Parent Category: 2022 HFE

*By Pasquale Dottorato*

**Back-Scatter Field**

Given a generic object illuminated by an electromagnetic wave (far-field hypothesis), it is reasonable to expect that in general it will re-radiate the incident power in all directions of space, with different intensity in the individual directions in relation to the characteristics of the object (shape, size, materials, etc.). If this object coincides with an antenna, it is possible to study the characteristics of the diffused wave as a function of the characteristic parameters of the antenna (gain, polarization, etc.) and the adaptation relationship of the antenna with its load.

**Radar Cross Section: Definition**

The differential radar section or Radar Cross Section (RCS) is a quantity that measures the intensity of power diffused by a generic object in the direction θs, φs [1]; in particular, called S_{inc} the incident power density (we suppose by simplicity a plane or locally plane incident wave), we have:

(1)

where dP represents the power radiated by the object in the cone of infinitesimal amplitude dΩ centered around

the direction θs, φs. Starting from the differential RCS, the total RCS is defined as follows

(2)

While the differential RCS is linked to the directive properties of diffusion, the total RCS gives a measure of the overall power re-radiated by the object. As defined in equation (1), the unit of measurement of the differential RCS is m^{2} / srad; although the steradian is a unit of dimensional measurement (for which dimensionally σ is in m^{2}), it is sometimes preferred to slightly modify the definition in order to measure σ simply in m^{2}:

(3)

**Antenna Scattering**

In general, any object has a radar section, and therefore re-radiates the incident power according to its own specific pattern described by the σ (θ, φ) defined in the previous paragraph.

In electromagnetic terms, the phenomenon of diffusion is generally originated by the motion of the electrons inside the object due to the effect of the incident field. Limiting itself to the case of perfectly conductive objects, the electrons move in response to the stress of the incident field so as to create a field equal to and opposite to it (so that the electric field inside a perfect electric conductor is necessarily always zero). If the incident field is time-varying, the electrons move continuously in such a way as to guarantee, instant by instant, the null field inside the material [2]: the field generated by the motion of the electrons obviously represents the field scattered by the object.

Figure 1 • Circuit representation of the Antenna.

In relation to the properties of the materials, the shape and size of the object, the diffuse field will have different intensities in the different directions of space. Differential RCS takes into account these directional properties, but its evaluation is often very difficult precisely because it is linked to the specific geometry of the object (which can be quite complicated) and to the properties of the materials.

Particular attention deserves the case in which the illuminated object is an antenna (or consists of an antenna as a relevant part). Called Z_{L} the impedance that closes the antenna terminals, the incident field determines the presence of a voltage generator in the equivalent circuit of the receiving antenna, and therefore a current I through the load impedance (and therefore to the antenna terminals).

It is known that when the terminals of an antenna are crossed by current I, it radiates an appropriate electromagnetic field, and this is true not only when it is used in transmission, but also when it is used in reception. In particular, the field radiated of the current I at the terminals may be expressed as follows [2]

( 4)

1. Phase factor that, in “normal environment” defines the propagation of the electromagnetic wave and attenuation. For normal environment [3] we mean a media with these properties:

- Isotropic
- Homogeneous
- Non-Dispersive
- Without Loss

2. Far Field factor, it shown that a flat or locally flat electromagnetic wave is inversely proportional to the distance from the source.

When the antenna is used in transmission, it is obvious that (4) allows to evaluate the exact radiated field, since in this case the current imposed by the generator that feeds the antenna is the only physical reason that determines the radiation.

When the antenna is used for reception, however, equation (4) does not exactly describe the overall re-radiated (i.e. scattered) field; although in fact the current I depends on the incident field, it evidently does not exhaustively summarize in itself the motion of the electrons in response to the applied incident field. Just think, for example, of the case in which the antenna terminals are left open: if on the one hand there is no doubt that I is zero, on the other it is equally clear that the electrons respond as always to the stresses of the incident field, thus producing a distribution current on the antenna which in turn generates a diffuse field that is not zero according to the considerations illustrated at the beginning of the paragraph. We can then formally write[1]-[2]:

( 5)

The first term is obviously represented by (4), while the second appears in general not easy to evaluate, as it depends on the geometry of the object and on the properties of the materials.

Furthermore, the residual term should not be misleading, since it is not reasonable to expect that this contribution will in general always be negligible compared to the other. In relation to the value of the current I, the shape and dimensions of the antenna and the properties of the materials, each of the two terms may be more or less relevant. In cases where the residual contribution is zero (or negligible), the diffuse field, dependent only on I, will be radiated according to the radiation pattern of the antenna, and therefore the overall diffused power P_{s} can be substituted for the radiated power in the formula that defines the directivity function:

(6)

Considering equations (1) and (3), it results:

(7)

therefore:

(8)

where η represents the antenna efficiency.

In the following paragraph we will proceed to the evaluation of the overall scattered field by means of the so-called antenna scattering theorem.

**Scattering Antenna Theorems**

An antenna having impedance Z_{A} is considered to be illuminated by a known electromagnetic field (typically uniform locally plane wave). Let Z_{L} be the load impedance seen from the antenna terminals (If antenna and load are connected via transmission line, Z_{L} represents the line input impedance; if, on the other hand, the load is connected directly to the antenna terminals, Z_{L} obviously represents the user impedance). Let Es (r, θ, φ| Z_{L}) be the overall field re-radiated by the antenna when closed on the load Z_{L}. Now let Io be the current flowing through the antenna terminals when Z_{L} = 0 (short circuit), and let Eshort be the scattered field in this case. Similarly, let I_{m} and I_{m}* be the currents across the antenna terminals in the cases Z_{L} = Z_{A} and Z_{L} = Z_{A}*, respectively.

Let E_{a} also be the field radiated by the antenna when it is used in transmission and powered by a generator that imposes a current I_{a} at the terminals (i.e. on the impedance Z_{A}). It is as if we had an equivalent circuit of the antenna (V_{g}, Z_{g}) for these three cases closed on:

- short circuit (in it flow the short circuit current I
_{short}, referring to Figure 1 Z_{L}= 0) - Load Z
_{A}(in it flow the current I_{m}, referring to Figure 1 Z_{L}=Z_{A}) - Load Z
_{A}* (in it flow the current I_{m}* , referring to Figure 1 Z_{L}= Z_{A}*)

Finally, define the antenna reflection coefficient as follows:

(1)

The antenna scattering theorem then admits the following equivalent formulations [1]:

(2)

(3)

(4)

The theorem essentially states that the field re-radiated in the generic point (r, θ, φ) by an antenna illuminated by a known field can always be expressed as the field that the antenna would radiate in (r, θ, φ), with same incident field, should it be closed on a reference impedance ± a corrective term that depends on the reference condition considered. A first, interesting observation can be derived by considering the expression (2) of the theorem and assuming Z_{L} = ∞ (open circuit at the antenna terminals):

(5)

When the antenna terminals are open, it is reasonable to assume that the total diffuse field coincides with the so-called residual contribution only, since the other contribution, dependent on the current at the terminals, is obviously zero in the event of an open circuit. This supposition is evidently confirmed by the previous expression. In fact, with reference to the second member of the equation obtained, it is quite clear that by subtracting the contribution due to the short-circuit current from the total scattered field when the terminals are short-circuited, as previously stated, only the residual contribution to the scattered field is obtained.

The expressions (2) - (4), which formalize the antenna scattering theorem, can be summarized in a single formula by reasoning as follows: both generically Z_{REF} a generic reference impedance and both I_{REF} the current that crosses the terminals when the antenna is closed on Z_{REF} is illuminated by the incident field. Furthermore, let E_{REF} be the overall field re-radiated by the antenna in this condition. It turns out then:

(6)

where K_{REF} represents an appropriate complex coefficient whose value depends on the chosen reference condition. In particular, from (3) - (5) it is evident that:

(7)

Ea represents the field radiated by the antenna when powered in transmission by the current I_{a}, while E_{REF} indicates the total scattered field when the antenna is closed on the Z_{REF} impedance (and is illuminated by a known incident field).

Therefore, E_{a}/I_{a} differs from E_{REF} / I_{REF} in that E_{REF} is not proportional to I_{REF} as it also contains the residual contribution which, as stated several times, does not depend on the current at the terminals. It is important to note that we usually refer to the two addends to the second member of (6) calling them structural mode and antenna mode respectively. When the antenna is closed on a conjugate load, the field affecting it determines currents on the surface which, in turn, determine the structural term. This term is independent of the load impedance. The term Antenna depends on the power drawn into the load of an antenna without loss and the power radiated by it due to a load mismatch.

From the definition of Radar Cross Section and from equations (5)-(7) we can deduce a range of validity of the RCS [1] and [3]:

(8)

Where σ^{a} is due to antenna mode and σS is due to the structural mode. The minimum value occurring when the two RCSs are in-phase while the maximum occurs when they are out-phase.

**References**

[1] E.F. Knott, Radar Cross Section, 2nd edition, Artech House, Norwood, MA, 1993

[2] C.A. Balanis, Antenna Theory, analysis and design, 2nd edition, John Wiley & Sons Inc., 1997

[3] R.E Collins, Foundations for Microwave Engineering, 2nd edition, John Wiley & Sons Inc., 1992

**About the Author**

*Pasquale Dottorato*

I’m R&D Manager at LABID a Beontag Company. I received the “Laurea” degree cum Lode in electrical engineering and Ph.D. degree from University of Naples, Italy. After an interesting experience at National Research Council of Italy (section on the applied electromagnetism and electronic microdevices) in collaboration with “Electronics & Telecommunication” dpt. at University “Federico II” of Naples, I worked in the design of Radar and microwave equipment for a Defence and Aerospace company in Rome. I’m IEEE member. My interests include Radar Systems, Conformal, Phased Array and Beamforming Antenna, Inverse and Scattering Electromagnetic Problems, Biological effect from Radiowave, RF Microelectronic Devices, RFID and Signal Processing. pasquale.eng.dottorato@ieee.org

### March 2023

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