#### Radar Cross-Section

Figure 1: Experimentally determined relative radar cross section σ /σ_{0} of a B 26 aircraft at a frequency of 3 GHz (after Skolnik).

Figure 1: Experimentally determined relative radar cross section σ /σ_{0} of a B 26 aircraft at a frequency of 3 GHz (after Skolnik).

#### Radar Cross-Section

The Radar Cross Section σ (RCS) is an aircraft-specific quantity that depends on many factors. The computational determination of the RCS is only possible for simple bodies. The RCS of simple geometric bodies depends on the ratio of the structural dimensions of the body to the wavelength.

Practically, the RCS of a target depends on:

- the physical geometry and exterior features of the target,
- the direction of the illuminating radar,
- the radar transmitters frequency,
- the electrical properties of the target's surface.

Whereas in the design of passenger airplanes more attention is paid to effectiveness and safety, in the case of an aircraft used for military purposes, care is taken to ensure that this reflective surface is as small as possible. Measures to achieve this are referred to as stealth technology.

Figure 2: The reference for the radar cross section: a metallic sphere, which in the view offers a (projected) circle with an area of 1 m².

Figure 2: The reference for the radar cross section: a metallic sphere, which in the view offers a (projected) circle with an area of 1 m²

##### What does RCS mean for radar?

The RCS of any reflector can be seen as a ratio to an idealized reference reflector. The projected area of an equivalent isotropic reflector (that is: reflecting equally in all directions) has an RCS of exactly one square meter. Reflecting equally in all directions means: in practice, this is only performed by a spherical reflector with an ideally conducting surface. From a large distance, you cannot see the spherical shape, you can only see a circular area: the so-called projection, both with the same diameter. The diameter of this sphere must be approximately 1.128 m to be the size of one square meter. Such an equivalent isotropic reflector delivers the same power per unit measure of solid angle back to the radar, regardless of the aspect angle (i.e., regardless of the direction of the radar). Isotropic reflective does not mean that this sphere would distribute the power arriving from one direction equally in all directions. The shape and size of the distribution are different, but always the same relative to the aspect angle of the illuminating radar, regardless of the angle from which the radar illuminates this sphere. However, the reference reflector can re-radiate only the power it has received:

σS_{t} |
= S_{r}· r^{2} |
with |
S_{t} – power density of the transmitter at the radar target in [W/m²]S _{r} – scattered power density at the receiving site in [W/m²]σS _{t} – is the power received and re-radiated by the radar target (in watts)σS _{t}/4π – is this power per solid angle, i.e. divided by 4π steradian (in watts per steradian)r – radius of the sphere |
(1) |

4π_{ } |

This equation (1) expresses only a power balance: only the power that arrives at this reflecting object can be reflected, and this power is radiated in many directions. From this power, however, the radar can only receive a small part again. This part depends on the effective antenna area, the antenna aperture.

Since the power density generated by the radar transmitter and arriving at the reflection point is put into a ratio with the reflected power density arriving at the radar, all other influences, such as free space attenuation and distance to the radar, are eliminated. It is assumed that for a monostatic radar, the propagation conditions are the same on the outbound and return paths.

Let S_{r} be the power density at the receiving point of the radar,
which we now see from the perspective of the reflecting object.
This power density has the unit of measurement watts per unit area: W/m².
The receiving antenna of the radar has only one
effective antenna aperture A_{r} (this is an area and a part of the surface of a sphere).
The received power of the radar antenna is then S_{r}·A_{r},
which is the power density at the antenna multiplied by its effective aperture.

However, this antenna can receive only a very small portion of the power reflected from the object in all directions, because it occupies only a small part of the surface of the sphere. This area is proportional to the solid angle Ω occupied by the total reflected power distributed on a spherical surface:

Ω = A_{r} / r^{2}
(2)

Thus, a power density (power per solid angle) of S_{r}·A_{r} / Ω arrives at the receiving antenna.
The solid angle can be replaced with the expression from equation (2) and leads to

S_{r}·A_{r} / Ω = S_{r}·A_{r} / (A_{r} / r^{2}) = S_{r}·r^{2}
(3)

The expression S_{r}·r² thus stands for the received power per unit solid angle (in watts per steradian)
and corresponds to equation (1) above.
This can then be rearranged to the equation (4) used below.

##### Non-isotropic reference reflector

In contrast to the isotropic radiator mentioned in the antenna technology, such an isotropic reference reflector can very well be constructed in reality. It would only be very unwieldy because of its dimensions. Since the direction to the radar is known in a measurement setup, a calibrated corner reflector can also be used. As usual in antenna technology, this has a gain G compared to the isotropic reflector, which, however, can be calculated out of the measurement result later.

##### Computational determination of the reflective area

In the following formula, the radar cross section indicates an effective area that captures the incoming wave and re-radiates it into space. Thus, only a power density caused by the surface of the sphere (4π r²) arrives at the receiving antenna of the radar. The radar cross section σ is defined as:

σ = 4π r^{2}· |
S_{r} |
where |
σ: apparent area in [m²], measure of the backscattering ability. S _{t}: power density of the transmitter at the radar target in [W/m²]S _{r}: scattered power density at the receiving location in [W/m²] |
(4) |

S_{t} |

The following formulas for calculation of the radar cross section are valid under the condition of optical, i.e., frequency-independent reflection at bodies that are much further away from the radar than the wavelength and which are much larger than the used wavelength of the radar.

Reflection from a sphere |
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Reflection at a cylinder |
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Reflection at a flat plate |
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Reflection at an inclined plate | …actually like the previous example, if the projection of the plate to the radar is used as the surface. Only: the reflected energy is reflected in a different direction. So the scanning radar cannot receive this energy. That is why there are bistatic radars, where the emitter and the receivers are spatially separated. |

Table 1: Reflection from geometric bodies

##### Radar cross sections for point-like targets

Targets | RCS [m²] | RCS [dB] |
---|---|---|

bird | 0.01 | -20 |

man | 1 | 0 |

cabin cruiser | 10 | 10 |

automobile | 100 | 20 |

truck | 200 | 23 |

corner reflector | 20379 | 43.1 |

Table 2: RCS for Point-Like Targets

In radar technology, point-like targets are targets whose geometric dimensions are smaller than the pulse volume of the radar set. In contrast to volumetric targets (occurring mainly in weather radar), they do not completely fill the pulse volume. In radar signal processing they occupy one resolution cell (maximum two if they are exactly on the boundary).

In reality, the radar cross section is composed of the sum of many small partial powers, which are located at different points of the reflecting object. Depending on the angle from which this object is illuminated, these partial areas have more or less influence, they may be obscured or their distance to the radar may differ by several multiples of half the wavelength so that they overlap partly constructively and partly destructively (see interference). The RCS is thus strongly dependent on the aspect angle and can no longer be easily calculated geometrically. It is usually a result of extensive practical measurements either on the original or with a model scaled down to the wavelength currently used in the measurement.

Some targets mentioned as examples have according to their geometrical extension a very large amount of the RCS and reflect therefore a rather large amount of the transmitted energy. The table on the right gives some examples of reflective surfaces in the X-band.

(Table from: M. Skolnik, “Introduction to radar systems”, 2nd Edition, McGraw-Hill, Inc 1980, page 44.

The reflective area of the corner reflector is given for an example with edge lengths of 1.5 m).