**3. Experimental validation of the CPFDTD modeling approach**

The full-wave CPFDTD technique has been employed due to its flexibility for modeling complex structures. A versatile 3-D CPFDTD algorithm, combined with a thin-wire approximation, is implemented is developed with the formulation whereby different cell sizes can be chosen along the three orthogonal axes. Geometrical symmetry is implemented using the Neumann boundary condition. The input impedance of the antenna is obtained by exciting the coaxial cable port using a Gaussian pulse where the width and delay are determined from the resonant frequency and desired bandwidth. The appropriate electric and magnetic field components are integrated at the feed port to yield the voltage across and current flowing in the inner conductor which subsequently allows the computation of the input impedance in the frequency domain using Fourier transformation. To evaluate the magnitude and phase of the far-fields a sinusoidal signal where the frequency is chosen at the resonant mode of the structure and the simulations are ramped up to steady-state value over a number of cycles to ensure convergence to the desired accuracy.

The CPFDTD model is verified against the measured resonant frequency and arfield radiation patterns. A simple prototype is simulated and constructed using only two edges bent down by an angle of 45° as shown in **Figure 3**.

These electromagnetic boundary conditions are enforced on the PEC conducting patch and ground plane which are not oriented along the Cartesian planes as follows. The electric and magnetic field components which are adjacent to the interface between two boundaries are updated using an integral approach applied to the fields that surround the contour of the appropriate cell [22]. This approach is better suited to conform to the 3-D surfaces under consideration to reduce numerical dispersion caused by traditional stair-step FDTD approximation.

As shown in **Figure 4**, the drooped surfaces are discretized using three different meshing schemes to suit a given bend angle. The vertical to horizontal mesh steps Δ*x* and Δ*z* are varied to conform to the drooping surface.

The measured resonant frequency is 1.55 GHz as compared to 1.575 GHz obtained from the CPFDT model. The measured phase response in the *E*-plane is displayed in **Figure 5**. The far-field patterns reveal asymmetry which is caused by the offset in the

*Three-Dimensional Microstrip Antennas for Uniform Phase Response or Wide-Angular Coverage… DOI: http://dx.doi.org/10.5772/intechopen.108338*

**Figure 3.**

*Prototype with two drooped edges; all dimensions are in cm.*

**Figure 4.** *Approximation of shallow, intermediate, and steep surfaces.*

**Figure 5.** *Measured and calculated phase pattern in the E-plane (x-z plane) for the antenna shown in Figure 4.*

**Figure 6.** *(a) Amplitude and (b) phase of the measured versus simulated far-field patterns in the H plane.*

**Figure 7.** *Measured and simulated elevation patterns for the 30° drooped antenna, εr = 2.2.*

antenna mount inside the anechoic chamber which is necessary in order to install the bends and cable on the antenna mount. For comparison, we referenced the calculated far-field patterns to the same offset origin. Results shown in **Figure 5** along with the amplitude and phase of the normalized *H*-plane patterns displayed in **Figure 6** show good agreement between measured and simulated results which validates our simulation approach.

Next, the bends are duplicated on the remaining two sides to arrive at the D3DMA geometry. We constructed a downward DMA to experimentally validate the final CPFDTD model. The prototype consists of a 62 × 62 mm patch, *L*f = 40 mm, *ϕ* = 60° printed on a substrate with *L*s = 100 mm, *h* = 1.5 mm, and *ε*r = 4.2. As shown in **Figure 7**, excellent agreement is seen between the simulated and measured far-field radiation patterns in the *E*- and *H*-planes.
