Abstract:
A method for identifying impedance related parameters in a wireless power transfer (WPT) system including n coils is disclosed. Said method includes determining optimum values of the impedance related parameters based on a set of measured input impedance by applying an evolutionary algorithm to solve optimum solutions. Wherein the set of measured input impedance includes an input impedance vector {right arrow over (Z)}=(Z 1 , Z 2 , . . . , Z m-1 , Z m ), each input impedance in the vector (Z k ) measured at different frequencies f k , (k=1, 2, . . . m); and the impedance related parameters includes d ll+1  representing a distance between the l-th coil and the l+1 coil (l=1, 2, . . . n−1) and C i  representing a capacitance of the capacitor connected to the i-th coil (i=1, 2, . . . n).

Description:
REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application claims priority to and the benefit of U.S. Provisional Patent Application Ser. No. 61/862,627, which was filed Aug. 6, 2013, entitled Methods for Parameter Identification, Load Monitoring and Output Power Control for Wireless Power Transfer Systems, the entirety of which application is hereby incorporated by reference. 
     
    
     BACKGROUND OF THE INVENTION 
       [0002]    1. Technical Field 
         [0003]    The present disclosure relates to the control of the output power for wireless power transfer systems. 
         [0004]    2. Description of Related Art 
         [0005]    Wireless power transfer based on the magnetic resonance and near-field coupling of two loop resonators was reported by Nicola Tesla a century ago. N. Tesla, “Apparatus for transmitting electrical energy,” U.S. Pat. No. 1,119,732, (Dec. 1, 1914). As pioneered by Tesla, wireless power transfer can be radiative or non-radiative depending on the energy transfer mechanisms. Radiative power can be emitted from an antenna and propagates through a medium (such as vacuum or air) over long distance (i.e. many times larger than the dimension of the antenna) in the form of electromagnetic waves. However, due to the omni-directional nature of the radiative power emission, the energy efficiency of power transmission is very low. Non-radiative wireless power transfer relies on the near-field magnetic coupling of conductive loops and can be classified as short-range and mid-range applications. Herein the term mid-range applications refer to the situation where the transmission distance between the power source and the load is larger than the dimension of the coil-resonators. 
         [0006]    It should be noted that wireless power transfer has been applied extensively in ac machines, which were also pioneered by Tesla. See, Robert Lomas, “The man who invented the twentieth century—Nikola Tesla—Forgotten Genius of Electricity,” Headline (1999), ISBN 0 7472 6265 9, p. 146. Using a cage induction machine as an example, energy is transferred from the excited stator windings across the air gap to the rotor cage. Energy transfer via coupled windings is the basic principle used in electric machines. Therefore, wireless power systems can be mathematically described by electric circuit theory for magnetically coupled circuits. 
         [0007]    Wireless power transfer has been an active research topic for transcutaneous energy systems for medical implants since the 1960&#39;s. See, J. C. Schuder, H. E. Stephenson, and J. F. Townsend, “High level electromagnetic energy transfer through a closed chestwall,”  IRE Int. Conv. Rec., pt.  9, vol. 9, pp. 119-126, (1961); W. H. Ko, S. P. Liang, and C. D. F. Fung, “Design of rf-powered coils for implant instruments,”  Med. Biol. Eng. Comput ., vol. 15, pp. 634-640, (1977); E. Hochmair, “System optimization for improved acuracy in transcutaneous signal and power transmission”,  IEEE Trans. Biomedical Engineering , vol. BME-31, no. 2, pp. 177-186, (February 1984); B. Choi, J. Nho, H. Cha, T. Ahn and S. Choi, “Design and implementation of low-profile contactless battery charger using planar printed circuit board windings as energy transfer device,”  IEEE Trans. Industrial Electronics , vol. 51, no. 1, pp. 140-147, (February 2004); and Y. Jang and M. M. Jovanovic, “A contactless electrical energy transmission system for portable-telephone battery chargers”,  IEEE Trans. Industrial Electronics , vol. 50, no. 3, pp. 520-527, (June 2003). 
         [0008]    This research has also involved induction heaters since the 1970&#39;s. See, W. G. Hurley and J. Kassakian, “Induction heating of circular ferromagnetic plates”,  IEEE Trans. Magnetics , vol. 15, no. 4, pp. 1174-1181, (July 1979). For modern short-range applications, the inductive power transfer (IPT) systems have attracted much attention since the 1990&#39;s. A. W. Green and J. T. Boys, “10 kHz inductively coupled power transfer-concept and control,”  Proc. ICPE - VSD , (1994), pp. 694-699; J. T. Boys, G. A. Covic and A. W. Green, “Stability and control of inductively coupled power transfer systems”,  Proc. Electric Power Applications , (2000), vol. 147, no. 1, pp. 37-43; J. T. Boys, A. P. Hu and G. A. Covic, “Critical Q analysis of a current-fed resonant converter for ICPT applications,”  Electronics Letters , vol. 36, no. 17, pp. 1440-1442, (2000); G. A. J. Elliott, G. A. Covic, D. Kacprzak and J. T. Boys, “A New Concept: Asymmetrical Pick-Ups for Inductively Coupled Power Transfer Monorail Systems,”  IEEE Trans. Magnetics,  vol. 42, no. 10, pp. 3389-3391, (2006); and M. L. G. Kissin, J. T. Boys and G. A. Covic, “Interphase Mutual Inductance in Polyphase Inductive Power Transfer Systems,”  IEEE Trans. Industrial Electronics , vol. 56, no. 7, pp. 2393-2400, (2009). The wireless charging systems for portable equipment, such as mobile phones have attracted much attention since the 2000&#39;s. B. Choi, J. Nho, H. Cha, T. Ahn and S. Choi, “Design and implementation of low-profile contactless battery charger using planar printed circuit board windings as energy transfer device,”  IEEE Trans. Industrial Electronics , vol. 51, no. 1, pp. 140-147, (February 2004); Y. Jang and M. M. Jovanovic, “A contactless electrical energy transmission system for portable-telephone battery chargers,”  IEEE Trans. Industrial Electronics , vol. 50, no. 3, pp. 520-527, (June 2003); C.-G. Kim, D. H. Seo, J. S. You, J. H. Park and B. H. Cho, “Design of a contactless battery charger for cellular phone,”  IEEE Trans. Industrial Electronics , vol. 48, no. 6, pp. 1238-1247, (December 2001); S. Y. R. Hui and W. C. Ho, “A new generation of universal contactless battery charging platform for portable Consumer Electronic equipment,”  IEEE Trans. Power Electronics , vol. 20, no. 3, pp. 620-627, (May 2005); X. Liu and S. Y. R. Hui, “Simulation Study and Experimental Verification of a Contactless Battery Charging Platform with Localized Charging Features,”  IEEE Trans. Power Electronics , vol. 22, no. 6, pp. 2202-2210, (November 2007); and S. Y. R. Hui, “Planar Inductive Battery Charging System”, U.S. Pat. No. 7,576,514, 2009. Wireless charging technology for portable electronic devices has reached the commercialization stage through the launch of the “Qi” Standard by the Wireless Power Consortium, now comprising over 135 companies worldwide. See the Wireless Power Consortium Website, available at: http://www.wirelesspowerconsortium.com. 
         [0009]    The launch of the world first wireless power standard Qi by the Wireless Power Consortium for portable electronics products has sped up research and development activities in wireless power transfer. Recent wireless power research activities focus on both short-range applications and mid-range applications. In general, wireless power transfer systems (WPTS) can be classified as 2-coil systems, 4-coil systems, systems with relay resonators and wireless power domino-resonator systems. S. Y. R. Hui, W. X. Zhong and C. K. Lee, “A critical review on recent progress of mid-range wireless power transfer,”  IEEE Transactions on Power Electronics  (in press) 
         [0010]    At present, a great deal of research is focused on improving the performance of wireless power transfer systems in order to increase the transfer distance, improve the efficiency and widen the operating frequency. All of these purposes are based on one thing, a well-known wireless power transfer system in which we know the topology of the system, all of the parameters of the components, the characteristics of the load, and the positions and directions of each coil. If so, it is easy to find the maximum efficiency operating point or maximum power transfer point, or the optimal operating point for other purposes. The biggest difficulty is how to find out the exact value of all the parameters of a given system, since as an extremely high order system, the slight difference between the predicted value and the real value of the parameters may lead to totally different performance at a given operating point, and some of these parameters cannot be measured precisely in an easy way. 
         [0011]    On the other hand, even if we know all the parameters of the WPT system, the load may be dynamic and will change at any time. In order to operate the system always at the optimal point, monitoring of the impedance of the load at real time is required. Previously, a research team reported the use of a wireless communication method to transmit the load conditions as feedback information to the input power controller. N. Y. Kim, K. Y. Kim, J. Choi and C. W. Kim, “Adaptive frequency with power-level tracking system for efficient magnetic resonance wireless power transfer,”  Electronics Letters , Vol. 48, No. 8, (April 2012), page(s): 452-454. This is a traditional approach that needs a wireless communication system, which may increase the cost of such overall system. In addition, the reported method does not involve any system parameter identification. 
       SUMMARY OF THE INVENTION 
       [0012]    The present disclosure is directed methods for identifying the system parameters and monitoring (including controlling the power of) the load of a wireless power transfer system without using any wired or wireless communication system from the load side for feedback control. 
         [0013]    According to the first aspect of the present disclosure, it provides a method for identifying impedance related parameters in a wireless power transfer (WPT) system including n coils, including: determining optimum values of the impedance related parameters based on a set of measured input impedance by applying an evolutionary algorithm to solve optimum solutions. Wherein the set of measured input impedance includes an input impedance vector {right arrow over (Z)}=(Z 1 , Z 2 , . . . , Z m-1 , Z m ), each input impedance in the vector (Z k ) measured at different frequencies f k , (k=1, 2, m); and the impedance related parameters includes d ll+1  representing a distance between the l-th coil and the l+1 coil (l=1, 2, n−1) and C i  representing a capacitance of the capacitor connected to the i-th coil (i=1, 2, . . . n). 
         [0014]    According to the second aspect of the present disclosure, it provides a method for monitoring a load Z L  in a wireless power transfer (WPT) system including n coils, comprising: sensing an input voltage U 1  and an input current I 1  of a transmitter coil of the n coils; determining a set of measured input impedance based on the sensed input voltage U 1  and the sensed input current I 1 ; identifying impedance related parameters according to the method of the first aspect of the present disclosure; and determining the load only based on the sensed input voltage U 1 , the sensed input current I 1 , and the identified impedance related parameters. 
         [0015]    According to the third aspect of the present disclosure, it provides a method for controlling output power in a wireless power transfer (WPT) system including n coils, comprises: (a) sensing an input voltage U 1  and an input current I 1  of a transmitter coil of the n coils; (b) determining whether impedance related parameters in the WPT system are known, wherein the impedance related parameters includes d ll+1  representing the distance between the l-th coil and the l+1 coil (l=1, 2, . . . , n−1) and C n  representing the capacitance of the capacitor connected to the i-th coil (i=1, 2, . . . n): (b.1) if the result of the step of determining is yes, the method further comprises: based on the sensed input voltage U 1 , the sensed input current I 1 , and the known impedance related parameters, estimating load Z L  of the WPT system, output current of the WPT system I n , power P out  of the load Z L , output voltage U o  of the WPT system, and efficiency η of the WPT system according to the method of the second aspect of the present disclosure; (b.2) if the result of the step of determining is no, the method further comprises: determining a set of measured input impedance based on the sensed input voltage U 1  and the sensed input current I 1 , identifying impedance related parameters in the WPT system according to one of the method of the first aspect of the present disclosure, based on the sensed input voltage U 1 , the sensed input current I 1 , and the identified impedance related parameters, estimating load Z L  of the WPT system, output current of the WPT system I n , power P out  of the load Z L , output voltage U o  of the WPT system, and efficiency η of the WPT system according to the method of the second aspect of the present disclosure; (c) generating feedback information based on the estimated parameters in step (b.1) or (b.2); and (d) controlling the operations of the transmitter coil based on the generated feedback information. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0016]    The foregoing and other features of the present disclosure will be more readily apparent from the following detailed description and drawings of illustrative embodiments of the disclosure wherein like reference numerals refer to like parts throughout the various figures unless otherwise specified and in which: 
           [0017]      FIG. 1  is a schematic of an n-Ring system for wireless power transfer; 
           [0018]      FIGS. 2( a ) and 2( b )  are graphs of experimental and simulation results for the input impedance of coil arrangements No.  1  and No.  2 , respectively, in Table 1 for 3-coil domino systems; 
           [0019]      FIGS. 3( a ) and 3( b )  are graphs of experimental and simulation results for the input impedance of coil arrangements No.  3  and No.  4 , respectively, in Table 1 for 3-coil domino systems; 
           [0020]      FIGS. 4( a ) and 4( b )  are graphs of experimental and simulation results for the input impedance of coil arrangements No.  1  and No.  2 , respectively, in Table 2 for 8-coil domino systems; 
           [0021]      FIGS. 5( a ) and 5( b )  are graphs of experimental and simulation results for the input impedance of coil arrangements No.  3  and No.  4 , respectively, in Table 2 for 8-coil domino systems; 
           [0022]      FIG. 6  is a schematic of a loaded resonator with the load connected in parallel with the resonant capacitor; 
           [0023]      FIG. 7  is a schematic of a control system for a WPTS with known system parameters; and 
           [0024]      FIG. 8  is a schematic of a control system for a WPTS with unknown system parameters. 
       
    
    
     DETAILED DESCRIPTION 
       [0025]    One aspect of this disclosure focuses on parameter identification for a wireless power transfer system. 
         [0026]    For a domino wireless power transfer system with n-coils as shown in  FIG. 1 , if we assume L i  is the self-inductance of the i th  coil, and M ij  is mutual-inductance between the i th  coil and the j th  coil, then the system could be described using equation (1). Note in  FIG. 1  that the load resistor is connected in series with the last coil-resonator in the following formulation. But the principle of this disclosure can also apply to the case where the load is connected across the capacitor of the last LC resonator. 
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         [0027]    A. Parameter Identification 
         [0028]    If we know all the parameters in the matrix in equation (1), then we can calculate the input impedance of the system for each frequency along the frequency axis, i.e., we can get a set of impedance values at different frequencies: Z f1 , Z f2 , . . . Z fm , where f i  is one of the different frequencies, and Z fi  is the input impedance at f i . Then we have equation (2): 
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         [0029]    Since the coils in our domino system are identical to each other and the self-inductance could be accurately calculated, we can treat L 1  through L n , and the coil resistance, R 1  through R n  as constant values. Meanwhile, we can treat the mutual inductances M 12 , M 23 , M (n−1)n  as functions of the distances between each coil pair, d 12 , d 23 , . . . , d (n−1)n ). Then equation (2) may be replaced by equation (3) as follows: 
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                               , 
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                                
                               
                                   
                               
                               , 
                               
                                 C 
                                 
                                   n 
                                   - 
                                   1 
                                 
                               
                               , 
                               
                                 C 
                                 n 
                               
                               , 
                             
                           
                         
                         
                           
                             
                               R 
                               load 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0000]    For a given system, the input impedance set (Z 1 , Z 2 , . . . , Z m-1 , Z m ) could easily be measured experimentally. 
         [0030]    Genetic Algorithm (GA) approach is used to search for the optimal values of d 12 , d 23 , . . . , d (n-1)n , C 1 , C 2 , . . . , C (n-1) , C n  and R load . With the help of a set of measured input impedance (Z 1 , Z 2 , . . . , Z m-1 , Z m ) at different frequencies, the GA can be used to obtain the optimum solutions for the following optimum problem: 
         [0000]    
       
         
           
             
               
                 
                   
                     J 
                      
                     
                       ( 
                       
                         
                           
                             
                               
                                 d 
                                 12 
                               
                               , 
                               
                                 d 
                                 23 
                               
                               , 
                               … 
                                
                               
                                   
                               
                               , 
                               
                                 d 
                                 
                                   
                                     ( 
                                     
                                       n 
                                       - 
                                       2 
                                     
                                     ) 
                                   
                                    
                                   
                                     ( 
                                     
                                       n 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                               , 
                               
                                 d 
                                 
                                   
                                     ( 
                                     
                                       n 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                    
                                   n 
                                 
                               
                               , 
                             
                           
                         
                         
                           
                             
                               
                                 C 
                                 1 
                               
                               , 
                               
                                 C 
                                 2 
                               
                               , 
                               … 
                                
                               
                                   
                               
                               , 
                               
                                 C 
                                 
                                   n 
                                   - 
                                   1 
                                 
                               
                               , 
                               
                                 C 
                                 n 
                               
                               , 
                             
                           
                         
                         
                           
                             
                               R 
                               load 
                             
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     min 
                      
                     
                       [ 
                       
                         
                           ∑ 
                           1 
                           m 
                         
                          
                         
                           ( 
                           
                             
                               
                                 ( 
                                 
                                   
                                      
                                     
                                       Z 
                                       i 
                                     
                                      
                                   
                                   - 
                                   
                                      
                                     
                                       Z 
                                       i 
                                       * 
                                     
                                      
                                   
                                 
                                 ) 
                               
                               2 
                             
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     θ 
                                     i 
                                   
                                   - 
                                   
                                     θ 
                                     i 
                                     * 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                           ) 
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Where |Z i | and θ i  are magnitude and angular value of the measured input impedance, while |Z i *| and θ i * are the simulated magnitude and angular value of the input impedance. 
         [0031]    In a representative example of the present disclosure for identifying the parameters for wireless power transfer system by using evolutionary algorithm, a 3-coil-domino system and an 8-coil-domido system were used. For each domino system, different coil distance and different load resistances were used to check whether this method could be used at different conditions. The different experimental conditions are listed in Table 1 and Table 2. The simulation results are listed in Table 3 and Table 4. 
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 experiments for 3-coil-domino system 
               
             
          
           
               
                 No. 
                 c1 
                 c2 
                 c3 
                 d12 
                 d23 
                 Rload 
               
               
                   
               
               
                 1 
                 #1 
                 #2 
                 #3 
                 0.20 m 
                 0.20 m 
                 11.8 Ohm 
               
               
                 2 
                 #1 
                 #2 
                 #3 
                 0.20 m 
                 0.25 m 
                 11.8 Ohm 
               
               
                 3 
                 #1 
                 #2 
                 #3 
                 0.20 m 
                 0.20 m 
                 49.9 Ohm 
               
               
                 4 
                 #1 
                 #2 
                 #3 
                 0.20 m 
                 0.25 m 
                 49.9 Ohm 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 2 
               
             
             
               
                   
               
               
                 experiments for 8-coil-domino system 
               
             
          
           
               
                   
                 c1 
                 c2 
                 c3 
                 c4 
                 c5 
                 c6 
                 c7 
                 c8 
               
               
                   
                   
               
             
          
           
               
                 No. 1 
                 #4 
                 #5 
                 #6 
                 #7 
                 #8 
                 #9 
                 #10 
                 #11 
               
               
                   
                 d12 
                 d23 
                 d34 
                 d45 
                 d56 
                 d67 
                 d78 
                 Rload 
               
               
                   
                 0.15 m 
                 0.15 m 
                 0.15 m 
                 0.15 m 
                 0.15 m 
                 0.15 m 
                 0.15 m 
                 11.8 Ohm 
               
               
                 No. 2 
                 #4 
                 #5 
                 #6 
                 #7 
                 #8 
                 #9 
                 #10 
                 #11 
               
               
                   
                 d12 
                 d23 
                 d34 
                 d45 
                 d56 
                 d67 
                 d78 
                 Rload 
               
               
                   
                 0.15 m 
                 0.15 m 
                 0.15 m 
                 0.15 m 
                 0.15 m 
                 0.15 m 
                 0.25 m 
                 11.8 Ohm 
               
               
                 No. 3 
                 #4 
                 #5 
                 #6 
                 #7 
                 #8 
                 #9 
                 #10 
                 #11 
               
               
                   
                 d12 
                 d23 
                 d34 
                 d45 
                 d56 
                 d67 
                 d78 
                 Rload 
               
               
                   
                 0.15 m 
                 0.15 m 
                 0.15 m 
                 0.15 m 
                 0.15 m 
                 0.15 m 
                 0.15 m 
                 49.9 Ohm 
               
               
                 No. 4 
                 #4 
                 #5 
                 #6 
                 #7 
                 #8 
                 #9 
                 #10 
                 #11 
               
               
                   
                 d12 
                 d23 
                 d34 
                 d45 
                 d56 
                 d67 
                 d78 
                 Rload 
               
               
                   
                 0.15 m 
                 0.15 m 
                 0.15 m 
                 0.15 m 
                 0.15 m 
                 0.15 m 
                 0.25 m 
                 49.9 Ohm 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
             
           
               
                 TABLE 3 
               
             
             
               
                   
               
               
                 simulation results for 3-coil-domino system 
               
             
          
           
               
                   
                 No. 
                 c1 
                 c2 
                 c3 
                 d12 
                 d23 
               
               
                   
                   
               
               
                   
                 1 
                 1.0191 
                 1.0040 
                 1.0197 
                 0.2021 
                 0.1984 
               
               
                   
                 2 
                 1.0180 
                 1.0040 
                 1.0195 
                 0.2016 
                 0.2494 
               
               
                   
                 3 
                 1.0192 
                 1.0031 
                 1.0215 
                 0.1890 
                 0.2167 
               
               
                   
                 4 
                 1.0192 
                 1.0036 
                 1.0211 
                 0.2042 
                 0.2531 
               
               
                   
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 4 
               
             
             
               
                   
               
               
                 simulation results for 8-coil-domino system 
               
             
          
           
               
                   
                 c1 
                 c2 
                 c3 
                 c4 
                 c5 
                 c6 
                 c7 
                 c8 
               
               
                   
                   
               
             
          
           
               
                 No. 1 
                 1.0166 
                 1.0187 
                 1.0187 
                 1.0174 
                 0.9695 
                 1.0080 
                 1.0026 
                 1.0233 
               
               
                   
                 d12 
                 d23 
                 d34 
                 d45 
                 d56 
                 d67 
                 d78 
               
               
                   
                 0.1523 
                 0.1480 
                 0.1413 
                 0.1571 
                 0.1504 
                 0.1477 
                 0.1520 
               
               
                 No. 2 
                 1.0116 
                 1.0170 
                 1.0202 
                 1.0203 
                 0.9700 
                 1.0081 
                 1.0062 
                 1.0177 
               
               
                   
                 d12 
                 d23 
                 d34 
                 d45 
                 d56 
                 d67 
                 d78 
               
               
                   
                 0.1529 
                 0.1493 
                 0.1414 
                 0.1586 
                 0.1494 
                 0.1460 
                 0.2493 
               
               
                 No. 3 
                 1.0183 
                 1.0184 
                 1.0160 
                 1.0190 
                 0.9670 
                 1.0092 
                 1.0044 
                 1.0124 
               
               
                   
                 d12 
                 d23 
                 d34 
                 d45 
                 d56 
                 d67 
                 d78 
               
               
                   
                 0.1464 
                 0.1516 
                 0.1530 
                 0.1488 
                 0.1442 
                 0.1487 
                 0.1468 
               
               
                 No. 4 
                 1.0165 
                 1.0160 
                 1.0143 
                 1.0121 
                 0.9701 
                 1.0191 
                 1.0099 
                 1.0024 
               
               
                   
                 d12 
                 d23 
                 d34 
                 d45 
                 d56 
                 d67 
                 d78 
               
               
                   
                 0.1471 
                 0.1513 
                 0.1519 
                 0.1465 
                 0.1482 
                 0.1495 
                 0.2506 
               
               
                   
               
             
          
         
       
     
         [0032]      FIG. 2  and  FIG. 3  are the experimental and simulation input impedance comparison for a 3-coil-domino system.  FIG. 4  and  FIG. 5  are the experimental and simulated input impedance comparison for 8-coil-domino system. Also the simulated efficiencies are plotted in these figures. In particular,  FIGS. 2( a ) and 2( b )  show the experimental and simulation input impedance comparison of coil parameters No. 1 and No. 2, respectively, from the experiment in Table 1 for the 3-coil-domino systems.  FIGS. 3( a ) and 3( b )  show the experimental and simulation input impedance comparison of coil parameters No. 3 and No. 4, respectively, from the experiment in Table 1 for the 3-coil-domino systems.  FIGS. 4( a ) and 4( b )  show the experimental and simulation input impedance comparison of coil parameters No. 1 and No. 2, respectively, from the experiment in Table 2 for the 8-coil-domino systems. Finally,  FIGS. 5( a ) and 5( b )  show the experimental and simulation input impedance comparison of coil parameters No. 3 and No. 4, respectively, from the experiment in Table 2 for the 8-coil-domino systems. 
         [0033]    B. Load Monitoring without Direct Output Information Feedback 
         [0034]    Another aspect of the present disclosure focuses on load monitoring for wireless power transfer system. 
         [0035]    In equation (1), since all the parameters, L 1  through L n , R 1  through R n , M 12 , M 23 , . . . M (n-1)n  and C 1 , C 2 , . . . , C n  are identified in the first aspect, the load impedance R 1  is the only obstacle which prevents simulation of the system and finding the optimal operating point, such as maximum power transfer or maximum efficiency of the system for a given load. It is not possible to assume the load always has the same impedance. Thus, the load must be continually sensed and the load impedance value renewed for calculating a new operating frequency in order to ensure the system always operates at the optimal point. 
         [0036]    An easy way to solve this problem involves rewriting equation (1) as equation (5) and equation set (6), and then equation set (7), where Z ij  is the function of R 1  through R n , M 12 , M 23 , . . . , M (n-1)n , and C 1 , C 2 , . . . , C n , Z L  is the impedance of the load at a given frequency. 
         [0000]    
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             U 
                             1 
                           
                         
                       
                       
                         
                           0 
                         
                       
                       
                         
                           ⋮ 
                         
                       
                       
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               Z 
                               11 
                             
                           
                           
                             
                               Z 
                               12 
                             
                           
                           
                             … 
                           
                           
                             
                               Z 
                               
                                 1 
                                  
                                 
                                   ( 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               Z 
                               
                                 1 
                                  
                                 n 
                               
                             
                           
                         
                         
                           
                             
                               Z 
                               21 
                             
                           
                           
                             
                               Z 
                               22 
                             
                           
                           
                             … 
                           
                           
                             
                               Z 
                               
                                 2 
                                  
                                 
                                   ( 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               Z 
                               
                                 2 
                                  
                                 n 
                               
                             
                           
                         
                         
                           
                             ⋮ 
                           
                           
                             ⋮ 
                           
                           
                             ⋱ 
                           
                           
                             ⋮ 
                           
                           
                             ⋮ 
                           
                         
                         
                           
                             
                               Z 
                               
                                 
                                   ( 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                                  
                                 1 
                               
                             
                           
                           
                             
                               Z 
                               
                                 
                                   ( 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                                  
                                 2 
                               
                             
                           
                           
                             … 
                           
                           
                             
                               Z 
                               
                                 
                                   ( 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                                  
                                 
                                   ( 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               Z 
                               
                                 
                                   ( 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                                  
                                 n 
                               
                             
                           
                         
                         
                           
                             
                               Z 
                               
                                 n 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                           
                           
                             
                               Z 
                               
                                 n 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                           
                           
                             … 
                           
                           
                             
                               Z 
                               
                                 n 
                                  
                                 
                                   ( 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 Z 
                                 nn 
                               
                               + 
                               
                                 Z 
                                 L 
                               
                             
                           
                         
                       
                       ] 
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               I 
                               1 
                             
                           
                         
                         
                           
                             
                               I 
                               2 
                             
                           
                         
                         
                           
                             ⋮ 
                           
                         
                         
                           
                             
                               I 
                               
                                 n 
                                 - 
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               I 
                               n 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       U 
                       1 
                     
                     = 
                     
                       
                         
                           Z 
                           11 
                         
                          
                         
                           I 
                           1 
                         
                       
                       + 
                       
                         
                           Z 
                           12 
                         
                          
                         
                           I 
                           2 
                         
                       
                       + 
                       … 
                       + 
                       
                         
                           Z 
                           
                             1 
                              
                             
                               ( 
                               
                                 n 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                         
                          
                         
                           I 
                           
                             ( 
                             
                               n 
                               - 
                               1 
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           Z 
                           
                             1 
                              
                             n 
                           
                         
                          
                         
                           I 
                           n 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     0 
                     = 
                     
                       
                         
                           Z 
                           21 
                         
                          
                         
                           I 
                           1 
                         
                       
                       + 
                       
                         
                           Z 
                           22 
                         
                          
                         
                           I 
                           2 
                         
                       
                       + 
                       … 
                       + 
                       
                         
                           Z 
                           
                             2 
                              
                             
                               ( 
                               
                                 n 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                         
                          
                         
                           I 
                           
                             ( 
                             
                               n 
                               - 
                               1 
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           Z 
                           
                             2 
                              
                             n 
                           
                         
                          
                         
                           I 
                           n 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   ⋮ 
                    
                   
                     
 
                   
                    
                   
                     0 
                     = 
                     
                       
                         
                           Z 
                           
                             
                               ( 
                               
                                 n 
                                 - 
                                 1 
                               
                               ) 
                             
                              
                             1 
                           
                         
                          
                         
                           I 
                           1 
                         
                       
                       + 
                       
                         
                           Z 
                           
                             
                               ( 
                               
                                 n 
                                 - 
                                 1 
                               
                               ) 
                             
                              
                             2 
                           
                         
                          
                         
                           I 
                           2 
                         
                       
                       + 
                       … 
                       + 
                       
                         
                           Z 
                           
                             
                               ( 
                               
                                 n 
                                 - 
                                 1 
                               
                               ) 
                             
                              
                             
                               ( 
                               
                                 n 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                         
                          
                         
                           I 
                           
                             ( 
                             
                               n 
                               - 
                               1 
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           Z 
                           
                             
                               ( 
                               
                                 n 
                                 - 
                                 1 
                               
                               ) 
                             
                              
                             n 
                           
                         
                          
                         
                           I 
                           n 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     0 
                     = 
                     
                       
                         
                           Z 
                           
                             n 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                          
                         
                           I 
                           1 
                         
                       
                       + 
                       
                         
                           Z 
                           
                             n 
                              
                             
                                 
                             
                              
                             2 
                           
                         
                          
                         
                           I 
                           2 
                         
                       
                       + 
                       … 
                       + 
                       
                         
                           Z 
                           
                             n 
                              
                             
                               ( 
                               
                                 n 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                         
                          
                         
                           I 
                           
                             ( 
                             
                               n 
                               - 
                               1 
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           Z 
                           nn 
                         
                          
                         
                           I 
                           n 
                         
                       
                       + 
                       
                         
                           Z 
                           L 
                         
                          
                         
                           I 
                           n 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         U 
                         1 
                       
                       - 
                       
                         
                           Z 
                           11 
                         
                          
                         
                           I 
                           1 
                         
                       
                     
                     = 
                     
                       
                         
                           
                             Z 
                             12 
                           
                            
                           
                             I 
                             2 
                           
                         
                         + 
                         … 
                         + 
                         
                           
                             Z 
                             
                               1 
                                
                               
                                 ( 
                                 
                                   n 
                                   - 
                                   1 
                                 
                                 ) 
                               
                             
                           
                            
                           
                             I 
                             
                               ( 
                               
                                 n 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             Z 
                             
                               1 
                                
                               n 
                             
                           
                            
                           
                             I 
                             n 
                           
                         
                          
                         
                           
 
                         
                         - 
                         
                           
                             Z 
                             21 
                           
                            
                           
                             I 
                             1 
                           
                         
                       
                       = 
                       
                         
                           
                             Z 
                             22 
                           
                            
                           
                             I 
                             2 
                           
                         
                         + 
                         … 
                         + 
                         
                           
                             Z 
                             
                               2 
                                
                               
                                 ( 
                                 
                                   n 
                                   - 
                                   1 
                                 
                                 ) 
                               
                             
                           
                            
                           
                             I 
                             
                               ( 
                               
                                 n 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             Z 
                             
                               2 
                                
                               n 
                             
                           
                            
                           
                             I 
                             n 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       ⋮ 
                        
                       
                         
 
                       
                       - 
                       
                         
                           Z 
                           
                             
                               ( 
                               
                                 n 
                                 - 
                                 1 
                               
                               ) 
                             
                              
                             1 
                           
                         
                          
                         
                           I 
                           1 
                         
                       
                     
                     = 
                     
                       
                         
                           
                             Z 
                             
                               
                                 ( 
                                 
                                   n 
                                   - 
                                   1 
                                 
                                 ) 
                               
                                
                               2 
                             
                           
                            
                           
                             I 
                             2 
                           
                         
                         + 
                         … 
                         + 
                         
                           
                             Z 
                             
                               
                                 ( 
                                 
                                   n 
                                   - 
                                   1 
                                 
                                 ) 
                               
                                
                               
                                 ( 
                                 
                                   n 
                                   - 
                                   1 
                                 
                                 ) 
                               
                             
                           
                            
                           
                             I 
                             
                               ( 
                               
                                 n 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             Z 
                             
                               
                                 ( 
                                 
                                   n 
                                   - 
                                   1 
                                 
                                 ) 
                               
                                
                               n 
                             
                           
                            
                           
                             I 
                             n 
                           
                         
                          
                         
                           
 
                         
                         - 
                         
                           
                             Z 
                             
                               n 
                                
                               
                                   
                               
                                
                               1 
                             
                           
                            
                           
                             I 
                             1 
                           
                         
                       
                       = 
                       
                         
                           
                             Z 
                             
                               n 
                                
                               
                                   
                               
                                
                               2 
                             
                           
                            
                           
                             I 
                             2 
                           
                         
                         + 
                         … 
                         + 
                         
                           
                             Z 
                             
                               n 
                                
                               
                                 ( 
                                 
                                   n 
                                   - 
                                   1 
                                 
                                 ) 
                               
                             
                           
                            
                           
                             I 
                             
                               ( 
                               
                                 n 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             Z 
                             nn 
                           
                            
                           
                             I 
                             n 
                           
                         
                         + 
                         
                           
                             Z 
                             L 
                           
                            
                           
                             I 
                             n 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0037]    Equation set (7) could be rewritten in matrix form as equation (8), 
         [0000]    
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               U 
                               1 
                             
                             - 
                             
                               
                                 Z 
                                 11 
                               
                                
                               
                                 I 
                                 1 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               - 
                               
                                 Z 
                                 21 
                               
                             
                              
                             
                               I 
                               1 
                             
                           
                         
                       
                       
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             
                               - 
                               
                                 Z 
                                 
                                   
                                     ( 
                                     
                                       n 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                    
                                   1 
                                 
                               
                             
                              
                             
                               I 
                               1 
                             
                           
                         
                       
                       
                         
                           
                             
                               - 
                               
                                 Z 
                                 
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                             
                              
                             
                               I 
                               1 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               Z 
                               12 
                             
                           
                           
                             … 
                           
                           
                             
                               Z 
                               
                                 1 
                                  
                                 
                                   ( 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               Z 
                               
                                 1 
                                  
                                 n 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             
                               Z 
                               22 
                             
                           
                           
                             … 
                           
                           
                             
                               Z 
                               
                                 2 
                                  
                                 
                                   ( 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               Z 
                               
                                 2 
                                  
                                 n 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             ⋮ 
                           
                           
                             ⋮ 
                           
                           
                             ⋱ 
                           
                           
                             ⋮ 
                           
                           
                             ⋮ 
                           
                         
                         
                           
                             
                               Z 
                               
                                 
                                   ( 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                                  
                                 2 
                               
                             
                           
                           
                             … 
                           
                           
                             
                               Z 
                               
                                 
                                   ( 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                                  
                                 
                                   ( 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               Z 
                               
                                 
                                   ( 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                                  
                                 n 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             
                               Z 
                               
                                 n 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                           
                           
                             … 
                           
                           
                             
                               Z 
                               
                                 n 
                                  
                                 
                                   ( 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               Z 
                               nn 
                             
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               I 
                               2 
                             
                           
                         
                         
                           
                             
                               I 
                               3 
                             
                           
                         
                         
                           
                             ⋮ 
                           
                         
                         
                           
                             
                               I 
                               n 
                             
                           
                         
                         
                           
                             
                               
                                 Z 
                                 L 
                               
                                
                               
                                 I 
                                 n 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0038]    In equation (8), Z 11 , Z 12 , . . . Z nn  are a function of frequency and known parameters, U 1  and I 1  are the input voltage and input current vector to the transmitter and could be measured easily, only I 2 , I 3 , . . . I n , Z L I n  are unknowns. Then we get the solution of the matrix as equation (9). 
         [0000]    
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             I 
                             2 
                           
                         
                       
                       
                         
                           
                             I 
                             3 
                           
                         
                       
                       
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             I 
                             n 
                           
                         
                       
                       
                         
                           
                             
                               Z 
                               L 
                             
                              
                             
                               I 
                               n 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 Z 
                                 12 
                               
                             
                             
                               … 
                             
                             
                               
                                 Z 
                                 
                                   1 
                                    
                                   
                                     ( 
                                     
                                       n 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 Z 
                                 
                                   1 
                                    
                                   n 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               
                                 Z 
                                 22 
                               
                             
                             
                               … 
                             
                             
                               
                                 Z 
                                 
                                   2 
                                    
                                   
                                     ( 
                                     
                                       n 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 Z 
                                 
                                   2 
                                    
                                   n 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               ⋮ 
                             
                             
                               ⋮ 
                             
                             
                               ⋱ 
                             
                             
                               ⋮ 
                             
                             
                               ⋮ 
                             
                           
                           
                             
                               
                                 Z 
                                 
                                   
                                     ( 
                                     
                                       n 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                    
                                   2 
                                 
                               
                             
                             
                               … 
                             
                             
                               
                                 Z 
                                 
                                   
                                     ( 
                                     
                                       n 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                    
                                   
                                     ( 
                                     
                                       n 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 Z 
                                 
                                   
                                     ( 
                                     
                                       n 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                    
                                   n 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               
                                 Z 
                                 
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                             
                             
                               … 
                             
                             
                               
                                 Z 
                                 
                                   n 
                                    
                                   
                                     ( 
                                     
                                       n 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 
                                   Z 
                                   nn 
                                 
                                  
                                 
                                     
                                 
                               
                             
                             
                               1 
                             
                           
                         
                         ] 
                       
                       
                         
                           - 
                           1 
                         
                          
                         
                             
                         
                       
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               
                                 U 
                                 1 
                               
                               - 
                               
                                 
                                   Z 
                                   11 
                                 
                                  
                                 
                                   I 
                                   1 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 - 
                                 
                                   Z 
                                   12 
                                 
                               
                                
                               
                                 I 
                                 1 
                               
                             
                           
                         
                         
                           
                             ⋮ 
                           
                         
                         
                           
                             
                               
                                 - 
                                 
                                   Z 
                                   
                                     1 
                                      
                                     
                                       ( 
                                       
                                         n 
                                         - 
                                         1 
                                       
                                       ) 
                                     
                                   
                                 
                               
                                
                               
                                 I 
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               
                                 - 
                                 
                                   Z 
                                   
                                     1 
                                      
                                     n 
                                   
                                 
                               
                                
                               
                                 I 
                                 1 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
         [0000]    It should be noted that the inverse matrix in (9) has the unique form: 
         [0000]    
       
         
           
               
             
               [ 
               
                 
                   
                     
                       Z 
                       12 
                     
                   
                   
                     … 
                   
                   
                     
                       Z 
                       
                         1 
                          
                         n 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     
                       Z 
                       22 
                     
                   
                   
                     … 
                   
                   
                     
                       Z 
                       
                         2 
                          
                         n 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     ⋮ 
                   
                   
                     ⋱ 
                   
                   
                     ⋱ 
                   
                   
                     ⋮ 
                   
                 
                 
                   
                     
                       Z 
                       
                         n 
                          
                         
                             
                         
                          
                         2 
                       
                     
                   
                   
                     … 
                   
                   
                     
                       Z 
                       nn 
                     
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
         [0000]    where the last column has zero elements except for the last element. 
         [0039]    Normally, the ac power source comes from the output of a power inverter (which is a dc-ac power converter) fed by a dc voltage source (U dc ) with very low source resistance. The dc power delivered by this dc voltage source can be considered as the input power P in . To solve (9), the power balance equation can be used, i.e., the scalar relationship of the input power can be used as follows: 
         [0000]        P   in   =U   dc   I   dc =η inv   U   1   I   1  cos(φ)  (10)
 
         [0000]    where η inv  is the energy efficiency of the power inverter, I dc  is the output current of the de voltage source and φ is the phase angle between U 1  and I 1 . U 1  is the fundamental component of the ac driving voltage of the transmitter coil if such driving voltage is not sinusoidal. 
         [0040]    The magnitude of U 1  and I 1  can be easily measured, e.g. with the use of peak detectors for their scale-down signals. Since the input information of Pin, U 1  and I 1  of equation (10) can be determined, cos(φ) and φ can be determined. 
         [0000]    
       
         
           
             
               
                 
                   φ 
                   = 
                   
                     
                       cos 
                       
                         - 
                         1 
                       
                     
                      
                     
                       ( 
                       
                         
                           P 
                           
                             i 
                              
                             
                                 
                             
                              
                             n 
                           
                         
                         
                           
                             η 
                             inv 
                           
                            
                           
                             U 
                             1 
                           
                            
                           
                             I 
                             1 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
         [0000]    This angle can be leading or lagging, which can be determined by comparing the scaled-down waveforms of U 1  and I 1  with a zero voltage reference in comparators. The rising voltage edges of the comparators for U 1  and I 1  can be used to determine whether I 1  is leading or lagging U 1 . 
         [0041]    If U 1  is used as the reference vector in the rotating frame, I 1  can be represented in complex form with respect to U 1  in equation (9). This is a very important point because only the magnitude and phase relationships of U 1  and I I  in equation (9) need to be known. Such a matrix equation can now be solved as a set of complex equations. It is not necessary to sample the instantaneous values of U 1  and I 1 , and therefore fast sampling and fast computational requirements are totally eliminated in this disclosure. For example, for an operating frequency of 500 kHz, if the instantaneous values of U 1  and I 1  are used in (9), the sampling frequency for U 1  and I 1  must be much higher than 500 kHz. However, if only the magnitudes of Pin, U 1  and I 1  are needed (as in one preferred approach explained previously), the sampling rate can be very low (e.g. 1 kHz). Therefore this approach greatly reduces the sampling frequency and costs and complexity of the control electronics. Of course, if fast computational controllers become available and economical, the sampled values of U 1  and I 1  can be used for solving (9). In this case, there is no needed to measure P in  and only measurements of U I  and I 1  are necessary. 
         [0042]    Finally, the solution of Z L  is: 
         [0000]    
       
         
           
             
               
                 
                   
                     Z 
                     L 
                   
                   = 
                   
                     
                       
                         Z 
                         L 
                       
                        
                       
                         I 
                         n 
                       
                     
                     
                       I 
                       n 
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Based on equations (9) and (10), the load power can then be determined from: 
         [0000]        P   out   =I   n   2   Re ( Z   L )  (13)
 
         [0000]    The output voltage is: 
         [0000]        U   o   =Z   L   I   n   (14)
 
         [0000]    Assuming that the power inverter has negligible power loss, the system energy efficiency is: 
         [0000]    
       
         
           
             
               
                 
                   
                     η 
                     = 
                     
                       
                         
                           P 
                           out 
                         
                         
                           
                             P 
                             out 
                           
                           + 
                           
                             
                               ∑ 
                               
                                 x 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                               
                                 I 
                                 x 
                                 2 
                               
                                
                               
                                 R 
                                 x 
                               
                             
                           
                         
                       
                        
                       
                           
                       
                        
                       or 
                     
                   
                    
                   
                     
 
                   
                    
                   
                     η 
                     = 
                     
                       
                         P 
                         out 
                       
                       
                         
                           U 
                           
                             i 
                              
                             
                                 
                             
                              
                             n 
                           
                         
                          
                         
                           I 
                           
                             i 
                              
                             
                                 
                             
                              
                             n 
                           
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         ϕ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where φ is the angle between U in  and I in . 
         [0043]      FIG. 6  is a schematic of the loaded resonator with the load connected in parallel with the resonant capacitor. If the loaded resonator has the load connected across the resonant capacitor as shown in  FIG. 6 , the load Z L  and the paralleled capacitor C n  are treated as a new load Z L ′, and all of the equations will be the same, except the Z nn  and Z L . Then equation (5) will be changed into equation (5a), equation (9) will be changed into equation (9a). Equation (12) will be changed into equation (12a), 
         [0000]    
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       [ 
                       
                         
                           
                             
                               U 
                               1 
                             
                           
                         
                         
                           
                             0 
                           
                         
                         
                           
                             ⋮ 
                           
                         
                         
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         [ 
                         
                           
                             
                               
                                 Z 
                                 11 
                               
                             
                             
                               
                                 Z 
                                 12 
                               
                             
                             
                               … 
                             
                             
                               
                                 Z 
                                 
                                   1 
                                    
                                   
                                     ( 
                                     
                                       n 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 Z 
                                 
                                   1 
                                    
                                   n 
                                 
                               
                             
                           
                           
                             
                               
                                 Z 
                                 21 
                               
                             
                             
                               
                                 Z 
                                 22 
                               
                             
                             
                               … 
                             
                             
                               
                                 Z 
                                 
                                   2 
                                    
                                   
                                     ( 
                                     
                                       n 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 Z 
                                 
                                   2 
                                    
                                   n 
                                 
                               
                             
                           
                           
                             
                               ⋮ 
                             
                             
                               ⋮ 
                             
                             
                               ⋱ 
                             
                             
                               ⋮ 
                             
                             
                               ⋮ 
                             
                           
                           
                             
                               
                                 Z 
                                 
                                   
                                     ( 
                                     
                                       n 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                    
                                   1 
                                 
                               
                             
                             
                               
                                 Z 
                                 
                                   
                                     ( 
                                     
                                       n 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                    
                                   2 
                                 
                               
                             
                             
                               … 
                             
                             
                               
                                 Z 
                                 
                                   
                                     ( 
                                     
                                       n 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                    
                                   
                                     ( 
                                     
                                       n 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 Z 
                                 
                                   
                                     ( 
                                     
                                       n 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                    
                                   n 
                                 
                               
                             
                           
                           
                             
                               
                                 Z 
                                 
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                             
                             
                               
                                 Z 
                                 
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                             
                             
                               … 
                             
                             
                               
                                 Z 
                                 
                                   n 
                                    
                                   
                                     ( 
                                     
                                       n 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 
                                   Z 
                                   nn 
                                   ′ 
                                 
                                 + 
                                 
                                   Z 
                                   L 
                                   ′ 
                                 
                               
                             
                           
                         
                         ] 
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 I 
                                 1 
                               
                             
                           
                           
                             
                               
                                 I 
                                 2 
                               
                             
                           
                           
                             
                               ⋮ 
                             
                           
                           
                             
                               
                                 I 
                                 
                                   n 
                                   - 
                                   1 
                                 
                               
                             
                           
                           
                             
                               
                                 I 
                                 n 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     5 
                      
                     a 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               I 
                               2 
                             
                           
                         
                         
                           
                             
                               I 
                               3 
                             
                           
                         
                         
                           
                             ⋮ 
                           
                         
                         
                           
                             
                               I 
                               n 
                             
                           
                         
                         
                           
                             
                               
                                 Z 
                                 L 
                                 ′ 
                               
                                
                               
                                 I 
                                 n 
                               
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   Z 
                                   12 
                                 
                               
                               
                                 … 
                               
                               
                                 
                                   Z 
                                   
                                     1 
                                      
                                     
                                       ( 
                                       
                                         n 
                                         - 
                                         1 
                                       
                                       ) 
                                     
                                   
                                 
                               
                               
                                 
                                   Z 
                                   
                                     1 
                                      
                                     n 
                                   
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 
                                   Z 
                                   22 
                                 
                               
                               
                                 … 
                               
                               
                                 
                                   Z 
                                   
                                     2 
                                      
                                     
                                       ( 
                                       
                                         n 
                                         - 
                                         1 
                                       
                                       ) 
                                     
                                   
                                 
                               
                               
                                 
                                   Z 
                                   
                                     2 
                                      
                                     n 
                                   
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 ⋮ 
                               
                               
                                 ⋮ 
                               
                               
                                 ⋱ 
                               
                               
                                 ⋮ 
                               
                               
                                 ⋮ 
                               
                             
                             
                               
                                 
                                   Z 
                                   
                                     
                                       ( 
                                       
                                         n 
                                         - 
                                         1 
                                       
                                       ) 
                                     
                                      
                                     2 
                                   
                                 
                               
                               
                                 … 
                               
                               
                                 
                                   Z 
                                   
                                     
                                       ( 
                                       
                                         n 
                                         - 
                                         1 
                                       
                                       ) 
                                     
                                      
                                     
                                       ( 
                                       
                                         n 
                                         - 
                                         1 
                                       
                                       ) 
                                     
                                   
                                 
                               
                               
                                 
                                   Z 
                                   
                                     
                                       ( 
                                       
                                         n 
                                         - 
                                         1 
                                       
                                       ) 
                                     
                                      
                                     n 
                                   
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 
                                   Z 
                                   
                                     n 
                                      
                                     
                                         
                                     
                                      
                                     2 
                                   
                                 
                               
                               
                                 … 
                               
                               
                                 
                                   Z 
                                   
                                     n 
                                      
                                     
                                       ( 
                                       
                                         n 
                                         - 
                                         1 
                                       
                                       ) 
                                     
                                   
                                 
                               
                               
                                 
                                   Z 
                                   nn 
                                   1 
                                 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                         
                           - 
                           1 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 
                                   U 
                                   1 
                                 
                                 - 
                                 
                                   
                                     Z 
                                     11 
                                   
                                    
                                   
                                     I 
                                     1 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   - 
                                   
                                     Z 
                                     12 
                                   
                                 
                                  
                                 
                                   I 
                                   1 
                                 
                               
                             
                           
                           
                             
                               ⋮ 
                             
                           
                           
                             
                               
                                 
                                   - 
                                   
                                     Z 
                                     
                                       1 
                                        
                                       
                                         ( 
                                         
                                           n 
                                           - 
                                           1 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                  
                                 
                                   I 
                                   1 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   - 
                                   
                                     Z 
                                     
                                       1 
                                        
                                       n 
                                     
                                   
                                 
                                  
                                 
                                   I 
                                   1 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                       
                   
                    
                   
                     
                       Z 
                       L 
                       ′ 
                     
                     = 
                     
                       
                         
                           Z 
                           L 
                           ′ 
                         
                          
                         
                           I 
                           n 
                         
                       
                       
                         I 
                         n 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     9 
                      
                     a 
                   
                   ) 
                 
               
             
           
         
       
     
         [0044]    Since it is known that Z 1  is the parallel impedance of Z L  and C n , the solution of Z L  is: 
         [0000]    
       
         
           
             
               
                 
                   
                     Z 
                     L 
                   
                   = 
                   
                     1 
                     
                       
                         1 
                         
                           Z 
                           L 
                           ′ 
                         
                       
                       - 
                       
                         j 
                          
                         
                             
                         
                          
                         ω 
                          
                         
                             
                         
                          
                         
                           c 
                           n 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     12 
                      
                     a 
                   
                   ) 
                 
               
             
           
         
       
     
         [0045]    To summarize the novel procedure for load monitoring without using direct feedback information, that is, based on the information of the input voltage (U in =U 1 ) and input current (Iin=I 1 ) only, the following is the procedure:
       1. Use the standard coupled circuit matrix equation (5) to mathematically describe the wireless power transfer system.   2. Note that only U 1  and I 1  are known, rearrange the matrix equation (5) to the form of equation (8) in which the column vector on the right-hand-side of the equation (8) consists of I 2  to I n  and Z L I n , where I 2  is the current in the second coil, I n  is the current in the receiver coil and Z L  is the load impedance.   3. Then create the inverse matrix of equation (8) and obtain the matrix equation in the form of equation (9), which is the column vector mentioned in the previous step as the subject of the equation. Note that the column vector on the right-hand-side of equation (9) consists of all the known information, including U 1 , I 1  and the system parameters. This crucial step ensures that any item in the column vector on the left-hand-side of equation (9) can be determined.   4. Determine the last two items of the column vector on the left-hand-side of equation (9). That is, obtain I n  and Z L I n . The item I n  is the load current (i.e. same as I out ), Z L I n  is the output voltage (U out ) of the receiver coil. I out  and/or U out  can be used in the control loop of  FIG. 7  and  FIG. 8 .   5. The load impedance Z L  can be determined by equation (12) or (16) and the output power P out  can be determined by equation (13).       
 
         [0051]    Regarding practical implementation, there are at least two possible approaches: 
         [0052]    (A) Using the Magnitudes of P in , U 1  and I 1 : 
         [0053]    If the instantaneously sampled magnitudes of the envelopes of the Pin, U 1  and I 1  waveforms are used for the proposed method, a sampling rate much lower than the operating frequency and a low-cost controller with limited computational power can be used. This is the preferred method as the sampling frequency for the system could be as low as 1 kHz typically. The method involves the following steps:
       1) Sample and measure the magnitude of P in  and the peak or root-mean-square magnitudes of U 1  and I 1  at a sampling frequency much lower than the frequency of U 1  and I 1 .   2) Determine the phase angle φ using equation (11).   3) Use U 1  as the reference vector for a rotating frame, then represent U 1  and I 1  in complex form (either in polar form or Cartesian form).   4) Solve equation (9) for I n  (=L out ) and Z L I n  with the U 1  and I 1  as complex numbers.   5) Determine U o  using equation (14), Z L  using equation (12), P out  using equation (13) and η (using equation (15)).   6) Then use the calculated U out , I out , Z L , P out , and η for an appropriate controller.       
 
         [0060]    (B) Using the Instantaneously Sampled Values of U 1  and I 1 : 
         [0061]    If the instantaneously sampled values of the waveforms of U 1  and I 1  are used for the proposed method, a sampling rate much higher than the operating frequency and a fast computational controller are needed. This is an alternative method and is suitable if very fast samplers and economical fast controllers are available. The alternative method involves the following steps:
       1) Sample and measure the instantaneous values of U 1  and I 1  at a sampling frequency much higher than the frequency of U I  and I 1 .   2) Solve equation (9) for the instantaneous values of I n (=I out ) and Z L I n .   3) Determine U o  using equation (14), Z L  using equation (12), P out  using equation (13) and η (using equation (15)).   4) Then use the calculated U out , I out , Z L , P out , and η for an appropriate controller.       
 
         [0066]    The experimental conditions and calculated self-inductances and mutual-inductances for an 8-coil-domino system are listed in Table 5 to Table 9. The measurement and calculation results are listed in Table 9. 
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 5 
               
             
             
               
                   
               
               
                 Self-inductances of each coil for 8-coil-domino system 
               
             
          
           
               
                   
                 L 1   
                 L 2   
                 L 3   
                 L 4   
                 L 5   
                 L 6   
                 L 7   
                 L 8   
               
               
                   
                   
               
             
          
           
               
                 Inductance 
                 8.204E−05 
                 8.204E−05 
                 8.204E−05 
                 8.204E−05 
                 8.204E−05 
                 8.204E−05 
                 8.204E−05 
                 8.204E−05 
               
               
                 (H) 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 6 
               
               
                   
               
               
                 Mutual-inductances of each pair of coils for 8-coil-domino system 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 M 12   
                 M 13   
                 M 14   
                 M 15   
                 M 16   
                 M 17   
                 M 18   
               
               
                   
               
               
                 Inductance 
                 9.210E−06 
                 2.588E−06 
                 1.004E−06 
                 4.733E−07 
                 2.589E−07 
                 1.555E−07 
                 9.936E−08 
               
               
                 (H) 
               
               
                   
               
               
                   
                 M 23   
                 M 24   
                 M 25   
                 M 26   
                 M 27   
                 M 28   
                 M 34   
               
               
                   
               
               
                 Inductance 
                 8.836E−06 
                 2.5544E−06 
                 9.887E−07 
                 4.734E−07 
                 2.584E−07 
                 1.537E−07 
                 9.048E−06 
               
               
                 (H) 
               
               
                   
               
               
                   
                 M 35   
                 M 36   
                 M 37   
                 M 38   
                 M 45   
                 M 46   
                 M 47   
               
               
                   
               
               
                 Inductance 
                 2.582E−06 
                 1.0126E−06 
                 4.813E−07 
                 2.589E−07 
                 8.964E−06 
                 2.616E−06 
                 1.020E−06 
               
               
                 (H) 
               
               
                   
               
               
                   
                 M 48   
                 M 56   
                 M 57   
                 M 58   
                 M 67   
                 M 68   
                 M 78   
               
               
                   
               
               
                 Inductance 
                 4.772E−07 
                  9.212E−06 
                 2.658E−06 
                 1.015E−06 
                 9.158E−06 
                 2.5878E−06  
                 8.885E−06 
               
               
                 (H) 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 7 
               
             
             
               
                   
               
               
                 Copper resistances of each coil for 8-coil-domino system 
               
             
          
           
               
                   
                 R 1   
                 R 2   
                 R 3   
                 R 4   
                 R 5   
                 R 6   
                 R 7   
                 R 8   
               
               
                   
                   
               
             
          
           
               
                 Resistance 
                 7.521E−01 
                 7.521E−01 
                 7.521E−01 
                 7.521E−01 
                 7.521E−01 
                 7.521E−01 
                 7.521E−01 
                 7.521E−01 
               
               
                 (Ohm) 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 8 
               
             
             
               
                   
               
               
                 Capacitances of each series capacitor for 8-coil-domino system 
               
             
          
           
               
                   
                 C 1   
                 C 2   
                 C 3   
                 C 4   
                 C 5   
                 C 6   
                 C 7   
                 C 8   
               
               
                   
                   
               
             
          
           
               
                 Capacitance 
                 1.020 
                 1.004 
                 1.009 
                 9.992 
                 9.960 
                 1.002 
                 1.012 
                 1.065 
               
               
                 (nF) 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
             
               
               
               
               
             
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
             
           
               
                 TABLE 9 
               
             
             
               
                   
               
               
                 Comparison of calculated impedances and experimental impedances 
               
             
          
           
               
                 Load 
                 Experimental Data from the transmitter 
                 Calculated load 
               
             
          
           
               
                 imped- 
                   
                 Phase 
                 impedances (Ohm) 
               
             
          
           
               
                 ance 
                 Frequency 
                   
                   
                 angle 
                 Real 
                 Imaginary 
               
               
                 (Ohm) 
                 (Hz) 
                 V in  (V) 
                 I in  (A) 
                 (Degree) 
                 part 
                 part 
               
               
                   
               
             
          
           
               
                 9.10 
                 539751.57 
                 1.5070 
                 0.0202 
                 4.05 
                 8.19 
                 1.03 
               
               
                 19.30 
                 522452.52 
                 1.2721 
                 0.0315 
                 −55.93 
                 18.58 
                 −4.71 
               
               
                 19.30 
                 535834.47 
                 1.0658 
                 0.0293 
                 −9.34 
                 18.43 
                 −4.68 
               
               
                 29.10 
                 520632.39 
                 1.3734 
                 0.0264 
                 −41.80 
                 27.69 
                 0.32 
               
               
                 30.00 
                 522739.37 
                 1.3266 
                 0.0285 
                 −47.74 
                 27.10 
                 −0.46 
               
               
                 39.30 
                 521899.22 
                 1.3136 
                 0.0264 
                 −34.75 
                 36.86 
                 −2.06 
               
               
                 49.30 
                 523702.25 
                 1.3132 
                 0.0258 
                 −31.07 
                 45.11 
                 −6.62 
               
               
                 59.20 
                 523180.83 
                 1.4107 
                 0.0233 
                 −25.78 
                 56.64 
                 −0.68 
               
               
                 69.60 
                 523512.07 
                 1.4578 
                 0.0221 
                 −22.15 
                 66.81 
                 −3.08 
               
               
                 79.50 
                 523707.33 
                 1.4909 
                 0.0210 
                 −19.16 
                 76.21 
                 −3.58 
               
               
                 99.60 
                 523813.39 
                 1.5872 
                 0.0189 
                 −17.07 
                 94.99 
                 −4.01 
               
               
                 119.40 
                 523887.33 
                 1.6689 
                 0.0172 
                 −14.97 
                 116.59 
                 −3.49 
               
               
                   
               
             
          
         
       
     
         [0067]    C. Output Power Control without Direct Output Information Feedback 
         [0068]    The present disclosure can be applied in two different situations. In the first situation the system parameters, i.e., all of the coil resistances, inductance and capacitance as well as mutual inductance between coils, are known. In that situation the measurable input current, input voltage and optionally input power are used to derive the necessary variables (U o , I o , Z L , P out  and efficiency) for feeding into the control loop based on equations (5) to (12). Thus, this can be done without using any direct measurement information or feedback from the output (load) side. The variables derived from this method of implementing the disclosure are accurate enough to control the system as if they were measured directly from the outputs. 
         [0069]    In the second situation, the system parameters are not known. These unknown parameters are derived from the measurable input current and input voltage only (from the 1st Coil) based on equations (1) to (4) and an intelligent algorithm (such as the Genetic Algorithm) which act to estimate the system parameters, such as the inductance and capacitance values. Once these values are accurately estimated, they are then used for values in the method according to the first situation where these values are known. 
         [0070]    (1) Assuming that the System Parameters of the WPTS are Known 
         [0071]    If the system parameters of the WPTS are known, the method described in Section B above can be used. The calculated U out , P L , I n , and Z L  values now offer information concerning output power control without using directly measured output information. Dynamically updated values of V o , P L , or I n  and Z L  can be used for output power control by controlling either the operating frequency or the input current of the driving coil on the input side. 
         [0072]    One example of the control system is illustrated in  FIG. 6 . Based on the measurable input voltage (U in ) and input current (I in ), the method of the present disclosure enables the determination of the information like a variable estimator. The information derived by the variable estimator includes, but is not limited to, the output load impedance (Z L ), output voltage (U out ), output current (I out ), output power (P out ) and the system efficiency (η). These variables can be used to fit into any control objective as required in a specific application. It should be noted that the feedback of the estimated efficiency is particularly useful for a wireless power transfer system, because the optimal frequency for maximizing the overall system efficiency can be load dependent. Therefore, the control signals may include both the magnitude of input voltage and/or input current and the frequency. 
         [0073]    The main feature of the disclosure is the use of the measurable input variables (namely input voltage and input current) of the first or Transmitter Coil of the wireless power transfer for output load monitoring and output power control, without using any direct measurements from the output load in the Receiver Coil. As such, the wireless power transfer system consists of 2 or more coils. 
         [0074]    Referring to  FIG. 7 , the parameters of the wireless power transfer system are presumed to be known, i.e., all coil resistance, inductance and capacitance as well as mutual inductance between coils are given. If the active source is a dc source  60 , an inverter  62  with an output filter is used to generate a sinusoidal voltage with controllable frequency and magnitude for driving the first coil (Transmitter Coil  63 ). Therefore, the input power is provided by energizing the Transmitter Coil and such power will be wirelessly transmitted to the last (Receiver) Coil  65  for powering the load  67 . Since according to the disclosure, measurements on the output load are to be eliminated; only the input voltage and the input current can be relied upon for output power control. Note that in the arrangement of  FIG. 7  the output load can be connected either in series with the last LC resonator or in parallel across the capacitor of the last LC resonator. 
         [0075]    A sensor block  64  in  FIG. 7  represents the use of the voltage and current sensors for obtaining such input voltage U in  and input current I in . The equations (9)-(15) are implemented in an Estimator Block  66 . Note that these equations require the input voltage and input current only. Estimator Block  66  may be a microprocessor programmed to execute equations (9)-(15) or some hardware device to perform the same function, such as a programmable gate array or application specific integrated circuit (ASIC). 
         [0076]    In the operation of the circuit of  FIG. 7 , the Estimator  66  solves equation (9) with the (known) measured values of U in , Iin in order to obtain I 2  to I n  and Z L I in . Then the Estimator generates the variables Z L , I o , P o  and η as follows:
       (i) Z L  from equation (12)   (ii) P o  from equation (13)   (iii) U o  from equation (14)   (iv) I o =I n  obtained already from equation (9)   (v) η from equation (15).
 
These become the outputs of the Estimator  66 .
       
 
         [0082]    The first four variables (i)-(iv) are calculated output information, obtained without using any direct output measurements. Together with the energy efficiency, they are calculated continuously at a high sampling rate (usually limited by the speed of the processor) to provide instantaneous output information for control and feedback information. Such calculated values can be fed into any control scheme to meet the specific control objectives of the wireless power transfer system. Based on the chosen control scheme, e.g., control objective  68 , the power inverter  62  is operated so that it generates the appropriate sinusoidal voltage at a controllable frequency and magnitude to meet the output power demand of the load  67  according to the control objective. 
         [0083]    (2) Assuming that the System Parameters of the WPTS are Unknown 
         [0084]    If the system parameters are not known, the methods described in both Section A and Section B above are needed and the control block is as shown in  FIG. 8 . In particular, an extra System Parameters Identification block  70  is included in this control. Here the input voltage and current are also used to determine the system parameters in a real time manner as explained in Section A. As a result, the input voltage and the input current are used to (i) estimate the system parameters dynamically and (ii) provide the control variables for the control system. Thus, unit 70 uses the input voltage U in  and current I in  to generate the system parameters, i.e., all coil resistance, inductance and capacitance as well as mutual inductance between coils. This is done using equations (1)-(4) and the intelligent algorithm (Genetic Algorithm) to determine the system parameters in the system matrix in equation (1). This method requires only the input voltage and current information. 
         [0085]    As an alternative, instead of using the input voltage and current, optionally the input power P in  is used. This is shown in dotted line in  FIG. 8 . If P in  is used, the method can be implement easily with a low sampling rate for the envelopes of the input voltage and input current waveforms (without the need for fast sampling of the instantaneous values of the input voltage and current waveforms). This provides a significant savings in computing power. 
         [0086]    Once the system parameters are determined, based either on input voltage and current, or input power, the System Parameters Identification unit  70  provides the parameters to the WPTS Model, like the Estimator  66  in  FIG. 7 , determines the variables Z L , U o , I o , P o  and η using equations (5) to (12) as explained above for  FIG. 7 . These variables are applied to compensator/controller  72 , which in part functions in the same way as the control objective  68  in  FIG. 7 . However, in  FIG. 8  the frequency and magnitude compensation are shown as performed in a separate unit  74 , whose outputs control the inverter  62 . 
         [0087]    It should be noted that the output load can be connected either in series with the last LC resonator or in parallel across the capacitor of the last LC resonator. 
         [0088]    In summary,  FIG. 7  shows a circuit according to the present disclosure which uses the Load Monitoring and Power Control method described in Part-B (for known system parameters).  FIG. 8  requires the Parameter Identification Method in Part A and the Load Monitoring &amp; Power Control in Part B, which is for the situation where the system parameters are unknown or are dynamically changing. 
         [0089]    The proposed methodology can be applied to a wide range of wireless power applications such as wireless power transfer and load monitoring of medical implants (when the number of coils is reduced to two, i.e. a transmitter coil and a receiver coil), of portable electronic products being charged on a wireless charging pad, and of WPT systems based on the relay resonators or domino-resonator systems (when the number of coils exceeds two). Therefore, the proposed methodology can be applied to any WPT system with 2 or more coils. 
         [0090]    Any reference in this specification to “one embodiment,” “an embodiment,” “exemplary embodiment,” etc., means that a particular feature, structure, or characteristic described in connection with the embodiment is included in at least one embodiment of the disclosure. The appearances of such phrases in various places in the specification are not necessarily all referring to the same embodiment. In addition, any elements or limitations of any disclosure or embodiment thereof disclosed herein can be combined with any and/or all other elements or limitations (individually or in any combination) or any other disclosure or embodiment thereof disclosed herein, and all such combinations are contemplated with the scope of the disclosure without limitation thereto. 
         [0091]    It should be understood that the examples and embodiments described herein are for illustrative purposes only and that various modifications or changes in light thereof will be suggested to persons skilled in the art and are to be included within the spirit and purview of this application. 
         [0092]    While certain exemplary techniques have been described and shown herein using various methods and systems, it should be understood by those skilled in the art that various other modifications may be made, and equivalents may be substituted, without departing from claimed subject matter. Additionally, many modifications may be made to adapt a particular situation to the teachings of claimed subject matter without departing from the central concept described herein. Therefore, it is intended that the claimed subject matter not be limited to the particular examples disclosed, but that such claimed subject matter may also include all implementations falling within the scope of the appended claims, and equivalents thereof.