ECE 6345ECE 6345Spring 2011Prof. David R. JacksonECE Dept.Notes 191OverviewOverviewIn this set of notes we introduce the Spectral Domain Immitance p(SDI) method, which is a powerful method for solving for the fields due to sources inside of layered media.  The basic idea is developed here by decomposing a finite current sheet into a set of infinite phased current sheets.  The fields are found from an infinite phased current sheet. The fields from the infinite current sheets are added together g(spectral integration) to recover the fields of the finite current sheet. 2Spectral Domain Immitance MethodSpectral Domain Immitance MethodWe initially consider a planar source inside of a layered structure. (The method can be extended to include vertical sources as well.)()The figure shows a single layer, but the method works for any number of layers. εz1ε(, )sJxy0ε0ε3SDI Method (cont.)SDI Method (cont.)z0εz1ε(, )sJxy0ε()(),(,)xyjkx kysxysJkk Jxye dxdy+∞ +∞+=∫∫Introduce Fourier transform pair:0ε()()()()()21,,2xyyjkx kyssxyxyJxyJ k k e dk dkπ−∞ −∞+∞ +∞−+=∫∫∫∫pConsidering the integrals as limits of it()2π−∞−∞()()()1xm ynjkxkyJJkkkk∞∞−+ΔΔ∑∑4sums, we can write()()()()2,,2xm ynjys s xm yn x ymnJxyJkkkkeπ=−∞ =−∞=ΔΔ∑∑SDI Method (cont.)SDI Method (cont.)()()()()21,,2xm ynjkxkys s xm yn x ymnJxy Jk k kkeπ∞∞−+=−∞ =−∞=ΔΔ∑∑ykxkΔykΔxk(),xmynkk5SDI Method (cont.)SDI Method (cont.)()()()()21,,2xm ynjkxkys s xm yn x ymnJxy Jkk kkeπ∞∞−+=−∞ =−∞=ΔΔ∑∑()1()Denote()()21,2mnsxm yn x yAJk k kkπ=ΔΔ()(),xm ynjk x k ysmnmnJxy Ae∞∞−+=−∞ =−∞=∑∑ThenThe finite-size current sheet is thus expressed as a superposition of 6pppinfinite phased current sheets.SDI Method (cont.)SDI Method (cont.)() (),,mnssmnJxy J xy∞∞∞∞=∑∑That is,mn=−∞=−∞where()(),xm ynjk x k ymnsmnJxyAe−+=z(, )mnsJxy7SDI Method (cont.)SDI Method (cont.)Consider a single sheet of current of the form:()0xyjkx kyppssJJe−+=The superscript p stands f“h d h ”zfor “phased current sheet.”(, )psJxyzzTM TE+We wish to determine the amplitude of the plane waves that this current source launches, and the field at any point inside the structure.8Note: TMzand TEzwaves reflect from the boundary and remain TMzand TEz, respectively.SDI Method (cont.)SDI Method (cont.)The current sheet launches a pair of plane waves that propagate in a transverse direction determined by the wavenumber.z(, )psJxyzzTM TE+ypwφxPWtktanypwxkkφ=top view9TMTMzzPWPW3D viewPWz3D viewThe upward-going TMzplane wave inside the yPWplayer is shown here.kHxEHytEtHtop viewPW10pwφxTMTMzzPW (cont.)PW (cont.)Denotekk⎛⎞ ⎛⎞ˆˆˆ ˆ ˆ ˆcos sinyxtpw pwttkkuk x y x ykkφφ⎛⎞ ⎛⎞== + = +⎜⎟ ⎜⎟⎝⎠ ⎝⎠andNote: These unit vectors depend on the values of (mn)ˆˆˆvzu=×anddepend on the values of (m,n).vzu=×yˆvyˆutk11pwφxTMTMzzPW (cont.)PW (cont.)For the TMzplane wave, we then haveFor the TMzplane wave, we then haveˆtuEuE=The “t” subscript means “transverse ˆtvHvH=(perpendicular) to the z direction.ytHtkφtE12pwφxTETEzzPWPWThe upward-going TEzplane wave is polarized as shown herezwave is polarized as shown here.ˆHHyPWˆˆtutvHuHEvE==kxEHykytEtHtk13pwφxtEWave ImpedancesWave ImpedancesThe wave impedances are calculated for waves traveling upward. TMz:ytHtkpwφxtETMuzvEkZHωε==TE:xytHtkTEz:φtEtHTEvEZHkωμ−==14pwφxuzHkTEN: TMTEN: TMzzPWPWConsider the TMzplane wave that gets launched by the current sheet:zzTM(, )psJxyzTMWe wish to use a TEN model to find the plane-wave field inside the layered structure.15TEN: TMTEN: TMzzPW (cont.)PW (cont.)We introduce the following modeling equations: ()()()000TMuuVzE,,zEz==ytHtk()()()()()()0000uuTMvvIzH,,zHz==The zero subscriptpwφxtETEN modelThe zero subscript indicates that the field has the exponential phase term suppressed.()TMI0TMZTMZsourceNote that the voltage (tangential ()TMIz+()1/ 22221ZTMsIg( gelectric field) must be continuous at the source location, so the source model is+-()TMVz()()1/ 22221/222zi i x yitkkkkkk=−−=−1TMZ16the source model is a parallel element.0TMZTEN: TETEN: TEzzPWPW()()()00TEVE EWe introduce similar modeling equations for the TEzcase: yHk()()()()()()000000TEvvTEuuVzE,,zEzIz H,,z Hz=− =−==φytEtHtkTEN modelpwφx0TEZsource()TEIz1TEZTEsIsource()TEVz+-()()1/ 22221/222zi i x yitkkkkkk=−−=−1TEZ170TEZSource ModelSource ModelConsider the current sheet in an infinite homogeneous medium:z1ε+ wave (TMz+TEz)(, )psJxy1ε- Wave (TMz+TEz)()()()(),, ,,00zzjkzttjkzHxyz Hxy eHxyz Hxy e++−−−+=()(),, ,,0zjttHxyz Hxy e=By symmetry,18yy y,()()0,,0,, yxHyxHtt−+−=Source Model (cont.)Source Model (cont.)DtDenote()()(),,,TM TEpp psssJxyJxyJxy=+The TM current is that part of the total current that launches lTMl()()()() () ()()ˆ,,,0,,0TM TM TMpttsJxyzHxy Hxy+−=× −only a TMzplane wave, while the TE current launches only a TEzplane wave.() ()()()()()ˆˆˆ,,0 ,,0ˆ,,0 ,,0vvvvzvHxy vHxyuH xyHxy+−+−=× −=− −()()()()()()()()()()()ˆ0, 0, 0 0, 0, 0ˆxyxyvvjkx k yvvjkx k yTM TMyyue H HII−++−−++−=− −()()()()()ˆxyxyTM TMjkx k yTMsueIzIzue I+−+=− −=−19Conclusion: The current that launches the TMzplane wave is polarized in the u direction.Source Model (cont.)Source Model (cont.)Hence() ()()ˆˆ,,TMppssJxy u uJ xy=⋅Similarly,() ()()ˆˆ,,TEppssJxyvvJxy=⋅yTMTEpJtkTMpsJsJ() ()()()()ˆˆ,,,TM TEpp pss sJxy u J xy v J xy=+φxt20Source Model (cont.)Source Model (cont.)We can now determine the source amplitude in the TEN model:()()ˆˆˆˆTMxyjkx k yppTMsssuJ uJ u ue I−+⋅=⋅ =⋅−From slide 19, so()sss()()ˆ,xyjkx kypTMssIuJxye+=− ⋅,or()ˆpTMIuJ=⋅or0ssIuJ=−⋅The zero subscript indicates that the current has the exponential phase Similarly,0ˆpTEssIvJ=⋅term suppressed.21TENTENThe TEN models are shown below.TEZ+−TMV+−TEV1TMZTMsITMI1TEZTEsITEI1TMZs1TEZs22ExampleExample(, )psJxyzxrεhr()Assume()()0ˆ,pjk x ysJxy xe−+=Find()yxEt,0≥zfor23Note: If we wanted to find Ez, we would need to find the transverse magnetic field first, and then apply Ampere’s law.Example (cont.)Example (cont.)()()0ˆ,pjk x ysJxy xe−+=kk=In this example we have00xykkkk==()1ˆˆˆuxy=+The unit vectors are()()21ˆˆ ˆˆˆ2uxyvzu xy=+=×= −+24()2Example (cont.)Example (cont.)()()ˆˆ1pTM pssJJuu=⋅yˆuˆv()()()001ˆ21ˆˆjkxyjkxyuexye−+−+=+x45o()()02jyxye=+xTE()()0ˆˆ1ˆpTE pssjk x