Computational Methods and Experimental Measurements XII 611 The analysis of microstrip antennas using the FDTD method M. Wnuk, G. Różański & M. Bugaj Faculty of Electronics, Military University of Technology, Poland Abstract The object of this article is to demonstrate the use of the FDTD method in analyzing planar antennas. Yee s cell as a basic element of the characteristic of the field in the FDTD method and Yee s algorithm are presented. The question of source modeling and stimulation of antenna, boundary conditions and method of analysis of planar antennas structure are also discussed. Some results of the use of this method have also been presented, as a confirmation of its accuracy and usefulness in the analysis of planar antennas. 1 Introduction Microstrip antennas have been one of the most innovative fields of antenna techniques in the last fifteen years. The main advantages of these antennas are: simplicity of production, small weight, narrow section, and easiness of integration of radiators with feeding system. The basic disadvantages are: narrow band, limited power capacity and insufficient efficiency of radiation. The basic configuration of microstrip antenna consists of a metallic strip printed on a thin, earthed dielectric base. The feeding is realized through concentric cable, run perpendicularly through substrate or strip line run on a substrate in the plane of the aerial. The methods of analysis and projection of microstrip antennas have developed simultaneously with the development of aerials. Nowadays, several methods of analysis of antennas on dielectric surface are being used. However, the most commonly used method is fullwave model that is based on Green function and moment method, where analysis relies on the solution of integral equation, concerning electric field, with regard to unknown currents flowing through elements of antenna and its feeding system.
612 Computational Methods and Experimental Measurements XII One of the most innovative methods concerning the numerical analysis of strip antennas is FDTD (Finite Difference Time Domain). There are many solutions and experimental works concerning this method. They cover various attitudes towards matters and whilst some of them only cover description of the method and analysis of simple strip structures, others present more complex structures. Regardless of the complexity of the structure, the method allows for efficient, exact and complex analysis covering all phenomena appearing in microstrip structures. 2 Yee s algorithm In 1966, Kane Yee [2] proposed a set of equations of finite differences for rotational Maxwell s equations constructed in time domain. Solution of these equations depends on following foundation: Instead of separately solving electric and magnetic fields using wave equations, simultaneously both in time and space are concerned with usage of joint rotational Maxwell s equations: It is an analogous approach to the formulation of integral equations of method moments MM where the border conditions for E and H are put on materials structure surface. The properties of tangential vectors H hard upon edges, corners and thin wires as well as properties of tangential vectors hard upon points, can be modeled separately if both magnetic and electric fields exist. The distribution of components of electric and magnetic fields in a cubic cell of Yee s mesh is presented in Fig. 1. E y E x Hz E y E z H x E z E x H y E z (i,j,k) E y E x Figure 1: Distribution of components of electric and magnetic fields in a cubic cell of Yee s mesh. In the figure, a mutual special stripping and localization of vectors of electric and magnetic field intensity can be observed. Each electric component is surrounded by four circulating magnetic components and similarly, each magnetic component is surrounded by four circulating electric components. Electric vectors are associated with the center of each edge of cube, whilst magnetic vectors are associated with the center of each plane being the wall of the elementary cell.
Computational Methods and Experimental Measurements XII 613 Three-dimensional space is filled with layers of matrices of Faraday s and Ampere s laws. The continuity of E and H fields changes is naturally supported in the area of interaction of different materials if the area of influence is parallel to the coordinates of the mesh. 3 Boundary conditions Use of the FDTD method is connected with numerous advantages due to the character of analysis. The method allows for analysis of complex structures and consideration of surface waves and other unfavorable properties of structures. The algorithm also has disadvantages; numerous problems in the description of the electromagnetic field need large calculation power, which becomes almost impossible. This means that concerned space has to be limited in a way allowing for minimal error. In order to minimize error in numerical applications, boundary conditions are being used. Solution based on first order boundary condition (Mur) having minimal error needs the limited area to be big enough to omit the effect of distraction of electromagnetic field. To obtain it, this progressive wave (in numerical form) moving along direction - x can be described as the following one-dimensional wave equation: where: 1 x vi E t tan = E tan - tangential component of vector of electric field intensity ν - velocity of electromagnetic wave. To fulfill the condition that flat waves will not reflect, the boundary condition is as follows: v t x E = E + E E v t+ x ( ) n+ 1 n i n+ 1 n 1 1 i where: E - tangential component of vector of intensity of electric field for k = E - tangential component of vector of intensity of electric field for k = 1. 1 The boundary Mur condition is precise only for plane waves incident at right angles, so reflection will occur only for waves incident at different angles. The case can be approximated by boundary conditions of higher orders. Among many other solutions concerning boundary conditions, conditions described as perfectly matched layer (PML) require some attention. The formula is a development of boundary conditions on the edge of two mediums and it allows for analysis of electromagnetic wave incident under any angle. It is achieved by diversion of electric and magnetic components of an electromagnetic field in the area of absorption and subordinating them with (1) (2)
614 Computational Methods and Experimental Measurements XII different lossiness for different directions. As an effect on external area, wave impedance depends on angle of incidence of wave and its frequency. During analysis of the electromagnetic field inside the stimulated area, zero value of all components of the electric and magnetic fields in t = moment is assumed. Gauss impulse is most commonly used for input function during the measuring of antenna characteristics. This allows for the selection of the bandwidth of antenna and many other frequency parameters by selection of width of impulse in a short time. Gauss impulse used in the process of analysis is presented in the following form: E f t e z 2 ( t t ) / T = () = (3) s In the case of simple microstrip structures fed by strip line in relation to source plane and in other planes, boundary conditions are applied. In many papers the plane of source represents the point of stimulation of the electromagnetic wave. Gauss impulse as a voltage stimulation of structure in moment t =, only possesses component E(z) of electric field intensity. Other coordinates are assumed to equal zero. Such a model of source has some errors. It is a result of the induction of components of electromagnetic field tangential to source plane which results in deformation of the inducted electromagnetic wave. An alternative solution to the problem is making some assumptions of the starting conditions. In t = o, E x and E z coordinates of electric field and tangential to them magnetic components transformed by ± y exist. What s more, H tan = H tan where H tan is the value of the magnetic component just before the surface of the source, while H is just after the plane. In order to select other components it is sufficient to use differential final equations. 4 Analysis of microstrip structure Application of the FDTD algorithm during the process of projection of microstrip antennas is connected with making a number of assumptions. In order to present planar structure in a precise, effective way and taking into consideration all phenomena influencing characteristics of antenna, it is necessary to use great calculation power. In order to save analysis time some simplifications have to be made however, the result must be precise enough. Moreover the analysis is not only limited to the structure of radiator, but the feeding of the antenna also has to be taken into consideration. It can complicate the process of projection and obtained results. Limitation of structure is connected with selection of proper boundary conditions, which reduce the area of analysis and simultaneously do not cause ambiguity or corrupt the results. The area, which will be analyzed in numerical calculations, is presented in Fig. 4. As one may observe in the figure, it has been presented as a rectangular planar radiator fed with a strip line. The limitation as
Computational Methods and Experimental Measurements XII 615 mentioned above, is connected with the limited computing power of today s computers. What s more, in order to obtain exact results it isn t necessary to analyze infinitely large areas. The beginning of a simulation requires imposition on an aerial a mesh of suitable dimensions. Exact modeling of an aerial requires selection of sizes x and y in such a way that the total number of cells fill the radiator (see Fig. 2), otherwise the calculations would be much more complicated. It would require separate analysis of cells placed on the border dielectric radiator. During the selection of dimensions of cells a condition of stability of the selected quantities should also be taken into account. z y x x y Figure 2: Example of a grid put on aerial ( x = y). While choosing sizes of cell we should also take into consideration the fulfillment of conditions of stability. 5 Project of selected microstrip aerial Problems connected with finite difference methods in the process of modeling of microstrip antennas arise as outlined above. Both the potential of the FDTD method in the analysis of planar structures and its limitations and system requirements have been presented. The method allows for setting basic parameters, frequency characteristics and the distribution of electrical and magnetic fields along the planar structure. In order to test its possibilities, an analysis of three microstrip antennas of a rectangular shape of radiator, resonance frequencies equal to 2.4 GHz, 5GHz and 7.5 GHz respectively and antenna with radiator in a shape of a circle for a frequency 7.5 GHz has been carried out. A program in Matlab describing analyzed structures modeled in FDTD method has been elaborated. With regard to time of implementation, size of available memory and computing power, modeled structures have been simplified. It does not however, limit the preciseness of analysis. 5.1 Model of analyzed planar aerial A modeled antenna has been presented in Fig. 3(a) & (b). For an analyzed planar structure the activity of the antenna is more important than the surrounding space, so the analyzed area is not too large. But it is large enough so that the
616 Computational Methods and Experimental Measurements XII results would reflect analyzed area precisely and strictly, taking into account the limited computing power of a computer. One of the most important parameters, which will be calculated, is S 11 whose value is dependent on the parameters of antenna. This is why the analyzed area has been limited in order to approximate it to the analyzed structure. The geometrical dimensions of each radiating structure have been approximated using suitable rectangular planar radiator (transmission narrowband model) and in case of radiator of round shape, the hilar model is used. W a L,764mm P1,9mm P1 2,46mm (a) With rectangular shape radiator. 2,46mm (b) With circular shape radiator. Figure 3: Analyzable microstrip aerial. The thickness of a dielectric and the width of feeding line according to literature [3] in the case of structures with resonance frequencies of up to 18 GHz, differs by less than a hundred parts of millimeter. As a consequence for all cases of antennas a constant thickness of dielectric h=.764 mm and width of feeding line 2.64 mm have been assumed. Analysis of the structure presented in Fig. 3 leads to the conclusion that the distribution of FDTD cells should be as little as possible in order to accurately describe the presented area. The limitation for us in each case is criteria of stability; the size of FTDT cell should not exceed λ/1. The lowest limit in this case has microstrip antenna with a resonance frequency of 7.5 GHz. Let s consider this radiating structure. In order to present this structure I have assumed three cell dimensions. So a step in direction of axis z is z =.265 mm and there are thirteen cells illustrating the surrounding space. In the two other directions I have taken a dimension in a way to exactly represent the radiating path, which in the case of microstrip antenna with a frequency of 7.5 GHz implicates 32 x 4 y where x =.389 mm, y =.4 mm. A solution obtained in this way is connected with the differentiation of the area and the selection of different quantities of cells in each direction. A similar method connected with presentation of such a situation was presented by Sheen et al. [6]. In order to fulfill the condition of stability (Courant condition) of algorithm the size of step is as follows:
Computational Methods and Experimental Measurements XII 617 z t = =. 441ps ε µ 2 (4) In order to simplify the considerations of the analyzed area a transformation has been made. All values have been related to the smallest size. The reason for this method is presented in next subsection. z 32 x x 4 y 12 y 14 z y 62 x Figure 4: Analyzed microstrip aerial. As a result, the area grew and this was caused by the growth of the path represented by the FDTD cells, - - 4 x 46 y for frequency f = 5GHz; - - 32 x 4 y for frequency f = 2.4 GHz; In Fig. 4, using an example of microstrip antenna with resonance frequency f=7.5 GHz, an area is analysed by final differences method in time domain. In the case of antenna with a round shape radiator I have used the same size cells in all directions. This method is the result of the necessity of obtaining an exact enough illustration of microstrip structure. In a program, boundary conditions type PML have been used because of the advantages. With these conditions there is no need to build other equations and take other assumptions due to mediums in an area of interest. As a result we may implement practical conditions as in the case of general solution, with only one difference. In case of k= there is no need to make conditions, because according to general laws the value of the electromagnetic field component equals zero. There is no need to rebuild boundary conditions with regard to this space. In order to enlarge the efficiency of the algorithm, the space connected with the given planes has been divided into three component parts. For example, dividing plane x-y is presented in Fig. 6. Two external elements require implementation of boundary conditions and a central element doesn t. It limits the number of needed equations and the size of matrix connected with given parameters. Using the presented algorithm, two structures presented in Fig. 5 have been analyzed. As a result of simulation, the following graphs present the distribution of field intensity (Figs. 5 & 6) and the parameter S 11. Figs. 6(a), (b) & (c) are representative of a rectangular structure and Fig. 6(d) is for a circular structure. In order to obtain the exact results of measurements in whole range of frequency, calculations have been done many times.
618 Computational Methods and Experimental Measurements XII Figure 5: Distribution of field electric intensity along feeding line (T = 3). (a) Aerial with resonance frequency 7.5 GHz. (b) Aerial with resonance frequency 5GHz. (c) Aerial with resonance frequency 2.4 GHz. (d) Circular planar radiator with resonance frequency 7.5 GHz. Figure 6: Distribution component E z the field electric intensity in moment T=5. 6 Conclusion Analysis of the characteristics presented in Fig. 7(a) allows for the determination of resonance frequency, which in this case is 7.5 GHz. What s more it is possible in an area of frequency close to 15 GHz for additional resonance responding to higher frequencies. Similarly, analysis of Fig. 7(b) determines the resonance frequency of the second antenna, which was equal to 5 GHz, while second
Computational Methods and Experimental Measurements XII 619 S11[dB] 5,1 2,4 4,7 7 9,3 11,6 13,9 16,2 18,5-5 -1-15 -2-25 f[ghz] S11[dB] 5,1 1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 9,1 1,1 11,1-5 -1-15 -2-25 f[ghz] (a) Aerial with resonance frequency 7.5 GHz. (b) Aerial with resonance frequency 5GHz. S11[dB] -5-1 -15-2.1.6 1.1 1.6 2.1 2.6 3.1 3.6 4.1 4.6 5.1 5.6 6.1 6.6 f[ghz] (c) Aerial with resonance frequency 2.4 GHz. S11[dB] 5,1 1,7 3,3 4,9 6,5 8,1 9,7 11,3 12,9 14,5 16,1 17,7 19,3-5 -1-15 -2-25 f[ghz] (d) Aerial with circular shape radiator. Figure 7: Frequency characteristic of entrance impedance (the parameter S 11 ). additional resonance occurred at a frequency of 1 GHz. In case of the third antenna [Fig. 7(c)] the resonance occurred in frequencies of 2.4 GHz and 4.8 GHz. In Fig. 7(d) there is a certain displacement of frequencies, probably resulting from limited dimensions of cells and illustrating the circular structure. The problem of discretisation, regardless of the time consuming nature, is one of the main disadvantages of analyzed method that may be minimized but not omitted. Obtained results besides those mentioned above where the antenna had a circular structure, correspond to the theoretical calculations. It should be highlighted that the limitation is the calculation power of the calculating instrument. Time of analysis is dependent on analyzed structure however, in each case the parameters of analysis did not change, with exception to the dimensions of space. This means that the time taken to do the calculations may be dependent on the dimensions of the analyzed space. Models of the antennas and measurements are being made. We may observe that the FDTD method used in the process of analysis allows accurate enough parameters of analyzed structure to be made. References [1] Munson, E., Conformal microstrip antennas and microstrip phased arrays. IEEE Trans. Antennas and Propagation, 28, pp. 74-78, 1974.
62 Computational Methods and Experimental Measurements XII [2] Yee, K.S., Numerical solution of initial boundary value problems involving Maxwell s equations in isotropic medias. IEEE Trans. Antennas and Propagation, 14, pp. 32 37, 1966. [3] Railton, C.J., Daniel, E.M., Paul, D.L. & McGreen, J.P., Optimized absorbing boundary conditions handicap the analysis of planar circuits using the finite difference time domain method. IEEE Trans. Antennas and Propagation, 41, pp. 29 296, 1993. [4] Bi, Z., Wu, K., Wu, Ch. & Litva, J., A dispersive boundary condition handicap microstrip component analysis using the FDTD method. IEEE Trans. Antennas and Propagation, 4, pp. 774 777, 1992. [5] Taflowe, A., Advance in computational electrodynamics The Finite - Difference Time Domain, Artech House Boston, 1995. [6] Sheen, D.M., Ali, S.M., Abouzahra, M.D. & Kong, J.A., Application of the three - dimensional finite difference time domain method, the analysis of planar microstrip circuits. IEEE Trans. Microwave Theory and Techniques, 38, pp. 849-857, 199. [7] Bi, Z., Wu, K., Wu, Ch. & Litva, J., Accurate characterization of planar printed antennas using finite difference time domain method. IEEE Trans. Antennas and Propagation, 4, pp. 526-533. [8] Reinex, A. & Jecko, B., Analysis of Microstrip Patch Anntenas Using Finite Difference Time Domain. IEEE Trans. Antennas and Propagation, 37, pp. 1361 1369, 1989.