# Far Fields and Receiving Antennas

Far fields and receiving antennas data consist of the electric far field data, RCS, gain, directivity and radiated power.

## Far Fields and Polarisation Types

VALUES OF THE SCATTERED ELECTRIC FIELD STRENGTH IN THE FAR FIELD in V Factor e^(-j*Re{BETA}*R)/R not considered LOCATION ETHETA EPHI directivity in dB ... THETA PHI magn. phase magn. phase vert. horiz. total ... 0.00 0.00 2.626E-16 -178.03 2.321E-16 22.06 -308.6129 -309.6881 -306.1070 ... 2.00 0.00 7.271E-02 104.04 0.000E+00 0.00 -19.7678 -999.9999 -19.7678 ... 4.00 0.00 1.449E-01 104.02 0.000E+00 0.00 -13.7772 -999.9999 -13.7772 ... POLARISATION axial r. angle direction 0.1758 138.76 RIGHT 0.0000 180.00 LINEAR 0.0000 180.00 LINEAR Gain is a factor of 1.00000E+00 ( 0.00 dB) larger than directivity The directivity/gain is based on an active power of 8.35911E-03 W and on a power loss of 0.00000E+00 W

For incident plane waves, the displayed values are the values of the scattered field, that is the incident field is not taken into account. However, for any other sources (such as elementary Hertzian dipoles or impressed radiation patterns), the fields radiated by the source are included.

In the far field a complex field strength ${E}_{\text{far}}$ is defined using the relation

In the .out file the $\vartheta $ (vertical) and $\phi $ (horizontal) components of ${E}_{\text{far}}$ are tabulated by magnitude and phase, that is ${E}_{\text{far,}\vartheta}$ and ${E}_{\text{far,}\phi}$ .

Using POSTFEKO results for other polarizations can be extracted. The corresponding formulas are as follows:

S-polarisation:

Z-polarisation:

left-hand circular polarisation:

right-hand circular polarisation:

## Radar Cross Section

For the radar cross section, the incident plane wave with complex amplitude ${E}_{0}$ carries a power density of

_{F0}denotes the wave impedance of the surrounding medium. The incident wave gets scattered on the object and a wave is reflected with the scattered power density

The radar cross section (RCS) $\sigma $ is then defined as follows:

## Gain and Directivity

For the gain a similar definition is used, except that the source power ${P}_{t}$ and not the radiated power ${P}_{r}$ is acting as reference as follows:

Between gain and directivity the following relation holds:

where $\eta $ represents the antenna efficiency.

## Polarisation and Axial Ratio

The coordinates are ${\overrightarrow{e}}_{r}$ , ${\overrightarrow{e}}_{\vartheta}$ and ${\overrightarrow{e}}_{\phi}$ , and the view is in the direction of the propagation of the wave ( ${\overrightarrow{e}}_{r}$ ).

To evaluate these quantities, the magnitude and phase of the far field components are defined as follows:

Using the abbreviation $\tau =\omega t-{\beta}_{0r}$ the temporal field strength vector in space can be written as:

This equation describes the polarisation ellipse depicted in the figure.

The minimum and maximum values of the field strength magnitude can be found at following times:

Let ${E}_{1}=|E({\tau}_{1})|$ and ${E}_{2}=|E({\tau}_{2})|$ and assume that ${E}_{1}>{E}_{2}$ , then according to the figure ${E}_{\mathrm{max}}={E}_{1}$ and ${E}_{\mathrm{min}}={E}_{2}$ .

The axial ratio (Minor/Major) is defined as

The axial ratio (Major/Minor) is defined as

A ratio (Minor/Major) of 0 means that the wave is a linearly polarised wave, but if the ratio (Minor/Major) has a value of 1 then it is a circularly polarised wave. The direction of rotation is right hand circular (RHC) for $0<\alpha -\beta <\pi $ and left hand circular (LHC) for $\pi <\alpha -\beta <2\pi $ .

Feko also computes and prints the polarisation angle $\gamma $ . It is the angle between the major axis of the polarisation ellipse and the unit vector ${\overrightarrow{e}}_{\vartheta}$ and can be computed using

## Poynting Vector and Radiated Power

If the far field request is set to request 2 or more points for both the theta and phi directions, then the Poynting vector is integrated over the specified sector (see the detailed discussion for the FF card). This integration provides the radiated power and is given below the field values.

It is also possible to set the far field request to only integrate the far field power without writing the field values to the output file.

VALUES OF THE SCATTERED ELECTRIC FIELD STRENGTH IN THE FAR FIELD in V Factor e^(-j*Re{BETA}*R)/R not considered Integration of the normal component of the Poynting vector in the angular grid DTHETA = 5.00 deg. and DPHI = 5.00 deg. (2701 sample points) angular range THETA angular range PHI radiated power -2.50 .. 182.50 deg. -2.50 .. 362.50 deg. 8.19957E-03 Watt 0.00 .. 180.00 deg. 0.00 .. 360.00 deg. 8.08720E-03 Watt Polarisation dependent radiated power: horizontal polarisation: 4.81599E-09 Watt ( 0.00 %) vertical polarisation: 8.08719E-03 Watt (100.00 %) S polarisation: 4.04360E-03 Watt ( 50.00 %) Z polarisation: 4.04360E-03 Watt ( 50.00 %) left hand circular pol.: 4.04360E-03 Watt ( 50.00 %) right hand circular pol.: 4.04360E-03 Watt ( 50.00 %)

- The first line gives the total power assuming that each specified point is located at the center of an incremental integration area. The effective area is therefore slightly larger than the area defined by the start and end angles.
- The second line gives the total power integrated over an area defined by the start and end angles.

For example, assuming a far field integration from $\phi =0\xb0$ to $\phi =350\xb0$ and $\vartheta =5\xb0$ and $\vartheta =175\xb0$ both in $10\xb0$ increments then the first total will give the total power through the sphere. It is also possible to set the request from $\phi =0\xb0$ to $\phi =360\xb0$ and $\vartheta =0\xb0$ to $\vartheta =180\xb0$ in which case the second total will give the correct power through the sphere.

The polarisation dependent power displayed in the second block of data is calculated according to the effective area of the second line.

## Receiving Antenna

Receiving antenna (far field pattern) with name: FarFieldReceivingAntenna1 RECEIVED POWER FOR IDEAL RECEIVING ANTENNA (FAR FIELD PATTERN) Received power (ideal match assumed): 2.6019E-03 W Relative phase of received signal: -8.6549E+01 deg.