Produção Nacional
Revisado por pares
Fast and accurate synthesis of electronically reconfigurable annular ring monopole antennas using particle swarm optimisation and artificial bee colony algorithms
2016; Institution of Engineering and Technology; Volume: 10; Issue: 4 Linguagem: Inglês
10.1049/iet-map.2015.0106
ISSN1751-8733
AutoresEduardo Jorge Brito Rodrigues, Hertz Wilton de Castro Lins, Adaildo G. D’Assunção,
Tópico(s)Satellite Communication Systems
ResumoIET Microwaves, Antennas & PropagationVolume 10, Issue 4 p. 362-369 Research ArticleFree Access Fast and accurate synthesis of electronically reconfigurable annular ring monopole antennas using particle swarm optimisation and artificial bee colony algorithms Eduardo Jorge Brito Rodrigues, Corresponding Author Eduardo Jorge Brito Rodrigues rodriguesejb@hotmail.com UFRN Department of Communications Engineering, Federal University of Rio Grande do Norte, Caixa Postal 1655, CEP 59072–970, Natal, RN, Brazil Anatel, National Agency of Telecommunications, CEP 58043–010, Joao Pessoa, PB, BrazilSearch for more papers by this authorHertz Wilton Castro Lins, Hertz Wilton Castro Lins UFRN Department of Communications Engineering, Federal University of Rio Grande do Norte, Caixa Postal 1655, CEP 59072–970, Natal, RN, BrazilSearch for more papers by this authorAdaildo Gomes D'Assunção, Adaildo Gomes D'Assunção UFRN Department of Communications Engineering, Federal University of Rio Grande do Norte, Caixa Postal 1655, CEP 59072–970, Natal, RN, BrazilSearch for more papers by this author Eduardo Jorge Brito Rodrigues, Corresponding Author Eduardo Jorge Brito Rodrigues rodriguesejb@hotmail.com UFRN Department of Communications Engineering, Federal University of Rio Grande do Norte, Caixa Postal 1655, CEP 59072–970, Natal, RN, Brazil Anatel, National Agency of Telecommunications, CEP 58043–010, Joao Pessoa, PB, BrazilSearch for more papers by this authorHertz Wilton Castro Lins, Hertz Wilton Castro Lins UFRN Department of Communications Engineering, Federal University of Rio Grande do Norte, Caixa Postal 1655, CEP 59072–970, Natal, RN, BrazilSearch for more papers by this authorAdaildo Gomes D'Assunção, Adaildo Gomes D'Assunção UFRN Department of Communications Engineering, Federal University of Rio Grande do Norte, Caixa Postal 1655, CEP 59072–970, Natal, RN, BrazilSearch for more papers by this author First published: 01 March 2016 https://doi.org/10.1049/iet-map.2015.0106Citations: 6AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract The synthesis of a simple electronically reconfigurable annular ring monopole antenna using a designing optimisation process based on particle swarm optimisation (PSO) and artificial bee colony (ABC) algorithms is proposed. Several antenna dimensions are selected as objective functions of PSO and ABC and their best solutions are considered as the optimised dimensions of the antenna geometry. Two radio-frequency p–i–n diodes, connecting the antenna feeding line to two microstrip stubs are used to change the antenna frequency response from ultra-wideband (UWB), from 3.1 to 10.6 GHz, to narrowband (NB) operation about 5.8 GHz. The antenna design and simulation are performed using Ansoft high-frequency structure simulator software. PSO and ABC algorithms are written in Java language. Thereafter, antenna prototypes are fabricated and measured for validation purposes. Simulation and measurement results are obtained showing good agreement. The measured optimised impedance bandwidths of the UWB and NB bands are up to 128 and 23%, respectively. Additionally, simulated and measured radiation patterns are very similar when the reconfigurable antenna is operating in OFF-state (UWB sensing antenna) and in ON-state (NB transmitting antenna) modes, indicating the proposed antenna geometry as a promising candidate for applications such as UWB and cognitive radio. 1 Introduction High-efficiency wireless networks standards that support chat applications, instant messaging and streaming require high-quality and small-size radio-frequency (RF) antennas [1]. Different shapes of microstrip antennas such as square or circular have been widely used due to their light weight, small size and low fabricating costs. Annular ring antennas have been studied experimental and theoretically, and are frequently used because of their interesting properties including improved surface wave efficiency, enhanced bandwidth (BW) and increased gain [2, 3]. Its main applications are in radar, satellite, indoor communications [4] and ultra-wideband (UWB), being most recently used in cognitive radio systems (CRSs) [5]. Antennas for CRS should be able to be wideband, capable to sense the wireless channel over a wide BW to identify idle bands, as well as be narrowband (NB) and adaptively adjust its frequency response, dynamically covering idle bands and avoiding the occupied bands [6]. The development of CRS and the need for frequency-agile front ends which will support frequency reconfigurable antennas has been receiving growing attention in the past decades [7]. Diverse methods of reconfiguring the antenna parameters have been used including mechanically changing antenna shape, changing the material properties of the antenna, using RF switches to activate or deactivate different parts of the antenna [8]. To implement reconfigurable antennas, different kinds of RF switches can be highlighted such as those using field-effect transistor (FET), RF micro-electromechanical system (MEMS), RF p–i–n diodes and varactor diodes [9-12]. In [9], an L-shaped slot reconfigurable antenna was proposed. RF p–i–n diodes and lumped capacitors located at specific positions were used to create short circuits across the slot. In [10], the antenna reconfiguration was achieved using gallium arsenide (GaAs) FET switches to connect multiple stubs to antenna feed line. In [11], a reconfigurable UWB antenna based on p–i–n and varactor diodes was presented. The selection of an operating filtering band was achieved by tuning the varactor and switching the p–i–n diode to the slot antenna. In [12], an RF MEMS-based antenna was presented. The reconfiguration was achieved by MEMS switch, which enables changing the resonance frequency. The RF p–i–n diode technique has been widely used in recent works [13-18]. Thanks to advantages such as fast switching time, low-cost and intrinsic properties such as ease to handle in laboratories and integrate to microstrip line circuits the RF p–i–n diode technique is pointed out in this paper. Characteristics of printed microwave structures can be improved using evolutionary computing, inspired on the abilities of, e.g. social insects. Recently, these techniques are receiving increasing research attention. A vast number of studies have reported on the success of using bioinspired algorithms for solving electromagnetic problems such as synthesis and optimisation of printed microwave circuits [19-25]. The use of particle swarm optimisation (PSO) algorithm in the optimisation of Yagi–Uda antenna design parameters such as element spacing and lengths was presented in [20], clearly showing the robustness of PSO in antenna optimisation. An overview of optimisation algorithms for antennas and microwave circuit was presented in [22], pointing out results such as antenna gain, return loss optimisation as well as measured results. In [23], design of conceptual frequency reconfigurable wideband E-shaped antenna using PSO was presented. Ideal switches were used illustrating the reconfiguration as concept proof, presenting good agreement between simulations and measurements. Artificial bee colony (ABC) algorithm was employed to synthesise linear antenna arrays in [24]. The element spacing determined by the algorithm reduced the side-lobe level of the array to satisfactorily low values. In [25], the design and optimisation of a planar spiral antenna for RF identification application using ABC algorithm was presented. Antenna minimisation and gain maximisation are some optimisation goals. Obtained results show that ABC can be efficiently applied to antenna design problems. This paper proposes the synthesis of an electronically reconfigurable annular ring antenna using PSO and ABC swarm intelligence optimisation algorithms. The main targets of the algorithms are reducing S11 and enhancing antenna BW. The proposed antenna is able to operate over the UWB range, from 3.1 to 10.6 GHz, as well as NB, about 5.8 GHz centre frequency. The rejection of the undesired frequencies occurs by changing from OFF-state to ON-state two RF p–i–n diodes connected to metallic stubs, placed around antenna feed line. Antenna designing dimensions are selected as objective functions of the algorithms. The best solutions given by the metaheuristics are considered as the optimised dimensions for the synthesised antennas. Design and simulations are performed using ANSOFT high-frequency structure simulator (HFSS) software. PSO and ABC algorithms are written in Java language. According to obtained dimensions, optimised antennas are designed using HFSS and prototypes are fabricated. Actual resistors, inductors and RF p–i–n diodes are embedded on antenna prototypes for measures. Compared with other related studies on reconfigurable antenna optimisation [13, 23, 25], this paper presents as particular advantage the simple layout of the electronically reconfigurable sensing and transmitting antenna for UWB and CRS, giving simulated and measured results of synthesised actual reconfigurable antennas (with RF p–i–n diodes). 2 Reconfigurable antenna structure Fig. 1 presents the layout of the proposed antenna that is designed on a single-layered substrate circuit board. The annular element, tuning stubs, DC bias and microstrip 50 Ω feeding line are on the top of the substrate as shown in Fig. 1a. The antenna side view is depicted in Fig. 1b. The bottom face, Fig. 1c, contains the truncated ground plane with a slit. The antennas are fabricated on FR-4 fibreglass substrate with ɛr of 4.4, loss tangent (tan δ) of 0.02 and a thickness of 1.6 mm. The antenna dimensions are [in millimetres (mm)]: W = L = 40.15; W1 = 14.80; W2 = 1.0; W3 = 2.34; L1 = 20.37; L2 = 2.0; a = 5.5, b = 9.2; Wb1 = 16.67; Wb2 = 1.1; Wb3 = 0.7; Lb1 = 9.5; Lb2 = 4.4; and Lb3 = 2.95. Design dimensions W4, L3 and L4 are selected to be optimised using bioinspired algorithms as described in next sections. These dimensions consist a solution space for HFSS parametric analyses, assumed to be within the intervals (in mm) W4 from 2.30 to 2.38; L3 from 2.55 to 2.91; and L4 from 19.15 to 19.45. Fig. 1Open in figure viewerPowerPoint Proposed annular ring reconfigurable antenna geometry a Top view b Side view c Bottom view To enable electronical reconfiguration, two commercial surface mountable devices (SMDs) SKYWORKS® RF p–i–n diodes, model SMP1340–011LF, are embedded on the prototypes in order to connect the microstrip feeding line to the parasitic stubs. Moreover, two current regulating SMD 47 Ω resistors and two commercial SMD Coilcraft® choke inductors, model 0604HQ-2N6XJL, are embedded in the proposed antenna geometry. In Fig. 1b, hmax is 2.8 mm, taking into account antenna thickness plus the SMD maximum heights. In the proposed model, the RF p–i–n diodes are used to behave as ‘ON/OFF’ RF switch. By reverse-biasing the p–i–n diode, the RF switches are in OFF-state. Otherwise, by forward-biasing the p–i–n diodes, the switches are in ON-state. The resulting solid state switch has a switching time much faster than any mechanical switch or relay. This model can be extended to other RF structures such as frequency selective surfaces (FSSs) or RF filters. In ON-state, the RF p–i–n diode has an equivalent circuit corresponding to an inductor in series with a resistor, while in OFF-state, the equivalent circuit consists of an inductor in series with a parallel capacitor and a resistor [9, 13]. The RF p–i–n diodes are modelled for numerical simulation as: series resistance under forward bias RS = 1.2 Ω; reverse parallel resistance RP = 5000 kΩ; diode total capacitance at zero or reverse bias CT = 0.14 pF; and lead inductance L = 0.45 nH (the same value for ON-state and OFF-state). 3 PSO algorithm The PSO is a stochastic computation technique based on the movement of natural groups of animals such as flocks of flying birds or fish schooling [26]. As in other algorithms, a population of individuals exists and the PSO algorithm is based on population survey. Each individual from the population is called ‘particle’ and represents a potential solution to a problem. In this sense, the particles can evolve by cooperation and competition among the individuals themselves through adjusting its flying, according to its own flying experience and the flying experience of some of its neighbours [19, 27]. As described in [22], the number of particles in the swarm is called the population size. When the algorithm starts running, each particle i moves as if they are in the nature through the search space , where n corresponds to the number of optimisation variables. Each particle is treated as a point in . Initially, particles from random places with random velocities looking for the best solutions are observed. The parameters of the solution i are encoded as the position of the particle i, . There are at least three important parameters of a particle: velocity, (in each iteration, a particle i changes its position according to the velocity); the best position of the particle, (position where the best solution was found by particle i so far) and the global best position of all particles i, (best overall value so far). The implemented PSO algorithm works as following. First, the number of particles is determined. Then, position and velocity of each particle are assigned as random values. Next, initial values of pbest and pglobal are defined. The particles are evaluated through the objective functions of the algorithm to determine initial pglobal. The particles exchange information about the results they obtained and according to these information they accelerate in the direction of their own pbest and simultaneously toward the pglobal, so the trajectory of the particle is alternated between these two goals. In this way, the particles explore entire search space to find the point that gives better result and, in the end, the trajectory to the pglobal. The particle velocity vi is updated according to [27] (1)where ω is called inertia weight that controls the value of old velocity vi. Parameters c1 and c2 are named accelerate coefficients. Both r1 and r2 are stochastic numbers, uniformly distributed between [0, 1]. In (1), notation ‘:=’ indicates that the value of vi will be updated in the next iteration. After the velocity is updated, each particle moves to another position pi according to its velocity, based on the following equation (2)Immediately after the movement of each particle, the overall amount of particles are evaluated once again through the optimisation functions of the algorithm in order to update, if necessary, the values of pbest and pglobal. The routine can be finished according to user settings such as the number of iterations. PSO algorithm can be applied to solve engineering and electromagnetic problems including antenna synthesis and RF circuit design as presented in [19, 20, 22, 23]. 4 ABC algorithm Honey bees have fantastic natural organisational abilities. For this reason, they are some of the most extensively studied social insects. Swarm intelligence ABC algorithm was introduced in [28] and has been used to solve and model different problems. The algorithm simulates the food foraging behaviour of honey bees in nature. An ABC consists of three groups of bees: employed, onlookers and scout. In ABC algorithm, first half of the colony consists of employed artificial bees and the second half constitutes the onlookers. The number of the employed bees or the onlooker bees is equal to the number of solutions in the population. An onlooker bee is the one who waits on the dance area for making decision to choose a food source. The bees that come into the hive and share the nectar information with the bees waiting on the dance area of the hive are called employed. For each food source, there is only one employed bee, i.e. the number of employed bees is equal to the number of food sources. The solutions of the optimisation problem in the algorithm are represented by the position of the food sources. The quality of the associated solution corresponds to the nectar amount of a food source. At the beginning of the process, a randomly distributed initial population P(C = 0), which contains a number of food sources (SN) equal to the number of employed bees (BN), is generated. Each solution of the algorithm xi, where i = {1, 2, …, SN}, is a D-dimensional vector, where D is the number of optimisation parameters [21]. After initialisation, the population of the positions start the search for food sources through repeated cycles C = {1, 2, …, MCN}, where MCN is the maximum number of cycles. An artificial employed or onlooker bee probabilistically produces a modification on the position in its memory for finding a new food source and test the nectar amount of the new source (solution). As presented in [28], the artificial bees randomly select a food source position and produce a modification on the one existing in their memory as described in equation below (3)where j ∈ {1, 2, …, D} and k ∈ {1, 2, …, SN} are random indexes. In (3), a candidate solution vi from the old solution xi can be generated. Although k is determined randomly, it has to be different from i. The parameter ϕij is uniformly distributed random number within [−1, 1]. Once the nectar amount of the new source is higher than that of the previous food source, the bee memorises the new position and forgets the old one. Otherwise, it keeps the position of the previous food source. The next step occurs when the employed bees complete the sharing of position information and the nectar information of the food sources with the onlooker bees on the dance area. In the nature, they perform the so called waggle dance to transmit this information. Since information about all the best sources is available to an onlooker bee on the dance area, it will probably watch numerous dances and chooses to employ itself at the most profitable source [29]. In a similar way to that carried out by the employed bee, the onlooker produces a modification on the position in its memory and checks the nectar amount of the candidate source. Providing that its nectar is higher than that of the previous one, the bee memorises the new position and forgets the old one. When the food source of an employed bee is exhausted by the employed and onlooker bees (i.e. MCN comes up to its limit value) and some food source cannot be improved, this food source is assumed to be abandoned and its onlooker bee becomes a scout [28]. In ABC algorithm, a food source is selected by an onlooker bee on the probability value pi associated with that food source, according to the following expression [28] (4)where fiti is the fitness value of the solution i, evaluated by its employed bee and proportional to the nectar amount of the food source in the position i. This algorithm can be used to antenna design optimisation as presented in [21, 24, 25]. 5 Details of simulations and measurements Initially, the design of a no optimised antenna is carried out in order to validate the design of the proposed antenna. The top layer dimensions in Fig. 1a are taken into account for simulations. The initial bottom layer dimensions are (in mm): W4 = 2.38, L3 = 2.91 and L4 = 19.45. Switching the RF p–i–n diodes from OFF-state to ON-state, implies on changing the antenna operation from UWB (sensing antenna over the range 3.1–10.6 GHz) to NB (centre frequency at 5.8 GHz). Electromagnetic simulations are carried out by ANSOFT HFSS. Measurements are done using Agilent E5071C-2K5. All the prototypes are fabricated on FR-4 fibreglass substrates. The overall dimension can reach 40.15 × 40.15 × 2.8 mm3. The first step toward implementing the optimisation is to define the parameters to be optimised. The antenna design dimensions W4, L3 and L4 are chosen because they are important impedance matching parameters that directly influence on the antenna impedance BW (BW %) and reflection coefficient S11 magnitude. Then, a set of values that consists the solution space are established to these bottom layer dimensions. These values are determined to perform the second step of the proposed analyses: the parametric simulations carried out by HFSS. The dimensions, in mm, are assumed to be within the intervals: 2.30 ≤ W4 ≤ 2.38; 2.55 ≤ L3 ≤ 2.91; and 19.15 ≤ L4 ≤ 19.45. Then, each proposed simulation scenario provides a set of solutions that forms the search space, where there is the optimum solution for the optimisation problem. This solution should be found after the numerical analysis performed by the optimisation algorithms. It is important to say that the presented optimisation model can easily be extended to other patch antenna layouts such as fractal or square formats. Once the search space is established, the PSO and ABC algorithms performances are tested and compared by a convergence test. The following step of the process is the application of PSO and ABC algorithms. The metaheuristics have as objective search for optimal values of W4, L3 and L4 in order to find maximum antenna BW and minimum S11. The results given by the algorithms will represent the points where there is the best impedance matching between the antenna feed line and the rectangular slit placed in the antenna ground plane. Once the PSO and ABC optimisation results are obtained, new optimised antenna designs are implemented and simulated in HFSS. Furthermore, modified and synthesised antenna prototypes are fabricated. At this point, the main expectation of this paper is that the synthesised antennas have enhanced performance, i.e. higher impedance BW and better S11, in comparison with the no optimised design. The fabricated PSO optimised antenna is shown in Fig. 2. Fig. 2Open in figure viewerPowerPoint Fabricated PSO optimised antenna (top layer left) and bottom view (right) At the time of their execution, both ABC and PSO algorithms generate new individuals according to their own heuristics. The evaluation function receives from the search space new values of BW and S11 through the interpolation method used for numerical analysis called apache commons math, present in Java library. The interpolation by bivariate and trivariate functions, from this same Java library, is used to implement PSO and ABC numerical analysis in this paper. The Tricubic Spline Interpolating Function is used in both cases. The spline function implements a real function in the construction of the PSO and ABC algorithms with the variations minimum and maximum values of the design parameters W4, L3 and L4. In Table 1, is presented an example of interpolation of one candidate solution from the values of W4, L3 and L4. Table 1. Sample interpolation performed in the optimisation process W4, mm L3, mm L4, mm BW, MHz S11, dB 2.34656 2.85507 19.26459 944.5745245145599 −19.10 The objective function of the optimisation problem can be mathematically formulated according to the following expression (5)In (5), the f function is an application of the concept of distance measures, widely used in various applications with high data dimensionality, computational intelligence techniques and time series. In this paper, it is used the Euclidean distance that is one of most straightforward similarity measure for time series [30, 31]. In PSO algorithm, the neighbourhood topology of the particles is related to how efficient is the algorithm in the exploitation of the search space [32]. The implemented PSO algorithm has a swarm with 50 particles in a global neighbourhood topology that is characterised by the simultaneous sharing of information such as velocity and position of each particle between all particles. Other PSO coefficients are 0.45 for the inertia weight, 2.80 for particle social trust and 0.45 for particle individual trust. Table 2 presents important parameters of the developed PSO and ABC algorithms. Table 2. Parameters of the developed PSO and ABC algorithms PSO ABC number of interactions/cycles 50 50 population size, particles/bees 50, 100, 150 50, 100, 150 6 Results Simulated and measured results from the no optimised antenna were obtained and demonstrated that the proposed design is able to cover the UWB band (3.1–10.6 GHz), when both RF p–i–n diodes were in OFF-state, whereas the antenna covers an NB about 5.8 GHz, when both RF p–i–n diodes were in ON-state. These results were stored to be compared with those for the optimised antennas geometries. Then, HFSS parametric simulations were performed based on investigating the ground plane dimensions of the no optimised antenna, taken into account the following ranges (in mm): 2.30 ≤ W4 ≤ 2.38; 2.55 ≤ L3 ≤ 2.91; and 19.15 ≤ L4 ≤ 19.45. The obtained BW and S11 results were obtained and saved to form the search space that will be explored by the optimisation algorithms. A convergence performance test of PSO and ABC algorithms was done. Considering a sample run of PSO and ABC, it was observed that the PSO algorithm finds its best ratings results relatively faster than ABC algorithm. However, the evaluation values are almost similar in the iteration number 100. The ABC has MCN and SN as its only common control parameters. As consequence, ABC algorithm is as simple and flexible as PSO employing less control parameters. After testing the metaheuristics and its performances, PSO and ABC routines were applied to scrutinise the lines and columns of the constructed search space. Then, each algorithm found its global optimum values for W4, L3 and L4. When the routines were finished, the PSO and ABC algorithms gave as results the optimised dimensions of the antenna's bottom layer. The PSO results were (in mm): W4 = 2.36, L3 = 2.88 and L4 = 19.15. The ABC algorithm results were (in mm): W4 = 2.33, L3 = 2.86 and L4 = 19.15. On the basis of these results, novel HFSS simulations were performed and the fabrication of the optimised prototypes was done. Thus, simulated and measured results were finally obtained and can be compared. Fig. 3 shows a comparison between simulated and the measured OFF-state (UWB) antenna S11 (decibels) as function of frequency (gigahertz). In Fig. 3a, the simulated values of S11 are below −10 dB in most part of the frequencies when in OFF-state. This is due to the relatively low insertion loss of the p–i–n diodes, ∼0.8–1 dB for each diode, when comparing with ideal metallic pads instead of actual devices. The measured results in Fig. 3b confirm that these insertion losses do not affect the final performance of the antennas which in most of the measured frequency band from 2 to 12 GHz, the S11 is lower than −10 dB. When the RF p–i–n diodes are in OFF-state, the measured impedance BW is about 128% (PSO optimisation) and 122% (ABC optimisation). When comparing the frequency responses depicted in Figs. 3a and b, it is observed good agreement between simulated and measured results. Fig. 3Open in figure viewerPowerPoint S11 (decibels) versus frequency (GHz) for both p–i–n diodes in OFF-state a Simulated results b Measured results Fig. 4 shows the obtained results of the RF p–i–n diodes in ON-state. The measured frequency response presented in Fig. 4b implies correspondent input impedance of ∼49 Ω at the frequency 5.8 GHz to the no optimised proposed antenna. The ON-state measured impedance BWs are 11.74 and 23.81% for PSO and ABC optimised antennas, with centre frequencies about 5.87 and 5.75 GHz, respectively. Simulated results are reasonably close to the measured ones, though the observed differences. The ABC optimised antenna presented the higher measured BW (1.37 GHz), 26% greater than its corresponding simulation result (1.01 GHz) and about 49% higher than measured PSO optimised result. Fig. 4Open in figure viewerPowerPoint S11 (decibels) versus frequency (GHz) for both p–i–n diodes in ON-state a Simulated results b Measured results Table 3 presents a summary of the simulated and measured results of the OFF-state (UWB) antennas, whereas Table 4 presents the results for the ON-state (NB) antennas. Parameters such as centre frequency, BW, impedance BW and S11 at the centre frequency are highlighted. Table 3. Summary of simulated and measured results for the OFF-state Simulated results – HFSS Measured results – Agilent E5071C-2K5 No optimisation PSO optimised ABC optimised No optimisation PSO optimised ABC optimised centre frequency, GHz 7.26 7.23 7.28 6.81 7.31 6.94 BW, GHz 9.26 9.54 9.43 8.13 9.37 8.53 BW, % 127.55 131.95 129.44 119.47 128.09 122.82 S11, dB at centre frequency −12.99 −13.83 −14.51 −11.01 −12.95 −20.28 Table 4. Summary of simulated and measured results for the ON-state Simulated results – HFSS Measured results – Agilent E5071C-2K5 No optimisation PSO optimised ABC optimised No optimisation PSO optimised ABC optimised centre frequency, GHz 5.95 5.985 6.015 5.72 5.875 5.755 BW, GHz 0.94 0.95 1.01 0.78 0.69 1.37 BW, % 15.80 15.87 16.79 13.64 11.74 23.81 S11 (decibels) at centre frequency −21.33 −20.27 −18.73 −27.5 −14.52 −21.1 The simulated radiation patterns obtained for the OFF-state are presented in Fig. 5. The no optimised azimuthal diagrams, XY (φ = 0°) and elevation diagrams XZ (φ = 90°) are compared with the PSO and ABC optimised results
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