Eddy Current Model Variational Current and Voltage Excitations for the Formulations Eddy Current Model Coupling Fields and Circuits Ralf Hiptmair and Oliver Sterz Generator Currents oliver.sterz@iwr.uni-heidelberg.de Excitation IWR-TS University of Heidelberg by Contacts Nonlocal Excitations Summary O. Sterz, IWR Simulation in Technology, University of Heidelberg MACSI-net Workshop 2003 – p.1/28
Eddy Current Approximation Eddy Current material laws Model Variational = = ǫ E curl H J D Formulations = − ∂ t B = µ H curl E B Coupling Fields div B = 0 = σ E + J g J and Circuits div D = ρ Generator Currents Excitation E : electric field strength ǫ : permittivity by Contacts H : magnetic field strength µ : permeability Nonlocal D : dielectric displacement σ : conductivity Excitations B : magnetic induction Summary J : current density ρ : charge density O. Sterz, IWR Simulation in Technology, University of Heidelberg MACSI-net Workshop 2003 – p.2/28
Eddy Current Setting Eddy Current Ω C : union of all conductors Ω C,i Ω I Model Ω I : insulator J G Ω G ∂ Ω Variational Ω C, 1 Ω = Ω C ∪ Ω I Ω C, 2 Formulations Ω G = supp J G Coupling Fields ∂ Ω = ∂ Ω e ∪ ∂ Ω h , and Circuits ∂ Ω e ∩ ∂ Ω h = ∅ a typical eddy current setting Generator Currents boundary conditions: Excitation n × E = f on ∂ Ω e ⊂ ∂ Ω n × H = g on ∂ Ω h ⊂ ∂ Ω and by Contacts Nonlocal important spaces: Excitations H ( curl ; Ω) := { u ∈ L 2 (Ω) , curl u ∈ L 2 (Ω) } Summary H 0 ( curl ; Ω) := { u ∈ L 2 (Ω) , curl u ∈ L 2 (Ω) , n × u | ∂ Ω = 0 } O. Sterz, IWR Simulation in Technology, University of Heidelberg MACSI-net Workshop 2003 – p.3/28
H -based Formulation (magnetic) Eddy Current V ( J g , g ) := { H ′ ∈ H ( curl ; Ω) , curl H ′ = J g in Ω I , n × H ′ = g on ∂ Ω h } Model Variational Formulations V 0 := V (0 , 0) Coupling Fields and Circuits Find H ∈ C 1 (]0 , T [ , V ( J g , g )) , such that for all H ′ ∈ V 0 Generator Currents � � Excitation 1 σ curl H · curl H ′ d x + ∂ t ( µ H ) · H ′ d x by Contacts Ω C Ω Nonlocal � � Excitations 1 σ J G · curl H ′ d x + ) · H ′ dS . ( n × E = � �� � Summary = f Ω C ∂ Ω e (initial value skipped here) O. Sterz, IWR Simulation in Technology, University of Heidelberg MACSI-net Workshop 2003 – p.4/28
A -based Formulation (electric) div B = 0 in R 3 = ⇒ B = curl A Eddy Current = ⇒ E = − ∂ t A − grad v ( v : scalar potential) Model ⇒ E = − ∂ t A “temporal gauge” = � Variational W ( f ) := { A ′ ∈ H ( curl ; Ω) , n × A ′ = − Formulations f dt on ∂ Ω e } Coupling Fields and Circuits Find A ∈ C 1 (]0 , T [ , W ( f )) , such that for all A ′ ∈ W (0) Generator Currents � � Excitation 1 µ curl A · curl A ′ d x + σ∂ t A · A ′ d x by Contacts Ω Ω C Nonlocal � � Excitations J G · A ′ d x − ) · A ′ dS . ( n × H = � �� � Summary = g Ω ∂ Ω h (again initial value skipped) O. Sterz, IWR Simulation in Technology, University of Heidelberg MACSI-net Workshop 2003 – p.5/28
A -based Formulation (electric) II Eddy Current remark on the uniqueness Model Variational “ungauged” formulation = ⇒ Formulations in Ω I A and E = − ∂ t A are only unique modulo an “electrostatic part” Coupling Fields and Circuits curl E and thus the magnetic field H is unique Generator for uniqueness: fix conductor charges and div A in Ω I Currents in most situations the “electrostatic part” is of no interest Excitation by Contacts Nonlocal don’t care about non-uniqueness Excitations Summary O. Sterz, IWR Simulation in Technology, University of Heidelberg MACSI-net Workshop 2003 – p.6/28
Coupling Quantities Desirable: coupling by U and I Eddy Current Model U (?), I Variational circuit Formulations eddy current equations model Coupling Fields and Circuits U, I Generator in eddy current model: Currents � I = J · n dS Excitation Σ by Contacts � however U γ = E · d s depends on path γ ! Nonlocal γ Excitations Summary Using U γ for coupling fields and define voltage through power circuits cannot be accomplished. O. Sterz, IWR Simulation in Technology, University of Heidelberg MACSI-net Workshop 2003 – p.7/28
Coupling by I and P y—Circuit View Eddy Current Do coupling by conservation of current I and power P : Model Variational P , I Formulations circuit eddy current Coupling Fields equations model and Circuits P, I Generator Currents circuit view: Excitation eddy current problem is seen as a one (or multi) port from by Contacts circuit model Nonlocal define voltage drop at (every) port by Excitations Summary U = P I O. Sterz, IWR Simulation in Technology, University of Heidelberg MACSI-net Workshop 2003 – p.8/28
Coupling by I and P —Eddy Current View eddy current view: Eddy Current power balance implied by the eddy current model Model (magneto-quasistatic Poynting theorem): Variational Formulations P mag + P Ohm = P = P Ω + P ∂ Ω Coupling Fields and Circuits with � � Generator σ | E | 2 d x P mag := ∂ t B · H d x , P Ohm := Currents Excitation Ω Ω C � � by Contacts P Ω := − E · J G d x , P ∂ Ω = − ( E × H ) · n dS . Nonlocal Ω ∂ Ω Excitations Summary sources are generator current distributions or inhomo- geneous boundary conditions O. Sterz, IWR Simulation in Technology, University of Heidelberg MACSI-net Workshop 2003 – p.9/28
Eddy Current Model Variational Formulations Coupling Fields Now look at several different variational and Circuits formulations for coupling... Generator Currents Excitation by Contacts Nonlocal Excitations Summary O. Sterz, IWR Simulation in Technology, University of Heidelberg MACSI-net Workshop 2003 – p.10/28
Generator Current Settings (a) (b) Eddy Current Model Ω C J G Variational Ω I Formulations Ω I Coupling Fields J G and Circuits Ω C Generator Currents Excitation by Contacts (a) closed current loops in Ω I , that is supp J g ⊂ Ω I , which model Nonlocal coils with known currents Excitations (b) current sources adjacent to conductors, i.e., supp J g ∩ Ω C � = ∅ Summary we only consider supp J g ⊂ Ω I , n × E = 0 on ∂ Ω for simplicity O. Sterz, IWR Simulation in Technology, University of Heidelberg MACSI-net Workshop 2003 – p.11/28
H -based Current Excitationg � J G · n dS Eddy Current choose a J G such that I = Model Σ choose H G ∈ C 1 (]0 , T [ , H ( curl ; Ω)) such that curl H G = J G Variational Formulations for all times (e.g. by the Biot-Savart law) Coupling Fields variational formulation Seek H ∈ H G + C 1 (]0 , T [ , V 0 ) such that for all H ′ ∈ V 0 and Circuits Generator � � Currents 1 σ curl H · curl H ′ d x + ∂ t ( µ H ) · H ′ d x = 0 . Excitation Ω C Ω by Contacts Nonlocal power: � � � Excitations σ | curl H | 2 d x + 1 P = ∂ t ( µ H ) · H d x = ∂ t ( µ H ) · H G d x , Summary Ω C Ω Ω � voltage is given by U · I = ∂ t ( µ H ) · H G d x Ω O. Sterz, IWR Simulation in Technology, University of Heidelberg MACSI-net Workshop 2003 – p.12/28
H -based Voltage Excitation use scaled quantities to represent a unit current source Eddy Current � Model J 0 · n dS = 1 , J G = I J 0 , curl H 0 = J 0 Σ Variational Formulations voltage can be written as Coupling Fields � and Circuits U = ∂ t ( µ H ) · H 0 d x Ω Generator Currents variational formulation for voltage excitation: Seek Excitation H ∈ C 1 (]0 , T [ , V 0 ) and I ∈ C 1 (]0 , T [) such that for all H ′ ∈ V 0 by Contacts � � 1 Nonlocal σ curl H · curl H ′ d x + ∂ t ( µ ( H + I H 0 )) · H ′ d x = 0 Excitations Ω C � Ω Summary ∂ t ( µ H ) · H 0 d x = U . Ω O. Sterz, IWR Simulation in Technology, University of Heidelberg MACSI-net Workshop 2003 – p.13/28
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