Matching design and mismatching analysis towards radar absorbing coatings based on conducting plate

Materials and Design 24 (2003) 391–396

Matching design and mismatching analysis towards radar absorbing coatings based on conducting plate Maosheng Caoa,c,*,1, Ruru Qinb, Chengjun Qiuc, Jing Zhua a

School of Materials Science and Engineering, Tsinghua University, Beijing 100084, PR China b Department of Applied Physics, Harbin Institute of Technology, Harbin 150001, PR China c Department of Materials Science and Engineering, Harbin Engineering University, Harbin 150001, PR China Received 1 July 2002; accepted 31 October 2002

Abstract Through a detailed investigation on the interaction of electromagnetic waves with the radar absorbing coating (RAC), we propose two design rules for single layer RAC. According to the rules, we may find the suitable electromagnetic parameters so as to satisfy the practical requirements of applied background of the coating. On the basis of the rules, a significant deduction has been gained, which represents the selecting law of the electromagnetic parameters of the coating when no reflecting wave has occurred at the interface between coating and free space to the incident wave. Combining our previous researches about quasi-standing wave and RAM design results, the applications of the rules have been presented and the interaction characteristic of a series of related parameters has been studied. It shows that the design rules have important guiding effects both in the material research and the single layer RAC design. Especially, the effect of various mismatching conditions for RAC performance has been given. 䊚 2002 Elsevier Science Ltd. All rights reserved. Keywords: Radar absorbing coatings (RAC); Electromagnetic parameters; Matching conditions

1. Introduction As previously known, it is important to suppress microwave reflection from metal structure targets in some special fields w1,2x. For example, microwave in many frequency bands is used for broadcasting, communications and radar. But how to design an ‘invisible’ target and how to design a better radar absorber is a developing work. In fact, the main problem for the design of a magnetic absorber is related to the choice of the materials w3x. While electromagnetic property depends on magnetic or dielectric parameters such as permeability, permittivity and loss constants w4–6x. Therefore, the search for design curves and equation for radar absorbing materials has been the subject of theoretical efforts for a long time. However, no general design rules and formula for the optimum electromag*Corresponding author. E-mail address: [email protected] (M. Cao). 1 The National Fund supported this work. The China Fund for Post Doctor also supported this work.

netic parameters and matching thickness of the absorbing materials seem to be available w6x. Actually, the most basic absorbing structure, a singlelayer homogeneous absorber (SLHA) backed by a perfect conducting plate has been noticed in literature since 1938 w7x. Although many approaches have been developed thereafter, the theories and rules related to generalized matching designs for bandwidth are not yet available. This paper presents new design rules for single layer and homogeneous absorber (SLHA). According to the rules, the matching parameters for high performance RAC can be determined. Furthermore, the corresponding coating parameter relationships have not yet been deduced. The work in this paper is taken as rules of generalized matching design. 2. Interactions between electromagnetic wave and the coatings Assume that the permeability and permittivity of macro electromagnetic parameters are ´s´9qi´0 and msm9qim9, respectively. We may take ´ and m as the

0261-3069/03/$ - see front matter 䊚 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 1 - 3 0 6 9 Ž 0 2 . 0 0 1 1 9 - X

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netic loss angle. If b is given, li of the coating inner can be written as follows: lis2pyb

(6)

In Eqs. (2)–(5), the constant term are as follows: Cs Fig. 1. Configuration of single-layer RAC based on conducting plate.

basic parameters corresponding to their interaction between the electromagnetic field and materials. The corresponding absorbing structure is shown in Fig. 1. For normal incident wave, the planar polarized wave can be written as follows: ExsE0eyg0z,

Hys(E0 yZ0)=eyg0z (z-0).

(1)

Here, g0siv(´0m0)1y2 yc stands for propagation constant of the wave in free space; while Z0s(m0 y ´0)1y2 stands for character impedance of free space. According to electromagnetic field theory, in the range of z-0 zone, the wave field is as follows:

(2)

In Eq. (2), R is the reflection index of coating and the second term is the reflection index of the interface between free space and the coating. While, the inner field of the coating is as follows: ExsE0(CeygzqDegz) HysE0 yZ0(CeygzyDegz) (0-z-d).

(3)

In Eq. (2), g0siv(´0 m0 )1y2 yc stands for propagation constant of the wave in coating. Z stands for the character impedance of the coating. Here, we can use gsaqib to represent electromagnetic propagation constant, and a and b are the attenuation factor and phase factor of the coating, respectively. The analytical expressions are as follows:

y 12 wtgd tgd y1qy(1qtg d )(1qtg d )x e

2

m

e

2

m

(4) v bs ym9´9 c

y 12 w1ytgd tgd qy(1qtg d )(1qtg d )x e

m

(7)

DsyCey2gd

(8)

R0s(Z1yZ0)y(Z1qZ0), Z1sZtanhgd

(9)

Generally speaking, the unit of reflection index is dB, and R is written as: RdBs20 lg±R±.

(10)

3. Two design rules for single layer RAC 3.1. The first principle

ExsE0(eyg0zqReg0z) HysE0 yZ0=(eyg0zyReg0z) (z-0).

v as ym9´9 c

2Z 1 Z0yZ , R0s (Z0qZ) 1qR0exp(y2gd) Z0qZ

2

e

2

m

(5) Where, tgdes´0, tgdmsm0y m9 they, respectively, stand for the tangent of dielectric loss angle and mag-

For given application limit, the corresponding electromagnetic parameters, frequency and coating thickness should satisfy following conditions: d5li y4spy2b, tgdestgdmstgd.

(11)

Here, the first principle is based on propagation characteristics of the wave in lossy medium. As previously known, the components of electricity and magnetism are orthogonal to each other and their phases are the same. While, for a lossy medium, the phase mentioned above is different. In other words, there are differences between electricity and magnetic components in wave phase. Therefore, Eq. (11) is a limit condition to realize the orthogonality of components of electricity and magnetism and their phases are the same for lossy medium. Furthermore, in order to obtain the best wave attenuation in inner coating, the quasi-standing wave must be formed in the coating w8x. This is the main physical picture for Eq. (11). Considering Eq. (11), the first principle can be written as:

y´9m9s

pc . 2dv

(12)

And therefore, Eq. (4) and Eq. (5) can be written as: as(py2d)=tgd

(13)

M. Cao et al. / Materials and Design 24 (2003) 391–396

v p bs y´9m9s . c 2d

(14)

For short, it is important that the quasi-standing wave in the coating is a necessary condition in order to realize the matching. This is a key in the first principle. 3.2. The second principle Certain applications demand the coating response A (dB) attenuation, then the corresponding electromagnetic parameters; frequency and coating thickness, should satisfy the following condition:

y

Bp E 1y10yAy20 m9 stanhC tgdF . yAy20 D2 G 1q10 ´9

(15)

For practical coating, we must improve the limit condition in order to design RAC effectively. The key point with the second principle is that the matching relationship for the electromagnetic parameter is presented for certain application limits.

m9s

Bp E pc tanhC tgdF D2 G 2dv

m0sm9=tgd.

393

(22)

(23)

Actually, when A™` in Eqs. (16)–(19), the infer can be formed. This is a matching condition of electromagnetic parameters. On the contrary, we can study the various laws of electromagnetic parameters and electromagnetic field characteristics in coating by means of matching conditions. The detail thoughts are as follows. Using Eq. (20), Eq. (22) and ZsZ06(m y ´), we can gain character impedance inner coating, that is: ZsZ0=tanh(py2=tgd). Using Eq. (7), Eq. (8), Eq. (13) and Eq. (14), we also can obtain the constant C and D, which are as follows: Cs1y(1qeyptgd)

(24)

4. Matching conditions for certain limits

Dseyptgd y(1qeyptgd).

(25)

Based on above two principles, the electromagnetic parameters corresponding to A (dB) attenuation are as follows:

From the above equations, we can see that Z, C and D are all real numbers. According to Eq. (3), we can easily obtain the electromagnetic field inner coating which is given as follows:

´9s

Bp E 1y10yAy20 pc ctanhC tgdF yAy20 D2 G 1q10 2dv

´0s´9=tgd

(16)

B

1 eyptgd gzE eygzq e F yptgd D 1qe 1qeyptgd G

ExsE0C

(17) Hys

m9s

Bp E 1q10yAy20 pc tanhC tgdF yAy20 D2 G 1y10 2dv

m0sm9=tgd.

(18)

(19)

According to the above two principles, we can deduce one substantial infer. For the coating of certain thickness d, the adequate corresponding conditions which have no reflection towards normal incident waves are as follows:

´9s

Bp E pc ctanhC tgdF D2 G 2dv

´0s´9=tgd

(26)

(20)

(21)

Bp E E0 ctanhC tgdF D2 G Z0 B

1 eyptgd gzE eygzy e F. yptgd D 1qe 1qeyptgd G

=C

(27)

Here, gs(p y2d)=tgdqi=(p y2d). From Eq. (26) and Eq. (27), we can see that the amplitudes of electricity and magnetic field for inner coating change with the position of the z-axis, the detail tendency is shown in Fig. 2. For electricity field, the amplitude at the interface between the coating and conducting plate is always zero, while the amplitude at the interface between the coating and free space is always the max value (E0). For magnetic field, the amplitude at the interface between the coating and free space is always E0 yZ0, while the amplitude at the interface between the coating and conducting plate changes with the tangent value of electromagnetic loss angle (tgd). Moreover, the amplitude of the electromag-

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Fig. 2. The amplitude of ExyEo vs. zyd in the coating. Fig. 3. The amplitude of Hyy(EoyZ0) vs. zyd in the coating.

netic inner coating always changes with tgd and the field responses quasi-standing wave. Considering the impeding characteristic inner coating, the impedance changes gradually with the position of Z, which is shown in Fig. 3. Moreover, the impedance at the interface between the coating and free space is Z0, which matches with free space impedance. The amplitude at the interface between the coating and conducting plate is always zero. Considering the phases of electromagnetic field, the phase of electricity field does not equal that of the magnetic field shown in Fig. 4. From Fig. 4, we can see that the phase of magnetic field is delayed with the phase angle compared with that of the electricity field. The differences between electricity and magnetic phase gradually increase with the position of z. It can also be seen that the phase difference may arrive at a maximum value at the interface between the coating and conducting plate (approx. 908).

5.1. Analysis of electromagnetic parameter dispersion under the condition of tgdestgdmstgd In this case, we may define two dispersion parameters, they can be written as follows:

´9sk1

Bp E pc ctanhC tgdF D2 G 2dv

´0s´9=tgd

m9sk2

Bp E pc tanhC tgdF D2 G 2dv

(28)

(29)

(30)

5. Mismatching analysis In the above discussions, the electromagnetic parameters of the coating determined by means of Eqs. (20)–(23) are always ideal. The condition may realize absolutely no reflection towards incident radar wave so that the target is absolutely stealthy. Therefore, we call it the matching condition. Unfortunately, the practical coating satisfied the matching condition only in some frequency points. In other words, at the range of broad frequency band, the electromagnetic parameters of coating tend to mismatch. Most importantly is that we should understand the differences resulting from the mismatching. It is very significant in RAC designs. The following discussions are two opinions towards mismatching.

Fig. 4. The wave impedance vs. location zyd in the coating.

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Fig. 6. The influence of electromagnetic parameter dispersion (k1 and k2) on the reflection index R.

Fig. 5. The phase differences of Ex and Hy vs. the location of zyd in the coating.

m0sm9=tgd.

(31)

Here, k1 and k2 are dispersion parameters corresponding to the permeability and permittivity of coating. The physical significances are as follows. For k1sk2s1, it shows that there is an ideal matching. While for k1/k2/1, the electromagnetic parameters of coating tend to mismatch. Moreover, The larger k1 or k2 (compared with 1), the higher the dispersion. From Fig. 5, we can see that the absorbing performances of the coating decrease with the rising mismatching. Furthermore, both k1 and k2 affect the absorbing performance of RAC.

´9sk1

Bp E pc ctanhC tgdeF D2 G 2dv

´0s´9=tgd

m9sk2

Bp E pc tanhC tgdmF D2 G 2dv

m0sm9=tgd.

(32) (33) (34) (35)

6. Conclusions We propose two design rules for a single layer RAC. The rules are simple and practical in high performance

5.2. Analysis of electromagnetic parameter dispersion under the condition of tgde/tgdm In this case, the tangent values of electricity and magnetic loss angles vary, while the electromagnetic parameters of RAC meet the following formulas. This is a more popular case. From Fig. 6, we can see that the electromagnetic parameters of RAC correspond to the matching case when tgdestgdm. Hence, RAC usually shows a better absorbing performance. But the absorbing performance will worsen with increasing differences between tgde and tgdm. Here, attention must be paid to one key point, there are two branches at tgdestgdm interface in every line of Figs. 6 and 7. The slope of the left curve (tgde)tgdm) varies fast, while the slope of the right curve (tgde-tgdm) varies relatively slowly. This means that the larger tgdm is, the more beneficial it is in controlling the absorbing performance of the RAC for certain ±(tgdeytgdm)±. In addition, the larger tgde is, the better absorbability for certain tgdm is.

Fig. 7. The influence of tgde and tgdm on the reflection index R.

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RAC designs. According to the rules, we deduced the matching equations of the electromagnetic parameters towards single layer RAC. Moreover, these equations relate to the matching parameters and curves applied to the background of RAC. Combined previous results of authors who have studied quasi-standing wave and RAM design results shows the applications of the rules have been presented. The most important thing is that the design rules have substantial guiding effects both on the material research and the single layer RAC design. In addition, the mismatching curves corresponding to various conditions are also for guiding RAC design. References w1x Stonier RA. Stealth aircraft and technology from World War II to the Gulf part I. SAMPE J 1991;27(4):9 –17.

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