This page describes how to calculate the directivity of an antenna, defined by IEEE as the ratio of the radiation intensity in a given direction from the antenna to the radiation intensity averaged over all direction, in FDTD by using a set of monitors and the far field projection functions built into the directivity analysis group. The user will learn how to obtain the azimuth and elevation directivity patterns, radiated power, and maximum directivity of an "ideal" dipole source using the directivity analysis group.

## Projections onto a Spherical Surface

The directivity analysis group is a natural extension of the Projections from a monitor box analysis group in which surface equivalence is once again invoked to determine the far fields radiated outside of a closed box from the radiating sources (Es , Hs) located inside the box, as depicted in the above figure. These farfields (Er , Eθ , Eϕ) are projected onto a sphere of a sufficiently large radius (in this case, 100meters) using the farfieldexact script command and related to the angular distribution of radiation intensity (U) - the power radiated from an antenna per unit solid angle.This allows us to determine the directivity (D) which in mathematical terms is written as \(D = 4 \pi U /P_{rad} \) in which Prad is the power radiated out of the monitor box. Often the maximum directivity is reported for an antenna, which is defined as maximum (Dmax) over all angles.

## Infinite Ground Plane

Illustration of a set of monitors above an infinite ground plane and their virtual images.

In many instances, it is appropriate to consider the antenna as mounted on top of a metallic ground plane. When the ground plane is sufficiently large and we are operating at low enough frequencies, it may be modeled as an infinitely large perfect electrical conductor (PEC). The presence of the PEC ground predictably alters the radiation properties of the antenna system. For instance, any energy radiating from the antenna directed towards the ground plane undergoes a reflection and reflects back, and the antenna will setup currents on the ground plane which radiate themselves.

Image theory is used in the directivity analysis group to account for the infinite PEC ground plane. The antenna over the ground plane is replaced with an equivalent system of the antenna and its virtual image, which provides the correct field distribution above the ground plane. Below the ground plane, the fields are zero. This technique is achieved in the analysis group by projecting the set of monitors above the ground plane into the negative half-space (-z), which is equivalent to having a set of monitors below the ground plane and projecting them into the positive half space (+z). This procedure allows us to use a finite-sized box to capture all the radiated fields, even though induced currents on the infinite PEC ground plane contribute to the radiation outside of the box.

## User Input and Results

After inserting the directivity analysis group, a set of monitors forming a box will be inserted into the simulation domain. In the Setup Tab of the analysis group, the user has must specify the span of the monitor (x span, y span, z span) and the down sampling of the fields on the monitors. Keeping down sample at 1 results in no down sampling. If an infinite ground plane is in the simulation, the variable inf gp should be set to 1.

Note: Monitor and Antenna Setup For Infinite Ground Plane When the user specifies an infinite ground plane, the analysis group assumes the ground plane’s surface normal is in the z direction and is located on the z=0 plane. Furthermore, the z span variable specifies the span of the monitor starting from z=0 to z=z span. |

If the power from the antenna’s feeding source passes through the box monitor and is not to be included in the directivity calculation, the window source variable should be set to 1. The user must specify which monitor the window should be located on by setting the window surface variable to x1, x2, y1, y2, z1, and z2. The size and center position of the monitor (window center (i, j), window width, window height) fix the size and center position of the window. For example, if window surface=z1, window center (i,j) should be entered as the x-axis position and y-axis center position of the window and window width and height as the x span and y span of the window. The variables coordinate entries (i,j) and the window's surface normal form a right handed triplet.

Under the Analysis Tab of the directivity analysis group, the user can specify the far-field resolution in θ and ϕ (in degrees) and the frequency to obtain the far-field results at.

Note: Directivity in Freespace In many antenna applications, the directivity is measured and/or calculated in free space and not inside other dielectric mediums. The directivity analysis group assumes that the radiation is occurring into either unbounded free space or, in the case of the infinite ground plane, a bounded half-space. |

Note: Resolution vs analysis time The resolution can significantly affect the time it takes to compute the directivity. Reducing the resolution can noticeable speed up the analysis, but can have a significant impact on the calculated radiated power and the absolute value of the directivity. Care should be taken to make sure any fine features in the far field are properly resolved with the specified resolution. |

The results view of the analysis group provides the θ (Dθ) and ϕ (Dϕ) polarizations of the directivity in linear units and the total radiated power (Prad). The total directivity (Dtot) is simply the sum of the two polarizations.

### Radiated Power

The calculation of Prad assumes that all the power was captured by the box of monitors and that this captured power is from the radiating antenna element itself and not from external sources from outside the box. However, in practice the antenna will need to be excited using an external source (termed an antenna feed), which must pass through the box. The power carried by the antenna feed will significantly influence the calculated Prad. To correct for this, we allow the user to specify a window around the antenna’s feed which subtracts out the power carried by that feed line. The size of this windows must be carefully chosen to capture only the input power and not the radiated power from the antenna. In most cases, the location and size of the source window and source should match (see the Rectangular Probe Antenna example for additional details). In cases where the antenna feed itself contributes to the total radiation, the impact of the window's size on the directivity and radiated power should be tested.

Note: Radiated Power (Prad) Normalization The calculation of radiated power is similar to the approach used in the Power transmission box analysis group. The transmission is measured through each monitor (subtracting our the window if needed) and then the radiated power is calculated with the formula \(P_{rad} = trans \times sourcepower\). When using the mode source or ports, the input power into a waveguide or transmission line may not equal the sourcepower which can introduce a normalization error into Prad. One way to correct for this error is to re normalize the radiated power to the source's/port's input power, as done in the Quarter-wave Monopole example. The calculation of radiated power is similar to the approach used in the Power transmission box analysis group. The transmission is measured through each monitor (subtracting our the window if needed) and then the radiated power is calculated with the formula \( P_{rad} = trans \times sourcepower\). When using the mode source or ports, the input power into a waveguide or transmission line may not equal the sourcepower which can introduce a normalization error into Prad. One way to correct for this error is to re normalize the radiated power to the source's/port's input power, as done in the Quarter-wave Monopole example. |

## Ideal Dipole Source

In Projections from a monitor box, the far-field patterns are found for two dipoles surrounded by the box monitors. Here, we use the directivity analysis group to investigate the radiation performance of a single dipole radiating into unbounded free space and a single dipole radiating into free space bounded by a PEC infinite ground plane for z ≤ 0. The associated files for this section, dipole_directivity.lsf and dipole_directivity.lms are used to find the directivity patterns and radiated power of the dipole antenna. A perspective view of the two simulation models is shown below. The yellow box shows the edges of the monitors, which are grouped together in the analysis group. Whereas the image on the left for the unbounded antenna reveals the box completely encloses the antenna, the images on the right for the bounded antenna reveals the box only partially encloses the antenna, in which the bottom edges of the monitors touch the PEC ground plane.

Running the script runs the simulation file for both bounded and unbounded simulation setups and generates the plot results shown below. It is evident there is a strong match between the theoretical (derived from [1] Balanis) and the FDTD results. Although only Dθ is shown, Dϕ from FDTD is well below than -120 dB, which is well within numerical error to the theoretical value of \(-\infty\) dB.

=======Unbounded Dipole=========

Incident Power: 10.015 fW

Radiated Power: 9.996 fW

Radiation Efficiency: 99.8 %

Directivity: 1.761 dB

=========Bounded Dipole=========

Incident Power: 5.008 fW

Radiated Power: 4.998 fW

Radiation Efficiency: 99.8 %

Directivity: 4.771 dB

The script also generates the radiation performance of the unbounded and bounded antenna. As expected, in the absence of loss the radiated and incident power are almost equal, resulting in a radiation efficiency of nearly 100%. Since the dipole source is used, the incident power is calculated from the dipolepower script command at λ=0.45μm. The calculated directivity of the unbounded and bounded antennas is extremely close to their theoretical directivity of 1.761dB and 4.761dB, respectively.

### Related publications

[1] C. A. Balanis, Antenna Theory and Design, 4th Edition. John Wiley & Sons (2016).