A) Basic relations for an infinitesimal dipole

The basic approach to estimate the EM field radiated by or scattered from a finite-size linear wire structure is to subdivide it into a number of infinitesimal dipoles of length dz′ [Reference Balanis4]. The electric and magnetic field components in the spherical coordinate system of the field radiated by an infinitesimal dipole of length dz′ positioned along the z-axis at z′ (see Fig. 1) in the far field are given as [Reference Balanis4]

(1) $$d{E_\theta } \simeq j\eta \displaystyle{{k{I_e}\lpar x^{\prime}\comma \; y^{\prime}\comma \; z^{\prime}\rpar \, {e^{ - jkR}}} \over {4\pi R}}\sin \theta \, dz^{\prime}\comma \;$$

(2) $$d{E_r} \simeq d{E_\phi }=d{H_r}=d{H_\theta }=0\comma \;$$

(3) $$d{H_\phi } \simeq j\displaystyle{{k{I_e}\lpar x^{\prime}\comma \; y^{\prime}\comma \; z^{\prime}\rpar \, {e^{ - jkR}}} \over {4\pi R}}\sin \theta \, dz^{\prime}\comma \;$$

where I _e (x′, y′, z′) is the electric current strength at the infinitesimal dipole, the wavenumber k is k = 2π/λ, η is the intrinsic impedance of propagation medium, and R is the distance from any point z′ on the wire to the observation point, given by $R=\sqrt {{r^2}+\lpar - 2rz^{\prime}\cos \theta+{{z^{\prime}}^2}\rpar } $ , where r is the distance from the center of coordinates to the observation point. The harmonic factor e ^−jωt in equations (1)–(3) and their derivatives are omitted.

Fig. 1. The geometry of the components of the electric field, radiated by infinitesimal dipole (after [4]).

Using the far-field approximation, when R can be replaced by R ≃ r − z′cos θ for the phase terms and by R ≃ r for the amplitude terms, equation (1) can be written as

(4) $$d{E_\theta }=j\eta \displaystyle{{k{I_e}\lpar x^{\prime}\comma \; y^{\prime}\comma \; z^{\prime}\rpar \, {e^{ - jkr}}} \over {4\pi r}}\sin \theta \, {e^{+jkz^{\prime}\cos \theta }}dz^{\prime}.$$

A similar equation can be derived for the magnetic field component (3). As all other radiated field components are equal to zero (2), an infinitesimal dipole is a pure polarization-anisotropic object. For such an object the generation of cross-polarized components of (re-)radiated field completely defined by the orientation of the dipole in the polarization basis of the radar's antenna system. As soon as in this paper we study the temporal behavior of the Doppler spectra of rotating blades, we will exclude geometric depolarization from consideration, and analyze the case when the radar is placed in the rotation plane of the structure and causing polarization of the radar signals in the rotation plane only.

B) Radiated field of finite length linear wire structure

Summing up the contributions from all the infinitesimal elements over a wire structure with a length L one derives

(5) $${E_\theta }=\int_L {d{E_\theta }}=j\eta \displaystyle{{k\, {e^{ - jkr}}} \over {4\pi r}}\sin \theta \, \left[{\int_L {{I_e}\lpar x^{\prime}\comma \; y^{\prime}\comma \; z^{\prime}\rpar \, {e^{jkz^{\prime}\cos \theta }}dz^{\prime}} } \right].$$

The radiated field is defined by two factors – the term outside the brackets is usually called as the element factor and the one within the brackets as the space factor. For analyzed linear wire structure, the element factor is defined as the field of a unit length infinitesimal dipole located at a reference point (the origin). In this study, we assume this point to be collocated at the center of the wind-turbine rotation. In general, this factor depends on the type of current and its direction. The space factor is a function of the current distribution along the linear wire structure.

If we assume that the wire structure is rotated, the elevation angle changes in time proportionally to the rotation frequency Ω as θ(t) = θ ₀ + Ωt, and for the most practically important cases Ω ≪ ω. This allows to simulate directly the scattered from rotating wire structure radar signals in time domain using equation (5) with following conversion to the Doppler frequency domain.

A combination of n linear wires produces the total scattered field as:

(6) $$\eqalign{{E_\theta} &= \sum\limits_n \int_{L_n} {dE_\theta} = j\eta \displaystyle{{k\, e^{- jkr}} \over {4\pi r}} \cr & \quad \times \sum\limits_n {\sin {\theta _n} \left[{\int_{{L_n}} {{I_e}\lpar x^{\prime}\comma \; y^{\prime}\comma \; z^{\prime}\rpar \, {e^{jkz^{\prime}\cos {\theta_n}}}dz^{\prime}} } \right]}.}$$

In this equation, we assume that there is no EM coupling between different elements of analyzed wire structure. This assumption, which is not correct from pure electrodynamics point of view, allows us to keep the analysis of complex wire constructions simple and is still useful for the interpretation of experimental results.

In the case of monostatic radar, the single-scattered on every infinitesimal piece of wire field is defined by the induced current I _ind (x′, y′, z′), which is proportional to the tangential component of the incident radar signal's field. The phase of this field in every specific location can be defined using the phase shift of the incident field relatively to the selected at the distance r phase (and rotation) center of the wire structure. As result, the induced on a wire structure current defined as

(7) $${I_{inc}}\lpar x^{\prime}\comma \; y^{\prime}\comma \; z^{\prime}\rpar \sim {E_\theta }\lpar r\rpar \cdot \sin \theta \cdot {e^{+jkz^{\prime}\cos \theta }}.$$

The resulting backscattered field is proportional to

(8) $$dE_\theta ^{BS} \lpar \theta\comma \; z^{\prime}\rpar \sim j\eta \displaystyle{{k{e^{ - jkr}}} \over {4\pi r}}{E_\theta }\lpar r\rpar \cdot {\sin ^2}\theta \cdot {e^{+j2kz^{\prime}\cos \theta }}dz^{\prime}.$$

A) Radiating long dipole antenna

To verify theoretically the model proposed and to create a reference for further analysis, we start with micro-Doppler simulation of a very long symmetrical dipole transmit antenna, which is fed at and rotates around its central point. In this case the current distribution along the wire is given by

(9) $$\eqalign{{I_e}\lpar x^{\prime} & = 0\comma \; y^{\prime}=0\comma \; z^{\prime}\rpar ={\hat{\bf a}_z}{I_0} \sin \left[{k\left({\displaystyle{l \over 2} - \vert {z^{\prime}} \vert } \right)} \right] \cr & \quad - l/2 \leq z^{\prime} \leq l/2\comma}$$

where I ₀ is the amplitude of the current in the feeding point. By substituting this equation into equation (5) and performing numerical integration one obtains the transmitted far-field patterns of such a dipole as a function of the changing in time orientation angle in the rotation plane. Comparison of the calculated radiation patterns for a finite-length dipole antenna [Reference Balanis4] with their analytical representation was used for the numerical algorithm validation.

The computed Doppler spectrogram of the received signals in the rotation plane with the initial elevation angle θ ₀ = 0° are presented in Figs. 2 and 3. The maximum Doppler shift appears at θ = ± 90°, when the dipole becomes perpendicular to the observer's line of sight. The intensity of the zero-Doppler component strongly depends on the long dipole's side lobes patterns, which are defined by the relation of antenna length to wavelength. In case of L = (n ± 1/2)λ the presence of the broadside lobes results in complete suppression of the zero-Doppler signals.

Fig. 2. Amplitude Doppler spectrogram (in dB) of the rotating symmetrical radiating dipole antenna with the total length L = 300λ, the rotation frequency Ω = 0.1 Hz, and the initial elevation angle θ ₀ = 0°.

Fig. 3. Amplitude Doppler spectrogram (in dB) of the rotating symmetrical radiating dipole antenna with the total length L = 300.5λ, the rotation frequency Ω = 0.1 Hz, and the initial elevation angle θ ₀ = 0°.

B) Asymmetrical long wire as a radar reflector

In case when the rotation center coincides with one of the wire end, the resulting Doppler spectrogram is presented in Fig. 4. The maximal Doppler shift occurs when the rotated wire becomes perpendicular to the radar's line of sight and all Doppler frequencies from zero till maximum value are presented in the spectrum with the same amplitude. Much lower amplitudes of temporal harmonic-like reflections are produces by the side lobes of reradiated by the wire field. The weakly modulated zero-Doppler reflections are presented for any length of the wire.

Fig. 4. Amplitude Doppler spectrogram (in dB) of a rotating asymmetrical linear wire with the length L = 300λ, the rotation frequency Ω = 0.1 Hz, and the initial elevation angle θ ₀ = 0° (the structure reflects radar signals in the rotation plane).

C) Symmetrical long wire as a radar reflector

The shift of the rotation center to the middle of the wire results in full symmetrization of the observed Doppler spectrum (Fig. 5). For the given example the total length of the wire structure is the same as in the previous asymmetrical case (Fig. 4), as result the maximum Doppler shift expectedly decreased in two times.

Fig. 5. Amplitude Doppler spectrogram (in dB) of a rotating symmetrical linear wire with the length L = 300λ, the rotation frequency Ω = 0.1 Hz, and the initial elevation angle θ ₀ = 0° (the structure reflects radar signals in the rotation plane).

D) Wired model of a wind turbine with three blades

A combination of three linear wires, which are progressively shifted by 120° with respect to each other in the rotation plane, creates a simple wire model of typical high-efficient wind turbine.

The simulated temporal dependence of the micro-Doppler spectral pattern for such wire structure is presented in Fig. 6. It represents quite well the main feature of real Doppler spectrogram of wind turbines with three blades.

Fig. 6. Amplitude Doppler spectrogram (in dB) of a rotating structure with three linear wires with the length L = 300λ, progressively shifted in the rotation plane by 120°, the rotation frequency Ω = 0.1 Hz, and the initial elevation angle θ ₀ = 0° (the structure reflects radar signals in the rotation plane).

Now we apply proposed model for the detailed interpretation of the real radar measurements of the Enercon E82-2.3 MW wind turbine [5, 6] near Etten-Leur, the Netherlands. The micro-Doppler pattern (temporal Doppler spectrogram) of such turbine measured with the PARSAX S-band radar [Reference Krasnov, Babur, Wang, Ligthart and van der Zwan7] is shown in the upper image in Fig. 7. For this a continuous LFM waveform with PRF 1 kHz and the bandwidth of 10 MHz was used. The Doppler spectrum pattern was calculated for sequential bursts of 128 sweeps using Hamming windowing without any noise and clutter suppression, and has been incoherently integrated over five range resolution elements to cover the most parts of wind-turbine construction.

Fig. 7. Amplitude micro-Doppler pattern (amplitude Doppler spectrogram, in dB) of the Enercon E82-2.3 MW wind-turbine near Etten-Leur, the Netherlands: measured with the PARSAX S-band radar (top) and simulated on the base of proposed algorithm for the simple three parallel wires per blade structure (middle) and for more complex structure when every blade is modeled as a bunch of wires (bottom).

In the turbine model with three-wires as described above the wires lengths of 82 m is used based on the available wind-turbine documentation [5, 6]. The blades rotation frequency (Ω = 0.03 Hz) has been determined from the periodicity of the real radar spectrogram (upper image in Fig. 7). After the blades rotation frequency selection, the simulated observation azimuth (angle between radar's line of sign and blades rotation plane) has been determined using the matching of the maximum observed Doppler speed with maximum expected linear speed of the end point of blade for the given blade's length and rotation speed. This turning was necessary as soon as, unfortunately, for this observation case the ground-truth data about the orientation of the blades rotation plane were not available. To limit the dynamic range of simulated spectrogram the Gaussian noise with a level, defined from the signal-to-noise ratio of the real radar data, was injected into the model during the simulation.

The simulated micro-Doppler pattern is presented in the middle image in Fig. 7. To represent visible on the real radar spectrogram temporal widening of the main Doppler spread lines and some harmonic-like patterns every blade was presented in the model as electromagnetically independent combination of three parallel wires with different lengths.

To represent more details of the real micro-Doppler pattern a few different corrections of simulated wire structure have been done. Finally, we found the way to reproduce the exponentially like decreasing in time pattern, which is located on micro-Doppler spectrogram after the main Doppler spread line – as it is shown in bottom image in Fig. 7. In this simulation, each blade has been modeled as a bunch of wires, separated in orientation in 3°, and their lengths are decreasing with the parabolic law. The simulated and real micro-Doppler patterns looks quite similar, and the resulting multi-wire structure also looks much more similar to the real turbine's blades.

A) Slowly rotating objects

For the case of slowly rotating wire structure, when the radar observation time Δt is much shorter than rotation period (ΩΔt ≪ 1), from the known serial expansion [Reference Abramowitz and Stegun8]:

(10) $$\eqalign{& \cos \theta \lpar t\rpar =\cos \lpar {\theta _0}+x\rpar =\cos {\theta _0} \cr & \quad - x\sin {\theta _0} - \displaystyle{{{x^2}} \over {2!}}\cos {\theta _0}+\displaystyle{{{x^3}} \over {3!}}\sin {\theta _0}+\cdots \comma}$$

where x = Ωt, follows that

(11) $${e^{jkz^{\prime}\cos \theta \lpar t\rpar }}{e^{j\omega t}} \approx {e^{jkz^{\prime}\cos {\theta _0}}}{e^{j\lpar {\omega - \Omega kz^{\prime}\sin {\theta_0}} \rpar t}}.$$

The second term in the right part of the equality (11) can be seen as the classical Doppler shift, which depends on the wire structure's rotation frequency as well as the orientations and positions of integrated infinitesimal dipoles relatively to the center of rotation.

B) Fast rotating objects

As soon as the wire's rotation period becomes comparable with the radar observation interval, not only linear terms in the serial expansion (10) have to be taken into consideration, and this expansion becomes not convenient to use for the close-form analysis. In this case, we can use relation (9.1.4, [Reference Abramowitz and Stegun8])

(12) $${e^{[\lpar 1/2\rpar z\lpar y - \lcub 1/y\rcub \rpar]}}=\sum\limits_{k=- \infty }^\infty {{y^k}{J_k}\lpar z\rpar }\comma \; \quad y \ne 0\comma \;$$

where J _k (z) are the Bessel functions of integer order k. Defining in equation (12) y = e ^jΩt , assuming without loss of generality θ ₀ = 0, and using the fact that cos x = sin(x + π/2), we come to the following result

(13) $${e^{jkz^{\prime}\cos \Omega t}}{e^{j\omega t}}={e^{j\omega t}}\sum\limits_{k=- \infty }^\infty {{e^{j\lpar \pi n/2\rpar }}{J_k}\lpar kz^{\prime}\rpar } \, {e^{j\Omega kt}}.$$

This equation represents the symmetrical distribution of an infinite amount of harmonics around the center frequency, in many senses similar to the spectrum of the frequency-modulated with harmonic signal of frequency Ω. It means that in case of fast rotated objects or long radar observation time, when radar observation time becomes compatible or longer then the rotation period, the Doppler model of spectrum interpretation is not working anymore. An observer will measure the continuous symmetrical spectrum with separated by Ω harmonics.

C) Simulated examples

The three wires model of wind turbine has been used for simulation of spectrograms for cases with different relation between the wind-turbine rotation period and the radar observation time (the burst duration of the Doppler radar). The initial case when the blades are rotating with Ω = 0.1 Hz and radar measures Doppler spectrum using 128 pulses with PRF 1 kHz is shown in Fig. 8(a). Figure 8(b) represents a case when blades rotate ten times faster and radar uses the same settings as before. In this case for a better visual compatibility the sizes of wired-blades are decreased ten times to keep the maximum observed Doppler frequency constant. It is clear that a faster rotation during the same observation time makes micro-Doppler pattern smoother. Further increase of the blades rotation speed in ten times (and in ten times decrease of their lengths) for the fixed radar setting results (Fig. 8(c)) in the complete destruction of the Doppler pattern and conversion of the Doppler spectrum according to equation (13). Decreasing the radar observation time of such fast-rotating object in eight times (switching to burst length 16) restores the visibility of the micro-Doppler pattern (Fig. 8(d)) and returns the observables in the Doppler velocity domain.

Fig. 8. Dependence of amplitude micro-Doppler patterns (in dB) from the relation between the rotation period of three-blades wind turbine and the radar's observation time.

These results, being supported with analytical derivations, give a method for optimal selection of the Doppler radar observation time for many types of rotating objects – not only wind turbines but also helicopters and propelled aircraft.