Antenna theory: Finite-Difference Maxwell’s Equations

Consider Maxwell’s curl equations in the time domain, including lossy materials:

  (1)

Equations (1) contain the source currents J and M and include losses due to conducting dielectric material σ and magnetic material losses σ∗. Both equations have the same form, with only an interchange of symbols. Expanding the curl operator, we get the following equation for the x-component of the magnetic field:

  (2)

The x-component of the electric field has the same form but with the interchanges . You obtain the equations for the y- and z-components by a cyclic variation (repeating pattern of interchanges) x → y → z → x → y, and so on. For example, the equations are reduced to two dimensions by leaving out the y-component.

FDTD calculates the field at discrete times and locations on a grid. The fields can be represented as an indexed function using integers:

Because we use central differences, for derivatives, and the magnetic (electric) field is found from the space derivative of the electric (magnetic) field, the magnetic and electric fields need to be spaced a half-space interval apart. The time derivative becomes

and means that the electric and magnetic components are interspersed at  t/2 times that which produces a leapfrog algorithm. We substitute these ideas into Eq. (2) to derive the time-stepping equation for one component:

  (3)

FDTD uses similar equations for the other components.

Yee’s Cell Figure 1 shows one cubic cell and the components of the fields. When we consider the upper face, we see that the magnetic field components are spaced a half space interval from the central electrical field and the arrows show the direction of fields. Although it would appear that the electric field is different on the upper and lower face along the z-axis, the method assumes that the field is constant throughout the cell. The magnetic fields shown are at the center of adjoining cells.

FIGURE 1 Unit cell of a Yee space lattice showing time and space separation of electric and magnetic fields in a cell.

A leapfrog solution uses stored values of the electric fields to calculate the magnetic fields at a half time interval later and stores these values. In the second step the solution takes another half time step and uses the stored values of the magnetic fields to calculate the electric fields. The method gains stability by using the half time steps and by solving for both electric and magnetic fields. Although the fields are a half time step out of synch, we can average between the two half time steps to produce simultaneous fields at a point, but we only need to do this when calculating equivalent currents on the surface used for far-field pattern calculations.