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chap2_V2 - Chapter 2 1-D Computational Electromagnetics In...

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Chapter 2 1-D Computational Electromagnetics In this chapter we discuss some typical numerical methods for one dimension. This will familiarize the reader with the computational methods that will be discussed in future chapters for two- and three-dimensional problems. In one-dimensional problems, the medium and fields depend on only one coordinate direction, say x , and independent of all other directions. 2.1 Finite-Difference Time-Domain Method In one-dimensional problems, the medium and fields only depend on one coordinate direction, say x , and independent of all other directions. In this case, Maxwell’s equation can be decoupled into two sets of problems, one with ( E y , H z ) and the other with ( E z , H y ) as their nonzero field components. For example, it can be shown easily that ( E y , H z ) are governed by ∂E y ∂x = - μ ∂H z ∂t - σ m H z - M z (2.1) ∂H z ∂x = - ∂E y ∂t - σ e E y - J y (2.2) where J y is the y component of the electric current density. Similarly, if there is a z component of the electric current density, the corresponding equations are ∂E z ∂x = μ ∂H y ∂t + σ m H y + M y (2.3) ∂H y ∂x = ∂E z ∂t + σ e E z + J z (2.4) Note that there is no coupling between the above two sets of equations. Our objective in this section is to develop a finite-difference time-domain method to solve equations (2.1) and (2.2). The solution of equation (2.3) and (2.4) is similar. 2.1.1 Finite-Difference Schemes Finite-difference schemes are Forward differencing scheme: ∂f ( x, t ) ∂x = f ( x + Δ x, t ) - f ( x, t ) Δ x + O x ) (2.5) 23
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CHAPTER 2. 1-D COMPUTATIONAL ELECTROMAGNETICS Backward differencing scheme: ∂f ( x, t ) ∂x = f ( x, t ) - f ( x - Δ x, t ) Δ x + O x ) (2.6) Central differencing scheme: ∂f ( x, t ) ∂x = f ( x + Δ x 2 , t ) - f ( x - Δ x 2 , t ) Δ x + O x 2 ) (2.7) The order of the error terms can be easily verified by Taylor expansions. Similar approximation can be written for the time derivatives. In the FDTD method, we usually use the 2nd-order, central differencing scheme through a staggered grid. 2.1.2 The Finite-Difference Time-Domain Method To solve equations (2.1) and (2.2), we first discretize the electric and magnetic fields at staggered spatial points and temporal points. Specifically, assume that the domain L = b - a is uniformly divided into I cells so that Δ x = b - a I , and the grid points for E y are at x e i = a + ( i - 1)Δ x , i = 1 , · · · , I + 1. Then the grid points for the magnetic field H z are located at x h i = x e i + 1 2 Δ x , i = 1 , · · · , I . We denote E n i E y ( x e i , n Δ t ) , H n + 1 2 i H z ( x h i , ( n + 1 2 t ) (2.8) Note that here we sample E y at the integer spatial and temporal points, and H z at half-integer spatial and temporal points. This particular choice can be reversed. As will be seen later, depending on the outer boundary conditions, one of these choices may be easier.
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