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161_1_Class14_No_vioice

# 161_1_Class14_No_vioice - EE161 Electromagnetic Waves...

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EE161 Electromagnetic Waves Spring, 2010 Prof. Y. Ethan Wang Electrical Engineering Dept. UCLA

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Lesson 14 Numerical Methods for Electromagnetics Finite Difference Time-Domain (FDTD)Method 2-D FDTD Formulation Parallel Plate Waveguide Example
Numerical Methods Finding the electromagnetic field distribution is a 3-D space is equivalent to solve Maxwell’s equations, which can be done numerically using computers Integral equations Maxwell’s equations Differential equations + Boundary conditions Linear equations or Matrix equations Time-domain approaches: solving for the time-varying field variable Frequency-domain approaches: solving for the phasor of the field FDTD, Time-domain finite element, Time-domain integral equation Finite element method, Method of Moments, mode matching Numerical methods obtain the field value at certain spatial points, time steps or the coefficients of basis functions upon which the field is expanded

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Finite Difference Time Domain (FDTD) •Emulate the electromagnetic waves in time-domain •From the curl equations of Maxwell equations •First proposed by Yee in 1966 Didn’t receive much attention until 1980’s No need to invert a big matrix!!! The result of FDTD method is the sampled electromagnetic field values on those spatial nodes and at different time steps FDTD methods are carried out based on a discretized time-space, e.g. time steps and spatial nodes on rectangular meshes.
3-D Formulation of FDTD H E E × = + t ε σ E H × -∇ = t μ Based on the curl equations: Using central difference operator to approximate any differential operator For example, if E (N) is to represent the electric field at time step N and E (N+1) represent the field at time step N+1, the time differential operator at the time step N+1/2 would now be, ) ( 2 ) ( ) 1 ( ) 2 / 1 ( t O t E E t E N N N + - 2245 + + Time-step size Infinite precision can be obtained if the time- step and grid size approach to zero

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3-D Formulation of FDTD - = y E z E t H z y x μ 1 The time-domain curl equations lead to: - = z E
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161_1_Class14_No_vioice - EE161 Electromagnetic Waves...

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