Numerical Analysis of Electromagnetic Fields in Multiscale Model

Transcription

1 Commun. Theor. Phys. 63 (205) Vol. 63, No. 4, April, 205 Numerical Analysis of Electromagnetic Fields in Multiscale Model MA Ji ( ), FANG Guang-You (ྠ), and JI Yi-Cai (Π) Key Laboratory of Electromagnetic Radiation and Sensing Technology, Institute of Electronics, Chinese Academy of Sciences, Beijing 0090, China (Received February 6, 205; revised manuscript received February 3, 205) Abstract Modeling technique for electromagnetic fields excited by antennas is an important topic in computational electromagnetics, which is concerned with the numerical solution of Maxwell s equations. In this paper, a novel hybrid technique that combines method of moments (MoM) with finite-difference time-domain (FDTD) method is presented to handle the problem. This approach employed Huygen s principle to realize the hybridization of the two classical numerical algorithms. For wideband electromagnetic data, the interpolation scheme is used in the MoM based on the dyadic Green s function. On the other hand, with the help of equivalence principle, the scattered electric and magnetic fields on the Huygen s surface calculated by MoM are taken as the sources for FDTD. Therefore, the electromagnetic fields in the environment can be obtained by employing finite-difference time-domain method. Finally, numerical results show the validity of the proposed technique by analyzing two canonical samples. PACS numbers: 4.20.Jb, Cb Key words: electromagnetic fields, method of moments (MoM), finite-difference time-domain (FDTD), hybrid technique Introduction The objective of computational electromagnetics (CEM) is to model the behavior of electromagnetic fields excited by radiation sources, which can reduce the high experimental cost in terms of equipment and manpower. Furthermore, CEM usually utilizes numerical technique to solve Maxwell s equations and can provide fundamental insights into electromagnetic problems through the power of visualization. In the past decade, with the development of computer science, numerical methods have been widely used to analyze radiation of antennas in complex environments. Among these approaches, the finite-difference time-domain (FDTD) technique [ 8] is suitable to treat the inhomogeneous background. Since FDTD is the direct solution of Maxwell s equations, the complexity has no distinct difference by applying this algorithm to simulate the electromagnetic wave traveling in various media. However, in the case of antennas above heterogeneous grounds, a small size of the FDTD cell must be employed to model the antenna precisely. This fact makes it difficult to solve this problem with FDTD. On the one hand, if a uniform mesh is used in the entire region, the memory requirements and calculation time may be very intensive. On the other hand, although a number of improved FDTD versions based on nonuniform mesh have been presented recently, the accuracy and stability are discussed in the electromagnetic community. Under this situation, the hybrid technique which combines integral equation method with FDTD seems to be a powerful tool for handling the problem. As we know, the method of moments (MoM) [9 5] based on electric field integral equation (EFIE) has natural advantages in analyzing antennas since it only needs mesh discretization on the surface of targets. Therefore, MoM in conjunction with FDTD seems to be a good choice. In the ongoing effort, one way to hybridize MoM with FDTD is decomposing the original model into two subgeometries according to Schelkunoff equivalence principle firstly. By exchanging field values on the interface between two regions, an iterative scheme must be employed due to the utilization of free-space Green s functions in the MoM region. However, only when the couplings between antennas and dielectric body are relatively weak, the iterative process can be convergent rapidly. And this hybrid technique is usually utilized to analyze antennas in environments working at one single frequency point. In this paper, to model the behavior of the electromagnetic waves traveling in the environments, a hybrid method which combines MoM based on dyadic Green s functions with FDTD is presented. In this technique, the Huygen s surface is applied to create equivalent sources in the environments. As couplings between the antenna and dielectric body are taken into account in MoM, the iterative scheme can be avoided. Furthermore, with the help of interpolation algorithm for frequency response, the fields in the environment can be observed over a broadband. Numerical results are shown to demonstrate the accuracy of this method. Supported in part by China Postdoctoral Science Foundation under Grant No. 20M550839, and in part by the Key Research Program of the Chinese Academy of Sciences under Grant No. KGZD-EW-603 Corresponding author, jima@mail.ie.ac.cn c 205 Chinese Physical Society and IOP Publishing Ltd

2 506 Communications in Theoretical Physics Vol Method Description 2. Method of Moments For the surface of a perfect electric conductor (PEC), the electric boundary condition is written as ˆn (E i + E s ) = 0, () where ˆn denotes the surface normal, E i is the incident electric field and E s is the scattered field which can be obtained from the induced surface current J via E s (r) = jk 0 η 0 Ḡ(r, r ) J(r )dω. (2) Ω In Eq. (2), Ω represents for the surface of the conductor. k 0 and η 0 are wave number and wave impedance of free space, respectively. r and r stand for the source and observation points, respectively. Ḡ is the well-known dyadic Green s function. Substitute the expression for E s from Eq. (2) into Eq. (), the electric field integral equation (EFIE) formulation of Maxwell s equations can be derived as ˆn Ω Ḡ(r, r ) J(r )dω = jk 0 η 0 ˆn E i (r). (3) Let current J be expanded with RWG basis functions and apply the Galerkin s procedure to yield a system of linear equations as ZI = V, (4) where Z, I and V stand for the impedance matrix, unknown coefficient vector and excitation vector, respectively. 2.2 Finite Difference in Time-Domain (FDTD) The Maxwell s equations in the Cartesian coordinate system can be written as H s z y Hs y z H s x z Hs z = ε E s x x = ε E s y + (ε ε 0 ) Ei x + (ε ε 0 ) Ei y + σ (E i x + E s x), + σ (E i y + E s y), 2.3 Hybrid Algorithm The concept of the proposed hybrid technique is illustrated in Fig.. It can be seen that a sub-domain is surrounded by a closed surface. The algorithm begins by evaluating the electromagnetic fields on this surface with MoM based on dyadic Green s functions. With the help of Huygen s equivalent theorem, that fields calculated by MoM are treated as sources for the interested computational domain in FDTD scheme. The detailed procedure is implemented as follows, (i) For a given frequency band f [f a, f b ] corresponding to the wave-number k [k a, k b ], determine electric currents on the antenna at some selected frequencies using the following matrix equation, Z(k i )I(k i ) = V (k i ), (k a k i k b, i =, 2,...). (7) Note that k i is the wave-number relating to the selected frequency over the broadband and couplings between the antenna and background media are taken into account by applying dyadic Green s functions to the impedance matrix Z. (ii) Use the impedance matrix interpolation scheme to obtain the frequency response of electric currents on the antenna. This step can avoid constructing impedance matrix repeatedly for wideband analysis with MoM. The detail about the selection of frequencies and interpolation technique is described in Ref. []. (iii) Calculate electric and magnetic fields radiated by the antenna at observation points on the Hyugen s surface. The observation point is determined by the center on a surface of Yee s cell which is used to model the subdomain for FDTD. (iv) Convert the radiation fields at the observation points into that in time domain by inverse Fast Fourier Transformation (IFFT). (v) Apply the equivalent sources and perfect matched layer (PML) to the sub-domain, the scattered fields can be evaluated by the FDTD algorithm. Therefore, the behavior of the waves traveling in this domain can be observed. H s y x Hs x y = ε E s z +(ε ε 0 ) Ei z +σ (E i z + E s z),(5) E s z y Es y z = µ H s x (µ µ 0 ) Hi x, E s x z Es z x = µ H s y (µ µ 0 ) Hi y E s y x Es x y = µ H s z (µ µ 0 ) Hi z, (6) where σ, ε and µ are the electric conductivity, permittivity and permeability of the background media, respectively, while ε 0 and µ 0 have similar definitions associated with free space. The finite-difference algorithm can be implemented by employing Yee s cells and field-advance equations. Fig. Concept of the hybrid technique. Overall, the flow chart of this algorithm can be summarized as the following figure,

3 No. 4 Communications in Theoretical Physics 507 Fig. 4 Electric field E x on an x-direction line. Fig. 2 Flow chart of the hybrid algorithm. 3 Numerical Results In this section, two canonical examples are analyzed by the proposed method to show the validity of the hybrid technique. In the first case, a simple thin wire dipole in free space is considered. The length of each arm of the wire is 0.25 m and the radius is mm. The working frequency band is 00 MHz 900 MHz with an interpolation step of 50 MHz. The dipole is discretized into segments. For the FDTD sub-domain, cells are used to model this region. To check the accuracy of the presented method, some observer points are located inside the Hyugen s surface. Figure 3 shows the frequency response of the electric field E x at observer, which is located 80 cm below the dipole. It can be seen that the results obtained by the hybrid method agree well with that by MoM. The curves in Fig. 4 are the x component of the electric field on an x-direction line, 80 cm away from the dipole with operating frequency 600 MHz. These curves show that the presented hybrid technique can indeed obtain correct numerical results. Fig. 5 Bowtie antenna above the ground. Frequency response of electric field E x at ob- Fig. 3 server. Fig. 6 Frequency response of electric field E x at observer. Another example considered in this paper is a bowtie antenna 0.26 m in length L, located 0.4 m above the ground whose relative dielectric permittivity is ε r = 6.0, as seen in Fig. 5. The electromagnetic fields inside a region which is demonstrated by the dashed line in Fig. 5 have been investigated. The size of the region is m m 0.8 m and two observers are fixed in the domain. Observer is located 0.5 m below the antenna, while observer 2 is located 0.2 m below the antenna. Figure 6 shows the frequency responses of the electric field

4 508 Communications in Theoretical Physics Vol. 63 E x at observer calculated by the MoM/FDTD hybrid method and MoM. The operating frequency of the antenna is from 50 MHz to 900 MHz and a good agreement between the two results can be seen. To study the characteristic of fields in this example, the antenna in the free-space is also analyzed as a reference. In fact, when the ground is absent, the antenna is in the free-space. Assume that the electromagnetic fields on the equivalent surface in frequency domain are converted into the BHW waveform. Figure 7(a) shows the transient electric field at observer. Since the relative dielectric permittivity of the ground is larger than that of the air, the traveling speed of the field in the ground is slower than that in the air. On the other hand, Fig. 7(b) shows the transient electric field at observer 2. It can be seen the magnitude of the field at the observer above the ground is slightly larger than that in the free-space due to the reflection of the ground. Finally, Fig. 8 shows the magnetic fields in the YoZ plane when the operating frequency is 300 MHz. And Fig. 9 shows those for 500 MHz. Due to the presence of the ground, a distinguishing feature of the magnetic field can be observed compared with the free-space situation. Note that values for the magnetic fields are demonstrated in Fig. 0. Fig. 7 Transient fields at (a) observer and (b) observer 2. Fig. 8 (Color online) Magnetic fields in the YoZ plane (300 MHz). Fig. 9 (Color online) Magnetic fields in the YoZ plane (500 MHz).

5 No. 4 Communications in Theoretical Physics 509 Fig. 0 Values (db) of fields for Figs. 8 and 9. 4 Conclusion The electromagnetic fields in environments have been studied by a hybrid technique which combines method of moments and finite-difference time-domain algorithm. With the help of Schelkunoff principle, the equivalent sources are created by MoM firstly. Then, FDTD is applied to calculate the electromagnetic fields in the interested computational region. Since the matrix interpolation is used in MoM, the presented technique can realize frequency sweeping analysis. Finally, some numerical results are demonstrated in the paper. Acknowledgments The authors thank Dr. Bo Zhao from Wave Computation Technologies, Inc. for the helpful technical support and suggestions. References [] K.S. Yee, IEEE Trans. Antennas Propagat. 4 (966) 302. [2] Z.S. Sacks, D.M. Kingsland, K.M. Lee, et al., IEEE Trans. Antennas Propagat. 43 (995) 460. [3] S.D. Gendey, IEEE Trans. Antennas Propagat. 44 (996) 630. [4] A. Taflove, Advances in Computational Electromagnetics: The FDTD Method, Artech House, Norwood (998). [5] D.M. Sullivan, Electromagnetic Simulation Using the FDTD Method, IEEE Press, New York (2000). [6] A. Taflove and S.C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed., Artech House, Norwood (2005). [7] M. Ha and M. Swqminathan, IEEE Microw. Wireless Compon. Lett. 2 (20) 225. [8] M. Yi, M. Ha, Z. Qian, et al., IEEE Trans. Microw. Theory Techn. 6 (203) [9] R.F. Harrington, Field Computation by Moment Methods, New York, MacMillan, New York (968). [0] S.M. Rao, D.R. Wilton, and A.W. Glisson, IEEE Trans. Antennas Propag. 30 (982) 409. [] K.L. Virga and Y.R. Samii, IEEE Trans. Antennas Propagat. 47 (999) 65. [2] K.N. Ramli, et al., Progress In Electromagnetics Research 33 (203) 7. [3] M.A. Francavilla, F. Vipiana, G. Vecchi, et al., IEEE Antennas Wireless Propagat. Lett. (202) 378. [4] F. Vipiana, M. Bercigli, M.A.E. Bautista, et al., IEEE Trans. Electromagn. Compat. 56 (204) 707. [5] M.A.E. Bautista, M.A. Francavilla, F. Vipiana, et al., IEEE Trans. Antennas Propag. 62 (204) 523.