Source Models for FEM S. Gedney EE699 – The Finite Element Method for Electromagnetics 1/19 Source Excitations in FEM • In order to simulate real problems necessary for engineering design using the FEM, we need to be able to represent physical sources • The source model required is often driven by the parameterization of the device under test that we are trying to extract. • How the parameterization is extracted also dictates the level of accuracy required of the source model: o Exact representation of a true physical source o Approximate representation  Non-ideal nature of source is de-embedded o Non-physical source  Post-processing of data renders exact source un-necessarySource Models for FEM S. Gedney EE699 – The Finite Element Method for Electromagnetics 2/19 Source Models • Impressed current source o Discrete point-dipole  This can only be approximated in a FEM simulation o Current density  Line current (again, approximated)  Surface current  Volume current • Plane Wave source o Plane wave injected into problem domain on a boundary o Exterior coupling (e.g., FE-BI) • Waveguide mode excitation o Approximate, aperture coupled o Exact modal excitation (Exterior coupling problem) • Discrete Lumped source model o Approximate Thevenin/Norton source model o Port mode (scattering parameter extraction) o Transmission line source model (approximate 1D modal excitation)Source Models for FEM S. Gedney EE699 – The Finite Element Method for Electromagnetics 3/19 Impressed Current Density • Recall the wave equation derived with impressed current densities: 2011ˆ1jma tot a tot a totrrra imp a impoorE E k E E d E E ndsjk E J d E M dεµµηµΩΓΩΩ⎛⎞∇× ⋅ ∇× − ⋅ Ω− × ∇× ⋅ =⎜⎟⎝⎠−⋅Ω−⋅∇×Ω∫∫∫∫     o The impressed current densities are assumed to be distributed over finite volumes jΩ and mΩ • Assume that there is an electric current density that is reduced to a surface. Then, this expression can be reduced to: 2011ˆja tot a tot a tot a improorrEEkEEdE EndsjkEJdsεηµµΩΓΓ⎛⎞∇× ⋅ ∇× − ⋅ Ω− × ∇× ⋅ =− ⋅⎜⎟⎝⎠∫∫∫   Source Models for FEM S. Gedney EE699 – The Finite Element Method for Electromagnetics 4/19 Plane Wave Source Injection • Assume that you have a finite dimensional object under test that is illuminated by a plane wave source • The object lies completely within the domain Ω, bound by Γ o The object under test is assumed to be a heterogeneous composition of conductors and penetrable materials ΩΓ • The fields modeled via the FEM method are the total field intensities o The total fields satisfy the expected boundary conditions on material/conductor surfaces 2011ˆ0a tot a tot a totrrrEEkEEdE EndsεµµΩΓ⎛⎞∇× ⋅ ∇× − ⋅ Ω− × ∇× ⋅ =⎜⎟⎝⎠∫∫  Source Models for FEM S. Gedney EE699 – The Finite Element Method for Electromagnetics 5/19 Scattered Field Formulation • The total field can be expressed as a superposition of the incident and scattered fields: o ,tot inc scat tot inc scatEEE HHH=+ = +    o Where, ,inc incEH are the incident plane wave which are propagating through the homogeneous free space in the absence of the of the object under test. o The incident electric field satisfies the weak-form equation: ()20ˆ0a inc a inc a incEEkEEdE EndsΩΓ∇× ⋅∇× − ⋅ Ω− ×∇× ⋅ =∫∫    o Expanding the total field as a function of the incident and scattered fields: ()()()2011ˆ0a inc scat a inc scatrraincscatrEEEkEEEdEEEndsεµµΩΓ⎛⎞∇× ⋅ ∇× + − ⋅ + Ω⎜⎟⎝⎠−×∇× + ⋅ =∫∫ o The incident field equation is subtracted from this, leading to:Source Models for FEM S. Gedney EE699 – The Finite Element Method for Electromagnetics 6/19 () ( )20201ˆ1ˆ11 1a scat a scat a scatroora inc a inc a incroorrEEkEEdjkEHndsE E k E E d jk E H ndsεηµεηµµΩ ΓΩ Γ⎛⎞∇× ⋅ ∇× − ⋅ Ω+ × ⋅⎜⎟⎝⎠⎛⎞⎛⎞=∇×⋅− ∇× − ⋅− Ω+ × − ⋅⎜⎟⎜⎟⎝⎠⎝⎠∫∫∫∫   • Note that the excitation appears in the form of volume current sources in material regions • Notes on Γ: o Near the exterior boundary, typically rµ = 1, and the boundary term is 0  Exception can be for layered media of infinite planar slabs. • Boundary term included, and the scattered field requires correct ABC o PEC surfaces:  Dirichlet boundary condition: ˆˆscat incnE nE×=− × o PMC surfaces:  Neuman boundary condition: ˆˆscat incnH nH×=− × ()ˆˆˆ1a scat a inc a incoo oo r oo rjk E H nds jk E H nds jk E H ndsηηµηµΓΓ Γ∴×⋅− ×− ⋅=− × ⋅∫∫ ∫    Source Models for FEM S. Gedney EE699 – The Finite Element Method for Electromagnetics 7/19 Total-Field/Scattered-Field Formulation totΩΓpwΓscatΩ • The domain Ω can be broken up into two regions, a scattered field region, and a total field region. These are separated by the boundary pwΓ. o In the total field region: 2011ˆˆ0tot pw obja tot a tot a tot a totrr rE E k E E d E E nds E E ndsεµµΩΓΓ⎛⎞∇× ⋅ ∇× − ⋅ Ω− ×∇× ⋅ − × ∇× ⋅ =⎜⎟⎝⎠∫∫∫    o In the scattered field region: ()20ˆˆ0scat pwa scat a scat a scat a scatE E k E E d E E nds E E ndsΓΩΓ∇× ⋅∇× − ⋅ Ω− ×∇× ⋅ − ×∇× ⋅ =∫∫∫    • On the boundary pwΓ, we must choose either a scattered field, or a total field as the unknown. Here, we will choose the scattered field. Thus, on pwΓ o ,tot scat inc tot scat incEE EHH H=+ = +   Source Models for FEM S. Gedney EE699 – The Finite Element Method for Electromagnetics 8/19 o This leads to: ()()20202011ˆˆˆtotpwpw objpw pwatotatotrrascatascatascat a totra inc a inc a incooEEkEEdEEkEEdE E nds E E ndsE E k E E d jk E H ndsεµµηΩΩΓΓΩΓ⎛⎞∇× ⋅ ∇× − ⋅ Ω+⎜⎟⎝⎠∇× ⋅∇× − ⋅ Ω−×∇× ⋅ − × ∇× ⋅ =−∇×⋅∇× − ⋅ Ω− × ⋅∫∫∫∫∫∫      o Where, the integral over pwΩ implies the