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 
     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 
     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. 
     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. 
     An embodiment describes a high efficiency antenna for the specific characteristics and environment for the power being transmitted and received. 
     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. 
     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 
     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 
       In the Drawings: 
         FIG. 1  shows a diagram of a wireless power circuit; 
         FIG. 2  shows an equivalent circuit; 
         FIG. 3  shows a diagram of inductive coupling; 
         FIG. 4  shows a plot of the inductive coupling; and 
         FIG. 5  shows geometry of an inductive coil. 
     
    
    
     DETAILED DESCRIPTION 
       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. 
     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. 
       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 . 
     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 . 
     Considering these values, the equations for mutual inductance are as follows:
 
 U   2   =jωMI   1  
 
 U   1   =jωMI   2  
 
where:
 
               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             
The Mesh equations are:
 
     
       
         
           
             
                 
             
             ⁢ 
             
               
                 
                   
                     
                       
                         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 
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
     where: 
     Source power:
 
 P   1 =Re{ U   s   ·I*   1   }=U   s ·Re{ I*   1 }for avg{ U   s }=0
 
     Power into load:
 
 P   2   =I   2   ·I*   2 Re{ z   L   }=|I   2 | 2 ·Re{ z   L   }=|I   2 | 2   ·R   L  
 
     Transfer efficiency: 
     
       
         
           
             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 
                       
                       * 
                     
                   
                 
                 } 
               
             
           
         
       
     
     Overall transfer Efficiency is therefore: 
             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       =         (     jω   ⁢           ⁢   M     )     2     =       -     ω   2       ⁢     M   2               
A Transfer efficiency equation can therefore be expressed as:
 
                   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       )                                           
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
 
                 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:
 
             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 :
 
             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 :
 
             η   =         P   2       P   1       ≅           ω   0   2     ⁢     M   2         4   ⁢     R   1     ⁢     R   2         ⁢           ⁢     (     weak   ⁢           ⁢   coupling     )               
where:
 
Mutual inductance:
 
     M=k√{square root over (L 1 L 2 )} where k is the coupling factor 
     Loaded Q factors: 
               Q     1   ,   L       =           ω   ⁢           ⁢     L   1           R   s     +     R   1         ⁢           ⁢     Q     2   ,   L         =       ω   ⁢           ⁢     L   2           R   L     +     R   2                 
Therefore, the transfer efficiency in terms of these new definitions:
 
A) when ω=ω 0 
 
             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):
 
                                
D) ω=ω 0 , R L =R 2 , (R s =0)
 
               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     )                                     
Q UL : Q unloaded
 
                 Q     1   ,   UL       =       ω   ⁢           ⁢     L   1         R   1         ;       Q     2   ,   UL       =       ω   ⁢           ⁢     L   2         R   2               
This shows that the output power is a function of input voltage squared
 
     
       
         
           
             
               
                 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 
     
       
         
           
             
                 
             
             ⁢ 
             
               
                 
                   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 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
             
           
         
       
     
     Therefore, when at or near the resonance condition: 
               ω   =       ω   0     =       ω   2     =         ω   0     →     X   1       =   0           ,       X   2     =   0                   P   2     =           X   M   2     ⁢       R   L     ·     U   s   2               R   1   ′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                                   
Showing that the power transfer is inversely proportional to several variables, including series resistances.
 
     Mutual inductance in terms of coupling factors and inductions: 
     
       
         
           
             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 
                     
                   
                 
               
             
             
               
                 
                     
                 
               
             
           
         
       
     
     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: 
                 P     1   ,   max       =         U   s   2         R   s     +     R     in   ,   min           =     Re   ⁢     {       U   s     ·     I   1   *       }           ;         
R in,min : min. permissible input resistance
 
Efficiency relative to maximum input power:
 
               η   ′     =         P   2       P     1   ,   max         =         P   2     ⁡     (     U   s   2     )         P     1   ,   max                 
Under resonance condition ω=ω 1 =ω 2 =ω 0 :
 
                     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                                   
Equation for input power (P 1 ) under the resonance condition is therefore:
 
     
       
         
           
             
               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 
                   
                 
                 ) 
               
             
           
         
       
     
     The current ratio between input and induced currents can be expressed as 
                 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           
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)
 
               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                                   
Output current I 2 : (under resonance condition)
 
     
       
         
           
             
               
                 
                   
                     I 
                     2 
                   
                   = 
                   
                     
                       
                         jω 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         M 
                       
                       
                         
                           
                             ( 
                             
                               
                                 R 
                                 s 
                               
                               + 
                               
                                 R 
                                 1 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 R 
                                 L 
                               
                               + 
                               
                                 R 
                                 2 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             ω 
                             0 
                             2 
                           
                           ⁢ 
                           
                             M 
                             2 
                           
                         
                       
                     
                     · 
                     
                       U 
                       s 
                     
                   
                 
               
             
             
               
                 
                     
                 
               
             
           
         
       
     
     Maximizing transfer efficiency and output power (P 2 ) can be calculated according to the transfer efficiency equation: 
             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       )                     
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)
 
η(L 2 , C 2 , X L ) reaches maximum for
 
                 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:
 
             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)
 
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:
 
                 (       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 ):
 
                 ⅆ   η       ⅆ     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)
 
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
 
                           ⁢             ⅆ   η       ⅆ     R   L         =   0     ;     ⁢     
     ⁢           ⁢         ⅆ     ⅆ     R   L         ·         ω   0   2     ⁢     M   2     ⁢     R   L               (         R   s     +     R   1       ︸     )       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                                         
Weak coupling condition ω 0   2 M 2 &lt;&lt;(R s +R 1 )R 2  
 
 R   L,opt   ≅R   2  
 
Conclusion 4)
 
There exists an optimum R L &gt;R 2  maximising η
 
Output power P 2 :
 
               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)
 
Output power P 2 (X 1 , X 2 ) reaches maximum for
 
               X   1     =         ω   ⁢           ⁢     L   1       -     1     ω   ⁢           ⁢     C   1         +     X   s       =   0                   X   2     =         ω   ⁢           ⁢     L   2       -     1     ω   ⁢           ⁢     C   2         +     X   L       =   0           
that is, when there is a resonance condition at both the primary and the secondary.
 
Output power P 2  wide resonance condition:
 
               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)
 
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:
 
                 (       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)
 
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 :
 
                           ⁢             ⅆ     P   2         ⅆ     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)
 
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.
 
     Inductive coupling is shown with reference to  FIGS. 3 ,  4   
       FIG. 5  illustrates the Inductance of a multi-turn circular loop coil 
     
       
         
           
             
               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 
                       
                       · 
                       
                         ℘ 
                         2 
                       
                       · 
                       
                         N 
                         1 
                       
                       · 
                       
                         10 
                         
                           - 
                           6 
                         
                       
                     
                     
                       
                         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 
                 
               
             
           
         
       
     
     In standard form: 
     
       
         
           
             
               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 
             
           
         
       
     
     The inductance of a single-turn circular loop is given as: 
               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           
where:
 
     R m : mean radius in m 
     b: wire radius in m, 
     For a Numerical example: 
     R 1 =0.13 m 
     R 0 =0.14 m 
     ω=0.01 m 
     N=36 
     →L=0.746 mH 
     The measured inductance 
     L meas =0.85 mH 
     The model fraction of Wheeler formula for inductors of similar geometry, e.g, with similar radius and width ratios is: 
     
       
         
           
             
               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 
             
           
         
       
     
     Using a known formula from Goddam, V. R., which is valid for w&gt;(R 0 −R 1 ) 
             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           ⌋             
1w H,m units:
 
     
       
         
           
             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 
               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.] 
     
       
         
           
             
               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 
             
           
         
       
     
     where Goddam, V. R. is the Thesis Masters Louisiana State University, 2005, and Terman, F. is the Radio Engineers Handbook, McGraw Hill, 1943. 
     Any of these values can be used to optimize wireless power transfer between a source and receiver. 
     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. 
     One embodiment, determines both maximum efficiency, and maximum received power, and determines which one to use, and/or how to balance between the two. 
     In one embodiment, rules are set. For example, the rules may specify: 
     Rule 1—Maximize efficiency, unless power transfer will be less than 1 watt. If so, increase power transfer at cost of less efficiency. 
     Rule 2—Maximize power transfer, unless efficiency becomes less than 30%. 
     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. 
     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. 
     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.