- Email: [email protected]
Time domain optimization of wire antennas loaded with passive linear elements using GA
Engineering Analysis with Boundary Elements 27 (2003) 345–349 www.elsevier.com/locate/enganabound
Time domain optimization of wire antennas loaded with passive linear elements using GA M. Ferna´ndez Pantojaa, A. Rubio Bretonesa, A. Monorchiob, R. Go´mez Martı´na,* a
Dpto de Electromagnetismo, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain b Department of Information Engineering, University of Pisa, Via Diotisalvi 2, I-56126 Pisa, Italy Received 9 May 2002; revised 29 May 2002; accepted 29 June 2002
Abstract In this communication Genetic algorithms (GAs) optimizers are applied, directly in the time domain, to the design of broadband thin-wire antennas. The broadband characteristics of the antennas are achieved by loading the wires with combinations of passive loads, trying to maximize the fidelity of the response while keeping the efficiency as high as possible. Multi-objective GAs are used in conjunction with a computer code based on the solution, in the time domain, of the electric field integral equation for wires with arbitrarily connected loads. Representative results are presented. q 2002 Elsevier Science Ltd. All rights reserved. Keywords: Genetic algorithms; Broadband antennas; Time domain; Integral equation methods; Thin-wire antennas
1. Introduction It is well known that the broadband characteristics of thin-wire antennas can be controlled by using resistive loads. Appropriate values of resistors located along the antenna geometry reduce unwanted reflections of the current at the ends of the wires enhancing the travelingwave component of the current and therefore extending the bandwidth. One set of widely used resistors follows a continuous profile, denoted the Wu –King profile [1]. As resistively loaded antennas suffer from low efficiency, several authors have proposed broadband designs including capacitors, constructed as gaps along the wires [2], combinations of resistive and capacitive loading [3] and even other reactive elements [4] to mitigate the problem. In this communication we propose to apply Genetic algorithms (GAs) techniques [5] to select the appropriate values of passive linear loads, to be located at specific points along thin-wire antennas to achieve a maximum bandwidth with the minimum possible losses. The loads can be either resistors (R ), capacitors (C ) or a combination of these elements. The GA optimization of wire antennas has been addressed in the literature quite extensively in * Corresponding author. Tel.: þ 34-958-243224; fax: þ 34-958-242353. E-mail address: [email protected] (R. Go´mez Martı´n).
the frequency-domain [5] and recently, in the time domain [6] for the case of resistively loaded antennas. The contribution of this work is to extend the method proposed in Ref. [6] to include the possibility of using both resistive and reactive loads. GAs are applied directly over the transient response of the antennas. To work explicitly in the time domain provides wide-band information from a single execution of the marching-on-in-time procedure and allows us to optimize the entire operating spectrum of the antenna in a single run, as opposed to the frequency domain formulation which repeatedly calls for the electromagnetic solver at many frequencies within the band of interest. The formulation employed is based on the solution of the electric field integral equation (EFIE) for thin-wire antennas with arbitrarily connected passive linear elements using a marching-on-in-time procedure to obtain the transient response of the antennas [7]. The main characteristics of the GAs are a real coded formulation and the capability of multi-objective optimization of both the input fidelity, defined as a time domain parameter that accounts for the input impedance, and the energy radiated by the wire antenna. A key point in the application of the GAs is that the range of values allowed for R and C should result in realistic designs. Next some more details are given on the specific implementation of the GAs and on the numerical method
0955-7997/03/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 9 5 5 - 7 9 9 7 ( 0 2 ) 0 0 1 2 2 - 4
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M.F. Pantoja et al. / Engineering Analysis with Boundary Elements 27 (2003) 345–349
applied to solve the EFIE. Finally, representative results are presented and discussed.
2. Numerical formulation The study has been carried out using a computer code that calculates, in the time domain, the currents induced on a thin-wire antenna when it is excited by an arbitrary electromagnetic signal. It is based on the solution of the EFIE using the method of moments in the time domain (MoMTD) [8]. For the case of a thin wire with passive loads connected in series at specific points along the wire, the EFIE which is deduced in detail in Ref. [7] is given by " # ~ ›Iðs0 ;t0 Þ R ~ R s^ ð s^0 ›Iðs0 ;t0 Þ 0 0 s^·E ðs;tÞ¼ ds0 2 3 qðs ;t Þþ 2 4p10 Cðs0 Þ cR2 ›s0 R c R ›t 0 ~i
› 1 ðt þIðs;tÞRl ðsÞþLl ðsÞ Iðs;tÞþ Iðs;tÞdt ›t Cl ðsÞ 0
ð1Þ
where C(s0 ) is the wire axis contour, s^0 and s^ are the unitary vectors over the source and field points, separated ~ I (s0 ,t0 ) and q(s0 ,t0 ) denote the unknown by a distance R; current and linear charge distribution at source point s0 at ~ i ðs;tÞ is the field applied to retarded time t0 ¼ t 2 R/c; E the observation point s and Rl(s ), Ll(s ), Cl(s ) are the values of the resistance, inductance and capacitance per unit length, respectively, located at the observation point s. The charge can be expressed in terms of the current by means of the equation of continuity. Eq. (1) is solved using the point matching form of the MoMTD. The wire length is approximated by straight segments and the time is divided into temporal intervals. Then, the unknown current distribution is expanded in terms of a set of two-dimensional, Lagrangian interpolation basis functions and the resulting equation is enforced at the center of the space-time intervals applying delta of Dirac functions as weighting functions. This transforms Eq. (1) into a matrix equation at each time step which is solved for the current I(s0 ,t ) by marching on in time [7]. When the passive RLC loads are arbitrarily connected at any point on the wire for example in parallel, Eq. (1) is not valid, but it has been demonstrated in Ref. [7], reasoning on the discretized version of Eq. (1), that at each time step any arbitrary connection of loads can be replaced by an equivalent configuration consisting of a resistor in series with an extra voltage source, the values of which depend on previously calculated currents on the wire. This allows the application of the same procedure described above for the calculation of the unknown currents. The parameters we need for the specific multi-objective GAs adopted in this work are the field radiated at any
particular direction, the calculation of which is straightforward from the current distribution, and the fidelity in the response, which is computed from the cross-correlation between input voltage and output current [9] by using the expression ls^VI ðtÞl ffi rVI ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi ð2Þ s^V ð0Þs^I ð0Þ max where sˆVI, sˆV and sˆI are the cross-and auto-covariances of both the input voltage and output current signals, respectively, computed directly in the time domain. From the field calculated at the direction of maximum radiation (umax, wmax) the normalized radiated energy is evaluated as X rad lE ðumax ; wmax ; tÞl2 ð3Þ 1ðu; wÞ ¼ t
Real coded GAs are employed so one chromosome, which characterizes one individual, is represented by a set of real numbers (or genes). As an example, for an antenna subdivided into Ns segments, which is loaded in all the segments with both resistive and capacitive elements, the GA needs to select the value of 2 Ns parameters, that is, the value of R and C at all the possible locations on the wire. If the antenna is excited symmetrically around its center, a chromosome will be considered comprising only Ns genes. Apart from the real codification previously described, GA uses a single-point crossover operator with a probability of crossover of 80%. For the mutation operator, it uses an inverse error function with a typical deviation of 0.1 times the range of the gene, and a probability of 0.4%. In the optimization of one parameter, the best individual in each generation is selected by a weighted roulette wheel scheme (a flowchart of the optimization process can be viewed in Fig. 1). For multi-objective optimizations the selection operator is based on the Pareto domination concept, with a triangular niching function added to guarantee the diversity in the population [5].
3. Results We now present some illustrative numerical results to demonstrate the effectiveness of the numerical procedure presented. As a starting point in the design of a broadband thin-wire antenna using only capacitors, we have considered the monopole antenna proposed in Ref. [2]. The wire antenna has a length of 163 mm and its radius is 1.5 mm. It is loaded, in the original design, with an inductance of L ¼ 1.257 £ 1028 H at the feed point, and four capacitances C1 ¼ 2.122 £ 10212 F, C2 ¼ 1.273 £ 10212 F and C3 ¼ C4 ¼ 6.366 £ 10212 F, which are located at 32.1, 62.3, 90.5 and 140.9 mm, respectively, from the feed point. GA optimization schemes have been applied to select other
M.F. Pantoja et al. / Engineering Analysis with Boundary Elements 27 (2003) 345–349
347
Fig. 2. Temporal evolution of the current at the feed point for the capacitively loaded monopole. Inset: broadband wire monopole.
Fig. 1. Flowchart of GA-MoMTD.
possible values for the capacitors at the same points as with the original monopole so that the fidelity of the antennas is improved. The values allowed for the capacitors range from C min ¼ 0.1 £ 10212 F to C max ¼ 20 £ 10212 F which roughly correspond to 0.1C2 – 10C1. The antenna is excited at its center with a voltage 2
V i ðtÞ ¼ cosð2pf0 tÞe2g ðt2tm Þ
2
where f0 ¼ 0.9 GHz, g ¼ 6 £ 108 s21 and tm ¼ 3.577 ns. This specific temporal dependence was chosen because its frequency spectrum has components in the interval between 0.6 and 1.2 GHz, which is where the antenna presents broadband performance according to the results shown in Ref. [2]. The antenna was discretized by using 16 segments (4 genes in each chromosome), and 512 temporal intervals of duration 33.958 ps were computed in every case. The optimization seeks a maximum fidelity response at the feeding point, and the selected final values, after 400 generations of 40 individuals each, are C 1 ¼ 0.1597 £ 10212 F, C 2 ¼ 13.612 £ 10212 F, C3 ¼ 3.562 £ 10212 F and C4 ¼ 15.363 £ 10212 F. A fixed inductance of L ¼ 2 £ 1028 H is added at the feeding point in order to compensate the negative values obtained for the reactance. In Fig. 2 we show the temporal evolution of the current at the feed point of
the antenna and compare it to that of the original monopole. Fig. 3 shows the real and imaginary parts of the input impedance of the monopole against frequency of both the original and the optimized monopole, calculated via Fourier transform from the temporal response. As the fidelity parameter is 0.8763 in the original design, and 0.9575 in the GA optimized monopole, we conclude that, regarding its broadband characteristics, the new design has a better performance than the one proposed in Ref. [2]. As a second example, we consider a multi-objective optimization of a 0.5 m long monopole antenna. The design specifications aim for the maximum broadband input impedance response up to 2.5 GHz, and also seeks to maximize the energy of the radiated electric field in the broadside direction. The antenna, with a radius of 3.8 mm, is fed at the center with a Gaussian, delta-gap, voltage source of 2
V i ðtÞ ¼ e2g ðt2tm Þ
2
Fig. 3. Input impedance of the monopole antennas loaded with capacitors.
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M.F. Pantoja et al. / Engineering Analysis with Boundary Elements 27 (2003) 345–349
Fig. 4. Multi-objective optimization diagram for the 100 and 9500 generations.
where g ¼ 5 £ 109 s21 and tm ¼ 0.4292 ns. For the MoMTD analysis, the monopole is discretized by using 16 segments. The GA scheme selects, at each segment except the delta-gap feeding, the optimal value of a RC series circuit (30 genes per chromosome). The resistance values allowed for the GA optimization have been selected as a discrete set from the Wu –King distributed resistance profile [1], that is, from R ¼ 0 to 446.125 V, and the capacitances have been chosen so that their corresponding impedances in the range of frequencies of interest are also contained in the Wu – King resistive distribution which gives Cmin ¼ 3.125 £ 10215 F and Cmax ¼ 62.5 £ 10212 F. Fig. 4 shows a diagram of the niched Pareto GA distribution after 100 and 9500 generations of 100 chromosomes each. Although the program was stopped
Fig. 5. Temporal evolution of the electric field radiated at u ¼ 90 by two different RC loaded monopoles compared with that of the Wu– King antenna.
after the 9500th generation, there was no appreciable evolution of the diagram once generation number 5000 was reached. The corresponding monopole loaded with a Wu – King resistive profile [1] is added to the diagram as an
Table 1 Values of resistors and capacitors for the individuals A and B Segment
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
POINT A
POINT B
Resistance (V)
Capacitance (nF)
Resistance (V)
Capacitance (nF)
0 0 0.10 0 0 0.97 7.76 126.61 114.35 23.98 3.64 0 0.32 0.36 0
55.83 30.96 0 32.03 0 61.28 0 37.46 26.10 35.44 20.13 0 48.98 30.24 43.56
0 0 1.18 39.38 28.16 6.56 65.62 77.96 91.26 21.44 3.65 0 0.28 0.34 1.68
55.83 31.79 2.08 39.97 0.14 63.82 8.53 37.65 26.02 38.34 20.13 0.01 48.82 28.19 48.40
Fig. 6. Input impedance of the monopole antennas loaded with resistors and capacitors compared with that of the Wu–King antenna.
M.F. Pantoja et al. / Engineering Analysis with Boundary Elements 27 (2003) 345–349
example of a high fidelity-low efficiency antenna. With the aid of this Pareto distribution, the designer is able to choose the individual that best fits the specifications compromising between low losses and wide bandwidth. In particular we have chosen the two individuals marked as A and B, respectively, in Fig. 3, the first one with fidelity 0.7115 and energy 0.2433 and the second one with fidelity 0.8626 and energy 0.1050. The specific values of R and C corresponding to each segment of individuals A and B are given in Table 1. Fig. 5 shows the temporal behavior of the electric field radiated at the broadside direction compared to that of the same antenna loaded with the Wu – King resistive profile. The input impedance of the previous cases is presented in Fig. 6a and b. It can be observed that both antennas (A and B) produce higher radiated energy than the Wu –King case at the expense of losing some broadband properties. Note that none of the antennas found by GA and represented in Fig. 3 has improved the fidelity of the Wu – King monopole as a consequence of having used the Pareto optimization. As it is explained in Ref. [5], Pareto schemes improve the front line of optimization in several parameters, resulting in a slow convergence in the solution. Searching in a single direction reduces the computational time. That single direction of search can be calculated weighting the multiple parameters of optimization with the aid of the Pareto front.
4. Conclusions In this paper the possibility of getting broadband information from a single run using time domain methods is exploited to optimize, via GA, the transient response of passively loaded thin-wire antennas. In particular the method has been applied to design, using arbitrarily connected loads, broadband antennas with the aid of the Pareto multi-objective method. As a result antennas with
349
better performance than the Wu –King design have been found. Apart from these particular results the goal of this paper is to emphasize the possibility of using time-domain methods in conjunction with GA to improve the broadband behavior of loaded antennas.
Acknowledgements This work was supported in part by the Spanish Ministerio de Ciencia y Tecnologı´a-FEDER, under the Project TIC 2001-3236-C02-01 and TIC 2001-2364-C03-03.
References [1] Wu TT, King RWP. The cylindrical antenna with nonreflecting resistive loading. IEEE Trans Antennas Propag 1965;13(5):369–73. [2] Popovic BD, Dragovic MB, Djordjevic AR. Optimization of antenna admittance. Chichester: Research Studies Press; 1982, chapter 8. [3] Montoya TP, Smith GS. A study of pulse radiation from several broadband loaded monopoles. IEEE Trans Antennas Propag 1996;44: 1172–82. [4] Halpern B, Mittra R. Broadband whip antennas for use in HF communications. IEEE AP-S Int Symp, Vancouver, Canada June 1985;II:763–7. [5] Rahmat-Samii Y, Michielssen E. Electromagnetic optimization by genetic algorithms. New York: Wiley; 1999. [6] Ferna´ndez Pantoja M, Monorchio A, Rubio Bretones A, Go´mez Martı´n R. Direct GA-based optimization of resistively loaded wire antennas in the time domain. Electron Lett 2000;36:1988– 90. [7] Ferna´ndez Pantoja M, Rubio Bretones A, Go´mez Martı´n R. Time domain analysis of thin-wire loaded antennas using integral equations. IEEE Proc Microwave Antennas Propag 1997;45:343– 53. [8] Rubio Bretones A, Go´mez Martı´n R, Salinas A. DOTIG1, a timedomain numerical code for the study of the interaction of electromagnetic pulse with thin wires. Int J Comput Math Electr Electron Engng (COMPEL) 1989;8:39–61. [9] Allen OE, Hill DA, Ondreajka AR. Time-domain antenna characterizations. IEEE Trans Electromagnet Compat 1993;35(3):339 –46.
Time domain optimization of wire antennas loaded with passive linear elements using GA M. Ferna´ndez Pantojaa, A. Rubio Bretonesa, A. Monorchiob, R. Go´mez Martı´na,* a
Dpto de Electromagnetismo, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain b Department of Information Engineering, University of Pisa, Via Diotisalvi 2, I-56126 Pisa, Italy Received 9 May 2002; revised 29 May 2002; accepted 29 June 2002
Abstract In this communication Genetic algorithms (GAs) optimizers are applied, directly in the time domain, to the design of broadband thin-wire antennas. The broadband characteristics of the antennas are achieved by loading the wires with combinations of passive loads, trying to maximize the fidelity of the response while keeping the efficiency as high as possible. Multi-objective GAs are used in conjunction with a computer code based on the solution, in the time domain, of the electric field integral equation for wires with arbitrarily connected loads. Representative results are presented. q 2002 Elsevier Science Ltd. All rights reserved. Keywords: Genetic algorithms; Broadband antennas; Time domain; Integral equation methods; Thin-wire antennas
1. Introduction It is well known that the broadband characteristics of thin-wire antennas can be controlled by using resistive loads. Appropriate values of resistors located along the antenna geometry reduce unwanted reflections of the current at the ends of the wires enhancing the travelingwave component of the current and therefore extending the bandwidth. One set of widely used resistors follows a continuous profile, denoted the Wu –King profile [1]. As resistively loaded antennas suffer from low efficiency, several authors have proposed broadband designs including capacitors, constructed as gaps along the wires [2], combinations of resistive and capacitive loading [3] and even other reactive elements [4] to mitigate the problem. In this communication we propose to apply Genetic algorithms (GAs) techniques [5] to select the appropriate values of passive linear loads, to be located at specific points along thin-wire antennas to achieve a maximum bandwidth with the minimum possible losses. The loads can be either resistors (R ), capacitors (C ) or a combination of these elements. The GA optimization of wire antennas has been addressed in the literature quite extensively in * Corresponding author. Tel.: þ 34-958-243224; fax: þ 34-958-242353. E-mail address: [email protected] (R. Go´mez Martı´n).
the frequency-domain [5] and recently, in the time domain [6] for the case of resistively loaded antennas. The contribution of this work is to extend the method proposed in Ref. [6] to include the possibility of using both resistive and reactive loads. GAs are applied directly over the transient response of the antennas. To work explicitly in the time domain provides wide-band information from a single execution of the marching-on-in-time procedure and allows us to optimize the entire operating spectrum of the antenna in a single run, as opposed to the frequency domain formulation which repeatedly calls for the electromagnetic solver at many frequencies within the band of interest. The formulation employed is based on the solution of the electric field integral equation (EFIE) for thin-wire antennas with arbitrarily connected passive linear elements using a marching-on-in-time procedure to obtain the transient response of the antennas [7]. The main characteristics of the GAs are a real coded formulation and the capability of multi-objective optimization of both the input fidelity, defined as a time domain parameter that accounts for the input impedance, and the energy radiated by the wire antenna. A key point in the application of the GAs is that the range of values allowed for R and C should result in realistic designs. Next some more details are given on the specific implementation of the GAs and on the numerical method
0955-7997/03/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 9 5 5 - 7 9 9 7 ( 0 2 ) 0 0 1 2 2 - 4
346
M.F. Pantoja et al. / Engineering Analysis with Boundary Elements 27 (2003) 345–349
applied to solve the EFIE. Finally, representative results are presented and discussed.
2. Numerical formulation The study has been carried out using a computer code that calculates, in the time domain, the currents induced on a thin-wire antenna when it is excited by an arbitrary electromagnetic signal. It is based on the solution of the EFIE using the method of moments in the time domain (MoMTD) [8]. For the case of a thin wire with passive loads connected in series at specific points along the wire, the EFIE which is deduced in detail in Ref. [7] is given by " # ~ ›Iðs0 ;t0 Þ R ~ R s^ ð s^0 ›Iðs0 ;t0 Þ 0 0 s^·E ðs;tÞ¼ ds0 2 3 qðs ;t Þþ 2 4p10 Cðs0 Þ cR2 ›s0 R c R ›t 0 ~i
› 1 ðt þIðs;tÞRl ðsÞþLl ðsÞ Iðs;tÞþ Iðs;tÞdt ›t Cl ðsÞ 0
ð1Þ
where C(s0 ) is the wire axis contour, s^0 and s^ are the unitary vectors over the source and field points, separated ~ I (s0 ,t0 ) and q(s0 ,t0 ) denote the unknown by a distance R; current and linear charge distribution at source point s0 at ~ i ðs;tÞ is the field applied to retarded time t0 ¼ t 2 R/c; E the observation point s and Rl(s ), Ll(s ), Cl(s ) are the values of the resistance, inductance and capacitance per unit length, respectively, located at the observation point s. The charge can be expressed in terms of the current by means of the equation of continuity. Eq. (1) is solved using the point matching form of the MoMTD. The wire length is approximated by straight segments and the time is divided into temporal intervals. Then, the unknown current distribution is expanded in terms of a set of two-dimensional, Lagrangian interpolation basis functions and the resulting equation is enforced at the center of the space-time intervals applying delta of Dirac functions as weighting functions. This transforms Eq. (1) into a matrix equation at each time step which is solved for the current I(s0 ,t ) by marching on in time [7]. When the passive RLC loads are arbitrarily connected at any point on the wire for example in parallel, Eq. (1) is not valid, but it has been demonstrated in Ref. [7], reasoning on the discretized version of Eq. (1), that at each time step any arbitrary connection of loads can be replaced by an equivalent configuration consisting of a resistor in series with an extra voltage source, the values of which depend on previously calculated currents on the wire. This allows the application of the same procedure described above for the calculation of the unknown currents. The parameters we need for the specific multi-objective GAs adopted in this work are the field radiated at any
particular direction, the calculation of which is straightforward from the current distribution, and the fidelity in the response, which is computed from the cross-correlation between input voltage and output current [9] by using the expression ls^VI ðtÞl ffi rVI ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi ð2Þ s^V ð0Þs^I ð0Þ max where sˆVI, sˆV and sˆI are the cross-and auto-covariances of both the input voltage and output current signals, respectively, computed directly in the time domain. From the field calculated at the direction of maximum radiation (umax, wmax) the normalized radiated energy is evaluated as X rad lE ðumax ; wmax ; tÞl2 ð3Þ 1ðu; wÞ ¼ t
Real coded GAs are employed so one chromosome, which characterizes one individual, is represented by a set of real numbers (or genes). As an example, for an antenna subdivided into Ns segments, which is loaded in all the segments with both resistive and capacitive elements, the GA needs to select the value of 2 Ns parameters, that is, the value of R and C at all the possible locations on the wire. If the antenna is excited symmetrically around its center, a chromosome will be considered comprising only Ns genes. Apart from the real codification previously described, GA uses a single-point crossover operator with a probability of crossover of 80%. For the mutation operator, it uses an inverse error function with a typical deviation of 0.1 times the range of the gene, and a probability of 0.4%. In the optimization of one parameter, the best individual in each generation is selected by a weighted roulette wheel scheme (a flowchart of the optimization process can be viewed in Fig. 1). For multi-objective optimizations the selection operator is based on the Pareto domination concept, with a triangular niching function added to guarantee the diversity in the population [5].
3. Results We now present some illustrative numerical results to demonstrate the effectiveness of the numerical procedure presented. As a starting point in the design of a broadband thin-wire antenna using only capacitors, we have considered the monopole antenna proposed in Ref. [2]. The wire antenna has a length of 163 mm and its radius is 1.5 mm. It is loaded, in the original design, with an inductance of L ¼ 1.257 £ 1028 H at the feed point, and four capacitances C1 ¼ 2.122 £ 10212 F, C2 ¼ 1.273 £ 10212 F and C3 ¼ C4 ¼ 6.366 £ 10212 F, which are located at 32.1, 62.3, 90.5 and 140.9 mm, respectively, from the feed point. GA optimization schemes have been applied to select other
M.F. Pantoja et al. / Engineering Analysis with Boundary Elements 27 (2003) 345–349
347
Fig. 2. Temporal evolution of the current at the feed point for the capacitively loaded monopole. Inset: broadband wire monopole.
Fig. 1. Flowchart of GA-MoMTD.
possible values for the capacitors at the same points as with the original monopole so that the fidelity of the antennas is improved. The values allowed for the capacitors range from C min ¼ 0.1 £ 10212 F to C max ¼ 20 £ 10212 F which roughly correspond to 0.1C2 – 10C1. The antenna is excited at its center with a voltage 2
V i ðtÞ ¼ cosð2pf0 tÞe2g ðt2tm Þ
2
where f0 ¼ 0.9 GHz, g ¼ 6 £ 108 s21 and tm ¼ 3.577 ns. This specific temporal dependence was chosen because its frequency spectrum has components in the interval between 0.6 and 1.2 GHz, which is where the antenna presents broadband performance according to the results shown in Ref. [2]. The antenna was discretized by using 16 segments (4 genes in each chromosome), and 512 temporal intervals of duration 33.958 ps were computed in every case. The optimization seeks a maximum fidelity response at the feeding point, and the selected final values, after 400 generations of 40 individuals each, are C 1 ¼ 0.1597 £ 10212 F, C 2 ¼ 13.612 £ 10212 F, C3 ¼ 3.562 £ 10212 F and C4 ¼ 15.363 £ 10212 F. A fixed inductance of L ¼ 2 £ 1028 H is added at the feeding point in order to compensate the negative values obtained for the reactance. In Fig. 2 we show the temporal evolution of the current at the feed point of
the antenna and compare it to that of the original monopole. Fig. 3 shows the real and imaginary parts of the input impedance of the monopole against frequency of both the original and the optimized monopole, calculated via Fourier transform from the temporal response. As the fidelity parameter is 0.8763 in the original design, and 0.9575 in the GA optimized monopole, we conclude that, regarding its broadband characteristics, the new design has a better performance than the one proposed in Ref. [2]. As a second example, we consider a multi-objective optimization of a 0.5 m long monopole antenna. The design specifications aim for the maximum broadband input impedance response up to 2.5 GHz, and also seeks to maximize the energy of the radiated electric field in the broadside direction. The antenna, with a radius of 3.8 mm, is fed at the center with a Gaussian, delta-gap, voltage source of 2
V i ðtÞ ¼ e2g ðt2tm Þ
2
Fig. 3. Input impedance of the monopole antennas loaded with capacitors.
348
M.F. Pantoja et al. / Engineering Analysis with Boundary Elements 27 (2003) 345–349
Fig. 4. Multi-objective optimization diagram for the 100 and 9500 generations.
where g ¼ 5 £ 109 s21 and tm ¼ 0.4292 ns. For the MoMTD analysis, the monopole is discretized by using 16 segments. The GA scheme selects, at each segment except the delta-gap feeding, the optimal value of a RC series circuit (30 genes per chromosome). The resistance values allowed for the GA optimization have been selected as a discrete set from the Wu –King distributed resistance profile [1], that is, from R ¼ 0 to 446.125 V, and the capacitances have been chosen so that their corresponding impedances in the range of frequencies of interest are also contained in the Wu – King resistive distribution which gives Cmin ¼ 3.125 £ 10215 F and Cmax ¼ 62.5 £ 10212 F. Fig. 4 shows a diagram of the niched Pareto GA distribution after 100 and 9500 generations of 100 chromosomes each. Although the program was stopped
Fig. 5. Temporal evolution of the electric field radiated at u ¼ 90 by two different RC loaded monopoles compared with that of the Wu– King antenna.
after the 9500th generation, there was no appreciable evolution of the diagram once generation number 5000 was reached. The corresponding monopole loaded with a Wu – King resistive profile [1] is added to the diagram as an
Table 1 Values of resistors and capacitors for the individuals A and B Segment
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
POINT A
POINT B
Resistance (V)
Capacitance (nF)
Resistance (V)
Capacitance (nF)
0 0 0.10 0 0 0.97 7.76 126.61 114.35 23.98 3.64 0 0.32 0.36 0
55.83 30.96 0 32.03 0 61.28 0 37.46 26.10 35.44 20.13 0 48.98 30.24 43.56
0 0 1.18 39.38 28.16 6.56 65.62 77.96 91.26 21.44 3.65 0 0.28 0.34 1.68
55.83 31.79 2.08 39.97 0.14 63.82 8.53 37.65 26.02 38.34 20.13 0.01 48.82 28.19 48.40
Fig. 6. Input impedance of the monopole antennas loaded with resistors and capacitors compared with that of the Wu–King antenna.
M.F. Pantoja et al. / Engineering Analysis with Boundary Elements 27 (2003) 345–349
example of a high fidelity-low efficiency antenna. With the aid of this Pareto distribution, the designer is able to choose the individual that best fits the specifications compromising between low losses and wide bandwidth. In particular we have chosen the two individuals marked as A and B, respectively, in Fig. 3, the first one with fidelity 0.7115 and energy 0.2433 and the second one with fidelity 0.8626 and energy 0.1050. The specific values of R and C corresponding to each segment of individuals A and B are given in Table 1. Fig. 5 shows the temporal behavior of the electric field radiated at the broadside direction compared to that of the same antenna loaded with the Wu – King resistive profile. The input impedance of the previous cases is presented in Fig. 6a and b. It can be observed that both antennas (A and B) produce higher radiated energy than the Wu –King case at the expense of losing some broadband properties. Note that none of the antennas found by GA and represented in Fig. 3 has improved the fidelity of the Wu – King monopole as a consequence of having used the Pareto optimization. As it is explained in Ref. [5], Pareto schemes improve the front line of optimization in several parameters, resulting in a slow convergence in the solution. Searching in a single direction reduces the computational time. That single direction of search can be calculated weighting the multiple parameters of optimization with the aid of the Pareto front.
4. Conclusions In this paper the possibility of getting broadband information from a single run using time domain methods is exploited to optimize, via GA, the transient response of passively loaded thin-wire antennas. In particular the method has been applied to design, using arbitrarily connected loads, broadband antennas with the aid of the Pareto multi-objective method. As a result antennas with
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better performance than the Wu –King design have been found. Apart from these particular results the goal of this paper is to emphasize the possibility of using time-domain methods in conjunction with GA to improve the broadband behavior of loaded antennas.
Acknowledgements This work was supported in part by the Spanish Ministerio de Ciencia y Tecnologı´a-FEDER, under the Project TIC 2001-3236-C02-01 and TIC 2001-2364-C03-03.
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