Abstract

Algorithms for a computational method of electromagnetics based on the integral form of Maxwell's equations are introduced. The algorithms are supported by the lowest- and next-to-the-lowest-order approximations of integrals over a cell surface and edge of the equations. The method supported by the lowest-order approximation of the integrals coincides with the original finite-difference timedomain (FDTD) method, a well-known computational method of electromagnetics based on the differential form of Maxwell's equations. The method supported by the next-to-the-lowest-order approximation can be considered a correction to the FDTD method. Numerical results of an electromagnetic wave propagating in a two-dimensional slab waveguide using the original and the corrected FDTD methods are also shown to compare them with an analytical result. In addition, the results of the corrected FDTD method are also shown to be more accurate and reliable than those of the original FDTD method.

Keywords: Maxwell's equations, integral form, finite-difference time-domain method, the lowest-order approximation, next-to-the-lowest-order approximation, computational method

## 1. Introduction

Maxwell's equations are considered the fundamental equations of an electromagnetic field. They consist of laws of Faraday, Ampére-Maxwell, and Gauss for magnetic and electric flux densities

$$
\partial\_l \mathbf{B} = -\nabla \times \mathbf{E},
\tag{1}
$$

$$
\partial\_t \mathbf{D} = \nabla \times \mathbf{H} - \mathbf{i},\tag{2}
$$

$$
\nabla \cdot \mathbf{B} = \mathbf{0},
\tag{3}
$$

$$
\nabla \cdot \mathbf{D} = \rho,\tag{4}
$$

where E and H are electric and magnetic fields, respectively, D and B are electric and magnetic flux densities, respectively, i is current density, ρ is charge density, <sup>∂</sup><sup>t</sup> <sup>f</sup> is the time derivative of field <sup>f</sup>, <sup>∇</sup> � <sup>A</sup> is the rotation of vector field <sup>A</sup>, <sup>∇</sup> � <sup>A</sup> is the divergence of vector A, and ρ is the electric charge density. Taking the

divergence of both sides of Eq. (2) and using Eq. (4), law of charge conservation is derived:

$$
\partial\_t \rho + \nabla \cdot \mathbf{i} = \mathbf{0}.\tag{5}
$$

Maxwell's equations combined with law of Lorentz are the foundation of electronics, optics, and electric circuits used to understand the physical structure dependence of an electromagnetic field distribution, the interaction between the structure and field, and other relevant characteristics. However, situations having analytical solutions of them are rare. Thus, computational method of electromagnetics is important.

For computational methods of electromagnetics, there are two major types, time domain and frequency domain. In a time-domain method, time is discretized. The field distribution of a particular time step is determined by Maxwell's equations and by the distribution of the previous time step. In a frequency-domain method, the time derivative is replaced by iω, where i is the imaginary unit and ω is the angular frequency. Thus, Maxwell's equations are solved. A user chooses a method by considering the analysis object, calculation accuracy, specifications of his or her computer, and other relevant factors.

The finite-difference time-domain (FDTD) method is a time-domain method used to analyze high-frequency electromagnetic phenomena in optical devices, antennae, and similar devices [1]. Its algorithm is based on the laws of Faraday (1), Ampére-Maxwell (2), and charge conservation (5). In the FDTD method, Gauss's laws (3) and (4) are not considered except for the initial condition. The reason can be easily understood by taking divergence of both sides of Eq. (1) and (2), and combining the charge conservation law (5) yields

$$
\partial\_t \nabla \cdot \mathbf{B} = \mathbf{0},
\tag{6}
$$

of its edge, and the component normal to the surface of the electric field is at the

where t<sup>0</sup> is a particular time and Δt is the time step. Let us now consider the top surface of the cell. At the center of the surface whose coordinates are x0; y0; z<sup>0</sup>

the magnetic field at <sup>t</sup> <sup>¼</sup> <sup>t</sup><sup>0</sup> and the electric field at <sup>t</sup> <sup>¼</sup> <sup>t</sup><sup>0</sup> <sup>þ</sup> <sup>1</sup>

Electro-magnetic Simulation Based on the Integral Form of Maxwell's Equations

DOI: http://dx.doi.org/10.5772/intechopen.81338

¼ �∂xEy <sup>t</sup>; <sup>x</sup>0; <sup>y</sup>0; <sup>z</sup><sup>0</sup>

<sup>∂</sup>tBz <sup>t</sup>; <sup>x</sup>0; <sup>y</sup>0; <sup>z</sup><sup>0</sup>

<sup>¼</sup> Bz <sup>t</sup><sup>0</sup> <sup>þ</sup> <sup>Δ</sup>t; <sup>x</sup>0; <sup>y</sup>0; <sup>z</sup><sup>0</sup>

<sup>2</sup> <sup>Δ</sup>t; <sup>x</sup><sup>0</sup> <sup>þ</sup> <sup>1</sup>

<sup>2</sup> <sup>Δ</sup>t; <sup>x</sup>0; <sup>y</sup><sup>0</sup> <sup>þ</sup> <sup>1</sup>

<sup>2</sup> Δt. Then, Bz t<sup>0</sup> þ Δt; x0; y0; z<sup>0</sup>

The yellow cell is used to calculate the magnetic field at time t ¼ t<sup>0</sup> þ Δt using

where the variables x, y, and z of B and E represent the x-, y-, and z-components of the B and E fields, respectively, and ∂<sup>x</sup> and ∂<sup>y</sup> represent the partial derivatives in the x- and y-directions, respectively. Replacing the partial derivatives by the

� Bz <sup>t</sup>0; <sup>x</sup>0; <sup>y</sup>0; <sup>z</sup><sup>0</sup>

Δt

Δy

<sup>2</sup> Δx; y0; z<sup>0</sup> � Ey <sup>t</sup><sup>0</sup> <sup>þ</sup> <sup>1</sup>

<sup>2</sup> Δy; z<sup>0</sup> � Ex <sup>t</sup><sup>0</sup> <sup>þ</sup> <sup>1</sup>

<sup>þ</sup> <sup>∂</sup>yEx <sup>t</sup>; <sup>x</sup>0; <sup>y</sup>0; <sup>z</sup><sup>0</sup>

<sup>2</sup> <sup>Δ</sup>t; <sup>x</sup><sup>0</sup> � <sup>1</sup>

 <sup>Δ</sup><sup>x</sup> <sup>þ</sup> <sup>O</sup>ð Þ <sup>Δ</sup><sup>y</sup> <sup>2</sup>

<sup>2</sup> <sup>Δ</sup>t; <sup>x</sup>0; <sup>y</sup><sup>0</sup> � <sup>1</sup>

are derived from Eqs. (9), (10),

<sup>2</sup> Δt by applying Eq. (1),

, (8)

<sup>þ</sup> <sup>O</sup>ð Þ <sup>Δ</sup><sup>t</sup> <sup>2</sup>

<sup>2</sup> Δx; y0; z<sup>0</sup>

<sup>2</sup> Δy; z<sup>0</sup>

,

, (9)

(10)

<sup>þ</sup> <sup>O</sup>ð Þ <sup>Δ</sup><sup>x</sup> <sup>2</sup>

(11)

,

,

center of its surface.

Yee lattice used in the FDTD method.

Figure 1.

Eq. (1) becomes

central differences yields

<sup>∂</sup>tBz <sup>t</sup>; <sup>x</sup>0; <sup>y</sup>0; <sup>z</sup><sup>0</sup>

where <sup>t</sup> <sup>¼</sup> <sup>t</sup><sup>0</sup> <sup>þ</sup> <sup>1</sup>

and (11) as the following:

<sup>¼</sup> Ey <sup>t</sup><sup>0</sup> <sup>þ</sup> <sup>1</sup>

<sup>¼</sup> Ex <sup>t</sup><sup>0</sup> <sup>þ</sup> <sup>1</sup>

<sup>∂</sup>xEy <sup>t</sup>; <sup>x</sup>0; <sup>y</sup>0; <sup>z</sup><sup>0</sup>

<sup>∂</sup>yEx <sup>t</sup>; <sup>x</sup>0; <sup>y</sup>0; <sup>z</sup><sup>0</sup>

65

$$
\partial\_t(\nabla \cdot \mathbf{D} - \rho) = \mathbf{0},\tag{7}
$$

the time derivatives of Eq. (3) and (4), respectively. This means that Gauss's laws of electric and magnetic flux densities are always satisfied when they are initially satisfied.

In the next section, an algorithm of the original FDTD method is shown. Next, a corrected algorithm of the FDTD method based on the integral form of Maxwell's equations is shown [2, 3]. Then, a numerical result of the propagation of electromagnetic waves in a two-dimensional slab waveguide is shown. In the subsequent section, the accuracy of the original and corrected FDTD methods is compared by showing the differences between the computational and analytical methods. The analytical method is shown in the appendix. The last section is devoted to conclusions.
