Bistatic RCS calculation for propellers at near-resonant frequencies

Abstract

Scattering from propellers is investigated using the 3D Finite Element Method (FEM), alongside a simpler integral equation (IE) method using 1D discretization. It is shown that under certain conditions and near the propellers’ resonant frequency, results using the two methods are in agreement, justifying the use of the IE method; this is advantageous because of smaller computation times, and for providing a physical insight to the scattering phenomena at hand. Anechoic chamber measurements are performed to validate the numerical results.

Keywords

Finite element method method of moments propeller radar detection unmanned aerial vehicle

1. Introduction

Lately, detection of Unmanned Aerial Vehicles (UAVs)—drones—is a widely researched topic because of their commercial availability and their potential for misuse or abuse. (An overview of the detection methods is given, e.g., in [1].) Because of the difficulty of detecting them with conventional radar systems owing to their small RCS (Radar Cross Section), it is necessary to more closely investigate the patterns in the signals reflected from UAVs [2,3].

A material commonly used in the manufacture of propellers for drones is Carbon-Fiber Reinforced Plastic (CFRP); for high frequency scattering problems, it has been measured to have good conductive properties [4,5]. Another advantage of detecting at this wavelength is that Digital Video Broadcasting-Terrestrial (DVB-T) transmitters also operate in this frequency band, raising the possibility of observation with passive radar systems. When a propeller is investigated near its resonant frequency ( ${\lambda}\approx \frac{l}{2}$ ), the maximum of the induced current will result in a strong scattered field; which in addition is strongly dependent on the propeller’s orientation [3]; for the rotating motion of a propeller, this characteristic periodical dependence makes it advantageous to discuss the problem in the frequency domain [6,7].

The rotation of the propeller causes the modulation of the scattered field (this is the so-called micro-Doppler effect). Often times, physical optics models are used in similar problems [6,8,9], however, this method is inapplicable for shorter propellers with lengths approximately equal to the wavelength. In order to solve the problem, it is necessary to solve the full set of Maxwell’s equations using the appropriate numerical methods.

The Finite Element Method (FEM) is commonly used in scattering calculations, because of its general formulation and flexibility. One of its disadvantages is the the associated high computational cost, due to the need for volume discretization of the scatterer and its surroundings.

It is also possible to model scattering phenomena using an integral equation method (the Method of Moments (MoM)) [10]. The usage of the MoM is advantageous in the case of thin propellers (as shown earlier in [11]), because a 1-dimensional discretization is able to accurately describe the problem, as only the longitudinal component of the induced current is significant. As a consequence, one can work with a much smaller number of degrees of freedom when compared to the 3D FEM (∼10 basis functions for the MoM vs. ∼10⁴ to 10⁵ degrees of freedom for the FEM), where, in addition to the propeller itself, the surrounding air must also be discretized (a possible FEM discretization of the propeller is illustrated in Fig. 1).

The main contribution of the present work is the extension of the scattering model for CFRP propellers. For this purpuse, we conduct a thorough investigation of the conductivity of CFRP propellers, by means of the proposed simulation framework (Section 2). In addition, we include a detailed description of the proposed reduced order MoM model (Section 3). Lastly, we perform a numerical and experimental analysis of bistatic scattering measurements for different geometrical arrangements (Section 4).

Fig. 1.

While the FEM model requires a 3D discretization of the propeller and the surrounding medium, in the MoM, a 1D discretization of the scatterer with appropriately chosen parameters (the radius a and length of a cylinder l) and a sinusoidal basis of approximately 10 functions yields similar results.

2. Investigation of the material properties of CFRP propellers

Electrical properties of carbon-fiber composites used in aviation have been studied extensively [4,12,13]; however, the exact material parameters are usually not included with a drone propeller, and thus are unknown to the user. In general, carbon-fiber composites have anisotropic conductive properties, with the conductance of the carbon tubing being generally higher “in-plane”, and being lower “out-of-plane”, with the exact values strongly dependent on the amount of carbon used. Because of favorable mechanical properties, the blades of the propeller will be oriented “in-plane”; consequently, we expect the propeller to behave as a good conductor. Additionally, since the induced currents mostly flow in a single direction (along the propeller’s length), it is reasonable to perform calculations using a scalar conductivity corresponding to this direction.

In order to determine the electrical conductivity of the propeller’s material, the resonant frequency (the frequency at which the strongest scattered field was observed) of the available 10 inch carbon-fiber composite propeller was measured to be f ₀ = (535 ±5) MHz (more details on the measurement setup will be given in Section 4).

Afterwards, we performed a frequency sweep FEM simulation; we were examining the dependence of the RCS and the resonant frequency on the electrical conductivity of the propeller’s material. Comparing the resonant frequencies (see Fig. 2), there are two possible values at which the resonant frequency is near the measured value: σ₁ ≈ 40 Sm⁻¹ and σ₂ ⪆ 10⁴ Sm⁻¹. Taking into consideration the measured and calculated RCS values (also shown in shown in Fig. 2), it is clear that the CFRP used has relatively good conductive properties (σ > 10⁴ Sm⁻¹); for our purposes it is largely indistinguishable from a perfect conductor. This was validated experimentally as well, by replacing the propeller with a copper coated slab of similar size; resulting in almost identical spectra as in the case of the CFRP propeller.

This procedure can possibly be generalized to other materials, of which a slab of similar shape can be manufactured, allowing for the measurement of high frequency electromagnetic material parameters using relatively small samples.

Fig. 2.

Dependence of the propeller’s resonant frequency and the monostatic radar cross section (at the measured resonant frequency of 535 MHz) on the electrical conductivity of the propeller’s material.

3. Calculation of the scattered field using the IE method

3.1. Governing equations

As mentioned beforehand, the use of the integral equation method is advantageous because its formulation leads to a problem with a smaller number of degrees of freedom. The key fact enabling the use of a 1D discretization is the fact that the non-longitudinal components of the induced current are negligible, so it is possible to calculate the equivalent (in terms of scattering) thin wire corresponding to a given propeller geometry. This is done by fitting the maximal value of the monostatic RCS (i.e, at “normal” incidence—when the wavevector of the incident wave is perpendicular and the polarization of the electric field is parallel to the propeller): $\begin{eqnarray}\displaystyle \text{RCS}_{\text{max}}=\lim _{r\rightarrow \infty }4{\pi}r^{2}\frac{|\mathbf{E}_{\text{max}}^{s}|^{2}}{|\mathbf{E}^{i}|^{2}}, & & \displaystyle\end{eqnarray}$ (1) with respect to the frequency characteristic (here, r is the distance between the observation point and the propeller, $\mathbf{E}_{\text{max}}^{s}$ is the scattered field and E _i is the incident field). We perform the fit by the minimization of the error function defined as the difference between RCS values calculated using the two methods in the following metric: $\begin{eqnarray}\displaystyle d(\text{RCS}_{\text{1D}},\text{RCS}_{\text{3D}})=\frac{1}{N{\Delta}f}\mathop{\sum }_{n=0}^{N}|(\text{RCS}_{\text{1D}}^{2}(l,a,f_{1}+n{\Delta}f)-\text{RCS}_{\text{3D}}^{2}(f_{1}+n{\Delta}f))|, & & \displaystyle\end{eqnarray}$ (2) where f ₁ and f ₁ + NΔf = f ₂ are appropriately chosen frequencies below and above the resonance frequency f ₀, and N is a division of the interval (for the investigated propeller, f ₀ = 535 MHz, and f ₁ = 450 MHz, f ₂ = 590 MHz were chosen), a radius and l length are the parameters of the thin wire model). The definition of the metric is based on the least squares method; however, one is free to choose any metric in accordance to the situation. The minimization is performed using the Matlab fmincon routine [14].

The application of the MoM starts with calculating the currents induced by the incident EM wave; this is done using the Electric Field Integral Equation (EFIE) for thin wires [10]: $\begin{eqnarray}\displaystyle \frac{-\text{j}}{{\omega}{\mu}}[\hat{\mathbf{t}}(\mathbf{r})\cdot \mathbf{E}^{i}(\mathbf{r})]=\hat{\mathbf{t}}(\mathbf{r})\cdot \left[1+\frac{1}{k^{2}}{\nabla}{\nabla}\cdot \right]\int _{L}I(\mathbf{r}^{\prime })\hat{\mathbf{t}}(\mathbf{r}^{\prime })G(\mathbf{r},\mathbf{r}^{\prime })\text{d}r^{\prime }, & & \displaystyle\end{eqnarray}$ (3) where ω is the angular frequency of the illuminating EM wave, 𝜇 is the permeability of the surrounding medium, E ⁱ is complex amplitude of the incident electric field, I (r ^′) is the line current along the thin wire, t̂(r ^′) is the tangential unit vector along the wire, k is the wavenumber, $G(\mathbf{r},\mathbf{r}^{\prime })=\frac{\exp (-jk|\mathbf{r}-\mathbf{r}^{\prime }|)}{4{\pi}|r-r^{\prime }|}$ is the free-space Green’s function, and the integration is performed along the length of the wire.

In the MoM, we approximate the current as a weighted sum of basis functions; the use of global basis functions is advantageous in some cases. For open ended thin wires, zero current is required at the ends of the wire. The same functions are used as basis and testing functions (Galerkin method). A sequence of global, harmonic functions satisfying these conditions is: $\begin{eqnarray}\displaystyle f_{n}(l^{\prime })=\sqrt{\frac{2}{l}}\sin \left(n\frac{{\pi}l^{\prime }}{L}\right),\quad n=1,2,3,\ldots & & \displaystyle\end{eqnarray}$ (4) where l ^′ is the position on the propeller of length l such that l ^′∈ [0, L]; these functions fulfill the zero current requirement automatically. From the induced currents, components of the scattered electric far field can be calculated [15]: $\begin{eqnarray}\displaystyle \mathbf{E}^{s}(\mathbf{r})=\left[(\mathbf{r}-\mathbf{r}_{\mathbf{0}})\times \left(\frac{-\text{j}{\omega}{\mu}\text{e}^{-\text{j}\mathbf{k}_{s}\cdot (\mathbf{r}-\mathbf{r}_{\mathbf{0}})}}{4{\pi}|\mathbf{r}-\mathbf{r}_{\mathbf{0}}|}\int _{L}\text{e}^{-\text{j}\mathbf{k}_{s}\mathbf{r}^{\prime }}I(\mathbf{r}^{\prime })\cdot \hat{\mathbf{t}}(\mathbf{r}^{\prime })\text{d}r^{\prime }\right)\right]\times \frac{\mathbf{r}-\mathbf{r}_{\mathbf{0}}}{|\mathbf{r}-\mathbf{r}_{\mathbf{0}}|^{2}}, & & \displaystyle\end{eqnarray}$ (5) where r and r ₀ are the position of the observation point and the center of the propeller, respectively, and k _s is the far field wavevector of the scattered wave, pointing from the scatterer towards the observation point. As the transmitter and receiver antennas are linearly polarized, it is useful to define the polarimetric RCS: $\begin{eqnarray}\displaystyle \text{RCS}_{yy}=\lim _{r\rightarrow \infty }4{\pi}r^{2}\frac{|{E_{y}^{s}|}^{2}}{|E_{y}^{i}|^{2}}. & & \displaystyle\end{eqnarray}$ (6) Here, the transmitter and receiver antennas are y polarized.

3.2. Application for a 2-blade propeller

In all the simulations, the polarization of the incident wave and the orientation of the propeller are both vertical. For the investigated 10 inch two-blade drone propeller shown in Fig. 1, the values minimizing are l = 25 cm ≈ 9.8 inches and a = 5.1 mm (FEM simulations of a wire of these dimensions have confirmed that the thin wire approximation is appropriate for our purposes). The frequency characteristic of the maximal RCS is shown in Fig. 3.

Fig. 3.

Comparison of frequency characteristics calculated by the FEM and the MoM for the 10 inch two-blade propeller.

Fig. 4.

Placement of the transmitting (1) and receiving (2 and 3) antennas for two orientations (a and b) of the propeller’s axis of rotation.

4. Bistatic measurements in the anechoic chamber

Our numerical calculations have been validated with measurements performed in an anechoic chamber (a sketch and a photograph of the measurement setup is included in Fig. 4 and Fig. 5, respectively).

Log-periodic antennas are used for illumination and detection; their gain has been measured along with the cable losses and other parameters before the experiment. Measurements are performed at the calculated and measured resonant frequency of 535 MHz.

4.1. Comparison of computational and experimental methods

From the amplitude and orientation of the incident electric field near the propeller; the magnitude of an y polarized wave is [16]: $\begin{eqnarray}\displaystyle E_{y}^{i}=\sqrt{\frac{P_{t}G_{t}Z_{0}}{2{\pi}R^{2}}}, & & \displaystyle\end{eqnarray}$ (7) where G _t and P _t are the antenna gain and radiated power of the dipole, Z ₀ is the characteristic impedance of vacuum and R is the distance between the dipole and the propeller. Using this relation, we can numerically calculate scattered field in any direction; the results can then be compared with results from the receiver antenna.

It has been shown, that for a rotating object illuminated by a plane wave, the spectrum of the scattered field will consist of discrete spectral components [7]: $\begin{eqnarray}\displaystyle \mathbf{E}_{s}(\mathbf{r},t)=\mathop{\sum }_{m=-\infty }^{\infty }\mathbf{E}_{m}(\mathbf{r})\text{e}^{\text{j}({\omega}+m{\Omega})t}. & & \displaystyle\end{eqnarray}$ (8)

The scattered spectra calculated using the so-called quasi-static approximation (detailed in [11]). Experimental results are shown in Fig. 6 (because results from the FEM and MoM calculations are within 0.5 dBm, they are not shown separately).

Fig. 5.

Photograph of the anechoic chamber measurements for a bistatic angle of 𝛽 =180° and a vertical orientation of the propeller’s axis of rotation (a/3).

Fig. 6.

Comparison of experimental and computational results for different arrangements; the plots are labeled with the notation used in Fig. 4 (for example, a/3 means the propeller is in orientation a) and receiver antenna 3 is used). Calculations accurately predict the intensity of the first even side peaks, while the central peak is more pronounced due to the miscellaneous scattering from the environment (the reflection from the walls of the anechoic chambers is dominant because of the small size of the propeller).

From the results, it is clear that the numerical methods used are appropriate for treating the problem; the scattered field has the predicted discrete spectrum, and the intensities of the first even side peaks are accurately predicted for all the investigated geometries.

The difference between theory and experiment in the case of the further side peaks is most likely caused by imperfections in the anechoic chamber walls and the lobe structure of the antennas used. These effects can cause significant reflections from the chamber walls; because the side peaks are of relatively low intensity (note the logarithmic scale in Fig. 6), even a weak disturbance can have a pronounced effect on their intensities. The asymmetry of the propeller and the imperfection of its rotating motion may also cause noticeable disturbances on a logarithmic scale (for example, odd peaks are predicted to be of zero intensity because of the propeller’s rotational symmetry, however, the measured intensities, while low, are non-zero).

Possible ways for the improvement of the accuracy of the measurement are currently being considered, including simultaneous measurements with multiple antennas, and a closer analysis of the detected signal.

5. Conclusions

Scattering from a propeller near its resonant frequency is investigated using the FEM, as well as an efficient integral equation model. Firstly, the material properties of the CFRP propellers are investigated; we introduce a novel approach for the measurement of a material’s high frequency conductive properties. Secondly, it is shown that for the given geometry of a propeller, it is possible to find an equivalent thin wire model with much fewer degrees of freedom. Lastly, the obtained results are validated experimentally by means of bistatic anechoic chamber measurements similar to a passive radar detection scenario.

Among our future aims are investigation of other propeller geometries and the development of numerical models capable of describing more complex systems (multiple rotating scatterers and their environment) at a manageable computational cost. Experimentally, it would be desirable to increase the precision of the measurement in order to allow the calculation of the orientation of the propeller’s axis of rotation from the measured spectra and other parameters of the measurement.

Footnotes

Acknowledgements

The work was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences.

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