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Unformatted text preview: Chapter 2 1D 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 threedimensional problems. In onedimensional problems, the medium and fields depend on only one coordinate direction, say x , and independent of all other directions. 2.1 FiniteDifference TimeDomain Method In onedimensional 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 = ² ∂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 = ² ∂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 finitedifference timedomain method to solve equations (2.1) and (2.2). The solution of equation (2.3) and (2.4) is similar. 2.1.1 FiniteDifference Schemes Finitedifference schemes are • Forward differencing scheme: ∂f ( x, t ) ∂x = f ( x + Δ x, t ) f ( x, t ) Δ x + O (Δ x ) (2.5) 23 CHAPTER 2. 1D COMPUTATIONAL ELECTROMAGNETICS • Backward differencing scheme: ∂f ( x, t ) ∂x = f ( x, t ) f ( x Δ x, t ) Δ x + O (Δ 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 (Δ 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 2ndorder, central differencing scheme through a staggered grid. 2.1.2 The FiniteDifference TimeDomain 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 )Δ t ) (2.8) Note that here we sample E y at the integer spatial and temporal points, and H z at halfinteger 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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This note was uploaded on 01/16/2011 for the course ECE 277 taught by Professor Qingliu during the Fall '09 term at Duke.
 Fall '09
 QingLiu
 Electromagnet

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