Department of Electrical Engineering CHALMERS UNIVERSITY OF TECHNOLOGY Gothenburg, Sweden 2017 Modeling multicopter radar return A study in discrimination of multicopter UAVs from birds using the micro-Doppler effect Master’s thesis in Applied Physics BJÖRN KARLSSON Master’s thesis 2017 Modeling Multicopter Radar Return A Study in Discrimination of Multicopter UAVs from Birds Using the Micro-Doppler Effect BJÖRN KARLSSON Department of Electrical Engineering Division of Signal processing and Biomedical engineering Chalmers University of Technology Gothenburg, Sweden 2017 iv Modeling Multicopter Radar Return A Study in Discrimination of Multicopter UAVs from Birds Using the Micro-Doppler Effect BJÖRN KARLSSON © BJÖRN KARLSSON, 2017. Supervisors: Patrik Dammert, Saab Surveillance Björn Engström, Saab Surveillance Examiner: Thomas Rylander, Electrical Engineering Master’s Thesis 2017 Department of Electrical Engineering Division of Signal processing and Biomedical engineering Chalmers University of Technology SE-412 96 Gothenburg Telephone +46 31 772 1000 Cover: Left: simulated micro-Doppler signature at X-band of a hexacopter. Right: hexacopter in flight. From [1]. CC-BY-SA. Gothenburg, Sweden 2017 v Modeling Multicopter Radar Return A Study in Discrimination of Multicopter UAVs from Birds Using the Micro-Doppler Effect BJÖRN KARLSSON Department of Electrical Engineering Chalmers University of Technology ABSTRACT There is an emerging need for detection of multicopter unmanned aerial vehicles (UAVs) and a radar solution is considered for this. However, multicopters and birds have very similar radar cross-sections thus a radar application is not easily realised. Although, the fast moving propellers of the multicopter suggests that, using the micro-Doppler effect, a distinction between the two can be made. This thesis investigates the expected radar return of different multicopter UAVs through simulation techniques and how a monostatic, pulsed Doppler radar may be designed to discriminate them from birds. The part of the multicopter which contributes the most to its micro-Doppler signature is the propeller. Therefore, calculations of the radar return are focused on the propellers. The static radar return is calculated in the simulation software Ansys HFSS using a 3D model of a propeller. In this simulated environment, different scenarios are created where properties such as polarisation, dielectric constant, length, carrier frequency and angle of illumination are varied and the effect of using ducted propellers is studied. The studied radar bands are L-, S-, C- and X-band. The results of these calculations are then compared to suggested mathematical models that predict the radar cross-section of propellers and very good agreement is observed. Thereafter, the dynamic radar return is simulated, where inverse synthetic aperture radar (ISAR) imaging techniques are tested and micro-Doppler signatures are generated. A simple technique to extract discriminative features from the micro-Doppler signature based on singular value decomposition (SVD) is also analysed and it can reveal both spectrum width and periodicity. Based on the simulations, the impact of different radar design parameters such as the pulse repetition frequency (PRF), illumination time, carrier frequency and polarisation on the detection of multicopters is investigated and conclusions are drawn on how these parameters are chosen for optimal detection. Keywords: UAV, multicopter, birds, micro-Doppler effect, radar, simulation vi vii ACKNOWLEDGEMENTS First of all, I would like to thank my supervisors at Saab Surveillance, Patrik Dammert and Björn Engström, for our many interesting discussions and for your encouragement and guidance throughout this project. Also, I thank my supervisor and examiner at Chalmers University of Technology, Thomas Rylander, for our in-depth, theoretical discussions about radar and electromagnetic computations in general. Lastly, a big thank you to employees at Saab for taking time for me whenever I had questions and also for making me feel welcome with kindness and by inviting me to activities such as “fika-Fridays”. Also, warmest of thanks to my fellow Master’s thesis students at Saab for your great ideas and support during this project, and for greatly contributing to the overall joyful experience of these past months. Björn Karlsson, Gothenburg, August 2017 viii ix CONTENTS LIST OF FIGURES ................................................................................................................... xi LIST OF TABLES ................................................................................................................... xv LIST OF ABBREVIATIONS ................................................................................................ xvii 1 INTRODUCTION .............................................................................................................. 1 1.1 Demand for detection of multicopters ......................................................................... 1 1.2 Objective ...................................................................................................................... 2 1.2.1 Scope .................................................................................................................... 2 1.3 Previous work .............................................................................................................. 2 1.4 Report outline .............................................................................................................. 3 2 MULTICOPTER STRUCTURE AND RADAR THEORY .............................................. 5 2.1 Multicopter structure and function .............................................................................. 5 2.1.1 Structural properties ............................................................................................. 6 2.1.2 Propeller structure ................................................................................................ 7 2.1.3 Characteristic behaviour ....................................................................................... 7 2.2 Pulsed Doppler radar working principle ...................................................................... 8 2.3 Radar return ............................................................................................................... 10 2.3.1 Radar cross-section ............................................................................................ 10 2.3.2 Electromagnetic simulation models ................................................................... 13 2.3.3 Range- and cross-range profiles ......................................................................... 14 2.3.4 Micro-Doppler signature .................................................................................... 14 2.3.4.1 Extraction of features using SVD ............................................................... 15 3 STATIC RADAR RETURN OF A SINGLE PROPELLER ............................................ 17 3.1 RCS calculations using Ansys HFSS ........................................................................ 17 3.1.1 Setup of parameters ............................................................................................ 18 3.1.2 Effects of PML reflections and error estimation ................................................ 19 3.2 Propeller characterisation and property variations .................................................... 21 3.2.1 Basic case: HH- and VV-polarisation ................................................................ 22 3.2.2 Variation of material and carrier frequency ....................................................... 23 3.2.3 Sensitivity to frequency variation....................................................................... 26 3.2.4 Elevation angle variation .................................................................................... 27 3.2.5 Ducted propeller ................................................................................................. 28 3.2.6 Variation of propeller length .............................................................................. 29 x 3.3 Approximation models .............................................................................................. 31 3.4 Discussion of choice of investigated cases ................................................................ 32 4 SIMULATION OF MULTICOPTER AND DISCRIMINATION FROM BIRDS ......... 35 4.1 Dynamic simulations using Matlab ........................................................................... 35 4.2 Multicopter model ..................................................................................................... 36 4.3 Range and cross-range profiling ................................................................................ 36 4.4 Micro-Doppler signature ........................................................................................... 38 4.4.1 Single propeller .................................................................................................. 39 4.4.2 Multiple propellers ............................................................................................. 41 4.4.3 Integration time and PRF ................................................................................... 42 4.5 Using SVD for detection algorithms ......................................................................... 44 4.6 Radar return of birds and bird discrimination ........................................................... 46 5 RESULTS AND DISCUSSION ....................................................................................... 49 5.1 Choice of radar design parameters ............................................................................ 49 5.2 Error analysis ............................................................................................................. 50 5.3 Societal consequences ............................................................................................... 51 5.4 Continued work ......................................................................................................... 51 6 CONCLUSIONS ............................................................................................................... 53 BIBLIOGRAPHY .................................................................................................................... 55 A Comparison of VV-polarised calculations ........................................................................... I B Details of RCS characteristics .......................................................................................... III C Duct with propeller ............................................................................................................ V D Matlab script for calculating micro-Doppler signatures .................................................. VII xi LIST OF FIGURES Fig. 2-1 Three examples of commercial multicopter models: a) Parrot AR Drone 2.0; b) DJI Phantom 2 Vision+; and c) Yuneec Typhoon H. Features such as camera equipment, landing gear and protective frame can be added to many models. From [15, 16, 17]. CC-BY-SA........ 5 Fig. 2-2. Propeller configurations for quadcopters: a) X4-configuration; b) H4-configuration; and c) +4-configuration. Each pair of opposite propellers rotate in the same direction, i.e. clockwise or counter-clockwise. ................................................................................................ 6 Fig. 2-3. a) Spherical coordinate system definition with unit vectors R, θ and φ, and values: distance R, elevation θ and azimuth φ. b) RCS of a perfectly conducting cylinder, oriented along the y-axis with radius r=5mm and length L=200 mm, given in polar coordinates for different azimuth angles -180°<φ<180° and for elevation angle θ=90°. The radar waveform uses HH-polarisation and a frequency f=10 GHz. .................................................................... 11 Fig. 2-4. Illustration of reflections at the PML. For simplicity, the illustration shows reflection of an optic ray whereas the actual wave behaviour may be much more complicated. Instead of being damped out by the PML, the wave reflected from the object is reflected at the boundary and it can interact with the object again. .................................................................................. 13 Fig. 2-5. The joint time-frequency representation of the micro-Doppler spectrum for a simulated rotating cylinder with r=5mm and L=200 mm, rotating at 1 Hz. The sinusoidal envelope is distinct and is due to that Doppler shift only occurs in the radial direction. When the rod is observed at broadside, flashes appear as many points on the rod are either approaching or receding. Two flashes are seen per complete revolution. ............................... 15 Fig. 3-1. An illustration of the setup used for calculations of the static radar return of a propeller in HFSS. .................................................................................................................... 18 Fig. 3-2. The impact of the minimum distance D from the propeller to the air box boundary on the RCS, with εr=8.0 and f=10 GHz. 0° is at broadside of the propeller. There is not much variation observed in the main lobes. The largest variation is seen in the deep minima as they are sensitive to changes due to the low RCS values. ............................................................... 19 Fig. 3-3. The setup of the RCS calculation reference case. The two spheres have a dielectric constant of εr=1.5-0.02/(j2πfε0) and the wave vector is aligned with the straight line through the centres of the two spheres. ................................................................................................. 20 Fig. 3-4. a) Test of the setup in HFSS in comparison to reference data and b) the absolute value of the difference between the two RCS datasets. ........................................................... 21 Fig. 3-5. The 9x4.7” CAD-model used in most simulations and its orientation in simulations, viewed from: a) 3D view; b) top view; c) side view; and d) front view. ................................ 22 Fig. 3-6. Calculated RCS of the 9x4.7” propeller with εr=8.0 for HH- and VV-polarisation. VV-polarisation is notably smaller than HH-polarisation by on average 8 dB and by 17.2 dB at broadside. ............................................................................................................................. 23 xii Fig. 3-7. RCS of the 9x4.7” propeller at different frequency bands for dielectric constants: a) εr=3.4; b) εr=5.7; and c) εr=8.0. For 1, 3 and 6 GHz the broadside RCS is much larger than end-fire RCS as expected. At 10 GHz however, end-fire RCS is comparable to, or even larger than, broadside RCS. ................................................................................................................ 24 Fig. 3-8. RCS of the 9x4.7” propeller at different frequency bands for the two special cases of dielectric constants: a) εr=2.0; and b) εr=∞. The amplitude of the RCS for εr=2.0 is lower than -40 dBm 2 at all angles and frequencies. For εr=∞, the pattern is much different from other dielectric constants and is seen to increase with decreasing frequency until it at 1 GHz behaves much like a dipole antenna. ........................................................................................ 25 Fig. 3-9. Test of the sensitivity to frequency changes for a 9x4.7” propeller with εr=8.0. A change in frequency of +10% was applied to frequencies: a) 1 GHz; b) 3 GHz; c) 6 GHz; and d) 10 GHz. An extra frequency was tested for 10 GHz for better visualisation. At 6 GHz, the end-fire lobes are increasing significantly by 14.0 dB which is attributed to the problem being in the Mie region. For 10 GHz it is seen that the broadside lobe at 4° merges with the lobe at 10° as frequency is increased. .................................................................................................. 26 Fig. 3-10. RCS of a 9x4.7” propeller with εr=8.0 and f=6 GHz for different elevation angles 50°≤θ≤130° and two different polarisations: a) HH-polarisation; and b) VV-polarisation. With HH-polarisation there is only a slight difference whereas for VV-polarisation the RCS at end-fire is increased at high angles due to the polarisation being along the blade length. There is also a large increase of RCS at broadside which is due to the problem being in the Mie region. ....................................................................................................................................... 27 Fig. 3-11. RCS of a 9x4.7” propeller with εr=8.0 and f=10 GHz for different elevation angles 50°≤θ≤130° and two different polarisations: a) HH-polarisation; and b) VV-polarisation. HH-polarisation shows a complex behaviour, although with rather similar amplitude levels, except at end-fire for high elevation angles. VV-polarisation shows a higher broadside RCS for high elevations which is due to the geometry of the propeller. The end-fire RCS is also higher at low angles θ=90±20°. ................................................................................................ 28 Fig. 3-12. The HFSS-design model used in calculations of the radar return from ducts. The 23 cm propeller is sometimes used in calculations as well. ..................................................... 29 Fig. 3-13. RCS of ducts with dielectric constants: a) εr=8.0; and b) εr=1.03. The variations are due to the duct being split up into 45 segments. The scale has been chosen to show the variations more clearly. ............................................................................................................ 29 Fig. 3-14. Variation of propeller length with εr=8.0 and models: a) 9x4.7”; b) 10x4.7”; and c) 11x4.7”. The broadside lobe width is decreasing with propeller length as expected. ......... 30 Fig. 3-15. Lobe widths ρ for different propeller lengths calculated in HFSS compared to the theoretical prediction ρ=λ/(2L). Dashed lines are calculated values from HFSS and solid lines are predicted values. The large deviation at 6 GHz and 11x4.7” is due to two peaks merging together in RCS. ....................................................................................................................... 31 xiii Fig. 4-1. Cross-range profile for frequency variation with εr=3.4. The propeller length of 23 cm matches the width of 3, 6 and 10 GHz while 1 GHz predicts it more poorly. .............. 37 Fig. 4-2. Cross-range profile at f=10 GHz. a) εr is varied for a 23 cm propeller. The width of the IFT is constant and predicts the length. b) Length is varied with εr=8.0. The widths of the IFTs are 22.3 cm, 24.8 cm and 27.8 cm which follow the tested lengths. ............................... 37 Fig. 4-3. Range profiles calculated for different εr. εr=3.4 and εr=2.0 show sinusoidal curves with amplitudes ±1 cm which correspond to the width of the propeller. However, other values of εr do not predict this as well. ................................................................................................ 38 Fig. 4-4. Micro-Doppler spectra at f=10 GHz for different dielectric constants: a) 2.0; b) 3.4; c) 5.7; d) 8.0; and e) ∞. The theoretical Doppler shift of the tip at this frequency is indicated as dashed lines. Note the different scales in amplitude as they are chosen to emphasize relative difference within each figure. ...................................................................................... 40 Fig. 4-5. Micro-Doppler spectra for εr=3.4 for different carrier frequencies: a) 10 GHz; b) 6 GHz; c) 3 GHz; and d) 1 GHz. Dashed lines indicate the theoretical Doppler shift of the tip at that carrier frequency. Note the different scales in amplitude as they are chosen to emphasize relative difference within each figure. .................................................................... 41 Fig. 4-6. Simulated micro-Doppler spectrum of a four propeller multicopter at f=10 GHz. The Doppler shift of the tip roughly matches the calculated value of fD=6.0 kHz (dashed lines) corresponding to the mean rotation rate frot=125 Hz used in this simulation. .......................... 42 Fig. 4-7. Comparison of STFTs applied to long and short time intervals. a) and c) show the short time intervals for STFTs for a single propeller and four propellers rotating at mean rate frot=125 Hz. b) and d) show the long time intervals for the same simulations. For the single propeller, long STFT reveals the rotation rate as the distance between lines is mostly 2frot=250 Hz whereas the four propeller case does not reveal correct rotation rate as the lines are seen to be more arbitrarily spaced out. ............................................................................... 45 Fig. 4-8. SVD analysis on four 23 cm propellers with mean rotation rate frot=125 Hz. The first three U-vectors, corresponding to the three largest singular values, are plotted, of which the 2 nd and 3 rd vector couple quite good to the width of the Doppler shifts. A clear periodicity is seen in the first V-vector which reveals a rotation rate of nearly 125 Hz when measured between marked peaks. The 10 first singular values are shown; however, as the first singular value is much larger than the others, the inset shows the following 9 singular values. ........... 46 Fig. 4-9. Measured micro-Doppler signature of a large sea bird seen at: a) broadside; and b) head-on. The used radar is vertically polarised and operates at X-band. Wing beats are hardly seen at broadside and are more prominent at head-on. The letters show position of the wing in its cycle where U=up, M=middle and D=down. From [4], © 2014 IEEE. Reproduced with permission. ....................................................................................................................... 47 xiv xv LIST OF TABLES Tab. 2-1. Frequency bands commonly used in search and track radar and their respective operating frequency ranges. ..................................................................................................... 10 Tab. 3-1. Calculated RCS in dBm 2 for a few identified angles where the change in RCS is large, as the distance D from the propeller to the boundary of the air box is increased. The largest change is seen for φ=-73° where the RCS changes by 1.45 dB from D=0.6λ to D=0.8λ. .................................................................................................................................................. 20 Tab. 3-2. The calculated broadside lobe widths for the five dielectric constants in comparison to the theoretical value based on antenna theory. A good agreement is seen at higher frequencies, whereas 1 GHz has the highest deviation for all εr. ............................................. 25 Tab. 3-3. Broadside RCS with HH-polarisation from HFSS compared to values calculated with (2.11), using radius r=a=3 mm and length L=23 cm. The difference shows how much the cylinder model overestimates the RCS in comparison to the calculated values for a propeller. Average differences at each frequency f={1; 3; 6; 10} GHz are {-3.1; -3.1; 4.0; 11.4} dB respectively. .............................................................................................................................. 32 Tab. 3-4. A summary of the investigated cases. As is evident from this, there are many combinations of parameters that have not been investigated. Although, the same trends shown in the investigated cases may be applicable to other variations as well. .................................. 33 Tab. 4-1. Minimum PRF needed to resolve three plausible tip velocities at different carrier frequencies. The flash times are listed for the highest tip velocity as the pulse width is assumed to be much smaller than the flash time. Also, unambiguous range is shown for the highest PRF at each frequency. These calculated values follows [14]. ................................... 43 Tab. 4-2. Minimum integration time needed for different propeller lengths and tip velocities based on time for one revolution. Lengths of 60-80 cm are more likely for small helicopter UAV rather than multicopter. ................................................................................................... 44 xvi xvii LIST OF ABBREVIATIONS CAD – Computer aided design CCW – Counter-clockwise CW – Clockwise FEM – Finite element method ICA – Independent component analysis IFFT – Inverse fast Fourier transform IFT – Inverse Fourier transform ISAR – Inverse synthetic aperture radar PCA – Principal component analysis PEC – Perfect electric conductor PML – Perfectly matched layer PRF – Pulse repetition frequency PRI – Pulse repetition interval RCS – Radar cross-section RPM – Revolutions per minute STFT – Short time Fourier transform SVD – Singular value decomposition UAV – Unmanned aerial vehicle xviii 1 1 INTRODUCTION In this chapter, the problem being studied in this thesis is defined. The background as to why the study has been conducted is first given, followed by a description of what the aim of the study is and how the scope is set. A brief summary of what has previously been done in the research field is given to put the contribution of this study in perspective to current theoretical knowledge. 1.1 Demand for detection of multicopters It is noticeable that over recent years, the popularity of small multirotor unmanned aerial vehicles (UAVs), or multicopters, has reached new heights and the market for small UAVs is also predicted to increase over coming years [2]. The multicopter is a rather complex structure and it is not an easy task to stabilise this aerial vehicle as there needs to be good communication between the individual rotors with good control models. However, technical advancements within the field have made this technology cheaper and available to the general public and multicopters can nowadays be bought as ready-to-fly models or built in a more customised manner from manufactured modules. The complete vehicle is also not too difficult to use and civilians can therefore use multicopters for leisure activities such as aerial photography or drone racing. Other potential applications with a more professional approach that can be seen for these UAVs are to use them in farming, monitoring wildlife or industrial logistics [2]. This increased availability of the technology to the public has also introduced a number of complications. As multicopters are easily obtained they may be used by people to perform illegal activities or to disturb order and as a result, legislation is being made in countries around the world to regulate dangerous activities [3]. Possible uses could be to disrupt air traffic at airports, smuggle goods into prisons or deploy poisonous biological or chemical substances in crowded areas such as big events. Additionally, a usually long operating distance makes it necessary to detect the UAV in the air and on a long enough distance to be able to react with countermeasures. Therefore, in order to be better prepared against these activities, a demand for good detection of multicopters has arisen and one way to realise detection and classification is through the use of a radar solution. However, it is not an easy task to utilise radar for this purpose. Multicopters are typically of small sizes and their electromagnetic response is in amplitude comparable to that of birds. Thus, lowering the detection thresholds of a radar system in order to find these smaller targets leads most likely to many, unwanted detections from nearby birds and as a result of this, the system may become saturated. However, one way of discriminating multicopters from birds may be to use the micro-Doppler effect which describes the internal motion pattern of the target, which gives it a distinct and unique signature. To be able to use the radar as a means of detection of multicopters, it is necessary that the target classification is fast and robust and, thus, the different design parameters of the radar system need to be optimised for the target of interest. This report investigates what radar returns (i.e. the signal received at the radar after it has scattered from a target) may be expected from multicopters and birds and how classification is affected by radar design parameters. 2 1.2 Objective The aim of this study is to use simulation techniques to investigate the radar return of multicopters for different radar waveforms and for different multicopter models, and how the radar return then can be used to detect small multicopters using a monostatic, pulsed Doppler radar. To achieve the aim, the work is focused on the following aspects:  Simulate the radar cross-section of different types of multicopters using two simulation tools Matlab and HFSS with appropriate numerical models such as the point scatter model and finite element method.  Calculate and analyse the micro-Doppler signature generated by the simulated radar returns and suggest ways to detect multicopters and distinguish them from nearby clutter or nuisance objects such as birds.  Analyse how radar design parameters such as carrier frequency, pulse repetition frequency and illumination time affect detection for different targets and distances. 1.2.1 Scope The focus of this study lies on the detection of multicopters from the electromagnetic response that they give. Therefore, hardware structure of the radar system is not part of the scope of this thesis and it is assumed that a certain investigated waveform can be generated. However, relevant waveforms that are commonly used in radar applications receive focus in order to optimise parameters that affect detection. There are many types of radars that can be used, but in this study only the monostatic, pulsed Doppler radar is considered. Other types, such as bistatic-/multistatic radars or continuous wave radar are not part of the thesis. The conditions are assumed to be perfect in the simulated environment, i.e. clutter other than birds is not considered. The effect of missing data points is also not considered. The type of target that is detected is limited to small multicopter UAVs that are available easily to the public. Even though larger multicopters are not investigated in this project, they most likely have a similar behaviour as smaller ones. This relation is however not thoroughly investigated or considered. 1.3 Previous work The increased demand for fast and robust radar detection of multicopters has led to many research groups investigating the area. As a result of this, rather recent publications that show good results can be found on the subject. There have been several studies of the radar cross-section (RCS) and micro-Doppler signature of birds. In [4], a vertically polarised X-band Synthetic Aperture Radar (SAR) is used for detection of gannets and it is concluded that extracted micro-Doppler spectrograms vary greatly depending on aspect angle and is best observed at head-on, even though RCS of birds is larger at broadside than on head-on/tail-on incidence [5]. This was believed to be due to the 3 low RCS of the wings, thus generating hardly noticeable Doppler shifts in comparison to the body. It has been seen that the polarisation has an effect on the detection of birds, where horizontal polarisation yields 2.5 times greater reflectivity than vertical polarisation when migrating birds are illuminated at broadside [6]. Studies have shown that discrimination of multicopters and birds can be achieved in a robust way. A classification rate of 92% using a singular value decomposition (SVD) classification method in combination with machine learning has been reported, although tested at a short distance to target of less than 30 m [7]. The discrimination of multicopters from birds has also been thoroughly examined by Torvik, where the focus is on detection at radar frequency bands L- and S-band [8]. The radar return of multicopters has been investigated theoretically and experimentally, where the experiments have involved measurements in field tests as well as in anechoic chamber [9, 10, 11, 12]. There have been a few studies on the properties of the propellers and their impact on the radar return, where a limited selection of propellers were investigated [11, 13]. A commercial, low-cost radar that is able to detect multicopters has been developed by the company QinetiQ based on one of these studies [11]. An initial study has also been conducted at Saab Surveillance, where the radar return is numerically calculated for a propeller and it is compared to thin cylinder models [14]. Although it has been shown that a working radar system can be developed, there are few publications that show extensive research on how the radar return varies for different models of multicopter UAV depending on target properties such as size, shape, material etc. The PhD thesis in [8] is extensive, although mainly focusing on two frequency bands. The dependence of radar return on these properties is important knowledge for designing a radar system with optimised radar design parameters for the specific target discrimination. 1.4 Report outline Chapter 2 describes the theoretical background of the project as well as the terminology used throughout this report. This includes details about the target of interest, the multicopter, and the radar system with its associated phenomena. An explanation of the models used in the electromagnetic simulations and their approximations are given here as well. In Chapter 3, the simulation software used and the relevant parameters that can be changed are described first. Details about the static electromagnetic response of a single propeller are thereafter presented, where the impact of the variation of parameters such as material and carrier frequency is investigated. The propeller is subsequently analysed dynamically in Chapter 4, where a full multicopter is modeled based on the single propeller model. A simple method of extracting discriminative features from the radar return is also studied in this chapter and a comparison with the expected radar return of birds is given. Finally, the results from the calculations are presented and compared in an analysis of radar design parameters and discussed from a broader perspective in Chapter 5 and a summary of the conclusions of the study is given in Chapter 6. 4 5 2 MULTICOPTER STRUCTURE AND RADAR THEORY In order to be able to detect a certain target, a clear understanding of the characteristics of the target is necessary. Therefore, the multicopter is first described in terms of its typical size, structure and behaviour. The structure and properties of the propellers are also discussed in a detailed way as they have a significant impact on the radar return and thus can be used as a classification feature. Aside from information about the target, it is also important to have knowledge about the radar waveform being used for detection and what kind of information that can be extracted from its echo from the target. As the simulated radar system is a monostatic, pulsed Doppler radar, this type of radar is further described along with different phenomena and radar terminology such as the radar cross-section and micro-Doppler effect associated with this kind of radar. The simulation of radar returns is a central part of this study, thus numerical simulation models for calculating radar returns and the approximations being made when they are used are also discussed. 2.1 Multicopter structure and function A multicopter is a multirotor aerial vehicle that uses the lift force of rotating propellers to gain or maintain height in the air. Unlike the helicopter that uses one propeller mainly for lift force and steering, and one propeller for eliminating torque, the multicopter uses multiple propellers that each is used for generating lift force and to steer the vehicle. The multicopter can be designed in many different ways. They may have different number of propellers, varying from the three propeller tricopter to the eight propeller octocopter. The body of the multicopters can be structured differently as well with different added features. Such features include a protective frame or duct around the propellers, landing gear, camera equipment etc. Each part of the multicopter may also be of very different materials such as plastics, glass reinforced plastics, carbon fibre or metals. Three examples of typical multicopter design for commercial models are shown in Fig. 2-1, where two quadcopters and one hexacopter are shown with their different individual features. Multicopters can therefore be very different from each other in terms of size, shape, structure and material and this makes it difficult to characterise and detect them using radar techniques. Fig. 2-1 Three examples of commercial multicopter models: a) Parrot AR Drone 2.0; b) DJI Phantom 2 Vision+; and c) Yuneec Typhoon H. Features such as camera equipment, landing gear and protective frame can be added to many models. From [15, 16, 17]. CC-BY-SA. a) b) c) 6 2.1.1 Structural properties A common denominator of all multicopters is the use of rotating propellers that can vary greatly in shape, size and materials. In addition, the propellers can also be arranged in a number of ways with different benefits and drawbacks. A very common multicopter is the quadcopter with four propellers. In quadcopters, the propellers can mainly be arranged in three different ways: X4-, H4- or +4-configuration. Here, the X4-configuration is the most common as it allows for a better camera placement without obstruction of the vehicle frame in the forward direction. The three configurations of the propellers of a quadcopter are illustrated in Fig. 2-2. Two of the rotors are rotating clockwise (CW) and two rotors are rotating counter-clockwise (CCW) as the quadcopter is viewed from above. This yields a zero net torque generated from the spinning propellers, which makes the vehicle keep its orientation during hover. Fig. 2-2. Propeller configurations for quadcopters: a) X4-configuration; b) H4-configuration; and c) +4-configuration. Each pair of opposite propellers rotate in the same direction, i.e. clockwise or counter-clockwise. Other common multicopters are the hexacopter with six propellers and the octocopter with eight propellers. The propeller configurations are typically structured as equally spaced out propellers at a fixed distance from the vehicle body with alternating CW and CCW rotation. Less common configurations are the Y6 and X8 configurations. These configurations use three and four pairs of propellers, respectively, where each pair consists of one CW rotating and one CCW rotating propeller on top of each other. This is much like the tricopter (or Y3-configuration) with three propellers, which is popular when making lightweight multicopters, and the X4 quadcopter. However, the increased number of propellers allows for a larger payload to be carried by the vehicle at the cost of a lower motor efficiency. The reduced efficiency is due to the fact that the added propeller is rotating in much more turbulent air. The size of the multicopter may vary quite largely. The larger multicopters, often intended for aerial photography or carrying different kinds of equipment, are of sizes around 1000-1500 mm in wingspan, when measured diagonally across the vehicle and including the propellers. The smallest models available can fit in one’s hand. Commercial quadcopters for leisure activities are typically in the range of 250-1000 mm in measured wingspan. The material of the body is of a lightweight and durable material, which is usually plastic or carbon fibre. a) b) c) 7 Some models may have protective frames or ducts around the propellers. These can either be in the form of a covering cylinder or duct as shown for the multicopter in Fig. 2-1a) or as a frame mounted underneath the propeller that is stretching out beyond the propeller diameter. A study has shown that propellers in ducts yield almost the same flight performance as non- ducted propellers but the use of ducts also introduces a decrease in endurance of the vehicle [18]. Protective frames of this type may be of different types of lightweight materials such as carbon fibre or plastics and frames designed for indoor usage may consist of plastic foams such as Styrofoam. 2.1.2 Propeller structure The propeller of the multicopter is an especially interesting part as it is a common denominator of all multirotor aircraft. The standard for describing its geometric properties is to provide the diameter and the pitch in inches, e.g. 3x4.5” or 3045 would mean a 3” diameter and a 4.5” pitch. The diameter is the diameter of the circle that the propeller creates as it spins about its centre and it is effectively the length of the propeller. The pitch is the distance along the propeller axis that the propeller moves in one revolution, should it rotate in a soft solid, like e.g. a screw in wood. Thus, it is a measure of how aggressive the propeller is as it is rotating in the air and is also an indication of its shape, a higher pitch results in a faster multicopter given a fixed number of revolutions per minute (RPM) at the expense of using more power. Propellers come in all sizes and shapes. However, typical geometrical properties for normal or large sized multicopters vary from about 5” to 20” in diameter and about 3” to 7” in pitch. A propeller can also consist of more than two blades, which is the most common arrangement. Three blades are rather common for multicopters, whereas four or more are not used widely. Adding an extra blade introduces more lift force and uses more power as the added extra blade needs to cut through the air. The propeller has a curved shape to generate the lift force and each blade of the propeller is at an angle such that the upper edge with a larger radius of curvature is facing in the rotational direction. This means that CW and CCW propellers are mirrored with respect to each other. The radius of curvature is greatest near the centre hub and is gradually decreasing towards the tips. For aerodynamic reasons, a larger radius of curvature generates equal lift as comparable to points located closer to the tip of the propeller, which have a smaller radius of curvature but move faster than points near the centre. Thus the shape of the blade varies to achieve equal lift force for all points along the propeller. The material of the propeller needs to be stiff and durable while preferably also being lightweight. The most common alternatives to achieve these specifications are plastics, such as ABS or nylon, and composite materials, such as carbon fibre. However, materials can vary quite largely from lightweight metals to glass fibre reinforced plastics or even wood. 2.1.3 Characteristic behaviour Multicopter UAVs are difficult to detect with radar systems. They fall under the LSS category of radar targets which stands for low, slow and small as UAVs in this category typically fly on a low altitude, move slowly and have a small RCS. The small RCS is related to their small physical size. Using the NATO classification guide, that was established on the JCGUAV 8 meeting in September 2009 [19], commercial multicopters fall into the Class I Micro and Class I Mini categories which include UAVs weighing less than 2 kg and 20 kg, respectively. Although the translational speed of the vehicle body is relatively slow, the rotational speed of the individual propellers is typically very fast. The rotational rate of a propeller can exceed 10000 RPM which, using the relation 𝑣𝑡𝑖𝑝 = 𝜔𝑟 = 2𝜋 ∙ 𝑅𝑃𝑆 ∙ 𝑟 = 2𝜋 𝑅𝑃𝑀 60 𝑟 = 𝜋 30 ∙ 𝑅𝑃𝑀 ∙ 𝑟 (2.1) where r is half the diameter of the propeller, corresponds to a propeller tip speed of vtip=133 m/s for a propeller diameter of 10”. Tip speed is limited by the pitch and diameter of the propeller and the motors, and it also depends on the momentary acceleration. It is also not exceeding the speed of sound as it would cause too much strain on the material. It can however be assumed to be around 100 m/s (or around 7500 RPM) on average while hovering. A smaller propeller also generally needs to rotate faster than a larger one in order to generate enough lift force. Thus, the RPM is typically slightly decreasing with increased propeller diameter for hover operation. The rotor and its rapid rotation influences the RCS with a clearly defined periodicity over small time intervals. This is a characteristic of multicopters that is much different from birds which have a wing beat frequency of 0-20 Hz [20], which yields much slower variations in the RCS or even no beat frequency as birds may glide at times. As a multicopter starts accelerating, it tilts the body in the direction it is headed, which allows the thrust of the propellers to generate forward motion. This tilt can be quite large and a tilt angle of about 40° is plausible, which could reveal propellers that are otherwise hidden by the vehicle body or ducts. 2.2 Pulsed Doppler radar working principle A well-known physical phenomenon within the field of wave theory is the Doppler effect. It states that a moving source of an emitted wave changes the frequency of the emitted signal when it is perceived by an external observer. If the source moves towards the observer (or, equivalently, the observer moves towards the source), the frequency of the perceived signal is shifted to a higher frequency than the emitted signal and if the source moves away from the observer, the frequency is shifted to a lower frequency than the emitted signal. The magnitude of the frequency shift is called the Doppler shift and it is positive for approaching targets and negative for receding targets. This phenomenon can commonly be observed in for example that the sound of an approaching ambulance siren has a higher pitch whereas an ambulance siren that moves away from the observer has a lower pitch. In radar applications, the Doppler effect can be used to acquire velocity information about a target by analysing the frequency content of a received wave and when using a pulsed Doppler radar, one gets information about the distance as well as the velocity. This is achieved by sending short pulses of microwave waveforms and, as they reflect against the target and are received at the radar, the time difference gives the distance and the Doppler shift gives the radial velocity. The radial velocity is measured along the radar line of sight. 9 Thus, if a target is moving strictly perpendicular to the line of sight of the radar it does not yield any Doppler shift. If we assume that the target’s velocity v is much smaller than the speed of light, i.e. v ≪ c , we can neglect any relativistic effects and the Doppler shift measured by a monostatic Doppler radar is then given by 𝑓𝐷 = 2𝑣 𝑓𝑡 𝑐 = 2𝑣 𝜆 , (2.2) where ft is the transmitted frequency and λ is the transmitted wavelength. An important design parameter of the pulsed Doppler radar is the pulse repetition frequency (PRF), which is the frequency at which the pulses are sent out from the radar system. The PRF influences the distance and velocity measurements. The maximum radial velocity is dependent on the maximum Doppler shift, which depends on the sampling frequency. Given the Nyquist sampling theorem from signal theory, the sampling frequency needs to be twice the maximum frequency content of the measured signal to avoid aliasing. The sampling frequency of a Doppler radar is the PRF. Thus, (2.2) with the Nyquist criterion gives us the maximum velocity 𝑣𝑚𝑎𝑥 = 𝑐𝑓𝐷,𝑚𝑎𝑥 2𝑓𝑡 = 𝑐 ∙ 𝑃𝑅𝐹 4𝑓𝑡 = 𝜆 ∙ 𝑃𝑅𝐹 4 . (2.3) The maximum distance on the other hand is given by the time it takes for a radiated wave to travel the distance to the target and back to the radar system before another wave has been emitted. This is expressed as 𝑟𝑚𝑎𝑥 = 1 2 ∆𝑡𝑐 = 𝑐 2 ∙ 𝑃𝑅𝐹 , (2.4) where Δt is the time between pulses, also called the pulse repetition interval (PRI). From (2.3) and (2.4) it is concluded that there must be a trade-off between maximum distance and velocity when choosing the PRF for the radar system and this is known as the Doppler dilemma. The Doppler dilemma can be expressed as a multiplication of the maximum velocity with the maximum distance, which becomes 𝑣𝑚𝑎𝑥𝑟𝑚𝑎𝑥 = 𝜆 ∙ 𝑃𝑅𝐹 4 𝑐 2 ∙ 𝑃𝑅𝐹 = 𝜆𝑐 8 . (2.5) It is seen in (2.5) that the product is a constant which is only dependent on the transmitted wavelength of the radar system and is independent of the PRF. Thus, an increase in the maximum velocity results in a decrease in the maximum distance and vice versa. The waveform used in pulsed Doppler radars is apart from the PRF also characterised by the carrier frequency, pulse width, bandwidth, polarisation and illumination time. The carrier frequency is the frequency used by the radar system and it is, for radar search and track applications, commonly in the range from about 1 GHz to 12 GHz [21]. This frequency range is divided into four different radar bands, which are listed in Tab. 2-1 with their respective frequency ranges. The pulse width is the length of one emitted pulse in time and is 10 approximately 1-100 µs. The bandwidth is the added frequency content (in addition to the carrier frequency) that emerges from that the emitted pulse has a finite width. The frequency content of the signal forms a spectrum in the frequency domain, centred on the carrier frequency and the bandwidth is the width of this spectrum and it is in the order of 100 MHz for radar systems. The emitted wave also has a polarisation which can be horizontal, vertical, linear or circular. The polarisation may be chosen for easier detection of the target. When a radar system is designed to emit and receive horizontally polarised waves, it is called HH-polarisation and, for vertically polarised waves, it is called VV-polarisation. The illumination time, also called integration time, of a target is the time that the radar main radiating lobe is illuminating the target. Tab. 2-1. Frequency bands commonly used in search and track radar and their respective operating frequency ranges. Frequency band L-band S-band C-band X-band Frequency range 1-2 GHz 2-4 GHz 4-8 GHz 8-12 GHz 2.3 Radar return When designing a radar system for detection of a certain target, it is important to know the target’s response to electromagnetic radiation. This response is called the radar return and it is characterised mainly by the RCS of the target. However, when using a Doppler radar, as is investigated in this study, additional information can be extracted from the target’s generated frequency spectrum that arises from the micro-Doppler effect. 2.3.1 Radar cross-section The RCS of an object describes how much of incoming radiation that is reflected back towards the observer. This is for most objects highly dependent on the angle of incidence since a complex structure may reflect differently in different directions and wave phenomena such as creeping waves or scattering from sharp edges can affect the RCS greatly. The RCS is most often described in a diagram in polar coordinates and it shows the dependence of the RCS on the azimuth angle φ for a fixed elevation angle θ in a spherical coordinate system, which is shown in Fig. 2-3a). An example of the HH-polarisation of the RCS is shown in Fig. 2-3b) for a perfectly conducting cylinder of radius r=5 mm and length L=200 mm. Here, the frequency is 10 GHz and θ=90°. It is clear that the RCS varies rapidly with the azimuth angle φ. The RCS is often denoted σ and is defined [22] as 𝜎 = lim 𝑅→∞ 4𝜋𝑅2 |𝐸𝑠 ⃗⃗⃗⃗ | 2 |𝐸𝑖 ⃗⃗ ⃗| 2 , (2.6) where R is the distance from the scatterer to the field point where the scattered field Es ⃗⃗ ⃗ is calculated. Here, Ei ⃗⃗ ⃗ is the incident electric field of a plane wave. The scattered electric field depends on the frequency of the incident wave, or, expressed differently, the relation of the 11 wavelength in comparison to the size of the scattering object and other length scales that describes its geometry. If such a length scale a is much larger than the wavelength, λ ≪ a, then it is within the optical region and the RCS is independent of the frequency for some simple scatterers such as the metal sphere. The frequency dependence of the RCS in this region may be much more complicated for more complicated geometries or dielectric scatterers. When λ ≫ a, it is within the Rayleigh Region and the RCS is rapidly decreasing with increased wavelength. Lastly, when the length scale a is comparable to the wavelength, λ ≈ a , it is within the Mie Region and the RCS has in general a rather complicated dependence with respect to frequency. Fig. 2-3. a) Spherical coordinate system definition with unit vectors R̂, θ ̂ and φ̂, and values: distance R, elevation θ and azimuth φ. b) RCS of a perfectly conducting cylinder, oriented along the y-axis with radius r=5mm and length L=200 mm, given in polar coordinates for different azimuth angles -180°<φ<180° and for elevation angle θ=90°. The radar waveform uses HH-polarisation and a frequency f=10 GHz. For dielectric scatterers, the radar return is also affected by the dielectric permittivity ε of the material, which can be expressed as a complex number = ′ − 𝑗 ′′. (2.7) Here, the real part ε’ describes the medium’s ability to store electric energy and the imaginary part ε” is related to the losses in the medium. Both the real and imaginary part of the permittivity may depend on the frequency and it can be very complicated, which requires accurate measurements for each individual material and frequency to be considered. For example, water is very transparent to visible light but is highly opaque at both higher and lower frequencies in the electromagnetic spectrum. One way to describe power losses in a medium is to use the loss tangent tan 𝛿. The loss tangent is given by tan 𝛿 = ′′ ′ (2.8) and gives the ratio between ε” and ε’. φ y x z a) b) θ ̂ φ̂ R̂ R θ 12 The half-power lobe width is the sector of opening angle Δφ in radians where the RCS is above half (-3 dB) of its maximum value. A suggested model to approximate the lobe width of a small propeller illuminated around broadside is ∆𝜑 ≅ 𝜆 2𝐿 , (2.9) where L is the total length or diameter of the propeller. This model is in contrast to the expression found in [23] where the lobe width is approximated as Δφ=λ/(L/2) for a helicopter rotor blade, where L is the diameter. Another approximation is to use the broadside lobe width of a thin, circular cylinder with a radius r much smaller than its length L, i.e. r ≪ L. An expression for the RCS of a thin cylinder near broadside is developed in [24] and based on that expression, a broadside lobe width of approximately Δφ=λ/(3L) is obtained, where L is the length of the cylinder. The expression in (2.9) is suggested as a better approximation since a multicopter propeller has a different geometry than both a helicopter blade and a cylinder. The amplitude of the RCS may be approximated using the backscattering from an infinitely long, circular cylinder with a radius a, chosen such that the volume of a cut of length L of the infinite cylinder, corresponding to the length of the propeller, encloses the same volume as the propeller [14]. An infinitely long structure does not have a RCS as the infinite geometry yields an infinite area. The scattering is instead expressed as a scattering width, which is RCS per unit length. A conversion can be made from the scattering width (σSW) of an infinite structure to the RCS (σRCS) of a finite cut of length l of the structure using the relation [25] 𝜎𝑅𝐶𝑆 ≅ 𝜎𝑆𝑊 2𝑙2 𝜆 . (2.10) Expressions for the amplitude of the broadside scattering width from an infinitely long, horizontally oriented cylinder have been developed [22] and are given for HH- and VV-polarisation by 𝜎𝐻𝐻(0) = 4 𝑘0 |∑ 𝜖𝑛(−1)𝑛𝐴𝑛 ∞ 𝑛=0 | 2 (2.11) 𝜎𝑉𝑉(0) = 4 𝑘0 |∑ 𝜖𝑛(−1)𝑛𝐵𝑛 ∞ 𝑛=0 | 2 (2.12) where ϵn is a constant with values ϵ0=1; ϵ1= ϵ2= ϵ3=…=2, k0 is the wavenumber in vacuum. An and Bn are coefficients given by the rather intricate expressions 𝐴𝑛 = − (𝑘1 𝜇1⁄ )𝐽𝑛(𝑘0𝑎)𝐽𝑛 ′ (𝑘1𝑎) − (𝑘0 𝜇0⁄ )𝐽𝑛 ′ (𝑘0𝑎)𝐽𝑛(𝑘1𝑎) (𝑘1 𝜇1⁄ )𝐻𝑛 (1)(𝑘0𝑎)𝐽𝑛′ (𝑘1𝑎) − (𝑘0 𝜇0⁄ )𝐻𝑛 ′(1)(𝑘0𝑎)𝐽𝑛(𝑘1𝑎) (2.13) 𝐵𝑛 = − (𝑘1 1⁄ )𝐽𝑛(𝑘0𝑎)𝐽𝑛 ′ (𝑘1𝑎) − (𝑘0 0⁄ )𝐽𝑛 ′ (𝑘0𝑎)𝐽𝑛(𝑘1𝑎) (𝑘1 1⁄ )𝐻𝑛 (1)(𝑘0𝑎)𝐽𝑛′ (𝑘1𝑎) − (𝑘0 0⁄ )𝐻𝑛 ′(1)(𝑘0𝑎)𝐽𝑛(𝑘1𝑎) (2.14) 13 where 𝑘0 2 = 𝜔2𝜇0 0 and 𝑘1 2 = 𝜔2𝜇1 1 𝜔 = 2𝜋𝑓 (2.15) for a cylinder with dielectric constant ε1 and permeability μ1, and a transmitted frequency f. Jn and Hn in the above equations are Bessel functions and Hankel functions respectively of the n-th order and primes denote the derivatives. 2.3.2 Electromagnetic simulation models For analytic RCS solutions to Maxwell’s equations to be feasible, the geometry of the scatterer must be sufficiently simple and, in addition, other approximations may be necessary to arrive at a tractable problem. Thus, the analytical results presented above for the RCS of thin, circular cylinders provide only coarse models for the scattering from a real propeller. However, numerical solution of Maxwell’s equations is a powerful alternative and, in particular, the finite element method (FEM) is well suited for problems with complicated geometry. For 3D problems, the FEM typically divides space into an unstructured mesh of tetrahedrons, where the electric field is represented by piecewise low-order polynomials. A system of linear equations is derived by setting the weighted residual of the differential equation to zero. For scattering problems, the computational domain discretised by the FEM can be efficiently truncated by a so-called perfectly matched layer (PML) [26]. The PML absorbs the outward propagating scattered field efficiently and this allows for a perfect electric conductor (PEC) backing that serves as the outer boundary of the computational domain. The scattered field in the vicinity of the scatterer is transformed [27] to the far-field region, where the RCS is computed. The PML is the most efficient at absorbing electric fields at near normal incidence to the PML. However, a wave impinging on the PML with a near tangential line of propagation (near 0° grazing angle) may be reflected at the PEC backing and propagate back into the computational domain. Thus, radiation that should have been absorbed by the PML may interact with the scattering object again. This problem is more prominent in elongated structures as the long structure promotes the effect as illustrated in Fig. 2-4. Fig. 2-4. Illustration of reflections at the PML. For simplicity, the illustration shows reflection of an optic ray whereas the actual wave behaviour may be much more complicated. Instead of being damped out by the PML, the wave reflected from the object is reflected at the boundary and it can interact with the object again. PML Radiating object PEC backing 14 2.3.3 Range- and cross-range profiles A technique commonly used in inverse synthetic aperture radar (ISAR) is to create range- and cross-range profiles of a target that shows the structure of the target in radial or perpendicular direction. This technique is based on the point scatter distribution model where the location of scatter centres is derived from varying the illumination angle or the transmitted waveform and then applying the inverse Fourier transform (IFT) on the received signal. The cross-range profile is achieved by varying the angle with which the target is illuminated. The total received signal is a sum of the signal from each point scatter centre as 𝐸𝑠(𝜑) = ∑𝐴𝑖𝑒 −𝑗2𝑘 cos(𝜑)𝑥𝑖𝑒−𝑗2𝑘 sin(𝜑)𝑦𝑖 𝑃 𝑖=1 , (2.16) where P is the number of assumed point scatterers, k is the wavenumber of the transmitted wave, Ai is the amplitude of the field from scatterer i, and xi and yi are the radial and the perpendicular displacement of a point scatterer i from the centre of the target. If the radial displacement of point scatterers is near constant, the term cos(φ)xi is approximated as constant and (2.16) may be rewritten as 𝐸𝑠(𝜑) = ∑𝐵𝑖𝑒 −𝑗2𝑘 sin(𝜑)𝑦𝑖 𝑃 𝑖=1 = ∑𝐵𝑖𝑒 −𝑗2𝜋 𝑘 𝜋 sin(𝜑)𝑦𝑖 𝑃 𝑖=1 , (2.17) where 𝐵𝑖 = 𝐴𝑖𝑒 −𝑗2𝑘 cos(𝜑)𝑥𝑖 is a constant. In (2.17) there is a Fourier relationship between k 𝜋 sin(φ) and yi, thus the scattered field may be expressed as a function of y by taking the IFT of the total signal as 𝐸(𝑦) = ℱ−1{𝐸𝑠(𝜑)} (2.18) As E(y) essentially is equal to Bi, a hypothesis is that the range profile may then be obtained by extracting the phase content of E(y) and dividing by 2k. 2.3.4 Micro-Doppler signature The motion of an object induces a shift in frequency of a reflected wave due to the Doppler phenomenon. However, an object may be comprised of several parts with individual motion, where each part induces a specific Doppler shift. Examples of this are the rotating blades of a helicopter or the swinging arms and legs of a human. An incoming electromagnetic wave is therefore modulated by all the different Doppler shifts and the returned signal may then be analysed spectrally to find the frequency shifts of the individual parts. This effect is known as the micro-Doppler effect and the individual motion of parts is called micro motion [28]. As the micro motion often is a function of time, the micro-Doppler effect is best observed in a joint time-frequency representation as illustrated in Fig. 2-5, where the radar return of a simulated rotating cylinder is shown. There is a clear sinusoidal envelope emerging from the rotation which is due to the Doppler shifts only appearing when a point on the object is moving towards or away from the observer. In the joint time-frequency representation, 15 Doppler shifts appear as sidebands to the shift induced by the translational movement of the body and positive Doppler shifts indicates movement towards the observer and negative shifts indicates movement away. Therefore, one observes so-called flashes with wide spectrum width when the cylinder is observed at broadside as every point on the body is either moving towards or away from the observer. These flashes are observed at times t={0.25; 0.75; 1.25; 1.75} in Fig. 2-5. Fig. 2-5. The joint time-frequency representation of the micro-Doppler spectrum for a simulated rotating cylinder with r=5mm and L=200 mm, rotating at 1 Hz. The sinusoidal envelope is distinct and is due to that Doppler shift only occurs in the radial direction. When the rod is observed at broadside, flashes appear as many points on the rod are either approaching or receding. Two flashes are seen per complete revolution. To obtain the micro-Doppler signature, the short time Fourier transform (STFT) is used. The Fourier transform is computed for some finite time interval and the result is used as a representation of the frequency content for that time interval. Given a sequence of such time intervals the time-frequency signature is built up from many of these STFTs. The appearance of the signature can be changed depending on the length of the time interval of each STFT. If the time is short, the micro-Doppler signature shows the instantaneous frequency content whereas a longer time instead represents the overall frequency content which is an average over the time interval. To fully capture the motion, an overlap of the STFTs is necessary for a smooth representation. The amount of overlap needed depends on the movement being imaged and its velocity components. 2.3.4.1 Extraction of features using SVD In order to use the micro-Doppler signature for classification it is necessary to be able to extract relevant features from it in a way such that algorithms can process the data. There are several ways to accomplish this, e.g. by principal component analysis (PCA) or independent component analysis (ICA). One simple way is to use singular value decomposition (SVD) of the data. With SVD, a matrix A may be rewritten on the form 16 𝐴 = 𝑈𝛴𝑉𝑇 , (2.19) where Σ is a diagonal matrix containing the singular values of A in descending order, and U and V are unitary matrices. If SVD is applied to complex time-frequency data as the one presented in Fig. 2-5, the column vectors of matrix U are coupled to the frequency data and the column vectors of V are coupled to the time data. There is also a relation to the magnitude of the singular values in Σ, where if the first singular value is large then the first vectors have the most significant data points and so on. This is used in various areas such as image compression where data points coupled to the lowest singular values are removed and thus the most significant data points are kept and important features in the image are preserved while reducing the image size. 17 3 STATIC RADAR RETURN OF A SINGLE PROPELLER The use of multiple propellers is the common denominator for all multicopters and the fast speed of the propellers is a very distinct feature that has potential to be used in classification algorithms. It is therefore of great interest to know how the radar return of a single propeller may look like and whether there are models that can approximate its most important features. However, the static radar return must first be calculated to simulate a propeller in motion. The static radar return of a single propeller is investigated by varying different properties such as the dielectric constant, polarisation, angle of illumination and carrier frequency. The simulation software used for these static calculations is Ansys HFSS, which is a finite element solver and the software is firstly described along with the setup of the calculations and the relevant parameters that can be changed. Thereafter, the different investigated cases are stated followed by the results of each case. The numerical results are then compared to analytical results that describe the RCS. Lastly, the choice of investigated cases is summarised and discussed to show which cases are spanned by this study. 3.1 RCS calculations using Ansys HFSS HFSS is a numerical tool for the computation of electromagnetic fields in, e.g., microwave problems. It uses the three dimensional FEM where the computational domain is divided into an unstructured mesh of volume elements in the form of tetrahedrons. A solution for the electric field is computed in each volume element and the residual for each volume element is calculated. The software then uses adaptive mesh refinement in order to minimise the residuals. This means that complicated areas such as sharp edges or other singularities where the residual is large are automatically subdivided into more volume elements in order for the electric field solution to approach a continuous case. The solution reaches a steady-state when the residuals have been minimised which is based on a convergence criterion set by the user. Analysed structures can be drawn in the program by the user in its computer aided design (CAD) environment using simple geometries. CAD models can also be imported into the program for analysis. The propeller investigated herein is not created from scratch but is instead imported into HFSS from an online database [29], where users share their created 3D models. In the simulations, different parameters can easily be varied by the usage of variables and by adding parameter sweeps. The setup used for all static calculations in this study is illustrated in Fig. 3-1. Given a propeller (centred at the origin and aligned with the y-axis), HFSS is used to calculate the complex scattering amplitude F⃗⃗ , which gives the scattered electric field 𝐸𝑠 ⃗⃗⃗⃗ = 𝑒−𝑗𝑘𝑅 𝑅 𝐹 . (3.1) The RCS can then be computed based on (2.6). According to HFSS documentation, the distance D between the scattering object and the PML-air interface should be chosen such that D>0.25λ. However, a greater distance of D=0.4λ is advised from previous experience with the software. As the propeller has internal structure, the distance D is based on the smallest 18 distance from the propeller to the PML which is determined by enclosing the propeller in a non-model object which is not part of the calculations and is as small as possible while still containing the structure. At the outer boundary of the air box, a layer of PML is assigned as boundary condition to simulate an infinitely large air volume and the infinite sphere tool may then be used to get the far field radar return. The propeller is then illuminated at angles -90°<φ<90° with an angular resolution of 1°, with φ=0° being at broadside and φ=±90° being at end-fire. As the propeller is symmetric looking from the front and from the back of it, the full 360° RCS is obtained by mirroring the RCS for the front side. Fig. 3-1. An illustration of the setup used for calculations of the static radar return of a propeller in HFSS. 3.1.1 Setup of parameters One of the settings relevant for RCS calculations is the setting of the PML. Here, one can change two parameters: the minimum radiating distance and the PML thickness. The minimum radiating distance is the minimum distance from a radiating body to the PML and should be set accordingly as the distance from the non-model object to the PML-air interface, i.e. the distance D in Fig. 3-1. It is although advised to choose this value to be somewhat smaller than the exact distance and it is set to 0.9D instead. The thickness of the PML is chosen such that unwanted radiation is damped out and experience has shown that for many cases, a thickness of 0.4λ is sufficient. The material of the propeller is varied in the simulations. Many material parameters can be changed in HFSS ranging from different magnetic properties such as magnetic saturation and relative permeability, to dielectric constant and loss tangent. Magnetic properties are for most materials investigated herein not relevant as they are non-magnetic and are therefore neglected. The dielectric constant is set to be real only, thus neglecting losses in the material. The loss tangent in plastics is usually in the order of tan 𝛿 = 0.1 and this approximation would therefore likely not be too impactful on the obtained results. An incident plane wave is assigned to the problem. Polarisation of the wave is set through two parameters: “E0 Phi” and “E0 Theta”. This determines the amplitude of the electric field in the φ̂ and θ ̂ directions respectively and the relation between the two amplitudes decides the polarisation. With “E0 Phi” set to 1 and “E0 Theta” set to 0 the wave becomes horizontally polarised and with “E0 Phi” set to 0 and “E0 Theta” set to -1 the wave becomes vertically polarised. PML x y z Air Non-model object Propeller D 19 When all parameters have been set, a convergence criterion is set and the simulation is run. The criterion set for all simulations in this study is “Maximum Delta Energy”=0.005 which is a strong criterion that is believed to yield accurate results. When results are obtained from the simulations, the received polarisation can be changed. This is done through choosing data points “ComplexMonostaticRCSPhi” for the complex field received in horizontal polarisation and “ComplexMonostaticRCSTheta” for data received in vertical polarisation. 3.1.2 Effects of PML reflections and error estimation As the propeller is an elongated structure, it may cause reflections in the assigned PML boundary condition. This is therefore tested by increasing the smallest distance D between the propeller and the PML in three steps, D={0.4λ; 0.6λ; 0.8λ}. The different parameters of this test case are chosen such that they represent a typical propeller in the calculations presented later in this report so that the test may result in relevant error estimation. A representative dielectric constant and frequency are chosen for the test as εr=8.0 with frequency f=10 GHz as the material is rather reflective and the frequency gives the smallest wavelength and, therefore, yields the smallest values of D and smallest grazing angles α. Polarisation is chosen as HH-polarisation. A 9” or 23 cm diameter propeller is used which is 7.63λ long expressed in wavelengths at this frequency. This means that the near-grazing angles tested for these distances then are α={6.0°; 8.9°; 11.9°} for a reflection point at air-PML interface as it intersects the plane y=0. The result of the calculations can be seen in Fig. 3-2. Little difference is seen as D is increased and this suggests that the adaptive meshing or PML dampening of HFSS is taking care of the issue of PML reflections. A few angles are identified where the difference is the largest and the RCS for these angles are summarised in Tab. 3-1. The angles are chosen such that they are not located in deep minima since the largest fluctuations are expected at such angles due to the low RCS values. The largest change in RCS is 1.45 dB, whereas there is almost no difference for most azimuth angles. As the increased distance D did not show large fluctuations in RCS, D=0.4λ is deemed to be accurate, which also is in good agreement with previous experience of the software. Therefore, the smaller air volume with D=0.4λ is used for simulations carried out within this study. Fig. 3-2. The impact of the minimum distance D from the propeller to the air box boundary on the RCS, with εr=8.0 and f=10 GHz. 0° is at broadside of the propeller. There is not much variation observed in the main lobes. The largest variation is seen in the deep minima as they are sensitive to changes due to the low RCS values. 20 Tab. 3-1. Calculated RCS in dBm 2 for a few identified angles where the change in RCS is large, as the distance D from the propeller to the boundary of the air box is increased. The largest change is seen for φ=-73° where the RCS changes by 1.45 dB from D=0.6λ to D=0.8λ. φ = -73° φ = -45° φ = -40° φ = 47° φ = 77° D = 0.4λ -42.00 - 35.43 -35.69 -37.51 -35.72 D = 0.6λ -41.40 -34.96 -35.55 -37.10 -36.21 D = 0.8λ -42.85 -35.52 -35.17 -37.74 -34.76 As a final test, a reference case for RCS calculations consisting of two dielectric spheres of radius 15 cm and at a distance of 4 m apart is used. The dielectric spheres have the permittivity εr=1.5-0.02/(j2πfε0). An electromagnetic wave impinges on one of the spheres along the axis of the two spheres and the monostatic RCS is calculated at frequencies 5 MHz to 4 GHz in steps of 5 MHz with a T-matrix solution method including multiple scattering. The setup of the reference case is shown in Fig. 3-3. This scenario is modeled in HFSS and calculated for the relevant frequencies with the parameters D=0.4λ, minimum radiating distance of the PML as 0.9D and thickness of the PML as 0.4λ. However, the reference case features a distance of 4 m between spheres and the problem investigated herein is 23 cm long. Thus, it is necessary to scale the frequency in order to make the distance in wavelengths between the spheres equal to the length of the propeller in wavelengths. Rescaling the frequency with a factor of 0.2/4=1/20 gives the new frequencies f={50; 150; 300; 500} MHz for the reference case and this corresponds to f={1; 3; 6; 10} GHz, respectively, for a typical propeller considered in this thesis. In addition, the list of frequencies is extended such that f={45; 50; 55; 135; 150; 165; 270; 300; 330; 450; 500; 550} MHz. The result of the test is presented in Fig. 3-4 and it shows good agreement with the reference data with a deviation less than 1 dB for most frequencies. At 55 MHz, however, there is a difference of 5 dB and, also, the RCS is very low at this frequency. Thus, spurious reflections at the PML-air interface may change the RCS substantially. Fig. 3-3. The setup of the RCS calculation reference case. The two spheres have a dielectric constant of εr=1.5-0.02/(j2πfε0) and the wave vector is aligned with the straight line through the centres of the two spheres. 4 m 30 cm 30 cm k 21 Fig. 3-4. a) Test of the setup in HFSS in comparison to reference data and b) the absolute value of the difference between the two RCS datasets. 3.2 Propeller characterisation and property variations Multicopter propellers come in all different sizes and shapes. To study how different properties affect the radar return without having to simulate each possible configuration, a set of representative cases is defined. In order to show the dependence of RCS on the material of the propeller, five different dielectric constants are chosen. Inspiration is taken from [30] where dielectric constants of plastics have been measured at a low frequency of 1 MHz. Thus, dielectric constants εr=3.4 and εr=5.7 are chosen which are the dielectric constants at low frequencies for nylon 6,6 and a glass reinforced plastic which are materials that could be used in multicopter propellers. This should however not be interpreted as their dielectric constants at higher frequencies as they are only accurate at the measured frequency but they are chosen nevertheless as no data at relevant frequencies could be found. A higher dielectric constant of εr=8.0 is chosen as well. Inspiration is here taken from [31] where the dielectric properties have been measured once again at lower frequencies of up to 0.5 MHz for carbon fibre in epoxy resin composite material. The dielectric constant was observed to be high at about εr=8.0 for 15wt% of carbon fibre and increased with increased wt%. It has also been shown that carbon fibre materials are almost as reflective as aluminium at radar frequencies, despite being much less conductive [32], which indicates that a high εr may be expected at higher frequencies. Also, two extreme cases were picked with εr=2.0 and PEC which could be seen as having an infinite dielectric constant and is henceforth referred to as εr=∞. These dielectric constants could represent wood which is largely transparent and a metal which is highly reflective. Propellers made out of wood and metals are uncommon but are alternatives that are used in some cases. Frequency and polarisation are varied as well. The chosen radar operating frequency bands are the ones most common within search and track radar systems: L-, S-, C- and X-band. To limit the number of investigated frequency cases, one representative frequency per frequency band was chosen as 1 GHz, 3 GHz, 6 GHz and 10 GHz, corresponding to wavelengths 30 cm, 10 cm, 5 cm and 3 cm. Polarisation is chosen as VV- or HH-polarisation as they are common within these radar applications. The elevation angle is also varied to investigate whether the radar return varies with angle of illumination. a) b) 22 The CAD-models of the propellers are taken from an online database [29], thus sizes and shapes of the propellers are limited to what is freely available. Available sizes range from 5” to 26” and a 9x4.7”, CCW propeller model is chosen as the base for the computations as the model seems to be well made and accurate and of a typical size for small or medium sized multicopters. The CAD-model is shown with its orientation in the problem setup in Fig. 3-5 and this model is used in all the following simulations unless it is stated explicitly that another propeller model is used. The dimensions of the red, non-model box enclosing the CAD-model are 32x230x11 mm and roughly gives the dimensions of the propeller. Fig. 3-5. The 9x4.7” CAD-model used in most simulations and its orientation in simulations, viewed from: a) 3D view; b) top view; c) side view; and d) front view. 3.2.1 Basic case: HH- and VV-polarisation The two computed polarisations are HH-polarisation and VV-polarisation which are common in radar applications. This is calculated for dielectric constant εr=8.0 at 10 GHz and the results are shown in Fig. 3-6. Vertical polarisation RCS is approximately 8 dB smaller than the horizontal polarisation at most angles. Furthermore, the vertical polarisation yields a 17.2 dB smaller RCS at broadside as compared to the horizontal polarisation. This is expected since the propeller is much larger in the horizontal direction, with its total length of 23 cm than it is in the vertical direction with its height of approximately 1 cm. The vertical polarisation results in smaller RCS than horizontal for this kind of structure which is in good agreement with [13] and [14]. As there is such a great difference in RCS between the two polarisations, HH-polarisation is used in the following simulations if nothing else is stated since a larger RCS is beneficial for detection of a target with radar. A calculation was also performed with εr=3.0 to compare with a previously calculated case with VV-polarisation in [14]. The two cases did however not fully agree and more details about this comparison can be found in Appendix A. a) b) c) d) 23 Fig. 3-6. Calculated RCS of the 9x4.7” propeller with εr=8.0 for HH- and VV-polarisation. VV-polarisation is notably smaller than HH-polarisation by on average 8 dB and by 17.2 dB at broadside. 3.2.2 Variation of material and carrier frequency To see how different materials of propellers may look like at different radar bands, a variation of the two parameters is performed. The result of the three mid-range values of dielectric constant, εr={3.4; 5.7; 8.0}, is shown in Fig. 3-7. At lower frequencies 1, 3 and 6 GHz there is a distinct lobe at around 0° (broadside) and there is almost no energy radiated at ±90° (end-fire). This can be intuitively expected as the propeller is structurally much larger when viewed at broadside, which should also be reflected in its RCS at this angle. However, at the higher frequency of 10 GHz, significant lobes at end-fire are also observed which are comparable to, and may even be larger than, the lobe at broadside. The end-fire lobe is also shifted to 60° instead of 90° as εr is increased. The large end-fire lobe is predicted for long, thin bodies illuminated at end-on incidence with HH-polarised radiation, where backscattered energy appears to radiate from the rear of the body [22]. Another contributing factor may be that the propeller has become much longer than the wavelength. The length of the propeller expressed in wavelengths for each of the frequencies is 0.76λ, 2.29λ, 4.58λ and 7.63λ, where the latter approximately fulfils the condition L=7.63λ ≫ λ . As a consequence, four RCS flashes per full revolution of the propeller may be observed at 10 GHz as opposed to the two flashes normally seen for elongated structures such as the rotor blades of a helicopter. 24 Fig. 3-7. RCS of the 9x4.7” propeller at different frequency bands for dielectric constants: a) εr=3.4; b) εr=5.7; and c) εr=8.0. For 1, 3 and 6 GHz the broadside RCS is much larger than end-fire RCS as expected. At 10 GHz however, end-fire RCS is comparable to, or even larger than, broadside RCS. Looking at the broadside lobe for the different cases there are some general patterns that can be observed. The maximum of the lobe is not always aligned with 0° and this is due to the geometry of the object. From the amplitudes of the RCS, it is clear that the RCS is very low for 1 GHz, where it is approximately 25 dB lower than at other carrier frequencies for all three cases. The long wavelength suggests that the problem is approaching the Rayleigh region in all dimensions of the propeller at this frequency and the RCS may therefore be expected to decrease quickly with decreased frequency. The frequencies 6 and 10 GHz yield the highest amplitudes and their amplitudes are also comparable to each other at broadside, whereas 3 GHz gives a slightly smaller RCS with about 6 dB reduction. The broadside lobe widths are calculated as the half-power lobe width given that the angular resolution is 1°. An estimate with improved angular resolution is made by linearly interpolating between data points. The lobe width is seen to be decreasing with increased frequency, which is predicted in (2.9). The two special cases of εr={2.0; ∞} are also calculated and the results can be seen in Fig. 3-8. Unsurprisingly, the RCS values for εr=2.0 are very low at below -40 dBm 2 at all frequencies and all aspect angles, which is due to the low dielectric constant. For εr=∞ the a) b) c) 25 pattern becomes more complicated with many side lobes for lower frequencies as well. The end-fire RCS lobes are shifted from 90° to 60° (as in the case with εr=8.0 at 10 GHz) which can be observed at frequencies 3, 6 and 10 GHz. This behaves much like the RCS of a thin PEC cylinder, as studied in [14], where the same type of side lobes at 60° are seen. The amplitude is in this case increasing with decreasing frequency, which is the opposite behaviour from the cases in Fig. 3-7 and Fig. 3-8a). The increase in RCS at broadside between the frequencies 1 and 3 GHz is 0.8 dB, between 3 and 6 GHz is 2.7 dB and between 6 and 10 GHz is 2.7 dB. At 1 GHz, it shows almost the same characteristics as a half-wave dipole antenna which is due to the PEC material and the length of the propeller being 0.76λ at this frequency. Fig. 3-8. RCS of the 9x4.7” propeller at different frequency bands for the two special cases of dielectric constants: a) εr=2.0; and b) εr=∞. The amplitude of the RCS for εr=2.0 is lower than -40 dBm 2 at all angles and frequencies. For εr=∞, the pattern is much different from other dielectric constants and is seen to increase with decreasing frequency until it at 1 GHz behaves much like a dipole antenna. A more detailed characterisation of the radiation patterns of all material cases can be found in a table in Appendix B where the lobe widths and amplitudes are presented for the aspect angles with the largest lobes. A comparison of the calculated broadside lobe widths with the theoretical model in (2.9) is summarised in Tab. 3-2, where a good agreement is seen at all frequencies except for 1 GHz. Tab. 3-2. The calculated broadside lobe widths for the five dielectric constants in comparison to the theoretical value based on antenna theory. A good agreement is seen at higher frequencies, whereas 1 GHz has the highest deviation for all εr. Frequency [GHz] Calculated lobe width [°] Theoretical lobe width (Δφ=λ/(2L)) [°] εr=2.0 εr=3.4 εr=5.7 εr=8.0 εr=∞ 1 56.1 45.6 40.4 38.7 75.5 37.6 3 12.5 12.6 12.6 12.7 10.5 12.5 6 7.0 7.0 7.0 7.1 5.6 6.3 10 2.9 3.4 4.1 4.5 3.4 3.8 a) b) 26 3.2.3 Sensitivity to frequency variation It is evident from previous calculations that the radar return varies greatly with frequency. To capture how sensitive the RCS of the propeller is to a small frequency fluctuation, a test is performed where each of the tested frequencies were increased by 10%. The result of this test is shown in Fig. 3-9. An extra frequency calculation is added to 10 GHz as the 10% increase is such a large step, going from 10 GHz to 11 GHz. For the lower frequencies of 1 and 3 GHz, there is only a small increase in RCS with increased frequency. However, at 6 GHz, large end-fire lobes are forming with an increase of 14.0 dB at 90° from 6 to 6.6 GHz, although still considerably smaller than the RCS at broadside. This sensitivity is attributed to the problem being in the Mie region. Thus, RCS may change also for small changes in frequency. For the 10 GHz variation, the RCS amplitudes remain somewhat constant while different lobes in the radiation pattern are instead seen to shift to other aspect angles. The narrow lobe width for the broadside RCS is also becoming more difficult to distinguish as the nearest neighbouring side lobe at 10° starts to merge with the broadside lobe at 4°. Fig. 3-9. Test of the sensitivity to frequency changes for a 9x4.7” propeller with εr=8.0. A change in frequency of +10% was applied to frequencies: a) 1 GHz; b) 3 GHz; c) 6 GHz; and d) 10 GHz. An extra frequency was tested for 10 GHz for better visualisation. At 6 GHz, the end-fire lobes are increasing significantly by 14.0 dB which is attributed to the problem being in the Mie region. For 10 GHz it is seen that the broadside lobe at 4° merges with the lobe at 10° as frequency is increased. a) b) c) d) 27 3.2.4 Elevation angle variation As the multicopter may tilt as much as 40° during acceleration, it is interesting to analyse whether there is any change in RCS of a propeller as it is illuminated at higher elevation angles θ. Angles are therefore varied from 50°≤θ≤130° in steps of 20°, with θ=90° being the horizontal plane. This is also done for VV-polarised radiation as it is believed that this may yield larger RCS at end-fire as the length of the propeller is more in the vertical dimension at higher elevation angles. The result of this variation for εr=8.0 at 6 GHz is shown in Fig. 3-10. With HH-polarisation, there is only a slight difference that can be seen at different elevations. In VV-polarisation, the difference is more apparent at both broadside and end-fire. At end-fire, RCS is seen to be larger, as expected. At broadside however, RCS is also seen to increase at higher elevation angles. This is most likely due to the problem being in the Mie region. Fig. 3-10. RCS of a 9x4.7” propeller with εr=8.0 and f=6 GHz for different elevation angles 50°≤θ≤130° and two different polarisations: a) HH-polarisation; and b) VV-polarisation. With HH-polarisation there is only a slight difference whereas for VV-polarisation the RCS at end-fire is increased at high angles due to the polarisation being along the blade length. There is also a large increase of RCS at broadside which is due to the problem being in the Mie region. In Fig. 3-11, the result of the elevation variation at 10 GHz is presented. This shows a more complex appearance of the RCS for HH-polarisation at different elevation angles. However, the amplitude levels are roughly the same except at end-fire where the RCS decreases for the highest angles θ={50°; 130°}. For VV-polarisation, the same increase of RCS at broadside as seen at 6 GHz is evident. In this case, this may be attributed to the geometrical shape of the propeller being more important at this frequency such that an incoming wave impinges on a specular reflex on its curvature, returning more energy to the source. The hole, with which the propeller is fastened to a rotating axis, also becomes more visible at higher elevation angles and effects such as cavity resonances may occur. These cavity resonances are however not seen in a real case as the hole is closed. At end-fire, a decrease in RCS is seen at higher angles. This is attributed to the shorter wavelength used and therefore the internal structure is of more importance and may reflect radiated energy in unexpected directions. a) b) 28 Fig. 3-11. RCS of a 9x4.7” propeller with εr=8.0 and f=10 GHz for different elevation angles 50°≤θ≤130° and two different polarisations: a) HH-polarisation; and b) VV-polarisation. HH-polarisation shows a complex behaviour, although with rather similar amplitude levels, except at end-fire for high elevation angles. VV-polarisation shows a higher broadside RCS for high elevations which is due to the geometry of the propeller. The end-fire RCS is also higher at low angles θ=90±20°. 3.2.5 Ducted propeller As previously mentioned, ducts may be used on multicopter models for protecting the rotating propellers. This could potentially be an issue for radar detection as incoming radiation is scattered at the duct, without interacting with the rotating propeller, thus not generating a clear micro-Doppler signature. Therefore, the impact of the duct is calculated. The model used in the duct calculations is shown in Fig. 3-12 with the 23 cm propeller as a reference. The thickness of the cylinder which represents the duct is set to 1 cm, outer radius to 14 cm and height to 3 cm. These parameters were chosen in order to mimic the duct dimensions seen in the Parrot AR Drone 2.0, which can be seen in Fig. 2-1a). The cylinder is also divided into segments as experience shows that perfectly smooth, curved surfaces introduce errors in calculations in HFSS and the number of segments is set to 45. The material of the duct for the Parrot model is most likely Styrofoam as it is intended for indoor use. The dielectric constant of Styrofoam has been measured to around εr=1.03 at a near 10 GHz frequency [33] and this is the value used in these calculations as well. Furthermore, one can imagine a worst case scenario in which a duct is made out of a more reflective material, thus εr=8.0 is also studied. The radar return is also calculated both with and without a 23 cm propeller with εr=8.0 and, in all cases, the elevation angle θ for the incident wave is set to 90° which gives incidence in the xy-plane and the frequency is set to f=10 GHz. The results of the HFSS calculations can be seen in Fig. 3-13. The RCS of the εr=8.0 duct without propeller is rather high at -18±0.5 dBm 2 . The small deviation of the RCS is due to the segmented cylinder and each of the peaks is seen where a segment is located. This deviation could be removed by increasing the number of segments, however the rather large increase in computational volume of this problem makes convergence harder to achieve. In the case of εr=1.03 it is clear that the RCS is very low at -51 dBm 2 which is expected as the low dielectric constant is almost the same as that of air which is close to 1. Large deviations of a) b) 29 approximately 5 dB are observed at φ={-90°; 0°; 90°; 180°}. This is where the duct is at the closest distance from the surrounding box of air and these effects may be due to the complicated wave propagation with reflections between duct and PML in conjunction with the low RCS values where small differences in absolute value may yield large differences in decibel. Fig. 3-12. The HFSS-design model used in calculations of the radar return from ducts. The 23 cm propeller is sometimes used in calculations as well. Fig. 3-13. RCS of ducts with dielectric constants: a) εr=8.0; and b) εr=1.03. The variations are due to the duct being split up into 45 segments. The scale has been chosen to show the variations more clearly. The low RCS of the duct with εr=1.03 suggests that the propeller is still visible through the walls whereas the high RCS of the εr=8.0 duct would make it difficult to distinguish anything within it. Simulations were carried out with the duct and propeller simultaneously and the results of these tests confirm these theories and can be seen in Appendix C. The duct with the high dielectric constant is seen to almost completely shadow the propeller while the duct with the low dielectric constant reveals the propeller in detail. However, even though ducts may cover the propellers completely at horizontal incidence of an electromagnetic wave they are most likely visible as the multicopter accelerates as they may be substantially tilted. 3.2.6 Variation of propeller length The length of the propeller is also varied in order to further investigate the length dependence on the radar return. Two additional CAD-models were chosen for this test, a 10x4.7” model a) b) 30 and an 11x4.7” model being 25.5 cm and 28 cm long respectively. These models were chosen as they were made using the same method for shaping the propellers to get correct aerodynamic properties. In addition, the propeller models have the same pitch, which makes comparisons easier. The angular resolution of φ is here set to 0.5° instead of the previous 1° as narrower lobe widths are expected in these calculations. The results of the test are seen in Fig. 3-14. The radiation patterns are seen to vary, however with roughly the same shape. At 10 GHz for the 11” propeller, there is an unexpected decrease of radiation at around 0° and the largest broadside flash is seen at 11° in comparison to 4° and -1° for the 9” and 10” propellers respectively. There is also a change in the radiation at 6 GHz for the 11” propeller as the broadside flash is seen to merge with the nearest side lobe, thus getting a wider lobe width than expected. These unexpected deviations are attributed to the change in CAD-model which could introduce additional changes to the radiation pattern due to slightly different meshing of the models. Fig. 3-14. Variation of propeller length with εr=8.0 and models: a) 9x4.7”; b) 10x4.7”; and c) 11x4.7”. The broadside lobe width is decreasing with propeller length as expected. The lobe widths are compared to the theoretical prediction in (2.9) for frequencies f={3; 6; 10} GHz and illustrated in Fig. 3-15. The calculated lobe widths follow the pred