Matching conditions for a homogeneous absorbing layer

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Available online at www.sciencedirect.com Procedia Engineering 00 (2017) 000–000

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Procedia Engineering 216 (2017) 79–84

9th International International Conference Conference on on Materials Materials for for Advanced Advanced Technologies Technologies(ICMAT (ICMAT2017) 2017) 9th

Matching conditions for a homogeneous absorbing layer Konstantin N. Rozanova,* and Marina Y. Koledintsevab a aInstitute

Institute for theoretical and applied electromagnetics, Moscow, Russian Federation Electromagnetic Compatibility Design Engineering, Oracle, Santa Clara, CA 95054, USA

b bElectromagnetic

Abstract The paper discusses available analytical approaches for analyzing radar absorbing performance of composite materials. Most published reports on the microwave properties of magnetic composites refer to radar absorbers as possible applications. The conventional approach is based on measurements of the effective permittivity and permeability of a composite with certain concentrations of inclusions; calculations of reflection coefficient of the composite layer backed by a metal substrate for the normal incidence of electromagnetic wave; and selection of the layer thickness that produces deep minimum of the frequency dependence of the reflectivity. The obtained reflectivity minimum, typically of −30, −40 dB, or even −60 dB deep is conventionally considered as a proof for “a good absorbing ability” of the inclusions comprising the composite. In the paper, simple analytical approximations based on the series expansions of the reflection coefficient are suggested for the reflection coefficient of radar absorbers. It is shown that the problem of radar absorbing performance is not reduced to obtaining good absorbing ability, but is a problem of matching of parameters of the absorbing material. Therefore, the depth of reflection minimum is not a suitable measure for the characterization of radar absorbers. To describe the quality of an absorber, other quality criteria are needed. © 2017 The Authors. Published by Elsevier Ltd. © 2017 The Authors. Published by Elsevier Ltd. Selection and/or peer-review under responsibility of the scientific committee of Symposium 2017 ICMAT. Selection and/or peer-review under responsibility of the scientific committee of Symposium 2017 ICMAT. Keywords: radar absorbers; reflection coefficient; absorbing bandwidth

1. Introduction Currently, microwave properties of composites are under intensive studies. Most published reports in this field refer to radar absorbers as possible applications, see [1] for a recent review. Conventionally, a homogeneous layer deposited on an infinite plane conducting surface is under consideration, with the normal incidence of a monochromatic plane wave. The complex reflection coefficient R is then written as

* *

corresponding author, e-mail [email protected]

1877-7058 © 2017 The Authors. Published by Elsevier Ltd. Selection and/or peer-review under responsibility of the scientific committee of Symposium 2017 ICMAT.

1877-7058 © 2017 The Authors. Published by Elsevier Ltd. Selection and/or peer-review under responsibility of the scientific committee of Symposium 2017 ICMAT. 10.1016/j.proeng.2017.09.829

Konstantin N. Rozanov et al. / Procedia Engineering (2017) 79–84 Rozanov and Koledintseva / Procedia Engineering 00 (2017)216 000–000

280

R

   tanh 2i

 ,  d    1

  tanh 2i  d   1

(1)

where R is the amplitude ratio of the reflected and incident waves, ='i'' is the permittivity,  ='i'' is the permeability, and d is thickness of the layer, and � is the wavelength. For a good absorbing performance, the reflection coefficient module must be below a prescribed level, R0, in a desired frequency range, from fmin to fmax. The reflection coefficient module is frequently referred to as reflectivity; reflection loss (RL) may be also used that is the squared module of R expressed in dB and having an opposite sign. The conventional approach to the analysis of the radar absorbing performance is based on the measured microwave material parameters of a composite. From this, frequency dependences of reflection loss are calculated for the normally incident electromagnetic wave, a certain concentration of inclusions in the composite, and various layer thicknesses. Then, the layer thickness is chosen to minimize the reflection coefficient at a frequency of interest. The obtained deep minimum, typically of −30, −40 dB, or even −60 dB is considered as a proof for “a good absorbing ability” of the inclusions comprising the composite. Some recent examples of the approach are given in [2–6]. The paper discusses available analytical approaches for analyzing radar absorbing performance of materials. It is shown that the observed low reflectivity values are a result of good matching of parameters of the composite rather than of “good absorbing ability”. 2. Representations for the reflection coefficient module The standard problem of radar absorbing performance involves the reflection coefficient module. The reflectivity is to be deduced from Eq. (1) that is written in terms of complex reflection coefficient; since transcendent function is involved in Eq. (1), this is not an easy task. Basically, two approaches to calculate the reflectivity from the complex reflection coefficient produced by Eq. (1) are known. The first one, firstly discussed by Pottel [7], employs a series expansion of the tangent in Eq. (1) near the operating frequency. The resulting equations involve simple analytical functions, but have the validity limits restricted by a narrow frequency range. However, this does not matter in many cases, as the absorbing performance of magnetic composites is typically narrowband. The second approach is based on the expansion of the tangent into the real and imaginary parts that allows the reflectivity to be calculated directly [8, 9]. From the first sight, the results of the approach are not restricted by any frequency range. However, the absorbing performance requires energy loss, which, in turn, is inevitably related to the frequency dispersion of material parameters. The frequency dispersion of permittivity and permeability is hard to be quantified in a wide range of microwave frequencies. Therefore, the approach does not allow for obtaining the analytical results anyway, and its application is typically reduced to producing nomograms analogous to the Cole–Cole diagrams for the permeability to find conditions for appearing two matching frequencies in ferrites [10]. Below, the first approach is employed. Two distinct cases of the series expansion of the tangent function are to be considered, the first is when the argument is close to 0, and the other when the argument is close to π/2. The first case corresponds to the magnetic Salisbury screen, the second case to the quarter-wavelength (Dallenbach) absorber. For the magnetic screen, a pretty simple equation obtained by the series expansion of the tangent in Eq. (1) at d→0, R

2id   1 , 2id   1

(2)

is in common use. However, for radar absorbers, Eq. (2) must be used with care, notwithstanding the optical thickness of the layer is small. To obtain good absorbing performance, the value of 2π��/� must be close to unity, which results in a slow convergence of the series expansion. However, the magnetic screen is hard to obtain at microwaves. The permeability of materials is known to decrease rapidly with frequency over the microwave band [11], and this is the reason why efficient absorbers with the reflection coefficient governed by Eq. (2) are rare. Examples of these are sintered ferrites operating at megahertz frequencies [12], and also absorbers based on artificial microwave magnets [13]. Most microwave radar absorbers comprising composite



Konstantin N. Rozanov et al. / Procedia Engineering 216 (2017) 79–84 Author name / Procedia Engineering 00 (2017) 000–000

81 3

materials fall into the class of quarter-wavelength absorbers. 3. Representations for quarter-wavelength absorbers For the quarter-wavelength absorber, the series expansion is made near the argument of tangent close to π/2, with only the first term being kept. In this case, the tangent is represented as

tan x  

1 .  2 x

(3)

With Eq. (3) substituted, Eq. (1) for the reflection coefficient transforms to R

   4d

     2i

  4d      2i   







.

(4)

The first term in the numerator and the denominator of Eq. (4) is the resonance term, the real part of which defines the frequency dependence of the reflection coefficient. If the real part of the second term is neglected in consideration of the resonance behavior, the well-known condition for the frequency of the quarter-wavelength resonance is obtained,





4d Re     .

(5)

Equation (5) defines the wavelength corresponding to the quarter-wavelength resonance. The quality factor of the resonance is defined by the imaginary parts of both the first and the second terms in the numerator and denominator of Eq. (4), with the second term being now not negligible. To simplify the equations, the resonance condition (5) can be substituted to Eq. (4). The resulting condition for the zero reflection at the resonance is written as 2d Re 

 Re 





Im    1 .

(6)

Loss in the material yielding zero reflection at the resonance can be determined from Eq. (6). Near the quarter-wavelength resonance, the entire frequency dependence of the reflection coefficient module is governed by the Lorentzian dependence,

 Re  4d Re        2 Re 2

2

R 



 

 1 

2d



Re

 

 

2

 Re  Im    

 . 2 2   2d   Re  4d Re       2 Re  1  Re  Re  Im        







 



 



(7)

A comparison between the exact Eq. (1) and approximations given by Eqs. (4) and (7) is shown in Fig. 1(a), for the parameters of the layer being ε = 48, � = 2 – i 0.52, d=1 mm. It is seen from the figure that the results of Eq. (1) and approximation (4) agree well, while there is some discrepancy between approximation (7) and Eq. (1) due to assumptions made in the derivation. However, with the account of high quality factor of the resonance and low reflection, the agreement is still reasonable, and the equation can be used in the analysis of the radar absorbing performance. The obtained Eq. (7) is not convenient for use because it contains square roots of the material parameters. To avoid this, more sophisticated series expansion can be applied, where one more null of the tangent and a symmetrical pole are kept:

Konstantin N. Rozanov et al. / Procedia Engineering 216 (2017) 79–84 Rozanov and Koledintseva / Procedia Engineering 00 (2017) 000–000

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tan x  



2x . 2  x  2  x 

(8)

(a)

(b)

Fig. 1. (a) The module of the reflection coefficient for a layer with ε=48, �=2–i 0.52, d=1 mm calculated with Eq. (1) (exact result, the solid line), Eq. (4) (the result of the series expansion, the dashed line), and Eq. (7) (the approximation by the Lorentzian frequency dependence, the dotted line). (b) The calculation made with the same parameters of the layer but with Eq. (1) (exact result, the solid line), Eq. (9) (the result of the series expansion, the dashed line) and Eq. (12) (the approximation by the Lorentzian frequency dependence, the dotted line)

Equation (8) accounts for the quarter-wavelength reflection minimum at x=π/2, the null of the tangent at x=0, and the symmetrical pole at x= − π/2. The inclusion of the additional null and pole allows for retaining odd/even properties of the reflection coefficient as a function of �. This is of importance for estimating ultimate bandwidth of radar absorbers. The corresponding equation for the reflection coefficient is written as R  

16d 2  16id  2 16d 2  16id  2

.

(9)

After some simplifications, it is readily obtained that the wavelength, where the magnitude of the reflection coefficient has a minimum, is given by Eq. (5), the same as derived before [14]. The reflectivity is zero in this minimum when

d       1 . r

(10)

Ιf the reflection coefficient is not zero in its minimum, the location of the minimum is still given by Eq. (5) and the depth of the minimum Rmin is Rmin 

d d

         1 .          1

The Lorentzian dependence for the reflection coefficient module obtained in the same way as Eq. (7) is written as

(11)



Konstantin N. Rozanov et al. / Procedia Engineering 216 (2017) 79–84 Author name / Procedia Engineering 00 (2017) 000–000

 d         1     2   . R  2 2 2                   d d 16    16d 2         2    1        

16d



2

        2 

2



 16d       

2

83 5

2

(12)

In contrast to Eq. (7), Eq. (12) involves higher powers of the wavelength, but does not involve any square roots, which simplifies it greatly. The depth of the reflection minimum is related directly to the material parameters of the layer. The accuracy of this representation is reduced as compared to the case of retaining only π/2 pole, as is shown in Fig. 1(b). The reason is the asymmetry of the retained singularities. However, this is paid back by the simplicity of the resulting equation. Note that the deviations of the approximating results from Eq. (1) are clearly seen in Fig. 1 because the numerical example of the frequency dependence of the reflection coefficient is selected so that the quality factor of the quarterwavelength resonance is high, and the reflectivity minimum exhibits nearly zero reflection. The sensitivity of the depth of the reflectivity minimum is high in this case, as is clearly seen from Eq. (12). For this reason, the references cited in Introduction contain the calculated frequency dependences of the reflectivity rather than measured dependences. In the experiment, these deep values of the reflectivity can hardly be obtained due to inaccuracy in the layer thickness determination, inhomogeneity of the composite, and other unavoidable technological tolerances in the material properties. 4. Discussion The results obtained suggest that low reflection coefficient is defined by just good matching of the material parameters to optimal values. The more accurately Eqs. (5) and either (6) or (10) are held, the lower value of the reflection coefficient is. Material loss may be either dielectric, or magnetic, or both. The contributions from dielectric and magnetic loss are just additive to each other. The absorbing performance can hardly be attributed to synergetic effect of several types of inclusions in composite, as well as to synergy between dielectric and magnetic loss, as is frequently suggested in literature, see, e.g., [15–17]. Also, it is clearly seen from Eq. (10) that in composites filled with ferromagnetic particles, magnetic loss dominates, because the real permittivity is usually much larger than the real permeability. Therefore, engineering of a radar absorber is a problem of matching rather than obtaining “good absorbing ability”. To produce good absorption at a given frequency, real and imaginary parts of complex material parameters must have certain values. The equations obtained are valid for narrow-band absorbers only. Note that the analysis of the wideband absorbers is not correct because of the frequency dispersion of material parameters. The equations introduced may be applied as a good first approximation, and they also used to analyze radar absorbing performance of materials. For example, it is clearly seen from Eqs. (5) and (11) that the depth of the reflection minimum can vary with the layer thickness if only material parameters of the layer exhibit frequency dispersion. Otherwise, the dependence on d in the right-hand part of (11) is completely equalized by the wavelength, where the /4 minimum appears, and which, according to (5), is also proportional to d. Moreover, for any given Rmin, except for Rmin=0, two different solutions can be found by setting d         to be either larger or smaller than unity. Note that the pole at x=0 is kept in expansion (8). For this reason, Eq. (9) would seem to describe the magnetic screen as well. However, accuracy of this description is low. This is because for x = π/2, the agreement with the initial Eq. (1) is achieved for the numerator of Eq. (8) having the numerical factor of “2”, which is a residue responsible for all the neglected nulls and poles. For x=0, the residue is different by the factor of π2/8 ≈ 1.23. This value is close to unity, but if neglected, rather large deviation is obtained between the depth of reflection minima in cases when these minima are deep enough. 4. Discussion Based on the series expansion of the reflection coefficient, approximate equations for the reflection from a quarterwavelength absorbing layer are derived. Two sets of series expansions are used: one of them produces more accurate

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Konstantin N. Rozanov et al. / Procedia Engineering 216 (2017) 79–84 Rozanov and Koledintseva / Procedia Engineering 00 (2017) 000–000

approximation, and the other yields more intuitive result. As is seen from the above consideration, the depth of reflection minimum is not a suitable measure for the characterization of radar absorbers. To describe the quality of an absorber, other quality criteria are needed, such as characterization of bandwidth and of angular performance [14]. Acknowledgements The authors acknowledge financial support of the work provided by Russian Foundation for Basic Research (RFBR) according to Agreement no. 15-08-03535. References [1] F.M. Idris, M. Hashim, Z. Abbas, I. Ismail, R. Nazlan, I.R. Ibrahim, Recent developments of smart electromagnetic absorbers based polymercomposites at gigahertz frequencies, J. Magn. Magn. Mater. 405 (2016) 197–208. [2] I. Choi, D.Y. Lee, D.G. Lee, Radar absorbing composite structures dispersed with nano-conductive particles, Composite Struct. 122 (2015) 23–30. [3] H. Gargama, A.K. Thakur, S.K. Chaturvedi, Polyvinylidene fluoride/nanocrystalline iron composite materials for EMI shielding and absorption applications, J. Alloys Compounds 654 (2016) 209–215. [4] S.L. Wen, Y. Liu, X.C. Zhao, Facile chemical synthesis, electromagnetic response, and enhanced microwave absorption of cobalt powders with controllable morphologies, J. Chem. Phys. 143 (2015) 084707. [5] S.S. Maklakov, A.N. Lagarkov, S.A. Maklakov, Y.A. Adamovich, D.A. Petrov, K.N. Rozanov, I.A. Ryzhikov, A.Y. Zarubina, K.V. Pokholok, D.S. Filimonov, Corrosion-resistive magnetic powder Fe@SiO2 for microwave applications, J Alloys Compounds 706 (2017) 267–273. [6] G.L. Wu, YH. Cheng, Y.Y Ren, Y.Q. Wang, Z.D. Wang, H.J. Wu, Synthesis and characterization of g-Fe2O3@C nanorod-carbon sphere composite and its application as microwave absorbing material, J. Alloys Compounds 652 (2015) 346–350. [7] P. Pottel, Uber die erhohung der frequenzbandbreite danner “λ/4-schicht” absorber fur electromagnetische zentimmeterwellen, Zeitschrift fur Angewandte Physik 11 (1959) 46–51. [8] E.F. Knott, The thickness criterion for single-layer radar absorbents, IEEE Trans. Antennas Propagat. 27 (1979) 698–701. [9] A.Q. Valenzuela, F.A. Fernandez, General design theory for single-layer homogeneous absorber IEEE Trans. Antennas Propagat. 44 (1993) 822–826. [10] H.M. Musal and D.C. Smith, Universal design chart for specular absorbers, IEEE Trans. Magn. 26 (1990) 1462−1464. [11] K.N. Rozanov and M.Y. Koledintseva, Application of generalized Snoek's law over a finite frequency range: A case study, J. Appl. Phys. 119 (2016) 073901. [12] Y. Naito and K. Suetake, Application of ferrite to electromagnetic wave absorber and its characteristics, IEEE Trans. Microw. Theory Techn. 19 (1971) 65−72. [13] A.N. Lagarkov, V.N. Semenenko, V.N. Kisel, V.A. Chistyaev, Development and simulation of microwave artificial magnetic composites utilizing nonmagnetic inclusions, J. Magn. Magn. Mater. 258–259 (2003) 161–166. [14] C.A. Stergiou, M.Y. Koledintseva, K.N. Rozanov, Hybrid polymer composites for electromagnetic shielding in electronic industry. In: Hybrid polymer composite materials, Eds: V.K. Thakur, M.K. Thakur, and A. Pappu. Volume 4 – Applications. Chapter 3. Elsevier, 2017. pp. 53–106. [15] W.C. Li, X. Zhou, Y. Ying, X.J. Qiao, F.X. Qin, Q. Li, and S.L. Che, Polarization-insensitive wide-angle multiband metamaterial absorber with a doublelayer modified electric ring resonator array, AIP Adv. 5 (2015) 067151. [16] B. Zhao, G. Shao, DD. Fan, W.Y. Zhao, R. Zhang, Fabrication and enhanced microwave absorption properties of Al2O3 nanoflake-coated Ni core-shell composite microspheres, RSC Adv. 4 (2014) 57424−57429. [17] Z.J. Wang, L. Wu, J.G. Zhou, W. Cai, B.Z. Shen, and Z.H. Jiang, Magnetite Nanocrystals on Multiwalled Carbon Nanotubes as a Synergistic Microwave Absorber, J. Phys. Chem. 117 (2013) 5446−5452.