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An optimizing method for design of microwave absorbing materials
Materials & Design Materials and Design 27 (2006) 700–705 www.elsevier.com/locate/matdes
Short communication
An optimizing method for design of microwave absorbing materials Xiaoling Yu b
a,*
, Gang Lin a, Duanming Zhang a, Huahui He
b
a Department of Physics, Huazhong University of Science and Technology, Wuhan 430074, PR China Department of Electronics and Technology, Huazhong University of Science and Technology, Wuhan 430074, PR China
Received 2 September 2004; accepted 20 December 2004 Available online 7 April 2005
Abstract An optimizing procedure of microwave absorber is proposed based on the absorbing mechanism for one-layer microwave absorber in this paper. Amplitude–frequency characteristic of the material is obtained under optimal fitting electromagnetic parameters together by using the optimizing method. The influence of dispersion of electromagnetic parameters and the thickness on the performance of the absorbing material is analyzed. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Absorbing materials; Optimizing method; Permeability; Permittivity
1. Introduction Microwave absorbing materials, which possess special performance for absorbing electromagnetic wave, have been applied in both civil and military various fields and play an important role in camouflage technology. Wave-absorbing materials required to have large electric and magnetic loss in the frequency range of interest and be formed into thin plates. Monolayer absorbing material with metal underlay is an elementary functional material which has been followed with interest for a long time[1–3]. The usual methods for designing wave-absorbing materials are to find zero reflection condition[3–5,7,8] – perfect matching condition based on equation (12). For such single-layer structure, zero reflection occurs when the surface input impedance of the material is equal to that of free space. But this is an idea case which can not be achieved actually. So almost no materials with thin layer possess ideal waveabsorbing performance in width frequency band. The key to the question is that there is no finalized design *
Corresponding author. Tel.: +86 27 87792612. E-mail address: [email protected] (X. Yu).
0261-3069/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.matdes.2004.12.022
law for the matching of the parameters in the material design yet. The design laws proposed in literature [3–7] is just used as conditions of constraint in practical optimization. This paper gives an optimizing method of frequency tracking for electromagnetic parameters (e 0 , e00 , l 0 , l00 ) of absorbing materials. The method realizes finding the solution of the complex functions of electromagnetic parameters of microwave absorbing material. Using this method, each dispersion function of the parameters can be obtained by considering the matching relation among the parameters simultaneously. The method possessing simple data structure can be realized easily and is effective for guiding design absorbing material.
2. Theoretical background Considering a plane wave normally incident on the surface of a monolayer magnetic wave-absorbing material coated on perfect conductor shown in Fig. 1. The direction of the wave transmitting is along Z axis, so the wave can be expressed as follows: Ex ¼ E0 ðec0 z þ Rec0 z Þ
z < 0;
ð1Þ
X. Yu et al. / Materials and Design 27 (2006) 700–705
701
The reflectivity in db unit is defined as R ¼ 20 log jCj:
So, the reflectivity R can express the absorbing performance of the material and is a function of the complex permittivity and permeability of the material, and the frequency of the wave. The objective of the optimization is to obtain the minimum of reflectivity of the material in the band as width as possible.
Fig. 1. Designations of the monolayer and the interface. c0 z H y ¼ g1 Rec0 z Þ 0 E 0 ðe
z < 0;
ð2Þ
where
3. Optimizing method
pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi c0 ¼ jw l0 e0 g0 ¼ l0 =e0 : The wave in the absorbing material is given by Ex ¼ E0 ðAecz þ Becz Þ 0 < z < d;
ð3Þ
H y ¼ g1 E0 ðAecz Becz Þ
ð4Þ
0 < z < d;
where g is the impedance of the material and c is transmit coefficient of the wave in the material, and that rffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l l0 jl00 l0 1 jtgdm g¼ ; ð5Þ ¼ ¼ 0 00 e e je e0 1 jtgde where e 0 , e00 , l 0 , l00 are the relative value of the complex permittivity and permeability of the material, tgde ¼ e00 =e0
tgdm ¼ l00 =l0
ð6Þ
and c ¼ a þ jb
ð7Þ
pffiffiffiffiffi c ¼ jx le=c
ð8Þ
from (7) and (8), a, b can be obtained and be expressed as rffiffiffi x pffiffiffiffiffiffiffi 1 0 0 le a¼ c 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tgde tgdm 1 þ 1 þ tg2 de þ tg2 dm þ tg2 de tg2 dm ; ð9Þ rffiffiffi x pffiffiffiffiffiffiffi 1 0 0 le b¼ c 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 tgde tgdm þ 1 þ tg2 de þ tg2 dm þ tg2 de tg2 dm : ð10Þ According to the electromagnetics theory, the input impedance between the air and the material is Z i ¼ g tanhðc dÞ;
ð11Þ
where d is the thickness of the material. Thus the reflection coefficient C with respect to a normally incident plane wave is C¼
Zi 1 : Zi þ 1
ð13Þ
ð12Þ
The complex permittivity and permeability of the absorbing material are the basic parameters which reflect the interaction between the EM wave and the material. Considering the relative complex permittivity and permeability can be expressed as eðf Þ ¼ e0 ðf Þ je00 ðf Þ;
ð14Þ
lðf Þ ¼ l0 ðf Þ jl00 ðf Þ;
ð15Þ
where f represent the frequency of the electromagnetic wave. The performance and working bandwidth of the material are affected by the functions e(f), l(f) and matching relation among the e(f), l(f) and d the thickness of the material. The optimization problem of the material is just to obtain the extremum of the reflectivity R underdefinite conditions according to the absorbing mechanism as has been expressed. The reflectivity R of the material is determined by a function set: {e 0 (f), e00 (f), l 0 (f), l00 (f), d}. The matching relation among the functions is not independent. So, the optimization procedure is rather complex. To solve the problem, an optimizing procedure of frequency tracking for electromagnetic parameters of the material is performed. The specific arithmetic can be expressed as follows: first dividing the range of working frequency of the material into some discrete character frequency points, then for each frequency ransack the hyperspace of the given value ranges of the electromagnetic parameters for the value of the electromagnetic parameters which made the reflectivity minimum at the given thickness of the material. The optimal performance of the material at the definite frequency can be assured by the optimal electromagnetic parameters searched. Optimal parameters corresponding for every frequency can be found and these optimal electromagnetic parameters form a discrete distribution according to the frequency. Thus, the dispersion function of electromagnetic parameters and the matching relation can be obtained automatically. This is important for guiding design of the absorbing material. This method predigests the optimization process, decrease calculation quantity and find numerical solution
702
X. Yu et al. / Materials and Design 27 (2006) 700–705
of optimal function of the electromagnetic parameters of the material. The optimizing method of frequency tracking and its overall numerical results can satisfy practice design require and is effective for guiding the design and preparation techniques of the absorbing material.
the range imaginary part of the complex permeability: 1–3; the range of real part of the complex permeability: 1– 80; the range imaginary part of the complex permeability: 0–15; the range of the frequency: 2–18 GHz.
4. Numerical examples In this section, the optimization method is performed for single-layer microwave absorbing materials with twin-complex electromagnetic parameter and backed by perfect conductor. The value ranges of the parameters in the optimizing producer for absorbing material design are restricted as follows: the range of real part of the complex permeability: 1– 3.5;
The numerical results for theory optimization are shown in Figs. 2–5. The profile of reflectivity of the material vs. frequency is given in Fig. 2, which is for monolayer absorbing material with thickness being 1, 2, 3 mm, respectively. In Fig. 2, optimum matching is achieved among the thickness d, and the frequency and the electromagnetic parameters (e 0 , e00 , l 0 , l00 ) of the material. Even the optimum matching is realized, the performance of the absorbing material at low frequency of the material is
Fig. 2. The profile of reflectivity vs. frequency for d = 1 mm, d = 2 mm, d = 3 mm.
X. Yu et al. / Materials and Design 27 (2006) 700–705
703
still not very well when the thickness is too thin as shown in the Fig. 2(a), the reflectivity R at 2 GHz is just 3.5 dB for d = 1 mm. It shows that the attenuation of thin absorbing material for electromagnetic wave is limited. But the reflectivity R at 2 GHz fall to 37.5 dB rapidly when d = 3 mm. So, the absorbing performance at low frequency can be modified obviously by adding thickness properly as the application condition is permitted. Fig. 3 is the characteristics of the absorbing material with d = 1 mm and reflectivity as shown in Fig. 2(a). Namely the material with thickness d = 1 mm will has optimum absorbing result when the dispersion of the electromagnetic parameters (e 0 , e00 , l 0 , l00 ) satisfy the require as shown in Fig. 3. Now the reflectivity is below to 20 dB and bandwidth can reach 13 GHz (see in Fig. 2(a)). At the same dispersion of the electromagnetic parameters (e 0 , e00 , l 0 , l00 ) mentioned above, the absorbing performance shall be bad when the thickness d > 1 mm or d < 1 mm. These can be seen from Fig. 6. Fig. 6 is reflectivity contour for the material with different thickness and with the electromagnetic parameters shown in
Fig. 4. Modified eletromagnetic parameters matching curve for material thickness d = 1.0 mm.
Fig. 5. Amplitude vs. frequency characteristic of reflectivity for modified eletromagnetic parameters of absorbing material with d = 1.0 mm.
Fig. 3. Electromagnetic parameters matching curve for absorbing material with d = 1.0 mm.
Fig. 3. This indicates that optimum function for electromagnetic parameters and matching relation are different for the material with different thickness.
704
X. Yu et al. / Materials and Design 27 (2006) 700–705 contour dB
3
-12.1 -1 5.7
2.5
d (mm)
2
1.5
-12.1
-4.87 -15.7
-8.48 -19.3 -20.2 6 .6 -3 -2 3
1
-19.3
-19.3
-2 30.2 -2 6 .6 -3 -19.3 -26.6
-8.48
-23
-4.87
-19.3
0.5 2
4
6
8
10
12
14
16
x 10
9
f (Hz) Fig. 6. Contours of reflectivity for absorbing material with different thickness.
Although the relation among the parameters shown in Fig. 3 is too complex to realize in practical application, the reflectivity obtained by the optimization method is much lower than that required in practical application. When f > 7 GHz, R 6 40 dB, this condition equal to no reflection. In practical application, ‘‘R 6 20 dB’’ is enough, sometimes ‘‘R 6 10 dB’’ has reached the require in the main. Thus, the design can be performed according to the reflectivity given in practical application and designed by the method. The modifying result for R 6 10 dB target is shown in Figs. 4 and 5. It is not like that the frequency characteristic of the electromagnetic parameters (e 0 , e00 , l 0 , l00 ) is in inverse proportion to the frequency as expressed in literature [9], but present a complex matching relation (see Fig. 3 or Fig. 4). The reflectivity R will debase further as imaginary relative permeability (l00 ) raise at high frequency. In the optimization procedure, it has been found that the performance of the material is quite sensitive for the change of the permeability, especially the imaginary at low frequency, but for the change of the permittivity at high frequency. If the range of permeability is 1–15 (real and imaginary), the performance at low frequency can be improved greatly maintaining the good performance at high frequency simultaneously. In optimizing hyperspace, the reflectivity of the material is a many dimensions curved surface which has many ‘‘valley points’’ of the reflectivity. Some ‘‘valley points’’ are not optimum point of the reflectivity but satisfy the needs for application (such as Fig. 5), then these
points can be applied in practice. Thus various selections can be offer for design of the materialÕs permeability and permittivity and more flexible selection chance of material can be obtained.
5. Conclusion From the above analysis, it can be known that the performance of absorbing materials is determined by the complex permittivity, complex permeability, frequency and the thickness of the material commonly. The optimum function of the complex permittivity and complex permeability are different for the material with different thickness. The reflectivity performance at low frequency can be improved obviously by the imaginary of the complex permittivity increasing. And the imaginary of the complex permittivity will influence the performance at high frequency. That is to say that the frequency band of monolayer absorbing material can be expanded by increasing the imaginary of the complex permeability at low frequency range and the imaginary of the complex permittivity at high frequency range. The performance of the material can also be changed by the thickness varying of the material for different requirement. So considering the matching relation among the complex permittivity, complex permeability, frequency and the thickness of the material simultaneously is the fundamental approach for optimizing the absorbing material.
X. Yu et al. / Materials and Design 27 (2006) 700–705
The frequency tracking optimization method proposed in this paper just takes the minimum of reflectivity of the absorbing material as an objective and do not impose any specific restrictions on other parameters. Old design methods often performed for one frequency on the premise of complex permittivity and complex permeability being constant values [6–8]. Compared with the old methods, the optimization procedure possesses more extensive directive function for preparation of the radar absorbing materials.
References [1] Johnos MH. Use of absorbing materials. In: Ridenour LN, editor. Radar system engineering. New York: McGraw-Hill; 1947.
705
[2] Salisbury WW. Absorbing body for electromagnetic waves. US Patent 2 599 944, June 10, 1952. [3] Fernandez A, Valenzuela A. General solution for single-layer electromagnetic wave absorber. Electron Lett 1985;21(1):20–1. [4] Leendert J du Toit. The design of Jauman absorbers. IEEE Antennas Propag Mag 1994;36(6):17–25. [5] Zhao H, Qin B, Hu D, Qin R. The design rules for single-layer radar absorbings. J Harbin Univ Technol 1993;25(2):104–10 [Chinese]. [6] Mahmoud SF, Habib MK. Design of a two layer microwave absorber. J Electromagnet Wave 1998;12(8):1005–14. [7] Valenzuela AQ, Fernandez Anibal F. General design theory for single-layer homogeneous absorber. IEEE Trans Anttennas Propag 1996;44(7):822–6. [8] Musal Jr HM, Smith DC. Universal design chart for specular absorbers. IEEE Trans Magnetics 1990;26(5):1462–4. [9] Wang X, Lu H, Qian J. Study on frequency characteristic of electromagnetic parameters of radar absorbing material. Camouflage Technol 1998(1):23–8 [Chinese].
Short communication
An optimizing method for design of microwave absorbing materials Xiaoling Yu b
a,*
, Gang Lin a, Duanming Zhang a, Huahui He
b
a Department of Physics, Huazhong University of Science and Technology, Wuhan 430074, PR China Department of Electronics and Technology, Huazhong University of Science and Technology, Wuhan 430074, PR China
Received 2 September 2004; accepted 20 December 2004 Available online 7 April 2005
Abstract An optimizing procedure of microwave absorber is proposed based on the absorbing mechanism for one-layer microwave absorber in this paper. Amplitude–frequency characteristic of the material is obtained under optimal fitting electromagnetic parameters together by using the optimizing method. The influence of dispersion of electromagnetic parameters and the thickness on the performance of the absorbing material is analyzed. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Absorbing materials; Optimizing method; Permeability; Permittivity
1. Introduction Microwave absorbing materials, which possess special performance for absorbing electromagnetic wave, have been applied in both civil and military various fields and play an important role in camouflage technology. Wave-absorbing materials required to have large electric and magnetic loss in the frequency range of interest and be formed into thin plates. Monolayer absorbing material with metal underlay is an elementary functional material which has been followed with interest for a long time[1–3]. The usual methods for designing wave-absorbing materials are to find zero reflection condition[3–5,7,8] – perfect matching condition based on equation (12). For such single-layer structure, zero reflection occurs when the surface input impedance of the material is equal to that of free space. But this is an idea case which can not be achieved actually. So almost no materials with thin layer possess ideal waveabsorbing performance in width frequency band. The key to the question is that there is no finalized design *
Corresponding author. Tel.: +86 27 87792612. E-mail address: [email protected] (X. Yu).
0261-3069/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.matdes.2004.12.022
law for the matching of the parameters in the material design yet. The design laws proposed in literature [3–7] is just used as conditions of constraint in practical optimization. This paper gives an optimizing method of frequency tracking for electromagnetic parameters (e 0 , e00 , l 0 , l00 ) of absorbing materials. The method realizes finding the solution of the complex functions of electromagnetic parameters of microwave absorbing material. Using this method, each dispersion function of the parameters can be obtained by considering the matching relation among the parameters simultaneously. The method possessing simple data structure can be realized easily and is effective for guiding design absorbing material.
2. Theoretical background Considering a plane wave normally incident on the surface of a monolayer magnetic wave-absorbing material coated on perfect conductor shown in Fig. 1. The direction of the wave transmitting is along Z axis, so the wave can be expressed as follows: Ex ¼ E0 ðec0 z þ Rec0 z Þ
z < 0;
ð1Þ
X. Yu et al. / Materials and Design 27 (2006) 700–705
701
The reflectivity in db unit is defined as R ¼ 20 log jCj:
So, the reflectivity R can express the absorbing performance of the material and is a function of the complex permittivity and permeability of the material, and the frequency of the wave. The objective of the optimization is to obtain the minimum of reflectivity of the material in the band as width as possible.
Fig. 1. Designations of the monolayer and the interface. c0 z H y ¼ g1 Rec0 z Þ 0 E 0 ðe
z < 0;
ð2Þ
where
3. Optimizing method
pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi c0 ¼ jw l0 e0 g0 ¼ l0 =e0 : The wave in the absorbing material is given by Ex ¼ E0 ðAecz þ Becz Þ 0 < z < d;
ð3Þ
H y ¼ g1 E0 ðAecz Becz Þ
ð4Þ
0 < z < d;
where g is the impedance of the material and c is transmit coefficient of the wave in the material, and that rffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l l0 jl00 l0 1 jtgdm g¼ ; ð5Þ ¼ ¼ 0 00 e e je e0 1 jtgde where e 0 , e00 , l 0 , l00 are the relative value of the complex permittivity and permeability of the material, tgde ¼ e00 =e0
tgdm ¼ l00 =l0
ð6Þ
and c ¼ a þ jb
ð7Þ
pffiffiffiffiffi c ¼ jx le=c
ð8Þ
from (7) and (8), a, b can be obtained and be expressed as rffiffiffi x pffiffiffiffiffiffiffi 1 0 0 le a¼ c 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tgde tgdm 1 þ 1 þ tg2 de þ tg2 dm þ tg2 de tg2 dm ; ð9Þ rffiffiffi x pffiffiffiffiffiffiffi 1 0 0 le b¼ c 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 tgde tgdm þ 1 þ tg2 de þ tg2 dm þ tg2 de tg2 dm : ð10Þ According to the electromagnetics theory, the input impedance between the air and the material is Z i ¼ g tanhðc dÞ;
ð11Þ
where d is the thickness of the material. Thus the reflection coefficient C with respect to a normally incident plane wave is C¼
Zi 1 : Zi þ 1
ð13Þ
ð12Þ
The complex permittivity and permeability of the absorbing material are the basic parameters which reflect the interaction between the EM wave and the material. Considering the relative complex permittivity and permeability can be expressed as eðf Þ ¼ e0 ðf Þ je00 ðf Þ;
ð14Þ
lðf Þ ¼ l0 ðf Þ jl00 ðf Þ;
ð15Þ
where f represent the frequency of the electromagnetic wave. The performance and working bandwidth of the material are affected by the functions e(f), l(f) and matching relation among the e(f), l(f) and d the thickness of the material. The optimization problem of the material is just to obtain the extremum of the reflectivity R underdefinite conditions according to the absorbing mechanism as has been expressed. The reflectivity R of the material is determined by a function set: {e 0 (f), e00 (f), l 0 (f), l00 (f), d}. The matching relation among the functions is not independent. So, the optimization procedure is rather complex. To solve the problem, an optimizing procedure of frequency tracking for electromagnetic parameters of the material is performed. The specific arithmetic can be expressed as follows: first dividing the range of working frequency of the material into some discrete character frequency points, then for each frequency ransack the hyperspace of the given value ranges of the electromagnetic parameters for the value of the electromagnetic parameters which made the reflectivity minimum at the given thickness of the material. The optimal performance of the material at the definite frequency can be assured by the optimal electromagnetic parameters searched. Optimal parameters corresponding for every frequency can be found and these optimal electromagnetic parameters form a discrete distribution according to the frequency. Thus, the dispersion function of electromagnetic parameters and the matching relation can be obtained automatically. This is important for guiding design of the absorbing material. This method predigests the optimization process, decrease calculation quantity and find numerical solution
702
X. Yu et al. / Materials and Design 27 (2006) 700–705
of optimal function of the electromagnetic parameters of the material. The optimizing method of frequency tracking and its overall numerical results can satisfy practice design require and is effective for guiding the design and preparation techniques of the absorbing material.
the range imaginary part of the complex permeability: 1–3; the range of real part of the complex permeability: 1– 80; the range imaginary part of the complex permeability: 0–15; the range of the frequency: 2–18 GHz.
4. Numerical examples In this section, the optimization method is performed for single-layer microwave absorbing materials with twin-complex electromagnetic parameter and backed by perfect conductor. The value ranges of the parameters in the optimizing producer for absorbing material design are restricted as follows: the range of real part of the complex permeability: 1– 3.5;
The numerical results for theory optimization are shown in Figs. 2–5. The profile of reflectivity of the material vs. frequency is given in Fig. 2, which is for monolayer absorbing material with thickness being 1, 2, 3 mm, respectively. In Fig. 2, optimum matching is achieved among the thickness d, and the frequency and the electromagnetic parameters (e 0 , e00 , l 0 , l00 ) of the material. Even the optimum matching is realized, the performance of the absorbing material at low frequency of the material is
Fig. 2. The profile of reflectivity vs. frequency for d = 1 mm, d = 2 mm, d = 3 mm.
X. Yu et al. / Materials and Design 27 (2006) 700–705
703
still not very well when the thickness is too thin as shown in the Fig. 2(a), the reflectivity R at 2 GHz is just 3.5 dB for d = 1 mm. It shows that the attenuation of thin absorbing material for electromagnetic wave is limited. But the reflectivity R at 2 GHz fall to 37.5 dB rapidly when d = 3 mm. So, the absorbing performance at low frequency can be modified obviously by adding thickness properly as the application condition is permitted. Fig. 3 is the characteristics of the absorbing material with d = 1 mm and reflectivity as shown in Fig. 2(a). Namely the material with thickness d = 1 mm will has optimum absorbing result when the dispersion of the electromagnetic parameters (e 0 , e00 , l 0 , l00 ) satisfy the require as shown in Fig. 3. Now the reflectivity is below to 20 dB and bandwidth can reach 13 GHz (see in Fig. 2(a)). At the same dispersion of the electromagnetic parameters (e 0 , e00 , l 0 , l00 ) mentioned above, the absorbing performance shall be bad when the thickness d > 1 mm or d < 1 mm. These can be seen from Fig. 6. Fig. 6 is reflectivity contour for the material with different thickness and with the electromagnetic parameters shown in
Fig. 4. Modified eletromagnetic parameters matching curve for material thickness d = 1.0 mm.
Fig. 5. Amplitude vs. frequency characteristic of reflectivity for modified eletromagnetic parameters of absorbing material with d = 1.0 mm.
Fig. 3. Electromagnetic parameters matching curve for absorbing material with d = 1.0 mm.
Fig. 3. This indicates that optimum function for electromagnetic parameters and matching relation are different for the material with different thickness.
704
X. Yu et al. / Materials and Design 27 (2006) 700–705 contour dB
3
-12.1 -1 5.7
2.5
d (mm)
2
1.5
-12.1
-4.87 -15.7
-8.48 -19.3 -20.2 6 .6 -3 -2 3
1
-19.3
-19.3
-2 30.2 -2 6 .6 -3 -19.3 -26.6
-8.48
-23
-4.87
-19.3
0.5 2
4
6
8
10
12
14
16
x 10
9
f (Hz) Fig. 6. Contours of reflectivity for absorbing material with different thickness.
Although the relation among the parameters shown in Fig. 3 is too complex to realize in practical application, the reflectivity obtained by the optimization method is much lower than that required in practical application. When f > 7 GHz, R 6 40 dB, this condition equal to no reflection. In practical application, ‘‘R 6 20 dB’’ is enough, sometimes ‘‘R 6 10 dB’’ has reached the require in the main. Thus, the design can be performed according to the reflectivity given in practical application and designed by the method. The modifying result for R 6 10 dB target is shown in Figs. 4 and 5. It is not like that the frequency characteristic of the electromagnetic parameters (e 0 , e00 , l 0 , l00 ) is in inverse proportion to the frequency as expressed in literature [9], but present a complex matching relation (see Fig. 3 or Fig. 4). The reflectivity R will debase further as imaginary relative permeability (l00 ) raise at high frequency. In the optimization procedure, it has been found that the performance of the material is quite sensitive for the change of the permeability, especially the imaginary at low frequency, but for the change of the permittivity at high frequency. If the range of permeability is 1–15 (real and imaginary), the performance at low frequency can be improved greatly maintaining the good performance at high frequency simultaneously. In optimizing hyperspace, the reflectivity of the material is a many dimensions curved surface which has many ‘‘valley points’’ of the reflectivity. Some ‘‘valley points’’ are not optimum point of the reflectivity but satisfy the needs for application (such as Fig. 5), then these
points can be applied in practice. Thus various selections can be offer for design of the materialÕs permeability and permittivity and more flexible selection chance of material can be obtained.
5. Conclusion From the above analysis, it can be known that the performance of absorbing materials is determined by the complex permittivity, complex permeability, frequency and the thickness of the material commonly. The optimum function of the complex permittivity and complex permeability are different for the material with different thickness. The reflectivity performance at low frequency can be improved obviously by the imaginary of the complex permittivity increasing. And the imaginary of the complex permittivity will influence the performance at high frequency. That is to say that the frequency band of monolayer absorbing material can be expanded by increasing the imaginary of the complex permeability at low frequency range and the imaginary of the complex permittivity at high frequency range. The performance of the material can also be changed by the thickness varying of the material for different requirement. So considering the matching relation among the complex permittivity, complex permeability, frequency and the thickness of the material simultaneously is the fundamental approach for optimizing the absorbing material.
X. Yu et al. / Materials and Design 27 (2006) 700–705
The frequency tracking optimization method proposed in this paper just takes the minimum of reflectivity of the absorbing material as an objective and do not impose any specific restrictions on other parameters. Old design methods often performed for one frequency on the premise of complex permittivity and complex permeability being constant values [6–8]. Compared with the old methods, the optimization procedure possesses more extensive directive function for preparation of the radar absorbing materials.
References [1] Johnos MH. Use of absorbing materials. In: Ridenour LN, editor. Radar system engineering. New York: McGraw-Hill; 1947.
705
[2] Salisbury WW. Absorbing body for electromagnetic waves. US Patent 2 599 944, June 10, 1952. [3] Fernandez A, Valenzuela A. General solution for single-layer electromagnetic wave absorber. Electron Lett 1985;21(1):20–1. [4] Leendert J du Toit. The design of Jauman absorbers. IEEE Antennas Propag Mag 1994;36(6):17–25. [5] Zhao H, Qin B, Hu D, Qin R. The design rules for single-layer radar absorbings. J Harbin Univ Technol 1993;25(2):104–10 [Chinese]. [6] Mahmoud SF, Habib MK. Design of a two layer microwave absorber. J Electromagnet Wave 1998;12(8):1005–14. [7] Valenzuela AQ, Fernandez Anibal F. General design theory for single-layer homogeneous absorber. IEEE Trans Anttennas Propag 1996;44(7):822–6. [8] Musal Jr HM, Smith DC. Universal design chart for specular absorbers. IEEE Trans Magnetics 1990;26(5):1462–4. [9] Wang X, Lu H, Qian J. Study on frequency characteristic of electromagnetic parameters of radar absorbing material. Camouflage Technol 1998(1):23–8 [Chinese].