Reconfigurable Antenna Design and Optimization

Proposing Rules and Guidelines for Reconfigurable Antenna Design and Optimization Using Graph Models

Joseph Costantine, Sinan al-Saffar, Christos G. Christodoulou, Chaouki T. Abdallah Electrical and Computer Engineering Department, University of New Mexico, Albuquerque, NM, USA

Abstract- This paper proposes rules and guidelines for the that will lead to an optimal reconfigurable antenna design optimization of reconfigurable antennas. In this paper graph using graph components. models are presented as tools to optimize the design of reconfigurable antennas. We study the characteristics of II. RECONFIGURABLE ANTENNAS CATEGORIES reconfigurable antennas that are grouped, categorized and graph modeled according to suggested rules. Design steps are defined Reconfigurable antennas come in a large variety of and examples are given. The presence of redundancy in an different shapes and forms [2]. These antennas exhibit different antenna structure is investigated and optimal configurations are forms of reconfiguralibity. We group them into 4 main suggested. categories based on their reconfigurability function as: - A reconfigurable resonance frequency antenna I. INTRODUCTION Reconfigurability, when used in the context of - A reconfigurable radiation pattern antenna antennas, is the capacity to change an individual radiator’s fundamental operating characteristics through electrical, - A reconfigurable polarization antenna mechanical, or other means [1]. The reconfiguration of such an antenna is achieved through an intentional redistribution of the - Different combinations of the above stated currents or, equivalently, the electromagnetic fields of the categories antenna’s effective aperture, resulting in reversible changes in the antenna impedance and/or radiation properties [2]. Many In the case of reconfigurable resonance frequency techniques can be used to achieve the reconfiguration of an antennas, frequency tuning occurs for different antenna antenna. Most of these techniques resort to switches, diodes or configurations [5-23]. This frequency tuning is shown in capacitors. Other techniques resort to mechanical alterations resonance shifting in a return loss data. In the case of like a rotation or bending of a certain antenna part. reconfigurable radiation pattern antennas, radiation patterns Reconfigurable antennas are mostly used on systems change in terms of shape, direction or gain [24-26]. In the case that require changing from one application into another. Some of a reconfigurable polarization antenna, polarization types of the reconfigurable antenna applications reside in Multiple change for every antenna configuration [27-28]. In the last Input Multiple Output (MIMO) channels, in cognitive radio, on category, antennas exhibit many properties combined together laptops, in cellular phones and many other systems. to yield a reconfigurable return loss with reconfigurable In this paper we divide reconfigurable antennas into polarization etc… [29-32]. four main categories and then we further classify them into 7 Reconfigurable antennas can also be further classified groups, based on their reconfiguration technique. We use into 7 main groups based on their reconfiguration techniques: graphs to model these reconfigurable antennas. - Group 1: Antennas using switches Graph models are defined as a pervasive modeling - Group 2: Antennas using diodes abstraction implemented in data structures. They are widely - Group 3: Antennas using capacitors or varactors used in computer science and in the development of - Group 4: Antennas using physical angular networking algorithms [3]. Graphs also find applications in self alteration assembly robotics where they are used to model the physics of - Group5:Antennas using different biasing the particles by describing the outcomes of interactions among networks them [4]. Herein we use graph models to optimize the - Group 6: Antenna arrays structure of a reconfigurable antenna and to understand its - Group 7: Antennas using reconfigurable feeding physical behavior. We set specific rules for graph modeling of networks different types of reconfigurable antennas and conclude with a set of design steps that may be used to get an optimal III. INTRODUCTION TO GRAPHS reconfigurable antenna design. We present several examples A graph can be defined as the collection of vertices elaborating our rules and design steps and suggest formulas that may be connected together with lines called edges. A simple labeled graph is represented by G = (V, E) where V is a Valid for: set of vertices, E is a set of pairs or edges from V. A graph can This rule is valid for multi-part antennas of group 1. be either directed or undirected. The edges in a directed graph have a certain determined direction while this is not the case in Constraints: an undirected graph shown in Figure 1, where a graph of 5 The connection between each two parts has a distinctive vertices (A,B,C,D,E) are connected by 7 edges [3]. Vertices angular direction. The designer defines a reference axis that may represent physical entities and edges between them in the represents the direction that the majority of parts have with graph represent the presence of a function resulting from each other or with a main part. The connections between the connecting these entities. If one is proposing a set of parts are represented by the edges. The edges’ weights guidelines for antenna design, then a possible modeling rule represent the angles that the connections make with the may be to create an edge between two vertices whenever their reference axis. A weight W=1 is assigned to an edge physical connection results in a meaningful antenna function. representing a connection that has an angle 0˚ or 180˚ with the reference axis, otherwise a weight W=2 is assigned to the edge as shown in equation 1.

Wij  Pij (1) 1 A  0 or180  ij Where Pij  2 Otherwise  Fig. 1. An example of an undirected graph  Where Aij represents the angle that the connection i,j form with Edges may have weights associated with them to the reference axis. represent costs or benefits that are to be minimized or maximized. For example if a capacitor is connecting two end Example: points of a system and these end points are represented by two As an example we will take the antenna shown in vertices in a graph, then the edge connecting these two vertices Figure 2 [5-8] and model it into a graph following rule 1.a. The has a weight equal to the capacitance of that capacitor. The basic antenna is a 130˚ balanced bowtie. A portion of the weight of a path is defined as the sum of the weights of its antenna corresponds to a two-iteration fractal Sierpinski dipole. constituent edges. The remaining elements are added (three elements on each In some cases it is useful to find the shortest path side) to make the antenna a more generalized reconfigurable connecting two vertices. This notion is used in graph structure. algorithms in order to optimize a certain function. The shortest This antenna’s graph modeling follows rule 1.a, path distance in a non weighted graph is defined as the where the different parts of the antenna (triangles added) are minimum number of edges in any path from vertex s to vertex connected by MEMS switches as shown in Figure 2. The v, otherwise if the graph is weighted then the shortest path vertices in the graph model represent the triangles added. The corresponds to the least sum of weights in a particular path. In edges connecting these vertices represent the connection of the a reconfigurable antenna design a shorter path may mean a corresponding triangles by MEMS switches. If a switch is shorter current flow and thus a certain resonance associated activated to connect triangle T1 to triangle T’1 shown in Figure with it. A longer path may denote a lower resonance frequency 2 then an edge appears between the vertex T1 and the vertex than the shorter path. T’1 as shown in the 1st state of the graph model in Figure 3. The direction of each connection is very important to the antenna function. In this design the connection between T1 and IV. GRAPH MODELING RULES T2 , T2 and T4, T’1 and T’3 , T’3 and T’6 are collinear with Different graph mapping rules apply for different the reference axis and as a result the edges representing these reconfigurable antenna groups. The graph mapping of a certain connections are weighted with W=1 and W=2 for the other antenna is governed by its structure and the reconfigurability connections. A weight represents a cost or a benefit of a certain techniques used in that particular structure. connection. In our case the cost of connecting parts at the same Herein an antenna is called a multi-part antenna if it is direction is less (w=1) than connecting parts at a deviated composed of an array of identical elements (triangular, direction (w=2). The graph modeling of this antenna is shown rectangular,…parts), otherwise it is called a single-part in Figure 3. antenna. Rule 1.b: The graph modeling of a single part antenna with switches bridging over slots is undirected with non-weighted Rule 1.a: The graph modeling of a multi-part antenna whose edges connecting vertices that represent the end points of parts are connected by switches is undirected with weighted each switch. edges connecting the vertices that represent the different parts. Valid for: This rule is valid for single part antennas of group 1. Constraints: In the case of switches bridging over multiple slots in one antenna structure the graph model takes into consideration one slot at a time.

Example: As an example we will take the antenna shown in Figure 4 [9]. This antenna is a triangular patch antenna with 2 slots incorporated in it. The authors suggested 5 swicthes to bridge over each slot in order to achieve the desired required functions. The graph modeling of this antenna following rule 1.b is shown in Figure 4 where vertices represent the end points each switch and edges represent the connections between these end points. When switch 1 is activated an edge appears between N1 and N’1 representing the 2 end-points of switch 1 as shown in Figure 4. The graph model of Figure 4 represents each slot at a time. For example N1 represent end-point 1 for switch 1 in slot 1 and end-point 1 for switch 1 in slot 2.

Fig. 2. The antenna Structure in [5-8]

Fig. 4. Antenna structure in [9] with graph modeling

Constraints: In addition to the constraints discussed in rule 1.a, the current flow direction in this case follows the directed edges through the conveniently biased diodes.

Example: As an example we will take the antenna in [10], shown in Figure 5. The lengths of the dipole-arm strip, and therefore the geometry of the antenna, can be changed using two PIN diode switches. In this way, it is possible to define two configurations for the antenna, one when both of the switches are turned on (“long” configuration) and another when they are Fig. 3. Graph model for different configurations of the antenna turned off (“short” configuration). The graph modeling of the in [6] antenna follows rule 2.a and shown in Figure 6. The vertices are the different parts of the antenna and the edges representing Rule 2.a: The graph modeling of a multi-part antenna whose the connections between the different antenna parts are parts are connected by diodes is directed along the current directed according to the current flow direction. All the edges direction through the diodes with weighted edges connecting have the same weight W=1 since they are all collinear with the vertices that represent the different parts. reference axis as shown in Figure 5.

Valid for: This rule is valid for multi-part antennas of group 2. edges connecting vertices that represent the different parts connected.

Valid for: This rule is valid for multi-part antennas of group 3.

Constraints: The edges’ weights in this case are calculated according to equation 2. All the capacitances of the different capacitors connecting the parts should be transformed to the same unit and then they should be normalized with respect to the largest capacitance. The weights represent the addition of the

normalized capacitances values with the values of Pij as shown in equation 2. Pij was discussed in rule 1.a.

W  P  C (2) ij ij ij normalized Fig. 5. Antenna Structure in [10] 1 A  0 or180  ij Where Pij  2 Otherwise  

Where Aij represents the angle that the connection i,j form with the reference axis. Cij represents the normalized capacitance of the capacitor connecting parts i and j.

Fig. 6. Antenna Graph Modeling In the case of variable capacitance diodes connecting the various parts instead of regular capacitors, then rule 3.a Rule 2.b: The graph modeling of a single part antenna with applies with the exception that the edges are directed diodes bridging over slots is directed along the current direction through the diodes with non-weighted edges connecting vertices that represent the end points of each diode.

Valid for: This rule is valid for single part antennas of group 2.

Constraints: The same constraints as rule 1.b apply.

Example: As an example we will take the antenna shown in Figure 7 [30]. This antenna achieves a reconfigurable return loss and Fig.7. Antenna Structure in [30] polarization between RHCP and LHCP. A probe fed square patch antenna with a pair of tuning stubs is designed for circular polarization performance and two orthogonal switchable slots are incorporated into the patch to control the resonant frequency. The switching is achieved via PIN diodes bridging over the slots as shown in Figure 7. The frequency tuning happens for every configuration. The graph model follows rule 2.b where vertices are the different end points of the diodes. When diode 1 is active an edge connects N1 and Fig.8. Antenna Graph modeling N’1 in the graph model and when diode 2 is active an edge connects N2 and N’2 in the graph model. The graph model is Example: shown in Figure 8. In this case we take the antenna shown in Figure 9 [11]. The antenna is a 2x2 reconfigurable planar wire grid antenna Rule 3.a: The graph modeling of a multi-part antenna with designed to operate in free space. Variable capacitors were parts connected by capacitors is undirected with weighted placed in the centers of 11 of the 12 wire segments that comprise the grid. The center of the 12th segment, located on Valid for: the edge of the grid, was reserved for the antenna feed. The This rule is valid for single part antennas of group 3. values of the variable capacitors were constrained to lie Constraints: between 0.1pF and 1 pF. These capacitors were then adjusted The graph should be undirected and weighted where the using a robust Genetic Algorithm (GA) optimization technique weights are defined in equation 3. in order to achieve the desired performance characteristics for Wij  Cijnormalized (3) the antenna. The graph modeling of this antenna follows rule 3.a and is shown in Figure 10. The vertices in this graph Where Cij represents the normalized capacitance of the model represent the different parts of the lines that are capacitor connecting end-points i and j. The capacitances connected together via a capacitor. values are calculated as discussed in rule 3.a. In the case of multiple slots, rule 1.b applies with the addition of equation 3.

In the case where a varactor is used instead of a capacitor then rule 3.b applies with the exception of the directed edges.

Example: As an example we take the antenna shown in Figure 11 [12]. The antenna is fed with an off-centered open circuited microstrip line with a 50 Ω impedance. Different variable capacitance diodes (varactors) values are used, and these Fig.9. Antenna structure in [11] varactor values are obtained by changing the biasing voltages. The graph modeling of this antenna follows rule 3.b. where the The values of the capacitors after the genetic optimization were vertices represent the end points of the different varactors. The not specified by the authors however let’s assume that the edges are directed according to the current flow and they are values are: weighted with different varactor values. The graph model is C1=0.1pF, C2=0.3pF, C3=0.4pF, C4=0.2pF, C5=1pF, shown in Figure 12. C6=0.5pF, C7=0.1pF, C8=0.8pF, C9=0.9pF, C10=1pF, C11=0.7 pF. The adjacency matrix A which is the matrix presenting all the weights values in the graph is shown below, where the weights were calculated according to equation 2.

Fig.10. Graph model of the antenna in [11] Fig.11. Antenna structure in [12]

 0 2.1 0 0 0 1.9 0 0 0    2.1 0 2.3 0 1.7 0 0 0 0   0 2.3 0 1.4 0 0 0 0 0     0 0 1.4 0 3 0 0 0 1.2 A   0 1.7 0 3 0 2.5 0 2 0    1.9 0 0 0 2.5 0 1.1 0 0   0 0 0 0 0 1.1 0 0 0     0 0 0 0 2 0 0 0 2.8   Fig.12. Graph modeling for the antenna in [12]  0 0 0 1.2 0 0 0 2.8 0  Rule 4: The graph modeling of an antenna using angular Rule 3.b: The graph modeling of a single part antenna change in its structure is undirected with weighted edges where capacitors are bridging over slots is undirected with connecting vertices that represent the different angles of the weighted edges connecting vertices that represent the end physical action. points of each capacitor. Valid for: Bending from 0˚ to 45˚ has to pass by 15˚ then the path from This rule is valid for antennas of group 4. A1 to A3 has to pass by A2 as shown in Figure 14. Constraints: The graph modeling this type of antennas is undirected since the angular change (bending or rotation) is reversible. The vertices represent the angles of this physical action. The weighted edges connecting the vertices represent the rotation or the bending process that is the state change from one angle to another. The weights represent constraints related to the system controlling the angular change like the rotation or the bending process i.e. time of rotation etc... Fig. 14. Graph modeling of the antenna in [31] Example: a. No bending, b. bending from 0˚ to 15˚, c. bending As an example we take the antenna shown in Figure 13 [31]. from 0˚ to 45˚, d. bending from 0˚ to 90˚ The antenna was fabricated over a sacrificial layer residing on the substrate. A thin layer of magnetic material is then Rule 5: The graph modeling of an antenna using biasing electroplated on the antenna surface. By etching away the networks to attach additional parts to each other is sacrificial layer between the antenna and substrate, the antenna undirected with weighted edges connecting vertices that is released and connected only by its feed line. When an represent the parts of the whole antenna system. external field is applied, the flexible region created at the junction between the released and unreleased microstrip line is Valid for: plastically deformed and the structure is bent by an angle. This This rule is valid for antennas of group 5. antenna exhibited a return loss tuning and a reconfigurable radiation pattern as detailed in [31]. The graph modeling Constraints: follows rule 4 where the bending angles are considered as The same constraints as rule 1.a apply. vertices. The physical bending is occurring as a response to an external field applied then removed when the antenna reaches a Example: rest angle. The time it takes for an antenna to reach that rest As an example we take the antenna shown in Figure 15 [13]. angle is very important in the antenna’s applications. The The antenna’s reconfiguration is achieved by turning ON or edges’ weights which are the costs that a designer must pay OFF various sections, to change the active length of the may represent in this case the time of bending. The different assembled monopole antenna structure. The main antenna weights can be evaluated as in equation 5: monopole is metal (Alumina) and the parts added are plasma wij  T (Ai  Aj ) (5) islands biased by 4 biasing networks as shown in Figure 15.

The weight Wij in this case represents the time it takes to bend The graph modeling of this antenna follows rule 5 where the the antenna from position i into position j. The adjacency different parts are the vertices and the edges represent the matrix A shown below can be evaluated numerically however connection of these parts by the activation of the different the exact numerical values depend on the fabricated system. biasing networks. The antenna’s graph model is shown in Figure 16.  0 T(A1  A2 ) T(A1  A3 ) T (A1  A4 ) T (A  A ) 0 T (A  A ) T (A  A ) A   2 1 2 3 2 4  T (A3  A1) T (A3  A2 ) 0 T (A3  A4 )   T (A4  A1) T (A4  A2 ) T (A4  A3 ) 0  The antenna is shown in Figure 13 and the graph modeling is shown in Figure 14.

Fig.15. The antenna Structure in [14] Fig. 13. Antenna Structure in [31] Rule 6: The graph modeling of an antenna array where different antennas are excited at different times is undirected In the graph model of Figure 14 A1 represents 0˚, A2 with weighted edges connecting vertices that represent the represents 15˚, A3 represent 45˚ and A4 represents 90˚. different antennas forming the array. the vertices are the different antennas on the different cube faces and the edges between them occur when the corresponding antennas are activated at the same time simulating their connection by the same feeding network and their radiated field coupling connection. The exact mutual coupling values between each 2 antennas were not specified in [25] however the weights are calculated according to equation 4 and are shown in the adjacency matrix A below. The graph model is shown in Figure 18.

 1 2  M12 1 M13 2  M14  2  M 2 2  M 2  M  Fig.16. Antenna Graph Model A   21 23 24  1 M 31 2  M 32 1 2  M 34    Valid for: 2  M 2  M 2  M 2 This rule is valid for antenna of group 6.  41 42 43 

Constraints: In the case where different antennas in an array system are excited at different times, the vertices in the modeling graph represent the different antennas. Undirected edges connecting different vertices represent the excitation presence of the corresponding different antennas at the same time. The angle that the antennas form with each other is of importance in the array function. The corresponding graph should be undirected with weighted edges where weights correspond to the Fig. 17. The antenna array in [25] antennas’ positions relatively to each other in addition to the mutual coupling in the case where mutual coupling is accounted for. M is the mutual coupling. All of the mutual coupling values should be expressed in the same unit and then they should be normalized with respect to the highest value.

The weights are calculated according to equation 4: W  P  M (4) ij ij ij normalized Fig.18. Graph model of the array antenna in [25] 1 A  0 or180  ij Rule 7: The rules defined previously in this section apply for the graph modeling of a reconfigurable feeding antenna Where Pij  2 Otherwise  where the reconfiguration is achieved in the feeding network.  Aij represents the angle that the antennas have with each other. Valid for: Mij represents the normalized mutual coupling amount between This rule is valid for antennas of group 7. antennas i and j. If there isn’t any mutual coupling between the antennas i and j then Mij=0. Constraints: The graph components in this case represent the feeding Example: components. All the rules constraints defined previously apply As an example we take the antenna shown in Figure 17 [25]. correspondingly. This antenna is a 3-Dimensional model. Four cube faces are chosen to reside in the same plane, facing the + x-direction and Example: + y-direction in a Cartesian coordinate system. All four As an example we take the antenna shown in Figure 19 [24]. antennas are similarly oriented with respect to the four cube This antenna is based on the parasitic antenna concept and it faces, with the primary plane and primary polarization realizes pattern diversity. The prototype is a three-element coincident to the plane of integration. The bottom face (-z- parasitic antenna array where aperture-coupled square patches direction) is used to the feed the structure and the top face are used as radiating elements. The slot selection results either (+zdirection) is unused in this work. The antennas are in an E-plane or H-plane coupling of the central patch with the mechanically fastened to the structure using nylon screws and adjacent parasitic patches. By switching ON a diode while the appropriately tapped receptacles on the cube faces. This other is OFF, the antenna can switch between horizontal or antenna exhibits reconfigurable radiation pattern as detailed in vertical polarization states with a single feeding port. To [25]. The graph modeling of this antenna follows rule 6 where realize the pattern diversity, each of the slot pairs in the parasitic patches is loaded by a switchable stub. The stub surface current distribution by introducing a physical planar lengths are adjusted by pin diodes which allow four different change [5-23]. patterns for one of the polarization state [24]. Figure 20 shows the feeding configuration connected by different diodes. The Statement 2: In order to design an antenna with a graph model according to rule 7 leads us to rule 2.a. where the reconfigurable radiation pattern, the designer must alter the vertices are the different lines in the feeding network originating feeding fields or the existing fields [24-26]. connected together. The graph model is shown in Figure 21 where the edges are directed. The edges weights are calculated Statement 3: In order to design an antenna with reconfigurable according to equation1. We took into consideration 8 antenna polarized fields, the designer must alter the surface structure of states in the graph model shown in Figure 21. the antenna accordingly, or switch the polarization of the originating fields [27-28].

Statement 4: In order to design an antenna with joint reconfigurable properties, the designer must use all of the above principles simultaneously [29-32].

VI. RECONFIGURABLE ANTENNA DESIGN STEPS After the above observations and conclusions we set some steps that facilitate the design of a reconfigurable antenna. Step 1: Specify the reconfigurability property that needs to be Fig.19. The antenna structure in [24] obtained Step 2: Specify the antenna structure Step 3: Choose the reconfigurable technique by applying the statements in section V. Step 4: Graph model the structure using the rules in section IV Step 5: Fine tune the structure according to desired applications using simulations and testing. Step 6: Should simulations in step 5 reveals undesirable behavior such as redundant sub-configurations that made it into the graph model. Remove the redundant parts and repeat steps 4and5.

VII. AN EXAMPLE OF DESIGNING RECONFIGURABLE ANTENNAS USING GRAPH MODELS Step 1: The reconfigurability property desired in this design is Fig.20. The Feeding network of the antenna in [24] a reconfigurable return loss and a reconfigurable radiation pattern. Step2: This antenna is based on replacing the patch in a microstrip antenna by many open ended microstrip lines intersecting with each other with optimized lengths, widths and spacings. The antenna is built one line at a time. We want to design a multi-band antenna. We choose the minimum number of resonances between 2.5 GHz and 6 GHz to be 5. In order to obtain at least 5 resonances we need at least 4 microstrip lines and a mid-section patch. Four microstrip lines intersecting with each other as shown in Figure 22a will create a mid-section and will definitely give us a multi band antenna. Fig.21. The graph model for the antenna in [24] Step3 : Since this antenna needs to exhibit a reconfigurable return loss and a reconfigurable radiation pattern then A summary of all the previous rules is shown in Table 1. statement 4 of section V applies which suggests a combination V. ANTENNA RECONFIGUABILITY ANALYSIS of statements 1 and 2 in section V. We need to execute a surface current distribution alteration and an alteration of the Reconfiguration of an antenna can be achieved based on the existing radiating fields since the feeding chosen herein is following basic principles or statements. through a 50 Ω SMA connector. In order to change the surface current distribution and alter the existing radiating fields we Statement 1: In order to design an antenna with frequency as suggest the use of switches to achieve a change in the whole the reconfigurable parameter, the designer must alter the antenna structure. By using switches to connect and disconnect the mid-section length of 7 cm. The middle substrate has a dielectric constant from the microstrip lines, the whole antenna structure changes εr=3.9 and a height 0.16 cm. The upper layer is composed of 4 leading to a surface current distribution change and an microstrip lines intersecting with each other. The length of alteration of the existing fields. each line is 4.5 cm and the width is 0.3 cm .The spacing between the lines is 0.2 cm. The antenna was fabricated and Multi- Vertices Directed Weighted tested, a comparison between the simulated and tested return Part loss is shown in Figure 24. A good analogy is noticed between Switches YES Parts NO YES the measured and simulated data. The return loss was measured NO End- NO NO between 1 Ghz and 6 Ghz. 5 resonances are clearly shown Points between 2.5 Ghz and 6 Ghz. The antenna presents both a Diodes YES Parts YES YES reconfigurable return loss and a reconfigurable radiation NO End- YES NO pattern as shown in Figure 25 and 26. Points Step 6: Our objective is to obtain a reconfigurable return loss Capacitors YES Parts NO YES and radiation pattern antenna. This antenna has to resonate at NO End- NO YES least at 5 frequencies between 2.5 GHz and 6 GHz. If we want Points to optimize our design then five different configurations are Varactors YES Parts YES YES needed which means if we attach 2 lines from each side of the NO End- YES YES mid-section, we will end up by having 5 total parts while Points keeping the symmetry of the structure and conserving the Angular N/A Angles NO YES radiation pattern properties. Change Many Biasing YES Parts NO YES The graph model of the optimized antenna is shown in Figure Networks 27. The antenna was simulated with 2 parts from each side as shown in Figure 28. The optimization of the 2 parts led to lines Antenna N/A Antennas NO YES of 0.9 cm width and 1.15 cm length form each side of the mid- Arrays section. A comparison was made between the optimal antenna Reconfigurable YES Parts YES/NO YES and the old redundant antenna and the S11 results show Feeding Depending complete analogy as shown in Figure 29 which proves that the on parts removed were redundant and 4 switches were spared. technique used The radiation patterns of the non optimal and the optimal Table 1. Section IV rules summary antennas are compared in Figure 30 when the switches are not activated which proves that the removal of the redundant parts did not affect the radiation pattern properties. The optimal antenna is now fabricated and tested and great analogy was shown between the tested and simulated S11 results as shown in Figure 31. The optimized fabricated antenna and the original fabricated redundant antenna are shown in Figure 32 for comparison.

Fig.22. The proposed structure. a). Switches on, b). Switches off

Step 4: Graph model the structure using section IV rules. In this case rule 1.a should be used. We have a mid section that other parts will be added to it. The vertices will represent the parts. Let’s call the vertex representing the mid-section P0. In our case we are suggesting adding four parts from each side at Fig.23.Graph model showing all possible connections the same time. VIII. INVESTIGATING RELATIONSHIPS BETWEEN PATH These parts will be added symmetrically and at the same time REDUNDANCY IN A GRAPH MODEL AND COMPONENT REDUNDANCY as shown in Figure 22a& b. The edges between the vertices IN A RECONFIGURABLE ANTENNA will represent the connection of these parts to the mid-section. The graph model is shown in Figure 23. Reconfigurable antennas were categorized in section Step 5: The structure of the proposed antenna shown in Figure II into four main categories. According to statement 1 of 23 consists of 3 layers. The lower layer which constitutes the section V frequency tuning occurs when the surface current square ground plane covers the entire substrate and has a side distribution is altered. Redundant antenna elements in this case are possible. The same current distribution may occur with fewer elements having the suitable dimensions. According to statement 2 of section V an antenna exhibits a reconfigurable radiation pattern once its existing or originating fields are altered. The alteration of these fields requires a drastic change in the whole antenna structure or in the feeding network. Redundancy in this case is not an issue since any added or removed parts will alter the radiation pattern as long as the part is added or removed in a distinctive direction that will result in a drastic change of the whole antenna structure or feeding network. According to statement 3 of section V the surface structure of the antenna has to keep a certain shape corresponding to the required polarization which means that added parts are always a necessity to achieve this polarization Fig.27. The optimal graph mapping reconfiguration. According to statement 4 of section V the designer has to take into considerations the previous statements simultaneously.

Fig.28.The antenna Optimized

Fig.24. A comparison of the simulated and measured S11

Fig.29. Comparison between the S11 results for the non- optimal and the optimal antenna when the switches are activated Fig.25. S11 paramter for different antenna configuration

Fig.26. Radiation pattern in the H plane for different antenna Fig.30. The radiation pattern for the non optimal and the configurations at F=2.33 GHz optimal antenna when the switches are open. consideration the other antenna requirements i.e. a redundant part can be removed as long as its removal will not affect the polarization status of the antenna in a reconfigurable return loss and reconfigurable polarization antenna.

Fig.31. A comparison between the simulated and tested S11 results for the optimal antenna

Fig. 33. An example of all possible unique paths in a given graph

Formulating the problem: In this formulation we investigate reconfigurable antennas Fig.32. The optimized fabricated antenna using one reconfiguration technique. If an antenna uses more than one reconfiguration technique then each technique is Problem Statement: investigated separately. We suggest three suitable equations to The designer needs to take into consideration the exact antenna our suggested design techniques. requirements before setting up his design. The real problem exists in the reconfigurable return loss antenna, since a Equation VIII.1: N(N 1) designer might add a redundant part to his antenna without NAC   2 (VIII.1a) realizing the losses he is adding to the whole system. 2 How can a designer identify an antenna part as redundant? N 2  N  2NAC 1  0  Problem solution: Step 4 in section VI requires from the designer to 1 1 8(NAC 1)  graph model the antenna before simulating it. Step 6 requires N    (VIII.1b) the designer to check back if any redundant sub-configurations  2  appeared in the design. The designer has to compare the Where NAC represents the number of all possible antenna number of possible unique paths with the number of required configurations and N is the number of vertices in the antenna functions and so antenna configurations. corresponding graph model. Every path in every graph model should correspond to Equation VIII.1 is valid for multipart reconfigurable antennas a different antenna configuration. If the total number of unique of section II groups 1,2,3,5,6,7. For antennas of group 2 using paths existing in a graph model is more than the number of the diodes to achieve reconfiguration, the graph is directed and so antenna configurations required then redundant paths may exist equation VIII.1 applies only if the number of paths remains the in that graph model and so the antenna has redundant parts. same as an undirected graph. The designer should arrange the An example on counting the total number of unique paths in a diodes in a direction so that the number of total unique paths is given graph is shown in Figure 33. the same as for an undirected graph with the same connections. In the case where redundant antenna parts were found, they should be removed as long as their removal won’t affect Equation VIII.2: other antenna reconfigurability properties. If these parts were N removed their corresponding vertices and edges should be NAC  1 (VIII.2a) removed. It is very important to indicate that the edges 2 eliminated must have a weight W=1 or non-weighted since an edge with a distinctive weight would be impossible to N  2 (NAC 1) (VIII.2b) eliminate. When the designer is dealing with reconfigurable antennas verifying statement 4 of section V he must take into Where NAC represents the number of all possible antenna configurations and N is the number of vertices in the corresponding graph model. Equation VIII.2 is valid for single part reconfigurable antennas of section IV groups 1,2,3,5,6,7.

Equation VIII.3: NAC  N Where NAC represents the number of all possible antenna configurations and N is the number of vertices in the Fig. 35.Optimal graph model for the antenna in [9] following corresponding graph model. rule 1.b Equation VIII.3 is valid for reconfigurable antennas of section Applying Equation VIII.3 to the design shown in Figure 13 and IV group 4. detailed in [31] confirms that this design is optimal since the number of possible antenna configurations or functions is equal Testing the Problem Formulations: to the number of angles of bending executed. In the example given in section VII at least 5 resonances and a X. CONCLUSION reconfigurable radiation pattern were required. Conserving the radiation pattern properties from the redundant antenna This paper suggests guidelines for optimally structure enforces on the designer keeping the symmetry of the designing a reconfigurable antenna. By following these steps structure. The graph model of the optimal design in Figure 27 an antenna designer will have a tool in his hand allowing him has 5 vertices resulting in 12 possible antenna configurations. to be efficient and reducing his costs and losses. These Applying equation VIII.1b to that example proves that 5 guidelines will lead to an optimal reconfigurable antenna under vertices are the least acceptable number due to the design the constraints elaborated previously. constraints. This proof is shown below: A reconfigurable antenna designer needs to answer a The antenna requires at least 5 possible configurations with very important question: will he be able to achieve his design symmetric structure. objective in the most efficient and less expensive way? NAC  5 This paper tries to answer this rising question knowing that a designer has to always compromise between improved forNAC  5,6,7 performance of an antenna and an increased complexity in its structure. 1 1 8 (NAC 1)  N     4 Graph models are used in this paper to understand and  2  optimize the physical structure of a reconfigurable antenna as shown in the examples presented. Redundancy in an antenna Figure 34 shows a graph model following rule 1.a of section IV structure is investigated and optimal antenna reconfigurations with 4 vertices. are suggested and formulated. This paper tries to answer many questions regarding reconfigurable antenna design and it gives the antenna designers a handy tool that facilitates and simplifies their designs while meeting the same objectives. Fig.34. Graph model with 4 vertices XI. REFERENCES This graph model will be translated into 3 parts attached to the [1] J.T.Bernhard “Reconfigurable Antennas”,Morgan and Claypool mid-section resulting in an asymmetric structure which doesn’t Publishers, 2007. preserve the radiation pattern properties. 4 total parts [2] C.A.Balanis “Modern Antenna Handbook”,John Wiley and Sons, 2008. represented by 4 vertices are not a good solution for this [3] T.H.Cormen, C.E.Leiserson, R.L.Rivest, C.Stein “Introductions to Algorithms”,MIT press, 2nd edition, 2001. antenna. N has to be > 4. Taking N=5 leads to NAC [4] E. Klavins “Programmable Self Assembly” IEEE Control Systems =5*4/2+2=12 according to equation VIII.1a which proves that Magazine, Vol. 27, No. 4, pp. 43-56, Aug. 2007. the optimization of the design in example 2 in section VII was [5] D. E. Anagnostou, Z.Guizhen, M.T. Chrysomallis, J.C. Lyke, G.E. accurate. Ponchak, J. Papapolymerou, and C. G. Christodoulou “Design, fabrication and measurement of an RF-MEMS-based self –similar reconfigurable antenna” , Applying Equation VIII.2a to the graph model shown in Figure IEEE Transactions on Antennas and Propagation, Vol. 54,Issue 2,Part 1, pp. 4 gives us NAC= (20/2)+1=11 possible antenna configurations 422-432, Feb 2006. while the number of configurations required in [9] is only 5 so [6] A.Patnaik, D. E. Anagnostou, C.G. Christodoulou and J.C. Lyke “ by applying Equation VIII.2b we end up in N=2*(4)=8 Neurocomputational analysis of a multiband reconfigurable planar antenna” , IEEE Transactions on Antennas and Propagation, Vol. 53,Issue 11, pp. 3453- vertices. These 8 vertices represent the 8 end points of 4 3458, Nov. 2005. switches. The optimal design will include one slot with four [7] A.Patnaik, D. E. Anagnostou, C.G. Christodoulou and J.C. Lyke “ A switches bridging over it. The optimal graph model of this frequency reconfigurable antenna design using neural networks” , IEEE antenna is shown in Figure 35. Antennas and Propagation society international symposium, Vol. 2A, pp.409- 412, July 2005.

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