Abstract:
The disclosure provides systems, methods, and apparatus for wireless power transfer. In one aspect, an apparatus configured to receive wireless power from a transmitter is provided. The apparatus includes an inductor having an inductance value. The apparatus further includes a capacitor electrically connected to the inductor and having a capacitance value. The apparatus further includes an optimizing circuit configured to optimize transfer efficiency of power received wirelessly from the transmitter, provided that an amount of the power received wirelessly and provided to a load is greater than or equal to a received power threshold, or optimize the amount of the power received wirelessly from the transmitter, provided that the power transfer efficiency is greater than or equal to an efficiency threshold.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application is a continuation of U.S. patent application Ser. No. 12/394,033, filed on Feb. 26, 2009 which claims priority benefit from U.S. provisional patent application No. 61/032,061, entitled “Antennas and Their Coupling Characteristics for Wireless Power Transfer via Magnetic Coupling, filed Feb. 27, 2008. Each of the above referenced applications are hereby incorporated herein by reference in their entirety. 
     
    
     BACKGROUND 
       [0002]    Our previous applications and provisional applications, including, but not limited to, U.S. patent application Ser. No. 12/018,069, filed Jan. 22, 2008, entitled “Wireless Apparatus and Methods”, the disclosure of which is herewith incorporated by reference, describe wireless transfer of power. 
         [0003]    The transmit and receiving antennas are preferably resonant antennas, which are substantially resonant, e.g., within 10% of resonance, 15% of resonance, or 20% of resonance. The antenna is preferably of a small size to allow it to fit into a mobile, handheld device where the available space for the antenna may be limited. 
         [0004]    An embodiment describes a high efficiency antenna for the specific characteristics and environment for the power being transmitted and received. 
         [0005]    Antenna theory suggests that a highly efficient but small antenna will typically have a narrow band of frequencies over which it will be efficient. The special antenna described herein may be particularly useful for this kind of power transfer. 
         [0006]    One embodiment uses an efficient power transfer between two antennas by storing energy in the near field of the transmitting antenna, rather than sending the energy into free space in the form of a travelling electromagnetic wave. This embodiment increases the quality factor (Q) of the antennas. This can reduce radiation resistance (R r ) and loss resistance (R l ) 
       SUMMARY 
       [0007]    The present application describes the way in which the “antennas” or coils interact with one another to couple wirelessly the power therebetween. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0008]    In the Drawings: 
           [0009]      FIG. 1  shows a diagram of a wireless power circuit; 
           [0010]      FIG. 2  shows an equivalent circuit; 
           [0011]      FIG. 3  shows a diagram of inductive coupling; 
           [0012]      FIG. 4  shows a plot of the inductive coupling; and 
           [0013]      FIG. 5  shows geometry of an inductive coil. 
       
    
    
     DETAILED DESCRIPTION 
       [0014]      FIG. 1  is a block diagram of an inductively coupled energy transmission system between a source  100 , and a load  150 . The source includes a power supply  102  with internal impedance Z s    104 , a series resistance R 4    106 , a capacitance C 1   108  and inductance L 1   110 . The LC constant of capacitor  108  and inductor  110  causes oscillation at a specified frequency. 
         [0015]    The secondary  150  also includes an inductance L 2   152  and capacitance C 2   154 , preferably matched to the capacitance and inductance of the primary. A series resistance R 2   156 . Output power is produced across terminals  160  and applied to a load ZL  165  to power that load. In this way, the power from the source  102  is coupled to the load  165  through a wireless connection shown as  120 . The wireless communication is set by the mutual inductance M. 
         [0016]      FIG. 2  shows an equivalent circuit to the transmission system of  FIG. 1 . The power generator  200  has internal impedance Zs  205 , and a series resistance R 1   210 . Capacitor C 1   215  and inductor L 1   210  form the LC constant. A current I 1   215  flows through the LC combination, which can be visualized as an equivalent source shown as  220 , with a value U 1 . 
         [0017]    This source induces into a corresponding equivalent power source  230  in the receiver, to create an induced power U 2 . The source  230  is in series with inductance L 2   240 , capacitance C 2   242 , resistance R 2   244 , and eventually to the load  165 . 
         [0018]    Considering these values, the equations for mutual inductance are as follows: 
         [0000]      U 2 =jωMI 1  
 
         [0000]      U 1 =jωMI 2  
 
         [0000]    where: 
         [0000]    
       
         
           
             
               z 
               M 
             
             = 
             
               j 
                
               
                   
               
                
               ω 
                
               
                   
               
                
               M 
             
           
         
       
       
         
           
             
               z 
               1 
             
             = 
             
               
                 z 
                 s 
               
               + 
               
                 R 
                 1 
               
               + 
               
                 j 
                  
                 
                   ( 
                   
                     
                       ω 
                        
                       
                           
                       
                        
                       
                         L 
                         1 
                       
                     
                     - 
                     
                       1 
                       
                         ω 
                          
                         
                             
                         
                          
                         
                           C 
                           1 
                         
                       
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               z 
               2 
             
             = 
             
               
                 z 
                 L 
               
               + 
               
                 R 
                 2 
               
               + 
               
                 j 
                  
                 
                   ( 
                   
                     
                       ω 
                        
                       
                           
                       
                        
                       
                         L 
                         21 
                       
                     
                     - 
                     
                       1 
                       
                         ω 
                          
                         
                             
                         
                          
                         
                           C 
                           2 
                         
                       
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               z 
               s 
             
             = 
             
               
                 R 
                 s 
               
               + 
               
                 jX 
                 s 
               
             
           
         
       
       
         
           
             
               z 
               L 
             
             = 
             
               
                 R 
                 L 
               
               + 
               
                 jX 
                 L 
               
             
           
         
       
     
         [0000]    The Mesh equations are: 
         [0000]    
       
         
           
             
               
                 U 
                 s 
               
               + 
               
                 U 
                 1 
               
               - 
               
                 
                   z 
                   1 
                 
                  
                 
                   I 
                   1 
                 
               
             
             = 
             
               
                 0 
                 → 
                 
                   I 
                   1 
                 
               
               = 
               
                 
                   ( 
                   
                     
                       U 
                       s 
                     
                     + 
                     
                       U 
                       1 
                     
                   
                   ) 
                 
                  
                 
                   / 
                 
                  
                 
                   z 
                   1 
                 
               
             
           
         
       
       
         
           
             
               
                 U 
                 2 
               
               - 
               
                 
                   z 
                   2 
                 
                  
                 
                   I 
                   2 
                 
               
             
             = 
             
               
                 0 
                  
                 
                     
                 
                  
                 
                   I 
                   2 
                 
               
               = 
               
                 
                   U 
                   2 
                 
                  
                 
                   / 
                 
                  
                 
                   z 
                   2 
                 
               
             
           
         
       
       
         
           
             
               I 
               1 
             
             = 
             
               
                 
                   
                     
                       U 
                       s 
                     
                     + 
                     
                       
                         z 
                         M 
                       
                        
                       
                         I 
                         2 
                       
                     
                   
                   
                     z 
                     1 
                   
                 
                  
                 
                     
                 
                  
                 
                   I 
                   2 
                 
               
               = 
               
                 
                   
                     
                       
                         z 
                         M 
                       
                        
                       
                         I 
                         1 
                       
                     
                     
                       z 
                       2 
                     
                   
                    
                   
                     
 
                   
                   → 
                   
                     I 
                     2 
                   
                 
                 = 
                 
                   
                     
                       
                         z 
                         M 
                       
                        
                       
                         ( 
                         
                           
                             U 
                             s 
                           
                           + 
                           
                             
                               z 
                               M 
                             
                              
                             
                               I 
                               2 
                             
                           
                         
                         ) 
                       
                     
                     
                       
                         z 
                         1 
                       
                        
                       
                         z 
                         2 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           
                             z 
                             M 
                           
                            
                           
                             U 
                             s 
                           
                         
                         
                           
                             
                               z 
                               1 
                             
                              
                             
                               z 
                               2 
                             
                           
                           - 
                           
                             z 
                             
                               M 
                               2 
                             
                           
                         
                       
                        
                       
                         
 
                       
                       → 
                       
                         I 
                         1 
                       
                     
                     = 
                     
                       
                         
                           
                             z 
                             M 
                           
                           
                             z 
                             M 
                           
                         
                         · 
                         
                           I 
                           2 
                         
                       
                       = 
                       
                         
                           
                             z 
                             2 
                           
                            
                           
                             U 
                             s 
                           
                         
                         
                           
                             
                               z 
                               1 
                             
                              
                             
                               z 
                               2 
                             
                           
                           - 
                           
                             z 
                             
                               M 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0019]    where: 
         [0020]    Source power: 
         [0000]        P   1 =Re{ U   s   ·I*   1   }=U   s ·Re{ I*   1 } for avg{ U   s }=0
 
         [0021]    Power into load: 
         [0000]        P   2   =I   2   ·I*   2 Re{ z   L   }=|I   2 | 2 ·Re{ z   L   }=|I   2 | 2   ·R   L  
 
         [0022]    Transfer efficiency: 
         [0000]    
       
         
           
             η 
             = 
             
               
                 
                   P 
                   2 
                 
                 
                   P 
                   1 
                 
               
               = 
               
                 
                   
                     
                       I 
                       2 
                     
                     · 
                     
                       I 
                       2 
                       * 
                     
                   
                    
                   
                     R 
                     L 
                   
                 
                 
                   
                     U 
                     s 
                   
                    
                   Re 
                    
                   
                     { 
                     
                       I 
                       1 
                       * 
                     
                     } 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 I 
                 2 
               
               · 
               
                 I 
                 2 
                 * 
               
             
             = 
             
               
                 
                   z 
                   M 
                 
                  
                 
                   z 
                   M 
                   * 
                 
                  
                 
                   U 
                   s 
                   2 
                 
               
               
                 
                   ( 
                   
                     
                       
                         z 
                         1 
                       
                        
                       
                         z 
                         2 
                       
                     
                     - 
                     
                       z 
                       
                         M 
                         2 
                       
                     
                   
                   ) 
                 
                  
                 
                   ( 
                   
                     
                       
                         z 
                         1 
                         * 
                       
                        
                       
                         z 
                         2 
                         * 
                       
                     
                     - 
                     
                       z 
                       
                         M 
                         2 
                       
                       * 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               Re 
                
               
                 { 
                 
                   I 
                   1 
                   * 
                 
                 } 
               
             
             = 
             
               Re 
                
               
                 { 
                 
                   
                     
                       z 
                       2 
                       * 
                     
                      
                     
                       U 
                       s 
                     
                   
                   
                     
                       
                         z 
                         1 
                         * 
                       
                        
                       
                         z 
                         * 
                       
                     
                     - 
                     
                       z 
                       
                         M 
                         2 
                       
                       * 
                     
                   
                 
                 } 
               
             
           
         
       
     
         [0023]    Overall transfer Efficiency is therefore: 
         [0000]    
       
         
           
             η 
             = 
             
               
                 
                   P 
                   2 
                 
                 
                   P 
                   1 
                 
               
               = 
               
                 
                   
                     
                       U 
                       s 
                       2 
                     
                     · 
                     
                       R 
                       L 
                     
                   
                    
                   
                     z 
                     M 
                   
                    
                   
                     z 
                     M 
                     * 
                   
                 
                 
                   
                     ( 
                     
                       
                         
                           z 
                           1 
                         
                          
                         
                           z 
                           2 
                         
                       
                       - 
                       
                         z 
                         M 
                         2 
                       
                     
                     ) 
                   
                    
                   
                     ( 
                     
                       
                         
                           z 
                           1 
                           * 
                         
                          
                         
                           z 
                           2 
                           * 
                         
                       
                       - 
                       
                         z 
                         
                           M 
                           2 
                         
                         * 
                       
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 Def 
                 . 
                 
                   : 
                 
               
                
               
                   
               
                
               
                 z 
                 1 
               
             
             = 
             
               
                 
                   z 
                   1 
                 
                  
                 
                   z 
                   2 
                 
               
               - 
               
                 z 
                 
                   M 
                   2 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       
                         → 
                         η 
                       
                       = 
                       
                         
                           
                             P 
                             2 
                           
                           
                             P 
                             1 
                           
                         
                         = 
                         
                           
                             
                               
                                 R 
                                 L 
                               
                                
                               
                                 z 
                                 M 
                               
                                
                               
                                 z 
                                 M 
                                 * 
                               
                             
                             
                               
                                 z 
                                 ′ 
                               
                                
                               
                                 z 
                                 * 
                               
                                
                               Re 
                                
                               
                                 { 
                                 
                                   
                                     
                                       z 
                                       2 
                                       * 
                                     
                                      
                                     
                                       z 
                                       ′ 
                                     
                                   
                                   
                                     
                                       z 
                                       ′ 
                                     
                                      
                                     
                                       z 
                                       
                                         ′ 
                                         * 
                                       
                                     
                                   
                                 
                                 } 
                               
                             
                           
                           = 
                           
                             
                               
                                 R 
                                 L 
                               
                                
                               
                                 z 
                                 M 
                               
                                
                               
                                 z 
                                 M 
                                 * 
                               
                             
                             
                               Re 
                                
                               
                                 { 
                                 
                                   
                                     z 
                                     2 
                                     * 
                                   
                                   · 
                                   
                                     z 
                                     ′ 
                                   
                                 
                                 } 
                               
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                       
                         
                           
                             
                               R 
                               L 
                             
                              
                             
                               z 
                               M 
                             
                              
                             
                               z 
                               M 
                               * 
                             
                           
                           
                             Re 
                              
                             
                               { 
                               
                                 
                                   z 
                                   2 
                                   * 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         z 
                                         1 
                                       
                                        
                                       
                                         z 
                                         2 
                                       
                                     
                                     - 
                                     
                                       z` 
                                       M 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                               } 
                             
                           
                         
                         = 
                         
                           
                             
                               R 
                               L 
                             
                             | 
                             
                               z 
                               M 
                             
                              
                             
                               | 
                               2 
                             
                           
                           
                             Re 
                              
                             
                               { 
                               
                                 
                                   z 
                                   1 
                                 
                                 | 
                                 
                                   z 
                                   2 
                                 
                                  
                                 
                                   | 
                                   2 
                                 
                                  
                                 
                                   
                                     - 
                                     
                                       z 
                                       2 
                                       * 
                                     
                                   
                                    
                                   
                                     z 
                                     M 
                                     2 
                                   
                                 
                               
                               } 
                             
                           
                         
                       
                     
                   
                 
               
                
               
                 
 
               
               → 
               η 
             
             = 
             
               
                 
                   P 
                   2 
                 
                 
                   P 
                   1 
                 
               
               = 
               
                 
                   
                     R 
                     L 
                   
                   | 
                   
                     z 
                     M 
                   
                    
                   
                     | 
                     2 
                   
                 
                 
                   | 
                   
                     z 
                     2 
                   
                    
                   
                     | 
                     2 
                   
                    
                   
                     
                       
                         · 
                         Re 
                       
                        
                       
                         { 
                         
                           z 
                           1 
                         
                         } 
                       
                     
                     - 
                     
                       
                         z 
                         M 
                         2 
                       
                        
                       Re 
                        
                       
                         { 
                         
                           z 
                           2 
                           * 
                         
                         } 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               Re 
                
               
                 { 
                 
                   z 
                   1 
                 
                 } 
               
             
             = 
             
               
                 R 
                 s 
               
               + 
               
                 R 
                 1 
               
             
           
         
       
       
         
           
             
               Re 
                
               
                 { 
                 
                   z 
                   2 
                   * 
                 
                 } 
               
             
             = 
             
               
                 
                   
                     R 
                     L 
                   
                   + 
                   
                     R 
                     2 
                   
                 
                  
                 
                   
 
                 
                 | 
                 
                   z 
                   2 
                 
                  
                 
                   | 
                   2 
                 
               
               = 
               
                 
                   
                     
                       
                         ( 
                         
                           
                             R 
                             L 
                           
                           + 
                           
                             R 
                             2 
                           
                         
                         ) 
                       
                       2 
                     
                     + 
                     
                       
                         ( 
                         
                           
                             ω 
                              
                             
                                 
                             
                              
                             
                               L 
                               2 
                             
                           
                           - 
                           
                             1 
                             
                               ω 
                                
                               
                                   
                               
                                
                               
                                 C 
                                 2 
                               
                             
                           
                           + 
                           
                             X 
                             L 
                           
                         
                         ) 
                       
                       2 
                     
                   
                    
                   
                     
 
                   
                   | 
                   
                     z 
                     M 
                   
                    
                   
                     | 
                     2 
                   
                 
                 = 
                 
                   
                     ω 
                     2 
                   
                    
                   
                     M 
                     2 
                   
                 
               
             
           
         
       
       
         
           
             
               z 
               
                 M 
                 2 
               
             
             = 
             
               
                 
                   ( 
                   
                     j 
                      
                     
                         
                     
                      
                     ω 
                      
                     
                         
                     
                      
                     M 
                   
                   ) 
                 
                 2 
               
               = 
               
                 
                   - 
                   
                     ω 
                     2 
                   
                 
                  
                 
                   M 
                   2 
                 
               
             
           
         
       
     
         [0000]    A Transfer efficiency equation can therefore be expressed as: 
         [0000]    
       
         
           
             
               
                 
                   η 
                   = 
                   
                     
                       
                         P 
                         2 
                       
                       
                         P 
                         1 
                       
                     
                     = 
                     
                       
                         
                           ω 
                           2 
                         
                          
                         
                           
                             M 
                             2 
                           
                           · 
                           
                             R 
                             L 
                           
                         
                       
                       
                         
                           
                             ( 
                             
                               
                                 R 
                                 s 
                               
                               + 
                               
                                 R 
                                 n 
                               
                             
                             ) 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       R 
                                       L 
                                     
                                     + 
                                     
                                       R 
                                       2 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               + 
                               
                                 ( 
                                 
                                   
                                     ω 
                                      
                                     
                                         
                                     
                                      
                                     
                                       L 
                                       2 
                                     
                                   
                                   - 
                                   
                                     1 
                                     
                                       ω 
                                        
                                       
                                           
                                       
                                        
                                       
                                         C 
                                         2 
                                       
                                     
                                   
                                   + 
                                   
                                     X 
                                     L 
                                   
                                 
                                 ) 
                               
                             
                             ] 
                           
                         
                         + 
                         
                           
                             ω 
                             2 
                           
                            
                           
                             
                               M 
                               2 
                             
                              
                             
                               ( 
                               
                                 
                                   R 
                                   L 
                                 
                                 + 
                                 
                                   R 
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                     
                 
               
             
           
         
       
     
         [0000]    Which reduces in special cases as follows:
   A) when ω=ω 0 =1/√{square root over (L 2 C 2 )}, X L =0 or where   
 
         [0000]    
       
         
           
             
               
                 ω 
                  
                 
                     
                 
                  
                 
                   L 
                   2 
                 
               
               - 
               
                 1 
                 
                   ω 
                    
                   
                       
                   
                    
                   
                     C 
                     2 
                   
                 
               
               + 
               
                 
                   X 
                   L 
                 
                  
                 
                   ( 
                   ω 
                   ) 
                 
               
             
             = 
             0 
           
         
       
       
         
           
             
               
                 
                   η 
                   = 
                   
                     
                       
                         P 
                         2 
                       
                       
                         P 
                         1 
                       
                     
                     = 
                     
                       
                         
                           
                             ω 
                             0 
                             2 
                           
                            
                           
                             M 
                             2 
                           
                         
                         
                           [ 
                           
                             
                               
                                 ( 
                                 
                                   
                                     R 
                                     s 
                                   
                                   + 
                                   
                                     R 
                                     n 
                                   
                                 
                                 ) 
                               
                                
                               
                                 ( 
                                 
                                   
                                     R 
                                     L 
                                   
                                   + 
                                   
                                     R 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 ω 
                                 0 
                                 2 
                               
                                
                               
                                 M 
                                 2 
                               
                             
                           
                           ] 
                         
                       
                       · 
                       
                         
                           R 
                           L 
                         
                         
                           ( 
                           
                             
                               R 
                               L 
                             
                             + 
                             
                               R 
                               2 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
             
             
               
                 
                     
                 
               
             
           
         
       
       
         B) when ω=ω 0 , R s =0: 
       
     
         [0000]    
       
         
           
             η 
             = 
             
               
                 
                   P 
                   2 
                 
                 
                   P 
                   1 
                 
               
               = 
               
                 
                   
                     ω 
                     0 
                     2 
                   
                    
                   
                     M 
                     2 
                   
                    
                   
                     R 
                     L 
                   
                 
                 
                   
                     
                       
                         R 
                         1 
                       
                        
                       
                         ( 
                         
                           
                             R 
                             L 
                           
                           + 
                           
                             R 
                             2 
                           
                         
                         ) 
                       
                     
                     2 
                   
                   + 
                   
                     
                       ω 
                       0 
                       2 
                     
                      
                     
                       
                         M 
                         2 
                       
                        
                       
                         ( 
                         
                           
                             R 
                             L 
                           
                           + 
                           
                             R 
                             2 
                           
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
       
         C) when ω=ω 0 , R s =0 R L =R 2 : 
       
     
         [0000]    
       
         
           
             η 
             = 
             
               
                 
                   P 
                   2 
                 
                 
                   P 
                   1 
                 
               
               = 
               
                 
                   
                     ω 
                     0 
                     2 
                   
                    
                   
                     M 
                     2 
                   
                 
                 
                   
                     4 
                      
                     
                       R 
                       1 
                     
                      
                     
                       R 
                       2 
                     
                   
                   + 
                   
                     2 
                      
                     
                       ω 
                       0 
                       2 
                     
                      
                     
                       M 
                       2 
                     
                   
                 
               
             
           
         
       
       
         D) when ω=ω 0 , R s =0 R L =R 2  2R 1 R 2 &gt;&gt;ω 0   2 M 2 : 
       
     
         [0000]    
       
         
           
             η 
             = 
             
               
                 
                   P 
                   2 
                 
                 
                   P 
                   1 
                 
               
               ≅ 
               
                 
                   
                     
                       ω 
                       0 
                       2 
                     
                      
                     
                       M 
                       2 
                     
                   
                   
                     4 
                      
                     
                       R 
                       1 
                     
                      
                     
                       R 
                       2 
                     
                   
                 
                  
                 
                     
                 
                  
                 
                   ( 
                   
                     weak 
                      
                     
                         
                     
                      
                     coupling 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    where:
 
Mutual inductance:
 
         [0028]    M=k√{square root over (L 1 L 2 )} where k is the coupling factor 
         [0000]    Loaded Q factors: 
         [0000]    
       
         
           
             
               Q 
               
                 1 
                 , 
                 L 
               
             
             = 
             
               
                 
                   
                     ω 
                      
                     
                         
                     
                      
                     
                       L 
                       1 
                     
                   
                   
                     
                       R 
                       s 
                     
                     + 
                     
                       R 
                       1 
                     
                   
                 
                  
                 
                     
                 
                  
                 
                   Q 
                   
                     2 
                     , 
                     L 
                   
                 
               
               = 
               
                 
                   ω 
                    
                   
                       
                   
                    
                   
                     L 
                     2 
                   
                 
                 
                   
                     R 
                     L 
                   
                   + 
                   
                     R 
                     2 
                   
                 
               
             
           
         
       
     
         [0000]    Therefore, the transfer efficiency in terms of these new definitions:
   A) when ω=ω 0      
 
         [0000]    
       
         
           
             η 
             = 
             
               
                 
                   P 
                   2 
                 
                 
                   P 
                   1 
                 
               
               = 
               
                 
                   
                     
                       k 
                       2 
                     
                     · 
                     
                       
                         
                           ω 
                           0 
                         
                          
                         
                           L 
                           · 
                           
                             ω 
                             0 
                           
                         
                          
                         
                           L 
                           2 
                         
                       
                       
                         
                           ( 
                           
                             
                               R 
                               s 
                             
                             + 
                             
                               R 
                               1 
                             
                           
                           ) 
                         
                          
                         
                           ( 
                           
                             
                               R 
                               L 
                             
                             + 
                             
                               R 
                               2 
                             
                           
                           ) 
                         
                       
                     
                   
                   
                     1 
                     + 
                     
                       
                         k 
                         2 
                       
                       · 
                       
                         
                           
                             ω 
                             0 
                           
                            
                           
                             L 
                             · 
                             
                               ω 
                               0 
                             
                           
                            
                           
                             L 
                             2 
                           
                         
                         
                           
                             ( 
                             
                               
                                 R 
                                 s 
                               
                               + 
                               
                                 R 
                                 1 
                               
                             
                             ) 
                           
                            
                           
                             ( 
                             
                               
                                 R 
                                 L 
                               
                               + 
                               
                                 R 
                                 2 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
                 · 
                 
                   
                     R 
                     L 
                   
                   
                     
                       R 
                       L 
                     
                     + 
                     
                       R 
                       2 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   η 
                   = 
                   
                     
                       
                         
                           k 
                           2 
                         
                         · 
                         
                           Q 
                           
                             1 
                             , 
                             L 
                           
                         
                         · 
                         
                           Q 
                           
                             2 
                             , 
                             L 
                           
                         
                       
                       
                         1 
                         + 
                         
                           
                             k 
                             2 
                           
                           · 
                           
                             Q 
                             
                               1 
                               , 
                               L 
                             
                           
                           · 
                           
                             Q 
                             
                               2 
                               , 
                               L 
                             
                           
                         
                       
                     
                     · 
                     
                       
                         R 
                         L 
                       
                       
                         
                           R 
                           L 
                         
                         + 
                         
                           R 
                           2 
                         
                       
                     
                   
                 
               
             
             
               
                 
                     
                 
               
             
           
         
       
       
         C) when ω=ω 0 , R L =R 2 , (R s =0): 
       
     
         [0000]    
       
                 
         
             
             
         
       
       
         D) ω=ω 0 , R L =R 2 , (R s =0) 
       
     
         [0000]    
       
         
           
             
               2 
                
               
                 R 
                 n 
               
                
               
                 R 
                 2 
               
             
             &gt;&gt; 
             
               
                 
                   ω 
                   0 
                   2 
                 
                  
                 
                   M 
                   2 
                 
               
                
               
                 
 
               
               -&gt; 
               1 
             
             &gt;&gt; 
             
               
                 k 
                 2 
               
                
               
                 Q 
                 
                   1 
                   , 
                   UL 
                 
               
                
               
                 
                   Q 
                   
                     2 
                     , 
                     UL 
                   
                 
                 / 
                 2 
               
             
           
         
       
       
         
           
             
               
                 
                   η 
                   = 
                   
                     
                       
                         P 
                         2 
                       
                       Pn 
                     
                     ≅ 
                     
                       
                         
                           
                             k 
                             2 
                           
                            
                           
                             Q 
                             
                               1 
                               , 
                               UL 
                             
                           
                            
                           
                             Q 
                             
                               2 
                               , 
                               UL 
                             
                           
                         
                         4 
                       
                        
                       
                           
                       
                        
                       
                         ( 
                         
                           weak 
                            
                           
                               
                           
                            
                           coupling 
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                     
                 
               
             
           
         
       
     
         [0000]    Q UL : Q unloaded 
         [0000]    
       
         
           
             
               
                 Q 
                 
                   1 
                   , 
                   UL 
                 
               
               = 
               
                 
                   ω 
                    
                   
                       
                   
                    
                   
                     L 
                     1 
                   
                 
                 
                   R 
                   1 
                 
               
             
             ; 
           
         
       
       
         
           
             
               Q 
               
                 2 
                 , 
                 UL 
               
             
             = 
             
               
                 ω 
                  
                 
                     
                 
                  
                 
                   L 
                   2 
                 
               
               
                 R 
                 2 
               
             
           
         
       
     
         [0000]    This shows that the output power is a function of input voltage squared 
         [0000]    
       
         
           
             
               
                 P 
                 2 
               
               = 
               
                 
                   f 
                    
                   
                     ( 
                     
                       U 
                       s 
                       2 
                     
                     ) 
                   
                 
                 = 
                 
                   
                     
                       I 
                       2 
                     
                     · 
                     
                       I 
                       2 
                       * 
                     
                   
                    
                   
                     R 
                     L 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               I 
               2 
             
             = 
             
               
                 
                   z 
                   M 
                 
                  
                 
                   U 
                   s 
                 
               
               
                 
                   
                     z 
                     1 
                   
                    
                   
                     z 
                     2 
                   
                 
                 - 
                 
                   z 
                   
                     M 
                     2 
                   
                 
               
             
           
         
       
       
         
           
             
               P 
               2 
             
             = 
             
               
                 
                   
                     z 
                     M 
                   
                    
                   
                     z 
                     M 
                     * 
                   
                    
                   
                     R 
                     L 
                   
                 
                 
                   
                     ( 
                     
                       
                         
                           z 
                           1 
                         
                          
                         
                           z 
                           2 
                         
                       
                       - 
                       
                         z 
                         M 
                         2 
                       
                     
                     ) 
                   
                    
                   
                     ( 
                     
                       
                         
                           z 
                           1 
                           * 
                         
                          
                         
                           z 
                           2 
                           * 
                         
                       
                       - 
                       
                         z 
                         M 
                         
                           2 
                           * 
                         
                       
                     
                     ) 
                   
                 
               
               · 
               
                 U 
                 s 
                 2 
               
             
           
         
       
       
         
           
             
               P 
               2 
             
             = 
             
               
                 
                   
                      
                     
                       z 
                       M 
                     
                      
                   
                   2 
                 
                 · 
                 
                   R 
                   L 
                 
                 · 
                 
                   U 
                   s 
                   2 
                 
               
               
                 
                   
                     z 
                     1 
                   
                    
                   
                     z 
                     2 
                   
                    
                   
                     z 
                     1 
                     * 
                   
                    
                   
                     z 
                     2 
                     * 
                   
                 
                 + 
                 
                    
                   
                     z 
                     M 
                   
                    
                 
                 + 
                 
                   
                     
                        
                       
                         z 
                         M 
                       
                        
                     
                     2 
                   
                   · 
                   
                     ( 
                     
                       
                         
                           z 
                           1 
                         
                          
                         
                           z 
                           2 
                         
                       
                       + 
                       
                         
                           z 
                           1 
                           * 
                         
                          
                         
                           z 
                           2 
                           * 
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               P 
               2 
             
             = 
             
               
                 
                   
                      
                     
                       z 
                       M 
                     
                      
                   
                   2 
                 
                 · 
                 
                   R 
                   L 
                 
                 · 
                 
                   U 
                   s 
                   2 
                 
               
               
                 
                   
                      
                     
                       
                         z 
                         1 
                       
                        
                       
                         z 
                         2 
                       
                     
                      
                   
                   2 
                 
                 + 
                 
                   
                     
                        
                       
                         z 
                         M 
                       
                        
                     
                     2 
                   
                    
                   2 
                    
                   Re 
                    
                   
                     { 
                     
                       
                         z 
                         1 
                       
                        
                       
                         z 
                         2 
                       
                     
                     } 
                   
                 
                 + 
                 
                   
                      
                     
                       z 
                       M 
                     
                      
                   
                   4 
                 
               
             
           
         
       
       
         
           
             
               z 
               M 
             
             = 
             
               jω 
                
               
                   
               
                
               M 
             
           
         
       
       
         
           
             
               z 
               M 
               * 
             
             = 
             
               
                 - 
                 jω 
               
                
               
                   
               
                
               M 
             
           
         
       
       
         
           
             
                
               
                 z 
                 M 
               
                
             
             = 
             
               
                 ω 
                  
                 
                     
                 
                  
                 M 
               
               = 
               
                 
                   z 
                   M 
                 
                  
                 
                   z 
                   M 
                   * 
                 
               
             
           
         
       
       
         
           
             
               z 
               M 
               * 
             
             = 
             
               
                 
                   - 
                   
                     ω 
                     2 
                   
                 
                  
                 
                   M 
                   2 
                 
               
               = 
               
                 - 
                 
                   
                      
                     
                       z 
                       M 
                     
                      
                   
                   2 
                 
               
             
           
         
       
       
         
           
             
               z 
               M 
               
                 2 
                 * 
               
             
             = 
             
               
                 
                   - 
                   
                     ω 
                     2 
                   
                 
                  
                 
                   M 
                   2 
                 
               
               = 
               
                 
                   z 
                   M 
                   2 
                 
                 = 
                 
                   - 
                   
                     
                        
                       
                         z 
                         M 
                       
                        
                     
                     2 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 z 
                 M 
                 2 
               
               · 
               
                 z 
                 M 
                 
                   2 
                   * 
                 
               
             
             = 
             
               
                  
                 
                   z 
                   M 
                 
                  
               
               4 
             
           
         
       
       
         
           
             
                
               
                 
                   z 
                   1 
                 
                  
                 
                   z 
                   2 
                 
               
                
             
             = 
             
               
                  
                 
                   z 
                   1 
                 
                  
               
               · 
               
                  
                 
                   z 
                   2 
                 
                  
               
             
           
         
       
       
         
           
             
               
                 
                   z 
                   1 
                 
                  
                 
                   z 
                   2 
                 
               
               + 
               
                 
                   z 
                   1 
                   * 
                 
                  
                 
                   z 
                   2 
                   * 
                 
               
             
             = 
             
               2 
                
               Re 
                
               
                 { 
                 
                   
                     z 
                     1 
                   
                    
                   
                     z 
                     2 
                   
                 
                 } 
               
             
           
         
       
       
         
           
             
               
                  
                 
                   
                     z 
                     1 
                   
                   · 
                   
                     z 
                     2 
                   
                 
                  
               
               2 
             
             = 
             
               
                 
                    
                   
                     z 
                     1 
                   
                    
                 
                 2 
               
               · 
               
                 
                    
                   
                     z 
                     2 
                   
                    
                 
                 2 
               
             
           
         
       
     
       Definitions: 
       [0032]    
       
         
           
             
               
                 z 
                 1 
               
               = 
               
                 
                   R 
                   1 
                   ′ 
                 
                 + 
                 
                   jX 
                   1 
                 
               
             
             ; 
             
               
                 z 
                 2 
               
               = 
               
                 
                   
                     
                       R 
                       2 
                       ′ 
                     
                     + 
                     
                       jX 
                       2 
                     
                   
                    
                   
                     
 
                   
                   | 
                   
                     
                       z 
                       1 
                     
                      
                     
                       z 
                       2 
                     
                   
                    
                   
                     | 
                     2 
                   
                 
                 = 
                 
                   
                     
                       ( 
                       
                         
                           R 
                           1 
                           ′2 
                         
                         + 
                         
                           X 
                           1 
                           2 
                         
                       
                       ) 
                     
                      
                     
                       ( 
                       
                         
                           R 
                           2 
                           ′2 
                         
                         + 
                         
                           X 
                           2 
                           2 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         R 
                         1 
                         ′2 
                       
                        
                       
                         R 
                         2 
                         ′2 
                       
                     
                     + 
                     
                       
                         X 
                         1 
                         2 
                       
                        
                       
                         R 
                         2 
                         ′2 
                       
                     
                     + 
                     
                       
                         X 
                         2 
                         2 
                       
                        
                       
                         R 
                         1 
                         ′2 
                       
                     
                     + 
                     
                       
                         X 
                         1 
                         2 
                       
                        
                       
                         X 
                         2 
                         2 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               Re 
                
               
                 { 
                 
                   
                     z 
                     1 
                   
                    
                   
                     z 
                     2 
                   
                 
                 } 
               
             
             = 
             
               
                 
                   Re 
                    
                   
                     ( 
                     
                       
                         R 
                         1 
                         ′ 
                       
                       + 
                       
                         jX 
                         1 
                       
                     
                     ) 
                   
                 
                  
                 
                   ( 
                   
                     
                       R 
                       2 
                       ′ 
                     
                     + 
                     
                       jX 
                       2 
                     
                   
                   ) 
                 
               
               = 
               
                 
                   
                     
                       
                         R 
                         1 
                         ′ 
                       
                        
                       
                         R 
                         2 
                         ′ 
                       
                     
                     + 
                     
                       
                         X 
                         1 
                       
                        
                       
                         X 
                         2 
                       
                     
                   
                    
                   
                     
 
                   
                   | 
                   
                     z 
                     M 
                   
                   | 
                 
                 = 
                 
                   X 
                   M 
                 
               
             
           
         
       
       
         
           
             
               P 
               2 
             
             = 
             
               
                 
                   X 
                   M 
                   2 
                 
                  
                 
                   
                     R 
                     1 
                   
                   · 
                   
                     U 
                     s 
                     2 
                   
                 
               
               
                 
                   
                     R 
                     1 
                     ′2 
                   
                    
                   
                     R 
                     2 
                     ′2 
                   
                 
                 + 
                 
                   
                     R 
                     1 
                     ′2 
                   
                    
                   
                     X 
                     2 
                     2 
                   
                 
                 + 
                 
                   
                     R 
                     1 
                     ′2 
                   
                    
                   
                     X 
                     1 
                     2 
                   
                 
                 + 
                 
                   
                     X 
                     1 
                     2 
                   
                    
                   
                     X 
                     2 
                     2 
                   
                 
                 + 
                 
                   2 
                    
                   
                     X 
                     M 
                     2 
                   
                    
                   
                     R 
                     1 
                     ′ 
                   
                    
                   
                     R 
                     2 
                     ′ 
                   
                 
                 + 
                 
                   2 
                    
                   
                     X 
                     M 
                     2 
                   
                    
                   
                     X 
                     1 
                   
                    
                   
                     X 
                     2 
                   
                 
                 + 
                 
                   X 
                   M 
                   4 
                 
               
             
           
         
       
       
         
           
             
               P 
               2 
             
             = 
             
               
                 
                   X 
                   M 
                   2 
                 
                  
                 
                   
                     R 
                     L 
                   
                   · 
                   
                     U 
                     s 
                     2 
                   
                 
               
               
                 
                   
                     ( 
                     
                       
                         
                           R 
                           1 
                           ′ 
                         
                          
                         
                           R 
                           2 
                           ′ 
                         
                       
                       + 
                       
                         X 
                         M 
                         2 
                       
                     
                     ) 
                   
                   2 
                 
                 + 
                 
                   
                     R 
                     1 
                     ′2 
                   
                    
                   
                     X 
                     2 
                     2 
                   
                 
                 + 
                 
                   
                     R 
                     2 
                     ′2 
                   
                    
                   
                     X 
                     1 
                     2 
                   
                 
                 + 
                 
                   
                     X 
                     1 
                     2 
                   
                    
                   
                     X 
                     2 
                     2 
                   
                 
                 + 
                 
                   2 
                    
                   
                     X 
                     M 
                     2 
                   
                    
                   
                     X 
                     1 
                   
                    
                   
                     X 
                     2 
                   
                 
               
             
           
         
       
     
         [0033]    Therefore, when at or near the resonance condition: 
         [0000]    
       
         
           
             
               ω 
               = 
               
                 
                   ω 
                   0 
                 
                 = 
                 
                   
                     ω 
                     2 
                   
                   = 
                   
                     
                       
                         ω 
                         0 
                       
                       → 
                       
                         X 
                         1 
                       
                     
                     = 
                     0 
                   
                 
               
             
             , 
             
               
                 X 
                 2 
               
               = 
               0 
             
           
         
       
       
         
           
             
               P 
               2 
             
             = 
             
               
                 
                   
                     X 
                     M 
                     2 
                   
                    
                   
                     
                       R 
                       L 
                     
                     · 
                     
                       U 
                       s 
                       2 
                     
                   
                 
                 
                   
                     
                       R 
                       j 
                       ′2 
                     
                      
                     
                       R 
                       2 
                       ′2 
                     
                   
                   + 
                   
                     2 
                      
                     
                       X 
                       M 
                       2 
                     
                      
                     
                       R 
                       1 
                       ′ 
                     
                      
                     
                       R 
                       2 
                       ′ 
                     
                   
                   + 
                   
                     X 
                     M 
                     4 
                   
                 
               
               = 
               
                 
                   
                     
                       X 
                       M 
                       2 
                     
                      
                     
                       R 
                       L 
                     
                   
                   
                     
                       ( 
                       
                         
                           
                             R 
                             1 
                             ′ 
                           
                            
                           
                             R 
                             2 
                             ′ 
                           
                         
                         + 
                         
                           X 
                           M 
                           2 
                         
                       
                       ) 
                     
                     2 
                   
                 
                 · 
                 
                   U 
                   s 
                   2 
                 
               
             
           
         
       
       
         
           
             
               P 
               2 
             
             = 
             
               
                 
                   
                     ω 
                     0 
                     2 
                   
                    
                   
                     M 
                     2 
                   
                    
                   
                     R 
                     L 
                   
                 
                 
                   
                     
                       
                         ( 
                         
                           
                             R 
                             s 
                           
                           + 
                           
                             R 
                             1 
                           
                         
                         ) 
                       
                       2 
                     
                      
                     
                       
                         ( 
                         
                           
                             R 
                             1 
                           
                           + 
                           
                             R 
                             2 
                           
                         
                         ) 
                       
                       2 
                     
                   
                   + 
                   
                     2 
                      
                     
                       ω 
                       0 
                       2 
                     
                      
                     
                       
                         M 
                         2 
                       
                        
                       
                         ( 
                         
                           
                             R 
                             s 
                           
                           + 
                           
                             R 
                             1 
                           
                         
                         ) 
                       
                     
                      
                     
                       ( 
                       
                         
                           R 
                           L 
                         
                         + 
                         
                           R 
                           2 
                         
                       
                       ) 
                     
                   
                   + 
                   
                     
                       ω 
                       0 
                       4 
                     
                      
                     
                       M 
                       4 
                     
                   
                 
               
               · 
               
                 U 
                 s 
                 2 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     P 
                     2 
                   
                   = 
                   
                     
                       
                         
                           ω 
                           0 
                           2 
                         
                          
                         
                           M 
                           2 
                         
                          
                         
                           R 
                           L 
                         
                       
                       
                         
                           ( 
                           
                             
                               
                                 ( 
                                 
                                   
                                     R 
                                     s 
                                   
                                   + 
                                   
                                     R 
                                     1 
                                   
                                 
                                 ) 
                               
                                
                               
                                 ( 
                                 
                                   
                                     R 
                                     1 
                                   
                                   + 
                                   
                                     R 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 ω 
                                 0 
                                 2 
                               
                                
                               
                                 M 
                                 2 
                               
                             
                           
                           ) 
                         
                         2 
                       
                     
                      
                     
                       U 
                       s 
                       2 
                     
                   
                 
               
             
             
               
                 
                     
                 
               
             
           
         
       
     
         [0000]    Showing that the power transfer is inversely proportional to several variables, including series resistances. 
         [0034]    Mutual inductance in terms of coupling factors and inductions: 
         [0000]    
       
         
           
             M 
             = 
             
               k 
               · 
               
                 
                   
                     L 
                     1 
                   
                    
                   
                     L 
                     2 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     P 
                     2 
                   
                   = 
                   
                     
                       
                         
                           ω 
                           0 
                           2 
                         
                          
                         
                           k 
                           2 
                         
                          
                         
                           L 
                           1 
                         
                          
                         
                           
                             L 
                             2 
                           
                           · 
                           
                             R 
                             L 
                           
                         
                       
                       
                         
                           ( 
                           
                             
                               
                                 ( 
                                 
                                   
                                     R 
                                     s 
                                   
                                   + 
                                   
                                     R 
                                     1 
                                   
                                 
                                 ) 
                               
                                
                               
                                 ( 
                                 
                                   
                                     R 
                                     1 
                                   
                                   + 
                                   
                                     R 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 ω 
                                 0 
                                 2 
                               
                                
                               
                                 k 
                                 2 
                               
                                
                               
                                 L 
                                 1 
                               
                                
                               
                                 L 
                                 2 
                               
                             
                           
                           ) 
                         
                         2 
                       
                     
                     · 
                     
                       U 
                       s 
                       2 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       
                         
                           k 
                           2 
                         
                          
                         
                           
                             
                               ω 
                               0 
                             
                              
                             
                               L 
                               1 
                             
                              
                             
                               ω 
                               0 
                             
                              
                             
                               L 
                               2 
                             
                           
                           
                             
                               ( 
                               
                                 
                                   R 
                                   s 
                                 
                                 + 
                                 
                                   R 
                                   M 
                                 
                               
                               ) 
                             
                              
                             
                               ( 
                               
                                 
                                   R 
                                   1 
                                 
                                 + 
                                 
                                   R 
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           ( 
                           
                             1 
                             + 
                             
                               
                                 k 
                                 2 
                               
                                
                               
                                 
                                   
                                     ω 
                                     0 
                                   
                                    
                                   
                                     L 
                                     1 
                                   
                                    
                                   
                                     ω 
                                     0 
                                   
                                    
                                   
                                     L 
                                     2 
                                   
                                 
                                 
                                   
                                     ( 
                                     
                                       
                                         R 
                                         s 
                                       
                                       + 
                                       
                                         R 
                                         M 
                                       
                                     
                                     ) 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         R 
                                         1 
                                       
                                       + 
                                       
                                         R 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           ) 
                         
                         2 
                       
                     
                     · 
                     
                       
                         
                           U 
                           s 
                           2 
                         
                          
                         
                           R 
                           L 
                         
                       
                       
                         
                           ( 
                           
                             
                               R 
                               s 
                             
                             + 
                             
                               R 
                               1 
                             
                           
                           ) 
                         
                          
                         
                           ( 
                           
                             
                               R 
                               L 
                             
                             + 
                             
                               R 
                               2 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     P 
                     2 
                   
                   = 
                   
                     
                       
                         
                           k 
                           2 
                         
                         · 
                         
                           Q 
                           
                             L 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                         · 
                         
                           Q 
                           
                             L 
                              
                             
                                 
                             
                              
                             2 
                           
                         
                       
                       
                         
                           ( 
                           
                             1 
                             + 
                             
                               
                                 k 
                                 2 
                               
                               · 
                               
                                 Q 
                                 
                                   L 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                               · 
                               
                                 Q 
                                 
                                   L 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                             
                           
                           ) 
                         
                         2 
                       
                     
                     · 
                     
                       
                         R 
                         L 
                       
                       
                         
                           ( 
                           
                             
                               R 
                               s 
                             
                             + 
                             
                               R 
                               1 
                             
                           
                           ) 
                         
                          
                         
                           ( 
                           
                             
                               R 
                               L 
                             
                             + 
                             
                               R 
                               2 
                             
                           
                           ) 
                         
                       
                     
                     · 
                     
                       U 
                       s 
                       2 
                     
                   
                 
               
             
             
               
                 
                     
                 
               
             
           
         
       
     
         [0035]    The power output is proportional to the square of the input power, as described above. However, there is a maximum input power beyond which no further output power will be produced. These values are explained below. The maximum input power P 1max  is expressed as: 
         [0000]    
       
         
           
             
               
                 P 
                 
                   1 
                   , 
                   max 
                 
               
               = 
               
                 
                   
                     U 
                     s 
                     2 
                   
                   
                     
                       R 
                       s 
                     
                     + 
                     
                       R 
                       
                         in 
                         , 
                         min 
                       
                     
                   
                 
                 = 
                 
                   Re 
                    
                   
                     { 
                     
                       
                         U 
                         s 
                       
                       · 
                       
                         I 
                         1 
                         * 
                       
                     
                     } 
                   
                 
               
             
             ; 
           
         
       
     
         [0000]    R in,min : min. permissible input resistance
 
Efficiency relative to maximum input power:
 
         [0000]    
       
         
           
             
               η 
               ′ 
             
             = 
             
               
                 
                   P 
                   2 
                 
                 
                   P 
                   
                     1 
                     , 
                     max 
                   
                 
               
               = 
               
                 
                   
                     P 
                     2 
                   
                    
                   
                     ( 
                     
                       U 
                       s 
                       2 
                     
                     ) 
                   
                 
                 
                   P 
                   
                     1 
                     , 
                     max 
                   
                 
               
             
           
         
       
     
         [0000]    Under resonance condition ω=ω 1 =ω 2 =ω 0 : 
         [0000]    
       
         
           
             
               
                 
                   
                     η 
                     ′ 
                   
                   = 
                   
                     
                       
                         ω 
                         0 
                         2 
                       
                        
                       
                         M 
                         2 
                       
                        
                       
                         
                           R 
                           L 
                         
                          
                         
                           ( 
                           
                             
                               R 
                               s 
                             
                             + 
                             
                               R 
                               
                                 in 
                                 , 
                                 min 
                               
                             
                           
                           ) 
                         
                       
                     
                     
                       
                         [ 
                         
                           
                             
                               ( 
                               
                                 
                                   R 
                                   s 
                                 
                                 + 
                                 
                                   R 
                                   1 
                                 
                               
                               ) 
                             
                              
                             
                               ( 
                               
                                 
                                   R 
                                   1 
                                 
                                 + 
                                 
                                   R 
                                   2 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               ω 
                               0 
                               2 
                             
                              
                             
                               M 
                               2 
                             
                           
                         
                         ] 
                       
                       2 
                     
                   
                 
               
             
             
               
                 
                     
                 
               
             
           
         
       
     
         [0000]    Equation for input power (P 1 ) under the resonance condition is therefore: 
         [0000]    
       
         
           
             
               P 
               1 
             
             = 
             
               
                 
                   P 
                   2 
                 
                 η 
               
               = 
               
                 
                   
                     
                       ω 
                       0 
                       2 
                     
                      
                     
                       M 
                       2 
                     
                      
                     
                       
                         R 
                         L 
                       
                        
                       
                         [ 
                         
                           
                             
                               ( 
                               
                                 
                                   R 
                                   s 
                                 
                                 + 
                                 
                                   R 
                                   1 
                                 
                               
                               ) 
                             
                              
                             
                               ( 
                               
                                 
                                   R 
                                   1 
                                 
                                 + 
                                 
                                   R 
                                   2 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               ω 
                               0 
                               2 
                             
                              
                             
                               M 
                               2 
                             
                           
                         
                         ] 
                       
                     
                      
                     
                       ( 
                       
                         
                           R 
                           L 
                         
                         + 
                         
                           R 
                           2 
                         
                       
                       ) 
                     
                   
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 ( 
                                 
                                   
                                     R 
                                     s 
                                   
                                   + 
                                   
                                     R 
                                     2 
                                   
                                 
                                 ) 
                               
                                
                               
                                 ( 
                                 
                                   
                                     R 
                                     L 
                                   
                                   + 
                                   
                                     R 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 ω 
                                 0 
                                 0 
                               
                                
                               
                                 M 
                                 2 
                               
                             
                           
                           ] 
                         
                         2 
                       
                       · 
                       
                         ω 
                         0 
                         2 
                       
                     
                      
                     
                       M 
                       2 
                     
                      
                     
                       R 
                       L 
                     
                   
                 
                 · 
                 
                   U 
                   s 
                   2 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     P 
                     1 
                   
                   = 
                   
                     
                       
                         
                           R 
                           L 
                         
                         + 
                         
                           R 
                           2 
                         
                       
                       
                         
                           
                             ( 
                             
                               
                                 R 
                                 s 
                               
                               + 
                               
                                 R 
                                 1 
                               
                             
                             ) 
                           
                            
                           
                             ( 
                             
                               
                                 R 
                                 L 
                               
                               + 
                               
                                 R 
                                 2 
                               
                             
                             ) 
                           
                         
                          
                         
                           + 
                           0 
                           2 
                         
                          
                         
                           M 
                           2 
                         
                       
                     
                     · 
                     
                       U 
                       s 
                       2 
                     
                   
                 
               
             
             
               
                 
                     
                 
               
             
           
         
       
       
         
           
             
               For 
                
               
                   
               
                
               
                 ( 
                 
                   
                     R 
                     s 
                   
                   + 
                   
                     R 
                     M 
                   
                 
                 ) 
               
                
               
                 ( 
                 
                   
                     R 
                     L 
                   
                   + 
                   
                     R 
                     2 
                   
                 
                 ) 
               
             
             &gt;&gt; 
             
               
                 ω 
                 0 
                 2 
               
                
               
                 M 
                 2 
               
                
               
                 : 
               
             
           
         
       
       
         
           
             
               P 
               1 
             
             ≅ 
             
               
                 U 
                 S 
                 2 
               
               
                 ( 
                 
                   
                     R 
                     s 
                   
                   + 
                   
                     R 
                     1 
                   
                 
                 ) 
               
             
           
         
       
     
         [0036]    The current ratio between input and induced currents can be expressed as 
         [0000]    
       
         
           
             
               
                 I 
                 2 
               
               
                 I 
                 1 
               
             
             = 
             
               
                 
                   
                     z 
                     M 
                   
                   · 
                   
                     U 
                     s 
                   
                   · 
                   
                     ( 
                     
                       
                         
                           z 
                           1 
                         
                          
                         
                           z 
                           2 
                         
                       
                       - 
                       
                         z 
                         n 
                         2 
                       
                     
                     ) 
                   
                 
                 
                   
                     ( 
                     
                       
                         
                           z 
                           1 
                         
                          
                         
                           z 
                           2 
                         
                       
                       - 
                       
                         z 
                         M 
                         2 
                       
                     
                     ) 
                   
                    
                   
                     z 
                     2 
                   
                    
                   
                     U 
                     s 
                   
                 
               
               = 
               
                 
                   
                     z 
                     M 
                   
                   
                     z 
                     2 
                   
                 
                 = 
                 
                   
                     jω 
                      
                     
                         
                     
                      
                     M 
                   
                   
                     
                       R 
                       L 
                     
                     + 
                     
                       R 
                       2 
                     
                     + 
                     
                       j 
                        
                       
                         ( 
                         
                           
                             ω 
                              
                             
                                 
                             
                              
                             
                               L 
                               2 
                             
                           
                           - 
                           
                             1 
                             
                               ω 
                                
                               
                                   
                               
                                
                               
                                 C 
                                 2 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               at 
                
               
                   
               
                
               ω 
             
             = 
             
               
                 ω 
                 0 
               
               = 
               
                 1 
                 
                   
                     
                       L 
                       2 
                     
                      
                     
                       C 
                       2 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       I 
                       2 
                     
                     
                       I 
                       1 
                     
                   
                   = 
                 
               
               
                 
                   
                     jω 
                      
                     
                         
                     
                      
                     M 
                   
                   
                     
                       R 
                       1 
                     
                     + 
                     
                       R 
                       2 
                     
                   
                 
               
             
             
               
                 
                     
                 
               
               
                 
                     
                 
               
             
           
         
       
       
         
           
             
               avg 
               . 
               
                 { 
                 
                   
                     I 
                     2 
                   
                   
                     I 
                     1 
                   
                 
                 } 
               
             
             = 
             
               π 
               2 
             
           
         
       
     
         [0000]    Weak coupling: R 1 +R 2 &gt;|jωM|
   →I 2 &lt;I 1  
 
Strong coupling: R 1 +R 2 &lt;|jωM|
   →I 2 &gt;I 1  
 
Input current I 1 : (under resonance condition)
   
 
         [0000]    
       
         
           
             
               I 
               1 
             
             = 
             
               
                 
                   P 
                   1 
                 
                 
                   U 
                   s 
                 
               
               = 
               
                 
                   
                     ( 
                     
                       
                         R 
                         1 
                       
                       + 
                       
                         R 
                         2 
                       
                     
                     ) 
                   
                   · 
                   
                     U 
                     s 
                   
                 
                 
                   
                     
                       ( 
                       
                         
                           R 
                           s 
                         
                         + 
                         
                           R 
                           1 
                         
                       
                       ) 
                     
                      
                     
                       ( 
                       
                         
                           R 
                           L 
                         
                         + 
                         
                           R 
                           2 
                         
                       
                       ) 
                     
                   
                   + 
                   
                     
                       ω 
                       0 
                       2 
                     
                      
                     
                       M 
                       2 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     I 
                     1 
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             R 
                             L 
                           
                           + 
                           
                             R 
                             2 
                           
                         
                         ) 
                       
                       
                         
                           
                             ( 
                             
                               
                                 R 
                                 s 
                               
                               + 
                               
                                 R 
                                 1 
                               
                             
                             ) 
                           
                            
                           
                             ( 
                             
                               
                                 R 
                                 L 
                               
                               + 
                               
                                 R 
                                 2 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             ω 
                             0 
                             2 
                           
                            
                           
                             M 
                             2 
                           
                         
                       
                     
                     · 
                     
                       U 
                       s 
                     
                   
                 
               
             
             
               
                 
                     
                 
               
             
           
         
       
     
         [0000]    Output current I 2 : (under resonance condition) 
         [0000]    
       
         
           
             
               
                 
                   
                     I 
                     2 
                   
                   = 
                   
                     
                       
                         jω 
                          
                         
                             
                         
                          
                         M 
                       
                       
                         
                           
                             ( 
                             
                               
                                 R 
                                 s 
                               
                               + 
                               
                                 R 
                                 1 
                               
                             
                             ) 
                           
                            
                           
                             ( 
                             
                               
                                 R 
                                 L 
                               
                               + 
                               
                                 R 
                                 2 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             ω 
                             0 
                             2 
                           
                            
                           
                             M 
                             2 
                           
                         
                       
                     
                     · 
                     
                       U 
                       s 
                     
                   
                 
               
             
             
               
                 
                     
                 
               
             
           
         
       
     
         [0039]    Maximizing transfer efficiency and output power (P 2 ) can be calculated according to the transfer efficiency equation: 
         [0000]    
       
         
           
             η 
             = 
             
               
                 
                   P 
                   2 
                 
                 
                   P 
                   1 
                 
               
               = 
               
                 
                   
                     ω 
                     2 
                   
                    
                   
                     M 
                     2 
                   
                    
                   
                     R 
                     L 
                   
                 
                 
                   
                     
                       ( 
                       
                         
                           R 
                           s 
                         
                         + 
                         
                           R 
                           n 
                         
                       
                       ) 
                     
                      
                     
                       [ 
                       
                         
                           
                             ( 
                             
                               
                                 R 
                                 L 
                               
                               + 
                               
                                 R 
                                 2 
                               
                             
                             ) 
                           
                           2 
                         
                         + 
                         
                           
                             
                               ( 
                               
                                 
                                   ω 
                                    
                                   
                                       
                                   
                                    
                                   
                                     L 
                                     2 
                                   
                                 
                                 - 
                                 
                                   1 
                                   
                                     ω 
                                      
                                     
                                         
                                     
                                      
                                     
                                       C 
                                       2 
                                     
                                   
                                 
                                 + 
                                 
                                   X 
                                   L 
                                 
                               
                               ) 
                             
                             2 
                           
                            
                         
                       
                       ] 
                     
                   
                   + 
                   
                     
                       ω 
                       2 
                     
                      
                     
                       
                         M 
                         2 
                       
                        
                       
                         ( 
                         
                           
                             R 
                             L 
                           
                           + 
                           
                             R 
                             2 
                           
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
     
         [0000]    After reviewing this equation, an embodiment forms circuits that are based on observations about the nature of how to maximize efficiency in such a system. 
       Conclusion 1) 
       [0000]    
       
         η(L 2 , C 2 , X L ) reaches maximum for 
       
     
         [0000]    
       
         
           
             
               
                 ω 
                  
                 
                     
                 
                  
                 
                   L 
                   2 
                 
               
               - 
               
                 1 
                 
                   ω 
                    
                   
                       
                   
                    
                   
                     C 
                     2 
                   
                 
               
               + 
               
                 X 
                 L 
               
             
             = 
             0 
           
         
       
       
         That is, efficiency for any L, C, X at the receiver is maximum when that equation is met.
 
Transfer efficiency wide resonance condition:
 
       
     
         [0000]    
       
         
           
             η 
             = 
             
               
                 
                   
                     P 
                     2 
                   
                   
                     P 
                     1 
                   
                 
                  
                 
                    
                   
                     ω 
                     = 
                     
                       ω 
                       0 
                     
                   
                 
               
               = 
               
                 
                   
                     
                       ω 
                       0 
                       2 
                     
                      
                     
                       M 
                       2 
                     
                   
                   
                     [ 
                     
                       
                         
                           ( 
                           
                             
                               
                                 R 
                                 s 
                               
                                
                             
                             + 
                             
                               R 
                               n 
                             
                           
                           ) 
                         
                          
                         
                           ( 
                           
                             
                               R 
                               L 
                             
                             + 
                             
                               R 
                               2 
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           ω 
                           0 
                           2 
                         
                          
                         
                           M 
                           2 
                         
                       
                     
                     ] 
                   
                 
                 · 
                 
                   
                     R 
                     1 
                   
                   
                     ( 
                     
                       
                         R 
                         L 
                       
                       + 
                       
                         R 
                         2 
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
       Conclusion 2) 
       [0000]    
       
         To maximise η R S  should be R S &lt;&lt;R 1    
         That is, for maximum efficiency, the source resistance R S  needs to be much lower than the series resistance, e.g., 1/50, or 1/100 th  or less
 
Transfer efficiency under resonance and weak coupling condition:
 
       
     
         [0000]    
       
         
           
             
               
                 ( 
                 
                   
                     R 
                     s 
                   
                   + 
                   
                     R 
                     n 
                   
                 
                 ) 
               
                
               
                 ( 
                 
                   
                     R 
                     L 
                   
                   + 
                   
                     R 
                     2 
                   
                 
                 ) 
               
             
             &gt;&gt; 
             
               
                 ω 
                 0 
                 2 
               
                
               
                 M 
                 2 
               
             
           
         
       
       
         
           
             η 
             ≅ 
             
               
                 
                   ω 
                   0 
                   2 
                 
                  
                 
                   
                     M 
                     2 
                   
                   · 
                   
                     
                       R 
                       L 
                     
                      
                   
                 
               
               
                 
                   ( 
                   
                     
                       R 
                       s 
                     
                     + 
                     
                       R 
                       n 
                     
                   
                   ) 
                 
                  
                 
                   
                     ( 
                     
                       
                         
                           R 
                           L 
                         
                          
                       
                       + 
                       
                         R 
                         2 
                       
                     
                     ) 
                   
                   2 
                 
               
             
           
         
       
     
       Maximising η(R L ): 
       [0044]    
       
         
           
             
               
                  
                 η 
               
               
                  
                 
                   R 
                   L 
                 
               
             
             = 
             
               
                 
                   
                     
                       ω 
                       0 
                       2 
                     
                      
                     
                       M 
                       2 
                     
                   
                   
                     
                       R 
                       s 
                     
                     + 
                     
                       R 
                       1 
                     
                   
                 
                 · 
                 
                   
                     
                       ( 
                       
                         
                           R 
                           L 
                         
                         + 
                         
                           R 
                           2 
                         
                       
                       ) 
                     
                     - 
                     
                       2 
                        
                       
                         R 
                         L 
                       
                     
                   
                   
                     
                       ( 
                       
                         
                           R 
                           L 
                         
                         + 
                         
                           R 
                           2 
                         
                       
                       ) 
                     
                     3 
                   
                 
               
               = 
               
                 
                   0 
                   -&gt; 
                   
                     R 
                     L 
                   
                 
                 = 
                 
                   R 
                   2 
                 
               
             
           
         
       
     
       Conclusion 3) 
       [0000]    
       
         η reaches maximum for R L =R 2  under weak coupling condition. 
         That is, when there is weak coupling, efficiency is maximum when the resistance of the load matches the series resistance of the receiver. 
         Transfer efficiency under resonance condition. 
         Optimizing R L  to achieve max. η 
       
     
         [0000]    
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         
                           
                              
                             η 
                           
                           
                              
                             
                               R 
                               L 
                             
                           
                         
                         = 
                         0 
                       
                       ; 
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       
                         
                            
                           
                              
                             
                               R 
                               L 
                             
                           
                         
                         · 
                         
                           
                             
                               ω 
                               0 
                               2 
                             
                              
                             
                               M 
                               2 
                             
                              
                             
                               R 
                               L 
                             
                           
                           
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           R 
                                           s 
                                         
                                         + 
                                         
                                           R 
                                           1 
                                         
                                       
                                        
                                     
                                     ) 
                                   
                                   2 
                                 
                                 
                                   R 
                                   1 
                                 
                               
                                
                               
                                 
                                   ( 
                                   
                                     
                                       R 
                                       L 
                                     
                                     + 
                                     
                                       R 
                                       2 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                             + 
                             
                               
                                 ω 
                                 0 
                                 2 
                               
                                
                               
                                 
                                   M 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       R 
                                       L 
                                     
                                     + 
                                     
                                       R 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                        
                       
                         u 
                         v 
                       
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       
                         
                           
                             u 
                             · 
                             
                               v 
                               ′ 
                             
                           
                           - 
                           
                             v 
                             · 
                             
                               u 
                               ′ 
                             
                           
                         
                         
                           v 
                           2 
                         
                       
                       = 
                       0 
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       
                         
                           u 
                           · 
                           
                             v 
                             ′ 
                           
                         
                         - 
                         
                           v 
                           · 
                           
                             u 
                             ′ 
                           
                         
                       
                       = 
                       0 
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       
                         u 
                         = 
                         
                           
                             ω 
                             0 
                             2 
                           
                            
                           
                             
                               M 
                               2 
                             
                             · 
                             
                               R 
                               L 
                             
                           
                         
                       
                       ; 
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       
                         u 
                         ′ 
                       
                       = 
                       
                         
                           ω 
                           0 
                           2 
                         
                          
                         
                           M 
                           2 
                         
                       
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       v 
                       = 
                       
                         
                           
                             
                               R 
                               1 
                               ′ 
                             
                              
                             
                               ( 
                               
                                 
                                   R 
                                   L 
                                 
                                 + 
                                 
                                   R 
                                   2 
                                 
                               
                               ) 
                             
                           
                           2 
                         
                         + 
                         
                           
                             ω 
                             0 
                             2 
                           
                            
                           
                             
                               M 
                               2 
                             
                              
                             
                               ( 
                               
                                 
                                   R 
                                   1 
                                 
                                 + 
                                 
                                   R 
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       
                         v 
                         ′ 
                       
                       - 
                       
                         2 
                          
                         
                           
                             R 
                             1 
                             ′ 
                           
                            
                           
                             ( 
                             
                               
                                 R 
                                 L 
                               
                               + 
                               
                                 R 
                                 2 
                               
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           ω 
                           0 
                           2 
                         
                          
                         
                           M 
                           2 
                         
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             u 
                             · 
                             
                               v 
                               ′ 
                             
                           
                           - 
                           
                             v 
                             · 
                             
                               u 
                               ′ 
                             
                           
                         
                         = 
                           
                          
                         
                           
                             
                               ω 
                               0 
                               2 
                             
                              
                             
                               M 
                               2 
                             
                              
                             
                               
                                 R 
                                 L 
                               
                                
                               
                                 ( 
                                 
                                   
                                     2 
                                      
                                     
                                       
                                         R 
                                         1 
                                         ′ 
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             R 
                                             L 
                                           
                                           + 
                                           
                                             R 
                                             2 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   + 
                                   
                                     
                                       ω 
                                       0 
                                       2 
                                     
                                      
                                     
                                       M 
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           - 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             ( 
                             
                               
                                 
                                   
                                     R 
                                     1 
                                     ′ 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         R 
                                         1 
                                       
                                       + 
                                       
                                         R 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                                 2 
                               
                               + 
                               
                                 
                                   ω 
                                   0 
                                   2 
                                 
                                  
                                 
                                   
                                     M 
                                     2 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         R 
                                         L 
                                       
                                       + 
                                       
                                         R 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             ) 
                           
                            
                           
                             ω 
                             0 
                             2 
                           
                            
                           
                             M 
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         0 
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             2 
                              
                             
                               R 
                               1 
                               ′ 
                             
                              
                             
                               
                                 R 
                                 L 
                               
                                
                               
                                 ( 
                                 
                                   
                                     R 
                                     L 
                                   
                                   + 
                                   
                                     R 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               ω 
                               0 
                               2 
                             
                              
                             
                               M 
                               2 
                             
                              
                             
                               R 
                               L 
                             
                           
                           - 
                           
                             
                               
                                 R 
                                 1 
                                 ′ 
                               
                                
                               
                                 ( 
                                 
                                   
                                     R 
                                     L 
                                   
                                   + 
                                   
                                     R 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             2 
                           
                           - 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             ω 
                             0 
                             2 
                           
                            
                           
                             
                               M 
                               2 
                             
                              
                             
                               ( 
                               
                                 
                                   R 
                                   L 
                                 
                                 + 
                                 
                                   R 
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         0 
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             2 
                              
                             
                               R 
                               1 
                               ′ 
                             
                              
                             
                               R 
                               L 
                               2 
                             
                           
                           + 
                           
                             2 
                              
                             
                               R 
                               1 
                               ′ 
                             
                              
                             
                               R 
                               2 
                             
                              
                             
                               R 
                               L 
                             
                           
                           + 
                           
                             
                               ω 
                               0 
                               2 
                             
                              
                             
                               M 
                               2 
                             
                              
                             
                               R 
                               L 
                             
                           
                           - 
                           
                             
                               R 
                               1 
                               ′ 
                             
                              
                             
                               R 
                               L 
                               2 
                             
                           
                           - 
                           
                             2 
                              
                             
                               R 
                               1 
                               ′ 
                             
                              
                             
                               R 
                               2 
                             
                              
                             
                               R 
                               L 
                             
                           
                           - 
                           
                             
                               R 
                               1 
                               ′ 
                             
                              
                             
                               R 
                               2 
                               2 
                             
                           
                           - 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               ω 
                               0 
                               2 
                             
                              
                             
                               M 
                               2 
                             
                              
                             
                               R 
                               L 
                             
                           
                           - 
                           
                             
                               ω 
                               0 
                               2 
                             
                              
                             
                               M 
                               2 
                             
                              
                             
                               R 
                               2 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         0 
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               ( 
                               
                                 
                                   1 
                                    
                                   
                                     R 
                                     1 
                                     ′ 
                                   
                                 
                                 - 
                                 
                                   R 
                                   1 
                                   ′ 
                                 
                               
                               ) 
                             
                              
                             
                               R 
                               L 
                               2 
                             
                           
                           - 
                           
                             
                               R 
                               1 
                               ′ 
                             
                              
                             
                               R 
                               2 
                               2 
                             
                           
                           - 
                           
                             
                               ω 
                               0 
                               2 
                             
                              
                             
                               M 
                               2 
                             
                              
                             
                               R 
                               2 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         0 
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       R 
                       L 
                       2 
                     
                     = 
                     
                       
                         
                           
                             R 
                             1 
                             ′ 
                           
                            
                           
                             R 
                             2 
                             2 
                           
                         
                         + 
                         
                           
                             ω 
                             0 
                             2 
                           
                            
                           
                             M 
                             2 
                           
                            
                           
                             R 
                             2 
                           
                         
                       
                       
                         R 
                         1 
                         ′ 
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     R 
                     L 
                   
                   = 
                   
                     
                       ± 
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     R 
                                     s 
                                   
                                   + 
                                   
                                     R 
                                     1 
                                   
                                 
                                 ) 
                               
                                
                               
                                 R 
                                 2 
                                 2 
                               
                             
                             + 
                             
                               
                                 ω 
                                 0 
                                 2 
                               
                                
                               
                                 M 
                                 2 
                               
                                
                               
                                 R 
                                 2 
                               
                             
                           
                           
                             ( 
                             
                               
                                 R 
                                 s 
                               
                               + 
                               
                                 R 
                                 1 
                               
                             
                             ) 
                           
                         
                       
                     
                     = 
                     
                       
                         ± 
                         
                           R 
                           2 
                         
                       
                       · 
                       
                         
                           
                             
                               ( 
                               
                                 
                                   R 
                                   s 
                                 
                                 + 
                                 
                                   R 
                                   1 
                                 
                               
                               ) 
                             
                             + 
                             
                               
                                 ω 
                                 0 
                                 2 
                               
                                
                               
                                 
                                   M 
                                   2 
                                 
                                 / 
                                 
                                   R 
                                   2 
                                 
                               
                             
                           
                           
                             ( 
                             
                               
                                 R 
                                 s 
                               
                               + 
                               
                                 R 
                                 1 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       R 
                       
                         L 
                         , 
                         opt 
                       
                     
                     = 
                     
                       
                         R 
                         2 
                       
                        
                       
                         
                           1 
                           + 
                           
                             
                               
                                 ω 
                                 0 
                                 2 
                               
                                
                               
                                 M 
                                 2 
                               
                             
                             
                               
                                 ( 
                                 
                                   
                                     R 
                                     s 
                                   
                                   + 
                                   
                                     R 
                                     1 
                                   
                                 
                                 ) 
                               
                                
                               
                                 R 
                                 2 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
         [0000]    Weak coupling condition ω 0   2 M 2 &lt;&lt;(R s +R 1 )R 2    
         [0000]    
       
      
       R 
       L,opt 
       ≅R 
       2  
      
     
       Conclusion 4) 
       [0000]    
       
         There exists an optimum R L &gt;R 2  maximising η
 
Output power P 2 :
 
       
     
         [0000]    
       
         
           
             
               P 
               2 
             
             = 
             
               
                 
                   X 
                   M 
                   2 
                 
                  
                 
                   R 
                   1 
                 
                  
                 
                   U 
                   s 
                   w 
                 
               
               
                 
                   
                     ( 
                     
                       
                         
                           R 
                           1 
                           ′ 
                         
                          
                         
                           R 
                           2 
                           ′ 
                         
                       
                       + 
                       
                         X 
                         M 
                         2 
                       
                     
                     ) 
                   
                   2 
                 
                 + 
                 
                   
                     R 
                     1 
                     ′2 
                   
                    
                   
                     
                       X 
                       2 
                       2 
                     
                      
                   
                 
                 + 
                 
                   
                     R 
                     2 
                     ′2 
                   
                    
                   
                     
                       X 
                       1 
                       2 
                     
                      
                   
                 
                 + 
                 
                   
                     
                       X 
                       1 
                       2 
                     
                      
                     
                       X 
                       2 
                       2 
                     
                   
                    
                 
                 + 
                 
                   2 
                    
                   
                     X 
                     M 
                     2 
                   
                    
                   
                     
                       
                         X 
                         2 
                       
                        
                       
                         X 
                         2 
                       
                     
                      
                   
                 
               
             
           
         
       
     
       Conclusion 5) 
       [0000]    
       
         Output power P 2 (X 1 , X 2 ) reaches maximum for 
       
     
         [0000]    
       
         
           
             
               X 
               1 
             
             = 
             
               
                 
                   ω 
                    
                   
                       
                   
                    
                   
                     L 
                     1 
                   
                 
                 - 
                 
                   1 
                   
                     ω 
                      
                     
                         
                     
                      
                     
                       C 
                       1 
                     
                   
                 
                 + 
                 
                   X 
                   s 
                 
               
               = 
               0 
             
           
         
       
       
         
           
             
               X 
               2 
             
             = 
             
               
                 
                   ω 
                    
                   
                       
                   
                    
                   
                     L 
                     2 
                   
                 
                 - 
                 
                   1 
                   
                     ω 
                      
                     
                         
                     
                      
                     
                       C 
                       2 
                     
                   
                 
                 + 
                 
                   X 
                   L 
                 
               
               = 
               0 
             
           
         
       
     
         [0000]    that is, when there is a resonance condition at both the primary and the secondary.
 
Output power P 2  wide resonance condition:
 
         [0000]    
       
         
           
             
               P 
               2 
             
             = 
             
               
                 
                   
                     ω 
                     0 
                     2 
                   
                    
                   
                     
                       M 
                       2 
                     
                     · 
                     
                       R 
                       L 
                     
                   
                 
                 
                   
                     [ 
                     
                       
                         
                           ( 
                           
                             
                               
                                 P 
                                 s 
                               
                                
                             
                             + 
                             
                               R 
                               1 
                             
                           
                           ) 
                         
                          
                         
                           ( 
                           
                             
                               R 
                               1 
                             
                             + 
                             
                               R 
                               2 
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           ω 
                           0 
                           2 
                         
                          
                         
                           M 
                           2 
                         
                       
                     
                     ] 
                   
                   2 
                 
               
               · 
               
                 U 
                 s 
                 2 
               
             
           
         
       
     
       Conclusion 6) 
       [0000]    
       
         To maximize P 2 , R S  should be R S &lt;&lt;R 1  
 
Output power P 2  for the wide resonance and weak coupling condition:
 
       
     
         [0000]    
       
         
           
             
               
                 ( 
                 
                   
                     R 
                     s 
                   
                   + 
                   
                     R 
                     1 
                   
                 
                 ) 
               
                
               
                 ( 
                 
                   
                     R 
                     L 
                   
                   + 
                   
                     R 
                     2 
                   
                 
                 ) 
               
             
             &gt;&gt; 
             
               
                 ω 
                 0 
                 2 
               
                
               
                 M 
                 2 
               
             
           
         
       
       
         
           
             
               P 
               2 
             
             ≅ 
             
               
                 
                   
                     ω 
                     0 
                     2 
                   
                    
                   
                     M 
                     2 
                   
                    
                   
                     R 
                     L 
                   
                 
                 
                   
                     
                       ( 
                       
                         
                           R 
                           s 
                         
                         + 
                         
                           R 
                           1 
                         
                       
                       ) 
                     
                     2 
                   
                    
                   
                     
                       ( 
                       
                         
                           R 
                           L 
                         
                         + 
                         
                           R 
                           2 
                         
                       
                       ) 
                     
                     1 
                   
                 
               
               · 
               
                 U 
                 s 
                 2 
               
             
           
         
       
     
       Conclusion 7) 
       [0000]    
       
         P 2 (R L ) reaches maximum for R L =R 2  (see conclusion 3) 
         For each of the above, the &gt;&gt; or &lt;&lt; can represent much greater, much less, e.g., 20× or 1/20 or less, or 50× or 1/50 th  or less or 100× or 1/100 th  or less.
 
The value R L  can also be optimized to maximize P 2 :
 
       
     
         [0000]    
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         
                            
                           
                             P 
                             2 
                           
                         
                         
                            
                           
                             R 
                             L 
                           
                         
                       
                       = 
                       0 
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       
                         
                           
                             u 
                             · 
                             
                               v 
                               ′ 
                             
                           
                           - 
                           
                             v 
                             · 
                             
                               u 
                               ′ 
                             
                           
                         
                         
                           v 
                           2 
                         
                       
                       = 
                       0 
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       
                         u 
                         = 
                         
                           
                             ω 
                             0 
                             2 
                           
                            
                           
                             M 
                             2 
                           
                            
                           
                             R 
                             L 
                           
                         
                       
                       ; 
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       
                         u 
                         ′ 
                       
                       = 
                       
                         
                           ω 
                           0 
                           2 
                         
                          
                         
                           M 
                           2 
                         
                       
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       v 
                       = 
                       
                         
                           [ 
                           
                             
                               
                                 ( 
                                 
                                   R 
                                   1 
                                   ′ 
                                 
                                 ) 
                               
                                
                               
                                 ( 
                                 
                                   
                                     R 
                                     L 
                                   
                                   + 
                                   
                                     R 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 ω 
                                 0 
                                 2 
                               
                                
                               
                                 M 
                                 2 
                               
                             
                           
                           ] 
                         
                         2 
                       
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       
                         v 
                         ′ 
                       
                       = 
                       
                         2 
                         · 
                         
                           [ 
                           
                             
                               
                                 R 
                                 1 
                                 ′ 
                               
                                
                               
                                 ( 
                                 
                                   
                                     R 
                                     L 
                                   
                                   + 
                                   
                                     R 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 ω 
                                 0 
                                 2 
                               
                                
                               
                                 M 
                                 2 
                               
                             
                           
                           ] 
                         
                         · 
                         
                           R 
                           1 
                           ′ 
                         
                       
                     
                      
                     
                       
 
                     
                      
                     
                       
                         
                           
                             ω 
                             0 
                             2 
                           
                            
                           
                             
                               M 
                               2 
                             
                             · 
                             
                               R 
                               L 
                             
                             · 
                             
                               2 
                                
                               
                                 [ 
                                 
                                   
                                     
                                       R 
                                       1 
                                       ′ 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           R 
                                           1 
                                         
                                         + 
                                         
                                           R 
                                           2 
                                         
                                       
                                       ) 
                                     
                                   
                                   + 
                                   
                                     
                                       ω 
                                       0 
                                       2 
                                     
                                      
                                     
                                       M 
                                       2 
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                            
                           
                             R 
                             1 
                           
                         
                         - 
                         
                           
                             
                               
                                 [ 
                                 
                                   
                                     
                                       R 
                                       1 
                                       ′ 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           R 
                                           L 
                                         
                                         + 
                                         
                                           R 
                                           2 
                                         
                                       
                                       ) 
                                     
                                   
                                   + 
                                   
                                     
                                       ω 
                                       0 
                                       2 
                                     
                                      
                                     
                                       M 
                                       2 
                                     
                                   
                                 
                                 ] 
                               
                               2 
                             
                             · 
                             
                               ω 
                               0 
                               2 
                             
                           
                            
                           
                             M 
                             2 
                           
                         
                       
                       = 
                       0 
                     
                      
                     
                       
 
                     
                      
                     
                       
                         
                           2 
                            
                           
                             
                               R 
                               L 
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     R 
                                     1 
                                     ′2 
                                   
                                    
                                   
                                     R 
                                     L 
                                   
                                 
                                 + 
                                 
                                   
                                     R 
                                     1 
                                     ′2 
                                   
                                    
                                   
                                     R 
                                     2 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           1 
                            
                           
                             R 
                             L 
                           
                            
                           
                             ω 
                             0 
                             2 
                           
                            
                           
                             
                               M 
                               2 
                             
                             · 
                             
                               R 
                               1 
                               ′ 
                             
                           
                         
                         - 
                         
                           
                             [ 
                             
                               
                                 
                                   R 
                                   1 
                                   ′ 
                                 
                                  
                                 
                                   R 
                                   L 
                                 
                               
                               + 
                               
                                 
                                   R 
                                   1 
                                   ′ 
                                 
                                  
                                 
                                   R 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   ω 
                                   0 
                                   2 
                                 
                                  
                                 
                                   M 
                                   2 
                                 
                               
                             
                             ] 
                           
                           2 
                         
                       
                       = 
                       0 
                     
                      
                     
                       
 
                     
                      
                     
                       
                         
                           2 
                            
                           
                             R 
                             1 
                             ′2 
                           
                            
                           
                             R 
                             L 
                             2 
                           
                         
                         + 
                         
                           2 
                            
                           
                             R 
                             1 
                             ′2 
                           
                            
                           
                             R 
                             2 
                           
                            
                           
                             R 
                             L 
                           
                         
                         + 
                         
                           2 
                            
                           
                             ω 
                             0 
                             2 
                           
                            
                           
                             M 
                             2 
                           
                            
                           
                             R 
                             1 
                             ′ 
                           
                            
                           
                             R 
                             L 
                           
                         
                         - 
                         
                           
                             R 
                             1 
                             ′2 
                           
                            
                           
                             R 
                             L 
                             2 
                           
                         
                         - 
                         
                           
                             R 
                             1 
                             ′2 
                           
                            
                           
                             R 
                             2 
                             2 
                           
                         
                         - 
                         
                           
                             ω 
                             0 
                             2 
                           
                            
                           
                             M 
                             4 
                           
                         
                         - 
                         
                           2 
                            
                           
                             R 
                             1 
                             ′2 
                           
                            
                           
                             R 
                             2 
                           
                            
                           
                             R 
                             L 
                           
                         
                         - 
                         
                           2 
                            
                           
                             R 
                             1 
                             ′ 
                           
                            
                           
                             ω 
                             0 
                             2 
                           
                            
                           
                             M 
                             2 
                           
                            
                           
                             R 
                             L 
                           
                         
                         - 
                         
                           2 
                            
                           
                             R 
                             1 
                             ′ 
                           
                            
                           
                             R 
                             2 
                           
                            
                           
                             ω 
                             0 
                             2 
                           
                            
                           
                             M 
                             2 
                           
                         
                       
                       = 
                       
                         0 
                         = 
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     2 
                                      
                                     
                                       R 
                                       1 
                                       ′2 
                                     
                                   
                                   - 
                                   
                                     R 
                                     1 
                                     ′2 
                                   
                                 
                                 ) 
                               
                                
                               
                                 R 
                                 L 
                                 2 
                               
                             
                             - 
                             
                               
                                 R 
                                 1 
                                 ′2 
                               
                                
                               
                                 R 
                                 2 
                                 2 
                               
                             
                             - 
                             
                               2 
                                
                               
                                 R 
                                 1 
                                 ′ 
                               
                                
                               
                                 R 
                                 2 
                               
                                
                               
                                 ω 
                                 0 
                                 2 
                               
                                
                               
                                 M 
                                 2 
                               
                             
                             - 
                             
                               
                                 ω 
                                 0 
                                 2 
                               
                                
                               
                                 M 
                                 4 
                               
                             
                           
                           = 
                           
                             0 
                             = 
                             
                               
                                 
                                   
                                     R 
                                     1 
                                     ′2 
                                   
                                   · 
                                   
                                     R 
                                     L 
                                     2 
                                   
                                 
                                 - 
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           R 
                                           1 
                                           ′ 
                                         
                                          
                                         
                                           R 
                                           2 
                                         
                                       
                                       + 
                                       
                                         
                                           ω 
                                           0 
                                           2 
                                         
                                          
                                         
                                           M 
                                           2 
                                         
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                               
                               = 
                               0 
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       R 
                       L 
                       2 
                     
                     = 
                     
                       
                         
                           ( 
                           
                             
                               
                                 R 
                                 1 
                                 ′ 
                               
                                
                               
                                 R 
                                 2 
                               
                             
                             + 
                             
                               
                                 ω 
                                 0 
                                 2 
                               
                                
                               
                                 M 
                                 2 
                               
                             
                           
                           ) 
                         
                         2 
                       
                       
                         R 
                         1 
                         ′2 
                       
                     
                   
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       R 
                       
                         L 
                         , 
                         opt 
                       
                     
                     = 
                     
                       
                         
                           
                             
                               R 
                               1 
                               ′ 
                             
                              
                             
                               R 
                               2 
                             
                           
                           + 
                           
                             
                               ω 
                               0 
                               2 
                             
                              
                             
                               M 
                               2 
                             
                           
                         
                         
                           R 
                           1 
                           ′ 
                         
                       
                       = 
                       
                         
                           R 
                           2 
                         
                          
                         
                           ( 
                           
                             1 
                             + 
                             
                               
                                 
                                   ω 
                                   0 
                                   2 
                                 
                                  
                                 
                                   M 
                                   2 
                                 
                               
                               
                                 
                                   ( 
                                   
                                     
                                       R 
                                       s 
                                     
                                     + 
                                     
                                       R 
                                       1 
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   R 
                                   2 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         
                           
                             R 
                             
                               L 
                               , 
                               opt 
                             
                           
                           = 
                           
                             
                               R 
                               2 
                             
                             · 
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   
                                     
                                       ω 
                                       0 
                                       2 
                                     
                                      
                                     
                                       M 
                                       2 
                                     
                                   
                                   
                                     
                                       ( 
                                       
                                         
                                           R 
                                           s 
                                         
                                         + 
                                         
                                           R 
                                           1 
                                         
                                       
                                       ) 
                                     
                                      
                                     
                                       R 
                                       2 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           Weak 
                            
                           
                               
                           
                            
                           coupling 
                            
                           
                             : 
                           
                         
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         
                           
                             R 
                             
                               L 
                               , 
                               opt 
                             
                           
                            
                           
                             &gt; 
                             ≅ 
                           
                            
                           
                             R 
                             2 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
       Conclusion 8) 
       [0000]    
       
         There exists an optimum R L &gt;R 2  maximizing P 2 . This R 1opt  differs from the R 1,opt  maximizing η. 
         One embodiment operates by optimizing one or more of these values, to foam an optimum value. 
       
     
         [0056]    Inductive coupling is shown with reference to  FIGS. 3 ,  4   
         [0000]      FIG. 5  illustrates the Inductance of a multi-turn circular loop coil 
         [0000]    
       
         
           
             
               R 
               m 
             
             = 
             
               
                 
                   R 
                   0 
                 
                 + 
                 
                   R 
                   1 
                 
               
               2 
             
           
         
       
       
         
           
             
               
                 
                   Wheeler 
                    
                   
                       
                   
                    
                   formula 
                    
                   
                       
                   
                    
                   
                     ( 
                     empirical 
                     ) 
                   
                 
               
               
                 
                   [ 
                   
                     Wheeler 
                     , 
                     
                       H 
                       . 
                       
                           
                       
                        
                       A 
                       . 
                     
                     , 
                     
                       “ 
                       
                         Simple 
                          
                         
                             
                         
                          
                         inductance 
                       
                     
                   
                 
               
             
             
               
                 
                   L 
                   = 
                   
                     
                       0.8 
                        
                       
                         
                           R 
                           m 
                           2 
                         
                         · 
                         
                           N 
                           2 
                         
                       
                     
                     
                       
                         6 
                          
                         
                           R 
                           m 
                         
                       
                       + 
                       
                         9 
                          
                         w 
                       
                       + 
                       
                         10 
                          
                         
                           ( 
                           
                             ( 
                             
                               
                                 R 
                                 o 
                               
                               + 
                               
                                 R 
                                 1 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   
                     
                       
                         formulas 
                          
                         
                             
                         
                          
                         for 
                          
                         
                             
                         
                          
                         radio 
                          
                         
                             
                         
                          
                         coils 
                       
                       ” 
                     
                     . 
                     
                         
                     
                      
                     Proc 
                     . 
                     
                         
                     
                      
                     I 
                   
                    
                   
                       
                   
                    
                   R 
                    
                   
                       
                   
                    
                   E 
                 
               
             
             
               
                 
                     
                 
               
               
                 
                   
                     
                       Vol 
                        
                       
                           
                       
                        
                       16 
                     
                     , 
                     
                       
                         pp 
                         . 
                         
                             
                         
                          
                         1328 
                       
                        
                       
                         - 
                       
                        
                       1400 
                     
                     , 
                     
                       Oct 
                       . 
                       
                           
                       
                        
                       1928. 
                     
                   
                   ] 
                 
               
             
             
               
                 
                   Note 
                    
                   
                     : 
                   
                    
                   
                       
                   
                    
                   this 
                    
                   
                       
                   
                    
                   i 
                    
                   
                       
                   
                    
                   accurate 
                    
                   
                       
                   
                    
                   if 
                    
                   
                       
                   
                    
                   all 
                    
                   
                       
                   
                    
                   three 
                    
                   
                       
                   
                    
                   terms 
                    
                   
                       
                   
                    
                   in 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   denominator 
                    
                   
                       
                   
                    
                   are 
                    
                   
                       
                   
                    
                   about 
                    
                   
                       
                   
                    
                   
                     equal 
                     . 
                   
                 
               
               
                 
                   
                     [ 
                     L 
                     ] 
                   
                    
                   μ 
                    
                   
                       
                   
                    
                   H 
                 
               
             
             
               
                 
                     
                 
               
               
                 
                   
                     [ 
                     
                       
                         R 
                         m 
                       
                       , 
                       
                         R 
                         i 
                       
                       , 
                       
                         R 
                         0 
                       
                       , 
                       ω 
                     
                     ] 
                   
                   = 
                   inch 
                 
               
             
             
               
                 
                   
                     Conversion 
                      
                     
                         
                     
                      
                     to 
                      
                     
                         
                     
                      
                     H 
                   
                   , 
                   
                     m 
                      
                     
                         
                     
                      
                     units 
                      
                     
                       : 
                     
                   
                 
               
               
                 
                   
                     1 
                      
                     
                         
                     
                      
                     m 
                   
                   = 
                   
                     
                       
                         
                           10 
                           
                              
                             ℘ 
                           
                         
                          
                         00 
                       
                       
                         - 
                         154 
                       
                     
                     · 
                     inch 
                   
                 
               
             
             
               
                 
                   L 
                   = 
                   
                     
                       0.8 
                        
                       
                         
                           R 
                           m 
                           2 
                         
                         · 
                         
                           N 
                           2 
                         
                       
                     
                     
                       
                         6 
                          
                         
                           
                             R 
                             m 
                           
                           · 
                           ℘ 
                         
                       
                       + 
                       
                         9 
                         · 
                         w 
                         · 
                         ℘ 
                       
                       + 
                       
                         10 
                          
                         
                           ( 
                           
                             
                               R 
                               0 
                             
                             - 
                             
                               R 
                               1 
                             
                           
                           ) 
                         
                          
                         ℘ 
                       
                     
                   
                 
               
               
                 
                   
                     
                       
                         
                           1 
                            
                           
                               
                           
                            
                           H 
                         
                         = 
                         
                           
                             10 
                             6 
                           
                            
                           
                               
                           
                            
                           μ 
                            
                           
                               
                           
                            
                           H 
                         
                       
                     
                   
                   
                     
                       
                         
                           [ 
                           L 
                           ] 
                         
                         = 
                         H 
                       
                     
                   
                 
               
             
             
               
                 
                   L 
                   = 
                   
                     
                       0.8 
                       · 
                       
                         R 
                         m 
                         2 
                       
                       · 
                       
                         ℘ 
                         2 
                       
                       · 
                       
                         N 
                         2 
                       
                       · 
                       
                         10 
                         
                           - 
                           6 
                         
                       
                     
                     
                       
                         6 
                          
                         
                           R 
                           m 
                         
                       
                       + 
                       
                         9 
                          
                         w 
                       
                       + 
                       
                         10 
                          
                         
                           ( 
                           
                             
                               R 
                               0 
                             
                             - 
                             
                               R 
                               1 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   
                     [ 
                     
                       
                         R 
                         m 
                       
                        
                       
                         R 
                         0 
                       
                        
                       
                         R 
                         1 
                       
                        
                       ω 
                     
                     ] 
                   
                   = 
                   m 
                 
               
             
           
         
       
     
         [0057]    In standard form: 
         [0000]    
       
         
           
             
               L 
               = 
               
                 
                   
                     μ 
                     0 
                   
                   · 
                   
                     A 
                     m 
                   
                   · 
                   
                     N 
                     2 
                   
                 
                 
                   K 
                   c 
                 
               
             
             ; 
           
         
       
       
         
           
             
               A 
               m 
             
             = 
             
               π 
               · 
               
                 R 
                 m 
                 2 
               
             
           
         
       
       
         
           
             
               μ 
               0 
             
             = 
             
               4 
                
               
                 π 
                 · 
                 
                   10 
                   
                     - 
                     7 
                   
                 
               
             
           
         
       
       
         
           
             L 
             = 
             
               
                 0.8 
                 · 
                 ℘ 
                 · 
                 
                   10 
                   
                     - 
                     6 
                   
                 
                 · 
                 
                   
                     
                       π 
                        
                       
                           
                       
                        
                       R 
                     
                      
                   
                   m 
                   2 
                 
                 · 
                 
                   N 
                   2 
                 
                 · 
                 
                   
                     4 
                      
                     
                       π 
                       · 
                       
                         10 
                         
                           - 
                           7 
                         
                       
                     
                   
                    
                 
               
               
                 
                   π 
                   · 
                   4 
                 
                  
                 
                   π 
                   · 
                   
                     10 
                     
                       - 
                       7 
                     
                   
                 
                  
                 
                   ( 
                   
                     
                       6 
                        
                       
                         R 
                         m 
                       
                     
                     + 
                     
                       9 
                        
                       w 
                     
                     + 
                     
                       10 
                        
                       
                         ( 
                         
                           
                             R 
                             0 
                           
                           - 
                           
                             R 
                             1 
                           
                         
                         ) 
                       
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             L 
             = 
             
               
                 
                   
                     
                       μ 
                       0 
                     
                      
                     
                       
                         A 
                         m 
                       
                       · 
                       
                         N 
                         2 
                       
                       · 
                       0.8 
                     
                      
                     
                       
                         ℘10 
                         
                           - 
                           6 
                         
                       
                       · 
                       10 
                     
                   
                   
                     4 
                      
                     
                       
                         π 
                         2 
                       
                       · 
                       
                         10 
                         
                           - 
                           7 
                         
                       
                     
                      
                     
                       ( 
                       
                         
                           6 
                            
                           
                             R 
                             m 
                           
                         
                         + 
                         
                           9 
                            
                           w 
                         
                         + 
                         
                           10 
                            
                           
                             ( 
                             
                               
                                 R 
                                 0 
                               
                               - 
                               
                                 R 
                                 1 
                               
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                 
                  
                 
                   1 
                   
                     K 
                     c 
                   
                 
                  
                 
                     
                 
                  
                 
                   R 
                   m 
                 
               
               = 
               
                 
                   
                     A 
                     m 
                   
                   π 
                 
               
             
           
         
       
       
         
           
             
               K 
               c 
             
             = 
             
               
                 
                   
                     π 
                     2 
                   
                   · 
                   25.4 
                 
                  
                 
                   ( 
                   
                     
                       6 
                        
                       
                         
                           
                             A 
                             m 
                           
                           π 
                         
                       
                     
                     + 
                     
                       9 
                        
                       w 
                     
                     + 
                     
                       10 
                        
                       
                         ( 
                         
                           
                             R 
                             0 
                           
                           - 
                           
                             R 
                             1 
                           
                         
                         ) 
                       
                     
                   
                   ) 
                 
               
               
                 2 
                 · 
                 1000 
               
             
           
         
       
       
         
           
             
               K 
               c 
             
             ≅ 
             
               
                 1 
                 8 
               
               · 
               
                 ( 
                 
                   
                     6 
                      
                     
                       
                         
                           A 
                           m 
                         
                         π 
                       
                     
                   
                   + 
                   
                     9 
                      
                     w 
                   
                   + 
                   
                     10 
                      
                     
                       ( 
                       
                         
                           R 
                           0 
                         
                         - 
                         
                           R 
                           1 
                         
                       
                       ) 
                     
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             
               L 
               = 
               
                 
                   
                     μ 
                     0 
                   
                    
                   
                     A 
                     m 
                   
                    
                   
                     N 
                     2 
                   
                 
                 
                   K 
                   c 
                 
               
             
             ; 
           
         
       
       
         
           
             
               A 
               m 
             
             = 
             
               
                 
                   
                     ( 
                     
                       
                         ( 
                         
                           
                             R 
                             0 
                           
                           + 
                           
                             R 
                             1 
                           
                         
                         ) 
                       
                       2 
                     
                     ) 
                   
                   2 
                 
                 · 
                 
                   π 
                    
                   
                     
 
                   
                   [ 
                   L 
                   ] 
                 
               
               = 
               H 
             
           
         
       
     
         [0058]    The inductance of a single-turn circular loop is given as: 
         [0000]    
       
         
           
             
               K 
               c 
             
             = 
             
               
                 
                   R 
                   m 
                 
                 · 
                 π 
               
               
                 ℘ 
                  
                 
                   [ 
                   
                     
                       
                         8 
                          
                         
                           R 
                           m 
                         
                       
                       6 
                     
                     - 
                     2 
                   
                   ] 
                 
               
             
           
         
       
       
         
           
             
               L 
               = 
               
                 
                   
                     μ 
                     0 
                   
                    
                   
                     A 
                     m 
                   
                 
                 
                   K 
                   c 
                 
               
             
             ; 
           
         
       
       
         
           
             
               A 
               m 
             
             = 
             
               
                 
                   R 
                   m 
                   2 
                 
                 · 
                 
                   π 
                    
                   
                     
 
                   
                   [ 
                   L 
                   ] 
                 
               
               = 
               H 
             
           
         
       
     
         [0000]    where: 
         [0059]    R m : mean radius in m 
         [0060]    b: wire radius in m, 
         [0061]    For a Numerical example: 
         [0062]    R 1 =0.13 m 
         [0063]    R 0 =0.14 m 
         [0064]    ω=0.01 m 
         [0065]    N=36 
         [0066]    →L=0.746 mH 
         [0067]    The measured inductance 
         [0068]    L meas =0.85 mH 
         [0069]    The model fraction of Wheeler formula for inductors of similar geometry, e.g, with similar radius and width ratios is: 
         [0000]    
       
         
           
             
               K 
               c 
             
             = 
             
               
                 1 
                 8 
               
                
               
                 ( 
                 
                   
                     5 
                      
                     
                       
                         
                           A 
                           m 
                         
                         π 
                       
                     
                   
                   + 
                   
                     9 
                      
                     w 
                   
                   + 
                   
                     10 
                      
                     
                       ( 
                       
                         
                           R 
                           0 
                         
                         - 
                         
                           R 
                           1 
                         
                       
                       ) 
                     
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             D 
             = 
             
               
                 
                   W 
                   2 
                 
                 + 
                 
                   
                     ( 
                     
                       
                         R 
                         0 
                       
                       - 
                       
                         R 
                         1 
                       
                     
                     ) 
                   
                   2 
                 
               
             
           
         
       
       
         
           
             
               R 
               m 
             
             = 
             
               
                 
                   R 
                   0 
                 
                 + 
                 
                   R 
                   1 
                 
               
               2 
             
           
         
       
     
         [0070]    Using a known formula from Goddam, V. R., which is valid for w&gt;(R 0 −R 1 ) 
         [0000]    
       
         
           
             L 
             = 
             
               
                 0.03193 
                 · 
                 
                   R 
                   m 
                 
                 · 
                 
                   N 
                   2 
                 
               
                
               
                 ⌊ 
                 
                   
                     2.303 
                      
                     
                       ( 
                       
                         1 
                         + 
                         
                           
                             w 
                             2 
                           
                           
                             32 
                              
                             
                               R 
                               m 
                               2 
                             
                           
                         
                         + 
                         
                           
                             D 
                             2 
                           
                           
                             96 
                              
                             
                               R 
                               m 
                               2 
                             
                           
                         
                       
                       ) 
                     
                      
                     
                       log 
                        
                       
                         ( 
                         
                           
                             8 
                              
                             
                               R 
                               m 
                             
                           
                           D 
                         
                         ) 
                       
                     
                   
                   - 
                   ℘ 
                   + 
                   
                     
                       w 
                       
                         1 
                         / 
                         2 
                       
                       2 
                     
                     
                       16 
                        
                       
                         R 
                         m 
                         2 
                       
                     
                   
                 
                 ⌋ 
               
             
           
         
       
     
         [0000]    1w H,m units: 
         [0000]    
       
         
           
             L 
             = 
             
               
                 μ 
                 0 
               
                
               
                 
                   R 
                   m 
                 
                 · 
                 
                   
                     N 
                     2 
                   
                    
                   
                     [ 
                     
                       
                         
                           ( 
                           
                             1 
                             + 
                             
                               
                                 w 
                                 2 
                               
                               
                                 32 
                                  
                                 
                                   R 
                                   m 
                                   2 
                                 
                               
                             
                             + 
                             
                               
                                 D 
                                 2 
                               
                               
                                 96 
                                  
                                 
                                   R 
                                   m 
                                   2 
                                 
                               
                             
                           
                           ) 
                         
                          
                         
                           ln 
                            
                           
                             ( 
                             
                               
                                 8 
                                  
                                 
                                   R 
                                   m 
                                 
                               
                               D 
                             
                             ) 
                           
                         
                       
                       - 
                       ℘ 
                       + 
                       
                         
                           
                             w 
                             2 
                           
                            
                           
                             y 
                             2 
                           
                         
                         
                           16 
                            
                           
                             R 
                             m 
                             2 
                           
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
       EXAMPLE 1 
       [0071]    
       
         
           
             
               R 
               1 
             
             = 
             
               0.13 
                
               
                   
               
                
               m 
             
           
         
       
       
         
           
             
               R 
               0 
             
             = 
             
               0.14 
                
               
                   
               
                
               m 
             
           
         
       
       
         
           
             W 
             = 
             
               0.01 
                
               
                   
               
                
               m 
             
           
         
       
       
         
           
             N 
             = 
             36 
           
         
       
       
         
           
             L 
             = 
             
               757 
                
               
                   
               
                
               μH 
             
           
         
       
       
         
           
             
               Ratio 
                
               
                 : 
               
                
               
                   
               
                
               
                 W 
                 
                   
                     R 
                     0 
                   
                   - 
                   
                     R 
                     1 
                   
                 
               
             
             = 
             
               
                 1 
                  
                 
                     
                 
                 -&gt; 
                 
                   y 
                   1 
                 
               
               = 
               0.8483 
             
           
         
       
       
         
           
             
               y 
               2 
             
             = 
             0.816 
           
         
       
     
       From [Terman, F.] 
     EXAMPLE 2 
     Given in [Goddam, V. R.] 
       [0072]    
       
         
           
             
               R 
               0 
             
             = 
             
               8.175 
                
               
                   
               
                
               inches 
             
           
         
       
       
         
           
             
               R 
               1 
             
             = 
             
               7.875 
                
               
                   
               
                
               inches 
             
           
         
       
       
         
           
             W 
             = 
             
               2 
                
               
                   
               
                
               inches 
             
           
         
       
       
         
           
             N 
             = 
             57 
           
         
       
       
         
           
             
               y 
               1 
             
             = 
             0.6310 
           
         
       
       
         
           
             
               y 
               2 
             
             = 
             
               
                 0.142 
                  
                 
                   
 
                 
                 -&gt; 
                 L 
               
               = 
               
                 2.5 
                  
                 
                     
                 
                  
                 mH 
                  
                 
                     
                 
                  
                 
                   ( 
                   
                     2.36 
                      
                     
                         
                     
                      
                     mH 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 Ratio 
                  
                 
                   : 
                 
                  
                 
                     
                 
                  
                 
                   2 
                   
                     
                       R 
                       0 
                     
                     - 
                     
                       R 
                       1 
                     
                   
                 
               
               = 
               
                 
                   2 
                   0.3 
                 
                 = 
                 6.667 
               
             
              
             
                 
             
           
         
       
       
         
           or 
         
       
       
         
           
             
               
                 
                   R 
                   0 
                 
                 - 
                 
                   R 
                   1 
                 
               
               W 
             
             = 
             
               
                 0.3 
                 2 
               
               = 
               0.15 
             
           
         
       
     
         [0073]    where Goddam, V. R. is the Thesis Masters Louisiana State University, 2005, and Terman, F. is the Radio Engineers Handbook, McGraw Hill, 1943. 
         [0074]    Any of these values can be used to optimize wireless power transfer between a source and receiver. 
         [0075]    From the above, it can be seen that there are really two different features to consider and optimize in wireless transfer circuits. A first feature relates to the way in which efficiency of power transfer is optimized. A second feature relates to maximizing the received amount of power—independent of the efficiency. 
         [0076]    One embodiment, determines both maximum efficiency, and maximum received power, and determines which one to use, and/or how to balance between the two. 
         [0077]    In one embodiment, rules are set. For example, the rules may specify: 
         [0078]    Rule 1—Maximize efficiency, unless power transfer will be less than 1 watt. If so, increase power transfer at cost of less efficiency. 
         [0079]    Rule 2—Maximize power transfer, unless efficiency becomes less than 30%. 
         [0080]    Any of these rules may be used as design rules, or as rules to vary parameters of the circuit during its operation. In one embodiment, the circuit values are adaptively changes based on operational parameters. This may use variable components, such as variable resistors, capacitors, inductors, and/or FPGAs for variation in circuit values. 
         [0081]    Although only a few embodiments have been disclosed in detail above, other embodiments are possible and the inventors intend these to be encompassed within this specification. The specification describes specific examples to accomplish a more general goal that may be accomplished in another way. This disclosure is intended to be exemplary, and the claims are intended to cover any modification or alternative which might be predictable to a person having ordinary skill in the art. For example, other sizes, materials and connections can be used. Other structures can be used to receive the magnetic field. In general, an electric field can be used in place of the magnetic field, as the primary coupling mechanism. Other kinds of antennas can be used. Also, the inventors intend that only those claims which use the-words “means for” are intended to be interpreted under 35 USC 112, sixth paragraph. Moreover, no limitations from the specification are intended to be read into any claims, unless those limitations are expressly included in the claims. 
         [0082]    Where a specific numerical value is mentioned herein, it should be considered that the value may be increased or decreased by 20%, while still staying within the teachings of the present application, unless some different range is specifically mentioned. Where a specified logical sense is used, the opposite logical sense is also intended to be encompassed.