Wave Impedance Equivalent Circuit Model for Square Loop Frequency Selective Surfaces

C. Júnior, Everaldo Faustino da; Silva, Paulo Henique da F.; Gomes Neto, Alfredo

doi:10.1590/2179-10742024v23i2281143

Everaldo Faustino da C. Júnior

Group of Telecommunications and Applied Electromagnetism, GTEMA, Federal Institute of Paraíba, João Pessoa, PB, Brazil.

https://orcid.org/0009-0007-9969-4581

Paulo Henique da F. Silva

Group of Telecommunications and Applied Electromagnetism, GTEMA, Federal Institute of Paraíba, João Pessoa, PB, Brazil.

https://orcid.org/0000-0002-7170-8693

Alfredo Gomes Neto

Group of Telecommunications and Applied Electromagnetism, GTEMA, Federal Institute of Paraíba, João Pessoa, PB, Brazil.

https://orcid.org/0000-0001-5437-9093

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Fig. 1
Square loop FSS periodic arrangement.

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Fig. 2
Window in a rectangular waveguide: (a) side view; (b) front view; (c) equivalent circuit approximation.

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Fig. 3
FSS: (a) Representation of transmission in terms of impedances; (b) Equivalent circuit.

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Fig. 4
FSS: (a) Illustration of the scattering problem in an FSS using the input impedance concept; (b) Equivalent circuit.

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Fig. 5
FSS: (a) Representation in terms of equivalent impedance; (b) Associated equivalent circuit.

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Fig. 6
Comparison between simulated results of effective electric permittivity models.

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Fig. 7
Comparison between simulation results: (a) FSS 1; (b) FSS 2; (c) FSS 3.

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Fig. 8
Images: (a) Prototype; (b) Measurement arrangement.

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Fig. 9
Comparison between prototype simulation results.

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Fig. 10
Comparison between simulated and measured results of the prototype.

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TABLE I
PARAMETERS USED IN SIMULATIONS OF EFFECTIVE ELETRIC PERMITTIVITY MODELS.

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TABLE II
PARAMETERS OF SQUARE LOOP TYPE FSS IN OVERLAPPED CONFIGURATION ON A DIELECTRIC.

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TABLE IIII
NUMERICAL RESULTS OBTAINED FOR RESONANCE FREQUENCY.

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TABLE IV
NUMERICAL RESULTS OBTAINED FOR BANDWIDTH.

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TABLE V
PARAMETERS OF THE FSS PROTOTYPE ANALYZED.

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TABLE VI
NUMERICAL RESULTS OBTAINED FOR RESONANCE FREQUENCY.

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TABLE VII
NUMERICAL RESULTS OBTAINED FOR BANDWIDTH.

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Model	Resonance Frequency (GHz)	Physical Dimensions (mm)
Model	Resonance Frequency (GHz)	P	d	w	g	N
[⁹[9] F. Costa, A. Monorchio, G. Manara, "An overview of equivalent circuit modeling techniques of frequency selective surfaces and metasurfaces," in Applied Computational Electromagnetics Journal, vol. 29, pp. 960-976, 2014. DOI:10.1002/J.1538-7305.1975.TB03551.X. https://doi.org/10.1002/J.1538-7305.1975... ]	3.5	20	18	1.4	0.8	1.8
[¹⁰[10] M. A. R. Barrera, W. P. Carpes, "Numerical Model of the Effective Permittivity for Square-Loop Frequency Selective Surfaces," in IEEE Transactions on Magnetics, vol. 51, pp. 1-4, 2015.]						$ε_{r e f} = ε_{r h} + (1 - ε_{r h}) e^{- 2 d h / (p \sqrt{w g})}$
[²³[23] G. L. R. Araújo, A. L. P. S. Campos, A. M. Martins, "Improvement of the equivalent circuit method for analysis of frequency selective surfaces using genetic algorithms and rational algebraic models," Prog Electromagn Res Lett, vol. 55, pp. 67-74, 2015. DOI:10.2528/PIERL15060803. https://doi.org/10.2528/PIERL15060803.... ]						0.8832

FSS	Reference	Material	ɛ_r	Geometrical Parameters (mm)
FSS	Reference	Material	ɛ_r	p	w	d	g	h
1	[¹¹[11] R. J. Langley, E. A. Parker, "Equivalent circuit model for arrays of square loops," Electronics Letters, vol. 18, pp. 294-296, 1982. DOI:10.1049/EL:19820201. https://doi.org/10.1049/EL:19820201.... ]	Polyester	3.00	5.25	0.47	5.00	0.25	0.021
2	[²⁴[24] T. Areias, R. Carvalho, A. L. P. S. Campos, A. G. Neto, J. J. G. Neto, M. M. Silva, "Frequency Selective Surface Microwave Absorber for WLAN Applications," in Journal of Communication and Information Systems, vol. 35, pp. 208-216, 2020. DOI:10.14209/jcis.2020.23. https://doi.org/10.14209/jcis.2020.23.... ]	FR-4	4.40	42.50	2.00	31.95	10.55	1.60
3	[¹⁴[14] A. L. P. S. Campos, "Frequency Selective Surfaces: Analysis and Design," Superfícies Seletivas em Frequência: Análise e Projeto, IFRN Editora, vol. 1, pp. 196, 2009.]	FR-4	4.40	12.00	1.00	10.00	2.00	1.50

FSS	Resonance Frequency (GHz)				Deviation (%)
FSS	ECM	iECM	ECM-WI	Ansoft Designer^TM	ECM	iECM	ECM-WI
1	15.31	15.36	15.32	15.33	0.13	0.20	0.07
2	2.53	2.26	2.27	2.28	10.96	0.88	0.44
3	6.96	6.12	6.50	6.46	7.74	5.26	0.62
				Mean Deviation (%)	6.28	2.11	0.38

FSS	Bandwidth (GHz)				Deviation (%)
FSS	ECM	iECM	ECM-WI	Ansoft Designer^TM	ECM	iECM	ECM-WI
1	7.27	7.26	7.22	6.84	6.29	6.14	5.56
2	0.73	0.58	0.59	0.58	25.86	0.00	1.72
3	3.43	2.50	2.85	2.61	31.42	4.21	9.20
				Mean Deviation (%)	21.19	3.45	5.49

Material	ɛ_r	P (mm)	w (mm)	d (mm)	g (mm)	h (mm)
FR-4	4.40	9.12	1.00	7.12	2.00	1.50

resonance frequency (GHz)			Deviation (%)
ECM-WI	Ansoft Designer^TM	Measured	ECM-WI	Ansoft Designer^TM
10.00	10.00	10.26	2.56	2.56

Bandwidth (GHz)			Deviation (%)
ECM-WI	Ansoft Designer^TM	Measured	ECM-WI	Ansoft Designer^TM
3.94	5.12	3.83	2.79	33.57

G (p, q, λ, κ) = \frac{1}{2} \frac{{(1 - β^{2})}^{2} [(1 - \frac{β^{2}}{4}) (C_{+} + C_{-}) + 4 β^{2} C_{+} C_{-}]}{(1 - \frac{β^{2}}{4}) + β^{2} (1 + \frac{β^{2}}{2} - \frac{β^{4}}{8}) (C_{+} + C_{-}) + 2 β^{6} C_{+} C_{-}}

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Wave Impedance Equivalent Circuit Model for Square Loop Frequency Selective Surfaces

Abstract