Finite-Difference Time-Domain Technique, Research Paper Example

FDTD has one inherent advantage over other electromagnetic simulation techniques in the frequency-domain, it provides a solution for the Maxwell’s equations present in the time domain. This technique applies a progressive discrete periodical calculations of the electromagnetic field values in time [1]. This domain approach of the FDTD technique allows the generation of broadband output by executing the program at one instance. This is also associated to the high quality scaling performance with the increase in the complexity of the problem. Efficiency is employed by quick outpacing of alternative techniques as the number of variables increase.

Basics

The FDTD technique divides time and space into discrete segments. Box-shaped cells are used to segment space. However, wavelength is larger than these spaces. Magnetic fields are located on the faces of the boxes while the electric fields are positions on the edges. This is called the Yee cell field orientation which forms the foundation of the finite-difference time-domain technique [1].

This orientations quantizes times into steps that denote the amount of time a field requires to move from one cell to another. There is an offset of the field values with respect to change in time as a result of the offset in the space between the magnetic fields and the electric fields. A leap frog schematic updates the magnetic and electric fields with the magnetic fields computed after the computation of the electric fields relative to the change in time.

The FDTD mesh (or grid) is generated from the accumulation of numerous FDTD cells that together form a 3D (three-dimensional) volume [2]. Each FDTD cell is characterized by three electric fields starting at a common node that is associated with the cells. This is the result of the overlapping of the edges and faces of the FDTD cells that are adjacent to each other. The common node of the electric fields is the origin of the three magnetic fields that characterize the faces of cell adjacent to the node.

Manipulation of the equations that compute the fields at specific locations allows the addition of materials such as dielectrics and conductors within the FDTD generated mesh. A salient example is setting the electric field computing equation to zero when adding an efficiently conducting wire segment to a given cell. This is because efficiently conducting substance or wire has an electric field that is identically zero. A wire is formed by connecting the edges of numerous end-to-end cells that are defined as efficiently conducting materials.

This model allows for the addition of other different materials by applying similar manipulations to the electric and magnetic field calculations, relative to the characteristics of the materials elected.

The FDTD grid allows for the generation of a geometrical structure that results from the association of numerous cell edges with the respective materials, with every box representing a single FDTD cell [3]. The value a waveform is added at each step time to the corresponding field value. The introduced waveform will be efficiently propagated throughout the FDTD grid relative to the characteristics of each individual cell.

The FDTD simulation has one important constraint, the dimensions (size) of the small box. This is because the box represent the upper frequency limit for the calculation that has been employed, as well as the time step size. This technique employs a rule of thumb that determines the minimum resolution ate ten cells per wave length. This also serves as the upper frequency limit. While this is in theory, the cell size tends to characterize a much smaller resolution which is important in resolving the features and dimension of the simulated structure. These include the length or thickness of a wire.

One-Dimensional Simulation

The one-dimensional simulation using the FDTD technique is applied to the simplest problems such as the simulation of a pulse propagating in a single-dimensional free space. This technique employs the Maxwell curl equations in free space that are time-dependent [4]. While the initial vectors of the equation ( and) are three dimensional, the one dimensional approach is applied by orienting the equations of the magnetic field of the plane wave in the direction and the electric field in the direction.

The subscripts denote time as. Both equations are founded on the assumption that and are incorporated in both time and space. The field is assumed to be found between the values of the field. The superscript indicates between which values before or after it occurs. In the FDTD technique, time is implicit making calculations easier and more accurate.

The calculations highlighted above are interleaved for both space and time. This is evident in the fact that the new value of takes into account the most current values of as well as the previous value of. The one-dimensional simulation employs absorbing boundary conditions in order to mitigate the reflection of and fields back into the problem space [2]. This technique can also be employed in the simulation of propagation in a dielectric medium. This is achieved through the addition of the relative dielectric constant, represented by, into the Maxwell equations.

Two-Dimensional Simulations

2D simulations are usually conducted on either the transverse magnetic (TM) mode or the transverse electric (TE) mode. The TM Mode is comprised of,, fields. The TE Mode is made up of the,, fields. Two dimensional simulation also takes into account the interleaving of fields in the calculations.

Three-Dimensional Simulation

This type of simulation is similar to two-dimensional simulation with the only difference being the logistical problems that increase the complexity of the problem. This is because this simulation employs all vector fields, with each field in three dimensions. The simulation starts with Maxwell equations and assumes the notations as the general assumption of normalized values is applied [2].

The relationship between and is the same in one-dimensional, two-dimensional and three-dimensional analysis with the difference being the presence of three equations. Three dimensional analysis also employs the use of PML with the only difference being the introduction of three direction as opposed to two in the two-dimensional analysis. One of the key constraints of the FDTD three-dimensional analysis is the steps size for both space and time. These two elements influence accuracy, stability of the FDTD technique and the numerical dispersion.