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An effective methodology to design scale model for magnetic absorbing coatings based on ORL
Results in Physics 7 (2017) 1698–1704
Contents lists available at ScienceDirect
Results in Physics journal homepage: www.journals.elsevier.com/results-in-physics
An effective methodology to design scale model for magnetic absorbing coatings based on ORL Liming Yuan a,⇑, Bin Wang b, Wei Gao a, Yonggang Xu a, Xiaobing Wang a, Qilin Wu c a
Science and Technology on Electromagnetic Scattering Laboratory, Shanghai 200438, PR China Shanghai Radio Equipment Research Institute, Shanghai 200090, PR China c Key Laboratory of High Performance Fibers & Products, Ministry of Education, Donghua University, Shanghai 201620, PR China b
a r t i c l e
i n f o
Article history: Received 18 October 2016 Received in revised form 20 April 2017 Accepted 6 May 2017 Available online 10 May 2017 Keywords: A magnetic absorbing coating Scale measurement Reflective loss Monostatic RCS
a b s t r a c t The scale measurement has a great significance in studying electromagnetic scattering properties. But there still exist great difficulties in constructing an accurate scale model including nonmetallic materials such as magnetic absorbing coatings. Based on the reflective loss of a coating irradiated obliquely by plane microwave, a method is proposed to solve the problem of designing the scale coating. The commercial simulation software FEKO is used to investigate the use of the method. According to simulating monostatic RCS of coating plate models, coating sphere models and coating spherecone models, results reveal that the monostatic RCS of the designed scale model have a great agreement with that of the theoretical scale model in the whole incidence angle range. Furthermore, the coating SLICY models, which include many electromagnetic scattering mechanisms, are constructed to verify the proposed method. The simulated result of the designed scale SLICY model is very close to that of the theoretical scale SLICY model. All the simulated results indicate that the method proposed in this paper is valid. Ó 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Introduction Radar cross section (RCS) is one of the most important electromagnetic scattering properties. It has a great significance in radar system development, object recognition, electronic warfare technology and so on. RCS of a target could often be obtained by the full scale measurement or the scale measurement. For there exist many difficulties in the full scale measurement, such as it is difficult to acquire the real full-scale target, the controllability is poor, the cost is very expensive and so on, the scale measurement has been increasingly concerned and widely applied to study electromagnetic scattering properties [1]. On the basis of the linear theory of Maxwell’s equations, a classic scale theory was proposed by J.A. Stratton and J.A. Sinclair [2,3]. The basic principle of the theory is that the scale system has the same electrical size with the full scale system, and then there differs a constant value between the scale model RCS and the full scale model RCS. This principle requires that the ratio of the geometric length to the microwave wavelength in the scale system is the same with that in the full scale system; meanwhile, electromagnetic parameters of model materials are identical in the both systems. Table 1 gives the relationships of
⇑ Corresponding author. E-mail address: [email protected] (L. Yuan).
physical quantities between the full scale system and the scale system. In Table 1, p is the scale ratio, then there exists the following relation between the scale model RCS and the full scale model RCS.
r0 ¼ r þ 20 log p
ð1Þ
where r is the scale model RCS at scale frequency, and r is the full scale model RCS at full scale frequency. In the scale measurement, one of the keys is to construct an accurate scale model. For metallic object or perfect electric conductor (PEC), there is little difficulty in constructing a satisfactory model. However, engineering targets always contain electric or/ and magnetic materials. The permittivity or/and permeability of these materials are typically frequency dependent [4,5]. In particular, when a target contains a magnetic material, such as a magnetic absorbing coating on a stealth aircraft, for permeability of a regular material tends to be a constant for the existence of Snoek limitation [6,7], it would be very difficult to obtain the scale material which has the same permeability at scale frequency. In this study, a scale-designing method for a magnetic absorbing coating is proposed based on the reflective loss of a coating irradiated obliquely by plane microwave. The optimization procedure base on the method is introduced in details. Monostatic RCS of coating plate models, coating sphere models and coating spherecone models as well as coating SLICY models are evaluated by the 0
http://dx.doi.org/10.1016/j.rinp.2017.05.007 2211-3797/Ó 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
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L. Yuan et al. / Results in Physics 7 (2017) 1698–1704 Table 1 Relationships of physical quantities between the full scale system and the scale system. Physical quantities
Full scale system
Scale system
Length Frequency Wavelength Permittivity Permeability
l f k
pl f =p pk
e l
e l
the coating thickness; er and lr are the relative plural permittivity and permeability of the coating, respectively. When the incident microwave is transverse magnetic wave (TM wave), which means that the electric field component of the incident microwave is in the plane constituted of the microwave propagation direction and the normal direction of the coating surface. And the magnetic field component is perpendicular to the plane. ORL of the coating could be calculated by following formulae.
Z in ðTMÞ ¼ commercial simulation software FEKO. Simulated monostatic RCS of designed scale models are compared with that of theoretical scale models. Theoretical considerations According to high frequency method for electromagnetic scattering, an identical scattering of a target could be obtained under the condition of the same surface reflectivity. By utilizing this principle, the scale material of a magnetic absorbing coating could be designed by making the reflective loss (RL) of the scale coating as identical as possible with that of the full scale coating. And the electromagnetic parameters of the scale coating could be different from that of the full scale coating. Because of natural dispersion characteristic, it is always difficult to prepare the same electromagnetic parameters at scale frequency with that at full scale frequency. So this method has an important significance in engineering. Here RL is calculated as the coating is irradiated obliquely by plane microwave, and this RL is called as ORL. ORL is associated with microwave polarization modes, as shown in Fig. 1. The coming optimizing to ORL will be carried out in a wide incidence angle range. When the incident microwave is transverse electric wave (TE wave), which means that the electric filed component of the incident microwave is perpendicular to the plane constituted of the microwave propagation direction and the normal direction of the coating surface. And the magnetic field component is in the plane. ORL of the coating could be calculated by following formulae [8].
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0 2 2pfd er lr sin h Z 0 lr A Z in ðTEÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tanh @j C er lr sin2 h
ð2Þ
Z in ðTEÞ cos h Z 0 RLðTEÞ ¼ 20 lg Z in ðTEÞ cos h þ Z 0
ð3Þ
where Z 0 is the characteristic impedance in free space, the value is 377 X; f is the incident microwave frequency; C is the microwave propagation velocity in free space; h is the incidence angle; d is
er
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 2pfd er lr sin h A tanh @j C 0
PEC
ð4Þ
ð5Þ
Fig. 2 shows the flow sketch for designing the scale coating of a magnetic absorbing coating. It mainly includes four steps, i.e., inputting ORL of a full scale coating, collecting measured electromagnetic parameters, fitting measured electromagnetic parameters and optimizing ORL of the scale coating. More details could be seen as following Firstly, using the relative plural electromagnetic parameters and thickness of a full scale magnetic absorbing coating, calculate ORL in the whole incidence angle range by utilizing Eqs. (2) and (3) or (4) and (5) at full scale frequency, and set the calculated ORL as the input of the following ORL optimized process. Secondly, select spherical carbonyl iron particles (SCIPs) as the absorbent and epoxy resin as the binder, prepare samples with different SCIPs concentrations for measuring the relative plural permittivities and permeabilities, and build the permittivity selection and the permeability selection. Thirdly, Fit the measured permittivities and permeabilities by taking advantage of two-exponent phenomenological percolation equation (TEPPE) [9–11], it could be written as following.
pi
/i1=t /1=t e /1=t i
þ
A/1=t e
þ ð1 p i Þ
1=s /1=s m /e 1=s /m þ A/1=s e
¼0
ð6Þ
where A ¼ ð1 f c Þ=f c and f c denotes the percolation threshold; pi is the SCIPs concentration; /e is the effective electromagnetic parameters of the composite, /i and /m are the intrinsic electromagnetic parameters of SCIPs and the binder, respectively; t and s are fitting parameters, which could be obtained by fitting the measured permittivities or permeabilities. Meanwhile, the intrinsic electromagnetic parameters of SCIPs could be written as following.
e¼1
r xe0
l¼1þ
2 X kMns ðkHna þ ian xÞ 3 n¼1;2; ðkHn þ ian xÞ2 x2
ð7Þ
a
E
q
H
PEC
q
qk
qk (a)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi er lr sin2 h
Z in ðTMÞ Z 0 cos h RLðTMÞ ¼ 20 lg Z in ðTMÞ þ Z 0 cos h
E
H
Z0
(b)
Fig. 1. ORL calculated diagram for a coating irradiated by plane microwave of different polarization modes: TE wave (a) and TM wave (b).
ð8Þ
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L. Yuan et al. / Results in Physics 7 (2017) 1698–1704
Input ORL of a full scale coating
Collect measured permittivities and permeabilities
Fit measured permittivities and permeabilities
Set an absorbent concentration in a scale coating
Set a scale coating thickness
Calculate the scale coating ORL
Contrast the scale coating ORL with the full scale coating ORL
N
Be identical? Y
Obtain the optimized scale coating formulation Fig. 2. The flow sketch of designing the scale coating for a magnetic absorbing coating.
where e0 is the permittivity of free space, r is the electric conductivity, x is the angular frequency, k is the gyromagnetic ratio, Ms is the saturation magnetization, Ha is the magnetic anisotropy field and a is damping coefficient. Finally, optimize ORL. This procedure mainly includes four parts: (1) Set the loop number, the scale material thickness range, the thickness step and the incidence angle range; (2) Define a SCIPs concentration, calculate electromagnetic parameters of the corresponding composite by Eqs. (6); (3) Define a thickness and take advantage of the electromagnetic parameters obtained in step (2), calculate ORL in the whole incidence angle range at scale frequency, furthermore, use the sum-of-squared differences to quantify the deviation between ORL of the given coating and that of the full scale coating, calculate the deviation, record the deviation as well as the corresponding thickness and SCIPs concentration; (4) Loop step (2) and (3), search the minimum deviation and take the corresponding thickness and SCIPs concentration as the formulation parameters of the optimized scale coating.
Results and discussions Now an example is given to verify the validation of the introduced method for designing the scale coating of a magnetic absorbing coating. The full scale frequency is 2 GHz, the scale ratio is 1/5 and the scale frequency would be 10 GHz. The thickness of the magnetic absorbing coating is 2 mm, and the permittivity and permeability are 14.49–0.12j and 3.56–1.12j, respectively. Here just TE wave is considered. The designing procedure is introduced as follows.
Firstly, calculate ORL of the full scale coating by Eqs. (2) and (3) as the coating is irradiated by TE wave in the range from 0° to 90°. Secondly, prepare a series of coaxial samples composed of SCIPs and epoxy resin. SCIPs concentration is different from each other. The densities of SCIPs and epoxy resin are measured to be 7.67 g/ cm3 and 2.15 g/cm3, respectively. Then SCIPs concentration could be derived from the sample density. Here SCIPs concentrations of prepared samples are 5.48%, 10.31%, 15.26%, 20.72%, 25.34%, 30.22% and 35.71%. Electromagnetic parameters could be measured by transmission/reflection method. The collected permittivities at 10 GHz are 2.98–0.1j, 3.68–0.14j, 4.42–0.12j, 5.56–0.14j, 6.51–0.16j, 8.88–0.19j and 10.42–0.24j, and the collected permeabilities at 10 GHz are 1.05–0.14j, 1.1–0.25j, 1.17–0.36j, 1.19– 0.51j, 1.21–0.65j, 1.23–0.8j and 1.28–0.98j. Thirdly, fit collected permittivities and permeabilities by Eq. (6), and then determine t and s. More details of the fitting process could be seen in Ref. [10] and [11]. Finally, set loop number as 5000, the scale coating thickness range as 0.5 mm–2.0 mm, the thickness step as 0.1 mm. At 10 GHz, loop calculating ORL of a scale coating with a different thickness and SCIPs concentration and the deviation of ORL between the scale coating and the full scale coating, meanwhile, record the deviation as well as the thickness and the SCIPs concentration. After all loops finished, search the minimum deviation and extract the corresponding thickness and SCIPs concentration, which are 1.6 mm and 12.6%, respectively. The calculated permittivity and permeability of the optimized scale coating are 4.25– 0.16j and 1.15–0.27j. Fig. 3 plots ORL of the full scale coating at 2 GHz and ORL of the optimized scale coating at 10 GHz, as shown in Fig. 3, they are almost identical in the range of 0°–90°. In order to verify the optimized scale coating, models including coating plates, coating spheres and coating spherecones are constructed to simulate monostatic RCS by the commercial software FEKO, as shown in Fig. 4. The excitation is selected as TE wave. The frequency is 2 GHz for simulating full scale models and 10 GHz for scale models, respectively. For theoretical scale models, the scale coating thickness is 1/5 of the full scale coating thickness, i.e., 0.4 mm, and the electromagnetic parameters are the same with that of the full scale coating. For designed scale models, the thickness of the scale is equal to 1.6 mm, and the permittivity and permittivity are equal to 4.25–0.16j and 1.15–0.27j. Parameters of simulated models are set as Table 2. Plate models are constructed by a square plate, the region of the plate is coating material and the bottom face of the plate is set as PEC, z-axis is perpendicular to the plate, the incidence angle ranges from 0° to 90°. Sphere models are constructed by coating metal spheres with the full scale coating or the optimized scale coating, the incidence angle ranges from 0° to 90°. Spherecone models are constructed by coating metal spherecones with the full scale coating or the optimized scale coating, the incidence angle ranges from 0° to 180°. All simulated results are plotted in Fig. 5. Fig. 5(a) and (b) show the simulated results of coating plate models and coating sphere models, respectively. As shown in Fig. 5(a) and (b), it could be seen that the monostatic RCS of the designed 1/5th scale model have a great agreement with that of the theoretical 1/5th scale model in the whole incidence angle range. This indicates that it is valid to design the scale coating of a magnetic absorbing coating by the introduced method. Fig. 5(c) plots the simulated results of coating spherecone models. As shown in Fig. 5(c), it could be seen that monostatic RCS of the designed 1/5th scale model is generally close to that of the theoretical 1/5th scale model. However, there is a little deviation between them. The deviation is caused by more complicated scatterings on spherecones, such as edge diffraction, surface travelling wave and so on. Because the introduced method based on ORL is derived from Fresnel reflection, these complicated
L. Yuan et al. / Results in Physics 7 (2017) 1698–1704
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Fig. 3. The ORL of the full scale coating at 2 GHz and that of the optimized scale coating at 10 GHz.
Fig. 4. Simulated models: a coating plate (a), a coating sphere (b), a coating spherecone (c).
scatterings could not be taken into account. These scatterings are more obvious when the cone part is irradiated in front than the sphere part is irradiated in front. It explains that why the deviation in the range of 0°–90° is more obvious than that in the range of 90° –180°. Fig. 6 plots the statistical result of relative monostatic RCS deviation between the designed scale spherecone model and the theoretical scale spherecone model. As shown in Fig. 6(a), most of deviations in the whole angle range are less than 10%, and the maximal deviation reaches 58% at the incidence angle of 120°. Furthermore, Fig. 6(b) gives statistical possibility of relative deviation. It could be seen that monostatic RCS deviation is less than 10% for 90% of incidence angles. This indicates that the designed scale coating could meet the requirement of scale measurement for coating spherecone models. To further verify the proposed method for optimizing the scale coating, monostatic RCS simulations are carried out by constructing the SLICY model, which includes many electromagnetic scattering mechanisms such as specular-reflection, multiple-reflection, creeping wave scattering, cavity scattering and so on, and is often used as an example to analyzing electromagnetic scattering mechanisms. Fig. 7(a) gives the full scale SLICY model as well as its key geometric parameters. The excitation is still selected as TE wave.
The incidence pitch angle ranges from 90° to 90° and the azimuth is equal to 0°. The frequency is 2 GHz for simulating the full scale SLICY model and 10 GHz for scale SLICY models, respectively. The theoretical scale SLICY model has the same electromagnetic parameters with the full scale SLICY model, and the scale coating thickness is 1/5 of the full scale coating thickness. The permittivity and permittivity of the designed scale SLICY model are equal to 4.25–0.16j and 1.15–0.27j, and the scale coating thickness is equal to 1.6 mm, as shown in Table 2. Fig. 7(b) plots the simulated results of coating SLICY models. As shown in Fig. 7(b), it could be seen that monostatic RCS of the designed scale SLICY model is generally close to that of the theoretical scale SLICY model. The deviation is mainly caused by those scattering mechanisms except specular-reflection and multiplereflection. Fig. 8 plots the statistical result of relative monostatic RCS deviation. It could be seen that monostatic RCS deviation is less than 5% for 34.8% of incidence angles, 10% for 70.2% of incidence angles and 15% for 93.9% of incidence angles. According to the calculating results plotted in Fig. 7(b), Table 3 gives the mean monostatic RCS values of the theoretical scale SLICY model and the designed scale SLICY model in the angle ranges of 90° to 60°, 60° to 30°, 30° to 0°, 0°–30°, 30°–60° and 60°–90°, and the relative deviation is 6.65%, 11.08%, 2.77%, 5.14%,
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Table 2 Parameters of simulated models. Simulated models Plates
Full scale model Theoretical 1/5th scale model Designed 1/5th scale model
Spheres
Spherecones
Geometric parameters
Coating permittivity
Coating permeability
Coating thickness
Length: 500 mm Width: 500 mm Length: 100 mm Width: 100 mm Length: 100 mm Width: 100 mm
14.49–0.12j
3.56–1.12j
2.0 mm
14.49–0.12j
3.56–1.12j
0.4 mm
4.25–0.16j
1.15–0.27j
1.6 mm
Full scale model Theoretical 1/5th scale model Designed 1/5th scale model
Diameter: 500 mm Diameter: 100 mm
14.49–0.12j 14.49–0.12j
3.56–1.12j 3.56–1.12j
2.0 mm 0.4 mm
Diameter: 100 mm
4.25–0.16j
1.15–0.27j
1.6 mm
Full scale model
Height of 500 mm Radius of sphere: 200 mm Height of 100 mm Radius of sphere: 40 mm Height of 100 mm Radius of sphere: 40 mm
14.49–0.12j
3.56–1.12j
2.0 mm
14.49–0.12j
3.56–1.12j
0.4 mm
4.25–0.16j
1.15–0.27j
1.6 mm
Theoretical 1/5th scale model
Designed 1/5th scale model
the cone: the bottom
the cone: the bottom
the cone: the bottom
Fig. 5. Simulated monostatic RCS of different models irradiated by TE wave: coating plate models (a), coating sphere models (b) and coating spherecone models (c).
Fig. 6. The relative deviation between the monostatic RCS of the designed scale spherecone model and the theoretical scale spherecone model: the angle distribution (a) and the statistical possibility (b).
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Fig. 7. The full scale SLICY model as well as its key geometric parameters (a) and simulated monostatic RCS results (b).
Fig. 8. The relative deviation between the monostatic RCS of the designed scale SLICY model and the theoretical scale SLICY model: the angle distribution (a) and the statistical possibility (b).
Table 3 Mean monostatic RCS values in different angle ranges (Unit: dBsm).
The theoretical scale SLICY model The designed scale SLICY model
90° to 60°
60° to 30°
30° to 0°
0°–30°
30°–60°
60°–90°
16.40 17.49
20.12 22.35
13.72 14.10
15.76 16.57
16.84 18.30
19.78 19.96
8.67% and 0.91% in each angle range, respectively. All the relative deviations are within the RCS measurement tolerance. On the basis of the previous results, it can be concluded that the designed scale coating could meet the requirement of the scale measurement for models with a magnetic coating. Conclusions In this paper, a method based on ORL has been proposed to solve the problem of designing the scale coating for a magnetic absorbing coating. An example is given to verify the validation. According to simulated monostatic RCS of coating plate models, coating sphere models and coating spherecone models as well as coating SLICY models, results reveal that the monostatic RCS of the designed scale model have a great agreement with that of
the theoretical scale model in the whole incidence angle range. These indicate that the designed scale coating could meet the requirement of the scale measurement for models with a magnetic coating. It worthy to point out that the scale coating is designed for a magnetic absorbing coating just at a single frequency. Further research into designing the scale coating at a wide frequency band is still on the way. Acknowledgements This work was supported by the National Natural Science Foundation of China under Grant NO. 61471242 and Grant NO. 61601299, and Shanghai Municipal Science and Technology Commission under Grant 15ZR1439600.
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References [1] Saez de Adana F et al. Asymptotic method for analysis of RCS of arbitrary targets composed by dielectric and/or magnetic materials. IEEE Proc Radar Sonar Navig 2003;150:375–8. [2] Stratton JA. Electromagnetic Theory. New York: MC-Graw-Hill Book Company; 1941. [3] Sinclair G. Theory of models of electromagnetic systems. Proc IRE 1948;36:1364–70. [4] Mano JF. Modelling of thermally stimulated depolarization current peaks obtained by global and thermal cleaning experiments. J Phys D Appl Phys 1998;31:2898. [5] Lagarkov AN et al. High-frequency behavior of magnetic composites. J Magn Magn Mater 2009;321:2082–92.
[6] Snoek JL. Dispersion and absorption in magnetic ferrites at frequencies above one Mc/s. Physica 1948;14:207–17. [7] Acher O et al. Generalization of Snoek’s law to ferromagnetic films and composites. Phy Rev B 2008;77:104440. [8] Kim JW et al. Microwave absorbers of two-layer composites laminate for wide oblique incidence angles. Mater Des 2001;31:1547–52. [9] Mclachlan DS. Equations for the conductivity of macroscopic mixtures. J Phys C Solid State Phys 1986;19:1339–54. [10] Chen Ping et al. Complex permittivity and permeability of metallic magnetic granular composites at microwave frequencies. J Phys D Appl Phys 2005;38:2302–5. [11] Chen Ping et al. Frequency dispersive complex permittivity and permeability of ferromagnetic metallic granular composite at microwave frequencies. J Magn Magn Mater 2011;323:3081–6.
Contents lists available at ScienceDirect
Results in Physics journal homepage: www.journals.elsevier.com/results-in-physics
An effective methodology to design scale model for magnetic absorbing coatings based on ORL Liming Yuan a,⇑, Bin Wang b, Wei Gao a, Yonggang Xu a, Xiaobing Wang a, Qilin Wu c a
Science and Technology on Electromagnetic Scattering Laboratory, Shanghai 200438, PR China Shanghai Radio Equipment Research Institute, Shanghai 200090, PR China c Key Laboratory of High Performance Fibers & Products, Ministry of Education, Donghua University, Shanghai 201620, PR China b
a r t i c l e
i n f o
Article history: Received 18 October 2016 Received in revised form 20 April 2017 Accepted 6 May 2017 Available online 10 May 2017 Keywords: A magnetic absorbing coating Scale measurement Reflective loss Monostatic RCS
a b s t r a c t The scale measurement has a great significance in studying electromagnetic scattering properties. But there still exist great difficulties in constructing an accurate scale model including nonmetallic materials such as magnetic absorbing coatings. Based on the reflective loss of a coating irradiated obliquely by plane microwave, a method is proposed to solve the problem of designing the scale coating. The commercial simulation software FEKO is used to investigate the use of the method. According to simulating monostatic RCS of coating plate models, coating sphere models and coating spherecone models, results reveal that the monostatic RCS of the designed scale model have a great agreement with that of the theoretical scale model in the whole incidence angle range. Furthermore, the coating SLICY models, which include many electromagnetic scattering mechanisms, are constructed to verify the proposed method. The simulated result of the designed scale SLICY model is very close to that of the theoretical scale SLICY model. All the simulated results indicate that the method proposed in this paper is valid. Ó 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Introduction Radar cross section (RCS) is one of the most important electromagnetic scattering properties. It has a great significance in radar system development, object recognition, electronic warfare technology and so on. RCS of a target could often be obtained by the full scale measurement or the scale measurement. For there exist many difficulties in the full scale measurement, such as it is difficult to acquire the real full-scale target, the controllability is poor, the cost is very expensive and so on, the scale measurement has been increasingly concerned and widely applied to study electromagnetic scattering properties [1]. On the basis of the linear theory of Maxwell’s equations, a classic scale theory was proposed by J.A. Stratton and J.A. Sinclair [2,3]. The basic principle of the theory is that the scale system has the same electrical size with the full scale system, and then there differs a constant value between the scale model RCS and the full scale model RCS. This principle requires that the ratio of the geometric length to the microwave wavelength in the scale system is the same with that in the full scale system; meanwhile, electromagnetic parameters of model materials are identical in the both systems. Table 1 gives the relationships of
⇑ Corresponding author. E-mail address: [email protected] (L. Yuan).
physical quantities between the full scale system and the scale system. In Table 1, p is the scale ratio, then there exists the following relation between the scale model RCS and the full scale model RCS.
r0 ¼ r þ 20 log p
ð1Þ
where r is the scale model RCS at scale frequency, and r is the full scale model RCS at full scale frequency. In the scale measurement, one of the keys is to construct an accurate scale model. For metallic object or perfect electric conductor (PEC), there is little difficulty in constructing a satisfactory model. However, engineering targets always contain electric or/ and magnetic materials. The permittivity or/and permeability of these materials are typically frequency dependent [4,5]. In particular, when a target contains a magnetic material, such as a magnetic absorbing coating on a stealth aircraft, for permeability of a regular material tends to be a constant for the existence of Snoek limitation [6,7], it would be very difficult to obtain the scale material which has the same permeability at scale frequency. In this study, a scale-designing method for a magnetic absorbing coating is proposed based on the reflective loss of a coating irradiated obliquely by plane microwave. The optimization procedure base on the method is introduced in details. Monostatic RCS of coating plate models, coating sphere models and coating spherecone models as well as coating SLICY models are evaluated by the 0
http://dx.doi.org/10.1016/j.rinp.2017.05.007 2211-3797/Ó 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
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L. Yuan et al. / Results in Physics 7 (2017) 1698–1704 Table 1 Relationships of physical quantities between the full scale system and the scale system. Physical quantities
Full scale system
Scale system
Length Frequency Wavelength Permittivity Permeability
l f k
pl f =p pk
e l
e l
the coating thickness; er and lr are the relative plural permittivity and permeability of the coating, respectively. When the incident microwave is transverse magnetic wave (TM wave), which means that the electric field component of the incident microwave is in the plane constituted of the microwave propagation direction and the normal direction of the coating surface. And the magnetic field component is perpendicular to the plane. ORL of the coating could be calculated by following formulae.
Z in ðTMÞ ¼ commercial simulation software FEKO. Simulated monostatic RCS of designed scale models are compared with that of theoretical scale models. Theoretical considerations According to high frequency method for electromagnetic scattering, an identical scattering of a target could be obtained under the condition of the same surface reflectivity. By utilizing this principle, the scale material of a magnetic absorbing coating could be designed by making the reflective loss (RL) of the scale coating as identical as possible with that of the full scale coating. And the electromagnetic parameters of the scale coating could be different from that of the full scale coating. Because of natural dispersion characteristic, it is always difficult to prepare the same electromagnetic parameters at scale frequency with that at full scale frequency. So this method has an important significance in engineering. Here RL is calculated as the coating is irradiated obliquely by plane microwave, and this RL is called as ORL. ORL is associated with microwave polarization modes, as shown in Fig. 1. The coming optimizing to ORL will be carried out in a wide incidence angle range. When the incident microwave is transverse electric wave (TE wave), which means that the electric filed component of the incident microwave is perpendicular to the plane constituted of the microwave propagation direction and the normal direction of the coating surface. And the magnetic field component is in the plane. ORL of the coating could be calculated by following formulae [8].
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0 2 2pfd er lr sin h Z 0 lr A Z in ðTEÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tanh @j C er lr sin2 h
ð2Þ
Z in ðTEÞ cos h Z 0 RLðTEÞ ¼ 20 lg Z in ðTEÞ cos h þ Z 0
ð3Þ
where Z 0 is the characteristic impedance in free space, the value is 377 X; f is the incident microwave frequency; C is the microwave propagation velocity in free space; h is the incidence angle; d is
er
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 2pfd er lr sin h A tanh @j C 0
PEC
ð4Þ
ð5Þ
Fig. 2 shows the flow sketch for designing the scale coating of a magnetic absorbing coating. It mainly includes four steps, i.e., inputting ORL of a full scale coating, collecting measured electromagnetic parameters, fitting measured electromagnetic parameters and optimizing ORL of the scale coating. More details could be seen as following Firstly, using the relative plural electromagnetic parameters and thickness of a full scale magnetic absorbing coating, calculate ORL in the whole incidence angle range by utilizing Eqs. (2) and (3) or (4) and (5) at full scale frequency, and set the calculated ORL as the input of the following ORL optimized process. Secondly, select spherical carbonyl iron particles (SCIPs) as the absorbent and epoxy resin as the binder, prepare samples with different SCIPs concentrations for measuring the relative plural permittivities and permeabilities, and build the permittivity selection and the permeability selection. Thirdly, Fit the measured permittivities and permeabilities by taking advantage of two-exponent phenomenological percolation equation (TEPPE) [9–11], it could be written as following.
pi
/i1=t /1=t e /1=t i
þ
A/1=t e
þ ð1 p i Þ
1=s /1=s m /e 1=s /m þ A/1=s e
¼0
ð6Þ
where A ¼ ð1 f c Þ=f c and f c denotes the percolation threshold; pi is the SCIPs concentration; /e is the effective electromagnetic parameters of the composite, /i and /m are the intrinsic electromagnetic parameters of SCIPs and the binder, respectively; t and s are fitting parameters, which could be obtained by fitting the measured permittivities or permeabilities. Meanwhile, the intrinsic electromagnetic parameters of SCIPs could be written as following.
e¼1
r xe0
l¼1þ
2 X kMns ðkHna þ ian xÞ 3 n¼1;2; ðkHn þ ian xÞ2 x2
ð7Þ
a
E
q
H
PEC
q
qk
qk (a)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi er lr sin2 h
Z in ðTMÞ Z 0 cos h RLðTMÞ ¼ 20 lg Z in ðTMÞ þ Z 0 cos h
E
H
Z0
(b)
Fig. 1. ORL calculated diagram for a coating irradiated by plane microwave of different polarization modes: TE wave (a) and TM wave (b).
ð8Þ
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Input ORL of a full scale coating
Collect measured permittivities and permeabilities
Fit measured permittivities and permeabilities
Set an absorbent concentration in a scale coating
Set a scale coating thickness
Calculate the scale coating ORL
Contrast the scale coating ORL with the full scale coating ORL
N
Be identical? Y
Obtain the optimized scale coating formulation Fig. 2. The flow sketch of designing the scale coating for a magnetic absorbing coating.
where e0 is the permittivity of free space, r is the electric conductivity, x is the angular frequency, k is the gyromagnetic ratio, Ms is the saturation magnetization, Ha is the magnetic anisotropy field and a is damping coefficient. Finally, optimize ORL. This procedure mainly includes four parts: (1) Set the loop number, the scale material thickness range, the thickness step and the incidence angle range; (2) Define a SCIPs concentration, calculate electromagnetic parameters of the corresponding composite by Eqs. (6); (3) Define a thickness and take advantage of the electromagnetic parameters obtained in step (2), calculate ORL in the whole incidence angle range at scale frequency, furthermore, use the sum-of-squared differences to quantify the deviation between ORL of the given coating and that of the full scale coating, calculate the deviation, record the deviation as well as the corresponding thickness and SCIPs concentration; (4) Loop step (2) and (3), search the minimum deviation and take the corresponding thickness and SCIPs concentration as the formulation parameters of the optimized scale coating.
Results and discussions Now an example is given to verify the validation of the introduced method for designing the scale coating of a magnetic absorbing coating. The full scale frequency is 2 GHz, the scale ratio is 1/5 and the scale frequency would be 10 GHz. The thickness of the magnetic absorbing coating is 2 mm, and the permittivity and permeability are 14.49–0.12j and 3.56–1.12j, respectively. Here just TE wave is considered. The designing procedure is introduced as follows.
Firstly, calculate ORL of the full scale coating by Eqs. (2) and (3) as the coating is irradiated by TE wave in the range from 0° to 90°. Secondly, prepare a series of coaxial samples composed of SCIPs and epoxy resin. SCIPs concentration is different from each other. The densities of SCIPs and epoxy resin are measured to be 7.67 g/ cm3 and 2.15 g/cm3, respectively. Then SCIPs concentration could be derived from the sample density. Here SCIPs concentrations of prepared samples are 5.48%, 10.31%, 15.26%, 20.72%, 25.34%, 30.22% and 35.71%. Electromagnetic parameters could be measured by transmission/reflection method. The collected permittivities at 10 GHz are 2.98–0.1j, 3.68–0.14j, 4.42–0.12j, 5.56–0.14j, 6.51–0.16j, 8.88–0.19j and 10.42–0.24j, and the collected permeabilities at 10 GHz are 1.05–0.14j, 1.1–0.25j, 1.17–0.36j, 1.19– 0.51j, 1.21–0.65j, 1.23–0.8j and 1.28–0.98j. Thirdly, fit collected permittivities and permeabilities by Eq. (6), and then determine t and s. More details of the fitting process could be seen in Ref. [10] and [11]. Finally, set loop number as 5000, the scale coating thickness range as 0.5 mm–2.0 mm, the thickness step as 0.1 mm. At 10 GHz, loop calculating ORL of a scale coating with a different thickness and SCIPs concentration and the deviation of ORL between the scale coating and the full scale coating, meanwhile, record the deviation as well as the thickness and the SCIPs concentration. After all loops finished, search the minimum deviation and extract the corresponding thickness and SCIPs concentration, which are 1.6 mm and 12.6%, respectively. The calculated permittivity and permeability of the optimized scale coating are 4.25– 0.16j and 1.15–0.27j. Fig. 3 plots ORL of the full scale coating at 2 GHz and ORL of the optimized scale coating at 10 GHz, as shown in Fig. 3, they are almost identical in the range of 0°–90°. In order to verify the optimized scale coating, models including coating plates, coating spheres and coating spherecones are constructed to simulate monostatic RCS by the commercial software FEKO, as shown in Fig. 4. The excitation is selected as TE wave. The frequency is 2 GHz for simulating full scale models and 10 GHz for scale models, respectively. For theoretical scale models, the scale coating thickness is 1/5 of the full scale coating thickness, i.e., 0.4 mm, and the electromagnetic parameters are the same with that of the full scale coating. For designed scale models, the thickness of the scale is equal to 1.6 mm, and the permittivity and permittivity are equal to 4.25–0.16j and 1.15–0.27j. Parameters of simulated models are set as Table 2. Plate models are constructed by a square plate, the region of the plate is coating material and the bottom face of the plate is set as PEC, z-axis is perpendicular to the plate, the incidence angle ranges from 0° to 90°. Sphere models are constructed by coating metal spheres with the full scale coating or the optimized scale coating, the incidence angle ranges from 0° to 90°. Spherecone models are constructed by coating metal spherecones with the full scale coating or the optimized scale coating, the incidence angle ranges from 0° to 180°. All simulated results are plotted in Fig. 5. Fig. 5(a) and (b) show the simulated results of coating plate models and coating sphere models, respectively. As shown in Fig. 5(a) and (b), it could be seen that the monostatic RCS of the designed 1/5th scale model have a great agreement with that of the theoretical 1/5th scale model in the whole incidence angle range. This indicates that it is valid to design the scale coating of a magnetic absorbing coating by the introduced method. Fig. 5(c) plots the simulated results of coating spherecone models. As shown in Fig. 5(c), it could be seen that monostatic RCS of the designed 1/5th scale model is generally close to that of the theoretical 1/5th scale model. However, there is a little deviation between them. The deviation is caused by more complicated scatterings on spherecones, such as edge diffraction, surface travelling wave and so on. Because the introduced method based on ORL is derived from Fresnel reflection, these complicated
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Fig. 3. The ORL of the full scale coating at 2 GHz and that of the optimized scale coating at 10 GHz.
Fig. 4. Simulated models: a coating plate (a), a coating sphere (b), a coating spherecone (c).
scatterings could not be taken into account. These scatterings are more obvious when the cone part is irradiated in front than the sphere part is irradiated in front. It explains that why the deviation in the range of 0°–90° is more obvious than that in the range of 90° –180°. Fig. 6 plots the statistical result of relative monostatic RCS deviation between the designed scale spherecone model and the theoretical scale spherecone model. As shown in Fig. 6(a), most of deviations in the whole angle range are less than 10%, and the maximal deviation reaches 58% at the incidence angle of 120°. Furthermore, Fig. 6(b) gives statistical possibility of relative deviation. It could be seen that monostatic RCS deviation is less than 10% for 90% of incidence angles. This indicates that the designed scale coating could meet the requirement of scale measurement for coating spherecone models. To further verify the proposed method for optimizing the scale coating, monostatic RCS simulations are carried out by constructing the SLICY model, which includes many electromagnetic scattering mechanisms such as specular-reflection, multiple-reflection, creeping wave scattering, cavity scattering and so on, and is often used as an example to analyzing electromagnetic scattering mechanisms. Fig. 7(a) gives the full scale SLICY model as well as its key geometric parameters. The excitation is still selected as TE wave.
The incidence pitch angle ranges from 90° to 90° and the azimuth is equal to 0°. The frequency is 2 GHz for simulating the full scale SLICY model and 10 GHz for scale SLICY models, respectively. The theoretical scale SLICY model has the same electromagnetic parameters with the full scale SLICY model, and the scale coating thickness is 1/5 of the full scale coating thickness. The permittivity and permittivity of the designed scale SLICY model are equal to 4.25–0.16j and 1.15–0.27j, and the scale coating thickness is equal to 1.6 mm, as shown in Table 2. Fig. 7(b) plots the simulated results of coating SLICY models. As shown in Fig. 7(b), it could be seen that monostatic RCS of the designed scale SLICY model is generally close to that of the theoretical scale SLICY model. The deviation is mainly caused by those scattering mechanisms except specular-reflection and multiplereflection. Fig. 8 plots the statistical result of relative monostatic RCS deviation. It could be seen that monostatic RCS deviation is less than 5% for 34.8% of incidence angles, 10% for 70.2% of incidence angles and 15% for 93.9% of incidence angles. According to the calculating results plotted in Fig. 7(b), Table 3 gives the mean monostatic RCS values of the theoretical scale SLICY model and the designed scale SLICY model in the angle ranges of 90° to 60°, 60° to 30°, 30° to 0°, 0°–30°, 30°–60° and 60°–90°, and the relative deviation is 6.65%, 11.08%, 2.77%, 5.14%,
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Table 2 Parameters of simulated models. Simulated models Plates
Full scale model Theoretical 1/5th scale model Designed 1/5th scale model
Spheres
Spherecones
Geometric parameters
Coating permittivity
Coating permeability
Coating thickness
Length: 500 mm Width: 500 mm Length: 100 mm Width: 100 mm Length: 100 mm Width: 100 mm
14.49–0.12j
3.56–1.12j
2.0 mm
14.49–0.12j
3.56–1.12j
0.4 mm
4.25–0.16j
1.15–0.27j
1.6 mm
Full scale model Theoretical 1/5th scale model Designed 1/5th scale model
Diameter: 500 mm Diameter: 100 mm
14.49–0.12j 14.49–0.12j
3.56–1.12j 3.56–1.12j
2.0 mm 0.4 mm
Diameter: 100 mm
4.25–0.16j
1.15–0.27j
1.6 mm
Full scale model
Height of 500 mm Radius of sphere: 200 mm Height of 100 mm Radius of sphere: 40 mm Height of 100 mm Radius of sphere: 40 mm
14.49–0.12j
3.56–1.12j
2.0 mm
14.49–0.12j
3.56–1.12j
0.4 mm
4.25–0.16j
1.15–0.27j
1.6 mm
Theoretical 1/5th scale model
Designed 1/5th scale model
the cone: the bottom
the cone: the bottom
the cone: the bottom
Fig. 5. Simulated monostatic RCS of different models irradiated by TE wave: coating plate models (a), coating sphere models (b) and coating spherecone models (c).
Fig. 6. The relative deviation between the monostatic RCS of the designed scale spherecone model and the theoretical scale spherecone model: the angle distribution (a) and the statistical possibility (b).
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Fig. 7. The full scale SLICY model as well as its key geometric parameters (a) and simulated monostatic RCS results (b).
Fig. 8. The relative deviation between the monostatic RCS of the designed scale SLICY model and the theoretical scale SLICY model: the angle distribution (a) and the statistical possibility (b).
Table 3 Mean monostatic RCS values in different angle ranges (Unit: dBsm).
The theoretical scale SLICY model The designed scale SLICY model
90° to 60°
60° to 30°
30° to 0°
0°–30°
30°–60°
60°–90°
16.40 17.49
20.12 22.35
13.72 14.10
15.76 16.57
16.84 18.30
19.78 19.96
8.67% and 0.91% in each angle range, respectively. All the relative deviations are within the RCS measurement tolerance. On the basis of the previous results, it can be concluded that the designed scale coating could meet the requirement of the scale measurement for models with a magnetic coating. Conclusions In this paper, a method based on ORL has been proposed to solve the problem of designing the scale coating for a magnetic absorbing coating. An example is given to verify the validation. According to simulated monostatic RCS of coating plate models, coating sphere models and coating spherecone models as well as coating SLICY models, results reveal that the monostatic RCS of the designed scale model have a great agreement with that of
the theoretical scale model in the whole incidence angle range. These indicate that the designed scale coating could meet the requirement of the scale measurement for models with a magnetic coating. It worthy to point out that the scale coating is designed for a magnetic absorbing coating just at a single frequency. Further research into designing the scale coating at a wide frequency band is still on the way. Acknowledgements This work was supported by the National Natural Science Foundation of China under Grant NO. 61471242 and Grant NO. 61601299, and Shanghai Municipal Science and Technology Commission under Grant 15ZR1439600.
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