Equivalent Circuit Model for the Frequency- Selective ... - IEEE Xplore

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 65, NO. 2, FEBRUARY 2017

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Equivalent Circuit Model for the FrequencySelective Surface Embedded in a Layer With Constant Conductivity Mohsen Zahir Joozdani and Mohammad Khalaj Amirhosseini

Abstract— In this paper, an equivalent circuit model for a frequency-selective surface (FSS) embedded in a layer with constant conductivity is proposed and investigated. Because the medium surrounding the FSS is lossy, the equivalent model consists of not only inductance and capacitance elements but also resistive elements. The reflection response of a square patch FSS embedded in the conductive slab is simulated in a wideband of frequency. Then, all circuit elements are extracted by genetic algorithm optimization, when the reflection coefficient achieved from the equivalent circuit model is the same as the one obtained by numerical simulation. It is also found that the behavior of the model is consistent with that of the simulated one with a good accuracy. Values of the model elements are plotted by the variation of the conductivity and permittivity of layer, and the effects of these parameters are studied on the values of circuit elements. Moreover, it is shown that the obtained model can be applied to determine the model of FSS used in practical absorbers. Index Terms— Conductivity, equivalent frequency-selective surfaces (FSSs).

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I. I NTRODUCTION

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OWADAYS, absorbers are used in a variety of military and commercial applications such as radar cross-section (RCS) reduction in stealth design and shielding microwave and electronic devices from malfunctioning interference. One of the well-known structures is the Salisbury absorber consisting of a lossy layer placed a quarter of wavelength above a perfect electric conductor plane [1]. The Salisbury absorber has a narrow bandwidth, and therefore, the multilayered structure known as Jaumann absorber is introduced [2]. The Jaumann absorbers have a much wider absorption bandwidth, but they are more bulky as well. It is shown in [3] that there is a tradeoff between the bandwidth and thickness of such structures. However, studies are continued to manufacture wideband and thin absorbers. Frequency-selective surfaces (FSSs) and their applications have been the subject of intensive research for the past two decades. They can be used in antenna for gain and

Manuscript received February 11, 2016; revised November 13, 2016; accepted November 25, 2016. Date of publication December 1, 2016; date of current version February 1, 2017. The authors are with the Electrical Engineering Department, Iran University of Science and Technology, Tehran 1684613114, Iran (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2016.2633947

bandwidth enhancement, and many combinations of lossy FSS and dielectric layers are reported to make wideband and low RCS absorbers [4]–[6]. It also seems that embedding FSS in the lossy layer of an absorber can provide a proper solution to fabricate absorbers with improved properties. Magnetic materials such as ferrite composites are very common in the design of wideband absorbers [7]. Recently, some structures consisting of FSS embedded in magnetic substrate have also been introduced, which are wideband, thin, and lightweight [8]–[12]. In [13]–[17], ultrawideband absorbers are produced for normal and oblique incident waves by including an FSS in multilayered dielectric structures. By embedding a fractal FSS in a heterogeneous composite, a thin absorber in the X-band is proposed in [18]. Other nonmagnetic materials which can be chosen as the lossy layer and embedded with FSS are carbon composites such as carbon nanotubes, graphites, and carbon fibers [19], [20]. Measurement techniques show that a large group of carbon composites have constant conductivity from low to microwave frequencies [21]–[23]. Simulation and analytical methods are common in analyzing the absorber structures. In spite of full-wave methods which can be very time consuming, the transmission line model can provide an essential and quick solution, especially for 2-D problems [16], [17], [24]. To employ the transmission line model for designing an absorber structure made by inserting FSS in a conductive slab, the equivalent circuit model of FSS in such a situation is required. This model can be used in parallel with the transmission line model of layers around the FSS. However, the model of FSS embedded in a layer with constant conductivity has not been investigated yet. There are only a limited number of FSS shapes whose equivalent circuit models have been studied in [25]. The permittivity and permeability of the layer in which the FSS is embedded affect the value of the model elements [26], [27]. In [27] and [28], a formula for the equivalent circuit model of FSS embedded in dielectric slab is extracted according to the permittivity and thickness of the slab. Recently, a circuit model of FSS embedded in thick plasma layer is studied in [29]. Plasma is a lossy and dispersive medium in which the permittivity and conductivity are the specific and complex functions of frequency, plasma frequency, and collision frequency. It is demonstrated that the FSS embedded in plasma layer can be presented by a circuit model consisting of four elements, and the values of the elements are plotted by the variation of plasma and collision frequencies. However, the permittivity

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Fig. 1. (a) Incident plane wave impinging on patch FSS in the normal direction. (b) Equivalent circuit model for FSS in free space.

and conductivity of the layer are constant and independent of frequency in this paper. It is indicated that the circuit model of the FSS embedded in this medium can be introduced by three elements, and the values of these elements are plotted by the variation of permittivity and conductivity. Additionally, the permittivity of plasma is negative at low frequencies and increases to 1 at high frequencies. Thus, the behaviors of FSS circuit model elements are strongly affected by the permittivity. Nevertheless, in this paper, the effect of conductivity is more obvious on the elements’ values and is studied better from the physical point of view. In Section II, a square patch FSS is assumed and its model is obtained in free space. The model of FSS embedded in a conductive layer is proposed in Section III. By changing the conductivity parameter, circuit model elements of FSS are plotted and their behaviors are discussed in Section IV. In Section V, it is shown how this model can be employed in practical absorbers. Conclusions are drawn in Section VI. II. FSS M ODEL IN F REE S PACE In Fig. 1(a), a square patch FSS in free space is shown, illuminated by an incident wave in the normal direction. A layer with constant conductivity is a lossy medium, and for an FSS embedded in a conductive layer, the effect of the loss of the layer can be presented by a conductance element in the FSS model. For the sake of brevity and investigating the effect of conductivity on the circuit elements without involving FSS loss, the FSS is considered lossless. By this assumption, the model of the FSS in free space is purely imaginary and includes a series of inductance and capacitance elements as shown in Fig. 1(b). The unit cell simulation of full-wave Computer Simulation Technology (CST) Studio software is incorporated to obtain the reflection coefficient magnitude of the FSS in a particular frequency band. Then, by applying genetic algorithm (GA) optimization, different values are assigned to the capacitance and inductance to obtain the reflection coefficient similar to the one obtained by simulation in the same frequency band. Where there is minimum difference between the reflections obtained from the simulation and circuit model, the best values

Fig. 2. Reflection response of square patch FSS obtained by full wave simulation and circuit model. Parameters of the unit cell are D = 10 mm and W = 2.5 mm.

of the elements are extracted. The accuracy of this model is from zero frequency to the frequency at which higher order Floquet modes occur which are inherently the evanescent fields surrounding the FSS in this frequency band. The extent of these evanescent fields is increased by increasing the frequency and periodicity length of the FSS. Evanescent modes can turn to propagating modes out of this frequency band. Therefore, more than one propagating mode can exist and the circuit model is not valid. It is found that higher order Floquet modes appear approximately from the frequency at which the corresponding wavelength is equal to the cell periodicity length of the FSS [25]. In the inset of Figs. 1(a) and 2, the dimension parameters of the FSS unit cell are shown, in which D is the center-to-center distance of the adjacent patches. In addition, the reflection coefficients of field magnitude for normal incidence obtained by circuit model and simulation are plotted in Fig. 2. The results are compared from f = 1 GHz to f = 30 GHz because the periodicity length is selected as D = 10 mm. The values extracted by GA for inductance and capacitance are L 0 = 0.5 nH and C0 = 47 fF. The equivalent circuit model of FSS in free space can be used for normal and oblique incidences of plane wave. It is clear that circuit elements’ values are different for every incident angel. Moreover, there is a determined relation between circuit elements’ values at normal and oblique incidences [27]. For the sake of brevity, only the normal incidence is considered in this paper. When the FSS is embedded in a lossless dielectric or magnetic slab, the circuit model is the same as Fig. 1(b), while the values of circuit model elements can be different. For the dielectric slab, the inductance value remains unchanged, while the value of capacitance is proportional to the permittivity of dielectric and the thickness of the slab. When the thickness value is larger than the periodicity length of the FSS, the role of thickness in capacitance value is negligible. However, the effect of slab thickness should be accounted for thinner slabs. In [27] and [28], a correct factor is defined as the ratio of capacitance value of FSS embedded in dielectric slab to the one in free space. To find the actual value of capacitance, the capacitance value of FSS in free space has to be multiplied

JOOZDANI AND AMIRHOSSEINI: EQUIVALENT CIRCUIT MODEL

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Fig. 4. Comparison of reflection responses of circuit model of Fig. 3(b) and simulation. Parameters are σ = 0.2S/m, εr = 4, and d = 30 mm. Fig. 3. (a) FSS embedded in a conductive slab illuminated by a normal incident plane wave. (b) Equivalent model for the FSS embedded in the conductive slab.

by the correct factor. In addition, a formula is extracted for the correct factor according to the permittivity and thickness of the slab, and it is shown that this factor is dependent on the shape of the FSS. The correct factor for the FSS embedded in magnetic slab is also obtained in [26] to calculate the inductance value. However, for the FSS embedded in lossy layer, the configuration of the model may be changed and resistive elements should be added. III. FSS M ODEL E MBEDDED IN C ONDUCTIVE S LAB In Fig. 3(a), the FSS described in the previous section embedded in a conductive layer is shown, illuminated by a normal incident electromagnetic plane wave. The model of this FSS illustrated in Fig. 3(b) consists of three elements, and has one element more than the model of FSS in free space. The conductance element in a branch parallel with a series of inductance and capacitance represents energy dissipation in vicinity of surface and between the patches of FSS. Values of capacitance and inductance elements are changed by different values of slab conductivity. To find the values of the unknown elements, the GA method is incorporated on the reflection response of the structure. Because curve fitting is performed on numerous frequency points in a wide range, other responses such as transmission and absorption are not needed to be investigated. Nevertheless, they can help the optimization to reach higher accuracies. The slab in which the FSS is embedded has a constant permittivity and conductivity. The thickness of the slab (d) is assumed to be larger than the periodicity length of FSS (D). Therefore, the slab thickness has no significant effect on circuit elements’ values and only the role of conductivity parameter is studied. The incident plane wave impinging on the structure is assumed to have exp( j ωt) behavior in the time domain. Obviously, the plane wave propagating through the conductive slab obeys the following Maxwell’s equations: ∇ × E = − j ωμ0 H

(1)

∇ × H = (σ + j ωε0εr )E

(2)

in which εr and σ are relative permittivity and conductivity, respectively, assumed to be constant and nondispersive. Using ∇ · E = 0, the wave equation can be written as ω2 E. (3) c2 It is assumed that the FSS is extended in the xy plane. Thus, the propagating wave in the z-direction has the following form which is a solution to (3): ∇ 2 E = −(σ + j ωε0 εr )

E = E 0 exp( j ωt − γ z)

(4)

where γ is the complex propagating constant. When transmission line models of subslabs and FSS are available, the total transmission matrix can be obtained as A B T = = Tsubslab × TFSS × Tsubslab. (5) C D The definitions of transmission matrix of the subslab and FSS are as follows: cosh (γ d/2) Z c sinh (γ d/2) (6) Tsubslab = Z c−1 sinh (γ d/2) cosh (γ d/2) in which Z c = η0 /((εr − j (σ/ωε0 )))1/2 is the characteristic impedance of the conductive slab 1 0 TFSS = (7) Yp 1 where Y p is the total admittance of circuit model. The reflection of the structure can be achieved from the transmission matrix as (A − Z 0 C)Z 0 + (B − Z 0 D) (8) = (A + Z 0 C)Z 0 + (B + Z 0 D) in which Z 0 is the characteristic impedance of free space. For example, in Figs. 4 and 5, the reflection coefficients of the field magnitude of the structure given by full-wave simulation and circuit model are compared for two cases. In Figs. 4 and 5, conductive layers with the conductivity of σ = 0.2 S/m and σ = 1 S/m are considered, respectively. In both cases, the relative permittivity is εr = 4 and the thickness of the slab is d = 30 mm. It can be seen that the end of the simulation frequency band is at 15 GHz because the frequency

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Fig. 5. Comparison of reflection responses of circuit model of Fig. 3 (b) and simulation. Parameters are σ = 1S/m, εr = 4, and d = 30 mm.

Fig. 6. Capacitance value by variation of relative permittivity (εr ) for different conductivity (σ ) values.

Fig. 7. Capacitance value by variation of conductivity (σ ) for different relative permittivity (εr ) values.

Fig. 8. Inductance value by variation of relative permittivity (εr ) for different conductivity (σ ) values.

at √ which Floquet modes occur is directly proportional to 1/ εr . In addition, the simulation and circuit model results are in good agreement. To extract the values of circuit model elements, MATLAB optimization toolbox is used to minimize the error function defined as the mean of absolute difference between the reflection responses obtained by simulation and circuit model at more than 100 points in the frequency band. The values of equivalent circuit model for σ = 0.2 S/m and σ = 1 S/m among the other values of conductivity are demonstrated in Figs. 6–11. IV. R ESULTS AND D ISCUSSION To verify the proposed circuit model, two practical ranges of 0 ≤ σ ≤ 1 S/m and 2 ≤ εr ≤ 10 are considered for the conductivity and relative permittivity of the slab, respectively. Based on the result of the experimental determination of carbon composite characteristics, constant relative permittivity and conductivity can be a good estimation for some groups of carbon family, especially when εr ≤ 10 [19]–[23]. The conductivity and relative permittivity ranges are divided into 20 and 9 points, respectively, and the values of circuit elements are plotted via the variation of permittivity and conductivity parameters in Figs. 6–11. For the points shown by circles

Fig. 9. Inductance value by variation of conductivity (σ ) for different relative permittivity (εr ) values.

in Figs. 6–11, the mentioned FSS embedded in the conductive layer with corresponding σ and εr of that point is imposed by normal incident plane wave. Then, this structure is simulated in the frequency √ band from 1 GHz to the frequency equal to f = c/(D εr ). Then, the circuit elements’ values are extracted, and the curves are plotted by curve fitting.


Fig. 10. Conductance value by variation of relative permittivity (εr ) for different conductivity (σ ) values.

Fig. 11. Conductance value by variation of conductivity (σ ) for different relative permittivity (εr ) values.

Mean error function for all of the considered points is below 0.05, showing that the circuit model’s response is consistent with the simulated one. In what follows, behaviors of the capacitance, inductance, and conductance elements are studied and investigated by changing the conductivity and relative permittivity parameters. In Fig. 6, the capacitance of the circuit model is plotted versus the permittivity of the slab for different values of conductivity. It is obvious that the capacitance value is proportional directly to the permittivity of the medium around the FSS. Moreover, it can be seen that the red line for σ = 0.2 S/m is upon the blue line for σ = 0. However, the cyan and purple lines for σ = 0.6 S/m and σ = 1 S/m, respectively, are below the blue line. For a better view, the capacitance value is depicted in Fig. 7 by the variation of conductivity. It is demonstrated that the capacitance value is constant or increases by a little slop until σ = 0.2 S/m and then decreases by increasing the conductivity. Behavior of the inductance is the opposite of the capacitance behavior. The inductance value is plotted according to the variation of permittivity in Fig. 8. It is observed that the value of inductance is approximately constant for σ = 0S/m. In addition, amplitude of the inductance variation is much larger in a low permittivity than in a high permittivity.

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The inductance behavior versus conductivity is depicted in Fig. 9. It is shown that by increasing the conductivity from zero to σ = 0.2 S/m, the inductance value decreases and then increases. Figs. 10 and 11 show the conductance value of the equivalent circuit model according to the relative permittivity and conductivity of the slab, respectively. By increasing the conductivity of the layer, the conductance value increases with a linear relationship. It seems that the relative permittivity does not have an essential effect on the conductance value. The total admittance of the FSS embedded in the conductive slab can be written as 1 + G. (9) Yp = 1 j ωL + j ωC When the slab is lossless, circuit elements will be L = L 0 , C = εr C0 , and G = 0. The capacitance of the FSS model is made by the charges induced on the edges of adjacent patches by the incident plane wave. Moreover, the inductance of the model is produced by the current induced on the patches. By embedding the FSS in the lossless medium, there is no other path for the induced current. However, in the conductive slab, the medium surrounding the FSS can make a new path of current parallel with the patches. This current path can be represented by the conductance in the circuit model of FSS. A part of the current which flows on the FSS can go through the conductance branch and causes an increase and decrease in the inductance and capacitance values, respectively. It can be observed in Figs.6–9 that for a high amount of conductivity, the inductance value increases and the capacitance value decreases. However, for low values of conductivity, the inductance and capacitance values decrease and increase, respectively. By increasing the conductivity from low to high values, the inductance and capacitance values reach minimum and maximum points, respectively. Increasing the conductivity increases not only the parallel current, but also the current between the patches which, in turn, increases the induced charges on the edges of the patches and the capacitance value. The two mentioned currents have opposite effects on the circuit elements, and for conductivity values of more than σ = 0.2 S/m, the effect of the parallel current is dominant. After σ = 0.2 S/m, the inductance and capacitance values increase and decrease, respectively. The structure parameters determine at which conductivity value the extremum points occur. To investigate the effect of thickness on the extremum point location, the inductance and conductance values are plotted for the variation of slab thickness in Figs. 12 and 13, respectively. It is obvious in Fig. 12 that by reducing the slab thickness, the extremum location is shifted toward higher values of conductivity. The conductance value is directly proportional to the crosssectional area of parallel current flow which depends on the periodicity length and thickness of the slab. Therefore, reducing the thickness decreases the conductance value, as shown in Fig. 13. It also increases the current between the patches because it makes this current more confined on the FSS. Thus, for the thinner layer, the extremum point occurs at a higher conductivity. It is clear in Fig. 13 that at large thicknesses, the conductance values converge.

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Fig. 12. Inductance value by variation of conductivity for different thicknesses of the slab.

Fig. 13. Conductance value by variation of conductivity for different thicknesses of the slab.

The changes of elements’ values are investigated according to the variation of gap size at constant periodicity length of D = 10 mm and variation of periodicity length at constant gap size of W = 2.5 mm. It can be found that in spite of the change in FSS size, the extremum point remains approximately at σ = 0.2 S/m. Also, it can be observed that the conductance value is changed by varying the FSS size and is increased by decreasing the inductance value and increasing the capacitance value. It has been mentioned above that increasing the conductance value increases the parallel current. On the other hand, increasing the capacitance value increases the current between the patches. Moreover, it can be seen that the ratio of capacitance value to conductance values is approximately constant. Therefore, both currents increase with the same ratio, and as a result, the size of the FSS has a negligible effect on the location of the extremum point on the conductivity axes. V. P RACTICAL A BSORBER D ESIGN U SING THE M ODEL Planar absorbers widely used in shielding applications and anechoic chambers can be made of foam appended by carbon or ferrite composites to lighten the structure. Apparently, the implementation of lossless FSS embedded within such a structure is impractical. On the other hand, etching metallic

FSS on dielectric substrates with a very low tangent loss is inexpensive and easy to fabricate. Thus, a practical prototype can be fabricated by embedding FSS between a lossy layer and a dielectric substrate. Obtaining the equivalent circuit model of the FSS in this case helps to analyze the structure by the transmission line model. For instance, assume the FSS mentioned in Section II is embedded between a lossy layer with constant conductivity and a dielectric substrate. By assuming that the conductive and dielectric layers are thick, it can be shown that the FSS circuit model is the same as the one proposed in Section III. Values of the FSS elements can be approximately obtained by averaging the values of elements when the FSS is embedded within the dielectric and conductive layers individually. Suppose that the relative permittivity and conductivity of the conductive layer are 2 and 0.2 S/m, respectively, and this layer has a thickness of d1 = 15 mm. These parameters are taken from the foam of a commercial absorber appended by a mixture of glow and a type of graphite, and are obtained by waveguide measurement. The dielectric substrate is FR4 (εr = 4.4) with the thickness of d2 = 1.6 mm. For the FSS embedded in the 3.2-mm-thick dielectric layer, the circuit model parameters are as follows: C1 = εr C0 = 206.8 fF, L 1 = 0.5 uH, and G 1 = 0 S in which C0 is the capacitance value of FSS in free space. For the FSS embedded in the 30-mm-thick conductive layer, the elements’ values are extracted from Figs. 6–11 as C2 = 96 fF, L 2 = 0.372 uH, and G 2 = 1.313 × 10−3 S. By averaging these two cases, the circuit model parameters of the FSS embedded between the conductive and dielectric layers are as follows: C = (C1 + C2 )/2 = 151.4fF, L = (L 1 + L 2 )/2 = 0.436 nH, and G = (G 1 + G 2 )/2 = 0.66 × 10−3 S. In Fig. 14, the reflection responses obtained by the averaging method and simulation are compared. In addition, the following precise values are obtained by the GA method: C = 140 fF, L = 0.441 nH, and G = 0.737×10−3 S. The goal of this example is not designing an optimized absorber, and other layers or ground plane may still be required to reach a desirable level of absorption. However, it shows that the values achieved by the approximation of averaging method are in good agreement with those obtained from the GA method. To improve the absorption properties of the structure, a thicker conductive layer is needed. For example, in Fig. 15, the reflection coefficient of field magnitude is plotted when the thickness of conductive layer is chosen to be twice the previous design as d1 = 30 mm and the other parameters are the same as before. In this case, the values of circuit model elements are required when the FSS is embedded in the conductive layer with the thickness of d = 60 mm. It can be seen in Figs. 12 and 13 that circuit element values converge for the thickness and conductivity values of d ≥ 20 mm and σ ≤ 0.2 S/m, respectively. Therefore, the values of circuit model elements can be assumed the same as the previous design. Results obtained by full-wave simulation and circuit model analysis for the second design are compared in Fig. 15. It shows that the reflection coefficient is below −9 dB in the wide frequency range from 2 to 15 GHz. The averaging method is an approximation method because the individual circuit models in each medium do not take


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These currents flow parallel with the FSS and between the patches in the conductive layer. Finally, a practical example of using this theoretical model is presented in designing an absorber with a combination of FSS, dielectric, and conductive layers. R EFERENCES

Fig. 14. Comparison of reflection coefficient magnitude of FSS embedded between conductive and dielectric layers obtained by simulation and averaging method. Parameters are D = 10 mm, W = 2.5 mm, d1 = 15 mm, and d2 = 1.6 mm.

Fig. 15. Comparison of reflection coefficient magnitude of the second design. Parameters are D = 10 mm, W = 2.5 mm, d1 = 30 mm, and d2 = 1.6 mm.

into account the boundary conditions that must be satisfied by the fields along the discontinuity between the dielectric and conductive mediums. This fact results in a little difference between the approximation values obtained by the averaging method and the exact values achieved from the GA method. Nevertheless, the results in Figs. 14 and 15 show that these differences are negligible. VI. C ONCLUSION In this paper, an equivalent circuit model is introduced for an FSS embedded in a slab with constant permittivity and conductivity. This model is proposed for thick slabs and can predict the behavior of the FSS in a wide frequency band from low frequencies to the frequencies at which grating lobes and higher order modes appear. In comparison with the model of FSS in free space, this model has an additional parallel conductance. Values of the model elements are plotted by variation of relative permittivity and conductivity in the ranges of 2 ≤ εr ≤ 10 and 0 ≤ σ ≤ 1S/m, respectively. It is shown that the elements’ behavior can be described by two currents produced because of the conductivity of the layer.

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Mohsen Zahir Joozdani was born in Isfahan, Iran, in 1986. He received the B.Sc. degree in electrical engineering from the Isfahan University of Technology, Isfahan, in 2008, and the M.Sc. degree in communication engineering from the Iran University of Science and Technology, Tehran, Iran, in 2011. He is currently pursuing the Ph.D. degree at the same university. His current research interests include RCS reduction, antenna design with low RCS, and modeling of frequency selective surfaces.

Mohammad Khalaj Amirhosseini was born in Tehran, Iran, in 1969. He received the B.Sc., M.Sc., and Ph.D. degrees from the Iran University of Science and Technology (IUST) in 1992, 1994, and 1998, respectively, all in electrical engineering. He is currently a Professor with the College of Electrical Engineering, IUST. His current research interests include electromagnetic direct and inverse problems including microwaves, and antennas and propagation.