Current vectors can be in x-y-z directions Galerkin method used 10ġ1 3D Basis functions Rao-Wilton-Glisson (RWG) basis functions 2D picture, has 3D analog Linear roof top functions To avoid low frequency illconditioning basis functions are decomposed into curl-free and divergence free parts At low frequencies the problem decomposes into an electrostatic and magnetostatic problems Finding loops (J.D.

Green s functions and Hankel transforms Text books: J.Mosig, W.Chew Krylov Subspace Iterative solvers Yale, Bell Labs Reduced order modeling CMU, Bell Labs, MIT 8ĩ 3D Formulation 3D volume integral formulation Mixed-potential Integral Equation (MPIE) Time harmonic Electric field Ohmic loss Vector Potential Scalar Potential 9ġ0 Discretization 3D volume formulation Charge: Discretized into triangular and rectangular surface charge elements Currents Rectangular and triangular prisms that carry volume currents. Current 7Ĩ Influencing work Drawn from the best numerical techniques of last decades FMM Yale, NYU (Rokhlin, Greengard) Fast Solvers MIT (White and students), Bell Labs (Kapur, Long).

Benchmark is always comparison to measurement. Wipro on TSMC 90nm 5Ħ Technology 65nm technology node 40 dielectric layers Layered lossy substrate lossless to 1000 S/m 10 layers of metal 0.1um width/thickness for interconnect metals Special thin film capacitors Ultra thick metals 3um thick top metals for low loss components like inductors 6ħ Accuracy considerations Final solution accuracy required is about 1% All circuits and devices are built to operate within 3 sigma variation of about 10% The solver formulation is optimized for this 3 significant digit accuracy realm The physics captures reality very well. 5 Full circuits Full circuit 1 VCO 3 : inductors + interconnect+ capacitor bank Diplexer 2 1.

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