther Lecture #10 Ver 09/26/06Fourier Representation Implications (Cont.)• Real nature of signals gives analytical properties to spectrum in the complex plane• Representation for R and X contain only even and odd powers of ω• Same is true for ε(ω)() () ()()()122***)()()()()()(+∑∑==+=+===−−=−nnnnXXRRjoddevenjXRZZIVIVZωωωωωωωωωωωωωCopyright 2006 Regents of University of California8EE 210 Applied EM Fall 2006, Neureuther Lecture #10 Ver 09/26/06Linear Dispersive Media• Constitutive relationship• Real function constraint• Substitute definitions using complex conjugate • Split into two equal parts• Make narrowband approximation() ()[]()[]()[]() ()[]()[][]() (){}()[]{} ( )()( ) ()()() ()()( ) ()()()()titieEddiEddtDEddiEiiEEiEeEiEddtDEωωωωωωωεωωωωεωωωωωωεωωωωεωωωεωωωεωωωωεωωωωεωωω′−−′−−⎥⎦⎤⎢⎣⎡⋅⎥⎦⎤⎢⎣⎡′−−′−′=∂∂⋅⎥⎦⎤⎢⎣⎡′−−→⋅′′+−′→⋅−′→⋅−′−′=∂∂⋅∫∫∫∫*****ImIm2)(21Copyright 2006 Regents of University of California9EE 210 Applied EM Fall 2006, Neureuther Lecture #10 Ver 09/26/06Linear Dispersive Media (Cont.)()()()()() ()()() ()()()()()()txHtxHtxEtxEEJStutxHtxHddtxEtxEddututxHtxHtxEtxEtBHtDEeffeffeff,,)(Im,,)(Im,,)(Re21,,)(Re21,,)(Im,,)(Im0000000000⋅−⋅−⋅−=⋅∇+∂∂⋅⎥⎦⎤⎢⎣⎡+⋅⎥⎦⎤⎢⎣⎡=∂∂+⋅+⋅=∂∂⋅+∂∂⋅ωµωωεωωωωµωωωεωµωωεωCopyright 2006 Regents of University of California10EE 210 Applied EM Fall 2006, Neureuther Lecture #10 Ver 09/26/06Time-Harmonic Fields• E is represented by a complex number called a phasor (when it rotates)• Products have a time independent (time-avereage) and a double frequency part()()[]() ()[]titixitiEeJEJEJtixxEeexEtxEeωωφωωφ2*00Re21)cos(Re),(−−−⋅+⋅=⋅−==Copyright 2006 Regents of University of California11EE 210 Applied EM Fall 2006, Neureuther Lecture #10 Ver 09/26/06Time-Harmonic Poynting’s Theorem• Real Part = Time-average• Imy Part is double frequency()()()()022141412133***=⋅+−+⋅⋅=⋅=×=∫∫∫ndaSxdwwixEdJHBwDEwHESSVmeVmeωCopyright 2006 Regents of University of California12EE 210 Applied EM Fall 2006, Neureuther Lecture #10 Ver 09/26/06Impedance from Poynting’s Theorem•Siis surface for signal feed and S is the outside surface• Take Real and Imy parts for R and X()ndaSxdwwixEdJiXRjXRiXRZIVndaSVIiiSSVmeVS⋅+−+⋅=−+=−==⋅−=∫∫∫∫−242133**ωVΙCopyright 2006 Regents of University of California13EE 210 Applied EM Fall 2006, Neureuther Lecture #10 Ver 09/26/06Foster’s Reactance Theorem• Start with div of E cross derivative of H plus derivative of E cross H; use Div theorem• Result: The derivative of the reactance and the susceptance with respect to ω is always positive• (There may be an alternative derivation using the time derivative of the expression for the impedance)()()**44VVwwBIIwwXmeme+=∂∂+=∂∂ωωVΙCollin 4.3Copyright 2006 Regents of University of California14EE 210 Applied EM Fall 2006, Neureuther Lecture #10 Ver 09/26/06Lorentz Reciprocity Theorem• Start with Lorenz reciprocity statement• put in integral form; substitute for J• use Div theorem• argue integral at infinity is zero due to same outgoing