IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 5, 2006 191 Calculation of Small Antennas Quality Factor Using FDTD Method S. Collardey, A. Sharaiha, Member, IEEE, and K. Mahdjoubi, Member, IEEE Abstract The complexity in the evaluation of the radiation quality factor Q of small antenna lies in the calculation of the stored energy. Previous authors proposed several analytical expressions for simple antennas to obtain the Q, which is of practical importance because of its relationship to the antenna bandwidth. This letter presents a general numerical method to calculate the Q using finite-difference time-domain (FDTD) method giving us accurate results and can be applied to complex antennas. Examples of a linear dipole antenna, a loop antenna and an inverted L-shaped antenna along with their results are presented and compared to analytical results of the literature. Index Terms Electrically small antennas, quality factor. I. INTRODUCTION THE performance properties of small antennas include the antenna s resonant frequency, resonant radiation resistance, efficiency and the quality factor Q. The quality factor is of practical interest when designing small antennas because of its relation to the antenna bandwidth. Previous works done by several authors [1] [6] starting from Chu [2] predict a lower bound on the radiation Q of small antennas enclosed in the smallest circumscribing sphere (as defined by Wheeler [1]). This lower bound of Q defined by previous authors is usually too low since it doesn t take into account the energy stored inside the sphere. We can see this in Fig. 1 where we present the Q calculated by McLean [6] versus of an electrically small dipole antenna enclosed in a sphere of radius a where is the wave number defined by where is the wavelength. Recently, Thiele et al. [7] have used the far field approach to obtain a more realistic bound of Q as well as McLean using an oblate spheroidal volume [8]. And Geyi [9] has also proposed analytical formulas for simple antennas taking into account the energy stored in all the space. We note that Geyi s [9] formulas give more exact value of Q as shown in Fig. 1. More recently, another approach exists based on the derivative of the antenna input impedance developed by Yaghjian [10], to obtain the Q and the bandwidth. In all these analytical studies except for the one done by Yaghjian, the authors assume that the antenna will be resonated with a lossless matching circuit so as to obtain a real input Fig. 1. Comparison between the quality factors calculated by McLean, Thiele and Geyi. (Note: a is the Wheeler sphere radius and k is the wave number wave.) impedance. Therefore, the quality factor Q can be given by this definition where is either the time average stored electric or the magnetic one and is the average radiated power and is the radian frequency. In this letter, we propose a simple technique to compute numerically the lower bound of the antenna radiation Q using the finite-difference time-domain (FDTD) method. The FDTD software based on a numerical method allows to take taking into account all the energy that can be stored by the antenna into the smallest sphere and consequently to obtain a realistic value of the Q based on the following fundamental definition [11]: (1) (2) Manuscript received December 16, 2005 revised February 3, 2006. S. Collardey is with the IETR (Institut d Electronique et de Télécommunications de Rennes, UMR CNRS 6164), IUT Saint-Malo, Rennes, France (e-mail: sylvain.collardey@univ-rennes1.fr). A. Sharaiha and K. Mahdjoubi are with the IETR, University of Rennes 1, Rennes, France. Digital Object Identifier 10.1109/LAWP.2006.873947 where is the time average energy stored in the system (3) (4) 1536-1225/$20.00 2006 IEEE
192 IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 5, 2006 This method can be applied to all kinds of antennas in any environment and it becomes useful when one calculates or designs antennas of certain complexity (noncanonical). In the Section II, we present the computation of Q using the FDTD method. In Section III, we compare our new approach with the analytical results obtained previously by McLean [6] and Geyi [9] for an electrically small dipole antenna, an inverted L-shaped antenna as well as a loop antenna. II. CALCULATION OF Q USING THE FDTD METHOD The FDTD method consists in the numerical computing of the electric and magnetic field components in every cell of a discretised volume at every moment of the discretised time. The FDTD computational volume is discretised into elementary cubic cells called Yee cells. The maximum values of fields components and are computed in each cell [12]. To evaluate the radiation Q, we first calculate the average total electric and magnetic energies using these following wellknown formulas: (5) Fig. 2. Dipole antenna considered in the FDTD method. where is the volume of the unit cell and is equal to. We also calculate the average radiated electric and magnetic energies by the following expressions: (6) (7) Fig. 3. Method of calculation. (8) where and are the radiated far fields. The average stored energy is then obtained by subtracting the total radiated field energy away from the total energy. Hence, the expression used to calculate Q here is (9) Knowing the radiated power in the far field, we can easily calculate the quality factor. III. RESULTS As an example, we consider a small dipole of size that fits a cubic volume of radius (see Fig. 2). The dipole antenna has a length of 27 mm and a radius equal to 0.1 mm. The antenna is fed by a voltage source with 50 impedance. The FDTD spatial steps are equal to 1 mm and the total cube dimension is equal to 92 mm. The nonpropagating energy is plotted to estimate the quantity stored in the near field of the antenna. The total and radiated energies are determined for each discrete volume (see Fig. 3). Fig. 4. Variations of stored energy when ka is equal to 0.85. Then, we compute the nonpropagating energy in. The plotted value denotes the supplementary stored energy when the dimension volume increases from 0 to 92 mm. In Fig. 4, the variations of the average stored electric and magnetic energies are plotted versus the distance for. We can see that the nonpropagating energy is mainly stored in the near field of the antenna and tends to vanish for. Consequently, the quality factor value calculated previously by Chu [2], McLean [6] and others will surely be too low since the main stored energy is not taken into account.
COLLARDEY et al.: CALCULATION OF SMALL ANTENNAS QUALITY FACTOR USING FDTD METHOD 193 Fig. 7. Current distributions versus distance. Fig. 5. Q of dipole antenna. Fig. 8. method. Inverted L-shaped antenna characteristics considered in the FDTD angular, as Geyi has supposed in his Q computation. However, if is greater than, the current distribution becomes sinusoidal. The two previous remarks explain why the Q obtained by Geyi becomes slightly different to the one calculated by the FDTD method when increases. In another example, we consider a small inverted L-shaped antenna shown in Fig. 8, which has also been studied by Geyi [9] and proposed the following analytical expression of Q: (10) Fig. 6. Difference between stored and total energy. The Q calculated using the FDTD method and based on the (9) is greater than the Q obtained by McLean and almost equivalent to the one obtained by Geyi [9], as shown in Fig. 5. The difference between the two curves is due to the fact that in Geyi s analysis he neglects the radiated energy and his formula based on a triangular current distribution is more accurate when. To confirm this, we plot the total electric energy in the volume as well as the total radiated energy versus (Fig. 6). We can notice that for small antennas with, the value of the total radiated energy is too small and can actually be neglected. Furthermore, as shown in Fig. 7, the current distribution on the antenna dipole depends on the frequency of interest. In this figure, the current distribution is plotted versus the distance from the antenna center, for different values of. We can notice that if remains lower than, the current distribution is nearly tri- where is the radius of the antenna, is the length of the vertical part of the antenna and is the length of the horizontal section. The FDTD spatial steps are equal to 1 mm and the cubic volume dimension is equal to 92 mm. Here again the Q calculated by FDTD is similar to the one calculated by Geyi. The difference is due to the same reasons indicated above (see Fig. 9). Finally, as a last example, we take a small loop antenna, which was also examined by Geyi. The analytical expression in this case is (11) where is the radius of the loop and is the radius of the wire. The quality factor computed by the FDTD method is close to the one obtained by Geyi (see Fig. 10).
194 IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 5, 2006 IV. CONCLUSION In this letter, a new and simple method is presented to calculate numerically the antenna radiation Q by using the FDTD method. The results obtained for different kind of antennas shows a good agreement between FDTD and the analytical approaches proposed by Geyi. The main advantage for using FDTD is that we can obtain more accurate results of Q factor for small antennas and can be easily used to characterize more complex structures. REFERENCES Fig. 9. Inverted L-shaped antenna Q. Fig. 10. Loop antenna Q. [1] H. A. Wheeler, Fundamental limitations of small antennas, Proc. IRE, vol. 35, pp. 1479 1484, Dec. 1947. [2] L. J. Chu, Physical limitations on omni-directional antennas, J. Appl. Phys., vol. 19, pp. 1163 1175, Dec. 1948. [3] R. E. Collin and S. Rothschild, Evaluation of antenna Q, IEEE Trans. Antennas Propagat., vol. AP-12, pp. 23 27, Jan. 1964. [4] R. L. Fante, Quality factor of general idea antennas, IEEE Trans. Antennas Propagat., vol. AP-17, pp. 151 155, Mar. 1969. [5] R. C. Hansen, Fundamental limitations in antennas, Proc. IEEE, vol. 69, pp. 170 182, Feb. 1981. [6] J. S. McLean, A re-examination of the fundamental limits on the radiation Q of electrically small antennas, IEEE Trans. Antennas Propagat., vol. 44, pp. 672 676, May 1996. [7] G. A. Thiele, P. L. Detweiler, and R. P. Penno, On the lower bound of the radiation Q for electrically small antennas, IEEE Trans. Antennas Propagat., vol. 51, pp. 1263 1269, Jun. 2003. [8] H. D. Holtz and J. S. McLean, Limits on the radiation Q of electrically small antennas restricted to oblong bounding regions, in Proc. IEEE Antennas Propagat. Int. Symp., vol. 4, Jul. 11 16, 1999, pp. 2702 2705. [9] W. Geyi, A method for the evaluation of small antenna Q, IEEE Trans. Antennas Propagat., vol. 51, pp. 2124 2129, Aug. 2003. [10] A. D. Yaghjian and S. R. Best, Impedance, bandwidth, and Q of antennas, IEEE Trans. Antennas Propagat., vol. 53, pp. 1298 1324, Apr. 2005. [11] H. G. Booker, Energy in Electromagnetism, ser. IEE Electromagnetic Wave Series 13. Stevenage, U.K.: Peter Peregrinus, 1982. [12] S. Collardey, A. Sharaiha, and K. Mahdjoubi, Evaluation of antenna radiation Q using FDTD method, Electron. Lett., vol. 41, no. 12, Jun. 2005.
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