# 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

If the far field is calculated, the following block in this form is displayed:
      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 
Note: In the above table the spacing between columns and the number of significant digits were reduced to facilitate convenient line breaks and rendering of the data.

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

(1) $\underset{R\to \infty }{\mathrm{lim}}E\left(r\right)\text{\hspace{0.17em}}=\frac{{e}^{-j{\beta }_{0}R}}{R}{E}_{\text{far}}$

with a large distance $R=|r|$ which tends to infinity (and which in the Feko calculations is identical to infinity).
Note: The dimension of ${E}_{\text{far}}$ is voltage due to this extra distance factor R.

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:

(2) ${E}_{\text{far},\text{S}}=\frac{1}{\sqrt{2}}\left({E}_{\text{far},\phi }-{E}_{\text{far},\vartheta }\right)$

Z-polarisation:

(3) ${E}_{\text{far},\text{Z}}=\frac{1}{\sqrt{2}}\left({E}_{\text{far},\phi }+{E}_{\text{far},\vartheta }\right)$

left-hand circular polarisation:

(4) ${E}_{\text{far},\text{LHC}}=\frac{1}{\sqrt{2}}\left({E}_{\text{far},\phi }+j{E}_{\text{far},\vartheta }\right)$

right-hand circular polarisation:

(5) ${E}_{\text{far},\text{RHC}}=\frac{1}{\sqrt{2}}\left({E}_{\text{far},\phi }-j{E}_{\text{far},\vartheta }\right)$

If a plane wave is included, the results will include the radar cross section. In the case of other sources without a plane wave source, the gain or directivity is included.
Note: If a plane wave is combined with, for example, a voltage source, the active RCS is obtained, but the gain/directivity will not be computed.

## Radar Cross Section

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

(6) ${S}_{i}=\frac{1}{2}\cdot \frac{|{E}_{0}{|}^{2}}{{Z}_{F0}}$
where ZF0 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
(7) ${S}_{s}=\frac{1}{2}\cdot \frac{|{E}_{\vartheta }{|}^{2}+|{E}_{\phi }{|}^{2}}{{Z}_{F0}}$

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

(8) ${\sigma }_{Total}=\underset{R\to \infty }{\mathrm{lim}}4\pi {R}^{2}\frac{{S}_{s}}{{S}_{i}}=\underset{R\to \infty }{\mathrm{lim}}4\pi \frac{|R{E}_{\vartheta }{|}^{2}+|R{E}_{\phi }{|}^{2}}{|{E}_{0}{|}^{2}}=4\pi \frac{|{E}_{\text{far},\vartheta }{|}^{2}+|{E}_{\text{far},\phi }{|}^{2}}{|{E}_{0}{|}^{2}}$

(9) ${\sigma }_{Horizontal}=\underset{R\to \infty }{\mathrm{lim}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}4\pi \frac{|R{E}_{\phi }{|}^{2}}{|{E}_{0}{|}^{2}}=4\pi \frac{|{E}_{\text{far},\phi }{|}^{2}}{|{E}_{0}{|}^{2}}$

(10) ${\sigma }_{Vertical}=\underset{R\to \infty }{\mathrm{lim}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}4\pi \frac{|R{E}_{\vartheta }{|}^{2}}{|{E}_{0}{|}^{2}}=4\pi \frac{|{E}_{\text{far},\vartheta }{|}^{2}}{|{E}_{0}{|}^{2}}$

## Gain and Directivity

For antenna and general radiation problems Feko computes either the gain or the directivity depending far field request setting.
Note: The gain/directivity setting applies to the values tabulated in the .out file only. Any quantity can be selected in POSTFEKO.
Assume that ${P}_{t}$ is the source power and ${P}_{v}$ are losses in the structure (such as dielectric losses), then a power of ${P}_{r}={P}_{t}-{P}_{v}$ will be radiated. The directivity (relative to an isotropic point source) is then defined as follows:

(11) ${D}_{Total}=4\pi {R}^{2}\frac{{S}_{s}}{{P}_{r}}=\frac{2\pi }{{Z}_{F0}}\cdot \frac{|{E}_{\text{far},\vartheta }{|}^{2}+|{E}_{\text{far},\phi }{|}^{2}}{{P}_{r}}$

(12) ${D}_{Horizontal}=\frac{2\pi }{{Z}_{F0}}z\frac{|{E}_{\text{far},\phi }{|}^{2}}{{P}_{r}}$

(13) ${D}_{Vertical}=\frac{2\pi }{{Z}_{F0}}\cdot \frac{|{E}_{\text{far},\vartheta }{|}^{2}}{{P}_{r}}$

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:

(14) ${G}_{Total}=4\pi {R}^{2}\frac{{S}_{s}}{{P}_{t}}=\frac{2\pi }{{Z}_{F0}}\cdot \frac{|{E}_{\text{far},\vartheta }{|}^{2}+|{E}_{\text{far},\phi }{|}^{2}}{{P}_{t}}$

(15) ${G}_{Horizontal}=\frac{2\pi }{{Z}_{F0}}\cdot \frac{|{E}_{\text{far},\phi }{|}^{2}}{{P}_{t}}$

(16) ${G}_{Vertical}=\frac{2\pi }{{Z}_{F0}}\cdot \frac{|{E}_{\text{far},\vartheta }{|}^{2}}{{P}_{t}}$

Between gain and directivity the following relation holds:

(17) $\frac{G}{D}=\frac{{P}_{r}}{{P}_{t}}=\frac{{P}_{t}-{P}_{v}}{{P}_{t}}=\eta$

where $\eta$ represents the antenna efficiency.

## Polarisation and Axial Ratio

The last three columns of the far field output give the polarisation information of the scattered wave. In general the polarisation is elliptical as shown in the figure.

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

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

(18) ${E}_{\text{far},\vartheta }=A\cdot {e}^{j\alpha }\text{ }{E}_{\text{far},\phi }=B\cdot {e}^{j\beta }$

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

(19) $\stackrel{\to }{E}\left(\tau \right)=\frac{A}{r}\cdot \text{cos}\left(\tau +\alpha \right)\cdot {\stackrel{\to }{e}}_{\vartheta }+\frac{B}{r}\cdot \text{cos}\left(\tau +\beta \right)\cdot {\stackrel{\to }{e}}_{\phi }$

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:

(20) ${\tau }_{1}=-\frac{1}{2}\cdot \text{arctan}\frac{{A}^{2}\cdot \text{sin}\left(2\alpha \right)+{B}^{2}\cdot \text{sin}\left(2\beta \right)}{{A}^{2}\cdot \text{cos}\left(2\alpha \right)+{B}^{2}\cdot \text{cos}\left(2\beta \right)}$
and
(21) ${\tau }_{2}={\tau }_{1}+\frac{\pi }{2}$

Let ${E}_{1}=|E\left({\tau }_{1}\right)|$ and ${E}_{2}=|E\left({\tau }_{2}\right)|$ 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

(22) $v=\frac{{E}_{min}}{{E}_{max}}=\frac{{E}_{2}}{{E}_{1}}$

The axial ratio (Major/Minor) is defined as

(23) $v=\frac{{E}_{max}}{{E}_{min}}=\frac{{E}_{1}}{{E}_{2}}$

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 ${\stackrel{\to }{e}}_{\vartheta }$ and can be computed using

(24) $\gamma =\text{arctan}\frac{B\cdot \text{cos}\left({\tau }_{1}+\beta \right)}{A\cdot \text{cos}\left({\tau }_{1}+\alpha \right)}$

## 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.

When analyzing an antenna the source power (calculated from the input impedance) should equal the integral of the radiated power over the surface of a sphere minus any losses such as dielectric losses and finite conductivity.
Tip: Use the power integration as a partial validation of the result.

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.

The output file will produce the following output (a full 3D far field request was set to generate the below output):
      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 %)
Feko gives two values for the total power:
1. 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.
2. 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 $\phi =0°$ to $\phi =350°$ and $\vartheta =5°$ and $\vartheta =175°$ both in $10°$ increments then the first total will give the total power through the sphere. It is also possible to set the request from $\phi =0°$ to $\phi =360°$ and $\vartheta =0°$ to $\vartheta =180°$ 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

When using a receiving antenna, the received power and phase of the received signal is given as follows:
 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.