Abstract:
An electrodynamic method includes providing an electrodynamic structure with a periphery, providing power, collecting electrons, and emitting electrons. The emitting electrons and the collecting electrons utilizes at least 20% of the periphery of the electrodynamic structure. The method includes conducting current to provide at least one of electrodynamic propulsion and power generation.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    This Patent Application is a Divisional of and claims priority to co-pending U.S. patent application Ser. No. 11/372,508, filed on Mar. 10, 2006, which is incorporated herein by reference. 
       BACKGROUND 
       [0002]    The potential use of long electric conductors in space in a form of electrodynamic tethers was discovered in early 80&#39;s. Electrodynamic tethers in space have attracted a lot of attention in recent years. Many researchers have contributed to the theory of their behavior in orbit. Some flight experiments have provided data on the interaction between the electrodynamic tethers and the geomagnetic field and ionosphere. 
         [0003]    Typically, an electrodynamic tether is a long electrical conductor that can be used to generate power and/or propulsion as the tether orbits a celestial body with a magnetic field. Flight experiments have provided data on the interaction between electrodynamic tethers and the geomagnetic field and ionosphere of the Earth. 
         [0004]    In 1993, the Plasma Motor Generator (PMG) experiment was performed on a Delta rocket with a primary goal of testing power generation and thrust by means of an electrodynamic tether. In the PMG experiment, a 500 meter (m) long electrodynamic tether was deployed into the ionosphere. The tether included a conducting wire with hollow cathodes at each end. An electric current was produced in the tether, demonstrating the potential of this technique to generate power and propulsion that could be used by satellites or space stations in low Earth orbit (LEO). The PMG mission was an example of a propulsion system for space transportation that did not utilize propellant, but rather achieved propulsion by converting orbital energy into electrical energy (deorbit) or electrical energy into orbital energy (orbit boosting). 
         [0005]    Two Tethered Satellite System (TSS) missions were flown in 1992 and 1996. The TSS included a satellite, a conducting tether, and a tether deployment/retrieval system flown on the Space Shuttle. Objectives of the TSS missions were to understand the electro-magnetic interaction between the tether system and the ambient space plasma, investigate its dynamics, and demonstrate current collection from the ionosphere to further develop tether capabilities for future tether applications on the Space Shuttle and Space Station. In the TSS-1 mission of 1992, the tether was only partially deployed and the mission was aborted. 
         [0006]    The TSS-1R mission of 1996 was a re-flight of the TSS-1 mission. The tether was deployed to the length of 19.7 km when it was severed by an electrical arc. Nevertheless, it was a significant mission for tethered satellites because it showed that electrodynamic tethers were more efficient than theoretically predicted, providing valuable data on electrical performance of the system. Power generation of several kilowatts was demonstrated. 
         [0007]    “Tethers in Space Handbook,” Second Edition, NASA Office of Space Flight, NASA Headquarters, Washington, D.C., 1989, edited by P. A. Penzo and P. W. Ammann, provides summaries of various applications and features of electrodynamic tethers, including methods to change orbital elements with electrodynamic tether propulsion and methods to control attitude dynamics of tethers. 
         [0008]    Typically, electrodynamic tethers are very long and operate at high voltages. The electrodynamic tethers run the risk of arcing as in the TSS-1R mission. Also, the electrodynamic tethers are susceptible to damage from meteors and/or debris due to the length of the tethers. In addition, electrodynamic tethers are difficult to scale up to move heavy payloads. 
         [0009]    For these and other reasons, there is a need for the present invention. 
       SUMMARY 
       [0010]    One embodiment provides an electrodynamic method including providing an electrodynamic structure with a periphery, providing power, collecting electrons, and emitting electrons. The emitting electrons and the collecting electrons utilizes at least 20% of the periphery of the electrodynamic structure. The method includes conducting current to provide at least one of electrodynamic propulsion and power generation. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0011]    The accompanying drawings are included to provide a further understanding of the present invention and are incorporated in and constitute a part of this specification. The drawings illustrate the embodiments of the present invention and together with the description serve to explain the principles of the invention. Other embodiments of the present invention and many of the intended advantages of the present invention will be readily appreciated as they become better understood by reference to the following detailed description. The elements of the drawings are not necessarily to scale relative to each other. Like reference numerals designate corresponding similar parts. 
           [0012]      FIG. 1  is a diagram illustrating one embodiment of an electrodynamic structure according to the present invention. 
           [0013]      FIG. 2  is a diagram illustrating the flow of electrons in a two-dimensional, planar electrodynamic structure. 
           [0014]      FIG. 3  is a diagram illustrating one embodiment of electrical components in a portion of a periphery area. 
           [0015]      FIG. 4  is a diagram illustrating one embodiment of an open net-like collector structure. 
           [0016]      FIG. 5  is a diagram illustrating one embodiment of a circular, two-dimensional, planar electrodynamic structure that includes a mesh of conductors. 
           [0017]      FIG. 6  is a diagram illustrating one embodiment of a hexagonal, two-dimensional, planar electrodynamic structure that includes a triangular mesh of conductors. 
           [0018]      FIG. 7  is a diagram illustrating one embodiment of a rectangular, two-dimensional, planar electrodynamic structure that includes a mesh of conductors. 
           [0019]      FIG. 8  is a diagram illustrating one embodiment of a rectangular, two-dimensional, planar electrodynamic structure that has only partial periphery utilization for electron collection and electron emission. 
           [0020]      FIG. 9  is a diagram illustrating one embodiment of a cylindrical, three-dimensional electrodynamic structure. 
           [0021]      FIG. 10  is a diagram illustrating one embodiment of a curved, half-elliptical, three-dimensional electrodynamic structure. 
           [0022]      FIG. 11  is a diagram illustrating one embodiment of an enclosed, three-dimensional electrodynamic structure. 
       
    
    
     DETAILED DESCRIPTION 
       [0023]    In the following Detailed Description, reference is made to the accompanying drawings, which form a part hereof, and in which is shown by way of illustration specific embodiments in which the invention may be practiced. In this regard, directional terminology, such as “top,” “bottom,” “upper,” “lower,” “front,” “back,” etc., is used with reference to the orientation of the Figure(s) being described. Because components of embodiments of the present invention can be positioned in a number of different orientations, the directional terminology is used for purposes of illustration and is in no way limiting. It is to be understood that other embodiments may be utilized and structural or logical changes may be made without departing from the scope of the present invention. The following detailed description, therefore, is not to be taken in a limiting sense, and the scope of the present invention is defined by the appended claims. 
         [0024]      FIG. 1  is a diagram illustrating one embodiment of an electrodynamic structure  20  according to the present invention. Electrodynamic structure  20  is configured to generate power and/or propulsion as it orbits a celestial body having a magnetic field. In one aspect, electrodynamic structure  20  is an electrodynamic sail. In other embodiments, electrodynamic structure  20  can be part of any suitable system, such as a space station. 
         [0025]    Electrodynamic structure  20  is substantially a circular, two-dimensional, planar electrodynamic structure that includes a periphery area  22  and supports  26 . As illustrated, electrodynamic structure  20  is coupled to a payload  24 . Periphery area  22  is situated at the periphery of electrodynamic structure  20  and has width W. Payload  24  is disposed in the center of electrodynamic structure  20  and is mechanically coupled to periphery area  22  via supports  26 . An interior area  25  is defined between an interior edge of periphery area  22  and payload  24 . In one embodiment, interior area  25  includes open gaps (i.e., gaps that do not contain components or material) between each of the supports  26 . 
         [0026]    In some embodiments, some or all parts and components are mounted on a substrate, such as a plastic film, to provide mechanical support. In some embodiments, a portion or all of periphery area  22  is covered with a substrate. In some embodiments, a portion or all of interior area  25  is covered with a substrate. In one embodiment, the substrate is a reflective film. 
         [0027]    In the embodiment illustrated in  FIG. 1 , periphery area  22  includes electron collectors  28 , electron emitters  30 , solar arrays  32 , and one or more controllers  34 . As illustrated, electron collectors  28  and electron emitters  30  are situated at an outer rim of periphery area  22 . As illustrated, solar arrays  32  are situated at an interior rim of periphery area  22 . 
         [0028]    In embodiments of two- or three-dimensional electrodynamic structures, electron collectors, electron emitters, solar arrays, and controllers can be disposed in the periphery area, such as periphery area  22 . Additionally, in embodiments of two- or three-dimensional electrodynamic structures, electron collectors, solar arrays, and controllers can be disposed in the interior area, such as interior area  25 . 
         [0029]    Electron collectors  28 , electron emitters  30 , solar arrays  32 , and controllers  34  are electrically coupled via conductive paths formed with switches and conductors. The conductive paths are not shown in  FIG. 1  for clarity, but one embodiment of a portion of a periphery area including conductive paths formed with switches and conductors is illustrated in  FIG. 3  and discussed below. In one embodiment, electron collectors  28  are electrically coupled to each other via conductive paths formed with switches and conductors. In one embodiment, electron emitters  30  are electrically coupled to each other via conductive paths formed with switches and conductors. In one embodiment, solar arrays  32  are electrically coupled to each other via conductive paths formed with switches and conductors. In one embodiment, conductive paths include conductors that include insulated strips of aluminum foil. 
         [0030]    Supports  26  are members that mechanically couple payload  24  to periphery area  22 . In one embodiment, supports  26  are non-rigid supports. In one embodiment, supports  26  are mechanically stiffened supports. In one embodiment, supports  26  are conductive and payload  24  and electrical components in periphery area  22  are electrically coupled via the conductive supports  26 . 
         [0031]    In embodiments of two- or three-dimensional electrodynamic structures, payloads, such as payload  24 , can be any suitable payload, such as a satellite that orbits the Earth, an inter-planetary satellite, or a satellite that orbits any suitable celestial body. In one embodiment, electrical components in the periphery area, such as periphery area  22 , are electrically coupled to the payload (e.g., payload  24 ) via supports (e.g., support  26 ) and the payload receives power from electrical components in the periphery area. In one embodiment, the payload includes one or more controllers that control operation of the electrodynamic structure (e.g., electrodynamic structure  20 ). 
         [0032]    The electron collectors, such as electron collectors  28 , described herein are defined to collect electrons from the ambient plasma. In one embodiment, the electron collectors are bare aluminum coated surfaces. In other embodiments, the electron collectors can be any suitable type of electron collectors. 
         [0033]    The electron collectors can cover not only a relatively narrow band around the periphery (e.g., periphery area  22 ) of the electrodynamic structure, but also much wider areas, reaching into the interior (e.g., interior area  25 ) of the electrodynamic structure. In one embodiment, the entire surface of the electrodynamic structure is employed for electron collection. 
         [0034]    The electron emitters, such as electron emitters  30 , described herein are defined to emit electrons into the ambient plasma. In one embodiment, the electron emitters are field emitter array cathodes (FEACs). In other embodiments, the electron emitters can be any suitable type of electron emitters. 
         [0035]    The solar arrays, such as solar arrays  32 , described herein are defined to absorb solar energy, and provide power in the form of electricity. In one aspect, solar arrays are one type of power system. In one aspect, solar arrays are solar energy collection devices. In one embodiment, solar arrays are thin-film solar arrays. In embodiments of two- or three-dimensional electrodynamic structures, any suitable power system can be used to generate power and electricity. 
         [0036]    One or more controllers, such as controller  34 , control operation of embodiments of electrodynamic structures (e.g., electrodynamic structure  20 ) via the switches and conductors in the conductive paths.  FIG. 2  is a diagram illustrating the flow of electrons in a two-dimensional, planar electrodynamic structure, such as electrodynamic structure  20 . At  50 , electron collectors (e.g., collectors  28 ) on one side of the electrodynamic structure are controlled with one or more controllers to collect electrons from ambient plasma. Under the control of one or more controllers, the collected electrons are driven to electron emitters (e.g., emitters  30 ) on the opposite side of the electrodynamic structure via conductors in the conductive paths. The collected electrons are driven using power generated via the power system (e.g., solar arrays  32 ). In addition, the motion through the ambient magnetic field produces an electro-motive force (EMF) in the conductors of the conductive paths, which can also be used to drive electrons to the opposite side of the electrodynamic structure in the direction of the EMF. At  52 , electron emitters (e.g., emitters  30 ) on the opposite side of the electrodynamic structure are controlled by one ore more controllers to emit electrons into the ambient plasma. The currents flowing through the conductors of the conductive paths between the electron collectors and the electron emitters interact with the ambient magnetic field to produce distributed Ampere forces, which can be employed to change the orbit and the orientation of the electrodynamic structure. 
         [0037]    In one embodiment, the area of electron collection at  50  and the area of electron emission at  52  are determined dynamically, as the electrodynamic structure moves including as the electrodynamic structure rotates, in the magnetic field of the celestial body and as the electrodynamic structure moves including as the electrodynamic structure rotates around its center of mass. In the area of electron collection at  50 , electron emitters remain inactive and in the electron emission area at  52 , electron collectors remain inactive. As the electrodynamic structure moves, electron collectors and electron emitters are turned on (selected) and turned off (deselected) at switching line  54  via one or more controllers to maintain one side as the area of electron collection at  50  and the other side as the area of electron emission at  52 . 
         [0038]    Conductors employed in embodiments of two- or three-dimensional electrodynamic structures can operate bi-directionally in the sense that the electrical current can flow either in the direction of the EMF induced in the conductor, or in the reverse direction, depending on the conductor orientation with respect to the magnetic field, mission goals, available power, and other system parameters. The reverse current is driven by available power sources, such as solar arrays  32 . If the current is reversed, electron collection and electron emission locations are reversed. Electron collection and emission occurs at multiple locations at the same time, to allow better utilization of thrust and power generation capabilities and to better control dynamics and optimize orbital maneuvering. 
         [0039]    Electrodynamic structure  20  is substantially a two-dimensional, planar electrodynamic structure having a circular shape. Embodiments of two-dimensional electrodynamic structures can be any suitable physical shape, such as, elliptical (e.g., circular, elongated elliptical) or polygonal, (e.g., rectangular, square, hexagonal). 
         [0040]    Embodiments of two-dimensional electrodynamic structures (e.g., electrodynamic structure  20 ) can be spin stabilized, wherein the spin axis is normal to the plane of the figure, such as  FIG. 1 . Embodiments of electrodynamic structures can spin at angular rates of 2 to 36 times the orbital rate. In each application, the spin rate is chosen based on performance and design trade-offs. In one embodiment, the electrodynamic structure spins at an angular rate that is substantially 6 to 8 times the orbital rate. 
         [0041]    Embodiments of electrodynamic structures (e.g., electrodynamic structure  20 ) can adjust spin axis, spin rate, and/or spin phase by varying direction, duration, amount, and/or path-length of currents flowing through the conductors. The orientation of the spin axis with respect to the orbital plane is chosen as best suited for a particular mission. For example, the spin axis can be pointed toward the Sun to maximize solar energy collection. Throughout the mission, the evolution of the spin axis orientation, the average spin rate, and the spin phase are controlled by modulations of the electrodynamic torques produced by variations in the electric current through the conductive paths. The spin-control current variations can be chosen to optimize the performance of the electrodynamic structure, while maintaining stability. 
         [0042]    Due to the spinning of embodiments of two-dimensional electrodynamic structures, the EMF induced in the conductors of the conductive paths changes direction during every revolution. To optimize variations in orbital elements and/or to optimize power generation, switches are controlled to drive electrons in a different direction during each revolution of the electrodynamic structure. As the electrodynamic structure spins, the switches are controlled to switch from electron collection on one side of the electrodynamic structure to electron collection on the other side of the electrodynamic structure and to switch from electron emission on the other side of the electrodynamic structure to electron emission on the one side of the electrodynamic structure. 
         [0043]    Since, the respective orientations of embodiments of spinning two-dimensional electrodynamic structures to the magnetic field are continuously changing and the currents in the conductors change with the rotation of the electrodynamic structure, the long-term evolution of the orbital and spin parameters is defined by a cumulative effect of the Ampere forces over periods of time longer or much longer than the spin period. Short-term oscillations of the dynamic and electrical parameters of the system, with periods shorter or much shorter than the spin period, are superimposed on the long-term evolution. The overall performance of electrodynamic structures can be estimated by averaging over a period of many orbits, which takes into consideration several groups of factors, including factors that change with the spin period, factors that change with the orbital period, and factors that change with the magnetic field rotation of the celestial body. 
         [0044]    The performance level of a substantially two-dimensional electrodynamic, planar structure (e.g., electrodynamic structure  20 ) can be estimated via Equation I. 
         [0000]        F/M=K*U* ( I/M )* B *( S/P )  Equation I
 
         [0000]    wherein F/M is the average total propulsion force of the electrodynamic structure per unit mass of the electrodynamic structure;
   U is the fraction of the periphery of the electrodynamic structure utilized for electron collection and emission;   (I/M) is the average total current collected and emitted per unit mass of the electrodynamic structure;   B is the induction of the magnetic field of the celestial body;   P is the value of a structure perimeter, where the periphery of the electrodynamic structure defines the structure perimeter;   S is the area enclosed by the structure perimeter; and   K is a dimensionless coefficient based on the electrodynamic structure implementation.   
 
         [0051]    Parameters in Equation I related to two-dimensional electrodynamic structures, such as electrodynamic structure  20 , can be maximized to optimize the performance level of the electrodynamic structure. Such parameters include periphery utilization for electron collection and emission U, area to perimeter ratio (S/P), and electric current production per unit mass of the electrodynamic structure (I/M). In many embodiments, parameter K is close to 2. The magnetic induction B does not depend on characteristics of the electrodynamic structure. 
         [0052]    In some embodiments, to maximize U, the entire periphery of the electrodynamic structure (e.g., electrodynamic structure  20 ) is lined with electron collectors (e.g., collectors  28 ) and electron emitters (e.g., emitters  30 ), as illustrated in  FIG. 1 . This yields a utilization factor of U=1 or 100%. If half of the periphery of the electrodynamic structure includes electron collectors and electron emitters and the other half is empty, U=½ or 50%. It is overly inefficient for the electrodynamic structure to have periphery utilization U of less than ⅕ or 20%. 
         [0053]    The area to perimeter parameter (S/P) is maximized if the electrodynamic structure is circular such as electrodynamic structure  20 . Other shapes, provide smaller area to perimeter ratios (S/P). Square or hexagonal shapes provide smaller but comparable (S/P) ratios, while elongated shapes, such as elongated elliptical shapes and elongated rectangular shapes have substantially lower (S/P) ratios. It is overly inefficient for the electrodynamic structure to have an area to perimeter ratio (S/P) less than ¼ or 25% of the ratio for a circle with the same perimeter. 
         [0054]    The electric current production per unit mass of the electrodynamic structure (I/M) depends on the electron collection and electron emission technologies and on the weight of the support structures. In one embodiment, electrodynamic structure  20  includes thin film solar arrays  32 , thin foil electron collectors  28 , FEAC electron emitters  30 , and a lightweight flexible spin-stabilized structure. In one embodiment, maximum electric current production per unit mass of electrodynamic structure  20  in LEO is at least 1 Ampere per kilogram (A/kg). It is overly inefficient for the electrodynamic structure to have maximum electric current production per unit mass in LEO less than 0.1 A/kg. 
         [0055]    In one embodiment, an electrodynamic structure, such as electrodynamic structure  20 , operates without batteries and stores energy in orbital motion instead of batteries. In one embodiment, while the electrodynamic structure is in sunlight, it accumulates energy by gaining altitude. Then, in eclipse, the electrodynamic structure uses the EMF induced in some of its conductors to drive electric currents through other conductors to produce, for example, out-of-plane forces and to continue changing the orbit inclination even without direct solar energy input. 
         [0056]    In one embodiment, in eclipse, the conductors with largest EMF&#39;s can be used for power generation, while other conductors with favorable orientations with respect to the magnetic field can utilize this power to produce thrust components to change certain orbit elements. 
         [0057]    Embodiments of electrodynamic structures having two or three dimensions, such as electrodynamic structure  20 , have small dimensions compared to one-dimensional electrodynamic tether systems. Due to these small dimensions, embodiments of the electrodynamic structures operate with low voltages that reduce the risk of arcing. In one embodiment, electrodynamic structure 20 has a 400 meter (m) diameter and in LEO experiences only 80 V or less of EMF induced in its conductors. 
         [0058]    Also, embodiments of electrodynamic structures having two or three dimensions, such as electrodynamic structure  20 , provide more efficient electrodynamic propulsion and power generation and can be scaled to propel heavy payloads. In one embodiment, a 900-1000 kg electrodynamic structure that is 400 m in diameter can change the inclination of a 2,000 kg payload in LEO at a rate of up to 0.7 degrees per day. To match this performance, an electrodynamic tether would need to be 80 km long and be able to operate at a voltage of 16 kV. 
         [0059]    In addition, embodiments of electrodynamic structures having two or three dimensions, such as electrodynamic structure  20 , are inherently resistant to meteor and debris damage. This is due at least in part to distributed designs and the redundancy of components. In the case of damage to one or more of the electron collectors, electron emitters, solar arrays, and/or supports, the controllers (e.g., controller  34 ) can reconfigure the conductive paths to bypass the failed component(s) and reroutes currents. 
         [0060]      FIG. 3  is a diagram illustrating one embodiment of electrical components  60  in a portion of a periphery area, such as periphery area  22 . The electrical components  60  include electron collectors  28   a - 28   c,  electron emitters  30   a  and  30   b , solar arrays  32   a - 32   c,  and switches  62   a  and  62   b.  The conductive paths that electrically couple electron collectors  28 , electron emitters  30 , solar arrays  32 , and one or more controllers  34  include switches  62   a  and  62   b.    
         [0061]    Electron collector  28   a  is electrically coupled to switch  62   a  via conductive path  64 . Electron collector  28   b  is electrically coupled to switch  62   a  via conductive path  66  and to switch  62   b  via conductive path  68 . Electron collector  28   c  is electrically coupled to switch  62   b  via conductive path  70 . Solar array  32   a  is electrically coupled to switch  62   a  via conductive path  72 . Solar array  32   b  is electrically coupled to switch  62   a  via conductive path  74  and to switch  62   b  via conductive path  76 . Solar array  32   c  is electrically coupled to switch  62   b  via conductive path  78 . Electron emitter  30   a  is electrically coupled to switch  62   a  via conductive path  80  and electron emitter  30   b  is electrically coupled to switch  62   b  via conductive path  82 . Switch  62   a  is electrically coupled to switch  62   b  via conductive path  84 . Switch  62   a  is electrically coupled to other switches (not shown) via conductive paths  86  and  88  and switch  62   b  is electrically coupled to other switches (not shown) via conductive paths  90  and  92 . 
         [0062]    One or more controllers  34  control switches  62   a  and  62   b  and employ the power from solar arrays  32   a - 32   c  to drive electrons across electrodynamic structure  20 . 
         [0063]    If the electrical components  60  are in the area of electron collection at  50  (shown in  FIG. 2 ), one or more controllers  34  control switches  62   a  and  62   b  to turn on electron collectors  28   a - 28   c  and turn off electron emitters  30   a  and  30   b.  Also, one or more controllers  34  control switches  62   a  and  62   b  to deliver the voltage produced via solar arrays  32   a - 32   c  to drive the electrons across electrodynamic structure  20  via conductors  86 ,  88 ,  90 , and/or  92 . 
         [0064]    In one embodiment, in the area of electron collection at  50 , one or more controllers  34  control switches  62   a  and  62   b  to couple electron collectors  28   a - 28   c  together and the combined electrons are driven to the area of electron emission at  52  (shown in  FIG. 2 ) via conductive paths  86 ,  88 ,  90 , and/or  92 . In one embodiment, in the area of electron collection at  50 , one or more controllers  34  control switches  62   a  and  62   b  to drive electrons from each of the electron collectors  28   a - 28   c  independently to the area of electron emission at  52  via conductive paths  86 ,  88 ,  90 , and/or  92 . 
         [0065]    If the electrical components  60  are in the area of electron emission at  52 , one or more controllers  34  control switches  62   a  and  62   b  to turn on electron emitters  30   a  and  30   b  and turn off electron collectors  28   a - 28   c.  Also, one or more controllers  34  control switches  62   a  and  62   b  to deliver the voltage produced via solar arrays  32   a - 32   c  to drive the electrons as needed via conductive paths  86 ,  88 ,  90 , and/or  92 . 
         [0066]    In one embodiment, in the area of electron emission at  52 , one or more controllers  34  control switches  62   a  and  62   b  to couple electron emitters  30   a  and  30   b  together and drive electrons from the area of electron collection at  50  to both electron emitters  30   a  and  30   b  substantially together via conductive paths  86 ,  88 ,  90 , and/or  92 . In one embodiment, in the area of electron emission at  52 , one or more controllers  34  control switches  62   a  and  62   b  to drive electrons to electron emitters  30   a  and  30   b  independently from the area of electron collection at  50  via conductive paths  86 ,  88 ,  90 , and/or  92 . 
         [0067]      FIG. 4  is a diagram illustrating one embodiment of an open net-like collector structure  94  that can be used in any two- or three-dimensional electrodynamic structure. Net-like collector structure  94  includes column ligaments  96  and row ligaments  98  that intersect at cross points. In one embodiment, at least some of the column ligaments  96  and row ligaments  98  are narrow tapes with bare metallic surfaces. In one embodiment, at least some of the column ligaments  96  and row ligaments  98  are narrow tapes with bare aluminum surfaces. In other embodiments, column ligaments  96  and row ligaments  98  can be any suitable ligament material. 
         [0068]    The open net-like collector structure  94  is more efficient if column ligaments  96  and row ligaments  98  are spaced at distances S that are many times larger than the width WL of each of the column ligaments  96  and row ligaments  98 . The net-like collector structure  94  can cover not only a relatively narrow band around the periphery (e.g., periphery area  22 ) of an electrodynamic structure, such as electrodynamic structure  20 , but also much wider areas, including into the interior (e.g., interior area  25 ) of the electrodynamic structure. In one embodiment, the entire surface of the electrodynamic structure is used for electron collection via an open net-like collector structure, such as open net-like collector structure  94 . In one embodiment, at least a portion of column ligaments  96  and/or row ligaments  98  are simultaneously employed as collectors and conductors of the conductive paths. 
         [0069]      FIG. 5  is a diagram illustrating one embodiment of a circular, two-dimensional, planar electrodynamic structure  100  that includes a mesh of conductors  102  disposed in an interior area  105 . Electrodynamic structure  100  is configured to generate power and/or propulsion as it orbits a celestial body having a magnetic field. In one aspect, electrodynamic structure  100  is an electrodynamic sail. In one aspect, electrodynamic structure  100  is similar to electrodynamic structure  20 , with the exception that electrodynamic structure  100  includes the mesh of conductors  102 . In other embodiments, electrodynamic structure  100  can be part of any suitable system, such as a space station. 
         [0070]    Conductive paths in electrodynamic structure  100  include the mesh of conductors  102 . The mesh of conductors  102  includes conductors that intersect other conductors and switches situated at the cross points of the conductors. The switches are controlled to direct currents through the mesh of conductors  102  and across electrodynamic structure  100  and to direct currents in current loops in the mesh of conductors  102 . In some embodiments, meshes of conductors (e.g., the mesh of conductors  102 ) include conductors that include insulated strips of aluminum foil. In some embodiments, meshes of conductors (e.g., the mesh of conductors  102 ) include non-conductive supports. 
         [0071]    Electrodynamic structure  100  includes the mesh of conductors  102  and periphery area  104 . Electrodynamic structure  100  is coupled to a payload  106 . Periphery area  104  is situated at the periphery of electrodynamic structure  100  and has width W. Payload  106  is mechanically coupled to components in periphery area  104  via the mesh of conductors  102 . Interior area  105  is defined between an interior edge of periphery area  104  and payload  106 . In one embodiment, the mesh of conductors  102  in interior area  105  are not mounted on a substrate and are not covered with material. In some embodiments, the mesh of conductors  102  are mounted on a substrate. In one embodiment, the substrate is reflective. In some embodiments, interior area  105  is covered partially or completely with a net-like collector structure, such as net-like collector structure  94 . 
         [0072]    In the embodiment illustrated in  FIG. 5 , periphery area  104  includes electron collectors  108 , electron emitters  110 , solar arrays  112 , and one or more controllers  114 . As illustrated, electron collectors  108  and electron emitters  110  are situated at an outer rim of the periphery area  104 . As illustrated, solar arrays  112  are situated on an interior rim of periphery area  104 . 
         [0073]    Electron collectors  108 , electron emitters  110 , solar arrays  112 , and controllers  114  are electrically coupled via conductive paths formed with switches and conductors, including the mesh of conductors  102 . In one embodiment, electron collectors  108  are electrically coupled to each other via the conductive paths. In one embodiment, electron emitters  110  are electrically coupled to each other via the conductive paths. In one embodiment, solar arrays  112  are electrically coupled to each other via the conductive paths. 
         [0074]    The stabilizing mechanical property of the mesh of conductors  102  contributes to stabilizing electrodynamic structure  100 . In one embodiment, the mesh of conductors  102  is a non-rigid mesh. In one embodiment, the mesh of conductors  102  includes mechanically stiffened supports and/or mechanically stiffened conductors. 
         [0075]    In one embodiment, the mesh of conductors  102  electrically couples payload  106  to the components in periphery area  104 . In one embodiment, payload  106  receives power from electrical components in periphery area  104  via the mesh of conductors  102 . In one embodiment, payload  106  includes one or more controllers that control operation of electrodynamic structure  100  via control signals communicated on the mesh of conductors. 
         [0076]    One or more controllers  114  control operation of electrodynamic structure  100  via switches in the conductive paths, including switches in the mesh of conductors  102 . 
         [0077]    Electron flow for a two-dimensional electrodynamic structure, such as electrodynamic structure  100 , is described above and illustrated in  FIG. 2 . In electrodynamic structure  100  and other electrodynamic structures having meshes of conductors, the currents flowing through the conductive paths, including currents flowing through the mesh of conductors (e.g., the mesh of conductors  102 ), interact with the ambient magnetic field to produce distributed Ampere forces, which can be employed to change the orbit and the orientation of the electrodynamic structure. Also, as the mesh of conductors moves through the ambient magnetic field EMF is produced in the conductors, which can be used to drive electrons to the opposite side of the electrodynamic structure in the direction of the EMF. 
         [0078]    In addition, power from the power system (e.g., solar arrays  112 ) and/or EMF in the conductors can be used to drive closed-loop currents through selected closed-loop paths in the mesh of conductors (e.g., the mesh of conductors  102 ). This does not require electron collection or electron emission. The closed-loop currents interact with the ambient magnetic field to produce distributed Ampere forces and torques to control the attitude dynamics of the electrodynamic structures (e.g., electrodynamic structure  100 ). These closed-loop currents through selected closed-loop paths in the mesh of conductors can provide improved control of the attitude dynamics of the electrodynamic structure via the mesh of conductors. 
         [0079]    Conductors in the conductive paths, including meshes of conductors (e.g., the mesh of conductor  102 ) can operate bi-directionally in the sense that the electrical current can flow either in the direction of the EMF induced in the conductor, or in the reverse direction, depending on the conductor orientation with respect to the magnetic field, mission goals, available power, and other system parameters. The reverse current is driven by available power sources, such as solar arrays. If the current is reversed, electron collection and electron emission locations are reversed. Meshes of conductors operating in this bi-directional manner facilitate electron collection and emission occurring at multiple locations at the same time, to allow better utilization of thrust and power generation capabilities and to better control dynamics and optimize orbital maneuvering. 
         [0080]      FIG. 6  is a diagram illustrating one embodiment of a hexagonal, two-dimensional, planar electrodynamic structure  150  that includes a triangular mesh of conductors  152  disposed in an interior area  155 . Electrodynamic structure  150  is configured to generate power and/or propulsion as it orbits a celestial body having a magnetic field. In one aspect, electrodynamic structure  150  is an electrodynamic sail. In one aspect, electrodynamic structure  150  is similar to electrodynamic structure  100 , with the exceptions that electrodynamic structure  150  is hexagonal and the conductors in the mesh of conductors  152  form triangular shaped cells. In other embodiments, electrodynamic structure  150  can be part of any suitable system, such as a space station. 
         [0081]    Conductive paths in electrodynamic structure  150  include the mesh of conductors  152 . The mesh of conductors  152  includes conductors that intersect other conductors to form triangular shaped cells. Switches are situated at the cross points of the conductors to direct currents through the mesh of conductors  152  and across electrodynamic structure  150  and to direct currents through closed-loop paths in the mesh of conductors  152 . The triangular shaped cells increase the mechanical strength and stability of the mesh of conductors  152  and electrodynamic structure  150 . Also, closed-loop currents driven through selected closed-loop paths, including triangular shaped cells, in the mesh of conductors  152  can provide improved control of the attitude dynamics of electrodynamic structure  150 . In some embodiments, meshes of conductors (e.g., the mesh of conductors  152 ) include non-conductive supports. 
         [0082]    A mesh of conductors, such as the mesh of conductors  152 , which includes triangular shaped cells can be used in any suitable electrodynamic structure having two or three dimensions. Also, the mesh of conductors that includes triangular shaped cells can be used in any suitable electrodynamic structure having any suitable physical shape, including circular, elliptical, square, rectangular, or hexagonal. 
         [0083]    Electrodynamic structure  150  includes the mesh of conductors  152  and periphery area  154 . Electrodynamic structure  150  is coupled to a payload  156 . Periphery area  154  is situated at the periphery of electrodynamic structure  150  and has width W. Payload  156  is mechanically coupled to components in periphery area  154  via the mesh of conductors  152 . Interior area  155  is defined between an interior edge of periphery area  154  and payload  156 . 
         [0084]    As illustrated in  FIG. 6 , periphery area  154  includes electron collectors  158 , electron emitters  160 , solar arrays  162 , and one or more controllers  164 . As illustrated, electron collectors  158  and electron emitters  160  are situated at an outer rim of the periphery area  154 . As illustrated, solar arrays  162  are situated on an interior rim of periphery area  154 . 
         [0085]    Electron collectors  158 , electron emitters  160 , solar arrays  162 , and controllers  164  are electrically coupled via conductive paths formed with switches and conductors, including the mesh of conductors  152 . In one embodiment, electron collectors  158  are electrically coupled to each other via the conductive paths. In one embodiment, electron emitters  160  are electrically coupled to each other via the conductive paths. In one embodiment, solar arrays  162  are electrically coupled to each other via the conductive paths. 
         [0086]    The stabilizing mechanical property of the mesh of conductors  152  contributes to stabilizing electrodynamic structure  150 . In addition, the triangular shaped cells formed via the conductors of the mesh of conductors  152  improve the mechanical strength and stability of electrodynamic structure  150 . In one embodiment, the mesh of conductors  152  is a non-rigid mesh. In one embodiment, the mesh of conductors  152  includes mechanically stiffened supports and/or mechanically stiffened conductors. 
         [0087]    In one embodiment, the mesh of conductors  152  electrically couples payload  156  to electrical components in periphery area  154 . In one embodiment, payload  156  via the mesh of conductors  152  and payload  156  receives power from electrical components in periphery area  154  via the mesh of conductors  152 . In one embodiment, periphery area  154  is electrically coupled to payload  156  via the mesh of conductors  152  and payload  156  includes one or more controllers that control operation of electrodynamic structure  150  via control signals communicated on the mesh of conductors. 
         [0088]    One or more controllers  164  control operation of electrodynamic structure  150  via switches in the conductive paths, including switches in the mesh of conductors  152 . 
         [0089]      FIG. 7  is a diagram illustrating one embodiment of a rectangular, two-dimensional, planar electrodynamic structure  200  that includes a mesh of conductors  202  disposed in an interior area  205 . Electrodynamic structure  200  is configured to generate power and/or propulsion as it orbits a celestial body having a magnetic field. In one aspect, electrodynamic structure  200  is an electrodynamic sail. In one aspect, electrodynamic structure  200  is similar to electrodynamic structures  100  and  150 , with the exception that electrodynamic structure  200  is rectangular and the conductors in the mesh of conductors  202  form rectangular shaped cells. In other embodiments, electrodynamic structure  200  can be part of any suitable system, such as a space station. In one embodiment, electrodynamic structure  200  is square. 
         [0090]    Conductive paths in electrodynamic structure  200  include the mesh of conductors  202 . The mesh of conductors  202  includes conductors that intersect other conductors to form rectangular shaped cells. Switches are situated at the cross points of the conductors to direct currents through the mesh of conductors  202  and across electrodynamic structure  200  and to direct currents through closed-loop paths in the mesh of conductors  202 . The mesh of conductors  202  increases the mechanical strength and stability of electrodynamic structure  200 . Also, closed-loop currents driven through selected closed-loop paths in the mesh of conductors  202  can provide improved control of the attitude dynamics of electrodynamic structure  200 . In some embodiments, meshes of conductors (e.g., the mesh of conductors  202 ) include non-conductive supports. 
         [0091]    Electrodynamic structure  200  includes the mesh of conductors  202  and periphery area  204 . Electrodynamic structure  200  is coupled to a payload  206 . Periphery area  204  is situated at the periphery of electrodynamic structure  200  and has width W. Payload  206  is mechanically coupled to components in periphery area  204  via the mesh of conductors  202 . Interior area  205  is defined between an interior edge of periphery area  204  and payload  206 . 
         [0092]    As illustrated in  FIG. 7 , periphery area  204  includes electron collectors  208 , electron emitters  210 , solar arrays  212 , and one or more controllers  214 . As illustrated, electron collectors  208  and electron emitters  210  are situated at an outer rim of the periphery area  204 . As illustrated, solar arrays  212  are situated on an interior rim of periphery area  204 . 
         [0093]    Electron collectors  208 , electron emitters  210 , solar arrays  212 , and controller  214  are electrically coupled via conductive paths formed with switches and conductors, including the mesh of conductors  202 . In one embodiment, electron collectors  208  are electrically coupled to each other via the conductive paths. In one embodiment, electron emitters  210  are electrically coupled to each other via the conductive paths. In one embodiment, solar arrays  212  are electrically coupled to each other via the conductive paths. 
         [0094]    The stabilizing mechanical property of the mesh of conductors  202  contributes to stabilizing electrodynamic structure  200 . In one embodiment, the mesh of conductors  202  is a non-rigid mesh. In one embodiment, the mesh of conductors  202  includes mechanically stiffened supports and/or mechanically stiffened conductors. 
         [0095]    In one embodiment, the mesh of conductors  202  electrically couples payload  206  to electrical components in periphery area  204 . In one embodiment, payload  206  receives power from electrical components in periphery area  204  via the mesh of conductors  202 . In one embodiment, payload  206  includes one or more controllers that control operation of electrodynamic structure  200  via control signals communicated on the mesh of conductors. 
         [0096]    One or more controllers  214  control operation of electrodynamic structure  200  via the switches in the conductive paths, including the switches in the mesh of conductors  202 . 
         [0097]      FIG. 8  is a diagram illustrating one embodiment of a rectangular, two-dimensional, planar electrodynamic structure  250  that has periphery utilization for electron collection and electron emission U of less than 1. In one embodiment, electrodynamic structure  250  has periphery utilization for electron collection and electron emission U substantially equal to ½. In other embodiments, electrodynamic structure  250  is an elongated rectangle and the periphery utilization for electron collection and electron emission U is less than ½. 
         [0098]    Electrodynamic structure  250  is configured to generate power and/or propulsion as it orbits a celestial body having a magnetic field. In one aspect, electrodynamic structure  250  is an electrodynamic sail. In one aspect, electrodynamic structure  250  is similar to electrodynamic structure  200 , with the exception that electrodynamic structure  250  does not have electron collectors or electron emitters on the top and bottom, such that periphery utilization U has been reduced to less than 1. In other embodiments, electrodynamic structure  250  can be part of any suitable system, such as a space station. 
         [0099]    Electrodynamic structure  250  includes a mesh of conductors  252 , a first periphery area  254   a,  and a second periphery area  254   b.  Electrodynamic structure  250  is coupled to a payload  256 . First periphery area  254   a  is situated at one side periphery of electrodynamic structure  250  and has width W. Second periphery area  254   b  is situated at the other side periphery of electrodynamic structure  250  and has width W. The mesh of conductors  252  are disposed in an interior area  255  defined by interior edges of periphery areas  254   a  and  254   b  and interior edges of the top and bottom periphery of electrodynamic structure  250 . Interior area  255  is defined between these defining interior edges and payload  256 . Payload  256  is mechanically coupled to components in periphery areas  254   a  and  254   b  via the mesh of conductors  252 . In some embodiments, meshes of conductors (e.g., the mesh of conductors  252 ) include non-conductive supports. 
         [0100]    As illustrated in  FIG. 8 , first periphery area  254   a  includes electron collectors  258   a,  electron emitters  260   a,  solar arrays  262   a,  and one or more controllers  264 . Electron collectors  258   a  and electron emitters  260   a  are situated at an outer rim of first periphery area  254   a.  Solar arrays  262   a  are situated on an interior rim of first periphery area  254   a.    
         [0101]    As illustrated in  FIG. 8 , second periphery area  254   b  includes electron collectors  258   b,  electron emitters  260   b,  and solar arrays  262   b.  Second periphery area  254   b  can also include one or more controllers  264 . Electron collectors  258   b  and electron emitters  260   b  are situated at an outer rim of second periphery area  254   b.  Solar arrays  262   b  are situated on an interior rim of second periphery area  254   b.    
         [0102]    Electron collectors  258 , electron emitters  260 , solar arrays  262 , and controllers  264  are electrically coupled via conductive paths formed with switches and conductors, including the mesh of conductors  252 . In one embodiment, electron collectors  258   a  are electrically coupled to each other via the conductive paths. In one embodiment, electron collectors  258   b  are electrically coupled to each other via the conductive paths. In one embodiment, electron emitters  260   a  are electrically coupled to each other via the conductive paths. In one embodiment, electron emitters  260   b  are electrically coupled to each other via the conductive paths. In one embodiment, solar arrays  262   a  are electrically coupled to each other via the conductive paths. In one embodiment, solar arrays  262   b  are electrically coupled to each other via the conductive paths. 
         [0103]    The stabilizing mechanical property of the mesh of conductors  252  contributes to stabilizing electrodynamic structure  250 . In one embodiment, the mesh of conductors  252  is a non-rigid mesh. In one embodiment, the mesh of conductors  252  includes mechanically stiffened supports and/or mechanically stiffened conductors. In some embodiments, meshes of conductors (e.g., the mesh of conductors  252 ) include non-conductive supports. 
         [0104]    Conductive paths in electrodynamic structure  250  include the mesh of conductors  252 . The mesh of conductors  252  includes conductors that intersect other conductors to form rectangular shaped cells. Switches are situated at the cross points of the conductors to direct currents through the mesh of conductors  252  and across electrodynamic structure  250  and to direct currents through closed-loop paths in the mesh of conductors  252 . Closed-loop currents driven through selected closed-loop paths in the mesh of conductors  252  provide improved control of the attitude dynamics of electrodynamic structure  250 . 
         [0105]    In one embodiment, the mesh of conductors  252  electrically couples payload  256  to electrical components in first periphery area  254   a  and/or second periphery area  254   b.  In one embodiment, payload  256  receives power from electrical components in the first periphery area  254   a  and/or second periphery area  254   b  via the mesh of conductors  252 . In one embodiment, payload  256  includes one or more controllers that control operation of electrodynamic structure  250  via control signals communicated on the mesh of conductors. 
         [0106]    One or more controllers  264  control operation of electrodynamic structure  250  via the switches in the conductive paths, including switches in the mesh of conductors  252 . 
         [0107]    The performance level of electrodynamic structure  250  can be estimated via Equation I. For a periphery utilization of U=1, the entire periphery of the electrodynamic structure is lined with electron collectors and electron emitters, such as with the rectangular electrodynamic structure  200  illustrated in  FIG. 7 . Electrodynamic structure  250 , however, does not have electron collectors or electron emitters on the top and bottom sides. Thus, electrodynamic structure  250  has a periphery utilization for electron collection and electron emission U of less than 1. In one embodiment, electrodynamic structure  250  is substantially square and the periphery utilization for electron collection and electron emission U is substantially equal to ½. In other embodiments, electrodynamic structure  250  is an elongated rectangle having longer top and bottom sides and the periphery utilization for electron collection and electron emission U is less than ½. It is overly inefficient for the electrodynamic structure to have periphery utilization U of less than ⅕. 
         [0108]    The area to perimeter parameter (S/P) is maximized if the electrodynamic structure is circular. If electrodynamic structure  250  is square, the area to perimeter ratio (S/P) is slightly smaller. Also, if electrodynamic structure  250  is rectangular, the area to perimeter ratio (S/P) is lower. It is overly inefficient for the electrodynamic structure to have an area to perimeter ratio (S/P) of less than ¼ of the ratio for a circle with the same perimeter. 
         [0109]      FIG. 9  is a diagram illustrating one embodiment of a cylindrical, three-dimensional electrodynamic structure  300  that is configured to generate power and/or propulsion as it orbits a celestial body having a magnetic field. In one aspect, electrodynamic structure  300  is an electrodynamic sail. In one aspect, electrodynamic structure  300  is similar to circular, two-dimensional electrodynamic structure  20 , with the exception that electrodynamic structure  300  includes one or more periphery areas situated at the periphery of electrodynamic structure  300  and perpendicular to the plane of electrodynamic structure  300 . In other embodiments, electrodynamic structure  300  can be part of any suitable system, such as a space station. 
         [0110]    Electrodynamic structure  300  includes supports  302  and periphery area with sections  304   a - 304   d.  Electrodynamic structure  300  is coupled to a payload  306 . The sections of the periphery area  304   a - 304   d  are situated next to each other at the periphery of electrodynamic structure  300  and perpendicular to the plane of supports  302 . Payload  306  is mechanically coupled to components in one or more periphery area sections  304   a - 304   d  via supports  302 . An interior area  305  is defined between at least one interior edge of at least one of periphery area sections  304   a - 304   d  and payload  306 . In some embodiments, periphery area sections have stiffness elements to maintain their shape and orientation. 
         [0111]    In one embodiment, each of the periphery area sections  304   a - 304   d  includes electron collectors, electron emitters, and solar arrays, which are suitably arranged and electrically coupled via conductive paths formed with switches and conductors (not shown for clarity). In some embodiments, electron collectors and/or solar arrays are suitably arranged and electrically coupled in interior area  305 . In some embodiments, the electron collectors, electron emitters, and solar arrays are electrically coupled via conductive paths including a mesh of conductors. 
         [0112]    In one embodiment, supports  302  are non-rigid supports. In one embodiment, supports  302  include mechanically stiffened supports and/or mechanically stiffened conductors. In one embodiment, one or more of the supports  302  electrically couple payload  306  to electrical components in one or more periphery areas  304   a - 304   d.  In one embodiment, electrodynamic structure  300  includes a mesh of conductors in place of supports  302  to mechanically and electrically couple payload  306  to components in one or more periphery area sections  304   a - 304   d.    
         [0113]    In one embodiment, payload  306  receives power from electrical components in one or more periphery area sections  304   a - 304   d  via conductive paths. In one embodiment, payload  306  includes one or more controllers that control operation of electrodynamic structure  300 . 
         [0114]    Electrodynamic structure  300  includes one or more controllers that controls operation of electrodynamic structure  300  via the switches in the conductive paths. The one or more controllers control electron collectors on one side of electrodynamic structure  300  to collect electrons from the ambient plasma and electron emitters on the opposite side of electrodynamic structure  300  to emit the electrons into the ambient plasma. Also, the controller controls power generation via a power system (e.g., solar arrays) and/or EMF in the conductors to drive the collected electrons to the electron emitters on the opposite side of electrodynamic structure  300 . 
         [0115]    Similar to as described above for two-dimensional electrodynamic structures, in three-dimensional electrodynamic structures (e.g., electrodynamic structure  300 ), the currents flowing through the conductive paths interact with the ambient magnetic field to produce distributed Ampere forces. These forces can be used to change the orbit and the orientation of the three-dimensional electrodynamic structure. Also, the motion through the ambient magnetic field produces EMF in the conductors, which can be used to drive electrons to the opposite side of the three-dimensional electrodynamic structure. 
         [0116]    Electrodynamic structure  300  is substantially a three-dimensional electrodynamic structure that has a two-dimensional projection  308  from the periphery of the three-dimensional structure. Periphery area sections  304   a - 304   d  project to a corresponding two-dimensional projected periphery area  310 . Interior area  305  projects to a corresponding two-dimensional projected interior area  312 . 
         [0117]    If electrodynamic structure  300  is spin stabilized with the spin axis normal to the plane of supports  302 , spin characteristics and dynamics of electrodynamic structure  300  are similar to spin characteristics and dynamics described above for two-dimensional electrodynamic structures. 
         [0118]    The performance level of electrodynamic structure  300  can be estimated via Equation I as applied to electrodynamic structure  300  and the two-dimensional projection  308 . 
         [0119]    In some embodiments, all periphery area sections  304   a - 304   d  include electron collectors and electron emitters on the entire periphery of electrodynamic structure  300 , which yields a utilization factor of U=1. If half of the periphery of the electrodynamic structure  300  includes electron collectors and electron emitters and the other half is empty, U=½. 
         [0120]    The area to perimeter ratio (S/P) is calculated from the two-dimensional projection at  308 . The periphery of the three-dimensional structure  300  defines the perimeter of the two-dimensional projection  308 . The two-dimensional projection at  308  is circular, which provides the best area to perimeter ratio (S/P). Other shapes, such as square or hexagonal shapes, provide smaller area to perimeter ratios (S/P). Elongated shapes, such as elongated elliptical shapes and elongated rectangular shapes have even lower area to perimeter ratios (S/P). 
         [0121]    The electric current production per unit mass of the electrodynamic structure (I/M) depends on the electron collection and electron emission technologies and on the weight of the support structures. In one embodiment, electrodynamic structure  300  has a lower specific current production rate I/M due to the mass of additional support structures, such as stiffness elements, used to maintain the non-planar shape. 
         [0122]      FIG. 10  is a diagram illustrating one embodiment of a curved, half-elliptical, three-dimensional electrodynamic structure  400  that is configured to generate power and/or propulsion as it orbits a celestial body having a magnetic field. In one aspect, electrodynamic structure  400  is an electrodynamic sail. In one aspect, electrodynamic structure  400  is similar to two-dimensional electrodynamic structure  100 , with the exception that electrodynamic structure  400  is a curved, elliptical shape and includes multiple sections of the periphery area  404   a  and  404   b . In other embodiments, electrodynamic structure  400  can be part of any suitable system, such as a space station. 
         [0123]    Electrodynamic structure  400  includes a mesh of conductors  402  and periphery area sections  404   a  and  404   b.  Electrodynamic structure  400  is coupled to a payload  406 . Periphery area sections  404   a  and  404   b  are situated next to each other at the periphery of electrodynamic structure  400 . Payload  406  is mechanically coupled to components in one or more periphery area sections  404   a  and  404   b  via the mesh of conductors  402 . An interior area  405  is defined between an interior edge of periphery area section  404   b  and payload  406 . The mesh of conductors  402  are disposed in interior area  405 . In some embodiments, meshes of conductors (e.g., the mesh of conductors  402 ) include non-conductive supports. 
         [0124]    In one embodiment, each of the periphery area sections  404   a  and  404   b  includes electron collectors, electron emitters, and solar arrays, which are suitably arranged and electrically coupled via conductive paths formed with switches and conductors, including the mesh of conductors  402 . In some embodiments, electron collectors and/or solar arrays are suitably arranged and electrically coupled in interior area  405 . 
         [0125]    The stabilizing mechanical property of the mesh of conductors  402  contributes to stabilizing electrodynamic structure  400 . In one embodiment, the mesh of conductors  402  is a non-rigid mesh. In one embodiment, the mesh of conductors  402  includes mechanically stiffened supports and/or mechanically stiffened conductors. 
         [0126]    Conductive paths in electrodynamic structure  400  include the mesh of conductors  402 . The mesh of conductors  402  includes conductors that intersect other conductors and switches situated at the cross points of the conductors. The switches are controlled to direct currents through the mesh of conductors  402  and across electrodynamic structure  400  and to direct currents in current loops in the mesh of conductors  402 . 
         [0127]    In one embodiment, the mesh of conductors  402  electrically couples payload  406  to the electrical components in one or more periphery area sections  404 . In one embodiment, payload  406  receives power from electrical components in one or more periphery area sections  404  via the mesh of conductors  402 . In one embodiment, payload  406  includes one or more controllers that control operation of electrodynamic structure  400  via control signals communicated on the mesh of conductors  402 . 
         [0128]    Electrodynamic structure  400  includes one or more controller that controls operation of electrodynamic structure  400  via switches in the conductive paths, including switches in the mesh of conductors  402 . 
         [0129]    Electrodynamic structure  400  is substantially a three-dimensional electrodynamic structure that has a two-dimensional projection  408  from the periphery of the three-dimensional structure. Periphery area sections  404   a  and  404   b  project to a corresponding two-dimensional projected periphery area  410 . Interior area  405  projects to a corresponding two-dimensional projected interior area  412 . 
         [0130]    If electrodynamic structure  400  is spin stabilized with the spin axis normal to the plane of projection  408 , spin characteristics and dynamics of electrodynamic structure  400  are similar to spin characteristics and dynamics described above for two-dimensional electrodynamic structures. In addition, similar to as described above for two-dimensional electrodynamic structures having meshes of conductors closed-loop currents through selected closed-loop paths in the mesh of conductors  402  can provide improved control of the attitude dynamics of electrodynamic structure  400  via the mesh of conductors  402 . 
         [0131]    The performance level of electrodynamic structure  400  can be estimated via Equation I as applied to electrodynamic structure  400  and the two-dimensional projection  408 . 
         [0132]    In some embodiments, both periphery area sections  404   a  and  404   b  include electron collectors and electron emitters on the entire periphery of electrodynamic structure  400 , which yields a utilization factor of U=1. If half of the periphery of the electrodynamic structure  400  includes electron collectors and electron emitters and the other half is empty, U=½. 
         [0133]    The area to perimeter ratio (S/P) is calculated from the two-dimensional projection at  408 . The periphery of the three-dimensional structure  400  defines the perimeter of the two-dimensional projection  408 . The two-dimensional projection at  408  is circular, which provides the best area to perimeter ratio (S/P). Other shapes, such as square or hexagonal shapes, provide smaller area to perimeter ratios (S/P). Elongated shapes, such as elongated elliptical shapes and elongated rectangular shapes have even lower area to perimeter ratios (S/P). 
         [0134]    The electric current production per unit mass of the electrodynamic structure (I/M) depends on the electron collection and electron emission technologies and on the weight of the support structures. In one embodiment, electrodynamic structure  400  has a lower specific current production rate I/M due to the mass of additional support structures, such as stiffness elements, used to maintain the non-planar shape. 
         [0135]      FIG. 11  is a diagram illustrating one embodiment of an enclosed, three-dimensional electrodynamic structure  500  that includes an upper conical section  502  and a lower conical section  504  coupled at a line of connection  506 . Periphery area  508   a  is situated at the periphery of upper conical section  502 . Periphery area  508   b  is situated at the periphery of lower conical section  504 . Periphery areas  508   a  and  508   b  meet at the line of connection  506 . 
         [0136]    Electrodynamic structure  500  is configured to generate power and/or propulsion as it orbits a celestial body having a magnetic field. In one aspect, electrodynamic structure  500  is an electrodynamic sail. In other embodiments, electrodynamic structure  500  can be part of any suitable system, such as a space station. 
         [0137]    Electrodynamic structure  500  includes periphery areas  508   a  and  508   b,  an upper mesh of conductors  510 , and a lower mesh of conductors  512 . Electrodynamic structure  500  can be coupled to a payload situated inside electrodynamic structure  500  or at a point of one of the conical sections  502  and  504 . The payload can be mechanically coupled to the upper mesh of conductors  510  and/or the lower mesh of conductors  512 . In one embodiment, components in periphery areas  508   a  and  508   b  are coupled to each other at the line of connection  506 . The upper mesh of conductors  510  and the lower mesh of conductors  512  are three-dimensional meshes. In one embodiment, the upper mesh of conductors  510  and the lower mesh of conductors  512  are not mounted on a substrate and are not covered with material. In some embodiments, the upper mesh of conductors  510  and the lower mesh of conductors  512  are mounted on a substrate. In one embodiment, the substrate is reflective. In some embodiments, meshes of conductors (e.g., the mesh of conductors  510 ) include non-conductive supports. 
         [0138]    In one embodiment, each of the periphery areas  508   a  and  508   b  includes electron collectors, electron emitters, and solar arrays, which are suitably arranged and electrically coupled via conductive paths formed with switches and conductors, which include the upper and lower meshes of conductors  510  and  512 . In some embodiments, electron collectors and/or solar arrays are suitably arranged and electrically coupled on surfaces of conical sections  502  and  504  other than in the periphery areas  508   a  and  508   b.    
         [0139]    The stabilizing mechanical properties of the upper and lower meshes of conductors  510  and  512  contribute to stabilizing electrodynamic structure  500 . In one embodiment, each of the upper and lower meshes of conductors  510  and  512  is a non-rigid mesh. In one embodiment, each of the upper and lower meshes of conductors  510  and  512  includes mechanically stiffened supports and/or mechanically stiffened conductors. In one embodiment, electrodynamic structure  500  includes stiffness supports between conical sections  502  and  504  to maintain its non-planar shape. 
         [0140]    Conductive paths in electrodynamic structure  500  include the upper and lower meshes of conductors  510  and  512 . Each of the upper and lower meshes of conductors  510  and  512  includes conductors that intersect other conductors and switches situated at the cross points of the conductors. The switches are controlled to direct currents through the upper and lower meshes of conductors  510  and  512  and across electrodynamic structure  500  and to direct currents in current loops in each of the upper and lower meshes of conductors  510  and  512 . 
         [0141]    In one embodiment, one or more of the upper and lower meshes of conductors  510  and  512  electrically couples the payload to electrical components in one or more periphery areas  508   a  and  508   b.  In one embodiment, the payload receives power from electrical components in one or more periphery areas  508   a  and  508   b  via one or both of the upper and lower meshes of conductors  510  and  512 . In one embodiment, the payload includes one or more controllers that control operation of electrodynamic structure  500  via control signals communicated on one or both of the mesh of conductors  510  and  512 . 
         [0142]    Electrodynamic structure  500  includes one or more controllers that control operation of electrodynamic structure  500  via switches in the conductive paths, including switches in the upper and lower meshes of conductors  510  and  512 . 
         [0143]    Electrodynamic structure  500  is substantially a three-dimensional electrodynamic structure that has a two-dimensional projection  514  from the periphery of the three-dimensional structure. Periphery areas  508   a  and  508   b  project to a two-dimensional projected periphery area  516 . The portions of conical sections  502  and  504  that do not include periphery areas  508   a  and  508   b  project to a corresponding two-dimensional projected interior area  518 . 
         [0144]    If electrodynamic structure  500  is spin stabilized with the spin axis normal to the plane of projection  514 , spin characteristics and dynamics of electrodynamic structure  500  are similar to spin characteristics and dynamics described above for two-dimensional electrodynamic structures. In addition, similar to as described above for two-dimensional electrodynamic structures having meshes of conductors closed-loop currents through selected closed-loop paths in the upper and lower meshes of conductors  510  and  512  can provide improved control of the attitude dynamics of electrodynamic structure  500 . 
         [0145]    The performance level of electrodynamic structure  500  can be estimated via Equation I as applied to electrodynamic structure  500  and the two-dimensional projection  514 . 
         [0146]    In some embodiments, both periphery areas  508   a  and  508   b  include electron collectors and electron emitters on the entire periphery of electrodynamic structure  500 , which yields a utilization factor of U=1. If half of the periphery of the electrodynamic structure  500  includes electron collectors and electron emitters and the other half is empty, U=½. 
         [0147]    The area to perimeter ratio (S/P) is calculated from the two-dimensional projection at  514 . The periphery of the three-dimensional structure  500  defines the perimeter of the two-dimensional projection  514 . The two-dimensional projection at  514  is circular, which provides the best area to perimeter ratio (S/P). Other shapes, such as square or hexagonal shapes, provide smaller area to perimeter ratios (S/P). Elongated shapes, such as elongated elliptical shapes and elongated rectangular shapes have even lower area to perimeter ratio (S/P). 
         [0148]    The electric current production per unit mass of the electrodynamic structure (I/M) depends on the electron collection and electron emission technologies and on the weight of the support structures. In one embodiment, electrodynamic structure  500  has a lower specific current production rate I/M due to the mass of additional support structures, such as stiffness elements, used to maintain the non-planar shape. 
         [0149]    Two- and three-dimensional electrodynamic structures can fly a variety of missions, taking advantage of propellantless propulsion and virtually unlimited changes in velocity. Two- and three-dimensional electrodynamic structures can repeatedly go from orbit to orbit, with or without payloads, dramatically changing orbital elements in a matter of weeks or months, and keeping all inclinations within reach. 
         [0150]    If desired, the propulsion capabilities of the two- and three-dimensional electrodynamic structures can be augmented by mounting ion thrusters at various points of the electrodynamic structure and utilizing some of the energy collected by the solar arrays of the electrodynamic structure for ion propulsion. 
         [0151]    Although specific embodiments have been illustrated and described herein, it will be appreciated by those of ordinary skill in the art that a variety of alternate and/or equivalent implementations may be substituted for the specific embodiments shown and described without departing from the scope of the present invention. This application is intended to cover any adaptations or variations of the specific embodiments discussed herein. Therefore, it is intended that this invention be limited only by the claims and the equivalents thereof.