Abstract:
Disclosed is an antenna system and method to optimize the design of an antenna using resistive sheets. The system and method are operative to design a topology of a resistive sheet to mitigate the adverse effects caused by the inherent losses of resistive sheets while operating as antennas. The system is designed to reduce a plurality of radiofrequency current “hot spots” and “pinch points,” associated with the flow of a current on a resistive sheet, by a sufficient extent so as to enable radiation of electromagnetic waves at substantially higher radiation efficiency as compared with antennas designed using traditional design techniques.

Description:
CROSS REFERENCE TO RELATED APPLICATION 
     This application is based upon and claims priority from U.S. Provisional Patent Application Ser. No. 61/812,366 entitled “Planar Antenna and Design Method Thereof,” filed with the U.S. Patent and Trademark Office on Apr. 16, 2013, by the inventors herein, the specification of which is incorporated herein by reference. 
    
    
     FIELD OF THE INVENTION 
     The present invention relates to antenna systems and methods. More particularly, the present invention relates to antennas made of resistive materials and antenna design and manufacturing methods for overcoming adverse effects caused by losses in such resistive materials. 
     BACKGROUND OF THE INVENTION 
     A number of resistive sheet or resistive layer antenna designs and systems exist within various industries for providing a partly conductive and at the same time optically transparent layer of material for multiple applications. The sheet resistivity and the light transparency of the resistive sheet are the key factors to determine the implementation of a resistive sheet antenna. In general, an antenna made of a resistive transparent sheet, such as Indium tin oxide, experience losses several orders of magnitude larger than an antenna made of a conductive material such as Copper or Silver. Therefore, antennas are primarily made of a conductive material, if possible. However, conductive materials are opaque to light. As a result, in certain applications requiring the use of a transparent antenna, a conductive material cannot be used. 
     In recent years, the demand for transparent antennas has increasingly grown for touchscreen, mobile platform, and automobile applications. In particular, the implementation of antennas, made of a transparent conductive layer, on the display window of a portable communication device have been addressed in the prior art, as described in U.S. Pat. No. 7,983,721 to Ding et al., the specification of which is incorporated herein by reference in its entirety. However, these efforts have faced certain challenges and limitations. Particularly, attempts made to provide an antenna design sufficiently transparent to light and at the same time capable of performing at radiation efficiency levels set up by industry standards have not been successful. A major challenge is that as the sheet resistivity of a resistive sheet increases, making the resistive sheet more conductive, the optical transparency of the resistive sheet decreases. Likewise, as the sheet resistivity increases, the power dissipated as heat as a result of currents flowing on the resistive sheet increases too. Accordingly, the radiated power and the radiation efficiency of the resistive sheet are reduced, making it very challenging for resistive sheet antennas to meet radiation efficiency industry standards. 
     Therefore, a compromise is required between two conflicting goals. Firstly, making the resistive sheet as conductive as possible, which means less transparent; and secondly, making the antenna more optically transparent, which means a more resistive sheet having a larger sheet resistivity. Current technology offers optically transparent resistive sheets having a sheet resistivity larger than 10 Ohms per square. However, for these values of sheet resistivity, standard designs techniques used for antennas made of conductive materials notably fail. 
     Moreover, in placing an antenna close to conductive or resistive materials, electromagnetic coupling between the antenna and these materials also contributes to power losses that decrease the effective radiated power at a system level. In most touchscreen and mobile platform applications, the antenna is surrounded by a number of conductive and resistive materials that must be considered, especially when designing an antenna using resistive sheets, to maximize the overall radiated power. Accordingly, manufacturers intending to use a resistive sheet on the touchscreen area as an antenna experience either an unacceptable reduction in radiation efficiency or an unacceptable performance of the touchscreen. This leads manufacturers to implementation of antenna system that are costly, aesthetically unappealing, or more importantly, highly inefficient. 
     Previous efforts have been made to develop a method of improving the radiation efficiency of antennas made of transparent resistive sheet, as described in U.S. Pat. No. 7,233,296 to Song, et al. However, this method is primarily aimed at determining values for current density over the surface of the resistive sheet to identify regions having concentrated flow of currents. Then the antenna efficiency is improved by increasing the conductivity in such areas of high current concentration. 
     The method described in the patent to Song et al., has also faced severe challenges and limitations. In particular, the resulting resistive layer will not be optically homogeneous. In other words, there will be areas of the resistive layer having darker spots resulting from the increased conductivity. Thus, although the resistive layers may meet optical transparency functional requirements, the resistive layer will not be aesthetically appealing. Furthermore, the manufacturing process used to provide different regions with different conductivity increases costs. Moreover, and more importantly, the areas of high-current concentration will vary depending on the type of application, the user operation, and the surrounding areas to the resistive sheet. Accordingly, small areas of higher conductivity on the resistive sheet may not cover a shift of the high-current spots. Alternatively, increasing the size of the areas of higher conductivity (darker areas) on the resistive sheet may further compromise the aesthetics and the optical transparency of the resistive sheet. 
     A way to approach the disadvantages of the efforts attempted by the prior art is to design a resistive sheet antenna, based on the topology of the resistive sheet. In this way, it is possible to increase the radiation efficiency of the antenna by identifying and mitigating or eliminating the sources of losses experienced by the antenna as current flows on the resistive sheet. In particular, a uniform radio frequency (RF) current distribution over the topology of the resistive sheet may prevent RF current “hot spots” and pinch points,” resulting in substantial increase of radiation efficiency. 
     An RF current “hot spot” is characterized by a region of a material wherein a concentration of RF current is present having significantly larger current levels as compared to other regions having a more uniform current distribution and lower current levels. In particular, for a resistive sheet, a “hot spot” region dissipates a substantial amount of power as heat, significantly reducing the amount of radiated power. 
     Likewise, an RF current “pinch point” is characterized by a region of a material wherein the physical configuration of the material forces the RF current to converge creating high concentration of current levels. Thus, a narrow region of a material will have larger current densities as compared to a wider region of the same material. Accordingly, a “pinch point” in a resistive material will result in a substantial amount of power dissipated as heat, significantly reducing the amount of radiated power. Therefore, it is critical for a resistive sheet antenna to avoid RF current “hot spots” and “pinch points,” in order to be able to radiate power and operate as an antenna. 
     Currently, there is no well-established method of deterministically creating a topology configuration of a resistive sheet to provide acceptable values of radiation efficiency, especially for resistive sheets having a sheet resistivity greater than 10 Ohms per square. 
     Thus, there remains a need in the art for antenna system designs and methods, using resistive sheets, capable of operating at radiation efficiencies that avoid the problems of prior art systems and methods. 
     SUMMARY OF THE INVENTION 
     An antenna system and method of optimizing the design of an antenna using resistive sheets, or equivalently resistive layers, is disclosed herein. One or more aspects of exemplary embodiments provide advantages while avoiding disadvantages of the prior art. The system and method are operative to design a topology of a resistive sheet to mitigate the adverse effects caused by the inherent losses of resistive sheets while operating as antennas. The system is designed to reduce a plurality of RF current “hot spots” and “pinch points” associated with the flow of a current on a resistive sheet, by a sufficient extent so as to enable radiation of electromagnetic waves at a substantially higher radiation efficiency as compared with antennas designed using traditional design techniques. 
     An antenna system using a resistive sheet designed according to the method described herein is able to uniformly distribute the currents flowing on said resistive sheet to reduce power losses as heat. Accordingly, more power is radiated improving the radiation efficiency of the antenna system. This increased radiation efficiency is primarily dictated by the resistive sheet topology that provides wide areas and smooth edges wherein current flows to yield a more uniform current density distribution over the resistive sheet. In addition, wide areas of the resistive sheet contribute to prevent RF current “pinch points,” while smooth edges contribute to avoid RF current “hot spots,” especially at contracted, corners, junctions, bends, periphery, or sharp regions of said resistive sheet, where significant RF power is dissipated as heat instead of being radiated. 
     The determination of the topology configuration of the resistive sheet is based on determining an effective alpha function, which depends on the physical dimensions of the resistive sheet and is proportionally related to the radiation losses of the resistive sheet. Thus, lower values of the effective alpha function translate into lower power losses as heat, and higher power available for radiation. The resistive sheet is discretized into pixels or cell units, and the effective alpha function is calculated for each pixel. Then circuit network theory is used to compute the effective alpha function for the entire resistive sheet structure. The values of the effective alpha function over the entire structure provides a key guidance to determine which areas of the topology must be adjusted to further improve the design and avoid any “hot spots” and “pinch points.” Alternatively, electromagnetic simulation software may be used to compute the effective alpha function. 
     The method to design a resistive sheet antenna with significantly higher radiation efficiency as compared to standard techniques used to design an antenna made of conductive material includes the step of determining a reference effective alpha function for an initial topology that avoids RF current “hot spots” and “pinch points.” The method further includes the steps of designing alternative antenna topologies of said antenna, wherein RF currents flow uniformly over as much area as possible of said antenna, reducing electromagnetic coupling between said antenna and other materials, and reducing RF current “hot spots” and RF current “pinch points” on other resistive material. The method further includes the step of selecting the antenna topology most suitable for the intended application of said antenna, in terms of performance or other predetermined criteria. 
     By significantly reducing the losses caused by currents flowing over a resistive sheet by means of determining a suitable topology of the resistive sheet and by increasing the uniform distribution of the current density flowing on the resistive sheet, the antenna system and method are able to provide outcomes that significantly increase the radiation efficiency, as compared to antenna designs using standard techniques. This increase in radiation efficiency may be multiple times larger, resulting in designs that meet or exceed challenging industry standards, in terms of antenna radiation performance and optical transparency, for a resistive sheet antenna. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The numerous advantages of the present invention may be better understood by those skilled in the art by reference to the accompanying drawings in which: 
         FIG. 1  shows a diagram of an equivalent circuit of an antenna in transmit mode. 
         FIG. 2  shows a representation of a current density flowing on a resistive sheet. 
         FIG. 3  is a representation of  FIG. 2  showing a discretization into pixels of the resistive sheet. 
         FIG. 4  shows a graph of antenna radiation efficiency, as a function of effective alpha, of a resistive sheet antenna having a radiation resistance of 50 Ohms for different values of sheet resistivity. 
         FIG. 5  shows a graph of antenna radiation efficiency, as a function of effective alpha, of a resistive sheet antenna having a radiation resistance of 75 Ohms for different values of sheet resistivity. 
         FIG. 6  shows a graph of antenna radiation efficiency, as a function of sheet resistivity, of a resistive sheet dipole antenna of 40-mm length and 10-mm width. 
         FIG. 7  shows a graph of antenna radiation efficiency, as a function of sheet resistivity, of a resistive sheet dipole antenna of 40-mm length and 4-mm width. 
         FIG. 8  shows an exemplary embodiment of a model of a planar dual-band dipole antenna system with a parasitic element made of a resistive sheet. 
         FIG. 9  shows a model of a planar dual-band dipole antenna system with a traditional parasitic element design made of a conductive material. 
         FIG. 10  shows a model of a planar dual-band dipole antenna system with a traditional parasitic element design made of a resistive sheet. 
         FIG. 11  shows a graph of antenna radiation efficiency, as a function of frequency, of different planar dual-band dipole antenna systems. 
         FIG. 12  shows a model of a planar semi-elliptical antenna in accordance with another embodiment. 
         FIG. 13  shows a schematic view of a method for designing an antenna using a resistive sheet. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     The following description is of a particular embodiment of the invention, set out to enable one to practice an implementation of the invention, and is not intended to limit the preferred embodiment, but to serve as a particular example thereof. Those skilled in the art should appreciate that they may readily use the conception and specific embodiments disclosed as a basis for modifying or designing other methods and systems for carrying out the same purposes of the present invention. Those skilled in the art should also realize that such equivalent assemblies do not depart from the spirit and scope of the invention in its broadest form. 
       FIG. 1  shows a schematic diagram of an equivalent circuit  10  of an antenna (not shown) in transmit mode. In transmit mode, the antenna is assumed to be attached to a voltage source  14  having a source internal impedance  16 . The antenna is represented in circuit  10  by an antenna reactance  18  and an antenna resistance composed of an antenna radiation resistance  20  and an antenna loss resistance  22 . In general, the antenna radiation efficiency accounts for losses within the antenna structure. For an antenna made of a resistive sheet or a resistive layer of material, the losses within the antenna structure are due primarily to conductive or ohmic losses caused by the power dissipated as electric current flows on the resistive sheet. The antenna radiation efficiency is defined as the ratio of the power delivered to antenna radiation resistance  20  to the power delivered to the total antenna resistance, which includes antenna radiation resistance  20  and antenna loss resistance  22 . This means that the antenna radiation efficiency, e, can be written as: 
             e   =       R   ⁢           ⁢   rad       Ro   +     R   ⁢           ⁢   rad               
where Rrad is antenna radiation resistance  20 , and Ro is antenna loss resistance  22 .
 
     From the above expression of the antenna radiation efficiency, e, it is clear that either an increase of the value of antenna radiation resistance  20  or a decrease of the value of antenna loss resistance  22  will increase the antenna radiation efficiency, e. In particular, the radiation efficiency of an antenna can be significantly affected by the presence of a resistive material near the antenna. This is due to the electromagnetic coupling of energy between the antenna and the resistive material. The energy that is coupled into and dissipated by the resistive material cannot be further radiated, resulting in less effective power radiated by the antenna. The outcome is equivalent to having an antenna with larger antenna loss resistance  22 . A number of factors affect the amount of electromagnetic coupling between an antenna and a nearby resistive material, including the dimensions and relative location of the antenna and the resistive material, the antenna polarization and operating frequency range, and the distance between the antenna and the resistive material. 
       FIG. 2  shows a representation of a current density flowing on a generic resistive sheet  30  having an arbitrary shape. In  FIG. 2 , resistive sheet  30  is characterized by a material having a maximum length  36 , a maximum width  38 , and a thickness. The thickness of resistive sheet  30  is several magnitudes smaller than maximum length  36  and maximum width  38 . Resistive sheet  30  may be referred to using a Cartesian coordinate system consisting of an “X” axis  32  and a “Y” axis  34 . Axes  32  and  34  are selected such that resistive sheet  30  lies in the plane defined by “X” axis  32  and “Y” axis  34  (the XY plane). The magnitude of the current density flowing on resistive sheet  30  is represented by a gray scale plot in  FIG. 2 , wherein different tonalitues of gray scale correspond to different magnitude levels of current density. 
     For resistive sheet  30  operating as an antenna element of effective length, L eff , and effective width, W eff , loss resistance  22  is given by the sheet resistivity, Rs, multiplied by the effective length-to-effective width ratio, i.e.: 
               R   ⁢           ⁢   o     =     R   ⁢           ⁢   s   ⁢       L   eff       W   eff               
where effective length, L eff , and effective width W eff  correspond to the length and width of resistive sheet  30  where the current density is different from zero, respectively. Therefore, a current density uniformly flowing over the entire area of resistive sheet  30  will yield an effective length identical to the physical length of resistive sheet  30  and an effective width identical to the physical width of resistive sheet  30 . Likewise, for a current density with non-uniform distribution on the entire area of resistive sheet  30 , the effective length or the effective width will be smaller than the corresponding physical length or physical width of resistive sheet  30 , respectively.
 
     It is possible to introduce an effective alpha function, α eff , of resistive sheet  30 , given by the effective length-to-effective width ratio of resistive sheet  30 , i.e.: 
     
       
         
           
             
               α 
               eff 
             
             = 
             
               
                 L 
                 eff 
               
               
                 W 
                 eff 
               
             
           
         
       
     
     Thus, the effective alpha depends on both the current density distribution along the length of resistive sheet  30  and the current density distribution along the width of resistive sheet  30 . By reducing the effective alpha function, α eff , of resistive sheet  30 , it is possible to reduce loss resistance  22  of resistive sheet  30 , which increases the radiation efficiency, e, of resistive sheet  30 . 
     Because of the inherent losses of resistive sheet  30 , regions of resistive sheet  30  having high concentration of current density will cause larger ohmic losses, degrading the antenna radiation efficiency of resistive sheet  30 . Thus, where resistive sheet  30  is intended to minimize losses and maximize radiation efficiency, a current density uniformly flowing over the entire area of resistive sheet  30  is preferable. In addition, both RF current “hot spots” and RF current “pinch points” must be avoided, particularly at edges, corners, or narrower regions of resistive sheet  30  in which high concentration of RF currents may occur. 
     Therefore, most preferably, a peripheral boundary, enclosing an area of said resistive sheet  30  layer, defines a topology designed to carry a desired current density that uniformly and smoothly flows over the entire area of resistive sheet  30 . Accordingly, the topology of resistive sheet  30  is typically configured to have rounded instead of sharp corners or to be larger in dimensions, as compared to a counterpart made of conductive material, to avoid abrupt disruption of currents and allow currents to naturally decay to levels that may not create significant losses at the edges of resistive sheet  30 . 
       FIG. 3  shows a discretization into pixels of resistive sheet  30 . A pixel  40  is a discretized element of resistive sheet  30  representing a smaller area of a total area of resistive sheet  30 . A gray tonality in pixel  40  represents a magnitude of the current density flowing on pixel  40  of resistive sheet  30 . A size of pixel  40  is selected to have a maximum length  42  and a maximum width  44 . Maximum length  42  of pixel  40  is smaller than maximum length  36  of resistive sheet  30 . Maximum width  44  of pixel  40  is smaller than maximum length  36  of resistive sheet  30 . For computational convenience, a generally square geometrical shape may be preferred for pixel  40  of resistive sheet  30 . However, selecting a square pixel  40  is not always possible due to the shape irregularities of resistive sheet  30  and the selected dimensions of pixel  40 , as can be seen in  FIG. 3  around the curved edges of resistive sheet  30 . In general, the shape of each pixel  40  of resistive sheet  30  can have any geometrical shape. Those skilled in the art will recognize that maximum length  42  of pixel  40  and maximum width  44  of pixel  40  are typically selected to be equal or smaller than a tenth of a wavelength, at the lowest frequency of operation in the medium where resistive sheet  30  is located. However, other sizes of maximum length  42  of pixel  40  and maximum width  44  of pixel  40  may be selected. 
     Referring to the coordinate system shown in  FIG. 2 , an alpha function in Y-axis  34  direction, of pixel  40  located at the m-th row and the n-th column of discretized resistive sheet  30 , as depicted in  FIG. 3 , can be defined as: 
               α   y     m   ⁢           ⁢   n       =         ∫   0     Δ   ⁢           ⁢   y       ⁢         D   ⁡     (   y   )       2     ⁢     ⅆ   y             ∫   0     Δ   ⁢           ⁢   x       ⁢         B   ⁡     (   x   )       2     ⁢     ⅆ   x                 
where D(y) is the normalized current density distribution within pixel  40  of resistive sheet  30  in Y-axis  34  direction, and B(x) is the normalized current density distribution within pixel  40  of resistive sheet  30  in X-axis  32  direction for current flowing along Y-axis  34  direction.
 
     Likewise, an alpha function in X-axis  32  direction, of pixel  40  depicted in  FIG. 3 , located at the m-th row and the n-th column of discretized resistive sheet  30 , can be defined as: 
     
       
         
           
             
               α 
               x 
               
                 m 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 n 
               
             
             = 
             
               
                 
                   ∫ 
                   0 
                   
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     x 
                   
                 
                 ⁢ 
                 
                   
                     
                       B 
                       ⁡ 
                       
                         ( 
                         x 
                         ) 
                       
                     
                     2 
                   
                   ⁢ 
                   
                     ⅆ 
                     x 
                   
                 
               
               
                 
                   ∫ 
                   0 
                   
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     y 
                   
                 
                 ⁢ 
                 
                   
                     
                       D 
                       ⁡ 
                       
                         ( 
                         y 
                         ) 
                       
                     
                     2 
                   
                   ⁢ 
                   
                     ⅆ 
                     y 
                   
                 
               
             
           
         
       
     
     Therefore, for resistive sheet  30  operating as an antenna element and having a sheet resistivity, Rs, the loss resistance  22  within pixel  40  can be written as:
 
 Ro   y   mn   =Rsα   y   mn  
 
 Ro   x   mn   =Rsα   x   mn  
 
where Ro y   mn  and Ro x   mn  are the loss resistance within pixel  40  of resistive sheet  30  in Y-axis  34  direction and X-axis  32  direction, respectively.
 
     Thus, each pixel of resistive sheet  30  can be characterized by a first loss resistance in Y-axis  34  direction and a second loss resistance in X-axis  32  direction. Accordingly, the calculation of the overall effective alpha function, α eff , of resistive sheet  30 , can then be determined by means of circuit theory methods well-known to those skilled in the art. Alternatively, the overall effective alpha function, α eff , of resistive sheet  30 , having sheet resistance Rs, can be determined by calculating the antenna radiation efficiency, e, of resistive sheet  30  and the antenna radiation resistance of resistive sheet  30 , using electromagnetic simulation software, such as Ansys-HFSS commercial software. Correspondingly, the topology of resistive sheet  30  may be adjusted to reduce the overall effective alpha function, α eff , according to the physical dimensions of resistive sheet  30  and the distribution of the current density flowing over resistive sheet  30 . 
       FIG. 4  shows a graph of an antenna radiation efficiency, as a function of an effective alpha, of a resistive sheet antenna having a radiation resistance of 50 Ohms for different values of sheet resistivity. This graph shows how the radiation efficiency of a resistive sheet antenna increases as the effective alpha function, α eff , decreases, because the antenna resistance loss becomes smaller. Particularly in  FIG. 4 , a solid-circle curve  52 , a solid-square curve  54 , and a solid-triangle curve  56  correspond to the antenna radiation efficiency mathematically calculated for an antenna using a resistive sheet material of 50-Ohm per square sheet resistivity, a 25-Ohm per square sheet resistivity, and a 15-Ohm per square sheet resistivity, respectively. 
       FIG. 5  shows a graph of an antenna radiation efficiency, as a function of an effective alpha, of a resistive sheet antenna having a radiation resistance of 75 Ohms for different values of sheet resistivity. This graph shows how the radiation efficiency of a resistive sheet antenna increases as the effective alpha function, α eff , decreases, because the antenna resistance loss becomes smaller. Particularly in  FIG. 5 , a solid-circle curve  62 , a solid-square curve  64 , and a solid-triangle curve  66  correspond to the antenna radiation efficiency mathematically calculated for an antenna using a resistive sheet material of 50-Ohm per square sheet resistivity, a 25-Ohm per square sheet resistivity, and a 15-Ohm per square sheet resistivity, respectively. 
       FIG. 6  shows a graph of an antenna radiation efficiency, as a function of sheet resistivity, of a resistive sheet dipole antenna of 40-mm length and 10-mm width. This graph represents a validation of a planar dipole-strip antenna  70  design using a resistive sheet material, having a 40-mm length and a 10-mm width. In  FIG. 6 , a solid-circle curve  72  corresponds to the antenna radiation efficiency calculated by a well-known commercially available electromagnetic software (Ansys-HFSS), for different values of sheet resistivity. Also, in  FIG. 6 , a solid-square curve  74  corresponds to the antenna radiation efficiency determined by using the effective alpha function, for different values of sheet resistivity. In this case, the computed value of the effective alpha function, α eff , is equal to 3. In  FIG. 6  it is evident how close the results represented by curve  72  and curve  74  match. This matching is indicative of the validation of this antenna design using the effective alpha function introduced in our model. 
       FIG. 7  shows a graph of an antenna radiation efficiency, as a function of sheet resistivity, of a resistive sheet dipole antenna of 40-mm length and 4-mm width. This graph represents a validation of a planar dipole-strip antenna  80  design using a resistive sheet material, having a 40-mm length and a 4-mm width. In  FIG. 7 , a solid-circle curve  82  corresponds to the antenna radiation efficiency calculated by a well-known and commercially available electromagnetic software (Ansys-HFSS), for different values of sheet resistivity. Also in  FIG. 7 , a solid-square curve  84  corresponds to the antenna radiation efficiency determined by using the effective alpha function, for different values of sheet resistivity. In this case, the computed value of the effective alpha function, α eff , is equal to 6. In  FIG. 7  it is evident how close the results represented by curve  82  and curve  84  match. This is indicative of the validation of this antenna design using the effective alpha function introduced in our model. 
     The determination and use of an effective alpha function, α eff , for designing an antenna, using a resistive sheet, includes for each possible configuration of interest, taking into account the following considerations: first, the available area wherein the antenna would be located; second, the presence of resistive and conductive materials, including other antennas, within a radius of two wavelengths at the lowest frequency of operation in the medium wherein the antenna is operating; third, discretizing the area of the antenna into pixels having a maximum length and a maximum width no larger than a tenth of a wavelength at the lowest frequency of operation in the medium where the antenna is operating; fourth, determining the current density distribution on the antenna surface, corresponding to an excitation source in the presence of other resistive and conductive materials; fifth, normalizing the current density distribution on the antenna surface wherein the antenna lies; sixth, calculating the effective alpha function corresponding to each pixel into which the total area of the antenna has been discretized; seventh, creating an equivalent electrical circuit network, corresponding to the antenna surface, by means of the calculated effective alpha function corresponding to each pixel in the directions of a Cartesian coordinate system wherein the area of the antenna has been discretized; eighth, determining parameters of the equivalent electrical circuit network, particularly the resistance loss of the antenna, corresponding to the current density distribution on the antenna. Alternatively, the overall effective alpha function, α eff , can be determined by calculating the antenna radiation efficiency and the antenna radiation resistance of the antenna, by means of electromagnetic simulation software, such as Ansys-HFSS commercial software or other methods well-known by those skilled in the art; and ninth, adjusting the topology of the antenna to reduce the overall effective alpha function, α eff , according to the physical dimensions of the antenna and the distribution of the current density flowing over the resistive sheet. In particular, the topology of the antenna must avoid RF current “hot spots” and RF current “pinch points,” by increasing the uniform distribution of currents flowing over said resistive sheet, at the frequencies of interest, as a result of implementing one or more of the following design considerations: increasing the area of the resistive sheet wherein the currents flow, reducing the sheet resistivity of the resistive sheet, and smoothing out the edges and avoiding sharp corners of the topology in regions wherein the currents flow. 
     In accordance with certain aspects of an embodiment,  FIG. 8  shows an exemplary configuration of a planar dual-band dipole antenna system  90 , comprising a dipole antenna element  92  and a parasitic element  94 . Dipole element  92  and parasitic element  94  are coplanar and both are disposed on a nonconductive substrate  95 . Dipole antenna element  92  consists of a thin sheet made of conductive material, such as copper or silver, whereas parasitic element  94  consists of a thin sheet or thin layer made of a resistive film, comprising Indium tin oxide, having a sheet resistivity of approximately 10 Ohms per square. 
     Dipole antenna element  92  consists of a first arm  92   a  and a second arm  92   b  fed at feeding point  96 , each arm having a length of about 21.5 mm and a width of 1 mm. Substrate  95  consists of a piece of glass with a relative dielectric permittivity of about 7, a tangent loss in the order of 0.05, and approximate dimensions of 50-mm in length, 24-mm in width, and 0.7-mm in thickness. Those skilled in the art will recognize that substrate  95  may be implemented using different varieties of glass, plastic, sapphire, and polymers such as polyethylene terephthalate (PET), having different thickness and tangent loss. 
     The length of arms  92   a  and  92   b  primarily defines a first lower frequency region of operation of antenna system  90 . Likewise, the length of parasitic element  94  primarily defines a second higher frequency region of operation of antenna system  90 . Those skilled in the art will realize that there are a variety of ways to feed dipole element  92 , including by means of an electromagnetically-coupled feeding element, a coaxial cable, a coplanar waveguide or other types of transmission line known in the prior art. 
     Parasitic element  94  has a shape with a footprint that fits within a rectangle of approximately 11.7-mm in length and about 8-mm in width. Parasitic element  94  consists of a modified rectangle having a convex side  94   a , elliptically shaped according to an ellipse with a major axis of 4.8 mm and a major-to-minor axes ratio of 2.45; a concave side, opposing said convex side  94   a , consisting of a first edge  94   c , a second edge  94   d , and a third edge  94   f ; and two substantially parallel and opposing sides  94   b  and  94   e , each with an approximate dimension of 5.6 mm. Edge  94   f  of parasitic element  94  has a length of approximately 1 mm and is positioned equidistant from sides  94   b  and  94   e . Moreover, edge  94   c  and edge  94   d  of parasitic element  94  each follow an elliptical shape according to an ellipse with a major axis of 6 mm and a major-to-minor axes ratio of 2. In other words, edge  94   c  and edge  94   d  of parasitic element  94  each extend elliptically inward from sides  94   b  and  94   e  to define a midpoint notch that corresponds to edge  94   f.    
     Parasitic element  94  is placed adjacent to dipole antenna element  92  without physically touching each other. Instead, parasitic element  94  electromagnetically couples to dipole antenna element  92  to ultimately characterize a performance of antenna system  90 . In particular, an input impedance, a polarization characteristic and a frequency range of operation of antenna system  90  are influenced by the topology of parasitic element  94 , in terms of the physical dimensions, configuration, and location of parasitic element  94 . In this case, parasitic element  94  is positioned with a minimum separation from dipole element  92  of about 0.1 mm. Accordingly, dipole element  92  acts as a driver element or feeding element of parasitic element  94  by means of electromagnetic coupling. 
     In this embodiment, the topology of parasitic element  94  is configured to increase a uniform flow of RF currents on parasitic element  94  and to avoid RF current “hot spots” in order to increase the antenna radiation efficiency of antenna system  90 . Particularly, parasitic element  94  is wider than traditional parasitic element designs using conductive materials; has two rounded corners at the joint of side  94   a  with side  94   b  and at the joint of side  94   a  with side  94   e ; has one smoothed, convex side  94   a ; and has one smooth, concave side formed by edges  94   c ,  94   d , and  94   f.    
     Moreover, dipole element  92  is excited such that the RF current is maximum at feed point  96  and gradually reduces to a zero level at a distance from feed point  96  corresponding to the location of sides  94   b  and  94   e  of parasitic element  94 , at the intended frequency of operation of antenna system  90  primarily defined by parasitic element  94 . Accordingly, parasitic element  94  is configured to reduce electromagnetic coupling at edge  94   f , and to gradually increase electromagnetic coupling at edges  94   c  and  94   d  as edges  94   c  and  94   d  approach sides  94   b  and  94   e , respectively. The location and physical configuration of edges  94   c ,  94   d , and  94   f  and side  94   a  of parasitic element  94  allows a more uniform and smoother current density distribution on parasitic element  94  and avoids RF current “hot spots” to increase the antenna radiation efficiency of antenna system  90 . 
     In this embodiment, side  94   b , side  94   e , the corner at the joint of side  94   b  with edge  94   c , and the corner at the joint of edge  94   d  with side  94   e  of parasitic element  94  do not need to be smoothed out because the level of the nearby RF currents are significantly low at the intended frequency of operation, of antenna system  90 , primarily defined by parasitic element  94 . 
     The effect of the topology of parasitic element  94  is better illustrated by referring to  FIGS. 9 to 11 .  FIG. 9  shows a model of a planar dual-band dipole antenna system  90  comprising dipole element  92  and a parasitic element  94   a  using a typical design made of a conductive material. Dipole element  92  and parasitic element  94   a  are coplanar and both are disposed on nonconductive substrate  95 .  FIG. 10  shows a model of planar dual-band dipole antenna system  90  comprising dipole element  92  and a parasitic element  94   b . Parasitic element  94   b  is designed using a typical design with a resistive layer having a sheet resistivity of approximately 10 Ohms per square. Likewise, dipole element  92  and parasitic element  94   b  are coplanar and both are disposed on nonconductive substrate  95 . The dimensions and designs of parasitic elements  94   a  and  94   b  are identical. Moreover, the only difference between parasitic element  94   a  and parasitic element  94   b  is the type of material used to model parasitic element  94   a  (conductive material) and  94   b  (resistive layer with sheet resistivity of about 10 Ohms per square). 
       FIG. 11  shows a graph of antenna radiation efficiency, as a function of frequency, of the three different versions of planar dual-band dipole antenna system  90 , corresponding to parasitic elements  94 ,  94   a , and  94   b . In these three cases, the antenna radiation efficiency of antenna system  90  has been calculated by a well-known commercial electromagnetic software (Ansys-HFSS). In  FIG. 11 , a solid-square curve  100  corresponds to the antenna radiation efficiency of planar dual-band dipole antenna system  90  with parasitic element  94   a ; a solid curve  102  corresponds to the antenna radiation efficiency of planar dual-band dipole antenna system  90  with parasitic element  94   b ; and a solid-diamond curve  104  corresponds to the antenna radiation efficiency of planar dual-band dipole antenna system  90  with parasitic element  94 . 
     In  FIG. 11 , the solid-square curve  100  clearly shows two frequency regions of preferred operation. A first frequency region in the 2.2 GHz to 2.6 GHz range with antenna radiation efficiency approximately between 80% and 90%, and a second frequency region in the 5 GHz to 5.8 GHz frequency range with antenna radiation efficiency approximately between 50% and 67%. These two frequency regions of operation are indicative of the dual-band capability of antenna system  90  with parasitic element  94   a  and correspond to two key frequency bands for WiFi applications. 
     Also in  FIG. 11 , the solid curve  102  shows that the antenna radiation efficiency of antenna system  90  significantly reduces to values below 20%, within the 5 GHz to 5.8 GHz frequency range, when parasitic element  94   a  is replaced with parasitic element  94   b . In other words, replacing the parasitic element made of conductive material with an identical element made of a resistive sheet of approximately 10 Ohms per square cuts by more than two and a half times the antenna radiation efficiency of antenna system  90 . Moreover, the antenna radiation efficiency is reduced to levels below 25%, which is a typical minimum required value by industry standards for on-screen antennas for laptops and tablet applications. This drastic reduction in antenna radiation efficiency is indicative that traditional antenna design techniques based on conductive materials do not work when these materials are replaced with a resistive sheet. 
     However, in  FIG. 11 , the solid-diamond curve  104 , shows that the antenna radiation efficiency of antenna system  90  can be maintained at levels approximately between 35% and 42%, within the 5 GHz to 5.8 GHz frequency range, when parasitic element  94   a  is replaced with parasitic element  94 . Therefore, by configuring parasitic element  94   a  to the topology of parasitic element  94  in accordance with aspects of an embodiment of the invention, an increased uniform and smoother flow of RF currents on parasitic element  94  increases the antenna radiation efficiency of antenna system  90  to levels well above the minimum industry standards and significantly above the corresponding levels of a traditional design that just replace conductive material with a resistive sheet. 
       FIG. 12  shows another exemplary configuration of antenna system  90  in accordance with aspects of an embodiment of the invention, comprising a resistive layer  120 , consisting of a resistive Indium tin oxide-based film disposed on nonconductive substrate  95 ; a feeding element  122 ; and a transmission line  124 , shown in  FIG. 12  as a coaxial cable. The topology of resistive layer  120  has a semi-elliptical configuration, comprising a first edge  126 , primarily having a linear shape, and a second edge  128 , having an elliptical shape. Second edge  128  is elliptically shaped according to an ellipse with a major axis of 20 mm and a major-to-minor axes ratio of 1.15. Accordingly, first edge  126  and second edge  128  join at two regions, each defining a corner of resistive layer  120 . Moreover, each corner of resistive layer  120  is shaped to follow an elliptical shape according to an ellipse of major axis 2.2 mm and a major-to-minor axes ratio of 1.05. 
     Resistive layer  120  is fed by feeding element  122 , which is coupled to coaxial cable  124 . Feeding element  122  consists of an approximately 1-mm radius circular pad, made of conductive material, electrically connected to the center conductor of coaxial cable  124 . Preferably, feeding element  122  is attached to resistive layer  120  by means of a commercially available adhesive tape, because of inherent practical difficulties to solder directly into resistive layer  120 . Furthermore, the conductive film used has a protective thin layer of polymer on each surface. Thus, effectively, the coupling of feeding element  122  to resistive layer  120  is primarily through electromagnetic coupling, even though feeding element  122  is in physical contact with the protective polymer layer of resistive layer  120 . 
     In regards to the configuration shown in  FIG. 12 , simulation results of antenna system  90  show that a radiation efficiency in the order of 38% is achievable using a resistive sheet of 50 Ohms per square at a frequency of 3 GHz. Likewise, a radiation efficiency of approximately 50% is obtained when using a resistive sheet of 25 Ohms per square at a frequency of 3 GHz. As a way of comparison, a Copper-based antenna element provides a radiation efficiency of about 95%, whereas antennas designed using traditional antenna design techniques, using conductive antenna elements as opposed to resistive sheets, provide an antenna radiation efficiency below 10% at a frequency of 3 GHz. As a reference, typical industry standards of antenna systems used for mobile platform applications require a minimum antenna radiation efficiency of 30%. In other words, this configuration not only provides a substantially larger radiation efficiency, compared to antennas designed using resistive sheets based on traditional design techniques, but also meets typical industry standards. 
     Those skilled in the art will realize that other methods of implementing transmission line  124 , in addition to using a coaxial cable, include a coplanar waveguide, a microstrip line and other types of transmission lines known in the prior art, any of which may be used without departing from the spirit and scope of the invention. Likewise, those skilled in the art will recognize that feeding element  122  may be implemented by using conductive adhesive, soldering a conductive terminal, or other types of electromagnetically-coupled feeding elements known in the prior art. 
     Alternatively, other forms of the configurations described herein may include a resistive sheet having a topology with at least one smooth edge and at least one smooth corner. In another configuration, the topology of a resistive sheet may be configured to reduce electromagnetic coupling to other resistive or conductive materials. In yet another configuration, the topology of a resistive sheet may be configured to have a shape as wide as possible, to have at least one region wide enough to avoid RF current “pinch points.” Likewise, in any of the configurations described herein, a resistive sheet antenna may operate in an elliptical polarization, including a generally linear polarization and a generally circular polarization; in a single frequency band or multiple frequency bands; and as part of a single, diversity, multiple input multiple output (MIMO), reconfigurable or beam forming network system. 
     Those skilled in the art will realize that the resistive layer described in the different configurations of antenna system  90  may be implemented by means of a resistive film comprising a metal oxide compound, such as tin oxide, disposed on substrate  95 , or by application of a resistive coating directly to substrate  95  or to a thin layer of a substrate such as polyethylene terephthalate to be disposed on substrate  95 . 
     Regarding each of the above-described configurations, a method as depicted in  FIG. 13  for designing an antenna topology, using resistive sheets of material or in the presence of other resistive material, and for setting up the antenna dimensional and operational parameters may be performed according to the following: 
     1. At step  1310 , determining a reference effective alpha function, α eff , for the resistive sheet of an initial topology design of the antenna. In particular, the initial topology of the antenna must avoid RF current “hot spots” and RF current “pinch points,” by increasing the uniform distribution of RF currents flowing over the resistive sheet, at the frequencies of interest. 
     2. Next, at step  1320 , designing alternative antenna topologies of the antenna, wherein RF currents flow uniformly over as much area as possible of the antenna. This may include creating new topologies of the antenna or adjusting the initial topology of the antenna to reduce the reference effective alpha function, α eff , according to the physical dimensions of the antenna and the distribution of the current density flowing over the resistive sheet, as a result of implementing one or more of the following design considerations: increasing the area of the resistive sheet wherein the currents flow, reducing the sheet resistivity of the resistive sheet, and smoothing out the edges and avoiding sharp corners of the topology in regions wherein the currents flow. 
     3. Next, at step  1330 , reducing, as much as possible, any electromagnetic coupling between the antenna and other materials within a radius of two wavelengths at the lowest frequency of operation of the antenna in the medium wherein the antenna is intended to operate. Go to step  1310 , if necessary, to consider additional antenna topologies. 
     4. Next, at step  1340 , reducing as much as possible RF current “hot spots” and RF current “pinch points” on the other resistive material, and increasing the uniform flow of RF currents over as much area of the resistive material as possible, in case that the electromagnetic coupling between the antenna and other materials cannot be totally eliminated. This may include reconfiguring the topology of the resistive sheet antenna to reduce the electromagnetic coupling between the resistive sheet antenna and the other resistive material. 
     5. Next, at step  1350 , repeating steps  1310  to  1340  for other antenna topologies of interest. 
     6. Last, at step  1360 , selecting an antenna topology most suitable for an intended application of the antenna, in terms of performance or other predetermined criteria. 
     Those of ordinary skill in the art will recognize that the steps above indicated can be correspondingly adjusted for specific configurations and other constraints, including operating frequency band and bandwidth, radiation gain, polarization, radiation efficiency, input impedance matching, operational conditions, surrounding environment, available area and location for implementation of the antenna, method of antenna feeding, and type of transmission line used for a given application. 
     Preferably, the uniformity of RF currents flowing over the resistive sheet, RF current “hot spots,” RF current “pinch points,” the electromagnetic coupling between two materials, the overall effective alpha function, α eff , and other antenna performance parameters, including but not limited to electromagnetic fields, radiation efficiency, currents, radiation gain, input impedance, and polarization are determined by means of a computer-assisted simulation tool and electromagnetic simulation software, such as Ansys-HFSS commercial software or other methods well-known by those skilled in the art. 
     Most preferably, a data processing and decision making algorithm may be implemented to analyze antenna parameters or calculate a figure of merit of the antenna performance, including but not limited to electromagnetic fields, radiation efficiency, currents, radiation gain, input impedance, and polarization, to support or guide the antenna design process as described herein, as those skilled in the art will realize. Alternatively, a resistive sheet antenna may be designed by determining and reducing the total effective alpha function or equivalently by increasing the radiation resistance of the antenna. 
     The various embodiments have been described herein in an illustrative manner, and it is to be understood that the terminology used is intended to be in the nature of words of description rather than of limitation. Any embodiment herein disclosed may include one or more aspects of the other embodiments. The exemplary embodiments were described to explain some of the principles of the present invention so that others skilled in the art may practice the invention. Obviously, many modifications and variations of the invention are possible in light of the above teachings. The present invention may be practiced otherwise than as specifically described within the scope of the appended claims and their legal equivalents.