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Composites Science and Technology

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Prediction and validation of electromagnetic performance of curved radar-absorbing structures based on equivalent circuit model and ray trackingmethod

Dae-Sung Son, Jong-Min Hyun, Jung-Ryul Lee∗

Department of Aerospace Engineering, Korea Advanced Institute of Science and Technology, Daejeon, South Korea

A R T I C L E I N F O

Keywords:Radar absorbing structureFrequency selective surfaceEquivalent circuit modelRay tracking methodFree space measurement

A B S T R A C T

In this study, an equivalent circuit model and ray tracking method were used to predict the electromagneticcharacteristics of curved radar-absorbing structures (RASs) applied with a frequency selective surface (FSS).After performing an electromagnetic analysis of FSS using a unit cell model, an equivalent circuit model re-flecting the characteristics of the FSS was constructed. The equivalent circuit model of the FSS was applied to anequivalent circuit model for the RAS to evaluate the absorption performance as a function of the curvature andincident angle. Electromagnetic characteristics of the curvature structure were predicted by estimating the pathof the electromagnetic wave using the ray tracking method. The calculated results were compared with theexperimental results using free space measurements. Through this study, it was possible to estimate anequivalent circuit model reflecting the electromagnetic characteristics of the RAS with respect to the incidentangle and curvature of the FSS. In addition, the electromagnetic performance of the entire curved structure wasevaluated using the ray tracking method.

1. Introduction

A frequency selective surface (FSS) can transmit or reflect a specificfrequency band. Therefore, many studies are being conducted, espe-cially in the microwave engineering and optics fields, with a particularfocus on topics like radio wave absorption, RCS reduction, and stealthradome [1–6]. The FSS is determined by various parameters, such asthe shape and dimensions of the pattern, the permittivity of the sur-rounding material, its thickness, and the angle of the incident wave.The FSS has the same characteristics as a circuit with capacitance andinductance, and an equivalent circuit model can be used to analyze theelectromagnetic characteristics of the FSS [7].

Various methods have been proposed to obtain the required elec-tromagnetic performance and to analyze the electromagnetic char-acteristics. The electromagnetic performance of a multi-layer structureis predictable using simple equations [8]. The electromagnetic analysisof structures with specific patterns is performed by using electro-magnetic simulation software [9]. Optimal design has been performedto satisfy the required electromagnetic performance using these ana-lysis methods [10]. Filippo Costa [11] proposed a method for predictingthe electromagnetic performance of a multi-layered structure with FSSusing an equivalent circuit model. An FSS was proposed where the

resonance frequency of the FSS can be changed using a PIN [12] andvaractor diode [13]. Lee [14] proposed a frequency selective fabriccomposite fabricated by woven carbon and dielectric fibers. In theprevious studies, the electromagnetic characteristics of a flat plate wereanalyzed. However, since applied shapes have curvature, it is necessaryto analyze the electromagnetic characteristics in structures with cur-vature. B. Philips [15] used the ray tacking method to predict thetransmission loss in structures with curvature. Z. Sipus [16] analyzedthe structural curvature by approximating the curvature as a set ofoverlapping subarrays. Dazhi Ding [17] evaluated the curved structureby applying the multilevel fast multipole algorithm (MLFMA) to thevolume-surface integral equation.

To analyze the FSS, numerical analysis techniques like finite-dif-ference time-domain method and finite element method are primarilyused. These techniques have high degrees of freedom for approximatingcomplex shapes, but they require significant computational cost or areoften impossible if the model to be analyzed is large. Therefore, toimprove computational efficiency, an equivalent circuit model con-sisting of capacitance or inductance induced by the FSS shape can beconstructed, and the electromagnetic characteristics can be analyzedmore easily and quickly. In addition, if transmission line theory is ap-plied to the equivalent FSS circuit model, it is possible to analyze the

https://doi.org/10.1016/j.compscitech.2018.09.002Received 18 April 2018; Accepted 3 September 2018

∗ Corresponding author.E-mail address: leejrr@kaist.ac.kr (J.-R. Lee).

Composites Science and Technology 167 (2018) 547–554

Available online 05 September 20180266-3538/ © 2018 Elsevier Ltd. All rights reserved.

T

electromagnetic characteristics of the structure applied with FSS, suchas a radar-absorbing structure (RAS). However, most of the RAS isapplied to the curved structural configuration, and since electro-magnetic waves are incident at various angles, the equivalent circuitmodel should be designed by considering the incident angle.

In this study, an RAS applied with an FSS was designed and itselectromagnetic characteristics were predicted by using an equivalentcircuit model. An equivalent circuit model considering the incidentangle was proposed. Electromagnetic analysis of the structural curva-ture was performed by applying the equivalent circuit model to the raytracking method. In previous research into the ray tracking method[15], curved structural analysis was performed by simply applying thetransmission/reflection coefficient at the transmitting/reflecting inter-face between each layer. This method is difficult to use when con-sidering changes in the transmission/reflection coefficient due tothickness and curvature. Therefore, in this study, by using an equivalentcircuit model and the ray tracking method, the electromagnetic char-acteristics of a curved RAS applied with an FSS were predicted in astructure with known curvature.

2. Analysis of curved radar-absorbing structures

2.1. Radar-absorbing structural design

To analyze the electromagnetic characteristics of an RAS dependingon the curvature, an RAS with a square-loop-shaped FSS was designed,as shown in Fig. 1(a). Fig. 1(b) shows how a perfect electrical conductor(PEC) layer, a spacer layer, and a FSS are stacked, and the absorptionfrequency is 10.5 GHz. The analysis was performed using commerciallyavailable analysis software (HFSS), and a unit cell model was used toanalyze the FSS using the Floquet theorem. The result is shown inFig. 1(c).

2.2. Equivalent circuit model of FSS

An equivalent circuit model for an FSS with a square loop pattern iscomposed of a simple LC circuit. Since the designed FSS consists of twosquare loops, it is possible to convert an equivalent circuit model withtwo LC circuits connected in parallel, as shown in Fig. 2. The impedanceof this equivalent circuit model can be calculated using Eq. (1). It hastwo reflection frequencies ( f f,1 3) determined by two square loops, andit has one transmission frequency ( f2) due to the interaction betweenthe two square loops. Since the equivalent circuit model has the sameelectromagnetic characteristics as the FSS, it can be used as an analy-tical model that reflects the electromagnetic characteristics of the FSS.

= − −− + −

=− −

+ −η ω L C ω L C

jω C ω L C C ω L CL L f ω f ωjω L L f ω

(1 )(1 )[ (1 ) (1 )]

( )( )( )( )FSS

21 1

22 2

12

2 2 22

1 1

1 2 12

32

1 2 22

⎜ ⎟⎛⎝

= = ++

= ⎞⎠

Where fL C

f C CC C L L

fL C

, 1 ,( )

, 11

1 12

1 2

1 2 1 23

2 2 (1)

The proposed equivalent circuit model can be compared with theHFSS results to calculate the value of the LC component according tothe incident angle. However, in the case of FSS, the electromagneticcharacteristics are determined by the dielectric constant and the per-meability of the adjacent materials. According to previous research

Fig. 1. (a) FSS shape, (b) RAS cross section, and (c) RAS analysis result.

Fig. 2. Equivalent circuit model of the FSS with two square loops.

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[18], there is a result that states the adjacent materials are affected upto thickness of 0.4 times the unit cell size. Therefore, since the size ofthe FSS unit cell is 8 mm, the thickness of the adjacent material af-fecting the FSS is 3.2mm. To reflect the characteristics of the FSS, theelectromagnetic analysis of the FSS was performed using the unit cellmodel shown in Fig. 3(a). The inductance and capacitance values in theequivalent circuit model are calculated using the results of unit cellmodel, and the results are shown in Fig. 3(b) and (c). The proposedequivalent circuit model can be used to calculate the equivalent im-pedance of the FSS as a function of the incident angle. Transmission/reflection loss using the calculated equivalent impedance were com-pared with the results of the HFSS shown in Fig. 4. It was confirmedthat the two results showed very similar results. The resonance fre-quencies were almost the same, and the peak values showed somedifferences, but there is no considerable mismatch because the graphswere plotted in decibel scale.

2.3. Analysis of RAS applied with the FSS

Equivalent circuit model of the RAS applied with the FSS is shownin Fig. 5(a). Eqs. (2)–(5) can be obtained by applying transmission linetheory to this equivalent circuit model. ηFSS is the equivalent impedancevalue calculated from the equivalent circuit model of the FSS obtainedabove. Impedance (η) and propagation constant (β) changes wereconsidered according to incident TE and TM radiation at various angles.The unit cell model shown in Fig. 5(b) was used to compare the resultsof the equivalent circuit model and the unit cell model. Fig. 6 showsgood agreement between the two models.

=−+

Γη ηη η

in air

in air (2)

=+

ηη η

η ηinFSS b

FSS b (3)

=++

η ηη β t jη β tη β t jη β t

cos( ) sin( )cos( ) sin( )b PI

a PI PI PI PI PI

PI PI PI a PI PI (4)

=η η jβ ttanh( )a Poam Poam Poam

⎜ ⎟⎛⎝

= = ⎞⎠

Where β ωc

μ ε ημε

, ,r r0 (5)

2.4. Curved structural analysis

To understand the electromagnetic characteristics of the curvedstructure, it is necessary to consider the influence of the curvature aswell as the incident angle. A schematic of incident wave propagation inthe curved structure is shown in Fig. 7. The wave incident at θ1 istransmitted at θ2 at the first interface, but the angle of incidencechanges to ′θ2 due to the curvature at the second interface. Therefore, θ1and θ3 are different in the curved structure. Eqs. (6)–(9) are used tocalculate the reflection coefficient of the curved structure. The values ofL and ′θ2 in the reflection coefficient equation can be simply expressedas functions of curvature and thickness. To confirm the effects of theincident angle and curvature, electromagnetic characteristics of thecurved RAS can be obtained by applying the equations for the curvedstructure to the equivalent circuit model of the RAS.

=−+

Γη ηη ηTE

TE s

TE s

1

1 (6)

=+

+η η

η β L θ jη β L θ

η β L θ jη β L θ

cos( cos ) sin( cos )

cos( cos ) sin( cos )TE ss s

s s2

3 22

2 2'

22

2

2'

22

2 3 22

2

⎜

⎟

⎛⎝

= = =

= = = = ⎞⎠

Where η η θ η η θ η η θ

η η θ β ωc

μ ε ημε

ββ

θθ

, sec , sec , sec ,

sec , , , sinsin

s s s

s r r

1 1 1 2 2 2 2'

2 2'

3 3 30

1

2

2

1 (7)

=−

+Γ

η ηη ηTM

TM p

TM p

1

1 (8)

=+

+η η

η β L θ jη β L θ

η β L θ jη β L θ

cos( cos ) sin( cos )

cos( cos ) sin( cos )TM pp p

p p2

3 22

2 2'

22

2

2'

22

2 3 22

2

⎜

⎟

⎛⎝

= = =

= = = = ⎞⎠

Where η η θ η η θ η η θ

η η θ β ωc

μ ε ημε

ββ

θθ

, cos , cos , cos ,

cos , , , sinsin

p p p

p r r

1 1 1 2 2 2 2'

2 2'

3 3 30

1

2

2

1 (9)

Fig. 3. (a) Unit cell model of FSS (HFSS), (b) inductance, and (c) capacitanceversus the incident angle.

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3. Ray tracking method

In this study, the electromagnetic performance of a curved structureis measured using a reflection loss measurement setup with a standardhorn antenna. Therefore, the ray tracking method is required to un-derstand the electromagnetic performance of the curved structure be-cause the analysis in the previous sections was only able to obtain re-sults for a single unit cell. Before performing the ray tacking method, itis necessary to understand how the electromagnetic waves from thestandard horn antenna propagate. Therefore, near field analysis was

performed using electromagnetic software (FEKO), and the results areshown in Fig. 8. As a result, it can be confirmed that the emitted ra-diation spreads radially, and it is possible to calculate the incident angleof the electromagnetic wave on the specimen. As shown in Fig. 8 (b),the energy density of the incident electromagnetic wave can be ob-tained from the Poynting vector. The incident angle and the energydensity of the electromagnetic wave incident on the specimen wereobtained by near field analysis, and the results were applied to the raytracking method. The electromagnetic wave incident on the specimenwas divided and applied to the ray tacking method. Also, the

Fig. 4. Comparison between the unit cell model (HFSS) and the equivalent circuit model of the FSS.

Fig. 5. (a) Equivalent circuit model of RAS, (b) Unit cell model of RAS (HFSS).

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electromagnetic characteristics of the curved RAS were predicted bycalculating the reflection loss of the electromagnetic wave that returnedto the standard horn antenna. The electromagnetic wave reflected fromthe RAS was calculated using the analytical method of the curved RASproposed above.

4. Comparison of experimental and calculated results

To verify the calculated results for the curved RAS, a specimen wasmanufactured, as shown in Fig. 9. The size of the specimen was largerthan that of the standard horn antenna. The FSS was fabricated by aphotolithography process using a thin copper-coated polyimide film.

Copper tape was used for the PEC. Fig. 10 shows the reflection lossmeasurement setup using a standard horn antenna. If the curvaturespecimen is measured in the far field, it is difficult to accurately mea-sure the reflected electromagnetic waves because the amount of elec-tromagnetic waves returns to the receiver antenna is small because ofthe curvature. Therefore, the electromagnetic performance according tothe curvature was measured by reducing the measurement distance inthe near field. Fig. 11 shows the results of the analysis and the ex-perimental results. As shown in Fig. 11(a), it was confirmed that theabsorption frequency increased with as the curvature increased. Acurvature increase means that the ratio of the electromagnetic wavewith large incident angles is increased. As the incident angle increased,

Fig. 6. Comparison between the unit cell model (HFSS) and the equivalent circuit model of the RAS.

Fig. 7. Schematic diagram of incident wave propagation in the curved structure.

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the absorption frequency increased, as shown in Fig. 6. Therefore, theabsorption frequency increased as the curvature increased. In addition,it was confirmed that the reflection loss reduced as a whole because theelectromagnetic wave returning to the standard horn antenna was re-duced due to the curvature. As shown in Fig. 11(b), the calculated re-sults show the same characteristics as the experimental results. As thecurvature increased, the absorption frequency increased and the re-flection loss decreased.

5. Conclusion

In this study, a novel analysis method using an equivalent circuitmodel and ray tracking method was proposed, as shown in Fig. 12. Thismethod was used to estimate the electromagnetic performance ofcurved RAS with FSS. To analyze the FSS, it is necessary to predict theelectromagnetic characteristics according to the incidence angle, be-cause the resonance frequency of the FSS changes with the incidentangle. Therefore, the electromagnetic analysis of the FSS was performedusing electromagnetic software (HFSS). Since the electromagnetic

Fig. 8. Results of near field analysis for the standard horn antenna at 10 GHz. (a) Instantaneous magnitude of the E-field and (b) magnitude of the Poynting vector onthe specimen.

Fig. 9. RAS specimens.

Fig. 10. Reflection loss measurement setup with a standard horn antenna.

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characteristics of the FSS vary depending on adjacent materials, the FSSwas analyzed by reflecting the electromagnetic properties of adjacentmaterials. Then, the equivalent circuit model of the FSS was con-structed, and the capacitance and inductance were calculated usingresults from a unit cell model. The equivalent circuit model of the FSSwas applied to the equivalent circuit model of the RAS, and the elec-tromagnetic performance was predicted using transmission line theory.Near field analysis was also performed to understand how electro-magnetic waves interact with a standard horn antenna. The results wereapplied to the ray tracking method to predict the electromagneticperformance of the entire curved RAS.

To verify the predicted electromagnetic performance of the curvedRAS with an FSS, three specimens with different curvatures were fab-ricated and the electromagnetic performance was measured using areflection loss measurement setup with a standard horn antenna. Thiswas more appropriate for near field experiments for the curved struc-ture. Near field numerical analysis was performed using electro-magnetic software (FEKO) to determine the electromagnetic char-acteristics of the standard horn antenna in the near field. The results of

the magnitude and direction of the incident electromagnetic waves onthe specimen were applied to the proposed method to predict theelectromagnetic performance of the entire curved structure. As a result,it was confirmed that as the curvature increased, the absorption fre-quency increased and the reflection loss decreased as a whole. Thesecharacteristics agreed well with the experimental results, which showsthat the developed electromagnetic performance prediction method forthe curved RAS with an FSS is effective and accurate.

The analytical method proposed in this study has the advantage ofreducing the analysis time by using a unit cell model instead of ana-lyzing the entire curved structure. In addition, since the equivalentcircuit model of the RAS reflects the electromagnetic characteristicsaccording to the curvature and incident angle, it is possible to analyzecomplex and large shapes. Therefore, it is expected that the proposedmethod can shorten the existing computational time and analyze a largecurved structure that is physically impossible with the existing soft-ware.

Acknowledgements

This research was conducted with the support of the LeadingDefense Core Technology R&D project (UC160032JD) by the Agencyfor Defense Development (ADD) and was financially supported by theMinistry of Trade, Industry, and Energy (MOTIE), South Korea, underthe “Commercializing fuel cell electric vehicle component industry andR&D Support Program”(reference number R0006462) supervised by theKorea Institute for Advancement of Technology (KIAT).

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