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Analysis of equivalent antenna based on FDTD method
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Available online at www.sciencedirect.com
ScienceDirect Defence Technology 10 (2014) 304e307 www.elsevier.com/locate/dt
Analysis of equivalent antenna based on FDTD method Yun-xing YANG, Hui-chang ZHAO*, Cui DI School of Electronic and Optical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China Received 14 January 2014; revised 17 February 2014; accepted 19 March 2014 Available online 31 July 2014
Abstract An equivalent microstrip antenna used in radio proximity fuse is presented. The design of this antenna is based on multilayer multipermittivity dielectric substrate which is analyzed by finite difference time domain (FDTD) method. Equivalent iterative formula is modified in the condition of cylindrical coordinate system. The mixed substrate which contains two kinds of media (one of them is air)takes the place of original single substrate. The results of equivalent antenna simulation show that the resonant frequency of equivalent antenna is similar to that of the original antenna. The validity of analysis can be validated by means of antenna resonant frequency formula. Two antennas have same radiation pattern and similar gain. This method can be used to reduce the weight of antenna, which is significant to the design of missile-borne antenna. Copyright © 2014, China Ordnance Society. Production and hosting by Elsevier B.V. All rights reserved.
Keywords: Equivalent microstrip antenna; FDTD; Multi-permittivity dielectric
1. Introduction In recent years, antenna has drawn more and more attention with the increase in demand for military antennas [1,7]. Especially for missile-borne antenna, it is required to be small in size and light in weight. In 1953,Deschamps proposed the conception of microstrip antenna [2].The microstrip antenna has many advantages, such as light weight, small size, and planar structure. Lots of work have been done on its miniaturization [3,4].But few research focused on decreasing the weight of antenna and leaving its basic parameters unchanged under the conditions of same size and thickness. In this work, another model is proposed. The aims of this work are: a) to preserve the shape of antenna; b) to preserve the of antenna electromagnetic parameters; and c) to decrease
* Corresponding author. E-mail address: [email protected] (H.C. ZHAO). Peer review under responsibility of China Ordnance Society.
the weight of antenna. For this, a substrate with low permittivity and high permittivity media is taken the place of the original substrate, in which air is used as a lower permittivity media (high permittivity media can be any media as its permittivity is higher than original media). The introduction of the air media can effectively decrease the weight of antenna. The analysis of equivalent antenna is based on FDTD. 2. Equivalent antenna structure and FDTD analysis In1966, K.S. Yee proposed the theory of FDTD [5].FDTD is a novel method which can be used to intuitively and succinctly describe Maxwell's equations. In this method, the electric field and magnetic field are included in a threedimensional model (Yee cell). It is used for solving all the problem of electromagnetic field. A circular antenna is easily integrated into a warhead of bomb compared to rectangle antenna. Therefore the formulae in Refs. [6,7] should be amended in the cylindrical coordinate system (see Fig. 1).
http://dx.doi.org/10.1016/j.dt.2014.07.005 2214-9147/Copyright © 2014, China Ordnance Society. Production and hosting by Elsevier B.V. All rights reserved.
Y.X. YANG et al. / Defence Technology 10 (2014) 304e307
0
1
305
1
0
C 2ε sDt n B 1 C B 1 2Dt C C B Ernþ1 B @i þ 2; j; kA ¼ 2ε þ sDtEr @i þ 2; j; kA þ 2ε þ sDt 3 2 1 1 1 1 1 1 1 1 nþ12 nþ12 nþ12 nþ12 H H4 H4 i þ ; j þ ; k Hz i þ ;j ;k i þ ; j; k þ i þ ; j; k 6 z 2 2 2 2 2 2 2 2 7 7 6 5 4 1 Dz i þ DrD4 2
0
1
0
ð1Þ
1
B 1 1C 1 1C 2Dt 1B C 2m sDt n12 B C Hrnþ2 B @i; j þ 2; k þ 2A ¼ 2m þ sDtHr @i; j þ 2; k þ 2A 2m þ sDt 3 2 1 1 1 1 1 Ezn i; j þ 1; k þ E4n i; j þ ; k þ 1 E4n i; j þ ; k Ezn i þ ; j; k þ 7 6 2 2 2 2 2 7 6 5 4 Dz DrD4
Two medium exist in Yee grid (the permittivities are εr1 ; εr2 ). From Ampere circuital theorem, we have, H
vE H*dl ¼ ∬ ε dS þ EdS vt c s
ð3Þ
Assuming that the magnetic conductivities of two media are the same. It can be known from the boundary condition of electromagnetic field that the normal component of magnetic field on the Interface between the two media is continuous, In cylindrical coordinate system, Eq. (3) could be rewritten as I H*dl ¼ Hrnþ0:5 ði; j þ 0:5; kÞrDq H4nþ0:5 ði; j þ 0:5; kÞ ð4Þ c
In FDTD iteration. 1 1 Ernþ2 i; j; k ¼ Ernþ1 i; j; k þ Ern i; j; k 2 The right side of Eq (5) can be extended to
Dr Hrnþ0:5 ði; j 0:5; kÞrDq þ H4nþ0:5 ði; j þ 0:5; kÞDr
Fig. 1. Yee cell.
ð2Þ
Fig. 2. Structure of substrate.
ð5Þ
306
Y.X. YANG et al. / Defence Technology 10 (2014) 304e307
m X n X Ernþ1 i; j; k Ern i; j; k vE vE ∬ ε dS ¼ ∬ εij dSij ¼ vt vt Dt s i¼1 j¼1 ij 2 3 6 7 4ε11 r1 Dr1 Dq1 þ ε12 r1 Dr1 Dq2 :::::: þ εmn rm Drm Dqn 5 ð6Þ ∬ sEdS ¼ s
2
m X n X i¼1
j¼1
Ernþ1 i; j; k Ern i; j; k ∬ sij EdSij ¼ 2 ij 3
6 7 4s11 r1 Dr1 Dq1 þ s12 r1 Dr1 Dq2 :::::: þ smn rm Drm Dqn 5 ð7Þ Substituting Eqs. (4), (6) and (7) into Eq. (3), we have,
1
0
Fig. 3. Resonant frequencies of three antennas.
1
0
C 2εeff seff Dt n B 1 C B 1 2Dt C C B Ernþ1 B @i þ 2; j; kA ¼2εeff þ seff DtEr @i þ 2; j; kA þ 2εeff þ seff Dt 3 2 1 1 1 1 1 1 1 1 1 1 1 1 Hznþ2 i þ ; j þ ; k Hznþ2 i þ ; j ; k H4nþ2 i þ ; j; k þ H4nþ2 i þ ; j; k 6 2 2 2 2 2 2 2 2 7 7 6 5 4 1 Dz i þ DrD4 2
After the simplification,the equivalent permittivity can be obtained ε11 r1 Dr1 D41 þε12 r1 Dr1 D42::::::::::::::::::::þ εmn rm Drm D4n ð9Þ rdrd4 Through the analysis mentioned above, the structure of antenna substrate can be designed, as shown in Fig. 2. εeff ¼
Fig. 3 shows that the resonant frequency of original antenna is 3.10 GHz, and the resonant frequencies of the equivalent antennas A and B are also remain unchanged. It can be concluded from the resonant frequency formula that the permittivity is the only factor which has effect on resonant frequency when the radius of radiation patch and the working mode remain the same. So two media can be treated as a single
3. Discussion of simulation Through the above analysis, FR4 is chosen for original substrate. The basic parameters of antenna are εr ¼ 4:4, R ¼ 15 mm, h ¼ 2 mm, r ¼ 13 mm. Eq (10) shows the relationship between two medium .Table 1 lists the parameters of equivalent antennas εr1 R21 þ εr2 R22 ¼ εr R2 ð10Þ R 1 þ R2 ¼ R
Table 1 Parameters of equivalent antennas.
A B
εr1
εr2
R1
R2
12.9 9.8
1 1
8.57 9.9
6.43 5.1
ð8Þ
Fig. 4. S11 with different feed feint.
Y.X. YANG et al. / Defence Technology 10 (2014) 304e307
307
deteriorated compared to the original antenna. So the equivalent antenna is not exactly equal to the original antenna. The parameters can be improved by changing the position of feed point. Fig. 4 shows the S11 parameters after changing the position of feed point. Fig. 5 and 6 show the radiation patterns of equivalent antenna and original antenna. It can be seen from Fig. 5 and 6 that the direction and gain of equivalent antenna are similar to those of the original one. This shows that the equivalent antenna can replace the original antenna. 4. Conclusions
Fig. 5. Original radiation pattern.
A method for decreasing the weight of antenna was proposed in the paper. By means of analysis based on FDTD, air medium and a higher permittivity medium are introduced to take the place of original substrate. The simulation results show that resonant frequency, radiation pattern and gain of equivalent antenna did not change. The air media can be used to reduce the weight of antenna. Acknowledgement The author would like to thank the National Natural Science Foundation of China (Grant No. 6111168) for the support. References
Fig. 6. Equivalent radiation pattern.
medium which has a same permittivity as that of original medium. The results prove the accuracy of FDTD analysis. Because the position of feed point was not changed in simulation, the S11 parameters of the equivalent antenna
[1] Chen N, Ammann MJ, Qing X, Wu XH, See TSP, Cai A. Planar. IEEE Microw Mag 2006;7(6):63e7. [2] Krstic M, Kanell Akopoulos I, Kokotovic P. Nonlinear and adaptive control design. New York: Wiley; 1995. [3] Wong KL. Compact and broadband microstrip antennas. John Wiley & Sons Inc; 2002. [4] Yee KS. Numerical solution of initial boundary value problems involving Maxwell equations in isotropic media. IEEE Trans Antennas Propagate 1966;14(3):302e7. [5] Taflove A, Hagness SC. Computational electrodynamics: the finite difference time domain method. 2nd ed. Norwood, MA: Artech Hoose; 2000. [6] Greengard L, Rokhlin V. A fast algorithm for particle simulation. J Comp Phys 1987;73:325e48. [7] Wu X, Tan YL, Xu J, Fang A. Research on the model of multilayed multiconductor system with equivalent permittivity, 32 (2). Journal of Zhejiang University of Technology; 2006. p. 179e82.
Available online at www.sciencedirect.com
ScienceDirect Defence Technology 10 (2014) 304e307 www.elsevier.com/locate/dt
Analysis of equivalent antenna based on FDTD method Yun-xing YANG, Hui-chang ZHAO*, Cui DI School of Electronic and Optical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China Received 14 January 2014; revised 17 February 2014; accepted 19 March 2014 Available online 31 July 2014
Abstract An equivalent microstrip antenna used in radio proximity fuse is presented. The design of this antenna is based on multilayer multipermittivity dielectric substrate which is analyzed by finite difference time domain (FDTD) method. Equivalent iterative formula is modified in the condition of cylindrical coordinate system. The mixed substrate which contains two kinds of media (one of them is air)takes the place of original single substrate. The results of equivalent antenna simulation show that the resonant frequency of equivalent antenna is similar to that of the original antenna. The validity of analysis can be validated by means of antenna resonant frequency formula. Two antennas have same radiation pattern and similar gain. This method can be used to reduce the weight of antenna, which is significant to the design of missile-borne antenna. Copyright © 2014, China Ordnance Society. Production and hosting by Elsevier B.V. All rights reserved.
Keywords: Equivalent microstrip antenna; FDTD; Multi-permittivity dielectric
1. Introduction In recent years, antenna has drawn more and more attention with the increase in demand for military antennas [1,7]. Especially for missile-borne antenna, it is required to be small in size and light in weight. In 1953,Deschamps proposed the conception of microstrip antenna [2].The microstrip antenna has many advantages, such as light weight, small size, and planar structure. Lots of work have been done on its miniaturization [3,4].But few research focused on decreasing the weight of antenna and leaving its basic parameters unchanged under the conditions of same size and thickness. In this work, another model is proposed. The aims of this work are: a) to preserve the shape of antenna; b) to preserve the of antenna electromagnetic parameters; and c) to decrease
* Corresponding author. E-mail address: [email protected] (H.C. ZHAO). Peer review under responsibility of China Ordnance Society.
the weight of antenna. For this, a substrate with low permittivity and high permittivity media is taken the place of the original substrate, in which air is used as a lower permittivity media (high permittivity media can be any media as its permittivity is higher than original media). The introduction of the air media can effectively decrease the weight of antenna. The analysis of equivalent antenna is based on FDTD. 2. Equivalent antenna structure and FDTD analysis In1966, K.S. Yee proposed the theory of FDTD [5].FDTD is a novel method which can be used to intuitively and succinctly describe Maxwell's equations. In this method, the electric field and magnetic field are included in a threedimensional model (Yee cell). It is used for solving all the problem of electromagnetic field. A circular antenna is easily integrated into a warhead of bomb compared to rectangle antenna. Therefore the formulae in Refs. [6,7] should be amended in the cylindrical coordinate system (see Fig. 1).
http://dx.doi.org/10.1016/j.dt.2014.07.005 2214-9147/Copyright © 2014, China Ordnance Society. Production and hosting by Elsevier B.V. All rights reserved.
Y.X. YANG et al. / Defence Technology 10 (2014) 304e307
0
1
305
1
0
C 2ε sDt n B 1 C B 1 2Dt C C B Ernþ1 B @i þ 2; j; kA ¼ 2ε þ sDtEr @i þ 2; j; kA þ 2ε þ sDt 3 2 1 1 1 1 1 1 1 1 nþ12 nþ12 nþ12 nþ12 H H4 H4 i þ ; j þ ; k Hz i þ ;j ;k i þ ; j; k þ i þ ; j; k 6 z 2 2 2 2 2 2 2 2 7 7 6 5 4 1 Dz i þ DrD4 2
0
1
0
ð1Þ
1
B 1 1C 1 1C 2Dt 1B C 2m sDt n12 B C Hrnþ2 B @i; j þ 2; k þ 2A ¼ 2m þ sDtHr @i; j þ 2; k þ 2A 2m þ sDt 3 2 1 1 1 1 1 Ezn i; j þ 1; k þ E4n i; j þ ; k þ 1 E4n i; j þ ; k Ezn i þ ; j; k þ 7 6 2 2 2 2 2 7 6 5 4 Dz DrD4
Two medium exist in Yee grid (the permittivities are εr1 ; εr2 ). From Ampere circuital theorem, we have, H
vE H*dl ¼ ∬ ε dS þ EdS vt c s
ð3Þ
Assuming that the magnetic conductivities of two media are the same. It can be known from the boundary condition of electromagnetic field that the normal component of magnetic field on the Interface between the two media is continuous, In cylindrical coordinate system, Eq. (3) could be rewritten as I H*dl ¼ Hrnþ0:5 ði; j þ 0:5; kÞrDq H4nþ0:5 ði; j þ 0:5; kÞ ð4Þ c
In FDTD iteration. 1 1 Ernþ2 i; j; k ¼ Ernþ1 i; j; k þ Ern i; j; k 2 The right side of Eq (5) can be extended to
Dr Hrnþ0:5 ði; j 0:5; kÞrDq þ H4nþ0:5 ði; j þ 0:5; kÞDr
Fig. 1. Yee cell.
ð2Þ
Fig. 2. Structure of substrate.
ð5Þ
306
Y.X. YANG et al. / Defence Technology 10 (2014) 304e307
m X n X Ernþ1 i; j; k Ern i; j; k vE vE ∬ ε dS ¼ ∬ εij dSij ¼ vt vt Dt s i¼1 j¼1 ij 2 3 6 7 4ε11 r1 Dr1 Dq1 þ ε12 r1 Dr1 Dq2 :::::: þ εmn rm Drm Dqn 5 ð6Þ ∬ sEdS ¼ s
2
m X n X i¼1
j¼1
Ernþ1 i; j; k Ern i; j; k ∬ sij EdSij ¼ 2 ij 3
6 7 4s11 r1 Dr1 Dq1 þ s12 r1 Dr1 Dq2 :::::: þ smn rm Drm Dqn 5 ð7Þ Substituting Eqs. (4), (6) and (7) into Eq. (3), we have,
1
0
Fig. 3. Resonant frequencies of three antennas.
1
0
C 2εeff seff Dt n B 1 C B 1 2Dt C C B Ernþ1 B @i þ 2; j; kA ¼2εeff þ seff DtEr @i þ 2; j; kA þ 2εeff þ seff Dt 3 2 1 1 1 1 1 1 1 1 1 1 1 1 Hznþ2 i þ ; j þ ; k Hznþ2 i þ ; j ; k H4nþ2 i þ ; j; k þ H4nþ2 i þ ; j; k 6 2 2 2 2 2 2 2 2 7 7 6 5 4 1 Dz i þ DrD4 2
After the simplification,the equivalent permittivity can be obtained ε11 r1 Dr1 D41 þε12 r1 Dr1 D42::::::::::::::::::::þ εmn rm Drm D4n ð9Þ rdrd4 Through the analysis mentioned above, the structure of antenna substrate can be designed, as shown in Fig. 2. εeff ¼
Fig. 3 shows that the resonant frequency of original antenna is 3.10 GHz, and the resonant frequencies of the equivalent antennas A and B are also remain unchanged. It can be concluded from the resonant frequency formula that the permittivity is the only factor which has effect on resonant frequency when the radius of radiation patch and the working mode remain the same. So two media can be treated as a single
3. Discussion of simulation Through the above analysis, FR4 is chosen for original substrate. The basic parameters of antenna are εr ¼ 4:4, R ¼ 15 mm, h ¼ 2 mm, r ¼ 13 mm. Eq (10) shows the relationship between two medium .Table 1 lists the parameters of equivalent antennas εr1 R21 þ εr2 R22 ¼ εr R2 ð10Þ R 1 þ R2 ¼ R
Table 1 Parameters of equivalent antennas.
A B
εr1
εr2
R1
R2
12.9 9.8
1 1
8.57 9.9
6.43 5.1
ð8Þ
Fig. 4. S11 with different feed feint.
Y.X. YANG et al. / Defence Technology 10 (2014) 304e307
307
deteriorated compared to the original antenna. So the equivalent antenna is not exactly equal to the original antenna. The parameters can be improved by changing the position of feed point. Fig. 4 shows the S11 parameters after changing the position of feed point. Fig. 5 and 6 show the radiation patterns of equivalent antenna and original antenna. It can be seen from Fig. 5 and 6 that the direction and gain of equivalent antenna are similar to those of the original one. This shows that the equivalent antenna can replace the original antenna. 4. Conclusions
Fig. 5. Original radiation pattern.
A method for decreasing the weight of antenna was proposed in the paper. By means of analysis based on FDTD, air medium and a higher permittivity medium are introduced to take the place of original substrate. The simulation results show that resonant frequency, radiation pattern and gain of equivalent antenna did not change. The air media can be used to reduce the weight of antenna. Acknowledgement The author would like to thank the National Natural Science Foundation of China (Grant No. 6111168) for the support. References
Fig. 6. Equivalent radiation pattern.
medium which has a same permittivity as that of original medium. The results prove the accuracy of FDTD analysis. Because the position of feed point was not changed in simulation, the S11 parameters of the equivalent antenna
[1] Chen N, Ammann MJ, Qing X, Wu XH, See TSP, Cai A. Planar. IEEE Microw Mag 2006;7(6):63e7. [2] Krstic M, Kanell Akopoulos I, Kokotovic P. Nonlinear and adaptive control design. New York: Wiley; 1995. [3] Wong KL. Compact and broadband microstrip antennas. John Wiley & Sons Inc; 2002. [4] Yee KS. Numerical solution of initial boundary value problems involving Maxwell equations in isotropic media. IEEE Trans Antennas Propagate 1966;14(3):302e7. [5] Taflove A, Hagness SC. Computational electrodynamics: the finite difference time domain method. 2nd ed. Norwood, MA: Artech Hoose; 2000. [6] Greengard L, Rokhlin V. A fast algorithm for particle simulation. J Comp Phys 1987;73:325e48. [7] Wu X, Tan YL, Xu J, Fang A. Research on the model of multilayed multiconductor system with equivalent permittivity, 32 (2). Journal of Zhejiang University of Technology; 2006. p. 179e82.