Patent Publication Number: US-2012042949-A1

Title: Solar concentrator

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of U.S. Provisional Patent Application 61/178,069, filed 14 May, 2009, which is incorporated herein by reference. 
    
    
     FIELD OF THE INVENTION 
     The present invention relates generally to solar radiation, and specifically to concentrating the radiation. 
     BACKGROUND OF THE INVENTION 
     As electrical energy demand grows, there is an increased interest in efficiently converting solar radiation to electrical energy. Typically, photovoltaic cells implement the conversion, and systems which perform the conversion using non-concentrated as well as concentrated solar radiation are known in the art. Concentrating systems typically use one or more mirrors to effect the concentration. 
     The description above is presented as a general overview of related art in this field and should not be construed as an admission that any of the information it contains constitutes prior art against the present patent application. 
     SUMMARY OF THE INVENTION 
     An embodiment of the present invention provides apparatus, including: 
     a photovoltaic cell; 
     a concave primary reflector configured to focus a first portion of incoming radiation toward a focal point; 
     a secondary reflector, which is positioned between the concave primary reflector and the focal point so as to direct the focused radiation toward the photovoltaic cell, and which has a central opening aligned with the photovoltaic cell; and 
     a transmissive concentrator, positioned so as to focus a second portion of the incoming radiation through the central opening onto the photovoltaic cell. 
     Typically, at least one of the primary reflector and the secondary reflector include a plurality of curved segments. 
     In a disclosed embodiment the apparatus further includes a tracking device connected to the photovoltaic cell, the primary reflector, the secondary reflector, and the transmissive concentrator, wherein the primary reflector has an aperture, and wherein dimensions of the transmissive concentrator and the aperture differ by no more than a value determined in response to a tracking error of the tracking device. 
     The transmissive concentrator may have a concentrator-dimension larger than a largest dimension of the secondary reflector. Alternatively, the transmissive concentrator and the secondary reflector may have congruent external dimensions. 
     A shape of the transmissive concentrator may be geometrically similar to the central opening. 
     The apparatus may include a homogenizer, positioned between the secondary reflector and the photovoltaic cell, which may redirect at least some of the focused radiation onto the photovoltaic cell. The homogenizer may redirect at least some of the second portion of the radiation onto the photovoltaic cell. 
     Typically, the central opening is aligned and dimensioned within the secondary reflector so as to receive none of the focused radiation. 
     In an alternative embodiment the transmissive concentrator has a concentrator-dimension larger than a largest dimension of the central opening. 
     In one embodiment the concave primary reflector includes a paraboloidal reflector. 
     There is further provided a method, including: 
     stamping flat metal plates so as to create a plurality of segments having a predetermined curved shape; and 
     joining the curved segments together in order to create a curved reflector. 
     The method may include applying a reflective coating to the metal plates prior to stamping the plates. Typically, a deformation caused by stamping the flat metal plates is within a tolerance limit of the reflective coating. 
     The predetermined curved shape and the curved reflector may be sections of a common paraboloid. 
     There is further provided a method, including: 
     configuring a concave primary reflector to focus a first portion of incoming radiation toward a focal point; 
     positioning a secondary reflector between the concave primary reflector and the focal point so as to direct the focused radiation toward a photovoltaic cell; 
     aligning a central opening in the secondary reflector with the photovoltaic cell; and 
     positioning a transmissive concentrator to focus a second portion of the incoming radiation through the central opening onto the photovoltaic cell. 
     There is further provided apparatus, including: 
     a plurality of flat metal plates which are configured to form respective curved segments having respective predetermined curved shapes; and 
     at least one joint which holds the curved segments together in order to create a curved reflector. 
     The present invention will be more fully understood from the following detailed description of the embodiments thereof, taken together with the drawings, in which: 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIGS. 1A ,  1 B, and  1 C, schematically illustrate components of a solar collector, according to an embodiment of the present invention; 
         FIG. 2A  is a schematic diagram showing irradiance at the position of a secondary reflector of the solar collector, due to reflection from a primary reflector, and  FIG. 2B  is a graph of the irradiance vs. distance, according to embodiments of the present invention; 
         FIGS. 3A and 3B  illustrate components of a concentrating photovoltaic (CPV) system, according to an alternative embodiment of the present invention; 
         FIG. 4A  illustrates a design for a cell mount, and  FIG. 4B  illustrates an alternative design for the cell mount, according to embodiments of the present invention; 
         FIG. 5A  is a schematic diagram showing irradiance at the position of a secondary reflector of the CPV system of  FIGS. 3A and 3B , due to reflection from a primary reflector, and 
         FIG. 5B  is a graph of the irradiance vs. distance, according to embodiments of the present invention; 
         FIG. 6  is a schematic sectional side view of a matrix of CPV systems, according to an embodiment of the present invention; 
         FIGS. 7A ,  7 B, and  7 C are schematic front, back and side views respectively of a primary reflector, according to an embodiment of the present invention; 
         FIG. 8  is a schematic front view of a primary reflector, according to an alternative embodiment of the present invention; 
         FIG. 9A  is a schematic front view of segments of a primary reflector, and  FIG. 9B  is a schematic plan view of a plane sheet for producing some of the segments, according to an embodiment of the present invention; 
         FIG. 10A  and  FIG. 10B  illustrate the reduction in deformation achieved by constructing a paraboloidal reflector from smaller segments, according to embodiments of the present invention; and 
         FIG. 11  is a schematic, pictorial illustration showing assembly of the matrix of  FIG. 6 , according to an embodiment of the present invention. 
     
    
    
     DETAILED DESCRIPTION OF EMBODIMENTS 
     Overview 
     Some embodiments of the present invention provide improved methods for concentrating solar radiation in a concentrating photovoltaic (CPV) system. An arrangement of reflectors comprises a primary concave reflector which reflects incoming solar radiation towards a focus. The rays from the primary reflector are intercepted by a secondary reflector which directs the rays to a solar cell. 
     Incoming solar rays which would normally be shaded from the primary reflector by the secondary reflector are intercepted by a transmissive ray concentrator, typically a Fresnel or refractive lens. The concentrator converges the intercepted rays towards the secondary reflector. The inventors have determined that there is a central region of the secondary reflector which receives no rays from the primary reflector. An opening is provided in this central region, permitting the converged rays from the concentrator to pass through the secondary reflector to the solar cell. Because of its positioning in the central region of the secondary reflector, the opening does not prevent passage of rays from the primary reflector to the solar cell. Thus, all incoming solar rays may be concentrated onto the same solar cell. 
     One or both of the reflectors in the CPV system may be produced by joining a plurality of metal plates so as to create the required curved reflector shape. Each metal plate is typically formed by stamping respective plane metal sheets, so forming respective segments of the reflector being produced. By forming a reflector from a plurality of segments stamped from plane sheets, the overall deformation from a plane to the required curved shape is substantially reduced compared to the deformation engendered by stamping a single metal sheet. The plane metal sheets may be pre-covered with reflective material and then stamped into their required shape. By forming the reflector from segments, the deformation is sufficiently reduced to prevent degradation of the reflective properties of the sheeting by the stamping. 
     DETAILED DESCRIPTION 
     Reference is now made to  FIGS. 1A ,  1 B, and  1 C, which schematically illustrate components of a solar collector  20 , according to an embodiment of the present invention. Collector  20  acts as a concentrating collector, concentrating incoming solar radiation energy in the form of parallel solar rays  28  onto a cell  22 , the cell converting the radiation to another form of energy. Typically, and as assumed in the following description, cell  22  comprises a photovoltaic cell, which absorbs the concentrated radiation energy and generates electric power from a portion of the absorbed energy. Thus collector  20  together with cell  22  act as a concentrating photovoltaic (CPV) system  24 . In some embodiments cell  22  has a generally square outline, although there is no limitation on the shape of the cell. Suitable CPV cells for use in system  24  comprise, but are not limited to, a CTJ Photovoltaic Cell produced by Emcore Corporation of Albuquerque, N. Mex., or a CDO-100-C3MJ Concentrator Solar Cell produced by Spectrolab, Inc. of Sylmar Calif. 
     Typically, CPV system  24  is mounted on a tracking device  25 , which rotates the system so that axis  32  points towards the sun. For simplicity, some supports connecting elements of system  24  together and to the tracking device are not shown in  FIGS. 1A ,  1 B and  1 C. 
       FIG. 1A  is a schematic top view of components of CPV system  24 , and  FIG. 1B  is a sectional side view of the system taken on a line I-I of  FIG. 1A . For clarity, elements supporting the components of the system are not shown in the  FIGS. 1A and 1B .  FIG. 1C  schematically illustrates supports for some of the components of the system. System  24  comprises a primary concentrating concave reflector  26 , which is assumed to have an approximately square outline. Reflector  26  may have any concave shape, or combination of shapes, which concentrate incoming approximately parallel light to a focal region. Such shapes include, but are not limited to, spherical and aspherical shapes. By way of example, reflector  26  is assumed to be formed as a paraboloid having an axis of symmetry  32  and a focal point  30  on the axis. 
     Reflector  26  is typically formed with an aperture  34  symmetrically located at its center. 
     Incoming solar rays  28  are comprised of two groups of rays: a central group  27  of rays, and a peripheral group  29  of the rays. As explained in more detail below, central group  27  are diverted by a transmissive concentrator  54 . Peripheral group  29  transmit directly to the primary reflector and are redirected as reflected rays  36  towards focal point  30 . 
     Reflected rays  36  are intercepted by a secondary reflector  38  before they reach point  30 . The secondary reflector has an axis of symmetry that is substantially coincident with axis  32 . The secondary reflector is positioned to reflect rays  36  towards the primary reflector, so that the rays reflected from the secondary reflector converge to a focal region  42 . Region  42  is approximately centered on axis  32 , and is located between the primary and secondary reflectors. 
     Secondary reflector  38  may be plane or curved, and if curved, it may be concave or convex. Hereinbelow, by way of example, the secondary reflector is assumed to be spherically convex. 
     The secondary reflector has an opening  44 , symmetrically disposed with respect to axis  32 . As explained in more detail below, opening  44  allows central group  27  of incoming rays  28  to reach the cell. Typically opening  44  has the same shape as the transmissive concentrator, and in this embodiment is circular, although in some embodiments the opening may be non-circular. 
     Ray concentrator  54  is positioned above the secondary reflector to intercept central group  27  of incoming rays  28 . Concentrator  54  typically comprises a Fresnel lens or a converging lens made from glass or transparent plastic. The concentrator is configured to divert the central group of rays through opening  44 , to region  42 . 
     In some embodiments system  24  comprises a transparent window  56  above concentrator  54 . Window  56  serves to shield the other elements of system  24  from dust or other material that could reduce the efficiency of operation of the system. Concentrator  54  may be connected to the window using optical cement, so that the window acts as a support for the concentrator. 
     Because central group  27  of rays are diverted (by concentrator  54 ) towards region  42 , the central ray group does not directly transmit to reflector  26 . To accommodate tracking errors, aperture  34  is typically configured to have dimensions somewhat smaller than concentrator  54 . The reduction in dimensions is typically based on an expected error of the tracking system, and enables collection of rays that miss the concentrator because of the tracking error. 
     In order to collect the rays passing through region  42  onto cell  22 , solar concentrator  20  comprises a homogenizer  46 , which is typically formed as a solid element from an optically clear glass designed to direct the incoming radiation, by total internal reflection, onto the cell. 
     Alternatively, homogenizer  46  may comprise an open tubular element having an axis of symmetry that is generally coincident with axis  32 . In this case, the homogenizer has a reflective inner surface and it is typically configured to have a lower opening  50  that surrounds and mates with cell  22 . The homogenizer has an upper opening  52  that is larger than its lower opening. In some alternative embodiments homogenizer  46  is in the form of a hollow truncated cone or pyramid, and in one embodiment the homogenizer comprises a hollow truncated square pyramid, having upper and lower openings that are square. 
     It will be understood that without ray concentrator  54 , some of rays  27  would be shaded from the primary reflector by the secondary reflector. Concentrator  54  ensures that all of rays  27  are directed to homogenizer  46 . 
     In some embodiments cell  22  requires cooling in order to perform its energy conversion function efficiently. For example, semiconducting photovoltaic cells for use in CPV systems typically only convert about 40% of their incident radiant energy to electric energy, so that the remainder is converted to heat. The cooling provided to cell  22  may be passive cooling, typically relying on natural convection of air surrounding the cell and/or of air surrounding heat conducting fins that conduct the heat from the cell. Alternatively or additionally, the cooling provided to the cell may comprise active cooling, which typically uses forced flow of a fluid such as air or water over a rear surface of the cell. For clarity and simplicity, mechanisms for providing the cooling are not shown in  FIGS. 1A ,  1 B and  1 C, but mechanisms for providing such cooling are described below. It will be appreciated that locating cell  22  in or close to aperture  34 , so that a rear surface of the cell is easily accessible from the rear of the primary reflector, facilitates provision of active or passive cooling to the cell. 
       FIG. 1C  illustrates a side view and a top view of a mounting  60  for primary reflector  26  and secondary reflector  38 . Elements of mounting  60  are typically formed from plastic, using an injection molding process. The mounting comprises a lower skeleton-like section  62 , which has an interior shape that is paraboloidal. Section  62  retains primary reflector  26  and has dimensions corresponding to those of the reflector. Mounting  60  also comprises an upper section  64  having a convex shape. Section  64  retains secondary reflector  38  and has dimensions corresponding to those of the secondary reflector, including a hole corresponding to opening  44  of the reflector. Sections  62  and  64  are connected together by thin supports  66  in order to minimize shading losses. The supports serve to hold the two reflectors fixed in their correct positions and orientations with respect to each other. Typically, mounting  60  may be used in a production phase to pre-assemble and align the primary and secondary reflectors as a composite unit, for maximum accuracy, so that the unit (the mounting with its reflectors) is available for a final assembly phase. The final assembly phase typically includes incorporating the composite unit in a system mounting panel, such as is exemplified in the description of matrix  200  ( FIG. 6 ) below. 
     Table I below gives characteristics of components of a first exemplary embodiment of system  24 . The dimensions given in Table I are approximate. 
     
       
         
           
               
               
             
               
                 TABLE I 
               
               
                   
               
               
                 Component 
                 Characteristics 
               
               
                   
               
             
            
               
                 Concentrator 54 
                 Circular Fresnel lens, diameter 180 mm, focal 
               
               
                   
                 length 190 mm 
               
               
                 Secondary reflector 38 
                 Convex spherical mirror, square 92 mm × 92 mm, 
               
               
                   
                 radius 200 mm. Central opening 44 in the mirror 
               
               
                   
                 is circular, corresponding to the shape of 
               
               
                   
                 concentrator 54. Opening 44 has a diameter of 50 
               
               
                   
                 mm. 
               
               
                 Primary reflector 26 
                 Paraboloidal, square 250 mm × 250 mm, focal 
               
               
                   
                 length 127 mm. Central aperture 34 is circular 
               
               
                   
                 with diameter 180 mm (corresponding with the 
               
               
                   
                 shape and diameter of the Fresnel lens). 
               
               
                 Solar cell 22 
                 Square, 10 mm × 10 mm 
               
               
                 Homogenizer 46 
                 Square truncated pyramid, of BK7 optical glass. 
               
               
                   
                 Length 34 mm, upper dimensions 22 mm × 22 
               
               
                   
                 mm, lower dimensions 9.8 mm × 9.8 mm 
               
               
                   
                 (corresponding to, but slightly smaller than, the 
               
               
                   
                 solar cell dimensions). 
               
               
                   
               
            
           
         
       
     
     Table II below gives typical approximate distances between components of the first exemplary embodiment of system  24 . 
     
       
         
           
               
               
             
               
                 TABLE II 
               
               
                   
               
               
                 Components 
                 Distance between Components 
               
               
                   
               
             
            
               
                 Concentrator 54-secondary reflector 38 
                 153 mm 
               
               
                 Secondary reflector 38-top of 
                  44 mm 
               
               
                 homogenizer 46 
                   
               
               
                 Solar cell 22-aperture 34 
                  17 mm 
               
               
                   
               
            
           
         
       
     
     Table III below gives characteristics of components of a second exemplary embodiment of system  24 . The dimensions given in Table III are approximate. 
     
       
         
           
               
               
             
               
                 TABLE III 
               
               
                   
               
               
                 Component 
                 Characteristics 
               
               
                   
               
             
            
               
                 Concentrator 54 
                 Circular Fresnel lens, diameter 130 mm, focal 
               
               
                   
                 length 220 mm 
               
               
                 Secondary reflector 38 
                 Convex hyperboloidal mirror, square 90 mm × 90 
               
               
                   
                 mm, hyperbola defined as radius 200 mm at the 
               
               
                   
                 vertex and conic constant, K, equal to −4. Central 
               
               
                   
                 opening 44 in the mirror is circular, 
               
               
                   
                 corresponding to the shape of concentrator 54. 
               
               
                   
                 Opening 44 has a diameter of 38 mm. 
               
               
                 Primary reflector 26 
                 Paraboloidal, square 230 mm × 230 mm, focal 
               
               
                   
                 length 127 mm. Central aperture 34 is circular 
               
               
                   
                 with diameter 120 mm (corresponding with the 
               
               
                   
                 shape of the Fresnel lens). The diameter is 
               
               
                   
                 slightly less than the lens to allow for tracking 
               
               
                   
                 errors. 
               
               
                 Solar cell 22 
                 Square, 5.5 mm × 5.5 mm 
               
               
                 Homogenizer 46 
                 Square truncated compound parabolic 
               
               
                   
                 concentrator, of BK7 optical glass. Length 23 
               
               
                   
                 mm, upper dimensions 14.36 mm × 14.36 mm, 
               
               
                   
                 lower dimensions 5.4 mm × 5.4 mm 
               
               
                   
                 (corresponding to, but slightly smaller than, the 
               
               
                   
                 solar cell dimensions). 
               
               
                   
               
            
           
         
       
     
     Table IV below gives typical approximate distances between the components of system  24  listed in Table III. 
     
       
         
           
               
               
             
               
                 TABLE IV 
               
               
                   
               
               
                 Components 
                 Distance between Components 
               
               
                   
               
             
            
               
                 Concentrator 54-secondary reflector 38 
                 163.2 mm 
               
               
                 Secondary reflector 38-top of 
                  67.8 mm 
               
               
                 homogenizer 46 
                   
               
               
                 Secondary reflector 38-solar cell 22 
                  90.8 mm 
               
               
                   
               
            
           
         
       
     
     It will be understood that the characteristics and distances given in Tables I-IV are given by way of example. Those having ordinary skill in the art will be able to formulate other characteristics for the components, and distances between the components, without undue experimentation. Typically the formulation may be achieved using an optical simulation package such as ZEMAX software produced by ZEMAX Development Corporation, Bellevue, Wash. 
       FIG. 2A  is a schematic diagram showing irradiance at the position of secondary reflector  38  of system  24 , due to reflection from the primary reflector, and  FIG. 2B  is a graph of the irradiance vs. distance, according to embodiments of the present invention. The diagram and graph are for the first exemplary embodiment of system  24  given above. The irradiance for  FIG. 2A  is plotted over a square region corresponding to the dimensions of the secondary reflector, i.e., 92 mm×92 mm. The graph of  FIG. 2B  plots the irradiance vs. distance along a symmetry line  70  of  FIG. 2A . 
     All the reflected radiation from the primary reflector is incident on the secondary reflector. Shaded region  72  illustrates the region of the secondary reflector that receives the primary&#39;s reflected radiation. As shown by  FIG. 2A , all the reflected radiation is contained within region  72  that is bounded by an inner circle  74  and an outer square  76  having an edge of approximately 80 mm. There is no reflected radiation from the primary reflector within a central region  78  having an external bound corresponding to inner circle  74 . 
     Embodiments of the present invention take advantage of the absence of any reflected radiation in central region  78  by providing opening  44  in the secondary reflector, since such an opening causes no reduction in radiation at the primary reflector. Not only does opening  44  cause no reduction in radiation at the primary reflector, but it allows all central group  27  of incoming rays  28  to be converged through the opening onto the cell. 
       FIGS. 3A and 3B  illustrate components of an alternative CPV system  124 , according to an embodiment of the present invention.  FIG. 3A  is a top view of system  124  and  FIG. 3B  is a sectional side view. Apart from the differences described below, the operation of CPV system  124  is generally similar to that of CPV system  24  ( FIGS. 1A ,  1 B, and  1 C) and elements indicated by the same reference numerals in both systems  24  and  124  are generally similar in operation. Elements in system  124  having an apostrophe &#39; appended to the reference numeral may differ in dimensions from elements (of the exemplary embodiments of system  24  described above) having the same numeral. 
     In system  124 , a concentrator  54 ′ and an aperture  34 ′ have substantially the same dimensions as a secondary reflector  38 ′. In addition, in system  124  a thin tube  126  fixedly connects the back of the secondary reflector to window  56 . The tube acts as a support for the secondary reflector and has a minimal foot print to minimize shading losses. 
     In system  124  secondary reflector  38 ′ is flat, rather than being curved as in system  24 . 
     Also in contrast to system  24 , in system  124  cell  22  and a homogenizer  46 ′ are positioned above the interior surface of primary reflector  26  by the cell and homogenizer being fixedly mounted on a cell mount  128 . Mount  128  is configured to be sufficiently narrow so as to be completely in the shadow of secondary reflector  38 ′, so that none of peripheral group  29  of the incoming solar rays are prevented from reaching a primary reflector  26 ′.  FIGS. 4A and 4B , referred to below, illustrate alternative designs for mount  128  and elements contained by the mount (not shown in  FIG. 3B ). 
     Repositioning cell  22  and homogenizer  46 ′ (from the locations of the cell and homogenizer of system  24  to those of system  124 ) requires repositioning of focal region  42 . Region  42  may be repositioned by changing parameters, for example the focal lengths, of secondary reflector  38 ′ and concentrator  54 ′. Evaluation of such changes will be apparent to those having ordinary skill in the optical arts. 
     As for system  24 , peripheral group  29  of rays pass directly to the primary reflector, and central group  27  of rays are converged by the concentrator to pass through an opening  44 ′ in the secondary reflector. Thus all incoming rays  28  are focused on cell  22 . 
     Table V below gives characteristics of components of an exemplary embodiment of system  124 . To differentiate the exemplary embodiments of the two systems (system  24  and system  124 ), the exemplary embodiment of system  124  is referred to as the third exemplary embodiment. 
     
       
         
           
               
               
             
               
                 TABLE III 
               
               
                   
               
               
                 Component 
                 Characteristics 
               
               
                   
               
             
            
               
                 Concentrator 54′ 
                 Square Fresnel lens or refractor, 95 mm × 95 mm, 
               
               
                   
                 focal length 134 mm 
               
               
                 Secondary reflector 38′ 
                 Plane square mirror, 95 mm × 95 mm, i.e., 
               
               
                   
                 congruent with the dimensions of concentrator 
               
               
                   
                 54′. The mirror has a central square opening 
               
               
                   
                 (16 mm × 16 mm) to correspond with and be 
               
               
                   
                 geometrically similar to the shape of concentrator 
               
               
                   
                 54′. 
               
               
                 Primary reflector 26′ 
                 Paraboloidal, square 250 mm × 250 mm, focal 
               
               
                   
                 length 127 mm. Central aperture 34′ is a square 
               
               
                   
                 95 mm × 95 mm (corresponding with the 
               
               
                   
                 dimensions and shape of concentrator 54′ and 
               
               
                   
                 reflector 38′). 
               
               
                 Solar cell 22 
                 Square, 10 mm × 10 mm 
               
               
                 Homogenizer 46′ 
                 Square truncated pyramid of BK7 optical glass. 
               
               
                   
                 Length 47 mm, upper dimensions 14.1 mm × 14.1 
               
               
                   
                 mm, lower dimensions 9.8 mm × 9.8 mm 
               
               
                   
                 (corresponding to, but slightly smaller than, the 
               
               
                   
                 solar cell dimensions). 
               
               
                   
               
            
           
         
       
     
     Table IV below gives typical approximate distances between components of the third exemplary embodiment. 
     
       
         
           
               
               
             
               
                 TABLE IV 
               
               
                   
               
               
                 Components 
                 Distance between Components 
               
               
                   
               
             
            
               
                 Concentrator 54′-secondary  
                 101.7 mm 
               
               
                 reflector 38′ 
                   
               
               
                 Secondary reflector 38′-top of 
                   24 mm 
               
               
                 homogenizer 46′ 
                   
               
               
                 Solar cell 22-aperture 34′ 
                  17.7 mm 
               
               
                   
               
            
           
         
       
     
       FIG. 4A  illustrates a design for cell mount  128 , and  FIG. 4B  illustrates an alternative design for the cell mount, according to embodiments of the present invention. For clarity, cell mount  128  illustrated in  FIG. 4A  is referred to as cell mount  128 A, and cell mount  128  illustrated in  FIG. 4B  is referred to as cell mount  128 B. As illustrated in  FIGS. 4A and 4B , both cell mounts are assumed to be mounted over aperture  34 ′ of system  124 . 
     Cell mount  128 A ( FIG. 4A ) is an open structure, typically comprising two or more branches from cell  22  to the upper surface of the primary reflector. The open structure allows passage of air through the structure. A passive heat sink  130  is located in a space  132  of the mount, by being fixedly attached to the rear surface of cell  22 . Heat sink  130  is typically a finned structure having a cross-section  134 , and is formed from a good heat conductor such as copper or aluminum. 
     Cell mount  128 B ( FIG. 4B ) is a closed structure, typically in the form of a closed hollow conical shape forming an enclosed space  134  in the mount. A cooling fluid, typically a gas such as air or a liquid such as water, is directed via a tube  136  to the rear surface of cell  22 , exiting into space  132  after contacting the cell&#39;s rear surface. The cooling fluid exits from space  134  via an exit port  138 . 
       FIG. 5A  is a schematic diagram showing irradiance at the position of secondary reflector  38 ′ of alternative CPV system  124 , due to reflection from the primary reflector, and  FIG. 5B  is a graph of the irradiance vs. distance, according to embodiments of the present invention. The graph plots the irradiance vs. distance along a symmetry line  150  of  FIG. 5A . As is apparent from comparison of the diagram and graph of  FIGS. 5A and 5B  with the diagram and graph of  FIGS. 2A and 2B , irradiance features described above for the secondary reflector of system  24  are present in the secondary reflector of system  124 . Thus in system  124 , as for system  24 , there is no reflected radiation from the primary reflector within a central region of the secondary reflector, illustrated in  FIG. 5A  as a region  152 . Thus, in both system  24  and system  124  substantially all the reflected radiation from the primary reflector is incident on the secondary reflector of the system, and none is incident on a respective central region of the secondary reflector. 
       FIG. 6  is a schematic sectional side view of a matrix  200  of CPV systems  124 , according to an embodiment of the present invention. For clarity and simplicity, individual elements of systems  124  are not labeled in  FIG. 6 . By way of example, matrix  200  is assumed to comprise six systems  124 , arranged in a 2×3 rectangular array, so that the matrix is approximately 500 mm×750 mm. Alternatively, matrix  200  may comprise other numbers of systems  124 , such as 24 systems arranged in a 4×6 array covering approximately 1 m×1.5 m. 
     Typically, the elements of the systems comprised in matrix  200  are mounted on a common base system mounting panel  202  by vertical supports  204  for the windows of the systems, and by structures  206  for the primary reflector. Each structure  206  is typically similar to skeleton-like section  62  of mount  60  ( FIG. 1C ). Panel  202  is in turn connected to a tracking device similar to that described above for system  24 . In some embodiments, rather than having separate windows for each of CPV systems  124 , one window  208 , together with respective transmissive concentrators, covers all systems  124  in the matrix. 
     An electric junction box  210  may be attached to panel  202 . Box  210  is typically configured to allow the electric power output from systems  124  to be connected in series, in parallel, or in a combination of series and parallel, according to requirements of a user of the matrix. (Typically, a side cover protects the panel from dust and moisture.) 
     It will be understood that a number of systems  24  may be arranged in matrices as described for systems  124 . Furthermore, a mix of systems  24  and  124 , and other CPV systems using the principles of CPV systems described herein, may be combined to form a matrix of CPV systems similar to matrix  200 . One of these matrices may be used to replace a non-concentrating photovoltaic system of similar dimensions. For example, some non-concentrating photovoltaic systems have dimensions of approximately 1 m×1.5 m. 
       FIGS. 7A ,  7 B, and  7 C are schematic front, back and side views respectively of primary reflector  26 ′ of system  124 , according to an embodiment of the present invention. Primary reflector  26 ′ is made as a multi-segment reflector, which is formed by splitting the reflector into smaller curved segments for easy manufacturing and for better optical properties. The smaller segments are subsequently assembled to produce larger reflector  26 ′, with aperture  34 ′, as shown in these figures. By way of example, reflector  26 ′ is assumed to be made from four substantially identical segments  252 . 
     The small curved segments may be made from flat metal sheet, such as aluminum, by stamping, which is generally a fast, low-cost operation. The stamp itself typically has a smaller radius of curvature than the desired segment shape, to account for spring-back of the metal after stamping. The exact shape of the stamp depends on the specific sheet metal that is used, and can be optimized by simple trial and error. 
     Before stamping, the sheet metal may be pre-coated with a reflective layer, or a flat pre-coated film may be applied to the metal sheet. Suitable materials for this purpose include Alanod 4270GP, produced by ALANOD Aluminium-Veredlung GmbH &amp; Co, Ennepetal, Germany and ReflecTech Mirror Film, produced by ReflecTech Inc., Arvada, Colo. Data sheets for these materials may be found at www.alanod.de/opencms/export/alanod/Technik_gallery/datasheets/4270GP_E.pdf and www.reflectechsolar.com/pdfs/ReflecTechBrochuretoEmail22Aug08.pdf, and are incorporated herein by reference. Both materials are commercially available as reels of silver-coated film. 
     After stamping, the reflector segments are joined together to form a complete reflector assembly, as shown in the figures. For example, the segments can be glued on their back sides to a substrate, typically made of a low-cost material, which acts as a joint to hold the segments together. In the embodiment shown in  FIGS. 7B and 7C , an aluminum profile is extruded with the exact parabolic shape of reflector  26 ′, and is then sliced into small ribs  254  that can be glued to the back of the assembled reflector to hold and join the segments together and to form a mounting base. Other suitable methods of assembling the segments will be apparent to those having ordinary skill in the art, such as by including clips in ribs  254 , and attaching the segments to the ribs with the clips. All such methods are included in the scope of the present invention. 
       FIG. 8  is a schematic front view of primary reflector  26 ′, according to an alternative embodiment of the present invention. In the alternative embodiment, primary reflector  26 ′ is made as a multi-segment reflector by being formed from eight segments: four substantially similar square segments  256 , and four substantially similar rectangular segments  258 . It will be understood that the sizes of the square and rectangular segments are adjusted according to the dimensions of reflector  26 ′ and aperture  34 ′. 
       FIG. 9A  is a schematic front view of segments of primary reflector  26  of system  24 , and  FIG. 9B  is a schematic plan view of a plane sheet for producing some of the segments, according to an embodiment of the present invention. Reflector  26  is made as a multi-segment reflector comprised of 12 segments which are assembled to form the complete reflector, (substantially as described above with respect to the segments of reflector  26 ′ illustrated in  FIGS. 7A ,  7 B, and  7 C). For simplicity, in  FIG. 9A  only three segments of reflector  26  are shown, in a left quadrant  260  of the reflector, since the other nine segments (in the other three reflector quadrants) are typically reproductions of the three segments shown. For completeness, aperture  34  is shown in  FIG. 9A . 
     Quadrant  260  is divided into a square segment  262  and two segments  264 ,  266  which are mirror images of each other. Typically, to produce segments  262  from a rectangular sheet, the sheet may initially be cut with substantially no wastage of material. In order to efficiently produce segments  264  and/or  266  from a rectangular sheet, the segments may initially be cut as is illustrated in diagram  268  ( FIG. 9B ) for two segments  266 , labeled  266 A and  266 B in the diagram. 
     The descriptions above for  FIGS. 7A ,  7 B,  7 C,  8 ,  9 A, and  9 B assume that a 3-dimensional reflector, reflector  26  or reflector  26 ′, is created from segments smaller than the overall size of the reflector. Creating a 3-dimensional reflector by stamping a flat metal sheet introduces deformation, which can create voids within the reflective layer as well as between the reflective coating and the underlying metal, and thus reduce reflectivity and service life of the reflector. However, as explained below with reference to  FIGS. 10A and 10B , by splitting the reflector into smaller segments, the deformation of each of the reflector segments is reduced. 
       FIG. 10A  and  FIG. 10B  illustrate the reduction in deformation achieved by constructing a paraboloidal reflector such as reflector  26  or reflector  26 ′, from smaller segments, according to embodiments of the present invention. For simplicity, apertures in the reflector are not shown in the figures. 
       FIG. 10A  illustrates constructing the reflector from four segments. In a diagram  270  solid lines illustrate a schematic top view of a single plane sheet  280 , also herein termed sheet ABCD. Broken lines in the diagram show top views of edges of the segments producing the reflector. A diagram  272  shows schematic respective cross-sections  282 ,  284 , of the single sheet before and after deformation into its paraboloidal shape. Cross-section  282  is taken along diagonal AC of sheet ABCD. Assuming the reflector is produced from four plane square sheets that are deformed into segments which are then joined together, diagram  272  shows schematic respective cross-sections  286 ,  288 , of an exemplary one of the four sheets before deformation, and after deformation to its paraboloidal segment. Cross-section  286  is taken along the diagonal AE of the exemplary plane square sheet. 
       FIG. 10B  illustrates splitting the reflector into nine segments, although typically only eight segments are used since a central aperture replaces the central segment. A diagram  274  is a schematic top view of single plane sheet  280 . A diagram  276  shows cross-sections  282 ,  284 , of the single sheet before and after deformation into its parabolic shape (as in  FIG. 10A ). Broken lines in the diagram show top views of edges of the segments producing the reflector. The reflector may be produced from eight or nine plane square sheets that are deformed into segments which are then joined together. Diagram  276  shows schematic respective cross-sections  290 ,  292 , of a central segment  294  before and after deformation to its parabolic shape. Cross-section  290  is taken along the diagonal GH of the segment. 
     Details of the deformation calculations in the cases illustrated by  FIGS. 10A and 10B  are given in an Appendix of this disclosure. The calculations are for a square parabolic reflector having a focal length f equal to half the edge length 2a of a square. As is illustrated in the figures, for one plane sheet there is a depth change H 1  of the sheet to produce the complete reflector. As is shown in the Appendix, H 1 =0.5a, and the change in area from the square plane to the square paraboloidal reflector, i.e., the deformation produced in the sheet, is an increase of almost 12%. Splitting the reflector into four segments gives a depth change for each segment of H 4 , where H 4 ≈0.129a, and the deformation caused is an increase of only about 3% of the area of the plane segment. 
     Splitting the reflector into nine segments gives correspondingly smaller depth changes and deformations than the changes generated for four segments. Thus, central segment  294  has a depth change H 9 , where H 9 ≈0.05a, and in this case the deformation is reduced to an increase of a little over 1%. 
     As stated above, suitable materials exist for pre-coating sheet metal with reflective material. Assuming this pre-coating, the production of parabolic reflectors from multiple segments reduces the deformation of the material to within the tolerance limits of the reflective material being used. It is therefore possible first to place the reflective coating on the sheet metal, using a reel of reflective material, and then to bend the metal. This process is substantially simpler and less costly than coating a curved shape. Furthermore, when the reflective sheeting is applied flat and then bent with the metal sheet as described above, the coated layer is more even and therefore has generally better performance than a coating applied to surfaces that are already curved. 
       FIG. 11  is a schematic, pictorial illustration showing assembly of matrix  200  ( FIG. 6 ) of CPV systems  124 , according to an embodiment of the present invention. The diagram shows assembly of reflectors  26 ′ on panel  202 . The component parabolic reflectors are made from four segments  252 , which are then joined together on ribs  254 , as shown in  FIGS. 7A and 7B . The reflectors are mounted on base panel  202 , along with vertical supports  204  for supporting the window and other components (not shown in  FIG. 11 ). The CPV cells, homogenizers, windows and secondary reflectors of systems  124  are then assembled onto the base to complete the system, as shown in  FIG. 6 . 
     The description above refers generally to concentrators that concentrate incoming solar radiation onto a photovoltaic cell. However, it will be understood that solar concentrators such as are described herein may be used concentrate incoming solar radiation onto apparatus other then photovoltaic cells. For example, such an apparatus may comprise a thermocouple, or a plurality of thermocouples assembled as a thermopile, either of which systems may also be used to generate electricity. Furthermore, the apparatus receiving the concentrated solar radiation may be configured to convert the radiation to another energy form, such as chemical or thermal energy. 
     Although the description above includes forming a primary reflector from a number of smaller curved segments, it will be understood that substantially the same process may be applied to the formation of a secondary reflector. 
     It will thus be appreciated that the embodiments described above are cited by way of example, and that the present invention is not limited to what has been particularly shown and described hereinabove. Rather, the scope of the present invention includes both combinations and subcombinations of the various features described hereinabove, as well as variations and modifications thereof which would occur to persons skilled in the art upon reading the foregoing description and which are not disclosed in the prior art. 
     APPENDIX 
     Area Deformation Caused by Paraboloid Formation 
     The surface area S p  of a paraboloid (formed by the rotation of the parabola 
     
       
         
           
             
               y 
               = 
               
                 
                   x 
                   2 
                 
                 
                   4 
                    
                   f 
                 
               
             
             , 
           
         
       
     
     f the focus of the paraboloid, about the y-axis) is given by: 
     
       
         
           
             
               
                 
                   
                     S 
                     p 
                   
                   = 
                   
                     
                       ∫ 
                       0 
                       c 
                     
                      
                     
                       2 
                        
                       π 
                        
                       
                           
                       
                        
                       x 
                        
                       
                         
                           ( 
                           
                             1 
                             + 
                             
                               
                                 ( 
                                 
                                   
                                      
                                     y 
                                   
                                   
                                      
                                     x 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                           ) 
                         
                       
                        
                       
                          
                         x 
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     where c is the x-value of the edge of the paraboloid. 
     Assuming that the complete paraboloid is stamped from a square sheet having edge 2a, the largest x-value of the paraboloid is a√{square root over (2)}. This is the value of c in equation (1). 
     From the equation for the parabola 
     
       
         
           
             ( 
             
               y 
               = 
               
                 
                   x 
                   2 
                 
                 
                   4 
                    
                   f 
                 
               
             
             ) 
           
         
       
     
     
       
         
           
             
               
                 
                   
                     
                        
                       y 
                     
                     
                        
                       x 
                     
                   
                   = 
                   
                     x 
                     
                       2 
                        
                       f 
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     Substituting the expressions for 
     
       
         
           
             
                
               y 
             
             
                
               x 
             
           
         
       
     
     and c into equation (1), and rearranging, gives: 
     
       
         
           
             
               
                 
                   
                     S 
                     p 
                   
                   = 
                   
                     
                       π 
                       f 
                     
                      
                     
                       
                         ∫ 
                         
                           x 
                           = 
                           0 
                         
                         
                           x 
                           = 
                           
                             a 
                              
                             
                               2 
                             
                           
                         
                       
                        
                       
                         x 
                          
                         
                           
                             ( 
                             
                               
                                 4 
                                  
                                 
                                   f 
                                   2 
                                 
                               
                               + 
                               
                                 x 
                                 2 
                               
                             
                             ) 
                           
                         
                          
                         
                            
                           x 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     Substituting u=4f 2 +x 2  (so that du=2×dx) into equation (3) gives: 
     
       
         
           
             
               
                 
                   
                     S 
                     p 
                   
                   = 
                   
                     
                       π 
                       
                         2 
                          
                         f 
                       
                     
                      
                     
                       
                         ∫ 
                         
                           u 
                           = 
                           
                             4 
                              
                             
                               f 
                               2 
                             
                           
                         
                         
                           u 
                           = 
                           
                             
                               4 
                                
                               
                                 f 
                                 2 
                               
                             
                             + 
                             
                               2 
                                
                               
                                 a 
                                 2 
                               
                             
                           
                         
                       
                        
                       
                         
                           u 
                         
                          
                         
                            
                           u 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     Assuming that the plate is stamped so that f=a, equation (4) evaluates as: 
     
       
         
           
             
               
                 
                   
                     S 
                     p 
                   
                   = 
                   
                     
                       
                         
                           π 
                           
                             2 
                              
                             f 
                           
                         
                         [ 
                         
                           
                             2 
                             3 
                           
                            
                           
                             u 
                             
                               3 
                               2 
                             
                           
                         
                         ] 
                       
                       
                         4 
                          
                         
                           f 
                           2 
                         
                       
                       
                         6 
                          
                         
                           f 
                           2 
                         
                       
                     
                     = 
                     
                       
                         
                           
                             π 
                              
                             
                                 
                             
                              
                             
                               a 
                               2 
                             
                           
                           3 
                         
                         [ 
                         
                           
                             6 
                              
                             
                               6 
                             
                           
                           - 
                           8 
                         
                         ] 
                       
                       ≈ 
                       
                         7.013 
                          
                         
                           a 
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     A circular flat sheet of radius a√{square root over (2)} has a surface area S f  given by: 
       S f =2πa 2 ≈6.28a 2   (6)
 
     From equations (5) and (6) the percentage increase Δ 1  in surface area, when deforming the flat sheet to a paraboloid is given by: 
     
       
         
           
             
               
                 
                   
                     Δ 
                     1 
                   
                   = 
                   
                     
                       
                         
                           S 
                           p 
                         
                         - 
                         
                           S 
                           f 
                         
                       
                       
                         S 
                         f 
                       
                     
                     = 
                     
                       
                         
                           
                             7.013 
                              
                             
                               a 
                               2 
                             
                           
                           - 
                           
                             6.28 
                              
                             
                               a 
                               2 
                             
                           
                         
                         
                           6.28 
                            
                           
                             a 
                             2 
                           
                         
                       
                       = 
                       
                         11.67 
                          
                         % 
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     Approximation of Paraboloid by a Dome 
     From the equation 
     
       
         
           
             y 
             = 
             
               
                 x 
                 2 
               
               
                 4 
                  
                 f 
               
             
           
         
       
     
     the paraboloid has a height h (from its vertex) given by: 
     
       
         
           
             
               
                 
                   h 
                   = 
                   
                     
                       
                         
                           ( 
                           
                             a 
                              
                             
                               2 
                             
                           
                           ) 
                         
                         2 
                       
                       
                         4 
                          
                         f 
                       
                     
                     = 
                     
                       
                         a 
                         2 
                       
                        
                       
                         ( 
                         
                           
                             since 
                              
                             
                                 
                             
                              
                             a 
                           
                           = 
                           f 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     At its edge, the paraboloid forms a circle of radius a√{square root over (2)}. 
     In the following, we approximate the paraboloid to the curved surface of a dome (a section generated by a plane cutting a sphere). The dome has height h and radius r of the circle generated by the plane. In this case r=a√{square root over (2)}. 
     By applying the Pythagoras theorem, the radius R c  of the sphere from which the dome is formed is given by: 
     
       
         
           
             
               
                 
                   
                     R 
                     c 
                   
                   = 
                   
                     
                       
                         
                           h 
                           2 
                         
                         + 
                         
                           r 
                           2 
                         
                       
                       
                         2 
                          
                         h 
                       
                     
                     = 
                     
                       
                         
                           
                             h 
                             2 
                           
                           + 
                           
                             
                               ( 
                               
                                 a 
                                  
                                 
                                   2 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                         
                           2 
                            
                           h 
                         
                       
                       = 
                       
                         
                           9 
                            
                           a 
                         
                         4 
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     The area of the curved surface of a dome is given by: 
       A dome =2πR c h  (10)
 
     so that, substituting into equation (10) the values of R c  and h from equations (8) and (9), 
     
       
         
           
             
               
                 
                   
                     A 
                     dome 
                   
                   = 
                   
                     
                       
                         
                           9 
                            
                           π 
                         
                         4 
                       
                        
                       
                         a 
                         2 
                       
                     
                     ≈ 
                     
                       7.068 
                        
                       
                         a 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     The percentage error Δ 2  generated by using equation (10) instead of equation (3) is: 
     
       
         
           
             
               
                 
                   
                     Δ 
                     2 
                   
                   = 
                   
                     
                       
                         7.068 
                         - 
                         7.013 
                       
                       7.013 
                     
                     = 
                     
                       0.78 
                        
                       % 
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     Thus, the error between assuming that the area of the curved surface is spherical, compared to the paraboloidal shape of the surface, is less than 1%. 
     The error calculation is for r=a√{square root over (2)}. For smaller values of r, the error is even less. 
     Producing the Paraboloid in Four Segments 
     Considering  FIG. 10A , for the parabola 
     
       
         
           
             y 
             = 
             
               
                 x 
                 2 
               
               
                 4 
                  
                 f 
               
             
           
         
       
     
     (and assuming appropriate axes) section  286  has vertices (0,0) and 
     
       
         
           
             
               ( 
               
                 
                   a 
                    
                   
                     2 
                   
                 
                 , 
                 
                   
                     a 
                     2 
                   
                   
                     2 
                      
                     f 
                   
                 
               
               ) 
             
             . 
           
         
       
     
     Since a=f, the length of the section is 
     
       
         
           
             
               
                 3 
                  
                 a 
               
               2 
             
             . 
           
         
       
     
     This is the diameter of the dome plane circle, so that the radius is 
     
       
         
           
             
               3 
                
               a 
             
             4 
           
         
       
     
     Assuming 
     
       
         
           
             
               
                 R 
                 c 
               
               = 
               
                 
                   9 
                    
                   a 
                 
                 4 
               
             
             , 
           
         
       
     
     and using equation (9) with 
     
       
         
           
             r 
             = 
             
               
                 3 
                  
                 a 
               
               4 
             
           
         
       
     
     to solve for H 4 . the height of the dome, gives: 
     
       
         
           
             
               
                 
                   
                     H 
                      
                     
                         
                     
                      
                     4 
                   
                   = 
                   
                     
                       
                         
                           2 
                            
                           
                             R 
                             c 
                           
                         
                         ± 
                         
                           
                             
                               
                                 ( 
                                 
                                   2 
                                    
                                   
                                     R 
                                     c 
                                   
                                 
                                 ) 
                               
                               2 
                             
                             - 
                             
                               4 
                                
                               
                                 r 
                                 2 
                               
                             
                           
                         
                       
                       2 
                     
                     = 
                     
                       
                         
                           2 
                            
                           
                             
                               9 
                                
                               a 
                             
                             4 
                           
                         
                         ± 
                         
                           
                             
                               4 
                                
                               
                                 
                                   ( 
                                   
                                     
                                       9 
                                        
                                       a 
                                     
                                     4 
                                   
                                   ) 
                                 
                                 2 
                               
                             
                             - 
                             
                               4 
                                
                               
                                 
                                   ( 
                                   
                                     
                                       3 
                                        
                                       a 
                                     
                                     4 
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                         
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     Equation (13) simplifies to: 
     
       
         
           
             
               
                 
                   
                     H 
                      
                     
                         
                     
                      
                     4 
                   
                   = 
                   
                     
                       
                         
                           ( 
                           
                             9 
                             - 
                             
                               6 
                                
                               
                                 2 
                               
                             
                           
                           ) 
                         
                         4 
                       
                        
                       a 
                     
                     ≈ 
                     
                       0.129 
                        
                       a 
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     From equation (10) the area of the curved surface of the dome is: 
     
       
         
           
             
               
                 
                   
                     A 
                     dome 
                   
                   = 
                   
                     
                       2 
                        
                       π 
                        
                       
                           
                       
                        
                       
                         R 
                         c 
                       
                        
                       H 
                        
                       
                           
                       
                        
                       4 
                     
                     = 
                     
                       
                         2 
                          
                         π 
                          
                         
                             
                         
                          
                         
                           
                             9 
                              
                             a 
                           
                           4 
                         
                          
                         0.129 
                       
                       ≈ 
                       
                         1.824 
                          
                         
                           a 
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     The area of a flat sheet with radius 
     
       
         
           
             
               3 
                
               a 
             
             4 
           
         
       
     
     is: 
     
       
         
           
             
               
                 
                   
                     A 
                     flat 
                   
                   = 
                   
                     
                       
                         π 
                          
                         
                           ( 
                           
                             
                               3 
                                
                               a 
                             
                             4 
                           
                           ) 
                         
                       
                       2 
                     
                     ≈ 
                     
                       1.767 
                        
                       
                         a 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     From equations (15) and (16) the percentage increase Δ 4  in surface area, when deforming the flat sheet to a paraboloidal segment is given by: 
     
       
         
           
             
               
                 
                   
                     Δ 
                     4 
                   
                   = 
                   
                     
                       
                         
                           A 
                           dome 
                         
                         - 
                         
                           A 
                           flat 
                         
                       
                       
                         A 
                         flat 
                       
                     
                     = 
                     
                       
                         
                           
                             1.824 
                              
                             
                               a 
                               2 
                             
                           
                           - 
                           
                             1.767 
                              
                             
                               a 
                               2 
                             
                           
                         
                         
                           1.767 
                            
                           
                             a 
                             2 
                           
                         
                       
                       = 
                       
                         3.23 
                          
                         % 
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     Comparing equations (7) and (17), it is apparent that the deformation caused by the smaller paraboloidal segment decreases significantly. 
     Producing the Paraboloid in Nine Segments 
     Considering  FIG. 10B , section  290  has a length GH, which is equal to 
     
       
         
           
             
               a 
                
               
                 8 
               
             
             3 
           
         
       
     
     This is the diameter of the dome plane circle, so that the radius is 
     
       
         
           
             
               
                 a 
                  
                 
                   8 
                 
               
               6 
             
             . 
           
         
       
     
     Using this value of radius, and applying the same operations as equations (13) and (14) gives a value for H 9 : 
       H9≈0.05a  (18)
 
     Applying the same operations as equations (15) and (16) gives: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             A 
                             dome 
                           
                           = 
                           
                             
                               2 
                                
                               π 
                                
                               
                                   
                               
                                
                               
                                 R 
                                 c 
                               
                                
                               H 
                                
                               
                                   
                               
                                
                               9 
                             
                             = 
                             
                               
                                 2 
                                  
                                 π 
                                  
                                 
                                   
                                     9 
                                      
                                     a 
                                   
                                   4 
                                 
                                  
                                 0.05 
                                  
                                 a 
                               
                               ≈ 
                               
                                 0.706 
                                  
                                 
                                   a 
                                   2 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             A 
                             flat 
                           
                           = 
                           
                             
                               
                                 π 
                                 ( 
                                 
                                   
                                     a 
                                      
                                     
                                       8 
                                     
                                   
                                   6 
                                 
                                 ) 
                               
                               2 
                             
                             ≈ 
                             
                               0.698 
                                
                               
                                 a 
                                 2 
                               
                             
                           
                         
                       
                     
                   
                   } 
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
     From equations (19) the percentage increase Δ 9  in surface area, when deforming the central flat sheet to the central paraboloidal segment is given by: 
     
       
         
           
             
               
                 
                   
                     Δ 
                     9 
                   
                   = 
                   
                     
                       
                         
                           A 
                           dome 
                         
                         - 
                         
                           A 
                           flat 
                         
                       
                       
                         A 
                         flat 
                       
                     
                     = 
                     
                       
                         
                           
                             0.706 
                              
                             
                               a 
                               2 
                             
                           
                           - 
                           
                             0.698 
                              
                             
                               a 
                               2 
                             
                           
                         
                         
                           0.698 
                            
                           
                             a 
                             2 
                           
                         
                       
                       = 
                       
                         1.146 
                          
                         % 
                       
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
     A generally similar percentage increase in surface area occurs for the other eight paraboloidal segments, all increases being smaller than the value of 3.23% given by equation (17).