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by: Adaline Pollich


Marketplace > University of Kentucky > Electrical Engineering > EE 699 > TOPS IN EE PRINC PRACT OF TECH COMM
Adaline Pollich
GPA 3.81


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Class Notes
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This 19 page Class Notes was uploaded by Adaline Pollich on Friday October 23, 2015. The Class Notes belongs to EE 699 at University of Kentucky taught by Staff in Fall. Since its upload, it has received 48 views. For similar materials see /class/228324/ee-699-university-of-kentucky in Electrical Engineering at University of Kentucky.




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Date Created: 10/23/15
Source Models for FEM S Gedney Source Excitations in FEM o In order to simulate real problems necessary for engineering design using the FEM we need to be able to represent physical sources 0 The source model required is often driven by the parameterization of the device under test that we are trying to extract 0 How the parameterization is extracted also dictates the level of accuracy required of the source model 0 Exact representation of a true physical source 0 Approximate representation 39 Nonideal nature of source is deembedded o Nonphysical source 39 Postprocessing of data renders exact source unnecessary EE699 i The Finite ElememMelhodfor Electromagnetics ll9 Source Models for FEM S Gedney Source Models o Impressed current source 0 Discrete pointdipole I This can only be approximated in a FEM simulation 0 Current density I Line current again approximated 39 Surface current 39 Volume current 0 Plane Wave source 0 Plane wave injected into problem domain on a boundary 0 Exterior coupling eg FEBI o Waveguide mode excitation 0 Approximate aperture coupled o Exact modal excitation Exterior coupling problem 0 Discrete Lumped source model 0 Approximate TheveninNorton source model 0 Port mode scattering parameter extraction 0 Transmission line source model approximate 1D modal excitation EE699 i The Finite ElememMelhodfor Electromagnetics 219 Source Models for FEM S Gedney Impressed Current DensitV 0 Recall the wave equation derived With impressed current densities VXEQ iVgtltEm k02l7a 8rEt0tdQ JEQ xiVxEm ds Q r F Illr J39k g I Ea impala j Ea V xiMideo 9 Q r o The impressed current densities are assumed to be distributed over nite volumes 9 and Qm 0 Assume that there is an electric current density that is reduced to a surface Then this expression can be reduced to vxgva LVXE IOI k02Ea 8rEt0tdQJEaXiVXEmt I dSZk0770JEa Ijimpds Q r F 1 rj EE699 i The Finite ElememMelhodfor Electromagnetics 319 Source Models for FEM S Gedney Plane Wave Source Iniection 0 Assume that you have a nite dimensional object under test that is illuminated by a plane wave source 0 The object lies completely Within the domain Q bound by F o The object under test is assumed to be a heterogeneous composition of conductors and penetrable materials 0 The elds modeled Via the FEM method are the total eld intensities o The total elds satisfy the expected boundary conditions on materialconductor surfaces Vx w LVgtltEM k02 a 398rEmtJdQ JEQXLVXEM dS 0 Q P It ll r EE699 i The Finite ElememMethod for Electromagnetics 419 Source Models for FEM S Gedney Scattered Field Formulation 0 The total eld can be expressed as a superposition of the incident and scattered elds 0 Em Etna Escat7 tot im scat 0 Where Ema F1 are the incident plane wave Which are propagating through the homogeneous free space in the absence of the of the object under test 0 The incident electric eld satis es the weakform equation Van VgtltEmc k0227a Ei 0dQ JEQ XVXEW ds O Q P 0 Expanding the total eld as a function of the incident and scattered elds I Awem mead 1556 gr EscatdQ Q r J a xiVxEi c Swt ds 0 F r o The incident eld equation is subtracted from this leading to EE699 i The Finite ElememMethod for Electromagnetics 519 Source Models for FEM S Gedney VXEQ iVgtltEWt k02l7a 8rl7scatd j kon0j aXFIWI ds Q Iur F J VgtltEa 1 LJVXEW k02 quot l 8rE CdQ39kon0JEa gtltur lIIi C ds Q Iur F 0 Note that the excitation appears in the form of volume current sources in material regions 0 Notes on F 0 Near the exterior boundary typically ur l and the boundary term is 0 I Exception can be for layered media of in nite planar slabs 0 Boundary term included and the scattered eld requires correct ABC 0 PEC surfaces I Dirichlet boundary condition x Escat gtlt Einc o PMC surfaces 39 Neuman boundary condition x Hscat gtlt Him 0770an gtlt Hm ds k0n0JEa gtlt ur 1Hi c 450 k0n0JEa gtlt urHi ds F F F EE699 i The Finite ElememMethod for Electromagnetics 619 Source Models for FEM S Gedney TotalFieldScatteredField Formulation o The domain Q can be broken up into two regions a scattered eld region and a total eld region These are separated by the boundary F o In the total eld region IVaniVgtltEmt k E 3Emtd 2 IEaxVxE Ot ds Eaxiwrwaszo rob Ur Qtot r 1 pw 0 In the scattered eld region IVanVxEW k E Emtd 2 j E xVxEW ds IE xVxEW dszo Qxcat rpw 1quot 0 On the boundary PM we must choose either a scattered eld or a total eld as the unknown Here we will choose the scattered eld Thus on F O Etot Escat Emc Htot Hscat Hinc EE699 i The Finite ElememMethod for Electromagnetics 719 Source Models for FEM S Gedney o This leads to van iwgwr 4w 3Ewrdg Qtot y j v X E vX 4w Emm QPW IE XVXESW ds I E XLVXEW ds PM PM j VxE VxEW k E Ei cdQ jkon0 j QPW FPW r 0 Where the integral over 91 implies the integral inside 9 due to basis associated With subtopologies ie edges or faces on F 0 Also note that the boundary integral I E a x V x E at ds due to FPW complementary normals on either side of F EE699 i The Finite ElememMethod for Electromagnetics 819 Source Models for FEM S Gedney 0 To evaluate the incident eld terms we can treat the boundary as a Dirichlet boundary expand the incident electric eld Via the curl conforming basis on FM E inc pr 4 Z ClichVi PW i1 0 This can be formulated into a linear system of equations as 4 a pr a 4 I E Cds 20 I Wlds fl W i1 fl W 0 Similarly 4 a pr a 4 W H ds 2b W mds rpw i1 l pw 0 Once we solve for cf and bf these vectors Will be multiplied against the local element matrices to ll the righthandside of the system matrix EE699 i The Finite ElememMethod for Electromagnetics 919 Source Models for FEM S Gedney Plane Wave in a 3D Space 0 In spherical coordinates the plane wave can be said to have an angle of incidence of 49 a The plane wave is then propagating along the negative radial direction With a timeharmonic k Vector 4 k ksin 49quot cos 15 sin 49quot sin We cos W Z X EE699 i The Finite ElememMethod for Electromagnetics 1019 Source Models for FEM S Gedney 0 The incident eld can be described as a superposition of vertical and horizontally polarized waves which have timeharmonic electric elds En0p018 76inc 7Wm ElzcejkF inc Egipde 76inc 7Wm ElfcejkF inc where 49 cos 49quot cos 15 cos 49quot sin We sin 9 2 43 sin 45 cos We and 13 x 2 22 o The magnetic eld is derived from the electric eld via Faraday s law EV A 8 inc A inc inc inc 39kF inc HVW m9 e 770 a a EH Aim Hgip018r6mc 7 mcgt mg ejk 6 770 Where 770 is the characteristic impedance of the host medium presumably free space EE699 i The Finite ElememMethod for Electromagnetics lll9 Source Models for FEM S Gedney WaveGuide Mode ExcitationTermination 0 Consider the FEM modeling of a discontinuity in a waveguide o Q the interior domain E H o ng the exterior wave guide domain Ewg Flwg A inc gt i E wg Q gt Eggns W8 4 ref Ewg 0 The domains are separated by the surface ng 0 Note that the waveguides do not have to be the same 0 If the elds in the exterior domain are known the can be coupled to the interior elds Via the boundary conditions on ng 4 EE699 i The Finite ElememMethod for Electromagnetics 1219 Source Models for FEM S Gedney E inc wg Q W8 Eref W8 o In the exterior waveguide region only the incident elds are assumed to be known 0 Objectives 0 Unique solution for if in Q and Ewg1flwg in ng o Symmetric Formulation EE699 i The Finite ElememMethod for Electromagnetics 1319 Source Models for FEM S Gedney The Interior FEProblem ainc Ewg gt Errans W8 W8 4 ref Ewg o The interior problem is represented by the standard FEM formulation based on the weakform of the vector Helmholtz equation nga iVxE k02 a effijd I EaxiVxE ds20 u Q 13g Iur 0 In the boundary integral we can note that V x E jk0770urEI r J EaxiVxE ds2jkono I EaxH ds2 j39k0n0 I Ea XFIdS F r livg Fgg 0 Therefore this can be expressed as inaiVxE kangr1 dQ jkonoIEa xFIds20 1 Q r 13g EE699 i The Finite ElememMethod for Electromagnetics l4l9 Source Models for FEM S Gedney The Exterior ModalProblem 0 Consider the ith waveguide port The magnetic eld can be expanded into forward and backward traveling modes ngi HV Zgi Hng39 o where the forward and backward traveling waves can be expanded via a discrete modal expansion M M H Z He jkml Egg 2 th ry km ml ml o where I is the waveguide axis km is the axial wavenumber and Hf are the waveguide modes along the crosssection o The incident modes are assumed to be known The re ected modes are weighted by unknown coef cients hm o The electric elds can be similarly expanded Ml kl M7 kl ij 7 T7 1 Ewgi ZEme m Ewgi ZZmhmHme m ml ml o E is again assumed to be known and in is a tensor product relating the magnetic and electric elds for each mode EE699 i The Finite ElememMethodfor Electromagnetics 1519 S Gedney Source Models for FEM Exterior Formulation 0 On the waveguide boundary ng of the i th port we pose ft g MJr M7 A A A 39kl 39kl o an 7 ngtltH ngtlt 2He m thHmeJm ng ng m1 m1 M4r M A a A a A 39kl 39kl o an ngtltE ngtlt 21526 1quot ZZmhmHme m I l r Wg m1 quot11 ng 0 Inner products are then appropriately performed MJr a a M7 o Ea gtltHds 26 11le I Equot gtltHdsthe m1 m1 rag jkmlwg Ag A a I E ngtlt Hmds 13g Fag M 39kl A 39kl 61W I ann Edsequotnge m1 Firm 39kl A mg I ann Eds mg 0 39kl M7 39kl eJWthejmwg I anZ m1 I g m IIds EE699 i The Finite ElememMethod for Electromagnetics 1619 Source Models for FEM S Gedney Coupled Formulation o In Q away from the waveguide boundary j vanivxE kg a grijdgho 9 y o In Q near the waveguide boundary vx a ivxE k02 a grEdQ Q r k0770Mthejkmlwg I Equot xFIdsj39k0n0MZejkmlwg I Ea xflds m1 r 1 m wg 1quotwg 0 On the waveguide boundary M7 39kl A a 39kl 761 A k07706 W I ann Edsk077ej W E hme 5 I ann Zm Hmds FQE quot11 rag M janng jkmlwg A 4 k07706 E e an Emds 1 m ng EE699 i The Finite ElememMethod for Electromagnetics 1719 Source Models for FEM S Gedney Discretization In Q the electric eld is expanded Via Hpcurlconforming basis functions on threedimensional elements The test functions in Q are also expanded With the same basis function space On F the electric eld is also expressed Via the Hp curl conforming basis wg representative of the face and edge functions of the 3D cell that lie on ng The modal fields 131 are known exactly or approximately and have full support over the entire ng surface 0 System Matrix fog in 0 59 0 5 if F 5r bk 0 FT 7 be 0 Where Imeld l V x wt ly x w 02 gwjdo o r EE699 i The Finite ElememMethod for Electromagnetics 1819 Source Models for FEM S Gedney and KQI is similarly de ned with W supported on F similar for Km o The coupling matrix block is Plm jk07706jkmlwg I W gtlt Eliza s mg 0 The diagonal block based on the waveguide modes is Zn m j39konoejkquotlwgejkmlwg I 15 x 1 2m IIds mg 0 The righthand side vectors are computed as M jkmlwg a A a bmj k077026 I w ngtltHmds ml Fryg 39kl M 39k l a ang meg A ben jkonoe 26 I ann Emds ml rag 0 Note that System Matrix is symmetric providing that the block matrix Z is symmetric 0 Note in general due to orthogonality i should be a diagonal matrix EE699 i The Finite ElememMethod for Electromagnetics 1919


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