1ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityLecture 14Time Harmonic FieldsIn this lecture you will learn:• Complex mathematics for time-harmonic fields• Maxwell’s equations for time-harmonic fields • Complex Poynting vectorECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityE and H-fields for a plane wave are (from last lecture):()()rktEntrEorrrr.cosˆ, −=ω2222.ˆˆˆzyxzyxkkkkkkzkykxkk++==⇒++=rrr0ˆ. =nkrck=ω()()()rktEnktrHoorrrr.cosˆˆ, −×=ωηTime-Harmonic Fields• Fields for which the time variation is sinusoidal are called time-harmonic fields. • Plane waves are just one example of time-harmonic fields• In the rest of this course, 95% of the material will deal with time-harmonic fields2ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityTime-Harmonic Signals in Circuits – Sinusoidal Steady StateConsider an RC circuit driven by a sinusoidal voltage source:~+-RC() ( )tVtVoωcos=()tVR+-()tVC()[]tjoeVtVωRe=() ( )[]tjRReVtVωωRe=()RCjRCjVVoRωωω+=⇒1()tI() ( )[]tjeItIωωRe=()RCjCjVIoωωω+=1Time-average power dissipation in the resistor:()() () ()[]()()⎥⎥⎦⎤⎢⎢⎣⎡+==222*12Re21CRCRRVIVtItVoRRωωωωCjω1Remember phasors from ECE210…..ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversitySome useful trigonometric Identities to recall before we start:()2cosθθθjjee−+=()jeejj2sinθθθ−−=Expression for the E-field of a plane wave in complex notation:()()()()⎥⎦⎤⎢⎣⎡=+=−−−−rktjorktjrktjoeEneeEntrErrrrrrrr...ˆRe2ˆ,ωωω()()θθθsincos jej+=()()θθθsincos jej−=−Time-Harmonic Fields and Complex NotationBasic idea:If the time-variation of fields is known a-priori to be sinusoidal (i.e. the fields are known to be time-harmonic) then, in order to simplify the math, one may not carry around the time dependence explicitly in calculationsLets look at plane waves as an example to see how the complex notation can be used to factor out the sinusoidal time dependence()()rktEntrEorrrr.cosˆ, −=ω3ECE 303 – Fall 2007 – Farhan Rana – Cornell University()()⇒−= rktEntrEorrrr.cosˆ,ω()()⎥⎦⎤⎢⎣⎡=− rktjoeEntrErrrr.ˆRe,ωDo a little more manipulation …()()⎥⎦⎤⎢⎣⎡=⎥⎦⎤⎢⎣⎡=−−tjrkjorktjoeeEneEntrEωωrrrrrr..ˆReˆRe,() ()[]tjerEtrEωrrrrRe, =()rkjoeEnrErrrr.ˆ−=where:The quantity , which is a time-independent complex vector, is a vector phasor for the plane wave()rErrIn the book, the vector phasor has an additional under-line and written as:()rErrFor the E-field of a plane wave we had…Time-Harmonic Fields and Vector PhasorsECE 303 – Fall 2007 – Farhan Rana – Cornell University()()⇒−= rktEntrEorrrr.cosˆ,ω() ()[]tjerEtrEωrrrrRe, =• All time-harmonic fields (not just plane waves) can be written in the form:()rErr() ()[]tjerEtrEωrrrrRe, =For the E-field of a plane wave we had…Complex NotationNow generalize to all time-harmonic fields:• Given a vector phasor for a time-harmonic field, one can find the actual time-dependent field as follows:where is a complex time-independent vector phasor()rErr() ()[]tjerEtrEωrrrrRe, =Example: Suppose I give you the following vector phasor for a plane wave:()zkjeAxrE−=ˆrrThen you can find the actual time-dependent E-field as follows:() ()[][ ]()zktAxeeAxerEtrEtjzkjtj−===−ωωωcosˆˆReRe,rrrr4ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityMaxwell’s Equations for Phasors - ILet the time-harmonic E and H-fields be:() ()[]tjerEtrEωrrrrRe, =() ()[]tjerHtrHωrrrrRe, =Gauss’ Law:Now we substitute these expressions in Maxwell’s equations one by one()()()[]()[]() ()rrEererEtrtrEotjtjoorrrrrrrrrρερερεωω=∇=∇⇒=∇.Re.Re,,.Gauss’ Law for the Magnetic Field:()()0.0,. =∇⇒=∇ rHtrHoorrrrµµAssume that the time-variations of charge density and current density are also sinusoidal:() ()[]tjertrωρρrrRe, =() ()[]tjerJtrJωrrrrRe, =The only way the above can be true for all time is if:(1)(2)ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityMaxwell’s Equations for Phasors - IIThe time-harmonic E and H-fields are:() ()[]tjerEtrEωrrrrRe, =() ()[]tjerHtrHωrrrrRe, =() ()[]tjertrωρρrrRe, =() ()[]tjerJtrJωrrrrRe, =Faraday’s Law:()()()[]()[]tjotjoerHjerEttrHtrEωωµωµrrrrrrrr−=×∇⇒∂∂−=×∇ReRe,,The only way the above can be true for all time is if:()()rHjrEorrrrµω−=×∇(3)Ampere’s Law:() ()()()[]()()[]tjotjtjoerEjerJerHttrEtrJtrHωωωεωεrrvrrrrrrrrr+=×∇⇒∂∂+=×∇ReRe,,,The only way the above can be true for all time is if:()()()rEjrJrHorrvrrrεω+=×∇(4)5ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityMaxwell’s Equations for Phasors - IIILet the time-harmonic E and H-fields be given as:() ()[]tjerEtrEωrrrrRe, =() ()[]tjerHtrHωrrrrRe, =() ()[]tjertrωρρrrRe, =() ()[]tjerJtrJωrrrrRe, =Maxwell’s equations for the vector phasors of time-harmonic fields are then:()()rrEorrrρε=∇ .()0. =∇ rHorrµ()()rHjrEorrrrµω−=×∇()()()rEjrJrHorrvrrrεω+=×∇Gauss’ Law:Gauss’ Law for the Magnetic Field:Faraday’s Law:Ampere’s Law:ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityCalculations in the Complex NotationSuppose for a plane wave we know the E-field to be:() ()[]tjerEtrEωrrrrRe, =()rkjoeEnrErrrr.ˆ−=whereHow does one find the vector phasor for the H-field?Use Faraday’s law for time-harmonic fields:()()()()()()()()()()()rkjoorkjoorkjoorkjooooeEnkrHeEnkkrHeEnkjjrHeEnjrHrEjrHrHjrErrrrrrrrrrrrvrrrrrrrrrrrr....ˆˆˆˆˆˆ−−−−×=⇒×=⇒×−=⇒⎟⎠⎞⎜⎝⎛×∇=⇒×∇=⇒−=×∇ηµωµωµωµωµωkkkˆ=r() ()[]tjerHtrHωrrrrRe, =6ECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityComplex Poynting VectorSuppose for a plane wave we know the E-field and H-field phasors to be:()rkjoeEnrErrrr.ˆ−=()()rkjooeEnkrHrrrr.ˆˆ−×=ηHow does one find the time-average power per unit area carried by the wave?Define a complex Poynting vector as:() ()()rHrErSrrrrrr*×=Claim: The time-average power per unit area is one-half of the real part of the complex Poynting vectorCheck:() ()[]() ()[]()ooooEkEnknrHrErStrSηη2ˆˆˆˆRe21 Re21 Re21,22*=⎥⎥⎦⎤⎢⎢⎣⎡××=×==rrrrrrrrwhich is indeed the right answerECE 303 – Fall 2007 – Farhan Rana – Cornell UniversityMore Calculations in the Complex Notation - IExample:Consider a plane wave with E-field of amplitude Eoand pointing in a direction 45-degrees