10.5_WiresJunctions

10.5_WiresJunctions - Thin Wire Modeling Donald R. Wilton...

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Thin Wire Modeling Donald R. Wilton Nathan Champagne
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Thin Wire Assumptions Current has only an axial component Current is azimuthally invariant Kirchhoff’s law applies at junctions: Current vanishes at wire ends 0.01 a λ ⇒≤ 0 V + 2 a 1 I 3 I 2 I E i 2 0 ˆ ˆ () ( ) 2 , Ia a d a J π φπ ≡+ rJ r ρ r r on wire axis A A 123 III = + ˆ a ρ
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A Wire Is Modeled as Collection of Straight, Thin Tubular Conductors 11 2 2 12 , 1 ee ξξ =+ += rr r Wire axis parameterization: r 2 e r 1 e r e a () 1 1 e h ξ = r A Λ 22 2 2 e h = r A Λ 1 A 2 A 1 2 ˆ I A Maintain current continuity at each segment junction Maintain current continuity at each segment junction I I () () 1 2 2 ˆ , , N nn n e e e II = =∈ r (global) (local) ± A C Λ ΛΛ e e = ± ± C
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Discretization and Geometry Data Structure Discretized wire structure e e = ± ± C 7 6 5 2 4 3 1 y x C 1 C 2 C 3 C 4 C 6 C 5 1 2 3 4 5 6 7 1.0000 0.5000 0.0000 -0.5000 -0.8000 0.0000 0.0000 0.1500 -0 .0500 0.0000 -0.1500 -0.8000 0.4000 0.8000 Global Node Number Coordinates x y Data structure for element nodes 0.1000 0.0500 0.0000 0.0000 0.0000 0.0000 0.0000 z
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Element Connectivity Data Structure Element to node mapping 1 2 3 4 5 6 1 2 3 4 3 6 2 3 4 5 6 7 Element Number Global Node Numbers Local Node 1 Local Node 2 7 6 5 2 4 3 1 y x C 1 C 2 C 3 C 4 C 6 C 5 • In addition to the connection data, we must associate a radius a e with each segment
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Element DoF Data Local Indices, Element e 1 2 # DoF’s DoF Index # DoF’s DoF Index 0 0 1 +1 1 e 2 3 4 5 6 1 -1 2 +2,+4 1 -2 1 +3 1 -3 0 0 1 -4 1 +5 1 -5 0 0 1 1 e i σ + = if sign of global DoF corresponding to th DoF of element is positive, if sign of global DoF corresponding to th DoF of element is negative ie Local Element-to-Global DoF mapping View in next slide 7 6 5 2 4 3 y x C 1 C 2 C 3 C 4 C 6 C 5 0 1 2,4 3 5 0 0 1
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KCL Easily Enforced Using PWL Bases N J -1 independent currents at a junction of N J line segments • Select independent bases in N J -1 arms of junction and overlap them onto remaining arm 2 24 II + Current out of at junction = C Current Charge z -ax is C 2 C 3 C 5 1 3 4 2 y 5 2 I 1 I J N I 3 I # " 123 0 J N III I + +++ = "
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EFIE (Pocklington) Formulation for Total Wire Current [] ˆˆ [ ] [ ; ] , 1 ;, ,, si mn n m m m mn m n m n ef ij ZI V V Zj K K j Z ωμ ωε −⋅ =⋅ = =< > ⎡⎤ > +< > ⎣⎦ EE E i AA Λ Λ Λ ∇Λ Applying boundary condition on the wire surface leads to the EFIE with moment equation where with element matrix 1 [;] ef e f ij i j e i jK K j > > <> E i Λ Λ Λ and element excitation vector Obs. pt. integral is usually reduced to a one-point angular integration
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Kernel for Thin Wire Integral Equation a r Kernel is the potential at
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10.5_WiresJunctions - Thin Wire Modeling Donald R. Wilton...

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