### B

UDAPEST### U

NIVERSITY OF### T

ECHNOLOGY AND### E

CONOMICS### F

ACULTY OF### E

LECTRICAL### E

NGINEERING AND### I

NFORMATICS DEPARTMENT OFBROADBANDINFOCOMMUNICATIONS ANDELECTROMAGNETICTHEORY

**On scattering and radiation from moving and** **composite bodies**

*Author:*

Károly MARÁK

*Supervisor:*

Dr. Sándor BILICZ

### S

UMMARYApril 2, 2020

**1 Background**

Around the beginning of my doctoral studies, there was great interest in the
topic of*Unmanned Aerial Vehicles’*(UAVs, drones) detection through radar from
the Microwave Remote Sensing Laboratory of the department. The challenge in
this task comes from the small size of drones, which makes their observation by
radar difficult, as at lower frequencies, their*Radar Cross Section*(RCS) is quite
small, and at higher frequencies, they are difficult to distinguish from other ob-
jects. However, it was observed experimentally that the modulation of the scat-
tered electromagnetic field caused by the rotation is most pronounced near the
resonant frequency of the propeller, and this modulation causes a discrete scat-
tered spectrum [1]. My first task was therefore to create a model of backscatter-
ing (monostatic radar detection) from a rotating propeller, which can be used to
calculate the scattered field as a function of parameters such as the geometry,
materials used, the speed of the rotation, etc.

After I constructed this model and verified it through alternative calculation methods, I expanded my area of research to include more general cases, such as a bistatic scattering experiment or a coupled model of two nearby rotating propellers. All of these scenarios were validated experimentally with a setup constructed by fellow PhD student Tamás Pet˝o.

The next topic of interest arose out of the first, namely, during the investiga- tion of the effects of the propellers’ material on the scattered field. Simulations I had performed on scattering from conductive materials implied that by inves- tigating the results of the scattering experiments, some conclusions could be drawn about the material of the propellers. Much of the scattering theory was in place by this point, however, the inverse problem was yet to be formulated, and, as the typical material for propellers is some kind of carbon-fiber compos- ite, I also had to address the question of anisotopy. After all this was done, I designed the experiment and procured some samples. Again with the help of Tamás Pet˝o, the experiment was set up to verify the feasibility of the idea.

Work on third and final thesis point was started in late 2017, during a short-
term scientific stay at the TU in Prague. Within the framework of the COST Ac-
tion 1301, I were to investigate antenna array pattern synthesis in the context
of radiated*Wireless Power Transfer*(WPT). After implementing the vanilla algo-
rithm based on Dr. Kracek’s idea, I extended the basic formalism, and tested it
on different goal patterns.

Since that time, this problem has branched to different subtopics including multiple pattern synthesis, interaction between nearby antennas, interpolation and minimization schemes, which required coming up with and implementing new ideas, however, earlier results concerning the efficient modelling of inter- acting scatterers were also useful, together with the proficiency gained with nu- merical software.

**2 Results**

From the theoretical standpoint, it was important to gain an understanding
of, e.g., the formalism describing scattering from moving bodies [2], properties
of anisotropic scatterers [3] and radiation pattern synthesis theory [4]. From
the numerical point of view, I learned about and tried several computational
schemes (such as the*Integral Equation*(IE) [5] or the*Finite Element Method*
(FEM) [6]) and numerical software (such as Matlab or Comsol).

Fortunately, with regards to experimental work, I could rely on the hardware assembled by Tamás Pet˝o, in addition to his expertise in the subject, meaning that I could concentrate mainly on theoretical matters. However, I assisted in al- most all of the performed measurements whose data I used, encountering sev- eral questions such as noise filtration on the spectrum analyzer, the choice of appropriate samples, unwanted reflection from the environment and working around the frequency-dependent characteristics of certain antennas.

**2.1 Investigation of electromagnetic scattering from rotating propellers**
Recently, Unmanned Aerial Vehicles (UAVs)—drones—have gotten attention on
account of their difficulty of detection with conventional radar systems due to
their very small radar cross section (RCS). In order to develop more powerful
detection methods, it is necessary to more deeply investigate the patterns in the
signal reflected from UAVs.

It is well known that illuminating a body in uniform, linear motion causes the scattered wave to suffer a so-called Doppler shift, meaning a shift in fre- quency. A similar phenomenon can be observed in the case of more compli- cated (micro-) motion, such as the rotation of a propeller; this phenomenon is called the micro-Doppler effect.

In the literature [7, 8], most of the time, similar problems are solved using a small-wavelength approximation such as Physical Optics (PO) or Ray Tracing.

In the case when such approximations are not appropriate (as in our case, when
the size of the propeller is comparable to the wavelength of the illumination), it
is necessary to perform a*full wave simulation*with the appropriate considera-
tions as to the rotation of the observed object.

In other cases [9], a time domain simulation is constructed. Fundamen-
tally, it would be possible to use such a method in the simulations, however, it
is simpler (we get ride of the*time*variable) and more consistent with the the-
ory (we are dealing exclusively with harmonic time functions) to remain in the
frequency domain.

I outline the following approach in the thesis: by using the properties of the transformation to and from rotating frames, I show that the spectrum of the scattered field contains an infinite but countable number of frequency compo- nents

**E***s*(r,t)=
X∞
*m=−∞*

**E***m*(r)e^{j(}^{ω+m}^{Ω}^{)t}, (1)

*Figure 1: While the FEM model requires a 3D discretization of the propeller and*
*the surrounding medium, in the MoM, a 1D discretization of the scatterer with*
*appropriately chosen parameters (a and l ) and a sinusoidal basis of approxi-*
*mately 5-10 functions yields similar results.*

whereΩis the angular velocity of the propeller’s rotation and*ω*is the angular
frequency of the illuminating wave.

Afterwards, I show that in the case when*ω*andΩare several orders of mag-
nitude apart, a so-called*quasi-stationary*approach [10] is applicable. This is
analogous to*bandpass sampling*or*undersampling*known from signal theory
[11]. Using this simplification, I then describe the spectrum of the electromag-
netic field scattered in any direction by calculating the scattered field for a num-
ber of stationary positions of the propeller. Due to the large aspect ratio of a pro-
peller, I perform the calculation using an efficient 1D discretization scheme by
means of an integral equation formulation, making use of the propeller’s geo-
metrical properties; I also validate the results by means of a 3D FEM simulation
(the schemes are illustrated in Fig. 1).

This procedure can be generalized to an arbitrary scatterer whose dimen-
sions are comparable (or smaller than) the wavelength. Specially, if it performs
a time-periodic motion and its position can be described by parameterΩt=*ϕ,*
then the coefficients in (1) can be calculated as the Fourier coefficients of the
function**E***s*(r,*ϕ)*

**E***s*(r,*ϕ)*=
X∞
*m=−∞*

**E***m*(r)e^{jϕ}. (2)

I also investigate the more general case of multiple adjacent scatterers. The re- sulting spectrum is in the form of

**E**^{sec}* _{s}* (r,

*t)*= X∞

*m*1=−∞

X∞
*m*2=−∞

**E***m*1,m2(r)e^{j(ω+}^{m}^{1}^{Ω}^{1}^{+}^{m}^{2}^{Ω}^{2}^{)t}, (3)
whereΩ1andΩ2are the angular velocities of two adjacent propellers.

Using the experimental hardware/setup assembled by fellow Phd student
Tamás Pet˝o, him and I have validated the theoretical results in a number of ex-
periments:*monostatic*and*bistatic*scattering experiments using a single CFRP
propeller, and also performed experiments with*two adjacent propellers*with
adjustable orientation and relative position. I calculate the radar cross section

*Figure 2: Anechoic chamber measurement scenario for two adjacent propellers.*

*Figure 3: Measured spectrum for two adjacent propellers d*=300 mm*apart. The*
*red signs denote the source of the spectral components.*

(RCS) of a 11 inch CFRP propeller from the measurement results; it is found
that it agrees acceptably well with the theoretical expectations, especially when
considering the small size of the scatterer (the experiments imply a value of
RCS_{exp}≈0.4m^{2}compared to the theoretical RCS_{th}≈0.25m^{2}). The sketch and
results of the experimental setup for the investigation of scattering from two ad-
jacent rotating propellers can be seen in Figs. 2 and 3, while results of a bistatic
scattering measurement and their comparison with the calculated values are
shown in Fig. 4.

-300 -200 -100 0 100 200 300
f-f_{0} [Hz]

-140 -120 -100 -80 -60 -40

P[dBm]

**a/3**

measurement calculations

*Figure 4: Comparison of experimental and computational results of the spec-*
*trum measured in a bistatic scenario. Calculations predict the intensity of the*
*side peaks relatively accurately, while the central peak is more pronounced due to*
*the miscellaneous scattering from the environment and the interference between*
*the transmitter and receiver antennas.*

**Summary**

*Thesis 1: I devised a computationally efficient, simplified 1D integral equation*
*(IE) based method in order to describe scattering from rotating carbon fiber rein-*
*forced plastic (CFRP) propellers, the results of which I validated by means of 3D*
*finite element method (FEM) simulations describing the exact propeller geome-*
*try.*

• The theoretical framework for treating the scattering problem involves a frequency domain calculation method (such as the aforementioned IE or the FEM), coupled with the quasi-stationary formalism for calculating scattering from moving objects.

• Specifically, the simplified IE model was shown to cause a massive im- provement in performance, owing to the much lower number of Degrees of Freedom (DoF) when compared to the complex 3D model.

• The framework was applied to the problems of bistatic and monostatic
scattering from a single rotating propeller, as well as monostatic scat-
tering from two adjacent propellers. Together with fellow PhD student
*Tamás Pet˝o, we designed experiments corresponding to these problems.*

• An experimental setup for the purpose of validating the model’s results
was assembled by*Tamás Pet˝o*and the*Microwave Remote Sensing and*
*Radar Laboratory. It consists of an adjustable frame capable of setting*

the orientation of the rotation axis of the propellers, as well as their posi- tion, antenna(s) for the purpose of illumination (excited with a continu- ous wave signal) and reception (connected to a spectrum analyzer) of the scattered field. The movement of the propeller is controlled by a brush- less DC motor which can be regulated through a programmable micro- controller. Using this setup, we performed several anechoic chamber and free-range experiments corresponding to the aforementioned scattering scenarios. I discuss the accuracy of the measurement method, as well as the prospects for future applications.

**Related publications: [C1, C2, C3, C4], [J1, J2]**

**2.2 Micro-Doppler based conductivity measurement**

In microwave applications such as radar detection or antenna design, the*elec-*
*trical conductivity*(σ) is a material parameter of high importance. There are
several methods of its measurement, however, many are limited to lower fre-
quencies [12] (due to the errors introduced into the experimental setup by high
frequency effects, particularly at the contacts), and it can show a dependence
on frequency for certain materials. Another group of techniques is applicable
at higher frequencies [13, 14]. Generally, for the aforementioned reasons, con-
tactless methods are preferred, however, these methods generally require either
a large sample compared to the wavelength, or a very precise measurement of
EM field quantities.

Based on the results of Thesis 1, I make use of the micro-Doppler effect caused by the rotation of an elongated sample to filter out the background noise of miscellaneous scattering from the environment. From the backscattered spec- trum, it is then possible to draw conclusions regarding the conductive proper- ties of the sample’s material near the frequency of the investigation :

• By measuring the relative intensity of the backscattered field and compar- ing it with a reference sample’s field;

• and by investigating the resonant frequency of the sample.

I perform numerical simulations of the scattering to assess the accuracy of
the method; as seen in Fig. 5, I show that it performs the best for conductivities
ranging from 1 S/m to 10^{3}S/m. I extend the idea to anisotropic samples, with
slight modifications proposed to the experimental setup in case of a significant
anisotropy of the material inspected.

Lastly, in collaboration with Tamás Pet˝o, we perform experiments to demon- strate the viability of the procedure on samples with dimensions similar to those of a typical drone propeller, with the results and samples being shown in Figs. 6 and 7, respectively.

10^{0} 10^{1} 10^{2} 10^{3} 10^{4}
[S/m]

10^{-2}
10^{-1}

|Es r / Ei| [m]

*(a)* *Dependence of the intensity of the*
*backscattered field on the conductivity of*
*an elongated sample. E*^{i}*and E*^{s}*indicate*
*the intensities of the incident and scattered*
*fields, and r denotes the distance from the*
*sample at which the scattered field is calcu-*
*lated.*

10^{2} 10^{3} 10^{4}
[S/m]

620 630 640 650 660 670

fr[MHz]

*(b) Dependence of the resonant frequency*
*on the conductivity of an elongated sample.*

*Figure 5: Scattered intensity and resonance frequency as functions of conduc-*
*tivity; the dotted red lines signify the interval of calculated intensities which cor-*
*respond to a given conductivity value with*±10%*deviation from the nominal*
*value.*

560 580 600 620 640 660

f[MHz]

-114 -112 -110 -108 -106

received power[dB] sample 1

sample 2

*Figure 6: Experimental validation of the proposed measurement method: re-*
*ceived power at frequency*(f−2F)*(where f is the frequency of the illumination*
*and F is the frequency of the sample’s rotation), corresponding to the -1st peak*
*in the scattering spectrum of the rotating sample. The blue marks correspond to*
*sample 1, while the red marks to sample 2 of a lower conductivity.*

*Figure 7: Photograph of the samples used during the demonstrative experiments,*
*3D printed from a conductive filament, next to a carbon fiber reinforced plastic*
*drone propeller. On the top is sample 1 with an infill percentage of 100%, while*
*in the middle is sample 2 with a 10% infill percentage.*

**Summary**

*Thesis 2: I propose an experimental arrangement to measure high frequency elec-*
*tromagnetic material parameters of small, elongates samples; I performed nu-*
*merical simulations to analyze the accuracy of the method, and designed a demon-*
*strative experiment.*

• Based on the results of Thesis 1, I propose to employ the micro-Doppler effect caused by the rotation of an elongated sample to filter out the back- ground noise of miscellaneous scattering from the environment. From the backscattered spectrum, one can then draw conclusions about the conductive properties of the sample’s material near the investigation fre- quency:

1. By measuring the relative intensity of the backscattered field and comparing it with a reference sample’s field;

2. And by investigating the resonant frequency of the sample.

• I performed numerical simulations of the scattering to assess the accu-
racy of the method; I found that it performs the best for conductivities
ranging from 1 S/m to 10^{3}S/m. I extended the idea to anisotropic sam-
ples, with slight modifications proposed to the experimental setup in case
of a significant anisotropy of the material inspected.

• Lastly, I designed and, in cooperation with fellow PhD student Tamás Pet˝o, performed experiments to demonstrate the viability of the proce- dure.

**Related publications: [C5], [J3]**

**2.3 Array synthesis using an iterative algorithm**

Antenna array pattern synthesis is one of the challenges arising in the develop- ment of 5th generation (5G) communications [15]. Here, the goal is to increase the efficiency of communication between devices, which requires the concen- tration of the radiated power into a certain direction. Usually, arrays with fixed structure are used, however, this does not make use of the possibilities of the proper placement of the array elements.

Another field of interest is Wireless Power Transfer (WPT). In some appli- cations, devices cannot be powered using conventional galvanic contacts or near field WPT (such as some sensors and actuators, or vehicles on the move);

a possible solution could be provided by a radiated WPT system. Naturally, an appropriate design of the energy transmitting and receiving structures is of paramount importance [16].

Since a general problem of antenna array synthesis has a large number of variables [17] and the running time of numerical optimization increases expo- nentially with the number of variables, many synthesis methods have been de- veloped which reduce the number of variables by imposing some kind of con- straint on the array structure, such as linear or planar arrays with uniformly ori- ented and spaced antennas. The method of antenna placement I propose in the thesis allows for the relaxation of the geometric constraints on the anten- nas’ placement, which allows for more degrees of freedom and potentially bet- ter performance, while keeping the computational complexity reasonably low.

In order to measure the similarity of radiation patterns, I use an inner prod- uct of radiation patterns

**f**_{1},f_{2}®

= Z

Ω h

**f**_{1}(ˆ**k)**i_{∗}

·**f**_{2}(ˆ**k)dΩ,** (4)

where**f**_{1}(ˆ**k) corresponds to the radiation pattern in a direction corresponding to**
wavevector**k, and the integration is performed over the unit sphere.**

In the norm generated by this inner product, I define the difference of two patterns as the functional

kf1−**f**_{2}k^{2}=

**f**_{1}−**f**_{2},**f**_{1}−**f**_{2}®

. (5)

When applying the method to a given goal pattern**f*** _{A}*and a basic antenna
element with pattern

**f**0, we are seeking on the set of allowed geometrical oper- ations (e.g., a combination of translations and rotations) and excitation (magni- tude and phase) those values which minimize the difference of

**f**

*and*

_{A}**f**

_{0}. After some algebraic manipulation, I show that the number of optimization variables can be reduced by the two describing the excitation (phase and magnitude). Af- terwards, the method is recursively applied to the results of the previous step, e.g., the difference of the goal pattern and the fields of the previously placed elements.

Next, I introduce an alternative algorithm (Algorithm 1), which makes it pos-
sible to specify the desired number of array elements (I). In its*I*+1th step, in-
stead of placing a new antenna, we reposition the 1st antenna and recalculate its
optimal excitation. This procedure is then continued on all the array elements
a specified number of times, or until the desired accuracy is reached.

**Algorithm 1:**Synthesis of pattern**f*** _{A}*for given number

*I*of elements which stops if

*G*generations of array are created, or, alternatively, if error

*δ*

*n*de- creases below∆.

(A2.1) Place new elements until*i*=*I*
(A2.2) *g*=1;**f***S*=P_{I}

*j*=1*c*_{j}**f***E*(x* _{j}*)

**while**

*g*<

*G*

**do**

(A2.3) *i*=0;*g*+ +
**while***i*<*I*(or*δ**n*≥∆)**do**

(A2.4) *i*+ +

(A2.5) **f*** _{S}*=

**f**

*−*

_{S}*c*

_{i}**f**

*(x*

_{E}*) (A2.6) £*

_{i}**x*** _{i}*¤

=_{x}

¯

¯

¯ D

**f***E*(x),f*A*−f*S*
E¯

¯

¯

(A2.7) *c** _{i}*=
D

**f***E*(x* _{i}*),f

*A*−

**f**

*S*E

(A2.8) **f*** _{S}*=

**f**

*+*

_{S}*c*

_{i}**f**

*(x*

_{E}*) (A2.9)*

_{i}*n*=

*I g*+i;

*δ*

*n*=°

°**f*** _{A}*−

**f**

*°*

_{S}°

I extend the algorithm to be able to synthesize multiple target patterns with a single geometrical arrangement; in this case, the sum of the errors’ squares is minimized. This can for example be used to implement steering or changing the beamwidth when necessary.

For illustration, the algorithm is tested on the simultaneous synthesis of 8 circularly polarized patterns using 32 half-wave dipole antennas. The target pat- terns are given by the relations

**f**_{A,}_{+}(*`*,*ϑ*,*ϕ*)=*N** _{A}*sin

^{3}

^{`}*ϑ*cos

^{10}

^{`}*φ*

2( ˆ* ϑ*+j ˆ

*), (6)*

**φ****f**

_{A,}_{−}(

*`*,

*ϑ*,

*ϕ*)=

*N*

*sin*

_{A}^{3}

^{`}*ϑ*cos

^{10}

^{`}*φ*

2( ˆ* ϑ*−j ˆ

*), (7) where ˆ*

**φ***and ˆ*

**ϑ***denote unit vectors in the polar and azimuthal directions, and*

**φ***N*

*is the normalization factor. The patterns are characterized by*

_{A}*`*=2, 3, 4, 5.

In Fig. 8, we outline the algorithm’s speed of convergence, while in In Fig. 9, we illustrate the goal and synthesized field patterns.

δn2(error)

100 200 300 400 500 600

0.05 0.10 0.50 1

ℓ=2 ℓ=3 ℓ=4 ℓ=5

n(iteration number)

*Figure 8: Errorδ*^{2}*n* *(with n*=*I g*+*i ) of synthesized patterns***f**_{S}*with respect to*
*goal patterns***f**_{A,+}*for`*=2, 3, 4, 5. After i=32, the antennas already placed are
*repositioned once per generation. Generations are marked by white and gray bars*
*in background of graph.*

0 π π 2 3π

π

4 3π

2 4

0

0 π π 3π 2

π

4 3π

2 4

0

0 π π 2 3π

π

0

dB] 0

π π 3π 2

π

0

*Figure 9: Synthesized radiation patterns***f***S* *plotted for the real part of theϑ-*
*component (dashed) and the imaginary part of theϕ-component (dotted) for*

*`*=2*(first column) and`*=5*(second column) in planes corresponding toϑ*=*π*/2
*(first row) andϕ*=0*(second row). The finely dotted thin line indicates the goal*
*pattern***f**_{A}*, which is the same forϑ- andϕ-components. The graph is normalized*
*with respect to f*_{0}*– the maximal value of the goal pattern for`*=5.

**Summary**

*Thesis 3: I implemented an iterative algorithm for the purpose of antenna array*
*pattern synthesis and proved its convergence, furthermore, I extended the scope of*
*the algorithm to allow the synthesis of arrays with a specified number of elements*
*by repositioning, and multiple-goal patterns.*

• Based on Dr. Kracek’s idea, I lay down the theoretical foundations of a novel iterative method of antenna array radiation pattern synthesis. The method can be applied to an arbitrary type of a basic antenna element, and is applicable to any goal radiation pattern. The algorithm works by iteratively placing antenna array elements through the rotation and trans- lation of a basic element.

• I prove that the synthesized radiation pattern converges to the best ap- proximation of the goal pattern in the sense of the norm defined on a Hilbert space.

• I extend the algorithm for the correction of already placed elements and for multiple goal patterns.

• Finally, in an example, I design an array capable of producing multiple cir- cularly polarized radiation patterns. The algorithm is able to handle this relatively complicated optimization problem with acceptable accuracy.

An improvement of the method’s numerical implementation could be a pos- sible continuation of the work presented here, as I believe that the study of effi- cient minimization algorithms [18] and interpolation schemes [19] can further improve the results and speed of the algorithm.

Steering can be implemented through making use of the extension to mul- tiple goal patterns on planar arrays (with different goal patterns corresponding to different radiation directions).

**Related publications: [C6], [J4]**

**Publications unrelated to the Theses: [C7],[C8], [J5]**

**Journal articles**

[J1] K. Marák, T. Pet˝o, S. Bilicz, S. Gyimóthy, and J. Pávó, “Electromagnetic
simulation of rotating propeller blades for radar detection purposes,”*IEEE*
*Transactions on Magnetics, vol. 54, no. 3, pp. 1–4, 2018.*

[J2] ——, “Bistatic RCS calculation for propellers at near-resonant frequencies,”

*International Journal of Applied Electromagnetics and Mechanics, vol. 59,*
no. 1, pp. 19–26, 2019.

[J3] K. Marák, S. Bilicz, and J. Pávó, “Experimental technique for high-
frequency conductivity measurement,”*COMPEL-The international jour-*
*nal for computation and mathematics in electrical and electronic engineer-*
*ing, vol. 38, no. 5, pp. 1711–1722, 2019.*

[J4] K. Marák, J. Kracek, and S. Bilicz, “Antenna array pattern synthesis using an
iterative method,”*IEEE Transactions on Magnetics, vol. 56, no. 2, pp. 1–4,*
2020.

[J5] K. Marák, “Characterization of the inverse problem in critical dimension
measurement of diffraction gratings,”*Periodica Polytechnica. Electrical En-*
*gineering and Computer Science, vol. 60, no. 3, p. 187, 2016.*

**Conference articles**

[C1] K. Marák, “Calculation of the radar cross section of small propellers by
means of an efficient integral equation method,” in*Proceedings of the*
*Mesterpróba conference, Budapest, 2017, pp. 20–23.*

[C2] K. Marák, T. Pet˝o, S. Bilicz, S. Gyimóthy, and J. Pávó, “Electromagnetic sim-
ulation of rotating propeller blades for radar detection purposes,” in*Pro-*
*ceedings of Compumag 2017 Daejeon.* International Compumag Society,
2017, pp. 1–2.

[C3] K. Marák, S. Bilicz, S. Gyimóthy, J. Pávó, and T. Pet˝o, “Bistatic RCS calcula-
tion for propellers at near-resonant frequencies,” in*Proceedings of the In-*
*ternational Symposium on Electromagnetics and Mechanics (ISEM) 2017,*
*Chamonix, 2017, pp. 1–4.*

[C4] T. Pet˝o, K. Marák, S. Bilicz, and J. Pávó, “Experimental and numerical stud-
ies on scattering from multiple propellers of small UAVs,” in*Proceedings*
*of the 2018 European Conference on Antennas and Propagation (EuCAP).*

IEEE, 2018, pp. 1–4.

[C5] K. Marák, S. Bilicz, and J. Pávó, “Experimental technique for high fre-
quency conductivity measurement,” in*IGTE Symposium, Graz, 2018, pp.*

1–1.

[C6] K. Marák, S. Bilicz, and J. Kracek, “Antenna array pattern synthesis using
a novel iterative method,” in*Proceedings of Compumag, Paris.* Interna-
tional Compumag Society, 2019, pp. 1–2.

[C7] K. Marák, “Investigation of microchannel based heat sinks for vlsi circuits,”

in*Proceedings of the Mesterpróba conference, Budapest, 2016, pp. 1–2.*

[C8] S. Bilicz, S. Gyimóthy, J. Pávó, P. Horváth, and K. Marák, “Uncertainty
quantification of wireless power transfer systems,” in *Wireless Power*
*Transfer Conference (WPTC), 2016 IEEE. IEEE, 2016, pp. 1–3.*

**Further references**

[1] T. Pet˝o, L. Sz ˝ucs, S. Bilicz, S. Gyimóthy, and J. Pávó, “The radar cross section
of small propellers on unmanned aerial vehicles,” in*2016 10th European*
*Conference on Antennas and Propagation (EuCAP). IEEE, 2016, pp. 1–4.*

[2] J. Van Bladel,*Relativity and engineering.* Springer Science & Business Me-
dia, 2012, vol. 15.

[3] A. Mehdipour, T. A. Denidni, A. R. Sebak, C. W. Trueman, I. D. Rosca, and S. V.

Hoa, “Anisotropic carbon fiber nanocomposites for mechanically reconfig-
urable antenna applications,” in*2013 IEEE Antennas and Propagation Soci-*
*ety International Symposium (APSURSI). IEEE, 2013, pp. 384–385.*

[4] S. J. Orfanidis, “Electromagnetic waves and antennas, 2008,”*Online Book*
*Available at: http://www. ece. rutgers. edu/ orfanidi/ewa, 2008.*

[5] W. C. Gibson,*The method of moments in electromagnetics. CRC press, 2008.*

[6] J.-M. Jin,*Theory and computation of electromagnetic fields.* John Wiley &

Sons, 2011.

[7] V. C. Chen, F. Li, S.-S. Ho, and H. Wechsler, “Micro-Doppler effect in
radar: phenomenon, model, and simulation study,”*IEEE Transactions on*
*Aerospace and Electronic Systems, vol. 42, no. 1, pp. 2–21, 2006.*

[8] S. Y. Yang, S. M. Yeh, S. S. Bor, S. R. Huang, and C. C. Hwang, “Electromag-
netic backscattering from aircraft propeller blades,”*IEEE Transactions on*
*Magnetics, vol. 33, no. 2, pp. 1432–1435, Mar 1997.*

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