1Spring 2011Notes 24ECE 6345Prof. David R. JacksonECE Dept.2OverviewIn this set of notes we derive the SDI formulation using a more mathematical, but general, approach (we directly Fourier transform Maxwell’s equations).This allows for all possible types of sources to be treated in one derivation.3General SDI MethodˆitH J j Ezz     ˆˆtxyxy     ˆˆˆˆˆt x yxyttx jk y jkj x k ykjkjk u      whereStart with Ampere’s law:Assume a 2D spatial transform:4General SDI Method (cont.)Hence we haveˆˆitjk u z H J j Ez    ˆˆˆˆˆˆˆˆˆu v zz u vz v u       ˆˆˆˆ ˆ ˆ ˆˆˆu v zH u H v H zHu H u v H v z H z        Note thatTake the components of the transformed Ampere’s equationˆˆˆ,,z u vNext, represent the field as5Examine TMzfield:ˆˆˆit v z zivuuiut z v vz jk H J j EHu J j EzHv jk H J j Ez       ,,u v zE H EIgnore equationˆvit v z zivuujk H J j EHJ j Ez    (2)(1)General SDI Method (cont.)6TMzFieldsWe wish to eliminate . To do this, use Faraday’s law:zEˆˆiitE M j Hjk u z E M j Hz        Take the component of the transformed Faraday’s Law:ˆviut z v vEjk E M j Hz   (3)7Substitute from (1) into (3) to obtainPutting all the sources on the RHS:zE 1iiut z t v v vEjk J jk H M j Hjz     2iiu t tv v v zE k kH j H M Jzj       222222111tttzkjkjjkkjkj     Note thatTMzFields (cont.)8Hence2iiutzv v zEkkH M Jzj    (4)TMzFields (cont.)9Equations (2) and (4) are rewritten as2ivuuiiutzv z vHJ j EzEkkM J Hzj      TMzFields (cont.)10Define:2TMTM iuTMTM i itzvzIj V JzkkVI M Jzj      We then have    ,,,,TMu x yTMv x yV z E k k zI z H k k zTMzFields (cont.)11Telegrapher’s Equations dsiv v L z v zt     LzCzdsizdsvzdsviLvzt  Allow for distributed sourcesso v+v-+-+-zi+i-12Telegrapher’s Equations (cont.)dsVj LI Vz  Also, dsvi i C z i zt     dsivCizt  dsIj CV Iz  Hence, in the phasor domain, Hence, in the phasor domain, so13Telegrapher’s Equations (cont.)Compare field equations for TMzfields with TL equations:dsVj LI Vz  dsIj CV Iz    TMTM iuIj V Jz   22TMTM i itzvzkkVj I M Jz      14Telegrapher’s Equations (cont.)We then make the following identifications:Hence22zCkL222022TLzzzTLzzkk LC kkkLZC        0TLzzTLzkkkZor15Sources: TMzTMTMdisud i its v zIJkV M J  For the sources we have, for the TMzcase:16Then we haveAssume Similarly,   , , , ( )sJ x y z J x y zTM is suIJTM is svVMSpecial case: planar surface-current sources:   , , , ( )sM x y z M x y zSources: TMz(cont.)  TMTMdis sudisvI J zV M zThis is a lumped parallel current generator.This is a lumped series voltage generator.17IfFor a vertical planar electric current: , , ( , ) ( )izJ x y z f x y z ,TMts x ykV f k kThen we haveSources: TMz(cont.)z = 0Example: ( , ) ( ) ( )f x y x y(unit-amplitude vertical electric dipole) ,1xyf k k 18TEzFieldsUse duality:iiiiEHHEJMMJ2ivuuiiutzv z vEM j HzHkkJ M Ezj         2ivuuiiutzv z vHJ j EzEkkM J Hzj      TMzTEz19TEz(cont.)Define 22TETE iuTETE i itzvzVj I MzkkIj V J Mz           0TLzzTEzkkZk22zLkC    ,,,,TEv x yTEu x yV z E k k zI z H k k zWe then identify20TEz(cont.)For the sources, we haveTETEdisud i its v zVMkI J MSpecial case of horizontal surface currents:TE is suTE is svVMIJSpecial case of vertical planar currents:TEtskIg ( , ) ( )izM g x y z21SummaryTMuTMvTEvTEuVEIHVEIHTM is suTM is svIJVMTE is suTE is svIMVJSpecial case of horizontal surface currents:TETEdisud i its v zVMkI J MTMTMdisud i its v zIJkV M J  22Summary (cont.)Special case of vertical planar currents:TMtskVf ( , ) ( )izJ f x y z ( , ) ( )izM g x y