Electromagnetic Field Theory I (NFT I) Numerische Methoden der. 7th Lecture / 7. Vorlesung Dr.-Ing. René Marklein
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1 Numerical Methods of Electromagnetic Field Theory I (NFT I) Numerische Methoden der Elektromagnetischen Feldtheorie I (NFT I) / 7th Lecture / 7. Vorlesung Dr.-Ing. René Marklein marklein@uni-kassel.de Universität Kassel Fachbereich Elektrotechnik / Informatik (FB 6) Fachgebiet Theoretische Elektrotechnik (FG TET) Wilhelmshöher Allee 7 Büro: Raum 3 / 5 D-34 Kassel University of Kassel Dept. Electrical Engineering / Computer Science (FB 6) Electromagnetic Field Theory (FG TET) Wilhelmshöher Allee 7 Office: Room 3 / 5 D-34 Kassel
2 -D EM Wave Propagation -D FDTD Staggered Grid in Space / D EM Wellenausbreitung -D FDTD Versetztes Gitter im Raum Interleaving of the and H y field components in space and time in the -D FDTD formulation / Überlappung der -und H y -Feldkomponente in der D-FDTD-Formulierung im Raum und in der Zeit 3 n z n z n z + 3 n z + Time plane / Zeitebene n t H y ( n z ) ( n z ) ( z ) n ( ) n z + ( ) n t 3 n z n z n z + 3 n z + n t +
3 -D EM Wave Propagation FDTD Normalization / D EM Wellenausbreitung FDTD Normierung t t = ( nz, nt) ( nz, nt ) ( nz+ /, nt /) ( nz /, nt /) ( nz, nt /) Hy Hy Ex Ex Jm y µ 0 z µ 0 ( nz + /, nt+ /) ( nz+ /, nt /) t ( nz+, nt) ( nz, nt) t ( nz+ /, nt) x = x y y e x ε0 z ε0 E E H H J xref xref t = t ref t tref = t = t cref cref z = x ref z c= cref cˆ ε = εref ε µ = µ ref µ µ ref = µ 0 E ˆ x = Eref Ex ˆ E ref ε µ ε Hy = Href Hy Href = = Eref = Eref = c ref µ ref µ µ ε ref Jex = Jeref Je x Jeref = Eref t ref µ ref E ref Jm x = Jm ref Jm x Jm ref = Href = t t c ref ref ref ref ref ref ref ref ref ref E Z H = H t E E tj ( nz, nt) ( nz, nt ) ( nz+ /, nt /) ( nz /, nt /) ( nz, nt /) y y x x m y E = E t H H t J ( nz + /, nt+ /) ( nz+ /, nt /) ( nz+, nt) ( nz, nt) ( nz+ /, nt) x x y y e x 3
4 -D FDTD Staggered Grid in Space Global Node Numbering / D-FDTD Versetztes Gitter im Raum Globale Knotennummerierung ( nz, nt ) H y ( n z, n H y t ) ( nz, nt ) H y ( nz,, nt ) H y + ( nz 3/, nt /) + E ( nz /, nt+ /) ( nz /, nt /) x + + ( nz 3/, nt /) + + z 3 n z n z n z n z n z n z + n z + n z + 3 ( n, n t ) H y H y ( n, n t ) ( n, n t ) ( nn, ) H t y ( n, n t ) ( nn E, t ) x ( n, n t ) H y + ( n, n t ) + 3 n z n z n z n z n z n z + n z + n z + 3 z 4
5 -D FDTD Algorithm Flow Chart / D-FDTD-Algorithmus Flussdiagramm Start n t = Compute -D Faraday s FDTD equation: For all nodes n inside the simulation region: ( nn, t) ( nn, ) (, /) (, /) t nn t n n t Hy = Hy t Ex E x Compute -D Ampère-Maxwell s FDTD equation: For all nodes n inside the simulation region: = + + E E t H H ( nn, t /) ( nn, t /) ( n, nt) ( nn, t) x x y y Electric current density excitation: For all excitation nodes n: ˆ( nn, t+ /) ˆ( nn, t+ /) ˆ( nn, t) E = E tj y y e y Boundary condition: For all PEC boundary nodes n: ˆ ( nn, t + /) E y = 0 nt = nt + No Yes nt Nt Stop 5
6 -D FDTD Algorithm Flow Chart / D-FDTD-Algorithmus Flussdiagramm Start n t = Berechne die D-Faraday-FDTD-Gleichung: Für alle Knoten n im Simulationsgebiet: ( nn, t) ( nn, ) (, /) (, /) t nn t n n t Hy = Hy t Ex E x Berechne die D-Ampère-Maxwell-FDTD-Gleichung: Für alle Knoten n im Simulationsgebiet:: = + + E E t H H ( nn, t /) ( nn, t /) ( n, nt) ( nn, t) x x y y Elektrische Stromdichteanregung: Für alle Anregungsknoten n ˆ( nn, t+ /) ˆ( nn, t+ /) ˆ( nn, t) E = E tj y y e y Randbedingungen: Für alle IEL-Randknoten n ˆ ( nn, t + /) E y = 0 nt = nt + Nein Ja nt Nt Stopp 6
7 FDTD Solution of the First Two -D Scalar Maxwell s Equations / FDTD-Lösung der ersten beiden D skalaren Maxwell-Gleichungen Maxwell s equations / Maxwellsche Gleichungen 0 z Z Hy(,) z t = Ex(,) z t Jm y(,) z t for / für t µ 0 z µ 0 0 t T E (,) z t = H (,) z t J (,) z t t z x y e x ε0 ε0 Initial condition / Anfangsbedingung H ( z, t) = J ( z, t) = 0 t 0 y m y Ex( z, t) = Je x( z, t) = 0 t 0 J (,) z t = K ( z ) δ ( z z ) f() t t > 0 ex e0 0 0 Causality / Kausalität Hyperbolic initialboundary-value problem / Hyperbolisches Anfangs-Randwert- Problem Boundary condition for a perfectly electrically conducting (PEC) material / Randbedingung für ein ideal elektrisch leitendes Material Ex (0, t) = 0 Ex ( Z, t) = 0 t x (0, ) 0 E t = Z x (,) E z t x (, ) 0 E Z t = z = 0 z = Z 7
8 FDTD Solution of the First Two -D Scalar Maxwell s Equations / FDTD-Lösung der ersten beiden D skalaren Maxwell-Gleichungen Discrete -D FDTD equations / Diskrete D-FDTD-Gleichungen ( z, t) ( z, t ) ( /, /) ( /, /) (, /) z+ t z t z t n N Hy = Hy t Ex E x tjm y for / für n N n n n n n n n n n n z z ( n z+ /, nt+ /) ( n /, /) (, ) (, ) z+ nt n z+ nt n z nt Ex = Ex t Hy H y ( n + /, n tj ) e x z t t t Initial condition / Anfangsbedingung ( nz, nt) ( nz, nt) y m y t H = J = 0 n ( nz, nt) ( nz, nt) x = e x = t ( n, n ) ( n ) ( n n ) ( n ) ex ex δ t E J 0 n z0 z z J z t K 0 f t n = > Causality / Kausalität Boundary condition for a perfectly electrically conducting (PEC) material / Randbedingung für ein ideal elektrisch leitendes Material (, n ) E t x ( Nz, nt) = 0 = 0 n N t t (, ) n t = (, ) n t H y = 0 0 n z = (, n t ) H y (, n t ) Z = zn z nz nt n, z n E t x Hy (, ) (, ) ( N z, n t ) H y z z z z nz = Nz ( Nz, nt) x E = 0 Discrete hyperbolic initial-boundary-value problem / Diskretes hyperbolisches Anfangs-Randwert- Problem 8
9 FDTD Solution of the First Two -D Scalar Maxwell s Equations / FDTD-Lösung der ersten beiden D skalaren Maxwell-Gleichungen Excitation pulse: RC(t) Time Domain / Anregungsfunktion: RC(t) Zeitbereich Excitation pulse: RC(f) Frequency Domain / Anregungsfunktion: RC(f) Frequenzbereich 9 Magntiude RC(f) / Betrag RC(f) Amplitude RC(t) / Amplitude RC(t)
10 FDTD Solution of the First Two -D Scalar Maxwell s Equations / FDTD-Lösung der ersten beiden D skalaren Maxwell-Gleichungen 0
11 FDTD Solution of the First Two -D Scalar Maxwell s Equations / FDTD-Lösung der ersten beiden D skalaren Maxwell-Gleichungen
12 Implementation of Boundary Conditions / Implementierung von Randbedingungen Boundary condition for a perfectly electrically conducting (PEC) material / Randbedingung für ein ideal elektrisch leitendes Material (, n ) E t x ( Nz, nt) = 0 = 0 n N t t Absorbing/open boundary condition / Absorbierende/offene Randbedingung Space-time-extrapolation of the first order / Raum-Zeit-Extrapolation der ersten Ordnung For / Für t = 0.5 a plane wave needs two time steps, n t, to travel over one grid cell with the size / braucht eine ebene Welle zwei Zeitschritte, n t, um sich über eine Gitterzelle der Größe auszubreiten (, nt) (, nt ) Ex = E x ( Nz, nt) ( Nz, nt ) Ex = Ex n N t t Space-time-extrapolation of the first order / Raum-Zeit-Extrapolation der ersten Ordnung
13 FDTD Solution of the First Two -D Scalar Maxwell s Equations / FDTD-Lösung der ersten beiden D skalaren Maxwell-Gleichungen 3
14 FDTD Solution of the First Two -D Scalar Maxwell s Equations / FDTD-Lösung der ersten beiden D skalaren Maxwell-Gleichungen 4
15 FDTD Solution of the First Two -D Scalar Maxwell s Equations / FDTD-Lösung der ersten beiden D skalaren Maxwell-Gleichungen 5
16 FDTD Books / FDTD-Bücher Kunz, K. S., Luebbers, R. J.: The Finite Difference Time Domain Method for Electromagnetics. 993 Taflove, A. (Editor): Advances in Computational Electrodynamics: The Finite-Difference Time- Domain Method. Artech House, 998. Taflove, A. (Editor): Computational Electrodynamics: The Finite- Difference Time-Domain Method. Artech House, Boston, 995. Taflove, A. (Editor): Computational Electrodynamics: The Finite-Difference Time- Domain Method. nd Editon, Artech House, Boston,
17 FDTD Books / FDTD-Bücher Sullivan, D. M.: Electromagnetic Simulation Using the FDTD Method. IEEE Press, New York,
18 End of Lecture 7 / Ende der 7. Vorlesung 8
Numerical Methods of Electromagnetic Field Theory II (NFT II) Numerische Methoden der Elektromagnetischen Feldtheorie II (NFT II) /
umerical Methods of lectromagnetic Field Theory II (FT II) umerische Methoden der lektromagnetischen Feldtheorie II (FT II) / 7th Lecture / 7. Vorlesung Dr.-Ing. René Marklein marklein@uni-kassel.de http://www.tet.e-technik.uni-kassel.de
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